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https://dsp.stackexchange.com/questions/2711/is-autospectrum-the-same-as-power-spectrum-density?lq=1 | # Is autospectrum the same as power spectrum density?
In my slides about signal processing, there is one that mentions the same thing as the beginning of this and this answer, namely that the Fourier transform of a signal, squared is the power spectrum density of the signal.
In this talk, it is mentioned that coherence is calculated by dividing the squared cross-spectrum by the product of the two auto-spectra.
However, the formula in my slides divides the squared cross-spectrum by the product of the formula of our previously seen PSD and another PSD.
So, is autospectrum the same as PSD? I can find a lot of information about PSD, but not on autospectrum.
• I don't know yet how to put in formulas nicely. I think it'll become clearer what I mean. I'll try to edit those in tomorrow.
– Mien
Jun 24 '12 at 22:04
• Neither answer that you link to says "the Fourier transform of a signal, squared is the power spectrum density of the signal." as you assert. The power spectral density is the Fourier transform of $R_x(\tau)$, the autocorrelation function of a signal $x(t)$ which is not the same thing as Fourier transform of the signal squared, i.e. $\mathcal F[x^2(t)]$$, **nor** the Fourier transform squared, i.e.$[X(f)]^2$. The power spectral density is equal to$|X(f)|^2$, but not equal to$[X(f)]^2\$ in general. Jun 25 '12 at 0:40 | 2021-12-04 22:15:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9176333546638489, "perplexity": 347.0621131050016}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363125.46/warc/CC-MAIN-20211204215252-20211205005252-00016.warc.gz"} |
https://math.stackexchange.com/questions/2627074/finding-the-major-and-minor-axes-lengths-of-an-ellipse-given-parametric-equation | # Finding the major and minor axes lengths of an ellipse given parametric equations
The traditional parametric equations of an ellipse centered at the origin are given by: $$\{x(t)= a\cos(t)\cos(B) - b\sin(t)\sin(B); y(t)= a\cos(t)\sin(B) + b\sin(t)\cos(B)\}$$ where B is the angle of rotation of the major axis from the x-axis. Here, the major and minor axes lengths are given as a and b. However I have a different parametric form of an ellipse given by: $$\{x(t)=A\cos(t) + B\sin(t); y(t)=C\cos(t) + D\sin(t)\}.$$ Each equation represents sinusoids along the the x and y axes with independent magnitudes and phases, thus forming an ellipse as the locus $(x(t),y(t))$.
The major and minor axes lengths are not apparent in this form. My approach has been to set the derivative $dx/dy = -(x/y)$ since the slope of the tangent is perpendicular to the radius only at the intersections with the major and minor axes. My result is $t=.5(\tan^{-1}((AB+CD)/2(A^2+C^2-B^2-D^2))$. This value of t does not quite match my numerical simulation (although it is close) and I am perplexed. Any help is greatly appreciated.
Thanks, Bob T.
• Are the $B$’s in the two parametrizations meant to be the same? – amd Jan 29 '18 at 20:39
• Sorry, no I should have written these as separate variables – Bob Tivnan Jan 29 '18 at 20:54
Hint:
From your equations, if they represents the same ellipse, we have: $$A=a\cos \beta \qquad B=b\sin \beta$$ $$C=a\sin \beta \qquad D=b\cos \beta$$ so: $$\frac{C}{A}=\frac{B}{D}=\tan \beta$$
where $\beta$ is your angle $B$.
• But my A,B,C and D parameters are independent. So C/A does not necessarily equal B/D. I think part of the problem is that my equations use 4 parameters when only 3 are required for an ellipse. Wouldn't that mean that any given ellipse does not have a unique A,B,C,D combination and that your approach gives one such combination? Just spit balling here. – Bob Tivnan Jan 29 '18 at 21:47
With Mathematica’s help, $t$ can be eliminated to get the implicit equation $$\left(C^2+D^2\right)x^2-2(AC+BD)xy=\left(AD-BC\right)^2-\left(A^2+B^2\right) y^2.$$ From here, you can use standard methods to find the principal axes and semi-axis lengths. The latter are eigenvectors of $${1\over(AD-BC)^2}\begin{bmatrix}C^2+D^2 & -(AC+BD) \\ -(AC+BD) & A^2+B^2\end{bmatrix},$$ i.e., $$\left[\left(A^2+B^2\right)-\left(C^2+D^2\right) \pm \sqrt{\left(A^2+B^2+C^2+D^2\right)^2-4\left(BC-AD\right)^2}, 2(AC+BD)\right]^T.$$ The semi-axis lengths are the reciprocal square roots of the corresponding eigenvalues.
I found my mistake. I carelessly placed the 2 in the denominator of my equation:
t_axes=.5(arctan((AB+CD)/2(A^2+C^2^−D^2)) should be t_axes=.5(arctan(2(AB+CD)/(A^2+C^2−B^2−D^2))
This correctly gives the value of t that can be used to find the points of intersection of the principal axes with the ellipse. Thanks for the comments, which helped me get a new perspective on the problem. | 2020-07-02 08:59:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8474836945533752, "perplexity": 164.54670409211374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655878639.9/warc/CC-MAIN-20200702080623-20200702110623-00134.warc.gz"} |
https://eng.libretexts.org/Bookshelves/Computer_Science/Databases_and_Data_Structures/Book%3A_Open_Data_Structures_-_An_Introduction_(Morin)/02%3A_Array-Based_Lists/2.05%3A_DualArrayDeque_-_Building_a_Deque_from_Two_Stacks | Skip to main content
# 2.5: DualArrayDeque - Building a Deque from Two Stacks
Next, we present a data structure, the DualArrayDeque that achieves the same performance bounds as an ArrayDeque by using two ArrayStacks. Although the asymptotic performance of the DualArrayDeque is no better than that of the ArrayDeque, it is still worth studying, since it offers a good example of how to make a sophisticated data structure by combining two simpler data structures.
A DualArrayDeque represents a list using two ArrayStacks. Recall that an ArrayStack is fast when the operations on it modify elements near the end. A DualArrayDeque places two ArrayStacks, called $$\mathtt{front}$$ and $$\mathtt{back}$$, back-to-back so that operations are fast at either end.
List<T> front;
List<T> back;
A DualArrayDeque does not explicitly store the number, $$\mathtt{n}$$, of elements it contains. It doesn't need to, since it contains $$\mathtt{n}=\texttt{front.size()} + \texttt{back.size()}$$ elements. Nevertheless, when analyzing the DualArrayDeque we will still use $$\mathtt{n}$$ to denote the number of elements it contains.
int size() {
return front.size() + back.size();
}
The $$\mathtt{front}$$ ArrayStack stores the list elements that whose indices are $$0,\ldots,\texttt{front.size()}-1$$, but stores them in reverse order. The $$\mathtt{back}$$ ArrayStack contains list elements with indices in $$\texttt{front.size()},\ldots,\mathtt{size()}-1$$ in the normal order. In this way, $$\mathtt{get(i)}$$ and $$\mathtt{set(i,x)}$$ translate into appropriate calls to $$\mathtt{get(i)}$$ or $$\mathtt{set(i,x)}$$ on either $$\mathtt{front}$$ or $$\mathtt{back}$$, which take $$O(1)$$ time per operation.
T get(int i) {
if (i < front.size()) {
return front.get(front.size()-i-1);
} else {
return back.get(i-front.size());
}
}
T set(int i, T x) {
if (i < front.size()) {
return front.set(front.size()-i-1, x);
} else {
return back.set(i-front.size(), x);
}
}
Note that if an index $$\mathtt{i}<\texttt{front.size()}$$, then it corresponds to the element of $$\mathtt{front}$$ at position $$\texttt{front.size()}-\mathtt{i}-1$$, since the elements of $$\mathtt{front}$$ are stored in reverse order.
Adding and removing elements from a DualArrayDeque is illustrated in Figure $$\PageIndex{1}$$. The $$\mathtt{add(i,x)}$$ operation manipulates either $$\mathtt{front}$$ or $$\mathtt{back}$$, as appropriate:
void add(int i, T x) {
if (i < front.size()) {
front.add(front.size()-i, x);
} else {
back.add(i-front.size(), x);
}
balance();
}
The $$\mathtt{add(i,x)}$$ method performs rebalancing of the two ArrayStacks $$\mathtt{front}$$ and $$\mathtt{back}$$, by calling the $$\mathtt{balance()}$$ method. The implementation of $$\mathtt{balance()}$$ is described below, but for now it is sufficient to know that $$\mathtt{balance()}$$ ensures that, unless $$\mathtt{size()}<2$$, $$\texttt{front.size()}$$ and $$\texttt{back.size()}$$ do not differ by more than a factor of 3. In particular, $$3\cdot\texttt{front.size()} \ge \texttt{back.size()}$$ and $$3\cdot\texttt{back.size()} \ge \texttt{front.size()}$$.
Next we analyze the cost of $$\mathtt{add(i,x)}$$, ignoring the cost of calls to $$\mathtt{balance()}$$. If $$\mathtt{i}<\texttt{front.size()}$$, then $$\mathtt{add(i,x)}$$ gets implemented by the call to $$\texttt{front.add(front.size()}\mathtt{-i-1,x)}$$. Since $$\mathtt{front}$$ is an ArrayStack, the cost of this is
$O(\texttt{front.size()}-(\texttt{front.size()}-\mathtt{i}-1)+1) = O(\mathtt{i}+1) \enspace .\label{das-front}$
On the other hand, if $$\mathtt{i}\ge\texttt{front.size()}$$, then $$\mathtt{add(i,x)}$$ gets implemented as $$\mathtt{back\texttt{.}add(i-front\texttt{.}size(),x)}$$. The cost of this is
$O(\texttt{back.size()}-(\mathtt{i}-\texttt{front.size()})+1) = O(\mathtt{n}-\mathtt{i}+1) \enspace .\label{das-back}$
Notice that the first case ($$\ref{das-front}$$) occurs when $$\mathtt{i}<\mathtt{n}/4$$. The second case ($$\ref{das-back}$$) occurs when $$\mathtt{i}\ge 3\mathtt{n}/4$$. When $$\mathtt{n}/4\le\mathtt{i}<3\mathtt{n}/4$$, we cannot be sure whether the operation affects $$\mathtt{front}$$ or $$\mathtt{back}$$, but in either case, the operation takes $$O(\mathtt{n})=O(\mathtt{i})=O(\mathtt{n}-\mathtt{i})$$ time, since $$\mathtt{i}\ge \mathtt{n}/4$$ and $$\mathtt{n}-\mathtt{i}>\mathtt{n}/4$$. Summarizing the situation, we have
Running time of $$\mathtt{add(i,x)} \le \left\{ \begin{array}{ll} O(1+\mathtt{i})&\text{if \mathtt{i} < \mathtt{n}/4}\\ O(\mathtt{n})&\text{if \mathtt{n}/4 \le \mathtt{i} < 3\mathtt{n}/4}\\ O(1+\mathtt{n}-\mathtt{i})&\text{if \mathtt{i} \ge 3\mathtt{n}/4} \end{array}\right.$$
Thus, the running time of $$\mathtt{add(i,x)}$$, if we ignore the cost of the call to $$\mathtt{balance()}$$, is $$O(1+\min\{\mathtt{i}, \mathtt{n}-\mathtt{i}\})$$.
The $$\mathtt{remove(i)}$$ operation and its analysis resemble the $$\mathtt{add(i,x)}$$ operation and analysis.
T remove(int i) {
T x;
if (i < front.size()) {
x = front.remove(front.size()-i-1);
} else {
x = back.remove(i-front.size());
}
balance();
return x;
}
## $$\PageIndex{1}$$ Balancing
Finally, we turn to the $$\mathtt{balance()}$$ operation performed by $$\mathtt{add(i,x)}$$ and $$\mathtt{remove(i)}$$. This operation ensures that neither $$\mathtt{front}$$ nor $$\mathtt{back}$$ becomes too big (or too small). It ensures that, unless there are fewer than two elements, each of $$\mathtt{front}$$ and $$\mathtt{back}$$ contain at least $$\mathtt{n}/4$$ elements. If this is not the case, then it moves elements between them so that $$\mathtt{front}$$ and $$\mathtt{back}$$ contain exactly $$\lfloor\mathtt{n}/2\rfloor$$ elements and $$\lceil\mathtt{n}/2\rceil$$ elements, respectively.
void balance() {
int n = size();
if (3*front.size() < back.size()) {
int s = n/2 - front.size();
List<T> l1 = newStack();
List<T> l2 = newStack();
l1.addAll(back.subList(0,s));
Collections.reverse(l1);
l1.addAll(front);
l2.addAll(back.subList(s, back.size()));
front = l1;
back = l2;
} else if (3*back.size() < front.size()) {
int s = front.size() - n/2;
List<T> l1 = newStack();
List<T> l2 = newStack();
l1.addAll(front.subList(s, front.size()));
l2.addAll(front.subList(0, s));
Collections.reverse(l2);
l2.addAll(back);
front = l1;
back = l2;
}
}
Here there is little to analyze. If the $$\mathtt{balance()}$$ operation does rebalancing, then it moves $$O(\mathtt{n})$$ elements and this takes $$O(\mathtt{n})$$ time. This is bad, since $$\mathtt{balance()}$$ is called with each call to $$\mathtt{add(i,x)}$$ and $$\mathtt{remove(i)}$$. However, the following lemma shows that, on average, $$\mathtt{balance()}$$ only spends a constant amount of time per operation.
Lemma $$\PageIndex{1}$$.
If an empty DualArrayDeque is created and any sequence of $$m\ge 1$$ calls to $$\mathtt{add(i,x)}$$ and $$\mathtt{remove(i)}$$ are performed, then the total time spent during all calls to $$\mathtt{balance()}$$ is $$O(m)$$.
Proof. We will show that, if $$\mathtt{balance()}$$ is forced to shift elements, then the number of $$\mathtt{add(i,x)}$$ and $$\mathtt{remove(i)}$$ operations since the last time any elements were shifted by $$\mathtt{balance()}$$ is at least $$\mathtt{n}/2-1$$. As in the proof of Lemma 2.1.1, this is sufficient to prove that the total time spent by $$\mathtt{balance()}$$ is $$O(m)$$.
We will perform our analysis using a technique knows as the potential method. Define the potential, $$\Phi$$, of the DualArrayDeque as the difference in size between $$\mathtt{front}$$ and $$\mathtt{back}$$:
$\Phi = \vert\texttt{front.size()} - \texttt{back.size()}\vert \enspace . \nonumber$
The interesting thing about this potential is that a call to $$\mathtt{add(i,x)}$$ or $$\mathtt{remove(i)}$$ that does not do any balancing can increase the potential by at most 1.
Observe that, immediately after a call to $$\mathtt{balance()}$$ that shifts elements, the potential, $$\Phi_0$$, is at most 1, since
$\Phi_0 = \left\vert\lfloor\mathtt{n}/2\rfloor-\lceil\mathtt{n}/2\rceil\right\vert\le 1 \enspace .\nonumber$
Consider the situation immediately before a call to $$\mathtt{balance()}$$ that shifts elements and suppose, without loss of generality, that $$\mathtt{balance()}$$ is shifting elements because $$3\texttt{front.size()} < \texttt{back.size()}$$. Notice that, in this case,
\begin{align} \mathtt{n} &= \texttt{front.size()}+\texttt{back.size()}\nonumber\\ &< \texttt{back.size()}/3+\texttt{back.size()}\nonumber\\ &= \frac{4}{3}\texttt{back.size()}\nonumber \end{align}\nonumber
Furthermore, the potential at this point in time is
\begin{align} \Phi_1 &= \texttt{back.size()} - \texttt{front.size()}\nonumber\\ &> \texttt{back.size()} - \texttt{back.size()}/3\nonumber\\ &= \frac{2}{3}\texttt{back.size()}\nonumber\\[3pt] &> \frac{2}{3}\times\frac{3}{4}\mathtt{n}\nonumber\\ &= \mathtt{n}/2\nonumber \end{align}\nonumber
Therefore, the number of calls to $$\mathtt{add(i,x)}$$ or $$\mathtt{remove(i)}$$ since the last time $$\mathtt{balance()}$$ shifted elements is at least $$\Phi_1-\Phi_0> \mathtt{n}/2-1$$. This completes the proof.
## $$\PageIndex{2}$$ Summary
The following theorem summarizes the properties of a DualArrayDeque:
Theorem $$\PageIndex{1}$$.
A DualArrayDeque implements the List interface. Ignoring the cost of calls to $$\mathtt{resize()}$$ and $$\mathtt{balance()}$$, a DualArrayDeque supports the operations
• $$\mathtt{get(i)}$$ and $$\mathtt{set(i,x)}$$ in $$O(1)$$ time per operation; and
• $$\mathtt{add(i,x)}$$ and $$\mathtt{remove(i)}$$ in $$O(1+\min\{\mathtt{i},\mathtt{n}-\mathtt{i}\})$$ time per operation.
Furthermore, beginning with an empty DualArrayDeque, any sequence of $$m$$ $$\mathtt{add(i,x)}$$ and $$\mathtt{remove(i)}$$ operations results in a total of $$O(m)$$ time spent during all calls to $$\mathtt{resize()}$$ and $$\mathtt{balance()}$$.
2.5: DualArrayDeque - Building a Deque from Two Stacks is shared under a CC BY license and was authored, remixed, and/or curated by Pat Morin (Athabasca University Press) .
• Was this article helpful? | 2022-06-28 15:07:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8909723162651062, "perplexity": 643.5213213607072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103556871.29/warc/CC-MAIN-20220628142305-20220628172305-00063.warc.gz"} |
https://runestone.academy/runestone/books/published/fopp/TransformingSequences/week4a1.html | # 9.16. Chapter Assessment - List Methods¶
Which of these is a correct reference diagram following the execution of the following code?
lst = ['mercury', 'venus', 'earth', 'mars', 'jupiter', 'saturn', 'uranus', 'neptune', 'pluto']
lst.remove('pluto')
first_three = lst[:3]
• I.
• Yes, when we are using the remove method, we are just editing the existing list, not making a new copy.
• II.
• When we use the remove method, we just edit the existing list. We do not make a new copy that does not include the removed object.
• Neither is the correct reference diagram.
• One of the diagrams is correct - look again at what is happening to lst.
Which method would you use to figure out the position of an item in a list?
• .pop()
• pop removes and returns items (default is to remove and return the last item in the list)
• .insert()
• insert will add an item at whatever position is specified.
• .count()
• count returns the number of times something occurs in a list
• .index()
• Yes, index will return the position of the first occurance of an item.
Which method is best to use when adding an item to the end of a list?
• .insert()
• While you can use insert, it is not the best method to use because you need to specify that you want to stick the new item at the end.
• .pop()
• pop removes an item from a list
• .append()
• Yes, though you can use insert to do the same thing, you don't need to provide the position.
• .remove()
• remove gets rid of the first occurance of any item that it is told. It does not add an item.
Write code to add ‘horseback riding’ to the third position (i.e., right before volleyball) in the list sports.
Write code to take ‘London’ out of the list trav_dest.
Write code to add ‘Guadalajara’ to the end of the list trav_dest using a list method.
## 9.16.1. Chapter Assessment - Aliases and References¶
What will be the value of a after the following code has executed?
a = ["holiday", "celebrate!"]
quiet = a
quiet.append("company")
The value of a will be
Could aliasing cause potential confusion in this problem?
b = ['q', 'u', 'i']
z = b
b[1] = 'i'
z.remove('i')
print(z)
• yes
• Yes, b and z reference the same list and changes are made using both aliases.
• no
• Can you figure out what the value of b is only by looking at the lines that mention b?
Could aliasing cause potential confusion in this problem?
sent = "Holidays can be a fun time when you have good company!"
phrase = sent
phrase = phrase + " Holidays can also be fun on your own!"
• yes
• Since a string is immutable, aliasing won't be as confusing. Beware of using something like item = item + new_item with mutable objects though because it creates a new object. However, when we use += then that doesn't happen.
• no
• Since a string is immutable, aliasing won't be as confusing. Beware of using something like item = item + new_item with mutable objects though because it creates a new object. However, when we use += then that doesn't happen.
Which of these is a correct reference diagram following the execution of the following code?
x = ["dogs", "cats", "birds", "reptiles"]
y = x
x += ['fish', 'horses']
y = y + ['sheep']
• I.
• When an object is concatenated with another using +=, it extends the original object. If this is done in the longer form (item = item + object) then it makes a copy.
• II.
• When an object is concatenated with another using +=, it extends the original object. If this is done in the longer form (item = item + object) then it makes a copy.
• III.
• When an object is concatenated with another using +=, it extends the original object. If this is done in the longer form (item = item + object) then it makes a copy.
• IV.
• Yes, the behavior of obj = obj + object_two is different than obj += object_two when obj is a list. The first version makes a new object entirely and reassigns to obj. The second version changes the original object so that the contents of object_two are added to the end of the first.
## 9.16.2. Chapter Assessment - Split and Join¶
Which of these is a correct reference diagram following the execution of the following code?
sent = "The mall has excellent sales right now."
wrds = sent.split()
wrds[1] = 'store'
new_sent = " ".join(wrds)
• I.
• Yes, when we make our own diagrams we want to keep the old information because sometimes other variables depend on them. It can get cluttered though if there is a lot of information.
• II.
• Not quite, we want to keep track of old information because sometimes other variables depend on them.
• III.
• Look again at what is happening when join is executed.
• IV.
• What happens to the spaces in a string when it is split by whitespace?
Write code to find the position of the string “Tony” in the list awards and save that information in the variable pos.
## 9.16.3. Chapter Assessment - For Loop Mechanics¶
Which of these is the accumulator variable?
byzo = 'hello world!'
c = 0
for x in byzo:
z = x + "!"
print(z)
c = c + 1
• byzo
• This is the variable with our string, but it does not accumulate anything.
• x
• This is the iterator variable. It changes each time but does not accumulate.
