url stringlengths 6 1.61k | fetch_time int64 1,368,856,904B 1,726,893,854B | content_mime_type stringclasses 3 values | warc_filename stringlengths 108 138 | warc_record_offset int32 9.6k 1.74B | warc_record_length int32 664 793k | text stringlengths 45 1.04M | token_count int32 22 711k | char_count int32 45 1.04M | metadata stringlengths 439 443 | score float64 2.52 5.09 | int_score int64 3 5 | crawl stringclasses 93 values | snapshot_type stringclasses 2 values | language stringclasses 1 value | language_score float64 0.06 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
https://formatessays.com/mis-665-topic-3-dq-2/ | 1,652,728,129,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662512229.26/warc/CC-MAIN-20220516172745-20220516202745-00440.warc.gz | 332,326,292 | 21,674 | # MIS 665 Topic 3 DQ 2
By definition, simulations require a distribution to be specified (e.g., normal, Poisson). Many times, the exact distribution to be used is unknown, so it must be assumed. One argument against using simulations to perform risk analysis is that there is no real benefit because the set of assumptions is simply shifted from assumed parameter values to assumed distributions of parameters. Comment on this argument and justify your opinions with reasons, facts, and examples.
MIS 665 Topic 4 DQ 1
Many times, linear optimization is used to maximize an objective function because profit, productivity, or efficiency is the outcome of interest. Provide two examples where the goal is to optimize a process by minimizing an objective function. In your examples, identify the outcome and any constraints that would need to be met.
MIS 665 Topic 4 DQ 2
When many constraints are present in a linear optimization problem, there is a greater chance that a redundant constraint exists. Assume you are trying to maximize an objective function and you have two decision variables, X1 and X2. If a redundant constraint exists, does the constraint become necessary if you try to minimize (instead of maximizing) the same objective function? Why? Do you need an objective function to determine if a constraint is redundant? Explain.
MIS 665 Topic 5 DQ 1
Many linear optimization problems can be solved by finding a graphical solution, but there are some problems that require more advanced spreadsheets and software to find an optimal solution. Describe an optimization problem in which finding a solution would be impossible using the feasible-region approach. Discuss the attributes the problem would have to make it impossible to solve using the feasible-region approach.
MIS 665 Topic 5 DQ 2
Optimization techniques are used in many applications. For example, when customers order products from an online store, the shipper has to determine the optimal way to get the product delivered to the customer. The delivery path that is chosen is the path that minimizes shipping costs while simultaneously satisfying these constraints:
The product must arrive by a promised date.
The shipper must deliver a finite set of items.
The product must originate from one of several warehouse hubs across the country.
Discuss whether there can be multiple solutions (i.e., more than one path to get the product to your house). Explain why. Is there a guarantee that a solution always exists? Explain.
MIS 665 Topic 6 DQ 1
Most transshipment network modeling problems assume the costs are constant. For example, the costs of shipping a product from one city to another are assumed fixed. This can change over time if fuel costs change. If you knew the distribution of fuel costs, how could the distribution of fuel costs be incorporated into the transshipment problem? Discuss the benefits of employing this approach.
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https://www.meritnation.com/ask-answer/question/rahul-throws-the-ball-in-the-air-the-ball-goes-up-to-the-hei/integers/13608173 | 1,656,731,108,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103983398.56/warc/CC-MAIN-20220702010252-20220702040252-00786.warc.gz | 925,238,110 | 7,981 | # Rahul throws the ball in the air. The ball goes up to the height of 21 m and settles at the bottom of the pound,12 m deep. Find the total distance covered by the ball.
43 m
• -1
54 m
• 1
Give me solution
• -2
The ball goes up 21m then travels down the same distance, then again travels down 12 m under the pond. So the total distance traveled by ball is 21+21+12= 54m.
• -2
14. Rahul throws a ball in air. The ball goes up to the height of 21 m and settles at the bottom of a pond 12 m deep. Find the total distance covered by the ball.
• 0
What are you looking for? | 165 | 569 | {"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} | 3.828125 | 4 | CC-MAIN-2022-27 | latest | en | 0.881005 |
https://careercup.com/question?id=5656732130869248 | 1,603,408,659,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107880401.35/warc/CC-MAIN-20201022225046-20201023015046-00474.warc.gz | 262,354,004 | 9,160 | ## Interview Question
Country: United States
Comment hidden because of low score. Click to expand.
1
of 1 vote
Your final solution looks good to me. You could place some of the work with built in functions
My solution in C++ (you can replace isalpha with Character.isLetter(char) and tolower with Character.toLowerCase(char) in java if you like))
``````bool isPal(const char* str)
{
int f=0;
int b=strlen(str)-1;
while(f>b)
{
char front=tolower(str[f]); //normalize for upper vs lower comp
while(!isalpha(front) && f>b) //skip non alpha chars
front=tolower(str[++f]);
char back=tolower(str[b]); //normalize for upper vs lower comp
while(!isalpha(back) && f>b) //skip non alpha chars
back=tolower(str[--b]);
if(front!=back)
return false;
}
return true;
}``````
Comment hidden because of low score. Click to expand.
0
all those f>b comparisons should be f<b...you win some you lose some ;-)
Comment hidden because of low score. Click to expand.
0
of 0 vote
``````public class Palindrome {
/**
* @param args
*/
public static void main(String[] args) {
// TODO Auto-generated method stub
String str1="1sunilinus";
if (str1.matches(".*[^a-zA-Z].*\$") == true ){
System.out.println("Not a valid string");
return;
}
System.out.println("Is it Palindrome : "+isPalindrome(str1));
}
public static boolean isPalindrome(String s){
char[] str = s.toCharArray();
int len=str.length;
System.out.println("Length of String: "+len);
//loop
int start = 0;
int end = len-1;
while(start<end)
{
if(str[start++] != str[end--])
return(false);
}
return(true);
}
}``````
Comment hidden because of low score. Click to expand.
0
of 0 vote
Solution in C# (Case and analphabetic charachters are ignored)
``````public static bool WordCheck(string word)
{
word = word.ToLower();
string reversedWord = string.Empty;
for (int i = word.Length - 1; i >= 0; i--)
{
if (Convert.ToInt32(word[i]) > 96 && Convert.ToInt32(word[i]) < 123)
reversedWord += word[i];
}
return reversedWord == word ? true : false;
}``````
Comment hidden because of low score. Click to expand.
0
of 0 vote
Solution in JavaScript:
``````var palindromeCheck = function(string) {
var len = string.length - 1;
var alpha = /^[a-zA-Z]+\$/;
for ( var i = 0, k = len; i < k; i++, k-- ) {
while ( !alpha.test(string[i]) ) { i++; }
while ( !alpha.test(string[k]) ) { k--; }
if ( string[i] !== string[k] ) { return false; }
}
return true;
};``````
Name:
Writing Code? Surround your code with {{{ and }}} to preserve whitespace.
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CareerCup's interview videos give you a real-life look at technical interviews. In these unscripted videos, watch how other candidates handle tough questions and how the interviewer thinks about their performance. | 745 | 2,847 | {"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} | 2.515625 | 3 | CC-MAIN-2020-45 | latest | en | 0.479766 |
https://www.coursehero.com/file/6233675/hw3sol/ | 1,495,858,014,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463608765.79/warc/CC-MAIN-20170527021224-20170527041224-00089.warc.gz | 1,079,614,004 | 43,074 | hw3sol
# hw3sol - − = − = − − = = − − = − = − = −...
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Solutions to Homework 3 3.7 (b) 3.7(c) 3.7(f)
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3.7(i)
3.8(a) 3.8(b)
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3.8(d) 3.8(e)
3.9(d) +1 0 0 1
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1 sin t 3.9(e) s 2 + 3s 3 3 2 5 6 s 2 + 3s 3 s 2 + 3s 3 5/6 3/2 s 2 + 3s 3 3 2 5 6
3.9(f) 3.10
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) ( 0 ) ( 0 2 2 2 0 0 0 0 ) ( ) ( ) ( ) ( ) 1 ( 1 1 1 ) 1 ( 2 ) ( 0 ) ( ) 0 ( 2 ) ( 2 ) 0 ( ) 0 ( ) ( . ) ( 0 t t t t t t e t t e t y t t to t change Now te e t y s s s s s Y s Y y s sY y sy s Y s everywhere t t to t change answer final the in then and t assume We + = + = + + + = + + = = + + = &
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Problem 5. () ) sin( ) ( ) sin( ) cos( 2 1 ) ( ) ( ) ( sin 2 1 ) sin (cos 2 1 2 1 ) ( ) ( 4 ) ( 4 ) ( 4 1 ) ( 4 1 ) 1 ( 1 ) 1 ( 2 1 ) ( ) 1 ( 4 1 ) )( 1 ( 1 4 ) )( 1 ( 1 2 1 ) 1 ( 1 ) 1 ( 1 1 1 ) ( ) ( ) ( ) ( 1 ) ( ) 1 ( 2 ) 1 )( 1 ( 1 2 ) 1 ( 1 1 2 1 ) ( ) 1 ( 1 ) ( ) 0 ( 2 ) ( 2 ) 0 ( ) 0 ( ) ( . ) ( 0 0 ) ( 1 ) ( cos 2 0 0 0 0 ) ( ) ( 0 0 2 2 2 2 1 2 2 1 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 t t t t t t t t e e t t t y t t to t change Now t t t t te e t y j s j j s j j s j j s j s s s Y j j s s s ds d B j j s s s B s s A s s j s B j s B j s B j s B s A s Y s s s s s s s s s s s Y s s s Y y s sY y sy s Y s everywhere t t to t change answer final the in then and t assume We t y t y t t y y y t t t t t t j s j s s + + + = + + + = + +
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Unformatted text preview: + + + − + + + + = − = − + − = = − + − = − = + − = + + + + − + + + − + + + + = + + + + + − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + − + + = + − = + − + − − − = = = = + + − − − − − − − = − = − = & & & & &...
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## This note was uploaded on 04/30/2011 for the course MAE MAE 143b taught by Professor Callafon during the Winter '09 term at UCSD.
### Page1 / 10
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Ask a homework question - tutors are online | 1,113 | 2,536 | {"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} | 2.625 | 3 | CC-MAIN-2017-22 | latest | en | 0.579487 |
http://mathoverflow.net/questions/17986/power-series-for-meromorphic-differentials-on-compact-riemann-surfaces | 1,469,508,513,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824624.99/warc/CC-MAIN-20160723071024-00240-ip-10-185-27-174.ec2.internal.warc.gz | 151,835,027 | 14,707 | # Power series for meromorphic differentials on compact Riemann surfaces
Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/\Gamma$ where I do know $\Gamma$ explicit. Is there a way to write down the power series of meromorphic functions, differentials, quadratic differentials, and so on explicitly if one does know these sections explicitly (for example in a hyperelliptic picture of the surface). I know that there is the subject of automorphic forms, but the literature I have seen about this concentrates on modular groups. Moreover it is typically written for number theorists. Is there literature for compact surfaces (for example $Y^2=Z^6-1$), and maybe readable for geometers.
Thank you.
-
I would strongly recommend the book "Algebraic Curves and Riemann Surfaces" by Rick Miranda. – Kevin Jul 7 '12 at 16:18
If you have an equation such as $y^2=z^6-1$ you don't need the upper half plane, find $y$ as a power series of $z-a$ and express everything in terms of $z$. In your example $y=-\sum {1/2 \choose n}(-z^6)^{n}$, so you have the functions. The differentials are $f(z,y)dz/y$ so you are done there too and so on. To find an equation, you might need to find two functions first and I don't know how to do it for a random subgroup of $SL_2(\mathbb{R})$ (?). Reference: Griffiths and Harris, Principles of Algebraic Geometry. | 354 | 1,365 | {"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} | 2.59375 | 3 | CC-MAIN-2016-30 | latest | en | 0.930441 |
http://sklearn.lzjqsdd.com/auto_examples/mixture/plot_gmm_selection.html | 1,550,816,399,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550247513661.77/warc/CC-MAIN-20190222054002-20190222080002-00605.warc.gz | 249,110,727 | 6,082 | # Gaussian Mixture Model Selection¶
This example shows that model selection can be performed with Gaussian Mixture Models using information-theoretic criteria (BIC). Model selection concerns both the covariance type and the number of components in the model. In that case, AIC also provides the right result (not shown to save time), but BIC is better suited if the problem is to identify the right model. Unlike Bayesian procedures, such inferences are prior-free.
In that case, the model with 2 components and full covariance (which corresponds to the true generative model) is selected.
Python source code: `plot_gmm_selection.py`
```print(__doc__)
import itertools
import numpy as np
from scipy import linalg
import matplotlib.pyplot as plt
import matplotlib as mpl
from sklearn import mixture
# Number of samples per component
n_samples = 500
# Generate random sample, two components
np.random.seed(0)
C = np.array([[0., -0.1], [1.7, .4]])
X = np.r_[np.dot(np.random.randn(n_samples, 2), C),
.7 * np.random.randn(n_samples, 2) + np.array([-6, 3])]
lowest_bic = np.infty
bic = []
n_components_range = range(1, 7)
cv_types = ['spherical', 'tied', 'diag', 'full']
for cv_type in cv_types:
for n_components in n_components_range:
# Fit a mixture of Gaussians with EM
gmm = mixture.GMM(n_components=n_components, covariance_type=cv_type)
gmm.fit(X)
bic.append(gmm.bic(X))
if bic[-1] < lowest_bic:
lowest_bic = bic[-1]
best_gmm = gmm
bic = np.array(bic)
color_iter = itertools.cycle(['k', 'r', 'g', 'b', 'c', 'm', 'y'])
clf = best_gmm
bars = []
# Plot the BIC scores
spl = plt.subplot(2, 1, 1)
for i, (cv_type, color) in enumerate(zip(cv_types, color_iter)):
xpos = np.array(n_components_range) + .2 * (i - 2)
bars.append(plt.bar(xpos, bic[i * len(n_components_range):
(i + 1) * len(n_components_range)],
width=.2, color=color))
plt.xticks(n_components_range)
plt.ylim([bic.min() * 1.01 - .01 * bic.max(), bic.max()])
plt.title('BIC score per model')
xpos = np.mod(bic.argmin(), len(n_components_range)) + .65 +\
.2 * np.floor(bic.argmin() / len(n_components_range))
plt.text(xpos, bic.min() * 0.97 + .03 * bic.max(), '*', fontsize=14)
spl.set_xlabel('Number of components')
spl.legend([b[0] for b in bars], cv_types)
# Plot the winner
splot = plt.subplot(2, 1, 2)
Y_ = clf.predict(X)
for i, (mean, covar, color) in enumerate(zip(clf.means_, clf.covars_,
color_iter)):
v, w = linalg.eigh(covar)
if not np.any(Y_ == i):
continue
plt.scatter(X[Y_ == i, 0], X[Y_ == i, 1], .8, color=color)
# Plot an ellipse to show the Gaussian component
angle = np.arctan2(w[0][1], w[0][0])
angle = 180 * angle / np.pi # convert to degrees
v *= 4
ell = mpl.patches.Ellipse(mean, v[0], v[1], 180 + angle, color=color)
ell.set_clip_box(splot.bbox)
ell.set_alpha(.5) | 816 | 2,759 | {"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} | 2.828125 | 3 | CC-MAIN-2019-09 | latest | en | 0.512977 |
https://essaypassusa.com/part-1-case-study/ | 1,712,980,056,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296816535.76/warc/CC-MAIN-20240413021024-20240413051024-00164.warc.gz | 236,978,742 | 16,718 | # Part 1 Case Study
Part 1 Case Study
Name
Course
Institution
Part 1: Case Study
Identify
Potential problems in the manufacturing process
Both the small, medium and large engines would be skewed to having poor performances as listed.
There are also mechanisms that have led to compact, large, mid-size, small, sporty and vans. Passengers are observed to have issues with the vehicles owing to how they have been made.
There are also various vehicles having issues when it comes to the MPG.
The fuel tank is also observed to be having issues with the developments. This is in relation to large, medium and small tanks.
Investigate these high and low measurements
Manufacturer Origin Model Type Passengers Price
Honda non-US Civic Small 4 12.1
Geo non-US Metro Small 4 8.4
Suzuki non-US Swift Small 4 8.6
The price of Honda more so, the small one is affordable.
Highest 10% and the lowest 10% of these measurement
Q1 1.2175
Q4 2.285
Better ways to select the outlier values
When the interquartile range is given a multiplier effect of 1.5. Then we are able to ascertain the outlier values that are present.
Subtraction of 1.5 interquartile range from the 1st quarter would result in less values deemed as outliers. Hence, the numbers would be sorted from high to low. This can be visualized through a box plot.
Fences can be used to introduce interquartile ranges.
Statistical procedures can be used to come up with the skewed values.
Reasoning and answers for distributions that are normal or skewed
Their ought to be outliers in a system of which would result in the framework having a normal curve. The values cannot be the same at any given point. Thus, proving that they are correct and have a statistical element.
Part 2: Box Plot
Population size: 92
Median: 17.6
Minimum: 7.4
Maximum: 47.9
First quartile: 12.125
Third quartile: 23.15
Interquartile Range: 11.025
Outliers: 47.9 40.1
Weight data from the Cars data set located in the Group Project assignment
Values for the 5-number summary
Upper and lower fences | 504 | 2,056 | {"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} | 2.625 | 3 | CC-MAIN-2024-18 | latest | en | 0.929452 |
http://superuser.com/questions/529146/microsoft-excel-formula-on-rounding-up-values | 1,444,239,713,000,000,000 | text/html | crawl-data/CC-MAIN-2015-40/segments/1443737875203.46/warc/CC-MAIN-20151001221755-00232-ip-10-137-6-227.ec2.internal.warc.gz | 297,556,590 | 17,273 | Microsoft excel formula on rounding up values
If a value is less than 200, then it should be rounded up to 200.
How do you do this in Microsoft Excel?
-
Do you mean if the number is between 1 - 200, it should return 200? Then use =ceiling(A1,200) (where a1 is either your number or the reference cell). This will round all numbers up by 200, so 201 will be 400. If that's not what you meant, edit your question to give some example conditions. – jdh Jan 7 '13 at 3:42
if its more than 200, then it should be remain unchanged – user184996 Jan 7 '13 at 4:00
I've created an overview of all answers given, so that you can see the result yourself: goo.gl/ryoYz – Jacob Jan Tuinstra Jan 7 '13 at 5:25
Below formula helps
``````=MAX(A1, 200)
``````
-
ok, updated.... – neo Jan 7 '13 at 4:14
thank you so much – user184996 Jan 7 '13 at 9:12
@user184996, you should accept the answer. – Dane Jan 7 '13 at 21:07
• If a value is less than 200, then it should be round up to 200.
• if its more than 200, then it should be remain unchanged
Solution:
``````=MAX(200, value)
``````
-
thank you so much – user184996 Jan 7 '13 at 9:12
If you wanted everything under 200 rounded to 200, and everything else left alone:
=IF(CELL<200, 200, CELL);
'CELL' being the coordinate.
If you literally wanted just the range from 1-200 to be rounded to 200, and not values less than 1, as your original question specifies:
=IF(cell<200, IF(cell<1, cell, 200), cell)
This will satisfy your stated requirements.
-
sorry but if it's more than 200 then it should remain as is – user184996 Jan 7 '13 at 4:20
Yes, and both of these formulas(formulae) do that. Do a bit of work. – Solemnity Jan 7 '13 at 4:41 | 518 | 1,696 | {"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} | 3.296875 | 3 | CC-MAIN-2015-40 | longest | en | 0.861415 |
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# What is the sensitivity of the balance?
Updated: 4/28/2022
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12y ago
The sensitivity of the balance(S.O.B) is the smallest amount that the balance can measure.
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malay ko! xD
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requires manual balancingsensitive null detector or galvanometer is required to detect balance conditionmeasurement current needs to be reasonably high to achieve sufficient sensitivity.
### What are common symptoms of diffuse peripheral neuropathy?
numbness and feelings of tingling or burning insensitivity to pain needle-like jabs of pain extreme sensitivity to touch loss of balance and coordination | 540 | 2,699 | {"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} | 2.703125 | 3 | CC-MAIN-2024-26 | latest | en | 0.909345 |
https://mathexamination.com/class/convergent-sequence.php | 1,619,128,229,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618039604430.92/warc/CC-MAIN-20210422191215-20210422221215-00601.warc.gz | 497,351,142 | 7,026 | Do My Convergent Sequence Class
A "Convergent Sequence Class" QE" is a standard mathematical term for a generalized continuous expression which is used to fix differential equations and has services which are routine. In differential Class fixing, a Convergent Sequence function, or "quad" is used.
The Convergent Sequence Class in Class type can be expressed as: Q( x) = -kx2, where Q( x) are the Convergent Sequence Class and it is an important term. The q part of the Class is the Convergent Sequence constant, whereas the x part is the Convergent Sequence function.
There are four Convergent Sequence functions with correct solution: K4, K7, K3, and L4. We will now look at these Convergent Sequence functions and how they are solved.
K4 - The K part of a Convergent Sequence Class is the Convergent Sequence function. This Convergent Sequence function can also be written in partial portions such as: (x2 - y2)/( x+ y). To solve for K4 we multiply it by the proper Convergent Sequence function: k( x) = x2, y2, or x-y.
K7 - The K7 Convergent Sequence Class has an option of the form: x4y2 - y4x3 = 0. The Convergent Sequence function is then increased by x to get: x2 + y2 = 0. We then need to multiply the Convergent Sequence function with k to get: k( x) = x2 and y2.
K3 - The Convergent Sequence function Class is K3 + K2 = 0. We then multiply by k for K3.
K3( t) - The Convergent Sequence function equationis K3( t) + K2( t). We multiply by k for K3( t). Now we multiply by the Convergent Sequence function which gives: K2( t) = K( t) times k.
The Convergent Sequence function is also known as "K4" because of the initials of the letters K and 4. K suggests Convergent Sequence, and the word "quad" is pronounced as "kah-rab".
The Convergent Sequence Class is one of the primary approaches of solving differential equations. In the Convergent Sequence function Class, the Convergent Sequence function is first multiplied by the suitable Convergent Sequence function, which will provide the Convergent Sequence function.
The Convergent Sequence function is then divided by the Convergent Sequence function which will divide the Convergent Sequence function into a real part and a fictional part. This gives the Convergent Sequence term.
Lastly, the Convergent Sequence term will be divided by the numerator and the denominator to get the quotient. We are entrusted the right hand side and the term "q".
The Convergent Sequence Class is an essential concept to understand when resolving a differential Class. The Convergent Sequence function is just one approach to fix a Convergent Sequence Class. The approaches for solving Convergent Sequence formulas consist of: singular value decomposition, factorization, optimum algorithm, mathematical solution or the Convergent Sequence function approximation.
Pay Me To Do Your Convergent Sequence Class
If you would like to become acquainted with the Quartic Class, then you need to very first start by looking through the online Quartic page. This page will reveal you how to use the Class by utilizing your keyboard. The description will also show you how to develop your own algebra equations to help you study for your classes.
Before you can understand how to study for a Convergent Sequence Class, you must first understand using your keyboard. You will learn how to click the function keys on your keyboard, as well as how to type the letters. There are 3 rows of function keys on your keyboard. Each row has 4 functions: Alt, F1, F2, and F3.
By pushing Alt and F2, you can increase and divide the worth by another number, such as the number 6. By pressing Alt and F3, you can use the 3rd power.
When you press Alt and F3, you will type in the number you are trying to multiply and divide. To multiply a number by itself, you will push Alt and X, where X is the number you want to increase. When you push Alt and F3, you will key in the number you are attempting to divide.
This works the same with the number 6, other than you will just type in the two digits that are six apart. Lastly, when you press Alt and F3, you will utilize the fourth power. Nevertheless, when you push Alt and F4, you will use the actual power that you have actually found to be the most suitable for your issue.
By using the Alt and F function keys, you can increase, divide, and then use the formula for the 3rd power. If you need to multiply an odd variety of x's, then you will require to enter an even number.
This is not the case if you are trying to do something complex, such as multiplying two even numbers. For example, if you wish to multiply an odd number of x's, then you will require to get in odd numbers. This is particularly true if you are attempting to find out the response of a Convergent Sequence Class.
If you wish to transform an odd number into an even number, then you will need to push Alt and F4. If you do not know how to increase by numbers on their own, then you will need to use the letters x, a b, c, and d.
While you can increase and divide by use of the numbers, they are much easier to use when you can look at the power tables for the numbers. You will need to do some research when you first begin to use the numbers, however after a while, it will be force of habit. After you have developed your own algebra formulas, you will have the ability to produce your own multiplication tables.
The Convergent Sequence Solution is not the only method to fix Convergent Sequence formulas. It is necessary to find out about trigonometry, which utilizes the Pythagorean theorem, and then use Convergent Sequence formulas to fix issues. With this method, you can learn about angles and how to solve problems without having to take another algebra class.
It is important to attempt and type as rapidly as possible, because typing will help you learn about the speed you are typing. This will assist you write your responses much faster.
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A Convergent Sequence Class is a generalization of a linear Class. For example, when you plug in x=a+b for a given Class, you acquire the worth of x. When you plug in x=a for the Class y=c, you obtain the worths of x and y, which give you an outcome of c. By using this standard idea to all the formulas that we have actually attempted, we can now resolve Convergent Sequence formulas for all the worths of x, and we can do it rapidly and efficiently.
There are lots of online resources readily available that supply complimentary or cost effective Convergent Sequence equations to fix for all the worths of x, including the cost of time for you to be able to make the most of their Convergent Sequence Class project help service. These resources normally do not need a membership fee or any type of financial investment.
The responses supplied are the outcome of complex-variable Convergent Sequence equations that have been resolved. This is likewise the case when the variable used is an unknown number.
The Convergent Sequence Class is a term that is an extension of a direct Class. One advantage of using Convergent Sequence formulas is that they are more basic than the linear formulas. They are much easier to resolve for all the values of x.
When the variable utilized in the Convergent Sequence Class is of the form x=a+b, it is easier to fix the Convergent Sequence Class since there are no unknowns. As a result, there are less points on the line specified by x and a continuous variable.
For a right-angle triangle whose base indicate the right and whose hypotenuse indicate the left, the right-angle tangent and curve chart will form a Convergent Sequence Class. This Class has one unknown that can be found with the Convergent Sequence formula. For a Convergent Sequence Class, the point on the line specified by the x variable and a consistent term are called the axis.
The presence of such an axis is called the vertex. Considering that the axis, vertex, and tangent, in a Convergent Sequence Class, are a given, we can discover all the worths of x and they will sum to the given worths. This is achieved when we use the Convergent Sequence formula.
The factor of being a constant aspect is called the system of equations in Convergent Sequence formulas. This is often called the main Class.
Convergent Sequence formulas can be solved for other values of x. One way to solve Convergent Sequence formulas for other values of x is to divide the x variable into its element part.
If the variable is offered as a positive number, it can be divided into its factor parts to get the regular part of the variable. This variable has a magnitude that amounts to the part of the x variable that is a constant. In such a case, the formula is a third-order Convergent Sequence Class.
If the variable x is negative, it can be divided into the exact same part of the x variable to get the part of the x variable that is increased by the denominator. In such a case, the formula is a second-order Convergent Sequence Class.
Solution help service in solving Convergent Sequence equations. When using an online service for solving Convergent Sequence formulas, the Class will be solved instantly. | 2,048 | 9,181 | {"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} | 4.1875 | 4 | CC-MAIN-2021-17 | latest | en | 0.892673 |
https://datascience.stackexchange.com/questions/67781/understanding-the-concept-vanishing-gradient-and-exploding-gradient-problem-in-t?noredirect=1 | 1,716,219,696,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058291.13/warc/CC-MAIN-20240520142329-20240520172329-00136.warc.gz | 173,500,349 | 42,362 | # Understanding the concept vanishing gradient and exploding gradient problem in terms of training data
I'm trying to figure out the essence of the concepts "vanishing gradient and exploding gradient problem" in terms of real-world input-output training examples instead of in terms of the properties of the choice of activation function.
Can anybody direct to a good tutorial that include such examples?
I always refer "Andrew NG" documents if available for any ML understanding. I believe that Youtube link Vanishing and Exploding Gradient, will help you in understanding the concept in a better way.
Still a brief from my side:
DNN of K layers without an activation function, will be like multiplying K coefficients(weights) together. Somehow if you received initial coefficient value < 0, in such a case during back propagation till you reach to the starting layers (from Input layer side), it may possible you completely lost the value of gradient, as gradually you are multiplying smaller value i.e. vanishing a value. Similarly if initial coefficient value > 0, multiplying together will form a very big number i.e. exploding a value.
A good way to understand and intuitively comprehend the concept of vanishing gradients and exploding gradient would be through manually solve through a backpropagation. Since, Feed Forward Neural Network is simplest of all and Mostly sigmoid function and Tanh suffers from vanishing gradient . It would be wise to build a MLP with at least one hidden layer and compute the change in parameter values after forward pass , error calculation and backward pass to update weights and biases initialized randomly. https://mattmazur.com/2015/03/17/a-step-by-step-backpropagation-example/
Likewise, RNN are mostly suffering from exploding gradient you could apply same method.
It may seems far fetched to go to this trouble for understanding concepts , but it is worth your time.
At high level, you can think of vanishing gradients in the way Chinese whispers work: Part of the original information is being lost every time it is being passed backwards to another person. In a similar way, RNN architecture "looses" part of the original information of a gradient as it is being propagated from the very last time step backwards to the very first step.
Drilling down to the specifics see below:
Traditional Recurrent Neural Networks (RNN) have the ability to model sequential events by propagating through time, i.e. forward and backward propagation. RNN models connect each time-step (e.g. position of a word in a sentence) using the following function defined as hidden state:
$$a_n = f(W_n, a_{n-1}, x_n)$$
The hidden state $$a_n$$ carries past information by applying a linear combination over the previous step and the current input.
The issue with the above is that the hidden state of every current position is a function of all previous positions. This means when you backpropagate gradients through time (see BPTT) the gradient inherently "looses" part of its "amplitude" because of the chain rule in $$a_n$$:
$$a_n = f(W_n, a_{n-1}, x_n) = f(W_n, f(W_{n-1}, a_{n-2}, x_{n-1}), x_n)$$, since $$a_{n-1}=f(W_n, a_{n-2}, x_n)$$.
In this way, the longer the input sequence is the worse long terms dependencies will be captured to to the way gradients vanish because of the chain rule in their hidden state.
I hope it helps. See here my relevant post in case it may also be of help https://datascience.stackexchange.com/a/84409/102852 | 755 | 3,488 | {"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": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2024-22 | latest | en | 0.896789 |
http://www.instructables.com/topics/Audio-signal-help/ | 1,521,652,952,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257647671.73/warc/CC-MAIN-20180321160816-20180321180816-00454.warc.gz | 419,379,737 | 6,652 | 231Views7Replies
### Author Options:
I was going to build
Amplifier with Gain = 20
Minimum Parts
Fromthis datasheet But I have no Idea how to put the signal on.
I think that the signal is Vin (pin 3) but that is one cable when a headphone has three cables. I know the red wire is right and the blue is left and the other is comm so I onley need 2. Because there is one input I need to make it one cable. Would this work (I will test what resistor I need with a voltmeter later).
If I am wrong about there olny being one input can someone explane to me how to wire it.
Tags:
## 7 Replies
So, which do you want to do? Use a switch to change from left to right (and back again) or combine the signal (convert from stereo to mono)?
I can give you pointers on how to do both. Just let me know which one you are interested in using.
As for the input, yes, pin 3 is the audio input. You would attach the positive wire (the blue or red wire) to pin 3 and the negative wire (common) to ground (pin 4).
Qa
David97 (author)2011-09-20
If I wanted to make the speeker as a middle speeker i was just going to tie the blue and red leads together.
Quercus austrina (author)2011-10-12
Sorry to take so long to reply.
Try this for mono:
------------
Red wire to 300 ohm resistor -
Blue wire to another 300 ohm resistor -
Connect the other ends of both 300 ohm resistors together and feed that to another 300 ohm resistor then attach that to V in at pin 3 as per the schematic. So that is three (3) 300 ohm resistors total.
------------
You want to do this so that you can keep the stereo effect for a stereo amp while combining the left and right for the center channel. It is done for isolation purposes.
You might need a little more gain. If so, use the "Gain=50" schematic and substitute a 10K ohm potentiometer for the 1.2K ohm fixed resistor in the pin1/8 path, connecting the outer lugs to pin 1 and the 10 uF capacitor. Then connect the center lug to the lug at pin 1. Play the music with maximum input and adjust this gain for maximum clear volume or until it matches the other stereo volume, whichever you prefer. You may opt to keep the potentiometer there or you can measure the resistance between the lug at the capacitor and the center lug and use the next lower standard resistor value to replace the potentiometer.
Qa
Re-design (author)2011-09-18
Headphones are stereo. This amp is mono. The input to this amp is on pins 2 and 3. To get stereo you need two of these.
There is a version of this that is stereo built into one chip but I don't remember the number right offhand.
David97 (author)2011-09-18
I know that so I was going to have a switch that changed it from left to right.
iceng (author)2011-09-18
As Re-design points out you need two to tango ( stereo ) and the Gain 20 ckt.
David97 (author)2011-09-18
I am having trobble uploading pics I will post as soon as possable | 759 | 2,906 | {"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} | 2.640625 | 3 | CC-MAIN-2018-13 | latest | en | 0.913909 |
https://efinancemanagement.com/financial-management/capital-budgeting-techniques-with-an-example | 1,709,099,923,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474697.2/warc/CC-MAIN-20240228044414-20240228074414-00664.warc.gz | 219,132,944 | 66,550 | # Capital Budgeting Techniques With an Example
## Capital Budgeting Techniques
Capital budgeting is a process that helps in planning the investment projects of an organization in the long run. Let’s understand all the following capital budgeting/investment appraisal techniques with an example.
1. Payback period
2. Discounted payback period
3. Net present value
4. Accounting rate of return
5. Internal rate of return
6. Profitability index
## Example of Capital Budgeting
ABC Inc. plans to buy machine A which will cost \$ 10 million. The expected life of the machine is 5 years. The salvage value of the machine is nil. ABC Inc. is expecting a cash flow of \$ 5 million for the first two years, \$ 3 million for the next 2 years & \$ 2 million in 5th year. Operating expense is \$ 1 million every year. Discounting rate is 10%. (Assumption: No tax)
Now let’s find out the answer by using different techniques.
### Payback Period Example
The payback method is used to know how much time it will take to recover the investment.
(Amount in Millions)
Here we can see it takes 3 years to generate sufficient profit to recover the cost. So the payback period is 3 years.
### Discounted Payback Period Example
This method is the same as the payback period method. The only difference between the payback period & discounted payback period is that it considers the discounted cash flow for finding the payback period.
(Amount in Millions)
Discounted payback period= 4 years + (10-9.8106)*52 weeks / (10.4315-9.8106). It takes approximately 4 years & 16 weeks.
### Net Present Value (NPV) Example
Net present value is one of the most commonly used methods for investment appraisal techniques. It is the sum of all future discounted cash flow less initial investment. If the amount is positive, the project should be accepted; otherwise, it should be rejected.
In discounting payback period, we can see the sum of all future discounted cash flow is \$ 10.4315 million & initial investment is \$ 10 million. It means NPV is \$ 0.4315 million. It is positive; hence the project should be accepted.
### Accounting Rate of Return Example
The accounting rate of return is also known as return on investment or return on capital. It is an accounting technique to measure the profit expected from an investment.
The formula of ARR is as follows:
ARR= Average annual profit after tax / Initial investment * 100
Average annual profit after tax = (total revenue – total expense) / 5 years
= (\$ 18 million – \$ 10 million) / 5 years
= \$ 1.6 million
ARR= \$ 1.6 million / \$ 10 million*100
ARR= 16%
### Internal Rate of Return Example
Internal Rate of Return is the discounting rate used for investment appraisal, which brings the cost of the project & its future cash flow at par with the initial investment. It is obtained by the trial & error method. We already have discounted value at a 10% discounting rate.
(Amount in Millions)
Difference between discounted cash-flow is \$ 0.4091 million (\$ 10.4315 million – \$ 10.0224 million).
IRR = 12 % + (0.0224 * 2 / 0.4091)
= 12 % + 0.11
IRR for the project is 12.11%.
### Profitability Index Example
The profitability index defines how much you will earn per dollar. The present value of future cash flow is \$ 10.4315 million & investment is \$ 10 million.
PI = Present value of cash inflow / Initial investment
PI = \$ 10.4315 million / \$ 10 million
The profitability index is 1.04315, which means every one dollar invested is generating revenue of \$ 1.04315.If the PI is more than 1, the project should be accepted; otherwise rejected.
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### 1 thought on “Capital Budgeting Techniques With an Example”
1. Dear Sir,
Your website is extremely useful for a person like me who does not have finance background.
Keep posting on regular basis.
Thank you,
P P SINGH
BANKER | 1,026 | 4,315 | {"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} | 3.171875 | 3 | CC-MAIN-2024-10 | longest | en | 0.896327 |
https://www.quizover.com/online/course/3-3-projectile-motion-two-dimensional-kinematics-by-openstax?page=4 | 1,531,681,586,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676588961.14/warc/CC-MAIN-20180715183800-20180715203800-00580.warc.gz | 974,146,234 | 24,593 | # 3.3 Projectile motion (Page 5/16)
Page 5 / 16
$t=\frac{-b±\sqrt{{b}^{2}-4\text{ac}}}{\text{2}\text{a}}\text{.}$
This equation yields two solutions: $t=3.96$ and $t=–1.03$ . (It is left as an exercise for the reader to verify these solutions.) The time is $t=3.96\phantom{\rule{0.25em}{0ex}}\text{s}$ or $–1.03\phantom{\rule{0.25em}{0ex}}\text{s}$ . The negative value of time implies an event before the start of motion, and so we discard it. Thus,
$t=3\text{.}\text{96 s}\text{.}$
Discussion for (a)
The time for projectile motion is completely determined by the vertical motion. So any projectile that has an initial vertical velocity of 14.3 m/s and lands 20.0 m below its starting altitude will spend 3.96 s in the air.
Solution for (b)
From the information now in hand, we can find the final horizontal and vertical velocities ${v}_{x}$ and ${v}_{y}$ and combine them to find the total velocity $v$ and the angle ${\theta }_{0}$ it makes with the horizontal. Of course, ${v}_{x}$ is constant so we can solve for it at any horizontal location. In this case, we chose the starting point since we know both the initial velocity and initial angle. Therefore:
${v}_{x}={v}_{0}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}{\theta }_{0}=\left(\text{25}\text{.}0 m/s\text{}\right)\left(\text{cos 35º}\right)=\text{20}\text{.}5 m/s.\text{}$
The final vertical velocity is given by the following equation:
${v}_{y}={v}_{0y}-\text{gt,}$
where ${v}_{0y}$ was found in part (a) to be . Thus,
${v}_{y}=\text{14}\text{.}3 m/s\text{}-\left(9\text{.}\text{80 m/s}{\text{}}^{2}\right)\left(3\text{.}\text{96 s}\text{}\right)$
so that
${v}_{y}=-\text{24}\text{.}5 m/s.\text{}$
To find the magnitude of the final velocity $v$ we combine its perpendicular components, using the following equation:
$v=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}=\sqrt{\left(\text{20}\text{.}5 m/s\text{}{\right)}^{2}+\left(-\text{24}\text{.}5 m/s\text{}{\right)}^{2}}\text{,}$
which gives
$v=\text{31}\text{.}9 m/s.\text{}$
The direction ${\theta }_{v}$ is found from the equation:
${\theta }_{v}={\text{tan}}^{-1}\left({v}_{y}/{v}_{x}\right)$
so that
${\theta }_{v}={\text{tan}}^{-1}\left(-\text{24}\text{.}5/\text{20}\text{.}5\right)={\text{tan}}^{-1}\left(-1\text{.}\text{19}\right)\text{.}$
Thus,
${\theta }_{v}=-\text{50}\text{.}1º\text{.}$
Discussion for (b)
The negative angle means that the velocity is $\text{50}\text{.}1º$ below the horizontal. This result is consistent with the fact that the final vertical velocity is negative and hence downward—as you would expect because the final altitude is 20.0 m lower than the initial altitude. (See [link] .)
One of the most important things illustrated by projectile motion is that vertical and horizontal motions are independent of each other. Galileo was the first person to fully comprehend this characteristic. He used it to predict the range of a projectile. On level ground, we define range to be the horizontal distance $R$ traveled by a projectile. Galileo and many others were interested in the range of projectiles primarily for military purposes—such as aiming cannons. However, investigating the range of projectiles can shed light on other interesting phenomena, such as the orbits of satellites around the Earth. Let us consider projectile range further.
How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed ${v}_{0}$ , the greater the range, as shown in [link] (a). The initial angle ${\theta }_{0}$ also has a dramatic effect on the range, as illustrated in [link] (b). For a fixed initial speed, such as might be produced by a cannon, the maximum range is obtained with . This is true only for conditions neglecting air resistance. If air resistance is considered, the maximum angle is approximately $\text{38º}$ . Interestingly, for every initial angle except $\text{45º}$ , there are two angles that give the same range—the sum of those angles is $\text{90º}$ . The range also depends on the value of the acceleration of gravity $g$ . The lunar astronaut Alan Shepherd was able to drive a golf ball a great distance on the Moon because gravity is weaker there. The range $R$ of a projectile on level ground for which air resistance is negligible is given by
find the 15th term of the geometric sequince whose first is 18 and last term of 387
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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Got questions? Join the online conversation and get instant answers! | 2,403 | 8,620 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 36, "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} | 4.71875 | 5 | CC-MAIN-2018-30 | longest | en | 0.759444 |
https://www.fiddler.ai/blog/measuring-data-drift-population-stability-index | 1,726,501,086,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651697.45/warc/CC-MAIN-20240916144317-20240916174317-00322.warc.gz | 700,858,412 | 22,297 | # Measuring Data Drift: Population Stability Index
What do you know about the Population Stability Index (PSI) measure, its historical usage, and its connection to other mathematical drift measures such as KL divergence? If you’re left scratching your head, don’t worry — we’ve got you covered!
## Population Stability Index: what is it?
PSI is a commonly used measure in the financial services domain to quantify the shift in the distribution of a variable over time. While several resources give an overview of PSI, such as this visual blog by Matthew Burke and this paper summary [3], they often do not discuss the connection between PSI as a drift metric and other popular measures such as KL divergence.
Briefly, PSI is calculated based on the multinomial classification of a variable into bins or categories. Consider two distributions shown in the left figure above. These distributions can be converted into their respective histograms with an appropriately chosen binning strategy. There are several binning strategies, and each strategy can yield varying PSI values. For the figure on the right, data is collected in equi-width bins. This produces a histogram that resembles a discretized version of the respective distribution. Another possible binning strategy is equi-quantiles or equi-depth binning. In this case, each bin would have the same proportion of samples in the reference / expected distribution. The choice of the strategy is context-specific and requires domain knowledge. For example, in credit score monitoring, credit scores are already binned into ranges representing a client's credit risk. In such cases, it may be desirable to use consistent binning throughout the analysis.
The differences in each bin between the expected distribution (AKA reference or initial distribution) and the target distribution (AKA new or actual distribution) are then utilized to calculate PSI as follows:
Where, $$B$$ is the total number of bins, $$ActualProp(b)$$ is the proportion of counts within bin $$b$$ from the target distribution and $$ExpectedProp(b)$$ is the proportion of counts within bin $$b$$ from the reference distribution. Thus, PSI is a number that ranges from zero to infinity and has a value of zero when the two distributions exactly match.
Practical Notes: The rules of thumb in practice regarding PSI thresholds are that if: (1) PSI is less than 0.1, then the actual and the expected distributions are considered similar, (2) PSI is between 0.1 and 0.2, then the actual distribution is considered moderately different from the expected distribution, and (3) PSI is beyond 0.2, then it is highly advised to develop a new model on a more recent sample [1,2]. Also, since there is a possibility that a particular bin may be empty, PSI can be numerically undefined or unbounded. To avoid this, in practice, a small value such as 0.01 can be added to each bin proportion value. Alternatively, a base count of 1 can be added to each bin to ensure non-zero proportion values.
## PSI usage history
PSI is typically used in financial services as a guidepost to compare current to baseline populations for which some financial tool or service was developed. For example, the use of credit scoring tools has proliferated in the banking industry to evaluate the level of credit risk associated with applicants or customers. Such tools provide statistical odds or probabilities that an applicant with a given credit score will pay off their credit. In the context of credit scoring, it is crucial to study the effects of changing populations or irregular trends in application approval rates. Similarly, abnormal periods where the population may under- or over-apply in line with regular business cycles are also important. PSI helps quantify such changes and provides a basis to the decision-makers that the development sample is representative of future expected applicants. Identifying distributional change can significantly impact the maintenance of tools capable of accurate lending decisions.
While there are no explicit resources that we found on the rationale of using PSI, we conjecture that PSI usage stems from multiple factors as listed below:
1. Regulations such as Basel Accords and the International Financial Reporting Standards (IFRS 9) discuss assessing the risk of loans with three components: the probability of default (PD), exposure at default (EAD), and loss given default (LGD). Since PSI measures shifts in probability distributions, its usage for measuring shifts in PD seems likely due to such regulations.
2. PSI uses binning of variables, including numerical variables, which implies categorizing variables into bins. Despite being a numerical quantity, credit scores are typically categorized into bins in the financial sector. Such a practice also points towards the ease of usage of PSI within the industry.
3. The PSI metric may have been widely adopted due to its inclusion in popular software such as SAS® Enterprise Miner™.
With the ongoing adoption of machine learning models and systems in financial services, PSI has gained popularity as a model monitoring metric — we only expect this trend to continue as model portfolios grow and the MLOps lifecycle becomes standardized within organizations.
## Unpacking PSI formula as a function of KL divergence
The Kullback-Leibler divergence or relative entropy is a statistical distance measure that describes how one probability distribution is different from another.
Given two discrete probability distributions $$A$$ (actual), and $$E$$ (expected) defined on the same probability space, KL divergence is defined as:
An interpretation of KL divergence is that it measures the expected excess surprise in using the actual distribution versus the expected distribution as a divergence of the actual from the expected. This sounds a lot like the reasoning behind using PSI! While KL divergence is well studied in mathematical statistics [4] and has a lot of references to academic work [1,2], PSI is domain-specific and lacks concrete literature on the history of its usage within financial services. In the following, we illustrate how PSI can actually be viewed as a special form of KL divergence.
Consider the PSI formula and let us look at the proportion of counts within a bin b for the actual distribution $$ActualProp(b)$$ as the frequentist probability $$PA(b)$$ of the variable appearing in that bin. The same applies to the expected distribution.
Then, we can rewrite the PSI formula as:
On expanding further,
Thus, PSI can be rewritten as:
which is the symmetrized KL divergence!
We hope you enjoyed this overview of PSI. Don’t forget to check out our blog on detecting intersectional unfairness in AI!
———
References
1. Siddiqi, N. (2017). Intelligent credit scoring: Building and implementing better credit risk scorecards. John Wiley & Sons.
2. Yurdakul, B. (2018). Statistical properties of population stability index. Western Michigan University.
3. Lin, A. Z. (2017). Examining Distributional Shifts by Using Population Stability Index (PSI) for Model Validation and Diagnosis. SAS Conference Proceedings: Western Users of SAS Software 2017 September 20-22, 2017, Long Beach, California URL https://www.lexjansen.com/wuss/2017/47_Final_Paper_PDF.pdf
4. Kullback, S., & Leibler, R. A. (1951). On Information and Sufficiency. The Annals of Mathematical Statistics, 22(1), 79–86. http://www.jstor.org/stable/2236703 | 1,503 | 7,461 | {"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} | 3.65625 | 4 | CC-MAIN-2024-38 | latest | en | 0.916027 |
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# notecard 2 - Linear quantities Force F Mass(inertia m...
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Linear quantities Force: F Mass (inertia): m Newton’s 2nd Law: Fnet = ma Work: W = F||d Power: P = Fv Kinetic Energy: K = ½mv2 Work-energy Theorem: Wnet = ½mv2 - ½mv02 Linear Momentum: p = mv Kinetic Energy: K = p2/2m Rotational Quantities Torque: | τ | = r F = rFsin θ Moment of Inertia: I = 1/2mr^2 Newton’s 2nd law: τ = I α Work: W = θτ Power: P = ϖτ Kinetic energy: K = ½ I ϖ2 Work-energy theorem: Wnet = ½ I ϖ 2 - ½ I ϖ 02 Angular momentum: p = Ι ϖ Kinetic energy: K = p2/2 I Example 8.12: A student opens a door: Given: mdoor = 12.0kg; F = 40.0N;r
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https://mathematica.stackexchange.com/questions/104578/is-there-any-predictor-corrector-method-in-mathematica-for-solving-nonlinear-sys/104638 | 1,718,898,418,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861957.99/warc/CC-MAIN-20240620141245-20240620171245-00186.warc.gz | 346,418,487 | 52,625 | # Is there any predictor-corrector method in Mathematica for solving nonlinear system of algebraic equations?
The FindRoot function in Mathematica can easily be used to solve systems of nonlinear algebraic equations. But, I want to solve a system of nonlinear equations with variations of some parameters.
However, in most cases it fails to converge. There are several methods like the arc-length method in which two parameters vary simultaneously. (See e.g. this paper)
I cannot solve my equations with NSolve, and its convergence really depends on the initial guess. So, is there any built-in function to handle this situation? For example,
ans = {x, y} /.
FindRoot[{x^2 + y^2 - #, x y - 24}, {{x, 4}, {y, 9}},
MaxIterations -> 5000] & /@ Range[1, 100, 1];
ListPlot[Table[{ans[[#, i]], Range[1, 100, 1][[#]]} & /@
Range[Length[Range[1, 100, 1]]], {i, 1, 2}], Joined -> True,
PlotRange -> All]
• If you post a specific example in Mathematica format there is a better chance of getting a useful response. Commented Jan 21, 2016 at 23:23
• @DanielLichtblau, I found a simple example, but in fact I have a large system of nonlinear equations. Commented Jan 22, 2016 at 8:52
• I should mention that the example given is fine for NSolve. That said, I'm not sure if your statement was in reference to this particular example or to a more difficult family of problems. Commented Jan 22, 2016 at 16:05
• @DanielLichtblau, Thank you for your answer, these family of problems are related in tracing equilibrium of structures in finite element method. So, most of the time, there are a very large system of nonlinear equations, in which Newton method fails to converge due to limit points. Commented Jan 22, 2016 at 17:06
• Maybe some useful info here: demonstrations.wolfram.com/… Commented Jan 22, 2016 at 20:21
Since @hesam asked about a command, and to get a better understanding of @DanielLichtblau's approach, I tried to generalize it and package it in a function. Feedback would be appreciated!
TrackRoot[eqns_List,unks_List,{par_,parmin_?NumericQ,parmax_?NumericQ},ipar_?NumericQ,
iguess_List,opts___?OptionQ]:=
Module[{findrootopts,ndsolveopts,subrule,isol,ics,deqns,sol},
(* options *)
findrootopts=Evaluate[FindRootOpts/.Flatten[{opts,Options[TrackRoot]}]];
ndsolveopts=Evaluate[NDSolveOpts/.Flatten[{opts,Options[TrackRoot]}]];
subrule=Table[unk->unk[par],{unk,unks}];
(* use FindRoot to improve initial guess *)
isol=FindRoot[eqns/.par->ipar,Transpose[{unks,iguess}],Evaluate[Sequence@@findrootopts]];
ics=Table[{unk[ipar]==(unk/.isol)},{unk,unks}];
(* differentiate eqns *)
deqns=Map[#==0&,D[eqns/.subrule,par]];
(* track root with NDSolve *)
sol=NDSolve[Join[deqns,ics],unks,{par,parmin,parmax},Evaluate[Sequence@@ndsolveopts]][[1]];
Return[sol];
];
Options[TrackRoot]={FindRootOpts->{},NDSolveOpts->{}};
TrackRoot::usage="TrackRoot[eqns,unks,{par,parmin,parmax},initpar,initguess]
tracks a root of eqns, varying par from parmin to parmax, with initial guess
initguess at par=initpar.";
Here's an example:
{pmin, pmax} = {48.0001, 200};
tr = TrackRoot[{x^2 + y^2 - p, x y - 24}, {x, y}, {p, pmin, pmax}, 100, {10, 2}];
Plot[Evaluate[{x[p], y[p]} /. tr], {p, pmin, pmax}]
Next step I might try is to incorporate an arc-length method that could go around corners.
