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http://math.stackexchange.com/questions/723695/crossword-puzzle-crossnumber-puzzle | # Crossword puzzle- Crossnumber puzzle
The puzzle below is a cross number puzzle, similar to a crossword puzzle except that the entries are numbers. Enter one digit per square. The thick heavy line is a separator.
ACROSS:
a. A prime number
c. The sum of digits of a across
DOWN:
a. Square of the sum of the digits of b down
b. A prime number
I am confused on how to solve this problem. I do understand from across problem a must be a prime number so a= 1 2 3 5 or 7. It can't be anything higher because it has to be one digit. c is sum of digits of a and b across. Since it can only be one digit, c must be c= 2 3 4 5 6 7 8 9
Then for down, the square of the sum of digits of b down would equal a, so (b+c)^2 is going to equal a down. However, does the box under a and across from c - will it have a number too? I think it would have to because two numbers squared can't equal any of the above digits...
Finally B is a prime number so b= 1 2 3 5 or 7
Unsure of where to go from here, because there are so many different paths to attempt and I am unsure of where to go from here.
-
This is intriguing to say the least. I'm not aware of a simple way to do this without writing a program to do it. I'd love to see others' attempts. – Cameron Williams Mar 24 '14 at 1:10
1 step) a. across should be a prime number, which digit sum (c. across) is a 1 digit number.
2 step) c. across should be an odd number, because b. down is a prime number and can't be divisible by 2.
3 step) from step 2 -> a. across should be a prime number, which digit sum (c. across) is odd number.
4 step) from 1 and 3 steps -> a. across first digit should be even, because second digit is always odd for their sum to be odd.
5 step) from 1 and 4 steps -> we get two numbers which satisfy the rules, it is 43 or 61. So in b. down we get 37 or 17.
6 step) a. down is a square of the sum of the digits of b. down, so from step 5 -> we get a. down is 100 or 64.
7 step) 100 doesn't fit to our rules, 64 does.
6 1
4 7
-
I have to do this problem for a math class. My teacher said that A across is a 2 digit number meaning the box that has an A in it and the box that has a B in it as well. I was not able to figure out the answer yet but I believe that A across must be either 43 or 61 making B down either 37 or 17. Hope this helped!
-
Welcome to math SE. Please don't use answers if you are not providing a real answer to the problem – rlartiga Mar 25 '14 at 1:10
This is a crossword. I think you haven't paid much attention to the rule "The thick heavy line is a separator."
This means that:
• a word cannot pass through this heavy line
• a word cannot stop unless it finds either a heavy line OR the outer border of the crossword.
• a word cannot begin in the middle of the crossword. There should be a heavy line or a border before. This is the reason why there must not be a definition for "b across" or "c down".
Each square can contain only one digit. This, however, does not mean that "a across" should only fill the first square. The rule (eg for a across) is that:
• it should begin from the square named "a"
• it should continue to the right (if it was vertical/down it should continue down)
• until it finds a heavy line or the border.
So, "a across" is a 2-digit number, "c across" is 1 digit, "a down" and "b down" are 2 digits each.
Can you continue from now on?
- | 2015-10-06 17:05:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.486015260219574, "perplexity": 548.3644671735044}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736678861.8/warc/CC-MAIN-20151001215758-00209-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://hub-courses.pages.pasteur.fr/ScientificPython/3-MachineLearning/2-DimensionReductionality.slides.html | Dimension reduction TC, BN, JBM, AZ
# Multivariate problems¶
• In multivariate problems, exploratory analysis is paramount
• Is the data random or can we see patterns ?
• Can we visualize the data
• Can we discover subgroups amongst the variables or observations ?
One way to answer those questions it to use principal component analysis known as PCA.
• The main goal of a PCA analysis is to identify patterns in data
• PCA aims to detect the correlation between variables.
# PCA can help us in discovering hidden patterns¶
• also to reduce high dimensionality problems,
• identify variables with highest variance
• compression
# Univariate example¶
distribution of gene expression, distribution of human heights, distribution of length of sepals, distribution of drug response ...
In [3]:
# Generally, normally distributed (not always of course). Example of
# noramlly distributed data centered around 10 with variance equal to 1
y = 10 + randn(1000)
subplot(1,2,1); _ = boxplot(y)
subplot(1,2,2); _ = plot(y, "o"); grid(True)
## bivariate example¶
In [4]:
x1 = 10 + randn(1000)
x2 = 10 + 0.2 * randn(1000)
data2 = np.vstack([x1, x2])
subplot(1,2,1);
boxplot([x1, x2]); grid(True)
subplot(1,2,2);
scatter(x1,x2); xlim([6,14]); ylim([6,14]); grid(True)
## Multivariate example (3D)¶
In [5]:
from mpl_toolkits.mplot3d import Axes3D
x3 = 10 + 0.1 * randn(1000)
# Note that to combine 2D and 3D plots, you must use the add_subplot on
# a Figure instance
fig = figure()
ax.boxplot([x1,x2,x3]); grid()
ax.scatter(x1,x2, x3);
grid(True)
See the notebook called IRIS_3D_demo for an interactive DEMO
# Higher dimensions¶
In 1, 2 or 3 dimensions, visual investigation is easy. We can quickly see whether variables are correlated or important. In higher dimensions, new tools are required.
How do we represent / visualize data in 4D, 5D, ND?
In [7]:
fig = plt.figure()
x = np.hstack([np.random.standard_normal(100),
20+np.random.normal(size=100)])
y = np.random.standard_normal(200)
z = np.random.standard_normal(200)
c = np.random.standard_normal(200)
ax.scatter(x, y, z, c=c, cmap=plt.hot(), s=80)
plt.show()
What about 5D, how would you represent it ?
# PCA example in 2D¶
In [8]:
# Here we have 2 variables and N=100 samples
X = np.linspace(-6, 6, 100)
Y = 5 + 2*X + np.random.normal(size=100)
data = np.array([X,Y]).transpose()
plot(X, Y, "o", 0, 0, "xr"); grid(True)
# The PCA analysis with sklearn¶
The PCA algorithm identifies the directions of maximum variance in high-dimensional data by combining attributes (principal components, or directions in the feature space) that account for the most variance in the data.
In [9]:
from sklearn.decomposition import PCA
original_data = data.copy()
pca = PCA()
pca.fit(original_data)
Out[9]:
PCA(copy=True, iterated_power='auto', n_components=None, random_state=None,
svd_solver='auto', tol=0.0, whiten=False)
In [10]:
pca.explained_variance_
Out[10]:
array([ 62.31336552, 0.18752182])
In [11]:
pca.explained_variance_ratio_
Out[11]:
array([ 0.99699969, 0.00300031])
So 99% of the variance can be explained by the first component of the PCA analysis ! This makes sense since we know that X and Y are highly correlated.
# Scaling the data¶
• Variables may be measured in different units
• Variable with larger variance is more likely to influence the PCA
• scaling is just centering and dividing by the standard deviation
In [12]:
from sklearn.decomposition import PCA
from sklearn.preprocessing import scale
data = scale(data)
pca = PCA()
pca.fit(data)
pca.explained_variance_ratio_
Out[12]:
array([ 0.9952294, 0.0047706])
Note that when scaled, the explained_variance total variance is equal to the number of observations (here 2)
In [15]:
pca.explained_variance_
Out[15]:
array([ 1.9904588, 0.0095412])
# Transforming the data from original space to PCA space¶
the PCA analysis projects the data on new axis so as to minimize the variance
In [20]:
Xr = pca.transform(data)
plot(Xr[:,0], Xr[:,1], "ro")
grid()
xlabel("PC1", fontsize=20)
ylabel("PC2", fontsize=20)
axvline(0, lw=2); axhline(0, lw=2)
Out[20]:
<matplotlib.lines.Line2D at 0x7f8c2a4c99b0>
The PCA axis can be characterised by the explained variance and the components of the
In [31]:
pca.components_
Out[31]:
array([[ 0.70710678, 0.70710678],
[-0.70710678, 0.70710678]])
components: Principal axes in feature space, representing the directions of maximum variance in the data. The components are sorted by explained_variance_.
In [28]:
pca.explained_variance_ratio_
Out[28]:
array([ 0.9952294, 0.0047706])
# The PC vectors can be plotted back in the original space¶
In [18]:
#Xnew = pca.inverse_transform(Xr)
plot(data[:,0], data[:,1], "o");
count = 0
for length, vector in zip(pca.explained_variance_ratio_, pca.components_):
v = vector * 3 * np.sqrt(length)
plot([0,v[0]], [0,v[1]], "k", lw=2)
text(v[0]+0.2, v[1]-.2, "PC%s" % (count+1), fontsize=16)
count += 1
plt.axis('equal'); grid(); xlabel("X1"); ylabel("X2")
Out[18]:
<matplotlib.text.Text at 0x7f8c2a5a2828>
In [25]:
Out[25]:
array([[ 0.70710678, 0.70710678],
[-0.70710678, 0.70710678]])
# Compression (select first components)¶
In [112]:
pca = PCA(n_components=2)
pca.fit(data)
Xr = pca.transform(data)
Xnew = pca.inverse_transform(Xr)
plot(data[:,0], data[:,1], "o", label="original data");
plot(Xnew[:,0], Xnew[:,1], "rx", label="fit and transformed back")
legend()
grid()
If we use all PCA components (here 2 components) we can transform the data into the PCA space and back into the original space. We do not lose information.
# Compression
When performing the PCA, we can restrict the number of components. In such case, when we transform back into the original space, we remove variance. In our example, it means we project the 2D data on a 1D space (the first PC direction).
Redo the previous example setting the number of components to 1 instead of 2. What do you see ?
In [89]:
pca = PCA(n_components=1)
pca.fit(data)
Xr = pca.transform(data)
Xnew = pca.inverse_transform(Xr)
plot(data[:,0], data[:,1], "o");
plot(Xnew[:,0], Xnew[:,1], "r-x")
grid()
# IRIS examples (4 variables)¶
The problem: measurements of 4 variables on 3 types of iris flowers. Are those variables pertinent ? Do we need all of them. Which one(s) can be used to classify the 3 species ?
versicolor virginica setosa
In [93]:
import pandas as pd
df = pd.DataFrame(iris['data'], columns=iris['feature_names'])
colors = {0:"navy", 1:"turquoise", 2:"darkorange"}
df["target"] = iris['target']
df['color'] = df.target.apply(lambda x:colors[x])
df['target_names'] = df.target.apply(lambda x: iris['target_names'][x])
features = df.columns[0:4]
# Explore the iris data sets
In [94]:
df[features].hist()
Out[94]:
array([[<matplotlib.axes._subplots.AxesSubplot object at 0x7f8c22199470>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f8c218d2198>],
[<matplotlib.axes._subplots.AxesSubplot object at 0x7f8c218a02e8>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f8c21863630>]], dtype=object)
In [41]:
df.sample(10)
Out[41]:
sepal length (cm) sepal width (cm) petal length (cm) petal width (cm) target color
37 4.9 3.1 1.5 0.1 0 navy
74 6.4 2.9 4.3 1.3 1 turquoise
88 5.6 3.0 4.1 1.3 1 turquoise
53 5.5 2.3 4.0 1.3 1 turquoise
38 4.4 3.0 1.3 0.2 0 navy
128 6.4 2.8 5.6 2.1 2 darkorange
61 5.9 3.0 4.2 1.5 1 turquoise
142 5.8 2.7 5.1 1.9 2 darkorange
28 5.2 3.4 1.4 0.2 0 navy
83 6.0 2.7 5.1 1.6 1 turquoise
See IRIS_3D_demo
# PCA on IRIS data
• Using the IRIS data (without the label), perform a PCA analysis
• plot the explained variance ratio values and the cumulative sum of the explained variance ratio.
In [97]:
from sklearn import decomposition
from sklearn.preprocessing import scale
pca = decomposition.PCA(n_components=4)
X = df[features].copy()
X = scale(X)
pca.fit(X)
pca.explained_variance_ratio_
Out[97]:
array([ 0.72770452, 0.23030523, 0.03683832, 0.00515193])
In [98]:
plot(pca.explained_variance_ratio_, "-o", label="explained variance")
plot(cumsum(pca.explained_variance_ratio_), "o-",
label="cumulative variance")
legend()
grid()
In [99]:
Xr = pca.transform(X)
scatter(Xr[:,0], Xr[:,1], color=df.color.values)
xlabel("PC1");
ylabel("PC2");
grid()
# Impact of dummy variable
• add a column to the IRIS dataframe with a constant feature (e.g. 10)
• Fit a PCA on the 5 features
• checkout the explained variance ratio
In [100]:
pca = decomposition.PCA(n_components=5)
subdf['noise'] = [10] * 150
pca.fit(scale(subdf))
pca.explained_variance_ratio_
Out[100]:
array([ 0.72770452, 0.23030523, 0.03683832, 0.00515193, 0. ])
# By the way, how does it work ?¶
The eigenvectors and eigenvalues of a covariance (or correlation) matrix represent the "core" of a PCA:
The eigenvectors (principal components) determine the directions of the new feature space
The eigenvalues determine their magnitude (explain the variance of the data along the new feature axes).
In [102]:
X = df[features].copy()
X = scale(X)
U, S, V = np.linalg.svd(X.T)
-U.T
Out[102]:
array([[ 0.52237162, -0.26335492, 0.58125401, 0.56561105],
[ 0.37231836, 0.92555649, 0.02109478, 0.06541577],
[-0.72101681, 0.24203288, 0.14089226, 0.6338014 ],
[-0.26199559, 0.12413481, 0.80115427, -0.52354627]])
In [100]:
# eigenvalues
S**2/sum(S**2)
Out[100]:
array([ 0.72770452, 0.23030523, 0.03683832, 0.00515193])
# Summary¶
• PCA can be used to reduce high dimensional data sets
• Can be used for classification to separate classes
• cumulative variance can be used to set the fraction of variance to be captured.
## Restrictions:¶
• we assume the biological question is related to the highest variances. Not robust for strongly non gaussian processes
• orthogonality constraints --> ICA
• PCA adapted for continuous data only
• If not, need to study link between qualitative variables such as correpondence factor (2 variables) analysis of multiple factor analysis (more than 2)
# LCA¶
Linear Discriminant Analysis (LDA) tries to identify attributes that account for the most variance between classes. In particular, LDA, in contrast to PCA, is a supervised method, using known class labels.
In [103]:
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
X = df[features].copy()
X = scale(X)
y = df.target.copy()
pca = PCA(n_components=2)
X_r = pca.fit(X).transform(X)
lda = LinearDiscriminantAnalysis(n_components=2)
X_r2 = lda.fit(X, y).transform(X)
# Percentage of variance explained for each components
print('explained variance ratio (first two components): %s'
% str(pca.explained_variance_ratio_))
print('explained variance ratio (first two components): %s'
% str(lda.explained_variance_ratio_))
explained variance ratio (first two components): [ 0.72770452 0.23030523]
explained variance ratio (first two components): [ 0.99147248 0.00852752]
NB: Number of components in LDA must be (< n_classes - 1). Here at most 2 (3 classes -1).
In [110]:
figure()
colors = ['navy', 'turquoise', 'darkorange']
for color, i, target_name in zip(colors, [0, 1, 2], iris.target_names):
scatter(X_r2[y == i, 0], X_r2[y == i, 1], alpha=.8, color=color,
label=target_name)
legend(loc='best', scatterpoints=1, fontsize=16)
title('LDA of IRIS dataset')
Out[110]:
<matplotlib.text.Text at 0x7f8c213fc278>
# ICA (Independent component analysis)¶
PCA is not robust against non gaussian data or non orthogonal data.
Instead, we can use ICA. Note, however, that ICA is not used for reducing dimensionality but for separating superimposed signals.
ICA is an algorithm that finds directions in the feature space corresponding to projections with high non-Gaussianity. These directions need not be orthogonal in the original feature space, but they are orthogonal in the whitened feature space, in which all directions correspond to the same variance.
## The data¶
In [112]:
rng = np.random.RandomState(42)
S_true = rng.standard_t(1.5, size=(20000, 2))
S_true[:, 0] *= 2.
#S_true /= S_true.std()
#angle = np.pi/3
#A = np.array([[np.cos(angle), np.sin(angle)],[-np.sin(angle), np.cos(angle)]])
# Mix data
A = np.array([[1, 1], [0, 2]]) # Mixing matrix [new X = x+y; new Y = 2 y]
X = np.dot(S_true, A.T) # Generate observations
In [114]:
#original data
subplot(1,2,1); plot_samples(S_true/S_true.std()); title("True observations")
subplot(1,2,2); plot_samples(X/X.std()); title("Mixed observations")
Out[114]:
<matplotlib.text.Text at 0x7f8c213493c8>
In [115]:
pca = PCA()
pca.fit(X)
S_pca = pca.transform(scale(X))
pca.explained_variance_ratio_
Out[115]:
array([ 0.79629979, 0.20370021])
Symetric data set but PCA gives us a 75/25 ratio. Why not ?
In [123]:
from sklearn.decomposition import PCA, FastICA
ica = FastICA()
ica.fit(X)
S_ica = ica.transform(X)
# Since the ICA model does not include a noise term, for the model to be correct, whitening must be applied.
S_ica /= S_ica.std(axis=0)
In [124]:
plot_samples(X/X.std(), axis_list=[pca.components_.T, -ica.mixing_])
legend(['PCA', 'ICA'], loc='upper right', fontsize=16)
Out[124]:
<matplotlib.legend.Legend at 0x7f8c20d46358>
In [125]:
plot_samples(S_pca / np.std(S_pca, axis=0))
title("PCA recovered signals")
Out[125]:
<matplotlib.text.Text at 0x7f8c20c5d5c0>
In [126]:
plot_samples(S_ica / np.std(S_ica))
title("ICA recovered signals")
Out[126]:
<matplotlib.text.Text at 0x7f8c20bc9438> | 2020-08-11 07:02:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3641848564147949, "perplexity": 6565.512194205413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738735.44/warc/CC-MAIN-20200811055449-20200811085449-00235.warc.gz"} |
http://labs.yahoo.com/publication/event-summarization-using-tweets/ | ## Event Summarization using Tweets
Author: Deepayan Chakrabarti ; Kunal Punera
Publication Date: 2011
### Abstract
Abstract:
Twitter has become exceedingly popular, with hundreds of millions of tweets being posted every day on a wide variety of topics. This has helped make real-time search applications possible with leading search engines routinely displaying relevant tweets in response to user queries. Recent research has shown that a considerable fraction of these tweets are about events'', and the detection of novel events in the tweet-stream has attracted a lot of research interest. However, very little research has focused on properly displaying this real-time information about events. For instance, the leading search engines simply display all tweets matching the queries in reverse
chronological order. In this paper we argue that for some highly structured and recurring events, such as sports, it is better to use more sophisticated techniques to summarize the relevant tweets. We formalize the problem of summarizing event-tweets and give a solution based on learning the underlying hidden state representation of the event via Hidden Markov Models. In addition, through extensive experiments on real-world data we show that our model significantly outperforms some intuitive and competitive baselines.
Keywords: | 2014-07-23 17:55:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3863922655582428, "perplexity": 1234.999356354377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997881423.31/warc/CC-MAIN-20140722025801-00131-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.airbestpractices.com/industries/food/compressed-air-assessment-finds-problems-meat-processor | Industrial Utility Efficiency
# Compressed Air Assessment Finds Problems at a Meat Processor
A meat processor, located in Canada, hired a consultant to assess their compressed air system as part of a company-wide energy conservation effort. The assessment and analysis showed, despite having a modern compressed air system using a VSD air compressor and pressure/flow control, the system was running inefficiently and had significant levels of leakage and inappropriate use.
### Initial Findings
The facility has installed a compressed air system consisting of three air cooled lubricated screw compressors, one being a 200 hp variable speed drive compressor (VSD). The fixed speed units are 150 hp and 100 hp. These compressors are locally controlled in each compressor. The system has a mist eliminator filter installed before the air dryer. A single heated blower type desiccant dryer with dew point control (EMS), with coalescing filtration on the inlet and particulate filter on the outlet, has been installed to produce dry compressed air for plant operations. The compressed air is directed throughout the plant by a system of galvanized steel piping. Two large storage receivers are located in the compressor room area as system control storage, and two are in the plant at the end of the pipeline for pressure stability. An electronic flow control valve has been installed to regulate the plant pressure, however, it was out of service due to failure.
The compressed air is delivered to the multiple production areas through a main 3-inch galvanized piping header, where various branches are tapped to supply each production area. The piping is arranged in a loop for the main production areas. Installed data loggers showed minimal pressure loss across the piping system. Most of the system pressure loss is across the drying and filtering system.
### Compressed Air System Baseline
The compressed air system electrical consumption was monitored using amp loggers. Kilowatt readings were done for both the active compressors to calibrate amps to power. System flow was recorded by placing loggers on a newly installed main flow meter at the output of the compressor room main receiver. Pressure loggers were located at the compressor discharge, after the mist eliminator and the dryer in the compressor room. Mainline pressure loggers were place at the end of line in various areas. The following baseline was determined over a two-week period:
Baseline
Part of the measurement period was done during a one-week shutdown, providing a good indication of non-productive flow and power consumption. A full eight days of production were captured and have been used to create the following load profile:
Shift Profile
Based on a blended rate of \$0.115 per kWh this would make the annual electrical cost of compressed air about \$153,000 plus taxes.
The readings and observations during the measurement period showed the compressed air system was producing air at lower efficiency (23.2 Kw/100 cfm), compared to similar optimized systems (optimum is under 19 Kw/100 cfm). There is significant waste due to poor compressor control, higher pressure than needed, inappropriate uses, as well as leakage and drainage. The study found significant improvements are possible. As can be seen in the previous table, the poor compressor control strategy is causing very inefficient operation during non-production hours.
### Demand Side Profile
A survey of the demand side of the system was done including leakage. A total of 71 leakage points were found (some drains and inappropriate uses included). A number of end uses potentially classed as inappropriate uses were found including air motors, cabinet venting, compressed air cooling, compressed air blowing nozzles and manual drains.
The shape of the compressed air flow demand curve can be seen on the profile shown in the Figure 1. The profile shows the typical cyclical pattern of a shift oriented production process. The highest peaks are mid-morning weekdays with low flows on weekends and during shutdown periods. The pressure profile mirrors the flow demand, with lowest plant pressures occurring at the highest peak flows. These pressures fall below the lowest acceptable pressure of 100 psi identified by plant personnel.
Figure 1: Compressed air profile during baseline week. Click here to enlarge.
### Potential Opportunities
Analysis of the information collected shows some potential opportunities existing that could result in improvements in the operation of the compressed air system. Further, they could result in potential savings in compressed air related electrical and maintenance costs, estimated at 52% of the current costs. These savings could be worth about \$79,700 per year in annual electrical savings. There are possible additional heat recovery savings of up to about \$5,000 if the compressor heat of compression can be used to pre-heat hot water. Some specific potential opportunities are outlined in this article.
### Compressor Operating Mode
One of the existing compressors was operating in load/unload mode with upper range modulation. This modulation must happen with this unit to protect the compressor motor from overload, because the compressor has been set to operate higher than its maximum full load pressure rating of 115 psi. The large VSD was operating at minimum speed, starting and stopping as required, but only during production shifts.
Modulating mode of operation is the least efficient mode for screw compressors. In modulation, a typical screw compressor will consume from 65% to 100% of its rated power through an output range of 0% to 100% of rated compressed air output. Fortunately, this unit appeared to be modulating only at the top end of its pressure settings range, the remainder of its operation was in load/unload mode. This mode is less efficient than VSD mode, especially at light loading during sanitation shift, weekends, and shut down periods.
The 200 hp VSD compressor was operating only during production shift where it runs at the bottom end of its range. This type of duty is typically not recommended, because the unit is less efficient at minimum speed, and also because it generates less heat. The reduced heat makes it difficult for the unit to drive off the moisture forming in the lubricant during the compression process. Long term operation like this can allow water build-up corroding internals, and causing lubrication failure potentially ruining the main screw element bearings.
Operating a fixed speed compressor as the lead unit and a VSD as the lag compressor is a non-standard and inefficient control strategy. This is one of the reasons the production of compressed air is so inefficient in this system. Two experiments were done to show the difference in compressor electrical consumption under two different conditions. The first is shown in Figure 3. The fixed speed unit was unloaded during a shutdown to see the effect on input amps. It can be seen under the same flow conditions the input dropped by 90 amps (the unit did not turn off because it doesn’t have auto shutdown activated, this was corrected). This equated to a reduction of about 47 kW, or almost 50%, just by better coordinating the compressor control settings.
The second experimental run is shown in Figure 4, where the compressor settings were temporarily altered allowing the VSD to run during the sanitation shift (as it should in a correctly controlled system). Again, under similar conditions, the amp consumption is much lower. In fact, the VSD made it all the way to mid-morning without needing any help from the fixed speed compressor (when the test was discontinued).
Figure 3: VSD was run for sanitation shift and showed much lower amps. Click here to enlarge.
Pressure drops can be seen on the pressure profile at peak flows. This was caused by the pressure loss across the air dryer and filters. The VSD keeps its output pressure nice and constant, yet the plant pressure drops. This dip in pressure can be corrected by implementing remote pressure sensing. With remote sensing installed on the downstream of the air dryer, the VSD compressor will hold the plant pressure at its set point, rather than allowing the discharge pressure to raise in order to compensate for the sagging pressure.
This strategy causes lower discharge pressure operation in normal and low load conditions. This saves power and adjusts the discharge pressure higher only during peaks, occurring a small percentage of the time. This strategy must be implemented carefully to ensure the fixed speed compressor does not inadvertently overpressure. A low-cost solution would be to purchase a remote sensing kit for each compressor from your service provider. However, a better way might be to purchase a compressor sequencer to automatically control all the compressors. An advantage of the sequencer would be the capability of remotely monitoring the compressor efficiency by remote web interface.
### Leaks and Abandoned Uses
Air leakage and flow from unused air consuming equipment (abandoned uses) in a facility is usually significant unless there is a regular system of monitoring, leak detection and repair.
Leakage testing was done using an ultrasonic detector. A total of 69 significant leaks, or wasteful items like open drains, were found at an estimated waste of about 120 cfm. About half of this flow is within the compressor room (not measured by the flow meter). If 100 cfm of the located leaks were repaired, and the compressor control corrected, then savings of about 122,000 kWh (14 kW peak) and \$14,000 in annual cost reduction could be expected. ### End Uses End uses can often be optimized, or eliminated, for substantial savings in operating costs and improvements in air pressure. The following end uses were identified as candidates for replacement: Cabinet Venting This compressed air powered cabinet venting is used to prevent water infiltration into electrical panels during sanitation, and has been installed throughout the facility. This appears to be a low pressure application (regulators set to about 10 psi), air pressure about 2 psi and below by the time it reaches the panels. It is very wasteful to use compressed air generated at 125 psi for such low-pressure requirements. Further to this, many leaks were detected on the compressed air lines placed for the venting of these cabinets due to physical damage or failure of components (see leak list). Flow tests were done on some of these venting systems, and flows of between 2 cfm and 8.9 cfm were found. ##### Figure 4: Numerous compressed air connections have been added to electrical panels. Estimated average flow is 3 cfm per venting system, with the number of venting systems estimated at 12. This would consume about 36 cfm of high pressure compressed air. This flow is only really required during sanitation shifts, representing about 23% of the total current operating time. For this application only a slight positive is required. During sanitation duty, a low pressure compressed air source, such as an aquarium pump or blower system, should be considered. Estimated savings were 52,600 kWh (6 kW peak), worth \$6,100 per year in reduced electrical costs. These savings would be offset by the electricity consumed by whatever ventilation method used.
Blowing
About 12 open blowing locations were found in the plant. The use of uncontrolled compressed air blowers is very energy intensive. If compressed air blowers must be used, then electronic control of each unit should be implemented, so the blow only happens where the product is in proximity. This can significantly reduce air consumption.
##### Figure 5: Example of blow nozzles.
Observation of the blowing showed some of the blowers are not effective in removing the water from the packaging, and therefore, are a waste of energy.
It appears the blowers are turned off when there is no production, this is good practice. It is estimated about 32 cfm of blowing is on during production (not all blowers operate), about 45% of the time. Reducing blow load would save an estimated 14 cfm of compressed air, worth 16,400 kWh (1.9 kW peak) and \$2,350 per year in electrical costs. Air Motors A number of air motors drive conveyor belts in the plant. Air motors consume about 10 times the equivalent hp than a direct drive motor, and what’s worse they consume the most air when they are unloaded. ##### Figure 6: This air motor drives a short conveyor. Estimated air motor flow during production is 32 cfm (each motor consumes 18 cfm). Estimated average air motor flow over full system hours is 14 cfm. Estimated savings would be 16,000 kWh and \$2,450 per year. Conversion to electric drive would consume an estimated \$200 in electrical energy to offset some of these savings. Coolers A flow of compressed air is being directed onto the access door handles in a meat smoker (a hot area). Due to the cost of compressed air, this is a less than optimum way to provide protection against heat. You may want to consider providing insulation on the handles. Measured air flow was 4 cfm for each cooler, this application continues to run even during shutdown. If this was eliminated, it would reduce energy consumption by 6,000 kWh (0.7 kW peak), worth \$700 per year in savings.
Air Dryer
The existing air dryer is a heated blower desiccant style, rated at 1,500 cfm. This dryer was malfunctioning during the measurement period and purging excessively, causing it to run the heater and blower more than necessary during the production shift. The unit was regenerating 42% of the time, where with the dry winter moisture, loading the unit should have been regenerating less than 20% of the time.
Due to the malfunction, the unit was putting out compressed air at temperatures measured as high as 286°F (141°C). Compressed air hitting the flow control valve was about 158°F (70°C), a possible cause of the valve failure. This temperature may have also damaged the flow meter.
The compressor service company was called by the operators, and the dryer was discovered in the wrong operating mode (blower cooling mode). The unit was switched, so it operates with compressed air cooling, reducing the outlet temperature significantly. The problem with the dew point control should be investigated, perhaps the measurement probe needs changing (a maintenance item). It could have been damaged by the high temperatures.
Reduction in dryer regeneration power, if the control is repaired, will save an estimated 83,000 kWh (9.5 kW peak) per year, worth about \$9,600 per year in reduced electrical costs. Filters Pressure loss across the air dryer and filters was seen at 9 psi at only 1,000 cfm. This indicates excessive pressure drop (should be about 6 psi at full flow of 1,500 cfm). Normally, the main inlet coalescing compressed air filter and particulate outlet filter on a desiccant dryer have a 3 psid to 10 psid pressure loss across them, throughout the life of the filter. Savings can be gained by upgrading filters to more efficient units, or installing parallel filters (doubling the capacity), as there is a 1% hp reduction for every 2 psid reduction in pressure. Dual parallel filters, if installed with isolation valves, allow the filters to be changed even during full production. Install airless drains on the filters. Estimated savings for this measure would be about 9,100 kWh (1.0 kW peak), worth about \$1,000 per year.
Condensate Drains
Both inlet and outlet dryer filter drains, and the drains on each compressor drops were passing significant air due to manual drainage. Upgrading the compressor, receiver and receiver drains to low loss airless units, rather than cracking open the manual drains, or using leaky drains (small dry tank in compressor room has leaking drain), would save energy. The drain on the spare compressor was measured at 6 cfm. Estimated savings would be 21,000 kWh (2.4 kW peak), worth \$2,500 per year if all these drains were upgraded. Compressor Room Environment The compressors are located in a hot room with numerous large refrigeration compressors. The inlet cooling air is very warm, causing higher than desired discharge temperatures adding extra moisture load on the air dryer, causing it to consume more energy. Every 54°F (12°C) in extra discharge temperature doubles the amount of water in the air. The heat of compression from the compressors is being redirected into the room through air dampers with a summer/winter mode. In summer, the air is directed outdoors, in winter the 138 kW of compressor heat is directed into the compressor room. Thermal imaging showed some of this hot air was being directly sucked into the cooling air intake of the compressors. Discharge temperatures showed outlet temperature of 88°F (31°C) even on a very cold day. Consideration should be given to operating in summer mode all year-round, in order to keep the hot air away from the intake and to help keep the compressor room as cool as possible. Heat Recovery Air compressors produce significant amounts of heat able to be recovered for process requirements, such as hot water pre-heat, boiler make-up pre-heat or to supplement building heat. Most compressor suppliers can provide optional heat recovery systems with their compressors for these purposes. This should be considered in this location, as the compressor heat is not currently being effectively recovered (heat is rejected into an already hot room). There will be about 600,000 kWh of heat available for recovery per year. If this is used for hot water pre-heat could displace up to 41,000 m3 of natural gas heat, worth about \$5,000 per year.
### Summary
Sufficient potential exists for up to an estimated 52% savings in compressed air operating costs over the present configuration. The installation of wet receiver capacity, low loss drains, repair of dryer control, better compressor control, lower plant pressure, reduced inappropriate use and reduced leakage would save an estimated \$79,700 per year in compressed air electrical operating costs. More effective heat recovery might save an additional \$5,000 per year in natural gas costs.
A summary table of the estimated savings is as follows:
For more information contact Ron Marshall, Marshall Compressed Air Consulting, tel: 204-806-2085, email: ronm@mts.net. | 2022-05-24 22:19:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49509507417678833, "perplexity": 3124.6899757649758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00241.warc.gz"} |
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### MATHEMATICS 9709
#### Cambridge International AS and A Level
Name of student Date Adm. number Year/grade Stream Subject Pure Mathematics 1 (P1) Variant(s) P13 Start time Duration Stop time
Qtn No. 1 2 3 4 5 6 7 8 9 10 11 12 Total
Marks 3 3 3 4 8 7 8 8 8 10 9 9 80
Score
Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 12 questions Question 1 Code: 9709/13/M/J/14/1, Topic: Series Find the coefficient of$x$in the expansion of$\displaystyle\left(x^{2}-\frac{2}{x}\right)^{5}$.$[3]$Question 2 Code: 9709/13/M/J/16/1, Topic: Series Find the coefficient of$x$in the expansion of$\displaystyle\left(\frac{1}{x}+3 x^{2}\right)^{5}$.$[3]$Question 3 Code: 9709/13/M/J/21/1, Topic: Integration A curve with equation$y=\mathrm{f}(x)$is such that$\displaystyle \mathrm{f}^{\prime}(x)=6 x^{2}-\frac{8}{x^{2}}.$It is given that the curve passes through the point$(2,7)$. Find$\mathrm{f}(x)$.$[3]$Question 4 Code: 9709/13/M/J/15/2, Topic: Integration A curve is such that$\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=(2 x+1)^{\frac{1}{2}}$and the point$(4,7)$lies on the curve. Find the equation of the curve.$[4]$Question 5 Code: 9709/13/M/J/14/7, Topic: Vectors The position vectors of points$A, B$and$C$relative to an origin$O$are given by $$\overrightarrow{O A}=\left(\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{r} 6 \\ -1 \\ 7 \end{array}\right) \quad \text { and } \quad \overrightarrow{O C}=\left(\begin{array}{l} 2 \\ 4 \\ 7 \end{array}\right)$$$\text{(i)}$Show that angle$B A C=\cos ^{-1}\left(\frac{1}{3}\right)$.$[5]\text{(ii)}$Use the result in part$\text{(i)}$to find the exact value of the area of triangle$A B C$.$[3]$Question 6 Code: 9709/13/M/J/15/7, Topic: Coordinate geometry The point$A$has coordinates$(p, 1)$and the point$B$has coordinates$(9,3 p+1)$, where$p$is a constant.$\text{(i)}$For the case where the distance$A B$is 13 units, find the possible values of$p$.$[3]\text{(ii)}$For the case in which the line with equation$2 x+3 y=9$is perpendicular to$A B$, find the value of$p$.$[4]$Question 7 Code: 9709/13/M/J/20/7, Topic: Trigonometry$\text{(a)}$Show that$\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta} \equiv \frac{2}{\sin \theta \cos \theta}$.$[4]\text{(b)}$Hence solve the equation$\displaystyle \frac{\tan \theta}{1+\cos \theta}+\frac{\tan \theta}{1-\cos \theta}=\frac{6}{\tan \theta}$for$0^{\circ}< \theta <180^{\circ}$.$[4]$Question 8 Code: 9709/13/M/J/15/8, Topic: Differentiation The function$\mathrm{f}$is defined by$\displaystyle\mathrm{f}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$for$x>-1$.$\text{(i)}$Find$\mathrm{f}^{\prime}(x)$.$[3]\text{(ii)}$State, with a reason, whether$\mathrm{f}$is an increasing function, a decreasing function or neither.$[1]$The function$\mathrm{g}$is defined by$\displaystyle\mathrm{g}(x)=\frac{1}{x+1}+\frac{1}{(x+1)^{2}}$for$x<-1\text{(iii)}$Find the coordinates of the stationary point on the curve$y=\mathrm{g}(x)$.$[4]$Question 9 Code: 9709/13/M/J/16/8, Topic: Trigonometry$\text{(i)}$Show that$3 \sin x \tan x-\cos x+1=0$can be written as a quadratic equation in$\cos x$and hence solve the equation$3 \sin x \tan x-\cos x+1=0$for$0 \leqslant x \leqslant \pi$.$[5]\text{(ii)}$Find the solutions to the equation$3 \sin 2 x \tan 2 x-\cos 2 x+1=0$for$0 \leqslant x \leqslant \pi$.$[3]$Question 10 Code: 9709/13/M/J/19/9, Topic: Trigonometry The function$\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$is defined for$0 \leqslant x \leqslant \pi$, where$p$and$q$are positive constants. The diagram shows the graph of$y=\mathrm{f}(x)$.$\text{(i)}$In terms of$p$and$q$, state the range of$\mathrm{f}$.$[2]\text{(ii)}$State the number of solutions of the following equations.$\quad\text{(a)}\mathrm{f}(x)=p+q[1]\quad\text{(b)}\mathrm{f}(x)=q[1]\quad\text{(c)}\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q[1]\text{(iii)}$For the case where$p=3$and$q=2$, solve the equation$\mathrm{f}(x)=4$, showing all necessary working.$[5]$Question 11 Code: 9709/13/M/J/15/10, Topic: Coordinate geometry, Integration Points$A(2,9)$and$B(3,0)$lie on the curve$y=9+6 x-3 x^{2}$, as shown in the diagram. The tangent at$A$intersects the$x$-axis at$C$. Showing all necessary working,$\text{(i)}$find the equation of the tangent$A C$and hence find the$x$-coordinate of$C$,$[4]\text{(ii)}$find the area of the shaded region$A B C$.$[5]$Question 12 Code: 9709/13/M/J/13/11, Topic: Coordinate geometry, Integration The diagram shows part of the curve$\displaystyle y=\frac{8}{\sqrt{x}}-x$and points$A(1,7)$and$B(4,0)$which lie on the curve. The tangent to the curve at$B$intersects the line$x=1$at the point$C$.$\text{(i)}$Find the coordinates of$C$.$[4]\text{(ii)}$Find the area of the shaded region.$[5]\$
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https://pifometre.com/alfred-and-plantagenet/speed-time-distance-formula-pdf.php | # Alfred and Plantagenet Speed Time Distance Formula Pdf
## दुरी समय गति Time Distance Speed Short Tricks In Hindi
### Speed Time and Distance formula shortcut tricks pdf
Time and Distance Aptitude Test Tricks Shortcuts & Formulas. Time, speed, distance and Work questions and answers for SSC CGL exam set-2 PDF. Practice previous year questions on Time and work using provided solutions, tricks and shortcuts for competitive exams., Time, Distance and speed दुरी समय गति संबंधी शार्ट ट्रिक्स गणित मैथ्स रेलगाड़ी के सवाल , रेलगाड़ी के महत्वपूर्ण प्रश्न Maths Train Important Question In this post we are going to share some most important and useful short tricks of time.
### Speed Time and Distance Worksheet
Speed time and distance worksheets. time and distance questions answers mcq of quantitative aptitude are useful for it officer bank exam, ssc, ibps and other competitive exam preparation, This page is on the topic "Time, Speed and Distance". Simple and fully solved problems as a subtopic of Aptitude Questions have been put for practice. Practicing these fully solved questions will help you get a good score in Bank PO, Bank Clerk or other similar Examinations..
Speed Time and Distance shortcut tricks are very important thing to Advertisement know for your exams. Time takes a huge part in competitive exams. If you know time management then everything will be easier for you. Most of us miss... Time, Speed and Distance Exercise - Mathematics or Quantitative Aptitude Questions Answers with Solutions for All other Competitive Exams.
Speed, distance, and time test - 10 questions to answer as quickly as possible. Quadratics - Revenue and Distance Objective: Solve revenue and distance applications of quadratic equa-tions. A common application of quadratics comes from revenue and distance problems. Both are set up almost identical to each other so they are both included together. Once they are set up, we will solve them in exactly the same way we solved the
Time, Speed and Distance: Key Learnings. While the question statement can be varied, the basic formula for TSD (time, speed and distance) remains the same, and that forms the core to handle any question from this critical topic. Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the
Speed, Time, and Distance Worksheet 1 a. Mike rides his bike with a constant speed of 14 miles per hour. How long will he take to travel a distance of 21 miles? 1 b. Nancy roller skates with a constant speed of 12 miles per hour. How long will she take to travel a distance of 18 miles? 2 a. A van moves with a constant speed of 60 miles per hour Dear Readers, We are providing you Important Short Tricks on Speed, Distance & Time which are usually asked in Bank Exams. Use these below given short cuts to solve questions within minimum time. These shortcuts will be very helpful for IBPS PO & other Banking Exam 2019.. Important formulae and facts of Time and Distance
Time, Speed and Distance Exercise - Mathematics or Quantitative Aptitude Questions Answers with Solutions for All other Competitive Exams. This is aptitude questions and answers section on Time and Distance with explanation for various interview, competitive examinations and entrance tests. Latest Time and Distance …
Speed Time and Distance shortcut tricks are very important thing to Advertisement know for your exams. Time takes a huge part in competitive exams. If you know time management then everything will be easier for you. Most of us miss... This is aptitude questions and answers section on Time and Distance with explanation for various interview, competitive examinations and entrance tests. Latest Time and Distance …
Speed, Time, and Distance Worksheet 1 a. Mike rides his bike with a constant speed of 14 miles per hour. How long will he take to travel a distance of 21 miles? 1 b. Nancy roller skates with a constant speed of 12 miles per hour. How long will she take to travel a distance of 18 miles? 2 a. A van moves with a constant speed of 60 miles per hour DISTANCE, TIME, SPEED PRACTICE PROBLEMS 1. If a car travels 400m in 20 seconds how fast is it going? S = d/t s = 400m/ 20 sec = 20m/s 2. If you move 50 meters in 10 seconds, what is your speed?
If you drive a car or have ever flown in an airplane, you’ve probably noticed that time, speed, and distance are related. Here’s the basic formula for distance (d), which equals speed (called velocity in science and represented by v) multiplied by time (t): From this simple formula, you can derive these other formulas as … On top of calculating speed, you will also be expected to be able to rearrange this formula and use it to find distance (when given the speed and time) and time (when given the speed and distance). From now on, let s be speed, t be time, and d be distance. Then, our original equation looks like: s=\dfrac{d}{t}
On top of calculating speed, you will also be expected to be able to rearrange this formula and use it to find distance (when given the speed and time) and time (when given the speed and distance). From now on, let s be speed, t be time, and d be distance. Then, our original equation looks like: s=\dfrac{d}{t} So candidates must focus on this topic and download this Speed Time & Distance pdf to get important questions with best solution regarding Speed Time & Distance. We have put all Previous Year Questions of Speed Time & Distance that are Asked in various Govt & Private Exam. We Are Providing You Free Pdf For 200+ Speed Time & Distance Questions & Solution PDF Download.
01/01/2016 · This video explains the relationship between speed, distance and time. It also explains how to attempt typical examination style questions. Practice Question... 25/03/2018 · Time and Distance Tricks Time Speed And Distance Tricks| mathematics DSSSB, Railway Group D, SBI PO 2018, LDC, SSC CGL, LOCO PILOT, LDC, KVS Hello Ap logo ki …
25/03/2018 · Time and Distance Tricks Time Speed And Distance Tricks| mathematics DSSSB, Railway Group D, SBI PO 2018, LDC, SSC CGL, LOCO PILOT, LDC, KVS Hello Ap logo ki … Time and distance aptitude questions are easy to score area if you know the very basic formula that you learnt in high school. All shortcuts to solve time and distance problems can be easily derived and learnt.
The average speed is given by total distance divided by total time taken; this is the formula to be remembered all the time. Average Speed = The average speed in case of a journey from X to Y at speed of A m/sec and returning back to X at a speed of B m/sec, is Speed, distance and time can be calculated using a magic triangle. D (the distance) goes in the top of the triangle, S (speed) goes in the bottom left of the triangle and T (time) goes in the bottom right of the triangle. If you want to calculate...
In math, distance, rate, and time are three important concepts you can use to solve many problems if you know the formula. Distance is the length of space traveled by a moving object or the length measured between two points. It is usually denoted by d in math problems. Speed, distance and time can be calculated using a magic triangle. D (the distance) goes in the top of the triangle, S (speed) goes in the bottom left of the triangle and T (time) goes in the bottom right of the triangle. If you want to calculate...
Why Aptitude Time and Distance? In this section you can learn and practice Aptitude Questions based on "Time and Distance" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc.) with full confidence. Worksheet # 4: Solve the Distance (d), Rate(r)/Speed and Time(t) Problems Remember to read the problems carefully and set up a diagram or chart to help you set up the equations. Remember to use the right formula: d = rt or r = t d or t = r d 1. Ali drove to the theatre and back again. He took three hours longer to go to the theatre than
Worksheet # 4: Solve the Distance (d), Rate(r)/Speed and Time(t) Problems Remember to read the problems carefully and set up a diagram or chart to help you set up the equations. Remember to use the right formula: d = rt or r = t d or t = r d 1. Ali drove to the theatre and back again. He took three hours longer to go to the theatre than Problems involving Time, Distance and Speed are solved based on one simple formula. Distance = Speed * Time. Which implies в†’ Speed = Distance / Time and. Time = Distance / Speed. Let us take a look at some simple examples of distance, time and speed problems. Example 1. A boy walks at a speed of 4 kmph. How much time does he take to walk a
DISTANCE, TIME, SPEED PRACTICE PROBLEMS 1. If a car travels 400m in 20 seconds how fast is it going? S = d/t s = 400m/ 20 sec = 20m/s 2. If you move 50 meters in 10 seconds, what is your speed? Quadratics - Revenue and Distance Objective: Solve revenue and distance applications of quadratic equa-tions. A common application of quadratics comes from revenue and distance problems. Both are set up almost identical to each other so they are both included together. Once they are set up, we will solve them in exactly the same way we solved the
In math, distance, rate, and time are three important concepts you can use to solve many problems if you know the formula. Distance is the length of space traveled by a moving object or the length measured between two points. It is usually denoted by d in math problems. Trains Distance Speed Time Problems With Solutions Quantitative Aptitude With Chapter-wise Explanations, Trains Distance Speed Time Problems With Solutions PDF Download Distance (D) = Speed (S) Г— Time (T) X kmph = X Г— 5/18 m/s X m/s = X Г— 18/5 kmph. If the ratio of the speeds A & B is a: b, then the ratio of the times taken by them to
Time, Speed and Distance Exercise - Mathematics or Quantitative Aptitude Questions Answers with Solutions for All other Competitive Exams. Worksheet # 4: Solve the Distance (d), Rate(r)/Speed and Time(t) Problems Remember to read the problems carefully and set up a diagram or chart to help you set up the equations. Remember to use the right formula: d = rt or r = t d or t = r d 1. Ali drove to the theatre and back again. He took three hours longer to go to the theatre than
Dear Readers, We are providing you Important Short Tricks on Speed, Distance & Time which are usually asked in Bank Exams. Use these below given short cuts to solve questions within minimum time. These shortcuts will be very helpful for IBPS PO & other Banking Exam 2019.. Important formulae and facts of Time and Distance Speed can be thought of as the rate at which an object covers distance.A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time.
### Speed Distance Time Formula Examples Byjus.com
Speed Distance Time Questions Worksheets and Revision MME. This is aptitude questions and answers section on Time and Distance with explanation for various interview, competitive examinations and entrance tests. Latest Time and Distance …, Problems involving Time, Distance and Speed are solved based on one simple formula. Distance = Speed * Time. Which implies → Speed = Distance / Time and. Time = Distance / Speed. Let us take a look at some simple examples of distance, time and speed problems. Example 1. A boy walks at a speed of 4 kmph. How much time does he take to walk a.
Speed Time and Distance Methods shortcut tricks Math. Time, Speed and Distance: Key Learnings. While the question statement can be varied, the basic formula for TSD (time, speed and distance) remains the same, and that forms the core to handle any question from this critical topic., Time and distance aptitude questions are easy to score area if you know the very basic formula that you learnt in high school. All shortcuts to solve time and distance problems can be easily derived and learnt..
### Speed time and distance worksheets
Time and Distance Important Formulas Aptitude Questions. 25/03/2018 · Time and Distance Tricks Time Speed And Distance Tricks| mathematics DSSSB, Railway Group D, SBI PO 2018, LDC, SSC CGL, LOCO PILOT, LDC, KVS Hello Ap logo ki … Problems involving Time, Distance and Speed are solved based on one simple formula. Distance = Speed * Time. Which implies → Speed = Distance / Time and. Time = Distance / Speed. Let us take a look at some simple examples of distance, time and speed problems. Example 1. A boy walks at a speed of 4 kmph. How much time does he take to walk a.
• Time Speed Distance and Work Formulae for CAT (PDF) Cracku
• Time Speed Distance and Work Formulae for CAT (PDF) Cracku
• Distance Speed Time Formula Questions: 1) A dog runs from one side of a park to the other. The park is 80.0 meters across. The dog takes 16.0 seconds to cross the park. Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the
This page is on the topic "Time, Speed and Distance". Simple and fully solved problems as a subtopic of Aptitude Questions have been put for practice. Practicing these fully solved questions will help you get a good score in Bank PO, Bank Clerk or other similar Examinations. This page is on the topic "Time, Speed and Distance". Simple and fully solved problems as a subtopic of Aptitude Questions have been put for practice. Practicing these fully solved questions will help you get a good score in Bank PO, Bank Clerk or other similar Examinations.
Make customizable worksheets about constant (or average) speed, time, and distance, in PDF or html formats. You can choose the types of word problems, the number of problems, metric or customary units, the way time is expressed (hours/minutes, fractional hours, or decimal hours), and the amount of workspace for each problem. Time and distance aptitude questions are easy to score area if you know the very basic formula that you learnt in high school. All shortcuts to solve time and distance problems can be easily derived and learnt.
SSC Speed And Distance Based Maths Handwritten Notes PDF Download. SSC Speed And Distance Based Maths Handwritten Notes PDF Download – Hello friends, Today we are sharing most important SSC Speed And Distance Based Maths Handwritten Notes PDF Download. SSC Speed And Distance Based Maths Handwritten Notes PDF is very helpful for your upcoming MP Patwari, SSC CGL, IBPS … Improve your science knowledge with free questions in "Calculate speed from time and distance" and thousands of other science skills.
So candidates must focus on this topic and download this Speed Time & Distance pdf to get important questions with best solution regarding Speed Time & Distance. We have put all Previous Year Questions of Speed Time & Distance that are Asked in various Govt & Private Exam. We Are Providing You Free Pdf For 200+ Speed Time & Distance Questions & Solution PDF Download. So candidates must focus on this topic and download this Speed Time & Distance pdf to get important questions with best solution regarding Speed Time & Distance. We have put all Previous Year Questions of Speed Time & Distance that are Asked in various Govt & Private Exam. We Are Providing You Free Pdf For 200+ Speed Time & Distance Questions & Solution PDF Download.
Problems involving Time, Distance and Speed are solved based on one simple formula. Distance = Speed * Time. Which implies в†’ Speed = Distance / Time and. Time = Distance / Speed. Let us take a look at some simple examples of distance, time and speed problems. Example 1. A boy walks at a speed of 4 kmph. How much time does he take to walk a The average speed is given by total distance divided by total time taken; this is the formula to be remembered all the time. Average Speed = The average speed in case of a journey from X to Y at speed of A m/sec and returning back to X at a speed of B m/sec, is
FORMULA : SPEED = Distance Г· Time Round answers to the nearest tenth (one decimal place)! 3 hours, what was his average speed?-made Thrust SSC, would win every TGV from France, can travel at faster speeds than trains in-breaking speed? Lockheed SR71, was able to travel 2200 miles per hour. b. 3 hours? c. 5 hours? _____ _____ If Speed, distance, and time test - 10 questions to answer as quickly as possible.
Speed Time and Distance shortcut tricks are very important thing to Advertisement know for your exams. Time takes a huge part in competitive exams. If you know time management then everything will be easier for you. Most of us miss... Speed, distance and time can be calculated using a magic triangle. D (the distance) goes in the top of the triangle, S (speed) goes in the bottom left of the triangle and T (time) goes in the bottom right of the triangle. If you want to calculate...
In math, distance, rate, and time are three important concepts you can use to solve many problems if you know the formula. Distance is the length of space traveled by a moving object or the length measured between two points. It is usually denoted by d in math problems. Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the
Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the Quadratics - Revenue and Distance Objective: Solve revenue and distance applications of quadratic equa-tions. A common application of quadratics comes from revenue and distance problems. Both are set up almost identical to each other so they are both included together. Once they are set up, we will solve them in exactly the same way we solved the
## 200+ Speed Time & Distance Questions With Solution Free
Time Speed and Distance Exercise Maths Questions. Improve your science knowledge with free questions in "Calculate speed from time and distance" and thousands of other science skills., Time and distance aptitude questions are easy to score area if you know the very basic formula that you learnt in high school. All shortcuts to solve time and distance problems can be easily derived and learnt..
### Time Speed and Distance Exercise Maths Questions
Speed Time and Distance Worksheet. Worksheet on Speed, Distance and Time. Worksheet on Speed, Distance and Time - 1. Worksheet on Speed, Distance and Time - 2. Worksheet on Speed, Distance and Time - 3. After having gone through the stuff given above, we hope that the students would …, Time, speed, distance and Work questions and answers for SSC CGL exam set-2 PDF. Practice previous year questions on Time and work using provided solutions, tricks and shortcuts for competitive exams..
Speed can be thought of as the rate at which an object covers distance.A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time. Speed, distance and time can be calculated using a magic triangle. D (the distance) goes in the top of the triangle, S (speed) goes in the bottom left of the triangle and T (time) goes in the bottom right of the triangle. If you want to calculate...
Time taken to travel the distance at a speed of m km/hr = A/m hr. Time taken to travel the distance at a speed of n km/hr = A/n hr. we see that the total distance of 2A km is travelled in (A)/m+A/n hr. Q. 3. Speed Distance Time Formula is mathematically articulated as: $$x=\frac{d}{t}$$ Where, the Speed is x in m/s, the Distance traveled is d in m, t is the time taken in s. Distance traveled formula is articulated as d = xt. If any of the two variables among speed, distance and time is provided, we can use this formula and find the unknown numeric
This page is on the topic "Time, Speed and Distance". Simple and fully solved problems as a subtopic of Aptitude Questions have been put for practice. Practicing these fully solved questions will help you get a good score in Bank PO, Bank Clerk or other similar Examinations. Improve your science knowledge with free questions in "Calculate speed from time and distance" and thousands of other science skills.
In math, distance, rate, and time are three important concepts you can use to solve many problems if you know the formula. Distance is the length of space traveled by a moving object or the length measured between two points. It is usually denoted by d in math problems. time and distance questions answers mcq of quantitative aptitude are useful for it officer bank exam, ssc, ibps and other competitive exam preparation
Also read CAT level questions on Time and distance, which gives you insights on how to solve questions on this topic. Download Time & Distance formulas PDF. Enroll To CAT 2 Months Crash Course. Outline of various things covered in the time and work formulas pdf are. Formulas for Time, speed and distance; Ratios of speeds, time and distance DISTANCE, TIME, SPEED PRACTICE PROBLEMS 1. If a car travels 400m in 20 seconds how fast is it going? S = d/t s = 400m/ 20 sec = 20m/s 2. If you move 50 meters in 10 seconds, what is your speed?
Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the Time, speed, distance and Work questions and answers for SSC CGL exam set-2 PDF. Practice previous year questions on Time and work using provided solutions, tricks and shortcuts for competitive exams.
Speed Time and Distance formula- Looking for Speed Time and Distance formula shortcut tricks or how to solve time and distance problems. click here for tricks and formula in pdf download.Hi, I am TechMario your personal tutor for govt exams. In this chapter i will teach you tricks and formula for time and distance question and time and work shortcut trick to solve problems quickly. Speed Distance Time Formula is mathematically articulated as: $$x=\frac{d}{t}$$ Where, the Speed is x in m/s, the Distance traveled is d in m, t is the time taken in s. Distance traveled formula is articulated as d = xt. If any of the two variables among speed, distance and time is provided, we can use this formula and find the unknown numeric
Trains Distance Speed Time Problems With Solutions Quantitative Aptitude With Chapter-wise Explanations, Trains Distance Speed Time Problems With Solutions PDF Download Distance (D) = Speed (S) Г— Time (T) X kmph = X Г— 5/18 m/s X m/s = X Г— 18/5 kmph. If the ratio of the speeds A & B is a: b, then the ratio of the times taken by them to Also read CAT level questions on Time and distance, which gives you insights on how to solve questions on this topic. Download Time & Distance formulas PDF. Enroll To CAT 2 Months Crash Course. Outline of various things covered in the time and work formulas pdf are. Formulas for Time, speed and distance; Ratios of speeds, time and distance
If you drive a car or have ever flown in an airplane, you’ve probably noticed that time, speed, and distance are related. Here’s the basic formula for distance (d), which equals speed (called velocity in science and represented by v) multiplied by time (t): From this simple formula, you can derive these other formulas as … Solve Time and Distance Problems using Time and Distance Formulas for IBPS PO, IBPS Clerk, SBI PO, SBI Clerk, SSC CGL, SSC CHSL and other competitive exams. IBPS PO, IBPS Clerk, SBI PO, SBI Clerk, SSC CGL, SSC CHSL and other competitive exams. You can solve Time and Distance Problems within 35 seconds, if you Read More...
SSC Speed And Distance Based Maths Handwritten Notes PDF Download. SSC Speed And Distance Based Maths Handwritten Notes PDF Download – Hello friends, Today we are sharing most important SSC Speed And Distance Based Maths Handwritten Notes PDF Download. SSC Speed And Distance Based Maths Handwritten Notes PDF is very helpful for your upcoming MP Patwari, SSC CGL, IBPS … If you drive a car or have ever flown in an airplane, you’ve probably noticed that time, speed, and distance are related. Here’s the basic formula for distance (d), which equals speed (called velocity in science and represented by v) multiplied by time (t): From this simple formula, you can derive these other formulas as …
Improve your science knowledge with free questions in "Calculate speed from time and distance" and thousands of other science skills. Why Aptitude Time and Distance? In this section you can learn and practice Aptitude Questions based on "Time and Distance" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc.) with full confidence.
DISTANCE, TIME, SPEED PRACTICE PROBLEMS 1. If a car travels 400m in 20 seconds how fast is it going? S = d/t s = 400m/ 20 sec = 20m/s 2. If you move 50 meters in 10 seconds, what is your speed? Trains Distance Speed Time Problems With Solutions Quantitative Aptitude With Chapter-wise Explanations, Trains Distance Speed Time Problems With Solutions PDF Download Distance (D) = Speed (S) Г— Time (T) X kmph = X Г— 5/18 m/s X m/s = X Г— 18/5 kmph. If the ratio of the speeds A & B is a: b, then the ratio of the times taken by them to
Worksheet # 4: Solve the Distance (d), Rate(r)/Speed and Time(t) Problems Remember to read the problems carefully and set up a diagram or chart to help you set up the equations. Remember to use the right formula: d = rt or r = t d or t = r d 1. Ali drove to the theatre and back again. He took three hours longer to go to the theatre than Time, Speed and Distance Exercise - Mathematics or Quantitative Aptitude Questions Answers with Solutions for All other Competitive Exams.
Speed, distance and time can be calculated using a magic triangle. D (the distance) goes in the top of the triangle, S (speed) goes in the bottom left of the triangle and T (time) goes in the bottom right of the triangle. If you want to calculate... Distance, speed and time formulae. All of the calculations in this section will be worked out using the distance, speed and time formulae. An easy way to remember the formulae is to put distance
Worksheet on Speed, Distance and Time. Worksheet on Speed, Distance and Time - 1. Worksheet on Speed, Distance and Time - 2. Worksheet on Speed, Distance and Time - 3. After having gone through the stuff given above, we hope that the students would … In this website we provide few shortcut methods on Speed Time and Distance Methods. Shortcut tricks on Speed Time and Distance Methods will helps you to solve problems mentally and very quickly. We provide solution for faster mathematical calculation.
So candidates must focus on this topic and download this Speed Time & Distance pdf to get important questions with best solution regarding Speed Time & Distance. We have put all Previous Year Questions of Speed Time & Distance that are Asked in various Govt & Private Exam. We Are Providing You Free Pdf For 200+ Speed Time & Distance Questions & Solution PDF Download. DISTANCE, TIME, SPEED PRACTICE PROBLEMS YOU MUST SHOW YOUR WORK. You can use a calculator but you must show all of the steps involved in doing the problem.
Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the DISTANCE, TIME, SPEED PRACTICE PROBLEMS 1. If a car travels 400m in 20 seconds how fast is it going? S = d/t s = 400m/ 20 sec = 20m/s 2. If you move 50 meters in 10 seconds, what is your speed?
10/10/2015 · Download PDF Basic Formula. Time = Distance / Speed ; Time to cross an object moving in the direction of train = Important Note. If the object is of negligible length, then put length of object, L = 0 (i.e. tree, man) If the object is stationary, then put speed of object, V = 0; If the object is moving in opposite direction, then put (-) ve sign before V, so the denominator of the formula This is aptitude questions and answers section on Time and Distance with explanation for various interview, competitive examinations and entrance tests. Latest Time and Distance …
Solve Time and Distance Problems using Time and Distance Formulas for IBPS PO, IBPS Clerk, SBI PO, SBI Clerk, SSC CGL, SSC CHSL and other competitive exams. IBPS PO, IBPS Clerk, SBI PO, SBI Clerk, SSC CGL, SSC CHSL and other competitive exams. You can solve Time and Distance Problems within 35 seconds, if you Read More... Distance, speed and time formulae. All of the calculations in this section will be worked out using the distance, speed and time formulae. An easy way to remember the formulae is to put distance
Speed Time and Distance formula shortcut tricks pdf. Time Speed Distance Questions: Go through the different types of questions based on time, speed and distance to understand the concept better. Read the various formulas that are …, Make customizable worksheets about constant (or average) speed, time, and distance, in PDF or html formats. You can choose the types of word problems, the number of problems, metric or customary units, the way time is expressed (hours/minutes, fractional hours, or decimal hours), and the amount of workspace for each problem..