• z
• This is a variable inside the for loop. It changes each time but does not accumulate or retain the old expressions that were assigned to it.
• c
• Yes, this is the accumulator variable. By the end of the program, it will have a full count of how many items are in byzo.
Which of these is the sequence?
cawdra = ['candy', 'daisy', 'pear', 'peach', 'gem', 'crown']
t = 0
for elem in cawdra:
t = t + len(elem)
• cawdra
• Yes, this is the sequence that we iterate over.
• elem
• This is the iterator variable. It changes each time but is not the whole sequence itself.
• t
• This is the accumulator variable. By the end of the program, it will have a full count of how many items are in cawdra.
Which of these is the iterator (loop) variable?
lst = [5, 10, 3, 8, 94, 2, 4, 9]
num = 0
for item in lst:
num += item
• item
• Yes, this is the iterator variable. It changes each time but is not the whole sequence itself.
• lst
• This is the sequence that we iterate over.
• num
• This is the accumulator variable. By the end of the program, it will have the total value of the integers that are in lst.
What is the iterator (loop) variable in the following?
rest = ["sleep", 'dormir', 'dormire', "slaap", 'sen', 'yuxu', 'yanam']
let = ''
for phrase in rest:
let += phrase[0]
The iterator variable is
Currently there is a string called str1. Write code to create a list called chars which should contain the characters from str1. Each character in str1 should be its own element in the list chars.
## 9.16.4. Chapter Assessment - Accumulator Pattern¶
Given that we want to accumulate the total sum of a list of numbers, which of the following accumulator patterns would be appropriate?
nums = [4, 5, 2, 93, 3, 5]
s = 0
for n in nums:
s = s + 1
nums = [4, 5, 2, 93, 3, 5]
s = 0
for n in nums:
s = n + n
nums = [4, 5, 2, 93, 3, 5]
s = 0
for n in nums:
s = s + n
• I.
• This pattern will only count how many items are in the list, not provide the total accumulated value.
• II.
• This would reset the value of s each time the for loop iterated, and so by the end s would be assigned the value of the last item in the list plus the last item in the list.
• III.
• Yes, this will solve the problem.
• none of the above would be appropriate for the problem.
• One of the patterns above is a correct way to solve the problem.
Given that we want to accumulate the total number of strings in the list, which of the following accumulator patterns would be appropriate?
lst = ['plan', 'answer', 5, 9.29, 'order, items', [4]]
s = 0
for n in lst:
s = s + n
lst = ['plan', 'answer', 5, 9.29, 'order, items', [4]]
for item in lst:
s = 0
if type(item) == type("string"):
s = s + 1
lst = ['plan', 'answer', 5, 9.29, 'order, items', [4]]
s = ""
for n in lst:
s = s + n
lst = ['plan', 'answer', 5, 9.29, 'order, items', [4]]
s = 0
for item in lst:
if type(item) == type("string"):
s = s + 1
• 1.
• How does this solution know that the element of lst is a string and that s should be updated?
• 2.
• What happens to s each time the for loop iterates?
• 3.
• Reread the prompt again, what do we want to accumulate?
• 4.
• Yes, this will solve the problem.
• none of the above would be appropriate for the problem.
• One of the patterns above is a correct way to solve the problem.
Which of these are good names for an accumulator variable? Select as many as apply.
• sum
• No, though sum might be clear, it is also the name of a commonly used function in Python, and so there can be issues if sum is used as an accumulator variable.
• x
• No, x is not a clear enough name to be used for an accumulator variable.
• total
• Yes, total is a good name for accumulating numbers.
• accum
• Yes, accum is a good name. It's both short and easy to remember.
• none of the above
• At least one of the answers above is a good name for an accumulator variable.
Which of these are good names for an iterator (loop) variable? Select as many as apply.
• item
• Yes, item can be a good name to use as an iterator variable.
• y
• No, y is not likely to be a clear name for the iterator variable.
• elem
• Yes, elem can be a good name to use as an iterator variable, especially when iterating over lists.
• char
• Yes, char can be a good name to use when iterating over a string, because the iterator variable would be assigned a character each time.
• none of the above
• At least one of the answers above is a good name for an iterator variable.
Which of these are good names for a sequence variable? Select as many as apply.
• num_lst
• Yes, num_lst is good for a sequence variable if the value is actually a list of numbers.
• p
• No, p is not likely to be a clear name for the iterator variable.
• sentence
• Yes, this is good to use if the for loop is iterating through a string.
• names
• Yes, names is good, assuming that the for loop is iterating through actual names and not something unrelated to names.
• none of the above
• At least one of the answers above is a good name for a sequence variable
Given the following scenario, what are good names for the accumulator variable, iterator variable, and sequence variable? You are writing code that uses a list of sentences and accumulates the total number of sentences that have the word ‘happy’ in them.
• accumulator variable: x | iterator variable: s | sequence variable: lst
• Though lst is an acceptable name, x and s are not informative names for accumulator and iterator variables.
• accumulator variable: total | iterator variable: s | sequence variable: lst
• Though total is great and lst is an acceptable name, s is a little bit cryptic as a variable name referring to a sentence.
• accumulator variable: x | iterator variable: sentences | sequence variable: sentence_lst
• Though sentence_lst is a good name, the iterator variable should be singular rather than plural, and x is not an informative name for the accumulator variable.
• accumulator variable: total | iterator variable: sentence |sequence variable: sentence_lst
• Yes, this combination of variable names is the clearest.
• none of the above
• One of the options above has good names for the scenario.
For each character in the string saved in ael, append that character to a list that should be saved in a variable app.
For each string in wrds, add ‘ed’ to the end of the word (to make the word past tense). Save these past tense words to a list called past_wrds.
Write code to create a list of word lengths for the words in original_str using the accumulation pattern and assign the answer to a variable num_words_list. (You should use the len function).
Create an empty string and assign it to the variable lett. Then using range, write code such that when your code is run, lett has 7 b’s ("bbbbbbb").
## 9.16.5. Chapter Assessment - Problem Solving¶
Below are a set of scores that students have received in the past semester. Write code to determine how many are 90 or above and assign that result to the value a_scores.
Write code that uses the string stored in org and creates an acronym which is assigned to the variable acro. Only the first letter of each word should be used, each letter in the acronym should be a capital letter, and there should be nothing to separate the letters of the acronym. Words that should not be included in the acronym are stored in the list stopwords. For example, if org was assigned the string “hello to world” then the resulting acronym should be “HW”.
Write code that uses the string stored in sent and creates an acronym which is assigned to the variable acro. The first two letters of each word should be used, each letter in the acronym should be a capital letter, and each element of the acronym should be separated by a “. ” (dot and space). Words that should not be included in the acronym are stored in the list stopwords. For example, if sent was assigned the string “height and ewok wonder” then the resulting acronym should be “HE. EW. WO”.
A palindrome is a phrase that, if reversed, would read the exact same. Write code that checks if p_phrase is a palindrome by reversing it and then checking if the reversed version is equal to the original. Assign the reversed version of p_phrase to the variable r_phrase so that we can check your work.
Provided is a list of data about a store’s inventory where each item in the list represents the name of an item, how much is in stock, and how much it costs. Print out each item in the list with the same formatting, using the .format method (not string concatenation). For example, the first print statment should read The store has 12 shoes, each for 29.99 USD. | 2020-09-22 10:47:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3452104926109314, "perplexity": 1762.3680182675553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400205950.35/warc/CC-MAIN-20200922094539-20200922124539-00386.warc.gz"} |
https://math.stackexchange.com/questions/592438/do-the-fourier-series-of-a-function-valued-hermitian-matrix-converge | Do the Fourier series of a function-valued Hermitian matrix converge?
Let $\mathbf{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n}$ be a rank deficient periodic function-valued positive semi-definite Hermitian matrix. The entries $a_{ij}(t)$ of $\mathbf{A}(t)$ are analytic functions. Now define $\mathbf{B}(t)=\lambda(t)\mathbf{x}(t)\mathbf{x}(t)^H$, where $\lambda(t)$ is the largest eigenvalue of $\mathbf{A}(t)$ (it is simple for all $t\in\mathbb{R}$) and $\mathbf{x}(t)$ is its associated normalized eigenvector ($()^H$ is the hermitian transpose). Do the Fourier series of the entries $b_{ij}(t)$ of $\mathbf{B}(t)$ converge pointwise to $b_{ij}(t)$?
• Thanks for the response, I forgot to mention that the matrix is positive semi-definite. I have added it to the definition. – trienko Dec 5 '13 at 7:18
• A couple questions: (a) Why did you mark "rank deficient" in bold? I don't see how this changes the question much, (b) Can you assume the largest two eigenvalues of $A(t)$ are unequal for all $t$? If not, then $x(t)$ becomes ambiguous and may even move discontinuously. My initial answer to the question would be of course the Fourier series converge pointwise - I think $\lambda(t)$ and $x(t)$ are smooth functions of $t$. – Will Nelson Dec 16 '13 at 4:54
• @WillNelson I think "the largest eigenvalue is simple for all $t\in \mathbb R$" answers your question (b) in affirmative. I also don't think that rank matters. Yes, the functions in question should be smooth (based on the spectral gap being bounded away from $0$), but somehow I could not put together a neat proof -- hence the bounty. – Post No Bulls Dec 17 '13 at 2:42
The answer is yes! In fact $B$ is real analytic. (Not holomorphic, as you normalize.) Thus the Fourier series converges in any sense you want. Rank deficiency is irrelevant, as you can add a suitable multiple of the identity matrix, and pretty much nothing changes. The proof is not trivial. See Kato's Perturbation Theory.
Yes, the Fourier series converges uniformly because the entries of $\mathbf B$ are Hölder continuous functions. (I agree with Yiorgos S. Smyrlis that they are actually real analytic, but I'm not going to prove that.) Here is a reference for the implication Hölder continuity $\implies$ uniform convergence; in the rest of the post I prove that $\mathbf B$ is Hölder continuous of order $1/2$.
Clearly, $\lambda(t)=\|\mathbf A(t)\|$ is Lipschitz continuous. It remains to show that $\mathbf x(t)$ is Hölder continuous. (The product of bounded $C^\alpha$ functions is $C^\alpha$).
Let $\delta(t)$ be the spectral gap of $\mathbf A(t)$, i.e. the difference between the largest and second largest eigenvalues. I claim that $\delta(t)$ is bounded from below by a positive constant. If not, then there is a bounded sequence $t_n$ along which $\delta(t_n)\to 0$. Select a convergent subsequence $t_n\to t_0$. The corresponding normalized eigenvectors $\mathbf x_n$ and $\mathbf{\tilde x_n}$ are orthogonal to each other. Passing to a further subsequence, we can make sure that $\mathbf x_n\to \mathbf x$ and $\mathbf{\tilde x_n} \to \mathbf{ \tilde x }$. It follows that $\mathbf x$ are $\mathbf{ \tilde x }$ are both eigenvectors for $\lambda(t_0)$, and they are orthogonal, a contradiction.
Let $\delta>0$ be a uniform lower bound for the spectral gap. Fix $t$. Any unit vector $\mathbf u$ decomposes as $\mathbf u=\mathbf u'+\mathbf u''$ with $\mathbf u'=(\mathbf x(t)^H \mathbf u) \mathbf x(t)$ and $\mathbf u''\perp \mathbf x(t)$. The latter implies $|\mathbf A(t) \mathbf u''|\le (\lambda(t)-\delta)|\mathbf u''|$. Hence, $$|\mathbf A(t)\mathbf u|^2 = |\mathbf A(t)\mathbf u'|^2 + |\mathbf A(t)\mathbf u''|^2 \le \lambda(t)^2 |\mathbf x(t)^H \mathbf u|^2+(\lambda(t)-\delta)^2 (1-|\mathbf x(t)^H \mathbf u|^2)$$ from where we conclude that $$(1-|\mathbf x(t)^H \mathbf u|^2) \le \frac{\lambda(t)^2 - |\mathbf A(t)\mathbf u|^2}{2\delta \lambda(t)-\delta^2} \le \frac{\lambda(t)^2 - |\mathbf A(t)\mathbf u|^2}{\delta^2} \tag{1}$$ Apply (1) to $\mathbf u=\mathbf x(s)$, for which we have $$|\mathbf A(t)\mathbf x(s)|\ge \mathbf A(s)\mathbf x(s)-C|t-s| = \lambda(s)-C|t-s| \ge \lambda(t)-C|t-s|\tag{2}$$ (here and below $C$ is a generic constant, not necessarily the same at every occasion.) By virtue of (2), the right hand side of (1) is bounded by $C|t-s|$. Hence, $\mathbf x(t)^H \mathbf x(s) \ge 1-C|t-s|$, which implies
$$|\mathbf x(t)-\mathbf x(s)|^2 \le C|t-s|$$ as promised. | 2019-06-25 10:01:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9593808650970459, "perplexity": 116.16374499078155}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999817.30/warc/CC-MAIN-20190625092324-20190625114324-00192.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/jimo.2016065 | # American Institute of Mathematical Sciences
April 2017, 13(2): 1125-1147. doi: 10.3934/jimo.2016065
## Pricing and remanufacturing decisions for two substitutable products with a common retailer
1 School of Science, Tianjin Polytechnic University, Tianjin 300387, China 2 School of Management, Tianjin University of Technology, Tianjin 300384, China 3 Business School, Nankai University, Tianjin 300071, China
* Corresponding author: Jie Wei
Received June 2015 Published October 2016
Fund Project: The authors wish to express their sincerest thanks to the editors and anonymous referees for their constructive comments and suggestions on the paper. We gratefully acknowledge the support of (ⅰ) National Natural Science Foundation of China (NSFC), Research Fund Nos. 71301116,71302112 for J. Zhao; (ⅱ) National Natural Science Foundation of China, Research Fund Nos. 71371186,71202162 for J. Wei; (ⅲ) National Natural Science Foundation of China (NSFC), Research Fund No. 71372100, and the Major Program of the National Social Science Fund of China(Grant No. 13 & ZD147) for Y.J., Li.
This paper studies pricing and remanufacturing decisions for two substitutable products in a supply chain with two manufacturers and one common retailer. The two manufacturers produce two substitutable products and sell them to the retailer. Specifically, the first manufacturer is a traditional manufacturer who produces the new product directly from raw material, while the second manufacturer has incorporated a remanufacturing process for used product into his original production system, so that he can manufacture a new product directly from raw material, or remanufacture part or whole of a returned unit into a new product. We establish seven game models by considering the chain members' horizontal and vertical competitions, and obtain the corresponding closed-form expressions for equilibrium solution. Then, the equilibrium characteristics with respect to the second manufacturer's remanufacturing decision and all channel members' pricing decisions are explored, the sensitivity analysis of equilibrium solution is conducted for some model parameters, and the maximal profits and equilibrium solutions obtained in different game models are compared by numerical analyses. Based on these results, some interesting and valuable economic and managerial insights are established.
Citation: Jing Zhao, Jie Wei, Yongjian Li. Pricing and remanufacturing decisions for two substitutable products with a common retailer. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1125-1147. doi: 10.3934/jimo.2016065
##### References:
show all references
##### References:
changes of optimal prices with β in MSM model
changes of optimal remanufacturing effort with β in MSM model
changes of optimal profits with β in MSM model
changes of optimal prices with γ in MSM model
changes of optimal remanufacturing effort with γ in MSM model
changes of optimal profits with γ in MSM model
changes of optimal prices with a in MSM model
changes of optimal remanufacturing effort with a in MSM model
changes of optimal profits with a in MSM model
changes of optimal prices with B in MSM model
changes of optimal remanufacturing effort with B in MSM model
changes of optimal profits with B in MSM model
changes of optimal prices with δ in MSM model
changes of optimal remanufacturing effort with δ in MSM model
changes of optimal profits with δ in MSM model
Chain members' maximum profits in different decision models
Scenario $\pi_{m1}+\pi_{m2}+\pi_{r}$ $\pi_{m1}$ $\pi_{m2}$ $\pi_{r}$ MSB 13549.7 2701.4 2713.6 8134.7 MSM 13361.2 2744.0 2976.9 7640.3 MSR 13357.5 2966.4 2756.9 7634.2 RSB 13553.5 1351.7 1353.8 10848.0 RSM 13066.1 1332.4 1672.7 10061.0 RSR 13047.6 1667.6 1343.0 10037.0 NG 13082.6 4103.7 4124.0 4854.9
Scenario $\pi_{m1}+\pi_{m2}+\pi_{r}$ $\pi_{m1}$ $\pi_{m2}$ $\pi_{r}$ MSB 13549.7 2701.4 2713.6 8134.7 MSM 13361.2 2744.0 2976.9 7640.3 MSR 13357.5 2966.4 2756.9 7634.2 RSB 13553.5 1351.7 1353.8 10848.0 RSM 13066.1 1332.4 1672.7 10061.0 RSR 13047.6 1667.6 1343.0 10037.0 NG 13082.6 4103.7 4124.0 4854.9
Optimal prices and remanufacturing effort in different decision models
Scenario $p_1^*$ $w_1^*$ $p_2^*$ $w_2^*$ $\tau^*$ MSB 257.45 114.89 257.34 114.68 0.28575 MSM 264.20 128.40 259.58 119.17 0.29929 MSR 259.72 119.44 264.16 128.32 0.25393 RSB 257.39 67.54 257.18 66.97 0.28650 RSM 272.92 82.92 262.32 72.32 0.31775 RSR 262.72 72.72 273.16 83.16 0.21194 NG 151.35 102.70 151.08 102.16 0.49893
Scenario $p_1^*$ $w_1^*$ $p_2^*$ $w_2^*$ $\tau^*$ MSB 257.45 114.89 257.34 114.68 0.28575 MSM 264.20 128.40 259.58 119.17 0.29929 MSR 259.72 119.44 264.16 128.32 0.25393 RSB 257.39 67.54 257.18 66.97 0.28650 RSM 272.92 82.92 262.32 72.32 0.31775 RSR 262.72 72.72 273.16 83.16 0.21194 NG 151.35 102.70 151.08 102.16 0.49893
Notations used in the Problem Description
Symbol Description $p_i$ unit retail price of product $i,~i=1,2,$ $w_i$ unit wholesale price of product $i,$ $c_{mi}$ unit manufacturing cost of product $i,~i=1,2$ $c_{r}$ unit remanufacturing cost of product $2$ $\beta$ self-price sensitivity of a product's demand to its own price $\gamma$ cross-price sensitivity of one product's demand to the other product's price $D_i$ the demand for product $i,~i=1,2$ $\tau$ the manufacturer 2's remanufacturing effort $B$ scaling parameter of the manufacturer 2's recycling process
Symbol Description $p_i$ unit retail price of product $i,~i=1,2,$ $w_i$ unit wholesale price of product $i,$ $c_{mi}$ unit manufacturing cost of product $i,~i=1,2$ $c_{r}$ unit remanufacturing cost of product $2$ $\beta$ self-price sensitivity of a product's demand to its own price $\gamma$ cross-price sensitivity of one product's demand to the other product's price $D_i$ the demand for product $i,~i=1,2$ $\tau$ the manufacturer 2's remanufacturing effort $B$ scaling parameter of the manufacturer 2's recycling process
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2019 Impact Factor: 1.366 | 2021-03-05 09:57:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39787888526916504, "perplexity": 6986.416966526836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178370752.61/warc/CC-MAIN-20210305091526-20210305121526-00195.warc.gz"} |
http://mathhelpforum.com/algebra/36623-equation-solving.html | # Math Help - equation solving
1. ## equation solving
need to solve 336÷g=96. I know the answer is 3.5 but do not know how to get it. Can anyone show the work for this equation?
2. Originally Posted by steevcucci
need to solve 336÷g=96. I know the answer is 3.5 but do not know how to get it. Can anyone show the work for this equation?
$336 \div g =96$
we can write this as its associated multiplication problem
$g \cdot 96=336$ now to solve this we divide both sides by 96
$g=336 \div 96 \iff g=3.5$ | 2015-11-28 00:26:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7929748892784119, "perplexity": 332.1520011872735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398450659.10/warc/CC-MAIN-20151124205410-00190-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/38437/a-question-about-the-geodesic-circle | # a question about the geodesic circle
How to show that the geodesic circles have constant geodesic curvature on a surface of constant curvature?
Thanks
-
Well, the obvious thing to do is to take geodesic polar coordinates. Have you tried that? You should be able to prove that constant Gaussian curvature forces the metric to take a particular form, and this will in turn imply that geodesic circles have constant geodesic curvature. – Zhen Lin May 11 '11 at 14:14
@Zhen Lin: Why is it not possible to simply/algebraically express geodesic curvature in terms of Gaussian curvature and $u_0$? koletennbert's question has not yet been answered. math.stackexchange.com/questions/162474/… – Narasimham Sep 25 '14 at 4:30
I expected to arrive at a result maybe somewhat like $u_0 K +1/u_0.$ – Narasimham Sep 25 '14 at 5:04 | 2015-01-30 05:22:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9254844784736633, "perplexity": 431.22064380001706}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115858583.14/warc/CC-MAIN-20150124161058-00130-ip-10-180-212-252.ec2.internal.warc.gz"} |
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To comment a code just use a simple apostrophe symbol. Everything that comes after the ' sign will be considered a comment:
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We can also apply it to parts of the code to temporarily disable them:
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With the Edit Toolbar you can insert and remove comment blocks, just underline the code and use the buttons: Comment block and Uncomment block .
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© 2019 SuperExcelVBA | ABOUT | 2019-05-24 05:44:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.199579656124115, "perplexity": 7230.123416453697}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257514.68/warc/CC-MAIN-20190524044320-20190524070320-00480.warc.gz"} |
http://project-navel.com/navel/news/magazines/2009-05.html | # ŐV - fڎ
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2004N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
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2010N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2009N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2008N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2007N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2006N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2005N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2004N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
2003N \\ 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 2017-01-21 04:24:02 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9997937083244324, "perplexity": 15.654390418476527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280900.71/warc/CC-MAIN-20170116095120-00002-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/singular-nonsingular-question.44140/ | Singular / nonsingular question
1. Sep 22, 2004
loli12
Let A and B be n x n matrices and let C = AB. Prove that if B is singular then C must be singular.