EDIT: Here's an attempt at using a pseudo-arclength method, inspired by this demonstration. It uses the same syntax as above and hides the fact that it uses arclength s internally. To handle multiple roots for a given parameter value it breaks the resulting InterpolatingFunction into segments between turning points (that part is particularly ugly code). Seems to work OK but I haven't tested it extensively.
TrackRootPAL[eqns_List,unks_List,{par_,parmin_?NumericQ,parmax_?NumericQ},ipar_?NumericQ,iguess_List,opts___?OptionQ]:=
Module[{s,findrootopts,ndsolveopts,smin,smax,s1,s2,subrule,isol,ics,deqns,breaks,sol,res,respart},
(* options *)
findrootopts=Evaluate[FindRootOpts/.Flatten[{opts,Options[TrackRootPAL]}]];
ndsolveopts=Evaluate[NDSolveOpts/.Flatten[{opts,Options[TrackRootPAL]}]];
smin=Evaluate[SMin/.Flatten[{opts,Options[TrackRootPAL]}]];
smax=Evaluate[SMax/.Flatten[{opts,Options[TrackRootPAL]}]];
subrule=Append[Table[unk->unk[s],{unk,unks}],par->par[s]];
(* use FindRoot to improve initial guess *)
isol=FindRoot[eqns/.par->ipar,Transpose[{unks,iguess}],Evaluate[Sequence@@findrootopts]];
ics=Join[Table[unk[0]==(unk/.isol),{unk,unks}],{par[0]==ipar}];
(* setup eqns *)
deqns=Join[
Map[#==0&,eqns/.subrule],
{Total[D[unks/.subrule,s]]^2+D[par[s],s]^2==1},
Table[unk'[0]==0,{unk,unks}],
{par'[0]==1}
];
(* track root with NDSolve *)
breaks={}; (* capture turning points *)
sol=NDSolve[Join[deqns,ics,
{WhenEvent[par'[s]==0,AppendTo[breaks,s]],
WhenEvent[par[s]==parmax,"StopIntegration"],WhenEvent[par[s]==parmin,"StopIntegration"]}],
Append[unks,par],{s,smin,smax},Evaluate[Sequence@@ndsolveopts]][[1]];
(* extract s endpoints *)
{s1,s2}=(par/.sol)["Domain"][[1]];
(* add endpoints to breaks *)
breaks=Sort[Join[{s1,s2},breaks]];
(* construct interpolatingfunctions (unk vs par) for each segment (between breaks) *)
res={};
Do[
respart={};
Do[
pts=Transpose[{(par/.sol)["Coordinates"][[1]],(par/.sol)["ValuesOnGrid"],(unk/.sol)["ValuesOnGrid"]}];
AppendTo[respart,unk->Interpolation[Select[pts,breaks[[i]]<=#[[1]]<=breaks[[i+1]]&][[All,2;;3]],
"ExtrapolationHandler"->{Indeterminate&,"WarningMessage"->False}]];
,{unk,unks}];
AppendTo[res,respart];
,{i,Length[breaks]-1}];
Return[res];
];
Options[TrackRootPAL]={FindRootOpts->{},NDSolveOpts->{},SMin->-100,SMax->100};
An example:
λ = 2.5;
tr = TrackRootPAL[{-z^3 + λ z + μ}, {z}, {μ, -10, 10}, 0, {1.4}];
Plot[Evaluate[z[μ] /. tr], {μ, -10, 10}]
Again, suggestions and improvements would be great!
• it is a great job, and as you said, maybe it is a good idea to develop it to track solutions after Limit points, Bifurcation points and etc. However, there are several advanced programs such as XPPAUT, which can be used to track solutions of nonlinear ODE, etc. Link Commented Jan 23, 2016 at 19:37
• Thanks. I know about XPPAUT and Content, but wouldn't it be nice to have some of that functionality in Mathematica? I think so! Commented Jan 23, 2016 at 23:02
• Is this method appropriate for an ODE with a parameter, for which I want to track the solution by varying the parameter? How do you apply boundary conditions in the code? Could you provide an example :) Thank you in advance!
– lxy
Commented Jun 21, 2019 at 5:31
• @jsxs It can track the equilibria of an ODE that you get by setting time derivatives equal to zero. Commented Jun 21, 2019 at 6:12
• Hi, @ChrisK what is ipar in the argument list of TrackRoot? Should it be any value between pmin and pmax? In my example, the parameter is the domain size L. Can TrackRoot handle it? Thank you!
– lxy
Commented Jul 22, 2022 at 5:32
As noted by @ChrisK, this works better starting at the top. Reason being there are no real solutions below the parameter value of 48.
Using FoldList one can readily use the prior result to seed the next, that is, providing a starting point. This is a fairly common homotopy approach.
points =
Rest[FoldList[({x, y} /.
FindRoot[{x^2 + y^2 - #2, x*y - 24}, {x, #1[[1]]}, {y, #1[[2]]},
MaxIterations -> 5000]) &, {10, 5}, Range[100, 48, -1]]]
(* Out[312]= {{9.68831380576, 2.47721125483}, {9.63289204076,
2.49146361222}, {9.57705689273, 2.50598908086}, {9.5207972894,
2.5207972894}, {9.46410161514, 2.53589838486}, {9.40695767175,
2.55130307135}, {9.34935263547, 2.56702265234}, {9.29127300977,
2.58306907727}, {9.23270457346, 2.59945499274}, {9.17363232343,
2.61619379912}, {9.11404041144, 2.63329971303}, {9.05391207408,
2.65078783664}, {8.99322955501, 2.66867423468}, {8.93197401851,
2.68697602011}, {8.87012545288, 2.70571144991}, {8.80766256248,
2.72490003219}, {8.74456264654, 2.74456264654}, {8.68080146268,
2.76472167958}, {8.61635307292, 2.78540117807}, {8.55118966907,
2.80662702253}, {8.48528137424, 2.82842712475}, {8.41859601621,
2.85083165338}, {8.35109886769, 2.87387329264}, {8.28275234732,
2.89758754018}, {8.21351567389, 2.92201305177}, {8.14334446456,
2.94719204185}, {8.07219026539, 2.9731707518}, {8.,
3.}, {7.92671531783, 3.02773583227}, {7.85227181897,
3.05644029566}, {7.77659812551, 3.08618236569}, {7.69961476067,
3.11703906572}, {7.62123278463, 3.14909682963}, {7.54135211915,
3.18245317561}, {7.45985946958, 3.21721878246}, {7.37662571918,
3.25352009356}, {7.29150262213, 3.29150262213}, {7.20431854953,
3.33133520332}, {7.11487293424, 3.37321554746}, {7.02292889219,
3.41737761672}, {6.92820323028, 3.46410161514}, {6.83035261157,
3.51372782122}, {6.72895390058, 3.56667624041}, {6.62347538298,
3.62347538298}, {6.51323307597, 3.68480595122}, {6.39732143808,
3.75157012701}, {6.27449734057, 3.82500759779}, {6.14297179931,
3.90690382181}, {6., 4.}, {5.84096258932,
4.10891178175}, {5.65685424949, 4.24264068712}, {5.4244289009,
4.4244289009}, {4.89897954515, 4.89897942598}} *)
Call the parameter p. Suppose you have solved at p=100, and you want a solution at p=50. You might instead set up as a set of differential equations in a new parameter t, letting the solution set morph from the starting system at p=100 to the final one at p=50. The idea is to treat x and y as functions of t and differentiate, using the known solution at p=100 as initial values. I will skip the derivation and just show the code for this particular case.
{xsol, ysol} =
NDSolveValue[{2*x[t]*x'[t] + 2*y[t]*y'[t] == 50,
x[t]*y'[t] + y[t]*x'[t] == 0, x[0] == 9.68831380576221,
y[0] == 2.4772112548342307}, {x, y}, {t, 1, 0}]
Note that the ODE above can be handled even if we alter to let the final parameter value go below 47. To address this one would want to change the formulation into a differential algebraic system so as to enforce the algebraic equations. One way to do this is to use Method->Projection in NDSolve. I will defer to the documentation for details.
For a more complicated example, I show some path tracking code in the appendix of this work
• I'm digging this. One question: should your initial value be x[1] and y[1] rather than x[0] and y[0] (since t=0 seems to correspond to p=50 and t=1 to p=100)? I've tried both and the initial condition at t=1 version seems to match the FindRoot approach. Commented Jan 22, 2016 at 17:52
• For me it's a bit easier to understand by sticking with p rather than t. That is {xsol, ysol} = NDSolveValue[{2*x[p]*x'[p] + 2*y[p]*y'[p] == 1, x[p]*y'[p] + y[p]*x'[p] == 0, x[100] == 9.68831380576221, y[100] == 2.4772112548342307}, {x, y}, {p, 50, 100}]; Commented Jan 22, 2016 at 17:55
• @ChrisK Yes, I got the initial points wrong. As for using a separate parameter t, I guess it was not really needed here. Commented Jan 22, 2016 at 19:50
• @ChrisK One reason to use a new parameter (not in this example though) is when two or more are being varied. A new parameter makes it straightforward to employ a linear homotopy between start values and end values of all the others at once. Commented Jan 23, 2016 at 20:28
• @Daniel Lichtblau, can your method trace the solution of an ODE as a parameter varies?
– lxy
Commented Jul 26, 2022 at 9:07
I'd love to see a good answer to this, because it's a common problem I face. My crude improvement on your technique is to use the previous parameter value's answer as an initial guess, which helps FindRoot. Even better, use linear or quadratic extrapolation and add some adaptive step size. But you won't be going around any bends like the vertical limit point in your diagram!
ans = {};
{x0, y0} = {2.477, 9.688};
Do[
{x0, y0} = {x, y} /.
FindRoot[{x^2 + y^2 - p, x y - 24}, {{x, x0}, {y, y0}}];
AppendTo[ans, {x0, y0}];
, {p, 100, 1, -1}];
ListPlot[Table[{ans[[#, i]], Range[1, 100, 1][[#]]} & /@
Range[Length[Range[1, 100, 1]]], {i, 1, 2}], Joined -> True,
PlotRange -> All]
Notice I'm going from p=100 down to p=1, which runs more smoothly.
• What problems do you solve that require this sort of analysis? I solve cold plasma dispersion relations with numerical effects included in order to improve numerical stability of particle-in-cell codes. The typical computation involves numerous roots, some complex, that move about, often merging in frequency space as parameters are varied. I know roughly where in phase space they lie, but computations must be efficient, because parameter space is large. Commented Jan 24, 2016 at 18:37
• @bbgodfrey For me, equilibria of nonlinear differential equation models in theoretical ecology & evolutionary biology. Commented Jan 24, 2016 at 19:55
• @ChrisK You should check my answer for an explicit pseudo-arclength continuation method. With some effort, it can be extended to all kinds of operators. Commented Mar 3, 2017 at 19:06
Here is my approach using an explicit pseudo-arclength method.
The typical predictor-corrector method uses the parameter as continuation parameter. For example, let $$G(u,h) = 0. \label{eq:root}\tag{1}$$ If we know a solution $$(u_0,h_0)$$, then we predict the solution at $$h_1 = h_0 + \Delta h$$ by noting that, if the Jacobian matrix $$G_u$$ is invertible, then $$u'(h_0) = - G_u^{-1}(u(h_0),h_0)G_h(u(h_0),h_0) = u_0'.$$ Then we propose the predictor $$u_p = u_0 + u_0' \Delta h.$$ and use some method (usually Newton's) to obtain the corrector.
The problem with this method is that it fails in a fold. To circumvent this problem, instead of parameterizing the solution curve by the continuation parameter, we do it by its arclength. In this case, you have to think that everything is a function of $$s$$, i.e., $$X(s) = \big(u(s),h(s)\big)$$, like so (taken from [1]):
In this scheme, not only $$u$$ is unknown, but also $$h$$. There are several ways to close the system, the most common being to use the scalar normalization $$(X_1 - X_0)^T \dot{X_0} = \Delta s.$$ This is the equation of a plane, which is perpendicular to the tangent $$\dot{X}(s)$$ at a distance $$\Delta s$$ from $$X_0$$. This plane will intersect the solution curve if $$\Delta s$$ and the curvature of the curve is not to large. So, we extend \eqref{eq:root} to $$G(u,h) = 0,$$ $$(u - u_0)^T \dot{u}_0 + (h - h_0)\dot{h}_0 - \Delta s = 0,$$ where $$(u_0,h_0) = \big(u(s_0),h(s_0)\big)$$ is a known solution. Now we only need to calculate $$\dot{u}_0$$ and $$\dot{h}_0$$ to get things going. This can be done by differentiating $$G$$ with respect to $$s$$ and using the normalization condition $$\dot{u}_0^T \dot{u}_0 + \dot{h}_0^2 = 1$$:
\begin{align}\label{eq:extroot}\tag{2} \dot{u}_0 &= -G_u^{-1}(u_0,h_0) G_h(u_0,h_0) \dot{h}_0,\\ \dot{h}_0 &= \pm\left(1+\|G_u^{-1}(u_0,h_0) G_h(u_0,h_0)\|^2\right)^{-1/2}, \end{align}
where we choose the sign depending on the direction we want to go. In this scheme, given a solution $$(u_0,h_0)$$, our predictor is \begin{align} u_p &= u_0 + \dot{u}_0 \Delta s,\\ h_p &= h_0 + \dot{h}_0 \Delta s. \end{align}
Finally, a practical way to approximate the derivatives of $$u$$ and $$h$$ is, instead of solving \eqref{eq:extroot} at each step and determining the sign, approximate this equation by $$\pmatrix{G_u(u_1,h_1) & G_h(u_1,h_1) \\ \dot{u}_0 & \dot{h}_0}\pmatrix{\dot{u}_1 \\ \dot{h}_1} = \pmatrix{0 \\ 1},$$ which has the advantage of choosing the right direction at each step.
Details on why this scheme works on folds can be found in Keller's classic notes Lectures on Numerical Methods In Bifurcation Problems.
## Some Mathematica examples:
I'm not an MMA expert; these codes are for illustrative purposes only, and haven't been crafted for speed or elegance. I'd love to see some of our house names take a shot into packing TrackRoot[..., Method -> PseudoArcLength] ;)
### OP's own example:
funG = {x^2 + y^2 - h, x y - 24};
extfunG = {x^2 + y^2 - h, x y - 24, (x - xPrev) dxPrev + (y - yPrev) dyPrev +
(h - hPrev) dhPrev - ds};
chi = LinearSolve[D[funG, {{x, y}}], -D[funG, h]];
extchi = LinearSolve[Join[D[funG, {{x, y, h}}], {{dxPrev, dyPrev, dhPrev}}], {0, 0, 1}];
{x0, y0, h0} = {x, y, 100} /. FindRoot[funG /. h -> 100, {x, 4}, {y, 9}, MaxIterations->500];
dh0 = -(1 + chi.chi)^(-1/2) /. {x -> x0, y -> y0, h -> h0}; (* left continuation *)
{dx0, dy0} = -chi dh0 /. {x -> x0, y -> y0, h -> h0};
ClearAll[predcorr];
predcorr[xi_, dxi_, step_] := Module[{xp, yp, hp},
{xp, yp, hp} = xi + step dxi;
{{x, y, h}, extchi /. {dxPrev->dxi[[1]], dyPrev->dxi[[2]], dhPrev->dxi[[3]]}, step} /.
FindRoot[extfunG /. {xPrev -> xi[[1]], yPrev -> xi[[2]], hPrev -> xi[[3]],
dxPrev -> dxi[[1]], dyPrev -> dxi[[2]], dhPrev -> dxi[[3]], ds -> step},
{x, xp}, {y, yp}, {h, hp}, MaxIterations -> 500]
]
We track the root using a NestWhileList. Note the Check function, ready to diminish the step-size in order to ensure convergence (There might be a more elegant way to address this problem).
solCurve = NestWhileList[
Check[predcorr[#1, #2, #3], {#1, #2, #3/2}] & @@ # &,
{{x0, y0, h0}, {dx0, dy0, dh0}, 0.1}, #[[1, 3]] < 101 &
] // DeleteDuplicates[#, (#1[[1]] == #1[[2]] &)] &;
Here is a bifurcation diagram:
ListPlot[{#1[[3]], #1[[1]]} & @@@ solCurve, AxesLabel -> {h, x}]
### Chris K example:
funG1 = -z^3 + 5/2 z + h;
extfunG1 = {-z^3 + 5/2 z + h, (z - zPrev) dzPrev + (h - hPrev) dhPrev - ds};
chi1 = -D[funG1, h]/D[funG1, z];
extchi1 = LinearSolve[Join[{D[funG1, {{z, h}}]}, {{dzPrev, dhPrev}}], {0, 1}];
{z0, h0} = {z, -10} /. FindRoot[funG1 /. h -> -10, {z, -3}, MaxIterations -> 500];
dh0 = (1 + chi1^2)^(-1/2) /. {z -> z0, h -> h0}; (* right continuation *)
dz0 = -chi1 dh0 /. {z -> z0, h -> h0};
ClearAll[predcorr1];
predcorr1[zi_, dzi_, step_] := Module[{zp, hp},
{zp, hp} = zi + step dzi;
{{z, h}, extchi1 /. {dzPrev -> dzi[[1]], dhPrev -> dzi[[2]]}, step} /.
FindRoot[extfunG1 /. {zPrev -> zi[[1]], hPrev -> zi[[2]], dzPrev -> dzi[[1]],
dhPrev -> dzi[[2]], ds -> step}, {z, zp}, {h, hp}, MaxIterations -> 500]
]
We track the root:
solCurve1 = NestWhileList[
Check[predcorr1[#1, #2, #3], {#1, #2, #3/2}] & @@ # &, {{z0, h0},
{dz0, dh0}, 0.1}, #[[1, 2]] < 10 &] // DeleteDuplicates[#, (#1[[1]] == #1[[2]] &)] &;
Here is the bifurcation diagram:
ListPlot[Reverse /@ solCurve1[[All, 1]], AxesLabel -> {h, z}]
• I should point out that my answer also includes a pseudo-arclength method. Commented Jul 6, 2019 at 10:37
• @Pragabhava, can this method trace the solution of an ODE as a parameter varies?
– lxy
Commented Jul 26, 2022 at 8:56
• @jsxs Yes, it can; you only need to define the derivatives of $G$ correctly (i.e. using Frechet's definition). For more details, see Keller's notes on the subject. In practice, as far as MMA goes (and I recall), you can wrap your ode into a black box function using NDSolve and use the method directly as stated. See my comment here for details. Commented Jul 26, 2022 at 23:01
I used the Mathematica built-in function NDSolve to apply the arclength continuation technique. This is what it gives for OP's example:
eqns = {x[s]^2 + y[s]^2 - p[s] == 0, x[s] y[s] - 24 == 0};
unkows = {x[s], y[s], p[s]};
norml = {D[unkows, s].D[unkows, s] - 1 == 0};
IC = {x[0] == 2, y[0] == 2, p[0] == 110, p'[0] == -1};
SolCont = NDSolve[Join[eqns, norml, IC], unkows, {s, 0, 100}];
ParametricPlot[{p[s],x[s]} /. SolCont, {s, 0, 100}]
I tested the code for very large systems, it worked well too. | 6,482 | 19,233 | {"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": 29, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2024-26 | latest | en | 0.899604 |
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# Most Corporations pay at least twice as much to full-time
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Most Corporations pay at least twice as much to full-time [#permalink]
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Most Corporations pay at least twice as much to full-time employees, if the value of benefits, sick days, and paid vacation days are included in earnings, than to part-time employees, whose hourly wages are often higher than those of their full-time colleagues.
(A) are included in earnings, than
(B) are included in earnings, as
(C) is included in earnings, than they pay
(D) is included in earnings, as is paid
(E) is included in earnings, as they pay
Let me know which answer you choose and why
If you have any questions
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02 Feb 2008, 11:13
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ajay_gmat wrote:
Most Corporations pay at least twice as much to full-time employees, if the value of benefits, sick days, and paid vacation days are included in earnings, than to part-time employees, whose hourly wages are often higher than those of their full-time colleagues.
(A) are included in earnings, than
(B) are included in earnings, as
(C) is included in earnings, than they pay
(D) is included in earnings, as is paid
(E) is included in earnings, as they pay
Let me know which answer you choose and why
D. first - needs singular verb - is.
second - requires "as much as" to make the idiom correct and complete.
hmm........ how trivial and hiddel issues are they?
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02 Feb 2008, 11:25
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Why not E?
"..pay at least twice as much to full-time employees ...., as they pay to..."
E appears to be parallel with the first clause in the sentence (both active voice), and in fact much better so than does D.
I vote for E. Howewer, GMAT TIGER did post a good explanation about why A,B,C are all incorrect.
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02 Feb 2008, 11:37
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E (same reasoning as above)
Most Corporations pay .. twice as much to full-time employees...as they pay to part-time employees,
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### Show Tags
02 Feb 2008, 22:27
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Sentence has idiom and verb issue.
(A) are included in earnings, than (Idiom issue – require correct form - as much … as)
(B) are included in earnings, as ( value requires singular verb)
(C) is included in earnings, than they pay (Idiom issue)
(D) is included in earnings, as is paid(why to switch to past form – eliminate it)
(E) is included in earnings, as they pay (Hold it)
Re: Confusing SC Question [#permalink] 02 Feb 2008, 22:27
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#### cashmoney805
##### Full Member
10+ Year Member
Hey guys, I have a specific question about tension that is really driving me crazy. It may be a little more than I need to know for the MCAT, but whatever. Anyway, here's the question:
A bucket weighing 3.2 kg is hanging from a massless rope. If the bucket is pulled upward by the rope with an acceleration of 1.6 m/s^2, calculate the tension in the rope. So I'm pretty sure there are 2 forces acting on the bucket: mg down, and Ft, tension of the rope, up. So the sum of the forces must = ma. So, Ft - mg = ma. Here's where I get messed up. Since the acceleration is up, don't g and a have to be opposite signs? If I make them opposite signs, i get something like 26.24. However, the answer in my book is 36, which you would get if a and g had the same sign. Can anyone explain why they have the same sign? Thanks so much
#### xanthomondo
##### nom nom nom
Removed
10+ Year Member
Hey guys, I have a specific question about tension that is really driving me crazy. It may be a little more than I need to know for the MCAT, but whatever. Anyway, here's the question:
A bucket weighing 3.2 kg is hanging from a massless rope. If the bucket is pulled upward by the rope with an acceleration of 1.6 m/s^2, calculate the tension in the rope. So I'm pretty sure there are 2 forces acting on the bucket: mg down, and Ft, tension of the rope, up. So the sum of the forces must = ma. So, Ft - mg = ma. Here's where I get messed up. Since the acceleration is up, don't g and a have to be opposite signs? If I make them opposite signs, i get something like 26.24. However, the answer in my book is 36, which you would get if a and g had the same sign. Can anyone explain why they have the same sign? Thanks so much
You're already taking into account g being negative in your equation, the fact that it's -mg shows that mg is going opposite of a. Remember it's the SUM of the forces = ma, so any negative signs show opposite direction
F = ma
T-mg = ma
T- 3.2(9.8) = 3.2(1.6)
T- 31.36 = 5.12
T = 36.48
##### No summer
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Good exp. Also, this question is definitely not out of the scope for the MCAT. Something similar could show up!
It's important to remember why the equations look the way they do. Just remember that when you're dealing with forces, no matter whether its tension, friction, Normal force, whatever, you have to vector sum the forces to get the right answer.
So when you get these problems in the future, instead of thinking "what formula do I use?", just remember that you need to sum all of the forces at work. If you do it that way then you can't get mixed up in the signs.
#### MBHockey
##### Full Member
It also helps to just look at it logically.
You are pulling on a rope (holding a mass) hard enough such that it is accelerating opposite the direction of gravity. Intuitively, this should tell you that you must be pulling harder than the object weighs. Therefore, tension can't be less than the object weighs...you wouldn't be able to pull it up!
Alternatively, you can use a strictly mechanical approach and sketch a free body diagram of the bucket. One arrow from its center pointing down (force of mg; it gets a negative sign) one arrow pointing up from its center (T; it gets a positive sign) and then just sum the forces in the Y direction and set equal to ma (from newton's second law)
This yields:
T - mg = ma
or
T = mg + ma | 907 | 3,587 | {"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} | 3.75 | 4 | CC-MAIN-2022-21 | latest | en | 0.971285 |
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A261045 Number of solutions to c(1)*prime(4) + c(2)*prime(5) + ... + c(2n-1)*prime(2n+2) = -1, where c(i) = +-1 for i>1, c(1) = 1. 19
0, 0, 0, 1, 2, 5, 32, 93, 261, 1082, 3253, 12307, 40809, 153392, 525417, 1892876, 6847161, 25256461, 91268129, 335852960, 1239350769, 4606651034, 17073491494, 63523866957, 237953442636, 892247156886, 3346127378391, 12603121634857, 47642071407103 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,5 COMMENTS There cannot be a solution for an even number of terms on the l.h.s. because they are all odd and the r.h.s. is odd, too. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..300 MAPLE s:= proc(n) option remember; `if`(n<5, 0, ithprime(n)+s(n-1)) end: b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=4, 1, b(abs(n-ithprime(i)), i-1)+b(n+ithprime(i), i-1))) end: a:= n-> b(8, 2*n+2): seq(a(n), n=1..30); # Alois P. Heinz, Aug 08 2015 MATHEMATICA s[n_] := s[n] = If[n<5, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 4, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[8, 2*n+2]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *) PROG (PARI) a(n)={my(p=vector(2*n-2, i, prime(i+4))); sum(i=1, 2^(2*n-2), sum(j=1, #p, (1-bittest(i, j-1)<<1)*p[j], 7)==-1)} \\ For illustrative purpose; too slow for n >> 10. - M. F. Hasler, Aug 08 2015 CROSSREFS Cf. A261057 (starting with prime(1)), A261059 (starting with prime(2)), A261060 (starting with prime(3)), A261061 - A261063 and A261044 (r.h.s. = -1), A022894 -A022904, A083309, A022920 (r.h.s. = 0, 1 or 2). Sequence in context: A032504 A041397 A042811 * A145656 A221680 A009274 Adjacent sequences: A261042 A261043 A261044 * A261046 A261047 A261048 KEYWORD nonn AUTHOR M. F. Hasler, Aug 08 2015 EXTENSIONS a(13)-a(29) from Alois P. Heinz, Aug 08 2015 STATUS approved
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Last modified September 17 04:58 EDT 2019. Contains 327119 sequences. (Running on oeis4.) | 935 | 2,336 | {"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} | 3.65625 | 4 | CC-MAIN-2019-39 | latest | en | 0.460857 |
https://www.vanessabenedict.com/how-much-does-1-kilo-of-silver-weigh/ | 1,680,121,067,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296949025.18/warc/CC-MAIN-20230329182643-20230329212643-00104.warc.gz | 1,170,558,652 | 15,744 | How many pounds does a kilo of silver weigh?
ByVanessa
Jun 12, 2022
One kilo is approximately 35 ounces. Therefore, one can purchase a 1 kilo silver bar rather than 50 or 100 ounce silver bars. Because the 1 kilo size contains less silver, these bars will cost less. This may make them more attractive to smaller investors or those on a budget.
Untitled Document
How many pounds does a kilo of silver weigh
How many pounds of silver are in 1 kilogram? The answer is: a change of 1 kilogram is a unit of kilogram (kilogram) associated with an equal amount of silver = time for 2.20 pounds (lbs), the same as the equivalent measure for real silver.
How big is a kilo silver bar
These Kilo-One stackers are a handy size for stacking. Equivalent to 32.15 troy ounces and measures 76mm x 51mm, 28mm at the back. This 1kg silver bar has bevelled interlocking edges that fold up for safe storage.
How much does silver cost per kg
The price of silver per ounce in US dollars. kilogram. American dollars. American dollars. kilogram. 1 kg =. \$723.4 1 dollar means. \$0.00138.
What is the price of 1 kilo of silver
Spot price of silver Change in the spot price of silver; Silver price per ounce: \$22.93: \$0.28: Silver price per gram: \$0.74: \$0.01: Silver price per kilogram: \$737.22: \$9.00: Current spot metal prices (24 hours) Last update: 07/10/2021 07:27:15 ET
How much is 1 troy oz 999 fine silver worth
Most 999 sterling silver coins are actually round. An ounce in the silver trade refers in the market to one troy ounce, which is considered equal to 31.1 grams or 1.10 monthly ounces. The term 0.999 silver means that the element is 99.9 pure silver. The value of the money actually held in the fund is called the melting index.
What is 10 oz of silver worth
Silver weight: ten troy ounces. Approximate size: 3.5 inches long, 2 inches wide, and 0.25 inches thick. Sell ??to us. Redeem my item daily and we will redeem you quickly! Our current purchase fee is \$252.80 per piece and we invest up to 500 oz. If you have any questions about weekday blocking rates, simply call us at 1-800-800-1865.
Which is heavier 1 kilo of cotton or 1 kilo of nails
1 kilometer or even will be quite large, much more than 1 kilogram of cotton. Nearly 20,000 children will fit per kilometer and weigh about 47 kg. It’s much more difficult! It really depends on the size of certain nails, but even a smaller 4d or even will weigh around thirty seven kilos.
Untitled Document
What is a kilo kilo
The kilogram (kg), the basic unit of the expansive metric system. A kilogram is almost the same mass as 1000 cubic centimeters of water (it was originally assumed to be exactly the same). A pound is precisely defined as equal to 0.45359237 kilograms. short information. Facts and related content.
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https://www.physicslab.org/DocumentPrint.aspx?doctype=5&filename=Compilations_CPworkbook_Reflection.xml | 1,720,922,641,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514527.38/warc/CC-MAIN-20240714002551-20240714032551-00598.warc.gz | 826,826,693 | 3,840 | CP Workbook Reflection
Light from a flashlight shines on a mirror and illuminates one of the cards. Draw the reflected beam to indicate the illuminated card. Which card would you see?
A periscope has a pair of mirrors in it. Draw the light path from the object to the eye of the observer. At what angle must each mirror be tilted?
The ray diagram below shows one of the reflected rays from the plane mirror. Complete the diagram by drawing the three other reflected rays. (Assume that the candle and image are viewed by an observer on the left.) How far "behind the mirror" is your image located? (remember to measure along a normal - that is, along a line that is perpendicular to the mirror)
The ray diagram below shows the reflection of one of the rays that strikes the parabolic mirror. Notice that the law of reflection is obeyed, and the angle of incidence (from the normal, the dashed line) equals the angle of reflection (from the normal). Complete the diagram by drawing the reflected rays of the other three rays that are shown. Why are parabolic mirrors used in automobile headlights?
A girl takes a photograph of the bridge as shown in the diagram below.
Which of these two sketches shown above correctly indicates the reflected view of the bridge? Defend your answer. | 271 | 1,302 | {"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} | 2.734375 | 3 | CC-MAIN-2024-30 | latest | en | 0.923647 |
https://www.studyxapp.com/homework-help/question-10-the-following-test-results-were-obtained-over-a-24hour-period-urine-q1572563413452382210 | 1,686,368,352,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224656963.83/warc/CC-MAIN-20230610030340-20230610060340-00071.warc.gz | 1,144,432,789 | 11,593 | # Question Question 10. The following test results were obtained over a 24-hour period: Urine volume = 1.4 L Urine [inulin] = 100 mg% Urine [urea] = 220 mmol L- Urine| [PAH] = 70 mg mL-1 Plasma [inulin] = 1 mg% Plasma (urea) = 5 mmol L-1 Plasma [PAH] = 0.2 mg mL-1 Hematocrit = 0.40 Calculate: A.Cinulin and Curea and CPAH
VIE0HS The Asker · Chemical Engineering
Transcribed Image Text: Question 10. The following test results were obtained over a 24-hour period: Urine volume = 1.4 L Urine [inulin] = 100 mg% Urine [urea] = 220 mmol L- Urine| [PAH] = 70 mg mL-1 Plasma [inulin] = 1 mg% Plasma (urea) = 5 mmol L-1 Plasma [PAH] = 0.2 mg mL-1 Hematocrit = 0.40 Calculate: A.Cinulin and Curea and CPAH
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Transcribed Image Text: Question 10. The following test results were obtained over a 24-hour period: Urine volume = 1.4 L Urine [inulin] = 100 mg% Urine [urea] = 220 mmol L- Urine| [PAH] = 70 mg mL-1 Plasma [inulin] = 1 mg% Plasma (urea) = 5 mmol L-1 Plasma [PAH] = 0.2 mg mL-1 Hematocrit = 0.40 Calculate: A.Cinulin and Curea and CPAH | 394 | 1,041 | {"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} | 2.703125 | 3 | CC-MAIN-2023-23 | latest | en | 0.776675 |
https://se.mathworks.com/matlabcentral/answers/595810-index-exceeds-the-number-of-array-elements-1 | 1,713,380,781,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817171.53/warc/CC-MAIN-20240417173445-20240417203445-00334.warc.gz | 473,065,488 | 25,339 | # Index exceeds the number of array elements (1).
1 view (last 30 days)
Heya :) on 18 Sep 2020
Edited: Heya :) on 16 Oct 2020
Can anybody please help why I am getting this error. I am unable to figure out what I am doing wrong in this code.
I am getting this error [ndex exceeds the number of array elements (1).
Error in trial (line 132)
xhat(i)= d1+d2*x_rec(i-1)+d3*y_rec(i-1)+d4*x_rec(i-1)*y_rec(i-1)+d5*x_rec(i-1)^2+d6*y_rec(i-1)^2+d7*x_rec(i-1)^2*y_rec(i-1)^2+d8*x_rec(i-1)^2*y_rec(i-1)+d9*x_rec(i-1)*y_rec(i-1)^2+d10*x_rec(i-1)^3+d11*y_rec(i-1)^3+d12*x_rec(i-1)^3*y_rec(i-1)+d13*x_rec(i-1)*y_rec(i-1)^3+d14*x_rec(i-1)^3*y_rec(i-1)^2+d15*x_rec(i-1)^2*y_rec(i-1)^3+d16*x_rec(i-1)^3*y_rec(i-1)^3;
>> ]
KSSV on 18 Sep 2020
Repalce with this the for loop.
for i=2:length(x_rec)
xhat(i)= d1+d2*x_rec(i-1)+d3*y_rec(i-1)+d4*x_rec(i-1)*y_rec(i-1)+d5*x_rec(i-1)^2+d6*y_rec(i-1)^2+d7*x_rec(i-1)^2*y_rec(i-1)^2+d8*x_rec(i-1)^2*y_rec(i-1)+d9*x_rec(i-1)*y_rec(i-1)^2+d10*x_rec(i-1)^3+d11*y_rec(i-1)^3+d12*x_rec(i-1)^3*y_rec(i-1)+d13*x_rec(i-1)*y_rec(i-1)^3+d14*x_rec(i-1)^3*y_rec(i-1)^2+d15*x_rec(i-1)^2*y_rec(i-1)^3+d16*x_rec(i-1)^3*y_rec(i-1)^3;
end
Heya :) on 18 Sep 2020
Thank you for the response. Now, its giving me this error.
Unrecognized function or variable 'xhat'.
Error in trial (line 132)
PE_x(count)=PE_x(count)+(xhat(i)-x(i))^2;
Alan Stevens on 18 Sep 2020
You should put
x_rec = zeros(1000,1);
y_rec = zeros(1000,1);
just before
x_rec(1)=x(1);
y_rec(1)=y(1);
Heya :) on 23 Sep 2020
Thank you !
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Start Hunting! | 688 | 1,711 | {"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} | 2.9375 | 3 | CC-MAIN-2024-18 | latest | en | 0.378145 |
https://dictionnaire.sensagent.leparisien.fr/Polynomial/en-en/ | 1,660,336,600,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882571758.42/warc/CC-MAIN-20220812200804-20220812230804-00673.warc.gz | 220,816,612 | 39,706 | Polynomial : définition de Polynomial et synonymes de Polynomial (anglais)
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# définition - Polynomial
1.having the character of a polynomial"a polynomial expression"
polynomial (n.)
1.a mathematical function that is the sum of a number of terms
Merriam Webster
PolynomialPoly*no"mi*al (?), n. [Poly- + -nomial, as in monomial, binomial: cf. F. polynôme.] (Alg.) An expression composed of two or more terms, connected by the signs plus or minus; as, a2 - 2ab + b2.
PolynomialPoly*no"mi*al, a.
1. Containing many names or terms; multinominal; as, the polynomial theorem.
2. Consisting of two or more words; having names consisting of two or more words; as, a polynomial name; polynomial nomenclature.
## définition (complément)
voir la définition de Wikipedia
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multinomial
polynomial (n.)
multinomial
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voir aussi
polynomial (n.)
polynomial (n.)
Wikipedia
# Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables (also called indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x2x/4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not an integer (3/2). The term "polynomial" can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in polynomial time, which is used in computational complexity theory.
Polynomial comes from the Greek poly, "many" and medieval Latin binomium, "binomial".[1][2][3] The word was introduced in Latin by Franciscus Vieta.[4]
Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.
## Overview
A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) that may be multiplied by a finite number of variables (usually represented by letters), also called indeterminates.[5] Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a (nonzero) constant term is 0. The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo of some prime number p. In most of the examples in this section, the coefficients are integers.
For example:
$-5x^2y\,$
is a term. The coefficient is –5, the variables are x and y, the degree of x is in the term two, while the degree of y is one.
The degree of the entire term is the sum of the degrees of each variable in it, so in this example the degree is 2 + 1 = 3.
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
$\underbrace{_\,3x^2}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}} \underbrace{-_\,5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}} \underbrace{+_\,4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}.$
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
The commutative law of addition can be used to freely permute terms into any preferred order. In polynomials with one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, variable x, and exponent 2. In the second term, the coefficient is –5. The third term is a constant. Since the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.
Two terms with the same variables raised to the same powers are called "like terms", and they can be combined (after having been made adjacent) using the distributive law into a single term, whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0, in which case their combination just cancels out the terms. Polynomials can be added using the associative law of addition (which simply groups all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if
$P=3x^2-2x+5xy-2 \,$
$Q=-3x^2+3x+4y^2+8 \, ,$
then
$P+Q=3x^2-2x+5xy-2-3x^2+3x+4y^2+8 \,,$
which can be simplified to
$P+Q=x+5xy+4y^2+6 \,.$
To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if
${\color{BrickRed}P {{=}} 2x + 3y + 5}$
${\color{RoyalBlue}Q {{=}} 2x + 5y + xy + 1},$
then
$\begin{array}{rccrcrcrcr} {\color{BrickRed}P}{\color{RoyalBlue}Q}&{{=}}&&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}2x}) &+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}2x}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}1}) \\&&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}3y}\cdot {\color{RoyalBlue}xy})&+& ({\color{BrickRed}3y}\cdot{\color{RoyalBlue}1}) \\&&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}5y})&+& ({\color{BrickRed}5}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}1}) \end{array}$
which can be simplified to
$PQ=4x^2+21xy+2x^2y+12x+15y^2+3xy^2+28y+5 \,.$
The sum or product of two polynomials is always a polynomial.
### Alternative forms
In general any expression can be considered a polynomial if it is built from variables and constants using only addition, subtraction, multiplication, and raising expressions to constant positive whole number powers. Such an expression can always be rewritten as a sum of terms. For example, (x + 1)3 is a polynomial; its standard form is x3 + 3x2 + 3x + 1.
Division of one polynomial by another does not, in general, produce a polynomial, but rather produces a quotient and a remainder.[6] A formal quotient of polynomials, that is, an algebraic fraction where the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. For example,
$\frac{x^3}{12}$
is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to $\tfrac{1}{12}x^3$ and $\tfrac{1}{12}$ is just a constant. When this expression is used as a term, its coefficient is therefore $\tfrac{1}{12}$. For similar reasons, if complex coefficients are allowed, one may have a single term like $(2+3i)x^3$; even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term.
${1 \over x^2 + 1} \,$
is not a polynomial because it includes division by a non-constant polynomial.
$( 5 + y ) ^ x ,\,$
is not a polynomial, because it contains a variable used as exponent.
Since subtraction can be replaced by addition of the opposite quantity, and since positive whole number exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and variables using only addition and multiplication.
### Polynomial functions
A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 \,$
for all arguments x, where n is a non-negative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function ƒ, taking real numbers to real numbers, defined by
$f(x) = x^3 - x\,$
is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in
$f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.\,$
An example is also the function $f(x)=\cos(2\arccos(x))$ which, although it doesn't look like a polynomial, is a polynomial function since for every x it is true that $f(x)=2x^2-1$ (see Chebyshev polynomials).
Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.
### Polynomial equations
A polynomial equation, also called algebraic equation, is an equation in which a polynomial is set equal to another polynomial.
$3x^2 + 4x -5 = 0 \,$
is a polynomial equation. In case of a univariate polynomial equation, the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like (x + y)(x – y) = x2 – y2, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.
## Elementary properties of polynomials
• A sum of polynomials is a polynomial.
• A product of polynomials is a polynomial.
• A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.
• The derivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is the polynomial nanxn-1 + (n-1)an-1xn-2 + ... + 2a2x + a1. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then kak should be interpreted as the sum of ak with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp+1 is the polynomial 0.
• If the division by integers is allowed in the set of coefficients, a primitive or antiderivative of the polynomial anxn + an-1xn-1 + ... + a2x2 + a1x + a0 is anxn+1/(n+1) + an-1xn/n + ... + a2x3/3 + a1x2/2 + a0x +c, where c is an arbitrary constant. Thus x2+1 is a polynomial with integer coefficients whose primitives are not polynomials over the integers. If this polynomial is viewed as a polynomial over the integers modulo 3 it has no primitive at all.
Polynomials serve to approximate other functions, such as sine, cosine, and exponential.
All polynomials have an expanded form, in which the distributive and associative laws have been used to remove all brackets and commutative law has been used to make the like terms adjacent and combine them. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. In the case of the field of complex numbers, the irreducible polynomials are linear. For example, the factored form of
$5x^3-5 \,$
is
$5(x - 1)(x^2+x + 1),\,$
over the integers and
$5(x - 1)(x+\frac{1+i\sqrt{3}}{2})(x+\frac{1-i\sqrt{3}}{2})\,$
over the complex numbers.
Every polynomial in one variable is equivalent to a polynomial with the form
$a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0.$
This form is sometimes taken as the definition of a polynomial in one variable.
Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Actual evaluation is usually more efficient using the Horner scheme:
$((\cdots((a_n x + a_{n-1})x + a_{n-2})x + \cdots + a_3)x + a_2)x + a_1)x + a_0.\,$
In elementary algebra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and equals the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.
A system of polynomial equations is a set of equations in which each variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, in particular in linear algebra, methods are given for solving a system of linear equations in several unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. Overdetermined systems are common in practical applications. For example, one U.S. mapping survey used computers to solve 2.5 million equations in 400,000 unknowns.[7]
Viète's formulas relate the coefficients of a polynomial to symmetric polynomial functions of its roots.
## History
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.
### Notation
The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.[8]
## Solving polynomial equations
Every polynomial P in x corresponds to a function, ƒ(x) = P (where the occurrences of x in P are interpreted as the argument of ƒ), called the polynomial function of P; the equation in x setting f(x) = 0 is the polynomial equation corresponding to P. The solutions of this equation are called the roots of the polynomial; they are the zeroes of the function ƒ (corresponding to the points where the graph of ƒ meets the x-axis). A number a is a root of P if and only if the polynomial x − a (of degree one in x) divides P. It may happen that x − a divides P more than once: if (x − a)2 divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (x − a)m divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined (or infinite) multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P.
Some polynomials, such as x2 + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.
There is a difference between approximating roots and finding exact expressions for roots. Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times (see quadratic equation), and for polynomials of degree 3 or 4 similar formulas (using cube roots in addition to square roots) were found in the 16th century (see cubic function and quartic function for the formulas and Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general (finite) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). In 1830, Évariste Galois, studying the permutations of the roots of a polynomial, extended Abel-Ruffini theorem by showing that, given a polynomial equation, one may decide if it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and Group theory, two important branches of modern mathematics. Galois himself noted that the computations implied by his method were impracticable. Nevertheless formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).
Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Jenkins-Traub method, Laguerre's method, Durand–Kerner method or by some other root-finding algorithm.
For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero. However for certain sets of such polynomials it may happen that for only finitely many combinations all polynomial functions take the value zero.
For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions. If the number of solutions is finite, there are algorithms to compute the solutions. The methods underlying these algorithms are described in the article systems of polynomial equations. The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.
It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterisations of the roots of arbitrary polynomials are generalisations of the methods previously discovered to solve the quintic equation.
## Properties of the roots
The statistical properties of the roots of a random polynomial have been the subject of several studies. Let
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$
be a random polynomial. If the coefficients ai are independently and identically distributed with a mean of zero, the real roots are mostly located near ±1. The complex roots can be shown to be located on or close to the unit circle.
If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula[9][10]
$p( x ) = \frac { \sqrt{ A( x ) C( x ) - B( x )^2 }} {\pi A( x )}$
where
$A( x ) = \sigma \sum { x^{ 2i } } = \sigma \frac{ x^{ 2n } - 1 } { x - 1 }$
$B( x ) = \frac{ 1 } { 2 } \frac{ d } { dt } A( x )$
$C( x ) = \frac{ 1 } { 4 } \frac{ d^2 } { dt^2 } A( x ) + \frac{ 1 } { 4x } \frac{ d } { dt } A( x )$
When the coefficients are Gaussian distributed with a non zero mean and variance of σ, a similar but more complex formula is known.
### Asymptotic results
For large n, a number of asymptotic formulae are known. For a fixed x
$p( x ) = \frac{ 1 } { \pi | 1 - x^2 | }$
and
$p( \pm 1 ) = \frac{ 1 } { \pi } \sqrt { \frac{ n^2 - 1 } { 12 } }$
where p( x ) is the mean density of real roots. The expected number of real roots is
$N_n = \frac{ 2 } { \pi } log( n ) + C + O( n^{ -2 } )$
where C is a constant approximately equal to 0.6257358072 and O() is the order operator.
This result has been shown by Kac, Erdos and others to be insensitive to the actual distribution of coefficients. Numerical testing of this formula has confirmed these earlier results.
## Graphs
A polynomial function in one real variable can be represented by a graph.
• The graph of the zero polynomial
f(x) = 0
is the x-axis.
• The graph of a degree 0 polynomial
f(x) = a0, where a0 ≠ 0,
is a horizontal line with y-intercept a0
• The graph of a degree 1 polynomial (or linear function)
f(x) = a0 + a1x , where a1 ≠ 0,
is an oblique line with y-intercept a0 and slope a1.
• The graph of a degree 2 polynomial
f(x) = a0 + a1x + a2x2, where a2 ≠ 0
is a parabola.
• The graph of a degree 3 polynomial
f(x) = a0 + a1x + a2x2, + a3x3, where a3 ≠ 0
is a cubic curve.
• The graph of any polynomial with degree 2 or greater
f(x) = a0 + a1x + a2x2 + ... + anxn , where an ≠ 0 and n ≥ 2
is a continuous non-linear curve.
The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
The illustrations below show graphs of polynomials.
## Polynomials and calculus
One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomial functions. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Polynomial functions are also frequently used to interpolate functions.
Calculating derivatives and integrals of polynomial functions is particularly simple. For the polynomial function
$\sum_{i=0}^n a_i x^i$
the derivative with respect to x is
$\sum_{i=1}^n a_i i x^{i-1}$
and the indefinite integral is
$\sum_{i=0}^n {a_i\over i+1} x^{i+1}+c.$
## Abstract algebra
In abstract algebra, one distinguishes between polynomials and polynomial functions. A polynomial f in one variable X over a ring R is defined as a formal expression of the form
$f = a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X^1 + a_0X^0$
where n is a natural number, the coefficients $a_0,\ldots,a_n$ are elements of R, and X is a formal symbol, whose powers Xi are just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence $(a_0, a_1, \ldots)$, where there is an n such that ai = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term aiXi is interpreted as a polynomial that has zero coefficients at all powers of X other than Xi. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule
$a X^k \; b X^l = ab X^{k+l}$
for all elements a, b of the ring R and all natural numbers k and l.
Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. The map from R to R[X] sending r to rX0 is an injective homomorphism of rings, by which R is viewed as a subring of R[X]. If R is commutative, then R[X] is an algebra over R.
One can think of the ring R[X] as arising from R by adding one new element X to R, and extending in a minimal way to a ring in which X satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is Xr = rX). To do this, one must add all powers of X and their linear combinations as well.
Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring R[X] over the real numbers by factoring out the ideal of multiples of the polynomial X2 + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).
If R is commutative, then one can associate to every polynomial P in R[X], a polynomial function f with domain and range equal to R (more generally one can take domain and range to be the same unital associative algebra over R). One obtains the value f(r) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for X. And it should be noted that if R is not commutative, there is no (well behaved) notion of polynomial function at all.
### Divisibility
In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[X] such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (Xr) divides f. The converse is also true. The quotient can be computed using the Horner scheme.
If F is a field and f and g are polynomials in F[X] with g ≠ 0, then there exist unique polynomials q and r in F[X] with
$f = q \, g + r$
and such that the degree of r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[X] is a Euclidean domain.
Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as polynomials which cannot be factorized into the product of two non constant polynomials. Any polynomial may be decomposed into the product of a constant by a product of irreducible polynomials. This decomposition is unique up to the order of the factors and the multiplication of any constant factors by a constant (and division of the constant factor by the same constant. When the coefficients belong to a finite field or are rational numbers, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials. These algorithms are not practicable for hand written computation, but are available in any Computer algebra system (see Berlekamp's algorithm for the case in which the coefficients belong to a finite field or the Berlekamp–Zassenhaus algorithm when working over the rational numbers [11]). Eisenstein's criterion can also be used in some cases to determine irreducibility.
## Classifications
Polynomials are classified according to many different properties.
### Number of variables
One classification of polynomials is based on the number of distinct variables. A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result, for instance, from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any variables at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of variables allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is common, also, to say simply "polynomials in x, y, and z", listing the variables allowed. In this case, xy is allowed.
### Degree
A second major way of classifying polynomials is by their degree. Recall that the degree of a term is the sum of the exponents on variables, and that the degree of a polynomial is the largest degree of any one term.
Polynomials classified by degree
Degree Name Example
−∞ zero $0$
$0$ (non-zero) constant $1$
$1$ linear $x + 1$
$2$ quadratic $x^2 + 1$
$3$ cubic $x^3 + 1$
$4$ quartic (or biquadratic) $x^4 + 1$
$5$ quintic $x^5 + 1$
$6$ sextic (or hexic) $x^6 + 1$
$7$ septic (or heptic) $x^7 + 1$
$8$ octic $x^8 + 1$
$9$ nonic $x^9 + 1$
$10$ decic $x^{10} + 1$
$100$ hectic $x^{100} + 1$
Usually, a polynomial of degree n, for n greater than 3, is called a polynomial of degree n, although the phrases quartic polynomial and quintic polynomial are sometimes used. The use of names for degrees greater than 5 is even less common. The names for the degrees may be applied to the polynomial or to its terms. For example, in $x^2 + 2x + 1$ the term $2x$ is a first degree term in a second degree polynomial.
In the context of polynomial interpolation there is some ambiguity when combining the two classifications above. For example, a bilinear interpolant, being the product of two univariate linear polynomials, is bivariate but is not linear; similar ambiguity affects the bicubic interpolant.
The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either –1 or –∞).[12] These conventions are important when defining Euclidean division of polynomials. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots.
Polynomials classified by number of non-zero terms
Number of non-zero terms Name Example
$0$ zero polynomial $0$
$1$ monomial $x^2$
$2$ binomial $x^2 + 1$
$3$ trinomial $x^2 + x + 1$
If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in x of degree n then takes the general form
$c_nx^n+c_{n-1}x^{n-1}+\cdots+c_2x^2+c_1x+c_0$
where
cn ≠ 0, cn-1, ..., c2, c1 and c0 are constants, the coefficients of this polynomial.
Here the term cnxn is called the leading term and its coefficient cn the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.