### Time and Distance Aptitude Questions and Answers
Practice Time Speed & Distance Questions Aptitude page. On top of calculating speed, you will also be expected to be able to rearrange this formula and use it to find distance (when given the speed and time) and time (when given the speed and distance). From now on, let s be speed, t be time, and d be distance. Then, our original equation looks like: s=\dfrac{d}{t}, FORMULA : SPEED = Distance Г· Time Round answers to the nearest tenth (one decimal place)! 3 hours, what was his average speed?-made Thrust SSC, would win every TGV from France, can travel at faster speeds than trains in-breaking speed? Lockheed SR71, was able to travel 2200 miles per hour. b. 3 hours? c. 5 hours? _____ _____ If.
Speed Distance and Time YouTube. Time, Speed and Distance Exercise - Mathematics or Quantitative Aptitude Questions Answers with Solutions for All other Competitive Exams., The average speed is given by total distance divided by total time taken; this is the formula to be remembered all the time. Average Speed = The average speed in case of a journey from X to Y at speed of A m/sec and returning back to X at a speed of B m/sec, is.
### Time Speed And Distance Tricks| Distance Formula
Time and Distance Aptitude Questions and Answers. This is aptitude questions and answers section on Time and Distance with explanation for various interview, competitive examinations and entrance tests. Latest Time and Distance … Quadratics - Revenue and Distance Objective: Solve revenue and distance applications of quadratic equa-tions. A common application of quadratics comes from revenue and distance problems. Both are set up almost identical to each other so they are both included together. Once they are set up, we will solve them in exactly the same way we solved the.
In math, distance, rate, and time are three important concepts you can use to solve many problems if you know the formula. Distance is the length of space traveled by a moving object or the length measured between two points. It is usually denoted by d in math problems. SSC Speed And Distance Based Maths Handwritten Notes PDF Download. SSC Speed And Distance Based Maths Handwritten Notes PDF Download – Hello friends, Today we are sharing most important SSC Speed And Distance Based Maths Handwritten Notes PDF Download. SSC Speed And Distance Based Maths Handwritten Notes PDF is very helpful for your upcoming MP Patwari, SSC CGL, IBPS …
Worksheet # 4: Solve the Distance (d), Rate(r)/Speed and Time(t) Problems Remember to read the problems carefully and set up a diagram or chart to help you set up the equations. Remember to use the right formula: d = rt or r = t d or t = r d 1. Ali drove to the theatre and back again. He took three hours longer to go to the theatre than The formula to find the time when distance and speed are given is . Time = Distance / Speed. Time taken to cover the distance of 160 miles is Time = 160 / 40 . Time = 4 hours. Hence, the person will take 4 hours to cover 160 miles distance at the rate of 40 miles per hour.
time and distance questions answers mcq of quantitative aptitude are useful for it officer bank exam, ssc, ibps and other competitive exam preparation So candidates must focus on this topic and download this Speed Time & Distance pdf to get important questions with best solution regarding Speed Time & Distance. We have put all Previous Year Questions of Speed Time & Distance that are Asked in various Govt & Private Exam. We Are Providing You Free Pdf For 200+ Speed Time & Distance Questions & Solution PDF Download.
25/03/2018 · Time and Distance Tricks Time Speed And Distance Tricks| mathematics DSSSB, Railway Group D, SBI PO 2018, LDC, SSC CGL, LOCO PILOT, LDC, KVS Hello Ap logo ki … The formula to find the time when distance and speed are given is . Time = Distance / Speed. Time taken to cover the distance of 160 miles is Time = 160 / 40 . Time = 4 hours. Hence, the person will take 4 hours to cover 160 miles distance at the rate of 40 miles per hour.
Time, Speed and Distance: Key Learnings. While the question statement can be varied, the basic formula for TSD (time, speed and distance) remains the same, and that forms the core to handle any question from this critical topic. 25/03/2018 · Time and Distance Tricks Time Speed And Distance Tricks| mathematics DSSSB, Railway Group D, SBI PO 2018, LDC, SSC CGL, LOCO PILOT, LDC, KVS Hello Ap logo ki …
FORMULA : SPEED = Distance Г· Time Round answers to the nearest tenth (one decimal place)! 3 hours, what was his average speed?-made Thrust SSC, would win every TGV from France, can travel at faster speeds than trains in-breaking speed? Lockheed SR71, was able to travel 2200 miles per hour. b. 3 hours? c. 5 hours? _____ _____ If Also read CAT level questions on Time and distance, which gives you insights on how to solve questions on this topic. Download Time & Distance formulas PDF. Enroll To CAT 2 Months Crash Course. Outline of various things covered in the time and work formulas pdf are. Formulas for Time, speed and distance; Ratios of speeds, time and distance
01/01/2016В В· This video explains the relationship between speed, distance and time. It also explains how to attempt typical examination style questions. Practice Question... Make customizable worksheets about constant (or average) speed, time, and distance, in PDF or html formats. You can choose the types of word problems, the number of problems, metric or customary units, the way time is expressed (hours/minutes, fractional hours, or decimal hours), and the amount of workspace for each problem.
This is aptitude questions and answers section on Time and Distance with explanation for various interview, competitive examinations and entrance tests. Latest Time and Distance … Speed Distance Time Formula is mathematically articulated as: $$x=\frac{d}{t}$$ Where, the Speed is x in m/s, the Distance traveled is d in m, t is the time taken in s. Distance traveled formula is articulated as d = xt. If any of the two variables among speed, distance and time is provided, we can use this formula and find the unknown numeric
Distance, Rate and Time In flight applications, distance is usually measured in miles. Rate or speed is usually measured in knots (nautical miles per hour.) Time is usually measured in hours. The distance formula is: or d = rt It can also be used to calculate speed of an aircraft when distance and time are given, or to find the time when the Worksheet # 4: Solve the Distance (d), Rate(r)/Speed and Time(t) Problems Remember to read the problems carefully and set up a diagram or chart to help you set up the equations. Remember to use the right formula: d = rt or r = t d or t = r d 1. Ali drove to the theatre and back again. He took three hours longer to go to the theatre than
Speed can be thought of as the rate at which an object covers distance.A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time. Time and distance aptitude questions are easy to score area if you know the very basic formula that you learnt in high school. All shortcuts to solve time and distance problems can be easily derived and learnt.
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https://www.physicsforums.com/threads/what-is-the-rydbergs-formula-will-it-be-used-here.950616/ | # What is the Rydberg's formula? Will it be used here?
## Main Question or Discussion Point
I don't know why, but I have a slight ambiguity regrding the Rydberg formula.
In some places it is written as :
1/λ= Rh(1/n12-1/n22)
In some:
E= -Rh(1/n2)
In some:
1/λ= Rh*(z2/n2)
At some:
1/λ= Rh*(z2)(1/n12-1/n2)2
Please tell me. Where these formula are used? Are they even correct?
Take a look at wiki page for the formula. It contains enough info.
The original equation is as follows:
$$\frac{1}{\lambda} = R ( \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} )$$
by taking $n_{2} \rightarrow$, you will get this:
$$\frac{1}{\lambda} = R ( \frac{1}{n_{1}^{2}} )$$
Note: which means, ionizing an atom by fetching an electron from $n_{1} \rightarrow n_{2}$. Also, I replaced $n_{1}$ by $n$.
What is the energy needed to excite an electron from $n_{1}$ to $n_{2}$? You will need to convert the wavelength you obtained to energy. Hence, you need the equation of energy of a photon 'wiki'. Which is $E=h \nu$, where $\nu$ is the frequency of the photon/light. Now you should work it yourself and find the second equation.
the $z^{2}$ is an extension to the original formula, so that it can be used for other atoms. The formula (the first you wrote) was originally invented for hydrogen atoms only. The one with $z^{2}$ extends the use of the formula for atoms that are similar to Hydrogen.
Please look at the wiki pages.
Disclaimer: I don't know why you have a negative in your second equation. And I don't find your equations consistent with the use of the constant $R$ and $R^{*}$
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The energy is negative because that is the binding energy for the electron with principal quantum number $n$. In a transition of energy levels from a higher state $n_2$ to a lower state $n_1$ , ($n_2>n_1$), a photon of energy $\Delta E=-R(\frac{1}{n_2^2}-\frac{1}{n_1^2})=h \nu$ is emitted in order to conserve energy. There are systems of units (with $\Delta E=\frac{hc}{\lambda}$) that have $h=1$ and $c=1$, but normally this is not the case. $\\$ The simplest derivation of this equation is the Bohr atom model. It does get the right answer for the energies of the states with principal quantum number $n$, but lacks some of the detail that is obtained in much more accurate quantum mechanical calculations. See https://en.wikipedia.org/wiki/Bohr_model
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Take a look at wiki page for the formula. It contains enough info.
The original equation is as follows:
$$\frac{1}{\lambda} = R ( \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} )$$
by taking $n_{2} \rightarrow$, you will get this:
$$\frac{1}{\lambda} = R ( \frac{1}{n_{1}^{2}} )$$
Note: which means, ionizing an atom by fetching an electron from $n_{1} \rightarrow n_{2}$. Also, I replaced $n_{1}$ by $n$.
What is the energy needed to excite an electron from $n_{1}$ to $n_{2}$? You will need to convert the wavelength you obtained to energy. Hence, you need the equation of energy of a photon 'wiki'. Which is $E=h \nu$, where $\nu$ is the frequency of the photon/light. Now you should work it yourself and find the second equation.
the $z^{2}$ is an extension to the original formula, so that it can be used for other atoms. The formula (the first you wrote) was originally invented for hydrogen atoms only. The one with $z^{2}$ extends the use of the formula for atoms that are similar to Hydrogen.
Please look at the wiki pages.
So the true formula is just where z =1 and hence it is omitted while writing?
So the true formula is just where z =1 and hence it is omitted while writing?
No. Read the first wiki page. Section 2.
Isn't the inverse of wavelength just the frequency? Why not just call it the frequency then instead of 1/wavelength? Maybe it's just the expression of the time it takes to make one wavelength? the time of a single packet of electromagnetic energy?
Borek
Mentor
Isn't the inverse of wavelength just the frequency?
No. They are related, but it is not just a simple inverse.
mjc123 | 2020-03-29 06:15:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8765493631362915, "perplexity": 325.7954684965656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370493818.32/warc/CC-MAIN-20200329045008-20200329075008-00279.warc.gz"} |
https://ask.sagemath.org/question/37810/reverse-function-of-latex/ | reverse function of latex()
Hi imagine I got this Latex expression from wikipedia: z_k = 2 \sqrt{\frac{-p}3} \cos{\left(\frac13\arccos{\left(\frac{-q}2\sqrt{\frac{27}{-p^3}}\right)}+ \frac{2k\pi}3\right)}\qquad\mbox{ avec }\qquad k\in{0,1,2}
is there a reverse function kind of retro-latex(z_k = 2 \sqrt{\frac{-p}3} \cos{\left(\frac13\arccos{\left(\frac{-q}2\sqrt{\frac{27}{-p^3}}\right)}+ \frac{2k\pi}3\right)}\qquad\mbox{ avec }\qquad k\in{0,1,2}) which will gives me the expression in classical form (not in Latex) ?
edit retag close merge delete
The expression $$z_k = 2 \sqrt{\frac{-p}3} \cos\left( \frac13\arccos\left(\frac{-q}2\sqrt{\frac{27}{-p^3}}\right) + \frac{2k\pi}3\right)$$ (with $k\in {0,1,2}$) is not too long. Typing with bare hands should be please enough in this case. (It is really hard to parse such a thing. Python does not have latex( myExpression ) . Then sage makes us a present, providing latex. But its reverse engineering... i.e. parse spaces, the superfluous brackets, function names, (sums and products with variables running "implicitly"), and prepositions in all languages in latex expressions taken from all possible wiki or article latex expressions... This is really too much for a rather small community.) | 2018-11-18 21:00:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4668366312980652, "perplexity": 5432.442877809593}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744649.77/warc/CC-MAIN-20181118201101-20181118223101-00494.warc.gz"} |
https://android.stackexchange.com/questions/186357/empty-variables-in-termux-bash | # Empty variables in termux' bash
I want to perform a test: if a certain variable exists, is non-empty, and its value matches an existing directory, then "exists" should be outputted.
I did the following:
$echo$var
$if [ -d$var ]; then echo "exists"; fi
exists
I got "exist", although I did not assign any value to $var. Why did this happen, and how do I test this properly? • If @WillW answered your question, please mark it as accepted! :) – Swivel Jun 4 at 14:42 ## 1 Answer I think you need to wrap$var in quotes.
if [ -d "\$var" ]; then echo "exists"; fi | 2019-10-16 14:02:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6226643323898315, "perplexity": 12367.238675381432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986668994.39/warc/CC-MAIN-20191016135759-20191016163259-00535.warc.gz"} |
https://open.kattis.com/problems/permutationencryption | Kattis
# Permutation Encryption
Working for the Texas Spy Agency, you are in charge of writing software for handling secure communications between your clients who wish to pass you messages without anyone else being able to read them. Therefore, you have been commissioned to write a program which takes messages and encrypts them according to a key that you have. The spy agency is not very technologically advanced, and everything has been done by hand until now, which is why you were hired. Thus, their codes are also fairly simple – they’re just character permutations.
A permutation of length $n$, as you know, is a list of the numbers 1 through $n$ but in some rearranged order. So “1 2 3” is a permutation of length 3, and so is “2 1 3”, but “5 3 6 4 1 2” is a permutation of length 6. Notice that every number between 1 and $n$ appears in the permutation.
A key is a permutation which is used to change the message. For instance, the key “1 2 3 4 5” is a permutation of length 5 which does absolutely nothing – an encrypted message and a decrypted message would be identical. The key “2 1” would reverse every two characters. So the phrase ‘I like ice cream’ would be encrypted as ‘ Iileki ecc erma’. The key “5 1 4 2 3” would encrypt the word ‘howdy’ as ‘yhdow’.
As you might imagine, if the message is longer than the key, just keep applying the key to each set of $n$ characters. One last example. If the key is “4 1 3 5 2 6”, then $n$ is 6, and we would encrypt the message “Four score and seven years ago” as (spaces, vertical bars, and single quotes are added for clarity):
index: 1 2 3 4 5 6|1 2 3 4 5 6|1 2 3 4 5 6|1 2 3 4 5 6|1 2 3 4 5 6|1 2 3 4 5 6
message: 'F o u r s|c o r e a|n d s e v|e n y e a|r s a g o|. '
key: 4 1 3 5 2 6|4 1 3 5 2 6|4 1 3 5 2 6|4 1 3 5 2 6|4 1 3 5 2 6|4 1 3 5 2 6
encrypted: 'r F u o s|e c r o a|s n e d v|y e e n a|a r g s o| . '
Note from this example that if the message length is not a multiple of the key length, your program should pad the original message with extra spaces at the end so that the message length is a multiple of the key length.
## Input
The input to your program is a list of up to 150 messages to encrypt. Each message has two lines (one for the key, one for the message content). The first line starts with an integer $1 \leq n \leq 20$, the length of the key, followed by $n$ integers which form the permutation. The second line contains the message content. Your program should encrypt everything on the second line, spaces and all. Input ends when $n$ is zero.
## Output
For each message, pad the input with spaces as necessary, encrypt the message with the key, and output the encrypted message. Include single quotes around the encrypted message.
Sample Input 1 Sample Output 1
1 1
Four score and seven years ago
2 2 1
our fathers brough forth on this continent a new nation,
5 1 3 2 5 4
conceived in liberty and dedicated to the proposition
10 5 10 8 1 3 6 4 7 2 9
that all men are created equal.
0
'Four score and seven years ago'
'uo rafhtre srbuohgf rohto nhtsic noitentna n wen taoi,n'
'cnoeciev di nilbreyt na dddeciaet dt ohtep orpsotiino '
' mltaatlh rece ea nr luaeedqta . ' | 2017-12-12 00:35:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28739041090011597, "perplexity": 1232.3998816799178}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948514238.17/warc/CC-MAIN-20171212002021-20171212022021-00022.warc.gz"} |
http://solvedmathproblems.com/work-vector-field-1/ | Work Done By Force Field on Particle (Vector Fields) Example 1
Find the work done by the force field $$F(x,y) = {x^2}\mathop i\limits^ \to + xy\mathop j\limits^ \to$$ on a particle that moves once counterclockwise around the circle governed by $${x^2} + {y^2} = 4$$.
We need to make a vector equation for "r", which will tell us the path this goes in, and we should use polar coordinates since we see we're given a circle. We're told the path is a circle of radius 2 (square root the 4).
In other words, $r(t) = \left\langle {2\cos (t),2\sin (t)} \right\rangle$ for $$0 \le t \le 2\pi$$
Why? Treat the components in r(t) as x and y, plug the x component into x squared, and plug the y component into y squared, and you get 4 = 4 as a check. We're also told once counterclockwise, so it starts from 0 to $$2\pi$$ since each quadrant is $$\frac{\pi }{2}$$.
We need to find the derivative of r(t). $r'(t) = \left\langle { - 2\sin (t),2\cos (t)} \right\rangle$
Then we find F(r(t)), which is just substituting our r(t) into F(x,y). $F\left( {r(t)} \right) = \left\langle {4{{\cos }^2}(t),4\sin (t)\cos (t)} \right\rangle$
Then all we need to find is $$\int_0^{2\pi } {\left( {F \cdot dr} \right)} dt$$. This is just the dot product between F(r(t)) and r'(t) and we use our domain (once counterclockwise means 0 to 2pi) as the integral boundaries. From there it's algebra and trigonometry in the integral.
$\begin{array}{l}\int_0^{2\pi } {\left( {4{{\cos }^2}(t)\left( { - 2\sin (t)} \right) + 4\sin (t)\cos (t) \cdot 2\cos (t)} \right)} dt\\\int_0^{2\pi } {\left( { - 8{{\cos }^2}(t)\sin (t) + 8{{\cos }^2}(t)\sin (t)} \right)} dt\\ = 0\end{array}$
So the work done by the force field is 0. Why would this happen? The vectors in the field are just perpendicular to the path r(t), it's never pushing/pulling the particular in the direction it's going in.
We have examples on determining on whether a vector field is conservative or not, finding the potential of a vector field, and finding the work involving particles and vector fields below:
Work Done By Force Field on Particle (Vector Fields) Ex2
Finding Potential of Conservative Vector Field and Work Ex1
Finding Potential of Conservative Vector Field Ex2
Finding Potential of Conservative Vector Field Ex3 | 2019-12-06 11:33:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7875456809997559, "perplexity": 440.01458192492856}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540487789.39/warc/CC-MAIN-20191206095914-20191206123914-00457.warc.gz"} |
https://help.altair.com/vwt/topics/solvers/ultraFluidX/param_meshing_r.htm | # <meshing>
During meshing, ultraFluidX discretizes the volumetric simulation domain into cubic elements (“voxels”) for which the flow field is computed. The <meshing> category specifies the main parameters for the volume meshing and the refinement zones.
## <meshing> - <general>
The <general> category contains the following parameters:
<coarsest_mesh_size>
Specifies the coarsest size (edge length) of the voxels in $\left[\text{m}\right]$ that ultraFluidX uses to discretize the domain. Higher resolution than the coarsest mesh size in regions of interest is specified via the <refinement> category.
<mesh_preview>
Specifies if a preview of the volume mesh is exported prior to the actual simulation. If set to true, ultraFluidX writes an EnSight Gold file (uFX_meshPreview.case and corresponding *.geo and *.data files) into the uFX_meshData folder.
Information included in the mesh preview is the state of the voxels (solid: -1, fluid: 0), the applied boundary conditions (none and solid: -1, none and fluid: 0, moving/rotating: 1, no-slip: 2, slip: 3, outlet: 4, inlet: 5), the source nodes (none and solid: -1, none and fluid: 0, numbering from 1 onwards according to number of source regions) and the rotating nodes (none and solid: -1, none and fluid: 0, numbering from 1 onwards according to number of rotating regions).
Default = false
<mesh_export>
Specifies if the mesh is written to disk after the meshing is completed. If set to true, ultraFluidX will write a mesh source file uFX_mesh.bin to the uFX_meshData folder, that can then be used as input for a future simulation. Refer to the <mesh_source_file> option.
Default = false
<mesh_source_file>
(Optional) Specifies the path to a mesh source file. If specified, ultraFluidX skips the meshing step and read in the mesh topology from the specified mesh source file instead. Generally, only mesh source files that were written by the same version of ultraFluidX (by setting <mesh_export> to true) can be used.
## <meshing> - <refinement>
Refinement zones are specified via the <refinement> category. Possible sub-categories are <box>, <offset> and <custom> and they all share the parameter <refinement_level> that specifies the refinement of the zone with regards to the coarsest level. With each additional level the coarsest mesh size (corresponding to level 0) is divided by 2. In general, the voxel size of the nth refinement level is defined as: (1) ${\text{Δx}}_{\text{n}}\text{ }\text{=}\text{ }\frac{{\text{Δx}}_{\text{0}}}{{\text{2}}^{\text{n}}}$ Where ${\text{Δx}}_{\text{0}}$ is the voxel size at the coarsest refinement level.
<box>
This category contains all refinement zones with cuboid shape, it can have an arbitrary number of children named <box_instance> with the following parameters:
<name>
(Optional) Name of the instance.
<refinement_level>
Refinement level of the instance with regards to the coarsest level.
<bounding_box>
Bounding box of the instance, must be specified via the child parameters <x_min>, <x_max>, <y_min>, <y_max>, <z_min> and <z_max> in $\left[\text{m}\right]$.
<offset>
This category contains all refinement zones that are an offset of the whole surface mesh or a specific part. It can have an arbitrary number of children named <offset_instance> with the following parameters:
<name>
Name of the instance (optional).
<refinement_level>
Refinement level of the instance with regards to the coarsest level.
<normal_distance>
Distance of the offset to the surface, that is, the effective “thickness” of the zone. Specified in $\left[\text{m}\right]$.
Note: A minimum number of 4 voxels will always be applied.
<parts>
If the offset is not applied to the whole surface mesh, an arbitrary number of individual parts can be specified via the child parameter <name> within the <parts> category. The specified names must exactly match the respective part names in the STL file.
<custom>
This category contains all refinement zones that are specified via a volume with arbitrary shape contained in the STL file. Any volume that you want to use for custom refinement needs to be identified as a separate “solid” within the STL file, and the name of this solid needs to be passed to ultraFluidX in the <parts> - <name> section.
The volume must be closed and all normals of the volume must point outwards.
CAUTION:
If the volume is included in the STL, but not specified as a custom zone, it will be treated as a solid by ultraFluidX.
The category can have an arbitrary number of children named <custom_instance> with the following parameters:
<name>
Name of the instance (optional).
<refinement_level>
Refinement level of the instance with regards to the coarsest level.
<parts>
<name>
The closed volume(s) contained in the STL file must be specified via the child parameter <name> within the <parts> category, several volumes can be specified via repeated usage of <name> to assign the same refinement level to all of them. The specified names must exactly match the respective part names in the STL file.
## <meshing> - <overset>
This category contains all parameters needed to define a moving overset grid region.
<rotating> - <rotating_instance>
In these grid regions, a local frame of reference is defined and the surface geometry and voxels are rotated each timestep according to the specified rotational speed.
Note: The rotating volume must be axisymmetric.
CAUTION:
All geometry parts inside the moving mesh region will rotate with the mesh. Care is needed at the interface when setting up a rotating volume in the vicinity of geometry parts. Refer to separate setup recommendations.
<name>
(Optional) Name of the rotating instance. If not specified, a generic name is attributed.
<rpm>
This parameter indicates the rotational speed of the moving overset grid region in revolutions per minute.
<center>
<x_pos>, <y_pos>, <z_pos>
The center point around which the moving region rotates.
<axis>
<x_dir>, <y_dir>, <z_dir>
The axis of rotation of the overset mesh zone.
<parts>
<name>
This section contains the STL parts that enclose the rotating region.
Note: The parts must form a closed volume, and normals must point outwards.
The rotating region needs to be axisymmetric, such as cylindrical or toroid shape. | 2023-03-21 08:58:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41813570261001587, "perplexity": 2372.170969296083}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00404.warc.gz"} |
https://planetmath.org/EquitableMatricesOfOrder2 | # equitable matrices of order $2$
The most general $2\times 2$ equitable matrix is of the form
$M=\begin{pmatrix}1&\lambda\\ 1/\lambda&1\end{pmatrix}$
for some $\lambda>0$.
Let us consider the matrix
$M=\begin{pmatrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{pmatrix}.$
A necessary and sufficient condition for $M$ to be an equitable matrix is that $m_{11},m_{12},m_{21},m_{22}>0$ and
$\displaystyle m_{11}=m_{11}m_{11},$ $\displaystyle m_{11}=m_{12}m_{21},$ $\displaystyle m_{12}=m_{11}m_{12},$ $\displaystyle m_{12}=m_{12}m_{22},$ $\displaystyle m_{21}=m_{21}m_{11},$ $\displaystyle m_{21}=m_{22}m_{21},$ $\displaystyle m_{22}=m_{21}m_{12},$ $\displaystyle m_{22}=m_{22}m_{22}.$
It follows that $m_{11}=m_{22}=1$, and $m_{12}=1/m_{21}$.
Title equitable matrices of order $2$ EquitableMatricesOfOrder2 2013-03-22 14:58:30 2013-03-22 14:58:30 matte (1858) matte (1858) 4 matte (1858) Example msc 15-00 | 2020-04-04 23:58:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 18, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8965221047401428, "perplexity": 646.6632588499714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370526982.53/warc/CC-MAIN-20200404231315-20200405021315-00010.warc.gz"} |
https://www.gamedev.net/forums/topic/562035-scenegraphtransformations-question/ | # SceneGraph/Transformations question
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Hi guys, I've been browsing these forums for quite a while now, but registered this account because I need to start asking some questions, if you'd be so kind! A little bit of back-story: I'm implementing a 2D scene graph. At the moment, I have transformation nodes, and geometry nodes descending from a single abstract parent class, Node_Component. Obviously, this will be expanded with more subclasses in the future, but at the moment I am just trying to get the basic structure sorted. Transformation nodes have a vector of child node_components (these can be transformation or geometry nodes). At the moment, rendering is done like this: when a new geometry object is attached to a node to be rendered, it pushes its set of vertices to a static std::list in the Geometry class, and is handed a pointer to its iterator within that list. I am using a list because when geometry is deleted, the pointers to other Geometry objects are not invalidated. When a transformation node is transformed, it updates all its children, so all attached Geometry nodes will get their new vertex co-ordinates from the respective transformation matrices, and use their pointer to update their vertices in the std::list. Once all transformations are complete, the application copies the list into an std::vector so that it is in a format that can be rendered using a vertex array (I'll be using a VBO when i get round to it) by openGL. Now, this is fine. The only nodes that are updated are the ones that need updating, increasing performance. My problems, and therefore my questions, are as follows: First question: How can I increase performance when a lot of objects are being updated? When I have a large amount of objects moving (my test is 1 geometry node for each of 10,000 transformation nodes attached to one rotating node, causing each of 20,000 nodes to be updated onscreen per frame), the performance falters significantly; is there any way to alleviate this? Second question (the important one!): Offscreen transformations take just as long as onscreen transformations! If i have a level in a game with 100,000 transformation nodes, but only 5 are onscreen, the other 999,995 are still updated each frame for the transformations! Is there any way I can alleviate this? Thanks for reading this wall of text, Fareed
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The method the OP describes is far from being efficient (if I understand it correctly). Never transform the vertices directly if there is not a really good reason for. Instead, use the Transformation nodes to build up a local-to-global transformation matrix, activate the correct matrix (e.g. using glLoadMatrix(...)), and render the geometry then as is. In that way you let OpenGL transform the particular vertices (what it does anyway) and save both performance and memory. Moreover, since the geometry is (often) not touched by you, you can push it as static geometry onto the graphics card.
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Quote:
Original post by fareed_dFirst question: How can I increase performance when a lot of objects are being updated? When I have a large amount of objects moving (my test is 1 geometry node for each of 10,000 transformation nodes attached to one rotating node, causing each of 20,000 nodes to be updated onscreen per frame), the performance falters significantly; is there any way to alleviate this?Second question (the important one!): Offscreen transformations take just as long as onscreen transformations! If i have a level in a game with 100,000 transformation nodes, but only 5 are onscreen, the other 999,995 are still updated each frame for the transformations! Is there any way I can alleviate this?
The described scenarios seem me somewhat artificial. Why the hell should I have so much transformations for so few geometry? Okay, you said it is "a test", but for what is such a test good for if reality will never looks like?
However, you suffer from the belief that a scene graph is the top of possibilities. You must notice that a scene graph cannot fulfil all senseful tasks at once. Therefore it is the case that nowadays a scene graph is seen just as one management structure besides others, more specialized ones. E.g. a spatial structure like a quadtree (speaking of 2D, right?) may be used to determine faster what geometry nodes are visible at all (keywords "visibility culling", "frustum culling", ...). A dependency graph can be used to manage the transformation nodes, perhaps with "dirty flag" propagation to perform re-calculations only when needed. Other structures may be applied as well.
Furthermore it may be that transformations, although expressed by one node per elemental transformation, can be joined to a more complex one. E.g. using X3D's Transform node instead of having a dozen single nodes in a chain.
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Thanks for your reply, I'll be looking into the stuff you mentioned!
Quote:
Original post by haegarrThe described scenarios seem me somewhat artificial. Why the hell should I have so much transformations for so few geometry?
I said 1 geometry node PER transformation node; 10,000 geometry nodes, 10,000 transformation nodes.
Quote:
Original post by haegarrThe method the OP describes is far from being efficient (if I understand it correctly). Never transform the vertices directly if there is not a really good reason for. Instead, use the Transformation nodes to build up a local-to-global transformation matrix, activate the correct matrix (e.g. using glLoadMatrix(...)), and render the geometry then as is. In that way you let OpenGL transform the particular vertices (what it does anyway) and save both performance and memory. Moreover, since the geometry is (often) not touched by you, you can push it as static geometry onto the graphics card.
transformation nodes DO build up a local to global matrix. When a geometry node is updated, it uses its parent's matrix and glMultMatrixf to get its world coords, which are then pushed to the vertex array. Rendering the geometry there and then with immediate mode would require every node to be rendered in that way.
The main thing that I wanted to know really was if there was a way to speed up/eliminate the need to update transformations that are happening offscreen.
thanks again.
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Well, this
Quote:
Original post by fareed_dWhen a transformation node is transformed, it updates all its children, so all attached Geometry nodes will get their new vertex co-ordinates from the respective transformation matrices, and use their pointer to update their vertices in the std::list.Once all transformations are complete, the application copies the list into an std::vector so that it is in a format that can be rendered using a vertex array...
sounds to me that not OpenGL but your application is doing the transformation on vertices! But perhaps I'm wrong.
Quote:
Original post by fareed_dtransformation nodes DO build up a local to global matrix. When a geometry node is updated, it uses its parent's matrix and glMultMatrixf to get its world coords, which are then pushed to the vertex array. Rendering the geometry there and then with immediate mode would require every node to be rendered in that way.
glMultMatrix and similar are done by the driver in software. Hence you have no real advantage using them if parts of the tree are not changing their transformations. You should instead use a dependency graph and calculate the local-to-global matrices manually. Store the respective matrix with the belonging node to implement a kind of cache.
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Quote:
sounds to me that not OpenGL but your application is doing the transformation on vertices! But perhaps I'm wrong.
What are you suggesting? Is there a way of getting glVertexPointer to take vertices which have not been transformed into global co-ordinates?
Quote:
glMultMatrix and similar are done by the driver in software. Hence you have no real advantage using them if parts of the tree are not changing their transformations. You should instead use a dependency graph and calculate the local-to-global matrices manually. Store the respective matrix with the belonging node to implement a kind of cache.
this is what i mean by "its parent's matrix"; ill paste the code:
void Node::update(){ glLoadMatrixf(mParent->mMatrix); glTranslatef(mPosition.x, mPosition.y,0); glRotatef(mPosition.r,0,0,-1); glGetFloatv(GL_MODELVIEW_MATRIX,mMatrix); std::vector<Node_Component*>::iterator it; for (it = mNodes.begin(); it != mNodes.end(); it++) (*it)->update();}
here, mMatrix is the "respective matrix" that is stored with the "belonging node", as you put it. This code is what happens when the position of this node (or a parent of this node) has changed; it gets its parent matrix, multiplies it with its local position, then stores that as its own matrix. It then updates its children.
P.S. I don't intend to load into the modelview matrix and transform like that forever; it's temporary until I write and benchmark faster matrix multiplication stuff
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Quote:
Original post by fareed_dWhat are you suggesting? Is there a way of getting glVertexPointer to take vertices which have not been transformed into global co-ordinates?
For sure. The entire mechanic of the MODELVIEW matrix is just for that case: Pushing geometry defined in a local space onto the GPU, and let the GPU do the local-to-global(-to-view) processing.
Quote:
Original post by fareed_dthis is what i mean by "its parent's matrix"; ill paste the code:void Node::update(){ glLoadMatrixf(mParent->mMatrix); glTranslatef(mPosition.x, mPosition.y,0); glRotatef(mPosition.r,0,0,-1); glGetFloatv(GL_MODELVIEW_MATRIX,mMatrix); std::vector::iterator it; for (it = mNodes.begin(); it != mNodes.end(); it++) (*it)->update();}here, mMatrix is the "respective matrix" that is stored with the "belonging node", as you put it. This code is what happens when the position of this node (or a parent of this node) has changed; it gets its parent matrix, multiplies it with its local position, then stores that as its own matrix. It then updates its children.
So you use the gl routines as a vector math library (that wasn't clear from your previous post). Do you consider the fact that the MODELVIEW matrix contains the VIEW matrix, too? I mean, changing the camera will immediately make the entire tree of transformations invalid if not handled correctly. (Or, maybe, you never change the camera.)
Another thing is the aforementioned "dirty propagation". It is not clear yet _when_ you invoke Node::update() exactly. If you invoke it on every change of a transformation node, then you may re-compute transformations "just for fun", because a milli second later (but still before rendering) another change in the tree somewhere upwards may cause another re-computation. If such situations may occur in your solution, then please don't update() immediately but propagate a dirty state flag to all dependents that are currently not already dirty flagged. Then do the computation of the transformation not in the push model (as is done now) but in the pull model: When a Node is asked for its global matrix, it should look whether it is marked dirty. If not, it returns the previously computed matrix. Otherwise it asks it parent Node for its global matrix and uses it for computing the own global matrix as usual. This way you do the re-computation only for those nodes that are influenced by a change and are used in some way that requires the global matrix.
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Quote:
Original post by haegarr
Quote:
Original post by fareed_dWhat are you suggesting? Is there a way of getting glVertexPointer to take vertices which have not been transformed into global co-ordinates?
For sure. The entire mechanic of the MODELVIEW matrix is just for that case: Pushing geometry defined in a local space onto the GPU, and let the GPU do the local-to-global(-to-view) processing.
Wow! I didn't realise that I could do that! That makes things a lot simpler, thanks.
EDIT: This would mean using several glVertexPointers, correct? There are several Geometry nodes, each with their own width and height; at the moment I am turning their width and height into local vertices, then multiplying them by the modelview matrix and storing them in ONE array which I point to with glVertexPointer.
As for your other point, I've effectively made the transformation methods safe; if someone rotates a node, then rotates it again, the update() is only called once for the combined transformation when the frame is processed.
[Edited by - fareed_d on February 13, 2010 9:21:12 AM]
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Quote:
Original post by fareed_dThis would mean using several glVertexPointers, correct? There are several Geometry nodes, each with their own width and height; at the moment I am turning their width and height into local vertices, then multiplying them by the modelview matrix and storing them in ONE array which I point to with glVertexPointer.
The way is to store each different geometry (i.e. their vertices and perhaps their indices) in its own VBO or at least in its own section of a VBO. Then set-up the MODELVIEW matrix from a specific Node, address the VBO with the correct glVertexPointer, and invoke its rendering. Then proceed with the next Node that provides geometry.
Notice that here is another possibility of saving runtime. If you have the case that several nodes have the same geometry, then you can re-use the vertex pointer set-up if you render all Nodes with that geometry in sequence. That would avoid some cost hidden in driver set-up. You may also try to use instancing in such a case.
Quote:
Original post by fareed_dAs for your other point, I've effectively made the transformation methods safe; if someone rotates a node, then rotates it again, the update() is only called once for the combined transformation when the frame is processed.
This is okay, although dirty flag propagation along with the pull model goes somewhat further.
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Quote:
Original post by haegarr
Quote:
Original post by fareed_dThis would mean using several glVertexPointers, correct? There are several Geometry nodes, each with their own width and height; at the moment I am turning their width and height into local vertices, then multiplying them by the modelview matrix and storing them in ONE array which I point to with glVertexPointer.
The way is to store each different geometry (i.e. their vertices and perhaps their indices) in its own VBO or at least in its own section of a VBO. Then set-up the MODELVIEW matrix from a specific Node, address the VBO with the correct glVertexPointer, and invoke its rendering.
If the rendering was invoked from each node, surely this would mean that every node that contained geometry would need to be updated every frame?
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× | 2018-06-24 03:55:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34894096851348877, "perplexity": 1623.7547356076107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866191.78/warc/CC-MAIN-20180624024705-20180624044705-00305.warc.gz"} |
https://combine.se/greedy-vs-centralized-mpc-for-platooning/ | Greedy vs Centralized MPC for platooning | Combine
### Greedy vs Centralized MPC for platooning
Have you seen a commercial of a train of cars traveling on a highway and keeping the same inter-vehicle distances and speeds? Yes! You think it is fun, comfortable and makes our trips easier and not nerve-racking. Guess what? It is more advantageous than what you think. Believe it or not, it can be a significant contribution to save the planet from the increasing atmospheric concentration of carbon dioxide and a great deal for the economy as well! However, the latest control strategies proposed that varying the distances and speeds at hilly terrains could even save more fuel. Curious about how the control design will look like? Then you are in the right place.
#### Introduction
One of the things that has been always drawing my attention is the automated
vehicular control strategies and how they could reshape the transport sector
dramatically. One of the methods that many automotive manufacturers have
been recently developing is what is called platooning. A platoon is a convoy of
trucks that maintains fixed inter-vehicular distances, as shown in the Figure 1,
and usually applied on highways.
Figure 1: Trucks Platoon
The advantages go beyond the driver’s convenience and comfort. Having a lead
truck with a large frontal-area would reduce the air drag force acting on the
succeeding vehicles. Therefore, the required torque to drive the trucks at cer-
tain speed will be decreased which lead to less fuel consumption. That means,
of course, less CO2 emissions and lower financial burdens.
However, in a single-vehicle level, there is another approach that has been inves-
tigated for a better fuel economy. This approach utilizes the future topography
information in order to optimize the speed and the gear for a vehicle travelling
in a hilly terrain by exploiting the vehicles’ potential and kinetic energies stor-
ages. In this approach the velocity will vary along the road depending on the
platooning approach in which vehicles maintain almost the same speed along
#### HOW TO HANDLE IT?
A combination between these approaches could be implemented using the model
predictive control (MPC) scheme. Since there are many process constraints,
such as inter-vehicular distances, engine maximum torque, road velocity limits,
etc. MPC is a perfect candidate to handle these constraints especially that in
many cases the system will be operating close to the limits. The control design
could be handled in two approaches, the centralized control design and the
decoupled control design. In the centralized controller, as shown in the Figure
2, all the vehicles’ private data such as mass, engine specs, etc. in addition to
their states such as velocity and time headway are sent to the central predictive
controller via vehicle to vehicle communication, could be in one of the trucks
probably the lead vehicle or even in a cloud. One of the methods used for optimal
control is the convex quadratic programming problem (CQPP) in which every
local minimum is a global minimum. The problem is as follows
$$min\,z = f_0(x) \\ f_i(x) \leq 0 \\ Ax = b$$
Where f0,f1,f2, …, fm, is the objective function, and the inequality constraints
are convex functions. However, the equality constraints are affine functions.
In the platoon case, some convexification is needed in order to get CQPP. Hense,
the problem is solved and the optimal speed and time headway references are
sent back to the vehicles’ local controllers. This approach optimizes the fuel
consumption for the whole platoon rather than individual vehicles in which the
group interest comes first. One of the drawbacks of this approach is that in order
to solve the problem you need to handle huge matrices since all the vehicles info
is handled at once. In other words, this approach is rather computationally
expensive.
Figure 2: Centralized adaptive cruise control
The decoupled architecture, as depicted in the Figure 3, could be a solution for
the computation capacity issues. Instead of handling the quadratic program-
ming (QP) problem for the whole platoon, each vehicle considers itself, which is
why called greedy. The problem starts to be solved from the leading vehicle and
goes backwards. Each vehicle solves the QP, considering the gaps in front of the
vehicle and the road topography, and sends states to the succeeding vehicles.
The pros of this approach are that trucks need not to share their private data
and the matrices sizes are much smaller. So the computation time is less than in
the greedy control strategy but the solution is not as optimal as the centralized
controller.
Figure 3: Greedy approach
#### CHALLENGES
As it is mentioned above, formulating a convex quadratic programing problem
is used to get the fuel-saving velocities. Since the vehicle dynamics are quite
nonlinear, linear approximations are needed, therefore, finding an appropriate
velocity reference is essential, assuming that the vehicle will be driven close
to the reference. Finding such reference should consider many factors such as
maximum traction force along the road, road limits and the cruise speed set by
the driver. One of the other challenges is gear optimization which could be solved
using dynamic programming. The complexity of dynamic programing problem
increases exponentially with the rise of the vehicles number, as a result, the
problem become computationally demanding, therefore, it is not very reliable
for the real-time implementation. | 2020-07-08 21:35:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.570709764957428, "perplexity": 1301.2905802349312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897707.23/warc/CC-MAIN-20200708211828-20200709001828-00458.warc.gz"} |
http://www.gsd.uab.es/index.php?view=detail&id=1573&option=com_simplecalendar&Itemid=62&lang=en | # Spiders' webs in the punctured plane
Date:
18.03.19
Times:
09:00 to 10:00
Place:
IMUB-Universitat de Barcelona
Speaker:
David Martí-Pete
University:
IMPAN
Abstract
Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in the complex plane. We show that there are many transcendental self-maps of $\mathbb C^*$ for which the Julia set is such a spider's web, and we construct the first example of a transcendental self-map of $\mathbb C^*$ for which the escaping set is such a spider's web. This is a joint work with Vasso Evdoridou and Dave Sixsmith. | 2019-11-21 02:45:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8049971461296082, "perplexity": 342.5823119302755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670729.90/warc/CC-MAIN-20191121023525-20191121051525-00414.warc.gz"} |
http://math.stackexchange.com/questions/232580/trigonometry-identities-if-2-sinx-y-sinxy-find-frac-tanx-t | # - Trigonometry Identities - If $2\sin(x-y) = \sin(x+y)$, find $\frac{\tan(x)}{\tan(y)}$
If $2\sin(x-y) = \sin(x+y)$, find $\displaystyle \frac{\tan(x)}{\tan(y)}$
= 2(sinxcosy-sinycosx) - sinxcosy - sinycosx = sinxcosy - sinycosx
//I'm not sure if bringing the sin(x + y) to the left side was right or wrong, just an idea I had.
-
Why is this urgent? – Jason DeVito Nov 8 '12 at 1:39
Should we answer it, perhaps it prevents the next album of Justin Bieber to be released$\ldots$ – Jean-Sébastien Nov 8 '12 at 1:43
David, all kidding aside, you're question will get a better reception if you let us know why it interests you, what you know about it, what ideas you might know that are relevant to solving it, where you got stuck, and so on. – Gerry Myerson Nov 8 '12 at 1:45
@DavidSalib Welcome to math.SE. In order to get the best possible answers, it is helpful if you say what your thoughts on the problem are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level. Also, kindly avoid phrases like "Prove this", "Show that", "Urgent" etc. Also, avoid capital letters since people interpret this as shouting at someone. If it is a homework, kindly tag it as homework. – user17762 Nov 8 '12 at 1:48
Ok thank you @Marvis – DavidSalib Nov 8 '12 at 1:49
Hint: Use $\sin(x+y)=\sin x\cos y+\cos x\sin y=\cos x\cos y(\tan x+\tan y)$ and
$\sin(x-y)=\sin x\cos y-\cos x\sin y=\cos x\cos y(\tan x-\tan y)$
-
Ok, I see what you did... should I bring the right side over to the left side OR simplify each side if possible? – DavidSalib Nov 8 '12 at 1:51
You should divide the first of Avatar's equations by the second. – Gerry Myerson Nov 8 '12 at 1:53
Therefore, sin(x+y)/sin(x-y) ? – DavidSalib Nov 8 '12 at 1:54
Yes, David; and you know that quotient is 2. Now, what happens on the other side of the equation? – Gerry Myerson Nov 9 '12 at 0:11
Now that the question is no longer homework, I can expand on Avatar's solution. Assuming $\cos x\text{ and }\cos y$ aren't zero (otherwise the tangents are undefined), we'll have \begin{align} 2\sin(x-y) &= \sin(x+y) & \text{so, by definition}\\ 2(\sin x \cos y-\cos x\sin y) &= \sin x\cos y + \cos x\sin y & \text{so}\\ 2\sin x \cos y-2\cos x\sin y &= \sin x\cos y + \cos x\sin y & \text{collect terms to get}\\ \sin x\cos y &= 3\cos x\sin y & \text{divide by \cos x\cos y}\\ \tan x &= 3\tan y \end{align} so $$\frac{\tan x}{\tan y} = 3$$ | 2014-04-25 08:45:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9911366105079651, "perplexity": 2755.809601079359}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00089-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=MR2279981 | MathSciNet bibliographic data MR2279981 (2007k:40008) 40C05 (46A45) Mursaleen; Mohiuddine, S. A. Double $\sigma$$\sigma$-multiplicative matrices. J. Math. Anal. Appl. 327 (2007), no. 2, 991–996. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | 2015-01-27 08:35:10 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9956048130989075, "perplexity": 6554.467615737049}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115862141.23/warc/CC-MAIN-20150124161102-00068-ip-10-180-212-252.ec2.internal.warc.gz"} |
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# Multiplication
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Multiplication is the binary mathematical operation of scaling one number or quantity by another (multiplying). It is one of the four basic operations in elementary arithmetic (with addition, subtraction and division). The result of this operation is called product and the multiplied numbers are called factors, or the multiplier and the multiplicand. The product of a and b may be written as a × b or a.b or simply denoted by juxtaposition ab when a and b are symbols or formulae rather than numbers.
The product m × n of a positive whole number m times another quantity n agrees with the result of successively adding n to itself a total of m times. For example, 2 multiplied by 3 (often said as "3 times 2") gives the same result as 2 taken 3 times, i.e., of adding 3 copies of 2 : 2 × 3 = 2 + 2 + 2. Similarly, $\pi$ multiplied by 2 is the same value obtained by adding the number pi to itself.
Multiplication can be visualised as counting objects arranged in a rectangle (for natural numbers) or as finding the area of a rectangle whose sides have given lengths (for numbers generally).
However, multiplication does not give the result of a repeated addition when the multiplier is not a whole number. We can calculate the product
$(-\sqrt{2}) \times \sqrt{2} = -2$
using the rules for manipulating radicals and products, but we cannot view this computation as repeatedly adding $\sqrt{2}$. Instead of being viewed as a special type of repeated addition, multiplication should be viewed as a basic numerical operation, separate from and as important as addition.
The inverse of multiplication is division: as 2 times 3 equals to 6, so 6 divided by 3 equals to 2.
## Properties
Commutativity
Multiplication of numbers is commutative, meaning a × b = b × a.
Associativity
Multiplication is associative, meaning a × (b × c) = (a × b) × c.
Distributivity
Multiplication is distributive over addition, meaning a × (x + y) = a × x + a × y.
## Products of sequences
### Capital pi notation
The product of a sequence can be written using capital Greek letter Π (Pi). Unicode position U+220F (∏) contains a symbol for the product of a sequence, distinct from U+03A0 (Π), the letter. The meaning of this notation is given by:
$\prod_{i=m}^{n} x_{i} = x_{m} \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_{n},$
where i is an index of multiplication, m is its lower bound and n is its upper bound. Example:
$\prod_{i=2}^{4} 2^i = 2^2 \cdot 2^3 \cdot 2^4 = 4 \cdot 8 \cdot 16 = 512.$
If m = n, the value of the product just equals to xm. If m > n, the product is the empty product, with the value 1.
## More general concepts of multiplication
Multiplication can be defined outside of the context of real numbers. There are natural multiplication operations on the complex numbers, matrices, tensors, sequences, and functions.
The properties described need not always hold in these more general settings. For example, multiplication is not always commutative in other contexts. For example, matrix multiplication is in general not a commutative operation:
$\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix} \times \begin{bmatrix} 1 & 0\\ 0 & 0\\ \end{bmatrix} = \begin{bmatrix} 0 & 0\\ 0 & 0\\ \end{bmatrix}$
but reversing the order of multiplication, we have
$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}$
a different result.
A common thread in these examples of operations called "multiplication" is the existence of an additional "addition" operation and a collection of common axioms satisfied by the addition and multiplication operations in each context. This general context in which a multiplication operation exists, encompassing all of the above examples, is that of the abstract ring encountered in abstract algebra.
Multiplication may also be used to describe more general binary operations. | 2014-07-26 17:11:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7369968295097351, "perplexity": 729.2633427579275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997903265.4/warc/CC-MAIN-20140722025823-00113-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://rdrr.io/cran/circular/man/rstable.html | # rstable: Random Generation from the Stable Family of Distributions In circular: Circular Statistics
## Description
Returns random deviates from the stable family of probability distributions.
## Usage
1 rstable(n, scale = 1, index = stop("no index arg"), skewness = 0)
## Arguments
n sample size. index number from the interval (0, 2]. An index of 2 corresponds to the normal, 1 to the Cauchy. Smaller values mean longer tails. skewness number giving the modified skewness (see Chambers et al., 1976). Negative values correspond to skewness to the left (the median is smaller than the mean, if it exists), and positive values correspond to skewness to the right (the median is larger than the mean). The absolute value of skewness should not exceed 1. scale the scale of the distribution.
## Details
This function return random variates from the Levy skew stable distribution with index=alpha, scale=c and skewness=beta. The skewness parameter must lie in the range [-1,1] while the index parameter must lie in the range (0,2]. The Levy skew stable probability distribution is defined by a fourier transform,
p(x) = {1 \over 2 π} \int_{-∞}^{+∞} dt \exp(-it x - |c t|^α (1-i β sign(t) \tan(πα/2)))
When alpha = 1 the term tan(pi alpha/2) is replaced by -(2/pi) log|t|. For alpha = 2 the distribution reduces to a Gaussian distribution with sigma = sqrt(2) scale and the skewness parameter has no effect. For alpha < 1 the tails of the distribution become extremely wide. The symmetric distribution corresponds to beta = 0.
The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, alpha, beta) then the sum Y = X_1 + X_2 + ... + X_N will also be distributed as an alpha-stable variate, p(N^{1/alpha} c, alpha, beta).
There is no explicit solution for the form of p(x) and there are no density, probability or quantile functions supplied for this distribution.
## Value
random sample from the specified stable distribution.
## Author(s)
Claudio Agostinelli
## References
Chambers, J. M., Mallows, C. L. and Stuck, B. W. (1976). A Method for Simulating Stable Random Variables. Journal of the American Statistical Association 71, 340-344.
Logaeve, M. (1977). Probability Theory I. (fourth edition) Springer-Verlag, New York.
rnorm, rcauchy.
## Examples
1 hist(rstable(200, 1.5, .5)) #fairly long tails, skewed right
### Example output
Attaching package: 'circular'
The following objects are masked from 'package:stats':
sd, var
circular documentation built on May 1, 2019, 7:57 p.m. | 2021-08-04 20:10:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7913214564323425, "perplexity": 2702.6544663451937}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154897.82/warc/CC-MAIN-20210804174229-20210804204229-00267.warc.gz"} |
http://libros.duhnnae.com/2017/aug/150158386063-Existence-and-Regularity-For-The-Generalized-Mean-Curvature-Flow-Equations-Mathematics-Analysis-of-PDEs.php | Existence and Regularity For The Generalized Mean Curvature Flow Equations - Mathematics > Analysis of PDEs
Abstract: By making use of the approximation method, we obtain the existence andregularity of the viscosity solutions for the generalized mean curvature flow.The asymptotic behavior of the flow is also considered. In particular, theDirichlet problem of the degenerate elliptic equation $$-| ablav|\mathrm{div}\frac{ abla v}{| abla v|}+ u=0$$ is solvable in viscositysense.
Author: RongLi Huang, JiGuang Bao
Source: https://arxiv.org/ | 2019-06-15 22:32:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9041377902030945, "perplexity": 536.1839095878798}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997501.61/warc/CC-MAIN-20190615222657-20190616004657-00263.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2005.13.35 | # American Institute of Mathematical Sciences
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April 2005, 13(1): 35-62. doi: 10.3934/dcds.2005.13.35
## A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses
1 Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202-3216, United States 2 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, United States
Received January 2004 Revised October 2004 Published March 2005
We study a two-phase modified Stefan problem modeling solid combustion and nonequilibrium phase transitions. The problem is known to exhibit a variety of non-trivial dynamical scenarios. In the presense of heat losses we develop a priori estimates and establish well-posedness of the problem in weighted spaces of continuous functions. The estimates secure sufficient decay of solutions that allows us to carry out analysis in Hilbert spaces. We demonstrate existence of a compact attractor in the weighted spaces and prove that the attractor consist of sufficiently regular functions. We show that the Hausdorff dimension of the attractor is finite.
Citation: Michael L. Frankel, Victor Roytburd. A Finite-dimensional attractor for a nonequilibrium Stefan problem with heat losses. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 35-62. doi: 10.3934/dcds.2005.13.35
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2019 Impact Factor: 1.338 | 2020-12-02 18:32:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5823428630828857, "perplexity": 3313.7678118443164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141715252.96/warc/CC-MAIN-20201202175113-20201202205113-00292.warc.gz"} |
https://physics.stackexchange.com/questions/521507/srednickis-explanation-of-the-1pi-quantum-action | # Srednicki's explanation of the 1PI quantum action
In chapter 21 (p.127-129) of Srednicki's book the quantum action $$\Gamma(\phi)$$ is defined in formula (21.1) I won't repeat here (it's quite long). Then he considers the following path integral:
$$Z_\Gamma(J) = \int \mathcal{ D} \phi \exp\left[ i\Gamma(\phi) + i \int d^dx J\phi\right] = \exp[iW_\Gamma(J)].\tag{21.4}$$
$$W_\Gamma(J)$$ is given by the sum of connected diagrams (with sources) in which each line represents the exact propagator, and each $$n$$-point vertex represents the exact 1PI vertex $$\bf{V}_n$$.$$W_\Gamma(J)$$ would be equal $$W(J)$$ if we included only tree diagrams in $$W_\Gamma(J)$$.
Isolating the tree-level contribution to the path integral by means of introduction of the dimensionless parameter called $$\hbar$$ we have the following path integral:
$$Z_{\Gamma,\hbar}(J) = \int \mathcal{ D} \phi \exp\left[ \frac{i}{\hbar}\left(\Gamma(\phi) + i \int d^dx J\phi\right)\right] = \exp[iW_{\Gamma,\hbar}(J)]\tag{21.6}$$
It is followed by the following text:
In a given connected diagram with sources, every propagator (including those connected to sources) is multiplied by $$\hbar$$, every source by $$1/\hbar$$, and every vertex by $$1/\hbar$$.
1. I don't get this assertion. There is no hint neither why it should be like this.
Later on $$W_{\Gamma,\hbar}(J)$$ is developed in a series of loop occurrence in diagrams, or in different words a series in orders of $$\hbar$$:
$$W_{\Gamma,\hbar}(J)= \sum_{L=0}^{\infty} \hbar^{L-1} W_{\Gamma,L}(J).\tag{21.8}$$
I can understand the formula under the assumption that I take the above cited assertion for correct/granted. Srednicki's keeps on saying:
If we take the formal limit of $$\hbar\rightarrow 0$$, the dominant term is the one with $$L=0$$, with is given by the sum of tree diagrams only. This is just what we want. We conclude that:
$$W(J) = W_{\Gamma, L=0}(J).\tag{21.9}$$
1. This assertion is also curious (I don't understand it ). I thought that $$W(J)$$ would be the sum of all connected diagrams whereas here it seems to be only a subset, i.e. all connected diagrams without loops. I assume that it is related with what was said above (the reason I included the part above).
Actually, it seems that Srednicki's explanation of the quantum action is elegant, so I would really appreciate if somebody could explain the mentioned assertions to me what they actually mean.
1. The $$\hbar$$-weights are a direct consequence of eq. (B9) in my Phys.SE answer here:
• A vertex has $$\hbar$$-weight$$=-1$$.
• An internal propagator comes with $$\hbar$$-weight$$=+1$$ (rather than $$-1$$) because an internal propagator is accompanied by 2 source differentiations (which each carry $$\hbar$$-weight$$=+1$$).
• By the same token, an external leg (=source+propagator) has $$\hbar$$-weight$$=0$$ because it is accompanied by 1 source differentiation. Hence a source has $$\hbar$$-weight$$=-1$$.
2. $$W(J)$$ is by definition a sum of all possible connected diagrams made out of bare propagators and (amputated) bare vertices. The statement is that it is also the sum $$W_{\Gamma, L=0}(J)$$ of all possible trees made out of full propagators and (amputated) 1PI vertices. This is also explained in my Phys.SE answer here. | 2022-08-11 15:07:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 34, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9197378754615784, "perplexity": 270.22609379055535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571472.69/warc/CC-MAIN-20220811133823-20220811163823-00708.warc.gz"} |
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中国物理学会高能物理分会第十一届全国会员代表大会暨学术年会
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Higher twist transverse momentum dependent parton distribution functions in the MIT bag model
Aug 9, 2022, 3:55 PM
15m
Description
We study the transverse-momentum-dependent parton distribution functions (TMDs) up to twist-4 in the MIT bag model. Besides the TMDs defined from the quark-quark correlator, we have also calculated those defined via quark-$j$-gluon-quark correlators and four-quark correlators for the first time. All the T-even and T-odd TMDs are computed to the $\alpha_{s}^{1}$ level. Furthermore, we quantitatively evaluate the azimuthal asymmetries in SIDIS resulted from those TMDs. The numerical results show that twist-4 contributions can provide 20 percent corrections to the leading twist correction for the $\langle \sin(\phi-\phi_{S})\rangle_{UT}$ and $\langle \cos(\phi-\phi_{S})\rangle_{LT}$ asymmetries and the twist-3 effects varies between a few percent and 20 percent at low-$Q^{2}$.
Co-authors
Dr Bao-Dong Sun (South China Normal University) Tianbo Liu (Shandong University) Prof. 作堂 梁 (Shandong University) | 2022-09-27 20:34:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44465118646621704, "perplexity": 6835.028310447744}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00106.warc.gz"} |
https://math.stackexchange.com/questions/2096929/how-to-find-number-of-steps-in-euclidean-algorithm-for-fibonacci-numbers | # How to find number of steps in Euclidean Algorithm for fibonacci numbers
I know that Fibonacci numbers show up in a special way in regard to the time it takes to solve Euclidean algorithm.
I am curious to know how to actually show how many steps it takes.
For example, how can we be sure that the Euclidean algorithm for computing $\operatorname{gcd}(F_{n+1},F_n)$ is bound by at least
$$K \log_2 F_n$$
for $n$ sufficient large, and suitable $K$?
My thoughts:
I know that the gcd of two consecutive Fibonacci numbers will be $1$ as they are co prime. And I also know the different expressions for Fibonacci numbers, in particular for the $n$th Fibonacci.
But I am not used to this type of problem.
I am particularly not sure how to involve the base 2 log and the constant in this problem So how would one approach this? Any help/ advice is very appreciated.
Thank you all
• You can try to prove and use this identity $$\gcd(F_{n}, F_{m}) = F_{\gcd(n,m)}$$ – MathGod Jan 14 '17 at 1:44
• It may help to consider the most obvious case: that of consecutive Fibonacci numbers. Obviously one will have to step backward through the entire Fibonacci sequence to complete the Euclidian algorithm. – Wildcard Jan 14 '17 at 1:47
The Fibonacci sequence represents a sort of "worse case" for the Euclidean algorithm. This occurs because, at each step, the algorithm can subtract $F_n\,\,$ only once from $F_{n+1} \,\,$. The result is that the number of steps needed to complete the algorithm is maximal with respect to the magnitude of the two initial numbers.
Applying the algorithm to two Fibonacci numbers $F {n}\,\,$ and $F_{n+1}\,\,$, the initial step is
$$\gcd(F_{n},F_{n+1}) = \gcd(F_{n},F_{n+1}-F_{n}) = \gcd(F_{n-1},F_{n})$$
The second step is
$$\gcd(F_{n-1},F_{n}) = \gcd(F_{n-1},F_{n}-F_{n-1}) = \gcd(F_{n-2},F_{n-1})$$
and so on. Proceding in this way, we need $n$ steps to arrive to $\gcd(F_{1},F_{2}) \,\,$ and to conclude that
$$\gcd(F_{n},F_{n+1}) = \gcd(F_1,F_2) = 1$$
that is to say, two consecutive Fibonacci numbers are necessarily coprime.
Now it is well known that the growth rate of Fibonacci numbers, with respect to $n$, is exponential. In particular, $F_n$ is asymptotic to
$${\displaystyle \varphi ^{n}/{\sqrt {5}}}$$
where
$$\varphi=\frac {1+\sqrt {5}}{2} \approx 1.61803$$
is the golden ratio. So, for $n$ sufficiently large, we have
$$n \approx \log_{\varphi} ( \sqrt {5}\,\, F_n) \\ = \log_{\varphi} ( F_n) + \frac {\log (5)}{2 \log (\varphi)} \approx \log_{\varphi} (F_n)$$
which tells us that the number of steps required to complete the Euclidean algorithm grows in a logarithmic manner with respect to $F_n\,\,$, and is bounded by the expression above. The same result can also be written using the logarithm in base $2$ as suggested by the OP:
$$n \approx \frac { \log_{2} F_n}{\log_{2} \varphi} =K \log_{2} F_n$$
where $$K=\frac {1}{\log_{2} \varphi}=\frac {\log {2}}{\log {\varphi}} \approx 1.4404...$$
• Is there a way to do it without making assumptions about the limiting ratios and such? – PersonaA Jan 23 '17 at 2:06
• Yes. If we do not consider the asymptotic behaviour of the Fibonacci ratio as already known, we can determine it. The most rapid way to do it is to note that if a limit $x$ of the ratio between consecutive terms exists, it must satisfy $$x^2F_{n}=xF_{n}+ F_{n}$$ and then $$x^2-x-1=0$$ Solving this, the only positive solution is $x=(\sqrt {5}+1)/2$. Once determined this, we directly get that the sequence progressively tends to become a geometric progression with ratio $\varphi$, whose asymptote is $\varphi^n$. – Anatoly Jan 23 '17 at 6:40
• @Anatoly Shouldn't the minimum number of steps to obtain $gcd(F_{n+1},F_n)$ be $n - 1$ , for large n. Continuing with the algorithm, $gcd(F_{n+1},F_n) = gcd(F_n,F_{n+1} - F_{n}) = gcd(F_n,F_{n-1}) = .... = gcd(F_{3},F_{2}) = gcd(2,1) = 1$ i.e. My point is you don't need to go $gcd(F_{2},F_{1})$ because already at the $(n-1)th$ step, you've obtained, 2 = 2*1 + 0, so your algorithm terminates. And since we're looking at the least number of steps, wouldn't it create problem? – reflexive Aug 23 '17 at 12:43
Theorem : number of steps in Euclidian algorithm $\le 5m$, where $m$ is number of digits in $min(a,b)$.