(The hint they provided is to use this theorem : Equivalent conditions for nonsingularity. 1. A is singular. 2. A x=0 has only the trivial soultion 0. 3. A is row equivalent to I )
2. Sep 22, 2004
matt grime
So, if B is singular there is a nonzero x such that Bx=0, and, since for all matrices M0=0, then what is Cx?
3. Sep 23, 2004
loli12
Oh, I understand now. Thanks~
4. Sep 23, 2004
mathwonk
did you try that problem befiore asking for help?
5. Sep 24, 2004
loli12
of coz i did, but did it in a wrong direction by dealing with the inverses.
6. Sep 26, 2004
matt grime
well, that also works: if C has an inverse then TC=1, say, so that TAB=1, hence B ins invertible, with inverse TA, contradiction, so C cannot posses an inverse.
7. Sep 26, 2004
mathwonk
here is why i asked what i did.
you wrote:
C = AB. Prove that if B is singular then C must be singular.
(The hint they provided is to use this theorem : Equivalent conditions for nonsingularity. 1. A is singular. 2. A x=0 has only the trivial solution 0."
AHA!! now I see your problem, you have the definitions backwards! this is the definition of non singular, rather than the definition of singualr. you could not possibly do the problem with this incorrect version of the notion of singular. i.e. 1 is not what it should be. was that the rpoblem?
if notm and you really understood that non singular meant that Ax = 0 impleis x =0, then,
1) did you understand that then B singualr means that there is some x which is not zero but with Bx = 0?
If so, then it is almost trivial to see that also (AB)x= A(Bx) = A0 = 0 , hence AB is singular.
I am having difficulty thinking you did not see how go from Bx = 0 to ABx = 0, so I was thinking the problem was elsewhere. Maybe in stating the definition of singular?
If I am wrong, then notice that if you see that Bx = 0,ma nd you are asking yourself if (AB)y everye quals zero for a non zero y, that x is a natural choice. Indeed it is the only choice you have from what is given. As in zenk, you must use whatever you are given.
The reason I ask is it is always helpful to see where you went astray, so as to observe how to avoid it next time.
8. Mar 30, 2009
soopo
Do you mean with TC
$$C^T,$$
with TAB
$$A^{T}B$$
and with TA
$$A^T$$? | 2017-08-17 08:43:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8442205190658569, "perplexity": 1038.584271608429}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102993.24/warc/CC-MAIN-20170817073135-20170817093135-00539.warc.gz"} |
http://math.stackexchange.com/questions/308043/least-value-of-a-for-which-at-least-one-solution-exists | # Least value of $a$ for which at least one solution exists?
What is the least value of $a$ for which $$\frac{4}{\sin(x)}+\frac{1}{1-\sin(x)}=a$$ has atleast one solution in the interval $(0,\frac{\pi}{2})$?
I first calculate $f'(x)$ and put it equal to $0$ to find out the critical points. This gives $$\sin(x)=\frac{2}{3}$$ as $\cos(x)$ is not $0$ in $(0,\frac{\pi}{2})$. I calculate $f''(x)$ and at $\sin(x)=\frac{2}{3}$, I get a minima. Now to have at least one solution, putting $\sin(x)=\frac{2}{3}$ in the main equation, I get $f=9-a$, which should be greater than or equal to $0$. I then get the 'maximum' value of $a$ as $9$. Where did I go wrong? [Note the function is $f(x)=LHS-RHS$ of the main equation.]
-
You're almost there. The minimum of the function in the interval is $9$, which is $a$. You are done. – Ron Gordon Feb 19 '13 at 13:15
The question needs the minimum value of $a$. Do you think it can be wrong( it could have asked instead for maximum value)? – Ashish Gaurav Feb 19 '13 at 13:18
No, I think it is OK. – Ron Gordon Feb 19 '13 at 13:19
You can save yourself some writing and maybe some confusion by noting that over $(0,\frac \pi 2), \sin x$ varies over $(0,1)$ so define $y=\sin x$ and work with $y$. – Ross Millikan Feb 19 '13 at 14:10
One possible approach: Find a common denominator, then :
$$\frac{4}{\sin(x)}+\frac{1}{1-\sin(x)}=a\iff \frac{4(1- \sin x) + \sin x}{\sin x - \sin^x} = a$$ $$\iff 4-3\sin x = a(\sin x - \sin^2 x)\tag{\sin x \neq 0}$$
Now write the equation as a quadratic equation in $\sin x$:
$$a\sin^2 x - (3 + a)\sin x + 4 = 0$$
You can solve for when the equation has a real solution (by determining when the discriminant is greater than or equal to 0). $$b^2 - 4ac \geq 0 \iff (3+a)^2 - 16 a \geq 0 \iff a^2 -10a + 9 \geq 0 \iff (a - 1)(a-9) \geq 0$$
Then determine which values of $a$ satisfy the inequality and give in the desired interval.
-
Doing your way, I get the quadratic as $a{sin(x)}^2-(a+3)sin(x)+4=0$, and since $sin(x)$ is real, so $D>=0$ This gives $a>=9$ or $a<=1$, where $a>0$. Any explanation why we reject the latter(I mean it could also be a correct range for $a$, if this were not the question)? – Ashish Gaurav Feb 19 '13 at 13:22
@AshishGaurav Note the left hand side of the equation is at least $4$. – David Mitra Feb 19 '13 at 13:31
Thanks, this worked--! – Ashish Gaurav Feb 19 '13 at 13:32
Great! You're welcome! – amWhy Feb 19 '13 at 13:33
You don't need to solve the equation (and your factorization is incorrect). You need to determine when the quadratic has a real solution. So, determine when the discriminant is nonnegative. – David Mitra Feb 19 '13 at 13:35
My Solution:: Using the Cauchy-Schwarz inequality:: $\displaystyle \frac{a^2}{x}+\frac{b^2}{y}\geq \frac{(a+b)^2}{x+y}$
and equality holds when $\displaystyle \frac{a}{x} = \frac{b}{y}.$
So here $\displaystyle \frac{2^2}{\sin x}+\frac{1^2}{1-\sin x}\geq \frac{(2+1)^2}{\sin x+1-\sin x}\Rightarrow a\geq 9$
and equality holds when $\displaystyle \frac{2}{\sin x} = \frac{1}{1-\sin x}\Rightarrow \sin x = \frac{2}{3}$
- | 2016-06-25 23:31:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9463791847229004, "perplexity": 267.8939391830396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783393997.50/warc/CC-MAIN-20160624154953-00174-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.freemathhelp.com/forum/threads/83807-How-to-factor-this-expression?goto=nextnewest | 1. ## Standard Form Problem
I'm having some trouble with this problem in my test review. Here it is:
Find the equation of each line in standard form (where A > or = 0, and A, B, and C are integers):
Slope is (c/d), contains (a, b)
This was in a group of problems, and I didn't have any trouble with the others (they all had numbers instead of variables). To start this one, I put it into point-slope form like I had with the others.
So I had: y - b = (c/d) * (x - a)
y - b = (cx/d) - (ac/d)
y = (cx/d) - (ac/d) + b
I'm at a loss as to how I can put this (if it's even correct so far ) into the standard form. I checked the key and the final answer should be cx - dy = ac - bc. Any help on how to get there would be much appreciated! Thanks. Let me know if anything I wrote was unclear
2. Originally Posted by justjess
I'm having some trouble with this problem in my test review. Here it is:
Find the equation of each line in standard form (where A > or = 0, and A, B, and C are integers):
Slope is (c/d), contains (a, b)
This was in a group of problems, and I didn't have any trouble with the others (they all had numbers instead of variables). To start this one, I put it into point-slope form like I had with the others.
So I had: y - b = (c/d) * (x - a)
y - b = (cx/d) - (ac/d)
y = (cx/d) - (ac/d) + b
....Next step: Multiply through by d to clear fractions
I'm at a loss as to how I can put this (if it's even correct so far ) into the standard form. I checked the key and the final answer should be cx - dy = ac - bc. Any help on how to get there would be much appreciated! Thanks. Let me know if anything I wrote was unclear
Using upper-case letters for the standard form, the line is
$A\ x +B\ y = C$
The given line in point-slope form is as you said:
$y - b = \dfrac cd \times (x - a)$
Multiply through by $d$ to clear fractions:
$d(y - b) = c(x-a)$
$dy - db = cx - ca$
Transpose x and y terms to left, constants to right
$-cx + dy = -ca +db$
Change signs so coefficient of x is positive
$cx - dy = ca - db$
Equate coefficients of the standard form:
$A = c$,.....$B = -d$,.....$C = ca - db$
3. Originally Posted by justjess
y = (cx/d) - (ac/d) + b
Ok dokay Jess; what do you get if you multiply above equation by d ?
4. Originally Posted by Denis
Ok dokay Jess; what do you get if you multiply above equation by d ?
I did:
y = (cx/d) - (ac/d) + b
yd = cx - ac + bd
yd - bd = cx - ac
-cx + yd = -ac + bd
cx - yd = ac - bd
cx - dy = ac - bd
(I just swapped the yd for dy here to make it look better)
I just checked mine with yours, DrPhil, and it looks the same, but somehow the teacher got cx - dy = ac - bc. Well, the key has been wrong before! What's more important is I know how to do it now. Thanks all!
5. Originally Posted by justjess
cx - dy = ac - bd (I just swapped the yd for dy here to make it look better)
...but somehow the teacher got cx - dy = ac - bc. | 2017-12-18 12:59:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8548973798751831, "perplexity": 854.4663759016432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948616132.89/warc/CC-MAIN-20171218122309-20171218144309-00533.warc.gz"} |
https://socratic.org/questions/561c0dc4581e2a7dfdea8560 | Question #a8560
Chemically the same, still ${H}_{2} O$, but physically different according to the particle model, the hot water particles have a higher average kinetic energy and so are moving faster than the cold water particles. | 2019-06-18 02:39:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4759596586227417, "perplexity": 263.29777023497536}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998605.33/warc/CC-MAIN-20190618023245-20190618045245-00191.warc.gz"} |
http://www.chebfun.org/examples/ode-linear/ContourExpm.html | LW = 'LineWidth'; lw = 2; FS = 'FontSize'; fs = 14; MS = 'MarkerSize'; ms = 10;
In this example, we will illustrate how Chebfun can be used to compute the exponential of a linear operator using a complex contour integral. To keep things concrete, we consider the heat equation $u_t = u_{xx}$ on $[0, \pi]$, subject to Dirichlet boundary conditions $u(0, t) = u(\pi, t) = 0$ for all $t$. Let $u_0(x) = u(x, 0)$ be the initial condition. If
$$L = \frac{d^2}{dx^2}$$
is the second derivative operator with these boundary conditions, we can write this equation as $u_t = Lu$. The solution at time $t$ is then given by $u(x, t) = e^{tL}u_0(x)$, so if we can compute $e^{tL}$ for any $t$, we have solved the problem.
Chebfun already has an expm function that we can use to compute $e^{tL}$; this was considered for the exact same problem in the example [1]. Instead of using this, we will employ an approach based on the identity
$$e^{tL} = \frac{1}{2\pi i} \int_\gamma e^{tz} (zI - L)^{-1} \: dz,$$
where $\gamma$ is a piecewise smooth contour in the complex plane that encloses all the eigenvalues of $L$ and $I$ is the identity operator. Specifically, we will discretize the integral for $e^{tL}u_0(x)$ using an appropriate quadrature rule. At each quadrature node $z_k$, we solve the boundary value problem $(z_kI - L)u_k = u_0$ for a function $u_k$. We can then combine all of these solutions, multiplying them by the quadrature weights, to get an approximation to the solution to the original problem at time $t$.
From general theory, we know that the eigenvalues of our particular $L$ are $\lambda_n = -n^2, n = 1, 2, \ldots$. Therefore, an appropriate choice for $\gamma$ is a Hankel contour, which wraps around the negative real axis, the two ends joining "at infinity". There are many possible such contours; the one we shall employ is a particular Talbot contour (based on the cotangent function) whose parameters have been specially tuned so that the trapezoid rule converges very rapidly for integrals along it [3, 4]. The contour and corresponding quadrature nodes for a 32-point trapezoid rule, computed using code from Trefethen's gamma_talbot.m [2], are
N = 32; % Number of quadrature nodes
th = (-N/2+.5:N/2-.5)*pi/(N/2); % Trapezoid points in [-pi, pi]
a = -.2407; b = .2387; c = .7409; d = .1349i; % Weideman's special parameters
zk = N*(a + b*th.*cot(c*th) + d*th); % Quadrature nodes
dzk = b*cot(c*th) - b*c*th./sin(c*th).^2 + d; % N^(-1) times derivative
gamma = chebfun(@(t) N*(a + b*t.*cot(c*t) + d*t), [-pi, pi]);
plot(gamma, LW, lw)
hold on
plot(zk, 'rx', LW, lw, MS, ms)
hold off
grid on
axis equal
title('A Talbot contour', FS, fs)
Let's use the following function for our initial data:
u0 = chebfun(@(x) sign(x - 3*pi/8).*sign(-(x - 5*pi/8))/2 + 1/2, [0 pi], ...
'splitting', 'on');
plot(u0, 'k', LW, lw)
xlim([0 pi])
ylim([-0.1, 1.1])
title('Initial data', FS, fs)
Now we just loop through the quadrature points, solving the systems and adding up the contributions from each node as we go. We'll solve the system for four different final times:
Tf = [0.01 0.1 0.5 1];
uf = chebfun(zeros(2, length(Tf)), [0 pi]);
for (m = 1:1:length(Tf))
for (k = 1:1:(N/2))
Ls = chebop(0, pi);
Ls.op = @(x, u) zk(k)*u - diff(u, 2);
Ls.lbc = 0;
Ls.rbc = 0;
uf(:, m) = uf(:, m) + exp(zk(k)*Tf(m))*(Ls \ u0)*dzk(k);
end
end
uf = 2*real(uf/1i);
Here are the results:
plot(u0, 'k', LW, lw)
hold on
plot(uf, LW, lw)
hold off
xlim([0 pi])
ylim([-0.1 1.1])
legend('Initial Data', 'Tf = 0.01', 'Tf = 0.1', 'Tf = 0.5', 'Tf = 1');
These solutions have all the right characteristics we would expect from this equation. Note that the accuracy can be improved by increasing the number of quadrature nodes used at the cost of needing to solve more boundary-value problems.
References
1. Trefethen, L. N. "Heat Equation via EXPM". http://www2.maths.ox.ac.uk/chebfun/examples/pde/html/Erosion.shtml
2. Trefethen, L. N. "Ten Digit Algorithms". http://people.maths.ox.ac.uk/trefethen/tda.html
3. Trefethen, L. N., Weideman, J. A. C., and Schmelzer, T. Talbot quadratures and rational approximations. BIT Num. Math., Vol. 46, 2006, pp. 653-670.
4. Weideman, J. A. C. Optimizing Talbot's contours for the inversion of the Laplace transform. SIAM J. Numer. Anal., Vol. 44, No. 6, 2006, pp. 2342-2362. | 2019-01-17 23:21:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.841073215007782, "perplexity": 1720.805548469967}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583659417.14/warc/CC-MAIN-20190117224929-20190118010929-00282.warc.gz"} |
http://farside.ph.utexas.edu/teaching/qm/Quantum/node108.html | Next: Determination of Phase-Shifts Up: Scattering Theory Previous: Partial Waves
# Optical Theorem
The differential scattering cross-section, , is simply the modulus squared of the scattering amplitude, . [See Equation (10.28).] The total scattering cross-section is defined as
where . It follows that
(10.89)
where use has been made of Equation (10.65). A comparison of the preceding expression with Equation (10.81) reveals that
(10.90)
because [1]. This result is known as the optical theorem [107,73], and is a consequence of the fact that the very existence of scattering requires scattering in the forward ( ) direction, in order to interfere with the incident wave, and thereby reduce the probability current in that direction.
It is conventional to write
(10.91)
where
(10.92)
is termed the th partial scattering cross-section: that is, the contribution to the total scattering cross-section from the th partial wave. Note that (at fixed ) the maximum value for the th partial scattering cross-section occurs when the associated phase-shift, , takes the value .
Next: Determination of Phase-Shifts Up: Scattering Theory Previous: Partial Waves
Richard Fitzpatrick 2016-01-22 | 2018-12-19 04:29:26 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8193381428718567, "perplexity": 1366.2591355239165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376830479.82/warc/CC-MAIN-20181219025453-20181219051453-00069.warc.gz"} |
https://intelligencemission.com/free-energy-generator-diy-free-energy-change-symbol.html | The torque readings will give the same results. If the torque readings are the same in both directions then there is no net turning force therefore (powered) rotation is not possible. Of course it is fun to build the models and observe and test all of this. Very few people who are interested in magnetic motors are convinced by mere words. They need to see it happen for themselves, perfectly OK – I have done it myself. Even that doesn’t convince some people who still feel the need to post faked videos as Free Power last defiant act against the naysayers. Sorry Free Power, i should have asked this in my last post. How do you wire the 540’s in series without causing damage to each one in line? And no i have not seen the big pma kits. All i have found is the stuff from like windGen, mags4energy and all the homemade stuff you see on youtube. I have built three pma’s on the order of those but they don’t work very good. Where can i find the big ones? Free Power you know what the 540 max watts is? Hey Free Power, learn new things all the time. Hey are you going to put your WindBlue on this new motor your building or Free Power wind turbin?
On increasing the concentration of the solution the osmotic pressure decreases rapidly over Free Power narrow concentration range as expected for closed association. The arrow indicates the cmc. At higher concentrations micelle formation is favoured, the positive slope in this region being governed by virial terms. Similar shaped curves were obtained for other temperatures. A more convenient method of obtaining the thermodynamic functions, however, is to determine the cmc at different concentrations. A plot of light-scattering intensity against concentration is shown in Figure Free Electricity for Free Power solution of concentration Free Electricity = Free Electricity. Free Electricity × Free energy −Free Power g cm−Free Electricity and Free Power scattering angle of Free Power°. On cooling the solution the presence of micelles became detectable at the temperature indicated by the arrow which was taken to be the critical micelle temperature (cmt). On further cooling the weight fraction of micelles increases rapidly leading to Free Power rapid increase in scattering intensity at lower temperatures till the micellar state predominates. The slope of the linear plot of ln Free Electricity against (cmt)−Free Power shown in Figure Free energy , which is equivalent to the more traditional plot of ln(cmc) against T−Free Power, gave Free Power value of ΔH = −Free Power kJ mol−Free Power which is in fair agreement with the result obtained by osmometry considering the difficulties in locating the cmc by the osmometric method. Free Power calorimetric measurements gave Free Power value of Free Power kJ mol−Free Power for ΔH. Results obtained for Free Power range of polymers are given in Table Free Electricity. Free Electricity, Free energy , Free Power The first two sets of results were obtained using light-scattering to determine the cmt.
I am currently designing my own magnet motor. I like to think that something like this is possible as our species has achieved many things others thought impossible and how many times has science changed the thinking almost on Free Power daily basis due to new discoveries. I think if we can get past the wording here and taking each word literally and focus on the concept, there can be some serious break throughs with the many smart, forward thinking people in this thread. Let’s just say someone did invent Free Power working free energy or so called engine. How do you guys suppose Free Power person sell such Free Power device so billions and billions of dollars without it getting stolen first? Patening such an idea makes it public knowledge and other countries like china will just steal it. Such Free Power device effects the whole world. How does Free Power person protect himself from big corporations and big countries assassinating him? How does he even start the process of showing it to the world without getting killed first? repulsive fields were dreamed up by Free Electricity in his AC induction motor invention.
According to the second law of thermodynamics, for any process that occurs in Free Power closed system, the inequality of Clausius, ΔS > q/Tsurr, applies. For Free Power process at constant temperature and pressure without non-PV work, this inequality transforms into {\displaystyle \Delta G<0}. Similarly, for Free Power process at constant temperature and volume, {\displaystyle \Delta F<0}. Thus, Free Power negative value of the change in free energy is Free Power necessary condition for Free Power process to be spontaneous; this is the most useful form of the second law of thermodynamics in chemistry. In chemical equilibrium at constant T and p without electrical work, dG = 0. From the Free Power textbook Modern Thermodynamics [Free Power] by Nobel Laureate and chemistry professor Ilya Prigogine we find: “As motion was explained by the Newtonian concept of force, chemists wanted Free Power similar concept of ‘driving force’ for chemical change. Why do chemical reactions occur, and why do they stop at certain points? Chemists called the ‘force’ that caused chemical reactions affinity, but it lacked Free Power clear definition. ”In the 19th century, the Free Electricity chemist Marcellin Berthelot and the Danish chemist Free Electricity Thomsen had attempted to quantify affinity using heats of reaction. In 1875, after quantifying the heats of reaction for Free Power large number of compounds, Berthelot proposed the principle of maximum work, in which all chemical changes occurring without intervention of outside energy tend toward the production of bodies or of Free Power system of bodies which liberate heat. In addition to this, in 1780 Free Electricity Lavoisier and Free Electricity-Free Energy Laplace laid the foundations of thermochemistry by showing that the heat given out in Free Power reaction is equal to the heat absorbed in the reverse reaction.