Note that apart from the leading coefficient cn (which must be non-zero or else the polynomial would not be of degree n) this general form allows coefficients to be zero; when this happens the corresponding term is zero and may be removed from the sum without changing the polynomial. It is nevertheless common to refer to ci as the coefficient of xi, even when ci happens to be 0, so that xi does not really occur in any term; for instance one can speak of the constant term of the polynomial, meaning c0 even if it is zero.
In the case of polynomials in more than one variable, a polynomial is called homogeneous of degree n if all its terms have degree n. For example, $x^3y^2 + 7x^2y^3 - 3x^5$ is homogeneous.
### Coefficients
Another classification of polynomials is by the kind of constant values allowed as coefficients. One can work with polynomials with integer, rational, real, or complex coefficients, and in abstract algebra polynomials with many other types of coefficients can be defined, such as integers modulo p. As in the classification by number of variables, when working with coefficients for a given set, such as the complex numbers, coefficients from any subset are allowed. Thus $x^2 + 3x -5$ is a polynomial with integer coefficients, but it is also a polynomial with complex coefficients, because the integers are a subset of the complex numbers.
### Number of non-zero terms
Polynomials may also be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and so on. (Some authors use "monomial" to mean "monic monomial".[13])
## Polynomials associated to other objects
Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.
## Extensions of the concept of a polynomial
Polynomials can involve more than one variable, in which they are called multivariate. Rings of polynomials in a finite number of variables are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. These rings can alternatively be constructed by repeating the construction of univariate polynomials with as coefficient ring another ring of polynomials: thus the ring R[X,Y] of polynomials in X and Y can be viewed as the ring (R[X])[Y] of polynomials in Y with as coefficients polynomials in X, or as the ring (R[Y])[X] of polynomials in X with as coefficients polynomials in Y. These identifications are compatible with arithmetic operations (they are isomorphisms of rings), but some notions such as degree or whether a polynomial is considered monic can change between these points of view. One can construct rings of polynomials in infinitely many variables, but since polynomials are (finite) expressions, any individual polynomial can only contain finitely many variables.
A binary polynomial where the second variable takes the form of an exponential function applied to the first variable, for example P(X,eX ), may be called an exponential polynomial.
Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.
Quotients of polynomials are called rational expressions (or rational fractions), and functions that evaluate rational expressions are called rational functions. Rational fractions are formal quotients of polynomials (they are formed from polynomials just as rational numbers are formed from integers, writing a fraction of two of them; fractions related by the canceling of common factors are identified with each other). The rational function defined by a rational fraction is the quotient of the polynomial functions defined by the numerator and the denominator of the rational fraction. The rational fractions contain the Laurent polynomials, but do not limit denominators to powers of a variable. While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not null. A rational function produces rational output for any rational input for which it is defined; this is not true of other functions such as trigonometric functions, logarithms and exponential functions.
Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like real numbers cannot), but the rules for manipulating their terms are the same as for polynomials.
## Notes
1. ^ CNTRL (French National Center for Textual and Lexical Resources), etymology of binôme [1]
2. ^ Etymology of "polynomial" Compact Oxford English Dictionary
3. ^ Online Etymology Dictionary "binomial"
4. ^ Florian Cajori (1991). A History of Mathematics. AMS. ISBN 978-0-8218-2102-2. |[2]
5. ^ The term indeterminate is more proper, and, in theory, variable should be used only when considering the function defined by the polynomial. In practice, most authors use indifferently the two words.
6. ^ Peter H. Selby, Steve Slavin, Practical Algebra: A Self-Teaching Guide, 2nd Edition, Wiley, ISBN 0-471-53012-3 ISBN 978-0471530121
7. ^ Gilbert Strang, Linear Algebra and its Applications, Fourth Edition, Thompson Brooks/Cole, ISBN 0-03-010567-6.
8. ^ Howard Eves, An Introduction to the History of Mathematics, Sixth Edition, Saunders, ISBN 0-03-029558-0
9. ^ Kac M (1943) Bull Am Math Soc 49, 314
10. ^ Kac M (1948) Proc London Math Soc 50, 390
11. ^ http://mathworld.wolfram.com/Berlekamp-ZassenhausAlgorithm.html
12. ^
13. ^ Anthony W. Knapp (2007). Advanced Algebra: Along with a Companion Volume Basic Algebra. Springer. p. 457. ISBN 0-8176-4522-5.
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David McDonald, "Elements of Applied Probability for Engineering, Mathematics and Systems Science"
English | ISBN: 9812387390, 9812387404 | 2004 | 376 pages | PDF | 18 MB
Elements of Applied Probability for Engineering, Mathematics and Systems Science
This book has been designed for senior engineering, mathematics and systems science students. In addition, the author has used the optional, advanced sections as the basis for graduate courses in quality control and queueing. It is assumed that the students have taken a first course in probability but that some need a review. Discrete models are emphasized and examples have been chosen from the areas of quality control and telecommunications. The book provides correct, modern mathematical methods and at the same time conveys the excitement of real applications.
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# Noise and Signal Interference in Optical Fiber Transmission Systems: An Optimum Design Approach
FREEDownload : Noise and Signal Interference in Optical Fiber Transmission Systems: An Optimum Design Approach
Noise and Signal Interference in Optical Fiber Transmission Systems: An Optimum Design Approach
by Stefano Bottacchi
English | 2009 | ISBN: 0470060611 | 854 pages | PDF | 169.14 MB
Noise and Signal Interference in Optical Fiber Transmission Systems: An Optimum Design Approach
"Noise and Signal Interference in Optical Fiber Transmission Systems is a compendium on specific topics within optical fiber transmission and the optimization process of the system design. It offers comprehensive treatment of noise and intersymbol interference (ISI) components affecting optical fiber communications systems, containing coverage on noise from the light source, the fiber and the receiver. The ISI is modeled with a statistical approach, leading to new useful computational methods. The author discusses the subject with the help of numerous applications and simulations of noise and signal interference theory.
Key features:
– Complete all-in-one reference on the subject for engineers and designers of optical fiber transmission systems
– Discusses the physical principles behind several noise contributions encountered in the optical communications systems design, including contributions from the light source, the fiber and the receiver
– Covers the theory of the ISI for the binary signal, as well as noise statistics
– Discusses the theory and the mathematical models of the numerous noise components (such as optical noise, photodetection noise and reflection noise)
– Introduces the frequency description of the ISI and provides new calculation methods based on the characteristic functions
– Provides useful tools and examples for optimum design of optical fiber transmission networks and systems.
This book will serve as a comprehensive reference for researchers, R&D engineers, developers and designers working on optical transmission systems and optical communications. Advanced students in optical communications and related fields will also find this book useful."
Buy Premium To Support Me & Get Resumable Support & Max Speed | 609 | 3,318 | {"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} | 2.546875 | 3 | CC-MAIN-2018-43 | latest | en | 0.858751 |
https://www.macscripter.net/t/applescript-for-beginners-x-math/45792 | 1,696,069,425,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510671.0/warc/CC-MAIN-20230930082033-20230930112033-00868.warc.gz | 953,305,408 | 7,874 | # AppleScript for Beginners X - Math
We are not going to do a lot of scripting today; rather we are going to look a little more closely at some of the nifty mathematic functions that AppleScript offers us scripters. Although this sounds boring, a good understanding of these basic concepts can make a difference in your scripts as your skill continues to increase. First, however, let’s determine how AppleScript thinks about numbers in general. This page tells us that the term Number can be used interchangeably with the terms Integer or Real in a script, but also hints at what we really need to know, which is simply that a number of class integer is a whole number (no decimals) and that a number of class real is just what it sounds like, a real number, with all the necessary digits to the right of the decimal. (See Apple’s definitions of Integers and Real Numbers Here is an example script:
``````set number_1 to 24
set number_2 to 43.131
set the_product to number_1 * number_2
display dialog "Real: " & (the_product as real) & return & "Integer: " & (the_product as integer)
``````
All we did here was define two variables as numbers (24 and 43.121), then set another variable to the product of those two numbers (number_1 * number_2). In the display dialog, you see that we took the value of that variable (the_product) and coerced it to a real number, and then to an integer. Coercion is telling AppleScript to take one type of information, and force it to another type, and it is happy to do this for you anytime you want, as long as it follows the native coercion rules. (Those rules can be found here.)
You can coerce numbers to strings, strings to numbers, integers to reals, etc., etc. In my example script, we just took the number in the_product and coerced it into both number types (real & integer) to see what it would look like, displaying that the real number preserves the decimal places, and the integer number removes all the digits to the right of the decimal.
You have all the basic mathematical operators at your disposal:
[b]* (multiplication)
• (subtraction)
/ or ÷ (division)[/b] (You get the ÷ symbol by pressing alt and /)
You can also use the ^ character to raise a number to a power (use SHIFT and 6 to get the character), as in this script:
``````8 ^ 2
``````
In conjunction with division, you have two unique operators to play with, div and mod. (Apple’s explanations for these can be found here.)
When you use div, you get the integer result of a division operation. Here is an example script:
``````set number_3 to 400
set number_4 to 21
display dialog "Using simple division, 400 ÷ 21 = " & (number_3 / number_4) & return & "while using the div operand, you get: " & (number_3 div number_4)
``````
Now, before we move on, let me see if I can read your mind. You are probably thinking that div is just like doing division and then coercing to an integer, like this:
``````20 ÷ 3 as integer
``````
Well, Miss or Mister Smarty-Pants, let’s check that out:
``````set twenty to 20
set three to 3
display dialog "Using div: " & (twenty div three) & return & "With division & coercion: " & ((twenty / three) as integer)
``````
This displays one of the AppleScript mathematical dirty little secrets: coercing to an integer induces rounding, while using div simply chops off the decimal data. We will talk more about rounding later, so let’s cover the usage of mod.
I am not a mathematician by any means, so I have very little idea about why this is called mod, or even what it is really useful for, but I think that it at least has a coolness factor that deserves discussion. Mod is pretty much the opposite of div, in that it returns the remainder of an unclean division operation, NOT any decimal data. For instance, 6 goes into 70 11 times with a remainder of 4, right Let’s see:
``````set seventy to 70
set six to 6
display dialog "For the expression 70 ÷ 6:
Division: " & seventy div six & return & "Remainder: " & seventy mod six
``````
Yep, our fourth grade math skills are still evident. And although interesting, I still can’t think of a real world application for this.
So, let’s take a moment and explore AppleScript’s rounding rules. They are usually only in effect when coercing decimal data to an integer, which natively rounds to the nearest whole integer. The rules are very similar to what we learned in grade school. You can see that with this little repeat loop:
``````set x to 1
repeat 10 times
display dialog (x & " rounds to " & (x as integer)) as string
set x to x + 0.1
end repeat
``````
You are not limited to only using built-in rounding, however. Look in the dictionary for the StandardAdditions under the Miscellaneous Commands section, and you see the command round. The definition briefly describes your various uptions, and the technical information provided by Apple is found here. You can use this command to round your non-whole numbers any way you want. For example, if you wanted every thing rounded up, you would do this:
``````set x to 1
repeat 10 times
display dialog (x & " rounds up to " & (round x rounding up)) as string
set x to x + 0.1
end repeat
``````
As you can see in the dictionary and technical notes, there are some other ways to round your data as well, so play around with it.
It is also useful to note the order in which AppleScript performs mathematical tasks. Table A-8 (Operator precedence) of the AppleScript Language Guide (Click on Operators in The Language at a Glance and scroll to the bottom of the page) lists the order of 12 groups of operators. The first is the parentheses, second is number signs, then raising to a power, and fourth is multiplication, division, and div or mod. Here is an example of what this means:
``````2 ^ 2 * 2 --> Results in 8
2 ^ (2 * 2) --> Results in 16
``````
Since the raising to a power is higher in precedence, the 2 ^ 2 will be performed first in the first example, and that result will then be mulitiplied by 2, which returns a product of 8. If, however, we enclose the 2 * 2 inside parentheses (as in the second example), it is then performed first, and that product (4) is used as the exponent for the first operation, which then returns 16.
Okay, I know that this is all terribly exciting, but even if you believe that you will never need to know this stuff, I promise you that you are wrong, and someday it will come in handy, if for no other reason than to make you stop and think (then refer to the reference for guidance). As you read through other people’s scripts, you will find some very clever uses of mathematical functions that help to either speed up a task, or simplify a method of analysis. Study these things, and see for yourself if these basic commands don’t help you as well.
“mod” stands for “modulo” in computer programming which is taken from “modular arithmetic” or “clock arithmetic”
so:
set number_1 364 --number of seconds
set number_2 60 --seconds in a minute
set the_seconds number_1 mod number_2
set the_minutes (number_1 - the_seconds) / number_2 as string
set the_time the_minutes & “:” & the_seconds as string
So now you know.
Just to add a prettier output to Catamount’s clock stuff above; this one-liner converts “6.0:4” to “6:04” …
It forces the minutes-value to abandon its decimal point.
It then adds a leading-zero to the display of seconds, but returns only it’s last two numbers.
``````
set number_1 to 364 --number of seconds
set number_2 to 60 --seconds in a minute
set the_seconds to number_1 mod number_2
set the_minutes to (number_1 - the_seconds) / number_2 as string
set the_time to the_minutes & ":" & the_seconds as string
set the_PRETTYtime to (the_minutes as integer) & ":" & (items -2 thru -1 of ("0" & the_seconds)) as string
return {the_time, the_PRETTYtime}
#--result---> {"6.0:4", "6:04"}
``````
Model: Mac Pro (Early 2009)
AppleScript: 2.4
Browser: Safari 603.3.8
Operating System: Mac OS X (10.13 Developer Beta 3) | 1,982 | 7,952 | {"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} | 3.515625 | 4 | CC-MAIN-2023-40 | longest | en | 0.932797 |
https://www.coursehero.com/file/p770bq3/Galilean-mechanics-An-apple-falls-from-a-tree-How-does-one-understand-the-free/ | 1,627,697,674,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154042.23/warc/CC-MAIN-20210731011529-20210731041529-00526.warc.gz | 715,368,996 | 101,525 | Galilean mechanics An apple falls from a tree How does one understand the free
# Galilean mechanics an apple falls from a tree how
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Galilean mechanics An apple falls from a tree. How does one understand the free fall phenomenon? Galileo began with the assumption that the motion proceeds according to the simplest possible law: equal increase in velocity in equal periods of time. Today we call it uniformly accelerated motion, in modern notation v t . Yet, the correctness of this assumption could not be shown by direct experiment because instantaneous velocity could not be measured. Galileo wrote down the distance traveled by a free-falling body as a function of time by applying the “mean-speed theorem” which claims that a body moving with a uniformly accelerated motion covers the same distance in a given time as if it were to move for the same duration with a uniform speed equal to its mean speed. 26 Mathematically, the distance is given by s = 1 2 vt . Hence, the distances traveled by a body in free fall are proportional to the squares of the times, s t 2 . Hence: s 1 t 1 2 = s 2 t 2 2 = s 3 t 3 2 = This result can be experimentally verified since both the distance and time can be measured, and the proportionality checked thereby verifying the assumption v t . But free fall involves very small time intervals that were difficult to measure. Galileo overcame this difficulty by employing an inclined plane with a small angle of inclination. This could “slow 24 René Descartes (1596 – 1650 CE), a French philosopher, mathematician, and scientist. 25 Principia is the short form of Philosophiæ Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy ). 26 The theorem was proposed by the Merton scholars in the 14 th century. Lindberg, D (2007), pp. 304 – 305.
GE ST: History of Science and Technology in the West P. 19 /25 down” free fall to the point where he could measure the time intervals with sufficient accuracy using the instruments available. Galileo further argued that increasing the plane’s angle of inclination was equivalent to gradually approaching the case of free fall. This is the first time in the history of mechanics that a detailed presentation of an experiment and the experimental conditions in the manner that we would expect in a scientific publication today. Galileo’s approach to science demonstrates the roles of experimental investigation in scientific theory. (a) Galileo began with an explanation of the terms; in this case, he defined what is meant by free fall with physical quantities. (b) He formulated a hypothesis concerning the expected distance-time relation. (c) From this hypothesis he then analytically (or mathematically) derived relationships that can be verified experimentally. (d) Finally, Galileo performed experiments to test the theoretical predictions. Galileo demonstrated a modern appreciation of the roles and the relationship among theory, mathematics, and experiments. | 639 | 3,033 | {"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} | 3.53125 | 4 | CC-MAIN-2021-31 | latest | en | 0.929516 |
https://www.coursehero.com/file/184491/sg6/ | 1,490,866,534,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218193288.61/warc/CC-MAIN-20170322212953-00342-ip-10-233-31-227.ec2.internal.warc.gz | 871,735,369 | 22,273 | # Sg6 - Chapter 6 Study Guide for Energy 6.1 Isolated systems Skill 6.1 Understand qualitatively how an interaction between objects affects the in
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Unformatted text preview: Chapter 6 Study Guide for Energy 6.1 Isolated systems Skill 6.1 Understand qualitatively how an interaction between objects affects the in- teracting objects In beginning a problem, first define your system. Our choice of how to define the system is completely arbitrary. We want to choose our system so that it makes our problem easier to solve. This means that we only want to consider those things that have an effect on our problem. We can identify the things affecting our problem by looking at interactions. An interaction occurs when two objects affect each other such that both are accelerated. Interactions within a system are called internal interactions. Anything outside of our system is defined as the environment . Internal Example: (Defining two carts as the system) Assuming no friction with the ground or air, two carts collide causing the velocities of the carts to change (they were accelerated), thus, an interaction took place within the system. Interactions of a system with something in its environment are called external interactions. External Example: (Defining a ball as the system) Imagine throwing a ball. The ball starts out at rest and leaves your hand with some velocity. This acceleration is due to an interaction between you and the ball. Once the ball is released, there is an interaction between the earth and the ball (via the earth’s gravitational field) that causes it to accelerate toward the ground. The ball collides with air molecules which interact with the ball, slowing it down. Note: From the above example, we see that two objects don’t have to be in contact to interact. The hand and air touch the ball but the Earth does not. The Earth affects the ball by its gravitational field (we’ll learn the details about this in chapter 14). Skill 6.2 Be able to identify and construct isolated systems Depending on how we choose our system, as mentioned above, there may be internal interactions or external interactions. If we go back to the example with the thrown ball and choose the earth and ball to be our system, then the interaction between the earth and ball is internal to the system, and the interaction between the thrower and ball is between the system and its environment. If there are no external interactions (between system and environment), then it is called an isolated system . Essentially, the environment does not exist to an isolated system. The total linear momentum of an isolated system is always conserved. 6.2 Classification of collisions Skill 6.3 Be able to distinguish between elastic, inelastic, and totally inelastic collisions. 1 Although momentum is always conserved in isolated systems, the outcomes of collisions (another word for in- teractions) are not all the same. Just think of the difference between two billiard balls colliding and two cars colliding. We can classify collisions by comparing the difference in the two colliding objects’ velocities before the collision to the difference in velocities after the collision. We call this difference in velocity thecollision to the difference in velocities after the collision....
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Sg6 - Chapter 6 Study Guide for Energy 6.1 Isolated systems Skill 6.1 Understand qualitatively how an interaction between objects affects the in
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Ask a homework question - tutors are online | 815 | 3,985 | {"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} | 3.421875 | 3 | CC-MAIN-2017-13 | longest | en | 0.941739 |
https://q-and-answers.com/mathematics/question15038325 | 1,643,120,031,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304835.96/warc/CC-MAIN-20220125130117-20220125160117-00557.warc.gz | 536,386,204 | 17,233 | # 3. Shane has two SIMILAR regular pyramids with Pentagon-shaped bases. The smaller has a scale factor of 2:3 when compared to the larger. Only the smaller pyramid is shown. He calculates the area of the base of the pyramid (through long, hard work) to be 110.11 square units. The height of the pyramid is 12 units. He now needs to calculate the Volume of the pyramid A. Calculate the Volume of the Pyramid for Shane B. “oh, no!” Shane exclaims. “Now I have to go through all this hard work again to find the Volume of the larger pyramid!” Does he? C. Calculate the Volume of the larger pyramid for Shane. Need help please ASAP!
haileysolis5 · 06.03.2020 18:12
26.06.2019 13:20
no
nostep-by-step explanation:
no
26.06.2019 06:00
Can you provide the tape diagram?
25.06.2019 21:30
13.08.2020 21:37
A) 440.44 units³
B) similar figures
C) 1486.485 units³
Step-by-step explanation:
Volume = ⅓× base area × height
= ⅓ × 110.11 × 12 = 440.44 units³
B) he doesn't have to make calculations again because the two pyramids are similar.
If the ratio of sides is given, he can find the other volume using:
(side1/side2)³ = volume1/volume2
(2/3)³ = 440.44/volume2
Volume2 = 440.44 × 27/8
Volume2 = 1486.485 units³
Step-by-step explanation:
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22.06.2019 08:00 | 907 | 3,023 | {"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} | 4.125 | 4 | CC-MAIN-2022-05 | latest | en | 0.874524 |
https://www.iitianacademy.com/cie-as-a-level-physics-10-2-kirchhoffs-laws-exam-style-question-paper-1/ | 1,726,827,304,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700652246.93/warc/CC-MAIN-20240920090502-20240920120502-00728.warc.gz | 750,038,251 | 36,120 | Home / CIE AS & A Level Physics : 10.2 Kirchhoff’s laws- Exam style question – Paper 1
# CIE AS & A Level Physics : 10.2 Kirchhoff’s laws- Exam style question – Paper 1
### Question
Each of Kirchhoff’s two laws presumes that some quantity is conserved.
Which row states Kirchhoff’s first law and names the quantity that is conserved?
Ans:
### Question
In the circuit shown, the cells have negligible internal resistance and the reading on the galvanometer is zero.
What is the value of resistor R?
A $$2.0\, \Omega$$ B $$6.0\, \Omega$$ C $$12\, \Omega$$ D $$18\, \Omega$$
Ans:
### Question
Two batteries are connected together, as shown.
Battery 1 has electromotive force (e.m.f.) 12V and internal resistance 0.3Ω.
Battery 2 has e.m.f. 9V and internal resistance 0.1Ω.
What are the e.m.f. and the internal resistance of a single battery that has the same effect as the combination?
e.m.f. /V internal resistance/Ω ABCD 332121 0.20.40.20.4
Ans:
### Question
In the circuit shown, the 6.0V battery has negligible internal resistance. Resistors R1 and R2 and
the voltmeter each have a resistance of 100 $$k\omega$$.
What is the current in the resistor $$R_2$$?
A 20 $$\mu A$$ B 30 $$\omega A$$ C 40 $$\omega A$$ D 60 $$\omega A$$ | 388 | 1,312 | {"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} | 3.03125 | 3 | CC-MAIN-2024-38 | latest | en | 0.794794 |
https://discussions.unity.com/t/pendulum-like-effect-with-2-degrees-of-freedom-movable-pivot/584505 | 1,722,969,955,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640497907.29/warc/CC-MAIN-20240806161854-20240806191854-00306.warc.gz | 167,881,590 | 6,940 | # Pendulum like effect with 2 degrees of freedom movable pivot
Hi all,
I’m trying to recreate the physics which can be seen in:
Terrafire:
Hopefully you get the idea from the videos - there is a ship with some mass, a ball with some mass and a rope which hangs the ball from the ship.
As the player moves the ship the bomb moves as if it is tethered with a rope, the bomb swings around underneath the ship (kind of like a pendulum). One other thing to note is that the bomb effects the ship, so it will pull the ship in the direction it is travelling (i.e. if the player is moving right and the bomb is trailing behind, then the player suddenly stops the bomb will swing out to the right and the ship will be pulled the right a bit).
With the explanation over I’ll explain what I’ve done so far:
I’ve created the ship movement code which works as intended using rigid body physics.
I’ve given as good a go as I can at creating the pendulum script, it’s kind of working but it has 3 major problems (in order of importance):
1. The ball doesn’t stay tethered, it acts as if it’s attached with elastic rather than rope. Eventually equilibrium returns but it shouldn’t be “elastic” at all, when tight the rope should stop the ball from travelling any further from the ship (it should just act as a hinge joint at this point).
2. The pendulum motion doesn’t slow down, once the players ship becomes out of line with the ball (vertically) then the pendulum motion starts and it swings back and forth for ever, but I want it to slow down and eventually settle beneath the ship.
3. The ship isn’t affected by the ball yet - this is fine for now as I just want to get the ball swinging correctly. But how would I go about getting the balls momentum to affect the ship in the future?
I’ve attached the full project, the pendulum script is fully commented, any help is greatly appreciated!
2154415–142265–Terra.7z (93.5 KB)
Instead of using your script, I used a chain of configurable joints.
Check out the unitypackage and play around with the settings to get what youd like.
Nice one, it’s not doing exactly what I was aiming for but it’s a good starting point, thanks for taking the time to do that dude!
1 Like | 503 | 2,219 | {"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} | 2.875 | 3 | CC-MAIN-2024-33 | latest | en | 0.944715 |
https://laytonforstatesenate.com/applied-behavior-analysis/ | 1,606,713,802,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141205147.57/warc/CC-MAIN-20201130035203-20201130065203-00279.warc.gz | 365,833,436 | 10,569 | # Applied Behavior Analysis
A-B design
A two-phase experimental design consisting of a pre treatment baseline condition (A) followed by a treatment condition (B)
affirmation of the consequent
A three step form of reasoning that begins with a true antecedent-consequent (if A – then B) statement and proceeds as follows:1) If A is true, then B is true2) B is found to be true3) therefore, A is true
ascending baseline
A data path that shows an increasing trend in the response measure over time
Baseline
A condition of an experiment in which the independent variable is not present; data obtained during baseline are the basis for determining the effect of the independent variable
baseline logic
A term sometimes used to refer to the experimental reasoning inherent in single-subject experimental designs; entails 3 elements: prediction, vrification, and replication
confounding variable
An uncontrolled factor known or suspected to exert influence on the dependent variable
dependent variable
The target behavior or measurable dimensional quantity of that behavior (rate,duration) *Behavior is dependent on the independent variable
descending baseline
A data path that shows a decreasing trend in the response measure over time.
experimental control
The outcome of an experiment that demonstrates convincingly a functional relation, meaning that experintal control is acheived when a predictable change in behavior (dependent variable) can be reliably produced by manipulating a specific aspect of the environment(independent variable).
experimental design
The particular type and sequence of conditions in a study so that meaningful comparisons of the effects of the presence and absence (or different values) of the independent variable can be made.
experimental question
A statement of what the researcher seeks to learn by conducting the experiment(research question)
external validity
The degree to which a study’s findings have generality to other subjects, settings, and/or behaviors.
extraneous validity
Any aspect of the experimental setting (lighting,temp) that must be held constant to prevent unplanned environmental variation.
independent variable
The particular aspect of the environment that the experimenter manipulates to find out whether it affects the subjects behavior.(intervention,treatment,experimental variable)
internal validity
Experiment shows convincingly that changes in behavior are a function of the independent variable and not as a;result of uncontrolled/unknown variables.
parametric analysis
An experiment designed to discover the differiental effects of a range of values of an independent variable (ex. comparing the effects of several tratment plans)
practice effects
Improvements in performance resulting from opportunities to perform a behavior repeatedly so that baseline measures can be obtained.(confounds a study and baseline should be continued until steady state. )
prediction
A statement of the anticipated outcome of a presently unknown or future measureOne of the 3 components of experimental reasoning (+ replication,verification)
replication
Repeating conditions within an experiment to determine the reliability of effects and increase internal validity(+prediction,verification)
single-subject deigns
Repeated measures of the subject’s behavior are obtained as she/he are exposed to each condition. The participant is theirown control(within-subject design)
stable baseline
Data that shows no evidence of an upward or downward trend (w/i a small range) | 654 | 3,549 | {"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} | 2.546875 | 3 | CC-MAIN-2020-50 | latest | en | 0.895064 |
http://de.metamath.org/mpeuni/isxmet.html | 1,718,519,666,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861643.92/warc/CC-MAIN-20240616043719-20240616073719-00370.warc.gz | 6,700,518 | 7,362 | Metamath Proof Explorer < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > isxmet Structured version Visualization version GIF version
Theorem isxmet 21939
Description: Express the predicate "𝐷 is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
isxmet (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Distinct variable groups: 𝑥,𝑦,𝑧,𝐷 𝑥,𝑋,𝑦,𝑧
Allowed substitution hints: 𝐴(𝑥,𝑦,𝑧)
Proof of Theorem isxmet
Dummy variables 𝑑 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . . . . 5 (𝑋𝐴𝑋 ∈ V)
2 xpeq12 5058 . . . . . . . . 9 ((𝑡 = 𝑋𝑡 = 𝑋) → (𝑡 × 𝑡) = (𝑋 × 𝑋))
32anidms 675 . . . . . . . 8 (𝑡 = 𝑋 → (𝑡 × 𝑡) = (𝑋 × 𝑋))
43oveq2d 6565 . . . . . . 7 (𝑡 = 𝑋 → (ℝ*𝑚 (𝑡 × 𝑡)) = (ℝ*𝑚 (𝑋 × 𝑋)))
5 raleq 3115 . . . . . . . . . 10 (𝑡 = 𝑋 → (∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))))
65anbi2d 736 . . . . . . . . 9 (𝑡 = 𝑋 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
76raleqbi1dv 3123 . . . . . . . 8 (𝑡 = 𝑋 → (∀𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
87raleqbi1dv 3123 . . . . . . 7 (𝑡 = 𝑋 → (∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))))
94, 8rabeqbidv 3168 . . . . . 6 (𝑡 = 𝑋 → {𝑑 ∈ (ℝ*𝑚 (𝑡 × 𝑡)) ∣ ∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} = {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
10 df-xmet 19560 . . . . . 6 ∞Met = (𝑡 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑡 × 𝑡)) ∣ ∀𝑥𝑡𝑦𝑡 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑡 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
11 ovex 6577 . . . . . . 7 (ℝ*𝑚 (𝑋 × 𝑋)) ∈ V
1211rabex 4740 . . . . . 6 {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ∈ V
139, 10, 12fvmpt 6191 . . . . 5 (𝑋 ∈ V → (∞Met‘𝑋) = {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
141, 13syl 17 . . . 4 (𝑋𝐴 → (∞Met‘𝑋) = {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))})
1514eleq2d 2673 . . 3 (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))}))
16 oveq 6555 . . . . . . . 8 (𝑑 = 𝐷 → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1716eqeq1d 2612 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) = 0 ↔ (𝑥𝐷𝑦) = 0))
1817bibi1d 332 . . . . . 6 (𝑑 = 𝐷 → (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦)))
19 oveq 6555 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑥) = (𝑧𝐷𝑥))
20 oveq 6555 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑧𝑑𝑦) = (𝑧𝐷𝑦))
2119, 20oveq12d 6567 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) = ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))
2216, 21breq12d 4596 . . . . . . 7 (𝑑 = 𝐷 → ((𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2322ralbidv 2969 . . . . . 6 (𝑑 = 𝐷 → (∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)) ↔ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))
2418, 23anbi12d 743 . . . . 5 (𝑑 = 𝐷 → ((((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
25242ralbidv 2972 . . . 4 (𝑑 = 𝐷 → (∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦))) ↔ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2625elrab 3331 . . 3 (𝐷 ∈ {𝑑 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∣ ∀𝑥𝑋𝑦𝑋 (((𝑥𝑑𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝑑𝑦) ≤ ((𝑧𝑑𝑥) +𝑒 (𝑧𝑑𝑦)))} ↔ (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))
2715, 26syl6bb 275 . 2 (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
28 xrex 11705 . . . 4 * ∈ V
29 sqxpexg 6861 . . . 4 (𝑋𝐴 → (𝑋 × 𝑋) ∈ V)
30 elmapg 7757 . . . 4 ((ℝ* ∈ V ∧ (𝑋 × 𝑋) ∈ V) → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3128, 29, 30sylancr 694 . . 3 (𝑋𝐴 → (𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ↔ 𝐷:(𝑋 × 𝑋)⟶ℝ*))
3231anbi1d 737 . 2 (𝑋𝐴 → ((𝐷 ∈ (ℝ*𝑚 (𝑋 × 𝑋)) ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
3327, 32bitrd 267 1 (𝑋𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋𝑦𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Colors of variables: wff setvar class Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 class class class wbr 4583 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 0cc0 9815 ℝ*cxr 9952 ≤ cle 9954 +𝑒 cxad 11820 ∞Metcxmt 19552 This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-xr 9957 df-xmet 19560 This theorem is referenced by: isxmetd 21941 xmetf 21944 ismet2 21948 xmeteq0 21953 xmettri2 21955 imasf1oxmet 21990 pstmxmet 29268
Copyright terms: Public domain W3C validator | 4,782 | 5,575 | {"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} | 3.515625 | 4 | CC-MAIN-2024-26 | latest | en | 0.218305 |
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## Complex Variables Pdf
The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit n -sphere , one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation.
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He can go to class without preparation. Flanigan treats this most important field of contemporary mathematics in a most unusual way. While all the material for an advanced undergraduate or first-year graduate course is covered, discussion of complex algebra is delayed for pages, until harmonic functions have been analyzed from a real variable viewpoint. Students who have forgotten or never dealt with this material will find it useful for the subsequent functions. In addition, analytic functions are defined in a way which simplifies the subsequent theory. Complex Variables.
Complex analysis is a beautiful, tightly integrated subject. It revolves around complex analytic functions. These are functions that have a complex derivative. Unlike calculus using real variables, the mere existence of a complex derivative has strong implications for the properties of the function. Complex analysis is a basic tool in many mathematical theories. By itself and through some of these theories it also has a great many practical applications.
Copies of the classnotes are on the internet in PDF format as given below. The "Proofs of Theorems" files were prepared in Beamer and they contain proofs of results which are particularly lengthy shorter proofs are contained in the notes themselves. These notes and supplements have not been classroom tested and so may have some typographical errors. Chapter 1. Complex Numbers. Chapter 2. Analytic Functions.
This book may well be timely and useful to the readers it is intended for: working scientists, students, and engineers. The topics contained are quite broad. It is noteworthy that a glossary is included that provides the reader with a useful guide to terminology and basic concepts. Other valuable features are: 1 a discussion of the available computer packages that can do some complex analysis such as Maple and Mathematica, 2 a pictorial catalog of conformal well-known maps, and 3 tables of Laplace transforms. Though Krantz warns that this handbook contains no theory
### Analytic function
In mathematics , an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable , but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about x 0 converges to the function in some neighborhood for every x 0 in its domain.
Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. To get the free app, enter your mobile phone number. A caution to mathematics professors: Complex Variables does not follow conventional outlines of course material. One reviewer noting its originality wrote: "A standard text is often preferred [to a superior text like this] because the professor knows the order of topics and the problems, and doesn't really have to pay attention to the text.
Complex Variables Pdf. It is both intrinsically beautiful and useful not only in mathematics but also. Complex Variables with Applications, 3rd ED.
### Complex Variables Pdf
Complex Variables Pdf. Person specification. Complex hybrid projective synchronization of complex-variable dynamical networks via open-plus-closed-loop control J. Brown and R. Complex Variables Joseph L. Course Outcomes: At the end of the Course, Student will be able to: CO 1 Compute improper integrals using beta and gamma functions and discuss the properties of the Legendre. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.
In mathematics , a holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain , complex differentiable in a neighborhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series analytic. Holomorphic functions are the central objects of study in complex analysis. Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function real, complex, or of more general type that can be written as a convergent power series in a neighbourhood of each point in its domain.
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27.08.2020 at 11:24 | 1,475 | 7,204 | {"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} | 3.3125 | 3 | CC-MAIN-2022-21 | latest | en | 0.928127 |
http://2904738546.srv042217.webreus.net/tag/625-square-root-508a2f | 1,679,807,118,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00731.warc.gz | 541,220 | 10,940 | Aero Bar Calories 100g, Lean Ux: Designing Great Products With Agile Teams Pdf, Mexican Clay Oven, Thinkpad E14 I5, Christophe Robin Baby Blonde Mask, Samsung S7 Edge 6gb Ram Price In Pakistan, Armour Corned Beef, Service Level Agreement Template, Nikon D6 Release Date, " /> Aero Bar Calories 100g, Lean Ux: Designing Great Products With Agile Teams Pdf, Mexican Clay Oven, Thinkpad E14 I5, Christophe Robin Baby Blonde Mask, Samsung S7 Edge 6gb Ram Price In Pakistan, Armour Corned Beef, Service Level Agreement Template, Nikon D6 Release Date, " />Aero Bar Calories 100g, Lean Ux: Designing Great Products With Agile Teams Pdf, Mexican Clay Oven, Thinkpad E14 I5, Christophe Robin Baby Blonde Mask, Samsung S7 Edge 6gb Ram Price In Pakistan, Armour Corned Beef, Service Level Agreement Template, Nikon D6 Release Date, " />
## 625 square root
Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. Is 625 a Perfect Square? 25 terms. The only square root of zero is zero. Square Root of 4. Volume to (Weight) Mass Converter for Recipes, Weight (Mass) to Volume to Converter for Recipes, Manually calculate the square root of a number with Javascript. A square root of a number 'a' is a number x such that x 2 = a, in other words, a number x whose square is a. The nearest previous perfect square is 576 and the nearest next perfect square is 676 . Use a factor tree. A whole number with a square root that is also a whole number is called a perfect square. The calculators will also shows acres based on the square feet or dimensions A radical is also in simplest form when the radicand is not a fraction. Square Root of 625. This is useful for estimating the size of a house, yard, park, golf course, apartment, building, lake, carpet, or really anything that uses an area for measurement. If you know a square root already to a few digits, such as sqrt(2)=1.414, a single cycle of divide and average will give you double the digits (eight, in this case). ... Square Root of 576. Accelerated Flashcards. With the help of this shortcut on how to find the square root of a number, you will be able to find out the square root of any number within seconds. Answer. 2. 625 is an odd number, because it is not evenly divisible by 2.. Find out more: What is an even number? The square root of 625 is a quantity (q) that when multiplied by itself will equal 625. ਚਿੱਤਰ ਵਿੱਚ ਦਿਖਾਏ ∠a ਅਤੇ ∠b ਦਾ ਆਪਸੀ ਸਬੰਧ ਕੀ ਹੈ? 3. The radicand no longer has any square factors. The Work . The principal square root function () = (usually just referred to as the "square root function") is a function that maps the set of nonnegative real numbers onto itself. looking at 625, you might know that 5^4 = 625 … Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. If you need to explain it step-by-step, I’ll do that. 6. Parity of 625. For example, 2 is the square root of 4, because 2x2=4. We have to find the factors of the number to be sure. We call this the square root of 625 in radical form. Method of prime factorization makes it more clear, i.e. What is the square root of 625 simplified in radical form? In that case we could think "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 is a good start. A square root of a number 'a' is a number x such that x 2 = a, in other words, a number x whose square is a. Question:What's the value of Square Root of 625? √ 625 = q × q = q 2 Answer: 625 is a perfect square because 25 * 25 = 625 Is 625 a prime number? If you need to use only one method … See, below on this web page, details on how to calculate this square root using the Babylonian Method. Find the square root, or the two roots, including the principal root, of positive and negative real numbers. To complete this part of the simplification we take the squre root of the factors which are to be extracted. 8. Find square root of 625 by ... maths. A square root of a number 'a' is a number x such that x2 = a, in other words, a number x whose square is a. Calculate the positive principal root and negative root of positive real numbers. For example, 625 = 5 x 125 = 5 x 5 x 25 = 5 x 5 x 5 x 5. The division method to obtain square root of 13.625 is given below: In this view square root of 13.625 upto 3 decimal digit is 3.691. I don’t know why you posted this question on Quora, when you could have done it by your self if you would have gathered information on how to find square root. For example, 25 is the square root of 625 because 252 = 25•25 = 625, -25 is square root of 625 because (-25)2 = (-25)•(-25) = 625. View Answer. What is square root? About Number 6. Six is the smallest composite number with two distinct prime factors, and the third triangular number. Use SQRT Function to find the SQUARE Root of a Number. 2 Answers C. McCracken Apr 24, 2017 25. boudreauava. The justification for taking out the square root of any number is this theorem to help simplify √a*b = √a * √b. 7. You can use division method to find the root or you can just memorize this one as root of 625 is very common.-1 ; this can be done by using long division mathod of finding square as well as of cube root.-2 ; It is 25-2 ; it is 25-2 ; square root of 625 is 25. 1. For example, 25 is the square root of 625 because 25 2 = 25•25 = 625, -25 is square root of 625 because (-25) 2 = (-25)•(-25) = 625. A number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. 25. Square Root of 625 square root of 625 . In addition to giving a way to find square roots by hand, this method can be used if all you have is a cheap 4-function calculator. Explanation: #sqrt625 = sqrt (25*25) =sqrt (25^2)=25# Also, let's not forget that -25 works too! The result is about 3.906 Step 3: Take the average of 4 and 3.906, which is 3.953. Step one: Find the nearest perfect square. In geometrical terms, the square root function maps the area of a square to its side length.. 25 split into prime factors sqrt625 625=color(red)(5)xxcolor(blue)(125) 625=color(red)(5)xxcolor(blue)(5xx25) 625=color(red)(5xx5)xxcolor(blue)(5xx5)=255^4 to square root half the power sqrt625=sqrt5^4=5^2=25# In mathematics, a square root of a number a is a number y such that y² = a, in other words, a number y whose square (the result of multiplying the number by itself, or y * y) is a. Square Root Of 625? The square root of a number is equal to the number of the square roots of each factor. The square root of 324 is a quantity (q) that when multiplied by itself will equal 324. (ii) Inside the square root, for every two same numbers multiplied, one number can be taken out of the square root. Square Root of 625. square root of 625: Square Root of 625. See below how to calculate the square root of 625 step-by-step using the Babylonian Method also known as Hero's Method. We attempt to show the different possible widths of a 625 square feet space. 625 = 5 4 To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent. Definition of square root. For example, 25 is the square root of 625 because 25 2 = 25•25 = 625, -25 is square root of 625 because (-25) 2 = (-25)•(-25) = 625. Square Root of 9. Is 625 a perfect square number? For example, 4 has two square roots: 2 and -2. ... √625 = take last 2 digits 25 and remaining is 6. Square root of 625 definition The square root of 625 in mathematical form is written with the radical sign like this √625. Square Root of 16. I believe you are referring to the technique of estimating a Square root by division. Square root of 625 = 6251/2 = 54/2 = 52 = 25. 24. SHANKAR IAS CURRENT AFFAIRS 2020. Is 625 a composite number? Answer:To find the Square Root We Need To split into small factors 625=√25x25=25. Thew following steps will be useful to find square root of a number by prime factorization. EASY. Algebra Radicals and Geometry Connections Simplification of Radical Expressions. Because there are 4 fives, and we are looking for the square root, (5 x 5)(5 x 5) = 625. Asked on December 27, 2019 by Amrin Kamboj. answer is 125. without a calculator, some logic might help you find the solution. Find the consecutive perfect squares between which the following numbers lie: 88888. ... Find the square root of the following number by division method: 18769. The square root of six hundred and twenty-five √625 = 25. Use the square root calculator below to find the square root of any imaginary or real number. What is the Square Root of 625 in simplest radical form? The square root of 625 is 25 Answer: YES, 4 is in the list of digital roots that are always perfect squares. Is 625 an even number? A square root of a number is a number that, when it is multiplied by itself (squared), gives the first number again. The Square root of 625 is of course 25.. All radicals are now simplified. 5. We call this the square root of 324 in radical form. While every effort is made to ensure the accuracy of the information provided on this website, neither this website nor its authors are responsible for any errors or omissions, or for the results obtained from the use of this information. The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. See also in this web page a Square Root Table from 1 to 100 as well as the Babylonian Method or Hero's Method. = 1 * 2 * 3, which is remarkable, because there is no other three numbers whose product is equal to their sum. Factoring. Here is the answer to questions like: Square root of 625 or what is the square root of 625? Is 4 in the list of digital roots that are always a square root (1, 4, 7 or 9)? Square root of 324 definition The square root of 324 in mathematical form is written with the radical sign like this √324. Square Root of 64. (like this 6 , 25 ) ( 1sq=1 , 2sq=4 , 3sq=9 , 4sq=16 , 5sq=25 ,6sq=36 , 7sq=49, The square root of six hundred and twenty-five √625 = 25 How To Calculate Square Roots In mathematics, a square root of a number a is a number y such that y² = a, in other words, a number y whose square (the result of multiplying the number by itself, or y * y) is a. Also tells you if the entered number is a perfect square. √ 324 = q × q = q 2 toppr. Is 625 a rational number? Factors which will be extracted are : 625 = 5 4 No factors remain inside the root !! All information in this site is provided “as is”, with no guarantee of completeness, accuracy, timeliness or of the results obtained from the use of this information. in your calculator you would take 625 to the 3d power and then take the 4th root of that. Check out the work below for reducing 625 into simplest radical form . OK, so now we know that 625 could be a perfect square. This is an approximation of the square root of 15.625. Only numbers bigger than or equal to zero have real square roots. What is the prime factorization of 625? The 4th of April 2016 is a Square Root … Square Root Day. It is the smallest perfect number: 6 = 1 + 2 + 3 and the faculty of 3 is 6 = 3! Find square root of 6 2 5 by long divison method. What is square root? (i) 9801We know that If a number ends with 1 if square root can end with 1 or 9∴ Possible unit digits = 1 or 9 Ex 6.3, 1 What could be the possible ‘one’s’ digits of the square root … Ex 6.3, 1 What could be the possible ‘one’s’ digits of the square root of each of the following numbers? Square Root of 25. 625. The square root radical is simplified or in its simplest form only when the radicand has no square factors left. (i) Decompose the number inside the square root into prime factors. What are the multiples of 625? 625 = 5x5x5x5 = 54. A number bigger than zero has two square roots: one is positive (bigger than zero) and the other is negative (smaller than zero). The radicand is the number or expression underneath the radical sign, in this example 9. For example, 4 and -4 are square roots of 16 because 4² = (-4)² = 16. you could also take the 4th root of 625 and then take that to the 3d power. square root of 625 is 25. Therefore the square root of 625 … For example, the principal square root of 9 is 3, denoted √9 = 3, because 32 = 3 ^ 3 = 9 and 3 is non-negative. (iii) Combine the like square root terms using mathematical operations. 16 in this case. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Square roots from 1 to 100 rounded to the nearest thousandth. Answered By . Square Root of 49. Is 625 an odd number? 4. MEDIUM. The term whose root is being considered is known as the radicand. Square Root of 36. USING OUR SERVICES YOU AGREE TO OUR USE OF. Kaitlin's Math Cards!!!!! The square root of 625 is 25, which is the smallest square number that can be written as sum of two consecutive squares (3 2 + 4 2). We can conclude that 625 could be a perfect square! step 2: Divide the number (15.625) by 4 (the square root of 16). Square Root of 1. Square root of 625 by division method - 4572362 ਪ੍ਰ:3. Square root calculator and perfect square calculator. Definition of square root. | 3,624 | 13,459 | {"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} | 2.671875 | 3 | CC-MAIN-2023-14 | latest | en | 0.858538 |
http://mathhelpforum.com/algebra/133527-mixed-numbers-gone-bad.html | 1,498,289,773,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128320227.27/warc/CC-MAIN-20170624064634-20170624084634-00709.warc.gz | 277,636,446 | 11,241 | 1. ## Mixed Numbers gone bad...
I've worked and re-worked this problem over the last couple of days and cannot understand where I'm going wrong. This is also the first time I'm attempting to use LaTeX so forgive me if I do it wrong.
Change the following in to common fractions:
$a^2+ab-b^2-\frac{a^3-2b^3}{a-2b}$
Right away I determine the LCD = a-2b
So I then re-write the equation:
$\frac{a^2(a-2b)}{a-2b}+\frac{ab(a-2b)}{a-2b}-\frac{b^2(a-2b)}{a-2b}-\frac{a^3-2b^3}{a-2b}$
After doing the multiplication I get:
$\frac{a^3-2a^2b+a^2b-2ab^2-ab^2+2b^3-a^3-2b^3}{a-2b}$
Combining like terms I get (I hope I've managed to keep all the information correct in the latex formatting):
$\frac{-a^2b-3ab^2}{a-2b}$
The answer the book shows is:
$-\frac{a^2b+3ab^2-4b^3}{a-2b}$
Any help would be greatly appreciated and much thanked in advance as always. Thanks!
E
2. Originally Posted by ejanderson
After doing the multiplication I get:
$\frac{a^3-2a^2b+a^2b-2ab^2-ab^2+2b^3-a^3-2b^3}{a-2b}$
So close. Check the very last sign in the numerator. Be careful!
Since the last fraction $-\frac{a^3-2b^3}{a-2b}$ already contains the LCD, I don't do a thing to the numerator which should keep the $-a^3-2b^3$ the same... $-a-2b^3$.
Now, looking back to the third fraction $-\frac{b^2(a-2b)}{a-2b}$ that becomes: $-\frac{ab^2+2b^3}{a-2b}$
The part of the answer I'm missing is the $-4b^3$ which should come from combining like terms from the last two fractions. However, in the way I'm doing it the $+2b^3$ in the third fraction gets cancelled out by the $-2b^3$ in the last fraction.
I "know" that my problem is coming from one of the last two fractions in the way I'm multiplying or combining terms but I just don't see it.
4. I would rewrite the last fraction as:
$
\frac{-(a^3-2b^3)}{a-2b}
$
Which will clearly become:
$
\frac{-a^3+2b^3}{a-2b}
$
Which, I think, should give the correct answer?
5. -(a^2 - 2b^3) = -a^3 +2b^3
6. Originally Posted by ejanderson
I "know" that my problem is coming from one of the last two fractions in the way I'm multiplying or combining terms but I just don't see it.
Please learn the Distributive Property of Multiplication over Addition. There is some confusion over the dual identity of that subtraction. Write it the grade school way and you will see it.
a - b = a + (-b)
It's ugly, but that's the way it is.
You can also run a test.
$5 - \frac{7-3}{2}$
$\frac{10 - 7 - 3}{2}\;\;$ or $\;\;\frac{10 - 7 + 3}{2}$ | 846 | 2,465 | {"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": 19, "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} | 3.921875 | 4 | CC-MAIN-2017-26 | longest | en | 0.893254 |
http://hackage.haskell.org/package/math-functions-0.1.6.0/docs/Numeric-Polynomial-Chebyshev.html | 1,561,063,261,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999273.24/warc/CC-MAIN-20190620190041-20190620212041-00107.warc.gz | 73,095,538 | 2,453 | math-functions-0.1.6.0: Special functions and Chebyshev polynomials
Copyright (c) 2009 2011 Bryan O'Sullivan BSD3 bos@serpentine.com experimental portable None Haskell98
Numeric.Polynomial.Chebyshev
Contents
Description
Chebyshev polynomials.
Synopsis
# Chebyshev polinomials
A Chebyshev polynomial of the first kind is defined by the following recurrence:
t 0 _ = 1
t 1 x = x
t n x = 2 * x * t (n-1) x - t (n-2) x
Arguments
:: Vector v Double => Double Parameter of each function. -> v Double Coefficients of each polynomial term, in increasing order. -> Double
Evaluate a Chebyshev polynomial of the first kind. Uses Clenshaw's algorithm.
Arguments
:: Vector v Double => Double Parameter of each function. -> v Double Coefficients of each polynomial term, in increasing order. -> Double
Evaluate a Chebyshev polynomial of the first kind. Uses Broucke's ECHEB algorithm, and his convention for coefficient handling. It treat 0th coefficient different so
chebyshev x [a0,a1,a2...] == chebyshevBroucke [2*a0,a1,a2...]
# References
• Broucke, R. (1973) Algorithm 446: Ten subroutines for the manipulation of Chebyshev series. Communications of the ACM 16(4):254–256. http://doi.acm.org/10.1145/362003.362037
• Clenshaw, C.W. (1962) Chebyshev series for mathematical functions. National Physical Laboratory Mathematical Tables 5, Her Majesty's Stationery Office, London. | 391 | 1,387 | {"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} | 2.625 | 3 | CC-MAIN-2019-26 | latest | en | 0.727911 |
dailysandesh.com | 1,726,457,345,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651668.29/warc/CC-MAIN-20240916012328-20240916042328-00773.warc.gz | 165,070,360 | 26,439 | # Mathematics in Natural Language Processing
Mathematics in Natural Language Processing
Are you someone who is looking for some concepts to understand the basics of Theoretical Statistics. Yes, you have come to the right place. In today’s blog, we will have an overview on the various mathematical concepts of Statistics used in Natural Language Processing.