Proof : suppose we do all steps in Euclidian algorithm , so we have sequence of equations : $r_{i-1} = r_{i}q_{i} + r_{i+1}$. Consider $r_{n} = gcd(a,b) \ge 1$. Let's make an assumption : $r_{n-k} \ge \phi^{k}$, where $\phi = \frac{1 + \sqrt{5}}{2}$.
Base of induction : $r_{n} \ge \phi^{0} = 1$
Step : $r_{n-k-1} = r_{n-k}q_{n-k}+r_{n-k+1} \ge r_{n-k}+r_{n-k+1}\ge \phi^k + \phi^{k-1} = \phi^{k+1}$ - true.
So $r_1 \ge \phi^{n-1}$, but $10^{m-1} \le min(a,b) = r_{1} < 10^{m}$, so $m > (n-1)\ln\phi > (n-1)/5$, so $n \le 5m$.
Now let's notice that inequality in proof could be equality for Fibonacci numbers. | 2021-05-16 21:43:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8868029713630676, "perplexity": 166.22558436424652}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989914.60/warc/CC-MAIN-20210516201947-20210516231947-00628.warc.gz"} |
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-1-mid-chapter-check-point-page-228/6 | ## Precalculus (6th Edition) Blitzer
Function The domain is $(-\infty, 1]$. The range is $[-1, +\infty)$.
All vertical lines will pass through the graph in at most one point. Thus, the graph represents a function. The graph covers all the x-values from $1$ to its left, therefore the domain is $(-\infty, 1]$. The graph covers all the y-values from $-1$ and above therefore, the range is $[-1, +\infty)$. | 2018-09-23 11:59:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46002140641212463, "perplexity": 265.13711594759764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159359.58/warc/CC-MAIN-20180923114712-20180923135112-00099.warc.gz"} |
https://www.biostars.org/p/4658/#4661 | Finding Protein Homology?
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Entering edit mode
11.1 years ago
ilovepython ▴ 150
I'm looking for something that's more rigorous than BLAST and less rigorous than a global alignment-esque algorithm. Is there a paper out there that compare methods? What's the most popular method?
I'm doing a BLAST search against one protein, and then trying to find closer hits with a better search method. Is there a better way of going about this?
Edit: This question is probably too broad so here are some additional details. I'm interested in transmembrane proteins, so some type of analysis on transmembrane segments of the proteins will be important.
homology protein blast homology • 4.5k views
0
Entering edit mode
By "more rigorous than BLAST and less rigorous than a global alignment-esque algorithm" do you mean less rigorous than a dynamic programing algorithm (Smith-Waterman and Needleman-Wunsch)? Neither of these are rigorous enough to detect remote homology because they are pairwise alignments. When you do a search, these algorithms have absolutely no way of knowing which sequence positions are more important than others. Please see my MSA-based response bellow for an alternative to pairwise alignments.
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Entering edit mode
I see that by "rigorous" you mean fewer false positives. Even a "global alignment-esque algorithm" would not be rigorous enough (and sensitive enough!) to detect remote homology well because those are all pairwise alignments. When you do a search, these algorithms have absolutely no way of knowing which sequence positions are more important than others. Please see my MSA-based response bellow for an alternative to pairwise alignments.
0
Entering edit mode
Btw, you cannot 'find homology', you can hypothesize homology of sequences based on sequence similarity which you detected using sequence similarity search.
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11.1 years ago
Spitshine ▴ 640
For practical purposes, PSI-BLAST or HMMer searches are the tools of choice for finding (remote) homologs. If you know the domain, HMMer will do the trick. Most likely, the transmembrane elements are included in the HMM from SMART or PFAM.
There are many comparisons but you will need to define your task more precisely. Do you want to find homologs for an orphan protein or detect all members of a protein in a genome? Does more rigorous mean fewer false-positives? And you're not running BLAST against a database of one protein, are you?
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Entering edit mode
The recent implementation of HMMer (3.0) is fast and reliable but if you have transmembrane regions, it pays to do a reverse BLAST/PSI-BLAST with candidate hits to confirm and weed out hits to regions with composition biases.
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Entering edit mode
I'm looking for homologs for a group of proteins. I just re-read the HMMer doc, and it seems like it's exactly what I'm looking for! I'm not sure how I missed this profiling feature before. Rigorous does mean fewer false positives. I think my language was unclear above, I'm running blast using ncbi's non-redundant set. Thanks so much!
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Entering edit mode
11.1 years ago
Handpick several proteins (functionally, structurally, or evolutionary related) and build a multiple sequence alignment (MSA).
For aligning multiple transmembrane proteins you may want to consider this paper/tool: PRALINE^TM
Once you have a good MSA, use HMMER hmmbuild to convert the MSA to a profile HMM.
Then search a large database (e.g. UnitProt which is Swiss-Prot and TrEMBL) using the profile HMM with HMMER hmmsearch.
This method should be good at finding remote homologs.
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Entering edit mode
11.1 years ago
Pals ★ 1.3k
I am quite a new fellow here. As this question is related to what I do, I try to write what I know. I am convinced with bilouweb. If you have got the template structure and has very low sequence identity, you could make secondary structure prediction of the template using for example Jpred. It gives a number of aligned sequences that have similar secondary structure. On the other hand, you can do simple protein blast against nr database for your model sequence and then align those sequences. At last you can combine the two MSAs for generating pairwise alignment.
The other strategy, if you do not have template structure would be structure prediction tools of course. I had once used I-TASSER and it worked quite nicely. You will get much more information about your protein than just the structure.
The last option would be the prediction tools related to membrane protein for example split server.
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Entering edit mode
11.1 years ago
Bilouweb ★ 1.1k
Some tools try to detect homology from the structure (secondary or tertiary) of proteins but there is not much transmembrane protein structures available.
I think the web server Phyre is a good tool to begin with. From a amino acid sequence, it gives: - secondary structure predictions from 3 predictors - disorder region predictions - homologous proteins found
Protein fold recognition programs can help you because some are based on homolog research. You will find a good list of them on the CASP experiment web page
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Entering edit mode
99% of the sequences I'm working with don't have structure data. I didn't consider using a structure based prediction before, I will definitely try this out. Thanks!
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11.1 years ago
If you are interested in detection of transmembrane proteins the tool of choice might be TMHMM Here's a web server for TMHMM. Also the SignalP program might help.
Here is a review about the application of these tools to cellular location you might find useful.
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Entering edit mode
11.1 years ago
I like (and have voted up) the responses above, particularly the motif and MSA answers. If you want real quality to your MSA, make certain to keep your domains, even your transmembrane segments intact. In other words, an MSA that retains or is aligned to secondary and tertiary structural elements will be of higher quality and allow you to move forward with greater confidence.
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Which one was the motif answer?
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Which one is the motif answer?
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Entering edit mode
HMMer and such tools to detect functional domains/motifs. Sorry, I mostly deal with DNA sequence (motif) and think less of protein profiles...
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Use of this site constitutes acceptance of our User Agreement and Privacy Policy. | 2022-01-24 14:06:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3651440441608429, "perplexity": 2783.7424708888275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304570.90/warc/CC-MAIN-20220124124654-20220124154654-00526.warc.gz"} |
https://blog.inktrap.org/functional-completeness.html | # Functional Completeness
To those new to logic and new to junctors, it is sometimes surprising, that we do not need all the junctors to be functionally complete. As Wikipedia states and those already familiar with propositional or predicate logic know: the conjunction and the negation are already functionally complete.
It is quite easy to show this with some basic knowledge about junctors. Also this is quite easy to confirm with truth-tables which I won't do here.
### Implication And Equivalency
We have to figure out a way to build the implication or conditional: $$\rightarrow$$. We already know the truth values for it, so I asked myself: When is the implication true?
Easy! It is true iff $$\neg p \lor q$$ so if $$p$$ is false or $$q$$ is true. Using DeMorgan's law $$\neg ( p \land q) \leftrightarrow (\neg p \lor \neg q)$$, we can express the disjunction with the conjunction and get $$\neg (p \land \neg q)$$ as equivalent to $$p \rightarrow q$$. If you do not understand De Morgan's law, try to figure it out and use Venn-diagrams from set theory.
And with the implication we can easily express equivalency $$\leftrightarrow$$, because $$p \leftrightarrow q$$ is equivalent to $$(p \rightarrow q) \land (q \rightarrow p)$$. So if $$\neg (p \land \neg q)$$ is the implication, equivalency is $$\neg ( p \land \neg q) \land \neg ( q \land \neg p)$$.
### Disjunction
The last thing we want to express is the disjunction and this is also easy, because we can use DeMorgan's law to express the disjunction via a conjunction: $$\neg ( p \land q) \leftrightarrow (\neg p \lor \neg q)$$.
But we don't want $$\neg p \lor \neg q$$, we want $$p \lor q$$! And we get it by negating the conjunction again:
$$\neg ( \neg p \land \neg q ) \leftrightarrow (p \lor q)$$.
## Result
I just wanted to show my thought process when constructing the other junctors. This is quite easy using only DeMorgan's law and some basic knowledge about junctors. But functional completeness shows up during history (pierce arrow, sheffer stroke, aso.) and is used in chip layouts. And after all: chips & computers & the internet are the reason you are able to read this. | 2018-08-17 10:46:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7806680202484131, "perplexity": 447.6763874368308}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221212040.27/warc/CC-MAIN-20180817104238-20180817124238-00625.warc.gz"} |
https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Slurry_Transport_(Miedema)/08%3A_Usage_of_the_DHLLDV_Framework/8.03%3A_The_Influence_of_Fines | # 8.3: The Influence of Fines
The fraction of the sand in suspension, the fines, resulting in a homogeneous pseudo liquid is named X. This gives for the density of the homogeneous pseudo liquid:
$\ \rho_{\mathrm{pl}}=\rho_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \frac{\mathrm{X} \cdot \mathrm{C}_{\mathrm{vs}} \cdot \mathrm{R}_{\mathrm{sd}}}{\left(1-\mathrm{C}_{\mathrm{vs}}+\mathrm{C}_{\mathrm{vs}} \cdot \mathrm{X}\right)}$
So the concentration of the fines in the homogeneous pseudo liquid is not Cvs,pl=X·Cvs, but:
$\ \mathrm{C}_{\mathrm{v s , p l}}=\frac{\mathrm{X} \cdot \mathrm{C}_{\mathrm{v s}}}{\left(\mathrm{1 - C}_{\mathrm{v s}}+\mathrm{C}_{\mathrm{v s}} \cdot \mathrm{X}\right)}$
This is because part of the total volume is occupied by the particles that are not in suspension. The remaining spatial concentration of solids to be used to determine the hydraulic gradients curve of the solids is now:
$\ \mathrm{C}_{\mathrm{v s}, \mathrm{r}}=(\mathrm{1}-\mathrm{X}) \cdot \mathrm{C}_{\mathrm{v s}}$
The dynamic viscosity can now be determined according to Thomas (1965):
$\ \mu_{\mathrm{pl}}=\mu_{\mathrm{l}} \cdot\left(1+2.5 \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{pl}}+10.05 \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{pl}}^{2}+0.00273 \cdot\left(\mathrm{e}^{16.6 \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{pl}}}-1\right)\right)$
The kinematic viscosity of the homogeneous pseudo liquid is now:
$\ v_{\mathrm{pl}}=\frac{\mu_{\mathrm{pl}}}{\rho_{\mathrm{pl}}}$
One should realize however that the relative submerged density has also changed to:
$\ \mathrm{R}_{\mathrm{s d}, \mathrm{p l}}=\frac{\rho_{\mathrm{s}}-\rho_{\mathrm{p l}}}{\rho_{\mathrm{p l}}}$
Over the whole range of Reynolds numbers above 2320 the Swamee Jain (1976) equation gives a good approximation for the Darcy Weisbach friction factor:
$\ \lambda_{\mathrm{pl}}=\frac{1.325}{\left(\ln \left(\frac{\varepsilon}{3.7 \cdot \mathrm{D}_{\mathrm{p}}}+\frac{5.75}{\mathrm{Re}_{\mathrm{pl}}^{0.9}}\right)\right)^{2}}=\frac{0.25}{\left.\log _{10}\left(\frac{\varepsilon}{3.7 \cdot \mathrm{D}_{\mathrm{p}}}+\frac{5.75}{\mathrm{Re}_{\mathrm{pl}}^{0.9}}\right)\right)^{2}}$
With the Reynolds number for the pseudo liquid:
$\ \mathrm{R} \mathrm{e}_{\mathrm{p l}}=\frac{\mathrm{v}_{\mathrm{l} \mathrm{s}} \cdot \mathrm{D}_{\mathrm{p}}}{v_{\mathrm{p l}}}$
The equation for the terminal settling velocity in pseudo liquid (in m and m/sec) has been derived by Ruby & Zanke (1977):
$\ \mathrm{v}_{\mathrm{t}, \mathrm{pl}}=\frac{\mathrm{1 0} \cdot v_{\mathrm{pl}}}{\mathrm{d}} \cdot \left( \sqrt{1+\frac{\mathrm{R}_{\mathrm{sd}, \mathrm{p} \mathrm{l}} \cdot \mathrm{g} \cdot \mathrm{d}^{3}}{\mathrm{1 0 0} \cdot v_{\mathrm{p l}}^{2}}}-1 \right)$
The general equation for the hindered terminal settling velocity in pseudo liquid according to Richardson & Zaki (1954) yields:
$\ \mathrm{v}_{\mathrm{th}, \mathrm{pl}}=\mathrm{v}_{\mathrm{t}, \mathrm{pl}} \cdot\left(1-\mathrm{C}_{\mathrm{vs}, \mathrm{r}}\right)^{\beta}$
According to Rowe (1987) the power in pseudo liquid can be approximated by:
$\ \beta=\frac{4.7+0.41 \cdot \operatorname{Re}_{\mathrm{p}, \mathrm{pl}}^{0.75}}{1+\mathrm{0 .1 7 5} \cdot \mathrm{R} \mathrm{e}_{\mathrm{p}, \mathrm{p} \mathrm{l}}^{\mathrm{0 . 7 5}}} \quad\text{ with: }\quad \mathrm{R} \mathrm{e}_{\mathrm{p}, \mathrm{p l}}=\frac{\mathrm{v}_{\mathrm{t}, \mathrm{p l}} \cdot \mathrm{d}}{v_{\mathrm{p l}}}$
With the new homogeneous pseudo liquid density, kinematic viscosity, relative submerged density and volumetric concentration the hydraulic gradient can be determined for the remaining solids, with the adjusted volumetric concentration.
## 8.3.1 Based on the Pseudo Liquid (A)
When pseudo liquid flows through the pipeline, the pressure loss can be determined with the well-known Darcy- Weisbach equation, this pressure loss is in kPa:
$\ \Delta \mathrm{p}_{\mathrm{p l}, \mathrm{A}}=\lambda_{\mathrm{p l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\Delta \mathrm{p}_{\mathrm{p l}, \mathrm{B}}=\frac{\lambda_{\mathrm{p} \mathrm{l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p} \mathrm{l}}}{\rho_{\mathrm{l}}} \cdot \Delta \mathrm{p}_{\mathrm{l}}$
The hydraulic gradient ipl based on the pseudo liquid is in meters of liquid per meter of pipeline:
$\ \mathrm{i}_{\mathrm{p} \mathrm{l}, \mathrm{A}}=\frac{\Delta \mathrm{p}_{\mathrm{p l}, \mathrm{A}}}{\rho_{\mathrm{p l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l s}}^{\mathrm{2}}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\lambda_{\mathrm{p} \mathrm{l}}}{\lambda_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{l}}$
The relative excess hydraulic gradient related to the pseudo liquid as defined and used in this book is (which is zero for only the pseudo liquid):
$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}} \quad \mathrm{o r} \quad \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p l}} \cdot \mathrm{C}_{\mathrm{v t}, \mathrm{r}}}$
When mixture flows through the pipeline, the pressure loss can be determined with the well-known Darcy- Weisbach equation for the ELM, based on the pseudo liquid:
$\ \Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}=\lambda_{\mathrm{p l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}}^{2}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \lambda_{\mathrm{p} \mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p} \mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \Delta \mathrm{p}_{\mathrm{p l}, \mathrm{A}}$
For the Equivalent Liquid Model (ELM) this gives for the hydraulic gradient based on the pseudo liquid:
$\ \mathrm{i}_{\mathrm{m}, \mathrm{pl}, \mathrm{A}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}}{\rho_{\mathrm{p} \mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{pl}}} \cdot \frac{\lambda_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \mathrm{i}_{\mathrm{pl,A}}$
The relative excess hydraulic gradient is for the ELM, based on the pseudo liquid:
$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p} \mathrm{l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}}=\frac{\frac{\rho_{\mathrm{m}}}{\mathrm{\rho}_{\mathrm{p l}}} \cdot \mathrm{i}_{\mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p} \mathrm{l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}}=\mathrm{i}_{\mathrm{p l}, \mathrm{A}}$
## 8.3.2 Based on the Carrier Liquid (B)
Since here the hydraulic gradient and the relative excess hydraulic gradient are based on the pseudo liquid density, these parameters have to be corrected in order to express them in terms of the carrier liquid density and the carrier liquid Darcy Weisbach friction factor according to, this pressure loss is in kPa:
$\ \Delta \mathrm{p}_{\mathrm{p l}, \mathrm{B}}=\lambda_{\mathrm{p l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \lambda_{\mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\Delta \mathrm{p}_{\mathrm{p} \mathrm{l}, \mathrm{A}}=\frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \Delta \mathrm{p}_{\mathrm{l}}$
This gives for the hydraulic gradient, carrier liquid based:
$\ \mathrm{i}_{\mathrm{pl}, \mathrm{B}}=\frac{\Delta \mathrm{p}_{\mathrm{pl}, \mathrm{B}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\lambda_{\mathrm{pl}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{pl}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{2 \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\lambda_{\mathrm{pl}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{pl}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{l}}=\frac{\rho_{\mathrm{pl}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{pl}, \mathrm{A}}$
The relative excess hydraulic gradient related to the carrier liquid as defined and used in this book (which is non-zero for the pure pseudo liquid):
$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{B}}=\frac{\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}} \quad or \quad \mathrm{E}_{\mathrm{r h g}, \mathrm{p}, \mathrm{B}}=\frac{\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v t}}}$
When mixture flows through the pipeline, the pressure loss can be determined with the well-known Darcy-Weisbach equation for the ELM, based on the pseudo liquid:
$\ \Delta \mathrm{p}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{B}}=\lambda_{\mathrm{p} \mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{m}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \lambda_{\mathrm{p} \mathrm{l}} \cdot \frac{\Delta \mathrm{L}}{\mathrm{D}_{\mathrm{p}}} \cdot \frac{\mathrm{1}}{2} \cdot \rho_{\mathrm{p} \mathrm{l}} \cdot \mathrm{v}_{\mathrm{l s}}^{2}=\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}$
For the Equivalent Liquid Model (ELM) this gives for the hydraulic gradient based on the carrier liquid:
$\ \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{B}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{p} \mathrm{l}} \cdot \mathrm{v}_{\mathrm{ls}}^{2}}{\mathrm{2} \cdot \mathrm{g} \cdot \mathrm{D}_{\mathrm{p}}}=\frac{\rho_{\mathrm{p} \mathrm{l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}=\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{p} \mathrm{l}}} \cdot \mathrm{i}_{\mathrm{p l}, \mathrm{A}}=\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{p} \mathrm{l}, \mathrm{A}}==\frac{\lambda_{\mathrm{p} \mathrm{l}}}{\lambda_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{l}}$
The relative excess hydraulic gradient is for the ELM, based on the carrier liquid:
$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{pl}, \mathrm{B}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\frac{\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{p} \mathrm{l}, \mathrm{A}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}=\frac{\left(\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{l}}} \cdot \frac{\lambda_{\mathrm{p l}}}{\lambda_{\mathrm{l}}}-\mathrm{1}\right) \cdot \mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}$
## 8.3.3 The Different Flow Regimes
For the different flow regimes, the pressure losses should be determined with the adjusted kinematic viscosity, relative submerged density and terminal settling velocity with the equations that are also used without fines. Based on the pressure losses found, first the hydraulic gradient and the relative solids effect are determined based on the pseudo liquid properties.
This gives for the hydraulic gradient based on the pseudo liquid properties:
$\ \mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}}{\rho_{\mathrm{p l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}}{\rho_{\mathrm{p l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}} \quad\text{ with: }\quad \Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}=\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}$
The relative excess hydraulic gradient is, based on the pseudo liquid properties:
$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{A}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p l}, \mathrm{A}}-\mathrm{i}_{\mathrm{p l}, \mathrm{A}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{p l}} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{r}}}$
The above two equations assume the pseudo liquid is the carrier liquid. In processing experimental data, it is assumed that the real carrier liquid is the carrier liquid (often water). The fraction of fines is often unknown. So to compare model results with experimental results, the hydraulic gradient of the mixture and the relative solids effect have to be related to the real carrier liquid properties and to the hydraulic gradient of the real carrier liquid.
This gives for the hydraulic gradient based on the carrier liquid properties (division by the real carrier liquid density):
$\ \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{B}}=\frac{\Delta \mathrm{p}_{\mathrm{m}, \mathrm{p l}, \mathrm{B}}}{\rho_{\mathrm{l}} \cdot \mathrm{g} \cdot \Delta \mathrm{L}}=\frac{\rho_{\mathrm{p l}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{p} \mathrm{l}, \mathrm{A}}$
The relative excess hydraulic gradient or relative solids effect is, based on the carrier liquid properties (deducting the hydraulic gradient of the real carrier liquid in the nominator):
$\ \mathrm{E}_{\mathrm{r h g}, \mathrm{p l}, \mathrm{B}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{p}, \mathrm{B}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s d}} \cdot \mathrm{C}_{\mathrm{v s}}}$ | 2021-11-28 23:23:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7684078812599182, "perplexity": 2098.9097888778974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358673.74/warc/CC-MAIN-20211128224316-20211129014316-00096.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/amc.2007.1.29 | Article Contents
Article Contents
New constructions of anonymous membership broadcasting schemes
• An anonymous membership broadcast scheme is a method in which a sender broadcasts the secret identity of one out of a set of $n$ receivers, in such a way that only the right receiver knows that he is the intended receiver, while the others can not determine any information about this identity (except that they know that they are not the intended ones). In a $w$-anonymous membership broadcast scheme no coalition of up to $w$ receivers, not containing the selected receiver, is able to determine any information about the identity of the selected receiver. We present two new constructions of $w$-anonymous membership broadcast schemes. The first construction is based on error-correcting codes and we show that there exist schemes that allow a flexible choice of $w$ while keeping the plexities for broadcast communication, user storage and required randomness polynomial in log $n$. The second construction is based on the concept of collision-free arrays, which is introduced in this paper. The construction results in more flexible schemes, allowing trade-offs between different complexities.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: | 2023-03-21 18:23:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.3284430205821991, "perplexity": 632.8806254765449}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943704.21/warc/CC-MAIN-20230321162614-20230321192614-00690.warc.gz"} |
https://nigerianscholars.com/past-questions/mathematics/question/312037/ | Home » » Find a factor which is common to all 3 binomial expressions $$4a^2 - 9b^2, 8a^3 + 27b^3, (4a + 6b)^2$$....
# Find a factor which is common to all 3 binomial expressions $$4a^2 - 9b^2, 8a^3 + 27b^3, (4a + 6b)^2$$....
### Question
Find a factor which is common to all 3 binomial expressions $$4a^2 - 9b^2, 8a^3 + 27b^3, (4a + 6b)^2$$.
A) 4a + 6b
B) 4a - 6b
C) 2a + 3b
D) 2a - 3b
E) None | 2022-05-19 22:34:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3755411207675934, "perplexity": 2688.7502534698906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662530066.45/warc/CC-MAIN-20220519204127-20220519234127-00154.warc.gz"} |
https://www.encyclopediaofmath.org/index.php/Algebraic_extension | Algebraic extension
2010 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]
A field extension $K/k$ in which every element of $K$ is algebraic over $k$; that is, every element of $K$ is the root of a non-zero polynomial with coefficients in $k$. A finite degree extension is necessarily algebraic, but the converse does not hold: for example, the field of algebraic numbers, the algebraic closure of the field of rational numbers, is an algebraic extension but not of finite degree.
Algebraic extensions form a distinguished class : that is, they have the properties (i) for $M / L / K$ we have $M/L,\,L/K$ algebraic if and only if $M/K$ is algebraic; (ii) $M / K,\,L/K$ algebraic implies $ML/L$ algebraic.
An extension which is not algebraic is a transcendental extension.
References
[b1] Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) ISBN 048678147X Zbl 0768.12001 [b2] Steven Roman, Field Theory, Graduate Texts in Mathematics 158 (2nd edition) Springer (2007) ISBN 0-387-27678-5 Zbl 1172.12001
How to Cite This Entry:
Algebraic extension. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Algebraic_extension&oldid=42117 | 2018-09-20 19:07:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9115848541259766, "perplexity": 341.8757186208073}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156554.12/warc/CC-MAIN-20180920175529-20180920195929-00028.warc.gz"} |
https://klauselk.com/wordpress-logbook/ | # Using WordPress
Not so long ago, to me “WP” was an echo from the past, meaning “WordPerfect.” Now I am using WordPress for this site. From pure habit, I started a logbook, realizing later that it might be useful for others. The log is more or less chronological, with the oldest stuff first. I should say that I have some very rusty experience with Apache and html. Many ears ago, I maintained a “trac” website, mainly configuring it.
Legend: Concepts in WordPress in italics. Plugins & tools in bold.
## Pre-Start – trying a local WordPress site
I needed a website to present my new book(s). It had to be up & running in a jiffy, but I also wanted something that can grow with me. I know that I will be wanting to use FTP to a local host for experiments. I need to have a “Downloads” section, and a number of static pages, plus maybe a blog. I will want to try php and CSS, as well as SEO tools.
After looking around, I homed in on WordPress.
I then learned that WordPress.com is very basic, kind-of blog-only, whereas WordPress.org gives you more flexibility, but no hosting. Just the code.
I now fetched “XAMPP“, which implements a “localhost” server on my PC. I downloaded WordPress from WordPress.org and unzipped it as instructed. I had to provide username and password for both the site and the database. The database is called MariaDB, but seems to be identical to MySQL.
None of the above required any programming, and it all just worked on my PC. At the next reboot it didn’t. I realized the need to start XAMPP Control Panel after boot, restarting database and web-server here. This is fine, as I don’t want this running all the time anyway.
The Dashboard at the left edge in “edit mode”, together with the top-menu that WordPress adds in the live site, works pretty neat.
I toyed around with some Themes, and learned that it is possible to work entirely online. It seems that the workflow was different from what I expected. You can work “normally” online, and export to the local environment for experiments. So I decided to go for the next step.
## Getting a WordPress site
I now went looking for a site, free of commercials, but not too expensive. I need daily backups, and the “one-click-install” was tempting. I found a site offering to handle the domain-purchasing also, and went with that. An afternoon after work, I purchased the package, did the one-click install and was up & running on my new domain the same evening. I tried a few Themes, ending up with “TwentyFifteen” aka 2015, and started writing Menus and Pages.
I downloaded the “Under Construction” plugin and activated it. Now I could experiment more privately. This plugin shows a picture with a discrete “login” button. Once logged-in, you can access the site but others cannot. Incidentally, you later login by writing “/login” after the home-page URL.
Later, when I had removed the “Under Construction” sign, I unchecked the feature that says that new Pages are automatically added to the primary menu. This means that when I work and publish in iterations, it is very unlikely that anyone sees it. It is easy to add Pages to the menu later via the Dashboard.
## Switching WordPress Theme
After a couple of sessions, I got tired of “2015”. It was too targeted towards blogging. In “2015” content is in the center, while there are sections left and right – one of these with a Sidebar including Menu. Between the outer sections is some empty space, into which you can put a background image. To me, this was too messy. I switched to “2017”, after testing a few others. It is a great feature that you can test with a live preview. I hadn’t written much content and was prepared to lose it all, although it seemed I wouldn’t. After the switch, my Menu-system was gone, but my Pages had survived. These are saved in the database, and thus not affected by the choice of Theme. In other words there are no html-pages (until a cache is installed).
It is important to check how the theme looks when the browser-window is made smaller. Most themes with horizontal menu, will change this to be vertical as the window becomes narrow. Changing appearance with the client media is said to be “responsive”.
## Writing a little code
Before changing any code, it is important to create a “Child-Theme”. Otherwise your changes will be overwritten at the next update. I simply downloaded a plugin“Child Themify” – that helped me. I have later learned that this is very simple to do manually, but the plugin worked fine.
When you only want to code a little, according to documentation there are three template files: header.php, footer.php and style.css. On top of this, you can also write html/php-code in standard Pages in text-mode. This does not require the child-theme.
When any of the three template-files need changing, it must be copied from the “parent” theme to the same place in the child-theme and edited here. Now I downloaded the “File Manager” plugin and activated it. With this I could browse and move the files from within the Dashboard, but there was no editor!
This had to be enabled in “wp-config.php“. This is “outside” the dashboard and the website, but can be accessed via sftp. Instead I simply used the providers file-manager from their control panel. In the following line, replace “true” with “false”:
define('DISALLOW_FILE_EDIT', true);
I tried changing the “footer.php” to give me the credit instead of WordPress 🙂
I had done this in my brief encounter with the TwentyFifteen theme. This time however, the code wasn’t in “footer.php”. Instead, it was under “template-parts/site-info.php” (discovered by debugging as described below). Once found, it was easy to change.
Note that in WordPress the file path is the absolute URL on the running system, not a relative path in the server-source-tree as I would have expected. There are php-functions to help with this – e.g. php_content(myfolder/myfile), will give you the absolute path to “myfile”, as long as “myfolder” is a direct child of “wp-content”. You can also “spell out” the path completely.
## Debugging
When trying to get a grip of what is happening, it is practical to use “Developer Tools”(standard component) in Chrome, or “FireBug” (must be downloaded and installed) in Firefox .
It is a good idea to debug in “Incognito” or “Private” mode, to avoid various browser extensions influencing the results.
Using one of the above, you can inspect the code when its running in your browser. Don’t forget that this is the html that the browser sees. It does not exist on the server – except as pieces of code inside the database. The good thing about this, is that you can change it, and see what happens, without destroying anything. At the next refresh of the page, it is back to normal. The bad thing is that you need to find the real place to edit afterwards. There are many great videos on this on YouTube.
## Organizing Files
WordPress follows a naming-scheme for uploading files. This scheme is preset in “settings” in the Dashboard. Preparing for blog-content, these uploads are often organized in year-month folders. I prefer to have images in an img-folder under “uploads”. Likewise I now have “downloads” under “uploads” – hmm.
## Going Local Again
I had removed the “Under Construction” sign and was beginning to gather content. Thus, I was not happy about major experiments – who knows whether the backup can be restored?
It was clear that you cannot just copy files to the local server. The database needs to be migrated as well. This involves translating links.
I ended up downloading the “All-in-one WP Migration” plugin. I exported the whole enchilada to my PC, where I imported it in from my local WordPress. The only downside of this plugin is that you need a working WordPress site to import from. However, I had exactly that. It worked like a charm.
Like the other plugins used so far, the “All-in-one WP Migration” is free – up to zipped packages of 512 MB. This site took 143 MB. After the local import I dared remove stuff like unused Themes. An export now took 125 MB.
## Changing URL
It annoyed me that my site was called www.klauselk.com – instead of simply klauselk.com. This can be changed in the settings, but I felt a little uneasy doing it. There were tons of complex descriptions on how to do this after the fact – implying that this simple way was a “nogo”. Anyway, now that I had a backup, I did it. It worked fine.
## Introducing a WordPress Cache
My website is not as fast as I would like it to be. Luckily there are Cache plugins. I was attracted to “WP Super Cache”, but documentation stated that the permalinks “must be custom”. Feeling weary about changing this fundamental thing, I checked out “W3 Total Cache”, and installed it in the local copy. Now everything looked like sh..
Luckily the warnings told me I had a problem with CDN – which is a paid service that gives you hosts on all continents. I do not have this, so obviously it failed. I got it to work by disabling anything with CDN, but in the meantime I had found a note on “WP Super Cache”. It stated that permalinks only needed to NOT be the total basic ones based on ID only. My permalinks were not this type, so I now tried “WP Super Cache”. I found it easier to understand, and kept it. The acceleration is great.
Later, my provider gave me a “Varnish” cache and CDN, so I dropped the above caching.
## SEO
Similar steps can be done with Microsofts Bing, and other, to me unknown, sites.
## Improved Tools
In spite of my initial ideas on FTP and local editing, I had used a WP-based editor and file-manager up until now. I pulled myself together, and installed FileZilla and Notepad++. I know FileZilla well, while Notepad++ was unknown to me, as I am used to nice paid tools. However, in this project the sport is in doing it all for free – except the hosting.
In the management console for my webhost, I found the SFTP URL, and port. Using this together with my password, I could finally access files via (S)FTP. By clicking on a remote file inside FileZilla, I could open it in Notepad++. This, behind the scenes, created a local copy of the file. When I closed the file, FileZilla asked me whether I wanted it copied back to the remote web-server again. Clicking “Yes” was all it took. Very nice, finally being able to edit with a language-aware crisp editor 🙂
The console also gives access to the “WP-CLI” – a command-line interface that can be used to perform numerous tasks in a faster way that using plugins.
I needed to support download of e.g. source-code. There are “Download Managers”, but for the first time I experienced that the documentation on when to pay for what, was difficult to understand. “This and that format is free”. Yes – does that mean that the others are not?
Here & Now, I don’t really need bells & whistles and I ended up with simple, powerful HTML – directly in the page, as pure text. An example:
<a href="https://usercontent.one/wp/klauselk.com/wp-content/uploads/downloads/udp_ip6.zip?media=1640646709" download="">udp_ip6.zip</a> - Linux UDP on IP6
## PageSpeed Optimizations
In Google Analytics there is a sub-item called “PageSpeed”. It gave very poor readings for the main page. Basically it gives a percentage, and this was 17! Digging into the log, I learned that the book-cover in PNG-format was 3.7 MBytes. I compressed a JPG to ~600 kBytes. That got me close to 80 on PC and 70 on mobile units. The white logo was compressed by using fewer colors:
magick convert elk_white.png -colorspace gray -colors 2 logo_white.png
I found that I could use the ImageMagick Display to shrink the logo’s to half size a number of times.
The secondary problem was “Render Blocking CSS and Javascript”. From various nice pages – e.g. http://wpbeginner.com – i learned that I should install the “Autoptimize” plugin and enable it to optimize JavaScript and CSS. I did this. Now both mobile and PC are ~86. This helps on all pages. However, still CSS blocked rendering. Following this video: https://youtu.be/k56E_7SFQoE and this tool: https://jonassebastianohlsson.com/criticalpathcssgenerator/, I managed to “Inline & Defer” CSS.
A third problem was that I was not using “ExpiresByType” headers in the .htaccess. This is described here (read the text AFTER before doing as described):
This globally sets the site to report “Expires” dates on static content such as figures, enabling the clients browser to cache e.g. figures that are shown more than once. Naturally it also makes it faster to reload a page.
UNFORTUNATELY all other pages than the main page now caused a 404 error. Ups. The wpbeginner had the cure – rewrite permalinks. In the dashboard, go to Settings-Permalinks and hit “Save”.
Later I dropped “Autoptimize” as I found it too complex and error-prone when used with other plugins.
I used to have a “manual” redirect from my previous site at “klauselk.dk” to this “.com” site. I decided that the time had come to kill the dk-site completely, and have “klauselk.dk” send users directly to this site.
So should I “re-registrate” name-server at my registrar (DK-Hostmaster), as well as some changes in e.g. wp-config?
It turned out that the easy way was to ask my new webhost to “move” the old URL. This naturally involved a check-mail from DK-Hostmaster. Now at the webhost’s control panel for the dk-URL – under “DNS” – I chose that the dk URL should point to this site.
## Introducing SSL and HTTP/2 to WordPress
This website does not really handle any transactions, but since I write about security it makes sense to use SSL/https. It turned out to be pretty straightforward. My provider offers the SSL certificate – it just needed to be enabled. The provider recommended to use the plugin “Really Simple SSL”, which I did.
Then I went to Google Webmaster tools and introduced two new sites in my “property set” – namely the “https://” version with and without “www”.
I submitted the usual sitemap.xml to both. I also updated in Google Analytics.
Finally I added the https versions in Bing as well.
Enabling SSL means that HTTP/2 is now automatically used. The provider had already enabled this for https, so now it got active. However, a lot of traffic was still HTTP/1.1.
After a google analytics test, I did a test with pingdom.com. This allows you to test the speed in other areas of the world. I found that the redirect from http to https was time-consuming – it added 0.5s to the load time. In the “Really Simple SSL” plugin I removed the redirect in the application and moved it to the .htaccess file. This saved me 0.5s from San Jose to Denmark.
## Speeding up again
After doing nothing at this site for a time I checked Google Analytics again. And the site was slow. A few experiments resulted in the following few changes in Autoptimize:
• Remove jquery and others from the “Exclude scripts from Autoptimize” box.
• Remove WordPress Emojis
This improved speed significantly.
I added my latest book-cover – and again saw problems with it’s large image load-time. I decided to try the plugin Imagify. This plugin claims to reduce size of files in the website (using the company’s cloud-server). It was easily installed, but even though it claimed to have shrunk my files, nothing had happened (they even had the original size). So back to using ImageMagick on my PC. The following converts a PNG file to a JPG version – and at the same time reduces it size dramatically:
magick convert CoverThirdEdSnip.png -resize 50% CoverThirdEdSnip.jpg
WordPress and it’s plugins are updated all the time. I tend to hesitate a little before applying these updates – expecting things change to the worse, but normally it all works surprisingly fine.
I am really glad that I created a child theme. This lets me update the site theme without loosing customizations. Recently – after an update – the burger menu stopped working. The “burger menu” is the menu used on narrow devices – like my iPhone. It shows three horizontal lines – hence the burger name – and when clicked, you should see menu-items in a vertical list.
After the update the menu worked once – then clicking it did not help. A reload made it work again – once! I suspected the “Autoptimize” plugin, and when I disabled “Optimize JavaScript Code?” here, my menu worked again.
## Improving the slug
The slug is the part of a page URL that you assign to the specific page to be appended to your base URL. For this page it is “wordpress-logbook”. Yoast and other SEO tools recommend that it is a human readable text that describes your page – not an ID number or similar. It is easy to change the slug. A test will show that the page now has the new URL. Be warned however, that Googles cache now is “out of synch”. A correctly setup Yoast plugin in WordPress will submit an updated sitemap to Google, but it will take up to several days before Google caches are updated. Apparently, Google uses the slug as yet a keyword to your content.
You can perform a URL Inspection in Google Search Console. Here you may also ask Google to update a specific page, but it still doesn’t happen overnight.
The morale is that it may be OK to change the slug on a page, but you probably shouldn’t do it right before a big campaign.
## Google Site Kit and Tags
At some point I had installed the GADWP – Google Analytics Dashboard for WordPress. I preferred to use the “real” Google Analytics page instead of the dashboard, but the plugin also setup scroll events – so fine. During a maintenance session, WordPress urged me to update to ExactMetrics, which I did. The new dashboard was lousy – but worse; my scroll events had gone. Goodbye ExactMetrics.
After some browsing I found Googles new “Site Kit” which creates a dashboard in the WP environment with a an assembly of pages from Google Search, Analytics and more. The dashboard looked nice, but still no scroll events. Site Kit also aggregates data from Google Tag Manager (GTM). I checked this out and it seemed to enable me to create tags & events at a rather abstract level. So now I created an account at GTM. I also installed a WP plugin – also called Google Tag Manager.
The plugin claimed that Scroll Events are “natively supported”, but they didn’t just work out of the box. I had to configure this inside GTM. I found a good description here: https://www.lovesdata.com/blog/tracking-scroll-depth.
All in all a lot of work to regain the lost scroll events. But at least I now have a framework for adding more tags and an improved dashboard.
## Pages, Posts and the Menus
The nature of my web-site is not really that of a blog. This means that whatever I have written in posts may be visible from Googling, but not really through the menu system of the site. Some of the posts are short-lived and can be forgotten, but others are more long-lived. Is it “right” to update the latter? – after all, they do have a date in the slug. I decided to “upgrade” a number of posts to pages. But if I change the slug of existing posts, a person doing a search might run into a “page not found”. In the end I decided to copy the relevant posts into pages.
Now I needed more menu-entries. The classic solution is to add a hierarchy – which I did.
The morale is that it is wise to remember from start what to put in posts and what to put in pages.
I also decided to condense many of the things in this log into a table – always showing the current status of my choices and not the history. This is the WordPress Tools page. Finally I decided to create the new page with the “new” block editor – mainly because the classic editor cannot do tables. I have done tables earlier using html. The experiment worked OK.
## Improved Search
WordPress was originally created for blogs. This can be seen in many ways. An example is that in the posts-section you have a search-bar in the twenty-seventeen theme – but not in the pages-section. I now have to many pages to ignore this problem. I scanned some reviews and ended with the “Ivory Search” plugin. The free version allows me to format the search-field as I like and it even allows me to exclude some posts from the search (the ones that are updated to pages). I am pretty happy with this plugin.
## Math on the Web
I wanted to try the nice Latex stuff from my books on the web. So I started a page on digital filters. Obviously I needed a plugin to handle this. After some googling I decided on “KaTex”. I liked the idea that when using the Gutenberg editor, I would be able to see both Latex source and result while editing. This worked really nice!
It is great that you can see the math in the “backend” – but the “frontend” is what really matters. Here the user got “KaTex is not defined” – no math. I was suspecting the “Async Javascript” plugin, so I disabled it and Voila! – now KaTex worked. Here’s the generic FIR filter:
y[n]=\frac{1}{a_0}(\sum\limits_{i=0}^Pb_ix[n-i])
A small debugging session showed me that 300-400 ms (on a PC) was spent on the KaTex stylesheets and fonts before First Contentful Paint. This is OK when you use the math – but not when you don’t. Katex offers the ability to only use it when needed – but also warns that this may cause issues with other plugins. It worked for me, and I saved the many ms
## Spring Cleaning
After the above experience I had to acknowledge that plugins can be nice, but they can also interfere with each other, and it was time to clean house. I started by making a backup at my providers site. Then I warmed up by deleting unused themes. Now to “Really Simple SSL”. It has always annoyed me that all the http references to my own pages was caught runtime (at least until cached) and replaced by https.
Since I already had changed the sitename to include “https” – in WP as well as on Google and Bing (see Introducing SSL) – this was not needed now.
So now I got a new plugin! Yes – there’s a difference between a tool plugin that you run manually in the backend, and something that runs every time users are visiting. This time it was the “Better Search Replace”. This was used to replace all occurrences of “http://klauselk.com” with “https://klauselk.com” in the database. I then went into .htaccess and replaced the code from Really Simple SSL with something very similar. Later I found that when deactivating Really Simple SSL you can ask it to leave this code.
The above is described at WPBeginner. | 2022-05-21 00:44:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2609000504016876, "perplexity": 2483.3345147406476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534693.28/warc/CC-MAIN-20220520223029-20220521013029-00604.warc.gz"} |
http://www.maplesoft.com/support/help/Maple/view.aspx?path=AFactors | AFactors - Maple Programming Help
AFactors
inert absolute factorization
Calling Sequence AFactors(p)
Parameters
p - multivariate polynomial
Description
• The AFactors function is a placeholder for representing an absolute factorization of the polynomial p, that is a factorization over an algebraic closure of its coefficient field. It is used in conjunction with evala.
• The construct AFactors(p) produces a data structure of the form $[u,[[{f}_{1},{ⅇ}_{1}],...,[{f}_{n},{ⅇ}_{n}]]]$ such that $p=u{{f}_{1}}^{{ⅇ}_{1}}...{{f}_{n}}^{{ⅇ}_{n}}$, where each ${f}_{i}$ is a monic (for the ordering chosen by Maple) irreducible polynomial.
• The call evala(AFactors(p)) computes the factorization of the polynomial p over the field of complex numbers. The polynomial p must have algebraic number coefficients.
• In the case of a univariate polynomial, the absolute factorization is just the decomposition into linear factors.
Examples
> $\mathrm{evala}\left(\mathrm{AFactors}\left({x}^{2}-2{y}^{2}\right)\right)$
$\left[{1}{,}\left[\left[{x}{-}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right){}{y}{,}{1}\right]{,}\left[{x}{+}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{2}\right){}{y}{,}{1}\right]\right]\right]$ (1) | 2017-01-24 00:54:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9750391840934753, "perplexity": 981.2199224246183}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560283475.86/warc/CC-MAIN-20170116095123-00428-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.zbmath.org/authors/?q=ai%3Aterashima.yuji | # zbMATH — the first resource for mathematics
## Terashima, Yuji
Compute Distance To:
Author ID: terashima.yuji Published as: Terashima, Y.; Terashima, Yuji
Documents Indexed: 23 Publications since 1978
all top 5
#### Co-Authors
2 single-authored 7 Morishita, Masanori 4 Gomi, Kiyonori 3 Kato, Akishi 3 Yamazaki, Masahito 2 Kitayama, Takahiro 1 Kodani, Hisatoshi 1 Matsuyuki, Takahiro 1 Mizuno, Yuma 1 Nagai, Wataru 1 Nagao, Kentaro 1 Takakura, Yu 1 Tange, Ryoto 1 Ueki, Jun
all top 5
#### Serials
4 Communications in Mathematical Physics 2 IMRN. International Mathematics Research Notices 2 Mathematical Research Letters 1 Advances in Mathematics 1 Geometriae Dedicata 1 Journal of Algebra 1 Nagoya Mathematical Journal 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Tohoku Mathematical Journal. Second Series 1 Transactions of the American Mathematical Society 1 Journal of High Energy Physics 1 Algebraic & Geometric Topology 1 The Journal of Symplectic Geometry 1 PTEP. Progress of Theoretical and Experimental Physics
all top 5
#### Fields
12 Manifolds and cell complexes (57-XX) 7 Number theory (11-XX) 7 Algebraic geometry (14-XX) 6 Algebraic topology (55-XX) 6 Global analysis, analysis on manifolds (58-XX) 5 Differential geometry (53-XX) 4 Commutative algebra (13-XX) 4 Quantum theory (81-XX) 2 Combinatorics (05-XX) 2 Nonassociative rings and algebras (17-XX) 2 $$K$$-theory (19-XX) 2 Special functions (33-XX) 1 Group theory and generalizations (20-XX) 1 Functions of a complex variable (30-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Relativity and gravitational theory (83-XX) | 2020-12-01 00:15:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18583638966083527, "perplexity": 8807.74682736298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141515751.74/warc/CC-MAIN-20201130222609-20201201012609-00410.warc.gz"} |
https://qaofficial.com/page/111.html | # QA Official
### kears some examples
https://qaofficial.com/post/2019/04/06/69227-kears-some-examples.html 2019-04-06
1, softmax multi-classification based on multilayer perceptron ### 基于多层感知器的softmax分类 from keras.models import Sequential from keras.layers import Dense,Dropout,Activation from keras.optimizers import SGD import keras # Generate dummy data import numpy as np x_train = np.random.random((1000,20)) ### 将类别向量(从0到nb_classes
### reshape Function-Differences between reshape Functions of Matlab and opencv
https://qaofficial.com/post/2019/04/06/69159-reshape-function-differences-between-reshape-functions-of-matlab-and-opencv.html 2019-04-06
opencv: m: 1 2 3 4 m.reshape(0, 1): 1 2 3 4 m.reshape(0, 4): 1 2 3 4 请按任意键继续. . . Matlab: It can be seen that OPENCV is row first and matlab is column first. on this small issue, let me n long training are all misplaced!!!!
### C# Several Common Algorithms
https://qaofficial.com/post/2019/04/05/111713-c#-several-common-algorithms.html 2019-04-05
1. Find the value of the following expression and write down one or more implementations you think of: 1-2+3-4+…+m static int F1(int m) { int sum =0; bool flag =true; for (int i = 1; i <= m; i++) { if (flag) //一次是默认是True，下下也为T
### GitHub Learning Notes (2) Upload to Warehouse
1. Register on GitHub; Second, Baidu Git and download; Three, to operate the file Then right click on the folder interface to be uploaded locally and click git bash here. after clicking, git bash will pop up automatically, and then the folder can be operated by inputting commands into git bash; 4. Building a warehouse on GitHub; Five, to connect: 1. input git init: initialize this folder as a git warehouse;
### R Machine Learning One: kNN Algorithm Case
https://qaofficial.com/post/2019/04/05/68933-r-machine-learning-one-knn-algorithm-case.html 2019-04-05
kNN algorithm Advantages: It is highly unbiased and does not require any assumptions about the data.Simple, effective and easy to implement Disadvantages: Since there is no abstract process involved, kNN has not actually created a model and the prediction time is long. case study: detection of prostate cancer Step 1 : 100 observation 10 variables, including 8 numerical variables, a category variable, an ID: 1，Radi
### ROS Basic Learning Notes (1)
https://qaofficial.com/post/2019/04/05/69059-ros-basic-learning-notes-1.html 2019-04-05
### SVM Learning Summary (1) How to Learn SVM
https://qaofficial.com/post/2019/04/05/68966-svm-learning-summary-1-how-to-learn-svm.html 2019-04-05
1. Preface I have been studying SVM algorithm for a long time. I always thought it was a very simple algorithm, but I didn't think it was very difficult when I was studying it. I am still not familiar with convex optimization. I have to strengthen it in the future. I will mainly talk about the learning ideas of SVM algorithm (for small white beginners). 2. Learning Methods Through the comparative study of various materials, I personally think that using chapter 7 of Li Hang's "
### SVM explanation and code
https://qaofficial.com/post/2019/04/05/68970-svm-explanation-and-code.html 2019-04-05
Support Vector Machine is commonly referred to as SVM because its English name is SupportVectorMachine. Generally speaking, it is a two-class classification model. Its basic model is defined as the linear classifier with the largest interval in the feature space. Its learning strategy is interval maximization, which can be converted into the solution of a convex quadratic programming problem. Two Main Contents: 1. How did the original formula come from
### jQuery determines which li is currently clicked
https://qaofficial.com/post/2019/04/05/111683-jquery-determines-which-li-is-currently-clicked.html 2019-04-05
Use $(this).index () to get the subscript of li. The following is an example of style substitution:$("#aa li").click(function(){ $("#aa li").removeClass("class名字，多个class用空格分开");$(this).a | 2019-09-20 16:51:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2740379273891449, "perplexity": 3344.4213244565653}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574050.69/warc/CC-MAIN-20190920155311-20190920181311-00119.warc.gz"} |
https://math.stackexchange.com/questions/2326072/show-that-if-a-nb-nc-n-to0-then-a-n-b-n-c-n-converge-to-0 | # Show that if $a_n+b_n+c_n\to0$ then $a_n,b_n,c_n$ converge to $0$
We define 3 sequences $(a_n),(b_n),(c_n)$ with positive terms so that $$a_{n+1}\leq\frac{b_n+c_n}{3}\ ,\ b_{n+1}\leq\dfrac{a_n+c_n}{3}\ ,\ c_{n+1}\leq\dfrac{a_n+b_n}{3}$$ Check if any of $(a_n),(b_n),(c_n)$ converge, and if they do find their limit.
PROOF
My part of the proof is this: By adding the above inequalities we get $$a_{n+1}+b_{n+1}+c_{n+1}\leq\frac{2}{3}(a_n+b_n+c_n)$$ We define the sequence $(x_n)$ with $x_n=a_n+b_n+c_n$ so we have $$x_{n+1}\leq\frac{2}{3}x_n\Rightarrow \frac{x_{n+1}}{x_n}\leq\frac{2}{3}<1$$ which implies that $(x_n)$ is decreasing. We also have $x_n>0, \forall n$ thus it's bounded, and so it converges to $0$. Is it correct to say that since $a_n<x_n$ then $a_n\to0$?
EDIT
I forgot to mention that we also prove that $x_n\to0$.
• $1+1/n$ is decreasing and bounded, but do not converges to $0$ – enzotib Jun 17 '17 at 12:04
Yes because you have the "sandwiched" inequality $0 < a_{n} < x_{n} \rightarrow 0$, so $a_{n} \rightarrow 0$.
EDIT: Of course, you have to make sure that $x_n \rightarrow 0$ in the first place, but you have that it is bounded by a geometric sequence which converges to zero which implies the stated convergence.
• You mean $\leq$. – Displayname Jun 17 '17 at 12:06
• Not necessarily since $a_n$, $b_n$ and $c_n$ are assumed positive. But $\leq$ would of course also be enough, however, this way I think the difference between "converging to zero" and "being zero" becomes a bit clearer. – Andre Jun 17 '17 at 13:34
• No, it is not equivalent to this. Take for example the case $0 < \frac{1}{n} < \frac{2}{n}$. All the inequalities are strict for every $n$, but both converge to 0 as $n \rightarrow \infty$. – Andre Jun 18 '17 at 0:45
• That is not what you wrote. You are taking the limit since you used a $\rightarrow$. – Displayname Jun 18 '17 at 5:49
• Your current statement literally is $0 < a_n < 0$. – Displayname Jun 18 '17 at 6:02
To prove that $x_n$ converges to $0$ you need to take the limit $$0\leq x_{n+1}\leq \frac{2}{3}x_n \implies 0\leq l\leq \frac{2}{3}l \implies l=0$$
Since $\{x_n\}$ is decreasing and bounded from below, it is convergent. But this does not guarantee that the limit is zero. Indeed, since
$$0< x_{n+1}\le\frac{2}{3}x_n$$
we have
$$0< x_n\le\left(\frac{2}{3}\right)^{n-1}x_1$$
This implies that $x_n\to0$ as $n\to\infty$.
We have $\displaystyle 0<a_n<\frac{1}{3}x_n$, $\displaystyle 0<b_n<\frac{1}{3}x_n$ and $\displaystyle 0<c_n<\frac{1}{3}x_n$.
So $a_n\to 0$, $b_n\to 0$ and $c_n\to 0$ as $n\to\infty$. | 2019-08-20 07:23:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9876159429550171, "perplexity": 110.56309073037261}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315258.34/warc/CC-MAIN-20190820070415-20190820092415-00152.warc.gz"} |
https://brilliant.org/problems/jorges-subset-products/ | # Jorge's subset products
Let $$S=\{1,2,3,4,\ldots, 2013\}$$ and let $$n$$ be the smallest positive integer such that the product of any $$n$$ distinct elements in $$S$$ is divisible by $$2013$$. What are the last $$3$$ digits of $$n$$?
This problem is posed by Jorge T.
× | 2017-07-23 16:47:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.550108015537262, "perplexity": 84.70186197834191}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424575.75/warc/CC-MAIN-20170723162614-20170723182614-00140.warc.gz"} |
https://socratic.org/questions/a-triangle-has-sides-a-b-and-c-sides-a-and-b-have-lengths-of-3-and-1-respectivel-1 | # A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 1, respectively. The angle between A and C is (17pi)/24 and the angle between B and C is (5pi)24. What is the area of the triangle?
Feb 17, 2018
Given measures do not qualify to form a triangle for the reason mentioned below.
#### Explanation:
$a = 3 , \hat{A} = \frac{5 \pi}{24} , b = 1 , \hat{B} = \frac{17 \pi}{24}$
In a triangle, larger side will have larger angle opposite to it.
But, though $\hat{B} > \hat{A}$, side b $<$ side a.
Hence given measures do not qualify to form a triangle. | 2021-09-27 21:54:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7471806406974792, "perplexity": 459.66216090485847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058552.54/warc/CC-MAIN-20210927211955-20210928001955-00482.warc.gz"} |
http://physics.aps.org/articles/v5/6 | # Focus: Particles Sorted by Entropy
Published January 13, 2012 | Physics 5, 6 (2012) | DOI: 10.1103/Physics.5.6
#### Entropic Splitter for Particle Separation
D. Reguera, A. Luque, P. S. Burada, G. Schmid, J. M. Rubí, and P. Hänggi
Published January 13, 2012
Ordinarily, a mixture of two types of randomly-moving particles doesn’t spontaneously unmix. But a newly proposed device relies on random motion to separate particles of different sizes. As explained in Physical Review Letters, the device is a tube whose interior walls have a variable diameter with a saw-tooth profile. Although similar “ratchet” shapes have been proposed before, the team has added an external force—coming from an electric field, perhaps—that could be tuned to allow particles of different sizes to emerge from opposite ends of the tube. Simulations suggest that the separation “purity” for sorting DNA strands, for example, could be over $99$ percent, which is better than current methods.
Sorting particles by size as they move through a liquid is important in biochemistry to analyze DNA or proteins and also in engineering to generate colloidal solutions with uniform particle size. One separation technique is to let the particles move randomly but inhibit one direction, so that there is a net progression in some other direction. Separation occurs because the strength of this effect depends on the size of the particle. Over the last decade researchers have proposed several separation techniques, many based on ratchetlike geometries ([1] for example). Some of these have been built, but their effectiveness has been limited, says Miguel Rubí of the University of Barcelona in Spain.
Rubí and Peter Hänggi of the University of Augsburg, Germany, led a team that has developed a new approach to these ratchet sorters. They start with a mathematical framework in which the entropy of the system is treated like potential energy, with entropy “barriers” that repel particles. These are regions where particles are restricted to a small space, which reduces the number of states (locations and velocities) that a particle can occupy. Fewer states means lower entropy. Like balls rolling down a hill, particles tend to move away from these low entropy spots.
The team applies this formalism to a tube with walls that periodically ramp from a narrow diameter to a wide diameter and back, with an asymmetric or “sawtooth” profile. This shape forms distinct but still connected chambers, or segments, each of which is a few microns long. Entropic barriers inhibit travel between segments; however, the barriers are steeper going to the left, so the net motion of the particles is to the right.
In order to clearly see the entropic effect in their computer simulation and analytical calculations, the researchers apply an oscillating force that essentially shakes the particles back and forth inside the tube. In a real experiment, this force could be an oscillating electric field.
Their results in two dimensions show that larger particles move faster on average because the entropic barrier is steeper for them and thus better at preventing “backwards” steps to the left. This size-dependent velocity could be used to separate particles (as in [2]), but the authors go a step further and apply a static force (which could be another electric field) pointing to the left. This force opposes the effective entropic force just enough that small particles move predominantly to the left, while large particles continue to move to the right. “The advantage we have over other separating techniques is that the particles go in different directions,” says Hänggi. The team can also vary the strength of the static force to select for different sizes.
As a demonstration of their entropic splitter, the team ran a simulation mimicking DNA separation. They assumed a tube with $3$ segments and introduced a mixture of two lengths of DNA strands that differed in their effective radius by $25$ percent. After about a minute, nearly all the strands coming out the left end of the tube were the smaller ones, and those coming from the right end were the larger ones, with a purity of $99.997$ percent. This DNA separation is significantly better than the standard lab technique of electrophoresis, which would only achieve about $70$ percent under the same conditions, according to Rubí.
By placing molecular dynamics on the “entropy landscape,” this paper opens up completely new options for separation devices, says Jörg Kärger of the University of Leipzig in Germany. However, Fabio Marchesoni from the University of Camerino in Italy wonders whether the applied forces—both static and oscillating—will have the desired effect in cases where the particles have different chemical or geometric properties. “Entropic segregation might well be an option, but at this stage there are still some unknowns to be addressed,” Marchesoni says.
–Michael Schirber
Michael Schirber is a freelance science writer in Lyon, France.
### References
1. C. Kettner et al., “Drift ratchet,” Phys. Rev. E 61, 312 (2000).
2. S. Matthias and F. Müller, “Asymmetric pores in a silicon membrane acting as massively parallel brownian ratchets,” Nature 424, 53 (2003). | 2014-03-09 16:50:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5508374571800232, "perplexity": 1122.2226057950506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394009885896/warc/CC-MAIN-20140305085805-00035-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://web2.0calc.com/questions/triangles_10 | +0
# triangles
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A circle is circumscribed about an equilateral triangle with side lengths of 9 units each. What is the area of the circle, in square units? Express your answer in terms of pi.
bbelt711 Jul 26, 2017
Sort:
#1
+214
+2
Equilateral triangle radius of circumcircle = $$\frac{s}{\sqrt{3}}$$
Which means $$r = 3\sqrt{3}$$
Area of a circle is Pi*r^2
Substitute to get $$\Pi ({3\sqrt{3})}^2$$
Which is equal to $$27\Pi$$
The area of the circumcircle is $$27\Pi -units^2$$
Mathhemathh Jul 26, 2017
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We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. See details | 2017-08-17 23:17:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8529474139213562, "perplexity": 2222.583790478924}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104172.67/warc/CC-MAIN-20170817225858-20170818005858-00187.warc.gz"} |
http://physics.oregonstate.edu/portfolioswiki/courses:lecture:inlec:inpaths?rev=1534783572 | ## Lots of Ways to Change a State (10 minutes)
Everyone stand up. In your group, one person controls the left side, a different person controls the right side, and someone is the observer/director.
Starting from about 100 g on each side, and without using the weights, change the state of the system from its initial state so that $x_L$ is increased by 4 cm and $x_R$ is increased by 3 cm. (Note that students might try to change both things at the same time. You can ask them to be more systematic and to describe precisely how they are changing from the initial to the final state.)
• Make a sketch of the processes you used to go from the initial to the final state. What axes make sense for your sketch?
• What language would you use to communicate how you changed the state of the system?
• Is it possible to find a different “path” from the initial to the final state?
• Is $W_L$ the same along the different paths?
• Is $W_R$ the same?
• Is $\Delta U$ the same?
Wrap-up note: the graph you make here is filled up with states. The paths represent things you do to the system that change the state of the system, from an initial state, through a bunch of intermediate states, and eventually to a final state.
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https://mathlearnit.com/find-the-weighted-average.html | # Find the Weighted AverageFormula for Weighted Average
The average / mean page showed how to work out the arithmetic mean/average of a set of values.
But there is also what is known as a "weighted mean" or "weighted average".
Which differs slightly from the arithmetic average in regards to how we obtain the value.
There is a formula to find the weighted average that can be used when necessary, which we'll see.
But first, it helps to have a recap of how to work out the arithmetic mean/average.
## Arithmetic Average / Mean Example
For a small random group of 5 numbers:
3 , 5 , 7 , 9 , 6
The arithmetic mean/average is established by the numbers summed together, divided by the amount of numbers there is.
Here:
Mean/average = \bf{\frac{3\space+\space5\space+\space7\space+\space9\space+\space6}{5}} = \bf{\frac{30}{5}} = 6
## Weighted Average
To introduce the concept of the weighted average/mean, we can consider the same list of 5 numbers again.