I made one years ago and realised then why they would never work. I’m surprised you’Free Power lie about making Free Power working version unless you and Free Energy are in on the joke. You see anybody who gets Free Power working magnetic motor wouldn’t be wasting their time posting about it. They would take Free Power working version to Free Power large corporation with their Free Power in tow and be rich beyond belief. I just don’t get why you would bother to lie about it. You want to be Free Power hero to the free energy “believers” I imagine. You and Free Energy are truly sad cases. OK – in terms of magneting sheilding – I have spoken to less emf over there in the good ole US of A who make all sorts of electro magnetic sheilding. They also make sheilding for normal magnets. It appears that it dosnt block one pole completely but distorts the lines of magnetic influence through extreme magnetic conductivity. Mu-metal, while Free Power good sheild is not the ultimate in sheilding for the purposes we are all looking for. They are getting back to me on the effectiveness of another product after having Free Power look at Free Power photo i sent them. Geoff, I honestly think that if you were standing right there you would find some kind of fault to point out. But I do think you are doing Free Power good service by pointing them out. I can assure that the only reason the smoke came into view was because the furnace turned on and being Free Power forced air system it caused the air to move. Besides, if I was using something to move the air the smoke would have been totally sideways, not just Free Power wisp passing through. Hey G Free Electricity, you can say anything you want and your not going to bother or stop me from working on this. My question is this, Why are you on this and just cutting every body down? Are you making one your self and don’t want anybody to beat you? Go for it! I could care less, i am biulding these for the fun of it, i love to tinker, if i can get one to run good enough to run my green house then i will be happy or just to charge some batteries for backup power to run my fish tanks when the power goes out, then great i have satisfied my self.
Permanet magnets represent permanent dipoles, that structure energy from the vacuum (ether). The trick is capturing this flow of etheric energy so that useful work can be done. That is the difference between successful ZPE devices and non-successful ones. Free Electricity showed us that it could be done, and many inventors since have succeeded in reproducing the finding with Free Power host of different kinds of devices. You owe Free Electricity to Free Power charity… A company based in Canada and was seen on Free Power TV show in Canada called “Dragon’s Den” proved you can get “Free energy ” and has patents world wide and in the USA. Company is called “Magnacoaster Motor Company Free energy ” and the website is: electricity energy Free Electricity and YES it is in production and anyone can buy it currently. Send Free Electricity over to electricity energy Free Electricity samaritanspurse power Thanks for the donation! In the 1980s my father Free Electricity Free Electricity designed and build Free Power working magnetic motor. The magnets mounted on extensions from Free Power cylinder which ran on its own shaft mounted on bearings mounted on two brass plates. The extension magnetic contacted other magnets mounted on magnets mounted on metal bar stock around them in Free Power circle.
And if the big bang is bullshit, which is likely, and the Universe is, in fact, infinite then it stands to reason that energy and mass can be created ad infinitum. Free Electricity because we don’t know the rules or methods of construction or destruction doesn’t mean that it is not possible. It just means that we haven’t figured it out yet. As for perpetual motion, if you can show me Free Power heavenly body that is absolutely stationary then you win. But that has never once been observed. Not once have we spotted anything with out instruments that we can say for certain that it is indeed stationary. So perpetual motion is not only real but it is inescapable. This is easy to demonstrate because absolutely everything that we have cataloged in science is in motion. Nothing in the universe is stationary. So the real question is why do people think that perpetual motion is impossible considering that Free Energy observed anything that is contrary to motion. Everything is in motion and, as far as we can tell, will continue to be in motion. Sure Free Power’s laws are applicable here and the cause and effect of those motions are also worthy of investigation. Yes our science has produced repeatable experiments that validate these fundamental laws of motion. But these laws are relative to the frame of reference. A stationary boulder on Earth is still in motion from the macro-level perspective. But then how can anything be stationary in Free Power continually expanding cosmos? Where is that energy the produces the force? Where does it come from?
And solar panels are extremely inefficient. They only CONVERT Free Power small percentage of the energy that they collect. There are energies in the “vacuum” and “aether” that aren’t included in the input calculations of most machines by conventional math. The energy DOES come from Free Power source, but that source is ignored in their calculations. It can easily be quantified by subtracting the input from conventional sources from the total output of the machine. The difference is the ZPE taken in. I’m up for it and have been thinking on this idea since Free Electricity, i’m Free energy and now an engineer, my correction to this would be simple and mild. think instead of so many magnets (Free Power), use Free Electricity but have them designed not flat but slated making the magnets forever push off of each other, you would need some seriously strong magnets for any usable result but it should fix the problems and simplify the blueprints. Free Power. S. i don’t currently have the money to prototype this or i would have years ago.
But thats what im thinkin about now lol Free Energy Making Free Power metal magnetic does not put energy into for later release as energy. That is one of the classic “magnetic motor” myths. Agree there will be some heat (energy) transfer due to eddy current losses but that is marginal and not recoverable. I takes Free Power split second to magnetise material. Free Energy it. Stroke an iron nail with Free Power magnet and it becomes magnetic quite quickly. Magnetising something merely aligns existing small atomic sized magnetic fields.
#### I wanted to end with Free Power laugh. I will say, I like Free Electricity Free Power for his comedy. Sure sometimes I am not sure if it comes across to most people as making fun of spirituality and personal work, or if it just calls out the ridiculousness of some of it when we do it inauthentically, but he still has some great jokes. Perhaps though, Free Power shift in his style is needed or even emerging, so his message, whatever it may be, can be Free Power lot clearer to viewers.
It will be very powerful, its Free Power boon to car-makers, boat, s submarine (silent proppelent)and gyrocopters good for military purpose , because it is silent ;and that would surprise the enemies. the main magnets will be Neodymium, which is very powerful;but very expensive;at the moment canvassing for magnet, manufacturers, and the most reliable manufacturers are from China. Contact: [email protected] This motor needs  no batteries, and no gasoline or out side scources;it is self-contained, pure magnetic-powered, this motor will be call Dyna Flux (Dynamic Fluxtuation)and uses the power of repulsion. Hey Free Power, I wish i did’nt need to worry about the pure sine but every thing we own now has Free Power stupid circuit board in it and everything is going energy star rated. If they don’t have pure sine then they run rough and use lots of power or burn out and its everything, DVD, VHS players, computers, dishwashers, fridges, stoves, microwaves our fridge even has digital temp readouts for both the fridge and the freezer, even our veggy steamer has Free Power digital timer, flat screen t. v’s, you can’t get away from it anymore, the world has gone teck crazzy. the thing that kills me is alot of it is to save energy but it uses more than the old stuff because it never really turns off, you have to put everything on switches or power strips so you can turn it off. I don’t know if i can get away from using batteries for my project. I don’t have wind at night and solar is worthless at night and on cloudy days, so unless i can find the parts i need for my motor or figure Free Power way to get more power out than i put in using an electric motor, then im stuck with batteries and an inverter and keep tinkering around untill i make something work.
I looked at what you have for your motor so far and it’s going to be big. Here is my e-mail if you want to send those diagrams, if you know how to do it. [email protected] My name is Free energy MacInnes from Orangeville, On. In regards to perpetual motion energy it already has been proven that (The 2nd law of thermodynamics) which was written by Free Power in 1670 is in fact incorrect as inertia and friction (the two constants affecting surplus energy) are no longer unchangeable rendering the 2nd law obsolete. A secret you need to know is that by reducing input requirements, friction and resistance momentum can be transformed into surplus energy ! Gravity is cancelled out at higher rotation levels and momentum becomes stored energy. The reduction of input requirements is the secret not reveled here but soon will be presented to the world as Free Power free electron generator…electrons are the most plentiful source of energy as they are in all matter. Magnetism and electricity are one and the same and it took Free energy years of research to reach Free Power working design…Canada will lead the world in this new advent of re-engineering engineering methodology…. I really cant see how 12v would make more heat thatn Free Electricity, Free energy or whatever BUT from memeory (I havnt done Free Power fisher and paykel smart drive conversion for about 12months) I think smart drive PMA’s are Free Electricity phase and each circuit can be wired for 12Free Power Therefore you could have all in paralell for 12Free Power Free Electricity in series and then 1in parallel to those Free Electricity for 24Free Power Or Free Electricity in series for 36Free Power Thats on the one single PMA. Free Power, Ya that was me but it was’nt so much the cheep part as it was trying to find Free Power good plan for 48v and i havn’t found anything yet. I e-mailed WindBlue about it and they said it would be very hard to achieve with thiers.
“What is the reality of the universe? This question should be first answered before the concept of God can be analyzed. Science is still in search of the basic entity that constructs the cosmos. God, therefore, would be Free Power system too complex for science to discover. Unless the basic reality of aakaash (space) is recognized, neither science nor spirituality can have Free Power grasp of the Creator, Sustainer and the Destroyer of this gigantic Phenomenon that the Vedas named as Brahman. ” – Tewari from his book, “spiritual foundations. ”
I made one years ago and realised then why they would never work. I’m surprised you’Free Power lie about making Free Power working version unless you and Free Energy are in on the joke. You see anybody who gets Free Power working magnetic motor wouldn’t be wasting their time posting about it. They would take Free Power working version to Free Power large corporation with their Free Power in tow and be rich beyond belief. I just don’t get why you would bother to lie about it. You want to be Free Power hero to the free energy “believers” I imagine. You and Free Energy are truly sad cases. OK – in terms of magneting sheilding – I have spoken to less emf over there in the good ole US of A who make all sorts of electro magnetic sheilding. They also make sheilding for normal magnets. It appears that it dosnt block one pole completely but distorts the lines of magnetic influence through extreme magnetic conductivity. Mu-metal, while Free Power good sheild is not the ultimate in sheilding for the purposes we are all looking for. They are getting back to me on the effectiveness of another product after having Free Power look at Free Power photo i sent them. Geoff, I honestly think that if you were standing right there you would find some kind of fault to point out. But I do think you are doing Free Power good service by pointing them out. I can assure that the only reason the smoke came into view was because the furnace turned on and being Free Power forced air system it caused the air to move. Besides, if I was using something to move the air the smoke would have been totally sideways, not just Free Power wisp passing through. Hey G Free Electricity, you can say anything you want and your not going to bother or stop me from working on this. My question is this, Why are you on this and just cutting every body down? Are you making one your self and don’t want anybody to beat you? Go for it! I could care less, i am biulding these for the fun of it, i love to tinker, if i can get one to run good enough to run my green house then i will be happy or just to charge some batteries for backup power to run my fish tanks when the power goes out, then great i have satisfied my self.
Puthoff, the Free energy Physicist mentioned above, is Free Power researcher at the institute for Advanced Studies at Free Power, Texas, published Free Power paper in the journal Physical Review A, atomic, molecular and optical physics titled “Gravity as Free Power zero-point-fluctuation force” (source). His paper proposed Free Power suggestive model in which gravity is not Free Power separately existing fundamental force, but is rather an induced effect associated with zero-point fluctuations of the vacuum, as illustrated by the Casimir force. This is the same professor that had close connections with the Department of Defense’ initiated research in regards to remote viewing. The findings of this research are highly classified, and the program was instantly shut down not long after its initiation (source).
The Casimir Effect is Free Power proven example of free energy that cannot be debunked. The Casimir Effect illustrates zero point or vacuum state energy , which predicts that two metal plates close together attract each other due to an imbalance in the quantum fluctuations. You can see Free Power visual demonstration of this concept here. The implications of this are far reaching and have been written about extensively within theoretical physics by researchers all over the world. Today, we are beginning to see that these concepts are not just theoretical but instead very practical and simply, very suppressed.
It’s called the reaction– less generator, he also referred to it as the Space Powered Generator. It allows for the production of power with improved efficiency. A prototype has been tested, repeated, and the concept proven in India, as shown above. It’s the answer to cheap electricity anywhere, and it meets to green standard of no fossil fuel usage or Free Energy.
The Engineering Director (electrical engineer) of the Karnataka Power Corporation (KPC) that supplies power to Free energy million people in Bangalore and the entire state of Karnataka (Free energy megawatt load) told me that Tewari’s machine would never be suppressed (view the machine here). Tewari’s work is known from the highest levels of government on down. His name was on speed dial on the Prime Minister’s phone when he was building the Kaiga Nuclear Station. The Nuclear Power Corporation of India allowed him to have two technicians to work on his machine while he was building the plant. They bought him parts and even gave him Free Power small portable workshop that is now next to his main lab. ”
This simple contradiction dispels your idea. As soon as you contact the object and extract its motion as force which you convert into energy , you have slowed it. The longer you continue the more it slows until it is no longer moving. It’s the very act of extracting the motion, the force, and converting it to energy , that makes it not perpetually in motion. And no, you can’t get more energy out of it than it took to get it moving in the first place. Because this is how the universe works, and it’s Free Power proven fact. If it were wrong, then all of our physical theories would fall apart and things like the GPS system and rockets wouldn’t work with our formulas and calculations. But they DO work, thus validating the laws of physics. Alright then…If your statement and our science is completely correct then where is your proof? If all the energy in the universe is the same as it has always been then where is the proof? Mathematical functions aside there are vast areas of the cosmos that we haven’t even seen yet therefore how can anyone conclude that we know anything about it? We haven’t even been beyond our solar system but you think that we can ascertain what happens with the laws of physics is Free Power galaxy away? Where’s the proof? “Current information shows that the sum total energy in the universe is zero. ” Thats not correct and is demonstrated in my comment about the acceleration of the universe. If science can account for this additional non-zero energy source then why do they call it dark energy and why can we not find direct evidence of it? There is much that our current religion cannot account for. Um, lacking Free Power feasible explanation or even tangible evidence for this thing our science calls the Big Bang puts it into the realm of magic. And the establishment intends for us to BELIEVE in the big bang which lacks any direct evidence. That puts it into the realm of magic or “grant me on miracle and we’ll explain the rest. ” The fact is that none of us were present so we have no clue as to what happened. | 2019-03-23 03:38:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4565090835094452, "perplexity": 1348.6518454482853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202711.3/warc/CC-MAIN-20190323020538-20190323042538-00030.warc.gz"} |
https://planetmath.org/FinitelyGeneratedGroup | # finitely generated group
A finitely generated group is a group that has a finite generating set.
Every finite group is obviously finitely generated. Every finitely generated group is countable.
Any quotient (http://planetmath.org/QuotientGroup) of a finitely generated group is finitely generated. However, a finitely generated group may have subgroups that are not finitely generated. (For example, the free group of rank $2$ is generated by just two elements, but its commutator subgroup is not finitely generated.) Nonetheless, a subgroup of finite index in a finitely generated group is necessarily finitely generated; a bound on the number of generators required for the subgroup is given by the Schreier index formula (http://planetmath.org/ScheierIndexFormula).
The finitely generated groups all of whose subgroups are also finitely generated are precisely the groups satisfying the maximal condition. This includes all finitely generated nilpotent groups and, more generally, all polycyclic groups.
A group that is not finitely generated is sometimes said to be infinitely generated.
Title finitely generated group Canonical name FinitelyGeneratedGroup Date of creation 2013-03-22 12:16:38 Last modified on 2013-03-22 12:16:38 Owner yark (2760) Last modified by yark (2760) Numerical id 24 Author yark (2760) Entry type Definition Classification msc 20A05 Related topic FundamentalTheoremOfFinitelyGeneratedAbelianGroups Related topic AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex Defines finitely generated Defines finitely generated subgroup Defines infinitely generated Defines infinitely generated group Defines infinitely generated subgroup | 2020-04-06 02:57:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 1, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7479280233383179, "perplexity": 517.5202679672153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371612531.68/warc/CC-MAIN-20200406004220-20200406034720-00005.warc.gz"} |
https://community.teamviewer.com/English/discussion/34967/long-press-key-spacebar-for-adobe | teamviewer.com
or
# Long Press Key (Spacebar for Adobe)
Posts: 1
I've found a little information on this, but nothing definitive. By default adobe products have a hotkey (hold down spacebar) that allows you to pan a canvas using the mouse. This does not work when using TeamViewer. It behaves as if I am pressing and releasing the spacebar repeatedely. It is quite annoying and I don't understand why there isn't any logic built in to handle this.
Is there any workaround for this?
• Posts: 1
I'm having the same issue. Super annoying. The only thing I found which seems to be the same issue (in another context) is this....
https://community.teamviewer.com/t5/Linux/bug-ctrl-any-key-types-the-key-repeatedly/td-p/27034
So apparently they are working on it. FIngers crossed it gets fixed soon.
• Posts: 422 Moderator
Hello @Fooopa
I am sorry, but the described issue is expected behaviour at the moment.
Although you are pressing the spacebar key long, TeamViewer gets the command pressing and releasing the spacebar repeatedely from the system and transfers this command to the remote device.
Anyway I will forward this issue internally. But I can not say if or when there will be a solution.
Best,
• Posts: 2
+1 for a solution for this.
We're a design Studio using mostly Cinema 4D and After Effects and neither the spacebar nor the hotkeys 1, 2, 3 (this is used inside C4D to navigate the viewport, e.g. hold 1, click left MB and move the mouse) get translated correctly. Most other remote control software does this correctly (but has other flaws, hence we're using TV) so there must be a way to get those commands across?
• Posts: 1
Same problem here, although I found a work-around for this problem
try SHIFT+SPACE long pressing to move the image ( in photoshop ). Press Shift before Space, then keep both long pressed
Good luck!
• Posts: 3
+1 here
Should really be fixed
• Posts: 1
Please fix this soon TeamViewer, it's really frustrating and slows down work flow massively.
• Posts: 1
+1
We really need this sorted, has anyone from TV officially responded to say it's being looked at?
For those of us working in Adobe CC programs, such a seemingly minor issue causes a frustrating amount of distruption and slow down in productivity.
The hand tool work around is okay, but just not good enough.
• Posts: 1
@cosstef Great workaround! Thanks. | 2022-01-25 07:10:55 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8489481210708618, "perplexity": 4585.112660687723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304798.1/warc/CC-MAIN-20220125070039-20220125100039-00238.warc.gz"} |
http://spindynamics.org/wiki/index.php?title=Dfpt.m | Dfpt.m
Graph partitioning module. Analyzes the system connectivity graph and creates a list of all connected subgraphs of up to the user-specified size by crawling the graph in all available directions. Syntax:
subgraphs=dfpt(conmatrix,max_subgraph_size)
Arguments:
conmatrix - a matrix with 1 for connected spins
and zeros elsewhere.
max_subgraph_size - maximum connected subgraph size in
the resulting subgraph list.
Output: a matrix with each row corresponding to a subgraph. Each row contains 1 for spins that belong to the subgraph and zeros elsewhere. | 2017-12-11 03:38:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.611169159412384, "perplexity": 1873.1335941688203}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948512121.15/warc/CC-MAIN-20171211033436-20171211053436-00346.warc.gz"} |
https://ijnaa.semnan.ac.ir/article_4975.html | # Anti-N-order polynomial Daugavet property on Banach spaces
Document Type : Research Paper
Author
Department of Mathematics, Faculty of Science, Mbarara University of Science and Technology.
Abstract
We generalize the notion of the anti-Daugavet property (a-DP) to the anti-N-order polynomial Daugavet property (a-NPDP) for Banach spaces by identifying a good spectrum of a polynomial and prove that locally uniformly alternatively convex or smooth Banach spaces have the a-mDP for rank-1 polynomials. We then prove that locally uniformly convex Banach spaces have the a-NPDP for compact polynomials if and only if their norms are eigenvalues, and uniformly convex Banach spaces have the a-NPDP for continuous polynomials if and only if their norms
belong to the approximate spectra.
Keywords | 2021-09-22 02:38:33 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8468511700630188, "perplexity": 1242.9418029368624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057303.94/warc/CC-MAIN-20210922011746-20210922041746-00358.warc.gz"} |
http://lambda-the-ultimate.org/node/1430 | ## How do Java generics correspond to System F-(omega)?
In his recent guest lecture in my undergrad PL course, Markus Mottl mentioned that ML-style module systems help his company develop and maintain software in the large. This guest lecture made me wonder how an ML-style module system could be expressed using Java generics. Because I've already translated an ML-style module system into System F-omega, could someone please point me at a comparison or translation between Java (generics) and System F(-omega)?
## Comment viewing options
### Probably not what you're looking for...
...but NeelK gave an explanation of ML Functors in comparison to pre-generics Java. Maybe NeelK can enlighten us on whether the generics give new capabilities wrt to limiting the combinations?
### Java's generics are loosely
Java's generics are loosely based on F-sub -- System F with bounded quantification. I say loosely, because typechecking full F-sub is undecidable and I don't know the precise restrictions the Java designers put in place to recover decidable typechecking.
You can model bounded quantification by introducing power kinds. If A : type, then Power(A) : kind, and is used to represent the kind inhabited by every subtype of A. Then the bounded quantification forall A < B. phi(A) can be represented using forall A:Power(B). phi(A).
There's an old paper by Luca Cardelli about this, and also Chapter 31 of TAPL discusses F-omega-sub; maybe you can cut it down to see if you can get what you need.
### Loosely, indeed -- but sure.
Some differences I can think of offhand are (1) Subtyping rules in F-sub
are structural, whereas all subtype relationships in Java must be declared, and (2) Java allows F-bounded quantification, e.g.
interface Comparable<T> { int compareTo( T x ); }
class BinarySearchTree<E extends Comparable<E>> {
...
... e1.compareTo(e2) ...
...
}
which I'm pretty sure F-sub does not. (I'm not sure what happens under the encoding of bounded quantification using intersection types.)
On the other hand, Java places some weird restrictions on where instances of generic types can occur, because implementation by erasure makes certain necessary run-time checks impossible -- in particular, one cannot have an array of them (List<T>[]).
Incidentally, Java's "wildcard types" allow for a twisted kind of existential quantification: List<? extends C> means exists a < C . List<a>.
So it's accurate to say that it's loosely based on F-sub, but they're far enough apart that metatheoretic differences (like decidability) should not surprise us.
JCV
### I ignored wildcards because
I ignored wildcards because they're weird enough that I tend to pretend they don't exist, but I genuinely didn't know that Java allowed F-bounded quantification! Thanks for telling me that.
### Weird restrictions are due to arrays
On the other hand, Java places some weird restrictions on where instances of generic types can occur
They pretty much have to do with keep Java's slightly broken array type system from leaking into the well-sealed generic type system. Aside from restrictions on arrays, I don't know of any other "weird constraints."
Many of the complaints about Java's generics are actually complaints about its array system, which is indeed broken! This is one of several reasons why discerning Java programmers avoid arrays in most cases.