The two mathematical concepts which are widely used in the theoretical concepts of Statistics and Probability and tend to be most useful in understanding variables are what we know as Covariance and Correlation. Both the concepts are generally used in the field of natural language processing for comparing data samples from different populations, where covariance determines how much two variables change randomly to each other, and correlation, which is a normalized version of covariance, determines the change in one variable as it affects another variable.
Before diving into details about covariance and correlation, let us first try to understand what is meant by variance and standard deviation.
What does it mean by variance and standard deviation? What is their mathematical representation? Let’s read them out.
Variance is referred to as the measure of variability i.e., it is calculated by taking the average of the squared deviation of a random variable from its mean. In other words, it measures how far a set of numbers are spread out from their average value which states the more spread the data, the larger the variance is in relation to the mean. The mathematical representation of variance is:
where,
S2 = sample variance
xi = value of one observation
x = mean value of all observations
n = number of observations
Standard deviation is a statistical term that measures the amount of dispersion or variation for a set of values relative (absolute variability of a random variable) to its mean as it is calculated as the square root of the variance. A low standard deviation indicates that the values tend to be nearer to the mean of the set, while a high standard deviation indicates that the values are further from the mean over a wider range, thus, the more spread of the data, the higher the standard deviation.
The general mathematical formula to find standard deviation for a given dataset is as follows:
So now, what do you mean by covariance and correlation?
In simple words, Covariance is a measure to indicate the extent to which two random variables change in tandem. Whereas Correlation is a measure used to represent how strongly two random variables are related to each other.
In a more explanatory sense, Covariance is defined as a quantitative measure of the extent to which the deviation of one variable from its mean matches the deviation of the other from its mean. It is actually a statistical technique that shows whether and how strongly pairs of variables are related. For example, how height and weight are related while describing taller people and shorter people and who is heavier.
In order to understand its mathematical representation, let us suppose we have two variables X and Y, then we represent the covariance between these two variables as Cov(X, Y). Now, if Σ(X) and Σ(Y) are the expected values of the variables, the covariance formula can be represented as:
On the contrary, correlation works primarily for quantifiable data where numbers hold much value and meaning. It cannot be used for purely categorical data, such as brands or goods purchased, gender, price of some items or maybe favorite color. The word correlation is used in our everyday life to denote some form of association between two quantitative variables like we might observe a correlation between the foggy days and attacks of wheezing.
So, how can we compare both? There is not much difference between the two but it is quite important to be clear with the theoretical concepts when we are discussing both at a time. What makes them apart is the fact that correlation values are standardized values whereas covariance values are not. A simple method to obtain the correlation coefficient of two variables is by dividing the covariance of these variables by the product of the standard deviations of the same values.
Both the terms are related to the linear relationship between variables, i.e., if one variable goes on increasing, then the other variable also moves in the same direction which means a positive correlation. On the other hand, if both the variables are in the opposite direction then correlation is negative. When there is no relationship, there are no changes.
A Correlation Matrix is used as a term that investigates the dependence between multiple variables at one go. There are three main applications of a correlation matrix some of which includes to diagnose or check other analysis, to input into other analyses, and to summarize large amounts of data.
On what kind of datasets do we normally use covariance and correlation for? A sample is randomly chosen from the population, and so we calculate covariance and correlation on samples rather than on the complete population.
Now that we have got a brief idea about the mathematical theory of covariance and correlation, let us explore how and where we can apply it in the field of data analytics. Principal Component Analysis is one such application. PCA can be defined as the dimensionality-reduction method that is often used to reduce the dimensionality of large data sets, by transforming a large set of variables into a smaller one that contains most of the information in the large set.
So how do we decide what to use? Correlation matrix or the covariance matrix? Let’s try to understand this with the help of examples, and to showcase agility of implementation across technologies, I shall execute this example in Python.
We will consider the ‘iris’ data-set for the same.
Step1:
Foremostly, we have to import the required libraries and then load the iris dataset. After that, we have to create a dataframe and drop empty records. Not to worry, we will check the code of each and every line below:
#importing necessary libraries
from sklearn import datasets
import pandas as pd
import numpy as np
# Since this is a bunch, create a dataframe
iris_df=pd.DataFrame(iris.data)
iris_df[‘class’]=iris.target
iris_df.columns=[‘sepal_len’, ‘sepal_wid’, ‘petal_len’, ‘petal_wid’, ‘class’]
iris_df.dropna(how=”all”, inplace=True) # remove any empty lines
#selecting only first 4 columns as they are the independent(X) variable
# any kind of feature selection or correlation analysis should be first done on these
iris_X=iris_df.iloc[:,[0,1,2,3]]
print(iris_X)
Step 2:
#This data-set will now be standardized using the inbuilt function.
from sklearn.preprocessing import StandardScaler
# let us now standardize the dataset
iris_X_std = StandardScaler().fit_transform(iris_X)
print(iris_X_std)
Step 3:
# I have then computed 3 matrices:
# covariance matrix on standardized data
mean_vec = np.mean(iris_X_std, axis=0)
cov_matrix = (iris_X_std – mean_vec).T.dot((iris_X_std – mean_vec)) / (iris_X_std.shape[0]-1)
print(‘Covariance matrix \n%s’ %cov_matrix)
# Correlation matrix on standardized data
cor_matrix = np.corrcoef(iris_X_std.T)
print(‘Correlation matrix using standardized data\n%s’ %cor_matrix)
# Correlation matrix on unstandardized data
cor_matrix2 = np.corrcoef(iris_X.T)
print(‘Correlation matrix using base unstandardized data \n%s’ %cor_matrix2) | 1,509 | 7,475 | {"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} | 4.375 | 4 | CC-MAIN-2024-38 | latest | en | 0.944084 |
http://ts-web.info/hsa-525-homework-week-3-31/ | 1,585,857,793,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585370507738.45/warc/CC-MAIN-20200402173940-20200402203940-00389.warc.gz | 185,900,843 | 9,047 | ### HSA 525 HOMEWORK WEEK 3
The unit receives revenue from four major payers. Designate the responsibility centers and the support centers for the organization selected. Detail should be extensive enough to present a challenge. What would your summary of these losses look like? Compute the straight line depreciation. How much difference would that make?
One of your first tasks is to conduct an internal financial analysis of the organization. Study Table 6—1, Table 6—2, and review the chapter text describing how the indirect cost is allocated. Decide how many cost centers should be used for the above expenses at the Center. Set up a worksheet for actual overhead costs and budget variance with a column for Routine Services Nursing and a second column for Laboratory 2. Which alternative is more desirable at this interest rate? Explain the potential asymmetries that may exist where leasing may be beneficial to both the lessors and the lessee. Prepare a rationale for the structure you have designed.
Using the monthly utilization information presented here, and omitting the community college training packs, find the fixed and variable portion of costs through the high—low method.
Increase the number of infusion chairs to hsz, and add another nurse for either four or six hours per day. Assignment Exercise 14—3 The head of your department is a prominent researcher. Set up a worksheet for the liquidity ratios. Make up your own organization chart for other employee levels within the function you have chosen.
CHURCHGOERS CLASSIFICATION ESSAY
Given the current state of this market segment: Wenn Sie sicher sind, dass der Ersteller dieser Karte jemandes oder Ihr Urheberrecht verletzt hat, teilen Sie uns dies bitte mit. Set up a worksheet for the profitability ratios.
Certain situations concerning the Intensive Care Unit and the Laboratory are described below. Designate the responsibility centers and the support centers for the organization selected. The new allocation bases are:. Compute the unadjusted rate of return using the average investment method. One of your first tasks is to conduct an internal financial analysis of the organization. Which alternative do you believe would be best? As the CFO, suggest at least one 1 way that you might minimize the impact of the trend on the organization.
The function I chose from the organizational chart is Finance. Provide support for your prediction. Write a four to five page paper in which you: Get your expert answer on EssayDons.
Next, determine one 1 key driver of health care cost escalation. Mehrere neue Karten Anzahl neue Karten: Comment on why these items are important for benchmarking purposes.
## HSA 525 WEEK 3 DISCUSSION
Homweork the name of each component and its amount s from the appropriate MHS financial statement. Be typed, double spaced, using Times New Roman font size 12with one-inch margins on all sides; citations and references must follow APA or school-specific format. Speculate on the likely reaction to the financial statements from various stakeholder groups employee, investors, shareholders. Verschieben Verschiebe die Karte in einen anderen Kartensatz. Also homeqork whether the percentage of annual interest on debt is revealed in the notes to the financial statements.
ATTICUS ESSAY WPP
# HSA Week 3 Discussion | HWACER
Depreciation Set up a purchase scenario of your own and compute the depreciation with and without salvage value.
Recommend a revenue strategy for the organization in the scenario to improve its revenue cycle management. Total the new results. John Whitten is still figuring out his equipment fund. I would very much recommend this site to my friends.
Using the existing budget, create a new budget for the next homewwork year. Help Center Find new research papers in: If so, do you believe the interest rate is fair and equitable?
Karte an Position verschieben.
Although his travel expenses are being funded by the foundation, he will still need to take along some personal money. | 807 | 4,026 | {"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} | 2.59375 | 3 | CC-MAIN-2020-16 | latest | en | 0.857684 |
https://codecrucks.com/question/data-structures-question-set-09/ | 1,721,505,157,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763517515.18/warc/CC-MAIN-20240720174732-20240720204732-00289.warc.gz | 154,478,876 | 22,618 | # Data Structures: Question Set – 09
#### What is a heap?
A heap is a specific kind of binary tree that fulfils the heap property, which can be either the max-heap property or the min-heap property, depending on the particular heap property being considered. It is often used in the process of implementing a priority queue.
#### What is the difference between a max-heap and a min-heap?
A max-heap is a type of binary tree in which the value of each node is larger than or equal to the values of its children, and the value that has the highest value is stored in the node that is the tree’s root. A min-heap is a type of binary tree in which the value of each node is either less than or equal to the values of its children, and the value with the lowest value is stored in the node that is directly beneath the root node.
#### What is the key distinction between a binary search tree and a binary heap?
A binary search tree is a binary tree in which the value of each node is greater than or equal to the values of its left subtree and less than or equal to the values of its right subtree, whereas a binary heap is a complete binary tree in which the value of each node is greater than or equal to (or less than or equal to) the values of its children. A binary heap differs from a binary search tree in that a binary search tree is not a complete binary tree.
#### The term “self-balancing binary search tree” refers to what exactly?
A binary search tree that automatically adapts its structure to maintain balance and prevent degeneration into a linked list is called a self-balancing binary search tree. This type of binary search tree is also known as a balanced binary search tree. AVL tree and Red-Black tree are two examples of trees that can be used.
#### What is the advantage of using a priority queue?
When processing needs to be prioritised based on importance or urgency, such as when scheduling tasks or processing network traffic, a priority queue is useful because it enables efficient access to the element with the highest priority. This makes it useful in applications where processing needs to be prioritised.
#### What is a linked list?
A linked list is a type of data structure that consists of a series of nodes, each of which has a pointer (reference) to the node after it in the sequence.
#### What distinguishes a linked list from an array?
A linked list is different from an array in that it does not store data in contiguous memory locations. Instead, each node in a linked list is dynamically allocated and can be found anywhere in memory.
#### What advantages do linked lists have over arrays?
The primary benefit of choosing a linked list over an array is that the former can expand dynamically while the latter has a set size. Linked lists are also more efficient at inserting and removing elements than arrays are.
#### What kinds of linked lists are there?
Singly linked lists, double linked lists, and circular linked lists are some of the several varieties of linked lists.
#### What is the time complexity of inserting an element into a linked list?
The time complexity of inserting an element into a linked list depends on the location of the insertion. If the insertion is at the beginning of the list, the time complexity is O(1). If the insertion is at the end of the list, the time complexity is O(n), where n is the number of elements in the list. If the insertion is at a specific position in the list, the time complexity is also O(n). | 724 | 3,504 | {"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} | 3.046875 | 3 | CC-MAIN-2024-30 | latest | en | 0.942238 |
http://www.singular.uni-kl.de/Manual/latest/sing_709.htm | 1,553,529,021,000,000,000 | text/html | crawl-data/CC-MAIN-2019-13/segments/1552912204077.10/warc/CC-MAIN-20190325153323-20190325175323-00479.warc.gz | 355,778,080 | 4,131 | # Singular
##### 7.5.17.0. doubleExt
Procedure from library `purityfiltration.lib` (see purityfiltration_lib).
Usage:
doubleExt(R,i), R matrix representing the left Module M=D^p/D^q(R^t) over a ring D
int i, less or equal the left projective dimension of M
Return:
matrix P, representing the double ext module
Purpose:
computes a matrix P, which represents the left module ext^i(ext^i(M,D))
Example:
```LIB "purityfiltration.lib"; ring D = 0,(x,y,z),dp; matrix R[7][3]= 0 ,0,1, 1 ,-4*x+z,-z, -1,8*x-2*z,z, 1 ,0 ,0, 0 ,x-y,0, 0 ,x-y,y, 0 ,0 ,x; // coker(R) is 2-pure, so all doubleExt are zero print(doubleExt(transpose(R),0)); ==> 1 print(doubleExt(transpose(R),1)); ==> 1,0,0, ==> 0,1,0, ==> 0,0,1 print(doubleExt(transpose(R),3)); ==> 1 // except of the second print(doubleExt(transpose(R),2)); ==> 4y-z,4x-z ``` | 292 | 820 | {"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} | 2.59375 | 3 | CC-MAIN-2019-13 | latest | en | 0.765363 |
http://www.transtutors.com/homework-help/finance/international-financial-management/investment-analysis-and-portfolio-management/historical-risk/measures-of-historical-rates-of-return/ | 1,513,525,858,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948596115.72/warc/CC-MAIN-20171217152217-20171217174217-00149.warc.gz | 470,937,738 | 16,730 | ## Measures Of Historical Rates Of Return
For any security the most important factor is the rate of return which is derived from the instrument. In some way or the any security has its base in the return which is derived. There are basically two forms of return which can be measured – historical return and prospective return. Homework help and assignment help section at Transtutors.com provides clear concept of all topics related with finance.
Historical return is the measure of the return which is earned from the security in the past periods. This can be used to assess the future performance of the security. On the other hand prospective return of the stock is the return which may be derived from the investment in the near future. At Transtutors.com we provide complete doubt removal assistance at our homework help and assignment help section.
The total return which is to be derived from any particular investment may be derived with the help of the following formula –
Total return = cash received + Change in price
Initial Price
As there are two elements involved in this and the change in price may be negative due to various factors, the total return derived from the stock may be negative. The total return of the stock is the combination of two forms of return, these are dividend yield and capital yield. Dividend yield is the proportion of dividend to the current price of the stock. Capital yield is the change in the price of the stock overtime. However if there is no change on the price of the security then the total return of the stock is equal to the dividend yield of the stock. Our team of tutors are available to provide homework help and assignment help on any topic related to finance.
The formal presentation of this formula is as under –
R = C + (P1 – P0)
P0
Where C = cash flow
P1= Price at the end of period
P0 = Price at the beginning of the period
For Example,
If a stock is presently trading at a price of \$120 at the end of one year the price of the stock is \$140. During this period the company declares a dividend of \$5 per share.
The total return in this situation can be derived as under –
R = C + (P1 – P0)
P0
= 5 + (140-120)
120
= 20.83%
However sometime return of the stock has to be calculated in a different manner as compared to the traditional method specified above. This is particularly true when a cumulative wealth index or a geometric mean has to be calculated. Our tutors at Transtutors.com are expert in finance from years and years, we provide excellent finance homework and assignment help
Return relative is defined as –
Return relative = C+ PE
PB
Return relative = 1 + total return in decimal
## Related Topics
All Finance Topics
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# Count of distinct possible strings after performing given operations
Given a numeric string S consisting of only three types of characters 0, 1, and 2 initially and following two operations:
• The occurrence of two consecutive 1 can be replaced by 3.
• The occurrence of two consecutive 2 can be replaced by 4.
The given task is to find the total number of different strings that can be formed by using the operations.
Examples:
Input: S = “0110000022”
Output:
Explanation:
There can be four different formed by using the operations, the four strings are
“0110000022”, “030000022”, “03000004”, “011000004”
Input: S = “111”
Output:
Approach:
In order to solve this problem, we are using a dynamic programming approach. We define an array dp to store the count of possible strings of length equal to its respective indices.
• Initialize dp[0] = dp[1] = 1.
• For any index i between [1, N-1], if the character at that index is ‘1’ or ‘2’ and is equal to that of its previous index, add the possible strings that can be formed of length i-1 and i-2. Otherwise, it is the same as that of length i-1.
dp[i + 1] = dp[i] + dp[i-1] if S[i] == S[i-1] where S[i] is either ‘1’ or ‘2’.
dp[i + 1] = dp[i] otherwise.
• Return dp[n] as the count of different strings possible.
Below is the implementation of the above approach:
## C++
`// C++ implementation of``// the above approach``#include``using` `namespace` `std;` `// Function that prints the``// number of different strings``// that can be formed``void` `differentStrings(string s)``{`` ``// Computing the length of`` ``// the given string`` ``int` `n = s.length();` ` ``vector dp(n + 1);` ` ``// Base case`` ``dp[0] = dp[1] = 1;` ` ``// Traverse the given string`` ``for` `(``int` `i = 1; i < n; i++) {` ` ``// If two consecutive 1's`` ``// or 2's are present`` ``if` `(s[i] == s[i - 1]`` ``&& (s[i] == ``'1'`` ``|| s[i] == ``'2'``))` ` ``dp[i + 1]`` ``= dp[i] + dp[i - 1];` ` ``// Otherwise take`` ``// the previous value`` ``else`` ``dp[i + 1] = dp[i];`` ``}` ` ``cout << dp[n] << ``"\n"``;``}` `// Driver Code``int` `main()``{`` ``string S = ``"0111022110"``;` ` ``differentStrings(S);` ` ``return` `0;``}`
## C
`#include ``#include ``#include ``void` `differentStrings(``char``* s)``{`` ` ` ``// Computing the length of`` ``// the given string`` ``int` `n = ``strlen``(s);` ` ``int` `dp[n + 1];` ` ``// Base case`` ``dp[0] = dp[1] = 1;` ` ``// Traverse the given string`` ``for` `(``int` `i = 1; i < n; i++) {` ` ``// If two consecutive 1's`` ``// or 2's are present`` ``if` `(s[i] == s[i - 1]`` ``&& (s[i] == ``'1'`` ``|| s[i] == ``'2'``))` ` ``dp[i + 1] = dp[i] + dp[i - 1];` ` ``// Otherwise take`` ``// the previous value`` ``else`` ``dp[i + 1] = dp[i];`` ``}` ` ``printf``(``"%d\n"``, dp[n]);``}` `// Driver Code``int` `main()``{`` ``char` `S[] = ``"0111022110"``;` ` ``differentStrings(S);` ` ``return` `0;``}` `// This code is contributed by phalashi.`
## Java
`// Java implementation of the above approach``import` `java.io.*;` `class` `GFG{` `// Function that prints the``// number of different strings``// that can be formed``static` `void` `differentStrings(String s)``{`` ` ` ``// Computing the length of`` ``// the given string`` ``int` `n = s.length();` ` ``int``[] dp = ``new` `int``[n + ``1``];` ` ``// Base case`` ``dp[``0``] = dp[``1``] = ``1``;` ` ``// Traverse the given string`` ``for``(``int` `i = ``1``; i < n; i++)`` ``{`` ` ` ``// If two consecutive 1's`` ``// or 2's are present`` ``if` `(s.charAt(i) == s.charAt(i - ``1``) &&`` ``(s.charAt(i) == ``'1'` `||`` ``s.charAt(i) == ``'2'``))`` ``dp[i + ``1``] = dp[i] + dp[i - ``1``];`` ` ` ``// Otherwise take the`` ``// previous value`` ``else`` ``dp[i + ``1``] = dp[i];`` ``}`` ``System.out.println(dp[n]);``}` `// Driver code``public` `static` `void` `main(String[] args)``{`` ``String S = ``"0111022110"``;` ` ``differentStrings(S);``}``}` `// This code is contributed by offbeat`
## Python3
`# Python3 implementation of``# the above approach` `# Function that prints the``# number of different strings``# that can be formed``def` `differentStrings(s):` ` ``# Computing the length of`` ``# the given string`` ``n ``=` `len``(s)` ` ``dp ``=` `[``0``] ``*` `(n ``+` `1``)` ` ``# Base case`` ``dp[``0``] ``=` `dp[``1``] ``=` `1` ` ``# Traverse the given string`` ``for` `i ``in` `range` `(``1``, n):` ` ``# If two consecutive 1's`` ``# or 2's are present`` ``if` `(s[i] ``=``=` `s[i ``-` `1``] ``and`` ``(s[i] ``=``=` `'1'` `or`` ``s[i] ``=``=` `'2'``)):` ` ``dp[i ``+` `1``] ``=` `dp[i] ``+` `dp[i ``-` `1``]` ` ``# Otherwise take`` ``# the previous value`` ``else``:`` ``dp[i ``+` `1``] ``=` `dp[i]`` ` ` ``print` `(dp[n])` `# Driver Code``if` `__name__ ``=``=` `"__main__"``: `` ``S ``=` `"0111022110"`` ``differentStrings(S)`` ` `# This code is contributed by Chitranayal`
## C#
`// C# implementation of the above approach``using` `System;``class` `GFG{` `// Function that prints the``// number of different strings``// that can be formed``static` `void` `differentStrings(``string` `s)``{`` ` ` ``// Computing the length of`` ``// the given string`` ``int` `n = s.Length;` ` ``int``[] dp = ``new` `int``[n + 1];` ` ``// Base case`` ``dp[0] = dp[1] = 1;` ` ``// Traverse the given string`` ``for``(``int` `i = 1; i < n; i++)`` ``{` ` ``// If two consecutive 1's`` ``// or 2's are present`` ``if` `(s[i] == s[i - 1] &&`` ``(s[i] == ``'1'` `||`` ``s[i] == ``'2'``))`` ``dp[i + 1] = dp[i] + dp[i - 1];`` ` ` ``// Otherwise take the`` ``// previous value`` ``else`` ``dp[i + 1] = dp[i];`` ``}`` ``Console.Write(dp[n]);``}` `// Driver code``public` `static` `void` `Main()``{`` ``string` `S = ``"0111022110"``;` ` ``differentStrings(S);``}``}` `// This code is contributed by Code_Mech`
## Javascript
``
Output:
`12`
Time Complexity: O(N), as we are using a loop to traverse N times so it will cost us O(N) time
Auxiliary Space: O(N), as we are using extra space for DP array.
Efficient approach : Space optimization O(1)
In previous approach we the current value dp[i] is only depend upon the previous 2 values i.e. dp[i-1] and dp[i+1]. So to optimize the space we can keep track of previous and next values by the help of three variables prev and next which will reduce the space complexity from O(N) to O(1).
Implementation Steps:
• Create 2 variables prev1 and prev2 to keep track o previous values of DP.
• Initialize base case prev = curr = 1.
• Create a variable curr to store current value.
• Iterate over subproblem using loop and update curr.
• Create variable next to keep track of next value of DP.
• After every iteration update prev and curr for further iterations.
• At last return curr.
Implementation:
## C++
`#include ``using` `namespace` `std;` `// Function that prints the``// number of different strings``// that can be formed``void` `differentStrings(string s)``{`` ``// Computing the length of`` ``// the given string`` ``int` `n = s.length();` ` ``// Base case`` ``int` `prev = 1, curr = 1;` ` ``// Traverse the given string`` ``for` `(``int` `i = 1; i < n; i++) {` ` ``// If two consecutive 1's`` ``// or 2's are present`` ``if` `(s[i] == s[i - 1]`` ``&& (s[i] == ``'1'`` ``|| s[i] == ``'2'``)) {`` ``int` `next = prev + curr;`` ``prev = curr;`` ``curr = next;`` ``}` ` ``// Otherwise take`` ``// the previous value`` ``else` `{`` ``prev = curr;`` ``}`` ``}` ` ``cout << curr << ``"\n"``;``}` `// Driver Code``int` `main()``{`` ``string S = ``"0111022110"``;` ` ``differentStrings(S);` ` ``return` `0;``}`
## Java
`import` `java.util.*;` `public` `class` `Main {` ` ``// Utility function to find the number of different strings`` ``// that can be formed`` ``public` `static` `void` `differentStrings(String s)`` ``{`` ` ` ``// Computing the length of the given string`` ``int` `n = s.length();` ` ``// Base case`` ``int` `prev = ``1``, curr = ``1``;` ` ``// Traverse the given string`` ``for` `(``int` `i = ``1``; i < n; i++) {` ` ``// If two consecutive 1's or 2's are present`` ``if` `(s.charAt(i) == s.charAt(i - ``1``)`` ``&& (s.charAt(i) == ``'1'` `|| s.charAt(i) == ``'2'``)) {`` ``int` `next = prev + curr;`` ``prev = curr;`` ``curr = next;`` ``}` ` ``// Otherwise take the previous value`` ``else` `{`` ``prev = curr;`` ``}`` ``}` ` ``System.out.println(curr);`` ``}` ` ``// Driver Code`` ``public` `static` `void` `main(String[] args) {`` ``String S = ``"0111022110"``;` ` ``differentStrings(S);`` ``}``}`
## Python3
`def` `different_strings(s: ``str``) ``-``> ``None``:`` ``# Computing the length of the given string`` ``n ``=` `len``(s)` ` ``# Base case`` ``prev, curr ``=` `1``, ``1` ` ``# Traverse the given string`` ``for` `i ``in` `range``(``1``, n):`` ``# If two consecutive 1's or 2's are present`` ``if` `s[i] ``=``=` `s[i ``-` `1``] ``and` `(s[i] ``=``=` `'1'` `or` `s[i] ``=``=` `'2'``):`` ``next` `=` `prev ``+` `curr`` ``prev ``=` `curr`` ``curr ``=` `next` ` ``# Otherwise take the previous value`` ``else``:`` ``prev ``=` `curr` ` ``print``(curr)` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:`` ``S ``=` `"0111022110"`` ``different_strings(S)`
Output:
`12`
Time Complexity: O(N), as we are using a loop to traverse N times so it will cost us O(N) time
Auxiliary Space: O(1)
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https://en.wikipedia.org/wiki/Principal_Components_Analysis | 1,495,855,571,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463608765.79/warc/CC-MAIN-20170527021224-20170527041224-00268.warc.gz | 932,702,503 | 69,136 | # Principal component analysis
(Redirected from Principal Components Analysis)
PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean.
Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components (or sometimes, principal modes of variation). The number of principal components is less than or equal to the smaller of the number of original variables or the number of observations. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.
PCA was invented in 1901 by Karl Pearson,[1] as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s.[2] Depending on the field of application, it is also named the discrete Kosambi-Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (Golub and Van Loan, 1983), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of [3]), Eckart–Young theorem (Harman, 1960), or Schmidt–Mirsky theorem in psychometrics, empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.
PCA is mostly used as a tool in exploratory data analysis and for making predictive models. PCA can be done by eigenvalue decomposition of a data covariance (or correlation) matrix or singular value decomposition of a data matrix, usually after mean centering (and normalizing or using Z-scores) the data matrix for each attribute.[4] The results of a PCA are usually discussed in terms of component scores, sometimes called factor scores (the transformed variable values corresponding to a particular data point), and loadings (the weight by which each standardized original variable should be multiplied to get the component score).[5]
PCA is the simplest of the true eigenvector-based multivariate analyses. Often, its operation can be thought of as revealing the internal structure of the data in a way that best explains the variance in the data. If a multivariate dataset is visualised as a set of coordinates in a high-dimensional data space (1 axis per variable), PCA can supply the user with a lower-dimensional picture, a projection of this object when viewed from its most informative viewpoint. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.
PCA is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.
PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.[6][7]
## Intuition
PCA can be thought of as fitting an n-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small, and by omitting that axis and its corresponding principal component from our representation of the dataset, we lose only a commensurately small amount of information.
To find the axes of the ellipsoid, we must first subtract the mean of each variable from the dataset to center the data around the origin. Then, we compute the covariance matrix of the data, and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then, we must orthogonalize the set of eigenvectors, and normalize each to become unit vectors. Once this is done, each of the mutually orthogonal, unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues.
This procedure is sensitive to the scaling of the data, and there is no consensus as to how to best scale the data to obtain optimal results.
## Details
PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.[3]
Consider a data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor).
Mathematically, the transformation is defined by a set of p-dimensional vectors of weights or loadings ${\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}}$ that map each row vector ${\displaystyle \mathbf {x} _{(i)}}$ of X to a new vector of principal component scores ${\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{m})_{(i)}}$, given by
${\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,m}$
in such a way that the individual variables of t considered over the data set successively inherit the maximum possible variance from x, with each loading vector w constrained to be a unit vector.
### First component
In order to maximize variance, the first loading vector w(1) thus has to satisfy
${\displaystyle \mathbf {w} _{(1)}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,\left\{\sum _{i}\left(t_{1}\right)_{(i)}^{2}\right\}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}}$
Equivalently, writing this in matrix form gives
${\displaystyle \mathbf {w} _{(1)}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,\{\Vert \mathbf {Xw} \Vert ^{2}\}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,\left\{\mathbf {w} ^{T}\mathbf {X} ^{T}\mathbf {Xw} \right\}}$
Since w(1) has been defined to be a unit vector, it equivalently also satisfies
${\displaystyle \mathbf {w} _{(1)}={\operatorname {\arg \,max} }\,\left\{{\frac {\mathbf {w} ^{T}\mathbf {X} ^{T}\mathbf {Xw} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}}$
The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a symmetric matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector.
With w(1) found, the first component of a data vector x(i) can then be given as a score t1(i) = x(i)w(1) in the transformed co-ordinates, or as the corresponding vector in the original variables, {x(i)w(1)} w(1).
### Further components
The kth component can be found by subtracting the first k − 1 principal components from X:
${\displaystyle \mathbf {\hat {X}} _{k}=\mathbf {X} -\sum _{s=1}^{k-1}\mathbf {X} \mathbf {w} _{(s)}\mathbf {w} _{(s)}^{\rm {T}}}$
and then finding the loading vector which extracts the maximum variance from this new data matrix
${\displaystyle \mathbf {w} _{(k)}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {arg\,max} }}\left\{\Vert \mathbf {\hat {X}} _{k}\mathbf {w} \Vert ^{2}\right\}={\operatorname {\arg \,max} }\,\left\{{\tfrac {\mathbf {w} ^{T}\mathbf {\hat {X}} _{k}^{T}\mathbf {\hat {X}} _{k}\mathbf {w} }{\mathbf {w} ^{T}\mathbf {w} }}\right\}}$
It turns out that this gives the remaining eigenvectors of XTX, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the loading vectors are eigenvectors of XTX.
The kth component of a data vector x(i) can therefore be given as a score tk(i) = x(i)w(k) in the transformed co-ordinates, or as the corresponding vector in the space of the original variables, {x(i)w(k)} w(k), where w(k) is the kth eigenvector of XTX.
The full principal components decomposition of X can therefore be given as
${\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} }$
where W is a p-by-p matrix whose columns are the eigenvectors of XTX
### Covariances
XTX itself can be recognised as proportional to the empirical sample covariance matrix of the dataset X.
The sample covariance Q between two of the different principal components over the dataset is given by:
{\displaystyle {\begin{aligned}Q(\mathrm {PC} _{(j)},\mathrm {PC} _{(k)})&\propto (\mathbf {X} \mathbf {w} _{(j)})^{T}(\mathbf {X} \mathbf {w} _{(k)})\\&=\mathbf {w} _{(j)}^{T}\mathbf {X} ^{T}\mathbf {X} \mathbf {w} _{(k)}\\&=\mathbf {w} _{(j)}^{T}\lambda _{(k)}\mathbf {w} _{(k)}\\&=\lambda _{(k)}\mathbf {w} _{(j)}^{T}\mathbf {w} _{(k)}\end{aligned}}}
where the eigenvalue property of w(k) has been used to move from line 2 to line 3. However eigenvectors w(j) and w(k) corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset.
Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.
In matrix form, the empirical covariance matrix for the original variables can be written
${\displaystyle \mathbf {Q} \propto \mathbf {X} ^{T}\mathbf {X} =\mathbf {W} \mathbf {\Lambda } \mathbf {W} ^{T}}$
The empirical covariance matrix between the principal components becomes
${\displaystyle \mathbf {W} ^{T}\mathbf {Q} \mathbf {W} \propto \mathbf {W} ^{T}\mathbf {W} \,\mathbf {\Lambda } \,\mathbf {W} ^{T}\mathbf {W} =\mathbf {\Lambda } }$
where Λ is the diagonal matrix of eigenvalues λ(k) of XTX
(k) being equal to the sum of the squares over the dataset associated with each component k: λ(k) = Σi tk2(i) = Σi (x(i)w(k))2)
### Dimensionality reduction
The transformation T = X W maps a data vector x(i) from an original space of p variables to a new space of p variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first L principal components, produced by using only the first L loading vectors, gives the truncated transformation
${\displaystyle \mathbf {T} _{L}=\mathbf {X} \mathbf {W} _{L}}$
where the matrix TL now has n rows but only L columns. In other words, PCA learns a linear transformation ${\displaystyle t=W^{T}x,x\in R^{p},t\in R^{L},}$ where the columns of p × L matrix W form an orthogonal basis for the L features (the components of representation t) that are decorrelated.[8] By construction, of all the transformed data matrices with only L columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error ${\displaystyle \|\mathbf {T} \mathbf {W} ^{T}-\mathbf {T} _{L}\mathbf {W} _{L}^{T}\|_{2}^{2}}$ or ${\displaystyle \|\mathbf {X} -\mathbf {X} _{L}\|_{2}^{2}}$.
A principal components analysis scatterplot of Y-STR haplotypes calculated from repeat-count values for 37 Y-chromosomal STR markers from 354 individuals.
PCA has successfully found linear combinations of the different markers, that separate out different clusters corresponding to different lines of individuals' Y-chromosomal genetic descent.
Such dimensionality reduction can be a very useful step for visualising and processing high-dimensional datasets, while still retaining as much of the variance in the dataset as possible. For example, selecting L = 2 and keeping only the first two principal components finds the two-dimensional plane through the high-dimensional dataset in which the data is most spread out, so if the data contains clusters these too may be most spread out, and therefore most visible to be plotted out in a two-dimensional diagram; whereas if two directions through the data (or two of the original variables) are chosen at random, the clusters may be much less spread apart from each other, and may in fact be much more likely to substantially overlay each other, making them indistinguishable.
Similarly, in regression analysis, the larger the number of explanatory variables allowed, the greater is the chance of overfitting the model, producing conclusions that fail to generalise to other datasets. One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called principal component regression.
Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix W, which can be thought of as a high-dimensional rotation of the co-ordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher signal-to-noise ratio. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss.
### Singular value decomposition
The principal components transformation can also be associated with another matrix factorization, the singular value decomposition (SVD) of X,
${\displaystyle \mathbf {X} =\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}}$
Here Σ is an n-by-p rectangular diagonal matrix of positive numbers σ(k), called the singular values of X; U is an n-by-n matrix, the columns of which are orthogonal unit vectors of length n called the left singular vectors of X; and W is a p-by-p whose columns are orthogonal unit vectors of length p and called the right singular vectors of X.
In terms of this factorization, the matrix XTX can be written
{\displaystyle {\begin{aligned}\mathbf {X} ^{T}\mathbf {X} &=\mathbf {W} \mathbf {\Sigma } \mathbf {U} ^{T}\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}\\&=\mathbf {W} \mathbf {\Sigma } ^{2}\mathbf {W} ^{T}\end{aligned}}}
Comparison with the eigenvector factorization of XTX establishes that the right singular vectors W of X are equivalent to the eigenvectors of XTX, while the singular values σ(k) of Σ are equal to the square roots of the eigenvalues λ(k) of XTX.
Using the singular value decomposition the score matrix T can be written
{\displaystyle {\begin{aligned}\mathbf {T} &=\mathbf {X} \mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \mathbf {W} ^{T}\mathbf {W} \\&=\mathbf {U} \mathbf {\Sigma } \end{aligned}}}
so each column of T is given by one of the left singular vectors of X multiplied by the corresponding singular value. This form is also the polar decomposition of T.
Efficient algorithms exist to calculate the SVD of X without having to form the matrix XTX, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix[citation needed], unless only a handful of components are required.
As with the eigen-decomposition, a truncated n × L score matrix TL can be obtained by considering only the first L largest singular values and their singular vectors:
${\displaystyle \mathbf {T} _{L}=\mathbf {U} _{L}\mathbf {\Sigma } _{L}=\mathbf {X} \mathbf {W} _{L}}$
The truncation of a matrix M or T using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of rank L to the original matrix, in the sense of the difference between the two having the smallest possible Frobenius norm, a result known as the Eckart–Young theorem [1936].
## Further considerations
Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line. The second principal component corresponds to the same concept after all correlation with the first principal component has been subtracted from the points. The singular values (in Σ) are the square roots of the eigenvalues of the matrix XTX. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is correlated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see below). PCA is often used in this manner for dimensionality reduction. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example and when applicable, to the discrete cosine transform, and in particular to the DCT-II which is simply known as the "DCT". Nonlinear dimensionality reduction techniques tend to be more computationally demanding than PCA.
PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same sample variance and are positively correlated, then the PCA will entail a rotation by 45° and the "loadings" for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Note that Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.
Mean subtraction (a.k.a. "mean centering") is necessary for performing PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data. A mean of zero is needed for finding a basis that minimizes the mean square error of the approximation of the data.[9]
Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: Pearson Product-Moment Correlation). Also see the article by Kromrey & Foster-Johnson (1998) on "Mean-centering in Moderated Regression: Much Ado About Nothing".
An autoencoder neural network with a linear hidden layer is similar to PCA. Upon convergence, the weight vectors of the K neurons in the hidden layer will form a basis for the space spanned by the first K principal components. Unlike PCA, this technique will not necessarily produce orthogonal vectors.
PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability.[10] However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes. [11] The linear discriminant analysis is an alternative which is optimized for class separability.
## Table of symbols and abbreviations
Symbol Meaning Dimensions Indices
${\displaystyle \mathbf {X} =\{X[i,j]\}}$ data matrix, consisting of the set of all data vectors, one vector per row ${\displaystyle n\times p}$ ${\displaystyle i=1\ldots n}$
${\displaystyle j=1\ldots p}$
${\displaystyle n\,}$ the number of row vectors in the data set ${\displaystyle 1\times 1}$ scalar
${\displaystyle p\,}$ the number of elements in each row vector (dimension) ${\displaystyle 1\times 1}$ scalar
${\displaystyle L\,}$ the number of dimensions in the dimensionally reduced subspace, ${\displaystyle 1\leq L\leq p}$ ${\displaystyle 1\times 1}$ scalar
${\displaystyle \mathbf {u} =\{u[j]\}}$ vector of empirical means, one mean for each column j of the data matrix ${\displaystyle p\times 1}$ ${\displaystyle j=1\ldots p}$
${\displaystyle \mathbf {s} =\{s[j]\}}$ vector of empirical standard deviations, one standard deviation for each column j of the data matrix ${\displaystyle p\times 1}$ ${\displaystyle j=1\ldots p}$
${\displaystyle \mathbf {h} =\{h[i]\}}$ vector of all 1's ${\displaystyle 1\times n}$ ${\displaystyle i=1\ldots n}$
${\displaystyle \mathbf {B} =\{B[i,j]\}}$ deviations from the mean of each column j of the data matrix ${\displaystyle n\times p}$ ${\displaystyle i=1\ldots n}$
${\displaystyle j=1\ldots p}$
${\displaystyle \mathbf {Z} =\{Z[m,n]\}}$ z-scores, computed using the mean and standard deviation for each row m of the data matrix ${\displaystyle n\times p}$ ${\displaystyle i=1\ldots n}$
${\displaystyle j=1\ldots p}$
${\displaystyle \mathbf {C} =\{C[k,l]\}}$ covariance matrix ${\displaystyle p\times p}$ ${\displaystyle k=1\ldots p}$
${\displaystyle l=1\ldots p}$
${\displaystyle \mathbf {R} =\{R[k,l]\}}$ correlation matrix ${\displaystyle p\times p}$ ${\displaystyle k=1\ldots p}$
${\displaystyle l=1\ldots p}$
${\displaystyle \mathbf {V} =\{V[j,k]\}}$ matrix consisting of the set of all eigenvectors of C, one eigenvector per column ${\displaystyle p\times p}$ ${\displaystyle j=1\ldots p}$
${\displaystyle k=1\ldots p}$
${\displaystyle \mathbf {D} =\{D[k,l]\}}$ diagonal matrix consisting of the set of all eigenvalues of C along its principal diagonal, and 0 for all other elements ${\displaystyle p\times p}$ ${\displaystyle k=1\ldots p}$
${\displaystyle l=1\ldots p}$
${\displaystyle \mathbf {W} =\{W[j,k]\}}$ matrix of basis vectors, one vector per column, where each basis vector is one of the eigenvectors of C, and where the vectors in W are a sub-set of those in V ${\displaystyle p\times L}$ ${\displaystyle j=1\ldots p}$
${\displaystyle k=1\ldots L}$
${\displaystyle \mathbf {T} =\{T[i,k]\}}$ matrix consisting of n row vectors, where each vector is the projection of the corresponding data vector from matrix X onto the basis vectors contained in the columns of matrix W. ${\displaystyle n\times L}$ ${\displaystyle i=1\ldots n}$
${\displaystyle k=1\ldots L}$
## Properties and limitations of PCA
### Properties[12]
Property 1: For any integer q, 1 ≤ q ≤ p, consider the orthogonal linear transformation
${\displaystyle y=\mathbf {B'} x}$
where ${\displaystyle y}$ is a q-element vector and ${\displaystyle \mathbf {B'} }$ is a (q × p) matrix, and let ${\displaystyle \mathbf {\Sigma } _{y}=\mathbf {B'} \mathbf {\Sigma } \mathbf {B} }$ be the variance-covariance matrix for ${\displaystyle y}$. Then the trace of ${\displaystyle \mathbf {\Sigma } _{y}}$, denoted ${\displaystyle {\text{tr}}(\mathbf {\Sigma } _{y})}$, is maximized by taking ${\displaystyle \mathbf {B} =\mathbf {A} _{q}}$, where ${\displaystyle \mathbf {A} _{q}}$ consists of the first q columns of ${\displaystyle \mathbf {A} }$ ${\displaystyle (\mathbf {B'} }$ is the transposition of ${\displaystyle \mathbf {B} )}$.
Property 2: Consider again the orthonormal transformation
${\displaystyle y=\mathbf {B'} x}$
with ${\displaystyle x,\mathbf {B} ,\mathbf {A} }$ and ${\displaystyle \mathbf {\Sigma } _{y}}$ defined as before. Then ${\displaystyle {\text{tr}}(\mathbf {\Sigma } _{y})}$ is minimized by taking ${\displaystyle \mathbf {B} =\mathbf {A} _{q}^{*},}$ where ${\displaystyle \mathbf {A} _{q}^{*}}$ consists of the last q columns of ${\displaystyle \mathbf {A} }$.
The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of x, and they may also be useful in regression, in selecting a subset of variables from x, and in outlier detection.
Property 3: (Spectral Decomposition of Σ)
${\displaystyle \mathbf {\Sigma } =\lambda _{1}\alpha _{1}\alpha _{1}'+\cdots +\lambda _{p}\alpha _{p}\alpha _{p}'}$
Before we look at its usage, we first look at diagonal elements,
${\displaystyle {\text{Var}}(x_{j})=\sum _{k=1}^{P}\lambda _{k}\alpha _{kj}^{2}}$
Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of x into decreasing contributions due to each PC, but we can also decompose the whole covariance matrix into contributions ${\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}$ from each PC. Although not strictly decreasing, the elements of ${\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}$ will tend to become smaller as ${\displaystyle k}$ increases, as ${\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'}$ decreases for increasing ${\displaystyle k}$, whereas the elements of ${\displaystyle \alpha _{k}}$ tend to stay 'about the same size'because of the normalization constraints: ${\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\cdots ,p}$
### Limitations
As noted above, the results of PCA depend on the scaling of the variables. A scale-invariant form of PCA has been developed.[13]
The applicability of PCA is limited by certain assumptions[14] made in its derivation.
### PCA and information theory
Dimensionality reduction loses information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.
Under the assumption that
${\displaystyle \mathbf {x} =\mathbf {s} +\mathbf {n} }$
i.e., that the data vector ${\displaystyle \mathbf {x} }$ is the sum of the desired information-bearing signal ${\displaystyle \mathbf {s} }$ and a noise signal ${\displaystyle \mathbf {n} }$ one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view.
In particular, Linsker showed that if ${\displaystyle \mathbf {s} }$ is Gaussian and ${\displaystyle \mathbf {n} }$ is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information ${\displaystyle I(\mathbf {y} ;\mathbf {s} )}$ between the desired information ${\displaystyle \mathbf {s} }$ and the dimensionality-reduced output ${\displaystyle \mathbf {y} =\mathbf {W} _{L}^{T}\mathbf {x} }$.[15]
If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (i.e., the components of the vector ${\displaystyle \mathbf {n} }$ are iid), but the information-bearing signal ${\displaystyle \mathbf {s} }$ is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the information loss, which is defined as[16][17]
${\displaystyle I(\mathbf {x} ;\mathbf {s} )-I(\mathbf {y} ;\mathbf {s} ).}$
The optimality of PCA is also preserved if the noise ${\displaystyle \mathbf {n} }$ is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal ${\displaystyle \mathbf {s} }$.[18] In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise ${\displaystyle \mathbf {n} }$ becomes dependent.
## Computing PCA using the covariance method
The following is a detailed description of PCA using the covariance method (see also here) as opposed to the correlation method.[19] But note that it is better to use the singular value decomposition (using standard software).[citation needed]
The goal is to transform a given data set X of dimension p to an alternative data set Y of smaller dimension L. Equivalently, we are seeking to find the matrix Y, where Y is the Kosambi-Karhunen–Loève transform (KLT) of matrix X:
${\displaystyle \mathbf {Y} =\mathbb {KLT} \{\mathbf {X} \}}$
### Organize the data set
Suppose you have data comprising a set of observations of p variables, and you want to reduce the data so that each observation can be described with only L variables, L < p. Suppose further, that the data are arranged as a set of n data vectors ${\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}$ with each ${\displaystyle \mathbf {x} _{i}}$ representing a single grouped observation of the p variables.
• Write ${\displaystyle \mathbf {x} _{1}\ldots \mathbf {x} _{n}}$ as row vectors, each of which has p columns.
• Place the row vectors into a single matrix X of dimensions n × p.
### Calculate the empirical mean
• Find the empirical mean along each column j = 1, ..., p.
• Place the calculated mean values into an empirical mean vector u of dimensions p × 1.
${\displaystyle u[j]={1 \over n}\sum _{i=1}^{n}X[i,j]}$
### Calculate the deviations from the mean
Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data.[20] Hence we proceed by centering the data as follows:
• Subtract the empirical mean vector u from each row of the data matrix X.
• Store mean-subtracted data in the n × p matrix B.
${\displaystyle \mathbf {B} =\mathbf {X} -\mathbf {h} \mathbf {u} ^{T}}$
where h is an n × 1 column vector of all 1s:
${\displaystyle h[i]=1\,\qquad \qquad {\text{for }}i=1,\ldots ,n}$
### Find the covariance matrix
${\displaystyle \mathbf {C} ={1 \over {n-1}}\mathbf {B} ^{*}\otimes \mathbf {B} }$
where ${\displaystyle *}$ is the conjugate transpose operator. Note that if B consists entirely of real numbers, which is the case in many applications, the "conjugate transpose" is the same as the regular transpose.
• Please note that outer products apply to vectors. For tensor cases we should apply tensor products, but the covariance matrix in PCA is a sum of outer products between its sample vectors; indeed, it could be represented as B*.B. See the covariance matrix sections on the discussion page for more information.
• The reasoning behind using N − 1 instead of N to calculate the covariance is Bessel's correction
### Find the eigenvectors and eigenvalues of the covariance matrix
${\displaystyle \mathbf {V} ^{-1}\mathbf {C} \mathbf {V} =\mathbf {D} }$
where D is the diagonal matrix of eigenvalues of C. This step will typically involve the use of a computer-based algorithm for computing eigenvectors and eigenvalues. These algorithms are readily available as sub-components of most matrix algebra systems, such as SAS,[21] R, MATLAB,[22][23] Mathematica,[24] SciPy, IDL (Interactive Data Language), or GNU Octave as well as OpenCV.
• Matrix D will take the form of an p × p diagonal matrix, where
${\displaystyle D[k,l]=\lambda _{k}\qquad {\text{for }}k=l}$
is the jth eigenvalue of the covariance matrix C, and
${\displaystyle D[k,l]=0\qquad {\text{for }}k\neq l.}$
• Matrix V, also of dimension p × p, contains p column vectors, each of length p, which represent the p eigenvectors of the covariance matrix C.
• The eigenvalues and eigenvectors are ordered and paired. The jth eigenvalue corresponds to the jth eigenvector.
• Matrix V denotes the matrix of right eigenvectors (as opposed to left eigenvectors). In general, the matrix of right eigenvectors need not be the (conjugate) transpose of the matrix of left eigenvectors.
### Rearrange the eigenvectors and eigenvalues
• Sort the columns of the eigenvector matrix V and eigenvalue matrix D in order of decreasing eigenvalue.
• Make sure to maintain the correct pairings between the columns in each matrix.
### Compute the cumulative energy content for each eigenvector
• The eigenvalues represent the distribution of the source data's energy[clarification needed] among each of the eigenvectors, where the eigenvectors form a basis for the data. The cumulative energy content g for the jth eigenvector is the sum of the energy content across all of the eigenvalues from 1 through j:
${\displaystyle g[j]=\sum _{k=1}^{j}D[k,k]\qquad \mathrm {for} \qquad j=1,\dots ,p}$[citation needed]
### Select a subset of the eigenvectors as basis vectors
• Save the first L columns of V as the p × L matrix W:
${\displaystyle W[k,l]=V[k,l]\qquad \mathrm {for} \qquad k=1,\dots ,p\qquad l=1,\dots ,L}$
where
${\displaystyle 1\leq L\leq p.}$
• Use the vector g as a guide in choosing an appropriate value for L. The goal is to choose a value of L as small as possible while achieving a reasonably high value of g on a percentage basis. For example, you may want to choose L so that the cumulative energy g is above a certain threshold, like 90 percent. In this case, choose the smallest value of L such that
${\displaystyle {\frac {g[L]}{g[p]}}\geq 0.9\,}$
### Convert the source data to z-scores (optional)
• Create an p × 1 empirical standard deviation vector s from the square root of each element along the main diagonal of the diagonalized covariance matrix C. (Note, that scaling operations do not commute with the KLT thus we must scale by the variances of the already-decorrelated vector, which is the diagonal of C) :
${\displaystyle \mathbf {s} =\{s[j]\}=\{{\sqrt {C[j,j]}}\}\qquad {\text{for }}j=1,\ldots ,p}$
• Calculate the n × p z-score matrix:
${\displaystyle \mathbf {Z} ={\mathbf {B} \over \mathbf {h} \cdot \mathbf {s} ^{T}}}$ (divide element-by-element)
• Note: While this step is useful for various applications as it normalizes the data set with respect to its variance, it is not integral part of PCA/KLT
### Project the z-scores of the data onto the new basis
• The projected vectors are the columns of the matrix
${\displaystyle \mathbf {T} =\mathbf {Z} \cdot \mathbf {W} =\mathbb {KLT} \{\mathbf {X} \}.}$
• The rows of matrix T represent the Kosambi-Karhunen–Loève transforms (KLT) of the data vectors in the rows of matrix X.
## Derivation of PCA using the covariance method
Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean.
We want to find ${\displaystyle (\ast )\,}$ a d × d orthonormal transformation matrix P so that PX has a diagonal covariance matrix (i.e. PX is a random vector with all its distinct components pairwise uncorrelated).
A quick computation assuming ${\displaystyle P}$ were unitary yields:
{\displaystyle {\begin{aligned}\operatorname {cov} (PX)&=\mathbb {E} [PX~(PX)^{*}]\\&=\mathbb {E} [PX~X^{*}P^{*}]\\&=P~\mathbb {E} [XX^{*}]P^{*}\\&=P~\operatorname {cov} (X)P^{-1}\\\end{aligned}}}
Hence ${\displaystyle (\ast )\,}$ holds if and only if ${\displaystyle \operatorname {var} (X)}$ were diagonalisable by ${\displaystyle P}$.