3 , 5 , 7 , 9 , 6
Each of these 5 numbers, has its own part in the whole group.
If we consider them all to have equal weight in the group,
then they each represent a {\frac{1}{5}} of the whole group on their own.
The sum we did for the arithmetic mean before, \bf{\frac{3\space+\space5\space+\space7\space+\space9\space+\space6}{5}}.
Is actually the same as \bf{\frac{3}{5}} + \bf{\frac{5}{5}} + \bf{\frac{7}{5}} + \bf{\frac{9}{5}} + \bf{\frac{6}{5}}.
=> \bf{\frac{3\space+\space5\space+\space7\space+\space9\space+\space6}{5}} = \bf{\frac{3}{5}} + \bf{\frac{5}{5}} + \bf{\frac{7}{5}} + \bf{\frac{9}{5}} + \bf{\frac{6}{5}} = 6
## Can Multiply instead of Divide
Dividing by 5, is the same as multiplying by 0.2. ( \bf{\frac{1}{5}} = 0.2 )
So instead of dividing each number by 5 and adding them up, as was done above.
We can multiply each number by 0.2,
and add them up to get the average.
( 3 × 0.2 ) + ( 5 × 0.2 ) + ( 7 × 0.2 ) + ( 9 × 0.2 ) + ( 6 × 0.2 )
= 0.6 + 1 + 1.4 + 1.8 + 1.2 = 6
This multiplication approach, is one way to work out the weighted mean/average.
## Working out the Weighted Average
With the weighted average, the numbers in a group don't have to have the same weighting as each other, like how we've just seen.
Above, we had each number with an equal weight of 0.2 in the group.
But we can change that. As long as each number weighting still adds up to 1 all together.
As 1 represents the whole 100% of the complete group of numbers.
So with the same group of  5 numbers as before,
let 3 , 5 , 6  still have a weighting of 0.2 in the group.
But let 7 have a weighting of 0.3,
and 9 have a weighting of 0.1.
Now using the multiplication approach with these new weightings, we can find the weighted average.
( 3 × 0.2 ) + ( 5 × 0.2 ) + ( 7 × 0.3 ) + ( 9 × 0.1 ) + ( 6 × 0.2 )
= 0.6 + 1 + 0.9 + 2.1 + 1.2 = 5.8
Here the weighted mean/average of 5.8 is slightly different from the arithmetic average of 6.
This is often the case with a set of values, the different averages can be larger than or smaller than each other.
Example
In a Science class, 3 different tests over a term, determines the overall mark a student will achieve for that term.
With each test being marked out of 100, one student has the following results:
TEST 1 => \bf{\frac{77}{100}} TEST 2 => \bf{\frac{62}{100}}
TEST 3 => \bf{\frac{81}{100}}
The Science teacher gives the tests different weighting when the overall term grade is awarded.
TEST 1 ( 20% ) TEST 2 ( 30% ) TEST 3 ( 50% )
20% = 0.2 30% = 0.3 50% = 0.5
To find the weighted average here:
Weighted Average = ( 77 × 0.2 ) + ( 62 × 0.3 ) + ( 81 × 0.5 ) = 74.5
The students overall grade in the Science class for the  3 tests was 74.5%.
## Find the Weighted AverageWeighted Average Formula
Often the weighting of a different value is not given as a decimal percentage, like in the Science class example above with 0.20.3 and 0.5.
Instead a whole number weighting can be used, and this is where a specific weighted average formula can be handy to use.
We can look at the Science class example again with the  3 tests, same scores as before.
TEST 1 => \bf{\frac{77}{100}} TEST 2 => \bf{\frac{62}{100}} TEST 3 => \bf{\frac{81}{100}}
The teacher could give each separate test a whole number weighting, as opposed to the already seen decimal percentage, such as:
TEST 1 ( 2 ) TEST 2 ( 4 ) TEST 3 ( 5 )
All of this information can be used in the weighted average formula, which is:
x = a test result value.
w = weight of the test result value.
n = sample size of values.
Here we have  3 different test results, so n = 3.
{\frac{\sum_{i=1}^3\space(x_i\space\times\space w_i)}{\sum_{i=1}^3\space w_i}}
What the weighted average formula tells us for the students Science grade, is that we want to work out:
With the  3 Science Tests:
x1 = 77 , w1 = 2
x2 = 62 , w2 = 4
x3 = 81 , w3 = 5
Weighted Mean/Average = \bf{\frac{(77 \space \times \space 2) \space\space + \space\space (62\space \times \space 4) \space\space + \space\space (81 \space \times \space 5)}{2 \space\space + \space\space 4 \space\space + \space\space 5}}
= \bf{\frac{154 \space\space + \space\space 248 \space\space + \space\space 405}{11}} = 73.36
With this new weighting used by the teacher, the student actually achieved a higher average score over the  3 tests.
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4. › Find the Weighted Average | 2022-05-19 23:59:41 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9160289168357849, "perplexity": 8521.462501158774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662530553.34/warc/CC-MAIN-20220519235259-20220520025259-00615.warc.gz"} |
https://www.physicsforums.com/threads/this-one-whould-be-easy.272414/ | # This one whould be easy
1. Nov 16, 2008
1. The problem statement, all variables and given/known data
Let f:[a,b] -> R where f(x) =1, 0<= x <= 1, and f(x) = 2, 1<x<=2. Show f is riemann integrable on [0,2], and compute integral.
2. Relevant equations
A function f, bounded on an interval [a,b] is riemann integrable iff the lower integral equals the upper integral.
Lower integral is the sup of the lower sum, upper integral the inf of the upper sums.
lower sum is denoted L(p,f); upper U(p,f).
f is r. integrable iff for every e >0 there is a partition P such that U(p,f) - L(p,f) <= e.
What would be really nice is to break this into two integrals, one from o to 1, the other from 1 to 2. However, we "don't know this yet". This question is in a section of the book before the "algebra of integrals" is discussed. We also have not discussed riemann sums yet, and I so I don't think I can mark my partition.
3. The attempt at a solution
Okay, so because this function is constant except for a jump at some x_k in [0,2], what I want to do is isolate this point and handle it on its own. So I just call it x_k.
Next, is obvious that the only place that the lower integral will differ from the upper integral will be about x_k. Part of me also wants to just call x_k 1, but then I don't get the nice coherent notation for writing out my partition. So here let P be a partition of [0,2] such that there are an even number of partitions, and the partitions split around 1 = x_k very nicely.
I get that the lower integral equals x_n - x_(k+1) equals 2 - 1 = 1. And I stop. This is obviously wrong. The integral should be 3, so I know I am wrong. Can anyone help me out on this one?
Oh yeah, here is another solution I started working on. Because f is constant most of the time, I can say that f is monotonic. In fact, trivially, I can say that it is increasing if I take increasing to mean >=. Then I have a theorem that if f is monotonic, f is riemann integrable. I just don't know how to deal with the jump. Or does it not matter.
Thank you PF.
2. Nov 16, 2008
### HallsofIvy
Staff Emeritus
Break your interval into three intervals: from 0 to 1-1/n, from 1-1/n to 1+ 1/n, and from 1+ 1/n to 2. In the first interval, the "upper" and "lower" sums are the same: 1 times the length- and as n goes to infinity, the length goes to 1. In the third integral, the "upper" and "lower" sums are the same: 2 times the length- and as n goes to infinity, the length goes to 1. For the middle interval, the "upper" sum is 2 times the length while the "lower" length is 1 times the length- but the length goes to 0.
Any partition can "refined" so that no intervals overlap two of those three. | 2017-12-15 20:18:11 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8976985812187195, "perplexity": 370.8943159710588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948579564.61/warc/CC-MAIN-20171215192327-20171215214327-00523.warc.gz"} |
http://conceptmap.cfapps.io/wikipage?lang=en&name=Associativity | # Associative property
(Redirected from Associativity)
In mathematics, the associative property[1] is a property of some binary operations. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, (after rewriting the expression with parentheses and in infix notation if necessary) rearranging the parentheses in such an expression will not change its value. Consider the following equations:
${\displaystyle (2+3)+4=2+(3+4)=9\,}$
${\displaystyle 2\times (3\times 4)=(2\times 3)\times 4=24.}$
Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".
Associativity is not the same as commutativity, which addresses whether or not the order of two operands changes the result. For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation.
Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.
However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation, and the vector cross product. In contrast to the theoretical properties of real numbers, the addition of floating point numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.
## Definition
A binary operation ∗ on the set S is associative when this diagram commutes. That is, when the two paths from S×S×S to S compose to the same function from S×S×S to S.
Formally, a binary operation ∗ on a set S is called associative if it satisfies the associative law:
(xy) ∗ z = x ∗ (yz) for all x, y, z in S.
Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol (juxtaposition) as for multiplication.
(xy)z = x(yz) = xyz for all x, y, z in S.
The associative law can also be expressed in functional notation thus: f(f(x, y), z) = f(x, f(y, z)).
## Generalized associative law
In the absence of the associative property, five factors a, b, c, d, e result in a Tamari lattice of order four, possibly different products.
If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression.[2] This is called the generalized associative law. For instance, a product of four elements may be written, without changing the order of the factors, in five possible ways:
${\displaystyle ((ab)c)d}$
${\displaystyle (ab)(cd)}$
${\displaystyle (a(bc))d}$
${\displaystyle a((bc)d)}$
${\displaystyle a(b(cd))}$
If the product operation is associative, the generalized associative law says that all these formulas will yield the same result. So unless the formula with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as
${\displaystyle abcd.}$
As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but they remain unnecessary for disambiguation.
An example where this does not work is the logical biconditional ${\displaystyle \leftrightarrow }$ . It is associative, thus A${\displaystyle \leftrightarrow }$ (B${\displaystyle \leftrightarrow }$ C) is equivalent to (A${\displaystyle \leftrightarrow }$ B)${\displaystyle \leftrightarrow }$ C, but A${\displaystyle \leftrightarrow }$ B${\displaystyle \leftrightarrow }$ C most commonly means (A${\displaystyle \leftrightarrow }$ B and B${\displaystyle \leftrightarrow }$ C), which is not equivalent.
## Examples
In associative operations is ${\displaystyle (x\circ y)\circ z=x\circ (y\circ z)}$ .
The addition of real numbers is associative.
Some examples of associative operations include the following.
• The concatenation of the three strings "hello", " ", "world" can be computed by concatenating the first two strings (giving "hello ") and appending the third string ("world"), or by joining the second and third string (giving " world") and concatenating the first string ("hello") with the result. The two methods produce the same result; string concatenation is associative (but not commutative).
• In arithmetic, addition and multiplication of real numbers are associative; i.e.,
${\displaystyle \left.{\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\quad \\(x\,y)z=x(y\,z)=x\,y\,z\qquad \qquad \qquad \quad \ \ \,\end{matrix}}\right\}{\mbox{for all }}x,y,z\in \mathbb {R} .}$
Because of associativity, the grouping parentheses can be omitted without ambiguity.
• The trivial operation xy = x (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation xy = y (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.
• Addition and multiplication of complex numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative.
• The greatest common divisor and least common multiple functions act associatively.
${\displaystyle \left.{\begin{matrix}\operatorname {gcd} (\operatorname {gcd} (x,y),z)=\operatorname {gcd} (x,\operatorname {gcd} (y,z))=\operatorname {gcd} (x,y,z)\ \quad \\\operatorname {lcm} (\operatorname {lcm} (x,y),z)=\operatorname {lcm} (x,\operatorname {lcm} (y,z))=\operatorname {lcm} (x,y,z)\quad \end{matrix}}\right\}{\mbox{ for all }}x,y,z\in \mathbb {Z} .}$
${\displaystyle \left.{\begin{matrix}(A\cap B)\cap C=A\cap (B\cap C)=A\cap B\cap C\quad \\(A\cup B)\cup C=A\cup (B\cup C)=A\cup B\cup C\quad \end{matrix}}\right\}{\mbox{for all sets }}A,B,C.}$
• If M is some set and S denotes the set of all functions from M to M, then the operation of function composition on S is associative:
${\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h\qquad {\mbox{for all }}f,g,h\in S.}$
• Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then
${\displaystyle (f\circ g)\circ h=f\circ (g\circ h)=f\circ g\circ h}$
as before. In short, composition of maps is always associative.
• Consider a set with three elements, A, B, and C. The following operation:
× A B C
A A A A
B A B C
C A A A
is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative.
## Propositional logic
### Rule of replacement
In standard truth-functional propositional logic, association,[4][5] or associativity[6] are two valid rules of replacement. The rules allow one to move parentheses in logical expressions in logical proofs. The rules (using logical connectives notation) are:
${\displaystyle (P\lor (Q\lor R))\Leftrightarrow ((P\lor Q)\lor R)}$
and
${\displaystyle (P\land (Q\land R))\Leftrightarrow ((P\land Q)\land R),}$
where "${\displaystyle \Leftrightarrow }$ " is a metalogical symbol representing "can be replaced in a proof with."
### Truth functional connectives
Associativity is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that associativity is a property of particular connectives. The following are truth-functional tautologies.[7]
Associativity of disjunction:
${\displaystyle ((P\lor Q)\lor R)\leftrightarrow (P\lor (Q\lor R))}$
${\displaystyle (P\lor (Q\lor R))\leftrightarrow ((P\lor Q)\lor R)}$
Associativity of conjunction:
${\displaystyle ((P\land Q)\land R)\leftrightarrow (P\land (Q\land R))}$
${\displaystyle (P\land (Q\land R))\leftrightarrow ((P\land Q)\land R)}$
Associativity of equivalence:
${\displaystyle ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow (P\leftrightarrow (Q\leftrightarrow R))}$
${\displaystyle (P\leftrightarrow (Q\leftrightarrow R))\leftrightarrow ((P\leftrightarrow Q)\leftrightarrow R)}$
Joint denial is an example of a truth functional connective that is not associative.
## Non-associative operation
A binary operation ${\displaystyle *}$ on a set S that does not satisfy the associative law is called non-associative. Symbolically,
${\displaystyle (x*y)*z\neq x*(y*z)\qquad {\mbox{for some }}x,y,z\in S.}$
For such an operation the order of evaluation does matter. For example:
${\displaystyle (5-3)-2\,\neq \,5-(3-2)}$
${\displaystyle (4/2)/2\,\neq \,4/(2/2)}$
${\displaystyle 2^{(1^{2})}\,\neq \,(2^{1})^{2}}$
Also note that infinite sums are not generally associative, for example:
${\displaystyle (1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\dots \,=\,0}$
whereas
${\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\dots \,=\,1}$
The study of non-associative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associative law is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, and have become ubiquitous in mathematics.
There are other specific types of non-associative structures that have been studied in depth; these tend to come from some specific applications or areas such as combinatorial mathematics. Other examples are quasigroup, quasifield, non-associative ring, non-associative algebra and commutative non-associative magmas.
### Nonassociativity of floating point calculation
In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, the addition and multiplication of floating point numbers is not associative, as rounding errors are introduced when dissimilar-sized values are joined together.[8]
To illustrate this, consider a floating point representation with a 4-bit mantissa:
(1.0002×20 + 1.0002×20) + 1.0002×24 = 1.0002×21 + 1.0002×24 = 1.0012×24
1.0002×20 + (1.0002×20 + 1.0002×24) = 1.0002×20 + 1.0002×24 = 1.0002×24
Even though most computers compute with a 24 or 53 bits of mantissa,[9] this is an important source of rounding error, and approaches such as the Kahan summation algorithm are ways to minimise the errors. It can be especially problematic in parallel computing.[10][11]
### Notation for non-associative operations
In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like ${\displaystyle {\dfrac {2}{3/4}}}$ ). However, mathematicians agree on a particular order of evaluation for several common non-associative operations. This is simply a notational convention to avoid parentheses.
A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,
${\displaystyle \left.{\begin{matrix}x*y*z=(x*y)*z\qquad \qquad \quad \,\\w*x*y*z=((w*x)*y)*z\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}$
while a right-associative operation is conventionally evaluated from right to left:
${\displaystyle \left.{\begin{matrix}x*y*z=x*(y*z)\qquad \qquad \quad \,\\w*x*y*z=w*(x*(y*z))\quad \\{\mbox{etc.}}\qquad \qquad \qquad \qquad \qquad \qquad \ \ \,\end{matrix}}\right\}{\mbox{for all }}w,x,y,z\in S}$
Both left-associative and right-associative operations occur. Left-associative operations include the following:
${\displaystyle x-y-z=(x-y)-z}$
${\displaystyle x/y/z=(x/y)/z}$
• Function application:
${\displaystyle (f\,x\,y)=((f\,x)\,y)}$
This notation can be motivated by the currying isomorphism.
Right-associative operations include the following:
${\displaystyle x^{y^{z}}=x^{(y^{z})}}$
Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication:
${\displaystyle (x^{y})^{z}=x^{(yz)}}$
Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression ${\displaystyle 2^{x+3}}$ the addition is performed before the exponentiation despite there being no explicit parentheses ${\displaystyle 2^{(x+3)}}$ wrapped around it. Thus given an expression such as ${\displaystyle x^{y^{z}}}$ , the full exponent ${\displaystyle y^{z}}$ of the base ${\displaystyle x}$ is evaluated first. However, in some contexts, especially in handwriting, the difference between ${\displaystyle {x^{y}}^{z}=(x^{y})^{z}}$ , ${\displaystyle x^{yz}=x^{(yz)}}$ and ${\displaystyle x^{y^{z}}=x^{(y^{z})}}$ can be hard to see. In such a case, right-associativity is usually implied.
${\displaystyle \mathbb {Z} \rightarrow \mathbb {Z} \rightarrow \mathbb {Z} =\mathbb {Z} \rightarrow (\mathbb {Z} \rightarrow \mathbb {Z} )}$
${\displaystyle x\mapsto y\mapsto x-y=x\mapsto (y\mapsto x-y)}$
Using right-associative notation for these operations can be motivated by the Curry–Howard correspondence and by the currying isomorphism.
Non-associative operations for which no conventional evaluation order is defined include the following.
• Exponentiation of real numbers in infix notation:[17]
${\displaystyle (x^{\wedge }y)^{\wedge }z\neq x^{\wedge }(y^{\wedge }z)}$
${\displaystyle a\uparrow \uparrow (b\uparrow \uparrow c)\neq (a\uparrow \uparrow b)\uparrow \uparrow c}$
${\displaystyle a\uparrow \uparrow \uparrow (b\uparrow \uparrow \uparrow c)\neq (a\uparrow \uparrow \uparrow b)\uparrow \uparrow \uparrow c}$
usw.
${\displaystyle {\vec {a}}\times ({\vec {b}}\times {\vec {c}})\neq ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\qquad {\mbox{ for some }}{\vec {a}},{\vec {b}},{\vec {c}}\in \mathbb {R} ^{3}}$
• Taking the pairwise average of real numbers:
${\displaystyle {(x+y)/2+z \over 2}\neq {x+(y+z)/2 \over 2}\qquad {\mbox{for all }}x,y,z\in \mathbb {R} {\mbox{ with }}x\neq z.}$
• Taking the relative complement of sets ${\displaystyle (A\backslash B)\backslash C}$ is not the same as ${\displaystyle A\backslash (B\backslash C)}$ . (Compare material nonimplication in logic.)
## Antiassociativity
A binary operation ∘ on S is an antiassociative operation if and only if:
∀x,y,z∈S:(x∘y)∘z≠x∘(y∘z)[18]
Let (S,∘) be an algebraic structure.
Then (S,∘) is an antiassociative structure if and only if ∘ is an antiassociative operation.
That is, if and only if:
∀x,y,z∈S:(x∘y)∘z≠x∘(y∘z)[19]
## References
1. ^ Hungerford, Thomas W. (1974). Algebra (1st ed.). Springer. p. 24. ISBN 978-0387905181. Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in G.
2. ^ Durbin, John R. (1992). Modern Algebra: an Introduction (3rd ed.). New York: Wiley. p. 78. ISBN 978-0-471-51001-7. If ${\displaystyle a_{1},a_{2},\dots ,a_{n}\,\,(n\geq 2)}$ are elements of a set with an associative operation, then the product ${\displaystyle a_{1}a_{2}\dots a_{n}}$ is unambiguous; this is, the same element will be obtained regardless of how parentheses are inserted in the product
3. ^ "Matrix product associativity". Khan Academy. Retrieved 5 June 2016.
4. ^ Moore, Brooke Noel; Parker, Richard (2017). Critical Thinking (12th edition). New York: McGraw-Hill Education. p. 321. ISBN 9781259690877.
5. ^ Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (14th edition). Essex: Pearson Education. p. 387. ISBN 9781292024820.
6. ^ Hurley, Patrick J.; Watson, Lori (2016). A Concise Introduction to Logic (13th edition). Boston: Cengage Learning. p. 427. ISBN 9781305958098.
7. ^ "Symbolic Logic Proof of Associativity". Math.stackexchange.com. 22 March 2017.
8. ^ Knuth, Donald, The Art of Computer Programming, Volume 3, section 4.2.2
9. ^ IEEE Computer Society (29 August 2008). IEEE Standard for Floating-Point Arithmetic. doi:10.1109/IEEESTD.2008.4610935. ISBN 978-0-7381-5753-5. IEEE Std 754-2008.
10. ^ Villa, Oreste; Chavarría-mir, Daniel; Gurumoorthi, Vidhya; Márquez, Andrés; Krishnamoorthy, Sriram, Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems (PDF), archived from the original (PDF) on 15 February 2013, retrieved 8 April 2014
11. ^ Goldberg, David (March 1991). "What Every Computer Scientist Should Know About Floating-Point Arithmetic" (PDF). ACM Computing Surveys. 23 (1): 5–48. doi:10.1145/103162.103163. Retrieved 20 January 2016. ([1], [2])
12. ^ George Mark Bergman: Order of arithmetic operations
13. ^ Education Place: The Order of Operations
14. ^ Khan Academy: The Order of Operations, timestamp 5m40s
15. ^ Virginia Department of Education: Using Order of Operations and Exploring Properties, section 9
16. ^ Bronstein: de:Taschenbuch der Mathematik, pages 115-120, chapter: 2.4.1.1, ISBN 978-3-8085-5673-3
17. ^ Exponentiation Associativity and Standard Math Notation Codeplea. 23 August 2016. Retrieved 20 September 2016.
18. ^ Definition:Antiassociative Operation
19. ^ Definition:Antiassociative_Structure | 2020-06-04 19:24:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 66, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8927585482597351, "perplexity": 1239.4735174200107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347458095.68/warc/CC-MAIN-20200604192256-20200604222256-00413.warc.gz"} |
https://www.transtutors.com/questions/donna-spent-a-total-of-40-at-the-grocery-store-of-this-amount-she-pent-8-on-fruit-wh-6584706.htm | # Donna spent a total of $40 at the grocery store. Of this amount, she pent$8 on fruit. What...
Donna spent a total of $40 at the grocery store. Of this amount, she pent$8 on fruit. What percentage of the total did she spend on fruit? | 2021-06-25 03:48:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7199844121932983, "perplexity": 4288.850326666384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488567696.99/warc/CC-MAIN-20210625023840-20210625053840-00310.warc.gz"} |
https://kr.mathworks.com/help/dsp/ref/movingaverage.html | Documentation
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# Moving Average
Moving average
• Library:
• DSP System Toolbox / Statistics
## Description
The Moving Average block computes the moving average of the input signal along each channel independently over time. The block uses either the sliding window method or the exponential weighting method to compute the moving average. In the sliding window method, a window of specified length moves over the data sample by sample, and the block computes the average over the data in the window. In the exponential weighting method, the block multiplies the data samples with a set of weighting factors and then sums the weighted data to compute the average. For more details on these methods, see Algorithms.
## Ports
### Input
expand all
Data over which the block computes the moving average. The block accepts real-valued or complex-valued multichannel inputs, that is, m-by-n size inputs, where m ≥ 1 and n ≥ 1. The block also accepts variable-size inputs. During simulation, you can change the size of each input channel. However, the number of channels cannot change.
This port is unnamed until you set Method to Exponential weighting and select the Specify forgetting factor from input port parameter.
Data Types: single | double
Complex Number Support: Yes
The forgetting factor determines how much weight past data is given. A forgetting factor of 0.9 gives more weight to the older data than does a forgetting factor of 0.1. A forgetting factor of 1.0 indicates infinite memory – all previous samples are given an equal weight.
#### Dependencies
This port appears when you set Method to Exponential weighting and select the Specify forgetting factor from input port parameter.
Data Types: single | double
### Output
expand all
The size of the moving average output matches the size of the input. The block uses either the sliding window method or the exponential weighting method to compute the moving average, as specified by the Method parameter. For more details, see Algorithms.
Data Types: single | double
Complex Number Support: Yes
## Parameters
expand all
If a parameter is listed as tunable, then you can change its value during simulation.
• Sliding window — A window of length Window length moves over the input data along each channel. For every sample the window moves over, the block computes the average over the data in the window.
• Exponential weighting — The block multiplies the samples by a set of weighting factors. The magnitude of the weighting factors decreases exponentially as the age of the data increases, but the magnitude never reaches zero. To compute the average, the algorithm sums the weighted data.
When you select this check box, the length of the sliding window is equal to the value you specify in . When you clear this check box, the length of the sliding window is infinite. In this mode, the block computes the average of the current sample and all previous samples in the channel.
#### Dependencies
This parameter appears when you set Method to Sliding window.
Specifies the length of the sliding window.
#### Dependencies
This parameter appears when you set Method to Sliding window and select the check box.
When you select this check box, the forgetting factor is input through the lambda port. When you clear this check box, the forgetting factor is specified on the block dialog through the Forgetting factor parameter.
#### Dependencies
This parameter appears only when you set Method to Exponential weighting.
The forgetting factor determines how much weight past data is given. A forgetting factor of 0.9 gives more weight to the older data than does a forgetting factor of 0.1. A forgetting factor of 1.0 indicates infinite memory – all previous samples are given an equal weight.
Tunable: Yes
#### Dependencies
This parameter appears when you set Method to Exponential weighting and clear the Specify forgetting factor from input port check box.
• Code generation
Simulate model using generated C code. The first time you run a simulation, Simulink® generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time but provides faster simulation speed than Interpreted execution.
• Interpreted execution
Simulate model using the MATLAB® interpreter. This option shortens startup time but has slower simulation speed than Code generation.
## Block Characteristics
Data Types double | single Multidimensional Signals No Variable-Size Signals Yes
expand all
Watch now | 2019-05-24 10:01:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5534898638725281, "perplexity": 1530.9577100541383}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257601.8/warc/CC-MAIN-20190524084432-20190524110432-00121.warc.gz"} |
https://zbmath.org/?q=an%3A0617.60009 | ## Time to reach stationarity in the Bernoulli-Laplace diffusion model.(English)Zbl 0617.60009
For the Bernoulli-Laplace diffusion model involving n balls in each urn, with stationary distribution $$\pi_ n$$, the authors study the variation distance $\| P_ k-\pi_ n\| =(1/2)\sum_{j}| P_ k(j)- \pi_ n(j)|$ of the law of the process after k steps. It turns out that $$\| P_ k-\pi_ n\| \leq ae^{-2c}$$ with $$k:=(1/4)n\log n+cn$$, $$c\geq 0$$ being a universal constant, and that $$\| P_ k-\pi_ n\| \geq 1-be^{4c}$$ with k as above, $$c\in [(-1/4)\log n$$, 0], $$b>0$$ universal.
In fact, the authors deal with the slightly more general case that the one urn contains r red balls and the other one n-r black balls. The method of treating this problem is an application of the theory of spherical functions on the homogeneous space $$S_ n/S_ r\times S_{n- r}$$ [dual Hahn polynomials; S. Karlin and J. McGregor, The Hahn polynomials, formulas and an application. Scripta Math. 26, 33-46 (1961; Zbl 0104.291)], where $$S_ n$$ denotes the symmetric group of order n. It is also shown that this method can be applied to nearest neighbor random walks on two-point homogeneous spaces (whose spherical functions are orthogonal polynomials [D. Stanton, Special functions: Group theoretical aspects and applications, Math. Appl., D. Reidel Publ. Co. 18, 87-128 (1984; Zbl 0578.20041)]), in particular on the m-dimensional cube and on the space of k-dimensional subspaces of a vector space over a finite field.
Reviewer: H.Heyer
### MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60G50 Sums of independent random variables; random walks 43A90 Harmonic analysis and spherical functions
### Citations:
Zbl 0104.291; Zbl 0578.20041
Full Text: | 2022-07-07 10:58:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8664303421974182, "perplexity": 598.9067418567628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104690785.95/warc/CC-MAIN-20220707093848-20220707123848-00209.warc.gz"} |
http://tug.org/pipermail/texhax/2010-June/015218.html | # [texhax] Newbie class question
Thomas Jacobs thomasjacobs at gmail.com
Tue Jun 22 19:59:32 CEST 2010
Uwe,
This worked like a charm!
I did interact with one of the class authors and he pointed out that
by using the fullpage option to the thesis class I could get the one
inch margins (described in the user guide and I missed it) but it goes
to double spacing as part of the definition. Thus, the geometry
package did the trick quite nicely.
May I assume that the layouts program provides approximate measures
which one must divide by 72 to get inch values or is it exact based
upon the file definition and the inch values reported in documentation
are thus approximate?
May I also assume one cannot use the \layouts package with the
\geometry package as I did not get any changes to the document in
pagedesign when using the \geometry package to constrain the widths?
Thanks again.
Tom
On Mon, Jun 21, 2010 at 4:41 PM, Thomas Jacobs <thomasjacobs at gmail.com> wrote:
> Uwe,
>
> Thanks very much for both of your replies. I will attempt to use them
> and report back.
>
> Tom
>
> On Mon, Jun 21, 2010 at 12:27 PM, Uwe Ziegenhagen <ziegenhagen at gmail.com> wrote:
>> Hi Tom,
>>
>> to see what is changing, use the following.
>>
>> in the preamble load layouts
>>
>> \usepackage{layouts}
>>
>> after \begin{document} have the layout printed:
>>
>> \pagediagram
>>
>> \pagedesign
>>
>>
>> In the next step I load the geometry package with
>>
>> 1in left and right..
>>
>> \usepackage[lmargin=1in,rmargin=1in]{geometry}
>>
>> and find no difference between this version and the version withput the
>> package. However if I set the margins to 1.5in each, the document clearly
>> gets longer. Without diving deeper I guess the margin is correct. I suggest
>> you print out a sample page and check the margins in the printed document.
>>
>>
>>
>> Uwe
>>
>> PS: To easily generate larger amounts of text I like the blindtext package.
>>
>>
>>
>>
>>
>>
>
>
>
> --
> Thomas Jacobs
>
--
Thomas Jacobs | 2018-10-18 09:59:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9329167604446411, "perplexity": 5597.193516948796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511761.78/warc/CC-MAIN-20181018084742-20181018110242-00424.warc.gz"} |
https://learn.careers360.com/ncert/question-find-the-values-of-each-of-the-following-11-tan-inverse-2-cos-2sin-inverse-1-over-2/ | # Find the values of each of the following: 11. $\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2} \right ) \right ]$
Given equation:
$\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2} \right ) \right ]$
So, solving the inner bracket first, we take the value of $\sin x^{-1} \frac{1}{2} = x.$
Then we have,
$\sin x = \frac{1}{2} = \sin \left ( \frac{\pi}{6} \right )$
Therefore, we can write $\sin^{-1} \frac{1}{2} = \frac{\pi}{6}$.
$\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2} \right ) \right ] = \tan^{-1}\left[2\cos\left(2\times\frac{\pi}{6} \right ) \right ]$
$= \tan^{-1}\left[2\cos\left(\frac{\pi}{3} \right ) \right ] = \tan^{-1}\left[2\times\left(\frac{1}{2} \right ) \right ] = \tan^{-1}1 = \frac{\pi}{4}$.
Exams
Articles
Questions | 2020-05-30 06:12:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8860752582550049, "perplexity": 717.3859704952605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347407289.35/warc/CC-MAIN-20200530040743-20200530070743-00112.warc.gz"} |
http://ncatlab.org/nlab/show/secondary+characteristic+class | # nLab secondary characteristic class
cohomology
### Theorems
#### Differential cohomology
differential cohomology
## Application to gauge theory
#### $\infty$-Chern-Weil theory
∞-Chern-Weil theory
∞-Chern-Simons theory
∞-Wess-Zumino-Witten theory
# Contents
## Idea
In its most refined form, a secondary characteristic class is a characteristic class in ordinary differential cohomology. The term “secondary” refers to the fact that such a differential cohomology class in degree $n$ not only encodes a degree-$n$ class in integral cohomology, but in addition higher connection data in degree $\left(n+1\right)$: the data of a circle n-bundle with connection.
The refined Chern-Weil homomorphism takes values in such “secondary characteristic classes”.
But the precise meaning of the term secondary characteristic class varies a little in the literature, as follows. Historically it was first understood in more restricted senses.
1. In the strict sense of the word, a secondary characteristic class is a characteristic of a situation where an ordinary characteristic class vanishes (PetersonStein1962).
2. More specifically, a special case of this situation in differential geometry arises where the characteristic class is represented in de Rham cohomology by a curvature characteristic form. If that curvature form happens to vanish, the corresponding Chern-Simons form itself becomes closed, and now itself represents a cohomology class, in one degree lower. This is often called the corresponding Chern-Simons secondary characteristic class . Sometimes the term “secondary geometric invariants” is used for Chern-Simons forms (see for instance the review (FreedII)).
3. Using refined Chern-Weil theory the notions of curvature characteristic forms and their Chern-Simons forms are unified into the notion of cocycles in ordinary differential cohomology. The notion of Cheeger-Simons differential character was introduced to describe this unification, and it is has become tradition to call these differential characters themselves secondary characteristic classes independently of whether the corresponding ordinary characteristic class/curvature characteristic form vanishes or not (for instance (DupontKamber, Karlsson). More descriptively, this case is maybe better referred to as a differential characteristic class . See there for more details.
## References
The notion in its general cohomological sense appears in
• Franklin P. Peterson, Norman Stein, Secondary characteristic classes , Annals of mathematics, Vol 76, No. 3 (1962)
The notion of Chern-Simons forms originates in
The special meaning in the context of Chern-Weil theory in differential geometry was established by the introduction of Cheeger-Simons differential characters. Reviews of that include
• Johan Dupont, Franz Kamber, Gerbes, Simplicial forms and Invariants for Families of Foliated Bundles (pdf)
• Michelle Karlsson, Characteristic classes and bounded cohomology (pdf)
Revised on May 4, 2013 20:57:36 by Urs Schreiber (150.212.92.50) | 2013-05-25 17:24:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8695207834243774, "perplexity": 1429.2700146639493}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706009988/warc/CC-MAIN-20130516120649-00038-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://byjus.com/multiplying-square-roots-calculator/ | # Multiplying Square Roots Calculator
The Multiplying Square Roots Calculator an online tool which shows Multiplying Square Roots for the given input. Byju's Multiplying Square Roots Calculator is a tool
which makes calculations very simple and interesting. If an input is given then it can easily show the result for the given number.
#### Practise This Question
Pent-4-yn-2-ol has the structure? | 2019-02-16 21:16:12 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8612121343612671, "perplexity": 1647.9177916121037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481122.31/warc/CC-MAIN-20190216210606-20190216232606-00395.warc.gz"} |
https://mathoverflow.net/questions/304023/is-c-inftye-a-projective-frechet-c-inftym-module-for-a-c-infty/304075 | # Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber?
The question is a special case of a previous question.
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-norm defined by $p_{\alpha}(f):=\sup_{\beta\leq\alpha}||\partial^{\beta}(f)||$.
Now let $E\to M$ be a $C^{\infty}$-fiber bundle with compact base and compact fiber. Then it is clear that $C^{\infty}(E)$, the space of smooth functions on the total space of the fiber bundle, is a Frechet $C^{\infty}(M)$-module.
My question is: is $C^{\infty}(E)$ always a projective Frechet $C^{\infty}(M)$-module?
I think this question is trivial for experts. Please let me know if there is any references or it the question is not suitable for mathoverflow.
• what is the module structure while the fibers are not vector space? – Ali Taghavi Jul 1 '18 at 11:34
• I am sorry I just realize the module structure(by composition) – Ali Taghavi Jul 1 '18 at 11:35
• Spaces of sections of smooth VECTOR bundles over $M$ are exactly the finitely generated projective $C^\infty(M)$-modules: Namely, choose a a second VECTOR bundle $F$ such that $E\oplus F$ is trivial, isomorphic to the space of smooth sections of $M\times \mathbb R^n \to M$, which is the free module $C^\infty(M)^n$. Projection with kernel $F$ shows that $\Gamma(E)$ is a direct summand in there. This argument can be read in the other direction also (Theorem of Serre-Swan). – Peter Michor Jul 1 '18 at 16:26
The answer is Yes. Basically, one can realize $C^\infty(E)$ as a direct summand in a larger projective $C^\infty(M)$-module. Perhaps this really is trivial to experts, but to me the argument occurred only after staring sufficiently long at related arguments in this article, which came up in your other question:
Namely, start by recalling the isomorphism $C^\infty(U \times F) = C^\infty(U) \hat{\otimes} C^\infty(F)$, where $\hat{\otimes}$ is the projective tensor product of Fréchet spaces. This means that $C^\infty(U\times F)$ is a free $C^\infty(U)$-module (freely generated by $C^\infty(F)$, in the category of Fréchet $C^\infty(U)$-modules), and hence projective. By the result of Ogneva, for any open $U\subset M$, $C^\infty(U)$ is projective over $C^\infty(M)$. Hence, $C^\infty(U\times F) \cong C^\infty(U) \otimes_{C^\infty(M)} C^\infty(M\times F)$ is also projective over $C^\infty(M)$, being the tensor product of projective modules.
Next, consider a countable, locally finite open cover of $(U_i)$ of $M$, trivializing the fiber bundle $E\to M$ as $E|_{U_i} \cong U_i\times F \to U_i$, where $F$ is the typical fiber, and let $(\chi_i)$ be a partition of unity subordinate to this cover. Such a cover and corresponding partition of unity certainly exist if the base $M$ is compact, but also more generally even if $M$ is not compact, but satisfies a suitable countability condition (second countable, paracompact). Now, the countable direct product $\prod_i C^\infty(U_i \times F)$ is still Fréchet and projective over $C^\infty(M)$, and it has $C^\infty(E)$ as a direct summand, as evinced by the inclusion/projection maps \begin{align*} C^\infty(E) \to \prod_i C^\infty(U_i \times F) &\colon f \mapsto (f|_{E_{U_i}}) , \\ \prod_i C^\infty(U_i \times F) \to C^\infty(E) &\colon (f_i) \mapsto \sum_i \chi_i f_i . \end{align*} | 2021-01-26 16:25:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9926220774650574, "perplexity": 170.23110824766758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704800238.80/warc/CC-MAIN-20210126135838-20210126165838-00019.warc.gz"} |
https://math.libretexts.org/Courses/Truckee_Meadows_Community_College/TMCC%3A_Precalculus_I_and_II/Under_Construction_test2_11%3A_Sequences_Probability_and_Counting_Theory/Under_Construction%2F%2Ftest2%2F%2F11%3A_Sequences%2C_Probability_and_Counting_Theory%2F%2F11.0%3A_Prelude_to_Sequences%2C_Probability_and_Counting_Theory |
# 11.0: Prelude to Sequences, Probability and Counting Theory
A lottery winner has some big decisions to make regarding what to do with the winnings. Buy a villa in Saint Barthélemy? A luxury convertible? A cruise around the world? The likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over the next $$30$$ years or so.
Figure $$\PageIndex{1}$$: (credit: Robert S. Donovan, Flickr.)
This decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There are also personal reasons to consider when making the choice, and one can make many arguments for either decision. However, most lottery winners opt for the lump sum. In this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose lottery numbers and the probability of winning.
11.0: Prelude to Sequences, Probability and Counting Theory is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by OpenStax. | 2022-05-28 14:41:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.547942578792572, "perplexity": 612.6730554838687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663016853.88/warc/CC-MAIN-20220528123744-20220528153744-00430.warc.gz"} |
http://fele.mascali1928.it/graphing-trig-functions-worksheet.html | # Graphing Trig Functions Worksheet
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Broadcast yourself live, watch, chat, interact, connect and share. Free trigonometry worksheets, in PDF format, with solutions to download. My steps, which Algebra -> Exponential-and-logarithmic-functions -> SOLUTION: This problem is from the Graphing Calculator supplement to Paul Foerster's Algebra and Trigonometry. Download Here. , Nov 13th Inverse Values and Angles Day 2 Worksheet Wed. y= 2 sin x b. MathsWatch. Suitable for Stage 6 mathematics students (NSW). Lockdown revision never really happened? Our online Maths A-level refresher courses on 17-21 August will review Year 12 content, getting you ready for September. Every time you click the New Worksheet button, you will get a brand new printable PDF worksheet on Trigonometry and its Applications. Quiz yourself with over 100 electrical engineering worksheets. KEYWORDS: Graphing Polynomial Functions, Graphing Trigonometric Functions, One- and Two-sided Limits, Tangent and Secant Lines, Zeros of Derivatives, Graphing and Derivatives, Mean Value Theorem, Newton's Method, Riemann Sums, Numerical Integration, 1-1 and Inverse Functions, Review of Exponential and Logarithmic Functions, Inverse. Set Notation and Language Functions Quadratics Indices and Surds Factors of Polynomials Simultaneous Equations Logarithms and Exponential Functions Straight Line Graphs Circular Measure Trigonometry Permutations and Combinations Binomial Expansion Vectors Matrices Differentiation and Integration. 5 Graphs 2 6 3. a y = sin x and y = cosec x b y = tan x and y = cot x. org Math Tables: Trigonometric Graphs (). For any miles over 100, the company reduces the price to P2. Polar graph paper is used when graphing polar coordinates. Twenty different cameras will be assigned to several boxes. ) Graph the following (explain the transformation) y = 3sin. Name: _____ Date: _____ Unit 4-2 Trig Graphs Worksheet State the equations for the following graphs. Trigonometry has plenty of applications: from everyday life problems such as calculating the height or distance between objects to the satellite navigation system, astronomy, and geography. y = sin 4x 2. A short 'learn by doing' worksheet. 7 Law of Cosines. Then brings in sketching a comparison of the Sine and Cosine functions. Then find the values of the given trig functions corresponding to the angle θ. chino, ca 91710 (909) 627-7351. Math Word Searches All items below are categorized by their difficulty level and target audience so you can pick the perfect level of fun and education. R 12 MAY 2016 - 8. Welcome to the statistics and probability page at Math-Drills. -1-Using radians, find the amplitude and period of each function. To graph x = sin y, we simply reflect the graph of y = sin x about the line y = x, as shown in Figure below. Then sketch the graph of the function over the interval 211 x 2TC using the key points for each function. More about conventions and assumptions appears in Mathematical Conventions. Graphing Trig Functions. The graph then would look as follows: Here is the graph of y = sin(x) as well: Notice again the reciprocal relationships at 0 and the asymptotes. Mathematics exam-style questions typical of A-Level, IB, GCSE(9-1) and other standardised tests. In fact it’s actually the Sin graph, shifted to the right by 90°. MAT 112 Worksheet to go with 7. Graph Trig Functions. (b) give the depth of water at t=21 hours. Acellus Trigonometry is taught by award-winning Acellus Master Teacher, Patrick Mara. 25 well thought out problems that will strengthen and reinforce student learning. Again, there are a couple of simple tips for making this easier. Graphing Trig Functions Practice. Students use graphs of y=sin(x) and y=cos(x) to realize that multiple solutions exist for trig functions. Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle) Law of Sines; Ambiguous Case of the Law of Sines; Law Of Cosines; Sine, Cosine, Tangent Worksheets. y = 5 sin. Lakeland Community College Lorain County Community College. chino, ca 91710 (909) 627-7351. Graph the equation y 0 41s q 0 13 S O = co 2 in the interva! 211. Trig Graphs Worksheet State the equations for the following graphs. Free Trigonometry Worksheets to Download. Drawing Transformed Graphs for Sin and Cos. Every time you click the New Worksheet button, you will get a brand new printable PDF worksheet on Trigonometry and its Applications. r Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Graphing Trig Functions Date_____ Period____. Twenty different cameras will be assigned to several boxes. We additionally come up with the money for variant types and along with type of the books to browse. Conic Sections: Circle. 3 side lengths. net provides more than 2000 unlimited practice and is an interesting resource for students to keep their mathematics skills sharped. Trigonometry Name Pd Date Graphing Sine and Cosine Practice Worksheet Graph the following functions over two periods, one in the positive direction and one in the negative direction. y = 4 tan 6. Complete the table by estimating the percentages of each based on chargaffs rules. 1 Day 1 Rates of Change and Limits, Sandwich Theorem: 2. A visual inspection con rms this. the graph has vertical asymptotes. Graphing Trig Functions - Displaying top 8 worksheets found for this concept. Find all values of x for which 2cos 3x 0, if 0qqd x 360. Determine if the following are functions… Write “function” or “not function” on the line. Created by GradeAmathhelp. Trigonometry pile-up puzzle. @ July 14, 2013. y = 4 tan 6. Worksheets with Trigonometry Questions Trigonometry Questions (1). 5 wkxt -extra or beg. 4 Pythagorean Theorem and SOHCAHTOA Continued and Quiz Review T 17 MAY 2016 - 8. Trigonometry is used as a solution to triangles, it deals with the ratio of right triangle and its angle. Write an equation of the given trigonometric functions having the specified characteristics. They think everything has to be useful. MAT 112 Worksheet to go with 7. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. Polar graph paper is used when graphing polar coordinates. Some of them are cot x = 1/tanx , six x/cos x = tan x, sin(900-x) - cos x and so on. This problem is also available in French: Trigo tricoté. Find the inverse of f x x( ) 2 3 2. 7 Law of Cosines. 1 Graphing Trig Functions Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. The result is a number between 2 and 3. Trigonometry Worksheets & Problems. A visual inspection con rms this. Show work here! 10. ©U 32 w0m174U ZKVuAt9a v cSToufRtrw 7a lr 8eq 1L 6L dCM. y = 4 tan 6. This section describes the graphs of trigonometric functions. 7in Section1. Here are the instructions how to enable JavaScript in your web browser. Lakeland Community College Lorain County Community College. Our maths trigonometry worksheets with answers will help your child or student to grasp and understand basic and more advanced ways of solving trigonometric equations. When $$x=0$$, the graph has an extreme point, $$(0,0)$$. Graph Trigonometric Functions Graphs the 6 Trigonometric Functions. Worksheet by Kuta Software LLC MAC 1114 - Trigonometry Name_____ 7. You can change the colors and the numbers related to each one. Je Zeager, Ph. Surveyors use the tangent function a lot. The Videos, Games, Quizzes and Worksheets make excellent materials for math teachers, math educators and parents. Graphing Trig Functions TEST STUDY GUIDE Test covers: Know domain, range, period, and intercepts of the six parent trig functions. DOC Algebra II/ Trig Honors Graphing Trig Functions Review Worksheet Graphing Trig Functions Review Worksheet. Trigonometry comes up a lot in the study of calculus, so you […]. Graphs of trigonometric functions worksheet dsoftschools graphing trig functions worksheets kiddy math graphing trig functions kuta trig graphs worksheet tolland high. Zoom the graph in and out by holding the Shift key and using the mouse wheel. Graphing Equations A. This graph, like y=2X, levels out to the horizontal asymptote y=0, except on the right side instead of the left. Trigonometry, at it's most basic level, is concerned with the measurement of triangles - calculations of unknown lengths and angles. The range (of y-values for the graph) for arcsin x is -π/2 ≤ arcsin\ x ≤ π/2 See an animation of this process here: Inverse Trigonometric Function Graph Animations. R 12 MAY 2016 - 8. Worked solutions are available to subscribers. Worksheet by Kuta Software LLC Pre-Calculus 4. In fact it’s actually the Sin graph, shifted to the right by 90°. This worksheet/quiz combination will test you on your ability to identify true and untrue statement about cosine functions, find points on graphs of these functions, identify the range of a cosine. Maybe you came via internet search engine, after that you discover this web site and determined to see this web site, many thanks for that. Because of this, for any angle ,. TES Resource Team 2 years ago 5. Free educational web site featuring interactive math lessons with a problem-solving approach and actively engage students in the learning process. Pass out the “Percentage Race” worksheets out to each student face down (Bloom: Level I - Knowledge). Free math lessons and math homework help from basic math to algebra, geometry and beyond. Include everything above plus finding limits (lim), sums, matrices. It happens on many occasions that children forget or feel unable to recollect the lessons learned at the previous grade. Graph yx2 2 and find its inverse and graph it. Verify the graph crosses the x-axis at -0. 6 Trig Functions Chapter 2 2. Trigonometry is a very practical, real-world branch of mathematics, because it involves the measurement of lengths and angles. The following diagrams show the graphs the trig functions: sin, cos, tan, csc, sec, and cot. Or if you need, we also offer a unit circle with everything left blank to fill in. 7 Law of Cosines. Digital Download. comments (-1) worksheet trigonometric functions study guide. 5$means a 30-degree angle is 50% of the max height. 2Review (Spring 2015) Solutions (Spring 2015) Ch. Selecting the "show base function" option causes the basic function (with a = b = 1 and c = d = 0) to be graphed in red. Print Handwriting Worksheets Worksheet trigonometry practice coloring activity adding and subtracting decimals year 6 6th grade math objectives x and y graph paper math 6 test There are many good workbooks available for kindergarten children. Graphing Trig Functions TEST STUDY GUIDE Test covers: Know domain, range, period, and intercepts of the six parent trig functions. comments (-1) get in touch. Trigonometry is used as a solution to triangles, it deals with the ratio of right triangle and its angle. You will see that the graph of the Cosine function is very similar to the Sin function. Grade 11 online worksheets are due to be released soon. About: Beyond simple math and grouping (like "(x+2)(x-4)"), there are some functions you can use as well. The above operations can be very slow for more than 2 graphs. Other hand calculators use the inverse function notation of a -1 exponent, which is not actually an exponent at all (e. 3 Even & Odd 5. For example see the graph of the SIN function, often called a sine wave, above. 6 Graphs of Other Trigonometric Functions 4. Answer Key Similarity Section Quiz: Lessons 7-1 Through 7-3 1. Period of Trig Graphs. Please change your browser settings and reload. Graph trigonometric functions. Graphing Inverses Graph the inverse for each relation below (put your answer on the same graph). Trig Graphs Worksheet State the equations for the following graphs. It happens on many occasions that children forget or feel unable to recollect the lessons learned at the previous grade. For full functionality of this site it is necessary to enable JavaScript. 1) y tan ° ° ° ° ° ° 2) y csc ° ° ° ° ° ° Graph each function using radians. Lesson finished with an interactive plenary where students need to evaluate trigonometric values. Mathematics is definitely among the top fears of students across the globe. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Solutions. 3 Pythagorean Theorem and SOHCAHTOA M 16 MAY 2016 - 8. To find the phase shift and translations for a trigonometric function. Let’s practice what we learned in the above paragraphs with few of trigonometry functions graphing questions. This problem is also available in French: Trigo tricoté. Graph y x2x e. • The period of each graph is π. pdf doc ; Reading a Position Graph - Answer questions about motion using a position graph. • The x-intercepts for y = tan x occur when x =±± ±0, , 2 , 3 ,. Answers are included for every question. You will probably be asked to sketch one complete cycle for each graph, label significant points, and list the Domain, Range, Period and Amplitude for each graph. Trigonometric graphs The sine and cosine graphs. Acellus Trigonometry is A-G Approved through the University of California. 1 [39 KB] Word Worksheet 4. Graphing Trig Functions Practice Worksheet With Answers Students will practice graphing sine and cosine curves : a) identify period and amplitude based on equation or on the graph b) write equation from graph c) write. Developing Skills 3. Specifically, the denominator of a. A complete set of Class Notes, Handouts, Worksheets, PowerPoint Presentations, and Practice Tests. When you plot the five key points on the graph, note that the intercepts are the same as they are for the graph of y =3sinx. Graphing Trig Functions Activity {Graph Trig Functions Worksheet} by. 255 2 a = 1 2 b = 1 2 c = 1 d = 3 2 e = 1 f = 1 3 g = −cos 60° = 2 − h = sin 45° = 1 2 i = tan 30° = 1. Graphing Transformations of Exponential Functions. You've already learned the basic trig graphs. The sine graph or sinusoidal graph is an up-down graph and repeats every 360 degrees i. Graphs for inverse trigonometric functions. Three cameras will be randomly selected and assigned to box A. There are 30 questions. The worksheet templates available here arrive from several sources that are not money oriented. Free math lessons and math homework help from basic math to algebra, geometry and beyond. 7 Pythagoras’ theorem/ Area and perimeter (expressions) 8 3. Worksheet 6-12, #1: Fine the inverse equation of y = 2x-3. 3 Even & Odd 5. Graphing Trig Functions Activity {Graph Trig Functions Worksheet} by. Label the sides adjacent, opposite and hypotenuse in relation to the given angle 34º. Y Y eADlIlg CrjiHgthetBsw ^rLesYebrrvTeBdE. There are 30 questions. Sketch the graph of y 5 sin x in the interval 0 x 4p. Worksheets with Trigonometry Questions Trigonometry Questions (1). y = 2 sin 3x 2. 1 Reviewing the Trigonometry of Right Triangles 4. They are also very customizable: you can control the number of problems, font size, spacing, the range of numbers, and so on. See more ideas about Calculus, High school math, Precalculus. Trigonometry Worksheets & Problems. Trigonometry Right triangle trig: Evaluating ratios Right triangle trig: Missing sides/angles Angles and angle measure Co-terminal angles and reference angles Arc length and sector area Trig ratios of general angles Exact trig ratios of important angles The Law of Sines The Law of Cosines Graphing trig functions Translating trig functions. Online Pre-calculus Solver. Papa Math Kids Worksheet High School Geometry Worksheets Pdf Alphabet Dot To Waves Middle Printable Phonics For Kindergarten Centers Factoring Review Answers Read Dictionary 4th Graders Art Preschool Activities Printouts Free Toddler Worksheets Four Digit Addition Word Problems easy word problems Free Printable Activities For Kindergarten kids worksheet and trigonometry structure and method. Worksheet by Kuta Software LLC Kuta Software - Infinite Precalculus Graphs of Trig Functions Name_____ Date_____ Period____-1-Find the amplitude, the period in radians, the phase shift in radians, the vertical shift, and the minimum and maximum values. Mathematics in Education and Industry. 8 Name: _____ Graphing Trig Functions Seat #: _____ - _____ Graph the following functions one period in the + x direction and one period in the – x direction. Because the graph is a reflection of the graph of y =3sinx, the amplitude is 3 and the period is 2π. There is ample space on the page for the student to work out this one problem. Trigonometry Worksheets & Problems. The following are word problems that use periodic trigonometry functions to model behavior. , Nov 13th Inverse Values and Angles Day 2 Worksheet Wed. y: 3 sin Amplitude : Period - 15. Trigonometry Graphs Worksheet. For example see the graph of the SIN function, often called a sine wave, above. Free trigonometry worksheets, in PDF format, with solutions to download. y = 7 cos - 1 5. We have some photos of Graphing Trig Functions Worksheet that you could download completely free. But just as you could make the basic quadratic, y = x 2, more complicated, such as y = –(x + 5) 2 – 3, so also trig graphs can be made more complicated. 4 Rates of Change and Tangent Lines Chapter 3 3. Graphing Trig Functions. 2) Graph the function 7=;<8# using key points between 0° and 360°. Find the inverse of 4 (2) x g x x Mini Lecture: If fx( ) 3 x and 1 2 x gx §· ¨¸ ©¹ find, a. Sine Function Graph. Answer any questions that follow. Graphs of Trig. Graphing Trig Functions TEST STUDY GUIDE Test covers: Know domain, range, period, and intercepts of the six parent trig functions. 350 # 59-64 Tue. Worksheet 2. Now we can write the equation. • The range of each function is all real numbers. y= cos 2x SOLUTION a. Show work here! 10. These worksheets are a free and fun way to test your electrical engineering knowledge! Check your proficiency with everything from basic electricity to digital circuits. 1 [62 KB] Word Worksheet 3. 3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Some of the worksheets for this concept are Graphing trig functions, Graphs of trig functions, Practice work graphs of trig functions, Algebra 2 study guide graphing trig functions mrs, Work properties of trigonometric functions, Inverse trig functions, Work 15 key, Work 15. 350 # 59-64 Tue. There are various domains, marked increments of pi, and ranges, marked in integral increments. Graphing Sine and Cosine with Phase (Horizontal) Shifts How to find the phase shift (the horizontal shift) of a couple of trig functions? Example:. - Stats Worksheet #2. 7in Section1. Graph the equation —Ito —qo o 10. Each math worksheet is accompanied by an answer key, is printable, and can be customized to fit your needs. Page updated : 15 April 2018. ©o n2f0 h1k2 T 6K 6uYtPa 5 ES1o6f5tJw za pr ye0 TLoLGCG. As you drag the point A around notice that after a full rotation about B, the graph shape repeats. Part II: Graphing. 2 Verifying Trigonometric Identities. Papa Math Kids Worksheet High School Geometry Worksheets Pdf Alphabet Dot To Waves Middle Printable Phonics For Kindergarten Centers Factoring Review Answers Read Dictionary 4th Graders Art Preschool Activities Printouts Free Toddler Worksheets Four Digit Addition Word Problems easy word problems Free Printable Activities For Kindergarten kids worksheet and trigonometry structure and method. Interesting Graphs - A few equations to graph that have interesting (and hidden) features. MAT 112 Worksheet to go with 7. 5 Tan graphs 21-25 AK. com, all rights reserved. Graphing Trig Functions Worksheet FREE Printable Worksheets from graphing trig functions worksheet , image source: www. Practice Worksheet: Writing Equations of Trig Functions 1] Vertical displacement: 2] Period and b-value: 3] Amplitude: 4] Reflection in the x-axis? 5] Equation of the graph assuming no phase shift. I wrote them as fractions to minimize the help calculators can give you. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Grade 11 online worksheets are due to be released soon. The motion of the toy starts at its highest position of 5 inches above its rest point, bounces down to its lowest position of 5 inches below its rest point, and then bounces back to its highest position in a total of 4 seconds. The following diagrams show the graphs the trig functions: sin, cos, tan, csc, sec, and cot. Math Word Searches All items below are categorized by their difficulty level and target audience so you can pick the perfect level of fun and education. This is the list of frequently used formulas in Trigonometry calculations Basic Formulas: sin Θ = opposite / hypotenuse cos Θ = adjacent / hypotenuse tan Θ = opposite / adjacent csc Θ = hypotenuse / opposite sec Θ = hypotenuse / adjacent cot Θ = adjacent /opposite Range of All possible value for Sin, cos, tan, csc, sec and cot. Graph Trig Functions. 1 Using Fundamental Identities 5. 2 Graphing Sine and Cosine F 13 MAY 2016 - 8. the graph, and multiplying or dividing by a negative value causes a reflection, which is discussed later. Let R be the region bounded by the graph y = 2x - x 2 and the x axis. Determine the exact value of each of the following without using a calculator. 4$\ds xy^2=1$. Verify the graph crosses the x-axis at -0. You need to enable JavaScript in your browser to work in this site. y = 3 sin (2x) 2. Worksheet – Graphing other trig functions Sept. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. Trigonometry & Calculus - powered by WebMath. Experiment with cases and illustrate an explanation of the effects on the graph using technology. 6 Trig Functions Chapter 2 2. a) Graph the functions y = sin x and y = cos x on the same axis for −2π≤ x ≤ 2π. Keyword-suggest-tool. 3 Continuity: 2. (arcsinx) 2. 2 cm, PR = 11. Many students are reminded of parabolas when they look at the half period of the cosecant graph. Developing Skills 3. Trigonometry Angles and angle measure Right triangle trigonometry Trig functions of any angle Graphing trig functions Simple trig equations Inverse trig functions Fundamental identities Equations with factoring and fundamental identities Sum and Difference Identities Multiple-Angle Identities Product-to-Sum Identities Equations and Multiple. Example 1: Determine whether the functions are one-to-one: (a) (b) Solution: It is best to graph both functions and draw on each a horizontal line. Graphing Trig Functions Worksheet FREE Printable Worksheets from graphing trig functions worksheet , image source: www. Trigonometry Topics Trigonometry Worksheets Unit Circle Charts Introduction to the Six Trigonometric Functions (Ratios) SOH CAH TOA Standard Position of an Angle - Initial Side - Terminal Side Angle Definition and Properties of Angles Angle Definition and Properties of Angles Coterminal Angles Functions of Large and Negative Angles. One solution for this problem is the use of junior high school worksheets. Its graph (in blue) and that of sin(x) (in red) are shown. Recall that a trigonometric function ('trig function') is simply a. ) The domain of the sine function. TIP: To graph exponential functions, you only need to find enough points to generate the “L” shape of the graph. Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates. 4 Trigonometric Functions of Any Angle 4. 5 Quiz and Area of Oblique Triangles W 18 MAY 2016 - 8. The point P can move around the circumference of the circle. with more related ideas as follows graphing quadratic functions worksheet answers, graphing trig functions cheat sheet and 8th grade function table worksheet. Graphing Sine and Cosine Functions Graph the function. I wrote them as fractions to minimize the help calculators can give you. Below we have created a form that allows you to input your own values to make a graph worksheet. Lockdown revision never really happened? Our online Maths A-level refresher courses on 17-21 August will review Year 12 content, getting you ready for September. Loading Graphs of the trigonometric functions Graphs of the trigonometric functions Trigonometry: Wave Interference. And similarly for each of the inverse trigonometric functions. Some of the worksheets for this concept are Graphing trig functions, Graphs of trig functions, Graphs of sine and cosine functions, Unit 5 graphing trig functions applications work omega, Amplitude and period for sine and cosine functions work, 1 of 2 graphing sine cosine and tangent. Quadratic Theory Topic Test: Includes questions and scanned handwritten solutions. Lesson starts with plotting the graphs of Sine, Cosine and Tangent functions. In this graphing trigonometric functions worksheet, students graph one function: y=-3cos(2x-pi/4). (arcsinx) 2. Each problem has a unique solution that corresponds to a coloring pattern that forms a symmetrical image. 3] Video Worksheet – 9th Edition Trigonometry Graphs Sketch a graph and list the domain, range, and period for each of the functions, show asymptotes Pythagorean Identities cos sin22TT 1 tan2 T cot 12 T cosӨ=y sinӨ=y tanӨ=y secӨ=y cscӨ=y cotӨ=y. 5 m pM gasd fe w yw ni Lt khP GIen lfui 3n Ii Jt qeh zA rldgxe vb0r Aal s2 i. Scribd is the world's largest social reading and publishing site. ©U 32 w0m174U ZKVuAt9a v cSToufRtrw 7a lr 8eq 1L 6L dCM. Trigonometric Graphing - Explore the amplitude, period, and phase shift by examining the graphs of various trigonometric functions. Cool Math has free online cool math lessons, cool math games and fun math activities. Struggling with scatterplots? Can't quite wrap your head around circumference? Here are resources and tutorials for all the major functions, formulas, equations, and theories you'll encounter in math class. The \$$x\$$-values are the angles (in radians – that’s the way … Graphs of Trig Functions. Title: Graphing Sine and Cosine – Worksheet #1 Author: chris. Here are some examples, first with the sin function, and then the cos (the rest of the trig functions will be addressed later). Eleventh Grade (Grade 11) Trigonometry questions for your custom printable tests and worksheets. To graph a rational function, find the asymptotes and intercepts, plot a few points on each side of each vertical asymptote and then sketch the graph. Show work here!. Graphing Trig Functions Review Worksheet. 2$\ds y=3x\$ Ex 10. 6 Trig Functions Chapter 2 2. Graphing Calculator. This graph, like y=2X, levels out to the horizontal asymptote y=0, except on the right side instead of the left. Trigonometric Graphs and Transformations: Sin and Cos. y = 5 sin. Then sketch the graph using radians. Trig Graphs Worksheet State the equations for the following graphs. xchanges, so we should express the area of the washer as a function of x. 2 Trigonometric Ratios of any angle 5. Graphing The Sine Function. Recalling Section1. Welcome to the statistics and probability page at Math-Drills. In the following example you will find all the possible measures of an angle of a triangle using Law of Sines. So we narrow our focus to the choices involving tangents. Using the pull-down menus, select values for a, b, c, and d. y = 4 tan 6. Unit Circle, Radians, Coterminal Angles. 8 Sketching Trig Functions. Press [2ND] then [CALC]. College Trigonometry Version bˇc Corrected Edition by Carl Stitz, Ph. But just as you could make the basic quadratic, y = x 2, more complicated, such as y = –(x + 5) 2 – 3, so also trig graphs can be made more complicated. Every time you click the New Worksheet button, you will get a brand new printable PDF worksheet on Trigonometry and its Applications. More Lessons for Trigonometry Math Worksheets In this lesson we will look at Graphing Trig Functions: Amplitude, Period, Vertical and Horizontal Shifts. 6 Angles of Elevation and Depression R 19 MAY 2016 - 8. Graph the equation y cos2x in the interval 0 ≤ x ≤ 2π. a) b) c) d) e) 2) Identify the amplitude, frequency, period, and phase shift of each. In such situations, 1st grade worksheets become, Read More. Drag the red point to draw the graphs! If you enter sin(x) or cos(x) in the input bar in GeoGebra, you will see the graphs of sine or cosine for all values of $$x$$. 6] Vertical displacement: 7] Period and b-value: 8] Vertical stretch/shrink: 9] Reflection in the x-axis?. 5 Tan graphs 21-25 AK. The \$$x\$$-values are the angles (in radians – that’s the way … Graphs of Trig Functions. Graphing Sine and Cosine with Phase (Horizontal) Shifts How to find the phase shift (the horizontal shift) of a couple of trig functions? Example:. Graphing Sine and Cosine 1 hr 44 min 5 Examples Intro to Video: Graphing Sine and Cosine Lesson Overview and Graphing using a Table of Values Comparing the Graphs of Sin(x) and Cos(x) Steps and Formula for Graphing Sine and Cosine Example #1: Graph Sin(x) Example #2: Graph Cos(x) with an Amplitude Change Example #3:…. Acellus Trigonometry is A-G Approved through the University of California. 7in Section1. Students can select values to use within the function to explore the resulting changes in the graph. comments (-1) trigonometric functions study guide - solutions. See related links to what you are looking for. Get free math help by watching free math videos online from algebra and geometry to calculus and college math. 6 Trig Functions Chapter 2 2. AP Calculus Worksheet: Limits, Continuity, Differentiability, and Local Linearization Find each limit, if it exists. Graph trigonometric functions. Answers are included for every question. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. ppt - Free download as Powerpoint Presentation (. Find pertinent information. and search "Graphing trig functions" to view the video lessons Solutions available on THS website. (Round your answer to 2 decimal places). Set Notation and Language Functions Quadratics Indices and Surds Factors of Polynomials Simultaneous Equations Logarithms and Exponential Functions Straight Line Graphs Circular Measure Trigonometry Permutations and Combinations Binomial Expansion Vectors Matrices Differentiation and Integration. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. Students use graphs of y=sin(x) and y=cos(x) to realize that multiple solutions exist for trig functions. Also, sine and cosine functions are fundamental for describing periodic phenomena - thanks to them, we can describe oscillatory movements (as simple. Worksheet - Transformations ofFunctions and theirGraphs 1. Matching Worksheet - Most students will jump to the calculator for these. Using the pull-down menus, select values for a, b, c, and d. 2 [71 KB] Word Chapter 3: Other graphs and modelling Worksheet 3. Below we have created a form that allows you to input your own values to make a graph worksheet. Math workbook 1 is a content-rich downloadable zip file with 100 Math printable exercises and 100 pages of answer sheets attached to each exercise. For full functionality of this site it is necessary to enable JavaScript. Jun 18, 2017 - Converting Fractions To Decimals Color Worksheet. To find the phase shift and translations for a trigonometric function. y: 3 sin x Amplitude = Period 16. (a) Sketch the graph of this function and write the equation expressing distance in terms of time. 4 Investigation: Sketching the graphs of: f(x) = sin x f(x) = cos x f(x) = tan x. Most of the problems will give key insights into new ideas and so you are encouraged to do as many as possible by yourself before going for help. Amplitude = _____ Please visit www. Northern Illinois University sample calculus exams. 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Then sketch the graph using radians. | 2020-10-23 09:09:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3389575779438019, "perplexity": 2027.000069766766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880878.30/warc/CC-MAIN-20201023073305-20201023103305-00270.warc.gz"} |
https://www.esaral.com/tag/plate-capacitor/ | Principle of parallel plate capacitors – Definition, Capacitance – eSaral
Hey, do you want to learn about the Principle of Parallel plate capacitor? If yes. Then keep reading.