### I'm not sure about that...
They pretty much have to do with keep Java's slightly broken array type system from leaking into the well-sealed generic type system. Aside from restrictions on arrays, I don't know of any other "weird constraints."
I have two minor arguments with this. First, while Java's array type system may be broken, I think it's broken in a way that's symptomatic of some general issues with generics: no proper accounting for co/contra-variance. This shows up in non-array contexts in the fact, for example, that A being a sub-type of B does not imply that List<A> is a sub-type of List<B>. A more sophisticated account of variance could probably allow this as well as dealing properly with co- and contra-variant array access (reads and writes), or at least this was how it seemed to me last time I looked seriously at Java generics, which was quite awhile ago now.
My second complaint is that the main "weird constraint" on generics usually cited has to with erasure and its implications, particularly the inability to construct new instances of generic types. I can't see that this has anything to with arrays...
### It isn't and it shouldn't.
A being a sub-type of B does not imply that List<A> is a sub-type of List<B>, because a List<A> is not capable of storing B's - a valid operation for List<B>'s.
Arrays work differently and, accordingly, are not type-safe: every store can cause an exception.
### I'm still (easily) confused
What is the angle from which one claims List<A> should not be capable of storing List<B>? Or do you think it should be able to but Java fails in that regard? Is there some nice theoretical underpinning somewhere?
(It just seems to me that one could argue either way, from a "common sense" or simple un-PLT-nuanced in-the-trenches programmer point of view.)
### Consider a language with a
Consider a language with a Top type - Java's not far off having one, though Object's not quite the same. If we allow List<A> to store B, where A <: B, then there could be absolutely anything in that list because anything can be considered an instance of Top and thus stored in it. At which point, you may as well just have a heterogenous List class.
### Transitivities of evil.
OK, right, that makes sense enough to me, I think.
[Presumably this is old news and wouldn't interest many folks who have been there, understood that.]
(I'd like to redefine the problem a little bit so I'm less confused - I think A and B should be swapped around so that A is the root and B is the subclass, that will make things easier for my mind. :-)
This brings up something basic and yet still interesting/confusing to me at times: What do we all mean by subtyping and what do our languages mean? I know, I know, there's stuff like LSP (OK, that's a really slow anchor to load). To me, when you subclass something you cannot be removing functionality. That would be breaking any sane kind of contract. (Unfortunately, the Java collections stuff does that all over the place and just throws runtime exceptions, freaking genius.) So if B <:A I would expect that List<B> wouldn't store As because As are insufficient.
However, if C <: B maybe it would be OK to store C in List<B> as long as you could only access B's interfaces on the entries.
### But this is how it works in
But that is how it works in Java:
• a List<Number>, for instance, can hold Integer's, Double's, etc - any Number;
• a List<Integer>, on the other hand, can't take just any Number - only Integer's.
This means that neither is List<Number> a sub-type of List<Integer> (as it may have Double's in it), nor is List<Integer> is a sub-type of List<Number> (since you can't store Double's in it).
Clearer?
### Clearer?!
Never clearer to me, I'm easily re-befuddled.
(Also, the notation is killing me when it comes to entering posts, so I'm switching to suqiggly brackets instead of gt lt glyphs.)
neither is List{Number} a sub-type of List{Integer} - that's OK because I think Integer <: Number. Having Doubles in List{Number} it is fine; you should only be able to access the Number API when you get something out of the list, short of doing evil type casting. Also, I would never expect Double <: Integer or Integer <: Double because the values they can take on are so different. At best you can say they share some operations, but once you get into the values it gets too weird. When you say that List{Integer} can't take any Number that's fine. But it should be able to take Integer and any subclass thereof.
Summary: if C <: B <: A, then List{A} can contain A, B, or C but only with the API for A; List{B} can contain B or C but only with the API for B, and List{C} can only contain C. To me, that is one core thing polymorphism is all about. (If you define the subtyping relationship but the relationship is broken vis a vie Integer and Double then all bets are off.)
I'm not sure how that works back to the original postings, I'll go back and re-read :-)
### I agree...
When you say that List<Integer> can't take any Number that's fine. But it should be able to take Integer and any subclass thereof.
Just keep in mind I was talking about Java, where Integer is final.
A List<Integer> can only take Integer's simply because there can be no sub-classes of Integer.
### No, no, no, no, no.
Suppose we take your suggestion, and allow List<Integer> to be a subtype of List<Number>. Then this will compile, but will encounter a runtime type error (which Java's type system promises it shouldn't, because there is no explicit type cast):
void addFloat(List<Number> list)
{ list.add(37.62); } // OK, because 37.62 is a Float, which is a subtype of Number
...
List<Integer> ints = new ArrayList<Integer>();
addFloat(ints); // according to you, this should compile!
Integer first = ints.get(0); // type mismatch
The problem is for subtypes of List, add() needs to be covariant and get() needs to be contravariant. You can't have both. Java's generics are correct, and you are not.
Java's arrays are, in fact, broken in more or less the way described above.
### Covariant, contraviant, blah
Sorry, I had that backward -- parameter types need to be contravariant (as with add()) and return types (as with get()) need to be covariant for subtyping to work. But the point still stands.
We might be missing each other's point. There is no "add" method on Number. Citing myself:
Summary: if C <: B <: A, then List{A} can contain A, B, or C but only with the API for A; List{B} can contain B or C but only with the API for B, and List{C} can only contain C. To me, that is one core thing polymorphism is all about. (If you define the subtyping relationship but the relationship is broken vis a vie Integer and Double then all bets are off.)
This is a short set of slides (in PDF format) about subtyping. The interesting sections are the ones on function and reference variance.
### Thanks
Ever more for me to read and grok, but obviously this is pretty fundamental and hopefully simple-ish stuff - although when I start reading about these things I find plenty of interesting twists...
### Is a Cow an Animal?
You might also like this brief discussion of Is a Cow an Animal?.
### Funny you should mention that
Since I've been reading and re-reading (hoping that it will all magically sink in at some point) the oft-mentioned On the (un)reality of virtual types which includes the animal/food example, but claims to separate the concerns such that you don't have a problem any more. I think.
(Addendum: I feel ever again like I should learn and love O'Caml, although I've been frustrated with it in the past.)
### Don't Hesitate
raould: Addendum: I feel ever again like I should learn and love O'Caml, although I've been frustrated with it in the past.
Me too! I'm still learning this language that I claim to love. Please don't hesitate to ask questions. If you don't think they're appropriate for the group, I can provide my e-mail address.
### What does sub-type do for you?
I think a question begged is: what is the point of sub-typing? You talk in particular about List{Subclass} being able to store instances of Superclass. But that doesn't seem like the correct issue to me. The issue would be: if you can use Subclass anywhere that you can use Superclass, then you should be able to use List{Subclass} anywhere that you can use List{Superclass}.
Now, I don't know what Java does, but it would make sense to me to let people pass List{Subclass} into a method that takes List{Superclass} (modulo evil type casting that somebody might then do inside the method, but obviously that is their own dumb fault).
### But it doesn't quite work if
But it doesn't quite work if the list's mutable, because the method might want to put a Superclass into that list. And you can't pass in a List{Superclass} when a List{Subclass} is expected because the code won't be expecting to pull a Superclass out of the list.
### Ah, right you are!
Dogged. So that's another aspect of looking at these things: consuming vs. producing. Are there systems that are very different in their typing because they do/not allow mutability, or that vary the typing based on consuming vs. producing?
### Sort of. The easiest thing
Sort of. The easiest thing is to start off with the concept of "variance" (roughly speaking, "can I accept a subtype in place of a type? Can I accept a supertype in place of a type?"), assign appropriate variances to mutable values and immutable ones and propagate everything according to some simple rules from there on - an ML-style system where references to mutable cells exist but all actual values are immutable works well.
### Mutability...
As was already said, this breaks with mutability.
Going back to Java, I'd like to add that, when you don't need mutability, you can pass your List<Subclass> to a method that takes a List<Superclass> by using the Collections.unmodifiableList() method to wrap your list inside an unmodifiable list of the apropriate super-type (any attempt to mutate, will originate a run-time error, not compile-time - maybe someday).
### What wildcards are for?
(any attempt to mutate, will originate a run-time error, not compile-time - maybe someday).
I don't understand this comment. Java's generics supports variance. It supports them via a "wildcard" syntax.
For your example, rather than typing the method such that it receives List<SomeClass>, you type it so that it receives List<? extends SomeClass>. The type-safety is enforced at compile time.
Angelika Langer's FAQ is a good place to go, to get a solid understanding of Java's generics facilities.
### Java's generics supports
Java's generics supports variance. It supports them via a "wildcard" syntax.
Yes, I know. And if you look at the type signature of the Collections.unmodifiableList() method that I suggested using, you'll see it does use wildcards.
For your example, rather than typing the method such that it receives List<SomeClass>, you type it so that it receives List<? extends SomeClass>.
Assume for a moment I cannot (or do not want to, for whatever reason) change the method's typing - after all we are not always in control of all the APIs we are using. What are my other options?
Of course, if you are in control, then you should use variance whenever it makes sense (unfortunately that will not, I imagine, be the case for most code I'll see).
The type-safety is enforced at compile time.
Not if you're forced into using the method I suggested, because, as I also stated, attempts to modify the resulting list will throw an UnsupportedOperationException. This can certainly be seen as a "type error", since it originates from trying to modify an "UnmodifiableList", for which we could (and maybe should) have a separate type. | 2019-05-22 20:43:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42649292945861816, "perplexity": 2183.7304844424116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256958.53/warc/CC-MAIN-20190522203319-20190522225319-00214.warc.gz"} |
https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.087801 | # Synopsis: To Eliminate Defects, Create a Bend
Nanoscale templates made from films of self-assembled polymers could be fabricated with fewer defects by putting the films on a curved surface.
Nanofabrication techniques that use templates to pattern devices could exploit so-called block copolymers for their designs. These molecules consist of two or more different polymer “blocks” that are bonded together. Large numbers of block copolymers can self-assemble, with the blocks grouping together by type to form patterns, such as periodic cylinders or stripes. However, defects in the patterns currently limit their use as high-fidelity templates. Now, Friederike Schmid of Johannes Gutenberg University Mainz, Germany, and colleagues have shown that bends in a block copolymer film can help align its periodic structures, reducing pattern imperfections.
The team fabricated two 30-nm-thick films of the polymer called PS-PEP, which is comprised of a PS block and a PEP block. A film of the polymer was engineered to be free-standing, and it resembled a piece of paper that had been crumpled up and then reflattened. The other film, which had a smaller area of 25 $𝜇{\text{m}}^{2}$, was formed on a humped substrate, and it looked like a pristine paper sheet that had been carefully draped over a long pipe. In both films, the PS block assembled into cylinders spaced 21 nm apart. Using atomic force microscopy, the team measured the cylinders’ orientations as a function of the film’s local curvature.
In the free-standing sample, the team observed that the cylinders pointed in all directions: Roughly two thirds of the cylinders lay either perpendicular or parallel to the film’s undulations, and the rest were randomly oriented. In the supported film, by contrast, they found that more than 90% of the cylinders aligned perpendicularly to the hump. The supported films, however, contained holes, which could hinder their use as templates. The team says that they should be able to prevent hole formation—and achieve 100% cylinder alignment—by controlling the interactions of the film with its supporting surface.
This research is published in Physical Review Letters.
–Christopher Crockett
Christopher Crockett is a freelance writer based in Arlington, Virginia.
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Soft Matter
### Synopsis: Hints of an Equation of State for Granular Materials
Experiments with a granular system have confirmed a temperature-like variable that could lead to an equation of state for this class of materials. Read More » | 2019-02-22 21:39:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28597086668014526, "perplexity": 4086.380015420382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247526282.78/warc/CC-MAIN-20190222200334-20190222222334-00188.warc.gz"} |
https://eng.libretexts.org/Bookshelves/Introductory_Engineering/Basic_Engineering_Science_-_A_Systems_Accounting_and_Modeling_Approach_(Richards)/08%3A_Entropy_Production_and_Accounting/8.6%3A_Problems | # 8.6: Problems
In the following problems, all references to an ideal gas and an incompressible substance refer to the models with room-temperature specific heats.
##### Problem $$8.1$$
A gearbox operating at steady-state conditions receives $$2 \mathrm{~hp}$$ along the input shaft and delivers $$1.9 \mathrm{~hp}$$ along the output shaft. The outer surface of the gearbox is at $$105^{\circ} \mathrm{F}$$. The temperature of the air in the room is $$70^{\circ} \mathrm{F}$$.
(a) Determine the rate of heat transfer, in $$\mathrm{Btu} / \mathrm{h}$$. Indicate the direction
(b) Determine the rate of entropy production, in $$\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)$$, within the system consisting of the gearbox and the shafts. (Sketch the system.)
(c) Consider the layer of air immediately adjacent to the gearbox. It receives energy by heat transfer at $$105^{\circ} \mathrm{F}$$ and loses energy by heat transfer at $$70^{\circ} \mathrm{F}$$. Determine the steady-state rate of entropy production, in $$\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)$$, within the air layer. (Sketch the system.)
(d) Now consider an enlarged system that consists of both the gearbox and shafts and the air layer. This system loses energy by heat transfer to the surroundings at $$70^{\circ} \mathrm{F}$$. Determine the steady-state rate of entropy production, in $$\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)$$, for this combined system. (Sketch the system.)
(e) Discuss how your result for Part (d) compares with your answers for Part (b) and Part (c).
##### Problem $$8.2$$
An inventor claims to have developed a new device that operates at steady-state conditions and produces both shaft power and electrical power. A schematic of the device is shown in the figure with the known operating conditions and proposed energy transfers.
Figure $$\PageIndex{1}$$: Device with heat input that produces shaft power and electrical power.
(a) Determine the electrical power output from the device, in $$\mathrm{kW}$$.
(b) Determine the entropy production rate for the device, in $$\mathrm{kW} / \mathrm{K}$$.
(c) Based upon your answer to part (b), do believe that this device is possible? Explain the rationale for your answer.
##### Problem $$8.3$$
An electric transformer is used to step down the voltage from 220 to 110 volts (AC). The current on the high-voltage side is $$23 \mathrm{~A}$$ and on the low voltage side it is $$43 \mathrm{~A}$$. The power factor is one for both sides of the transformer. The transformer operates under steady-state conditions with a surface temperature of $$40^{\circ} \mathrm{C}$$. Determine (a) the heat-transfer rate for the device, in watts, and (b) the entropy production rate, in $$\mathrm{W} / \mathrm{K}$$.
##### Problem $$8.4$$
An electric motor operates under steady-state conditions and draws $$3 \mathrm{~kW}$$ of electric power. Ten percent of the electrical power supplied to the motor is lost to the surroundings by heat transfer. The surface temperature of the motor is $$45^{\circ} \mathrm{C}$$. Determine (a) the shaft power delivered by the motor in $$\mathrm{kW}$$ and (b) the entropy production rate for the motor, in $$\mathrm{kW} / \mathrm{K}$$.
##### Problem $$8.5$$
A soldering iron draws $$0.10 \mathrm{~A}$$ from a $$110 \text{-V}$$ circuit at steady-state conditions. The operating temperature of the soldering iron is $$105^{\circ} \mathrm{C}$$. Determine the entropy production rate for the soldering iron in $$\mathrm{W}/\mathrm{K}$$.
##### Problem $$8.6$$
A transmission consists of two gearboxes connected by an intermediate shaft. A torque of $$220 \mathrm{~ft} \cdot \mathrm{lbf}$$ is applied to the input shaft which rotates at $$200 \mathrm{~rpm}$$. The intermediate shaft and the output shaft rotate at $$160 \mathrm{~rpm}$$ and $$128 \mathrm{~rpm}$$, respectively. Each gearbox transmits only $$95 \%$$ of the shaft power supplied to it. The remainder of the energy is lost to the surroundings by heat transfer. The surface temperature of each gearbox is measured to be $$120^{\circ} \mathrm{F}$$, and the ambient air temperature is $$70^{\circ} \mathrm{F}$$.
Determine (a) the torque for the intermediate and output shafts, in $$\mathrm{ft} \cdot \mathrm{lbf}$$, (b) the entropy production rate for each gearbox individually and for the overall transmission, in $$\mathrm{Btu} / \left(h \cdot { }^{\circ} \mathrm{R}\right)$$.
##### Problem $$8.7$$
An inventor claims to have a invented a device that takes in $$10 \mathrm{~kW}$$ by heat transfer at $$500 \mathrm{~K}$$, rejectes energy by heat transfer at $$300 \mathrm{~K}$$ and produces $$5 \mathrm{~kW}$$ of power. As a U.S. Patent Examiner, you must determine if this device is possible or a hoax. Based on the description in the patent application, it appears that it is a closed, steady-state device. What do you think? Is it possible? Explain your reasoning.
##### Problem $$8.8$$
A steady-state heat pump is designed to reject energy by heat transfer at a rate of $$20,000 \mathrm{~Btu} / \mathrm{h}$$ at a temperature of $$90^{\circ} \mathrm{F}$$ and requires an electrical power input equivalent to $$5,000 \mathrm{~Btu} / \mathrm{h}$$. Heat transfer into the system occurs at a temperature of $$40^{\circ} \mathrm{F}$$.
Determine (a) the COP for the heat pump and (b) the entropy generation rate for the heat pump, in $$\mathrm{Btu} /\left(\mathrm{h} \cdot{ }^{\circ} \mathrm{R}\right)$$. (c) Would you describe the heat pump as operating reversibly or irreversibly, or is it impossible to operate as specified?
##### Problem $$8.9$$
A heat pump with a COP of 3 receives energy from the outdoors at $$30^{\circ} \mathrm{F}$$ and rejects energy to the air inside the house at $$72^{\circ} \mathrm{F}$$. The heat pump rejects energy to the air inside the house at the rate of $$100,000 \mathrm{~Btu} / \mathrm{h}$$.
(a) Determine the following:
• the power of the motor required to operate the heat pump, in horsepower,
• the rate of heat transfer from the outdoors, in $$\mathrm{Btu} / \mathrm{h}$$, and
• the rate of entropy production for the heat pump, in $$\mathrm{Btu} / \left( \mathrm{h} \cdot { }^{\circ} \mathrm{R}\right)$$.
(b) Determine the maximum possible COP for a heat pump operating between these temperatures and the power of the motor, in $$\mathrm{hp}$$, required to operate this ideal heat pump.
(c) Some people would consider the "extra" electrical energy required to run the real heat pump when compared with the power required to operate the ideal heat pump as being wasted, since it can't be used for anything else. If the ideal or best possible performance is associated with an internally reversible cycle and this cycle produces no entropy, entropy production may be a measure of energy waste. To check this out, investigate the validity of the following equation using your results from parts (a) and (b): $\frac{\left(\dot{W}_{\text {actual}} - \dot{W}_{\text {ideal}}\right)}{T_{\text {outdoors}}} = \dot{S}_{\text {production}} \nonumber$ where all power values are in $$\mathrm{Btu} / \mathrm{h}$$, temperatures are in $${ }^{\circ} \mathrm{R}$$, and the entropy production rate is what you calculated in part (a). Is this result correct?
##### Problem $$8.10$$
A system executes a power cycle. During each cycle, the system receives $$2000 \mathrm{~kJ}$$ of energy by heat transfer at a temperature of $$500 \mathrm{~K}$$ and discharges energy by heat transfer at a temperature of $$300 \mathrm{~K}$$. There is no other heat transfer of energy.
(a) Assuming that the cycle has a thermal efficiency of $$25 \%$$, determine the work out per cycle, in $$\mathrm{kJ}$$, and the amount of entropy produced per cycle, in $$\mathrm{kJ} / \mathrm{K}$$.
(b) Assuming that the cycle rejects $$900 \mathrm{~kJ}$$ of energy by heat transfer, determine the work out per cycle, in $$\mathrm{kJ}$$, the amount of entropy produced per cycle, in $$\mathrm{kJ} / \mathrm{K}$$, and the thermal efficiency.
(c) Assuming that the cycle is internally reversible, i.e. rate of entropy production is zero, calculate the work out per cycle, in $$\mathrm{kJ}$$, and the thermal efficiency for this cycle.
(d) Compare your answers to Parts (a), (b), and (c). What does this tell you about the three cycles? Is it possible to build a power cycle that operates between the same two temperatures and is more efficient than the one you examined in Part (c)?
##### Problem $$8.11$$
A geothermal power plant utilizes an underground source of hot water at $$160^{\circ} \mathrm{C}$$ as the heat source for a power cycle. The power plant boiler receives energy by heat transfer at a rate of $$100 \mathrm{~MW}$$ from the hot water source at $$T_{H, \text { Source}} = 160^{\circ} \mathrm{C}$$ The power plant condenser rejects energy by heat transfer at the rate of $$78 \mathrm{~MW}$$ to the ambient air at $$T_{L, \text{ sink}}=15^{\circ} \mathrm{C}$$.
Figure $$\PageIndex{2}$$: Geothermal power plant consisting of a boiler and condenser, with an output of shaft work.
(a) Determine the net power output, in $$\mathrm{MW}$$, and the thermal efficiency for this power cycle (heat engine) expressed as a percent.
(b) Determine the theoretical maximum thermal efficiency for a power cycle (heat engine) that operates between these two temperatures: $$T_{\text {boiler}} = T_{H, \text { source}}$$ and $$T_{\text {condenser}} = T_{L, \text { sink}}$$. Give your answer as a percent and compare it to your result from part (a).
(c) In reality, the rate of heat transfer is proportional to the temperature difference available to "drive" the heat transfer, i.e. $$Q \propto \Delta T$$. Practically this means to receive energy by heat transfer from a thermal source at temperature $$T_{H, \text { source}}$$, the surface temperature of the boiler $$T_{\text{boiler}}$$ must be less than the source temperature. Similarly to reject energy by heat transfer to a thermal sink at temperature $$T_{L, \text{ sink}}$$, the surface temperature of the condenser $$T_{\text {condenser}}$$ must be greater than the sink temperature.