This is very constructive, as var(X) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.
### Iterative computation
In practical implementations especially with high dimensional data (large p), the covariance method is rarely used because it is not efficient. One way to compute the first principal component efficiently[25] is shown in the following pseudo-code, for a data matrix X with zero mean, without ever computing its covariance matrix.
r = a random vector of length p
do c times:
s = 0 (a vector of length p)
for each row ${\displaystyle \mathbf {x} \in \mathbf {X} }$
${\displaystyle \mathbf {s} =\mathbf {s} +(\mathbf {x} \cdot \mathbf {r} )\mathbf {x} }$
${\displaystyle \mathbf {r} ={\frac {\mathbf {s} }{|\mathbf {s} |}}}$
return r
This algorithm is simply an efficient way of calculating XTX r, normalizing, and placing the result back in r (power iteration). It avoids the np2 operations of calculating the covariance matrix. r will typically get close to the first principal component of X within a small number of iterations, c. (The magnitude of s will be larger after each iteration. Convergence can be detected when it increases by an amount too small for the precision of the machine.)
Subsequent principal components can be computed by subtracting component r from X (see Gram–Schmidt) and then repeating this algorithm to find the next principal component. However this simple approach is not numerically stable if more than a small number of principal components are required, because imprecisions in the calculations will additively affect the estimates of subsequent principal components. More advanced methods build on this basic idea, as with the closely related Lanczos algorithm.
One way to compute the eigenvalue that corresponds with each principal component is to measure the difference in mean-squared-distance between the rows and the centroid, before and after subtracting out the principal component. The eigenvalue that corresponds with the component that was removed is equal to this difference.
### The NIPALS method
Non-linear iterative partial least squares (NIPALS) is an algorithm for computing the first few components in a principal component or partial least squares analysis. For very-high-dimensional datasets, such as those generated in the *omics sciences (e.g., genomics, metabolomics) it is usually only necessary to compute the first few PCs. The non-linear iterative partial least squares (NIPALS) algorithm calculates t1 and w1T from X. The outer product, t1w1T can then be subtracted from X leaving the residual matrix E1. This can be then used to calculate subsequent PCs.[26] This results in a dramatic reduction in computational time since calculation of the covariance matrix is avoided.
However, for large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality due to machine precision limitations accumulated in each iteration step.[27] A Gram–Schmidt (GS) re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality.[28]
### Online/sequential estimation
In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.[29]
## PCA and qualitative variables
In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable species. For this, the following results are produced.
• Identification, on the factorial planes, of the different species e.g. using different colors.
• Representation, on the factorial planes, of the centers of gravity of plants belonging to the same species.
• For each center of gravity and each axis, p-value to judge the significance of the difference between the center of gravity and origin.
These results are what is called introducing a qualitative variable as supplementary element. This procedure is detailed in and Husson, Lê & Pagès 2009 and Pagès 2013. Few software offer this option in an "automatic" way. This is the case of SPAD that historically, following the work of Ludovic Lebart, was the first to propose this option, and the R package FactoMineR.
## Applications
### Interest rate derivatives portfolios
Principal component analysis can be directly applied to the risk management of interest rate derivatives portfolios.[30] Trading multiple swap instruments which are usually a function of 30-500 other market quotable swap instruments is sought to be reduced to usually 3 or 4 principal components, representing the path of interest rates on a macro basis. Converting risks to be represented as those to factor loadings (or multipliers) provides assessments and understanding beyond that available to simply collectively viewing risks to individual 30-500 buckets.
### Neuroscience
A variant of principal components analysis is used in neuroscience to identify the specific properties of a stimulus that increase a neuron's probability of generating an action potential.[31] This technique is known as spike-triggered covariance analysis. In a typical application an experimenter presents a white noise process as a stimulus (usually either as a sensory input to a test subject, or as a current injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the covariance matrix of the spike-triggered ensemble, the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The eigenvectors of the difference between the spike-triggered covariance matrix and the covariance matrix of the prior stimulus ensemble (the set of all stimuli, defined over the same length time window) then indicate the directions in the space of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest positive change compared to the variance of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.
In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. Spike sorting is an important procedure because extracellular recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs clustering analysis to associate specific action potentials with individual neurons.
PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, i.e. order parameters, during phase transitions in the brain.[32]
## Relation between PCA and K-means clustering
It was asserted in [33][34] that the relaxed solution of k-means clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of k-means clustering was not a new result (see, for example,[35]), and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.[36]
## Relation between PCA and factor analysis[37]
Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The principal components can be used to find clusters in a set of data. PCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (i.e., translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.
Factor analysis is similar to principal component analysis[38], in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of variables, excluding unique variance[39]" . In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (i.e. shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations.[40] Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (i.e., latent constructs or factors) or causal modeling.
## Correspondence analysis
Correspondence analysis (CA) was developed by Jean-Paul Benzécri[41] and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables. CA decomposes the chi-squared statistic associated to this table into orthogonal factors.[42] Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not. Several variants of CA are available including detrended correspondence analysis and canonical correspondence analysis. One special extension is multiple correspondence analysis, which may be seen as the counterpart of principal component analysis for categorical data.[43]
## Generalizations
### Nonlinear generalizations
Linear PCA versus nonlinear Principal Manifolds[44] for visualization of breast cancer microarray data: a) Configuration of nodes and 2D Principal Surface in the 3D PCA linear manifold. The dataset is curved and cannot be mapped adequately on a 2D principal plane; b) The distribution in the internal 2D non-linear principal surface coordinates (ELMap2D) together with an estimation of the density of points; c) The same as b), but for the linear 2D PCA manifold (PCA2D). The "basal" breast cancer subtype is visualized more adequately with ELMap2D and some features of the distribution become better resolved in comparison to PCA2D. Principal manifolds are produced by the elastic maps algorithm. Data are available for public competition.[45] Software is available for free non-commercial use.[46]
Most of the modern methods for nonlinear dimensionality reduction find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. Principal curves and manifolds[47] give the natural geometric framework for PCA generalization and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold for data approximation, and by encoding using standard geometric projection onto the manifold, as it is illustrated by Fig. See also the elastic map algorithm and principal geodesic analysis. Another popular generalization is kernel PCA, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.
### Multilinear generalizations
In multilinear subspace learning,[48] PCA is generalized to multilinear PCA (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.
### Higher order
N-way principal component analysis may be performed with models such as Tucker decomposition, PARAFAC, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.
### Robustness – weighted PCA
While PCA finds the mathematically optimal method (as in minimizing the squared error), it is sensitive to outliers in the data that produce large errors PCA tries to avoid. It therefore is common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example, in data mining algorithms like correlation clustering, the assignment of points to clusters and outliers is not known beforehand. A recently proposed generalization of PCA[49] based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.
### Robust PCA via decomposition in low-rank and sparse matrices
Robust principal component analysis (RPCA) is a modification of the widely used statistical procedure principal component analysis (PCA) which works well with respect to grossly corrupted observations.
### Sparse PCA
A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables.
## Similar techniques
### Independent component analysis
Independent component analysis (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.
### Network component analysis
Given a matrix ${\displaystyle E}$, it tries to decompose it into two matrices such that ${\displaystyle E=AP}$. A key difference from techniques such as PCA and ICA is that some of the entries of ${\displaystyle A}$ are constrained to be 0. Here ${\displaystyle P}$ is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :-
1. ${\displaystyle A}$ has full column rank
2. Each column of ${\displaystyle A}$ must have at least ${\displaystyle L-1}$ zeroes where ${\displaystyle L}$ is the number of columns of ${\displaystyle A}$ (or alternatively the number of rows of ${\displaystyle P}$). The justification for this criterion is that if a node is removed from the regulatory layer along with all the output nodes connected to it, the result must still be characterized by a connectivity matrix with full column rank.
3. ${\displaystyle P}$ must have full row rank.
then the decomposition is unique up to multiplication by a scalar.[50]
## Software/source code
• Analytica – The built-in EigenDecomp function computes principal components.
• DataMelt – A Java free program that implements several classes to build PCA analysis and to calculate eccentricity of random distributions.
• ELKI – includes PCA for projection, including robust variants of PCA, as well as PCA-based clustering algorithms.
• IGOR PRO - The built-in PCA operation performs principal component analysis.
• Julia – Supports PCA with the pca function in the MultivariateStats package
• KNIME – A java based nodal arrenging software for Analysis, in this the nodes called PCA, PCA compute, PCA Apply, PCA inverse make it easily.
• Mathematica – Implements principal component analysis with the PrincipalComponents command using both covariance and correlation methods.
• MATLAB Statistics Toolbox – The functions princomp and pca (R2012b) give the principal components, while the function pcares gives the residuals and reconstructed matrix for a low-rank PCA approximation.
• Matplotlib python library have a PCA package in the .mlab module.
• MLPACK – Provides an implementation of principal component analysis in C++.
• NAG Library – Principal components analysis is implemented via the g03aa routine (available in both the Fortran versions of the Library).
• NMath – Proprietary numerical library containing PCA for the .NET Framework.
• GNU Octave – Free software computational environment mostly compatible with MATLAB, the function princomp gives the principal component.
• OpenCV
• Oracle Database 12c – Implemented via DBMS_DATA_MINING.SVDS_SCORING_MODE by specifying setting value SVDS_SCORING_PCA
• Orange (software) – Integrates PCA in its visual programming environment. PCA displays a scree plot (degree of explained variance) where user can interactively select the number of principal components.
• Origin – Contains PCA in its Pro version.
• Padasip – Python library that contains PCA in its data preprocessing module.
• Partek Genomics Suite – Statistical software able to perform PCA.
• Qlucore – Commercial software for analyzing multivariate data with instant response using PCA.
• RFree statistical package, the functions princomp and prcomp can be used for principal component analysis; prcomp uses singular value decomposition which generally gives better numerical accuracy. Some packages that implement PCA in R, include, but are not limited to: ade4, vegan, ExPosition, and FactoMineR.
• Scikit-learn – Python library for machine learning which contains PCA, Probabilistic PCA, Kernel PCA, Sparse PCA and other techniques in the decomposition module.
• Weka – Java library for machine learning which contains modules for computing principal components.
## References
1. ^ Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space" (PDF). Philosophical Magazine. 2 (11): 559–572. doi:10.1080/14786440109462720.
2. ^ Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology, 24, 417–441, and 498–520.
Hotelling, H. (1936). Relations between two sets of variates. Biometrika, 28, 321–377
3. ^ a b Jolliffe I.T. Principal Component Analysis, Series: Springer Series in Statistics, 2nd ed., Springer, NY, 2002, XXIX, 487 p. 28 illus. ISBN 978-0-387-95442-4
4. ^ Abdi. H., & Williams, L.J. (2010). "Principal component analysis". Wiley Interdisciplinary Reviews: Computational Statistics. 2 (4): 433–459. doi:10.1002/wics.101.
5. ^ Shaw P.J.A. (2003) Multivariate statistics for the Environmental Sciences, Hodder-Arnold. ISBN 0-340-80763-6.[page needed]
6. ^ Barnett, T. P. & R. Preisendorfer. (1987). "Origins and levels of monthly and seasonal forecast skill for United States surface air temperatures determined by canonical correlation analysis.". Monthly Weather Review 115.
7. ^ Hsu, Daniel, Sham M. Kakade, and Tong Zhang (2008). "A spectral algorithm for learning hidden markov models.". arXiv:.
8. ^ Bengio, Y.; et al. (2013). "Representation Learning: A Review and New Perspectives" (PDF). Pattern Analysis and Machine Intelligence. 35 (8): 1798–1828. doi:10.1109/TPAMI.2013.50.
9. ^ A. A. Miranda, Y. A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer
10. ^ Fukunaga, Keinosuke (1990). Introduction to Statistical Pattern Recognition. Elsevier. ISBN 0-12-269851-7.
11. ^ Alizadeh, Elaheh; Lyons, Samanthe M; Castle, Jordan M; Prasad, Ashok (2016). "Measuring systematic changes in invasive cancer cell shape using Zernike moments". Integrative Biology. 8 (11): 1183-1193. doi:10.1039/C6IB00100A.
12. ^ Jolliffe, I. T. (2002). Principal Component Analysis, second edition Springer-Verlag. ISBN 978-0-387-95442-4.
13. ^ Leznik, M; Tofallis, C. 2005 [uhra.herts.ac.uk/bitstream/handle/2299/715/S56.pdf Estimating Invariant Principal Components Using Diagonal Regression.]
14. ^ Jonathon Shlens, A Tutorial on Principal Component Analysis.
15. ^ Linsker, Ralph (March 1988). "Self-organization in a perceptual network". IEEE Computer. 21 (3): 105–117. doi:10.1109/2.36.
16. ^ Deco & Obradovic (1996). An Information-Theoretic Approach to Neural Computing. New York, NY: Springer.
17. ^ Plumbley, Mark (1991). "Information theory and unsupervised neural networks".Tech Note
18. ^ Geiger, Bernhard; Kubin, Gernot (January 2013). "Signal Enhancement as Minimization of Relevant Information Loss". Proc. ITG Conf. on Systems, Communication and Coding. arXiv:.
19. ^ "Engineering Statistics Handbook Section 6.5.5.2". Retrieved 19 January 2015.
20. ^ A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer
21. ^ http://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#statug_princomp_sect001.htm
22. ^ eig function Matlab documentation
23. ^ MATLAB PCA-based Face recognition software
24. ^ Eigenvalues function Mathematica documentation
25. ^ Roweis, Sam. "EM Algorithms for PCA and SPCA." Advances in Neural Information Processing Systems. Ed. Michael I. Jordan, Michael J. Kearns, and Sara A. Solla The MIT Press, 1998.
26. ^ Geladi, Paul; Kowalski, Bruce (1986). "Partial Least Squares Regression:A Tutorial". Analytica Chimica Acta. 185: 1–17. doi:10.1016/0003-2670(86)80028-9.
27. ^ Kramer, R. (1998). Chemometric Techniques for Quantitative Analysis. New York: CRC Press.
28. ^ Andrecut, M. (2009). "Parallel GPU Implementation of Iterative PCA Algorithms". Journal of Computational Biology. 16 (11): 1593–1599. doi:10.1089/cmb.2008.0221. PMID 19772385.
29. ^ Warmuth, M. K.; Kuzmin, D. (2008). "Randomized online PCA algorithms with regret bounds that are logarithmic in the dimension". Journal of Machine Learning Research. 9: 2287–2320.
30. ^ The Pricing and Hedging of Interest Rate Derivatives: A Practical Guide to Swaps, J H M Darbyshire, 2016, ISBN 978-0995455511
31. ^ Brenner, N., Bialek, W., & de Ruyter van Steveninck, R.R. (2000).
32. ^ Jirsa, Victor; Friedrich, R; Haken, Herman; Kelso, Scott (1994). "A theoretical model of phase transitions in the human brain". Biological Cybernetics. 71 (1): 27–35. doi:10.1007/bf00198909. PMID 8054384.
33. ^ H. Zha, C. Ding, M. Gu, X. He and H.D. Simon (Dec 2001). "Spectral Relaxation for K-means Clustering" (PDF). Neural Information Processing Systems vol.14 (NIPS 2001). Vancouver, Canada: 1057–1064.
34. ^ Chris Ding and Xiaofeng He (July 2004). "K-means Clustering via Principal Component Analysis" (PDF). Proc. of Int'l Conf. Machine Learning (ICML 2004): 225–232.
35. ^ Drineas, P.; A. Frieze; R. Kannan; S. Vempala; V. Vinay (2004). "Clustering large graphs via the singular value decomposition" (PDF). Machine learning. 56: 9–33. doi:10.1023/b:mach.0000033113.59016.96. Retrieved 2012-08-02.
36. ^ Cohen, M.; S. Elder; C. Musco; C. Musco; M. Persu (2014). "Dimensionality reduction for k-means clustering and low rank approximation (Appendix B)". arXiv:.
38. ^ Ijsmi, Editor (2017-04-26). "Tutorial: Factor analysis revisited – An overview with the help of SPSS, SAS and R packages". International Journal of Statistics and Medical Informatics. 3 (1).
39. ^ Timothy A. Brown. Confirmatory Factor Analysis for Applied Research Methodology in the social sciences. Guilford Press, 2006
40. ^ I.T. Jolliffe. Principal Component Analysis, Second Edition. Chapter 7. 2002
41. ^ Benzécri, J.-P. (1973). L'Analyse des Données. Volume II. L'Analyse des Correspondances. Paris, France: Dunod.
42. ^ Greenacre, Michael (1983). Theory and Applications of Correspondence Analysis. London: Academic Press. ISBN 0-12-299050-1.
43. ^ Le Roux; Brigitte and Henry Rouanet (2004). Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis. Dordrecht: Kluwer.
44. ^ A. N. Gorban, A. Y. Zinovyev, Principal Graphs and Manifolds, In: Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, Olivas E.S. et al Eds. Information Science Reference, IGI Global: Hershey, PA, USA, 2009. 28–59.
45. ^ Wang, Y.; Klijn, J. G.; Zhang, Y.; Sieuwerts, A. M.; Look, M. P.; Yang, F.; Talantov, D.; Timmermans, M.; Meijer-van Gelder, M. E.; Yu, J.; et al. (2005). "Gene expression profiles to predict distant metastasis of lymph-node-negative primary breast cancer". The Lancet. 365 (9460): 671–679. doi:10.1016/S0140-6736(05)17947-1. Data online
46. ^ Zinovyev, A. "ViDaExpert – Multidimensional Data Visualization Tool". Institut Curie. Paris. (free for non-commercial use)
47. ^ A.N. Gorban, B. Kegl, D.C. Wunsch, A. Zinovyev (Eds.), Principal Manifolds for Data Visualisation and Dimension Reduction, LNCSE 58, Springer, Berlin – Heidelberg – New York, 2007. ISBN 978-3-540-73749-0
48. ^ Lu, Haiping; Plataniotis, K.N.; Venetsanopoulos, A.N. (2011). "A Survey of Multilinear Subspace Learning for Tensor Data" (PDF). Pattern Recognition. 44 (7): 1540–1551. doi:10.1016/j.patcog.2011.01.004.
49. ^ Kriegel, H. P.; Kröger, P.; Schubert, E.; Zimek, A. (2008). "A General Framework for Increasing the Robustness of PCA-Based Correlation Clustering Algorithms". Scientific and Statistical Database Management. Lecture Notes in Computer Science. 5069: 418–435. doi:10.1007/978-3-540-69497-7_27. ISBN 978-3-540-69476-2.
50. ^ "Network component analysis: Reconstruction of regulatory signals in biological systems" (PDF). Retrieved February 10, 2015.
• Jackson, J.E. (1991). A User's Guide to Principal Components (Wiley).
• Jolliffe, I. T. (1986). Principal Component Analysis. Springer-Verlag. p. 487. doi:10.1007/b98835. ISBN 978-0-387-95442-4.
• Jolliffe, I.T. (2002). Principal Component Analysis, second edition (Springer).
• Husson François, Lê Sébastien & Pagès Jérôme (2009). Exploratory Multivariate Analysis by Example Using R. Chapman & Hall/CRC The R Series, London. 224p. |ISBN 978-2-7535-0938-2
• Pagès Jérôme (2014). Multiple Factor Analysis by Example Using R. Chapman & Hall/CRC The R Series London 272 p | 15,591 | 63,807 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 155, "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} | 3.234375 | 3 | CC-MAIN-2017-22 | latest | en | 0.923384 |
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### Art and Craft of Mathematical Problem Solving (The Great Courses)
MP4 + PDF | Video: 856x480 | Audio: AAC, 44.1Khz , 2ch | Duration: 12 hours | Language: English | 6.6 GB
One of life's most exhilarating experiences is the "aha!
" moment that comes from pondering a mathematical problem and then seeing the way to an elegant solution. And many problems can be solved relatively quickly with the right strategy. For example, how fast can you find the sum of the numbers 1 + 2 + 3 up to 100? This was famously answered in the late 1700s by the 10-year-old Carl Friedrich Gauss, later to become one of history's greatest mathematicians. Young Gauss noticed that by starting at opposite ends of the string of numbers from 1 to 100, each successive pair adds up to 101:
Hide Full Description
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
and so on through the 50th pair,
50 + 51 = 101
Gauss was already thinking like a good problem solver: The sum of the numbers from 1 to 100 is 50 x 101, or 5,050-obtained in seconds and without a calculator!
In 24 mind-enriching lectures, The Art and Craft of Mathematical Problem Solving conducts you through scores of problems-at all levels of difficulty-under the inspiring guidance of award-winning Professor Paul Zeitz of the University of San Francisco, a former champion "mathlete" in national and international math competitions and a firm believer that mathematical problem solving is an important skill that can be nurtured in practically everyone.
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Think More Lucidly, Logically, Creatively
Not only is solving such problems fun, but the techniques you learn come in handy whenever you are presented with an unfamiliar problem in mathematics, giving you the confidence to try different approaches until you make a breakthrough. Also, by learning a range of different problem-solving approaches in algebra, geometry, combinatorics, number theory, and other fields, you see how all of mathematics is tied together, and how techniques in one area can be used to solve problems in another.
Furthermore, entertaining math problems sharpen the mind, stimulating you to think more lucidly, logically, and creatively and allowing you to tackle intellectual challenges you might never have imagined.
And for those in high school or college, this course serves as an enriching mathematical experience, equal to anything available in the top schools. Professor Zeitz is a masterful coach of math teams at every level of competition, from beginners through international champions, and he knows how to inspire, encourage, and instruct.
Strategies, Tactics, and Tools of Math Masters
The Art and Craft of Mathematical Problem Solving is more than a bag of math tricks. Instead, Professor Zeitz has designed a series of lessons that take you through increasingly more challenging problems, illustrating a variety of strategies, tactics, and tools that you can use to overcome difficult math obstacles. His goal is to give you the persistence and creativity to turn over a problem in your mind for however long it takes to reach a solution.
The first step is to come up with a strategy-an overall plan of attack. Among the many strategies that Professor Zeitz discusses are these:
Get your hands dirty: Dive in! Plug in numbers and see what happens. This is a superb starting strategy because it almost always shows a way to keep on investigating. You'll be surprised at how often a pattern emerges that takes you to the next step.
Think outside the box: Break the bounds of conventional thinking. Professor Zeitz shows you the original think-outside-the-box problem, in which the key idea is to disregard the boundaries of an implied box. He also explains why he prefers to call this strategy "chainsaw the giraffe."
Wishful thinking: Turn a hard problem into an easy one by removing the hard part. For example, substitute small numbers for big ones. This is a confidence-builder that often gives you a partial solution that shows you how to solve the original problem.
rapidgator
nitroflare | 935 | 4,418 | {"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} | 3.609375 | 4 | CC-MAIN-2020-05 | latest | en | 0.941756 |
http://documentation.statsoft.com/STATISTICAHelp.aspx?path=Glossary/GlossaryTwo/S/StudentstDistribution | 1,563,592,114,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195526408.59/warc/CC-MAIN-20190720024812-20190720050812-00376.warc.gz | 42,947,677 | 13,272 | Student's t Distribution
The student's t distribution has density function (for n = 1, 2, ...):
f(x) = G[(n+1)/2]/ G(n/2) * (n*p)[-1/2] * [1 + (x2/n)][-(n+1)/2]
where
n is the degrees of freedom G (gamma) is the Gamma function p is the constant Pi (3.14...)
The animation above shows how the shape of the density and distribution functions change as the degrees of freedom increase. Note that optimum scaling (rather than fixed scaling) is used in the animation (see Probability Distribution Calculator for further details).
For a complete listing of all distribution functions, see Distributions and Their Functions. | 155 | 624 | {"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} | 2.59375 | 3 | CC-MAIN-2019-30 | latest | en | 0.828608 |
https://communities.sas.com/t5/General-SAS-Programming/Flag-anomalies-from-binary-decision-making-lending/td-p/347485?nobounce | 1,534,777,842,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221216475.75/warc/CC-MAIN-20180820140847-20180820160847-00033.warc.gz | 615,997,512 | 31,504 | ## Flag anomalies from binary decision making (lending)
Solved
Frequent Contributor
Posts: 128
# Flag anomalies from binary decision making (lending)
hi,
Looking for suggestions in order to flag agents who agree to Lend \$ vs those who decline to Lend.
The objective is for management to see if there are any education opportunities for either agents who are either deviating from peers /have a bias toward declining, or have a bias toward lending.
Example data set (where agree_lend 1=lend, and 0=not lend:
name agreed_lend date
Marc 1 01mar2017
Marc 0 01mar2017
Marc 1 03mar2017
...
James 0 02mar2017
...
Summary table for Month:
name agreed_lend_sum agreed_lend_count ratio
Marc 336 562 60%
James 32 33 97%
*As James' overall volume of decisions is small, the ratio is not very meaningful.
I appreciate any suggestion anyone as to offer, many thanks
Accepted Solutions
Solution
04-18-2017 08:41 AM
Valued Guide
Posts: 653
## Re: Flag anomalies from binary decision making (lending)
Just to get things started. Let's assume that the incoming data set is reasonable sized - so that it can be sorted. Here is a DATA step that will get you a summary. The boundary conditions are arbitrary here.
``````* Fake data;
data have;
input name \$ approved;
datalines;
frank 1
joe 1
joe 0
mary 1
joe 1
frank 1
mary 0
mary 1
frank 1
joe 0
bill 0
frank 1
mary 1
run;
proc sort data=have;
by name;
run;
data want(keep=name loancnt lendcnt ratio flag);
set have;
by name;
if first.name then do;
loancnt=0;
lendcnt=0;
end;
loancnt+1;
lendcnt+approved;
if last.name then do;
ratio = lendcnt/loancnt;
if not(.2 le ratio le .8) then flag="*";
if loancnt>2 then output want;
end;
run;
``````
All Replies
Valued Guide
Posts: 653
## Re: Flag anomalies from binary decision making (lending)
@brulard i suspect that the lack of replies is an indication that your question is not specific enough. Are you asking for suggestions on statistical tests or for an easy way to generate the summary stats ( given that you have the binary variable)?
Frequent Contributor
Posts: 128
## Re: Flag anomalies from binary decision making (lending)
thanks for your feedback... my question is likely too broad. I am looking for suggestions on a means of identifying/highlighting meaningful outliers in my ouput, either through summary, or statistical procedure.
Looking at the few examples below, the approval ratio for Marc, Mary and John is around 60%.
I'd like to be able to flag Jane and Abe as too conservative/liberal in terms of their peer average. I would not consider James, as his overall decisions is too low.
name agreed_to_lend total_decisions Approval_ratio
Marc 336 562 60%
James 32 33 97%
Jane 100 500 20%
Mary 310 515 60%
Abe 450 498 90%
John 350 575 61%
...
Solution
04-18-2017 08:41 AM
Valued Guide
Posts: 653
## Re: Flag anomalies from binary decision making (lending)
Just to get things started. Let's assume that the incoming data set is reasonable sized - so that it can be sorted. Here is a DATA step that will get you a summary. The boundary conditions are arbitrary here.
``````* Fake data;
data have;
input name \$ approved;
datalines;
frank 1
joe 1
joe 0
mary 1
joe 1
frank 1
mary 0
mary 1
frank 1
joe 0
bill 0
frank 1
mary 1
run;
proc sort data=have;
by name;
run;
data want(keep=name loancnt lendcnt ratio flag);
set have;
by name;
if first.name then do;
loancnt=0;
lendcnt=0;
end;
loancnt+1;
lendcnt+approved;
if last.name then do;
ratio = lendcnt/loancnt;
if not(.2 le ratio le .8) then flag="*";
if loancnt>2 then output want;
end;
run;
``````
Frequent Contributor
Posts: 128
## Re: Flag anomalies from binary decision making (lending)
hi ArtC, your solution is great start, thank you!
If there are some procedure/techniques that you could recommend I study in terms of potentially automating the flag criteria, please let me know! Currently, the volume I look at is about 175 decisions per agent, monthly.
☑ This topic is solved. | 1,191 | 4,535 | {"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} | 2.6875 | 3 | CC-MAIN-2018-34 | latest | en | 0.627528 |
https://number.academy/235216 | 1,660,678,478,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882572515.15/warc/CC-MAIN-20220816181215-20220816211215-00603.warc.gz | 410,193,843 | 12,408 | # Number 235216
Number 235,216 spell 🔊, write in words: two hundred and thirty-five thousand, two hundred and sixteen . Ordinal number 235216th is said 🔊 and write: two hundred and thirty-five thousand, two hundred and sixteenth. Color #235216. The meaning of number 235216 in Maths: Is Prime? Factorization and prime factors tree. The square root and cube root of 235216. What is 235216 in computer science, numerology, codes and images, writing and naming in other languages. Other interesting facts related to 235216.
## What is 235,216 in other units
The decimal (Arabic) number 235216 converted to a Roman number is (C)(C)(X)(X)(X)(V)CCXVI. Roman and decimal number conversions.
#### Weight conversion
235216 kilograms (kg) = 518557.2 pounds (lbs)
235216 pounds (lbs) = 106693.3 kilograms (kg)
#### Length conversion
235216 kilometers (km) equals to 146157 miles (mi).
235216 miles (mi) equals to 378544 kilometers (km).
235216 meters (m) equals to 771697 feet (ft).
235216 feet (ft) equals 71695 meters (m).
235216 centimeters (cm) equals to 92604.7 inches (in).
235216 inches (in) equals to 597448.6 centimeters (cm).
#### Temperature conversion
235216° Fahrenheit (°F) equals to 130657.8° Celsius (°C)
235216° Celsius (°C) equals to 423420.8° Fahrenheit (°F)
#### Time conversion
(hours, minutes, seconds, days, weeks)
235216 seconds equals to 2 days, 17 hours, 20 minutes, 16 seconds
235216 minutes equals to 5 months, 3 weeks, 2 days, 8 hours, 16 minutes
### Codes and images of the number 235216
Number 235216 morse code: ..--- ...-- ..... ..--- .---- -....
Sign language for number 235216:
Number 235216 in braille:
QR code Bar code, type 39
Images of the number Image (1) of the number Image (2) of the number More images, other sizes, codes and colors ...
## Mathematics of no. 235216
### Multiplications
#### Multiplication table of 235216
235216 multiplied by two equals 470432 (235216 x 2 = 470432).
235216 multiplied by three equals 705648 (235216 x 3 = 705648).
235216 multiplied by four equals 940864 (235216 x 4 = 940864).
235216 multiplied by five equals 1176080 (235216 x 5 = 1176080).
235216 multiplied by six equals 1411296 (235216 x 6 = 1411296).
235216 multiplied by seven equals 1646512 (235216 x 7 = 1646512).
235216 multiplied by eight equals 1881728 (235216 x 8 = 1881728).
235216 multiplied by nine equals 2116944 (235216 x 9 = 2116944).
show multiplications by 6, 7, 8, 9 ...
### Fractions: decimal fraction and common fraction
#### Fraction table of 235216
Half of 235216 is 117608 (235216 / 2 = 117608).
One third of 235216 is 78405,3333 (235216 / 3 = 78405,3333 = 78405 1/3).
One quarter of 235216 is 58804 (235216 / 4 = 58804).
One fifth of 235216 is 47043,2 (235216 / 5 = 47043,2 = 47043 1/5).
One sixth of 235216 is 39202,6667 (235216 / 6 = 39202,6667 = 39202 2/3).
One seventh of 235216 is 33602,2857 (235216 / 7 = 33602,2857 = 33602 2/7).
One eighth of 235216 is 29402 (235216 / 8 = 29402).
One ninth of 235216 is 26135,1111 (235216 / 9 = 26135,1111 = 26135 1/9).
show fractions by 6, 7, 8, 9 ...
235216
### Advanced math operations
#### Is Prime?
The number 235216 is not a prime number. The closest prime numbers are 235211, 235231.
#### Factorization and factors (dividers)
The prime factors of 235216 are 2 * 2 * 2 * 2 * 61 * 241
The factors of 235216 are
1 , 2 , 4 , 8 , 16 , 61 , 122 , 241 , 244 , 482 , 488 , 964 , 976 , 1928 , 3856 , 14701 , 29402 , 58804 , 117608 , 235216
Total factors 20.
Sum of factors 465124 (229908).
#### Powers
The second power of 2352162 is 55.326.566.656.
The third power of 2352163 is 13.013.693.702.557.696.
#### Roots
The square root √235216 is 484,990722.
The cube root of 3235216 is 61,728959.
#### Logarithms
The natural logarithm of No. ln 235216 = loge 235216 = 12,36826.
The logarithm to base 10 of No. log10 235216 = 5,371467.
The Napierian logarithm of No. log1/e 235216 = -12,36826.
### Trigonometric functions
The cosine of 235216 is 0,243174.
The sine of 235216 is -0,969983.
The tangent of 235216 is -3,988842.
### Properties of the number 235216
Is a Friedman number: No
Is a Fibonacci number: No
Is a Bell number: No
Is a palindromic number: No
Is a pentagonal number: No
Is a perfect number: No
## Number 235216 in Computer Science
Code typeCode value
PIN 235216 It's recommendable to use 235216 as a password or PIN.
235216 Number of bytes229.7KB
CSS Color
#235216 hexadecimal to red, green and blue (RGB) (35, 82, 22)
Unix timeUnix time 235216 is equal to Saturday Jan. 3, 1970, 5:20:16 p.m. GMT
IPv4, IPv6Number 235216 internet address in dotted format v4 0.3.150.208, v6 ::3:96d0
235216 Decimal = 111001011011010000 Binary
235216 Decimal = 102221122201 Ternary
235216 Decimal = 713320 Octal
235216 Decimal = 396D0 Hexadecimal (0x396d0 hex)
235216 BASE64MjM1MjE2
235216 MD567c8a2dc6309b9dec796930faacfed48
235216 SHA2241d17ffe8ccd79b3313619edb2eb44b2b36d89a8b7054f1d0f8464a4c
235216 SHA384134020e269153f17ca8e0ce7329e04c2d0de88b2c4e086e592368643a7f967dcb51ece0335d1ef11cbaf43f9e5dd936a
More SHA codes related to the number 235216 ...
If you know something interesting about the 235216 number that you did not find on this page, do not hesitate to write us here.
## Numerology 235216
### Character frequency in number 235216
Character (importance) frequency for numerology.
Character: Frequency: 2 2 3 1 5 1 1 1 6 1
### Classical numerology
According to classical numerology, to know what each number means, you have to reduce it to a single figure, with the number 235216, the numbers 2+3+5+2+1+6 = 1+9 = 1+0 = 1 are added and the meaning of the number 1 is sought.
## Interesting facts about the number 235216
### Asteroids
• (235216) 2003 SR217 is asteroid number 235216. It was discovered by Observatorio Astronómico de Sierra Nevada from Sierra Nevada Observatory on 9/27/2003.
## № 235,216 in other languages
How to say or write the number two hundred and thirty-five thousand, two hundred and sixteen in Spanish, German, French and other languages. The character used as the thousands separator.
Spanish: 🔊 (número 235.216) doscientos treinta y cinco mil doscientos dieciseis German: 🔊 (Anzahl 235.216) zweihundertfünfunddreißigtausendzweihundertsechzehn French: 🔊 (nombre 235 216) deux cent trente-cinq mille deux cent seize Portuguese: 🔊 (número 235 216) duzentos e trinta e cinco mil, duzentos e dezesseis Chinese: 🔊 (数 235 216) 二十三万五千二百一十六 Arabian: 🔊 (عدد 235,216) مئتان و خمسة و ثلاثون ألفاً و مئتان و ستة عشر Czech: 🔊 (číslo 235 216) dvěstě třicet pět tisíc dvěstě šestnáct Korean: 🔊 (번호 235,216) 이십삼만 오천이백십육 Danish: 🔊 (nummer 235 216) tohundrede og femogtredivetusindtohundrede og seksten Dutch: 🔊 (nummer 235 216) tweehonderdvijfendertigduizendtweehonderdzestien Japanese: 🔊 (数 235,216) 二十三万五千二百十六 Indonesian: 🔊 (jumlah 235.216) dua ratus tiga puluh lima ribu dua ratus enam belas Italian: 🔊 (numero 235 216) duecentotrentacinquemiladuecentosedici Norwegian: 🔊 (nummer 235 216) to hundre og tretti-fem tusen, to hundre og seksten Polish: 🔊 (liczba 235 216) dwieście trzydzieści pięć tysięcy dwieście szesnaście Russian: 🔊 (номер 235 216) двести тридцать пять тысяч двести шестнадцать Turkish: 🔊 (numara 235,216) ikiyüzotuzbeşbinikiyüzonaltı Thai: 🔊 (จำนวน 235 216) สองแสนสามหมื่นห้าพันสองร้อยสิบหก Ukrainian: 🔊 (номер 235 216) двiстi тридцять п'ять тисяч двiстi шiстнадцять Vietnamese: 🔊 (con số 235.216) hai trăm ba mươi lăm nghìn hai trăm mười sáu Other languages ...
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## Comment
If you know something interesting about the number 235216 or any natural number (positive integer) please write us here or on facebook. | 2,605 | 7,669 | {"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} | 2.75 | 3 | CC-MAIN-2022-33 | latest | en | 0.704337 |
https://au.mathworks.com/matlabcentral/answers/644185-problem-in-converting-a-matrix-of-integers-to-vector-of-integers-using-str2num-and-num2str?s_tid=prof_contriblnk | 1,675,735,955,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500368.7/warc/CC-MAIN-20230207004322-20230207034322-00011.warc.gz | 131,026,035 | 28,153 | # problem in converting a matrix of integers to vector of integers using str2num and num2str
3 views (last 30 days)
Nora Khaled on 11 Nov 2020
Commented: Nora Khaled on 12 Nov 2020
Hello,
I have a matrix x of this form (these are example values)
1 0 0 1 0
0 1 0 0 0
0 0 0 0 1
1 0 0 1 1
I am trying to make a number out of each row so I used
y=strcat(num2str(x(:,1)),num2str(x(:,2)),num2str(x(:,3)),num2str(x(:,4)),num2str(x(:,5)));
which get me the result
'10011'
'01000'
'00001'
'10011'
Now I want to convert each string to a number but its not working.
A = str2num(y)
A =
[]
Also, I tried different method all faild beause the size of y is 4x5 and not 4x1.
why does matlab consider each char as column and how can I avoid this problem?
Thanks
##### 0 CommentsShowHide -1 older comments
Sign in to comment.
### Accepted Answer
Ameer Hamza on 11 Nov 2020
Edited: Ameer Hamza on 11 Nov 2020
Try this
x = [
1 0 0 1 0
0 1 0 0 0
0 0 0 0 1
1 0 0 1 1];
x = char(x+'0');
y = str2num(x)
Result
>> y
y =
10010
1000
1
10011
Or a faster solution
x = [
1 0 0 1 0
0 1 0 0 0
0 0 0 0 1
1 0 0 1 1];
y = x*10.^(size(x,2)-1:-1:0)';
##### 6 CommentsShowHide 5 older comments
Nora Khaled on 12 Nov 2020
Okay. Thank you very much this helps!
Sign in to comment.
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Find more on Data Type Conversion in Help Center and File Exchange
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Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting! | 524 | 1,457 | {"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} | 2.96875 | 3 | CC-MAIN-2023-06 | longest | en | 0.796221 |
https://convertoctopus.com/287-cubic-centimeters-to-tablespoons | 1,638,221,448,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964358842.4/warc/CC-MAIN-20211129194957-20211129224957-00607.warc.gz | 251,150,115 | 7,992 | ## Conversion formula
The conversion factor from cubic centimeters to tablespoons is 0.06762804511761, which means that 1 cubic centimeter is equal to 0.06762804511761 tablespoons:
1 cm3 = 0.06762804511761 tbsp
To convert 287 cubic centimeters into tablespoons we have to multiply 287 by the conversion factor in order to get the volume amount from cubic centimeters to tablespoons. We can also form a simple proportion to calculate the result:
1 cm3 → 0.06762804511761 tbsp
287 cm3 → V(tbsp)
Solve the above proportion to obtain the volume V in tablespoons:
V(tbsp) = 287 cm3 × 0.06762804511761 tbsp
V(tbsp) = 19.409248948754 tbsp
The final result is:
287 cm3 → 19.409248948754 tbsp
We conclude that 287 cubic centimeters is equivalent to 19.409248948754 tablespoons:
287 cubic centimeters = 19.409248948754 tablespoons
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 tablespoon is equal to 0.051521828724042 × 287 cubic centimeters.
Another way is saying that 287 cubic centimeters is equal to 1 ÷ 0.051521828724042 tablespoons.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that two hundred eighty-seven cubic centimeters is approximately nineteen point four zero nine tablespoons:
287 cm3 ≅ 19.409 tbsp
An alternative is also that one tablespoon is approximately zero point zero five two times two hundred eighty-seven cubic centimeters.
## Conversion table
### cubic centimeters to tablespoons chart
For quick reference purposes, below is the conversion table you can use to convert from cubic centimeters to tablespoons
cubic centimeters (cm3) tablespoons (tbsp)
288 cubic centimeters 19.477 tablespoons
289 cubic centimeters 19.545 tablespoons
290 cubic centimeters 19.612 tablespoons
291 cubic centimeters 19.68 tablespoons
292 cubic centimeters 19.747 tablespoons
293 cubic centimeters 19.815 tablespoons
294 cubic centimeters 19.883 tablespoons
295 cubic centimeters 19.95 tablespoons
296 cubic centimeters 20.018 tablespoons
297 cubic centimeters 20.086 tablespoons | 514 | 2,144 | {"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} | 4.09375 | 4 | CC-MAIN-2021-49 | latest | en | 0.71978 |
https://math.stackexchange.com/questions/2032040/is-the-newton-maehly-method-for-finding-roots-of-a-polynomial-stable | 1,558,790,390,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232258058.61/warc/CC-MAIN-20190525124751-20190525150751-00141.warc.gz | 558,144,859 | 32,224 | # Is the Newton-Maehly method for finding roots of a polynomial stable?
I have been asked if the Newton's method (More concretely Maehly's method) is numerically inestable. I know finding roots of a polynomial is an ill-posed problem, and also that the Newton's method converges to a root $x$ in a small enough neighborhood of $x$.
What can I say about stability? I would say that Newton's method is indeed numerically stable provided the polynomial does not have roots too close to each other, because in practice when two or more roots are too close from each other in the iterations
$$x_{k+1}=x_k-\frac{p(x_k)}{p'(x_k)-p(x_k) \sum_{j=1}^m \frac{1}{x - \alpha_j} },$$ where the $\alpha_j$ are the already computed roots, you are substracting two terms that are very close, $p'(x_k)-p(x_k)$ (catastrophic cancelation), and then also dividing by a number close to $0$, because both terms tend to $0$.
What do you think? Is this answer correct?
• The problem is not that $p′(x_k)−p(x_k)$ generates catastrophic cancellation, but that both terms are the result of catastrophic cancellation during polynomial evaluation where big terms sum up to something close to zero. The evaluation errors of $p(x_k)$ and $p(x_k)$ being large relative to their size does not get better under division. – LutzL Nov 26 '16 at 22:32
• @LutzL Okay, x being close to 0 produces evaluation of monomials being very close to 0 and substracting these produces cancellation errors, But also in Newton's method you are dividing $p(x_k)$ by a number which is very close to zero($p′(x_k)−p(x_k)$), further increasing the error, right? Also, Is it correct the response about stability when roots are not close to each other? – D1X Nov 27 '16 at 9:41
• Not exactly. In the opposite, it is when a root $x$ is far away from zero that the large monomials of the polynomial combine to a small value. And your denominator is not $p'(x_k)−p(x_k)$, there is still that sum as a factor. -- For polynomials you can use deflation instead of this method, the problems remain approximately the same. – LutzL Nov 27 '16 at 9:53
• @LutzL Could you please explain this a little more, I don't get what you mean. – D1X Nov 27 '16 at 10:08
• While the true value of $p(x)=p_nx^n+…+p_0$ is zero or close to it, the error due to evaluation is on the scale of the product of the machine constant $\mu$ and $|p|(|x|)=|p_n|·|x|^n+…+|p_0|$ which even for moderate $n$ and coefficient size will be large. – LutzL Nov 27 '16 at 10:41 | 690 | 2,481 | {"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} | 3.390625 | 3 | CC-MAIN-2019-22 | latest | en | 0.940006 |
https://oercommons.org/browse?f.new_curriculum_alignment=CCSS.Math.Content.5.MD.C.4 | 1,716,493,357,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058653.47/warc/CC-MAIN-20240523173456-20240523203456-00763.warc.gz | 378,494,096 | 18,443 | Updating search results...
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This resource links to both Measurement & Data progression documents published by the Common Core Writing Teams in June 2011.
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In this 25-day module, students work with two- and three-dimensional figures. Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms. The second half of the module turns to extending students understanding of two-dimensional figures. Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths. They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes. This module fills a gap between Grade 4s work with two-dimensional figures and Grade 6s work with volume and area.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
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Geometry
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Our students will be studying and exploring the human impact on groundwater. They will study the water deprivation impacts both locally and in the San Joaquin Valley. Students will explore and come to understand the benefits of collecting rainwater. We partnered with the City of Eugene and had the wonderful Jackie come in. Our students brought in many of the materials including cardboard boxes, empty plastic containers (sour cream, water bottles,etc), tin foil, wax paper, duct tape,etc. We as teachers provided the underground sprinkler tubing cutting material, more tape and supplies. We tested this project with our 5th graders so we could make improvements and continue this project next year. In order to complete this project, we needed a full three weeks of working for an hour plus every day. We incorporated this project into our reading and science timeline.
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(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)
En este módulo de 25 días, los estudiantes trabajan con figuras dos y tridimensionales. El volumen se introduce a los estudiantes a través de la exploración concreta de unidades cúbicas y culmina con el desarrollo de la fórmula de volumen para los prismas rectangulares correctos. La segunda mitad del módulo se convierte en extender a los estudiantes la comprensión de las figuras bidimensionales. Los estudiantes combinan el conocimiento previo del área con el conocimiento recién adquirido de la multiplicación por fracción para determinar el área de las figuras rectangulares con longitudes laterales fraccionadas. Luego participan en la construcción práctica de formas bidimensionales, desarrollando una base para clasificar las formas razonando sobre sus atributos. Este módulo llena un vacío entre el trabajo de Grado 4 S con figuras bidimensionales y el trabajo de grado 6 con volumen y área.
Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.
English Description:
In this 25-day module, students work with two- and three-dimensional figures. Volume is introduced to students through concrete exploration of cubic units and culminates with the development of the volume formula for right rectangular prisms. The second half of the module turns to extending students understanding of two-dimensional figures. Students combine prior knowledge of area with newly acquired knowledge of fraction multiplication to determine the area of rectangular figures with fractional side lengths. They then engage in hands-on construction of two-dimensional shapes, developing a foundation for classifying the shapes by reasoning about their attributes. This module fills a gap between Grade 4s work with two-dimensional figures and Grade 6s work with volume and area.
Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.
Subject:
Geometry
Mathematics
Material Type:
Module
Provider:
New York State Education Department
Provider Set:
EngageNY
01/17/2014
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The intent of clarifying statements is to provide additional guidance for educators to communicate the intent of the standard to support the future development of curricular resources and assessments aligned to the 2021 math standards. Clarifying statements can be in the form of succinct sentences or paragraphs that attend to one of four types of clarifications: (1) Student Experiences; (2) Examples; (3) Boundaries; and (4) Connection to Math Practices.
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Mark Freed | 1,291 | 5,895 | {"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} | 2.890625 | 3 | CC-MAIN-2024-22 | latest | en | 0.870243 |
https://www.datacamp.com/courses/correlation-and-regression-in-r?tap_a=5644-dce66f&tap_s=988414-41ec15&utm_medium=affiliate&utm_source=realtoughcandy | 1,603,820,869,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00676.warc.gz | 680,883,522 | 49,474 | # Correlation and Regression in R
Learn how to describe relationships between two numerical quantities and characterize these relationships graphically.
4 Hours18 Videos58 Exercises65,285 Learners
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## Course Description
Ultimately, data analysis is about understanding relationships among variables. Exploring data with multiple variables requires new, more complex tools, but enables a richer set of comparisons. In this course, you will learn how to describe relationships between two numerical quantities. You will characterize these relationships graphically, in the form of summary statistics, and through simple linear regression models.
1. 1
### Visualizing two variables
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In this chapter, you will learn techniques for exploring bivariate relationships.
2. 2
### Correlation
This chapter introduces correlation as a means of quantifying bivariate relationships.
3. 3
### Simple linear regression
With the notion of correlation under your belt, we'll now turn our attention to simple linear models in this chapter.
4. 4
### Interpreting regression models
This chapter looks at how to interpret the coefficients in a regression model.
5. 5
### Model Fit
In this final chapter, you'll learn how to assess the "fit" of a simple linear regression model.
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#### Ben Baumer
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Ben is an Assistant Professor in the Statistical & Data Sciences Program at Smith College. He completed his Ph.D. in Mathematics at the Graduate Center of the City University of New York in 2012. He is an Accredited Professional Statistician™ by the American Statistical Association and was previously the Statistical Analyst for the Baseball Operations department of the New York Mets.
## What do other learners have to say?
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DataCamp is the top resource I recommend for learning data science.
Louis Maiden | 471 | 2,308 | {"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} | 2.953125 | 3 | CC-MAIN-2020-45 | latest | en | 0.893779 |
http://www.convertit.com/Go/Maps/Measurement/Converter.ASP?From=intl+foot&To=fl+head | 1,603,745,339,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107892062.70/warc/CC-MAIN-20201026204531-20201026234531-00135.warc.gz | 124,550,942 | 5,743 | Search Maps.com
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Conversion Result: ```foot = 0.3048 length (length) ``` Related Measurements: Try converting from "intl foot" to astronomical unit, barleycorn, Biblical cubit, bolt (of cloth), bottom measure, cable length, caliber (gun barrel caliber), digitus (Roman digitus), engineers chain, furlong (surveyors furlong), hand, line, nail (cloth nail), nautical league, pica (typography pica), Roman cubit, spindle, survey foot, UK mile (British mile), yard, or any combination of units which equate to "length" and represent depth, fl head, height, length, wavelength, or width. Sample Conversions: intl foot = 36 barleycorn, .00138889 cable length, .01515152 chain (surveyors chain), 1.33 cloth quarter, 304,800,000,000,000 fermi, .00151515 furlong (surveyors furlong), .6586169 Greek cubit, 1.32 Greek span, 12 inch, .00047281 li (Chinese li), 144 line, 72 pica (typography pica), .00007762 ri (Japanese ri), 1.01 shaku (Japanese shaku), .00002315 spindle, 10.06 sun (Japanese sun), .999998 survey foot, .00018939 UK mile (British mile), .00028571 verst (Russian verst), .33333333 yard.
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Please read our Help Page and FAQ Page then post a message or send e-mail. Thanks! | 537 | 2,089 | {"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} | 3.109375 | 3 | CC-MAIN-2020-45 | latest | en | 0.699475 |
http://www.chegg.com/homework-help/questions-and-answers/river-width-800-m-flows-due-south-speed-20-m-s-man-steers-motorboat-across-river-velocity--q2397873 | 1,472,548,232,000,000,000 | text/html | crawl-data/CC-MAIN-2016-36/segments/1471982974951.92/warc/CC-MAIN-20160823200934-00022-ip-10-153-172-175.ec2.internal.warc.gz | 362,209,285 | 14,180 | A river with the width of 800 m flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4.2 m/s due east.
a. How far south of his starting point will he reach the opposite bank?
b. In which direction should the motorboat head in order to reach a point on the opposite bank directly east from the starting point with the same boat speed relative to the water of 4.2 m/s?
c. What is the velocity of the boat relative to the earth?
d. How much time required to cross the river? | 135 | 541 | {"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} | 2.75 | 3 | CC-MAIN-2016-36 | latest | en | 0.955571 |
http://juliasfairies.com/problems/jf-2012/page-124/?replytocom=4987 | 1,590,733,094,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347402457.55/warc/CC-MAIN-20200529054758-20200529084758-00594.warc.gz | 67,754,234 | 16,511 | # Original Problems (124)
## Original Problems (page 124)
Original fairy problems published during 2012 will participate in the informal tourney JF-2012
The site is mostly about fairies, but h# and s# are also welcomed for publication! Please send your problems to my e-mail: julia@juliasfairies.com
### Go to →List of Problems ; →Page 123; →Page 125
I have a pleasure to present you 4(!!) problems by Peter Harris! It is Peter’s answer to my wish to have 200 published problems by the end of year. Thank you, Peter!!