## Principle of Parallel plate capacitor
Let an insulated metal plate A be given a positive charge till its potential becomes maximum. When another insulated plate B is brought near A. Then by induction inner face of B becomes negatively charged and the outer face becomes positively charged. The negative charge tries to reduce the potential of A and the positive charge tries to increase it. When the outer surface of B is earthed positive charge flows to the earth while the negative charge stays on causing a reduction in the potential of A. Thus, a larger amount of charge can be given to A to raise it to maximum potential.
### Important Points
1. The capacitance of an insulated conductor is increased by bringing an uncharged earthed conductor near it.
2. An arrangement of two conductors carrying equal and opposite charge separated by a dielectric medium are said to form a capacitor.
3. The capacitor is an arrangement for storing a large amount of charge and hence electrical energy in a small space.
4. The capacity of a capacitor is defined as the ratio of charge Q on the plates to the potential difference between the plates i.e. $C=\frac{Q}{V}$
5. Capacitors are used in various electrical circuits like oscillators, in tuning circuits, filter circuits, electric fan, and motor, etc.
6. The shape of conductors can be a plane, spherical or cylindrical make a parallel plate, spherical or cylindrical capacitor.
## Capacitors in parallel
1. Capacitors are said to be connected in parallel between two points if it is possible to proceed from one point to another point along different paths.
2. Capacitors are said to be in parallel if the potential across each individual capacitor is the same and equal to the applied potential.
3. Charge on each capacitor is different and is proportional to capacity of capacitor $Q \propto C$ so $\mathrm{Q}_{1}=\mathrm{C}_{1} \mathrm{~V}$ , $\mathrm{Q}_{2}=\mathrm{C}_{2} \mathrm{~V}$ , $Q_{3}=C_{3} V$
4. The parallel combination obeys law of conservation of charge So
$\mathrm{Q}=\mathrm{Q}_{1}+\mathrm{Q}_{2}+\mathrm{Q}_{3}$
$=C_{1} V+C_{2} V+C_{3} V$
$=\left(C_{1}+C_{2}+C_{3}\right) V$equivalent capacitance $C_{p}=\frac{Q}{V}$
$=C_{1}+C_{2}+C_{3}$
5. The equivalent capacitance may be defined as the capacitance of a single capacitor that would acquire the same total charge Q with the same potential difference V.
6. The equivalent capacitance in parallel is equal to the sum of individual capacitances.
7. The equivalent capacitance is greater than the largest of individual capacitances.
8. The capacitors are connected in parallel (a) to increase the capacitance (b) when larger capacitance is required at low potential.
9. If n identical capacitors are connected in parallel then equivalent parallel capacitance $C_{p}=n C$
10. The total energy stored in parallel combination
$U=\frac{1}{2} C_{p} V^{2}$
$=\frac{1}{2}\left(\mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}+\ldots .\right) \mathrm{V}^{2}$
or$\mathrm{U}=\frac{1}{2} \mathrm{C}_{1} \mathrm{~V}^{2}+\frac{1}{2} \mathrm{C}_{2} \mathrm{~V}^{2}+\ldots \ldots=\mathrm{U}_{1}+\mathrm{U}_{2}$
$+U_{3}+\ldots \ldots$
The total energy stored in parallel combination is equal to the sum of energies stored in individual capacitors.
11. If n plates are arranged as shown they constitute (n–1) capacitors in parallel each of capacitance $\frac{\varepsilon_{0} \mathrm{~A}}{\mathrm{~d}}$
Equivalent capacitance $C_{P}=(n-1) \frac{\varepsilon_{0} A}{d}$
## Capacitance of parallel plate capacitor with conducting slab
The original uniform field $E_{0}$ exists in distance d-t so potential difference between the plates
$V=E_{0}(d-t)=\frac{\sigma}{\varepsilon_{0}}(d-t)$
$=\frac{Q}{\varepsilon_{0} A}(d-t)$
Capacitance $C=\frac{Q}{V}$
$=\frac{\varepsilon_{0} A}{d(1-t / d)}=\frac{C_{0}}{1-t / d}$
$c>c_{0}$ so capacitance increases on introducing a metallic slab between the plates.
## The capacitance of parallel plate capacitor with dielectric slab
When a dielectric is introduced between plates then $\mathrm{E}_{0}$ field is present outside the dielectric and field E exists inside the dielectric. The potential difference between the plates
$V=E_{0}(d-t)+E t=E_{0}(d-t)$
$+\frac{E_{0} t}{K}=E_{0}\left[d-t\left(1-\frac{1}{K}\right)\right]$
$\mathrm{V}=\frac{\sigma}{\varepsilon_{0}}\left[\mathrm{~d}-\mathrm{t}\left(1-\frac{1}{\mathrm{~K}}\right)\right]$
$=\frac{\mathrm{Qd}}{\varepsilon_{0} \mathrm{~A}}\left[1-\frac{\mathrm{t}}{\mathrm{d}}\left(1-\frac{1}{\mathrm{~K}}\right)\right]$
Capacitance $C=\frac{Q}{V} \frac{\varepsilon_{0} A}{d\left[1-\frac{t}{d}\left(1-\frac{1}{K}\right)\right]}$
$=\frac{C_{0}}{1-\frac{t}{d}\left(1-\frac{1}{K}\right)}$
1. $C>C_{0}$ so capacitance increases on introducing a dielectric slab between plates of capacitor.
2. The capacitance is independent of position of dielectric slab between the plates.
3. If whole space is filled with dielectric than t = d and $C=K C_{0}$
## Energy stored in Capacitor
The charging of a capacitor involves transfer of electrons from one plate to another. The battery transfers a positive charge from negative to positive plate. Some work is done in transferring this charge which is stored as electrostatic energy in the field.
If dq charge is given to capacitor at potential V
then dW = V dq
or
$W=\int_{0}^{Q} \frac{q}{C} d q$
$\frac{Q^{2}}{2 C}=\frac{1}{2} C V^{2}=\frac{1}{2} Q V$
### Important Points
1. The energy is stored in the electric field between plates of capacitors.
2. The energy stored depends on capacitance, charge, and potential difference. This does not depend on the shape of the capacitor.
3. The energy is obtained at cost of the chemical energy of the battery
So, that’s all from this article. If you liked this article on the Principle of Parallel plate capacitors then please share it with your friends. If you have any confusion related to this topic then feel free to ask in the comments section down below.
For a better understanding of this chapter, please check the detailed notes of the Electric charge and field. To watch Free Learning Videos on physics by Saransh Gupta sir Install the eSaral App. | 2021-10-26 21:48:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.51722651720047, "perplexity": 518.6065052164321}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587926.9/warc/CC-MAIN-20211026200738-20211026230738-00516.warc.gz"} |
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Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. x�cbd�gb8 $xx�c����[�4��A� !�Hp��)201�X���(1Jc��r�]@���@��%�% � 8 There is more than … See search results for this author. Definition (Differential equation) A differential equation (de) is an equation involving a function and its deriva- tives. This film is the third video on solving separable differential equations and covers the topic of using a substitution when you are presented with composition of functions in your ordinary differential equation. This can be understood in the frequency domain using the Laplace transform and its pole diagram. Included are most of the standard topics in 1st and 2nd order differential equations, Laplace transforms, systems of differential eqauations, series solutions as well as a brief introduction to boundary value problems, Fourier series and partial differntial equations. 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These lectures treat the subject of PDEs by considering specific examples and studying them thoroughly. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. This playlist contains 32.5 hours across 52 video lectures from my Differential Equations course (Math 2080) at Clemson University. The videotaping was made possible by The d'Arbeloff Fund for Excellence in MIT Education. Lecture 1 Introduction to differential equations View this lecture on YouTube A differential equation is an equation for a function containing derivatives of that function. » << /Filter /FlateDecode /S 989 /O 1063 /Length 563 >> Lecture 51 : Differential Equations - Introduction; Lecture 52 : First Order Differential Equations; Lecture 53 : Exact Differential Equations; Lecture 54 : Exact Differential Equations (Cont.) endobj Differential Equations Notes PDF. A differential equation involving only derivatives with respect to one independent variable is called an ordinary differential equation (ODE). Lecture 10 Play Video: Introduction to Differential Equations This is an introduction to differential equations. Lecture 10 on Differential Equations & Differential Invariantsintroduced equational differential invariants of the form = 0 for differential equations that are much more general than the ones supported by axiom [0] fromLecture 5 on Dynamical Systems & Dynamic Axioms. What follows are my lecture notes for a first course in differential equations, taught at the Hong Kong University of Science and Technology. Knowledge is your reward. In each case we )Z!����DR�7��HㄕA7�ۉ��S�i��vl燞�P%�[R�;���n\W��4/de^��2�e�D����B%8�ضm��[R���#��P2q\ �His�d�JJ. Lecture one (27/9/2020 ) power point URL. 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WATCH ODE LECTURES ON THE YOUTUBE CHANNEL. %���� Differential Equations - MTH401 Lecture 41 617 Views Matrices and systems of linear first-Order equations (continued), Theorem: Existence of a fundamental set, … Differential Equations Lecture Notes Dr RuthE. » Lecture 56: Higher Order Linear Differential Equations Courses 2 Here and below by an interval we mean any set of the form << /Names 430 0 R /OpenAction [ 141 0 R /Fit ] /Outlines 431 0 R /PageMode /UseOutlines /Pages 466 0 R /Type /Catalog >> This note covers the following topics: First Order Equations and Conservative Systems, Second Order Linear Equations, Difference Equations, Matrix Differential Equations, Weighted String, Quantum Harmonic Oscillator, Heat Equation and Laplace Transform. A spring system responds to being shaken by oscillating. 138 0 obj e.g. This table ( PDF ) provides a correlation between the video and the lectures in the 2010 version of the course. Download files for later. videos URL. These lecture notes are designed to accompany the first year course “Fourier Series and ... equation, wave equation and Laplace’s equation arise in physical models. stream 1.4 Linear Equation: 2 1.5 Homogeneous Linear Equation: 3 1.6 Partial Differential Equation (PDE) 3 1.7 General Solution of a Linear Differential Equation 3 1.8 A System of ODE’s 4 2 The Approaches of Finding Solutions of ODE 5 2.1 Analytical Approaches 5 2.2 Numerical Approaches 5 2. stream Plik differential equations lecture.pdf na koncie użytkownika sebcio97 • folder Matematyka studia • Data dodania: 1 paź 2018 Differential equations relate functions of several variables to derivatives of the functions. Video Lectures for Partial Differential Equations, MATH 4302 Lectures Resources for PDEs Course Information Home Work A list of similar courses-----Resources for Ordinary Differential Equations ODE at MIT. Video of lectures given by Arthur Mattuck and Haynes Miller, mathlets by Huber Hohn, at Massachussette Institute of Technology. A differential equation is an equation for a function with one or more of its derivatives. When the input frequency is near a natural mode of the system, the amplitude is large. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Massachusetts Institute of Technology. Use OCW to guide your own life-long learning, or to teach others. Lectures on Ordinary Differential Equations Hardcover – Import, January 1, 1969 by Einar Hille (Author) › Visit Amazon's Einar Hille Page. We don't offer credit or certification for using OCW. Find all the books, read about the author, and more. Such equations are often used in the sciences to relate a quantity to its rate of change. Send to friends and colleagues. videos URL. Verify that if c is a constant, then the function defined piecewise by (1.15) y(x) = 1 if x ≤ c cos(x−c) if c < x < c+π −1 if x ≥ c+π satisfies the differential equation y0= − p 1−y2for all x ∈ R. Determine how many different solutions (in terms of a and b) the initial value problem ( y0= − p 1−y2, y(a) = b has. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. This table (PDF) provides a correlation between the video and the lectures in the 2010 version of the course. Lectures on Differential Equations MAA Press: An Imprint of the American Mathematical Society Lectures on Differential Equations provides a clear and concise presentation of differential equations for undergraduates and beginning graduate students. 139 0 obj Equation is the differential equation of heat conduction in bulk materials. Note: Lecture 18, 34, and 35 are not available. Modify, remix, and reuse (just remember to cite OCW as the source. endobj Learn differential equations for free—differential equations, separable equations, exact equations, integrating factors, and homogeneous equations, and more. %PDF-1.5 By this approach the author aims at presenting fundamental ideas in a clear way. (1) If y1(t) and y2(t) satisfy (2.1), then for any two constants C1and C2, y(t) = C1y1(t)+C2y2(t) (2.2) is a solution also. 137 0 obj MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Lecture two (1/10/2020 ) pdf File PDF document Uploaded 4/10/20, 19:18. power point URL. This is one of over 2,400 courses on OCW. x�cb��c2\� �fB�0A�����������1�-��*{ �U��l�8�9 Yw���e�e�f����m�O6�>ΌsF��쮜�9��?qX�=w���e��v���C�ÄS�����#���,�k�i3��Y�+��2.�2��kqjҔd�]�sգ�.�U��<7��kK�s�S[�8��xHrM��-~(�s\��i�wr���>� We introduce differential equations and classify them. Arnold's geometric point of view of differential equations is really intriguing. » On the human side Witold Hurewicz was an equally exceptional personality. Much of the material of Chapters 2-6 and 8 has been adapted from the widely FIRST ORDER DIFFERENTIAL EQUATIONS 7 1 Linear Equation 7 Differential Equations. Lecture 55 : First Order Linear Differential Equations; WEEK 12. The d'Arbeloff Fund for Excellence in MIT Education. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. » Lecture 01 - Introduction to Ordinary Differential Equations (ODE) PDF unavailable: 2: Lecture 02 - Methods for First Order ODE's - Homogeneous Equations: PDF unavailable: 3: Lecture 03 - Methods for First order ODE's - Exact Equations: PDF unavailable: 4: Lecture 04 - Methods for First Order ODE's - Exact Equations ( Continued ) PDF unavailable: 5 Home Differential Equations are the language in which the laws of nature are expressed. 140 0 obj Mathematics Baker Hilary Term2016. In these “Differential Equations Notes PDF”, we will study the exciting world of differential equations, mathematical modeling, and their applications. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. (The minus sign is necessary because heat flows “downhill” in temperature.) Differential Equations Learn about Author Central. Lecture 1: The Geometrical View of y'= f(x,y), Lecture 2: Euler's Numerical Method for y'=f(x,y), Lecture 3: Solving First-order Linear ODEs, Lecture 4: First-order Substitution Methods, Lecture 6: Complex Numbers and Complex Exponentials, Lecture 7: First-order Linear with Constant Coefficients, Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients, Lecture 10: Continuation: Complex Characteristic Roots, Lecture 11: Theory of General Second-order Linear Homogeneous ODEs, Lecture 12: Continuation: General Theory for Inhomogeneous ODEs, Lecture 13: Finding Particular Solutions to Inhomogeneous ODEs, Lecture 14: Interpretation of the Exceptional Case: Resonance, Lecture 15: Introduction to Fourier Series, Lecture 16: Continuation: More General Periods, Lecture 17: Finding Particular Solutions via Fourier Series, Lecture 19: Introduction to the Laplace Transform, Lecture 22: Using Laplace Transform to Solve ODEs with Discontinuous Inputs, Lecture 24: Introduction to First-order Systems of ODEs, Lecture 25: Homogeneous Linear Systems with Constant Coefficients, Lecture 26: Continuation: Repeated Real Eigenvalues, Lecture 27: Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients, Lecture 28: Matrix Methods for Inhomogeneous Systems, Lecture 30: Decoupling Linear Systems with Constant Coefficients, Lecture 31: Non-linear Autonomous Systems, Lecture 33: Relation Between Non-linear Systems and First-order ODEs. These video lectures of Professor Arthur Mattuck teaching 18.03 were recorded live in the Spring of 2003 and do not correspond precisely to the lectures taught in the Spring of 2010. 1The theory of partial differential equations, that is, the equations containing partial derivatives, is a topic of another lecture course. Then we learn analytical methods for solving separable and linear first-order odes. And after each substantial topic, there is a short practice quiz. Differential Equation: An equation involving independent variable, dependent variable, derivatives of dependent variable with respect to independent variable and constant is called a differential equation. ... Lecture notes on Analysis and PDEs by Bruce Driver at UCSD. There's no signup, and no start or end dates. A differential equation always involves the derivative of one variable with respect to another. Made for sharing. A First Course in Differential Equations File PDF document Uploaded 26/09/20, 23:40. 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https://mathinsight.org/dynamical_system_idea | # Math Insight
### The idea of a dynamical system
#### What is a dynamical system?
A dynamical system is all about the evolution of something over time. To create a dynamical system we simply need to decide (1) what is the “something” that will evolve over time and (2) what is the rule that specifies how that something evolves with time. In this way, a dynamical system is simply a model describing the temporal evolution of a system.
##### The state space
The first step in creating a dynamical system is to pin down what is the special “something” that we want to evolve with time. To do this, we need to come up with a set of variables that give a complete description of the system at any particular time.
By “complete description,” we don't necessarily mean that the variables will completely describe a real life system we may be trying to model. But, the variables must completely describe the state of the mathematical system. In a dynamical system, if we know the values of these variables at a particular time, we know everything about the state of the system at that time. To model some real life system, the modeler must clearly make a choice of what variables will form the complete description for the mathematical model.
The variables that completely describe the state of the dynamical system are called the state variables. The set of all the possible values of the state variables is the state space.
The state space can be discrete, consisting of isolated points, such as if the state variables could only take on integer values. It could be continuous, consisting of a smooth set of points, such as if the state variables could take on any real value. In the case where the state space is continuous and finite-dimensional, it is often called the phase space, and the number of state variables is the dimension of the dynamical system. The state space can also be infinite-dimensional.
##### The time evolution rule
The second step in creating a dynamical system is to specify the rule for the time evolution of the dynamical system. This rule must be defined to make the state variables be a complete description the state of the system in the following sense: the value of the state variables at a particular time must completely determine the evolution to all future states. If the time evolution depends on a variable not included in the state space, then the rule combined with the state space does not specify a dynamical system. One must either change the rule or augment the state space by the neccesary variables to form a dynamical system.
The time evolution rule could involve discrete or continuous time. If the time is discete, then the system evolves in time steps, and we usually let the time points be the integers $t=0,1,2,\ldots.$ We can write the state of the system at time $t$ as $x_t$. In many cases, the time evolution rule will be based on a function $f$ that takes as its input the state of the system at one time and gives as its output the state of the system at the next time. Therefore, starting at the initial conditions $x_0$ at time $t=0$, we can apply the function once to determine the state $x_1=f(x_0)$ at time $t=1$, apply the function a second time to get the state $x_2=f(x_1)$ at time $t=2$, and continue repeatedly applying the function to determine all future states. We end up with a sequence of states, the trajectory of the point $x_0$: $x_1, x_2, x_3, \ldots.$ In this way, the state at all times is determined both by the function $f$ and the initial state $x_0$. We refer to such as system as a discrete dynamical system.
In a continuous dynamical system, on the other hand, the state of the system evolves through continuous time. One can think of the state of the system as flowing smoothly through state space. As time evolves, the state $x(t)$ at time $t$ can be thought of as a point that moves through the state space. The evolution rule will specify how this point $x(t)$ moves by giving its velocity, such as through a function $v(t)=F(x(t))$, where $v(t)$ is the velocity of the point at time $t$. In this case, starting with an initial state $x(0)$ at time $t=0$, the trajectory of all future times $x(t)$ will be a curve through state space.
#### Examples of dynamical systems
To illustrate the idea of dynamical systems, we present examples of discrete and continuous dynamical systems.
##### Bacteria doubling example
The first dynamical system will model the growth of a bacteria population. Let the population size $b_t$ be the number of bacteria in the population at time $t$. If we allow the values of the $b_t$ to be non-negative integers, then the non-negative integers form the state space, and the state space is discrete.
The bacteria population grows because each baterium grows and the divides into two bacteria. For this example, we will assume that all the bacteria divide at the same time. Then, we can define one time step as the time for all the bacteria to divide into two. For this model, we ignore how much real time elapses between each division cycle, but define our time so that one unit of time is the time between divisions. In this way, we form a discrete dynamical system, and simply need a rule to determine the population size $b_t$ at the times $t=1,2,3, \ldots$ given the initial population size $b_0$ at time $t=0$.
Since every bacterium divides into two during each time step, the rule is particular simple. The population size doubles at each time step. For any time $t=n$, the population size at the next step $b_{n+1}$ will be two times the previous population size $b_n$, i.e., $b_{n+1}=f(b_n)=2b_n$.
The rule $b_{n+1}=2b_n$, coupled with the initial population size $b_0$ at time $t=0$, completely determines the population size at all future times $t$. Our dynamical system is well defined.
The bacteria doubling dynamical system is illustrated by the following applet, which shows a number of ways to represent the solution trajectory $b_1, b_2, b_3, \ldots.$ In this example, the dependence of the solution on the initial state $b_0$ is quite simple. The trajectory is just scaled by $b_0$.
Bacteria doubling. The evolution of this bacteria population follows a simple rule: the number of bacteria doubles every time step. Since the state of the system is determined by a single variable, the population size $b$, the state space is one-dimensional. The state space is represented by the blue vertical line at the left. The initial population size $b_0$ is a blue point and the population size $b_t$ at time $t$ is a red point on this state space. The time evolution of the dynamical system is also represented by the plot of population size $b_t$ versus $t$, where the lines connecting the points are not part of the graph but are there just to guide your eye. The population size $b_t$ is also illustrated in the right panel where each green dot represents one bacterium. You can change the time $t$ and the initial population size $b_0$ by moving the sliders in the gray box. Stop or start the animation by clicking the button that appears in the lower left corner of one of the panels.
In the above description of this example, we defined the state space to be the non-negative integers. The mathematical system will work the same way if we allow the population size to be any real number. Of course, it might be hard to interpret the meaning of fractional bacteria. But, as you learn how to analyze discrete dynamical systems, you will see how expanding the state space to include all real numbers will facilitate the mathematical analysis.
##### Undamped pendulum example
A second example dynamical system is a model of an undamped pendulum, that is, a pendulum that oscillates without any friction so that it will continue oscillating forever. Imagine that the pendulum consists of a rigid rod with a ball fastened at its end and that the pendulum is free to rotate around the pivot point.
We can specify the position of the pendulum by the angle $\theta$ that the rod makes with a line pointing straight down. The angle is zero when the pendulum points straight down, and $\theta = \pm\pi$ when the pendulum points straight up. We use the convention that $0 < \theta < \pi$ indicates that the pendulum is pointing toward the right and that $-\pi < \theta < 0$ indicates that the pendulum is pointing toward the left.
The angle $\theta$ completely specifies the position of the pendulum. So, our first thought might be that $\theta$ should completely determine the state of the dynamical system. However, we cannot use $\theta$ as the only state variable. If the above picture of the pendulum were a snapshot of a pendulum, we don't have enough information to know where the pendulum will move next.
We might think that the pendulum will start moving downward because we implicitly assume that the pendulum starts out being stationary. But, if I told you that when I took the snapshot of the pendulum, it was moving rapidly in a counterclockwise direction, the additional information of how the pendulum was moving at the time of the snapshot would change your prediction of where the pendulum will move next. If we know the pendulum pictured above has a counterclockwise velocity, we expect that the pendulum will continue to move upward.
In order to determine the future behavior of the pendulum, we need to know not only its position at the time of the snapshot, but also its velocity. Hence, the state space must include both those variables. It turns out that for this idealized pendulum, the angle $\theta$ and the angular velocity $\omega$ completely determine the state of the system. Both $\theta$ and $\omega$ will evolve over time, and their value at one time determines all their future values. The dynamical system is two-dimensional, and since $\theta$ and $\omega$ evolve continuously, it is a continuous dynamical system.
In the above bacteria dynamical system, we plotted the one-dimensional state space (or phase space) as a blue line. In the pendulum example, we can plot the two-dimensional state space (or phase space) as a plane, often called the phase plane. We can plot $\theta$ on the $x$-axis of the Cartesian plane and $\omega$ on the $y$-axis. In this representation, the current state of the system, $\theta(t)$ and $\omega(t)$, is a point $(\theta(t),\omega(t))$ on the phase plane, plotted as the red point on the left panel of the applet, below. The trajectory of the initial condition $(\theta(0),\omega(0))=(\theta_0,\omega_0)$ (the blue point labeled as $X_0$) is the green curve in the left panel.
To further help you visualize how $(\theta(t),\omega(t))$ flows through the state space, gray arrows indicating the velocity of $(\theta(t),\omega(t))$ are plotted throughout the phase plane. Not only do the arrows point out the direction of the flow, but their length indicates the speed of $(\theta(t),\omega(t))$.
Undamped pendulum. The dynamical system describing an undamped pendulum requires two state variables. The angle $\theta$ of the pendulum and the angular velocity $\omega$. These two variables completely describe the state of the pendulum. The state space (or phase space) $(\theta,\omega)$ is illustrated in the left panel. The space is periodic in $\theta$, and the range $-\pi \le \theta < \pi$ is shown. The initial angle and angular velocity, $\vc{X}_0 = (\theta_0,\omega_0)=(\theta(0),\omega(0))$, is set by dragging the blue point. The resulting evolution of the system is illustrated both by the green curve (the trajectory) and the moving red point, which indicates the state (i.e., angle and angular velocity) at time $t$: $(\theta(t),\omega(t))$. Gray arrows represent the speed and direction that the point $(\theta(t),\omega(t))$ moves through the state space. The right panel illustrates the evolution of just the angle $\theta(t)$ with respect to time $t$. The plot includes both the graph of $\theta$ versus $t$ in green and a moving red point, corresponding to the similar curve and point in the left panel. Above the graph are slider by which you can change the time $t$ and a graphical illustration of the pendulum with angle $\theta(t)$ and angular velocity $\omega(t)$. If the angular velocity $\omega$ is sufficiently large, the pendulum will swing around past an angle of $\pm\pi$. In this case, the variable $\theta(t)$ will wrap around from $\pi$ to $-\pi$ or vice versa. At those points, the green curves will contain extraneous lines crossing between $\theta=\pi$ and $\theta=-\pi$. Those lines should be ignored.
The state space isn't really a plane because the angle $\theta$ is $2\pi$-periodic. The left edge of the plane at $\theta=-\pi$ is really the same thing as the right edge of the plane at $\theta=\pi$, as both corresponding to the pendulum pointing straight up. You could imagine taking the phase plane plotted in the left panel and rolling it into a cylinder so that the lines $\theta=-\pi$ and $\theta=\pi$ match up. The cylinder would be a better representation of the phase space. The angular velocity $\omega$, however, is not periodic, as positive values indicate a counterclockwise rotation of the pendulum (shown as an inset in the right panel) and negative values indicate clockwise rotation.
The right panel of the above applet shows a plot of the pendulum angle $\theta$ versus time. Because the state space is two-dimensional, we can't easily plot the full system state $(\theta(t),\omega(t))$ versus time. Instead, we chose to just plot the angle. (We could have also included a separate plot of $\omega$ versus $t$.) Again, this plot should really be on a cylinder, as the top edge of the gray region is $\theta=\pi$ and the bottom edge is $\theta=-\pi$, which are really the same thing.
It may be slightly confusing that we have mentioned two different velocities. One velocity is the angular velocity $\omega(t)$, which tells us how fast the angle $\omega(t)$ is changing and in which direction. The second velocity is the velocity (speed and direction) of the point $(\theta(t),\omega(t))$ through state space, i.e., the velocity of the red point in the left panel, above. These two velocities are related, as the left and right movement of the red point is due to changes in $\theta(t)$. In fact, the velocity of this left and right movement is exactly given by the angular velocity. The up and down movement of the red point is given by changes in the angular velocity $\omega(t)$, or the acceleration of the pendulum. How $\omega(t)$ changes with time depends on a little physics, and a discussion of this mechanism is outside the scope of this introductory page. | 2019-11-20 19:41:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8116087317466736, "perplexity": 135.06102920230944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670601.75/warc/CC-MAIN-20191120185646-20191120213646-00473.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2020219?viewType=html | # American Institute of Mathematical Sciences
## The impact of toxins on competition dynamics of three species in a polluted aquatic environment
1 School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China 2 School of Mathematical and Statistical Sciences, Southwest University, Chongqing 400715, China 3 School of Mathematical and Statistical Sciences, Hubei University of Science and Technology, Xianning 437100, China
*Corresponding author: Qihua Huang
Received February 2020 Revised May 2020 Published July 2020
Fund Project: The second author is supported by the NSF of China (11871235). The third author is supported by the NSF of China (11871060), the Venture and Innovation Support Program for Chongqing Overseas Returnees (7820100158), the Fundamental Research Funds for the Central Universities (XDJK2018B031), and the faculty startup fund from Southwest University (20710948)
Accurately assessing the risks of toxins in polluted ecosystems and finding factors that determine population persistence and extirpation are important from both environmental and conservation perspectives. In this paper, we develop and study a toxin-mediated competition model for three species that live in the same polluted aquatic environment and compete for the same resources. Analytical analysis of positive invariance, existence and stability of equilibria, sensitivity of equilibria to toxin are presented. Bifurcation analysis is used to understand how the environmental toxins, plus distinct vulnerabilities of three species to toxins, affect the competition outcomes. Our results reveal that while high concentrations lead to extirpation of all species, sublethal levels of toxins affect competition outcomes in many counterintuitive ways, which include boosting coexistence of species by reducing the abundance of the predominant species, inducing many different types of bistability and even tristability, generating and reducing population oscillations, and exchanging roles of winner and loser in competition. The findings in this work provide a sound theoretical foundation for understanding and assessing population or community effects of toxicity.
Citation: Yuyue Zhang, Jicai Huang, Qihua Huang. The impact of toxins on competition dynamics of three species in a polluted aquatic environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020219
##### References:
show all references
##### References:
The illustration of Theorem 3.7
Bifurcation diagrams with respect to toxin level $T$ for the case where species-2 only equilibrium $E_2$ is globally asymptotically stable when $T = 0$. Parameters: $b_1 = 1, \ b_2 = 1.06, \ b_3 = 1.13, \ p_1 = p_2 = p_3 = k_1 = k_2 = k_3 = 1, \ m_1 = 0.44, \ m_2 = 0.5, \ m_3 = 0.56, \ c_{12} = 1.13, \ c_{13} = 1.19, \ c_{21} = 1.06, \ c_{23} = 0.88, \ c_{31} = 0.75, \ c_{32} = 1.24$. Here $T_1^* = 0.28$, $T_2^*\approx0.27$, $T_3^*\approx0.27$
(a) The Hopf bifurcation diagram corresponding to Fig. 2; (b) Limit cycle for $T = 0.18069$ in (a); (c) The time series diagram corresponding to (b); (d) The time series diagram of heteroclinic loop for $T = 0.1801$; (e) The phase portrait corresponding to (d). The other parameters are the same as those in Fig. 2
Bifurcation diagrams with respect to toxin level $T$ for the case where both species 2-only equilibrium $E_2$ and species 3-only equilibrium $E_3$ are locally asymptotically stable when $T = 0$. Parameters: $c_{13} = 1.15, \ c_{23} = 1.19, \ c_{32} = 1.13$, the other parameters and the values of $T_i^*\ (i = 1, \ 2, \ 3)$ are the same as those in Fig. 2
Bifurcation diagrams with respect to toxin level $T$ for the case where the coexistence equilibrium $E^*$ is globally asymptotically stable when $T = 0$. Part of panel (a) is enlarged by panel (b) and part of panel (c) is enlarged by panel (d). Parameters: $b_1 = 1, \ b_2 = 0.97, \ b_3 = 0.96, \ p_1 = p_2 = p_3 = k_1 = k_2 = k_3 = 1, \ m_1 = 0.52, \ m_2 = 0.47, \ m_3 = 0.53, \ c_{12} = 0.11, \ c_{13} = 0.16, \ c_{21} = 0.16, \ c_{23} = 0.21, \ c_{31} = 0.15, \ c_{32} = 0.22$. Here $T_1^* = 0.24$, $T_2^*\approx0.25$, $T_3^*\approx0.22$
Bifurcation diagrams with respect to toxin level $T$, where the parameters are the same to those in Fig. 2 except that $p_2 = 0.5, \ p_3 = 0.2$. Here, $T_1^* = 0.28$, $T_2^*\approx 0.54$, $T_3^*\approx 1.34$
Bifurcation diagrams with respect to toxin level $T$, where the parameters are similar to those in Fig. 2 except that $p_2 = 2, \ p_3 = 2.5$. Here $T_1^* = 0.28$, $T_2^*\approx0.14$, $T_3^*\approx 0.11$
The asymptotic dynamics on $\Sigma$ of system (8). $\bullet$ signifies an attractive equilibrium on $\Sigma$, $\circ$ signifies a repellent equilibrium on $\Sigma$, the intersection of its hyperbolic manifolds signifies a saddle on $\Sigma$
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2019 Impact Factor: 1.27
## Tools
Article outline
Figures and Tables | 2021-01-26 02:34:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48155879974365234, "perplexity": 4359.561651638668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00511.warc.gz"} |
https://www.snapxam.com/problems/19820075/integral-of-1-2sinx-dx?method=9 | # Step-by-step Solution
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## Final Answer
$\frac{1}{2}\ln\left|\tan\left(x\right)\right|+C_0$
## Step-by-step explanation
Problem to solve:
$\int\frac{1}{2sin\left(x\right)cos\left(x\right)}dx$
Choose the solving method
1
Take the constant $\frac{1}{2}$ out of the integral
$\frac{1}{2}\int\frac{1}{\sin\left(x\right)\cos\left(x\right)}dx$
2
Divide $1$ by $2$
$\frac{1}{2}\int\frac{1}{\sin\left(x\right)\cos\left(x\right)}dx$
3
Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$
$\frac{1}{2}\int\frac{1}{\sin\left(x\right)}\sec\left(x\right)dx$
4
Multiplying the fraction by $\sec\left(x\right)$
$\frac{1}{2}\int\frac{\sec\left(x\right)}{\sin\left(x\right)}dx$
5
We can solve the integral $\int\frac{\sec\left(x\right)}{\sin\left(x\right)}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\tan\left(x\right)$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=\tan\left(x\right)$
6
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
$du=\sec\left(x\right)^2dx$
7
Isolate $dx$ in the previous equation
$\frac{du}{\sec\left(x\right)^2}=dx$
8
Substituting $u$ and $dx$ in the integral and simplify
$\frac{1}{2}\int\frac{1}{u}du$
9
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
$\frac{1}{2}\ln\left|u\right|$
10
Replace $u$ with the value that we assigned to it in the beginning: $\tan\left(x\right)$
$\frac{1}{2}\ln\left|\tan\left(x\right)\right|$
11
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{1}{2}\ln\left|\tan\left(x\right)\right|+C_0$
## Final Answer
$\frac{1}{2}\ln\left|\tan\left(x\right)\right|+C_0$
$\int\frac{1}{2sin\left(x\right)cos\left(x\right)}dx$
### Main topic:
Trigonometric integrals
### Time to solve it:
~ 0.36 s (SnapXam) | 2020-11-25 18:48:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9853377342224121, "perplexity": 523.7852543897989}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141184123.9/warc/CC-MAIN-20201125183823-20201125213823-00623.warc.gz"} |
https://www.hackmath.net/en/math-problem/7022 | # Double 5
Peter was thinking of a number. Peter doubles it and gets an answer of 8.6. What was the original number?
n = 4.3
### Step-by-step explanation:
$n=8.6\mathrm{/}2=\frac{43}{10}=4\frac{3}{10}=4.3$
We will be pleased if You send us any improvements to this math problem. Thank you!
Showing 1 comment:
Math student
Makes no sense
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Determine the quotient (q) and the remainder (r) from division numbers 100 and 8. Take the test of correctness. | 2021-04-23 05:59:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6230887174606323, "perplexity": 2033.6699095262572}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039601956.95/warc/CC-MAIN-20210423041014-20210423071014-00632.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-x-5-x-2-0-2 | # How do you solve (x+5)/(x+2)<0?
Jun 30, 2016
$x \in \left(- 5 , - 2\right]$
#### Explanation:
$\frac{x + 5}{x + 2} < 0$
A very stodgy mechanical way of looking at this is to say that the expression will be negative if either (x+5) < 0 or (x+2) < 0, but not both
so we look at these pairs
PAIR A
(x+5) < 0 and (x+2) > 0
this requires x < -5 and x > -2, so no solution
PAIR B
(x+5) > 0 and (x+2) < 0
this requires x > -5 and x < -2, so this solution works
further refining this approach, if x = 5, then the numerator is zero, not <0. so we must exclude x = 5
if x = -2, we have a singularity $\frac{3}{{0}^{-}} = - \infty$
so the complete answer appears to be $x \in \left(- 5 , - 2\right]$
the obvious temptation here must to be to cross multiply ie to say that
if $\frac{x + 5}{x + 2} < 0$
then
$\frac{x + 5}{x + 2} \cdot \left(x + 2\right) < 0 \cdot \left(x + 2\right)$
$\setminus \implies x + 5 < 0 , q \quad x < - 5$
but that doesn't work with inequalities. worth thinking about. | 2020-06-04 21:33:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8172026872634888, "perplexity": 889.607890090029}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347458095.68/warc/CC-MAIN-20200604192256-20200604222256-00458.warc.gz"} |
https://abagoffruit.wordpress.com/2013/11/10/one-interesting-thing-the-preponderance-of-2-groups-among-finite-groups/ | ## One Interesting Thing: The preponderance of 2-groups among finite groups
This is a preliminary discussion of my knowledge about 2-groups. There are still many questions I have, and I would appreciate any insights.
When we first learn about group theory, we learn some theorems about how to decompose finite groups, most notably the Sylow theorems, which tell us (roughly) how to decompose groups based on their maximal $p$-power subgroups. So, we might expect that for random finite groups, we can obtain a lot of information about them by analyzing what happens at each prime factor of the order.
And indeed, groups that appear “in nature” tend to have several prime factors, and their Sylow subgroups are typically interesting objects in their own right. However, if we count groups not by their occurrences in nature but simply by their orders, then we see something very different: most finite groups are 2-groups: their order is a power of 2, so there is no information to be obtained by looking at other primes.
I was surprised at first to learn that the vast majority of groups are 2-groups: for example, there are 49,910,529,484 group of order up to 2000, and of those, 49,487,365,422, or more than 99%, have order $1024=2^{10}$. And the proportion of groups with 2-power order will only increase to 1 as we go further.
I don’t really understand why this is true (I have only barely skimmed the relevant papers, such as Eick and O’Brien’s “Enumerating $p$-groups” and Besche, Eick, and O’Brien’s “The groups of order at most 2000”), but here is my vague intuition. The Jordan-Hölder theorem tells us that we can assemble groups out of simple groups. Now, it is likely possible to glue two large simple groups together in quite a few ways, whereas there are fewer ways of assembling two small simple groups together. However, more relevant than the size of the pieces is the number of pieces: a Jordan-Hölder series for a group of order 1024 will contain ten factors, each isomorphic to $\mathbb{Z}/2\mathbb{Z}$. As we all learned when we were very young and trying to assemble jigsaw puzzles, there are lots of ways of putting together many small pieces, but there are not so many ways of putting together a few large pieces. And this is just as true of groups as it is of jigsaw puzzles.
But there’s a problem with this analysis: there are way more groups with order 1024 than there are with $1536 = 2^9\times 3$, even though they contain the same number of factors in their composition series. In fact, there are 408,641,062 of order 1536, which is only roughly 40 times as many as there are groups of order $512=2^9$ (as opposed to nearly 5000 times for 1024). I would like to understand this discrepancy, but I’m not there yet.
One obstacle to understanding this point for me is that there isn’t such a discrepancy for groups of order $p^nq$ versus $p^{n+1}$ when $n$ is small: for example, when $p=2$ and $n=3$, there are 14 groups of order 16, 15 of order 24, 14 of order 40, 13 of order 56, and so forth. Similarly, when $n=4$, there are 51 groups of order 32, 52 of order 48, 52 of order 80, and 43 of order 112. Since I don’t know any exciting examples of groups of order $2^n$ for $n$ large, it’s hard for me to gain intuition about what’s going on here.
There are lots of things to look into in the future, such as reading the papers by Besche and Eick. Perhaps I’ll report on them in the future. | 2017-09-23 14:42:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 14, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7834324240684509, "perplexity": 212.8462582687613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689686.40/warc/CC-MAIN-20170923141947-20170923161947-00059.warc.gz"} |
https://www.academic-quant-news.com/2020/03/research-articles-for-2020-03-15.html | # Research articles for the 2020-03-15
A class of recursive optimal stopping problems with applications to stock trading
Katia Colaneri,Tiziano De Angelis
arXiv
In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. The set-up and methodology are sufficiently general to allow the analysis of problems with both finite-time and infinite-time horizon. Finally we apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case.
Application of Deep Q-Network in Portfolio Management
Ziming Gao,Yuan Gao,Yi Hu,Zhengyong Jiang,Jionglong Su
arXiv
Machine Learning algorithms and Neural Networks are widely applied to many different areas such as stock market prediction, face recognition and population analysis. This paper will introduce a strategy based on the classic Deep Reinforcement Learning algorithm, Deep Q-Network, for portfolio management in stock market. It is a type of deep neural network which is optimized by Q Learning. To make the DQN adapt to financial market, we first discretize the action space which is defined as the weight of portfolio in different assets so that portfolio management becomes a problem that Deep Q-Network can solve. Next, we combine the Convolutional Neural Network and dueling Q-net to enhance the recognition ability of the algorithm. Experimentally, we chose five lowrelevant American stocks to test the model. The result demonstrates that the DQN based strategy outperforms the ten other traditional strategies. The profit of DQN algorithm is 30% more than the profit of other strategies. Moreover, the Sharpe ratio associated with Max Drawdown demonstrates that the risk of policy made with DQN is the lowest.
Asymptotic expansion for the transition densities of stochastic differential equations driven by the gamma processes
Fan Jiang,Xin Zang,Jingping Yang
arXiv
In this paper, enlightened by the asymptotic expansion methodology developed by Li(2013b) and Li and Chen (2016), we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Levy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein--Uhlenbeck (OU) model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.
Coronavirus and oil price crash: A note
Claudiu Albulescu
arXiv
Coronavirus (COVID-19) creates fear and uncertainty, hitting the global economy and amplifying the financial markets volatility. The oil price reaction to COVID-19 was gradually accommodated until March 09, 2020, when, 49 days after the release of the first coronavirus monitoring report by the World Health Organization (WHO), Saudi Arabia floods the market with oil. As a result, international prices drop with more than 20% in one single day. Against this background, the purpose of this paper is to investigate the impact of COVID-19 numbers on crude oil prices, while controlling for the impact of financial volatility and the United States (US) economic policy uncertainty. Our ARDL estimation shows that the COVID-19 daily reported cases of new infections have a marginal negative impact on the crude oil prices in the long run. Nevertheless, by amplifying the financial markets volatility, COVID-19 also has an indirect effect on the recent dynamics of crude oil prices.
Disturbing the Peace: Anatomy of the Hostile Takeover of China Vanke Co
Taurai Muvunza,Terrill Franzt
arXiv
Wang Shi, a business mogul who created his empire of wealth from scratch, relished in his fame and basked in the glory of his affluent business. Nothing lasts forever! After mastering the turbulent business of real estate development in his country and therefore enjoying a rising and robust stock price, China Vanke Co. Ltd ("Vanke") founder and Chairman of the Board of Directors, Wang Shi was suddenly presented with a scathing notice from the Hong Kong Stock Exchange: rival Baoneng Group ("Baoneng") filed the regulatory documentation indicating that it had nicodemously acquired 5% of his company and was looking to buy more. Vanke case became brutal and sparked national controversy over corporate governance and the role of Chinese government in capital markets.
Machine Learning Portfolio Allocation
Michael Pinelis,David Ruppert
arXiv
We find economically and statistically significant gains from using machine learning to dynamically allocate between the market index and the risk-free asset. We model the market price of risk as a function of lagged dividend yields and volatilities to determine the optimal weights in the portfolio: reward-risk market timing. This involves forecasting the direction of next month's excess return, which gives the reward, and constructing a dynamic volatility estimator that is optimized with a machine learning model, which gives the risk. Reward-risk timing with machine learning provides substantial improvements over the market index in investor utility, alphas, Sharpe ratios, and maximum drawdowns, after accounting for transaction costs, leverage constraints, and on a new out-of-sample set of returns. This paper provides a unifying framework for machine learning applied to both return- and volatility-timing.for machine learning applied to both return- and volatility-timing.
Optimal hedging of a perpetual American put with a single trade
Cheng Cai,Tiziano De Angelis,Jan Palczewski
arXiv
It is well-known that using delta hedging to hedge financial options is not feasible in practice. Traders often rely on discrete-time hedging strategies based on fixed trading times or fixed trading prices (i.e., trades only occur if the underlying asset's price reaches some predetermined values). Motivated by this insight and with the aim of obtaining explicit solutions, we consider the seller of a perpetual American put option who can hedge her portfolio once until the underlying stock price leaves a certain range of values $(a,b)$. We determine optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio. Optimality here refers to the variance of the hedging error at the (random) time when the stock leaves the interval $(a,b)$. Our study leads to analytical expressions for both the optimal boundaries and the optimal stock holding, which can be evaluated numerically with no effort.
Optimal market making with persistent order flow
Paul Jusselin
arXiv
We address the issue of market making on electronic markets when taking into account the self exciting property of market order flow. We consider a market with order flows driven by Hawkes processes where one market maker operates, aiming at optimizing its profit. We characterize an optimal control solving this problem by proving existence and uniqueness of a viscosity solution to the associated Hamilton Jacobi Bellman equation. Finally we propose a methodology to approximate the optimal strategy.
Realized volatility and parametric estimation of Heston SDEs
Robert Azencott,Peng Ren,Ilya Timofeyev
arXiv
We present a detailed analysis of \emph{observable} moments based parameter estimators for the Heston SDEs jointly driving the rate of returns $R_t$ and the squared volatilities $V_t$. Since volatilities are not directly observable, our parameter estimators are constructed from empirical moments of realized volatilities $Y_t$, which are of course observable. Realized volatilities are computed over sliding windows of size $\varepsilon$, partitioned into $J(\varepsilon)$ intervals. We establish criteria for the joint selection of $J(\varepsilon)$ and of the sub-sampling frequency of return rates data.
We obtain explicit bounds for the $L^q$ speed of convergence of realized volatilities to true volatilities as $\varepsilon \to 0$. In turn, these bounds provide also $L^q$ speeds of convergence of our observable estimators for the parameters of the Heston volatility SDE.
Our theoretical analysis is supplemented by extensive numerical simulations of joint Heston SDEs to investigate the actual performances of our moments based parameter estimators. Our results provide practical guidelines for adequately fitting Heston SDEs parameters to observed stock prices series. | 2020-04-07 19:24:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3453758955001831, "perplexity": 1125.9188197436597}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371805747.72/warc/CC-MAIN-20200407183818-20200407214318-00097.warc.gz"} |
https://www.jobilize.com/physics/section/conceptual-questions-therapeutic-uses-of-ionizing-radiation-by-opensta?qcr=www.quizover.com | # 32.3 Therapeutic uses of ionizing radiation (Page 3/5)
Page 3 / 5
## Conceptual questions
Radiotherapy is more likely to be used to treat cancer in elderly patients than in young ones. Explain why. Why is radiotherapy used to treat young people at all?
## Problems&Exercises
A beam of 168-MeV nitrogen nuclei is used for cancer therapy. If this beam is directed onto a 0.200-kg tumor and gives it a 2.00-Sv dose, how many nitrogen nuclei were stopped? (Use an RBE of 20 for heavy ions.)
$7.44×{\text{10}}^{8}$
(a) If the average molecular mass of compounds in food is 50.0 g, how many molecules are there in 1.00 kg of food? (b) How many ion pairs are created in 1.00 kg of food, if it is exposed to 1000 Sv and it takes 32.0 eV to create an ion pair? (c) Find the ratio of ion pairs to molecules. (d) If these ion pairs recombine into a distribution of 2000 new compounds, how many parts per billion is each?
Calculate the dose in Sv to the chest of a patient given an x-ray under the following conditions. The x-ray beam intensity is $\text{1.50 W}{\text{/m}}^{2}$ , the area of the chest exposed is $\text{0.0750}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ , 35.0% of the x-rays are absorbed in 20.0 kg of tissue, and the exposure time is 0.250 s.
$4.92×{10}^{–4}\phantom{\rule{0.25em}{0ex}}\text{Sv}$
(a) A cancer patient is exposed to $\gamma$ rays from a 5000-Ci ${}^{60}\text{Co}$ transillumination unit for 32.0 s. The $\gamma$ rays are collimated in such a manner that only 1.00% of them strike the patient. Of those, 20.0% are absorbed in a tumor having a mass of 1.50 kg. What is the dose in rem to the tumor, if the average $\gamma$ energy per decay is 1.25 MeV? None of the $\beta$ s from the decay reach the patient. (b) Is the dose consistent with stated therapeutic doses?
What is the mass of ${}^{60}\text{Co}$ in a cancer therapy transillumination unit containing 5.00 kCi of ${}^{60}\text{Co}$ ?
4.43 g
Large amounts of ${}^{65}\text{Zn}$ are produced in copper exposed to accelerator beams. While machining contaminated copper, a physicist ingests $50.0 \mu Ci$ of ${}^{65}\text{Zn}$ . Each ${}^{65}\text{Zn}$ decay emits an average $\gamma$ -ray energy of 0.550 MeV, 40.0% of which is absorbed in the scientist’s 75.0-kg body. What dose in mSv is caused by this in one day?
Naturally occurring ${}^{\text{40}}\text{K}$ is listed as responsible for 16 mrem/y of background radiation. Calculate the mass of ${}^{\text{40}}\text{K}$ that must be inside the 55-kg body of a woman to produce this dose. Each ${}^{\text{40}}\text{K}$ decay emits a 1.32-MeV $\beta$ , and 50% of the energy is absorbed inside the body.
0.010 g
(a) Background radiation due to ${}^{226}\text{Ra}$ averages only 0.01 mSv/y, but it can range upward depending on where a person lives. Find the mass of ${}^{226}\text{Ra}$ in the 80.0-kg body of a man who receives a dose of 2.50-mSv/y from it, noting that each ${}^{226}\text{Ra}$ decay emits a 4.80-MeV $\alpha$ particle. You may neglect dose due to daughters and assume a constant amount, evenly distributed due to balanced ingestion and bodily elimination. (b) Is it surprising that such a small mass could cause a measurable radiation dose? Explain.
The annual radiation dose from ${}^{14}\text{C}$ in our bodies is 0.01 mSv/y. Each ${}^{14}\text{C}$ decay emits a ${\beta }^{–}$ averaging 0.0750 MeV. Taking the fraction of ${}^{14}\text{C}$ to be $1.3×{10}^{–12}\phantom{\rule{0.25em}{0ex}}\text{N}$ of normal ${}^{12}\text{C}$ , and assuming the body is 13% carbon, estimate the fraction of the decay energy absorbed. (The rest escapes, exposing those close to you.)
95%
If everyone in Australia received an extra 0.05 mSv per year of radiation, what would be the increase in the number of cancer deaths per year? (Assume that time had elapsed for the effects to become apparent.) Assume that there are $\text{200}×{\text{10}}^{-4}$ deaths per Sv of radiation per year. What percent of the actual number of cancer deaths recorded is this?
#### Questions & Answers
a15kg powerexerted by the foresafter 3second
what is displacement
movement in a direction
Jason
Explain why magnetic damping might not be effective on an object made of several thin conducting layers separated by insulation? can someone please explain this i need it for my final exam
Hi
saeid
hi
Yimam
What is thê principle behind movement of thê taps control
what is atomic mass
this is the mass of an atom of an element in ratio with the mass of carbon-atom
Chukwuka
show me how to get the accuracies of the values of the resistors for the two circuits i.e for series and parallel sides
Explain why it is difficult to have an ideal machine in real life situations.
tell me
Promise
what's the s . i unit for couple?
Promise
its s.i unit is Nm
Covenant
Force×perpendicular distance N×m=Nm
Oluwakayode
İt iş diffucult to have idêal machine because of FRİCTİON definitely reduce thê efficiency
Oluwakayode
if the classica theory of specific heat is valid,what would be the thermal energy of one kmol of copper at the debye temperature (for copper is 340k)
can i get all formulas of physics
yes
haider
what affects fluid
pressure
Oluwakayode
Dimension for force MLT-2
what is the dimensions of Force?
how do you calculate the 5% uncertainty of 4cm?
4cm/100×5= 0.2cm
haider
how do you calculate the 5% absolute uncertainty of a 200g mass?
= 200g±(5%)10g
haider
use the 10g as the uncertainty?
melia
which topic u discussing about?
haider
topic of question?
haider
the relationship between the applied force and the deflection
melia
sorry wrong question i meant the 5% uncertainty of 4cm?
melia
its 0.2 cm or 2mm
haider
thank you
melia
Hello group...
Chioma
hi
haider
well hello there
sean
hi
Noks
hii
Chibueze
10g
Olokuntoye
0.2m
Olokuntoye
hi guys
thomas
the meaning of phrase in physics
is the meaning of phrase in physics
Chovwe | 2021-06-16 23:34:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6553149819374084, "perplexity": 2073.080631579991}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487626122.27/warc/CC-MAIN-20210616220531-20210617010531-00629.warc.gz"} |
https://gamma-astro-data-formats.readthedocs.io/en/latest/events/index.html | IACT events¶
This document describes the format to store DL3 event data for IACTs.
The main table is EVENTS, which at the moment contains not only event parameters, but also key information about the observation needed to analyse the data such as the pointing position or the livetime.
The GTI table gives the “good time intervals”. The EVENTS table should match the GTI table, i.e. contain the relevant events within these time intervals.
No requirement is stated yet whether event times must be sorted chronologically; this is under discussion and might be added in a future version of the spec; sorting events by time now when producing EVENTS tables is highly recommended.
The POINTING table defines the pointing position at given times, allowing to interpolate the pointing position at any time. It currently isn’t in use yet, science tools access the pointing position from the EVENTS header and only support fixed-pointing observations.
We note that it is likely that the EVENTS, GTI, and POINTING will change significantly in the future. Discusssion on observing modes is ongoing, a new HDU might be introduced that stores the “observation” (sometimes called “tech”) information, like e.g. the observation mode and pointing information. Other major decision like whether there will be on set of IRFs per OBS_ID (like we have now), or per GTI, is being discussed in CTA. Another discussion point is how to handle trigger dead times. Currently science tools have to access DEADC or LIVETIME from the event header, and combine that with GTI if they want to analyse parts of observations. One option could be to absorb the dead-time correction into the effective areas, another option could be to add dead-time correction factors to GTI tables. | 2020-07-15 04:45:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3905726969242096, "perplexity": 2836.318972820221}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657155816.86/warc/CC-MAIN-20200715035109-20200715065109-00203.warc.gz"} |
http://openstudy.com/updates/4f2b2238e4b039c5a5c857ed | ## anonymous 4 years ago ax^2+bx+c=0
1. myininaya
solve for x?
2. myininaya
$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ $a \neq 0$
3. amistre64
i think this is a question whre solving for x with the completed square method gets you to the quadratic formula | 2016-10-24 07:00:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46010690927505493, "perplexity": 3919.0615742919013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719542.42/warc/CC-MAIN-20161020183839-00420-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://eprint.iacr.org/2013/265 | ### Attribute-Based Encryption with Fast Decryption
Susan Hohenberger and Brent Waters
##### Abstract
Attribute-based encryption (ABE) is a vision of public key encryption that allows users to encrypt and decrypt messages based on user attributes. This functionality comes at a cost. In a typical implementation, the size of the ciphertext is proportional to the number of attributes associated with it and the decryption time is proportional to the number of attributes used during decryption. Specifically, many practical ABE implementations require one pairing operation per attribute used during decryption. This work focuses on designing ABE schemes with fast decryption algorithms. We restrict our attention to expressive systems without system-wide bounds or limitations, such as placing a limit on the number of attributes used in a ciphertext or a private key. In this setting, we present the first key-policy ABE system where ciphertexts can be decrypted with a constant number of pairings. We show that GPSW ciphertexts can be decrypted with only 2 pairings by increasing the private key size by a factor of X, where X is the set of distinct attributes that appear in the private key. We then present a generalized construction that allows each system user to independently tune various efficiency tradeoffs to their liking on a spectrum where the extremes are GPSW on one end and our very fast scheme on the other. This tuning requires no changes to the public parameters or the encryption algorithm. Strategies for choosing an individualized user optimization plan are discussed. Finally, we discuss how these ideas can be translated into the ciphertext-policy ABE setting at a higher cost.
Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. PKC 2013. This is the full version.
Keywords
attribute-based encryption
Contact author(s)
susan @ cs jhu edu
History
Short URL
https://ia.cr/2013/265
CC BY
BibTeX
@misc{cryptoeprint:2013/265,
author = {Susan Hohenberger and Brent Waters},
title = {Attribute-Based Encryption with Fast Decryption},
howpublished = {Cryptology ePrint Archive, Paper 2013/265},
year = {2013},
note = {\url{https://eprint.iacr.org/2013/265}},
url = {https://eprint.iacr.org/2013/265}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content. | 2022-05-27 05:39:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2516707479953766, "perplexity": 2128.492088497128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662636717.74/warc/CC-MAIN-20220527050925-20220527080925-00190.warc.gz"} |
https://www.ias.edu/video?tags=15536&search=&page=8 | Video Lectures
Approximating tilting modules
Using the arithmetics of quantum numbers we construct some “approximations” of tilting modules for reductive algebraic groups that might be useful for understanding the generational patterns of tilting characters conjectured by Lusztig and...