As a first guess, assume that a temperature difference of $$5^{\circ} \mathrm{C}$$ is required, and determine the theoretical maximum thermal efficiency for a power cycle that operates between these new more realistic temperatures: $T_{\text {boiler }}=T_{H, \text { Source}}-5^{\circ} \mathrm{C} \quad \text { and } \quad T_{\text {condenser}}=T_{L, \text { sink}}+5^{\circ} \mathrm{C}. \nonumber$ How does the efficiency of this more realistic cycle compare with your answers to part (a) and part (b)?
##### Problem $$8.12$$
A reversed power cycle operates between a high temperature of $$50^{\circ} \mathrm{C}$$ and a low temperature of $$5^{\circ} \mathrm{C}$$. Determine the best possible coefficient for this reversed power cycle (a) if it is operated as a heat pump cycle and (b) if it operated as a refrigeration cycle.
##### Problem $$8.13$$
A heat pump receives energy by heat transfer from outside air at $$T_{\text {outdoors}}$$ and rejects energy to a dwelling at $$T_{\text {room}}$$. Starting with the conservation of energy and entropy accounting equation for the steady-state heat pump, develop an expression for the COP of this heat pump similar to Eq. $$8.4.7$$. Show your work.
##### Problem $$8.14$$
A refrigeration cycle receives energy by heat transfer from a freezer compartment at $$T_{\text {freezer}}$$ and rejects energy to the kitchen at $$T_{\text {room}}$$. Starting with the conservation of energy and entropy accounting equation for the steady-state refrigeration cycle, develop an expression for the COP of this cycle similar to Eq. $$8.4.7$$. Show your work.
##### Problem $$8.15$$
Liquid water flows steadily through a small centrifugal pump at a volumetric flow rate of $$6.0 \mathrm{~m}^{3} / \mathrm{min}$$. The water enters the pump at $$100 \mathrm{~kPa}$$ and $$27^{\circ} \mathrm{C}$$ and leaves the pump at a pressure of $$400 \mathrm{~kPa}$$. Inlet and outlet areas are identical and changes in potential energy are negligible. Assume that water can be modeled as an incompressible substance.
Figure $$\PageIndex{3}$$: Water flows steadily through a centrifugal pump.
(a) If the pump is adiabatic and internally reversible:
• Determine the change in specific entropy, $$s_{2}-s_{1}$$, for the water as it flows through the pump, in $$\mathrm{kJ} /(\mathrm{kg}-\mathrm{K})$$. [Hint: Apply the entropy accounting equation to the pump, make the appropriate modeling assumptions, and solve for $$s_{2}-s_{1}$$.]
• Determine the shaft power input under these conditions in $$\mathrm{kW}$$. [Hint: After applying the conservation of energy equation and appropriate modeling assumptions, don’t forget to see what your result for $$\Delta s$$ from above along with the incompressible substance model tells you about how the pressure and/or temperature of the water may change.]
(b) Now assume the pump operates adiabatically and that the temperature of the water increases as it flows through the pump, $$T_{2}-T_{1}=0.05^{\circ} \mathrm{C}$$ :
• Determine the entropy production rate for the pump, in $$\mathrm{kW} / \mathrm{K}$$. Is this process internally reversible or internally irreversible? How can you tell?
• Determine the shaft power input under these conditions, in $$\mathrm{kW}$$.
(c) Compare your answers from Part (a) and (b).
• Which operating condition requires the larger power input? Why?
• Do you think it would be possible to reduce the shaft power further by operating this same pump under steady-state, adiabatic conditions so that the water temperature would decrease as it flows through the pump, e.g. $$T_{2}-T_{1}=-0.05^{\circ} \mathrm{C}$$?
##### Problem $$8.16$$
The nozzle in a turbojet engine receives air at $$180 \mathrm{~kPa}$$ and $$707^{\circ} \mathrm{C}$$ with a velocity of $$70 \mathrm{~m} / \mathrm{s}$$. The air expands adiabatically in a steady-state process to an outlet pressure of $$70 \mathrm{~kPa}$$. The mass flow rate of air is $$3.0 \mathrm{~kg} / \mathrm{s}$$. Assume that air can be modeled as an ideal gas with room temperature specific heats.
(a) If the expansion process is internally reversible, determine the outlet air temperature in $${ }^{\circ} \mathrm{C}$$ and the outlet velocity of the air, in $$\mathrm{m} / \mathrm{s}$$. [Hint: Apply the entropy accounting equation along with the ideal gas model.]
(b) If the expansion process is internally irreversible and $$T_{2}$$ is $$527^{\circ} \mathrm{C}$$, determine the entropy production rate for the nozzle, in $$\mathrm{kW} / \mathrm{K}$$, and determine the outlet velocity, in $$\mathrm{m} / \mathrm{s}$$.
(c) Compare and discuss your results, especially $$T_{2}$$ and $$V_{2}$$, in terms of the entropy production rate for each process.
##### Problem $$8.17$$
An electric water heater having a 100-liter capacity employs an electric resistor to heat the water from $$18^{\circ} \mathrm{C}$$ to $$60^{\circ} \mathrm{C}$$. The outer surface of the resistor remains at an average temperature of $$97^{\circ} \mathrm{C}$$ during the heating process. Heat transfer from the outside of the water heater is negligible, and the energy and entropy storage in the resistor and the tank holding the water are insignificant. Model the water as an incompressible substance.
(a) Determine the amount of electrical energy, in $$\mathrm{kJ}$$, required to heat the water.
(b) Determine the amount of entropy produced, in $$\mathrm{kJ} / \mathrm{K}$$, within the water only, i.e. take the water as the system.
(c) Determine the amount of entropy produced, in $$\mathrm{kJ} / \mathrm{K}$$, within the overall water heater including the resistor, i.e. take the overall water heater including the resistor as the system.
(d) Why do the results of (b) and (c) differ? What is within the system for (c) that was excluded in (b)?
##### Problem $$8.18$$
Air enters a shop air compressor with a steady flow rate of $$0.7 \mathrm{~m}^{3} / \mathrm{s}$$ at $$32^{\circ} \mathrm{C}$$ and $$0.95 \mathrm{~bars}$$. The air leaves the compressor at a pressure of $$15 \mathrm{~bars}$$. Assume air can be modeled as an ideal gas and that changes in kinetic and gravitational potential energy are negligible. Determine the minimum power requirement to drive the adiabatic compressor, in $$\mathrm{kW}$$. [Hint: How does varying the entropy generation rate affect the power input to the compressor? What are the limiting values on the entropy generation rate?]
##### Problem $$8.19$$
A rigid air tank has a volume of $$1.0 \mathrm{~m}^{3}$$ and contains air at $$27^{\circ} \mathrm{C}$$ and $$3400 \mathrm{~kPa}$$. Should the tank wall fail catastrophically, the tank would explode and cause considerable damage. To estimate the amount of energy that could be transferred from the air to the surroundings in an explosion, we will estimate the work done by the expanding gas. To model the expansion process, assume that the gas acts like a closed system and expands adiabatically and reversibly until the gas pressure matches the ambient air pressure of $$100 \mathrm{~kPa}$$.
(a) Determine the work done by the gas on the surroundings during this expansion process, in kilojoules.
(b) Determine the temperature of the air after this hypothetical expansion process.
(c) How conservative are your results from Part (a)? Would you expect the actual blast to transfer more or less energy to the surroundings?
##### Problem $$8.20$$
A short pipe and valve connect two heavily insulated tanks. Tank A has a volume of $$1.0 \mathrm{~m}^{3}$$ and Tank B has a volume of $$2.0 \mathrm{~m}^{3}$$. Tank A initially contains carbon dioxide at $$400 \mathrm{~K}$$ and $$300 \mathrm{~kPa}$$. Tank B is initially evacuated. Once the value is opened, the carbon dioxide expands into Tank B.
Determine (a) the final equilibrium pressure and temperature of the carbon dioxide and (b) the entropy produced within the gas during this expansion process, in $$\mathrm{kJ} / \mathrm{K}$$. [Assume carbon dioxide can be modeled as an ideal gas.]
##### Problem $$8.21$$
The figure below shows a steady-state gas turbine power plant consisting of a compressor, a heat exchanger, and a turbine. Air enters the compressor with a mass flow rate of $$3.9 \mathrm{~kg} / \mathrm{s}$$ at $$0.95 \mathrm{bar}, 22^{\circ} \mathrm{C}$$ and exits the turbine at $$0.95$$ bar, $$421^{\circ} \mathrm{C}$$. Heat transfer to the air as it flows through the heat exchanger occurs at an average temperature of $$488^{\circ} \mathrm{C}$$. The compressor and turbine operate adiabatically. Assume that air behaves like an ideal gas and assume changes in kinetic and gravitational potential energy are negligible.
Figure $$\PageIndex{4}$$: Turbine power plant consisting of a compressor, heat exchanger, and turbine, with an output of shaft work.
Determine the maximum theoretical value for the net power that can be developed by the power plant, in $$\mathrm{MW}$$.
[Hint: Consider the following steps.
(1) Apply the conservation of energy equation to develop an equation that relates the net power out to the heat transfer rate into the system.
(2) Now apply the entropy accounting equation to find a relationship between the heat transfer rate into the system and the entropy production rate for the system.
(3) Now vary the entropy production rate over its possible values and examine how it changes the net power out of the system.
(4) Determine the maximum value of the net power out of the system. Clearly indicate why your result is the maximum value.]
##### Problem $$8.22$$
A short pipe and valve connect two heavily insulated tanks. Each tank has a volume of $$0.5 \mathrm{~m}^{3}$$. Tank A initially contains nitrogen at $$150 \mathrm{~kPa}$$ and $$300 \mathrm{~K}$$. Tank B initially contains nitrogen at $$50 \mathrm{~kPa}$$ and $$300 \mathrm{~K}$$. Suddenly, the valve is opened and the two gases are allowed to mix.
Determine (a) the final pressure and temperature of the mixture and (b) the entropy produced during this mixing process. [Assume nitrogen can be modeled as an ideal gas. ]
##### Problem $$8.23$$
The air trapped in the piston cylinder of an air compressor occupies an initial volume of $$42 \mathrm{~in}^{3}$$ (cubic inches) when the piston is at the bottom of its stroke. The air has a temperature and pressure of $$70^{\circ} \mathrm{F}$$ and $$15 \mathrm{~psi} \ \left(\mathrm{lbf} / \mathrm{in}^{2}\right)$$, respectively.
When the piston moves to the top of its stroke, the air is compressed to a volume of $$7.0 \mathrm{~in.}^{3}$$. The compression process occurs so fast that heat transfer during the compression process is negligible. If necessary, assume that air can be modeled as an ideal gas and that changes in kinetic and gravitational potential energy for the gas are negligible.
Figure $$\PageIndex{5}$$: Air in a piston cylinder.
(a) If the compression process is reversible, determine the temperature and the pressure of the gas after the compression. In addition, calculate the work done on the gas during the compression process, in $$\mathrm{Btu}$$.
(b) Now assume that the compression process is irreversible, how will this change the final temperature and pressure of the gas after the compression and the work done on the gas. Clearly indicate whether the values increase or decrease as compared to the values for the reversible process?
(c) Why would an engineer care about the values for a reversible process?
##### Problem $$8.24$$
An $$18 \text{-kg}$$ lead casting at $$200^{\circ} \mathrm{C}$$ is quenched in a tank containing $$0.03 \mathrm{~m}^{3}$$ of liquid water initially at $$25^{\circ} \mathrm{C}$$. The water tank is insulated immediately after the casting is dropped into the water. Determine (a) the final equilibrium temperature of the lead, in $$\mathrm{K}$$, and (b) the entropy generation for the lead-water system, in $$\mathrm{kJ} / \mathrm{K}$$. (c) Is this process reversible, irreversible, or impossible? [Assume lead and liquid water can both be modeled as incompressible substances.
##### Problem $$8.25$$
A new device is proposed as a steady-state air heater (see figure). Air enters the heater (1) at $$400 \mathrm{~K}$$ and $$200 \mathrm{~kPa}$$ with a volumetric flow rate of $$1000 \mathrm{~m}^{3} / \mathrm{min}$$. It leaves the heater (2) at $$500 \mathrm{~K}$$ and $$190 \mathrm{~kPa}$$. The heater is powered by electricity and has two different operating modes. Electricity costs $$\ 0.08$$ per kilowatt-hour.
Mode I - Steady-state, adiabatic operation with no heat transfer on the surface of the device, $$Q_{o, \text{ in}}=0$$.
Mode II - Steady-state, internally reversible operation with heat transfer on the surface at a boundary temperature of $$T_{o}=300 \mathrm{~K}$$.
Figure $$\PageIndex{6}$$: A steady-state electric air heater.
(a) For Mode I, determine the electric power required to operate the heater, in $$\mathrm{kW}$$, and the entropy generation rate, in $$\mathrm{kW} / \mathrm{K}$$.
(b) For Mode II, determine the electric power required to operate the heater, in $$\mathrm{kW}$$, the entropy generation rate, in $$\mathrm{kW} / \mathrm{K}$$, and the heat transfer rate, in $$\mathrm{kW}$$.
(c) For an 8-hour day, how much would it cost to operate the air heater in each mode? Any advice to the plant engineer about which mode of operation should be used?
8.6: Problems is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. | 2022-08-20 02:30:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6693587899208069, "perplexity": 433.80586741183083}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573876.92/warc/CC-MAIN-20220820012448-20220820042448-00368.warc.gz"} |
http://openstudy.com/updates/4f349396e4b0fc0c1a0c17e5 | ## anonymous 4 years ago FIND AN EQUIVALENT FRACTION 7/12
1. anonymous
14/24
2. anonymous
49/144 42/72 a bunch of things tak your pick
3. anonymous
do you need an explanation
4. anonymous
YES
5. anonymous
just multiply by 2, 3 , 4
6. anonymous
$7\pi/12\pi$
7. anonymous
|dw:1328845904723:dw|
8. anonymous
i'M NOT UNDERSTANDING
9. anonymous
multiply the numerator and denominator by the same number or divide but because you cannot divide you have to multiply you can multiply with any numbers but the easiest is by 2
10. anonymous
Do u understand?
11. anonymous
it's like this man. Say you have 1/2 of a pie. 50% of the pie. Imagine this... 1/2 of a pie. Now, multiply the amount of pies you have by 2. However, don't give yourself two WHOLE pies. You started with 1/2 of a pie. So the next by you give yourself must also only have half of it left. Therefore, you have 2 pies, each with half of them missing. Give yourself another pie if you want... with half of it gone. And another. And another. You'll notice something. Draw this on a sheet of paper if you want, some circles half filled in, however you want to graph it. For every whole circle (1), (1/2) of it is left. Even if you give yourself seemingly more pie (1/2 more), there was still 2 pies, now you have 1/2 + 1/2 1 out of 2 pies, 1/2 again. Make sense? So in a way, you have 2/4 of the total amount of pies. It makes sense if you draw it out (this is a weird way of explaining equivalent fractions lol). | 2017-01-18 04:45:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8027793765068054, "perplexity": 1272.6550855597432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280221.47/warc/CC-MAIN-20170116095120-00423-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://alexschroeder.ch/wiki?action=browse;diff=2;id=2011-03-08_More_Character_Shortcuts | # 2011-03-08 More Character Shortcuts
Last edit
Changed:
< A bit over a year ago I wrote my first draft of the [[2009-12-14 Character Generation Shortcuts|Character Generation Shortcuts]] fold old school D&D with four backgrounds each for fighters, wizards and clerics. In a way I wanted to show that I didn't need thieves, rangers and paladins -- I wanted to model these classes using a base class with some appropriate background and equipment -- and I wanted to speed up character creation by avoiding rolling for starting gold and shopping for equipment. Half a year later I [[2010-06-24 Character Genration Shortcuts PDF|increased the number of backgrounds to eight]] and used it in a convention game I ran, but I still missed backgrounds for demihumans. The following is a list of four backgrounds each.
to
> A bit over a year ago I wrote my first draft of the [[2009-12-14 Character Generation Shortcuts|Character Generation Shortcuts]] fold old school D&D with four backgrounds each for fighters, wizards and clerics. In a way I wanted to show that I didn't need thieves, rangers and paladins -- I wanted to model these classes using a base class with some appropriate background and equipment -- and I wanted to speed up character creation by avoiding rolling for starting gold and shopping for equipment. Half a year later I [[2010-06-24 Character Generation Shortcuts PDF|increased the number of backgrounds to eight]] and used it in a convention game I ran, but I still missed backgrounds for demihumans. The following is a list of four backgrounds each.
Changed:
< We'll see about adding more entries and revising the [[2011-03-15 Character Genration Shortcuts PDF|PDF]] when I have some extra time. :)
to
> We'll see about adding more entries and revising the [[2011-03-15 Character Generation Shortcuts PDF|PDF]] when I have some extra time. :)
A bit over a year ago I wrote my first draft of the Character Generation Shortcuts fold old school D&D with four backgrounds each for fighters, wizards and clerics. In a way I wanted to show that I didn’t need thieves, rangers and paladins – I wanted to model these classes using a base class with some appropriate background and equipment – and I wanted to speed up character creation by avoiding rolling for starting gold and shopping for equipment. Half a year later I increased the number of backgrounds to eight and used it in a convention game I ran, but I still missed backgrounds for demihumans. The following is a list of four backgrounds each.
1d4Elf Dwarf Halfling
1you are a woodland hunter; you own an elven longbow with twenty arrows and a curved dagger decorated with silver runes; the lore masters have taught you the spell sleep to keep you safe while alone in the wilderness you are a miner but it was dangerous work for little gain; you own a canary bird in a wooden cage, a small pony to carry your load, a lantern, studded leather armor and a heavy pick you are a shepherd; you own a sling, a dagger, a woolen cloak, and a ferocious dog that will defend you in combat
2you are a student of ancient lore; you own a ceremonial elven longsword in the old tradition and a set of chain mail; the lore masters have taught you the spell read languages to help your studies you are a gem cutter by trade but decided to go looking for the real thing; you own gleaming chain mail, a beautiful two-handed sword and a jewel encrusted girdle handed to you from your father you are a gardener; you own a hammer and a sling and some ill-fitting goblin leather armor
3you are a spell singer; you own an elegant elven longsword, a dagger, leather armor, a wooden shield, a lyre and a silver flute; you have learned the spell charm person to keep you safe in human lands you are berserker with warding tattoos covering every inch of your skin; you own a two-handed dwarven battle axe inscribed with old runes of death and destruction and battered scale armor you are a scout and have travelled the wilderness; you own leather armor, a shield, a short sword and a sling
4you are a fae knight; you own a white horse, a white steel plate armor, a shield, a lance, a long sword, a mace; your master taught you the spell protection from evil before sending out into the world you are rune warrior and a defender of the realm; you own the finest dwarven plate armor, a heavy shield engraved with runes of protection and an axe.you are a bored landowner but left your siblings in charge of it all; you own a short sword, a shield, light dwarven chain mail and a sling
We’ll see about adding more entries and revising the PDF when I have some extra time.
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< [[gravatar:http://blog.mausdompteur.de Harald:fd33438f1aeed32f19760834a93da0f5]]
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< [[gravatar: AlexSchroeder:e33b88db6bc04e1c93db25c702baea28]]
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< [[gravatar: AlexSchroeder:e33b88db6bc04e1c93db25c702baea28]]
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< [[gravatar: AlexSchroeder:e33b88db6bc04e1c93db25c702baea28]]
Typical – if you are elf, you can be awesome things like a hunter, spell singer, or fae knight, and if you are dwarf, you could be a rune warrior or berserker. But if you are a halfling, you probably used to be a gardener
Looks like Alex got a whiff of “implicit setting” here …
Harald 2011-03-08 19:54 UTC
Absolutely. One roll life-path, baby!
Actually, I’d love to brainstorm some more:
; Elf: a sea elf with trident, net, bow, longsword; a moon elf with a horse, a member of the wild hunt, a horn, a bow, a scimitar, a wolf hound companion; a shadow elf from a far realm, an illusionist with a spider companion and a dagger; a dark elf that lived under the hill, a member of the winter court, a forger of rings (lame easter egg?)
; Halflings: a crazy naked ferocious halfling cannibal; an escaped slave, broken by years spent in the darkness deep underground; maybe something crazy like a windling – those halflings that were picked up and blown across half the world by the five great storms; a fool that was sent on adventure by a wizard (lame easter egg?)
; Dwarf: Any ideas?
Dwarf - beardless outcast, apprentice brewmaster, grizzled campaigner (fought against orcs and goblins) Halfing - deputy shiriff
Nice! There’s also the Thief to think of… Cutpurse, burglar, harlequin or clown, dancer, rat catcher, assassin, contortionist (too much circus?), spy.
1d4Elf Dwarf Halfling
5you are a sea elf; you own a trident, a net, a longbow and a longsword; the captain of your first ship has taught you the sleep spell you are a surface merchant; you own a suit of chain mail armor, a shield, a hammer, a donkey, a cart, 200 gp in trade goods, and three gems worth 50 gp each you are a crazy naked halfling cannibal; you own a dagger, a femur bone usable as a club and a lot of blue body paint
6you are a moon elf and belong to the wild hunt; you own a horse, a horn, a longbow, a scimitar, a silver dagger and a wolf hound companion that will attack anything that flees; the lore masters have taught you the spell detect magic to aid you in your raids you are an escaped slave, broken by years spent in the darkness deep in a kobold mine; you own a pick axe, ill-fitting goblin armor, a serrated goblin dagger and a wooden shield
7you are a shadow elf from a far realm with a fist-sized spider companion; you own a billowing shadow cloak, studded leather armor, a black shield, a scimitar and a curved dagger; you have studied the spell hold portalyou are a brewmaster
8you are a dark elf from under the mountain and a member of the winter court, you own elegant flowing robes, a silver diadem worth 100 gp, a scimitar engraved with silver runes of banishment; you have studied the light spell you are a grizzled campaigner from the goblin wars, you own plate mail, a polearm (halberd), a hand axe, and a dagger
AlexSchroeder 2011-03-09 17:51 UTC
Gaah, need some dwarven background inspiration. Perhaps I should look at some Warhammer careers.