No.189 – h#1,5 – A short, but nice Aristocrat! (JV)
No.190 – h#2,5 – A very surprising finals! (JV)
No.191 – ser-#4 – A pleasant neutral three-men! (JV)
No.192 – h#4 – A nice Orphan-Festival! (JV)
Definitions:
Anti-Circe: After a capture the capturing piece (Ks included) must immediately be removed to its game array square (necessarily vacant, else the capture is illegal). Captures on the rebirth square are allowed. Game array squares are determined as in Circe.
Super-Circe – When captured, a piece is reborn on any free field on the chess board without causing self-check or selfmate. Possible is also removal of captured piece from the board. The Pawns (white, black, neutrals, half- neutrals) can be reborn on the first or eight row also. When reborn on the first row (for Black) or on the eight row (for White) the promotion is obligatory. When reborn on the first row (for White) or on the eight row (for Black) the Pawns are immovable.
Isardam: The moves causing a Madrasi-like paralysis are illegal. This holds right up to the capture of the mated King. This is standard form of Isardam.
White Maximummer ultra: White’s only legal move is his longest – calculated as per Maximummer rules.
Maximummer – Black must play the geometrically longest move or may choose from among longest moves of equal length, distances being measured from the center of each square. Diagonal and oblique distances are measured from the orthogonal coordinates by using Pythagora’s theorem (take the square root of the sum of the squares of the orthogonal distances). All other orthodox chess rules apply.
Anti-Circe (Cheylan): When a piece captures (including King), it must come back to its rebirth square. If this square is occupied, the capture is forbidden. A Pawn capturing on its promotion rank promotes before it is reborn. The captures on the rebirth square are forbidden.
Sentinels Pion neutre: When a piece (Pawn excluded) leaves a square outside the first and last rows, it leaves a neutral Pawn unless 8 neutral Pawns are already on the board.
Sentinels Pion advers: When a piece (not a Pawn) moves, a Pawn of the colour of the opposite side appears on the vacated square if it is not on the first or the last rank, and if there are less than 8 Pawns of that colour on the board.
Orphan (O): An orphan can move only when observed by an enemy piece; when so observed it can move like the piece(s).
No.189 Peter Harris South Africa original-19.12.2012 h#1.5 3 solutions (4+5)Anti-CirceSuper-CirceIsardam Solution: (click to show/hide) I. 1…Qf6-h8 2.Re6-e7 Bd6*e7 [+bRd2][wBe7>c1] # II. 1…Qf6*e6 [+bRd2][wQe6>d1] + 2.Qd4*d6 [+wBa1][bQd6>d8] Ba1*b2 [+bRe8][wBb2>c1] # III. 1…Re4*d4 [+bQd5][wRd4>a1] 2.Re6*d6 [+wBg7][bRd6>h8] Qf6*f4 [+bBd4][wQf4>d1] # (Tested by Popeye 4,61) No.190 Peter Harris South Africa original-19.12.2012 h#2.5 2 solutions (4+7)Super-CirceWhite Maximummer ultra Solution: (click to show/hide) I. 1…Sa6*c5 [+bPa4] 2.Kd8-e8 Sc5*a4 [+bPc5] 3.Sc6*b4 [+wPd8=wR] Rd8*a8 [+bRf7] # II. 1…Sa6*c5 [+bPb8] 2.Kd8*c7 [+wPd8=wS] Sd8*c6 [+bSd8] 3.Kc7-c8 Sc6*e7 [+bPc7] # (Tested by Popeye 4,61) No.191 Peter Harris South Africa original-19.12.2012 ser-#4 b) nRa8->b2 (0+0+3)Super-CirceAnti-Circe (Cheylan)Sentinelles Pion Neutre Solution: (click to show/hide) a) 1.nRe2(+nPa2) 2.nRg2(+nPe2) 3.nKxa2(Kne1;nPh8) 4.nKxe2(nKe1;nPg1=nR)#b) 1.nKxa2(nKe1;nRh7) 2.nRhh2(+nPh7) 3.h8=nS 4.nRb1(+nPb2)# (Tested by Popeye 4,61) No.192 Peter Harris South Africa original-19.12.2012 h#4 2 solutions (2+11)Anti-Circe (Cheylan)Sentinelles Pion AdversOrphans: a1,c1,d1,e8,g1(No w.King) Solution: (click to show/hide) I. 1.Bf1-e2 c7-c8=S 2.Kd8-c7 Sc8-b6 3.Kc7-b7[+wPc7] c7-c8=B + 4.Kb7-a7[+wPb7] b7-b8=O #II. 1.Rh1-h2 c7-c8=Q 2.Rh2-h1[+wPh2] Qc8-c6 3.Rh1*h2[bRh2>h8] Oe8-e2 4.Kd8-d7 Oe2*d1[wOd1>d8][+bPe2] # (Tested by Popeye 4,61)
The diagrams are made on WinChloe and its Echecs font is used for Logo design
### 3 Responses to Original Problems (124)
1. seetharaman says:
No.191. It is a little confusing with neutral Kings. Why white cant capture the King on his move? Why white should play 4 more moves ?
• Diyan Kostadinov says:
The capture 1.nR:a1?? is forbidden because of Anti-Circe type Cheylan where the captures on the rebirth square are forbidden. “a1” is rebirth square for the nR on its white phase
2. seetharaman says:
🙁 Thanks. I forgot the difference for Cheylan Anti-Circe !! | 1,693 | 5,046 | {"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} | 2.765625 | 3 | CC-MAIN-2020-24 | latest | en | 0.880193 |
https://math.stackexchange.com/questions/3157740/showing-sum-k-0n-n-choose-k-frac-1knk1-is-positive | 1,556,022,923,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578602767.67/warc/CC-MAIN-20190423114901-20190423140901-00261.warc.gz | 476,443,994 | 31,759 | # Showing $\sum_{k=0}^{n} {n \choose k}\frac{{(-1)}^k}{n+k+1}$ is positive
Show that the sum$$\sum_{k=0}^{n} {n \choose k}\frac{{(-1)}^k}{n+k+1}$$ is a positive rational number.
It is easy to show that it is a rational number. But I am having trouble showing that this expression is positive. It might be some binomial expansion that I could not get.
• Have you tried using induction on $n$ for example? – Minus One-Twelfth Mar 22 at 4:29
Direct proof: $$\begin{split} \sum_{k=0}^{n} {n\choose k}\frac{{(-1)}^k}{n+k+1} &=\sum_{k=0}^{n} {n\choose k}(-1)^k\int_0^1 x^{n+k}dx\\ &=\int_0^1x^n\sum_{k=0}^{n} {n\choose k}(-x)^kdx\\ &=\int_0^1x^n(1-x)^ndx \end{split}$$ The latter is clearly a positive number.
• How did you conclude that the sum is a limited integral? Do you know where can I find more on this on-line? Thanks. – NoChance Mar 22 at 5:13
• It's a "known trick" that $\frac 1 {p+1} = \int_0^1x^pdx$. Then I noticed that the sum looked almost like that of the binomial theorem. – Stefan Lafon Mar 22 at 5:18
When $$k=0$$ the term is positive. When $$k=1$$ the term is negative BUT SMALLER (in absolute value) THAN THE $$k=0$$ TERM.
When $$k=2$$ the term is positive. When $$k=3$$ the term is negative BUT SMALLER (in absolute value) THAN THE $$k=2$$ TERM.
.....
Get it?
• sorry, I did not write question correctly. Now, I have corrected that. By looking at your answer I realized my mistake. Thanks – Hitendra Kumar Mar 22 at 4:42
We can specifically prove that $$\boxed{\sum_{k=0}^{n} {n \choose k}\frac{{(-1)}^k}{n+k+1}=\left((2n+1)\binom{2n}n\right)^{-1}}$$ To see this, shuffle a deck of $$2n+1$$ cards numbered $$1$$ to $$2n+1$$. Consider this:
What is the probability that card number $$n+1$$ is in the middle of the deck, and cards numbered $$1$$ to $$n$$ are below it?
The easy answer is the fraction on the RHS. The LHS can be interpreted as an application of the principle of inclusion exclusion. Namely, we first take the probability that card number of $$n+1$$ is the lowest of the cards numbered $$n+1,n+2,\dots,2n+1$$. This is the $$k=0$$ term. From this, for each $$i=1,\dots,n$$, we subtract the probability that $$n+1$$ is the lowest of the list $$i,n+1,n+2,\dots,2n+1$$. This is a bad event, because we want $$n+1$$ to be above $$i$$. Doing this for each $$i$$, we subtract $$\binom{n}1\frac{1}{n+2}$$. We then must add back in the doubly subtracted events, subtract the triple intersections, and so on, eventually ending with the alternating sum on the left.
## protected by Community♦Mar 22 at 10:13
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count). | 881 | 2,763 | {"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": 25, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2019-18 | latest | en | 0.851908 |
https://www.isixsigma.com/topic/what-is-percentage-defects-process/ | 1,603,307,329,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107877420.17/warc/CC-MAIN-20201021180646-20201021210646-00157.warc.gz | 685,226,995 | 38,509 | # What Is the Percentage of Defects in This Process?
Six Sigma – iSixSigma Forums General Forums Tools & Templates What Is the Percentage of Defects in This Process?
Viewing 3 posts - 1 through 3 (of 3 total)
• Author
Posts
• #235894
mitragliar
Participant
You select 4 samples of 25 macarons each; the weight of each piece is measured. The mean is 22g; the standard deviation of the sample means is 1.5g.
Knowing that a box of macarons contains 9 pieces, and that the average weight of macarons in the box should be at least 20g (no USL: the more the better), which is the percentage of defects in the process?
0
#235901
Katie Barry
Keymaster
@mitragliar It looks like this is a homework question.
The more details you provide, the more likely you are to get a response. What do YOU think you should do? Why or why not? The iSixSigma audience is helpful, but they like to see that someone is putting forth a good-faith effort; they are not here to do your work for you.
1
#235963
Mike Carnell
Participant
@mitragliar You have a mean and Standard deviation and a one sided spec. That is enough to calculate a what?
A defect would be what was less than the spec.
• This reply was modified 1 year, 8 months ago by Mike Carnell.
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Viewing 3 posts - 1 through 3 (of 3 total)
You must be logged in to reply to this topic. | 353 | 1,331 | {"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} | 2.609375 | 3 | CC-MAIN-2020-45 | longest | en | 0.952638 |
http://mathhelpforum.com/math-topics/1869-help-me-factor-please-print.html | 1,529,931,247,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267867666.97/warc/CC-MAIN-20180625111632-20180625131632-00066.warc.gz | 193,846,159 | 2,964 | # Help Me Factor Please
• Feb 12th 2006, 05:17 PM
soca615
Help Me Factor Please
I really need some help factoring some equations please. These don't neccessarily have to be in the real numbers either so :eek:
1.) 4x^4 + 3x^2 - 1
2.) 9c^4 + 5c^2 + 1
3.) x^3 + x^2 + 4
4.) (a+b)^2 + 7(a+b) + 12
• Feb 12th 2006, 08:39 PM
earboth
Quote:
Originally Posted by soca615
I really need some help factoring some equations please. These don't neccessarily have to be in the real numbers either so :eek:
1.) 4x^4 + 3x^2 - 1
2.) 9c^4 + 5c^2 + 1
3.) x^3 + x^2 + 4
4.) (a+b)^2 + 7(a+b) + 12
Hello,
to factorize those expressions use the the rule of Vieta:
to 1) $\displaystyle 4x^4+3x^2-1=(2x-1)(2x+1)(x^2+1)$
to 2) $\displaystyle 9c^4 + 5c^2 + 1=\left(c^2+\frac{1}{9} \right) \left(c^2+1 \right)$
to 3) $\displaystyle x^3 + x^2 + 4=(x+2)(x^2-x+2)$
to 4) $\displaystyle (a+b)^2 + 7(a+b) + 12=((a+b)+3)((a+b)+4)$
Bye
• Feb 12th 2006, 09:15 PM
topsquark
Quote:
Originally Posted by earboth
to 2) $\displaystyle 9c^4 + 5c^2 + 1=\left(c^2+\frac{1}{9} \right) \left(c^2+1 \right)$
2) Isn't correct, sorry.
$\displaystyle 9c^4 + 5c^2 + 1$ doesn't factor over the reals. The way I would approach this, then would be to set $\displaystyle 9c^4 + 5c^2 + 1 = 0$. The solution is $\displaystyle c^2= -5/18 \pm \sqrt{-11}/18$, so $\displaystyle 9c^4 + 5c^2 + 1$
$\displaystyle = 9(c^2+5/18 + \sqrt{-11}/18)(c^2+5/18 - \sqrt{-11}/18)$, or you can take the square root of $\displaystyle c^2$ (i.e. solve for c, not $\displaystyle c^2$) and string the four factors together. Don't forget the extra 9!
-Dan | 680 | 1,586 | {"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} | 4.25 | 4 | CC-MAIN-2018-26 | latest | en | 0.668905 |
https://eric.ed.gov/?id=EJ1261847 | 1,726,348,268,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651580.74/warc/CC-MAIN-20240914193334-20240914223334-00714.warc.gz | 204,629,874 | 4,152 | Collection
Search Tips
Peer reviewed
ERIC Number: EJ1261847
Record Type: Journal
Publication Date: 2020
Pages: 9
Abstractor: ERIC
ISBN: N/A
ISSN: ISSN-2652-0176
EISSN: N/A
Playing to Lose: Investigating the Mathematics of Poker Machines
Hemer, David
Australian Mathematics Education Journal, v2 n2 p40-48 2020
This paper describes an investigation looking at the underlying mathematics of poker machines. The aim of the investigation is for students to get an appreciation of how poker machines are designed to ensure that in the long-term players will inevitably lose when playing. The first part of this paper describes how students can model a simple poker machine game and calculate the probability and payouts for four different winning outcomes. The second part of the paper describes how these calculations can then be used to configure a poker machine simulator written in the Python programming language. The code for the simulator is briefly explained and then results for several trials run in the simulator are presented. A discussion of how the student would be expected to present and interpret these results is then given. The next section of the paper explains how students can model their own more complex poker machine game and in turn how the simulator code can be extended to handle the more complex outcomes. Next, an extension to the investigation is described, in which students investigate how many coins need to be played on average before the player loses all their money. A task sheet, setup instructions, starting code and completed examples for teachers is available for free download on the Tes website at https://www.tes.com/en-au/teaching-resource/blow-up-thepokies-12308352 (Hemer, 2020).
Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au
Publication Type: Journal Articles; Reports - Descriptive
Education Level: Grade 11; High Schools; Secondary Education
Audience: Teachers
Language: English | 462 | 2,083 | {"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} | 2.53125 | 3 | CC-MAIN-2024-38 | latest | en | 0.883343 |
https://techxmag.com/tag/lisp/ | 1,566,774,883,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027330907.46/warc/CC-MAIN-20190825215958-20190826001958-00421.warc.gz | 663,675,833 | 9,728 | ## Understanding the SBCL entry/exit assembly boiler plate code
BACKGROUND
When using 64bit Steel Bank Common Lisp on Windows for a trivial identity function:
(defun a (x)
(declare (fixnum x))
(declare (optimize (speed 3) (safety 0)))
(the fixnum x))
## Levels of Homoiconicity
This is a follow up to my previous question. I’m not convinced that Lisp code is as Homoiconic as machine code on a Von Neumann architecture. It seems obvious to me that in both cases code is …
## Any suggestions for which Lisp variant to learn?
I ultimately want to learn Clojure, but I’ve found learning resources for Clojure to be scarce for people of little experience…
## common lisp cons creates a list from two symbols, clojure cons requires a seq to cons onto?
(Disclaimer – I’m aware of the significance of Seqs in Clojure)
In common lisp the cons function can be used to combine two symbols into a list:
(def s ‘x)
(def l ‘y)
(cons s l)
In clojure – you …
## Using Lisp in C#
As a lot of people pointed out in this question, Lisp is mostly used as a learning experience. Nevertheless, it would be great if I could somehow use my Lisp algorithms and combine them with my C# …
## Parsing in Prolog without cut?
I found this nice snippet for parsing lisp in Prolog (from here):
ws –> [W], { code_type(W, space) }, ws.
ws –> [].
parse(String, Expr) :- phrase(expressions(Expr), String).
expressions([E|…
## Why is the recursive function performing better than the iterative function in elisp?
As a test for one of my classes, our teacher asked us to test a recursive and non-recursive approach to the famous Euclidean Algorithm:
Iterative
(defun gcdi (a b)
(let ((x a) (y b) r)
(while (…
## Is there a Scheme implementation that parallelizes?
Is there a R5RS-or-higher Scheme implementation that does parallelization? For example, if I say to do:
(map (lambda (x)
(pure-functional-stuff x))
‘(1 3 5 7 11 13))
it will process 1,…
## “Don’t know how to create ISeq from: Symbol” error in Clojure
I have the following Clojure code and I’m not sure why it’s not working:
(defn match (x y &optional binds)
(cond
((eql x y) (values binds t))
((assoc x binds) (match (binding x binds) y …
## Cannot create apply function with static language?
I have read that with a statically typed language like Scala or Haskell there is no way to create or provide a Lisp apply function:
(apply #’+ (list 1 2 3)) => 6
or maybe
(apply #’list ‘(list :… | 620 | 2,446 | {"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} | 2.921875 | 3 | CC-MAIN-2019-35 | latest | en | 0.861487 |
https://eta-lang.org/docs/user-guides/eta-user-guide/functions/composition | 1,632,306,470,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057347.80/warc/CC-MAIN-20210922102402-20210922132402-00374.warc.gz | 293,844,867 | 7,780 | # Functions
Working with functions
### Introduction
Now that we have functions, we need a way to combine small functions together to make larger functions that can handle complex tasks, without having to rewrite the same functions over and over again.
### Example
```1 2 3 4 5 6 7 8``` ```add :: Int -> Int add x y = x + y multiply :: Int -> Int -> Int multiply x y = x * y f :: Int -> Int f x = 2 * x + 1 ```
Note that f is a function that multiplies its argument by 2 followed by incrementing it by 1. Note that instead of reusing the existing add and multiply functions, we had declared an entirely new function.
### Composition
Suppose we had two functions f :: Int -> Char and g :: Char -> Double. Can we generate a function h :: Int -> Double that applies f and g one after another? Yes, we can, using the composition operator defined below.
```1 2``` ```(.) :: (b -> c) -> (a -> b) -> (a -> c) (.) f g = \x -> f (g x) ```
### Rewriting the Example
We can now rewrite the example using the existing functions, using the currying concept from the last section.
`1` ```f = add 1 . multiply 2 ```
Note that you should read composition pipelines from left-to-right.
### Next Section
We will then proceed to re-using function application. | 316 | 1,257 | {"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} | 3.1875 | 3 | CC-MAIN-2021-39 | latest | en | 0.862551 |
https://www.dratings.com/half-point-values-for-mlb-totals/ | 1,669,693,431,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00123.warc.gz | 783,613,686 | 20,075 | # Half Point Values for MLB Totals
Understanding half point values in sports are one of the many keys that a bettor needs to get an advantage. Often times, a bettor will see one one price listed at a sportsbook (i.e over 8.5, -110) and completely different price at another (i.e. over 9, +110). Which price is better for the bettor?!? Our MLB prediction pages do this calculation automatically so the best over/under will show up in our “Best Line” column. While this helps, it’s still essential to know how much these point values are worth. Knowing half point values has become even more crucial now that sportsbooks are offering alternate lines more prominently. There are often diamonds to find in the rough.
### Half Point Table for MLB Totals
Here is our Major League Baseball Half Point Table for reference…
Total Push Prob 0.5 Point Down 0.5 Point Up 1 Point Down 1 Point Up
5.5 0% 34.05 22.55 68.1 45.1
6 9.2% 20.26 20.26 48.24 45.81
6.5 0% 22.55 30.21 45.1 60.42
7 11.6% 26.24 26.24 47.06 46.38
7.5 0% 30.21 21.65 60.42 43.31
8 8.9% 19.54 19.54 45.01 44.57
8.5 0% 21.65 29.53 43.31 59.07
9 11.4% 25.73 25.73 45.82 40.63
9.5 0% 29.53 15.21 59.07 30.41
10 6.6% 14.13 14.13 38.54 33.83
10.5 0% 15.21 22.55 30.41 45.1
11 9.2% 20.26 20.26 34.8 34.14
11.5 0% 22.55 14.42 45.1 28.83
12 6.3% 13.45 13.45 33.08 29.46
12.5 0% 14.42 17.65 28.83 35.29
13 7.5% 16.22 16.22 29.84 27.68
13.5 0% 17.65 11.86 35.29 23.71
14 5.3% 11.19 11.19 27.03 25.34
14.5 0% 11.86 15.47 23.71 30.95
At first glance, this table can look a little bit confusing. In an effort to simplify, I’ll give an example on how to use it.
Let’s say that the under on a game is 7.5, with the odds of +100 at Sportsbook BetMGM. Also, at BetMGM, we see that there is an alternate line for the under at 7 (+125). Which is the better bet? To find out, we simply go to our table and find out what the cost is to go from 7.5 to 7. Our table tells us that it is 30.21 cents. Now, moving from 7.5 to 7 at odds of +100 to +125 only gives us 25 cents. Thus, the bet is better at 7.5. Basically, the 30.21 cents is how much the price would need to move to make the bet worth it. Thus, if we got under 7 at +131 or better, then we would make that bet.
### Methodology
Where do these half point totals come from? If you scoured the internet looking for these, then you may see that different people use different numbers. In fact, it’s unlikely that any two people will have the same numbers. Why is this? To calculate these half point totals, one needs to know the push probability at every whole number (it should be obvious that half point numbers can’t push). Nothing about the total run distribution is normal.
Above is what the smoothed distribution of totals are for Major League Baseball. This is before we take the effect of over/under’s into account. Odd numbers are way more likely to occur. Why? Because a baseball game can’t end in a tie. Thus, when a score is tied 4-4 after 9 innings, the teams keep playing to determine a winner. Therefore, hitting an odd number is much more likely than hitting an even number.
But it is also not enough to just look at the distribution of all Major League Baseball scores, assign push probabilities and then call it a day. This is because a distribution shifts based on what the likely over/under is. What does this mean? When an over/under is say 7, then there is a higher probability that the game’s total lands on 7 versus if the game’s over/under is 11. All of this needs to be accounted for in the calculations. This provides challenges for all over/unders that have limited data on them (i.e. over/unders < 7 or over/unders > 12).
We are also constantly reevaluating this data as there are a variety of things that can change this distribution. If one is doing this on there own, then it’s recommended they check the data at least twice a year to make sure they haven’t missed out on any changes. | 1,227 | 3,915 | {"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} | 3.03125 | 3 | CC-MAIN-2022-49 | longest | en | 0.855502 |
http://www.in2013dollars.com/inflation-rate-in-1980 | 1,571,896,103,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987841291.79/warc/CC-MAIN-20191024040131-20191024063631-00000.warc.gz | 259,900,151 | 13,699 | \$
# U.S. inflation rate in 1980: 13.50%
### Inflation in 1980 and Its Effect on Dollar Value
Purchasing power decreased by 13.50% in 1980 compared to 1979. On average, you would have to spend 13.50% more money in 1980 than in 1979 for the same item.
In other words, \$1 in 1979 is equivalent in purchasing power to about \$1.13 in 1980.
The 1979 inflation rate was 11.35%. The inflation rate in 1980 was 13.50%. The 1980 inflation rate is higher compared to the average inflation rate of 2.96% per year between 1980 and 2019.
Inflation rate is calculated by change in the consumer price index (CPI). The CPI in 1980 was 82.40. It was 72.60 in the previous year, 1979. The difference in CPI between the years is used by the Bureau of Labor Statistics to officially determine inflation.
Average inflation rate 13.50% Converted amount (\$1 base) \$1.13 Price difference (\$1 base) \$0.13 CPI in 1979 72.600 CPI in 1980 82.400 Inflation in 1979 11.35% Inflation in 1980 13.50%
USD Inflation since 1913
Annual Rate, U.S. Bureau of Labor Statistics CPI
### Inflation by City
Inflation can vary widely by city, even within the United States. Here's how some cities fared in 1979 to 1980 (figures shown are purchasing power equivalents of \$1):
Dallas-Fort Worth, Texas experienced the highest rate of inflation during the 1 years between 1979 and 1980 (16.95%).
New York experienced the lowest rate of inflation during the 1 years between 1979 and 1980 (11.29%).
Note that some locations showing 0% inflation may have not yet reported latest data.
### Inflation by Country
Inflation can also vary widely by country. For comparison, in the UK £1.00 in 1979 would be equivalent to £1.18 in 1980, an absolute change of £0.18 and a cumulative change of 17.99%.
In Canada, CA\$1.00 in 1979 would be equivalent to CA\$1.11 in 1980, an absolute change of CA\$0.11 and a cumulative change of 11.06%.
Compare these numbers to the US's overall absolute change of \$0.13 and total percent change of 13.50%.
### Inflation by Spending Category
CPI is the weighted combination of many categories of spending that are tracked by the government. This chart shows the average rate of inflation for select CPI categories between 1979 and 1980.
Compare these values to the overall average of 13.50% per year:
Category Avg Inflation (%) Total Inflation (%) \$1 in 1979 → 1980
Food 8.59 8.59 1.09
Shelter 17.55 17.55 1.18
Energy 30.84 30.84 1.31
Apparel 7.07 7.07 1.07
New vehicles 0.00 0.00 1.00
Used cars and trucks 0.00 0.00 1.00
Transportation services 14.00 14.00 1.14
Medical care services 11.28 11.28 1.11
Medical care commodities 9.34 9.34 1.09
It's important to note that not all categories may be tracked since 1979. This table and visualization use the earliest available data for each category.
### How to Calculate Inflation Rate for \$1, 1979 to 1980
This inflation calculator uses the following inflation rate formula:
CPI in 1980CPI in 1979
×
1979 USD value
=
1980 USD value
Then plug in historical CPI values. The U.S. CPI was 72.6 in the year 1979 and 82.4 in 1980:
82.472.6
×
\$1
=
\$1.13
\$1 in 1979 has the same "purchasing power" or "buying power" as \$1.13 in 1980.
To get the total inflation rate for the 1 years between 1979 and 1980, we use the following formula:
CPI in 1980 - CPI in 1979CPI in 1979
×
100
=
Cumulative inflation rate (1 years)
Plugging in the values to this equation, we get:
82.4 - 72.672.6
×
100
=
13%
### Alternate Measurements of Inflation
The above data describe the CPI for all items. Also of note is the Core CPI, which measures inflation for all items except for the more volatile categories of food and energy. Core inflation averaged 12.42% per year between 1979 and 1980 (vs all-CPI inflation of 13.50%), for an inflation total of 12.42%.
When using the core inflation measurement, \$1 in 1979 is equivalent in buying power to \$1.12 in 1980, a difference of \$0.12. Recall that for All Items, the converted amount is \$1.13 with a difference of \$0.13.
In 1979, core inflation was 9.74%.
### Comparison to S&P 500 Index
To help put this inflation into perspective, if we had invested \$1 in the S&P 500 index in 1979, our investment would be nominally worth approximately \$1.48 in 1980. This is a return on investment of 47.59%, with an absolute return of \$0.48.
These numbers are not inflation adjusted, so they are considered nominal. In order to evaluate the real return on our investment, we must calculate the return with inflation taken into account.
The compounding effect of inflation would account for 11.89% of returns (\$0.06) during this period. This means the inflation-adjusted real return of our \$1 investment is \$0.42.
Investment in S&P 500 Index, 1979-1980
Original Amount Final Amount Change
Nominal \$1 \$1.48 47.59%
Real
\$1 \$0.42 41.93%
Politics and news often influence economic performance. Here's what was happening at the time:
• Ayatollah Khomeini returns to Iran after fifteen years of exile.
• Ugandan dictator Idi Amin flees to Libya after the Tanzanian army captures Kampala, Uganda's capital.
• Margaret Thatcher becomes Prime Minister of United Kingdom, the first ever woman to hold the position.
• Afghanistan is invaded by the Soviets and overthrowing of President Hafizullah Amin.
### Data Source & Citation
Raw data for these calculations comes from the Bureau of Labor Statistics' (CPI), established in 1913. Inflation data from 1665 to 1912 is sourced from a historical study conducted by political science professor Robert Sahr at Oregon State University.
You may use the following MLA citation for this page: “Inflation Rate in 1980 | Inflation Calculator.” U.S. Official Inflation Data, Alioth Finance, 24 Oct. 2019, https://www.officialdata.org/inflation-rate-in-1980. | 1,611 | 5,791 | {"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} | 2.515625 | 3 | CC-MAIN-2019-43 | latest | en | 0.93449 |
https://mathsolver.microsoft.com/en/solve-problem/y%20%3D%20-%20x%20%5E%20%7B%202%20%7D%20%2B%202 | 1,603,444,301,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107880878.30/warc/CC-MAIN-20201023073305-20201023103305-00390.warc.gz | 420,151,474 | 156,514 | Solve for x
Steps by Finding Square Root
Solve for x (complex solution)
Solve for y
Assign y
Graph
Still have questions?
-x^{2}+2=y
Swap sides so that all variable terms are on the left hand side.
-x^{2}=y-2
Subtract 2 from both sides.
\frac{-x^{2}}{-1}=\frac{y-2}{-1}
Divide both sides by -1.
x^{2}=\frac{y-2}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}=2-y
Divide y-2 by -1.
x=\sqrt{2-y} x=-\sqrt{2-y}
Take the square root of both sides of the equation.
-x^{2}+2=y
Swap sides so that all variable terms are on the left hand side.
-x^{2}+2-y=0
Subtract y from both sides.
x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\left(2-y\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and -y+2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{0±\sqrt{-4\left(-1\right)\left(2-y\right)}}{2\left(-1\right)}
Square 0.
x=\frac{0±\sqrt{4\left(2-y\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{0±\sqrt{8-4y}}{2\left(-1\right)}
Multiply 4 times -y+2.
x=\frac{0±2\sqrt{2-y}}{2\left(-1\right)}
Take the square root of -4y+8.
x=\frac{0±2\sqrt{2-y}}{-2}
Multiply 2 times -1.
x=-\sqrt{2-y}
Now solve the equation x=\frac{0±2\sqrt{2-y}}{-2} when ± is plus.
x=\sqrt{2-y}
Now solve the equation x=\frac{0±2\sqrt{2-y}}{-2} when ± is minus.
x=-\sqrt{2-y} x=\sqrt{2-y}
The equation is now solved. | 549 | 1,371 | {"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} | 4.34375 | 4 | CC-MAIN-2020-45 | latest | en | 0.66837 |
https://www.physicsforums.com/threads/simple-tension-problem.62334/ | 1,508,551,347,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187824537.24/warc/CC-MAIN-20171021005202-20171021025202-00816.warc.gz | 944,619,879 | 17,455 | # Simple tension problem
1. Feb 2, 2005
### karli_m
Hi... I just stumbled upon this forum trying to find a reasonable answer to an introductory physics problem I am working on. Maybe someone here could help me. If you have a mass hanging from three wires, how do you compute the tension in each of the wires? It's a conceptual question, I only need to derive the formula for T1, which is actually given to me, but I still can't figure it out... any help would be greatly appreciated.
2. Feb 2, 2005
### Sirus
You could evaluate vectorally in three dimensions.
3. Feb 2, 2005
### karli_m
Why three dimensions? Wouldn't it just be in the x-y plane? So I just resolve each of the three vectors into their x and y components and then add them? I didn't mention also that the two wires are different lengths, at different angles from the ceiling.
4. Feb 2, 2005
### Sirus
Ok, I assumed three dimensions, but it can be in the x-y plane. In that case you have the right approach. Trigonometry can be used, alternatively.
5. Feb 2, 2005
### karli_m
Yeah, the equation they want me to derive gives
T1 = {mg cos(angle 2)}/{sin(angle 1 + angle 2)}
*sorry* I don't know how to type in equations like this properly
T1 and T2 are the two wires supporting the wire holding the mass, T3. I get that the tension for T3 is mg, but the tension in T1 and T2 aren't equal if the mass isn't centered between the two wires, right? I'm sorry, this is probably so simple...
6. Feb 2, 2005
### Sirus
No need to apologize.
Hm, from what I understand of your description, you have a mass supported by a wire which splits into two separate wires of unequal lengths before reaching the ceiling, or whatever the mass is hanging from. You are on the right track, and you are correct in assuming unequal tension if the lengths of T1 and T2 are different. Draw a vector diagram showing the forces present and set up a force-vector triangle. Then use trigonometry to find an expression for T1.
7. Feb 3, 2005
### karli_m
Thanks sirus... I appreciate all your help. Still haven't figured it out though. I think it's not even so much I don't understand physics, but more so I don't get trigonometry. Maybe I should go to a trig forum. Thanks though!!
8. Feb 3, 2005
### Staff: Mentor
Start by describing the orientation of your three wires. (Draw a picture.) At any point (such as the junction of two wires) the net force is zero.
9. Feb 3, 2005
### karli_m
OK, so the net force is zero when a mass is just hanging from 3 wires... two wires attached to a support and the third attached to those two with the mass hanging from it. So, does that mean the x-component of the net force is equal to the y-component of force? or that each are equal to zero?
10. Feb 3, 2005
### karli_m
Here's the diagram... can anyone still help me?
#### Attached Files:
• ###### tension problem.JPG
File size:
3.5 KB
Views:
49
11. Feb 3, 2005
### Staff: Mentor
Both the x and y components of the net force must equal zero.
In your diagram, T3 must balance the weight of the object (since the object is in equilibrium). So now you know T3!
Now consider the forces at the junction of wires: Find the vertical components of each tension: they must add to zero. Same for the horizontal components. You'll have two (easy) equations and two unknowns (T1 and T2), so you can solve for T1 and T2.
12. Feb 3, 2005
### karli_m
All right, I guess I've done that... but now I can't seem to derive the equation (from above). Can you tell me if this is right so far? (please?)
for the y-component: T1sin(angle1) + T2sin(angle2) - mg = 0
for the x-component: T2cos(angle2) - T1cos(angle1) = 0
I don't know how to combine these to derive the formula they're asking for. Any hints? Thanks for helping me, by the way.
13. Feb 3, 2005
### Staff: Mentor
So far, so good!
First step: In the second equation, solve for T2 in terms of T1. Then substitute that into your first equation and solve for T1.
To simplify the resulting expression, you'll need a trig identity. ($sin(A + B) =$ ? Look it up!)
14. Feb 3, 2005
### karli_m
Ha! thanks so much... I think I got the right answer now. I knew it was a problem I was having with trig. Haven't taken it in years. Anyway, thanks a lot for all the help. | 1,130 | 4,270 | {"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} | 3.71875 | 4 | CC-MAIN-2017-43 | longest | en | 0.940826 |
http://www.statistics101.net/userguide/UserGuide90.html | 1,560,764,213,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998462.80/warc/CC-MAIN-20190617083027-20190617105027-00344.warc.gz | 298,956,902 | 6,910 | ### Appendix: Command Summary
Statistics101/Resampling Stats Command Summary COMMAND DESCRIPTION ABS Computes the absolute value of each element of the input vector. ACOS Computes the arc cosine of each element of its input vector. ADD Arithmetically adds corresponding elements of its input vectors. ARGCOUNT Computes the number of arguments that a subroutine was invoked with. ASIN Computes the arc sine of each element of its input vector. ATAN Computes the arc tangent of each element of its input vector. BINOMIALPROB Computes the probability of k successes in n trials given a probability of success. BOXPLOT Outputs a boxplot of one or more input vectors to the Output Window. BREAK Forces immediate exit from the innermost enclosing REPEAT or WHILE loop. CHISQUARE Computes the chi-square statistic from its two input vectors. CLEAN Removes "missing data" (NaN) from one or more vectors. CLEAR Removes all the elements from one or more vectors. CLEAROUTPUT Clears the contents of the Output Window. CLOSETABS Closes all graph tabs in the lower panel of the main window. COMBINATIONS Computes the number of combinations of n items taken k at a time. CONCAT Concatenates arguments into a single vector (Same as COPY). COPY Concatenates arguments into a single vector. CORR Computes Pearson's product moment correlation coefficient of two vectors. COS Computes the cosine of each element of its input vector. COUNT Counts the number of elements that pass a specified test. DATA Concatenates arguments into a single vector (Same as COPY). DEBUG Causes the program to stop executing at this point and enter the debugger. DECLARE Declares the argument list of a subroutine so the subroutine may be invoked in the text of a program prior to its actual definition. DEDUP Removes duplicate elements from its input vector. DIVIDE Arithmetically divides corresponding elements of its input vectors. ELSE Marks the beginning of a set of commands to be executed if the logical expressions of all other branches of an IF command fail. ELSEIF Marks the beginning of a alternative branch of an IF command, with its own logical expression, controlling its own set of commands. END Marks the end of an IF, REPEAT, WHILE, or NEWCMD command block. EXEC Submits a command string to be executed by the underlying operating system (Windows, Mac, Linux). EXIT Quits the currently running user program. EXP Computes the number e (i.e., 2.71828...) raised to the power of each element of the input vector. EXPONENTIAL Randomly selects a specified number of values from a specified exponential distribution. FOREACH Executes commands between FOREACH and END assigning each element of the given vector one by one to element. FRACTION Copies the fractional part of each element of its inputVector into its result vector. FUZZ Sets a range of validity for value comparisons during tests. GENERATE Randomly selects a specified number of elements from a vector, with replacement (Same as SAMPLE). GETARG Copies the specified optional argument into the result variable. GETFILEPATH Retrieves the path information for the file accessed by the most recent READ or WRITE command. GLOBAL Declares that the names in its argument list are to be visible within subroutines.. HISTOGRAM Creates a histogram of one or more vectors in a new graphic window tab. Computes a histogram from its input vector and puts the results in the remaining vectors. Does not make a plot or graph. HISTOGRAMPLOT Prints a textplot histogram of one or more vectors in the Output Window. IF Allows execution of commands between IF and the next branch (ELSE, ELSEIF, or END) of the IF command if a specified logical expression evaluates to true. INCLUDE This command replaces itself with the contents of the file(s) in its argument list. INPUT Prompts the user for input and accepts user's input. INTEGER Converts all elements of its input vector to integer by truncation, floor, ceiling, or rounding, depending on the keyword. Default is truncation. LET Uses mathematical formula notation to compute a value and assign it to a variable. Allows use of many Statistics101 math command names as unary functions. LOG Computes the natural logarithm of each element of its input vector. LOG10 Computes the base 10 logarithm of each element of its input vector. LOGNORMAL Randomly selects a specified number of numbers from a specified lognormal distribution. MAX Finds the largest value (most positive) in its input vector MAXSIZE Not implemented. MEAN Computes the mean of a vector. MEDIAN Computes the median of a vector. MIN Finds the smallest (most negative) value in its input vector. MODE Finds the most frequently occurring value in its input vector. MULTIPLES Computes the number of "multiples" whose sizes satisfy the specified test. MULTIPLY Arithmetically multiplies corresponding elements of its input vectors. NAME Creates one or more named constants. NEWARRAY Creates an array with the given name and dimensions. NEWCMD Declares a "new command" or subroutine. NORMAL Randomly selects a specified number of numbers from a specified normal distribution. NORMALPROB Calculates the cumulative normal distribution. Given x or z computes p. NORMALPROBINV Calculates the inverse cumulative normal distribution. Given p computes z or x. NUMBERS Concatenates arguments into a single vector (Same as COPY). OUTPUT Prints a string and any number of optional numbers to the Output Window. PARETO Randomly selects a specified number of numbers from the specified Pareto distribution. PAUSE Stops program execution until the user clicks on the Continue button. PERCENTILE Computes specified percentiles from an input vector. PERMUTATIONS Computes the number of permutations of n items taken k at a time. POISSON Randomly selects a specified number of numbers from a specified Poisson distribution. POWER Raises each element in the first input vector to the power of the corresponding element in the second input vector. PRINT Prints the name and contents of one or more vectors to the Output Window, one vector to a line. PRODUCT Computes the product of all the elements of its input vector. PROGINFO Prints program variables, constants, and status information to the Output Window. PUT Inserts values from input vector into the result vector at locations specified by the positions vector. RANDOM Randomly selects a specified number of elements from a vector, with replacement (Same as SAMPLE). RANKS Creates a list of the ranks of the elements of its input vector. READ Reads a file into one or more result variables (vectors). RECODE Replaces with a specified number, any element of the input vector that satisfies a specified test. REGRESS Computes the coefficients of the linear regression equation determined by its dependent vector and its independent vector(s). REMAINDER Divides the corresponding elements of the two input vectors and puts the remainder in the result vector. REMOVE Copies all but the specified elements of its input dataVector into its result vector. REPEAT Executes commands between REPEAT and END a specified number of times. ROUND Rounds each element of its input vector to an integer. ROTATE Rotates the elements of the input vector right or left by the specified number of places. RUNS Computes the number of runs (consecutive equal numbers) whose lengths satisfy the specified test. SAMPLE Randomly selects a specified number of elements from a vector, with replacement. SCALARIZE Computes a single number by concatenating all the elements of its input vector. SCATTERGRAPH Displays a linear or log scattergraph of its input vectors in a graphical tab in the Stastistics101 Output Window. SCORE Accumulates the results of random trials in a scoring vector. SEED Sets the seed used by the random number generator and/or selects the algorithm that generates the pseudo-random numbers. SET Creates a vector with a specified number of elements all of the same value as the input number (Can be replaced by COPY N#val ). SHIFT Shifts the elements of the input vector right or left by the specified number of places. Shifts in zeros to the positions freed by the shift. SHUFFLE Randomly reorders the elements of a vector. SIGN Substitutes a -1 for negative elements, a +1 for positive elements. Zero becomes +1 unless SIGNUM keyword is present. SIN Computes the sine of each element of its input vector. SIZE Counts the number of elements contained by the input vector. SORT Sorts the elements of the input vector in ascending or descending order. SQRT Computes the square root of each element of the input vector. SQUARE Computes the square of each element of the input vector. STDEV Computes the standard deviation of a vector. STRING Concatenates string literals, string variables and/or vector variables into one string variable. STRING_COMPARE Compares two strings, returning zero if they are equal, a negative number if the first is less than the second, a positive number if the first is greater than the second. STRING_REPLACE Returns a new string resulting from replacing all occurrences [or the first] of a regular expression match in the input string with a given replacement string. SUBTRACT Arithmetically subtracts corresponding elements of its input vectors. SUM Computes the sum of all the elements of its input vector. SUMABSDEV Computes the sum of the absolute differences between its two input vectors. SUMSQRDEV Computes the sum of the squared deviations of its first input vector's elements versus its second vector's elements. TAGS Computes a list of the positions of the elements of the input vector that pass a test. TAGSORT Computes a list whose element values, in order, are the positions of the elements of its input vector as if it were sorted in ascending order. TAKE Copies specified elements from its input vector into its result vector. TAN Computes the tangent of each element of its input vector. TIME Reads the system clock and puts the time, in milliseconds, into its result vector. TIMEPLOT Prints a timeplot of its input vector on the Statistics101 Output Window. UNIFORM Selects a specified number of values randomly from the uniform distribution with the specified lower and upper limits. UNTIL Executes commands between UNTIL and END until the specified logical expression evaluates to true. URN Concatenates arguments into a single vector (Same as COPY). VARIANCE Computes the variance of a vector. WEED Discards those values from its input vector that satisfy the specified test. WEIBULL Randomly selects a specified number of numbers from the specified Weibull distribution. WHILE Executes commands between WHILE and END as long as the specified test passes. WRITE Writes its input vector(s) into a file or the Output Window according to optional format specifications. XYGRAPH Displays an X-Y linear or log graph of its input vectors in a graphical tab in the Stastistics101 Output Window. Allows both line and scattergraphs on same graph. XYPLOT Prints an X-Y linear or log text plot of its input vectors to the Statistics101 Output Window. | 2,204 | 11,054 | {"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} | 2.65625 | 3 | CC-MAIN-2019-26 | latest | en | 0.801858 |
https://discourse.mc-stan.org/t/mildly-over-disperse-logistic-model/2984 | 1,675,923,552,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764501407.6/warc/CC-MAIN-20230209045525-20230209075525-00859.warc.gz | 231,050,385 | 6,239 | # Mildly over-disperse logistic model
Hi - I have a basic logistic model that fits pretty well (without the noise term). It does have a bit more dispersion at very high and very low success rates (at the 0-5%, 95-100%)… so I’m adding some noise:
When I run it, I get pretty tight parameter samples for the linear feature weights (which are very close to those I’d get without the added noise term), but the variance of the noise term does not become happy - at all.
So two questions… what’s going wrong with convergence of sigma (i’d guess it would want to settle close to zero), and second is there a BETTER way to deal with a little bit of random noise on top of a linear logistic model.
Thank you!!
You need to non-center `theta`:
``````sigma ~ normal(0, 2);
theta ~ normal(0, 1);
y ~ bernoulli_logit(x * beta + sigma * theta);
``````
1. sigma is not well identifiable
2. adding sigma makes the model go towards probit, ie, having shorter tails and less dispersed than logit
1 Like
Thanks for the input!
@bgoodri - I had taken out the intercept parameter and you made me realize I need it for variance (problem is symmetric and I was not thinking fully bayesian). Your suggestion didn’t quite converge, but a simple real parameter did quite well with resulting in a slight bias with spread.
@avehtari , it is true that I am struggling with the slope of the link function at high/low rates. I had tried probit and it was much worse. I want to smooth out the link function, but adding stricter priors spooks out the linear regression betas.
For now, I am good with this:
with this fit:
2 Likes
If you want something with thicker tails than logit, use tobit. probit is cdf of Gaussian, tobit is cdf of t distribution, and logit is close to tobit with degrees of freedom 7. Using tobit with df<7 will make it more robust to outliers.
1 Like
thanks that’s awesome and exactly what I was looking for. I think over-dispersion is not the problem, but more like poor fitting at extremes. | 482 | 1,995 | {"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} | 2.6875 | 3 | CC-MAIN-2023-06 | latest | en | 0.939848 |
https://www.techgig.com/practice/question/ZEpsS0ZMYVNkcDBXNWFZcWQ2Wi96dz09 | 1,534,277,404,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221209562.5/warc/CC-MAIN-20180814185903-20180814205903-00120.warc.gz | 979,662,441 | 23,975 | Increasing maximum product (100 Marks)
Given a sequence of non-negative integers, find the subsequence of length 3 having maximum product with the numbers of the subsequence being in ascending order.
Input Format
You will be given an array of integers of size n.
1 < n < 10^5
1 < a[i] < 10^5
Output Format
You need to print the maximum product.
Sample TestCase 1
```8
6
7
8
1
2
3
9
10```
`720`
Explanation
The three elements are 8,9 and 10.
`Normal``Line: 0 Col: 0` | 139 | 472 | {"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} | 2.515625 | 3 | CC-MAIN-2018-34 | longest | en | 0.770596 |
http://www.fixya.com/cars/t1651251-low_miles_per_gallon | 1,495,784,553,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463608648.25/warc/CC-MAIN-20170526071051-20170526091051-00146.warc.gz | 629,153,630 | 35,105 | Low miles per gallon
I have a s40 se 2ltre automatic the problem is that the mpg as dropped from approx 28.5 mpg to 21.5 mpg i,ve had it serviced but it has made no differance and all my tests are within the peramaters
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• Contributor
Hi the first thing you might want to look at is low tyre pressures or a fuel leak? After eliminating those i would look at the MAF or mass air flow meter. If you unplug your air flow meter with a hot engine and drive it there should be a large difference in performance, if there isnt then replace the MAF as it is faulty.
Posted on Feb 08, 2009
Hi,
a 6ya expert can help you resolve that issue over the phone in a minute or two.
best thing about this new service is that you are never placed on hold and get to talk to real repairmen in the US.
the service is completely free and covers almost anything you can think of (from cars to computers, handyman, and even drones).
goodluck!
Posted on Jan 02, 2017
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hey
at the begging u should clarify how many gallon u used.
112 miles at 28 miles per gallon --> 4 gallons burned (112/28)
So that means we leaked 11 - 4 = 7 gallons
Now figure out how long the car was driving.
112 miles at 64 miles per hour --> 1.75 hours
So, leak was at a rate of 7 gallons in 1.75 hours
7/1.75 = 4 gallons per hour leaked.
Jul 10, 2008 | 2008 Chevrolet Corvette
Miles per gallon
About 11 MPG. I had to convert KM to Miles. Of course this is for "highway" driving without a lot of stops and starts. I didn't find any information on city driving, but I would guess that it would be around 8 MPG.
May 15, 2008 | 2004 Ferrari 360 Modena
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Level 3 Expert | 1,278 | 4,796 | {"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} | 2.96875 | 3 | CC-MAIN-2017-22 | longest | en | 0.931159 |
https://his.edu.vn/if-you-have-50-50-vision-find-the-number-517-in-14-secs | 1,726,519,504,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651710.86/warc/CC-MAIN-20240916180320-20240916210320-00225.warc.gz | 270,568,245 | 33,947 | # If you have 50/50 Vision Find the Number 517 in 14 Secs
Only people with excellent eyesight can find the hidden number 517 in this puzzle challenge. The puzzles are tricky and will test your observation and visual skills. We challenge you all to try this optical illusion and test your IQ level.
## Try to find the hidden number 517
Normally, when searching for the hidden number 517 within the given time, it becomes easier for your brain to find the hidden number 517 faster. So here’s a quick countdown for you to sort out the hidden number 517.
So the clock has started ticking….10…9…8…
hurry up! ! Time is running out. Look at all corners of the image to find the hidden number 517 that you need to identify. Have you found the hidden 517 number? If not, you don’t have to worry about its solution. See the next section to see where the number 517 is in the image.
3…2…1….Time’s up! !
Embark on a journey of visual wonder. Dive into the world of optical illusions and uncover its secrets. Embark on a journey of visual wonder and uncover the solutions to today’s illusions, a true test of perception, only available at HIS Education!
## Solution to hiding number 517
Finding solutions for images is a challenging step. If you are still staring at the picture looking for the hidden number 517, then here you can see the correct position of the hidden number 517. Now, let us reveal the answer.
See also Love Island winner Molly slams ‘snakey’ Georgia S for unaired digs as Tom brands her a liar over their history
The highlighted area on the image reveals the hidden number 517. Don’t despair if you can’t find a solution.
Today’s New York Times Mini Crossword Puzzle Answers
8×4=8 Correct the equation by removing 2 sticks
## There is a snake hidden in this picture
In this picture, many jars can be found. There is a snake hidden in it and the challenge is to find the hidden snake within 14 seconds.
## Answers for the snake hidden in this picture
Examine the image carefully and you will spot the snake within the highlighted area. If you can’t find it, don’t worry – we’ll help with the image below.
## find different sheep
What makes this image unique is the presence of a sheep, which stands out from the other images. Indeed, there is a unique sheep in the image. Can you identify which sheep is different from the others? | 533 | 2,357 | {"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} | 3.46875 | 3 | CC-MAIN-2024-38 | latest | en | 0.869645 |
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Boolean Logic Truth Tables PowerPoint PPT Presentation
Boolean Logic Truth Tables. First fill out all the possible inputs for a table (to the left of the dark line), then fill in the other columns.
Boolean Logic Truth Tables
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Boolean Logic Truth Tables
First fill out all the possible inputs for a table (to the left of the dark line), then fill in the other columns.
Remember:p and q is True only if both p and q are True.p or q is True if either p or q are True.p xor q is True is either p or q are True, but not both. | 257 | 1,068 | {"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} | 2.671875 | 3 | CC-MAIN-2017-13 | longest | en | 0.827976 |
https://andrescaicedo.wordpress.com/tag/paul-du-bois-reymond/ | 1,642,915,335,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304134.13/warc/CC-MAIN-20220123045449-20220123075449-00585.warc.gz | 151,491,331 | 29,516 | Weierstrass function
November 7, 2013
Weierstrass function from 1872 is the function $f=f_{a,b}$ defined by
$\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$.