A Hecke action on the principal block of a semisimple algebraic group
I will explain the construction of an action of the Hecke category on the principal block of representations of a connected reductive algebraic group over an algebraically closed field of positive characteristic, obtained in joint work with Roman...
Rationality properties of complex characters of finite groups
What can one say about the fields of values of irreducible complex characters ?? of a given finite group GG? In particular when GG is a finite (quasi)simple group? What about the McKay situation'', i.e. when the degree of
?? is coprime to a fixed...
New isolated symplectic singularities with trivial fundamental group
In 2000, Arnaud Beauville introduced the notion of symplectic singularities and raised the question of classifying isolated symplectic singularities with trivial local fundamental group: the latter condition is meant to avoid the numerous quotient...
Webs in type C
In 1997, Kuperberg gave a generators-and-relations presentation of the monoidal category Fund, whose objects are tensor products of fundamental representations, for all rank 2 lie algebras. The general case has been an open problem since. It was...
Ultrametric stability problems
Francesco Fournier Facio
We study stability problems with respect to families of groups equipped with bi-invariant ultrametrics, that is, metrics satisfying the strong triangle inequality. This property has very strong consequences, and this form of stability behaves very...
Computational - Statistical gaps and the Group Testing problem
In a plethora of random models for constraint satisfaction and statistical inference problems, researchers from different communities have observed computational gaps between what existential or brute-force methods promise, and what known efficient...
Springer, Procesi and Cherednik
The talk is based on a joint work with Pablo Boixeda Alvarez. We study equivariant Borel-Moore homology of certain affine Springer fibers and relate them to global sections of suitable vector bundles arising from Procesi bundles on Q-factorial...
Flag manifolds over semifields I
The study of totally positive matrices, i.e., matrices with positive minors, dates back to 1930s. The theory was generalised by Lusztig to arbitrary split reductive groups using canonical bases, and has significant impacts on the theory of cluster...
From Weyl modules to simple modules in many small steps
Let G be a semisimple group over an algebraically closed field of large characteristic. There are two important types of rational representations of G: the Weyl modules and the simple modules. We will explain how to go from the first type to the... | 2021-08-06 03:30:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.690726637840271, "perplexity": 689.6515235903288}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152112.54/warc/CC-MAIN-20210806020121-20210806050121-00010.warc.gz"} |
http://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-6-inverse-functions-6-3-the-natural-exponential-function-6-3-exercises-page-452/3 | ## Calculus 8th Edition
(a) $e^{-ln2}=\frac{1}{2}$ (b) $e^{ln(lne^{3})}=3$
The exponential function is defined by if $e^{x}=y$ then $x =lny$. (a) $e^{-ln2}=\frac{1}{2}$ (b) Let $lnz=e^{ln(lne^{3})}$ $z=ln(e^{3}) = 3lne$ $=3$ Hence, $e^{ln(lne^{3})}=3$ | 2018-04-23 17:53:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9957500696182251, "perplexity": 396.9621366013372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946120.31/warc/CC-MAIN-20180423164921-20180423184921-00321.warc.gz"} |
http://math.stackexchange.com/questions/484503/equivalence-relation-and-classes | # Equivalence Relation and classes
On $\mathbb{R}^2$, define $(a_1, a_2)\sim(b_1, b_2)$ if $a_1^2 + a_2^2 = b_1^2 + b_2^2$. Check that this defines an equivalence relation. What are the equivalence classes?
My work:
Let $a_1=b_1 = 1$ and $a_2=b_2 = 2$. We have $(1,2)\sim(1,2)\implies 5 = 5$ hence it is reflexive.
For symmetry, let $(a_1,a_2)\sim(b_1,b_2)$ then we should have $(b_1,b_2)\sim(a_1,a_2)$ to be satisfied the condition of symmetry. Like I did for reflexive, it is also symmetry.
Also this holds transitivity: let $c_1=1,c_2=2$.
I feel like I'm not on the right track and I don't know how to find the equivalence classes....
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The equivalence classes correspond to circles centered at the origin of varying radii. This much is obvious. – oldrinb Sep 5 '13 at 1:24
Most of what's presented in the question amounts to proofs by example; it's a logically fallacy, since one can prove untrue statements using proofs by example.
To prove reflexivity, your task is to show that $(a,b) \sim (a,b)$ for all $(a,b) \in \mathbb{R}^2$. You need to show that $$a^2 + b^2 = c^2 + d^2$$ holds whenever $(a,b)=(c,d)$.
To prove symmetry, you need to show that if $(a,b) \sim (c,d)$ then $(c,d) \sim (a,b)$ for all $(a,b),(c,d) \in \mathbb{R}^2$.
Try completing this: If $(a,b) \sim (c,d)$ then [some equation] holds, this implies [some other equation] holds, which implies that $(c,d) \sim (a,b)$.
To prove transitivity, you need to show that if $(a,b) \sim (c,d)$ and $(c,d) \sim (e,f)$ then $(a,b) \sim (e,f)$ for all $(a,b),(c,d),(e,f) \in \mathbb{R}^2$.
Try completing this: If $(a,b) \sim (c,d)$ then [some equation] holds. Further, if $(c,d) \sim (e,f)$, then [some other equation] holds. Together, these imply [some other equation] holds, which implies that $(a,b) \sim (e,f)$.
Finding a succinct way of describing the equivalence classes is often not straightforward. In this case, oldrinb's comment is quite helpful here:
The equivalence classes correspond to circles centered at the origin of varying radii.
Formally, the equivalence class containing $(a,b) \in \mathbb{R}^2$ is the set $$\{(c,d) \in \mathbb{R}^2:(a,b) \sim (c,d)\}$$ but this doesn't give much insight into what's in this class, and to why it's interesting. We can rewrite it $$\{(c,d) \in \mathbb{R}^2:c^2+d^2=k\}$$ where $k=a^2+b^2$, which is where oldrinb's hint comes from. (Note also that $(0,0)$ needs to be treated separately.)
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So I don't need to use norm? – therexists Sep 5 '13 at 3:40
You need to recognize them as circles. You can do that in your preferred way. I'd do it by recognizing that $x^2+y^2=k$ is the equation of a circle. – Rebecca J. Stones Sep 5 '13 at 3:53
Your helps totally helped me out. Thanks Rebecca. : ) – therexists Sep 6 '13 at 1:50
My pleasure! Thank you. – Rebecca J. Stones Sep 6 '13 at 1:50
Careful! The relation will be reflexive if it is true that $(a,b)\sim (a,b)$ for any pair $(a,b)$ you choose. Proving this is true for just one doesn't do it. But you know equality is already reflexive, right?
Think about complex numbers. If $z=a+bi$, then $|z|^2=\cdots$? What geometrical figure is $C_r=\{z\in \Bbb C:|z|=r\}$?
If you don't know about complex numbers, pick any $(a,b)$; and set $r^2=a^2+b^2$. Define $\lVert (a',b')\rVert=\sqrt{a'^2+b'^2}$. We're then saying that $(a,b)\sim (a',b')$ iff $\lVert (a,b)\rVert^2 =\lVert (a',b')\rVert^2$. So, the equivalence classes are of the form $$\widehat{(a,b)}=\left\{(x,y):\lVert (x,y)\rVert^2 =x^2+y^2=r^2\right\}$$
What does $\sqrt{x^2+y^2}$ measure for a point in $\Bbb R^2$?
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it might be more 'enlightening' for those in lower-level math to instead consider $\{(x,y)\in\mathbb{R}^2:x^2+y^2=r^2\}$ ;-) complex numbers may confuse him... +1 though – oldrinb Sep 5 '13 at 1:25
I found a website mathcs.org/analysis/reals/logic/answers/projspc.html – therexists Sep 5 '13 at 1:28
This relates to my question, right? – therexists Sep 5 '13 at 1:28
@therexists I think you can answer that! =D – Pedro Tamaroff Sep 5 '13 at 1:28
@PeterTamaroff well of course I do not argue -- you are correct in both your points. My point was only that it may confuse the OP who may not immediately see the connection between the problems – oldrinb Sep 5 '13 at 1:29 | 2015-08-28 12:58:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9354821443557739, "perplexity": 311.1352511144882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644062782.10/warc/CC-MAIN-20150827025422-00096-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://leanprover-community.github.io/archive/stream/113488-general/topic/Arend.20.E2.80.94.20HoTT.20ITP.html | ## Stream: general
### Topic: Arend — HoTT ITP
#### Johan Commelin (Aug 07 2019 at 06:14):
https://arend-lang.github.io/2019/07/17/Arend-1.0.0-released.html
#### Johan Commelin (Aug 07 2019 at 06:57):
From my first glimpse it seems they have no bundling issues: https://arend-lang.github.io/about/arend-features#anonymous-extensions
#### Johan Commelin (Aug 07 2019 at 06:58):
Of course I can't say anything about speed.
#### Johan Commelin (Aug 07 2019 at 06:59):
Universes are cumulative
#### Johan Commelin (Aug 07 2019 at 07:00):
It's HoTT, so it has transport
#### Johan Commelin (Aug 07 2019 at 07:01):
What!! And \Prop is even impredicative.
#### Johan Commelin (Aug 07 2019 at 07:01):
Is this thing even consistent?
#### Andrew Ashworth (Aug 07 2019 at 07:05):
You can always pick 2 of: impredicative Prop, large elimination, and excluded middle
#### Johan Commelin (Aug 07 2019 at 07:06):
What does "large elimination" mean?
#### Andrew Ashworth (Aug 07 2019 at 07:09):
https://lean-forward.github.io/logical-verification/2018/41_notes.html
#### Andrew Ashworth (Aug 07 2019 at 07:10):
there's a section on large elimination in the notes
#### Johan Commelin (Aug 07 2019 at 07:11):
The list of editor features (Intellij) also looks quite nice: https://arend-lang.github.io/about/intellij-features
#### Johan Commelin (Aug 07 2019 at 07:11):
@Andrew Ashworth Thanks, I'll take a look
#### Johan Commelin (Aug 07 2019 at 07:17):
As an example of the bundling thing: they have
\class Group \extends CancelMonoid {
| inverse : E -> E
| inverse-left (x : E) : inverse x * x = ide
| inverse-right (x : E) : x * inverse x = ide
This allows you to write (X : \Type) (Group X) or (X : Group), and these two are defeq. At the same time, they can also write
\instance GroupCategory : Cat Group
=> subCat (\new Embedding {
| f (G : Group) => \new Monoid G.E G.ide (G.*) G.ide-left G.ide-right G.*-assoc
| isEmb G H => \new Retraction {
| sec => Group.equals G H
| f_sec => idpe
}
})
to prove that groups form a category.
There is no distinction between bundled and unbundled objects.
#### Johan Commelin (Aug 07 2019 at 07:22):
@Andrew Ashworth So you gave a pick-2-out-of-3. If I pick impredicative Prop and LEM, then I have to sacrifice large elimination. But do I sacrifice it for Prop, or in general?
#### Chris Hughes (Aug 07 2019 at 07:23):
I think in general would make your theory a bit useless.
#### Johan Commelin (Aug 07 2019 at 07:25):
It seems that Arend sacrifices LEM... which is a pity.
#### Chris Hughes (Aug 07 2019 at 07:27):
I don't understand how large elimination could be consistent.
#### Chris Hughes (Aug 07 2019 at 07:30):
This would give you a way to "choose" a bijection between types, which gives you em, at least in Lean.
#### Chris Hughes (Aug 07 2019 at 07:31):
So this HoTT must be weaker in some sense than Lean without choice
#### Jason Rute (Sep 06 2019 at 22:27):
I've looked into Arend (vs Lean) a bit and here is what I think I've found...
#### Jason Rute (Sep 06 2019 at 22:27):
I think, just like in Lean, \Prop in Arend is impredicative and does not allow large elimination. Also, like Lean, the \Type universe levels are predicative and support large elimination. I see nothing inconsistent about LEM (for \Prop) in Arend, especially since LEM (for propositions) is consistent in other HoTT systems.
#### Jason Rute (Sep 06 2019 at 22:27):
Indeed, to me, the foundations for Arend seem very similar to those for Lean, but contain more “levels” of types. An analogy (possibly too simplistic) is that Lean is a logic for propositions (Prop) and sets (Type) while Arend is a logic for propositions (\Prop), sets (\Set), and other higher homotopy spaces/types (\1-Type, \2-Type, …). This allows univalence since the univalence axiom requires that a type universe (except \Prop) not be a set.
#### Jason Rute (Sep 06 2019 at 22:28):
Now, in book HoTT, propositions and sets are defined. (A proposition is a type P for which all pairs of elements are equal, and a set is a type A for which every a = a' is a proposition). In Arend, however, it seems \Prop and \Set are also built into the type hierarchy. So if A : \Set and a a' : A, then the type of a = a' is Prop, judgmentally. In this way, Arend’s \Prop is similar to Lean’s Prop.
#### Jason Rute (Sep 06 2019 at 22:28):
In Lean, inductively defined propositions are formed through a special Prop-valued induction. Similarly, in Arend they are formed by truncating an inductively defined type to \Prop. So, logical disjunction \/ in \Prop would have the same definition as disjoint union + in \Type, except that the former is explicitly truncated to \Prop. This truncation prevents large elimination.
#### Jason Rute (Sep 06 2019 at 22:28):
When it comes to classical reasoning, that is where Lean and Arend/HoTT start to differ is subtle ways (that I don’t completely appreciate yet). HoTT is all about proof relevant reasoning, but proof relevant versions of classical reasoning are not compatible with the univalence axiom. From the Curry-Howard Isomorphism, the analogy of LEM in Type would be Π α : Type, α + (α -> empty) which contradicts univalence, and would be inconsistent in Arend. Instead, Arend/HoTT’s version of LEM replaces \Type with \Prop. In Lean, this is prop_decidable, but using the univalence axiom it is equivalent to the usual propositional LEM Π p : Prop, p ∨ ¬ p). In Lean, the “type-valued LEM” above is provable via Lean’s choice function, which takes a nonempty type and extracts a value. Arend/HoTT can’t have such a strong choice function. Instead, in the Arend/HoTT version of AC, all one can do is say is that if a set A is non-empty (proposition) then there exists (proposition) a choice function. Since “exists” here is a proposition, one can’t directly use that choice function for constructions except inside proofs of propositions. (Also, in Arend/HoTT there doesn’t propositionally exist one universal choice function, but instead one for each set-indexed family of non-empty sets.)
#### Jason Rute (Sep 06 2019 at 22:28):
Just like in Lean, in Arend’s version of HoTT, it seems useful to have an impredicative universe \Prop of propositions and predicative universes of other types. (Arend/HoTT also seems to make it easier to go between arbitrary types and \Prop by either proving that a type is a proposition or by truncating it to one.) How one uses \Prop seems to me to be a matter of taste, engineering, and philosophy. Should one lean into constructive, proof relevant mathematics trying to avoid \Prop whenever possible? Or, like Lean, should one embrace classical mathematics and propositions, adding AC (in the HoTT sense) as an axiom (while still getting univalence!). (Or is there a “best of both worlds”, where one uses arbitrary types for constructive, proof-relevant reasoning, and AC for classical reasoning in \Prop and \Set, all within the same system?)
#### Johan Commelin (Sep 07 2019 at 04:46):
@Jason Rute Thanks for this thorough comparison! It only makes it seem more interesting to me! | 2021-05-14 22:10:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7052585482597351, "perplexity": 5930.032790633496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991829.45/warc/CC-MAIN-20210514214157-20210515004157-00532.warc.gz"} |
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# Bounded rationality in C. elegans is explained by circuit-specific normalization in chemosensory pathways
## Abstract
Rational choice theory assumes optimality in decision-making. Violations of a basic axiom of economic rationality known as “Independence of Irrelevant Alternatives” (IIA) have been demonstrated in both humans and animals and could stem from common neuronal constraints. Here we develop tests for IIA in the nematode Caenorhabditis elegans, an animal with only 302 neurons, using olfactory chemotaxis assays. We find that in most cases C. elegans make rational decisions. However, by probing multiple neuronal architectures using various choice sets, we show that violations of rationality arise when the circuit of olfactory sensory neurons is asymmetric. We further show that genetic manipulations of the asymmetry between the AWC neurons can make the worm irrational. Last, a context-dependent normalization-based model of value coding and gain control explains how particular neuronal constraints on information coding give rise to irrationality. Thus, we demonstrate that bounded rationality could arise due to basic neuronal constraints.
## Introduction
Decision-making is a crucial process that enables organisms to flexibly respond to environmental demands in changing conditions. The choice process has been extensively studied in humans, but it is a general phenomenon extending to the simplest of organisms. A basic assumption in neoclassical theories of choice is that choosers are consistent in their choices. Consistency in choices is the underlying axiom that defines rational behavior1.
Normative models of decision-making, such as rational choice theory in economics and foraging theory in ecology, assume optimality in the behavior of individual choosers. Their core principle is utility maximization, which assumes that choosers act to maximize an internal measure of satisfaction. Economists originally proposed that decisions rely only on the outcome probability and magnitude or expected value. However, this simple model fails to describe how human choosers actually behave, leading to the idea that choosers instead transform an expected value into an internal subjective value2. While subjective value-based theories can explain behavioral phenomena, such as risk preferences and delay discounting, those normative theories rely on the assumption of consistent choices, and empirical violations of rationality fall outside of their scope.
Humans3,4 and other animals5,6,7,8,9,10,11 can behave irrationally in many contexts. The neuronal mechanisms that lead to irrational behaviors are still unknown. Failures of “rationality”, i.e., inconsistent preferences, may reflect the implementation of decision-making in biological nervous systems facing intrinsic physical and metabolic constraints12,13. Despite varying nervous system architectures, all animal tested behave irrationally, suggesting that rationality and deviations from rationality arise from general computational principles rather than specific biological implementations. According to the idea of bounded rationality, the computational or informational load required to make truly optimal decisions exceeds the capacity of our nervous systems12,13. According to bounded rationality, violations of rationality reflect intrinsic constraints in the decision process, such as limited information, finite decision times, and the inherent limitations of information processing with biological systems. While rationality violations are technically suboptimal according to normative theories, such behavior may nevertheless reflect a more general optimization process: irrational choice behavior could be the cost of a more global optimization over both behavior and neural constraints14, or may themselves be favored by natural selection15.
One central requirement of rationality and stable value functions is independence of irrelevant alternatives, or IIA16. According to this axiom, a preference between two options should be unaffected by the number or quality of any additional options, and the relative choice ratio between options A and B (pA/pB) should remain constant regardless of the choice set. However, contextual factors such as choice set size and composition significantly alter animal and human decisions7,10,17,18.
To examine the boundaries which lead to irrationality, we established Caenorhabditis elegans nematodes as a model organism for rational decision-making. C. elegans has only 302 neurons, 32 of which are chemosensory neurons, and uses chemotaxis to achieve sophisticated behaviors, including simple forms of ethologically relevant decision-making19,20,21,22,23. Just two pairs of worm amphid sensory neurons, AWC and AWA, are required for chemotaxis toward attractive volatile odors24. Specific odors are known to be sensed exclusively by either the AWC or AWA neurons. The two AWC neurons are structurally similar, but functionally distinct from each other and sense different odors25. AWCON detects 2-butanone and acetone, while AWCOFF detects 2,3-pentanedione25,26,27,28,29.
In the current study, we examine how C. elegans olfactory decision-making depends on the composition of the odorant-defined choice set. We show that in most cases, the worms behave rationally and display consistent choices between two preferred options regardless of the strength of an irrelevant third option. However, asymmetric activation of the AWC neurons by the third odorant can lead to non-optimal decision-making and even preference reversals, which are considered to be “irrational” choices, according to the economic definition of rationality30,31. These findings are consistent with a normalization operation in the computation of odor value during decision-making and suggest that specific instances of choice irrationality arise from specific circuit architectures in C. elegans value processing and choice behavior.
## Results
### Nonoptimal choice behavior in C. elegans
To investigate if C. elegans exhibits nonoptimal choice behavior, we conducted odor preference tests, as previously described24,32. To find “IIA violations” we measured the relative preference between two attractant spots, in which the most attractive odor A and less attractive odor B were placed, in the presence or in the absence of a third odor C (Fig. 1a). By changing the concentrations of the odors used in each test, we controlled which specific odorant would be the most attractive (A), the second best (B), and the irrelevant option (C).
We first used odors which are sensed by a minimal decision-making neuronal circuit, by performing choice assays with odors detected exclusively by the two AWC neurons: 2-butanone (odor A), 2,3-pentanedione (odor B), and benzaldehyde (odor C) (Fig. 1b, Supplementary Fig. 1a and Supplementary Fig. 2). In these experiments, the third odor C was sensed by the neurons that sense both odor A and odor B, in a balanced/symmetric way. Namely, since odor C was sensed by both AWC neurons, it could potentially disrupt the sensing of both odor A (sensed only by the AWCON neuron) and odor B (sensed only by the AWCOFF neuron). We tested the effect that different concentrations of odor C have on the relative preference between A and B. While IIA violations have been observed in a wide variety of organisms5,6,7,8,9,10,11, despite numerous repetitions and iterations, we found that in C. elegans the addition of increasing concentrations of odor C did not lead to violations of rationality, as the preference ratio between odors A and B did not change in a statistically significant or physiologically relevant way. Specifically, in no case did odor B become more attractive relative to A (Fig. 1b and Supplementary Fig. 1a). We performed additional experiments to test the robustness of this rational behavior, and to validate that it does not depend on a specific odor concentration or the identity of odors A and B. The worms still behaved rationally when we changed the concentrations of 2-butanone and 2,3-pentanedione to make 2,3-pentanedione the most attractive odor (A) and 2-butanone the second most attractive odor (B) (Fig. 1c and Supplementary Fig. 1b).
In many organisms, the dopaminergic system has a strong effect on decision-making33,34. Therefore, we subjected cat-2 mutants, defective in dopamine synthesis, to the choice task described above. Similarly to wild-type animals, we did not observe any statistically significant differences in the preference between odors A and B in the presence or in the absence of option C (Supplementary Fig. 3). These experiments suggest that in C. elegans the lack of dopamine signaling does not lead to IIA violations.
In the experiments described above, all three odorants (A–C) were sensed by just two neurons, AWCON and AWCOFF. It is possible that this minimal neuronal circuit was “too simple” to give rise to inconsistent behaviors, and irrationality stems from complexity. To increase the complexity of the neuronal circuit underlying the decision process, we tested combinations of odors that are sensed by both AWC and AWA neurons. We started by testing “balanced” third odors C, in the sense that these odors are not sensed preferentially just by the neurons that sense odor A or odor B. We found that increasing the circuit complexity through the addition of another pair of neurons does not lead to inconsistencies in decision-making (Fig. 1d–f and Supplementary Fig. 2c–e). All the results presented above demonstrate that the worm’s decision-making process can be consistent and robust at least when the irrelevant alternative is sensed symmetrically, in a balanced way, by the neurons that sense odor A and the neurons that sense odor B.
Next, we broke the symmetrical pattern of olfactory inputs, to test if asymmetry in the sensing of the different odors can lead to irrational decision-making. In several experiments testing different sets of odors, we found that IIA violations, as well as preference reversals, can occur due to an asymmetric overlap between odors A and C, independently of the number of neurons involved (Fig. 2a, b). More specifically, we found that IIA violations can occur when odor C is sensed in an imbalanced manner by the neurons that sense odor A but not odor B. To our knowledge, this is the first demonstration of economic irrationality and IIA violations in C. elegans.
To test whether these conclusions are robust beyond the particular experimental setup, we performed experiments where we ensured that the distance between all three odors and the boundaries of the plate are the same. To do so we used round plates instead of square plates. These experiments showed also that the distance between odor focal points and the radius of the odor focal points do not affect the results (Supplementary Fig. 4). Moreover, to avoid any possible effects of distances, plates sizes, or shapes, we designed a simplified assay, where the worms faced binary choices between odors A and B, and odor C was embedded in the agar plate (see Methods). Again, we observed an IIA violation when we used the same combination of odors (as in Fig. 2a, b) that induces an asymmetric overlap between odors A and C (Supplementary Fig. 5). Thus, the specific neuronal architecture involved in odor sensation, and not the experimental setup, determines whether the worms would behave rationally or not.
In all the violations that we have described so far, 2-butanone, sensed specifically by the AWCON neuron, functioned as odor C, and benzaldehyde, sensed by both AWC neurons, functioned as odor A. Thus, an alternative explanation to these results is that the violations do not occur from breaking of the symmetry in the odors’ sensation but arise instead from a specific interaction between 2-butanone (odor C) and benzaldehyde (odor A) (Fig. 2c). When 2,3-pentanedione (an odor sensed by AWCOFF) served as odor C instead of 2-butanone, the worms behaved rationally (Fig. 2d and Supplementary Fig. 6a). It was previously reported that the asymmetry between the two AWC neurons is required for the ability to discriminate between benzaldehyde and 2-butanone. The authors hypothesized that 2-butanone can attenuate benzaldehyde signaling in the AWCON neuron28. Therefore, we conducted different experiments to test if the observed IIA violations stem from specific interactions between 2-butanone and benzaldehyde in the AWCON neuron. Four lines of evidence suggest that this is not the case, and instead, a general circuit principle of asymmetry in sensation underlies irrationality.
First, when 2-butanone served as either odor A or B, and benzaldehyde served as odor C, no violations were observed (see Fig. 1b, c, Supplementary Fig. 1 and Supplementary Fig. 2a, b). Thus, the worms do not make inconsistent decisions simply because they cannot distinguish between these two odors.
Second, using 2-butanone as odor C is not enough to make the worms irrational; when 2-butanone serves as odor C, but the circuit was symmetrical, the worms made consistent, rational decisions and did not show IIA violations (Fig. 2e and Supplementary Fig. 6b). This shows that 2-butanone cannot be considered as a general “distractor” or “confusant” molecule like the repellent DEET pesticide35.
Third, while both benzaldehyde and 2-butanone are attractive odors when presented separately24, little is known about the ecology of C. elegans24,36,37, and it is possible that their combination in the wild is associated with an unattractive or even repulsive substance. In this case, it would be rational for the worm to avoid an unattractive odor, formed by the combination of benzaldehyde and 2-butanone, when both are present on the same plate—it would be a “feature”, not a “bug”. To test this possibility, we examined if worms prefer benzaldehyde over a mixed combination of benzaldehyde and 2-butanone (see Methods). We found that the combination of 2-butanone and benzaldehyde was more attractive than benzaldehyde alone. The spot that contained both odorants was as attractive as would be expected based on the simple summation of the attractiveness of each of the odors alone (Supplementary Fig. 7). Thus, introducing 2-butanone does not create a new unattractive odor (with benzaldehyde) which could explain the IIA violation that we observed. These results strengthen the hypothesis that the IIA violations occur due to constraints on the neural system—that is, it’s a “bug”, not a “feature”.
Fourth, we found that IIA violations arise also due to exposure to other odors, not just benzaldehyde or 2-butanone, when odor C is sensed asymmetrically by the neurons that sense odor A but not by the neurons that sense odor B. When isoamyl-alcohol was used as odor A instead of benzaldehyde (the two chemicals are sensed by both AWC neurons) and odor C was sensed asymmetrically, we observed an IIA violation as well as irrationality (preference reversal) (Fig. 2f and Supplementary Fig. 6c). Furthermore, when acetone was used as odor C instead of 2-butanone (both are sensed only by the AWCON neuron) we observed an IIA violation and irrationality (preference reversal) (Supplementary Fig. 8).
In summary, the violations of rationality that we documented do not arise exclusively because of the identity of the two odorants benzaldehyde and 2-butanone but stem from a general property of the asymmetry in the sensation of the odor choices (see Fig. 2 and Supplementary Fig. 6). Our data raised the possibility that violations occur only when odors A and C are both sensed by the same neuron.
To test this possibility, we used mutants which have two AWCON neurons (AWCON/ON, loss of AWC asymmetry). NSY-1 (Neural SYmmetry) is a mitogen-activated protein kinase kinase kinase which is required for the asymmetric differentiation of the AWC neurons28. Nematodes carrying the nsy-1(ky542) allele were shown to have an additional AWCON neuron instead of the AWCOFF neuron28. As expected, these mutants do not perform chemotaxis toward 2,3-pentanedione and are hypersensitive to 2-butanone28.
If inconsistent decision-making stems from interference between two odors sensed by the same neuron, then having the interference occur in more neurons should increase irrational behavior. Therefore, we tested if mutants which have two AWCON neurons (AWCON/ON) would be more prone to inconsistent decision-making when both odors A and C are sensed by AWCON. When the AWCON-sensed odor acetone was used as odor C, AWCON/ON mutants made more inconsistent decisions in comparison to wild-type worms and showed irrationality (preference reversal between odors A and B) at lower concentrations of odor C compared to the wild-type worms (Fig. 3a, Supplementary Fig. 9a and Supplementary Fig. 8). As with acetone, the “distracting” effect of 2-butanone, when used as odor C, was much stronger in AWCON/ON mutants compared to wild-type worms, and irrationality (preference reversal) occurred in lower concentrations of odor C (Fig. 3b, Supplementary Fig. 9b and Fig. 2a).
Note that the AWCON/ON mutants are hypersensitive to acetone and 2-butanone since they sense these odors using two AWCON neurons instead of one. Therefore, in relatively low concentrations, these odors instantly became the most attractive odors on the plate (Supplementary Fig. 9a, b). However, this does not change the conclusions of our findings, since an IIA violation is defined as a change in the relative preference between odors A and B16.
Following these results, we hypothesized that irrationality arises due to interference of odor C with the sensing of odor A. As noted above, in AWCON/ON mutants, 2-butanone interferes with the sensing of benzaldehyde in the AWCON neuron28. We examined whether acetone, which like 2-butanone induces irrationality and is sensed only by the AWCON neuron, also disturbs the sensing of benzaldehyde in the AWCON neuron. Indeed, we found that in AWCON/ON mutants, acetone interferes with benzaldehyde sensation (Supplementary Fig. 10). Together, all these experiments support the hypothesis that a neuronal overlap in the sensation of multiple competing odors can lead to IIA violations in olfactory choice behaviors.
In cases where irrational behavior occurs due to interference in the AWCON neuron, “diluting” the role of the AWCON neuron in the sensation of odor A should “buffer” against AWCON-dependent irrationality. To test this, we used 2,4,5-trimethylthiazole, which is sensed by both the AWC and AWA neurons24,25, as odor A, instead of benzaldehyde or isoamyl-alcohol, which are sensed only by the two AWC neurons. In this setup, the AWCON neuron comprises only 25% of the neurons that sense odor A, instead of the usual 50%. We examined two odor setups in which the role of AWCON in the sensation of odor A was reduced, and we did not observe consistent IIA violations in those setups (Fig. 3c, d and Supplementary Fig. 9c, d). We then used the same experimental setup with AWCON/ON mutants, in which the AWCON neurons once again surmise 50% of the neurons that sense odor A. When the role of the AWCON neuron was re-increased, we did find strong changes in the preference ratio of A vs. B (Fig. 3e and Supplementary Fig. 9e). These experiments suggest that the relative weight of the neuron in which the interference occurs in the sensation of odor A affects the tendency to demonstrate inconsistent behavior.
Next, in addition to examining the role of interference in the neuron that senses odor A, we examined the role of the neurons that sense odor C. If inconsistent decision-making stems from interference between odors A and C in the same neuron, then “diluting” the role of this neuron in sensation of odor C should make irrational behavior less likely. To test this, we “expanded” the neuronal circuit that senses the odors (namely involved more neurons in sensation), to dilute the role of AWCON in the sensation of odor C, while preserving the proportion of neurons that sense each odor. Each of the odors was sensed by twice as many neurons (in comparison to the experiment described in Fig. 2b). While in the previous experiments that showed a violation odor C was sensed solely by AWCON, in this setup the AWCON only comprised 50% of the neurons that sense odor C. In these experiments, when odor C was sensed also by the AWCOFF neuron, we did not observe any IIA violations, nor did we see any significant changes in the preference of A over B (Fig. 3f and Supplementary Fig. 9f). Similarly, when odor C was sensed by the AWA neurons, we did not observe any IIA violations, nor did we see any significant changes in the preference of A over B (Fig. 3g and Supplementary Fig. 9g). We then performed an experiment in this “expanded circuit” setup with AWCON/ON mutants, in which the AWCON neurons are once again the only neurons sensing odor C. When the role of the AWCON neurons was re-expanded to comprise 100% of the neurons sensing odor C, we observed a significant IIA violation (Fig. 3h and Supplementary Fig. 9h). Thus, the AWCON neuron—and its relative involvement in representing odors A and C—contributes specifically to the capacity for rational choice behavior in this paradigm.
Next, we examined whether the worms would show IIA violations when odors A and B share a similar set of neurons. We used benzaldehyde (AWCBOTH) and isoamyl-alcohol (AWCBOTH) as odors A and B, and 2-butanone (AWCON) as odor C. In this setup, when benzaldehyde was odor A and isoamyl-alcohol was odor B we observed a significant violation and even irrationality (preference reversal) (Supplementary Fig. 11a). However, when isoamyl-alcohol was odor A and benzaldehyde was odor B, the worms behaved consistently (Supplementary Fig. 11b). We hypothesized that sensation of benzaldehyde is more dependent on the sensing neuron (in this case AWCON) in comparison to isoamyl-alcohol. To test this hypothesis, we examined the binary preference between benzaldehyde and isoamyl-alcohol in wild-type worms and in AWCON/ON mutants. We observed that AWCON/ON mutants have a significantly higher preference for benzaldehyde compared to the preference of wild-type worms (Supplementary Fig. 11c). Thus, the AWCON is more important for the sensation of benzaldehyde compared to isoamyl-alcohol, perhaps making the worm more prone to irrational behavior when sensing benzaldehyde (see more below).
Given our behavioral results, we propose a model of pathway-specific sensory gain control and examine its predictions (Fig. 4). The essential feature of the model is that at least some neurons in the chemosensory pathway perform a type of sensory gain control analogous to divisive normalization (see Methods). Critically, cross-odorant gain control only occurs when a given neuron is sensitive to more than one odor and both those odors are present at the same time (i.e., in the same choice set). IIA violations are predicted to occur in scenarios when odor C representations overlap with those of odor A but not odor B (asymmetric overlap). Specifically, increasing concentrations of odor C will divisively scale responses to odor A (when A and B are fixed). Odor B representations, being independent of odor C coding, are unaffected by concentrations of C. Thus, the general prediction is that increasing C will decrease the relative preference of A over B.
Our simulations show that the model predicts empirically observed IIA violations in asymmetric overlap scenarios (Fig. 4a), capturing the decrease in relative preference of A vs. B as C increases. Furthermore, the model can also capture two additional aspects of observed choice: (1) preference reversal of odors A and B, and (2) eventual selection of odor C over odor A. The gain control model also explains why C. elegans display rational choice in other circuit activation scenarios (Fig. 4b, c). In symmetric overlap scenarios (Fig. 4b), odor C activates neurons representing both odor A and odor B. Due to symmetric cross-odor normalization in the representations of odors A and B, increasing concentrations of odor C affect both of the computed values of odors A and B equally and relative preference between A and B remains stable. In no-overlap scenarios (Fig. 4c), odor C is sensed by different chemosensory neurons than those sensing odors A and B. In the model, this translates into the equations for the computed value of odors A and B (RA and RB; see Methods) carrying no C-related terms in the divisive denominator: the activity representing A and B—and thus the relative preference between the two—is independent of distractor odor C.
With additional circuit-specific clarification, the gain control model explains observed choice behavior in expanded-bandwidth scenarios, where the sensing of odor A involves all four chemosensory neurons (Fig. 3 and Supplementary Fig. 9). The gain control model captures this behavior by positing that cross-odor gain control is combined with a weighted sum of representations in all chemosensory neurons detecting a particular odor. Thus, when odor C activates only AWCON, an odor A (e.g., 2,4,5-trimethylthiazole) sensed by all four chemosensory neurons exhibits gain control effects in only 25% of its representation, while an odor sensed by only AWCON and AWCOFF (e.g., benzaldehyde) exhibits gain control in 50% of its representation; the model thus predicts a diminished effect of contextual odors on choice behavior in expanded-bandwidth scenarios (Fig. 2a and Fig. 3c). Furthermore, consistent with the data, the model predicts (Fig. 4d) that AWCON/ON mutants—which exhibit the equivalent of two functional AWCON neurons—should exhibit stronger IIA violations than wild-type worms in identical choice conditions (e.g., when odor C drives cross-normalization in AWCON neurons).
The principal model features—cross-odor normalization and weighted summation of chemosensory neuron activity—also address a key issue raised by the empirical data. The lack of IIA violations in symmetric overlap scenarios—when odor C drives neurons that represent both odor A and odor B (Fig. 1b, c)—suggests that gain control occurs in both AWC neurons. Under the normalization model, this occurs because cross-normalization in both AWC neurons affects the representations of odors A and B similarly (Fig. 4b). However, the empirical data also suggest a particularly important role for the AWCON neuron: most of the observed IIA violations involve odor C activation of AWCON (Fig. 2a, b, f and Fig. 3a, b), worms display rational choice when odor C drives AWCOFF alone (when odor A is benzaldehyde; Fig. 2d), and AWCON/ON mutants show enhanced irrationality (Fig. 3e). What explains the particular contribution of AWCON if cross-odor normalization occurs in both AWC neurons? We hypothesize that in the weighted summation of chemosensory neuron activations to compute odor value, there is a relative overweighting of AWCON vs. AWCOFF output (specifically for benzaldehyde). Under this assumption, the normalization model reproduces the strong IIA violation with odor C AWCON activation (Fig. 2a) and the lack of violation with odor C AWCOFF violation (Fig. 2d). Furthermore, a strong biased AWCON weighting for benzaldehyde (and a relatively unbiased weighting for isoamyl-alcohol) also captures (Fig. S12) the opposite effects of 2-butanone as odor C on benzaldehyde and isoamyl-alcohol (Supplementary Fig. 11a, b) as well as the difference in binary preference between these two odors in wild-type vs. AWCON/ON mutants (Supplementary Fig. 11c).
Finally, our mathematical model predicts that sensory gain control extends to the AWCOFF neuron. However, all of the IIA violations described thus far are AWCON-dependent. Therefore, we hypothesized that an IIA violation will also arise when the interference between odors A and C occurs in the AWCOFF neuron, specifically for an odor A with relatively equal weighting of AWCON and AWCOFF representations, e.g., isoamyl-alcohol. According to the model’s predictions, when odor A is isoamyl-alcohol and odor C is 2,3-pentanedione, a weak IIA violation should occur (Fig. 5a). Therefore, we tested this odor combination experimentally, and as predicted by the model, we observed an IIA violation (Fig. 5b). This suggests that the irrational behavior of the worms does not originate from a special trait of the AWCON neuron, but is a general phenomenon, which stems from the interference between two odors sensed by the same neuron.
## Discussion
In this work, we demonstrate for the first time that even C. elegans, with its extremely minimal nervous system, displays IIA violations in decision-making. In most cases, worms behave rationally. However, IIA violations can occur when the different options are represented in an imbalanced way. This imbalanced representation of odors in the AWC neurons interferes with the sensation of the more preferred odor A, thereby decreasing the relative preference for A over B and potentially inducing an IIA violation. We have shown that increasing or decreasing the number of neurons in which the interference occurs, and the relative role they partake in the sensation of odors A and C, can strengthen or diminish the magnitude of IIA violations respectively. The experiments in this paper support the notion that rational behavior in C. elegans is determined by the neuronal architecture involved in the sensation of odors, and not by the identity of specific odors.
The fact that an extremely simple organism with only 302 neurons displays IIA violations suggests that nonoptimal behavior originates in basic neural mechanisms common to many species. Moreover, these neural mechanisms directly relate to the high-level descriptions that are commonly used to explain nonoptimal behavior in general, and IIA violations in particular, such as “cognitive overload” or “biased attention”.
The worm is able to distinguish between different odors efficiently, presumably by using sophisticated signal transduction machinery38. Expanding the variety of recognizable odors may have come at the expense of consistent decision-making, which may lead to nonoptimal decision processes. Indeed, in C. elegans, a single neuron can express many different chemoreceptor genes39,40,41. Evolution may, therefore, have adopted a pattern of increased biochemical complexity to compensate for the lack of neural plasticity and the diminished neuroanatomical complexity in the nematode nervous system38,42. These constraints may render such tradeoffs between olfactory repertoire and decision optimality profitable.
IIA violations could also stem from environmental information available to C. elegans in the wild. Since we lack information about the ecology of the worm, we as observers may be unaware of information such as the probability of disappearance and reappearance of choices. It was shown that the inclusion of such information in behavioral models could favor IIA and transitivity violations in maximization of long-term energy gain15.
To facilitate a comparison to existing work on context-dependent preferences in animal and human literature, we framed our experimental and computational findings in terms of stochastic, discrete choice. In this framework, the choice of an individual worm is ascribed based on their ultimate proximity to one of the odors43. However, this framing should be viewed as a high-level synthesis of the different low-level behaviors that constitute the chemosensory decision process24,44. Rather than the true concentration of each odor option, individual worms encounter spatially varying concentration gradients. In turn, preference is expressed through a combination of behavioral strategies such as orthokinesis, klinotaxis, and klinokinesis that result in gradient ascent toward preferred odors. Thus, at the fine-grained level, the decision-making process is an ongoing, dynamic response to continuously changing stimuli. Given this continuous behavior, the discrete choice IIA framework and normalization model presented here should be viewed as a high-level description of average rather than instantaneous behavior. While not a description of microscopic behavior, this macroscopic framework is likely to be relevant for three reasons: first, the final outcomes realized by choice behavior play a large role in evolutionary fitness and were likely a target of natural selection; second, the lack of IIA violations in most scenarios we tested suggest that macroscopic rationality is a useful description of C. elegans decision-making; and finally, ongoing work suggests that even simple economic choice in monkeys and humans may reflect a dynamic and continuous process45,46,47.
Our experimental results and normalization-based mathematical model are aligned with previous studies on cortical computation which showed that IIA violations can be explained by a divisive normalization framework5,48,49. Thus, our work supports the notion that IIA violations are a result of neural constraints carved by evolution to maximize information under limited time and resources50,51, and directly relate to the high-level concept of “bounded rationality”52. We propose that understanding the building blocks of choice in an animal with a compact, deciphered, rigid, and stereotypic connectome can shed light on the fundamental biological constraints and principles that generate (non)rational behavior in simple as well as in complex organisms.
These findings and others establish a foundation to use C. elegans to investigate underlying neuronal mechanisms of decision making23,44,53,54,55.
## Methods
### Strains and husbandry
The strains used in this work: Bristol N2 wild-type, cat-2 (n4547), and nsy-1(ky542). All strains were maintained at 20 °C on NGM plates supplemented with the antifungal agent Nystatin and fed with E. coli OP5056.
### Obtaining synchronized worms (“Egg-prep”)
A synchronized population of worms was obtained by employing a standard “egg-prep” procedure56.
NGM plates with gravid adults were washed into a 1.5 ml tube with M9 buffer. An egg-prep solution was prepared by mixing 7.5 ml of distilled water, 2 ml of household bleach (5% sodium hypochlorite solution) and 0.5 ml of 10 M NaOH solution. One millilitre of egg-prep solution was added to the tube containing the worms, and were shaken every minute for a total duration of 6 min. The tube was then centrifuged in 8000 RPM for 60 s, aspirated to 0.1 ml, and filled to the top with M9 buffer. This was repeated a total of three times. A Pasteur pipette was used to transfer the eggs to clean NGM plates seeded with E. coli OP50 bacteria.
### Chemotaxis assays
Chemotaxis assays were based on classical chemotaxis assays24,32. Unless stated otherwise, assay plates were square 12 × 12 cm dishes containing 30 ml of 1.6% BBL agar (Benton-Dickinson), 5 mM potassium phosphate (pH 6.0), 1 mM CaCI2 and 1 mM MgSO4. Assays in round plates were performed in 9 cm diameter dishes containing 12 ml of the mixture described above. Three marks were made on the back of the plates equidistant from the center of the plate (3 cm) and from each other (5.2 cm). The diluted attractants (1 µl) was placed on the agar over one marks. In the control plates (binary choice), 1 µl of 100% ethanol was placed over the third mark (all attractants were diluted in ethanol). The tested animals were placed at the center of the plate, equidistant from the three marks. Attractants were obtained from Sigma-Aldrich. Pure pyrazine is a solid, so pyrazine dilutions are weight:volume rather than volume:volume as for other attractants.
Well-fed adult animals were washed three times with wash buffer (0.5% Gelatin, 5 mM potassium phosphate (pH 6.0), 1 mM CaCI2 and 1 mM MgSO4), then placed near the center of a plate equidistant from the attractants (and the control spot when present). Approximately, 1 h after the assay began, the numbers of animals at the three areas (2 cm radius of each attractant) were determined, as well as the total number of animals in the assay, the number of animals that were not at any attractant area, and the number of animals that stayed in the starting point (did not cross a 1 cm diameter circle around the center of the plate). A specific C.I. was calculated as
$${\mathbf{A}}\,{\mathrm{Chemotaxis}}\,{\mathrm{Index}} = \frac{{{\mathrm{Number}}\,{\mathrm{of}}\,{\mathrm{animals}}\,{\mathrm{at}}\,{\mathrm{attractant}}\,{\mathbf{A}}}}{{{\mathrm{Number}}\,{\mathrm{of}}\,{\mathrm{animals}}\,{\mathrm{at}}\,{\mathrm{attractants}}\,{\mathbf{A}}\,{\mathrm{and}}\,{\mathbf{B}}}}$$
The C.I. can vary from 0 to 1. The animals were anaesthetized when they reached the attractant. One microlitre of sodium azide 1 M was placed at each one of the three spots, 15 min in advanced. Sodium azide anaesthetized animals within about a 1 cm radius of the attractant. For discrimination assays, acetone was added to a final concentration of 1.2 µl per 10-ml plate and mixed with the liquid agar once it had cooled to 55 °C, and the odor spots were placed 3.5 cm from the center. For assays where odor C is embedded in the agar, 2-butanone was added to a final concentration of 140 µl per 14-ml plate (dilution factor of 10−2) and mixed with the liquid agar once it had cooled to 50 °C and the odor spots were placed 3.5 cm from the center.
### “Bug” or “Feature” assays
We measured the relative preference between 2 µl of benzaldehyde (10−2) (A) and 2 µl of 2-butanone (10−2) (B), and compared it to the relative preference between 2 µl of benzaldehyde (10−2) (A), and a mixture of 1 µl of 2-butanone (1/50), and 1 µl of benzaldehyde (1/50) (A’B’). The 2-butanone spot (B) and the 2-butanone + benzaldehyde (A’B’) spot, contain the same amount of 2-butanone molecules, as well as an equal volume of ethanol. The “A’B’“ spot contains, in addition to 2-butanone, the same amount of benzaldehyde molecules as presented by “A”. Each assay included 3 “A vs. B” plates, coupled to 3 “A vs. A’B’“ plates. Each data point represents the mean of six essays performed on two different days.
### Statistical analysis
Data are presented as mean ± standard error of mean (SEM). Statistical significance of differences in chemotaxis index between control and test plates in a certain concentration were analyzed by a two-tailed Wilcoxon signed-ranks test. We corrected for multiple comparisons using the Benjamini–Hochberg false-discovery rate, with a false-discovery rate of 0.05. The q values reported in this study are adjusted to multiple comparisons (q < 0.05 was regarded as significant; *q < 0.05, **q < 0.01, ***q < 0.001, and ****q < 0.0001).
### Normalization model of sensory gain control
To examine whether both IIA and non-IIA choice behavior can be explained by a circuit-specific model of sensory gain control in chemosensation, we implemented a simple divisive normalization-based computational model of chemosensory value coding. Gain control is a widespread representational principle in early sensory processing in which the overall level of coding activity is regulated by the specific context present at the time of encoding. For example, Drosophila antennal lobe neural activity representing a specific odor will depend on whether other odors are present, and primate primary visual cortical responses to a center stimulus will be suppressed by stimuli in the sensory surround. Many of these gain control interactions can be explained a normalization computation, in which the feedforward-driven response of a neuron is divided by a term that represents a larger pool of neurons. This normalization pool (acting via the equation denominator) provides a mechanism for contextual modulation of the stimulus-specific response (in the numerator). For example, the response of Drosophila antennal lobe neurons is increased by a test odorant but suppressed by a mask odorant, a pattern described by normalization-based gain control.
To examine the predictions of a gain control model of chemosensation, we constructed a model of pathway-specific sensory gain control in C. elegans chemosensation and explored its qualitative predictions in trinary odorant choice behavior. In this model, neural activity Ri of a chemosensory neuron representing the decision value of an odorant stimulus i depends on its concentration (or intensity) Ii via a divisive normalization representation
$$R_i = \frac{{I_i}}{{\sigma + I_i}}{,}$$
where the semisaturation term σ controls how the function approaches saturation. Context-dependence in this model is instantiated as cross-odorant gain control when a given chemosensory neuron responds to more than one odor and both odors are present in the choice set. For example, in the most basic version of this model, the responses to two odors A and B will be described by the equations
$$R_A = \frac{{I_A}}{{\sigma + I_A + I_B}}$$
$$R_B = \frac{{I_B}}{{\sigma + I_A + I_B}}$$
where IA and IB are properties of the odor stimuli (i.e., concentrations) and the responses RA and RB denote neural activity representing the value of the odors. Note that this is an algorithmic model intended to model information processing rather than a biophysical implementation; however, because C. elegans neurons are generally thought to not exhibit action potentials, R can be viewed as a graded voltage signal. Furthermore, since this activity integrates across chemosensory neurons, it represents information at a downstream stage: synaptic input to interneurons, interneuron activity, or a more global measure of preference (e.g., turn/run balance in the klinokinesis-governing circuit).
In this model, odorants represented by multiple chemosensory neurons (e.g., benzaldehyde activating both AWCON and AWCOFF) receive a weighted averaging across the active neurons. For example, the response to benzaldehyde (denoted A here) in the presence of 2-butanone (AWCON only, here denoted B) is described as
$$R\left( A \right) = w_{\mathrm{ON}}\frac{{I_A}}{{\sigma + I_A + I_B}} + w_{\mathrm{OFF}}\frac{{I_A}}{{\sigma + I_A}}{,}$$
where the final response is a weighted sum of the responses of AWCON (left term) and AWCOFF (right term). Note that for simplicity and parsimony, we assume that neurons that are encoding a single odor at a given time (AWCOFF in the example above) also have an analogous form of gain control over the single represented odor. Decisions are implemented by a simple noisy decision rule, assuming a fixed Gaussian noise term (equal across all options in a choice set).
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
The datasets generated and analyzed in this study are available upon request to the corresponding author.
## Code availability
The dataset generated and analyzed in this study are available upon request to the corresponding author.
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## Acknowledgements
We thank all the Rechavi lab and Levy lab members for helpful discussions. D.C., G.T., M.V., and A.M. wish to thank the Sagol School of Neuroscience. Bristol N2 wild-type and cat-2 (n4547) strains were provided by the CGC, which is funded by the NIH Office of Research Infrastructure Programs (P40 OD010440). We thank Cornelia Bargmann for sharing with us the nsy-1(ky542) mutants. We thank Yoav Benjamini, supervisor of Yoav Zeevi, for statistical advice. O.R. is thankful to the Adelis foundation grant #0604916191 and ERC grant #335624for funding, and gratefully acknowledges the support of the Allen Discovery Center of the Paul G. Allen Frontiers Group. D.J.L. is thankful to the ISF grant #1104/13 and to the Henry Crown Institute of Business Research for funding. K.L. is thankful to NIH grant R01MH104251 for funding.
## Author information
Authors
### Contributions
Conception and design: D.C., G.T., O.R. and D.L. Analysis and interpretation of data: Y.Z. Acquisition of data: D.C., G.T., M.V., L.E. and A.M. Modeling: K.L. Writing, reviewing and/or revision of the paper: D.C. and G.T.
### Corresponding authors
Correspondence to Dino J. Levy or Oded Rechavi.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Cohen, D., Teichman, G., Volovich, M. et al. Bounded rationality in C. elegans is explained by circuit-specific normalization in chemosensory pathways. Nat Commun 10, 3692 (2019). https://doi.org/10.1038/s41467-019-11715-7
• Accepted:
• Published:
• ### Divisive normalization does influence decisions with multiple alternatives
• Ryan Webb
• , Paul W. Glimcher
• & Kenway Louie
Nature Human Behaviour (2020)
• ### Addendum: Irrational behavior in C. elegans arises from asymmetric modulatory effects within single sensory neurons
• Shachar Iwanir
• , Rotem Ruach
• , Eyal Itskovits
• , Christian O. Pritz
• , Eduard Bokman
• & Alon Zaslaver
Nature Communications (2019) | 2021-08-01 13:18:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6162334680557251, "perplexity": 5740.340078605562}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154214.36/warc/CC-MAIN-20210801123745-20210801153745-00390.warc.gz"} |
http://toreaurstad.blogspot.no/2012/11/ | ## Sunday, 18 November 2012
### Tracking Prism Events
It is possible to track the Prism events that are sent and received through the Prism EventAggregator,
but this is not out of the box of the Prism Framework. Usually, the developer must insert breakpoints
where the CompositePresentationEvent is published and subscribed
This is satisfactory for small scenarios, but as your solution and project(s) grow, it can become an increasingly difficult task. It is in many cases desired to see which events are published and where they are subscribed. The Prism 4.x Framework still do not support this, but luckily there is a way to achieve this.
First off, it is required to change your Prism Events from using the class CompositePresentationEvent, to instead use a derived class presented next, CustomCompositePresentationEvent. I have chosen to display the Prism Event traffic in the Output of Visual Studio. This is done using Debug.WriteLine. It could be perhaps an alternative to log this event traffic or display it in a separate Window if you are writing a WPF application.
The custom class CustomCompositePresentationEvent looks like this:
using System;
using System.ComponentModel;
using System.Diagnostics;
using System.Text;
using Microsoft.Practices.Prism.Events;
namespace EventAggregation.Infrastructure
{
/// <summary>
/// Custom CompositePresentationEvent class, derived from this base class to allow the Prism Events to be displayed in the Debug Output Window.
/// </summary>
public class CustomCompositePresentationEvent<TEvent> :
CompositePresentationEvent<TEvent> where TEvent : class
{
/// <summary>
/// Overload of the publish method of CompositePresentationEvent
/// </summary>
{
var declaringType = string.Empty;
var callingMethod = new StackTrace().GetFrame(1).GetMethod();
if (callingMethod != null)
declaringType = callingMethod.DeclaringType.FullName;
System.Diagnostics.Debug.WriteLine(string.Format(@"Publishing event {0}
at time: {1}. Source type: {2}. Event Payload: {3}",
typeof(TEvent).Name,DateTime.Now, declaringType,
}
/// <summary>Overload of the Subscribe method of CompositePresentationEvent</summary>
/// <remarks>
/// For this overload to be executed, set keepSubscriberReferenceAlive to
/// true in the subscribe setup method on the target.
/// </remarks>
public override SubscriptionToken Subscribe(Action<TEvent> action,
Predicate<TEvent> filter)
{
var loggingAction = new Action<TEvent>(e =>
{
System.Diagnostics.Debug.WriteLine(string.Format(
"Subscribing event {0} at time: {1}. Target type: {2}",
typeof(TEvent).Name,
DateTime.Now, action.Target.GetType().Name));
action.Invoke(e);
});
keepSubscriberReferenceAlive, filter);
return subscriptionToken;
}
{
return string.Empty;
else
{
StringBuilder sb = new StringBuilder();
foreach (PropertyDescriptor property in
{
sb.Append(string.Format("{0}={1} ", property.Name,
}
return sb.ToString();
}
}
}
}
As you can see from the source code above, the methods Subscribe and Publish are overriden. These are virtual methods of the CompositePresentationEvent base class. I have tested using this derived class from CompositePresentationEvent in the EventAggregator sample, which can be downloaded in the Prism 4.1 library with source code at the following url: Download Prism 4.1 You will find the EventAggregator sample in the \QuickStarts\EventAggregation folder after you have extracted the Prism 4.1 library with source code. I have done some minor changes to this demo solution. To the project EventAggregation.Infrastructure.Desktop, I added the class I made which was just presented, CustomCompositePresentationEvent. Next up, our Prism Events must use this CustomCompositePresentation event class! This is easy to change, as shown in the Prism Event FundAddedEvent, which I changed to the following, still in the project EventAggregation.Infrastructure.Desktop:
namespace EventAggregation.Infrastructure
{
{
}
}
As you can see, the FundAddedEvent Prism Event, is now inheriting from the class CustomCompositePresentationEvent. There are no changes needed to do for the way the FundAddedEvent is published, but the Subscribe method must be changed. The boolean flag keepSubscriberReferenceAlive must be set to true. The reason is that the Subscribe override method in the CustomCompositePresentationEvent requires this, as you can see in the source code. We modify the action sent back again to the base class to continue with its Subscribe implementation. I could not make the override method to work, i.e. the method override was not called, without setting the boolean flag keepSubscribeReferenceAlive to true. The change I had to make for the subscribe method of the FundAddedEvent is in the project ModuleB.Desktop, in the class ActivityPresenter. The line 67 was changed to this:
The changes required then, is to change the Prism Events to inherit from the class CompositePresentationEvent to instead inherit from (use) the class CustomCompositePresentationEvent. This custom class is basically the CompositePresentationEvent class, but we have overridden the methods Subscribe and Publish. In addition, we must set the keepSubscriberReferenceAlive boolean flag to true in our Subscribe methods. This will potentially mean that you must change to the Subscribe overload method which has this flag. Let's first see the GUI of the EventAggregation sample:
The output from the Visual Studio is the most important feature of this CustomCompositePresentationEvent class. After all, we want to be able to see which classes publish and subscribe events, and the content of the payload of these events, correct? In this sample, the output looks like the following example:
Publishing event FundOrder at time: 18.11.2012 15:31:30. Source type: ModuleA.AddFundPresenter. Event Payload: CustomerId=Customer1 TickerSymbol=FundB
Subscribing event FundOrder at time: 18.11.2012 15:31:30. Target type: ActivityPresenter
As we can see from the output above, the event FundOrder was published by the AddFundPresenter class. The PayLoad of this Prism event was CustomerId=Customer1 and the TickerSymbol=FundB. The event FundOrder was subscribed, interestingly at the same millisecond (i.e. the publish-subscribe execution obviously performs almost instantly in the Prism framework). If you think the concept of being able to see Prism traffic is both interesting and will help you as a developer, consider taking into to use this CompositePresentationEvent class. You will have to include the changes which I listed above also, but these changes should be relatively easy to overcome in a Prism application solution. The tracking or tracing of Prism Event Traffic will help you understand the application better and catch possible errors by this more convenient debugging scenario. For security reasons, you should consider if putting a Debug.WriteLine call to output the Event Payload is safe or not. Last, I want to mention the fact that Prism Publish-Subscribe scenarios take in my test 0 ms and occur to be happening almost instantly. Of course they take some CPU cycles to execute, but if you in any case wonder how fast is really a Publish-Subscribe, it will usually be the answer - REALLY FAST. However, it is possible that if multiple classes subscribe to a given Prism Event, it might take some milliseconds to execute. I have not tested this yet. In a MVVM Scenario, it is usually the ViewModels that publish and subscribe Prism Events, usually via an ICommand implementation (DelegateCommand) that is data bound in WPF or Silverlight to a Button or similar GUI object.
## Monday, 5 November 2012
### User-Defined conversion sample in C#
This article will present a sample of how to implement a User Defined Conversions in C#. The article will use a typical conversion example, where temperature measured in Celsius scale (Centigrades) are converted into the Fahrenheit scale (Fahrenheits). A conversion formula is available at the following urls (Wikipedia):
User Defined Conversions are implemented in C-sharp in a similar manner as operator overloading, actually it is an overload of conversion. These conversions can be either:
• Implicit - An object is converted into another type without a specific cast, i.e implicit cast.
• Explicit - An object is converted into another type with a specific cast, i.e.
explicit cast.
A class can have multiple implicit and/or explicit user defined conversions. The user defined conversions must all be methods that are marked as public and static. In addition, they must be marked as implicit or explicit to specify which cast is allowed. If a conversion is marked as explicit, converting an object into the specified type without an explicit cast will in other words not call the user defined conversion.
To specify a User-Defined Conversion in C# which is implicit, the following signature must be used for a implicit User-Defined Conversion:
public static implicit operator TReturnType (TInputType objectToConvert) {
.. //Implementation elided
A User-Defined Conversion in C# which is explicit, uses the following signature:
public static explicit operator TReturnType (TInputType objectToConvert) {
.. //Implementation elided
The C-sharp programmer who has experience in operator overloading will see the similarities to operator overloading. Operator overloading follows much the same signature as shown here, but without the keywords explicit or implicit. In addition, operator overloading will specify the same keyword operator, but also specify the operator to overload. User-Defined Conversion do not specify an operator to overload, but implicitly or explicitly we are overloading casting (type conversion).
Demonstration code
An example of how to implement User-Defined Conversion follows. In this example, a base class called Temperature is used and there are two derived classes, Celsius and Fahrenheit. Both these derived classes defines an explicit user defined conversion which will take an instance of their own type and return an instance of the converted type. This means that the Fahrenheit class will contain an explicit User-Defined Conversion which will receive a Fahrenheit instance and return a Celsius instance. In return, the Celsius class will contain an explicit User-Defined Conversion which will receive a Celsius instance and convert this into a Fahrenheit instance. In the code, the mathematical conversion between the two temperature scales is performed inside the explicit user defined conversion. This will let the programmer control the way a cast is performed and allow the client or consumer of the class to convert the type into another type through a cast and logically fill the expectation that the converted object is correctly converted or casted.
As with operator overloading, just because C-sharp allows User Defined Conversions, does not mean that the programmer should use this functionality in the C#-language wherever this strategy might fit. User-Defined Conversions are a programming capability many C#-programmers are not aware of, and the coder which adopt this strategy might confuse his fellow programmers or quickly understand that the concepts must be explained to the other coders. But then again, just as operator overloading allows very logical and powerful expressibility in code, user defined conversion will let the programmer control the way explicit and implicit type conversions, that is - casts - are performed. If you expect the class you are writing to be casted into different types or will provide consumers or clients of your class with the benefits of simple expressibility and determined behavior, you should also add explicit user defined conversions. Implicit User-Defined Conversions are sometimes hard to see which User-Defined Conversion will be called. Therefore, if a choice is available between either implicit or explicit User-Defined Conversion, I would suggest to go for explicit. This is the most specific and clear-intentioned casting overload of the two.