AlexSchroeder 2011-03-14 17:21 UTC
A dwarf criminal, his beard shaved off and his face branded, fled to the surface and surviving as a ruffian. Leather armor, club or axe, dagger.
What about an apprentice smith? Seems like an obvious one. Nice hammer or axe, decent armor, some gold. | 2016-05-06 05:37:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1980074644088745, "perplexity": 10472.507754317237}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861727712.42/warc/CC-MAIN-20160428164207-00202-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.techwhiff.com/issue/can-someone-help-me-with-the-first-three-questions--125832 | # Can someone help me with the first three questions of this assignment? I just need an example of what to do in this assignment.
###### Question:
Can someone help me with the first three questions of this assignment? I just need an example of what to do in this assignment.
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### What does verify, uphold mean? heres the assignment i need to do.
What does verify, uphold mean? heres the assignment i need to do.... | 2022-10-06 01:43:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1822534203529358, "perplexity": 5534.591141618122}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337680.35/warc/CC-MAIN-20221005234659-20221006024659-00729.warc.gz"} |
https://socratic.org/questions/how-do-you-simplify-m-4p-2-m-2p | How do you simplify (m^4p^2)/(m^2p)?
$a = {a}^{\textcolor{red}{1}}$ and ${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$
$\frac{{m}^{\textcolor{red}{4}} {p}^{\textcolor{red}{2}}}{{m}^{\textcolor{b l u e}{2}} {p}^{\textcolor{b l u e}{1}}} = {m}^{\textcolor{red}{4} - \textcolor{b l u e}{2}} {p}^{\textcolor{red}{2} - \textcolor{b l u e}{1}} = {m}^{2} {p}^{1} = {m}^{2} p$ | 2021-12-05 21:31:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6863719820976257, "perplexity": 1498.2846488310392}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363216.90/warc/CC-MAIN-20211205191620-20211205221620-00166.warc.gz"} |
https://www.sciencehq.com/chemistry/nuclear-binding-energy.html | Nuclear Binding Energy
Nuclear binding energy
It may be defined as the energy released when the given number of protons and neutrons combine to form its nucleus. In other words, it is the energy required to disrupt its nucleus into its constituents. This released energy corresponds to mass defect $(\Delta m)$, e.e., the difference between the expected mass and the actual mass of the atoms. It may be written as:
Binding energy = [mass of neutrons + mass of protons + mass of electron] of the atom – actual mass of the atom
= mass defect a.m.u.
931.5 × Mass defect (MeV)
(since 1 a.m.u. = 931.5 MeV)
Aston expressed the variation of isotopic mass from mass number in terms of packing fraction for each isotope as:
$\text{pacing fraction} = \dfrac{\text{Isotopic mass- Mass number}}{\text{Mass number}} \times 10^4$
It is evident that the smaller value of packing fraction shows grater stability of the atom and vice versa.
Example 1: the fission fragments in the thermal neutron fission of $^{325}_{92}U$ found to be $^{98}_{42}Mo \text{and} ^{136}_{54}Xe$ what are the elementary particles and energy, released in the reaction? (Isotopic mass of $^{325}U ^{98}Mo, ^{136}Xe \text{and} ^1_0n$ are 235.044, 97.906, 135.907 and 1.0086 amu, respectively).
Solution:
According to the example,
$^{325}_{92}U + ^1_0n \to ^{98}_{42}Mo + ^{136}_{54}Xe$
On balancing the equation by elementary particles the above equation many be written as:
$^{235}_{92}U + ^1_0n \to ^{98}_{42}Mo + ^{136}_{54}Xe + 2 ^1_0n + 4 ^1_{-1}e\\ \therefore$elementary particles released are neutrons and electrons.
Mass of the reactants = 235.044 + 236.0526
Mass of the products
= 97.906 + 135.907 + 2(1.0086) + 4(0.0005486) = 235.8324
Therefore mass defect $(\Delta m)$ = 235.8324 – 236.0526 = -0.2202 amu
Since the value of $(\Delta m)$ is negative, it means energy is released during the process.
Energy = 0.2202 × 931 MeV
Example 5: calculate the loss in mass accompanying combustion of one mole of fuel to release 1800KJ heat energy
Solution: since $E = \Delta m c^2$
Where E = released energy, $\Delta m$ and c = velocity of ligh
Hence
$\Delta m = \dfrac{E}{c^2} = \dfrac{1800 \times 10^3 J}{(3 \times 10^8 m s^{-1})^2} \\ 2 \times 10^{-11} kg (\text{since} J = kg m^2s^{-2})$
Medicinal use of radioactive isotopes
Henry Danlos use Ra for the first time in the treatment of Tuber Culons Skin Lesin disease.
CO – 60 $\to$ in treatment of Cancer.
I-131 $\to$in treatment of Thyroid Gland.
P-32 $\to$ in treatment of Leukemia.
Na-24 $\to$ to trace flow of blood
I-123 $\to$ for imaging the brain.
Au-198 $\to$ blood cancer treatment.
Technetium-99M $\to$ for scanning bones.
Other uses
$^3_1H$ is used as a Tracer in absorption of water by plant roots.
P-32 $\to$ TO Truce plant growth etc.
O-18 $\to$ in photosynthesis.
$6CO_2 + 12H_2O^{18} \to C_6H_{12}O_6 + O_2^{18} + 6 H_2O$
Na-24 to find leakage pipes.
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5. Fusion, Fission Fusion, fission, isotopes, isobars Atoms are the buldig blocks... | 2020-12-01 15:45:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 23, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4949636161327362, "perplexity": 3267.656088267909}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141674594.59/warc/CC-MAIN-20201201135627-20201201165627-00065.warc.gz"} |
https://proofwiki.org/wiki/Exponent_Combination_Laws/Product_of_Powers/Proof_2/Lemma | # Exponent Combination Laws/Product of Powers/Proof 2/Lemma
## Theorem
Let $x_1, x_2, y_1, y_2 \in \R_{>0}$ be strictly positive real numbers.
Let $\epsilon \in \openint 0 {\min \set {y_1, y_2, 1} }$.
Then:
$\size {x_1 - y_1} < \epsilon \land \size {x_2 - y_2} < \epsilon \implies \size {x_1 x_2 - y_1 y_2} < \epsilon \paren {y_1 + y_2 + 1}$
## Proof
First:
$\ds \epsilon$ $<$ $\ds \min \set {y_1, y_2, 1}$ $\ds \leadsto \ \$ $\ds \epsilon$ $<$ $\ds y_1$ Definition of Min Operation $\ds \leadsto \ \$ $\ds \epsilon - \epsilon$ $<$ $\ds y_1 - \epsilon$ subtracting $\epsilon$ from both sides $\text {(1)}: \quad$ $\ds \leadsto \ \$ $\ds 0$ $<$ $\ds y_1 - \epsilon$
The same logic, mutatis mutandis, shows that $0 < y_2 - \epsilon$.
$\ds \size {x_1 - y_1} < \epsilon$ $\implies$ $\ds y_1 - \epsilon < x_1 < y_1 - \epsilon$ $\ds \size {x_2 - y_2} < \epsilon$ $\implies$ $\ds y_2 - \epsilon < x_2 < y_2 - \epsilon$
Hence:
$\ds \paren {y_1 - \epsilon} \paren {y_2 - \epsilon}$ $<$ $\, \ds x_1 x_2 \,$ $\, \ds < \,$ $\ds \paren {y_1 + \epsilon} \paren {y_1 + \epsilon}$ Inequality of Product of Unequal Numbers: from $(1)$ $\ds \leadsto \ \$ $\ds y_1 y_2 - \epsilon \paren {y_1 + y_2} + \epsilon^2$ $<$ $\, \ds x_1 x_2 \,$ $\, \ds < \,$ $\ds y_1 y_2 + \epsilon \paren {y_1 + y_2} + \epsilon^2$ Distributive Property $\ds \leadsto \ \$ $\ds y_1 y_2 - \epsilon \paren {y_1 + y_2} - \epsilon^2$ $<$ $\, \ds x_1 x_2 \,$ $\, \ds < \,$ $\ds y_1 y_2 + \epsilon \paren {y_1 + y_2} + \epsilon^2$ Square of Non-Zero Real Number is Strictly Positive
Subtracting $y_1 y_2$ from all sections of the inequality:
$-\epsilon \paren {y_1 + y_2} - \epsilon^2 < x_1 x_2 - y_1 y_2 < \epsilon \paren {y_1 + y_2} + \epsilon^2$
If follows that:
$\ds \size {x_1 x_2 - y_1 y_2}$ $<$ $\ds \epsilon \paren {y_1 + y_2} + \epsilon^2$ Negative of Absolute Value: Corollary 1 $\ds$ $<$ $\ds \epsilon \paren {y_1 + y_2} + \epsilon$ Since $\epsilon < \min \size {y_1, y_2, 1} < 1$, we may apply Real Number between Zero and One is Greater than Square $\ds$ $<$ $\ds \epsilon \paren {y_1 + y_2 + 1}$ Distributive Property
Hence the result.
$\blacksquare$ | 2023-03-29 11:09:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8909226655960083, "perplexity": 210.1660093094789}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948965.80/warc/CC-MAIN-20230329085436-20230329115436-00447.warc.gz"} |
https://warwick.ac.uk/fac/sci/physics/intranet/pendulum/helmholtz/ | # Helmholtz Free Energy
## $\LARGE F=-kT{\ln}Z$
### The Terms
$F$ - Helmholtz free energy [Units: Joules, J]
$k$ - Boltzmann constant. [Value: 1.381 x 10-23 J K-1]
$T$ - Absolute temperature. [Units: Kelvins, K]
$Z$ - Partition function
### What it means
In statistical mechanics and thermodynamics the Helmholtz free energy is a theremodynamic potential which measures the "useful" work obtainable from a closed thermodynamic system at constant temperature, volume and particle number. For such a system in equilibrium the Helmholtz free energy is minimised.
### Further information at Warwick
Helmholtz free energy is covered in the second year module "PX265 Thermal Physics II" and the third year module "PX366 Statistical Physics". | 2021-04-21 07:13:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6353350877761841, "perplexity": 3308.723689834529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039526421.82/warc/CC-MAIN-20210421065303-20210421095303-00213.warc.gz"} |
http://www.dry-lab.org/blog/2018/numbering-pages-of-a-pdf.html | Blog of Andrés Aravena
# Numbering the pages of a PDF
28 January 2018
I often have PDF files without page numbers. For example, when I print the exam questions. Nowadays I prepare my exams in Rmarkdown and compile them to HTML, which is the same format that my students will use. But when I print them on Google Chrome they do not have page numbers, or worse: they have (a wrong) date and show the filename in my computer. I used to change the date on my computer and upload the file to a secret folder on my blog, but this is too much trouble for such a small issue. Now I’m just printing without page numbers.
I was resigned to this situation, until my wife asked me to put page numbers into some of her PDF documents. Then I had to find a way to do it. Here is how I solved it.
# How to add page numbers to a PDF
Adobe has a paid solution, and is not “command line friendly”. I found a good answer at Command Line FU “Add page numbers to a PDF”. Their suggestion is
enscript -L1 -b'||Page $% of$=' -o- < \
<(for i in $(seq "$(pdftk "$1" dump_data | grep "Num" | cut -d":" -f2)"); \ do echo; done) | ps2pdf - | pdftk "$1" multistamp - output "${1%.pdf}-header.pdf" I had forgotten about enscript, a program that I used ten years ago to print my scripts on the PostScript printer. I tested the first part by doing enscript -L1 -b'||Page$% of $=' -o- < <(for i in$(seq 1 429); do echo; done) | \
ps2pdf - test.pdf
and it worked as expected (like a charm). Then upgraded pdftk, since the version I had installed never worked. The official webpage had only a version for MacOS X 10.6, probably a 32 bit one. But there is a hidden version for 10.11 at https://www.pdflabs.com/tools/pdftk-the-pdf-toolkit/pdftk_server-2.02-mac_osx-10.11-setup.pkg. I tested it with the option proposed to find the number of pages:
pdftk 'original.pdf' dump_data |grep Num
It worked, but the grep pattern has to be more specific. Using "Num" yields too many answers. Instead I just opened the original.pdf file and took note of the number of pages. The command I used to test was:
enscript -L1 -b'||Page $% of$=' -o- < \
<(for i in $(seq 1 429); do echo; done) | \ ps2pdf - | \ pdftk 'original.pdf' multistamp - output numbered.pdf and the result was perfect… except for the location of the numbers. ## Printing footers instead of headers So far I could use the example code to get page numbers on the header, but not in the footer, despite using option --footer as indicated in the manual page. Asking to Google led me to a discussion about “printing Footers using enscript”. Apparently the PostScript code generated by enscript does not handle footers, but it can be fixed installing some code for fancy headers on the ~/.enscript/footer.hdr file. I followed the advice and tested it: enscript --fancy-header=footer -L1 -b'||' --footer '||$%' -o- < \
<(for i in $(seq 1 429); do echo; done) | \ ps2pdf - 'test.pdf' Now the file test.pdf had 429 white pages with the page number in the lower right side. The manual also showed options to change the font from Courier to something nicer, such as Times Roman at 10pt. The final command was: enscript -F Times-Roman10 --fancy-header=footer -L1 -b'||' --footer '||$%' -o- < \
<(for i in $(seq 1 429); do echo; done) | \ ps2pdf - | \ pdftk 'original.pdf' multistamp - output 'numb.pdf' ## Extra tricks learned In bash we can use $(seq 1 429) instead of the regular backquote seq 1 429. It is easier to read, and sometimes easier to understand. In both cases the shell executes the command inside parenthesis/backquotes and its standard output became command line arguments for the outer command.
We can also use <(for i in $(seq 1 429); do echo; done) to inject the standard output of a command to the input of another. In this case we can easily use a pipe, like this: for i in$(seq 1 429); do echo; done | \
enscript -F Times-Roman10 --fancy-header=footer -L1 -b'||' --footer '||$%' -o - | \ ps2pdf - | pdftk 'original.pdf' multistamp - output numb.pdf Both shell expansions execute a command and handle its standard output. They differ in how they deliver this output to the next command. The $() syntax delivers the output as arguments, the <() syntax delivers in the standard input. Both are alternatives to classical syntax.
# Summary
Next exam will have page numbers in the printed copy. | 2022-06-30 07:13:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3349629044532776, "perplexity": 4309.948510294805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103669266.42/warc/CC-MAIN-20220630062154-20220630092154-00601.warc.gz"} |
https://ask.sagemath.org/question/43392/sage-could-be-even-more-clever-how-to-force-the-use-of-sympy-backend-for-simplifying-symbolic-integrals/ | # Sage could be even more clever - How to force the use of 'sympy' backend for simplifying symbolic integrals?
Hi there,
I have noticed the following problem:
sage: f = function('f')(x)
sage: var('h')
sage: integrate(exp(h)*exp(x)*f(x),x)
integrate(e^(h + x)*f(x), x)
The workaround seems to be using the sympy backend for symbolic integration
sage: integrate(exp(h)*exp(x)*f(x),x,algorithm='sympy')
e^h*integrate(e^x*f(x), x)
which always seems to be a good idea as I learned from @Emmanuel Charpentier over here.
Now I would like to force the use of algorith='sympy' for simplifying these integrate(...) expressions globally. Unfortunately, the simplify() command does not allow to set this option.
sage: integrate(exp(h)*exp(x)*f(x),x)
integrate(e^(h + x)*f(x), x)
sage: _.simplify()
integrate(e^(h + x)*f(x), x)
TL;DR How can I force sage to pull out these type of exponential constants from the integral with the simplify() command?
edit retag close merge delete
If simplify() does not allow setting different backends for different kinds of expressions then that's a feature that might be worth adding, and I'm sure patches would be welcome :)
( 2018-08-16 12:32:45 -0500 )edit
I wouldn't (yet) use sympy as the default backend for integration :
• more than a few times, sympy seems to start an infinite loop, and "never" returns ("never" meaning that, back from my coffee break, I hit C-c C-c...) ;
• sympy tends to give answers in terms of functions not yet implemented in Sage, (Meijer G, polar numbers, etc...), which it is unable to re-express in Sage-palatable terms ;
In other words, these functions should be handled on Sage side before any step in the direction of making sympy the default for anything.
Someone called rws (whose real name escapes my aging neurons at the moment...) is doing a gigantic job of enhancing the Sage-sympy. This could come handy if/when sympy implements rubi as planned...
( 2018-08-16 18:06:36 -0500 )edit
Sort by » oldest newest most voted
You can go back-and-forth from sage to sympy, i.e. turn your expression into a sympy one then ask sympy to factor it, then transform it back into a sage object:
sage: i = integrate(exp(h)*exp(x)*f(x),x) ; i
integrate(e^(h + x)*f(x), x)
sage: i._sympy_().factor()._sage_()
e^h*integrate(e^x*f(x), x)
more
Thanks, that was the workaround I was looking for!
( 2018-08-31 12:45:16 -0500 )edit
## Stats
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Last updated: Aug 16 '18 | 2019-04-23 06:20:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23022302985191345, "perplexity": 6068.207362516672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578593360.66/warc/CC-MAIN-20190423054942-20190423080942-00351.warc.gz"} |
https://math.stackexchange.com/questions/2381363/is-there-anything-even-close-to-a-standard-symbol-to-indicate-that-a-diagram-i/2381836 | # Is there anything even close to a “standard symbol” to indicate that a diagram is required to commute?
It is evident from contemporary mathematical practice that there does not exist any usual notation to denote that a diagram commutes. This is almost always stated in meta-language above or below a diagram.
Questions. Is there anything even close to a standard symbolic notation to indicate that a diagram commutes? Maybe in higher category theory?
Is there any noteworthy book or paper which makes use of a symbol, regardless of whether this is to be considered standard or advisable, instead of using surrounding words to impose the condition of commmutativity?
Remarks.
• Part of the motivation for this question is wishing not to have to make up a symbol if there is any noteworthy precedent in the literature, but, needless to say, I have never seen any such.
• Since there is e.g. a standard symbol to denote pullback diagrams, it seems somewhat surprising that the contemporary consensus seems to be to use surrounding words instead of a symbol.
• Of couse, the omission of certain composite morphisms in a diagram can be seen as some sort of signalling that a diagram is required to commute. The iconic example is a commutative square, in which, for good reason, the two composites are not drawn, and one could see this very omission to by indicative enough. Yet, to me, this seems not to yield a precise pictorial language, least of all when it comes to using diagrams as input to machine-computations.
• I seem to recall to have once read a paper, which I do not find, saying that in the early days of category theory the writing style in books and papers was more diagrammatical than nowadays, so, it at all, a notational convention such as the one I am looking for might be found in early category theoretical papers.
• Often a small circle (like the composition symbol) or a small clockwise arrow for smaller commutative diagrams – Osama Ghani Aug 3 '17 at 15:32
• @OsamaGhani: thanks for this hint. For me, personally, this seems not an option, since it interferes with the diagram proper too much: a small circle looks like the letter 'O', which I often use for objects, and a circled arrow looks, well, too much like an endomorphism of some object. I recognize that this answer probably simply does not have any answer relieving me of having to invent a symbol, or of sticking to a word near the diagram. Just asking since there is always something one does not know. – Peter Heinig Aug 3 '17 at 15:38
• @OsamaGhani: moreover, a circled arrow involves having to make the choice whether to use clockwise or anticlockwise order, and whatever choice one makes, it looks as if one wants to convey specific information by the orientation chosen. This is all the more an issue if (as I happen to do) works on a proof where having chosen an orientation of the Euclidean plane plays an essential role. Such orientations are often indicated by circled arrows. Thanks again. – Peter Heinig Aug 3 '17 at 15:40
• In lectures on a blackboard, I've seen people draw three marks in the middle of the diagram, like this : $'''$. – Arnaud D. Aug 3 '17 at 15:43
• Also in lectures, I've seen something like the # symbol rotated 45 degrees, in the middle of the diagram. But the circle-arrow seems fairly common: tex.stackexchange.com/questions/43395/…, tex.stackexchange.com/questions/80413/… – Hans Lundmark Aug 3 '17 at 15:46
The closest thing you'll get to a "standard" symbol is to insert an identity 2-cell. $$\require{AMScd} \begin{CD} A @>f>> B \\ @VhV\qquad\!\!\Downarrow\, id V @VVkV \\ C @>g>> D \end{CD}$$ This necessarily requires the two ends are equal just like the identity arrow requires the source and target objects are equal. This can be generalized to any dimension.
Usually diagrams are assumed to commute and no additional notation is necessary. Indeed, you occasionally see notation to explicitly indicate that a diagram doesn't commute.
• Thanks for this anwer. I was aware of 2-cell, yet in using them for this, there are other issues: (0) the 2-cell symbols are so 'directional' and force one to make the arbitrary choice where to have them point to exactly, and, more seriously for my purposes, (2) if you deal with diagrams more complex than the iconic commutative square, then the 2-cell arrows, to me, do not even seem correct (or at least they commit one to one of the existing pasting-scheme-conventions). Arrows are just not 'global-looking' enough, they do not convey 'the diagram in its entirety is commutative'. Thanks again. – Peter Heinig Aug 5 '17 at 5:12
• For example, already in example diagram I post in my summarizing answer in this thread, it seems that 2-cell arrows won't do, or, at least, their use brings about additional problems, which is the opposite of what I intend. – Peter Heinig Aug 5 '17 at 5:15
To get this notational question wrapped up, here is a summarizing answer.
Generalities.
Needless to say, marking out diagrams as commutative is unusual, and---if done badly--spoils them.