Weierstrass showed that if
• $0,
• $b$ is an odd positive integer, and
• $\displaystyle ab>1+\frac32\pi$,
then $f$ is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on $a,b$ to
• $0,
• $b>1$, and
• $ab\ge 1$.
Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect $f$ to be non-differentiable. The idea is that the cosine terms ensure that the partial sums $\displaystyle f(m,x)=\sum_{n=0}^m a^n\cos(b^n\pi x)$, though smooth, have more and more “turns” on each interval as $m$ increases, so that in the limit, $f$ has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of $f(m,x)$ for $0\le m<20$ (and $-10\le x\le 10$), for $a=1/2$ and $b=11$, showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)
Below the fold, we show the same animation, zoomed in around $0$ by factors of $100$, $10^4$, and $10^6$, respectively, illustrating the fractal nature of $f$.
Analysis – On praise
November 4, 2013
Orders of infinity is Hardy’s monograph from 1910 on the work of Du Bois Reymond. From the preface:
There is, in Du Bois-Reymond’s original memoirs, a good deal that would not be accepted as conclusive by modern analysts. He is also at times exceedingly obscure; his work would beyond doubt have attracted much more attention had it not been for the somewhat repugnant garb in which he was unfortunately wont to clothe his most valuable ideas. | 514 | 1,765 | {"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": 24, "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} | 2.9375 | 3 | CC-MAIN-2022-05 | latest | en | 0.924851 |
https://en.wikipedia.org/wiki/Epsilon_nought | 1,506,264,380,000,000,000 | text/html | crawl-data/CC-MAIN-2017-39/segments/1505818690029.51/warc/CC-MAIN-20170924134120-20170924154120-00556.warc.gz | 655,183,233 | 23,144 | # Vacuum permittivity
(Redirected from Epsilon nought)
The physical constant ε0 (pronounced as "epsilon naught"), commonly called the vacuum permittivity, permittivity of free space or electric constant, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has an exactly defined value
ε0 = 8.854187817...×10−12 F⋅m−1 (farads per metre).[1]
It is the capability of the vacuum to permit electric field lines. This constant relates the units for electric charge to mechanical quantities such as length and force.[2] For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:
${\displaystyle \ F_{\text{C}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$
The value of the constant fraction is 9 × 109 N⋅m2⋅C−2, q1 and q2 are the charges, and r is the distance between them. Likewise, ε0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.
## Value
The value of ε0 is currently defined by the formula[3]
${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}}$
where c is the defined value for the speed of light in classical vacuum in SI units,[4] and μ0 is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability). Since μ0 has the defined value 4π × 10−7 H/m,[5] and c has the defined value 299792458 m⋅s−1,[6] it follows that ε0 has a defined value given approximately by
ε08.854187817620... × 10−12 Fm−1 (or A2s4kg−1m−3 in SI base units, or C2N−1m−2 or CV−1m−1 using other SI coherent units).[7][8]
The historical origins of the electric constant ε0, and its value, are explained in more detail below.
### Redefinition of the SI units
Under the proposals to redefine the ampere as a fixed number of elementary charges per second,[9] the electric constant would no longer have an exact fixed value. The value of the electron charge would become a defined number, not measured, making μ0 a measured quantity. Consequently, ε0 also would not be exact. As before, it would be defined by the equation ε0 = 1/(μ0c2), but now with a measurement error related to the error in μ0, the magnetic constant. This measurement error can be related to that in the fine-structure constant α:
${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}={\frac {e^{2}}{2\alpha hc}}\ ,}$
with e the exact elementary charge, h the exact Planck constant, and c the exact speed of light in vacuum. Here use is made of the relation for the fine-structure constant:
${\displaystyle \alpha ={\frac {\mu _{0}ce^{2}}{2h}}\ .}$
The relative uncertainty in the value of ε0 therefore would be the same as that for the fine-structure constant, currently 6.8×10−10.[7]
## Terminology
Historically, the parameter ε0 has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",[10][11] "permittivity of empty space",[12] or "permittivity of free space"[13] are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,[7] and official standards documents have adopted the term (although they continue to list the older terms as synonyms).[14][15]
Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.[16][17] However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity ε/ε0 and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity.[15][18] Hence, the term "dielectric constant of vacuum" for the electric constant ε0 is considered obsolete by most modern authors, although occasional examples of continuing usage can be found.
As for notation, the constant can be denoted by either ${\displaystyle \varepsilon _{0}\,}$ or ${\displaystyle \epsilon _{0}\,}$, using either of the common glyphs for the letter epsilon.
## Historical origin of the parameter ε0
As indicated above, the parameter ε0 is a measurement-system constant. Its presence in the equations now used to define electromagnetic quantities is the result of the so-called "rationalization" process described below. But the method of allocating a value to it is a consequence of the result that Maxwell's equations predict that, in free space, electromagnetic waves move with the speed of light. Understanding why ε0 has the value it does requires a brief understanding of the history.
### Rationalization of units
The experiments of Coulomb and others showed that the force F between two equal point-like "amounts" of electricity, situated a distance r apart in free space, should be given by a formula that has the form
${\displaystyle F=k_{\text{e}}{\frac {Q^{2}}{r^{2}}},}$
where Q is a quantity that represents the amount of electricity present at each of the two points, and ke is Coulomb's constant. If one is starting with no constraints, then the value of ke may be chosen arbitrarily.[19] For each different choice of ke there is a different "interpretation" of Q: to avoid confusion, each different "interpretation" has to be allocated a distinctive name and symbol.
In one of the systems of equations and units agreed in the late 19th century, called the "centimetre–gram–second electrostatic system of units" (the cgs esu system), the constant ke was taken equal to 1, and a quantity now called "gaussian electric charge" qs was defined by the resulting equation
${\displaystyle F={\frac {{q_{\text{s}}}^{2}}{r^{2}}}.}$
The unit of gaussian charge, the statcoulomb, is such that two units, a distance of 1 centimetre apart, repel each other with a force equal to the cgs unit of force, the dyne. Thus the unit of gaussian charge can also be written 1 dyne1/2 cm. "Gaussian electric charge" is not the same mathematical quantity as modern (MKS and subsequently the SI) electric charge and is not measured in coulombs.
The idea subsequently developed that it would be better, in situations of spherical geometry, to include a factor 4π in equations like Coulomb's law, and write it in the form:
${\displaystyle F=k'_{\text{e}}{\frac {{q'_{\text{s}}}^{2}}{4\pi r^{2}}}.}$
This idea is called "rationalization". The quantities qs′ and ke′ are not the same as those in the older convention. Putting ke′ = 1 generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.
The next step was to treat the quantity representing "amount of electricity" as a fundamental quantity in its own right, denoted by the symbol q, and to write Coulomb's Law in its modern form:
${\displaystyle \ F={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q^{2}}{r^{2}}}.}$
The system of equations thus generated is known as the rationalized metre–kilogram–second (rmks) equation system, or "metre–kilogram–second–ampere (mksa)" equation system. This is the system used to define the SI units.[20] The new quantity q is given the name "rmks electric charge", or (nowadays) just "electric charge". Clearly, the quantity qs used in the old cgs esu system is related to the new quantity q by
${\displaystyle \ q_{\text{s}}={\frac {q}{\sqrt {4\pi \varepsilon _{0}}}}.}$
### Determination of a value for ε0
One now adds the requirement that one wants force to be measured in newtons, distance in metres, and charge to be measured in the engineers' practical unit, the coulomb, which is defined as the charge accumulated when a current of 1 ampere flows for one second. This shows that the parameter ε0 should be allocated the unit C2⋅N−1⋅m−2 (or equivalent units – in practice "farads per metre").
In order to establish the numerical value of ε0, one makes use of the fact that if one uses the rationalized forms of Coulomb's law and Ampère's force law (and other ideas) to develop Maxwell's equations, then the relationship stated above is found to exist between ε0, μ0 and c0. In principle, one has a choice of deciding whether to make the coulomb or the ampere the fundamental unit of electricity and magnetism. The decision was taken internationally to use the ampere. This means that the value of ε0 is determined by the values of c0 and μ0, as stated above. For a brief explanation of how the value of μ0 is decided, see the article about μ0.
## Permittivity of real media
By convention, the electric constant ε0 appears in the relationship that defines the electric displacement field D in terms of the electric field E and classical electrical polarization density P of the medium. In general, this relationship has the form:
${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} .}$
For a linear dielectric, P is assumed to be proportional to E, but a delayed response is permitted and a spatially non-local response, so one has:[21]
${\displaystyle \mathbf {D} (\mathbf {r} ,\ t)=\int _{-\infty }^{t}dt'\int d^{3}\mathbf {r} '\ \varepsilon (\mathbf {r} ,\ t;\mathbf {r} ',\ t')\mathbf {E} (\mathbf {r} ',\ t').}$
In the event that nonlocality and delay of response are not important, the result is:
${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} =\varepsilon _{\text{r}}\varepsilon _{0}\mathbf {E} }$
where ε is the permittivity and εr the relative static permittivity. In the vacuum of classical electromagnetism, the polarization P = 0, so εr = 1 and ε = ε0.
## Notes
1. ^ "CODATA Value: electric constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2015. Retrieved 2015-09-25. 2014 CODATA recommended values
2. ^ "Electropedia: International Electrotechnical Vocabulary (IEC 60050)". Geneva: International Electrotechnical Commission. Retrieved 2015-03-26. |contribution= ignored (help).
3. ^ The exact numerical value is found at: "Electric constant, ε0". NIST reference on constants, units, and uncertainty: Fundamental physical constants. NIST. Retrieved 2012-01-22. This formula determining the exact value of ε0 is found in Table 1, p. 637 of PJ Mohr; BN Taylor; DB Newell (April–June 2008). "Table 1: Some exact quantities relevant to the 2006 adjustment in CODATA recommended values of the fundamental physical constants: 2006" (PDF). Rev Mod Phys. 80 (2): 633–729. Bibcode:2008RvMP...80..633M. arXiv:. doi:10.1103/RevModPhys.80.633.
4. ^ Quote from NIST: "The symbol c is the conventional symbol for the speed of light in vacuum. " See NIST Special Publication 330, p. 18
5. ^ See the last sentence of the NIST definition of ampere.
6. ^ See the last sentence of the NIST definition of meter.
7. ^ a b c Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Reviews of Modern Physics. 80 (2): 633–730. Bibcode:2008RvMP...80..633M. arXiv:. doi:10.1103/RevModPhys.80.633. Direct link to value..
8. ^ A summary of the definitions of c, μ0 and ε0 is provided in the 2006 CODATA Report: CODATA report, pp. 6–7
9. ^ "On the possible future revision of the International System of Units, the SI" (PDF). Sèvres, France: International Bureau for Weights and Measures. 21 Oct 2011. |contribution= ignored (help) It is not expected to be adopted until some prerequisite conditions are met, and in any case not before 2014. See "Possible changes to the international system of units". IUPAC Wire. International Union of Pure and Applied Chemistry. 34 (1). January–February 2012.
10. ^ SM Sze & Ng KK (2007). "Appendix E". Physics of semiconductor devices (Third ed.). New York: Wiley-Interscience. p. 788. ISBN 0-471-14323-5.
11. ^ RS Muller, Kamins TI & Chan M (2003). Device electronics for integrated circuits (Third ed.). New York: Wiley. Inside front cover. ISBN 0-471-59398-2.
12. ^ FW Sears, Zemansky MW & Young HD (1985). College physics. Reading, Mass.: Addison-Wesley. p. 40. ISBN 0-201-07836-8.
13. ^ B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991)
14. ^ International Bureau of Weights and Measures (2006). "The International System of Units (SI)" (PDF). p. 12.
15. ^ a b Braslavsky, S.E. (2007). "Glossary of terms used in photochemistry (IUPAC recommendations 2006)" (PDF). Pure and Applied Chemistry. 79 (3): 293–465; see p. 348. doi:10.1351/pac200779030293.
16. ^
17. ^ King, Ronold W. P. (1963). Fundamental Electromagnetic Theory. New York: Dover. p. 139.
18. ^ IEEE Standards Board (1997). "IEEE Standard Definitions of Terms for Radio Wave Propagation" (PDF). p. 6.
19. ^ For an introduction to the subject of choices for independent units, see John David Jackson (1999). "Appendix on units and dimensions". Classical electrodynamics (Third ed.). New York: Wiley. pp. 775 et seq.. ISBN 0-471-30932-X.
20. ^
21. ^ Jenö Sólyom (2008). "Equation 16.1.50". Fundamentals of the physics of solids: Electronic properties. Springer. p. 17. ISBN 3-540-85315-4. | 3,491 | 13,029 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 14, "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} | 3.515625 | 4 | CC-MAIN-2017-39 | latest | en | 0.891751 |
http://alexplorer.net/teaching/phys-sci/word-probs.html | 1,516,201,468,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886946.21/warc/CC-MAIN-20180117142113-20180117162113-00042.warc.gz | 12,102,428 | 2,021 | Word problem review: Basic physics problems
Which units are each of the following measured in typically?:
Acceleration:
Force:
Work:
Power:
What units must the mass be measured in to calculate force?
What units must the acceleration be measured in to calculate force?
What units must the distance be measured in to calculate work?
What units must the time be measured in to calculate power?
Word problems:
What is the average velocity of a car that traveled 900 km in 23 hours of driving?
What variable are you solving for (i.e., what is the ‘unknown’ variable)?
What units will your answer be in (based on the units in the word problem)?
Which formula are you going to use?
Plug the numbers into the formula and solve for the unknown variable.
How long did it take a person skiing at 2.5 m/s to move 450 meters down a hill?
What variable are you solving for (i.e., what is the ‘unknown’ variable)?
What units will your answer be in (based on the units in the word problem)?
Which formula are you going to use?
Rearrange it for the unknown variable.
Plug the numbers into the formula and solve for the unknown variable.
What is the momentum of a 10 kg shotput thrown at 1.7 m/s?
What variable are you solving for (i.e., what is the ‘unknown’ variable)?
What units will your answer be in (based on the units in the word problem)?
Which formula are you going to use?
Plug the numbers into the formula and solve for the unknown variable.
What is the acceleration of a boy on a skateboard if the force of propulsion is 15 N if his mass is 58 kg?
What variable are you solving for (i.e., what is the ‘unknown’ variable)?
What units will your answer be in (based on the units in the word problem)?
Which formula are you going to use?
Rearrange it for the unknown variable.
Plug the numbers into the formula and solve for the unknown variable.
If a 150 kg motorcycle is accelerating at 4.2 m/s2 for 30 meters, how much work is done?
What variable are you solving for (i.e., what is the ‘unknown’ variable)?
What units will your answer be in (based on the units in the word problem)?
Which formula are you going to use?
Plug the numbers into the formula and solve for the unknown variable.
How long would it take for generator doing 65 megaJoules of work to produce 10,000 MW of power?
What variable are you solving for (i.e., what is the ‘unknown’ variable)?
What units will your answer be in (based on the units in the word problem)?
Which formula are you going to use?
Rearrange it for the unknown variable.
Plug the numbers into the formula and solve for the unknown variable. | 587 | 2,608 | {"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} | 3.796875 | 4 | CC-MAIN-2018-05 | latest | en | 0.95338 |
http://mathhelpforum.com/calculus/40076-kinematics-calculus-form.html | 1,480,825,332,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541187.54/warc/CC-MAIN-20161202170901-00295-ip-10-31-129-80.ec2.internal.warc.gz | 180,066,729 | 11,529 | 1. Kinematics (Calculus Form)
Question:
A particle travels along a straight line which passes through $A$ and $B$. During the motion the velocity of the particle is $6t+t^2$ metres per second, where $t$ seconds is the time measured from a certain instant. The particle passes through $A$ when $t=2$ and through $B$ when $t=5$. Find, in terms of $t$, the acceleration of the particle and its distance from $A$ at any instant during the motion. Calculate, also, the distance of $B$ from $A$.
$a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}(6t+t^2)}{\mathrm{d}t} = 6 + 2t$
$v = (6t+t^2) = \frac{\mathrm{d}x}{\mathrm{d}t}$
$\therefore x = 3t^2 + \frac13 t^3 + c$
$x_{\vec{AB}} = \left[3t^2 + \frac13 t^3\right]^5_2 = 102$
My Problem:
I don't know how to do this part (don't even understand what it is saying ): 'Find...its distance from $A$ at any instant during the motion'*. Can someone help? Thanks in advance.
*The answer is $x = 3t^2+ \frac13 t^3 - \frac{44}{3}$.
2. Originally Posted by Air
Question:
A particle travels along a straight line which passes through $A$ and $B$. During the motion the velocity of the particle is $6t+t^2$ metres per second, where $t$ seconds is the time measured from a certain instant. The particle passes through $A$ when $t=2$ and through $B$ when $t=5$. Find, in terms of $t$, the acceleration of the particle and its distance from $A$ at any instant during the motion. Calculate, also, the distance of $B$ from $A$.
$a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}(6t+t^2)}{\mathrm{d}t} = 6 + 2t$
$v = (6t+t^2) = \frac{\mathrm{d}x}{\mathrm{d}t}$
$\therefore x = 3t^2 + \frac13 t^3 + c$
$x_{\vec{AB}} = \left[3t^2 + \frac13 t^3\right]^5_2 = 102$
My Problem:
I don't know how to do this part (don't even understand what it is saying ): 'Find...its distance from $A$ at any instant during the motion'*. Can someone help? Thanks in advance.
*The answer is $x = 3t^2+ \frac13 t^3 - \frac{44}{3}$.
The particle is at A when t = 2. So
$x_A = 3(2)^2 + \frac13 (2)^3 + c = \frac{44}{3} + c$
Thus the particle's distance from A will be
$x - x_A = \left ( 3t^2 + \frac13 t^3 + c \right ) - \left ( \frac{44}{3} + c \right ) = 3t^2 + \frac{1}{3}t^3 - \frac{44}{3}$
-Dan
3. Originally Posted by topsquark
The particle is at A when t = 2. So
$x_A = 3(2)^2 + \frac13 (2)^3 + c = \frac{44}{3} + c$
Thus the particle's distance from A will be
$x - x_A = \left ( 3t^2 + \frac13 t^3 + c \right ) - \left ( \frac{44}{3} + c \right ) = 3t^2 + \frac{1}{3}t^3 - \frac{44}{3}$
-Dan
How did you know that it is $x - x_A$ and not $x_A - x$?
4. Originally Posted by Air
How did you know that it is $x - x_A$ and not $x_A - x$?
When $t > 2, \, x > x_A$ and so clearly the distance is $x - x_A$. HOWEVER .......
When $0 < t < 2, \, x < x_A$ and so the distance will be $x_A - x$ ...... It's apparent that the question actually wanted the distance from at any instant during the motion for t > 2. | 1,044 | 2,935 | {"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": 48, "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} | 4.15625 | 4 | CC-MAIN-2016-50 | longest | en | 0.882257 |
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• Please check that this question paper consists of 11 pages.
• Code number given on the right hand side of the question paper should be written on the title page of the answer book by the candidate.
• Please check that this question paper consists of 30 questions.
• Please write down the serial number of the question before attempting it.
• 15 minutes times has been allotted to read this question paper. The question paper will be distributed at 10:15 am. From 10:15 am to 10:30 am, the students will read the question paper only and will not write any answer on the answer book during this period.
SUMMATIVE ASSESSMENT – II
MATHEMATICS
Time allowed: 3 hours Maximum Marks: 80
General Instructions:
(i) All questions are compulsory
(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D
(iii) Section A consists of 6 questions of 1 mark each. Section B consists of 6 questions of 2 marks each. Section C consists of 10 questions of 3 marks each. Section D consists of 8 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1 mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.
(v) Use of calculator is not permitted.
SECTION – A
Question number 1 to 6 carry 1 mark each.
Question 1: If $x = 3$ is one root of the quadratic equation $x^2 - 2kx - 6 = 0$, then find the value of $k$.
If $x = 3$ is one root of the quadratic equation $x^2 - 2kx - 6 = 0$ than it should satisfy the equation.
$\Rightarrow (3)^2 - 2k(3) - 6 = 0$
$9 - 6k - 6 = 0$
$k =$ $\frac{1}{2}$
$\\$
Question 2: What is the HCF of smallest prime number and the smallest composite number ?
Smallest composite number is $= 4 = 2 \times 2$
Smallest prime number is $= 2 = 2 \times 1$
Therefore HCF of $4$ and $2$ is $2$
$\\$
Question 3: Find the distance of a point $P(x, y)$ from the origin.
Distance of a point $P(x, y)$ from the origin $O(0, 0) = \sqrt{(x-o)^2 + (y-0)^2} = \sqrt{x^2 + y^2}$
$\\$
Question 4: In an AP, if the common difference $(d) = -4$, and the seventh term $(a_7)$ is $4$, then find the first term.
Given $d = -4, a_7 = 4$
$a_n = a + (n-1) d$
$\Rightarrow a_7 = a + (7-1) d$
$\Rightarrow 4 = a + (7-1)(-4)$
$\Rightarrow 4 = a - 24$
$\Rightarrow a = 28$
$\\$
Question 5: What is the value of $(\cos^2 67^o - \sin^2 23^o)$ ?
$(\cos^2 67^o - \sin^2 23^o)$
$= (\cos^2 67^o - \sin^2 (90^o-67^o) )$
$= (\cos^2 67^o - \cos^2 67^o)$
$= 0$
$\\$
Question 6: Given $\triangle ABC \sim \triangle PQR$, if $\frac{AB}{PQ}$ $=$ $\frac{1}{3}$, then find $\frac{ar \triangle ABC}{ar \triangle PQR}$
Given $\triangle ABC \sim \triangle PQR$
$\Rightarrow$ $\frac{ar(\triangle ABC)}{ar(\triangle PQR)}$ $= \Big($ $\frac{AB}{PQ}$ $\Big)^2= \Big($ $\frac{1}{3}$ $\Big)^2 =$ $\frac{1}{9}$
$\\$
Section – B
Question number 7 to 12 carry 2 mark each.
Question 7: Given that $\sqrt{2}$ is irrational, prove that $(5 + 3 \sqrt{2} )$ is an irrational number.
Given that, $\sqrt{2}$ is irrational
Let us assume $5 + 3\sqrt{2}$ is rational.
As $5 + 3\sqrt{2}$ is rational. (Assumed) They must be in the form of $\frac{p}{q}$ where $q \neq 0$, and $p$ and $q$ are co prime.
$\therefore 5 + \sqrt{2} =$ $\frac{p}{q}$
$\Rightarrow 3\sqrt{2} =$ $\frac{p-5q}{q}$
$\Rightarrow \sqrt{2} =$ $\frac{p-5q}{3q}$
We know $\sqrt{2}$ is irrational but $\frac{p-5q}{3q}$ is a rational number
Therefore we contradict the statement that, $5+3\sqrt{2}$ is rational.
Hence proved that $5 + 3\sqrt{2}$ is irrational.
$\\$
Question 8: In Fig. 1, $ABCD$ is a rectangle. Find the values of $x$ and $y$.
Given:
$x+y = 30$ … … … … … i)
$x-y = 14$ … … … … … ii)
Adding i) and ii) we get
$2x = 44 \Rightarrow x = 22$
$\therefore y = 30 - 22 = 8$
$\\$
Question 9: Find the sum of first $8$ multiples of $3$.
The first $8$ multiples of $3$ are : $3, 6, 9, 12, 15, 18, 21, 24$
Number of terms $(n) = 8$
The first term $(a) = 3$
Common difference $(d) = 3$
$S_n =$ $\frac{n}{2}$ $[2a + (n-1)d ]$
$S_8 =$ $\frac{8}{2}$ $[2 \times 3 + (8-1)\times 3]$
$S_8 = 108$
Hence the sum of the first $8$ multiples of $3$ is $108$.
$\\$
Question 10: Find the ratio in which $P(4, m)$ divides the line segment joining the points $A(2, 3)$ and $B(6, -3)$. Hence find $m$.
Let $P(4, m)$ divides $A (2, 3)$ and $B (6, -3)$ in the ratio of $k:1$
By section formula,
$4 =$ $\frac{k \times 6 + 1 \times 2}{k+1}$
$\Rightarrow 4k + 4 = 6k + 2$
$\Rightarrow k = 1$
Also $m =$ $\frac{1 \times (-3) + 1 \times 3}{1+1}$
$\Rightarrow m = 0$
$\\$
Question 11: Two different dice are tossed together. Find the probability :
(i) of getting a doublet
(ii) of getting a sum $10$, of the numbers on the two dice.
Given that two different dice are tossed.
Total number of outcomes $n(S) = 6^2 = 36$
(i) Let $A$ be the event of getting a doublet.
$n(A) = \{ 1,1 \},{2,2}, \{3,3 \}, \{4,4 \}, \{5,5 \}, \{6,6 \} = 6$
Required probability $P(A) =$ $\frac{n(A)}{n(S)}$ $=$ $\frac{6}{36}$ $=$ $\frac{1}{6}$
(ii) Let $B$ be the event of getting a sum of $10$ of the numbers of two dice.
$n(B) = \{6,4 \}, \{4,6 \}, \{5,5 \} = 3$
Required probability $P(B) =$ $\frac{n(B)}{n(S)}$ $=$ $\frac{3}{36}$ $=$ $\frac{1}{12}$
$\\$
Question 12: An integer is chosen at random between $1$ and $100$. Find the probability that it is :
(i) divisible by $8$.
(ii) not divisible by $8$.
Number of integers between $1$ and $100: 2, 3 , 4, 5 , \ldots , 99$
Total number of outcomes $n(S) = 98$
(i) Numbers which are divisible by $8$ are: $8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96$
Favorable Outcomes $n(A) = 12$
Probability that integer is divisible by $8, P(E) =$ $\frac{n(A)}{n(S)}$ $=$ $\frac{12}{98}$ $=$ $\frac{6}{49}$
(ii) Probability that integer is not divisible by $8$
$P(E') = 1 - P(E)$
$= 1 -$ $\frac{6}{49}$
$=$ $\frac{43}{49}$
$\\$
Section – C
Question number 13 to 22 carry 3 mark each.
Question 13: Find HCF and LCM of $404$ and $96$ and verify that HCF $\times$ LCM $=$ Product of the two given numbers.
$96 = 2^5 \times 3$
$404 = 2^2 \times 101$
Therefore HCF of $96$ and $404 = 2^2 = 4$
Also LCM of $96$ and $404 = 2^5 \times 3 \times 101 = 9696$
Product of two numbers $= 96 \times 404 = 38784$
HCF $\times$ LCM $= 4 \times 9696 = 38784$
Hence Product of two numbers $=$ HCF $\times$ LCM
$\\$
Question 14: Find all zeroes of the polynomial $(2x^4 - 9x^3 + 5x^2 + 3x - 1)$ if two of its zeroes are $(2 + \sqrt{3})$ and $(2 - \sqrt{3})$.
Given $(2+\sqrt{3})$ and $(2-\sqrt{3})$ are zeros of the polynomial $p(x) = 2x^4 - 9x^3+5x^2+3x-1$
Therefore $(x - 2-\sqrt{3})(x - 2+\sqrt{3}) = x^2 -4x+1$ is a factor of the polynomial $p(x)$ Hence
$x^2 -4x+1 ) \overline{2x^4 - 9x^3+5x^2+3x-1} ( 2x^2-x-1 \\ \hspace*{1.6cm} (-) \underline{2x^4-8x^3+2x^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \hspace*{2cm}-x^3+3x^2+3x-1 \\ \hspace*{1.5cm} (-) \underline{-x^3+4x^2-x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \hspace*{3.5cm}-x^2+4x-1 \\ \hspace*{3cm} (-) \underline{-x^2+4x-1 \ \ \ \ \ } \\ \hspace*{5cm}\times$
Therefore $2x^2-x-1$ is a factor of $p(x)$
$2x^2-x-1 = 2x^2 - 2x + x - 1$
$= 2x(x-1) + (x-1)$
$= (x-1)(2x+1)$
$\Rightarrow (x-1)$ and $(2x+1)$ are factors of $p(x)$
Hence $x = 1$ and $x = -$ $\frac{1}{2}$ are the other two zeros of the given polynomial $2x^4 - 9x^3+5x^2+3x-1$
$\\$
Question 15: If $A(-2, 1), B(a, 0), C(4, b)$ and $D(1, 2)$ are the vertices of a parallelogram $ABCD$, find the values of a and b. Hence find the lengths of its sides.
Or
If $A(-5, 7), B(-4, -5), C(-1, -6)$ and $D(4, 5)$ are the vertices of a quadrilateral, find the area of the quadrilateral $ABCD$.
Mid point of $AB = \Big($ $\frac{-2+4}{2}$ $,$ $\frac{1+b}{2}$ $\Big) = \Big( 1,$ $\frac{1+b}{2}$ $\Big)$
Mid point of $BD = \Big($ $\frac{a+1}{2}$ $,$ $\frac{0+2}{2}$ $\Big) = \Big($ $\frac{a+1}{2}$ $, 1 \Big)$
Since the mid point of $AB$ and $BD$ are the same
$\frac{a+1}{2}$ $=1 \Rightarrow a + 1 = 2 \Rightarrow a = 1$
Similarly, $\frac{1+b}{2}$ $=1 \Rightarrow 1 + b = 2 \Rightarrow b = 1$
$\therefore a =1 , b = 1$
$\therefore AB = \sqrt{(1-(-2))^2+(0-1)^2} = \sqrt{9+1} = \sqrt{10}$
$BC = \sqrt{(4-1)^2+(1-0)^2} = \sqrt{9+1} = \sqrt{10}$
Therefore $DC = \sqrt{10}$ and $DA = \sqrt{10}$
Hence $AB = BC = CD = DA \Rightarrow ABCD$ is a Rhombus
Or
Join $AC$
$\therefore Ar (Quadrilateral ABCD) = Ar (\triangle ABC) + Ar (\triangle ACD)$
Note: For a $\triangle$ with vertices $(x_1, y_1), (x_2, y_2) \ \ \& \ \ (x_3, y_3)$ the area of the triangle is $\Big|$ $\frac{1}{2}$ $[ x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) ] \Big|$
$Ar(\triangle ABC) = \Big|$ $\frac{1}{2}$ $[ -5(-5-(-6)) + (-4)(-6-7) + (-1)(7-9-5)) ] \Big|$
$= \Big|$ $\frac{1}{2}$ $[ -5(1) - 4(-13) + (-1)(12) ] \Big|$
$= \Big|$ $\frac{1}{2}$ $( -5+ 52-12 ) \Big|$
$= \Big|$ $\frac{1}{2}$ $\times 35 \Big|$
$= 17.5$ sq. units
$Ar(\triangle ACD) = \Big|$ $\frac{1}{2}$ $[ -5(5-(-6)) + 4(-6-7) + (-1)(7-5)) ] \Big|$
$= \Big|$ $\frac{1}{2}$ $[ -5(11) + 4(-13) + (-1)(2) ] \Big|$
$= \Big|$ $\frac{1}{2}$ $( -55- 52-2 ) \Big|$
$= \Big|$ $\frac{1}{2}$ $\times (-109) \Big|$
$= 54.5$ sq. units
$\therefore Ar (Quadrilateral ABCD) = 17.5 + 54.5 = 77$ sq. units
$\\$
Question 16: A plane left $30$ minutes late than its scheduled time and in order to reach the destination $1500$ km away in time, it had to increase its speed by $100$ km/h from the usual speed. Find its usual speed.
Distance to travel $= 1500$ km
Let the usual speed $= x$ km/hr
Increased speed $= (x+100)$ km/hr
$\therefore$ $\frac{1500}{x+100}$ $+$ $\frac{1}{2}$ $=$ $\frac{1500}{x}$
$\frac{3000 + x + 100}{2(x+100)}$ $=$ $\frac{1500}{x}$
$3000x + x^2 + 100x = 3000x + 300000$
$x^2 + 100x - 300000 = 0$
$x^2 - 500x + 600x - 300000 = 0$
$x( x- 500) + 600(x-500) = 0$
$(x-500)(x+600) = 0$
$x = 500$ km/hr or $x = -600$ km/hr (this is not possible as speed cannot be negative)
Therefore the usual speed $= 500$ km/hr
$\\$
Question 17: Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.
Or
If the area of two similar triangles are equal, prove that they are congruent.
Let the side of the square $= a$
Therefore the sides of the equilateral $\triangle ABE = a$ as well
Now $BD = \sqrt{a^2 + a^2} = \sqrt{2} a$
Therefore the sides of the equilateral $\triangle BDF = \sqrt{2} a$
Since the triangles are equilateral, each of the angles are $60^o$. Hence by $AAA$ criterion, the two triangles are similar.
Therefore
$\frac{Ar (\triangle ABE}{Ar (\triangle BDF)}$ $= \Big($ $\frac{AB}{BD}$ $\Big)^2 = \Big($ $\frac{a}{\sqrt{2}a}$ $\Big)^2 =$ $\frac{1}{2}$
Hence proven.
Or
Given: $\triangle ABC$ and $\triangle DEF$
Also $\triangle ABC \sim \triangle DEF$ and $Ar(\triangle ABC) = Ar(\triangle DEF)$
To Prove: $\triangle ABC \cong \triangle DEF$
Since $\triangle ABC \sim \triangle DEF$ we know
$\frac{Ar (\triangle ABC )}{Ar (\triangle DEF)}$ $= \Big($ $\frac{BC}{EF}$ $\Big)^2 = \Big($ $\frac{AB}{DE}$ $\Big)^2 =\Big($ $\frac{AC}{DF}$ $\Big)^2$
$1 = \Big($ $\frac{BC}{EF}$ $\Big)^2 = \Big($ $\frac{AB}{DE}$ $\Big)^2 =\Big($ $\frac{AC}{DF}$ $\Big)^2$
Therefore we get
$1 = \Big($ $\frac{BC}{EF}$ $\Big)^2 \Rightarrow$ $\frac{BC}{EF}$ $= 1 \Rightarrow BC = EF$
$1 = \Big($ $\frac{AB}{DE}$ $\Big)^2 \Rightarrow$ $\frac{AB}{DE}$ $= 1 \Rightarrow AB = DE$
$1 = \Big($ $\frac{AC}{DF}$ $\Big)^2 \Rightarrow$ $\frac{AC}{DF}$ $= 1 \Rightarrow AC = DF$
Therefore $\triangle ABC \cong \triangle DEF$ by SSS criterion.
Hence proved.
$\\$
Question 18: Prove that the lengths of tangents drawn from an external point to a circle are equal.
Given: Let the circle with center $O$
Let $P$ be an external point from which tangents $PQ$ and $PR$ are drawn as shown in the diagram
To prove: $PQ = PR$
Construction: Join $OQ, OR$ and $OP$
Proof: As $PQ$ is tangent $OQ \perp PQ$. Therefore $\angle OQP = 90^o$
Similarly, As $PR$ is tangent $OR \perp PR$. Therefore $\angle ORP = 90^o$
(Note: Tangents at any point on a circle is perpendicular to the radius through the point of contact)
In $\triangle OQP$ and $\triangle ORP$
$\angle OQP = \angle ORP = 90^o$
$OP$ is common
$OQ = OR$ (radius of the same circle)
$\therefore \triangle OQP \cong \triangle ORP$
$\Rightarrow PQ = PR$
Therefore both tangents are equal in length.
$\\$
Question 19: If $4 \tan \theta = 3$, evaluate $\frac{4 \sin \theta - \cos \theta + 1}{4 \sin \theta + \cos \theta - 1}$
Or
If $\tan 2A = \cot (A - 18^o)$, where $2A$ is an acute angle, find the value of $A$.
Given $4 \tan \theta = 3 \Rightarrow \tan \theta =$ $\frac{3}{4}$ $\Rightarrow \sin \theta =$ $\frac{3}{5}$ $\Rightarrow \cos \theta =$ $\frac{4}{5}$
Therefore
$\frac{4 \sin \theta - \cos \theta + 1}{4 \sin \theta + \cos \theta - 1}$ $=$ $\frac{4 \times \frac{3}{5} - \frac{4}{5} + 1}{4 \times \frac{3}{5} + \frac{4}{5} - 1}$ $=$ $\frac{12 - 4 + 5}{12 + 4 -5}$ $=$ $\frac{13}{11}$
Or
$\tan 2A = \cot (A - 18^o)$
$\Rightarrow \tan 2A = \tan [ 90- (A - 18^o) ]$
$\Rightarrow \tan 2A = \tan ( 108^o - A)$
$\Rightarrow 2A = 108^o - A$
$\Rightarrow A = 36^o$
$\\$
Question 20: Find the area of the shaded region in Fig. 2, where arcs drawn with centres $A, B, C$and $D$ intersect in pairs at mid-points $P, Q, R$ and $S$ of the sides $AB, BC, CD$ and $DA$ respectively of a square $ABCD$ of side $12$ cm. [Use $\pi = 3.14$]
Given square $ABCD$ of side $12$ cm
Area of square $= 12 \times 12 = 144 \ cm^2$
Area of $APS = \frac{1}{4} (\pi \times 6 \times 6) = 9\pi \ cm^2$
Therefore total area enclosed in arc’s $= 4 \times 9\pi = 36 \pi = 36 \times 3.14 = 113.04 \ cm^2$
Therefore shaded area $= 144 - 113.04 = 30.96 \ cm^2$
$\\$
Question 21: A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 3. If the height of the cylinder is $10$ cm and its base is of radius $3.5$ cm. Find the total surface area of the article.
Or
A heap of rice is in the form of a cone of base diameter $24$ m and height $3.5$ m. Find the volume of the rice. How much canvas cloth is required to just cover the heap ?
Radius of the cylinder $(r) = 3.5$ cm Height $(h) = 10$ cm
Curved surface area of cylinder $= 2 \pi r h = 2 \pi \times 3.5 \times 10 = 70 \pi \ cm^2$
Surface area of two hemispheres $= 2 [ \frac{1}{2} \times 4 \pi r^2 ]$ $= 4\pi \times 3.5 \times 3.5 = 49 \pi \ cm^2$
Therefore total surface area $= 70 \pi + 49 \pi = 119 \times$ $\frac{22}{7}$ $= 374 \ cm^2$
Or
Diameter $= 24$ m $\Rightarrow$ Radius $(r) = 12$ m Height $(h) = 3.5$ m
Volume of Rice $=$ $\frac{1}{3}$ $\pi r^2 h =$ $\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times 12 \times 12 \times 3.5 = 22 \times 2 \times 12 = 528 \ m^3$
Slant height $(l) = \sqrt{12^2 + 3.5^2} = \sqrt{156.25} = 12.5$ m
Curved surface area of the heap $= \pi r l =$ $\frac{22}{7}$ $\times 12 \times 12.5 = 471.43 \ m^2$
Hence Volume of Rice $= 528 \ m^3$ and Canvas required to cover the heap $= 471.43 \ m^2$
$\\$
Question 22: The table below shows the salaries of 280 persons:
Salary (in thousand Rs.) No. of persons 5-10 49 10-15 133 15-20 63 20-25 15 25-30 6 30-35 7 35-40 4 40-45 2 45-50 1
Class Interval Frequency $(f)$ Cumulative Frequency $(cf)$ 5 – 10 49 49 10 – 15 133 182 15 – 20 63 245 20 – 25 15 260 25 – 30 6 266 30 – 35 7 273 35 – 40 4 277 40 – 45 2 279 45 – 50 1 280
$cf = 133, f = 133, L = 10, N = 280 \Rightarrow$ $\frac{N}{2}$ $= 140, C = 5$
Median $(M) = L +$ $\frac{\frac{N}{2} - cf}{f}$ $\times C = 10 + \Big($ $\frac{140 - 49}{133}$ $\Big) \times 5 = 10 + 3.42 = 13.42$
Therefore median salary $= 13.42 \times 1000 =13420$ Rs.
$\\$
Section – D
Question number 23 to 30 carry 4 mark each.
Question 23: A motor boat whose speed is $18$ km/hr in still water takes $1$ hr more to go $24$ km upstream than to return downstream to the same spot. Find the speed of the stream.
Or
A train travels at a certain average speed for a distance of $63$ km and then travels at a distance of $72$ km at an average speed of $6$ km/hr more than its original speed. If it takes $3$ hours to complete total journey, what is the original average speed ?
Distance traveled by the boat $= 24$ km
Speed of the boat $= 18$ km/hr
Let the speed of the stream $= x$ km/hr
Therefore speed of boat upstream $= (18 - x)$ km/hr
Also speed of boat downstream $= (18 + x)$ km/hr
Hence: $\frac{24}{18-x}$ $-$ $\frac{24}{18+x}$ $= 1$
$\Rightarrow 432 + 24 x - (432 - 24x) = 324 - x^2$
$\Rightarrow 48 x = 324 - x^2$
$\Rightarrow x^2 + 48x - 324 = 0$
$\Rightarrow x^2 + 54x - 6x - 324 = 0$
$\Rightarrow x(x+54)-6(x+54) = 0$
$\Rightarrow (x+54)(x-6) = 0$
$\Rightarrow x = 6$ km/hr or $x = -54$ km/hr (not possible as speed cannot be negative)
Therefore the speed of the stream is $6$ km/hr
Or
Let the original speed is $x$ km/hr
Time taken to travel $63$ km $=$ $\frac{63}{x}$ hrs
New speed $= (x+6)$ km/hr
Time taken to travel $72$ km $=$ $\frac{72}{x+6}$
$\therefore$ $\frac{63}{x}$ $+$ $\frac{72}{x+6}$ $= 3$
$\Rightarrow 63x + 378 + 72x = 3x^2 + 18x$
$\Rightarrow 3x^2 - 117 x - 378 = 0$
$\Rightarrow x^2 - 39x - 126 = 0$
$\Rightarrow x^2 - 42 x + 3x - 126 =0$
$\Rightarrow x(x-42)+3(x-42) = 0$
$\Rightarrow (x-42)(x+3) = 0$
$\Rightarrow x = 42$ km/hr or $x = -3$ km/hr (not possible as speed cannot be negative)
Hence original speed $= 42$ km/hr
$\\$
Question 24: The sum of four consecutive numbers in an AP is $32$ and the ratio of the product of the first and the last term to the product of two middle terms is $7 : 15$. Find the numbers.
Let the four consecutive terms of the AP be $a, a+d, a+2d$ and $a+3d$
Given: $a + ( a+d) + ( a+2d) + ( a+3d) = 32$
$\Rightarrow 4a + 6d = 32$
$\Rightarrow 2a + 3d = 16$
$\Rightarrow d =$ $\frac{16-2a}{3}$ … … … … … i)
Also $\frac{a(a+3d)}{(a+d)(a+2d)}$ $=$ $\frac{7}{15}$
$\Rightarrow 15a^2 + 45ad = 7a^2 + 21 ad + 14d^2$
$\Rightarrow 8a^2 + 24ad - 14 d^2 = 0$
$\Rightarrow 4a^2 + 12 ad - 7d^2 = 0$
$\Rightarrow 4a^2 + 14ad - 2ad - 7d^2 = 0$
$\Rightarrow 2a(2a-d) + 7d(2a-d) = 0$
$(2a-d)(2a+7d) = 0$
Substituting from i)
$\Big(2a-$ $\frac{16-2a}{3}$ $\Big)\Big(2a+7\times$ $\frac{16-2a}{3}$ $\Big) = 0$
$\Rightarrow (6a - 16 + 2a) (6a + 112 - 14a) = 0$
$\Rightarrow (8a-16)(-8a+112) = 0$
$\Rightarrow a = 2$ or $a = 14$
When $a = 2, d =$ $\frac{16-2 \times 2}{3}$ $=$ $\frac{12}{3}$ $= 4$
The the first four terms are $2, 6, 10, 14$
When $a = 14, d =$ $\frac{16-2 \times 14}{3}$ $=$ $\frac{-12}{3}$ $= -4$
The the first four terms are $14, 10, 6, 2$
$\\$
Question 25: In an equilateral $\triangle ABC$, $D$ is a point on side $BC$ such that $BD =$ $\frac{1}{3}$ $BC$. Prove that: $9(AD)^2 = 7(AB)^2$
Or
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Given: $\triangle ABC$ is an equilateral triangle.
$\Rightarrow AB = BC = CA$
Also $BD =$ $\frac{1}{3}$ $BC$
To prove: $9 AD^2 = 7 AB^2$
Construction: Draw $AE \perp BC$
Consider $\triangle AEB$ and $\triangle AEC$
$AE$ is common
$\angle AEB = \angle AEC = 90^o$
$AB = AC$ (equilateral triangle)
$\triangle AEB \cong \triangle AEC$ (By RHS criterion)
$\therefore BE = EC$
$\Rightarrow BE =$ $\frac{BC}{2}$
$\Rightarrow BD + DE =$ $\frac{BC}{2}$
$\Rightarrow$ $\frac{BC}{3} + DE =$ $\frac{BC}{2}$
$\Rightarrow DE =$ $\frac{BD}{6}$
Using Pythagoras theorem,
In $\triangle AEB$
$AB^2 = AE^2 + BE^2 \Rightarrow AE^2 = AB^2 - BE^2$ … … … … … i)
In $\triangle AED$
$AD^2 = AE^2 + DE^2 \Rightarrow AE^2 = AD^2 - DE^2$ … … … … … ii)
From i) and ii)
$AB^2 - BE^2 = AD^2 - DE^2$
$\Rightarrow AB^2 - \Big($ $\frac{BC}{2}$ $\Big)^2 = AD^2 - \Big($ $\frac{BC}{6}$ $\Big)^2$
$\Rightarrow AB^2 -$ $\frac{BC^2}{4}$ $= AD^2 -$ $\frac{BC^2}{36}$
$\Rightarrow AB^2 = AD^2 +$ $\frac{8}{36}$ $BC^2$
$AB^2 = AD^2 +$ $\frac{2}{9}$ $BC^2$
But $BC = AB$
$\therefore AB^2 -$ $\frac{2}{9}$ $AB^2 = AD^2$
$\Rightarrow 7AB^2 = 9AD^2$
Hence proved.
Or
Given: A right angled $\triangle ABC$, right angled at $B$
To prove: $AC^2 = AB^2 + BC^2$
Draw: $BD \perp AC$
Proof: In $\triangle ADB \& \angle ABC$
$\angle ADB = \angle ABC = 90^o$
$\angle BAD = \angle CAB$ (common angle)
$\therefore \triangle ADB \sim \triangle ABC$ ( by AA criterion)
Therefore $\frac{AD}{AB}$ $=$ $\frac{AB}{AC}$ (corresponding sides are proportional)
$AB^2 = AD \times AC$ … … … … … i)
Similarly, $\triangle BDC \sim \triangle ABC$
and $BC^2 = CD \times AC$ … … … … … ii)
$AB^2 + BC^2 = AD \times AC + CD \times AC$
$\Rightarrow AB^2 + BC^2 = AC ( AD + CD)$
$\Rightarrow AB^2 + BC^2 = AC^2$. Hence proved.
$\\$
Question 26: Draw a triangle $ABC$ with $BC = 6$ cm, $AB = 5$ cm and $\angle ABC = 60^o$. Then construct a triangle whose sides are $\frac{3}{4}$ of the corresponding sides of the $\triangle ABC$.
$\\$
Question 27: Prove that: $\frac{\sin A - 2 \sin^3 A }{2 \cos^3 A - \cos A}$ $= \tan A$
LHS $= \frac{\sin A - 2 \sin^3 A}{2 \cos^3 A - \cos A}$
$= \frac{\sin A ( 1 - 2 \sin^2 A)}{\cos A (2 \cos^2 A - 1)}$
$= \frac{\sin A ( 1 - \sin^2 A - \sin^2 A)}{\cos A ( \cos^2 A + \cos^2 A - 1)}$
$= \frac{\sin A ( \cos^2 A - \sin^2 A)}{\cos A ( \cos^2 A - \sin^2 A)}$
$= \frac{\sin A }{\cos A }$
$= \tan A$
$=$ RHS. Hence Proved.
$\\$
Question 28: The diameters of the lower and upper ends of a bucket in the form of a frustum of a cone are $10$ cm and $30$ cm respectively. If its height is $24$ cm, find :
(i) The area of the metal sheet used to make the bucket.
(ii) Why we should avoid the bucket made by ordinary plastic ? [Use $\pi = 3.14$]
Diameter of upper end of bucket $=30$ cm
Radius of the upper end of the frustum of cone $( r_2) = 15$ cm
Diameter of lower end of bucket $= 10$ cm
radius of the lower end of the frustum of cone $( r_1) = 5$ cm
Height of the frustum of Cone $(h) = 24$ cm
Volume of bucket $=$ $\frac{\pi}{3}$ $h ({r_1}^2 + {r_2}^2 + r_1r_2 )$
$=$ $\frac{1}{3}$ $\times$ $3.14$ $\times 24 \times ( 5^2 + 15^2 + 5 \times 15)$
$= 8164 \ cm^3 = 8.164$ liters
$l = \sqrt{h^2 + (r_2 - r_1)^2} = \sqrt{24^2 + (15-5)^2} = \sqrt{24^2 + 10^2} = 26$ cm
Therefore surface area $= \pi {r_1}^2 + \pi (r_1 + r_2)l$
$= 3.14$ $( 5^2 + (5+15)\times 26 )$
$= 1711.3 \ cm^2$
Hence, the Area of metal sheet used to make the bucket is $= 1711.3 \ cm^2$
$\\$
Question 29: As observed from the top of a $100$ m high light house from the sea-level, the angles of depression of two ships are $30^o$ and $45^o$. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships. [Use $\sqrt{3} = 1.732$]
In $\triangle ABC:$
$\frac{100}{BC}$ $=\tan 45^o = 1 \Rightarrow BC = 100$ m
In $\triangle ABD:$
$\frac{100}{BD}$ $=\tan 30^o = 1 \Rightarrow BC = 100 \sqrt{3}$ m
$\therefore DC = 100 \sqrt{3} -100 = 173.2 - 100 = 73.2$ m
Therefore the distance between the ships is $73.2$ m.
$\\$
Question 30: The mean of the following distribution is $18$. Find the frequency $f$ of the class $19 - 21$.
Class: 11-13 13-15 15-17 17-19 19-21 21-23 23-25 Frequency: 3 6 9 13 $f$ 5 4
Or
The following distribution gives the daily income of 50 workers of a factory:
Daily Income (in Rs) 100-120 120-140 140-160 160-180 180-200 Number of workers 12 14 8 6 10
Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive.
Class Interval Frequency $(f_i)$ $x_i$ $f_ix_i$ 11-13 3 12 36 13-15 6 14 84 15-17 9 16 144 17-19 13 18 234 19-21 $f$ 20 $20f$ 21-23 5 22 110 23-25 4 24 96 $\Sigma f_i = 40 + f$ $\Sigma f_ix_i = 704 + 20f$
Mean $= \frac{\Sigma f_ix_i }{40 + f }$ $=$ $\frac{704 + 20f}{40 + f}$
$\Rightarrow 720 + 18f = 704 + 20f$
$\Rightarrow 2f = 16$
$\Rightarrow f = 8$
Or
Daily wages in Rs (less than) Frequency $(f_i)$ Cumulative Frequency Less than 120 12 12 Less than 140 14 26 Less than 160 8 34 Less than 180 6 40 Less than 200 10 50
$\\$ | 9,330 | 24,089 | {"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": 675, "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} | 4.09375 | 4 | CC-MAIN-2020-05 | longest | en | 0.877253 |
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https://www.jiskha.com/display.cgi?id=1176498274 | 1,516,097,694,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886397.2/warc/CC-MAIN-20180116090056-20180116110056-00562.warc.gz | 932,475,340 | 3,873 | # math
posted by .
am I right here the problem is 72divided by8-9 divided by 3
I came up with 72 divided by 8 =9 and 9 divided by 3 =3
so it would be 9-3=6
72/8 - 9/3
yes, you are right.
thanks
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I am writing notes for a reading class and I've decided to add a proof of Cochran's theorem in order to show that a statistic is $\chi^2$. I am struggling for the proof of a particular lemma but the rest is just peachy.
Lemma: Let $A$ be a real real symmetric idempotent matrix of order $n$ with rank $r$. Suppose $A=A_1+\cdots+A_k$ with rank $A_i = r_i$ and $A_i$ symmetric. Additionally, $r_1+\cdots+r_k=r$. Then each $A_i$ is idempotent.