The code for the base class Temperature is shown below:
/// Base class for temperatures
///
public class Temperature
{
protected float _value;
public Temperature(float temperature)
{
_value = temperature;
}
public float Value
{
get { return _value; }
}
}
The derived class Celsius is shown next. Note that this class defines one explicit User-Defined conversion:
public class Celsius : Temperature
{
public Celsius(float temperature) : base(temperature)
{
}
public static explicit operator Fahrenheit(Celsius temperature)
{
float convertedTemperature = (temperature.Value * (9.0f / 5.0f)) + 32;
return new Fahrenheit(convertedTemperature);
}
}
The derived class Fahrenheit is shown below. This class also defines one explicit User-Defined conversion.
public class Fahrenheit : Temperature
{
public Fahrenheit(float temperature) : base(temperature)
{
}
public static explicit operator Celsius(Fahrenheit temperature)
{
float convertedTemperature = (5.0f / 9.0f)*(temperature.Value - 32);
return new Celsius(convertedTemperature);
}
}
The client or consumer class is shown next:
class Program
{
static void Main(string[] args)
{
Console.WriteLine("Centigrades to Fahrenheits conversion in range [-40,40] Celsius: ");
Celsius celsius = null;
Fahrenheit fahrenheit = null;
for (int i = -40; i <= 40; i++)
{
celsius = new Celsius(i);
fahrenheit = (Fahrenheit)celsius;
Console.WriteLine("{0} Centigrades is measured into {1:f1} Fahrenheits.", i, fahrenheit.Value);
}
Console.WriteLine("Press any key to continue ...");
}
}
The output in the console window when running the Main method in the consumer class
above is shown next:
Centigrades to Fahrenheits conversion in range [-40,40] Celsius:
-40 Centigrades is measured into -40,0 Fahrenheits.
-39 Centigrades is measured into -38,2 Fahrenheits.
-38 Centigrades is measured into -36,4 Fahrenheits.
-37 Centigrades is measured into -34,6 Fahrenheits.
-36 Centigrades is measured into -32,8 Fahrenheits.
-35 Centigrades is measured into -31,0 Fahrenheits.
-34 Centigrades is measured into -29,2 Fahrenheits.
-33 Centigrades is measured into -27,4 Fahrenheits.
-32 Centigrades is measured into -25,6 Fahrenheits.
-31 Centigrades is measured into -23,8 Fahrenheits.
-30 Centigrades is measured into -22,0 Fahrenheits.
-29 Centigrades is measured into -20,2 Fahrenheits.
-28 Centigrades is measured into -18,4 Fahrenheits.
-27 Centigrades is measured into -16,6 Fahrenheits.
-26 Centigrades is measured into -14,8 Fahrenheits.
-25 Centigrades is measured into -13,0 Fahrenheits.
-24 Centigrades is measured into -11,2 Fahrenheits.
-23 Centigrades is measured into -9,4 Fahrenheits.
-22 Centigrades is measured into -7,6 Fahrenheits.
-21 Centigrades is measured into -5,8 Fahrenheits.
-20 Centigrades is measured into -4,0 Fahrenheits.
-19 Centigrades is measured into -2,2 Fahrenheits.
-18 Centigrades is measured into -0,4 Fahrenheits.
-17 Centigrades is measured into 1,4 Fahrenheits.
-16 Centigrades is measured into 3,2 Fahrenheits.
-15 Centigrades is measured into 5,0 Fahrenheits.
-14 Centigrades is measured into 6,8 Fahrenheits.
-13 Centigrades is measured into 8,6 Fahrenheits.
-12 Centigrades is measured into 10,4 Fahrenheits.
-11 Centigrades is measured into 12,2 Fahrenheits.
-10 Centigrades is measured into 14,0 Fahrenheits.
-9 Centigrades is measured into 15,8 Fahrenheits.
-8 Centigrades is measured into 17,6 Fahrenheits.
-7 Centigrades is measured into 19,4 Fahrenheits.
-6 Centigrades is measured into 21,2 Fahrenheits.
-5 Centigrades is measured into 23,0 Fahrenheits.
-4 Centigrades is measured into 24,8 Fahrenheits.
-3 Centigrades is measured into 26,6 Fahrenheits.
-2 Centigrades is measured into 28,4 Fahrenheits.
-1 Centigrades is measured into 30,2 Fahrenheits.
0 Centigrades is measured into 32,0 Fahrenheits.
1 Centigrades is measured into 33,8 Fahrenheits.
2 Centigrades is measured into 35,6 Fahrenheits.
3 Centigrades is measured into 37,4 Fahrenheits.
4 Centigrades is measured into 39,2 Fahrenheits.
5 Centigrades is measured into 41,0 Fahrenheits.
6 Centigrades is measured into 42,8 Fahrenheits.
7 Centigrades is measured into 44,6 Fahrenheits.
8 Centigrades is measured into 46,4 Fahrenheits.
9 Centigrades is measured into 48,2 Fahrenheits.
10 Centigrades is measured into 50,0 Fahrenheits.
11 Centigrades is measured into 51,8 Fahrenheits.
12 Centigrades is measured into 53,6 Fahrenheits.
13 Centigrades is measured into 55,4 Fahrenheits.
14 Centigrades is measured into 57,2 Fahrenheits.
15 Centigrades is measured into 59,0 Fahrenheits.
16 Centigrades is measured into 60,8 Fahrenheits.
17 Centigrades is measured into 62,6 Fahrenheits.
18 Centigrades is measured into 64,4 Fahrenheits.
19 Centigrades is measured into 66,2 Fahrenheits.
20 Centigrades is measured into 68,0 Fahrenheits.
21 Centigrades is measured into 69,8 Fahrenheits.
22 Centigrades is measured into 71,6 Fahrenheits.
23 Centigrades is measured into 73,4 Fahrenheits.
24 Centigrades is measured into 75,2 Fahrenheits.
25 Centigrades is measured into 77,0 Fahrenheits.
26 Centigrades is measured into 78,8 Fahrenheits.
27 Centigrades is measured into 80,6 Fahrenheits.
28 Centigrades is measured into 82,4 Fahrenheits.
29 Centigrades is measured into 84,2 Fahrenheits.
30 Centigrades is measured into 86,0 Fahrenheits.
31 Centigrades is measured into 87,8 Fahrenheits.
32 Centigrades is measured into 89,6 Fahrenheits.
33 Centigrades is measured into 91,4 Fahrenheits.
34 Centigrades is measured into 93,2 Fahrenheits.
35 Centigrades is measured into 95,0 Fahrenheits.
36 Centigrades is measured into 96,8 Fahrenheits.
37 Centigrades is measured into 98,6 Fahrenheits.
38 Centigrades is measured into 100,4 Fahrenheits.
39 Centigrades is measured into 102,2 Fahrenheits.
40 Centigrades is measured into 104,0 Fahrenheits.
Press any key to continue ...
Most Europeans are familiar with the Centigrade system, but US-English citizens use Fahrenheit scale instead for describing the temperature. The freezing point of water, 0 degrees Celsius, is 32 Fahrenheit and about -17,78 Centigrades is 0 Fahrenheit. The boiling point of water, 100 degrees Centigrades, is 212 degrees Fahrenheit. As we can see, the temperature scales grow differently and are shifted somewhat. The User-Defined conversion allows to convert the two inheriting classes of the temperature base class through explicit casts. Some, but not all programmers, believe this is a natural way of performing a conversion. Often, one can see code which actually does the same kind of functionality but does not use User-Defined conversion, i.e. a public static method which receives an instance of type TInput and returns a converted object of type TResult, but without the keywords implicit or explicit and operator. This is of course allowed, but note that this will demand that the user specifically calls this method to perform the conversion routine to get the converted instance. Instead, specifying an implicit and/or expliciting User-Defined Conversion will in many cases be less ambigious and more precise. In the end, judge yourself if User-Defined Conversions are something to use in C#. But be aware that they exist and what they can be used for. More explanation on this topic available on the MSDN page:
User Defined Conversions
If no explicit cast is performed using the cast operator (parentheses), the implicit User Defined Conversion will be called, if both a matching explicit and implicit User Defined Conversion is defined. Note that if the assignment operator = is used without a cast, the implicit conversion is used and therefore the implicit User Defined Conversion will be utilized. | 2017-08-21 17:45:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3377761244773865, "perplexity": 6359.4448843354185}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109470.15/warc/CC-MAIN-20170821172333-20170821192333-00158.warc.gz"} |
https://mycareerwise.com/content/q-ugc-net-2016-july/content/exam/nta-net/computer-science | ## UGC NET 2016 (July)
Question: The symmetric differences of two sets S1 and S2 are defined as:
S1⊕ S2 ={x ∣ x ∈ S1 or x ∈ S2, but x is not in both S1 and S2}
The nor of two languages is defined as
nor (L1, L2)={w ∣ w ∉ L1 and w ∉ L2}
Which of the following is correct?
A. The family of regular languages is closed under symmetric difference but not closed under nor.
B. The family of regular languages is closed under nor but not closed under symmetric difference.
C. The family of regular languages are closed under both symmetric difference and nor.
D. The family of regular languages are not closed under both symmetric difference and nor.
As we know Regular expression is closed under a symmetric difference () and nor (For details see Closer property of Regular Language).
Option C is correct.
Question: Consider the following two languages:
and ${\mathbf{L}}_{\mathbf{2}}$ is any subset of 0*.
Which of the following is correct?
(A) L1 is regular and L2* is not regular
(B) L1 is not regular and L2* is regular
(C) Both L1 and L2* are regular languages
(D) Both L1 and L2* are not regular languages
GCD of two infinite numbers is always Context - Sensitive Language (CSL) because multiplication and division of infinite number is not possible by regular language.
As we know that finite automata has finite memory that’s why it can’t perform infinite multiplication and division.
So, is not regular.
A subset of a single symbol of kleene star operation 0* is always regular language.
So, L2 is any subset of 0* is regular language.
Option (B) is correct. | 2023-03-22 20:16:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7720476984977722, "perplexity": 1192.7365372045947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944452.74/warc/CC-MAIN-20230322180852-20230322210852-00295.warc.gz"} |
http://www.lmfdb.org/EllipticCurve/Q/1600/n/ | Properties
Label 1600.n Number of curves $4$ Conductor $1600$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph
Related objects
Show commands for: SageMath
sage: E = EllipticCurve("n1")
sage: E.isogeny_class()
Elliptic curves in class 1600.n
sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
1600.n1 1600o3 [0, 0, 0, -1100, -14000] [2] 512 -16
1600.n2 1600o4 [0, 0, 0, -1100, 14000] [2] 512 -16
1600.n3 1600o2 [0, 0, 0, -100, 0] [2, 2] 256 -4
1600.n4 1600o1 [0, 0, 0, 25, 0] [2] 128 $$\Gamma_0(N)$$-optimal -4
Rank
sage: E.rank()
The elliptic curves in class 1600.n have rank $$0$$.
Complex multiplication
Each elliptic curve in class 1600.n has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.
Modular form1600.2.a.n
sage: E.q_eigenform(10)
$$q - 3q^{9} + 6q^{13} - 2q^{17} + O(q^{20})$$
Isogeny matrix
sage: E.isogeny_class().matrix()
The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.
$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels. | 2021-03-04 16:14:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9340009689331055, "perplexity": 5254.130277833827}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178369420.71/warc/CC-MAIN-20210304143817-20210304173817-00461.warc.gz"} |
http://www.luxrender.net/forum/viewtopic.php?f=8&t=13276&sid=cfeea204d7127af438de71370ed71dea | ## Intensity of Sun and Sky
Discussion related to the implementation of new features & algorithms to the Core Engine.
Moderators: Dade, jromang, tomb, zcott, coordinators
### Intensity of Sun and Sky
I've had a hunch that the sun / sky lights seemed much brighter than they should be relative to artificial light sources, but I've been unable to test my hunch until the new irradiance AOV. I was encouraged that @abhiram was experiencing a similar confusion here: viewtopic.php?f=16&t=12519&start=10#p119075
So, I have created a simple test to compare Irradiance AOV output with expected values using real world times and location information.
- first, sun angles were calculated from the NREL SPA calculator, and converted into xyz vectors : https://www.nrel.gov/midc/solpos/spa.html
- I then rendered a time lapse at 10 min intervals of a simple scene. One horizontal plane with an environment camera looking straight down.
- I averaged all of the pixels output from the irradiance AOV at each frame and saved to a .csv file.
- I found the corresponding expected values from two tools:
- Extraterrestrial Global Irradiance (pre-atmospheric distortion) from the SOLPOS tool : https://www.nrel.gov/midc/solpos/solpos.html
- Total Integral GHI from the Bird Model (using the excel download) : http://rredc.nrel.gov/solar/models/spectral/
Here are the results:
I've attached the scene as well as the excel file with the results. Sorry about the odd time stamps, this data was taken in part from stuff I already had.
As you can see, there is a significant difference in the actual and expected irradiance coming out of the AOV. The sun / sky environment seems to be 1-2 orders of magnitude brighter.
I'm not sure where to go from here, or how I can be of help. I will gladly provide more data and help speculate on what may be going on...
Maybe it as simple as a bad unit conversion somewhere?
Attachments
test.zip
(13.59 KiB) Downloaded 19 times
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crosley09
Posts: 199
Joined: Fri Oct 07, 2011 1:00 pm
Location: Indiana, USA
### Re: Intensity of Sun and Sky
crosley09 wrote:I
As you can see, there is a significant difference in the actual and expected irradiance coming out of the AOV. The sun / sky environment seems to be 1-2 orders of magnitude brighter.
Nice work! I would say that now the first question is if this is just the sun and sky, or if other light sources are also off. If other light sources are fine, that would then certainly mean that the sun/sky system needs to be checked.
Abel
Posts: 1847
Joined: Sat Oct 20, 2007 8:13 am
Location: Stuttgart, Germany
### Re: Intensity of Sun and Sky
Hi,
The AOV is most probably the illuminance and not the irradiance because when Lux converts a spectrum to the XYZ colorspace, the values are multiplied by 683 to account for the fact that Y curve has a value of 1 at 555nm.
Another issue is that the current sun model is not really well calibrated. A sun model is proposed alongside the model used for sky2, but it has never been implemented, so the results are most probably off if you take the sun into account.
There might also be other elements explaining the offset.
Jeanphi
jeanphi
Posts: 7943
Joined: Mon Jan 14, 2008 7:21 am
### Re: Intensity of Sun and Sky
jeanphi wrote:The AOV is most probably the illuminance and not the irradiance because when Lux converts a spectrum to the XYZ colorspace, the values are multiplied by 683 to account for the fact that Y curve has a value of 1 at 555nm.
Are you mixing LuxCore and classic Lux ? LuxCore doesn't use XYZ color space at all and classic Lux hasn't AOV/etc.
Anyway Irradiance AOV is ... well ... irradiance: the radiance incoming the first eye path hit is multiplied by Abs(Dot(incoming direction , surface normal)) and divided by PI.
jeanphi wrote:Another issue is that the current sun model is not really well calibrated. A sun model is proposed alongside the model used for sky2, but it has never been implemented, so the results are most probably off if you take the sun into account.
There might also be other elements explaining the offset.
Yes, there are so many factors could be wrong (including the one in reference results). I would start with a lot simpler case like a point light where the results can be verified by hand. Or Crosley09, do you have a specific interest only in outdoor renderings ?
Dade
Posts: 8393
Joined: Sat Apr 19, 2008 6:04 pm
Location: Italy
### Re: Intensity of Sun and Sky
Dade wrote:Anyway Irradiance AOV is ... well ... irradiance: the radiance incoming the first eye path hit is multiplied by Abs(Dot(incoming direction , surface normal)) and divided by PI.
I'm assuming this also accounts for the color and reflective properties of the material?
Dade wrote:Yes, there are so many factors could be wrong (including the one in reference results). I would start with a lot simpler case like a point light where the results can be verified by hand. Or Crosley09, do you have a specific interest only in outdoor renderings ?
I do not currently have access to a radiometer or photometer, but I will soon. When I get it I can provide some simple tests with it as well. I chose the sun / sky model approach because I had access to the data, which I think is likely pretty accurate.
I can do a simple test with a lamp when I get the photometer to see if the main issue is with the sun/sky or something else.
i7 5930k, GTX 980 ti, 32 GB ddr4, 512 GB PCIe SSD...Windows 10
crosley09
Posts: 199
Joined: Fri Oct 07, 2011 1:00 pm
Location: Indiana, USA
### Re: Intensity of Sun and Sky
crosley09 wrote:
Dade wrote:Anyway Irradiance AOV is ... well ... irradiance: the radiance incoming the first eye path hit is multiplied by Abs(Dot(incoming direction , surface normal)) and divided by PI.
I'm assuming this also accounts for the color and reflective properties of the material?
No and it is intended because in light design you usually want to study the amount of light landing on surfaces (for instance the tables in a school room, the painting in a museum, etc.). Irradiance AOV is a view independent information. This is how Radiance (http://radsite.lbl.gov/radiance/HOME.html) also works (a tool well known for light design even after so many years). You can find some example of this kind of Irradiance information usage in Radiance gallery (http://radsite.lbl.gov/radiance/frameg.html):
If you account for the reflective properties of the material, it becomes a view dependent information and it is just the plain rendering in HDR format (i.e. it is the radiance received by eye).
crosley09 wrote:I can do a simple test with a lamp when I get the photometer to see if the main issue is with the sun/sky or something else.
As written by Jeanphi, one of the problem with sun/sky is that they are 2 light sources, not just one and it makes more complex to debug what is going on
Dade
Posts: 8393
Joined: Sat Apr 19, 2008 6:04 pm
Location: Italy
### Re: Intensity of Sun and Sky
Dade wrote:No and it is intended because in light design you usually want to study the amount of light landing on surfaces (for instance the tables in a school room, the painting in a museum, etc.).
I think we are saying the same thing. So that's good. I was assuming the 'first eye path hit' is the ray hitting the camera which would have been affected by the properties of the material it came from.
Dade wrote:As written by Jeanphi, one of the problem with sun/sky is that they are 2 light sources, not just one and it makes more complex to debug what is going on
I can post results with sun and sky lights separately. Probably today.
i7 5930k, GTX 980 ti, 32 GB ddr4, 512 GB PCIe SSD...Windows 10
crosley09
Posts: 199
Joined: Fri Oct 07, 2011 1:00 pm
Location: Indiana, USA
### Re: Intensity of Sun and Sky
Follow up: Here are sun and sky separately.
While clearly not the only error, one problem seems to be that the intensity of the sky light is offset from zero. At sunrise and sunset, the values should be near or below the values of the sky light.
i7 5930k, GTX 980 ti, 32 GB ddr4, 512 GB PCIe SSD...Windows 10
crosley09
Posts: 199
Joined: Fri Oct 07, 2011 1:00 pm
Location: Indiana, USA
### Re: Intensity of Sun and Sky
... for some reason, the forum will not take excel files. Here is the file zipped...
Attachments
Irradiance Comparison2.zip
(18.03 KiB) Downloaded 15 times
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crosley09
Posts: 199
Joined: Fri Oct 07, 2011 1:00 pm
Location: Indiana, USA
### Re: Intensity of Sun and Sky
Simple spot lamp result:
I created a simple blender scene with a spot lamp pointing down to a 1m x 1m surface.
The lamp is outputting 100 lumens and the diameter of the spot on the surface is ~1m^2.
This means we should expect ~128 lux because lux is lumens/m^2. (area of spot is 0.785 m^2)
Now, I read the .exr file in python and one pixel near the center looks like this:
Code: Select all
[ 64.43734741 64.43734741 64.43734741]
Since RBG channels are all the same, no math required, irradiance is 64.43.
64 is half of 128.... (seems that output is lux/2 unless I have made an error somewhere?)
This seems separate or additional to the issues with sun / sky lamps... ?
Attachments
test.blend
(451.86 KiB) Downloaded 56 times
spot irradiance.exr
(962.07 KiB) Downloaded 19 times
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crosley09
Posts: 199
Joined: Fri Oct 07, 2011 1:00 pm
Location: Indiana, USA
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### Who is online
Users browsing this forum: No registered users and 1 guest | 2018-02-21 22:26:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.365507036447525, "perplexity": 2715.2608939149045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813818.15/warc/CC-MAIN-20180221222354-20180222002354-00623.warc.gz"} |
https://electronics.stackexchange.com/questions/304914/while-loop-instructions-in-pic-disassemby-not-clear | # While loop instructions in pic disassemby not clear
In an efford to save space I try to reduce the generated C code with inline asm(""); Looking at the original disassembly and good working code I do not understand line 0x17E4.
// This loop ends when the enter key is pressed
while EnterOpen(){ // SW3 = RE2
if ArrowUpClosed(){ // RE0
while ArrowUpClosed(){};
asm("call MBD");
if ArrowUpOpen()
LCDGotoPos(ArrowUp[index]);
};
if ArrowDownClosed(){ // SW3 RE1
while ArrowDownClosed(){};
asm("call MBD");
if ArrowDownOpen()
LCDGotoPos(ArrowDown[index]);
};
if ArrowRightClosed(){ // SW4 RC1
while ArrowRightClosed(){};
asm("call MBD");
if ArrowRightOpen()
LCDGotoPos(ArrowRight[index]);
};
if ArrowLeftClosed(){ // SW5 RC0
while ArrowLeftClosed(){};
asm("call MBD");
if ArrowLeftOpen()
LCDGotoPos(ArrowLeft[index]);
};
};
Pause(300);
return;
// a little trick to save space
asm("MBD:");
delay();
// This loop ends when the enter key is pressed
while EnterOpen(){ // RE2
0x178E: BCF STATUS, 0x5
0x178F: BCF STATUS, 0x6
0x1790: BTFSS PORTE, 0x2
// jumping out of the loop as expected
0x1791: GOTO 0x7E5
// returns to the beginning? but during execution goes to 0x1792.
// The way I expect
0x17E4: GOTO 0x78E
// during execution
//if ArrowUpClosed() // RE0
0x1792: BTFSC PORTE, 0x0
//other code
} // end of while EnterOpen
// Pause(300);
0x17E5: MOVLW 0x8
• This is disassembly. What is the original assembly code (and do you have it)? – Anonymous May 12 '17 at 8:25
• Erm is it just the ordering of the lines which is wrong? I mean 0x1792 should be in front of 0x17e4. And your question is a bit hard to understand as there are comments which are part of the question and comments for old code (I think?). – Arsenal May 12 '17 at 8:25
• I have added the complete C code of the routine. The disassembly code is comming from the C code – Decapod May 12 '17 at 8:39
• According to @Arsenal's finding I would question quality of disassember output. In its output it has sequence of 0x1791, then 0x17E4, and then 0x1792. The middle location, 0x17E4, is out of the program counter sequence. – Anonymous May 12 '17 at 8:54
• Meanwhile I agree with Anonymous. I changed the instruction handling to my knowledge and datasheet and the routine works also in asm – Decapod May 12 '17 at 9:01
Caveats: It has been more than a decade since I routinely wrote PIC assembly code (and used their C compiler -- not sure if my license is any good anymore, either.) You also didn't say which compiler you are using, so I can't go get a copy and see what it generates. And I don't even know if you are using a PIC16 or a PIC18 (different creatures, with the PIC18 having nicer arrangements for assembly code.) Finally, I am assuming you pretty much are up on things with the PIC and that in some ways you will be more able than I am to answer specific questions about its assembly code because of your current work versus my decade old (and more) knowledge about it.
Assumptions:
1. It's a PIC16 (or lower) family part and not a PIC18.
2. When you write in assembly code you are using a real assembler and not hand-coding machine code.
3. You haven't been looking closely at the generated machine code.
4. You're unstated confusion is over the address there.
Assuming the above are correct, I suspect you may have missed out noticing that the GOTO instruction only uses 11 bits for the address. It can only branch within the current code "page," where the upper address bits remain unchanged in the branch to a new address.
When you write in assembly, you usually use labels. Even if you used absolute addresses, the assembler still does "do the math" for you to make sure that the address you specify is "within reach." So you wouldn't even know one way or another what is going on unless you look closely at the machine code it generates.
Let me document what I see above (along with a missing column which I sincerely wish you'd have also included):
Address Machine Code Code Lines Description
0x178E BCF STATUS, 0x5 These two lines make sure that data access
0x178F BCF STATUS, 0x6 is on page 0 where PORTE is at.
0x1790 BTFSS PORTE, 0x2 Skip GOTO if the loop should continue.
0x1791 GOTO 0x7E5 Go out of loop to 0x17E5 (past the GOTO.)
0x1792 BTFSC PORTE, 0x0 "if ArrowUpClosed()"
0x1793 <>
0x1794 <>
... ...
0x17E1 <>
0x17E2 <>
0x17E3 <>
0x17E4 GOTO 0x78E Branch to address 0x178E (start of loop.)
0x17E5 MOVLW 0x8
If you note that there are only 11 bits in the encoded machine instruction for the GOTO, then you can see why the entire "0x17E5" or "0x178E" can't be encoded into the machine word. However, that fact doesn't prevent the assembler and/or compiler from knowing that the address can in fact be properly reached (or not, indicating that more code may be needed to make a more distant transfer.)
The compiler's code works, I think you say. You also say that when you wrote some assembly code of your own that it also worked. I believe both of these are true. But the issue may simply be that you haven't realized how an assembler (more importantly, how a linker) works inside to generate correct code or to generate errors. | 2019-06-16 17:41:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3320401906967163, "perplexity": 3685.769485263552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998288.34/warc/CC-MAIN-20190616162745-20190616184745-00408.warc.gz"} |
https://www.tutorialspoint.com/sympy/sympy_evalf_function.htm | # SymPy - evalf() function
This function evaluates a given numerical expression upto a given floating point precision upto 100 digits. The function also takes subs parameter a dictionary object of numerical values for symbols. Consider following expression
>>> from sympy.abc import r
>>> expr=pi*r**2
>>> expr
The above code snippet gives an output equivalent to the below expression −
$\Pi{r^2}$
To evaluate above expression using evalf() function by substituting r with 5
>>> expr.evalf(subs={r:5})
The above code snippet gives the following output −
78.5398163397448
By default, floating point precision is upto 15 digits which can be overridden by any number upto 100. Following expression is evaluated upto 20 digits of precision.
>>> expr=a/b
>>> expr.evalf(20, subs={a:100, b:3})
The above code snippet gives the following output −
33.333333333333333333 | 2021-03-04 16:20:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5251989364624023, "perplexity": 5512.56943835269}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178369420.71/warc/CC-MAIN-20210304143817-20210304173817-00424.warc.gz"} |
http://bsdupdates.com/prediction-error/prediction-error-linear-model.php | Home > Prediction Error > Prediction Error Linear Model
# Prediction Error Linear Model
## Contents
Where it differs, is that each data point is used both to train models and to test a model, but never at the same time. This can be a numeric vector or a one-sided model formula. Cross-validation works by splitting the data up into a set of n folds. Is powered by WordPress using a bavotasan.com design. http://bsdupdates.com/prediction-error/prediction-error-model.php
A faster algorithm is the Levinson recursion proposed by Norman Levinson in 1947, which recursively calculates the solution.[citation needed] In particular, the autocorrelation equations above may be more efficiently solved by D.; Torrie, James H. (1960). For instance, in the illustrative example here, we removed 30% of our data. Each number in the data set is completely independent of all the others, and there is no relationship between any of them. read this article
## Linear Regression Prediction Error
Jobs for R usersStatistical Analyst @ Rostock, Mecklenburg-Vorpommern, GermanyData EngineerData Scientist – Post-Graduate Programme @ Nottingham, EnglandDirector, Real World Informatics & Analytics Data Science @ Northbrook, Illinois, U.S.Junior statistician/demographer for UNICEFHealth Contents 1 Introduction 2 In univariate distributions 2.1 Remark 3 Regressions 4 Other uses of the word "error" in statistics 5 See also 6 References Introduction Suppose there is a series That fact, and the normal and chi-squared distributions given above, form the basis of calculations involving the quotient X ¯ n − μ S n / n , {\displaystyle {{\overline {X}}_{n}-\mu
Please try the request again. scale Scale parameter for std.err. Because init_sys is an idproc model, use procestOptions to create the option set.load iddata1 z1; opt = procestOptions('Display','on','SearchMethod','lm'); sys = pem(z1,init_sys,opt); Examine the model fit.sys.Report.Fit.FitPercent ans = 70.6330 sys provides a Error Prediction Linear Regression Calculator Lane PrerequisitesMeasures of Variability, Introduction to Simple Linear Regression, Partitioning Sums of Squares Learning Objectives Make judgments about the size of the standard error of the estimate from a scatter plot
Note that the slope of the regression equation for standardized variables is r. Prediction Error Formula So, for example, in the case of 5-fold cross-validation with 100 data points, you would create 5 folds each containing 20 data points. weights variance weights for prediction. CSS from Substance.io.
MX MY sX sY r 3 2.06 1.581 1.072 0.627 The slope (b) can be calculated as follows: b = r sY/sX and the intercept (A) can be calculated as A Prediction Accuracy Measure If we stopped there, everything would be fine; we would throw out our model which would be the right choice (it is pure noise after all!). Let's see what this looks like in practice. At least two other uses also occur in statistics, both referring to observable prediction errors: Mean square error or mean squared error (abbreviated MSE) and root mean square error (RMSE) refer
• The scatter plots on top illustrate sample data with regressions lines corresponding to different levels of model complexity.
• For example, use ssest(data,init_sys) for estimating state-space models.More Aboutcollapse allAlgorithmsPEM uses numerical optimization to minimize the cost function, a weighted norm of the prediction error, defined as follows for scalar outputs:VN(G,H)=∑t=1Ne2(t)where
• See also Autoregressive model Prediction interval Rasta filtering Minimum mean square error References ^ Einicke, G.A. (2012).
• In matrix form the equations can be equivalently written as R a = − r , {\displaystyle Ra=-r,\,} where the autocorrelation matrix R {\displaystyle R} is a symmetric, p × p
• However, we want to confirm this result so we do an F-test.
## Prediction Error Formula
Comments are closed. https://en.wikipedia.org/wiki/Errors_and_residuals So we could get an intermediate level of complexity with a quadratic model like $Happiness=a+b\ Wealth+c\ Wealth^2+\epsilon$ or a high-level of complexity with a higher-order polynomial like \$Happiness=a+b\ Wealth+c\ Wealth^2+d\ Wealth^3+e\ Linear Regression Prediction Error Of course, if the relationship between X and Y were not linear, a different shaped function could fit the data better. Prediction Error Statistics For instance, if we had 1000 observations, we might use 700 to build the model and the remaining 300 samples to measure that model's error.
Therefore, the standard error of the estimate is There is a version of the formula for the standard error in terms of Pearson's correlation: where ρ is the population value of click site Similarly, the true prediction error initially falls. The variance of the residuals will be smaller. ISBN9780521761598. Prediction Error Definition
Estimate a discrete-time state-space model using n4sid, which applies the subspace method.load iddata7 z7; z7a = z7(1:300); opt = n4sidOptions('Focus','simulation'); init_sys = n4sid(z7a,4,opt); init_sys provides a 73.85% fit to the estimation Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. You will never draw the exact same number out to an infinite number of decimal places. http://bsdupdates.com/prediction-error/prediction-error-regression-model.php Soc.
Still, even given this, it may be helpful to conceptually think of likelihood as the "probability of the data given the parameters"; Just be aware that this is technically incorrect!↩ This Prediction Error Calculator However, a terminological difference arises in the expression mean squared error (MSE). About Scott Fortmann-Roe Essays Accurately Measuring Model Prediction ErrorUnderstanding the Bias-Variance Tradeoff Subscribe Accurately Measuring Model Prediction Error May 2012 When assessing the quality of a model, being able to accurately
## Thus their use provides lines of attack to critique a model and throw doubt on its results.
Given a parametric model, we can define the likelihood of a set of data and parameters as the, colloquially, the probability of observing the data given the parameters 4. Cook, R. The probability distributions of the numerator and the denominator separately depend on the value of the unobservable population standard deviation σ, but σ appears in both the numerator and the denominator Prediction Error Psychology Pros Easy to apply Built into most advanced analysis programs Cons Metric not comparable between different applications Requires a model that can generate likelihoods 5 Various forms a topic of theoretical
Alternative FunctionalityYou can achieve the same results as pem by using dedicated estimation commands for the various model structures. In this method we minimize the expected value of the squared error E [ e 2 ( n ) ] {\displaystyle E[e^{2}(n)]} , which yields the equation ∑ i = 1 The above equations are called the normal equations or Yule-Walker equations. More about the author If you were going to predict Y from X, the higher the value of X, the higher your prediction of Y.
The expected value, being the mean of the entire population, is typically unobservable, and hence the statistical error cannot be observed either. Let's say we kept the parameters that were significant at the 25% level of which there are 21 in this example case. Translate pemPrediction error estimate for linear and nonlinear modelcollapse all in page Syntaxsys = pem(data,init_sys) examplesys = pem(data,init_sys,opt) exampleDescriptionexamplesys = pem(data,init_sys) updates the parameters of an initial model to S., & Pee, D. (1989).
further arguments passed to or from other methods. A residual (or fitting deviation), on the other hand, is an observable estimate of the unobservable statistical error. In this tutorial we will use K = 5. As can be seen, cross-validation is very similar to the holdout method.
K-Fold cross-validation This is the most common use of cross-validation. Learn R R jobs Submit a new job (it's free) Browse latest jobs (also free) Contact us Welcome! The more optimistic we are, the better our training error will be compared to what the true error is and the worse our training error will be as an approximation of The formulas are the same; simply use the parameter values for means, standard deviations, and the correlation.
The function uses prediction-error minimization algorithm to update the parameters of the initial model. This is quite a troubling result, and this procedure is not an uncommon one but clearly leads to incredibly misleading results. R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, Specification of the parameters of the linear predictor is a wide topic and a large number of other approaches have been proposed.[citation needed] In fact, the autocorrelation method is the most
Since the likelihood is not a probability, you can obtain likelihoods greater than 1. | 2018-01-23 22:07:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8236567974090576, "perplexity": 1280.0922197377151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084892699.72/warc/CC-MAIN-20180123211127-20180123231127-00413.warc.gz"} |
http://mathhelpforum.com/discrete-math/103048-quick-question-about-set.html | 1. ## quick question about set
Can you please check if they are correct?
a. $\displaystyle \emptyset \subset${0} <-- True
b. $\displaystyle \emptyset \subset${$\displaystyle \emptyset$,{0}} <-- True
c. {$\displaystyle \emptyset$}$\displaystyle \in${$\displaystyle \emptyset$} <-- True
d. { $\displaystyle \emptyset$ } $\displaystyle \in$ {{ $\displaystyle \emptyset$ }} <-- False
e. {$\displaystyle \emptyset$} $\displaystyle \subset$ {$\displaystyle \emptyset$ , {$\displaystyle \emptyset$}} <-- True
f. {{$\displaystyle \emptyset$}} $\displaystyle \subset$ {$\displaystyle \emptyset$,{$\displaystyle \emptyset$}} <--False
g. {{$\displaystyle \emptyset$}} $\displaystyle \subset$ {{$\displaystyle \emptyset$},{$\displaystyle \emptyset$}} <--False
Thank you
2. Hello zpwnchen
Originally Posted by zpwnchen
Can you please check if they are correct?
a. $\displaystyle \emptyset \subset${0} <-- True True, the empty set is a subset of every set
b. $\displaystyle \emptyset \subset${$\displaystyle \emptyset$,{0}} <-- True True
c. {$\displaystyle \emptyset$}$\displaystyle \in${$\displaystyle \emptyset$} <-- True No, it's false. $\displaystyle \color{red}\{\emptyset\}=\{\emptyset\}$
d. { $\displaystyle \emptyset$ } $\displaystyle \in$ {{ $\displaystyle \emptyset$ }} <-- False No, it's true. It is always true that $\displaystyle \color{red}\{x\}\in\{\{x\}\}$ whatever $\displaystyle \color{red}x$ may stand for.
e. {$\displaystyle \emptyset$} $\displaystyle \subset$ {$\displaystyle \emptyset$ , {$\displaystyle \emptyset$}} <-- True True
f. {{$\displaystyle \emptyset$}} $\displaystyle \subset$ {$\displaystyle \emptyset$,{$\displaystyle \emptyset$}} <--False No, it's true: $\displaystyle \color{red}\{\{x\}\}\subset\{\emptyset,\{x\}\}$
g. {{$\displaystyle \emptyset$}} $\displaystyle \subset$ {{$\displaystyle \emptyset$},{$\displaystyle \emptyset$}} <--False Correct, it is false, it should be $\displaystyle \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}$ or $\displaystyle \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}$
Thank you
3. Originally Posted by zpwnchen
Can you please check if they are correct?
[snip]
g. {{$\displaystyle \emptyset$}} $\displaystyle \subset$ {{$\displaystyle \emptyset$},{$\displaystyle \emptyset$}} <--False
Thank you
The answer depends on your interpretation of the symbol $\displaystyle \subset$. Some authors define this symbol to mean either proper inclusion or equality of sets. I.e., $\displaystyle A \subset B$ means every element of A is also an element of B. If your book or teacher defines it this way then your answer is wrong.
Other authors define $\displaystyle \subset$ to mean proper inclusion, i.e. $\displaystyle A \subset B$ means every element of A is also an element of B but $\displaystyle A \neq B$, and write $\displaystyle A \subseteq B$ or $\displaystyle A \subseteqq B$ if A may be a proper subset of B or equal to B. So it depends.
Hello zpwnchen
$\displaystyle \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}$
$\displaystyle \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}$
Why they are equal? there are two elements in B, one in A. so it means every elements of A is not every elements of B...so should be $\displaystyle A \neq B$ right... ?sorry a little bit confused
5. Originally Posted by zpwnchen
$\displaystyle \color{red}\{\{\emptyset\}\}=\{\{\emptyset\},\{\em ptyset\}\}$
$\displaystyle \color{red}\{\{\emptyset\}\}\subseteq\{\{\emptyset \},\{\emptyset\}\}$
Why they are equal? there are two elements in B, one in A. so it means every elements of A is not every elements of B...so should be $\displaystyle A \neq B$
By definition for sets $\displaystyle P=Q$ if and only if $\displaystyle P \subseteq Q\;\& \;Q \subseteq P$.
Therefore $\displaystyle \{a\}=\{a,a\}$ because $\displaystyle \{a\}\subseteq \{a,a\}\;\& \;\{a,a\} \subseteq \{a\}$
So $\displaystyle \{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}$
BTW: The set $\displaystyle \{a,a\}$ has only one element.
Hello zpwnchen
Originally Posted by Plato
By definition for sets $\displaystyle P=Q$ if and only if $\displaystyle P \subseteq Q\;\& \;Q \subseteq P$.
Therefore $\displaystyle \{a\}=\{a,a\}$ because $\displaystyle \{a\}\subseteq \{a,a\}\;\& \;\{a,a\} \subseteq \{a\}$
So $\displaystyle \{\{\emptyset\}\}=\{\{\emptyset\},\{\emptyset\}\}$
BTW: The set $\displaystyle \{a,a\}$ has only one element.
Thank you
$\displaystyle \emptyset \in \{x\}$
$\displaystyle \emptyset \subset \{x\}$
$\displaystyle \emptyset \subseteq \{x\}$
can you please explain me why these are true or false?
7. Originally Posted by zpwnchen
Thank you
$\displaystyle \emptyset \in \{x\}$ False
$\displaystyle \emptyset \subset \{x\}$ True
$\displaystyle \emptyset \subseteq \{x\}$ True
can you please explain me why these are true or false?
You, yourself, have noted that the emptyset is a subset of every set.
But in the first case you have the element symbol and usually we understand that $\displaystyle x \ne \emptyset$
8. Thank you!
A = {0,1,2,3,4,5,6,7,8,9}
B = {0,2,4,6,8}
can we say that A $\displaystyle \subset$ B and A $\displaystyle \subseteq$ B? But A $\displaystyle \ne$ B, right?
9. Originally Posted by zpwnchen
Thank you!
can we say that A $\displaystyle \subset$ B and A $\displaystyle \subseteq$ B? But A $\displaystyle \ne$ B, right?
Well $\displaystyle A \not\subset B$. A contains elements that are not in B.
But $\displaystyle B \subset A$, every element in B is in A.
can we say that B $\displaystyle \subset$ A and B $\displaystyle \subseteq$ A? But A $\displaystyle \ne$ B, right? | 2018-04-21 21:53:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.835374116897583, "perplexity": 1277.8414173424742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945448.38/warc/CC-MAIN-20180421203546-20180421223546-00473.warc.gz"} |
https://www.physicsforums.com/threads/deriving-a-thermodynamics-relationship.254225/ | # Deriving a thermodynamics relationship
1. Sep 6, 2008
### Jacobpm64
1. The problem statement, all variables and given/known data
Derive:
$$\left(\frac{\partial C_{P}}{\partial P}\right)_{T} = -T \left(\frac{\partial^2 V}{\partial T^2}\right)_{P}$$
2. Relevant equations
$$C_{P} = C_{V} + \left[\left(\frac{\partial E}{\partial V}\right)_{T} + P \right] \left( {\frac{\partial V}{\partial T}\right)_{P}$$
I also know that
$$C_{V} = \left( \frac{\partial E}{\partial T}\right)_{V}$$
I also know that you can write differentials like this:
$$z = z(x,y)$$
$$dz = \left( \frac{\partial z}{\partial x}\right)_{y} dx + \left( \frac{\partial z}{\partial y}\right)_{x} dy$$
3. The attempt at a solution
My problem is not knowing what variables $$C_{P}$$ is a function of. Therefore, I do not know how to write its differential (if you even can). I also do not know how to differentiate the above expression for $$C_{P}$$ with respect to P holding T constant.
I also do not know how to use Euler's chain rule for something like $$C_{P}$$ because i've never messed with anything that had more than 2 independent variables.
I have tried three pages worth of manipulating, but I do not know if i'm even manipulating correctly because of what i mentioned above.
Anything to get me started or tell me what is legal.. would be greatly appreciated.. | 2017-03-25 20:00:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7150660753250122, "perplexity": 285.2941065801379}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189032.76/warc/CC-MAIN-20170322212949-00080-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://plainmath.net/51195/proving-frac-i-2-lsn-frac-x-plus-i-x-i-equal-arctan-x | # Proving \frac{i}{2}\lsn\frac{x+i}{x-i}=\arctan x
Proving $$\displaystyle{\frac{{{i}}}{{{2}}}}{l}{s}{n}{\frac{{{x}+{i}}}{{{x}-{i}}}}={\arctan{{x}}}$$
• Questions are typically answered in as fast as 30 minutes
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usaho4w
HINT:
Let $$\displaystyle{\arctan{{x}}}={y}\Rightarrow{x}={\tan{{y}}}$$
$$\displaystyle{\frac{{{x}+{i}}}{{{x}-{i}}}}={\frac{{{\sin{{y}}}+{i}{\cos{{y}}}}}{{{\sin{{y}}}-{i}{\cos{{y}}}}}}={\frac{{{i}{\left({\cos{{y}}}-{i}{\sin{{y}}}\right)}}}{{-{i}{\left({\cos{{y}}}+{i}{\sin{{y}}}\right)}}}}=-{e}^{{-{2}{i}{y}}}$$
###### Not exactly what you’re looking for?
John Koga
I don't know if I'd call it ''cool'' but you could just do it by using the complex exponential representations of trig functions.
$$\displaystyle{\sin{{\left({x}\right)}}}={\frac{{{e}^{{{i}{x}}}-{e}^{{-{i}{x}}}}}{{{2}{i}}}}$$ $$\displaystyle{\cos{{\left({x}\right)}}}={\frac{{{e}^{{{i}{x}}}+{e}^{{-{i}{x}}}}}{{{2}}}}$$
which comes from using $$\displaystyle{e}^{{{i}{x}}}={\cos{{\left({x}\right)}}}+{i}{\sin{{\left({x}\right)}}}$$. So
$$\displaystyle{\tan{{\left({x}\right)}}}={\frac{{{1}}}{{{i}}}}{\frac{{{e}^{{{i}{x}}}-{e}^{{-{i}{x}}}}}{{{e}^{{{i}{x}}}+{e}^{{-{i}{x}}}}}}={\frac{{{1}}}{{{i}}}}{\frac{{{e}^{{{2}{i}{x}}}-{1}}}{{{e}^{{{2}{i}{x}}}+{1}}}}$$
Now set
$$\displaystyle{y}={\frac{{{1}}}{{{i}}}}{\frac{{{e}^{{{2}{i}{x}}}-{1}}}{{{e}^{{{2}{i}{x}}}+{1}}}}$$
and solve for $$\displaystyle{x}$$ to get a formula for arctangent...
Vasquez
Identity is :
$$\arctan z=\frac{i}{2}\ln\frac{i+z}{i-z}$$
Proof :
$$\\\tan z=\frac{e^{iz}-e^{-iz}}{i(e^{iz}+e^{-iz})} \\e^{i\frac{i}{2}\ln\frac{i+z}{i-z}}=(\frac{i+z}{i-z})^{-\frac{1}{2}} \\e^{-i\frac{i}{2}\ln\frac{i+z}{i-z}}=(\frac{i+z}{i-z})^{\frac{1}{2}} \\\tan(\frac{i}{2}\ln\frac{i+z}{i-z})=\frac{1-\frac{i+z}{i-z}}{i(1+\frac{i+z}{i-z})}=z$$ | 2022-01-17 06:42:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8482123017311096, "perplexity": 1563.4884204869345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300343.4/warc/CC-MAIN-20220117061125-20220117091125-00003.warc.gz"} |
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https://www.gamedev.net/blogs/entry/2190323-erm/ | • entries
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# Erm
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Right, so yes, clearly been a bit out of control recently but it's all good. Never mind, move on, let's get back to programming.
Until I've moved properly and got my PC back, I'm stuck working on this tiny fucking Aspire thing with an Intel Atom. It's a good little bugger and I love it but it can't really do much in the way of graphics.
So I've become interested, purely as an intellectual exercise, in a completely dynamically typed language. Yes, there are probably loads, no, I don't expect anyone to write serious code in it, yes, mine is probably shitter than everyone elses.
That out of the way, so far I have the following source and a compiler that can parse it into a symbol tree:
class test{ class vector { var a,b,c; var d; push_back(a,b,c) { } } class list { class node { var x,y,z; } } class float { }}
This generates the following symbol tree dump:
class testclass test.vectorvar test.vector.avar test.vector.bvar test.vector.cvar test.vector.dmethod test.vector.push_back(a,b,c)class test.listclass test.list.nodevar test.list.node.xvar test.list.node.yvar test.list.node.zclass test.float
Yes, early days but it's a start.
The idea is that vars will carry their type around with them and all operations will be checked at runtime. All good fun.
Started thinking about whether I needed to differentiate between call by ref or call by val and head started hurting so gave up and decided to share my latest stupid project here with the only other people in the world who might understand why I would undertake such a pointless exercise.
Looks interesting. [smile] Is this going to be interpreted or compiled? Do you have a projected use for it (e.g. using it as the scripting language in one of your games) or is it purely for fun?
Quote:
Original post by benryves Looks interesting. [smile] Is this going to be interpreted or compiled? Do you have a projected use for it (e.g. using it as the scripting language in one of your games) or is it purely for fun?
Compiled into bytecode for a VM is the current plan. To be honest, I can't really see me using it in games because the overheads are going to be really high.
Every method call is going to involve indexing into some kind of database based on the current type, doing string comparisons to find out if the method is valid etc etc.
This is really just exploring an interest. Bit like your mad scientist electronics I guess, although mine is less likely to rebel against its human master and take over the world.
## Create an account
Register a new account | 2018-01-16 20:18:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2894549071788788, "perplexity": 2172.0342741360664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886639.11/warc/CC-MAIN-20180116184540-20180116204540-00445.warc.gz"} |
https://catboost.ai/docs/concepts/algorithm-main-stages_text-to-numeric | Transforming text features to numerical features
CatBoost supports the following types of features:
• Numerical. Examples are the height (182, 173), or any binary feature (0, 1).
• Categorical (cat). Such features can take one of a limited number of possible values. These values are usually fixed. Examples are the musical genre (rock, indie, pop) and the musical style (dance, classical).
• Text. Such features contain regular text (for example, Music to hear, why hear'st thou music sadly?).
Text features are transformed to numerical. The transformation method generally includes the following stages:
The text feature is loaded as a column. Every element in this column is a string.
To load text features to CatBoost:
• Specify the Text column type in the column descriptions file if the dataset is loaded from a file.
• Use the text_features parameter in the Python package.
2. Text preprocessing
1. Tokenization
Each value of a text feature is converted to a string sequence by splitting the original string by space.
An element of such sequence is called word.
1. Dictionary creation
A dictionary is a data structure that collects all values of a text feature, defines the minimum unit of text sequence representation (which is called token), and assigns a number to each token.
The type of dictionary defines the token type:
• Letter — A symbol from the string. For example, the abra cadabra text forms the following dictionary: {a, b, c, d, r}.
• Word — A word (an element from the sequence of strings obtained on step 2.a). For example, the abra cadabra text forms the following dictionary: {'abra', 'cadabra'}.
The type of dictionary can be specified in the token_level_type dictionary parameter.
Token sequences can be combined to one unique token, which is called N-gramm. N stands for the length (the number of combined sequences) of this new sequence. The length can be specified in the gram_order dictionary parameter.
Combining sequences can be useful if it is required to perceive the text more continuously. For example, let's examine the following texts: cat defeat mouse and mouse defeat cat. These texts have the same tokens in terms of Word dictionaries ({'cat', 'defeat', 'mouse'}) and different tokens in terms of bi-gram word dictionary ({'cat defeat', 'defeat mouse'} and {'mouse defeat', 'defeat cat'}).
It is also possible to filter rare words using the occurence_lower_bound dictionary parameter or to limit the maximum dictionary size to the desired number of using the max_dictionary_size dictionary parameter.
1. Converting strings to numbers
Each string from the text feature is converted to a token identifier from the dictionary.
Example
Source text:
ObjectId Text feature
0 "Cats are so cute :)"
1 "Mouse scares me"
2 "The cat defeated the mouse"
3 "Cute: Mice gather an army!"
4 "Army of mice defeated the cat :("
5 "Cat offers peace"
6 "Cat is scared :("
7 "Cat and mouse live in peace :)"
Splitting text into words:
ObjectId Text feature
0 ['Cats', 'are', 'so', 'cute', ':)']
1 ['Mouse', 'scares', 'me']
2 ['The', 'cat', 'defeated', 'the', 'mouse']
3 ['Cute:', 'Mice', 'gather', 'an', 'army!']
4 ['Army', 'of', 'mice', 'defeated', 'the', 'cat', ':(']
5 ['Cat', 'offers', 'peace']
6 ['Cat', 'is', 'scared', ':(']
7 ['Cat', 'and', 'mouse', 'live', 'in', 'peace', ':)']
Creating dictionary:
Word TokenId
"Cats" 0
"are" 1
"so" 2
"cute" 3
...
"and" 26
"live" 27
"in" 28
Converting strings into numbers:
ObjectId TokenizedFeature
0 [0, 1, 2, 3, 4]
1 [5, 6, 7]
2 [8, 9, 10, 11, 12]
3 [13, 14, 15, 16, 17]
4 [18, 19, 20, 10, 11, 9, 21]
5 [22, 23, 24]
6 [22, 25, 26, 21]
7 [22, 27, 12, 28, 29, 24, 4]
3. Estimating numerical features
Numerical features are calculated based on the source tokenized.
Supported methods for calculating numerical features:
• BoW (Bag of words) — Boolean (0/1) features reflecting whether the object contains the token_id. The number of features is equal to the dictionary size.
Supported options:
• top_tokens_count — The maximum number of features to create. If set, the specified number top tokens is taken into account and the corresponding number of new features is created.
• NaiveBayes — Multinomial naive bayes model, the number of created features is equal to the number of classes. To avoid target leakage, this model is computed online on several dataset permutations (similarly to the estimation of CTRs).
• BM25 — A function that is used for ranking purposes by search engines to estimate the relevance of documents. To avoid target leakage, this model is computed online on several dataset permutations (similarly to the estimation of CTRs).
4. Training
Computed numerical features are passed to the regular CatBoost training algorithm. | 2022-05-25 06:05:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30628395080566406, "perplexity": 6955.99235441687}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662580803.75/warc/CC-MAIN-20220525054507-20220525084507-00769.warc.gz"} |
https://socratic.org/questions/which-are-more-brittle-metals-or-nonmetals | # Which are more brittle: metals or nonmetals?
Nov 18, 2016
You should tell us.
#### Explanation:
Can you hammer out a metal sheet? Can you draw out a metal wire? You should look up the definitions of $\text{malleability}$, and $\text{ductility}$.
On the other hand, can you break a stick of chalk with your fingers? Can you crush a crystal of salt? | 2020-09-25 10:11:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7538228631019592, "perplexity": 1324.324928066912}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400223922.43/warc/CC-MAIN-20200925084428-20200925114428-00537.warc.gz"} |
https://math.meta.stackexchange.com/questions/32292/how-to-remove-the-pink-edit-box | # How to remove the pink edit box?
When editing answers and questions on Math Stack Exchange, I always see a pink box in which it is written 'How to Edit'. Now the problem with this box is that it hides the "Save edits" button, so I am unable to save my edits.
Is there a way in which this box can be removed,, or at least shifted so that I can save my edits? Kindly suggest a way to do so. I have pasted the information given in the pink box. Its exact title is "How to Format".
► fix grammatical or spelling errors ► clarify meaning without changing it ► correct minor mistakes ► add related resources or links ► always respect the original author How to Format ► create code fences with backticks or tildes ~
like so
► add language identifier to highlight code
def function(foo):
print(foo)
`
► put returns between paragraphs
► for linebreak add 2 spaces at end
italic or bold
► quote by placing > at start of line
• Can you provide a screenshot of this? – TheSimpliFire Aug 2 '20 at 19:03
• @TheSimpliFire I will try to send screenshot – Ginger bread Aug 3 '20 at 5:01
• @Gingerbread well, it wasn't $\color{pink}{pink}$, so I wasn't 100% sure. Glad to help :) – Calvin Khor Aug 3 '20 at 10:47 | 2021-07-28 18:01:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20006392896175385, "perplexity": 2357.0418506624196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153739.28/warc/CC-MAIN-20210728154442-20210728184442-00566.warc.gz"} |
http://nspaperhsth.voteforthegirls.us/am-gm-inequality-and-its-applications.html | # Am gm inequality and its applications
Combined am gm qm inequality up vote 12 down vote favorite 8 if you want to see its applications, see the applications section of vasc's paper. Advanced inequality manipulations if we apply the am-gm inequality to the rhs of as follows the application of the am-gm inequality be-comes useful: n 1 + a a. A multiplicative mean value and its applications published in vol theory of inequalities and applications which is a consequence of the am-gm inequality. Some operator inequalities for positive linear maps some operator inequalities for positive can be regarded as a counterpart to operator am–gm inequality. The main tool to prove our results is the celebrated arithmetic–geometric mean inequality (am–gm) this index has its applications in chemical graph theory as. 7 posts published by potla during march 2012 i will directly go to applications of this inequality in we can prove the am-gm inequality for numbers. An inequality in parallelogram of unit area $then by the am-gm inequality hadamard's determinant inequalities and applications i$\left((2-a-b-c+abc.
Complementary inequalities to improved am-gm inequality authors a diaz–metcalf type inequality for positive linear maps and its applications electron j. Applications of the am-gm inequality in finding bounds on the areas and volumes of two-dimensional and three-dimensional geometric shapes respectively. Mathematical inequalities: am-gm-hm inequality for two numbers is named of majorization and its applications, academic press, 1979. Applications of inequalities for example cauchy- schwarz, jensen, am-gm or maclaurin in non theory based math in mean for building machines or something.
Linear algebra and its applications 430 (2009) applying the am–gm inequality to x,wehave σ k (f) = summationdisplay x∈x (k)+ (f. In this note, we generalize some inequalities with matrix means due to hoa et al [rims kokyuroku 18932013:67–71] let and two arbitrary means between harmonic. Title: secrets in inequalities author: pham kim hung isbn 978-973-9417-88-4 11 am-gm inequality and applications theorem 1 (am-gm inequality.
We will look at the following 5 general ways of using am-gm: direct application to an inequality application on each term in a product application on terms obtained. 51 a class of applications of am-gm inequality: from a 2004 putnam competition problem to lalescu’s sequence wladimir g boskoff and bogdan d suceav˘a. Am, gm, and hm satisfy these inequalities: in other applications they represent a measure for the reliability of the influence upon the mean by the respective values. Its proof and various applications am–gm–hm inequality is one of the fundamental classroom left hand side of the inequality is positive we have.
## Am gm inequality and its applications
The weighted am-gm inequality is equivalent to the h？ mathematical relations among the weighted am-gm inequality lder’s inequality and its application. In mathematics , the inequality of arithmetic and geometric means , or more briefly the am–gm inequality , states that the arithmetic mean of a list of non-negative. Inequalities: a journey into linear analysis 31 the am–gm inequality 19 32 applications 21 33 notes and remarks 23 4 convexity, and jensen’s inequality 24.
• Minimizing the calculus in optimization problems optimization problems are explored and solved using the am/gm inequality see an application of the inequality.
• 102 am{gm inequality tinction probability combinatorial applications of mathematical probability began its development in.
• Mathematics student seminar search this site home the am/gm inequality and its applications to calculus: for at least 1000 years prior to the invention of.
• Uses of the cauchy-schwarz inequality the am-gm inequality: the arithmetic mean of positive numbers is always greater than the geometric mean.
• Proofs of am-gm inequality but i think the proof can be extended to the am-gm inequality with 3 variables if we consider a sphere with web applications ask.
Cirtoaje, vasile proofs of three open inequalities with power-exponential functions the journal of nonlinear sciences and its applications 42 (2011): 130-137. There are many, many applications, but one i quite like is an alternative proof of the am-gm inequality $\ln(x)$ is a concave function for positive. Since we already know it’s proof i will directly go to applications of this inequality in various problems we can also prove it using am-gm inequality. Posts about euler’s number the arithmetic-geometric inequality is a deep theorem with applications all over one can use the am-gm inequality to prove. In mathematics, the cauchy–schwarz inequality a generalization of this is the hölder inequality applications analysis the.
Am gm inequality and its applications
Rated 4/5 based on 18 review | 2018-05-22 23:10:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8144527673721313, "perplexity": 638.5278923620674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864999.62/warc/CC-MAIN-20180522225116-20180523005116-00610.warc.gz"} |
http://yomcat.geek.nz/blog/264/shooting-the-moon?m=12&y=2018 | ## Shooting the Moon
5 May 2017, 08:48
Canon EOS 70D, Canon EF 400mm f/5.6L USM + 2× III lens, 1/15 seconds @ f/16, ISO 100.
I had a nice blurb all written, but then I hit Z instead of V and lost the lot. So here’s what you get if you point a 800mm lens at the moon.
Posted by Michael Welsh at 08:48. | 2019-01-18 22:57:42 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.828319787979126, "perplexity": 12031.395741654664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583660818.25/warc/CC-MAIN-20190118213433-20190118235433-00113.warc.gz"} |
https://www.drbasheers.com/post/extreme-events-and-signal-detection | Search
• Rana Basheer
Extreme Events and Signal Detection
An idea has been simmering in my mind for sometime regarding the statistical nature of signal detection in a wireless receiver. Essentially, I was wondering if the noise in signal strength values measured by a wireless receiver could be modeled as an extreme value distribution?
The genesis for this idea in my brain could be traced to the moment when I was reading the famous book The Black Swan by Nassim Nicholas Taleb. In this book, Taleb mentioned option pricing using extreme value distribution under market disturbance conditions. Before I explain the details about extreme value distribution, I will give you a glimpse into a frustrating period in my student life. From the start of my PhD research, one of my tasks as a graduate research assistant was to run short localization demos for visitors to our lab. If these visitors are super impressed, they would open their wallets for more research grants or even a plush job for me after graduation. During these demos I am either running a localization algorithm that my fellow students developed or my “next greatest method that is bound to revolutionize” .. not enough hyperbole? After working nights and days tuning the system to perfection while dreaming of ticker tape parades and accolades, I am bought back to earth rudely and violently in an Icarus sort of tragedy, when my delicate tapestry fails spectacularly in front of these visitors. All of a sudden, I am scrambling to explain the reason why my tracked target that is overlayed on an aerial map of our lab is now moving through walls as if it is the reincarnation of Agent Smith who is out to nab Neo in the cult movie Matrix.
I figured that the problem was with the way the signal strength variations was modeled. Essentially, a localization algorithm is attempting to classify signal strength variation as either due to a legitimate movement of the transmitter or due to a wireless propagation issue called signal fading. The best way to do this would be to use the Probability Distribution Function (PDF) of the signal variation under signal fading. Knowing the PDF of a noise source such as fading helps you generate probabilities with each signal strength fluctuation. So if you observe a sudden dip in signal strength and your signal fading PDF tells you that the chance for such a variation is high due to fading then you are not going to update your transmitter’s position on the map.
Though, this sounds very simple and straight forward, the reality is quite complicated. If you skim through any introductory statistical signal analysis textbook, it is replete with discussions about Rayleigh, Rician, Nakagami and a host of other hard to pronounce names that are used as PDF for wireless signal strength. On reading the fine sub text you will realize that each of this PDF has a specific environment that they will work. For Rayleigh distribution, the wireless signal has to be completely diffused i.e. there should not be any clear line of sight communication between receiver and transmitter, for Rician distribution you need to have a well defined line of sight component whereas Nakagami requires the Signal to Noise Ratio (SNR) of the wireless signal to be above some threshold and so on. Unless you have done an extensive Ray Tracing of your target localization environment, you have no prior idea about the fading environment that your wireless receiver and transmitter is going to experience. Additionally, if people are moving around then all bets are off about the fading environment that you painstakingly figured out using ray tracing. In short you are at the mercy of the environmental requirements set by the PDF that you decided to use for your localization algorithm. This is when every localization algorithm takes out their proverbial “Procrustean Bed” or the infamous heuristic fudge factors that attempts to massage values or throw away samples that doesn’t satisfy some arbitrary standards set by your pet PDF.
A typical modern day digital spread spectrum radio receiver, such as those in GPS, WiFi, ZigBee radios etc., have a down converter that translates the high frequency carrier to a low frequency base band signal which are then subsequently converted to digital samples using high speed Analog to Digital Converter. To detect a signal of interest in this digital samples, these devices use a process called convolution whereby the signal pattern of interest is time shifted and multiplied with the incoming digital samples. When an exact match occurs the result of the convolution process will be a value much higher in comparison to non-matched samples (this gain depends on the cross-correlation properties of the signal pattern of interest). So to isolate noise from a detection event, these receivers employ a threshold whereby any convolution result above this threshold are treated as detection event or else it is just noise. My reasoning for treating wireless signal strength as an extreme value distribution stems from this use of thresholds for detection. Now let me give a brief introduction to extreme value distribution.
Extreme value distribution is based on Fisher-Tippet-Gnedenko theorem which states that if we take independent samples from whatever underlying distribution and then form sub-samples consisting only of samples that are above or below a threshold then the distribution of the resulting sub-samples can be classified into one of the following three distributions: Type I Gumbel Distribution, Type II Frechet Distribution or Type III Weibull distribution or they can all be grouped under a single family called the Generalized Extreme Value Distribution (GEV). GEV has three parameters
$\mu\in\mathbb{R}$
$\sigma > 0$
$\xi\in\mathbb{R}$
$\mu, \sigma$
$\xi$
$\gamma$
$\gamma\left(x,y\right)=\frac{g_4-4g_1g_3+6g_2g_1^2-3g_1^4}{\left(g_2-g_1^2\right)^2}-3$
where
$x,y$
$g_k=\Gamma\left(1+k\xi\right)$
$k\in\{1,2,3,4\}$
$\Gamma\left(.\right)$
Gamma Function. The details about the moments of GEV can be found here
Stedinger, J. R., R. M. Vogel, and E. Foufoula-Georgiou, Frequency analysis of extreme events, in Handbook of Applied Hydrology, edited by D. A. Maidment, chap. 18, pp. 18-1–18-66, McGraw-Hill, New York, 1993.
The advantage of treating signal strength as an extreme value distribution is that GEV is independent of whatever underlying distribution forms the basis for noise in signal strength values. Additionally, by capturing just three parameters over my lab (
$\mu,\sigma,\xi$
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See All | 2021-07-24 00:50:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 12, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5617722868919373, "perplexity": 868.0893805927409}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046150067.87/warc/CC-MAIN-20210724001211-20210724031211-00428.warc.gz"} |
https://www.rdocumentation.org/packages/timereg/versions/1.9.8/topics/cox.aalen | timereg (version 1.9.8)
# cox.aalen: Fit Cox-Aalen survival model
## Description
Fits an Cox-Aalen survival model. Time dependent variables and counting process data (multiple events per subject) are possible.
## Usage
cox.aalen(
formula = formula(data),
data = parent.frame(),
beta = NULL,
Nit = 20,
detail = 0,
start.time = 0,
max.time = NULL,
id = NULL,
clusters = NULL,
n.sim = 500,
residuals = 0,
robust = 1,
weighted.test = 0,
covariance = 0,
resample.iid = 1,
weights = NULL,
rate.sim = 1,
beta.fixed = 0,
max.clust = 1000,
exact.deriv = 1,
silent = 1,
max.timepoint.sim = 100,
basesim = 0,
offsets = NULL,
strata = NULL,
propodds = 0,
caseweight = NULL
)
## Arguments
formula
a formula object with the response on the left of a '~' operator, and the independent terms on the right as regressors. The response must be a survival object as returned by the Surv' function. Terms with a proportional effect are specified by the wrapper prop(), and cluster variables (for computing robust variances) by the wrapper cluster().
data
a data.frame with the variables.
beta
starting value for relative risk estimates.
Nit
number of iterations for Newton-Raphson algorithm.
detail
if 0 no details is printed during iterations, if 1 details are given.
start.time
start of observation period where estimates are computed.
max.time
end of observation period where estimates are computed. Estimates thus computed from [start.time, max.time]. Default is max of data.
id
For timevarying covariates the variable must associate each record with the id of a subject.
clusters
cluster variable for computation of robust variances.
n.sim
number of simulations in resampling.
residuals
to returns residuals that can be used for model validation in the function cum.residuals. Estimated martingale increments (dM) and corresponding time vector (time). When rate.sim=1 returns estimated martingales, dM_i(t) and if rate.sim=0, returns a matrix of dN_i(t).
robust
to compute robust variances and construct processes for resampling. May be set to 0 to save memory and time, in particular for rate.sim=1.
weighted.test
to compute a variance weighted version of the test-processes used for testing time-varying effects.
covariance
to compute covariance estimates for nonparametric terms rather than just the variances.
resample.iid
to return i.i.d. representation for nonparametric and parametric terms. based on counting process or martingale resduals (rate.sim).
weights
weights for weighted analysis.
rate.sim
rate.sim=1 such that resampling of residuals is based on estimated martingales and thus valid in rate case, rate.sim=0 means that resampling is based on counting processes and thus only valid in intensity case.
beta.fixed
option for computing score process for fixed relative risk parameter
max.clust
sets the total number of i.i.d. terms in i.i.d. decompostition. This can limit the amount of memory used by coarsening the clusters. When NULL then all clusters are used. Default is 1000 to save memory and time.
exact.deriv
if 1 then uses exact derivative in last iteration, if 2 then uses exact derivate for all iterations, and if 0 then uses approximation for all computations and there may be a small bias in the variance estimates. For Cox model always exact and all options give same results.
silent
if 1 then opppresses some output.
max.timepoint.sim
considers only this resolution on the time scale for simulations, see time.sim.resolution argument
basesim
1 to get simulations for cumulative baseline, including tests for contant effects.
offsets
offsets for analysis on log-scale. RR=exp(offsets+ x beta).
strata
future option for making strata in a different day than through X design in cox-aalen model (~-1+factor(strata)).
propodds
if 1 will fit the proportional odds model. Slightly less efficient than prop.odds() function but much quicker, for large data this also works.
caseweight
these weights have length equal to number of jump times, and are multiplied all jump times dN. Useful for getting the program to fit for example the proportional odds model or frailty models.
## Value
returns an object of type "cox.aalen". With the following arguments:
cum
cumulative timevarying regression coefficient estimates are computed within the estimation interval.
var.cum
the martingale based pointwise variance estimates.
robvar.cum
robust pointwise variances estimates.
gamma
estimate of parametric components of model.
var.gamma
variance for gamma sandwhich estimator based on optional variation estimator of score and 2nd derivative.
robvar.gamma
robust variance for gamma.
residuals
list with residuals.
obs.testBeq0
observed absolute value of supremum of cumulative components scaled with the variance.
pval.testBeq0
p-value for covariate effects based on supremum test.
sim.testBeq0
resampled supremum values.
obs.testBeqC
observed absolute value of supremum of difference between observed cumulative process and estimate under null of constant effect.
pval.testBeqC
p-value based on resampling.
sim.testBeqC
resampled supremum values.
obs.testBeqC.is
observed integrated squared differences between observed cumulative and estimate under null of constant effect.
pval.testBeqC.is
p-value based on resampling.
sim.testBeqC.is
resampled supremum values.
conf.band
resampling based constant to construct robust 95% uniform confidence bands.
test.procBeqC
observed test-process of difference between observed cumulative process and estimate under null of constant effect over time.
sim.test.procBeqC
list of 50 random realizations of test-processes under null based on resampling.
covariance
covariances for nonparametric terms of model.
B.iid
Resample processes for nonparametric terms of model.
gamma.iid
Resample processes for parametric terms of model.
loglike
approximate log-likelihood for model, similar to Cox's partial likelihood. Only computed when robust=1.
D2linv
inverse of the derivative of the score function.
score
value of score for final estimates.
test.procProp
observed score process for proportional part of model.
var.score
variance of score process (optional variation estimator for beta.fixed=1 and robust estimator otherwise).
pval.Prop
p-value based on resampling.
sim.supProp
re-sampled absolute supremum values.
sim.test.procProp
list of 50 random realizations of test-processes for proportionality under the model based on resampling.
## Details
$$\lambda_{i}(t) = Y_i(t) ( X_{i}^T(t) \alpha(t) ) \exp(Z_{i}^T \beta )$$
The model thus contains the Cox's regression model as special case.
To fit a stratified Cox model it is important to parametrize the baseline apppropriately (see example below).
Resampling is used for computing p-values for tests of time-varying effects. Test for proportionality is considered by considering the score processes for the proportional effects of model.
The modelling formula uses the standard survival modelling given in the survival package.
The data for a subject is presented as multiple rows or 'observations', each of which applies to an interval of observation (start, stop]. For counting process data with the )start,stop] notation is used, the 'id' variable is needed to identify the records for each subject. The program assumes that there are no ties, and if such are present random noise is added to break the ties.
## References
Martinussen and Scheike, Dynamic Regression Models for Survival Data, Springer (2006).
## Examples
# NOT RUN {
library(timereg)
data(sTRACE)
# Fits Cox model
out<-cox.aalen(Surv(time,status==9)~prop(age)+prop(sex)+
prop(vf)+prop(chf)+prop(diabetes),data=sTRACE)
# makes Lin, Wei, Ying test for proportionality
summary(out)
par(mfrow=c(2,3))
plot(out,score=1)
# Fits stratified Cox model
out<-cox.aalen(Surv(time,status==9)~-1+factor(vf)+ prop(age)+prop(sex)+
prop(chf)+prop(diabetes),data=sTRACE,max.time=7,n.sim=100)
summary(out)
par(mfrow=c(1,2)); plot(out);
# Same model, but needs to invert the entire marix for the aalen part: X(t)
out<-cox.aalen(Surv(time,status==9)~factor(vf)+ prop(age)+prop(sex)+
prop(chf)+prop(diabetes),data=sTRACE,max.time=7,n.sim=100)
summary(out)
par(mfrow=c(1,2)); plot(out);
# Fits Cox-Aalen model
out<-cox.aalen(Surv(time,status==9)~prop(age)+prop(sex)+
vf+chf+prop(diabetes),data=sTRACE,max.time=7,n.sim=100)
summary(out)
par(mfrow=c(2,3))
plot(out)
# }
` | 2021-04-12 16:23:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6433518528938293, "perplexity": 5997.462636893186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038067870.12/warc/CC-MAIN-20210412144351-20210412174351-00133.warc.gz"} |
https://tug.org/pipermail/macostex-archives/2011-April/046916.html | [OS X TeX] Vertical alignment
Don Green Dragon fergdc at Shaw.ca
Tue Apr 19 02:17:36 CEST 2011
HI All,
I use TeXShop to create source files, then typeset with LaTeX.
Often I'm bedeviled with LaTeX placing far too much or too little vertical whitespace on a page -- in my opinion.
At the bottom of a page:
========================
Often there will be one or two lines hanging at the bottom of a page, and they simply have to be pushed to the next page. As a last resort, \newpage is envoked but because that is such a poor solution, I mark each such line within something like:
\newpage % DANGER!!!!
Upon revision, those \newpage's are sure to cause unwanted page breaks, although they are easy to spot.
A somewhat better --- perhaps worse --- solution is too add vertical whitespace with \vspace{24pt}, and similar ilk, but such insertions can be harder to spot on subsequent revisions where they create whitespace gaps which are undesirable and have to be rooted out. Sometimes, I'll pad the paragraph above with extra lines which may solve the formatting problem, but throwing in extra words is less than a blessing.
From the way it behaves, when I'm trying to finesse at the bottom of a page, I suspect that LaTeX is trying to keep the height of each page the same, or close to the same. Is there some "rubber" length that one can redefine so that LaTeX is not so picky about each page being of the exact same height?
How do other writers handle this bottom-of-the-page problem?
Near the top of a page:
=======================
Sometimes when I use \begin{align} or or ..... LaTeX places an absurd amount of whitespace above or possibly below, or possibly both. Too often I end up with stuff like
\ \\[-36pt]
to pull the following lines up and close a whitespace gap. Of course, a page break in an unexpected place can play havoc with such a solution. I'm wondering if there is a better way!
Don Green Dragon
fergdc at Shaw.ca | 2022-10-03 08:31:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9360295534133911, "perplexity": 2153.1164792508357}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337404.30/warc/CC-MAIN-20221003070342-20221003100342-00177.warc.gz"} |
https://codereview.stackexchange.com/questions/159705/real-time-udp-packet-decoder | # Real time UDP packet decoder
I have basic C++ lectures at a university and I don't really know how I can optimize this line of code for a real-time application (I call this function every 5 ms for UDP packet decoding).
• How can I improve its efficiency (in time processing rather than memory? (declare variable in the UDPmanager class, use static variable, ... ?)
• I am using openFrameworks for doing this project. Would it be better to use core C++ functions?
void UDPmanager::readPacket() {
string message=udpMessage;
static int count = 0;
if(message!="")
{
vector<string> strSensor = ofSplitString(message,"[/c]");
vector<string> strQoS = ofSplitString(strSensor[0],"|");
{
sleep(1);
return; // No need to use this packet. (= means it was duplicated in the network, < means it doesn't arrived in order)
}
strSensor.erase(strSensor.begin());
cout << "loss : " << packetLoss << endl;
cout << "counter : " << count++ << endl;
if (strSensor.size()!=sensors.size())
{
ofLog() << "The number of sensors changed in the system !" << endl;
}
for(unsigned int j =0 ; j<strSensor.size();j++)
// For each sensors start decoding
{
vector<string> strData = ofSplitString(strSensor[j],"[/p]");
for(size_t i=0; i<strData.size() ; ++i)
{
static vector<string> value = ofSplitString(strData[i],"|");
if (value[0] == "[/i]") // Duplicate initialization frame
{
break;
}
if (value[0] == "POS")
{
}
else
{
if(value[0] == "TOUCH")
{
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].touch = (atoi(value[i+1].c_str()))==1;
}
}
else if(value[0] == "TTHS")
{
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].tths = atoi(value[i+1].c_str());
}
}
else if(value[0] == "RTHS")
{
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].rths = atoi(value[i+1].c_str());
}
}
else if(value[0] == "FDAT")
{
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].fdat = atoi(value[i+1].c_str());
}
}
else if(value[0] == "BVAL")
{
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].bval = atoi(value[i+1].c_str());
}
}
else if(value[0] == "DIFF")
{
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].diff = atoi(value[i+1].c_str());
}
}
else {
ofLogError() << "Unrecognized key: " << value[0];
}
}
}
}
}
}
• Have you instrumented the code to find out where most time is being spent (i.e. stackoverflow.com/q/9608949/255803)? Which platform is this going to run on? What are the "std::cout << ..." statements doing on the realtime (embedded) platform? – AVH Apr 3 '17 at 12:42
• @Darhuuk I will look for that thank you ! For now it is on MacOS. Those cout are for debug reasons – Henri Koch Apr 3 '17 at 12:48
• The reason I asked about the platform is because it can have a large impact on how to optimize, e.g.: speed of multiplications, speed of loading from fixed address vs loading from a register-stored address + register-stored offset (compare PIC16 vs Cortex-M), ... Of course, the compiler would hopefully help you out in those areas, but it helps to know what's problematic in order to guide the compiler in the right direction. – AVH Apr 3 '17 at 13:13
• Well a simple optimisation would be to use std::string_view instead of std::string when splitting message. You do this so often and you probably allocate each time memory. This doesn't look like embedded at all. – Maikel Apr 3 '17 at 14:01
• You say "real time", but do you really mean it? Why is there a sleep(1) inside a real-time function? – 200_success Apr 3 '17 at 14:14
Welcome to CodeReview. If you are going to permanently keep your debug code around then you might want to you might want to embed it within ifdefs such as
#ifdef DEBUG
debug code ...
#endif
Generally, here in CodeReview, we frown on still having debug code within the source code.
Your debug code could be optimized by using "\n" in the first 2 print statements, std::endl performs a flush operation on the output stream (Context switching is not good in code that needs to perform well).
Prefer std::cout Over using std;
If you are coding professionally you should get out of the habit of using the "using std;" statement. The code will more clearly define where cout and other functions are coming from (std::atoi, std::endl). As you start using namespaces in your code it is better to identify where each function comes from because there may be function name collisions from different namespaces. The function cout you may override within your own classes.
Profile the Code
Use profiling tools to identify where the code is spending the most time. This StackOverflow question discusses a profiling tool that works on Linux and Unix (Mac OS X). Breaking the code and following the Single Responsibility Principle may help with profiling.
Clean Up the Indentation
I know that the indentation problems I see in the code may be cut and paste errors but it would make the code easier to read if it was all indented the same way.
An example of indentation problems, most of the following lines should start in the same column.
vector<string> strSensor = ofSplitString(message,"[/c]");
vector<string> strQoS = ofSplitString(strSensor[0],"|");
{
sleep(1);
return; // No need to use this packet. (= means it was duplicated in the network, < means it doesn't arrived in order)
}
Missing Code
Reviewing the code would be easier if more of the UDPmanager Class was included, for instance the packetLoss variable is not defined within the scope of the code.
The function ofSplitString is obviously a member of the UDPmanager, but we can't see what it does and the code calls it 4 times (2 times in the loop), that function may be part of the performance issue. It should definitely be part of the optimization.
First Optimization
If the message is empty why not just return from readPacket() immediately?
if (message=="") {
return
}
This would reduce the indentation in the rest of the code by one level.
Second Optimization
Reduce the number of string comparisons, in the lucky case of "POS" the code does only one string comparison, in the most unlucky case of "DIFF" the code performs 7 string comparisons.
One way to reduce the string comparisons in this code is to have another function that converts the string to an enum or integer constant. A function is not completely necessary see converting strings to enums. I suggested a separate function because the code is already too complex/long for one function, consider the Single Responsibility Principle.
If the code utilized an enum or symbolic constant, then the series of if then else if statements could be reduced to a switch/case statement:
fieldToUpdate = convertStringToEnum(value[0]);
switch (fieldToUpdate) {
case POS:
break;
case TOUCH:
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].touch = (atoi(value[i+1].c_str()))==1;
}
break;
case TTHS:
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].tths = atoi(value[i+1].c_str());
}
case RTHS:
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].rths = atoi(value[i+1].c_str());
}
break;
case FDAT:
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].fdat = atoi(value[i+1].c_str());
}
break;
case BVAL:
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].bval = atoi(value[i+1].c_str());
}
break;
case DIFF:
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].diff = atoi(value[i+1].c_str());
}
break;
default:
ofLogError() << "Unrecognized key: " << value[0];
break;
}
Magic Numbers: Use Symbolic Constants
There are 3 uses of numbers that aren't completely clear what they are, In the third line of the function 100000 is passed into udpConnection.Receive() as the second argument. In the "POS" code section the code divides by 10000.0 twice, what does this represent. Us symbolic constants that identify what the numbers actually mean. This allows you or someone else to come back to the code in 3 or more years and know exactly what is going on.
Use Iterators
I see indices used in a number of places, you might get performance enhancements by using iterators rather than indices.
Example of problem code
for(unsigned int i=0;i<sensors[j].data.size();i++)
{
sensors[j].data[i].tths = atoi(value[i+1].c_str());
}
Instead of value[i+1] use an iterator that points to the current value.
Use atof Instead of atoi In the following code it would be better to use std::atof instead of std::atoi
if (value[0] == "POS")
{ | 2021-06-13 20:29:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24170711636543274, "perplexity": 4019.8899960154417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487610841.7/warc/CC-MAIN-20210613192529-20210613222529-00056.warc.gz"} |
http://weblib.cern.ch/collection/Preprints?ln=de&as=1 | # Preprints
Letzte Einträge:
2018-12-18
10:36
Beam Instabilities in Circular Particle Accelerators / Metral, Elias (CERN) The theory of impedance-induced bunched-beam coherent instabilities is reviewed following Laclare's formalism, adding the effect of an electronic damper in the transverse plane. [...] CERN-ACC-SLIDES-2018-0003. - 2018. - 164 p.
2018-12-18
04:03
The Compact Linear $e^+e^-$ Collider (CLIC) -- 2018 Summary Report / CLICdp Collaboration The Compact Linear Collider (CLIC) is a TeV-scale high-luminosity linear $e^+e^-$ collider under development at CERN. [...] arXiv:1812.06018 ; CERN-2018-005-M. - Fulltext
2018-12-18
04:03
All-Sky Measurement of the Anisotropy of Cosmic Rays at 10 TeV and Mapping of the Local Interstellar Magnetic Field / HAWC Collaboration We present the first full-sky analysis of the cosmic ray arrival direction distribution with data collected by the HAWC and IceCube observatories in the Northern and Southern hemispheres at the same median primary particle energy of 10 TeV. [...] arXiv:1812.05682. - Fulltext
2018-12-18
04:03
Top quark properties / Van Mulders, Petra The multi-purpose experiments at CERN's Large Hadron Collider have a very rich programme in top quark physics. [...] arXiv:1812.05819. - 5 p. Fulltext
2018-12-18
04:03
Polarized backgrounds of relic gravitons / Giovannini, Massimo The polarizations of the tensor modes of the geometry evolving in cosmological backgrounds are treated as the components of a bispinor whose dynamics follows from an appropriate gauge-invariant action. [...] CERN-TH-2018-215 ; arXiv:1812.05635. - 11 p. Fulltext
2018-12-18
04:02
Neutrino flavour as a test of the explosion mechanism of core-collapse supernovae / Bar, Nitsan (Weizmann Inst.) ; Blum, Kfir (CERN ; Weizmann Inst.) ; D'Amico, Guido (CERN ; Stanford U.) We study the ratio of neutrino-proton elastic scattering to inverse beta decay event counts, measurable in a scintillation detector like JUNO, as a key observable for identifying the explosion mechanism of a galactic core-collapse supernova. [...] arXiv:1811.11178. - 28 p. Fulltext
2018-12-17
11:04
Search for Vector Like Quarks by CMS / Beauceron, Stephanie (Lyon, IPN) /CMS Collaboration We present results of searches for massive vector-like top and bottom quark partners using proton-proton collision data collected with the CMS detector at the CERN LHC at a center-of-mass energy of 13 TeV. Single and pair production of vector-like quarks are studied, with decays into a variety of final states, containing top and bottom quarks, electroweak gauge and Higgs bosons. [...] CMS-CR-2018-378.- Geneva : CERN, 2018 - 5 p. Fulltext: PDF; In : 25th Rencontres du Vietnam, Quy Nhon, Vietnam, 5 - 11 Aug 2018
2018-12-16
19:20
Measurement and interpretation of differential cross sections for Higgs boson production at $\sqrt{s}=$ 13 TeV Differential Higgs boson (H) production cross sections are sensitive probes for physics beyond the standard model. [...] CERN-EP-2018-304 ; CMS-HIG-17-028-003. - 2018. Additional information for the analysis - CMS AuthorList - Fulltext
2018-12-16
17:33
Search for a heavy resonance decaying to a top quark and a vector-like top quark in the lepton+jets final state in pp collisions at $\sqrt{s} =$ 13 TeV A search is presented for a heavy spin-1 resonance Z' decaying to a top quark and a vector-like top quark partner T in the lepton+jets final state. [...] CERN-EP-2018-313 ; CMS-B2G-17-015-003. - 2018. Additional information for the analysis - CMS AuthorList - Fulltext
2018-12-16
02:03
Inclusive search for supersymmetry in pp collisions at $\sqrt{s} =$ 13 TeV using razor variables and boosted object identification in zero and one lepton final states An inclusive search for supersymmetry (SUSY) using the razor variables is performed using a data sample of proton-proton collisions corresponding to an integrated luminosity of 35.9 fb$^{-1}$ , collected with the CMS experiment in 2016 at a center-of-mass energy of $\sqrt{s} =$ 13 TeV. [...] CERN-EP-2018-307 ; CMS-SUS-16-017-003. - 2018. Additional information for the analysis - CMS AuthorList - Fulltext
Suchen Sie auch nach: | 2018-12-18 13:25:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9142314791679382, "perplexity": 4514.969677524296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829399.59/warc/CC-MAIN-20181218123521-20181218145521-00209.warc.gz"} |
https://www.techwhiff.com/issue/lived-in-the-andes-mountains-capital-city-was-cuzco--281701 | # Lived in the Andes Mountains Capital city was Cuzco Practiced polytheistic religion What civilization is described by the characteristics listed above? Question 1 options: Inca Aztec Olmec Maya
###### Question:
Lived in the Andes Mountains
Capital city was Cuzco
Practiced polytheistic religion
What civilization is described by the characteristics listed above?
Question 1 options:
Inca
Aztec
Olmec
Maya
### Main Store Road in Fort Myers was damaged in a recent storm. Which level of government is responsible for repairing this road? A.state B.local C.federal D.national PLZ ANSWER NOW!!!!!!!!!!!!!!!!!!!!!
Main Store Road in Fort Myers was damaged in a recent storm. Which level of government is responsible for repairing this road? A.state B.local C.federal D.national PLZ ANSWER NOW!!!!!!!!!!!!!!!!!!!!!...
### 25.00 mL of an unknown concentration of acetic acid is titrated with 40.62 ml of the standard base (use molarity calculated from Q3 above). What is the molarity of this acid?
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### Choose the sentence that BEST summarizes the invitation. A) Parents can attend Prehistoric Giants from 8 A.M. until 3 P.M. B) Parents should report to the school gym between 9 A.M. and 3 P.M. to see the exhibit. C) Mrs. Casey's students are excited to share their projects because it took very hard work to do them. D) Mrs. Casey's students' parents can view dinosaur projects in her classroom on April 6th, between 9 A.M. and 3 P.M.
Choose the sentence that BEST summarizes the invitation. A) Parents can attend Prehistoric Giants from 8 A.M. until 3 P.M. B) Parents should report to the school gym between 9 A.M. and 3 P.M. to see the exhibit. C) Mrs. Casey's students are excited to share their projects because it took very hard...
### | Lord Montague adds that "many a morning hath he there been seen, with ____________ augmenting the fresh morning's dew, adding to clouds more clouds with his ________________________" (1.1.134-136).
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### I need help with quiz
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### Write briefly the third stage in the evolution of money?
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### Which three angles would be supplementary? Select ALL that apply. a 30, 60, 60 b 45, 45, 90 c 72, 28, 80 d 25, 75, 100 e 180, 180, 180
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### What is the best description of organisms in the precambrian era?A. Land plants and animalsB. Dinosaurs and primitive mammalsC. Single-celled organisms and soft, boneless animals.D. Ferns and flying reptiles
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### A car slowing down ata red light is what kind of acceleration
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### 90% of people marry there 7th grade love. since u have read this, u will be told good news tonight. if u don't pass this on nine comments your worst week starts now this isn't fake. apparently if u copy and paste this on ten comments in the next ten minutes you will have the best day of your life tomorrow. you will either get kissed or asked out in the next 53 minutes someone will say i love you
90% of people marry there 7th grade love. since u have read this, u will be told good news tonight. if u don't pass this on nine comments your worst week starts now this isn't fake. apparently if u copy and paste this on ten comments in the next ten minutes you will have the best day of your life to...
### Arccos 2x + arccos x = TT/2How to solve find x ?
arccos 2x + arccos x = TT/2How to solve find x ?... | 2023-03-31 10:22:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2179027646780014, "perplexity": 7468.92968616983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949598.87/warc/CC-MAIN-20230331082653-20230331112653-00519.warc.gz"} |
https://www.bionicturtle.com/forum/threads/p1-t3-200-market-and-limit-orders.5979/ | What's new
# P1.T3.200. Market and limit orders
#### David Harper CFA FRM
##### David Harper CFA FRM
Staff member
Subscriber
Questions:
200.1. You want to purchase 5,000 shares of a stock with the following real-time quote: bid $29.94 - ask$30.08 with bid-ask sizes of 3,000 shares (bid) and 10,000 shares (ask). The previous price close was $29.99. Which of the following is the most likely execution of your market buy order? a. You pay$29.94 for all 5,000 shares
b. You pay $29.99 for all 5,000 shares c. You pay$29.94 for 3,000 shares but you pay $30.08 for the remaining 2,000 shares d. You pay$30.08 for all 5,000 shares
200.2. You own 10,000 shares of a stock that is currently quoted with a bid price of $19.95 and an ask (offer) price of$20.03, with bid-ask sizes much larger than your holdings. You enter a limit sell order at $20.00. Which of the following executions of your LIMIT order is most likely? a. Your sale will be immediately executed at$20.00
b. Your sale will be immediately executed at $19.95 c. Your sale will be immediately executed at$20.03
d. Your sale may not be executed (or a partial fill may be executed)
200.3. Joe Smith owns 100,000 shares of ACME stock which currently trades at $50.00. He enters a stop-limit order to sell at$48.00 with a limit of $47.00. Which of the following is most likely? a. The order is not logical, the stop should be above the current$50.00
b. Conditional on the bid or ask falling to $48, the sell order is filled only if the sale can be executed at$47 or higher, but may not be filled
c. Conditional on the bid or ask falling to $48, the sell order will be immediately executed but the price will not be guaranteed d. The sell order immediately begins execution at$48 but ceases if the bid drops below $47 such that a partial fill is possible. Answers: #### Angelo86na ##### New Member 200.1 answer c (or the second b) 200.2 for esclusion answer d 200.3 answer b #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber @nychets & @Angelo86na: I notice you both answered (C) for 200.1. I hope my question is well stated but please note: the bid/ask is$29.94 (size 3,000) - $30.08 (size 10,000). As a buyer, the best offer by market sellers is$30.08, so I can likely get my buy order filled at \$30.08 or higher (most likely higher if i exceed the size).
Since this market buy order is within the size, my answer is (D). Please let me know if you think the question is imprecise?
Thanks!
#### Angelo86na
##### New Member
Sorry David, is my fault. Your question is precise, I wore the clothes of the dealer answering the question.
#### nychets
##### New Member
Doh! Question is fine, I can't imagine what I did there. I trade all day long, thank heaven I don't err like that every day, lol. I answered too quickly and carelessly. It's an excellent reminder for me to read the easy questions carefully on the Level II exam in June 2013!
#### David Harper CFA FRM
##### David Harper CFA FRM
Staff member
Subscriber
@nychets - thank you nychets ... phew ... I don't trade quite enough to be facile with the bid/ask so I'm glad it looks okay, thanks,
#### Niravparikh281
##### New Member
Hi..
Why not D over B for 3rd question. Investors can input any bid or ask they want but a trade must get executed at 48 for the stop-limit order to kick in. Not sure what I am missing | 2019-12-15 05:15:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3018537759780884, "perplexity": 5684.906945149986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541301598.62/warc/CC-MAIN-20191215042926-20191215070926-00191.warc.gz"} |
https://www.esaral.com/category/information/iit-jee/jee-mains/ | Difference Between Potentiometer and Voltmeter – Current Electricity || Class 12, JEE & NEET
The Potentiometer and the Voltmeter both are the voltage measuring devices. The main Difference Between Potentiometer and Voltmeteris that the potentiometer measures the emf of the circuit whereas voltmeter measures the end terminal voltage of the circuit. The other differences between the Potentiometer and the voltmeter are explained below in the comparison chart.
## Standardization of Potentiometer
The process of determination of potential gradient on wire of potentiometer is known as standardisation of potentiometer. A standard cell is one whose emf remains constant. Cadmium cell with emf 1.0186 V at $20^{\circ}$C is used as a standard cell. In laboratory a Daniel cell with emf 1.08 V is usually used as a standard cell.
If $\ell_{0}$ is the balancing length for standard emf $E_{0}$ then potential gradient $x=\frac{E_{0}}{l_{0}}$
### Key Differences:
The following are the key differences between Potentiometer and voltmeter.
1. The Potentiometer is an instrument used for measuring the emf, whereas the voltmeter is a type of meter which measures the terminal voltage of the circuit.
2. The Potentiometer accurately measures the potential difference because of zero internal resistance. Whereas, the voltmeter has a high internal resistance which causes the error in measurement. Thus the voltmeter approximately measures the voltage.
3. The sensitivity of the Potentiometer is very high, i.e. it can measure small potential differences between the two points. The voltmeter has low sensitivity.
4. The Potentiometer uses the null deflection type instrument whereas the voltmeter uses the deflection type instrument.
5. The Potentiometer has infinite internal resistance, whereas the Potentiometer has high measurable resistance.
Watch out the Video: Applications of Potentiometer & its Construction by Saransh Sir.
### Conclusion
The Potentiometer and voltmeter both measures the emf in volts. The Potentiometer is used in a circuit where the accurate value of voltage is required. For approximate calculation, the voltmeter is used.
Physics Revision Series by Saransh Sir (AIR 41 in IIT-JEE)
JEE Main April 2020 Application Form (7 February, 2020) | Important Details Check Here
JEE Main April 2020 Application form filling process is starting from 7th February, 2020. The last date to apply online for JEE Main April 2020 is March 6. JEE Main April 2020 Exam will be held from April 5 to April 11, 2020.
The application process of JEE Main 2020 includes registration, filling the application form, uploading scanned images, payment of application fee and taking the printout of the filled-in form. For JEE Main 2020 registration, candidates must check the eligibility criteria as prescribed by the NTA. Candidates who have missed the JEE Main 2020 January exam can appear for the JEE Main 2020 April session.
Joint Entrance Examination is organized by National Testing Agency to offer admission into courses like B.Tech / B.E., B.Arch, B.Plan. JEE Main is a computer based test that takes place at national level. The duration of the exam is 3 hours. It consists of multiple-choice questions and numerical value type questions. There is negative marking in the case of multiple choice questions. After qualifying this exam, the candidates can take admission in IIITs / CFTIs / NITs or can sit for JEE Advance exam.
JEE Main April 2020 Exam Application Form
### List of Documents Required for Filling Application Form
Candidate needs some documents to fill and submit the application form. Check the list of required documents here:
• scanned image of photograph
• scanned image of signature
• a valid e-mail id
• valid and active mobile number
• past academic mark sheets and certificates
The candidates have to upload Photo and Signatue in the application form of JEE Main 2020 and as per the given specifications:
### JEE Main April 2020 Important Points
Registration Process will be different for those who applied in January and for those haven’t applied in January JEE Main Exam.
If you have applied in JEE Main January 2020: No need to register. You can directly login with the same credentials and fill the application form for JEE Main April attempt.
If you haven’t applied in January 2020 JEE Main Exam: Then you have to fill the application form. First you need to register and then fill the correct details in the form, upload images in the application form & pay the application fee.
### JEE Main April 2020 Exam Pattern
As per the eligibility criteria for, a few changes in the pattern of the question paper(s) and number of question(s) for B.E./B.Tech has been approved by the JEE Apex Board (JAB) for the conduct of JEE (Main)-2020 Examination.
Paper Subject with Number of Questions Type of Questions TIMING OF THE EXAMINATION(IST) B.tech Mathematics – 25(20+5) Physics – 25(20+5) Chemistry – 25(20+5) 20 Objective Type – Multiple Choice Questions (MCQs) & 5 Questions with answer as numerical value, with equal weightage to Mathematics, Physics & Chemistry 1st Shift: 09:30 a.m. to 12:30 p.m. 2nd Shift: 02:30 p.m. to 05:30 p.m.
### JEE Main 2020 Dates
Application form availability February 7, 2020
Last date to apply March 6, 2020
Last date to upload images and pay application fee March 8, 2020
Issue of admit card March 16, 2020
Exam date April 5 to April 11, 2020
Release date of answer key and recorded responses Second week of April, 2020
Release of answer key Last week of April, 2020: Paper 1
Second week of May, 2020: Paper 2 & 3
Declaration of result April 30, 2020
Announcement of NTA score Third week of May, 2020
### Eligibility Criteria
There is no age limit for the candidates to appear in JEE Main 2020 Examination. The candidates who have passed their 12th Examination in 2018, 2019 and appearing in 2020 are eligible to JEE Main Examination 2020.
Those candidates who cleared the class 12 exam in 2017 or before 2017 are not eligible to appear for JEE Main 2020 exam.
#### Educational Qualification
Candidates must have at least 5 subjects in class 12 exam or equivalent exam. Where Math, Physics and Chemistry are the essential subjects.
### INFORMATION BULLETIN for JEE MAIN April 2020
NTA (National Testing Agency) publishes Information Bulletin for JEE MAIN April 2020 in PDF Form. Candidates can download JEE Main 2020 brochure in English and Hindi are available.
The JEE Main 2020 information brochure is that document which provides complete details of the Joint Entrance Examination (Main), officially provided by the exam conducting body NTA. It is recommended to read it first and then apply for the exam.
JEE Main April 2020 Information Bulletin in Hindi JEE Main April 2020 Information Bulletin in English DOWNLOAD PDF
### JEE Main 2020 Detailed Syllabus
If you want to improve %ile for JEE Main April 2020, then Join BOUNCE BACK CRASH COURSE FOR JEE 2020. Register Here!!!
FREE Revision Series For JEE 2020 | Quick Revision Videos
👉Physics Revision Series
👉Chemistry Revision Series
👉Mathematics Revision Series
JEE Main April 2020 Exam | Important Dates | Syllabus | Exam Pattern & Registration Process
JEE Main April 2020 Dates of Exam have been changed again by the National Testing Agency (NTA). JEE Main April 2020 Exam will be held from April 5 to April 11, 2020. Earlier, the Joint Entrance Examination Main April Attempt Exam was supposed to be held from 3rd – 9th April, 2020
According to the official notification, the online JEE Main 2020 application form for April Exam will release on February 7, 2020. Eligible candidates will be able to fill and submit their online application form until March 6, 2020. The scanned images can be uploaded and payment of the application fee can be done till March 8, 2020.
After the JEE Main 2020 April Attempt, the ranks of the qualified candidates will be released taking into consideration better of the both attempts of JEE Main 2020 scores of all candidates appeared in January 2020 and April 2020 examinations and will be made available on the official website of JEE Main.
### JEE Main 2020 Dates
Application form availability February 7, 2020
Last date to apply March 6, 2020
Last date to upload images and pay application fee March 8, 2020
Issue of admit card March 16, 2020
Exam date April 5 to April 11, 2020
Release date of answer key and recorded responses Second week of April, 2020
Release of answer key Last week of April, 2020: Paper 1
Second week of May, 2020: Paper 2 & 3
Declaration of result April 30, 2020
Announcement of NTA score Third week of May, 2020
NTA has declared JEE Main 2020 Result for January Session on January 17, 2020. Results for JEE Main April 2020 will be announced on April 30, 2020. Candidates can access their results by entering their date of birth and roll number, along with the declaration of the JEE Main Result, NTA has released the scorecard in online mode.
### JEE Main 2020 Exam Pattern
As per the eligibility criteria for, a few changes in the pattern of the question paper(s) and number of question(s) for B.E./B.Tech has been approved by the JEE Apex Board (JAB) for the conduct of JEE (Main)-2020 Examination.
Paper Subject with Number of Questions Type of Questions TIMING OF THE EXAMINATION(IST) B.tech Mathematics – 25(20+5) Physics – 25(20+5) Chemistry – 25(20+5) 20 Objective Type – Multiple Choice Questions (MCQs) & 5 Questions with answer as numerical value, with equal weightage to Mathematics, Physics & Chemistry 1st Shift: 09:30 a.m. to 12:30 p.m. 2nd Shift: 02:30 p.m. to 05:30 p.m.
There is no age limit for the candidates to appear in JEE Main 2020 Examination. The candidates who have passed their 12th Examination in 2018, 2019 and appearing in 2020 are eligible to JEE Main Examination 2020.
Those candidates who cleared the class 12 exam in 2017 or before 2017 are not eligible to appear for JEE Main 2020 exam.
#### Educational Qualification
Candidates must have at least 5 subjects in class 12 exam or equivalent exam. Where Math, Physics and Chemistry are the essential subjects.
### FREE Revision Series For JEE | Quick Revision Videos
👉Physics Revision Series
👉Chemistry Revision Series
👉Mathematics Revision Series
JEE MAIN 2020 Question Paper PDF Download | All Shifts (7th, 8th & 9th January)
NTA has successfully conducted JEE Main Jan 2020 exam in all the test centers and we understand that students are waiting for JEE MAIN 2020 Question Paper PDF format. Based on the reviews by the students, it is concluded that the exam is comparatively easy than the previous year. Furthermore, the questions asked in the exam were more conceptual based. So Question Papers PDF of all shifts are available here to download.
JEE Main 2020 Question Paper – Candidates can download JEE Main question paper for January 07, 08, 09, 2020, for Shift 1 and Shift 2 from this page.
The candidates can challenge the answer keys online. The window to raise objections will be available for a week. In case, the NTA finds the objections raised are incorrect, they will publish the result of JEE Main. The tentative result declaration date is by January 31.
### Candidates may check their JEE Main question papers from Here!!!
JEE Main 2020 Exam | January Attempt | Students’ Reactions and Reviews
## JEE MAIN 2020 Students Reactions | 7th January (SHIFT-1)
JEE MAIN, one of the biggest examination in the country was scheduled today. The difficulty level and selection ratio are the well-known aspects of this exam but the number of candidates that appear every year, is also the factor that makes it so special. Every year more than ten lakhs of aspirants apply and compete for just only 30-35 thousand seats (approximately) of National Institute of Technologies and IIITs. Today on 7th January 2020, it was the day, when JEE Main 2020 Exam was held in the morning shift from 9:30-12:30 across India.
Many of you are also appeared or going to appear in near future and want to know the overall level of today’s exam. So, dedicated to students and in order to help them, eSaral is again here, providing you the details about today’s exam. In this video, our team members have gathered the information about the questions, paper pattern and the difficulty level of the exam. We went to some of the centers in KOTA city, yes!! The coaching city.
Get to know about the students reviews and reactions regarding JEE Main 2020 Exam:
So, watch the video to know how they feel about the exam. We asked them regarding the easy to tough subjects based on the questions and topics, also the marking scheme of the numerical type questions and the ratio of 11th and 12th standards topics in the exam.
So if you have these kind of doubts or want to know about the detail watch the video and based on the conversation with the aspirants we developed a PDF too. You can download the same to know in detail.
## JEE MAIN 2020 Students Reactions | 7th January (SHIFT-2)
So, here are the students reviews on the Second shift of 7th January JEE Main Exam. As per the students reviews it can be concluded that both shifts of the day comprised of easy to moderate level questions. Watch out the video and comment your review of JEE Main Exam
## JEE MAIN 2020 Students Reactions | 8th January (SHIFT-2)
JEE Main 2020 Paper Analysis | Discussion with Students
## JEE MAIN 2020 Paper Analysis for 7th January SHIFT-1
Today on 7th January, the national level competition exam JEE MAIN was held across the nation and with the completion of the exam, there are many queries in the mind of aspirants. It is natural to feel a certain kind of anxiety after the exam like how others feel, the difficulty level of the exam and the expected cut-offs. Here, at eSaral we are providing you the first detailed analysis that is available to all the students. Watch the video to now about the questions’ level (according to eSaral students) and the topics wise questions from 11th and 12th standards.
The physics faculty at eSaral, Saransh Gupta sir and chemistry faculty Prateek Gupta sir are doing the detailed analysis of the JEE MAIN QUESTION PAPER as per the reviews by the students of eSaral. In the video, the questions from each subject and the topics from where they were asked are explained using prepometer, the tool designed by the eSaral. In the discussion session the students of eSaral told about the questions and difficulty level and the question that they faced in the examination.
JEE Main 2020 Paper Analysis for January 07th exam is updated here. Students are sharing their reviews of JEE Main Exam here. Watch out the complete video till the end to know in detail!
Know the Number of questions asked per chapter:
#### Watch the Reactions and Reviews of Students outside JEE Main Exam Center
In physics, it is found that many of the questions were direct and formula based related to the memory of the candidate. There were approximately 10 questions form the class 11th syllabus out of 25 questions.
Watch out the video if you want to know about the same kind of detailed analysis for chemistry and mathematics. You will also get some idea about specific questions and their solutions that were discussed in the analysis video.
So from our analysis we found the overall exam as easy to moderate level. Only a few questions were placed in the difficult to very difficult category and it was found that the question were not much confusing.
We have developed a PDF regarding the detailed analysis. Download the file and know about the exam.
If you think that this time you could not attempt your best attempt then don’t worry we are going to start our BOUNCE BACK CRASH COURSE for the April month JEE MAIN exam. Enroll and ace the exam.
The above PDF contains weightage of number of questions per chapter.
## JEE MAIN 2020 Paper Analysis for 7th January SHIFT-2
P
Know the Number of questions asked per chapter:
## JEE MAIN 2020 Paper Analysis for 9th January SHIFT-2
JEE Main 2020 Question Paper JEE Main question paper for January 07, 08, 09, 2020, for Shift 1 and Shift 2 are given below. Download or View from here!!!
### 👉 Click to Join Free Physics Revision Series by Saransh Gupta Sir (AIR-41, IIT-Bombay)
Liquid Solution – JEE Main Previous Year Question of with Solutions
JEE Main Previous Year Question of Chemistry with Solutions are available here. Practicing JEE Main Previous Year Papers Questions of Chemistry will help all the JEE aspirants in realizing the question pattern as well as help in analyzing their weak & strong areas.
Get detailed Class 11th &12th Chemistry Notes to prepare for Boards as well as competitive exams like IIT JEE, NEET etc.
eSaral helps the students in clearing and understanding each topic in a better way. eSaral is providing complete chapter-wise notes of Class 11th and 12th both for all subjects.
Besides this, eSaral also offers NCERT Solutions, Previous year questions for JEE Main and Advance, Practice questions, Test Series for JEE Main, JEE Advanced and NEET, Important questions of Physics, Chemistry, Math, and Biology and many more.
Simulator
Previous Years AIEEE/JEE Mains Questions
Mind Map of Magnetic Effects of Current for Class 12, JEE & NEET
Magnetic Effects od Current in Class 12th comprises variety of cases with important formulae and key points. So here is the mind map to help you in remembering all the formulae and important key concepts on finger tips.
JEE MAIN 2020 Admit Card for January Attempt | Download Hall Ticket for JEE Main 2020 Exam
NTA JEE Main 2020 Admit Card for January 2020 exam has released today on official website of JEE Main. The exam will be conducted from January 6 to 11, 2020. Candidates can download their admit card by logging in using their registration number and password.
When you Click on the above link, a new login page will open just like given below:
NOTE: Candidates must preserve the JEE MAIN 2020 admit card safely as it will be required at different stages of the admission and counselling process. Applicants who do not carry admit card will not be allowed to write the exam at the test center.
• Application number.
• Password or date of birth.
JEE Main 2020 Admit Card will look like the image given below:
### What to Carry in Examination Hall
On the exam day, candidate have to carry following documents to the exam hall:
1. JEE Main 2020 hall ticket – Print it on an A4 sheet and make sure all the information should be clear and correct.
2. One passport size photograph – Along with the JEE Main admit card, candidates also have to carry one passport size photograph. This should be the same photo as the one uploaded in the form.
3. Valid Original ID Proof – As per information brochure, candidates have to carry one id proof. However, it has to be carried in original and should contain photograph of the candidate.
It is better that candidates carry the same id proof, the details of which were entered in the application form. List of valid id proofs for JEE Main 2020 is as follows:
• PAN card.
• Voter id card.
• Ration card.
• Passport.
• 12th Class admit card with photograph.
• Bank passbook with photograph.
4. PwD Certificate – Candidates who applied for scribe facility have to carry PwD Certificate with the admit card.
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http://gujarattechnocastings.com/787hbe/page.php?id=d9c172-matrix-representation-of-a-relation | Lecture 13 Matrix representation of relations EQUIVALENCE RELATION Let A be a non-empty set and R a binary relation on A. R is an equivalence relation if, and only if, R is reflexive, symmetric, and transitive. We need to consider what the cofactor matrix … This follows from the properties of logical products and sums, specifically, from the fact that the product GikHkj is 1 if and only if both Gik and Hkj are 1, and from the fact that ∑kFk is equal to 1 just in case some Fk is 1. However, with a formal definition of a matrix representation (Definition MR), and a fundamental theorem to go with it (Theorem FTMR) we can be formal about the relationship, using the idea of isomorphic vector spaces (Definition IVS). In other words, each observation is an image that is “vectorized”. \PMlinkescapephraseorder They are applied e.g. We perform extensive characterization of perti- This is the first problem of three problems about a linear recurrence relation … We rst use brute force methods for relating basis vectors in one representation in terms of another one. A relation in mathematics defines the relationship between two different sets of information. The vectorization operator ignores the spatial relationship of the pixels. \PMlinkescapephraserelational composition , In this case it is the scalar product of the ith row of G with the jth column of H. To make this statement more concrete, let us go back to the particular examples of G and H that we came in with: The formula for computing G∘H says the following: (G∘H)ij=theijthentry in the matrix representation forG∘H=the entry in theithrow and thejthcolumn ofG∘H=the scalar product of theithrow ofGwith the jthcolumn ofH=∑kGikHkj. Lecture 13 Matrix representation of relations EQUIVALENCE RELATION Let A be a non-empty set and R a binary relation on A. R is an equivalence relation if, and only if, R is reflexive, symmetric, and transitive. One of the best ways to reason out what G∘H should be is to ask oneself what its coefficient (G∘H)ij should be for each of the elementary relations i:j in turn. You have a subway system with stations {1,2,3,4,5}. More generally, if relation R satisfies I â R, then R is a reflexive relation. Representing Relations Using Matrices To represent relationRfrom setAto setBby matrixM, make a matrix withjAjrows andjBjcolumns. and By definition, induced matrix representations are obtained by assuming a given group–subgroup relation, say H ⊂ G with [36] as its left coset decomposition, and extending by means of the so-called induction procedure a given H matrix representation D(H) to an induced G matrix representation D ↑ G (G). Some of which are as follows: 1. Example. In other words, each observation is an image that is “vectorized”. \PMlinkescapephraseOrder 17.5.1 New Representation. Then the matrix representation for the linear transformation is given by the formula R is reflexive if and only if M ii = 1 for all i. Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as being arranged in a lexicographic block of the following form: 1:11:21:31:41:51:61:72:12:22:32:42:52:62:73:13:23:33:43:53:63:74:14:24:34:44:54:64:75:15:25:35:45:55:65:76:16:26:36:46:56:66:77:17:27:37:47:57:67:7. When the row-sums are added, the sum is the same as when the column-sums are added. Therefore, we can say, ‘A set of ordered pairs is defined as a rel… Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations G and H. G=[0000000000000000000000011100000000000000000000000], H=[0000000000000000010000001000000100000000000000000]. m ij = { 1, if (a,b) Є R. 0, if (a,b) Є R } Properties: A relation R is reflexive if the matrix diagonal elements are 1. If R is a binary relation between the finite indexed sets X and Y (so R ⊆ X×Y), then R can be represented by the logical matrix M whose row and column indices index the elements of X and Y, respectively, such that the entries of M are defined by: Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The inverse of the matrix representation of a complex number corresponds to the reciprocal of the complex number. O The matrix representation of the relation R is given by 10101 1 1 0 0 MR = and the digraph representation of the 0 1 1 1 0101 e 2 relation S is given as e . The other two relations, £ L y; L z ⁄ = i„h L x and £ L z; L x ⁄ = i„h L y can be calculated using similar procedures. each relation, which is useful for “simple” relations. In either case the index equaling one is dropped from denotation of the vector. (That is, \+" actually means \_" (and \ " means \^"). \PMlinkescapephrasereflect 1We have also experimented with a version of LRE that learns to generate a learned matrix representation of a relation from a learned vector representation of the relation. Representation of Types of Relations. No single sparse matrix representation is uniformly superior, and the best performing representation varies for sparse matrices with different sparsity patterns. A symmetric relation must have the same entries above and below the diagonal, that is, a symmetric matrix remains the same if we switch rows with columns. \PMlinkescapephraseRelation It is served by the R-line and the S-line. For example, 2R4 holds because 2 divides 4 without leaving a remainder, but 3R4 does not hold because when 3 divides 4 there is a remainder of 1. 2 6 6 4 1 1 1 1 3 7 7 5 Symmetric in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. G∘H=[0000000000000000000000001000000000000000000000000], Generated on Sat Feb 10 12:50:02 2018 by, http://planetmath.org/RelationComposition2, matrix representation of relation composition, MatrixRepresentationOfRelationComposition, AlgebraicRepresentationOfRelationComposition, GeometricRepresentationOfRelationComposition, GraphTheoreticRepresentationOfRelationComposition. A: If the ij th entry of M(R) is x, then the ij th entry of M(R-bar) is (x+1) mod 2. Every logical matrix a = ( a i j ) has an transpose aT = ( a j i ). We determine a linear transformation using the matrix representation. Reflexive in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. In incidence geometry, the matrix is interpreted as an incidence matrix with the rows corresponding to "points" and the columns as "blocks" (generalizing lines made of points). 1 2 The outer product of P and Q results in an m à n rectangular relation: Let h be the vector of all ones. We describe a way of learning matrix representations of objects and relationships. \PMlinkescapephraserelation Relations can be represented in many ways. . Applying the rule that determines the product of elementary relations produces the following array: Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicities count as one, and this gives the ultimate result: With an eye toward extracting a general formula for relation composition, viewed here on analogy with algebraic multiplication, let us examine what we did in multiplying the 2-adic relations G and H together to obtain their relational composite G∘H. Consider the table of group-like structures, where "unneeded" can be denoted 0, and "required" denoted by 1, forming a logical matrix R. To calculate elements of R RT it is necessary to use the logical inner product of pairs of logical vectors in rows of this matrix. i Wikimedia Commons has media related to Binary matrix. As noted, many Scikit-learn algorithms accept scipy.sparse matrices of shape [num_samples, num_features] is place of Numpy arrays, so there is no pressing requirement to transform them back to standard Numpy representation at this point. We need to consider what the cofactor matrix … Mathematical structure. Definition: Let be a finite … Then U has a partial order given by. (a) A matrix representation of a linear transformation Let $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$, and $\mathbf{e}_4$ be the standard 4-dimensional unit basis vectors for $\R^4$. Proposition 1.6 in Design Theory[5] says that the sum of point degrees equals the sum of block degrees. j Suppose a is a logical matrix with no columns or rows identically zero. \PMlinkescapephraseRepresentation (1960) "Matrices of Zeros and Ones". \PMlinkescapephrasesimple \PMlinkescapephraseSimple. are two logical vectors. This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of logical arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction. Ryser, H.J. Find the matrix of L with respect to the basis E1 = 1 0 0 0 , E2 = 0 1 0 0 , E3 = 0 0 1 0 , E4 = 0 0 0 1 . By definition, ML is a 4×4 matrix whose columns are coordinates of the matrices L(E1),L(E2),L(E3),L(E4) with respect to the basis E1,E2,E3,E4. Relations: Relations on Sets, Reflexivity, Symmetry, and Transitivity, Equivalence Relations, Partial Order Relations Graphs and Trees: Definitions and Basic Properties, Trails, Paths, and Circuits, Matrix Representations of Graphs, Isomorphism’s of Graphs, Trees, Rooted Trees, Isomorphism’s of Graphs, Spanning trees and shortest paths. The defining property for the gamma matrices to generate a Clifford algebra is the anticommutation relation {,} = + =,where {,} is the anticommutator, is the Minkowski metric with signature (+ − − −), and is the 4 × 4 identity matrix.. ( These facts, however, are not sufficient to rewrite the expression as a complex number identity. ( . In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form: A moment’s thought will tell us that (G∘H)ij=1 if and only if there is an element k in X such that Gik=1 and Hkj=1. composition In a similar way, for a system of three equations in three variables, This defining property is more fundamental than the numerical values used in the specific representation of the gamma matrices. In other words, every 0 … We list the elements of … \PMlinkescapephraserepresentation . We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as logical matrices, and also to grasp the analogy between relational composition and ordinary matrix multiplication as it appears in linear algebra. , Matrix Representations - Changing Bases 1 State Vectors The main goal is to represent states and operators in di erent basis. Consequently there are 0's in R RT and it fails to be a universal relation. "[5] Such a structure is a block design. \PMlinkescapephraseRelational composition ... be the linear transformation from the $3$-dimensional vector space $\R^3$ to $\R^3$ itself satisfying the following relations. To fully characterize the spatial relationship, a tensor can be used in lieu of matrix. The relation R can be represented by the matrix M R = [m ij], where m ij = (1 if (a i;b j) 2R 0 if (a i;b j) 62R Reflexive in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. Taking the scalar product, in a logical way, of the fourth row of G with the fourth column of H produces the sole non-zero entry for the matrix of G∘H. In this if a element is present then it is represented by 1 else it is represented by 0. The relations G and H may then be regarded as logical sums of the following forms: The notation ∑ij indicates a logical sum over the collection of elementary relations i:j, while the factors Gij and Hij are values in the boolean domain ={0,1} that are known as the coefficients of the relations G and H, respectively, with regard to the corresponding elementary relations i:j. Between the resultant sparse matrix representations, specifically the difference in location of gamma... In R RT and it fails to be re exive, 8a ( a j i ). [ ]. Heights of ( 0, then the rows are orthogonal is thus.. Indexed sets for more detail from Y to Z no columns or rows zero... Any level and professionals in related fields this too makes it possible to treat relations as because... Gpu architecture, sparse matrix representation, multiple observations are encoded using a zero-one matrix let R a! Exchange is a matrix representation of a relation Design second solution uses a linear transformation ; a 2! Then the m à n rectangular relation: let h be the vector is question. Complement of a pair of finite sets.. matrix representation of a relation is,! Obtained by swapping all zeros and ones '' number corresponds to a binary relation between a pair 2-adic. Too makes it possible to treat relations as ob-jects because they both have vector representations orthogonal to loop small... The equivalent transformations using matrix operations ignores the spatial relationship, a can. Only if m = 1, 2, of objects and relationships degrees equals the of... Show the equivalent transformations using matrix operations ) has an transpose at = ( a i j ) an! Vectors in one representation in terms of another one a reflexive relation may notice that form! An transpose at = ( a j i ), i = 1, 2, applied component-wise *! To the reciprocal of the gamma matrices way to think about RoS: ( not a,... Matrices of zeroes and ones '' the following set is the set of pairs for which the relation R.! Is usually called a scalar product combination and linearity of linear transformation which is useful for matrix representation of a relation simple ”.! Us recall the rule for finding the relational composition of a pair of finite sets.. matrix,. Case the index equaling one is dropped from denotation of the complex.! ( Q j ) is a block Design n equals one, then R is reflexive if only... Gamma matrices instance is the set of ordered pairs, matrix and digraphs: ordered,... Sparse dataset main diagonal Widths and heights of ( 0, ). On the main diagonal from x to Y, and the S-line by way learning.. [ 2 matrix representation of a relation be computed in expected time O ( n2 ) [. When the row-sums are added, the sum of block degrees a subway system with stations { 1,2,3,4,5.... Location of the matrix representation is uniformly superior, and is thus finite to the reciprocal the... Complement of a complex number corresponds to the reciprocal of the pixels Ryser ( 1961 ) matrices of and. Either case the index equaling one is dropped from denotation of the complex number ” relations complement a. Is how to think about RoS: ( not a definition, just a way of disentangling this formula one! At any level and professionals in related fields of perti- let m R and S..! A row-sum is called its point degree and a column-sum is the set of ordered pairs, matrix and:! P i ). [ 2 ] are not sufficient to rewrite expression... ; with this matrix representation using the matrix index equaling one is dropped from denotation of the complex number 2! 4 ] a particular instance is the universal relation the numerical values used in lieu of matrix ordered! Makes it possible to treat relations as ob-jects because they both have vector representations by way of matrix... Learning matrix representations, specifically the difference in location of the relations R and m be its zero-one let! The operations and & or between two matrices applied component-wise of pairs for which relation. With different sparsity patterns matrix and digraphs: ordered pairs of x and Y are to., we study the inter-relation between GPU architecture, sparse matrix representation is uniformly superior, is! Re exivity { for R to be a relation from Y to Z we a! Gpu architecture, sparse matrix representations of the gamma matrices matrix representation of a relation instance is the set of ordered pairs – digraphs... Nite sets can be used in the matrix representation of the same element values represent.: double * a ; a ) 2 R what are advantages of matrix paper. Y to Z a column vector gamma matrices is 0, then the are... Boolean algebra with the operations and & or between two different sets of.! Using matrix operations Y, and is thus finite the vector of all logical m à n matrices orderings x... Will show the equivalent transformations using matrix operations other words, each observation is image... A number of more restricted special forms and digraphs: ordered pairs – of matrices of zeroes ones... 1, 2, the rows are orthogonal m or n equals one, then R is a vector! Brute force methods for relating basis vectors in one representation in terms of another one about RoS: not. ( 1961 ) Traces of matrices of zeros and ones '' a universal relation h.... The relations R and m S denote respectively the matrix representation index equaling one is dropped denotation... Possible to treat relations as ob-jects because they both have vector representations a number of more restricted forms! Or antisymmetric, from the matrix representation of a logical vector of x and Y are represented using ordered –. Superior, and if n = 1, 2,, we study the between., the sum of block degrees and \ means \^ '' ). [ 2 ] with... Note the differences between the resultant sparse matrix representation of the same element values a row,.: double * a ; with this matrix representation of the pixels a = ( i... The universal relation h hT when the row-sums are added they arise in a matrix! The following set is the block degree ; all matrices are with respect to these orderings 1961 ) matrices... The spatial relationship, a tensor can be represented using a zero-one matrix let R be a relation between pair! A subway system with stations { 1,2,3,4,5 } reflexive relation they both have vector.... A column-sum is the set of all ones R-line and the S-line extensive of... Relation h hT R to be re exive, symmetric or transitive just by looking at the same values. Relational composition of a pair of finite sets.. matrix representation a particular instance is the set of for... Reflexive in a variety of representations and have a number of distinct m-by-n binary matrices is equal to,. Served by the R-line and the sparse dataset its zero-one matrix let R be relation... The matrix ( Mi j ), j = 1 it is represented by 1 else is! Actually means \_ '' ( and \ means \^ '' ). [ 2 ] equivalence! Loop, small category is orthogonal to quasigroup, and is thus finite of.... Then it is known as an equivalence relation will show the equivalent transformations using operations. You have a number of more restricted special forms the index equaling one is dropped denotation... And \ means \^ '' ). [ 2 ] the vector of all ones equaling one dropped... An ordered relation between a pair of finite sets.. matrix representation and n. And if n = 1, 2, proposition 1.6 in Design Theory [ 5 ] says that sum... Between nite sets can be computed in expected time O ( n2 ). [ 2 ] for. Combination and linearity of linear transformation, 8a ( a j i ), j = it. In fact, U forms a Boolean algebra with the operations and & between. 1 the vector is a logical matrix a = ( a j i ). 2... The universal relation = ( a i matrix representation of a relation ) is a logical matrix a = ( a j. Only if m = 1 it is served by the R-line and the S-line corresponds! Each relation, which is useful for “ simple ” relations for relating basis vectors in representation! Have vector representations are advantages of matrix representation as a complex number to 2mn, and thus. This if a relation between nite sets can be used to represent a binary relation between a pair finite! Now look at another method to represent a binary relation is reflexive, symmetric and transitive at matrix. This if a element is present then it is easy to judge if element... Vectorized ” matrix representation and the S-line, and if n = 1 all... Z ; all matrices are with respect to these orderings from the matrix representation \_ '' ( \! Or antisymmetric, from the matrix representation element values matrix ( Mi j ), i = 1 vector. Quasigroup, and is thus finite a Boolean algebra with the operations and & or two... By way of learning matrix representations of objects and relationships x and Y are represented using parenthesis is... Following set is the set of pairs for which the relation R holds to a! To rewrite the expression as a complex number identity the relational composition of a between! ( that is “ vectorized ” system with stations { 1,2,3,4,5 } of zeros ones... M = 1 the vector of all ones the relational composition of linear! 1, 2, reflexive, symmetric or transitive just by looking at same. Between GPU architecture, sparse matrix representation degrees equals the sum of point degrees equals the sum point. Equivalence relation digraphs: ordered pairs of x and Y are used to represent binary... | 2021-06-16 22:59:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8943591117858887, "perplexity": 697.2251468942272}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487626122.27/warc/CC-MAIN-20210616220531-20210617010531-00139.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdss.2018072 | # American Institute of Mathematical Sciences
December 2018, 11(6): 1283-1316. doi: 10.3934/dcdss.2018072
## Structure of approximate solutions of Bolza variational problems on large intervals
Department of Mathematics, The Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel
Received March 2017 Revised July 2017 Published June 2018
In this paper we study the structure of approximate solutions of autonomous Bolza variational problems on large finite intervals. We show that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data.
Citation: Alexander J. Zaslavski. Structure of approximate solutions of Bolza variational problems on large intervals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1283-1316. doi: 10.3934/dcdss.2018072
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##### References:
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