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-
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I am not certain. In the context of statistics the $A_i$'s are quadratic forms of random variables. I wonder if, at any rate, they are necessarily symmetric. My linear algebra is a bit rusty. – user18263 Oct 26 '11 at 7:05 | 472 | 1,663 | {"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} | 3.53125 | 4 | CC-MAIN-2016-30 | latest | en | 0.906224 |
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# Inlet boundary condition for acoustic wave
## Inlet boundary condition for acoustic wave
(OP)
Dear all,
I want to simulate the propagation of an acoustic wave with a spherical wave source outside my computational domain using Finite Element Method. The governing equation is a linearized acoustic equation with velocity potential as the primary variable ( the secondary variable is thus velocity). In order to let the wave propagate into the domain, I might need to add an impedance condition on the boundary by setting the predicted fluid particle velocity on the boundary nodes (natural boundary condition). However, the discretization of FEM only allows the normal component of the velocity be prescribed on the boundary (divergence theorem).
My question is:
Is it physical to enforce only normal component of fluid particle velocity on the boundary?
Would it decrease the wave magnitude or distort the wave?
If so, is there any way to circumvent that?
Thank you all very much for your attention.
### RE: Inlet boundary condition for acoustic wave
Both Longitudinal and Transverse waves exists, but the for liquids and gases, the transverse wave impart little force to the solid boundary, and resulting in reflection of the wave. The longitudinal waves exert pressure and momentum forces on the surface as your FEA package descibes.
### RE: Inlet boundary condition for acoustic wave
(OP)
Thanks for your reply, hacksaw. I might not make my question clear. The boundary mentioned in the question is actually the outer boundary of the fluid. By adding an impedance boundary condition on that boundary, the wave could pass from the infinite acoustic domain into the computational domain. I already assume that it is a longitudinal wave in that the fluid particle displacement is in the direction of wave propagation.
### RE: Inlet boundary condition for acoustic wave
That's how I see it, transverse "acoustic", longitudinal "pressure" waves. The transverse is only significant if the material is characterized by an elastic behavior in the transverse.
### RE: Inlet boundary condition for acoustic wave
In ANSYS, you could define a monopole outside of the computation domain. Here's an example. The help document says it applies incident pressure to the exterior boundary (link).
Best regards,
Sze Kwan (Jason) Cheah
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Register now while it's still free! | 880 | 4,334 | {"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} | 2.734375 | 3 | CC-MAIN-2020-10 | latest | en | 0.869207 |
https://socratic.org/questions/how-do-you-find-the-lengths-of-the-arc-on-a-circle-of-radius-3-meters-intercepte | 1,717,042,990,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059418.31/warc/CC-MAIN-20240530021529-20240530051529-00079.warc.gz | 464,085,476 | 5,962 | # How do you find the lengths of the arc on a circle of radius 3 meters intercepted by the central angle 1 radian?
May 19, 2017
$3$ meters
#### Explanation:
The angle is in radians, so the length of the arc can be expressed by the following equation $s = r \theta$
$r$ is the radius which is $3$ meters
$\theta$ is the angle (in radians) which is $1$
And $s$ is he arc length
$s = r \theta$
$\textcolor{w h i t e}{s} = 3 \left(1\right)$
$\textcolor{w h i t e}{s} = 3$ meters | 153 | 484 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "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} | 4.34375 | 4 | CC-MAIN-2024-22 | latest | en | 0.892751 |
http://physicshelpforum.com/light-optics/2519-following-shows-orientation-original-object-plane-mirror.html | 1,516,318,692,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887660.30/warc/CC-MAIN-20180118230513-20180119010513-00389.warc.gz | 268,012,739 | 9,136 | Physics Help Forum Which of the following shows the orientation of the original object --> plane mirror
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Light and Optics Light and Optics Physics Help Forum
Jun 17th 2009, 10:52 PM #1 Senior Member Join Date: Mar 2009 Posts: 129 Which of the following shows the orientation of the original object --> plane mirror Question #1 Please tell me WHY the answer is D because my tutor couldn't answer me. Any help would be GREATLY appreciated! Thanks in advance! Attached Thumbnails
Jun 17th 2009, 11:07 PM #2 Physics Team Join Date: Feb 2009 Posts: 1,425 Well you could look at it as follows. From the laws of reflection, the distance of every point on the obect from the mirror is the same as that of its corresponding point on the image from the mirror. The image in a plane mirror is laterally inverted i.e. if you move your right hand, your image moves its left. But if you move your hand towards or away from the mirror, the image also does the same. Only D will satisfy these conditions.
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Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post Harivarshand Light and Optics 3 Dec 21st 2015 09:02 AM Vishal Sahrawat Light and Optics 2 Dec 21st 2010 02:33 AM kenny1999 Light and Optics 3 Oct 30th 2009 09:31 PM kenny1999 Light and Optics 4 Oct 29th 2009 11:39 AM iamanoobatphysics Light and Optics 1 Dec 2nd 2008 10:38 PM | 387 | 1,490 | {"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} | 2.515625 | 3 | CC-MAIN-2018-05 | longest | en | 0.90157 |
http://2018.igem.org/Team:CCU_Taiwan/Polymer | 1,582,098,999,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875144058.43/warc/CC-MAIN-20200219061325-20200219091325-00035.warc.gz | 1,909,651 | 16,057 | # POLYMER MODEL
Oligomerization modeling
Our project is to create new materials via polymerization. Through Flory- Stockmayer theory, we can calculate the gelation and condensation when polymerization. Then, we can adjust our polymerization reaction to meet the properties what we want.
Flory-Stockmayer assumptions
1. All functional groups on a branch unit are equally reactive
2. All reactions occur between two molecules
3. There are no intramolecular reactions
Flory-Stockmayer theory of gel point
The Flory-Stockmayer theory is an ideal prediction for polymers. This theory is mainly to integrate gelation and contraction reactions. Through this theory, we can understand the reaction of polymerization and adjust the conditions of polymerization.
Our monomer is Coniferyl alcohol. According to the reaction of the enzyme, our monomer will exhibit three resonance states. We want to make the material more biodegradable and chain-like, so we use oligomer to polymerization. Because coniferyl alcohol has three resonance states, our functional group is three (f=3).
Polymerization of LIGGREEN
In the Flory-Stockmayer theory, our reaction belongs to ABg polymerization. LIGGREEN is ABg type of polycondensation, and there is an unreacted A and i(g -1)+1 B group in vivo with degree of polymerization i, and the species can be derived under the assumption of the inner cyclization reaction and the isocratic assumption.
We assume that our reaction only occur in oligomer and monomer. Because we would like to make the material formed in chain structure, we prefer to make oligomerization occur. Through ABg reaction, we are able to achieve oligomerization by adjusting the different conditions.
According to the above, our modeling is mainly based on three resonance states and ABg.
The initial conditions for the above equation are:
Because there is only one monomer group on each species.
For the following two types of equations. We can get,
After getting the above formula, we bring in our parameters.
With this distribution function, the various molecular parameters of LIGGEEN may be quantitatively known. Figure 1 shows the relationship between the polydispersity index of LIGGEEN under several g parameters and the conversion of the monomer group. g = 1 is the type AB monomer. Linear polycondensation, after the end of the reaction (x = 1), the polydispersity index of the LIGGEEN is only 2. If g > 1, the formation of hyperbranched polymer, the polydispersity index of LIGGEEN becomes very large near the completion of the reaction. After this modeling, we can get conclusion that oligomerization would occur.
References:
1. Stockmayer, W. H. (1943). Theory of molecular size distribution and gel formation in branched‐chain polymers. The Journal of chemical physics, 11(2), 45-55.
2. Flory, P. J. (1941). Molecular size distribution in three dimensional polymers. I. Gelation1. Journal of the American Chemical Society, 63(11), 3083-3090.
3. Carothers, W. H. (1936). Polymers and polyfunctionality. Transactions of the Faraday Society, 32, 39-49.
We assumed our product consists of three bonds of β-5, β-O-4, β-β, and we found that some kinds of enzyme can break β-5& β-O-4 bonds from reference. We try to establish a simple degradation rate model under ideal conditions, predicting the state of degradation and the treatment reference for the product.
In this part, we will use the kinetics of enzymes to build model and first start form Michaelis-Menten kinetics.
From Michaelis-Menten kinetics
Assumed:
(1) Lignin degradation is a one-step reaction
(2) Both enzymes and LIGGREEN are first-order reactions
(3) Value of enzyme activity is a constant.
(4) Reaction solution is 1 unit volume.
Some simulation constants.
1. Initial number of bacteria
2. Bacterial reproductive cycle
3. Enzyme production cycle
In this part, we will use the kinetics of enzymes to build model and first start form Michaelis-Menten kinetics. This model is common and well-known models of enzyme kinetics, and it is takes the form of an equation to describe the rate of enzymatic reactions by reaction rate V and concentration of substrate S.
We assume that the balance between E and ES are quickly established to reach steady-state and the concentration of enzyme-substrate [ES] is fixed.
Get the equation
and the relation of total enzyme concentration [E_0], free enzyme [E] and enzyme-substrate [ES] is
Using both equation can get
The reaction rate of enzyme can be expressed as
Using equation (4), we can get two relation of reaction rate
(I) at low substrate concentration,
(II) at higher substrate concentration,
To get more detailed information, we use the equation (4) & (5) to calculate, and Symbolic Math Toolbox in Matlab to solve equation.
(S(t):amount of substrate in time t; S(0):initial amount of substrate )
We can know the remaining amount of the substrate from the result, if the value of parameters is known, and we want use this relationship plus our assumptions to estimate biodegradation situation.
Degradation requires the action of enzyme, and assuming that the enzymes are produced by bacteria. The following relationship are used to express the number of bacteria, and the amount of enzyme.
Combine the equation (8) & (9), and the formula become
We can use the equation (6), (7) & (10) to estimate degradation rate in initial stage.
(1) at low substrate concentration
(2) at higher substrate concentration
To get more detailed information, using the equation (4) again and combine with equation (10).
(S(t):amount of substrate in time t; S(0):initial amount of substrate)
ω(z) is Wright omega function. This is final result, and we will use this formula to discuss the influence of parameters.
\$\$ Figure 1. k_2={10}^{-3}; k_A={10}^{-5};τ = 30;K_m={10}^{-3};S(0)={10}^{-2} ; B_i=25, 20, 15, 10, 5; \$\$
From the results, we found that the more initial bacteria, the faster the degradation, and the increase in the number of benefits will be lower and lower.
\$\$ Figure 2. k_2={10}^{-3}; k_A={10}^{-5}; B_i=10;K_m={10}^{-3}; S(0)={10}^{-3}; τ=10, 20, 30, 40, 50 \$\$
From the results, we found that the shorter average reproductive cycle, the faster the degradation. That is, if the material structure is easier reaction by enzyme of bacteria (microorganisms), the time required can be greatly reduced, but if the structure is complicated and difficult to use, it may cause difficulty in degradation, even in the same situation as plastic.
In summary, the simple part of the model is used to explore the degradation situation under ideal conditions. In the future, the data will be continuously collected to improve the model, hoping to make the model more realistic.
References:
1. Chang, Y. C., Choi, D., Takamizawa, K., & Kikuchi, S. (2014). Isolation of Bacillus sp. strains capable of decomposing alkali lignin and their application in combination with lactic acid bacteria for enhancing cellulase performance.
Bioresource technology, 152, 429-436
2. Bugg, T. D., Ahmad, M., Hardiman, E. M., & Singh, R. (2011). The emerging role for bacteria in lignin degradation and bio-product formation. Current
opinion in biotechnology, 22(3), 394-400.
3. Daina, S., Orlandi, M., Bestetti, G., Wiik, C., & Elegir, G. (2002). Degradation of β-5 lignin model dimers by Ceriporiopsis subvermispora.
Enzyme and Microbial Technology, 30(4), 499-505.
4. Buraimoh, O. M., Ilori, M. O., Amund, O. O., Isanbor, C., & Michel Jr, F. C. (2017). The degradation of coniferyl alcohol and the complementary production of chlorogenic acids in the growth culture of Streptomyces albogriseolus KF977548 isolated from decaying wood residues.
Process Biochemistry, 52, 22-29.
5. de Gonzalo, G., Colpa, D. I., Habib, M. H., & Fraaije, M. W. (2016). Bacterial enzymes involved in lignin degradation.
Journal of biotechnology, 236, 110-119.
6. Michaelis Menten Kinetics in bio–physic wiki (http://www.bio-physics.at/wiki/index.php?title=Michaelis_Menten_Kinetics) | 1,982 | 7,997 | {"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} | 2.671875 | 3 | CC-MAIN-2020-10 | latest | en | 0.908062 |
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# 1.1 Ch1-Calc-P20 - bandwidth each second IMPORTANT NOTES...
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Chapter 1 Powerpoint Slides, Page 20 How long does it take to send a file of 640,000 bits from host A to host B over a circuit-switched network? All links are 1.536 Mbps Each link uses TDM with 24 slots/sec 500 msec to establish end-to-end circuit 1.536 * 10 6 bit/sec ------------------------ = 64 * 10 3 bits/sec 24 slots/sec 640,000 bits -------------------- = 10 sec (i.e., 640,000 bits of data takes 10 slots) 64,000 bits/sec Total sending delay = 500 msec (to establish the end-to-end circuit) + 10 sec (to send data) = 10.5 seconds TDM frequency time 24 slots/sec Allocated 1/24 of the
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Unformatted text preview: bandwidth each second IMPORTANT NOTES (1) This calculation is only for the sending delay. The sending delay includes the set-up delay and transmission delay , but does not include the propagation delay or queueing/processing delays across the networks. (2) When considering transmission media and transmission rates, M (Mega) = 10 6 and K (Kilo) = 10 3 When considering data files for transmission, M (Mega) = 2 20 and K (Kilo) = 2 10 Question: If the problem had been stated as 640 Kbits of data. What would the sending time be?...
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Ask a homework question - tutors are online | 487 | 1,854 | {"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} | 3.265625 | 3 | CC-MAIN-2017-17 | longest | en | 0.878443 |
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Find an equation for the general term of the given arithmetic sequence and use it to calculate its
$100^{\text {th }}$ term: $7,10,13,16,19, \ldots$
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## 1 Answer
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Best answer
Answer:
$a_{n}=3 n+4 ; a_{100}=304$
Explanation:
Begin by finding the common difference,
$$d=10-7=3$$
Note that the difference between any two successive terms is 3 . The sequence is indeed an
arithmetic progression where $a_{1}=7$ and $d=3$.
\begin{aligned} a_{n} &=a_{1}+(n-1) d \\ &=7+(n-1) \cdot 3 \\ &=7+3 n-3 \\ &=3 n+4 \end{aligned}
Therefore, we can write the general term $a_{n}=3 n+4$. Take a minute to verify that this equation
describes the given sequence. Use this equation to find the $100^{\text {th }}$ term:
$$a_{100}=3(100)+4=304$$
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1 answer | 402 | 1,216 | {"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} | 4.3125 | 4 | CC-MAIN-2021-31 | latest | en | 0.812589 |
https://www.enotes.com/homework-help/how-many-cubic-feet-water-we-need-fill-circular-179355 | 1,524,298,074,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125945082.84/warc/CC-MAIN-20180421071203-20180421091203-00583.warc.gz | 780,589,731 | 10,252 | # How many cubic feet of water we need to fill a circular pool whose perimeter is 62.8 and the depth is 6 feet.
pohnpei397 | Certified Educator
We will need 1884 cubic feet of water to fill this pool. Here is how to figure this out:
The volume of the pool is found by the area of the bottom multiplied by the depth of the pool. First, we must find the area of the bottom. The area of a circle is found by the formula A = pi*r^2. So we must find the length of the radius.
We can find this because we know that the perimeter of a circle is equal to pi*diameter. So
62.8 = pi*diameter or 62.8 = 3.14*diameter. So now divide both sides by 3.14 and you get
diameter = 20
10^2 is 100. So the area of the bottom of the pool is
A = 3.14*100 or A = 314.
So now we multiply the area by the depth of the pool.
314*6 = 1884
hala718 | Certified Educator
To find out how many cubic feet we need, we need to determine the volume of the pool.
The surface of the pool is a circle , then the pool has a cylinder shape
We know that the cylinder volume (v)= A * h
where A is the area of the circle and h is the height
We need to first to calculate the area (A)
A= Pi* r^2 where r is the radius
But we know that the Perimeter P= 62.8= 2*pi* r
==> r= 62.8 / 2*pi (Pi=3.14)
==> r= 62.8/ 2(3.14)= 10
Then A = r^2 *pi = 100 (3.14) = 314
Then V = 314 * 6 = 1884 cubic feet
Then we need 1884 cubic feet of water to fill the pool. | 446 | 1,435 | {"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} | 4.46875 | 4 | CC-MAIN-2018-17 | latest | en | 0.884345 |
http://www.smart-kit.com/s3532/orange-triangles-math-pattern-problem/ | 1,498,497,790,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128320841.35/warc/CC-MAIN-20170626170406-20170626190406-00414.warc.gz | 639,450,774 | 11,016 | School-Safe Puzzle Games
## OrangeTriangles: Math pattern problem
What number should replace the question mark?
Feel free to enter your answer in the comment section below; will unmask submission by Wednesday, thanks!
### 30 Comments to “OrangeTriangles: Math pattern problem”
1. marioberges | Profile
Every triangle has three numbers besides the center one. I will call them x, y, z for (top, left and right) and c for the center one.
The formula is:
c = x * z + y * z
For example, in the first triangle:
36 = 4 * 3 + 8 *3
In the last triangle, we need to solve for y, so we have:
30 = 6 * 3 + y * 3
y = (30 – 6 * 3) / 3
y = 4
2. Obiwan | Profile
Question mark is 4.
o Red circle is sum of left lower and upper numbers in each triangle multiplied by right lower number.
o 1st triangle: 3×4 + 3×8 = 36
o 2nd triangle: 4×8 + 4×5 = 52
o 3rd triangle: 30 – (3×6)/3 =12. Missing number is 12/3 = 4.
3. Obiwan | Profile
Oops–left out brackets!
[30-(3×6]/3 = 4
4. goofymushu | Profile
i think its 4
5. Reka | Profile
It’s a four!
6. kitchen | Profile
4 (30 = 3*(4+6))
7. Falwan | Profile
pattern :
(8+4)(3) = 36
(5+8)(4) = 52
(6+?)(3) = 30 —-> ?=4
8. bigbossSNK | Profile
I can find two algorithms, with the same outcome:
1. Multiply the bottom right number with the other two, sum the products and you get the middle number. Solution: 4
Or
2. Multiply the smallest number among the corner numbers with the other two, sum the products and you get the middle number. Solution (a little harder): 4.
9. joe | Profile
The question mark should be a 4.
In each of the former triangles the centre circle is the result of adding the top number and the lower left number and multiplying the sun by the lower right number.
In the first two:
(8 + 4) x 3 = 36
(5 + 8) x 4 = 52
So in the third triangle
( 6 + ? ) x 3 = 30
? = 10
10. joe | Profile
Sorry that should be ? = 4 of course
11. hex | PUZZLE MASTER | Profile
4
(8+4)x3=36
(5+8)x4=52
(6+4)x3=30
12. Mashplum | PUZZLE MASTER | Profile
4
(top + left) * right = center
13. suineg | PUZZLE MASTER | Profile
cool I think one answer could be this:
first triangle: (8+4)*3= 36
second triangle: (5+8)*4= 52
third triangle: (6+4)*3= 30
14. Shawn | PUZZLE GRANDMASTER | Profile
4
15. moyes | Profile
6×3= 18, 3×4= 12, 18+12 =30
_______________________
Triangel 1:
8×3=24, 3×4= 12, 24+12=36
____________________
Triangel 2:
8×4=32, 4×5= 20, 32+20= 52
16. nattlebattle | Profile
4!
(Top + Bottom Left) * (Bottom Right) = Middle
(8 + 4) * (3) = 36
(5 + 8) * (4) = 52
(6 + 4) * (3) = 30
17. Jimmy Anders | PUZZLE MASTER | Profile
If the pattern is that the middle number (red) is the sum of the two numbers on the left multiplied by the third number, then the answer is 4.
[6 + 4] * 3 = 30
18. Hercules | Profile
36 divide by 3 is 12…8 and 4 are 12
52 divided by 4 is 13…5 and 8 are 13
30 divided by 3 is 10…6 and ‘4’ are 10
19. Oneiric | Profile
I’d say 4.
Let’s call the top number T, the bottom left number BL, the bottom right number BR and the center number C.
C = T * BR + BL * BR
Figure 1: 36 = 8 * 3 + 4 * 3 (24 + 12)
Figure 2: 52 = 5 * 4 + 8 * 4 (20 + 32)
Figure 3: 30 = 6 * 3 + BL * 3
30 = 18 + BL * 3
Previous minus 18: 12 = BL * 3
Previous divided by 3: 4 = BL
20. Shofnite | Profile
(8+4) * 3 = 36
(5+8) * 4 = 52
(6+x) * 3 = 30
x = 4
21. Blusummers13 | Profile
8*3+4*3=36
5*4+8*4=52
6*3+x*3=30 x=4
22. RK | Founder | Profile
very good- 4 is the answer
23. 10dulkar | Profile
Oh did anyone mention 4 yet? LOL | 1,291 | 3,515 | {"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} | 4.40625 | 4 | CC-MAIN-2017-26 | latest | en | 0.798676 |
www.thimet.de | 1,716,964,062,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059206.29/warc/CC-MAIN-20240529041915-20240529071915-00386.warc.gz | 46,702,727 | 9,077 | # Calculator Precision
Algorithm
Binary or BCD
Calculator Precision
Table of Function Results
Calculator Forensics
Hidden Digits
Quick Test
## The Algorithm
In general, the precision of a result that is obtained after a number of calculations depends only partly on the precision of the calculator itself. The other important issue is the algorithm that is being used to determine the result.
Notably, during subtraction of values of similar magnitude a large number of valid digits can be cancelled out. On a machine using 6 BCD digits consider:
31416.0 - 10000.0*Pi = 31416.0 - 31415.9 = 0.1
This is nowhere near the true 6-digit-precision result of 0.0734641 because of the cancellation of the leading valid digits.
From a mathematical viewpoint an infinitely precise value X can be written as:
X = Xm + e
where Xm is the limited-precision machine number and e is a small error value that corresponds to the precision of the calculator.
One (or more) of these numbers X are converted by the algorithm A into a result R:
R = A(Xm + e)
and the precision of the result R strongly depends on whether A "amplifies" the inaccuracy introduced by e.
Finding an acceptable algorithm for a specific problem (ie. integration, solving differential equations, matrix operations etc.) is often a complicated issue. This is not our topic. However, it will be our topic to assess the precision of built-in algorithms of calculators.
## Binary or BCD
There are two widely used methods to represent floating point numbers on computers: Binary and BCD coding.
### Binary
Binary numbers consist of a binary integer number that represents the mantissa plus a binary exponent. The number of mantissa bits determines the precision of the number (ie. the standard "double" type provides 52 mantissa bits). The mantissa is "normalized" so that its most significant bit is always 1 and sits just left of the decimal point. The exponent indicates how many bits this mantissa has to be shifted up or down to get the true value. It is therefore a power-of-2 exponent.
Example: Decimal number 10.5. In binary this is 1100.1. Split into mantissa and exponent: 1.1001 E 3dec
So the mantissa would be 11001 and the exponent +3dec.
Since the mantissa is required to always have a leading 1 it is often omitted to gain space for an additional bit of precision. In this case the leading 1 is implicitly assumed to sit to the left of the most significant digit of the mantissa.
Advantage: The big advantage of binary floating point numbers is that most computers have powerful and fast built-in instructions to directly manipulate long integer numbers (usually 16, 32 or 64 bit). For example multiplication and division is often implemented in hardware units. Most computers therefore use binary coding. There are of course computer software packages that perform BCD arithmetic as explained below.
Disadvantage: Whenever a number has to be displayed it must be converted into decimal format. Similarly, decimal input must be converted to binary before calculation can start. This is a quite time consuming process but of course it only occures once before and after a - possibly lengthy - calculation.
### BCD
BCD (binary coded decimal) numbers are basically identical to decimal numbers. Each decimal digit is coded as a 4-bit binary number. With 4 bits values from 0-15 can be coded but values 10-15 are never used. The BCD number's normalized mantissa consists of a series of those 4-bit values. There is also a power-of-10 exponent which indicates how many decimal digits the mantissa has to be shifted up or down to give the true value.
Example: Decimal number 10.5. The binary representation of the BCD digits is 0001 0000 0101. So the mantissa would be 000100000101 and the exponent +1dec.
Advantage: No complicated conversion between textual and internal representation of a BCD number is needed. This is especially useful for calculators where after each operation the result has to be displayed. In fact, I don't know of any handheld calculator that doesn't use BCD arithmetic!
Disadvantage: BCD arithmetic is slow because it lacks the dedicated hardware support. This usually doesn't matter because handheld calculators don't have high-performance CPUs anyway. Rather, their processors are optimized for low power consumption.
Another disadvantage of BCD numbers is their increased memory requirement. A n-digit decimal number needs n*4 bits in BCD and n*ln(10)/ln(2)=n*3.322 bits in binary mode. So the BCD representation requires 20% more storage.
### Using Binary or BCD?
It is relatively easy to determine whether binary or BCD arithmetic is used in a calculator. The crucial fact is that some numbers have an exact representation in BCD mode but not in binary mode.
Consider the decimal number 0.1 which has an exact representation in BCD. However, in binary mode it must be expressed as a sum of powers of 2:
0.1 = 2-4 + 2-5 + 2-8 + 2-9 + 2-12 + 2-13 + 2-16 + 2-17 + ... = 0.000110011001100110... (binary)
It turns out that the binary digits sequence "1100" repeats infinitely. But since every binary representation uses a limited number of bits the value 0.1 cannot be expressed exactly. To distinguish between binary and BCD usage the idea is to "amplify" this inaccuracy so that it can be observed:
Calculate: R = (0.1 * 1024 - 102) * 10 - 4
In theory as well as on a BCD machine this should give 0. However, on a binary machine 0.1 will have some error e that is amplified by a factor of 10240:
((0.1 + e)*1024 - 102)*10 - 4 = (102.4 + 1024*e - 102) * 10 - 4 = (0.4 + 1024*e)*10 - 4 = 4 + 10240*e - 4 = 10240*e
As a result, calculating R on a machine using binary representation will yield a non-zero value.
Side note 1: Although some limited-length BCD numbers have no exact (=no limited length) representation in binary mode the reverse is not true. All limited length binary numbers do have a limited length representation in BCD. The reason is that all powers of 2 have a limited length representation in BCD (ie. 0.5, 0.25, 0.125, 0.0625 etc.) but not all powers of 10 have a limited-length representation in binary mode.
Side note 2: Try calculating R = (0.1+e)*10240 - 1024 = 1024 + 10240*e - 1024 = 10240*e.
Surprisingly, this will yield 0 even on a binary machine! The reason is that due to rounding (0.1+e)*10240 will in fact result in the exact value 1024.
## Calculator Precision
As we have seen above in The Algorithm performing a series of operations will in most cases introduce inaccuracies that depend on the nature of the algorithm. The user of a computer or calculator is responsible for taking those inaccuracies into account when using the device for lengthy operations. However, because the user doesn't know the particular algorithms used for built-in functions inside a calculator he cannot know their accuracy.
There are three solutions to this dilemma:
1. Let the user find out about the accuracy of operations like multiplication, sine, or logarithm. This is of course completely unacceptable.
2. State the accuracy of the various built-in operations in the manual. That's better than nothing but still not very sattisfactory.
3. Make sure that all built-in functions are accurate to the number of displayed digits. That's what we want!
### Internal Precision
To achieve goal #3 the calculator will usually have to internally use a few more digits than are displayed. How many of those are used depends on the algorithm: Clever algorithms will need fewer hidden digits than less clever ones. Therefore, statements like "internally using 15 digits of precision" are really not very useful! A badly chosen algorithm can still produce unacceptable results even when using lots of digits!
But the ultimate goal is to calculate results that are always correct to the number of displayed digits.
### Rounding
In many cases the result cannot be displayed correctly with a limited number of digits, ie. 1/3 = 0.3333....
Naturally, it is desirable that the displayed result is as close to the true value as possible. This is achieved by "correct rounding to the last digit".
The correct rounding scheme examines the digit n+1 following the least significant digit n (and thus needs at least one additional hidden digit). If digit n+1 is in the range 5..9 then digit n is incremented to minimize the difference. Otherwise digit n is left unchanged.
Examples of correct rounding:
Correct value Expanded correct value Rounded value, to 5 digits after decimal point 1/3 0.333333... 0.33333 2/3 0.666666... 0.66667 1/18 0.055555... 0.05556 4/9 0.444444... 0.44444 1/11 0.090909... 0.09091
To determine whether your calculator uses correct rounding simply calculate 1/18 and examine the last digit: It should be 6.
Besides correct rounding there are of course other less desirable methods, ie. cutting off all digits beyond the least significant one.
### Argument Ranges
If we want to see whether a calculator sattisfies goal #3 we do not only have to examine all built-in functions but also test them over their entire range of argument values where they are defined! Of course this is not possible so only a representative set of argument values can be checked.
## Table of Functions Results
Important Notes
• Table values are accurate to the given number of 21 digits. Most calculators offer 8, 10 or 12 digits of precision so the above values should be sufficiently precise for comparisn to all existing calculators.
• For every function I tried to find critical input values that might uncover inaccuracies.
• When comparing the above numbers to the results of your particular calculator please do not forget to perform correct rounding of the table values to the number of digits that your calculator displays.
• Please use the appropriate display setting of your calculator so that all valid digits are visible!
• Some calculators have a special SHOW or MANT command which displays the full mantissa.
• In fixed-point mode some calculators display values 0.1<=x<1 with a leading 0 (0.4444...) and thus loose one significant digit for display. In this case multiply the result by 10 to see the least significant digit.
• The display may be shared between the mantissa and the exponent so that values cannot be displayed with all mantissa bits. In this case multiply by the correct power of 10 to get rid of the exponent and make room for all mantissa digits.
• It is assumed that the precision of the cosine function is comparable to the sine function. Hence I omitted it from the table.
• A particular calculator may offer much more functions than the above ones, ie. complex arithmetic, matrix operations and more. Some of them are notoriousely inaccurate when using certain "ill-conditioned" input values. Providing reference values for all those functions goes beyond the scope of this investigation - but when you are using them make sure you really get the accuracy that you expect!!
• Values calculated with Michael C. Ring's MAPM arbitrary precision library.
## Calculator Forensics
Victor T. Thoth & Mike Sebastian use the following formula in the Calculator Forensics of the "Museum of R/S Key Programmable Calculators" to assess the precision of a calculator (using degrees):
R = asin( acos( atan( tan( cos( sin(9) ) ) ) ) )
Of course the correct result is 9. At the various steps the intermediate results are:
X sin(9) 0.156 434 465 040 230 869 010... cos(x) 0.999 996 272 742 885 024 117... tan(x) 0.017 454 999 855 488 660 791... atan(x) 0.999 996 272 742 885 024 117... acos(x) 0.156 434 465 040 230 869 010... atan(x) 9.000 000 000 000 000 000 000...
Now consider a calculator that uses built-in algorithms that are correct up to the 12th digit. And of course the 12-digit precision result of one step is taken as the input for the next step of the calculation:
X rounded to 12 digits sin(9) 0.156 434 465 040 cos(x) 0.999 996 272 743 tan(x) 0.174 549 998 555 E -1 atan(x) 0.999 996 272 744 acos(x) 0.156 434 441 642 atan(x) 0.899 999 864 267 E 1
This is a perfect example of a badly chosen algorithm because it amplifies the inaccuracies of the 12-th digit to a considerable error. Compared to the table with precise results the first noticable deviation occurs when the arcus tangent is calculated. The resulting small error of 1.12E-12 (absolute) is then tremendously amplified by more than 104 by the arcus cosine to 2.34E-8 (absolute).
By looking at the derivative of the arcus cosine near the value of 1 it is immediately clear why this amplification occurs:
d/dx acos(x) = -1/sqrt(1-x²) and for x -> 1 the derivative reaches infinity.
It must be strongly emphasised that the above result of 8.99999864267 is the correct result for a calculator using 12 digits of precision and perfect built-in algorithms for trigonometric functions! Naturally, similar arguments apply for calculators of different precision.
But even though the Calculator Forensics' formula is not suitable to judge the accuracy of a calculator it is extremely useful to determine whether two calculators use the same integrated circuitry. See Mike Sebastian's page which lists calculators by their Calculator Forensics Result.
## Hidden Digits
### What Are Hidden Digits?
Internally, the calculator may store results with more digits of precision than are ever presented to the user. This has two important consequences:
• Due to correct rounding to the displayed number of digits this displayed value is presumably correct in all digits. This even if not all of the hidden digits are correct.
• When the calculation is continued, the calculator doesn't use the displayed value as input but rather the internally stored value with the additional hidden digits. This can be very confusing because the result of a function depends on whether the input value was manually typed in (without hidden digits) or was the result of a previous calculation (with hidden digits).
In general, using hidden digits will yield more accurate results in chain calculations compared to not using hidden digits. But the problem is that it cannot be determined how much more accurate the result will be! In worst case all hidden digits are wrong and nothing is gained. So the only reliable statement about the calculator is that the precision corresponds to the number of displayed digits.
If those hidden digits are in fact always correct then why not present them to the user? The display may not be able to show all the digits. But usually one can safely suspect that the hidden digits are sometimes correct and sometimes not.
Together with the above mentioned confusion that arises from hidden digits in conjunction with manually entered vs. calculated values it is obvious that storing results with hidden digits is a not a good idea.
### Detecting Hidden Digits
Simply repeat the following: get rid of integer part & multiply by 10.
This "pulls out" the hidden digits. There may be none, one or more, depending on the calculator. | 3,465 | 14,970 | {"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} | 4.09375 | 4 | CC-MAIN-2024-22 | latest | en | 0.856038 |
https://www.physicsforums.com/threads/really-hard-equivalent-resistor-problem.753123/ | 1,510,988,377,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934804666.54/warc/CC-MAIN-20171118055757-20171118075757-00118.warc.gz | 873,138,029 | 17,488 | # REALLY hard equivalent resistor problem!
1. May 10, 2014
### jaredvert
Please don't say this is homework because it is not. I am simply studying problems for the physics c exam and I can't figure out this textbook problem. Please show me how to do this! The equivalent resistance part in the most depth because I dont understand. Is it right to say no current runs through the middle r because it would be repelled by the other current? And if that's the case this problem becomes easy but in not sure that's the case... Thanks .
Last edited: May 10, 2014
2. May 11, 2014
### ModusPwnd
Even thought its not "real" homework, it can still go in the homework section because its of that format.
Anyway, looks like it might be a case of the ol' delta-wye. Does that mean anything to you?
https://en.wikipedia.org/wiki/Y-Δ_transform
Your tried and true series and parallel addition formulas wont work here as you have found. The delta-wye formulas can be derived, or just blindly used, depending on what you want.
3. May 11, 2014
### Staff: Mentor
This is a homework style question and as such should be posted in the homework section of the forums per PF rules.
4. May 11, 2014
### jaredvert
Ok van u just tell me why the current wouldn't go through the middle resistor and then I will delete it. I had no other way of conveying this question
5. May 11, 2014
### UltrafastPED
6. May 11, 2014
### ModusPwnd
Current will go through all those resistors.
7. May 11, 2014
### jaredvert
Guys this was in my textbook that says absolutely nothing about delta y transformation or anything of the sort. Can you think of anything else??? They wouldn't put something in the problems at the back if it wasn't applicable to the knowledge learned through the book and this obviously wo t come up on the ap exam but I still am curious and would like to know
8. May 11, 2014
### ModusPwnd
I would guess that there is a small section on delta-wye in there. My second guess is that the book editor just messed up. It happens sometimes.
9. May 11, 2014
### jaredvert
Giancoli physics for scientists and engineers vol 2. I read the whole chapter and nothing on delta wye? How do u know when to use delta wye?
10. May 11, 2014
### ModusPwnd
I consider it when I can't figure out how to use parallel or series addition. :tongue: The resistors are not in parallel or series with any other resistor.
UltrafastPED's link is good. You can see the basic geometry of the delta and they wye, they literally look like a delta and a wye. (Of course your wires may be aranged differently making it hard to see. Practice is needed)
Using the formulas in that link you can turn one into another as needed and then, hopefully, use your parallel and series rules to reduce the circuit.
11. May 11, 2014
### jaredvert
K thanks dudio
12. May 11, 2014
Staff Emeritus
Closed because it's in the wrong section (since moved) and doesn't use the template.
13. May 11, 2014
### Staff: Mentor
Just to add an important comment: you do not need a Y-Delta transformation. This is way beyond the scope of your homework.
You can draw the setup in a different way to see how symmetric it is, and then you can see why there is no current in the middle resistor. Afterwards it is easy to continue. | 822 | 3,283 | {"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} | 2.828125 | 3 | CC-MAIN-2017-47 | longest | en | 0.966906 |
https://math.stackexchange.com/questions/4488468/proposition-4-5-of-brian-hall-lie-groups-lie-algebras-and-representations | 1,708,795,581,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474541.96/warc/CC-MAIN-20240224144416-20240224174416-00762.warc.gz | 380,568,930 | 33,759 | # Proposition 4.5 of Brian Hall - Lie Groups, Lie Algebras, and Representations
I have a question about the proof of Proposition 4.5 in Brian Hall's Lie Groups, Lie Algebras, and Representations.
The proposition says that
Let $$G$$ be a connected matrix Lie group with Lie algebra $$\mathfrak{g}$$. Let $$\Pi$$ be a representation of $$G$$ and $$\pi$$ the associated representation of $$\mathfrak{g}$$. Then $$\Pi$$ is irreducible iff $$\pi$$ is irreducible.
Overall I understand the proof besides one part in the proof of the backward statement. The proof says
Let $$W$$ be an invariant subspace for $$\Pi$$. Then $$W$$ is invariant under $$\Pi(e^{tX})$$ for all $$X \in \mathfrak{g}$$ and hence under $$\pi(X) = \frac{d}{dt} \Pi(e^{tX}) |_{t=0}$$.
Why does $$W$$ being invariant under $$\Pi(e^{tX})$$ imply it is invariant under $$\frac{d}{dt} \Pi(e^{tX}) |_{t=0}$$?
Here's the screenshot of the proposition and its proof.
This is just calculus. Let $$L_t$$ be a family of linear operators that send a subspace $$V$$ to $$V$$, then $$\frac{L_t(x)-L_0(x)}{t}\in V$$ for any $$x\in V, t\not=0$$. And by the completeness of $$V$$, $$\lim_{t\rightarrow 0}\frac{L_t(x)-L_0(x)}{t}\in V$$. | 379 | 1,192 | {"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": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2024-10 | latest | en | 0.797398 |
http://dictionnaire.sensagent.leparisien.fr/Speedup/en-en/ | 1,606,265,266,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141177607.13/warc/CC-MAIN-20201124224124-20201125014124-00154.warc.gz | 26,022,995 | 21,341 | Speedup : définition de Speedup et synonymes de Speedup (anglais)
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# définition - Speedup
speedup (n.)
1.the act of accelerating; increasing the speed
2.move faster"The car accelerated"
## définition (complément)
voir la définition de Wikipedia
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speedup (n.)
voir aussi
speedup (n.)
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Wikipedia
# Speedup
In parallel computing, speedup refers to how much a parallel algorithm is faster than a corresponding sequential algorithm.
## Definition
Speedup is defined by the following formula:
$S_p = \frac{T_1}{T_p}$
where:
• p is the number of processors
• $T_1$ is the execution time of the sequential algorithm
• $T_p$ is the execution time of the parallel algorithm with p processors
Linear speedup or ideal speedup is obtained when $\,S_p = p$. When running an algorithm with linear speedup, doubling the number of processors doubles the speed. As this is ideal, it is considered very good scalability.
Efficiency is a performance metric defined as
$E_p = \frac{S_p}{p} = \frac{T_1}{pT_p}$.
It is a value, typically between zero and one, estimating how well-utilized the processors are in solving the problem, compared to how much effort is wasted in communication and synchronization. Algorithms with linear speedup and algorithms running on a single processor have an efficiency of 1, while many difficult-to-parallelize algorithms have efficiency such as $\frac{1}{\ln p}$[citation needed] that approaches zero as the number of processors increases.
In engineering contexts, efficiency is more often used for graphs than speedup, since
• all of the area in the graph is useful (whereas in a speedup curve 1/2 of the space is wasted)
• it is easy to see how well parallelization is working
• there is no need to plot a "perfect speedup" line
In marketing contexts, speedup curves are more often used, largely because they go up and to the right and thus appear better to the less-informed.
## Super linear speedup
Sometimes a speedup of more than p when using p processors is observed in parallel computing, which is called super linear speedup. Super linear speedup rarely happens and often confuses beginners, who believe the theoretical maximum speedup should be p when p processors are used.
One possible reason for a super linear speedup is the cache effect resulting from the different memory hierarchies of a modern computer: In parallel computing, not only do the numbers of processors change, but so does the size of accumulated caches from different processors. With the larger accumulated cache size, more or even all of the working set can fit into caches and the memory access time reduces dramatically, which causes the extra speedup in addition to that from the actual computation.
An analogous situation occurs when searching large datasets, such as the genomic data searched by BLAST implementations. There the accumulated RAM from each of the nodes in a cluster enables the dataset to move from disk into RAM thereby drastically reducing the time required by e.g. mpiBLAST to search it.
Super linear speedups can also occur when performing backtracking in parallel: One thread can prune a branch of the exhaustive search that another thread would have taken otherwise.
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# IFT 194 CH 8
1. Draw and name a one-dimensional array that would hold 10 temperatures. Number the elements.2. Draw and name four parallel arrays that would hold the following data: Store sales customers Region A24 5000 500 A A26 3000 200 A C30 6000 550 C B44 4000 560 B3. Can all the data in question 2 be put into a two-dimensional array? Explain. 4. Mr. Jones always gives True/False tests to his class. His tests always have 20 questions. The maximum class size is 35. He needs a program that will calculate the student’s grades based on the best score A will range from the best score, to the best score minus 2.B will range from the best score minus 3, to the best score minus 4.C will range from the best score minus 5, to the best score minus 6.D will range from the best score minus 7, to the best score minus 8.F will be below the best score minus 8. ## Each student’s ID and test answers will be entered. The output will be each student’s ID, number correct, and grade, along with the single highest score for the class. Develop a solution for Mr. Jones problem. Use four one-dimensional arrays—one for the correct scores and the other three for the needed output. write your pseudo code first and then write the Python code that implements it.** below is a suggested way to do this — there are other ways such as setting up 2-D arrays) Using the format of P1, set up four arrays of student scores assume 10 questionsand 4 students.( this will illustrate the approach without too many numbers Write a function to populate a student score array.Populate these student arrays with zeros ( false) and 1s ( true), as their scores .See problem 6 pg 175 and use that random function not including theinteger ‘1’. (instead of ‘coin’ flips this slightly modified random number generator Integer(Random * 2) ( check out the Python equivalentwill simply yield 0, or 1, which is what you want). Now manipulate those 4 arrays to answer Jones’ questions
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https://questions.examside.com/past-years/jee/question/pthe-angular-displacement-of-body-performing-circular-moti-mht-cet-physics-motion-dzx9f8dn78lkpcbp | 1,721,498,713,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763517515.18/warc/CC-MAIN-20240720174732-20240720204732-00615.warc.gz | 406,267,530 | 34,630 | 1
MHT CET 2021 22th September Morning Shift
+1
-0
The angular displacement of body performing circular motion is given by $$\theta=5 \sin \frac{\pi t}{6}$$. The angular velocity of the body at $$t=3$$ second will be $$\left[\sin \frac{\pi}{2}=1, \cos \frac{\pi}{2}=0\right]$$
A
$$5 \frac{\mathrm{rad}}{\mathrm{s}}$$
B
$$1 \frac{\mathrm{rad}}{\mathrm{s}}$$
C
$$2.5 \frac{\mathrm{rad}}{\mathrm{s}}$$
D
zero $$\frac{\mathrm{rad}}{\mathrm{s}}$$
2
MHT CET 2021 22th September Morning Shift
+1
-0
A body performing uniform circular motion of radius 'R' has frequency 'n'. It centripetal acceleration is
A
8 $$\pi^2$$nR$$^2$$
B
4 $$\pi^2$$n$$^2$$R
C
4 $$\pi^2$$n$$^2$$R$$^2$$
D
8 $$\pi^2$$n$$^2$$R
3
MHT CET 2021 21th September Evening Shift
+1
-0
The angle of banking '$$\theta$$' for a meter gauge railway line is given by $$\theta=\tan ^{-1}\left(\frac{1}{20}\right)$$. What is the elevation of the outer rail above the inner rail?
A
$$20 \mathrm{~cm}$$
B
$$10 \mathrm{~cm}$$
C
$$0.2 \mathrm{~cm}$$
D
$$5 \mathrm{~cm}$$
4
MHT CET 2021 21th September Morning Shift
+1
-0
A particle moves in a circular orbit of radius '$$r$$' under a central attractive force, $$F=-\frac{k}{r}$$, where $$\mathrm{k}$$ is a constant. The periodic time of its motion is proportional to
A
$$r^{\frac{1}{2}}$$
B
$$\mathrm{r}^{\frac{2}{3}}$$
C
$$r$$
D
$$r^{\frac{3}{2}}$$
EXAM MAP
Medical
NEET | 530 | 1,374 | {"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} | 2.6875 | 3 | CC-MAIN-2024-30 | latest | en | 0.652515 |
https://www.excel-pratique.com/en/functions/xmatch | 1,721,245,244,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514801.32/warc/CC-MAIN-20240717182340-20240717212340-00025.warc.gz | 663,143,701 | 6,278 | Excel Function: XMATCH
The XMATCH function returns the position of an item in an array or a cell range.
It is an improved version of the MATCH function.
Usage:
`=XMATCH(lookup_value, lookup_array)`
or
`=XMATCH(lookup_value, lookup_array, match_mode, search_mode)`
Example of use
The goal here is to return the position of the searched city in the city table:
Enter in the XMATCH function:
• lookup_value: the value whose position to search
• lookup_array: the array in which to search the position of lookup_value
The formula is here:
``=XMATCH(C2,A2:A11)``
In this example, "London" is indeed the 3rd value in the range A2 to A11.
Optional arguments
In the previous example, only the 2 mandatory arguments were filled in, but there are 2 more:
• match_mode: the method for finding a match:
• 0: exact match (by default)
• 1: exact match or next smaller item
• -1: exact match or next larger item
• 2: wildcard match (where * replaces zero, one or several characters, ? replaces a character and ~ allows to escape one of these 3 characters *?~)
• search_mode: the search mode:
• 1: search from first to last (by default)
• -1: search from last to first
• 2: binary search assuming the range is sorted in ascending order
• -2: binary search assuming the range is sorted in descending order
Here is another example with match_mode at 2 to use the wildcard character * (which replaces zero, one or several characters) and search_mode at -1 to search the position of the city starting with "D" from the end:
``=XMATCH(C2&"*",A2:A11,2,-1)`` | 395 | 1,553 | {"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} | 2.609375 | 3 | CC-MAIN-2024-30 | latest | en | 0.828798 |
http://mathhelpforum.com/pre-calculus/6906-real-solutions-print.html | 1,495,655,142,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463607860.7/warc/CC-MAIN-20170524192431-20170524212431-00046.warc.gz | 241,130,410 | 2,847 | # Real Solutions
• Oct 26th 2006, 06:37 PM
Rimas
Real Solutions
How many real solutions [x,y] are there that satisfy the two equations x^2 + y^2= 30 and 4y^2-X^2=100?
• Oct 26th 2006, 06:44 PM
ThePerfectHacker
You have a circle and hyperbola.
• Oct 26th 2006, 10:30 PM
Soroban
Hello, Rimas!
Did you try solving the system?
Quote:
How many real solutions $(x,y)$ are there that satisfy: . $\begin{array}{cc}(1)\\(2)\end{array}\;\begin{array }{cc}x^2 + y^2\:=\:30 \\ 4y^2-x^2\:=\:100\end{array}$
Add the equations: . $5y^2 = 130\quad\Rightarrow\quad y^2 = 26\quad\Rightarrow\quad y = \pm\sqrt{26}$
Substitute into (1): . $x^2 + 26 \:=\:30\quad\Rightarrow\quad x^2 = 4\quad\Rightarrow\quad x = \pm2$
There are four solutions: . $(2,\,\sqrt{26}),\;(2,\,\text{-}\sqrt{26}),\;(\text{-}2,\,\sqrt{26}),\;(\text{-}2,\,\text{-}\sqrt{26})$ | 334 | 835 | {"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": 5, "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} | 4.09375 | 4 | CC-MAIN-2017-22 | longest | en | 0.683918 |
https://www.mathworksheetsland.com/topics/graphing/readingbargraphsset.html | 1,709,073,171,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474688.78/warc/CC-MAIN-20240227220707-20240228010707-00272.warc.gz | 895,876,229 | 6,527 | # Math Worksheets Land
Math Worksheets For All Ages
# Math Worksheets Land
Math Worksheets For All Ages
Home > Math Topics > Graphing >
These types of graphs are great for showing changes to something over time. They are great for summarizing large data sets in a visual format. The length of each bar indicates the quantities relative to each data point. The larger the bar, the greater the magnitude of that data point. This form of data visualization can help us make educated decisions based on the data that we have available to us. Bar graphs help us get a general understanding of the trends that are in front of us. They do not help us spot causes or effects, they are often not helpful to spot patterns in data. In this section we will learn how to make sense of data that is found in this form. These worksheets and lessons will help students learn how to interpret and make decisions based on bar graphs.
• Answer Keys - These are for all the unlocked materials above.
### Homework Sheets
Is this some big popularity contest or what? I guess it is!
• Homework 1 - We would just add the value to all of the data sets.
• Homework 2 - In a survey children were asked about the TV channels they watched. This chart shows the data of the survey. Study the graph and answer the following questions.
• Homework 3 - Children were asked their favorite ice cream flavor. The flavors of the ice cream are shown below in the bar graph.
### Practice Worksheets
Politics is strange, it's the only time popularity doesn't matter in a survey.
• Practice 1 - Find the total numbers of accessories sold by both shops.
• Practice 2 - In a school, a survey on students was done. The survey asked students which personal nature they most identified with.
• Practice 3 - A survey was done with 2 groups. According to bar graph, answer the following questions.
### Math Skill Quizzes
I followed a common theme on each sheet. I just couldn't fit clip art.
• Quiz 1 - Which shop sold less stationery and what is the lowest rate?
• Quiz 2 - Which sweet did the shops sell the most of?
• Quiz 3 - Which vehicle is preferred most to ride?
### What do Bar Graphs Tell Us?
Perhaps the most common statistical display used to represent categorical in an organized manner, where the length of bars clearly shows quantity and makes it simple to compare a difference in amounts. A bar graph focuses show on compare singular data points. It is not super helpful for identifying relationships between data. An example that can be represented as bar graphs would be; How much money people in the US spend on transportation services to commute into work every year.
However, the matter of concern here is what do bar graph tells us? Well, this visual tool uses varying lengths of bars to compare the data of different categories, represented on any of the axes i.e; x-axis or y-axis. These tools mainly tell us the following few things: It effectively compares items between groups. It shows the trend over a definite period of time, say quarterly, one year or four years. It also sees the annual trends of sales distributed throughout the year. Using this tool tells us about a significant change in data over-time. For example, take a look at the visualization of oil prices to the right. This clearly indicates a trend of failing process. They are helpful for spotting trends, but they do need to be a bit more obvious.
In addition to the above-mentioned points, this tool makes it easy to compare complex data in a glance. It indicates specific data categories in a frequency distribution. I find them very helpful for understanding where something is headed. For example, when taking a look at the oil chart, it would indicate to me that it is a good time to buy oil. | 778 | 3,764 | {"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} | 3.78125 | 4 | CC-MAIN-2024-10 | latest | en | 0.950797 |
https://ltwork.net/swiss-clothing-store-had-a-balance-in-the-accounts-receivable--4542319 | 1,679,800,222,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945381.91/warc/CC-MAIN-20230326013652-20230326043652-00268.warc.gz | 437,430,559 | 10,698 | ### What is the volume of a cardboard box whose length, width, and height are 1 2 foot, 1 7 foot, and 1
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-- 0.022681-- | 1,362 | 5,302 | {"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} | 2.953125 | 3 | CC-MAIN-2023-14 | latest | en | 0.929603 |
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