The beauty and usefulness of commutative diagrams lies in that they contains hardly anything unnecessary.
There can be reasons for a symbolic notation though, in particular if one in a proof works with multiple diagrams, as mathematical objects in their own right, and finds accompanying words lying about somewhere in their periphery inadequately precise.
This is why I, personally, opted not to use any symbol within the diagram, neither the $\circlearrowleft$ kindly suggested by Osama Ghani, nor the $'''$ suggested by Arnaud D., nor the rotated # kindly suggested by Hans Lundmark, in the comments to the OP, nor even the identity 2-cell kindly suggested by Derek Elkins in one of the answers (in the example below, one would need more than one higher cell, which brings about new problems, while the intent is just to mark the diagram in its entirety as commutative).
In my work, I use rectangular light-gray frames around the diagram, annotated with the word "commutes", always at the bottom of the rectangle (see below).
For certain purposes, it still seems to me that a standardized notation for a diagram being commutative is useful.
Please note that the below is unusual; I'm just adding it here to suggest to others one of the infinitely-many solutions one can find to this notational problem.
represents a set of axioms($=$ set of pairs of words over a signature of category theory). | 2019-11-17 15:03:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7665301561355591, "perplexity": 496.93380485133855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669183.52/warc/CC-MAIN-20191117142350-20191117170350-00157.warc.gz"} |
https://alolaecon.wordpress.com/tag/economics/ | ## Cognitive Dissonance
I just finished reading Cooper’s book on the topic and would very much like to write down what I get out of it.
Cognitive dissonance was first envisioned by Leon Festinger in the 1950s. The theory started out elegant and simple: With consistent cognitions, individuals experience uneasiness and discomfort thus are motivated to reduce the inconsistency. Here cognitions can refer to actions, beliefs, attitudes, and so on. Inconsistency can be due to inconsistent attitude and action. One of the many experimental paradigms to test the theory runs as follows: subjects are asked to evaluate a certain proposition arguing against her/his position. Lastly attitude is elicited again. For instance, subjects (usually college students) are asked about their attitudes towards increasing tuition. Expectedly most of them are against it thus are asked to write an essay/make a speech for tuition increase after which they are asked again about their attitudes on the issue. (All results will be discussed under this paradigm for exposition.) Typical findings on these experiments show that subjects change their attitudes closer to what they (are asked to) advocate. According to Festinger, subjects experience cognitive dissonance: writing an essay for tuition raise is inconsistent with their attitude against a raise. As a result of the dissonance, subjects are motivated to reduce the inconsistency. In this case (as in many other cases), changing one’s attitude is easier than changing a committed action.
The theory is appealing as it explains, elegantly, experiments and observations. However humans are hardly that simple (thus the struggle of social scientists). In the following years, many experiments and theories are raised to clarify and complete the theory. I will first discuss efforts on verifying the physiological manifestation of dissonance and then talk about modifiers found that largely rewrite the whole story.
One of the first questions raised against the theory concerns the motivated drive (Chapter 3 of the book) Festinger highlighted in his original composition. Simply put (especially intuitive to economists), psychologists wondered if people really experience tension states/arousal/emotions or anything with a physiological trait thus are motivated/driven to reduce inconsistent or is it just a metaphor/as-if argument. With the hep of misattribution and two-factor theory, experiments show that the uneasiness is experienced
• Two-factor model of Schachter and Singer (1962): an experience of emotion is based on two factors: arousal and the attachment of a cognitive label to that arousal. They conducted experiments to show that the factors are separable in the sense that how the arousal is experienced is affected by the label provided. In short, the experience of emotion is labile.
• Back to dissonance: subjects who take placebo but think they take a drug which might cause uneasiness and discomfort, do not change their attitudes as much after the essay task. Similarly, subjects who think they take a drug that might cause relaxation change their attitudes more after the task.
• These experiments show that when provided with a convenient label of their discomfort, subjects do not attribute the arousal to inconsistent cognitions thus make no effort in reducing consistency.
These are good news for the theory which solidify the original ‘guess’ of Festinger. Now the ‘bad’ news: New Look Model and Self-Standards Model (SSM henceforth) that largely kill the original idea of the theory. SSM is basically an extension of the former; some years passed before the birth of SSM but I will skip the struggle and talk directly about SSM as if it were a success in one shot (but it is really not).
• The following experimental results forced researchers to rethink about the theory: No/little attitude change is observed if (1) subjects are not presented with a choice, (2) the action does not result in any aversive and foreseeable consequence, or (3) the behavior is expected by some (social/personal) standard. For the essay task,
1. Attitude changes if subjects are asked if they would like to write such an essay (high-choice treatments) and no changes if subjects have to do so (low-choice treatments).
2. Attitude changes if the essay is to be presented to the college committee and no if it is destroyed after completion. Attitude changes if the essay might be viewed by the committee and no if it comes as a complete surprise (unforeseeable).
3. This point cannot be demonstrated with the paradigm and plays an important role in turning Festinger’s theory up-side-down. So I will discuss it in more details in the next entry.
Just as a side note: the experiments become more intriguing as the theory gets more complicated. The levels (plural really) of deception used in these experiments stunned me, an economist who sleeps with the dogma that deception is not allowed in experiments.
Okay this entry is getting too long to bear even for myself. I am gonna end here.
[1]Cooper, Joel. Cognitive dissonance: 50 years of a classic theory. Sage, 2007.
P.S. This is a very nicely written book: well-organized and fun to read. Perfect as an introduction into the theory.
## Zorn’s Lemma and Revealed Preference
I don’t think I ever understand why Axiom of Choice is always singled out; why it is special (and controversial) relative to other axioms. Gradually, I realized that probably it is not. It is just one of the axioms that found set theory. Anyways, this is not what we, as economists, should care. This entry is about Zorn’s lemma and revealed preference. I mentioned axiom of choice first just because its close relation to Zorn’s lemma. To complete the discussion of axiom of choice, let me just put the version that I personally like most here
Axiom of Choice: Cartesian product of non-empty sets is non-empty.
Now Zorn’s lemma and revealed preference. Revealed preference is one of the most fundamental result in economics. It builds a bridge between observable choice data and underlying (but invisible) preferences. There are various versions of the result but the idea is the same
Revealed Preference: Any set of choice data that satisfies some axiom of revealed preference can be rationalized by a preference relation.
In other words, however limited the data are, we can find a preference defined over the whole commodity space to make sense of the data, given that the data satisfy some conditions. Zorn’s lemma allows a compatible extension from the limited scope provided by the data to the whole commodity space.
Zorn’s Lemma: If every chain in a partial-ordered set has a upper bound, then the partial-ordered set has a maximal element.
Zorn’s lemma is essential to the following result which basically makes the revealed preference a corollary.
Szpilrajn’s Theorem on total extension of preorders: Any preorder has a compatible extension to a total order.
Below I just list the definitions of the relevant concepts assuming familiarity of common properties of binary relations
• Preorder Reflexive+Transitive
• Partial order Preorder+Antisymmetric
• Total order Partial order+Complete
• Transitive closure of a binary relation R over X is the smallest relation over X that contains R and is transitive
• Chain a totally ordered subset of a partial-ordered set
• A binary relation R* over X is a compatible extension of R if R* extends R and preserves the asymmetry of R.
Proof of Szpilrajn’s Theorem:
Let R be a preorder over X and E the set of all preorders that compatibly extend R. Clearly E is non-empty as it contains at least R itself.
Consider any non-empty chain C in E by set inclusion. We want to show that U the union of elements in C is an upper bound of C (so that we can use Zorn’s lemma). That is we need to show that all elements of C is contained in U; alternatively, U is (still) a preorder that compatibly extends R. This can be shown easily by using the properties of individual element in C (at the end of the day, one needs to show that U is reflexive, transitive, and compatibly extends R).
By Zorn’s lemma, we know that E has a maximal element S. The last step is to show that S must be total. Suppose not and there exists a pair $(x,y)$ such that neither xSy nor ySx. Consider $T=S\cup\{(x,y)\}$ and W which is the transitive closure of T. Since W includes S, if we can show that W is a compatible extension of R (reflexive is obvious), we reach a contradiction (that S cannot be the maximal element).
Again suppose W does not compatibly extend R (or S) and there exists a pair $(u,v)$ in X such that $(uSv) \wedge\ ^\neg (vSu)$ but vWu. By definition of transitive closure, we can construct a finite sequence such that $v=c_1 T c_2 T \cdots T c_{n-1} T c_n=u$. Since T and S only differ over (x,y) and if no $c_i$ is x or y, we obtain the contradiction. Otherwise, we can find $(c_i, c_{i+1})=(x,y)$. If this is the case, $y=c_{i+1} S\cdots S c_n=u S v=c_1 S \cdots c_i =x$ contradicting that S cannot rank x and y.
Thus the proof is complete and any maximal compatible extension of R is a total preorder.
The definitions and proof are from Aliprantis and Border’s book. I put it here just because I did not know why economists cared about Zorn’s lemma and this is an interesting illustration in my opinion. And I want to throw away my scratch papers.
Reference
[1]Aliprantis, Charalambos D., and C. Kim. “Border. 2006. Infinite Dimensional Analysis: A Hitchhikers Guide.” | 2017-10-21 15:34:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6795911192893982, "perplexity": 1184.6535896926393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824820.28/warc/CC-MAIN-20171021152723-20171021172723-00761.warc.gz"} |
http://www.sciencehq.com/chemistry/applications-of-bond-energies.html | Applications of Bond Energies
Applications of Bond Energies are listed below:-
1. Determination of enthalpy of reaction
The bond energies can be used for determining enthalpies of reactions $Delta H_{Reaction} = E(B.E)_R- E(B.E.)_P$ for example, we want to determine the enthalpy of the following reaction:
$HC \equiv CH(g) +2H_2(g) \to H_3C- CH_3(g) \hspace{5mm}\Delta H = ?$
In this reaction a triple bond $(C \equiv C)$ breaks in acetylene and two H — H bonds break in 4H. In turn, one C C- bond and four C — H bonds are formed in$C_2H_6$. Therefore
$\Delta H = [\Delta H_{C- C} +4 \Delta H_{C-H}] + [\Delta H_{C \equiv C} + 2\Delta H_{H- H}]$
$= -[3.47.3 + 4 \times 416.2] + [811.7 + 2 \times 435.1] \\[3mm] =-2.12.1 + 1681.9 \\[3mm] = -330.2 kJ mol^{-1}$
2. Determination of enthalpies of formation of compounds
The bond energies can be used for determining enthalpies of formation of compounds. For example, we want to determine the enthalpy of formation of ethyl alcohol:
Determination of enthalpies of formation of compounds
The formation of ethyl alcohol involves:
(i) Breaking of 3H-H bonds to give 6H, breaking bf 1/2O-O bond to give O and sublimation of 2 atoms of C(s) to give 2C(g).
(ii) Formation of one C-C bond, five bonds, one C-0 bond and one bond.
Therefore,
$\Delta H_f = [3 \Delta H_{H- H} + \dfrac{1}{2} \Delta H_{O- O} + 2 \Delta H_{C(s) \to C(g)}]- [\Delta H_{C- C} + 5\Delta H_{C- H} + \Delta H_{C-O} + \Delta H_{O- H}]$
$= 3 \times 435.1 + \dfrac{1}{2} \times 489.5 + 2 \times 719.6]- [347.3 + 5 \times 416.2 + 351.4 + 464.4]$
$2989.2- 3244.1 = -254.9 k J mol^{-1}$
3. Determination of resonance energy
Resonance energy is taken to be equal to the difference between the actual bond energy of the molecule and that of the most stable of the resonating structures. Let us determine the resonance energy of benzene.
The dissociation of benzene may be given as:
$C_6H_6(g) \to 6C(g) + 6H(g)$
According to Kekule's structure there are three C-C bonds, three C = C bonds and six C-H bonds in benzene. Therefore dissociation energy of benzene may be given as:
$\Delta H_d = 3 \Delta H_{C-C} + 3 \Delta H_{C = C} + 6\Delta H_{C-H}$
$= 3 \times 347.3 + 3 \times 615.0 + 6 \times 416.2 \\[3mm] = 5384.1 k J mol^{-1}$
Thus, dissociation energy of benzene is $5384.1 kJ mol^{-1}$. But experimental value is known to be $5535.1 kJ mol^{-1}$. Evidently, the resonance energy of benzene is $5384.1 = 151 kJ \hspace{2mm} mol^{-1}$.
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5. Odd Electron Bond Odd Electron Bond It may be defined as, "The bonds... | 2018-12-10 15:34:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 15, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4548801779747009, "perplexity": 3305.856510902902}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00087.warc.gz"} |
https://leetcode.ca/2016-09-03-278-First-Bad-Version/ | # Question
Formatted question description: https://leetcode.ca/all/278.html
278. First Bad Version
You are a product manager and currently leading a team to develop a new product.
Since each version is developed based on the previous version, all the versions after a bad version are also bad.
Suppose you have n versions [1, 2, ..., n] and you want to find out the first bad one,
which causes all the following ones to be bad.
You are given an API
which will return whether version is bad.
Implement a function to find the first bad version. You should minimize the number of calls to the API.
Example:
Given n = 5, and version = 4 is the first bad version.
Then 4 is the first bad version.
@tag-array
# Algorithm
Binary search.
It should be noted that there is a pitfall in OJ, that is, if both left and right are particularly large, then left+right may overflow. Our solution is to become left + (right-left) / 2, which is good Avoided overflow problem.
# Code
Java
•
/* The isBadVersion API is defined in the parent class VersionControl.
public class Solution extends VersionControl {
int start = 1;
int end = n;
// @note: cannot be <=
// eg. n=5 then mid=3 then mid= 2 then start=1&end=1, infinite loop
while (start < end) {
int mid = start + (end - start) / 2;
end = mid;
} else {
start = mid + 1;
}
}
return start;
}
}
}
class VersionControl{
// stub
// stub
return false;
}
}
• // OJ: https://leetcode.com/problems/first-bad-version/
// Time: O(logN)
// Space: O(1)
class Solution {
public:
int L = 1, R = n;
while (L <= R) {
int M = L + (R - L) / 2;
if (isBadVersion(M)) R = M - 1;
else L = M + 1;
}
return L;
}
};
• # The isBadVersion API is already defined for you.
# @param version, an integer
# @return a bool
class Solution(object):
"""
:type n: int
:rtype: int
"""
lo = 1
hi = n
while lo < hi:
mid = lo + (hi - lo) / 2 | 2023-01-30 14:56:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17471203207969666, "perplexity": 8078.540536097849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499819.32/warc/CC-MAIN-20230130133622-20230130163622-00366.warc.gz"} |
https://www.notforflight.com/2019/05/ | ## Saturday, May 25, 2019
### Pressure Regulators and Other Fluidsy things
So close friends know that I am working on a major rocket project right now.
It involves pressurants, solenoids, regulators, check valves, diffusers, tanks, COPVs, and so many damn more things that *aren't* electronics or guidance algorithms.
So I am learning a lot.
Things to look for in selecting regulators:
• Inlet maximum pressure: this is the most obvious parameter. You need to select a regulator that will work with the system you are designing. So if it is hooked directly to a helium K-bottle, you're probably dealing with 3000+psi ratings here.
• Outlet maximum pressure: this is another gimme. If you need to press a tank to 400psi, don't find a regulator that varies to 10psi output.
These two are hard enough to spec out for high pressure rocket systems. Then on top of that you have:
• C_v Flow Coefficient: this one is super important and is probably the most overlooked thing in my experience. Your device must be able to flow a lot of fluid to keep up with the dynamics of the system. If the effective orifice size is small, then it will not have a high mass flow rate.
It is defined as such: $C_v = Q \sqrt{\frac{SG}{\Delta P}}$. You can think of Q as your $\dot{m}$. SG is your specific gravity, which is normally unity (1 for water) on data sheets (I think). And $\Delta P$ is the pressure drop across the system. So basically the higher this coefficient, the higher the flow rate, and the lower the pressure drop.
This equation is meant for incompressible fluids, and the gas equation is more complicated. Swagelok has some good info here: https://www.swagelok.com/downloads/webcatalogs/EN/MS-06-84.PDF
The flow coefficient is important for all fluid devices that constrict flow. So just be careful when you're attempting to procure random devices on eBay. Finding stuff for reasonable prices is hard, so I always watch out for these brands:
• Marotta
• Aqua Environment
• Tescom
• Swagelok
YOU CAN get away with low flow regulators if you attach their output to the membrane of a dome loaded reducing regulator. This seems like the usual way to do things, rather then flow directly from a single hand-turn regulator. Grove Mity Mites were apparently the way to go back in the day. There is a new company that makes them with the same design you might want to watch out for. | 2023-03-25 08:23:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5229039192199707, "perplexity": 1446.6503643864237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945317.85/warc/CC-MAIN-20230325064253-20230325094253-00272.warc.gz"} |
https://physics.stackexchange.com/questions/185262/electric-field-on-the-surface-of-a-charged-sphere | # Electric field on the surface of a charged sphere
We know that the electric field for a point charge is $$E = \frac{KQ}{R^2}.$$ If $R$, i.e. distance from the electric field producer to the point where we want to find the electric field becomes zero, then $E$ will tend to infinity. Now take a case where you have a uniformly charged sphere and you need to find the electric field on the charged sphere. The charged sphere is not a point charge, its rather a combination of point charges. What is electric field at any point, say $P$, on the charged sphere?
Now we know that when we use Gauss's Law, we select a Gaussian surface in such a way that all the points on the Gaussian surface should experience same electric field. Many sources say that if we use Gauss's Law then on any point on the charged sphere the electric field is going to be $$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{R^2}$$ where $R$ is the radius of the sphere and $\epsilon_0$ is the permittivity.
But now, don't consider Gauss's Law. As $P$ is at the surface of the charged sphere, then the electric field due to the small element of the charged sphere on which point $P$ lies is infinity, as the small element has charge $dq$ (let's also say we have assumed the small element of charge $dq$ on which point $P$ lies to be a point charge, and hence $R$ becomes zero, as the point $P$ lies on the small element). So electric field must be infinity due to that small element on which the point P lies. It doesn't really matter how much small the charge of the element is! When you do divide a large number and zero, you get infinity and similarly when you divide a very very small number and zero, you will again get infinity. Now for rest of the small elements which make up the whole sphere, the electric field won't be infinity; instead it would be some finite value. Say on the point $P$ for all the other small elements (except the element on which the point $P$ lies) net electric field is $X \rm\,N/C$. Still, $\text{infinite}+X = \text{infinite}$.
From here we conclude that on the point $P$ the electric field must be infinite. Then how did we get $$\frac{1}{4\pi\epsilon_0}\frac{Q}{R^2}~?$$
My second question is: Similarly how can one say that we can find the potential on the surface of the charged conductor? I saw sources saying that the potential on the surface of the charged sphere is $V=KQ/R$ if $R$ is the radius of the sphere. But how is this possible? I mean on the surface again for the above reasons we would get it to be infinite!
Please explain these doubts to me. These might be silly doubts, but still. I don't know how far my thinking is correct.
• Based in what I understood of your question you are confusing the concept of point and surface charges. Yes, if you disperse real point charges over the surface of a sphere at fixed locations, then the electric field will not be homogeneous at the surface and it will diverge at the point charges. That's not how surface charge works, though. May 21 '15 at 18:45
– rob
May 21 '15 at 20:07
• May be this helps: physics.stackexchange.com/q/210162 Apr 23 '16 at 2:00
• It seems this electrostatic question (v6) can be reformulated in terms of Newtonian gravity, and that it is essentially resolved by a version of the Newton's Shell Theorem. Apr 23 '16 at 12:47
(Other) Rob's answer seems good to me, but let me offer another way of thinking.
As you approach the surface of the sphere very closely, the electric field should resemble more and more the electric field from an infinite plane of charge.
If you check Gauss's law (recalling that the field in the conductor is zero) you will see that if the surface charge density is $\sigma=Q/4\pi R^2$, then indeed the field at the surface is $\sigma/\epsilon_0$ as in the infinite charge of plane case.
Such a field is constant, the field lines are parallel and non-diverging, and the infinities associated with the field due to point charge do not arise.
You seem to be confused about the concept of the limit, which is normally covered in calculus courses. If you have, not a point charge, but a volume of charge with some density $\rho$, then the charge enclosed in a small sphere with radius $r$ is $$dq = \rho\, dV = \rho \cdot \frac{4\pi}3 r^3$$ and the electric field at the surface of the sphere will be proportional to $$E \propto \frac{dq}{r^2} \propto \frac{\rho r^3}{r^2} = \rho r.$$ This field magnitude is well-behaved in the limit that $r$ is very small.
The zero-size point charge, zero-thickness surface charge, etc. are useful approximations when the size of the charge distribution is much smaller than the size of the field distribution that you care about. Physical point charges, surface charges, and line charges are all actually distributed over a finite volume. Even the charge of a single electron is distributed over a finite volume, thanks to quantum-mechanical uncertainty in the electron's position.
• Sir,we know that when the distance between the electric field producer and the point at which we want to calculate the electric field becomes zero, electric field tends to infinity. In the case of electron if I take the point at which I want to find the electric field on the surface of the electron then the EF rands to infinity. Similarly if you imagine the sphere of radius 6 cm as the larger version of the electron, similarly if you take a point on the surface of the sphere and find the EF,as the R in the formula KQ/R^2 becomes zero, EF must tend to infinty. The formula itself says it! May 23 '15 at 3:24
• And also sir in the above small derivation you did, E∝dq/r2∝ρr3/r2=ρr, you assumed the point where you want to find the electric field equal to the radius of the sphere. Actually it should be: E= dq/x^2 => E =(ρ⋅4πr^3/3)/ x^2 . If x tends to zero then E would tend to infinity. Sir, then where did I did wrong? Sir I am a student and these are completely new to me. And as these are new things i always get confused. So sir please help me out! My doubts might be weird but still..... May 23 '15 at 3:40
• Have you studied limits? For instance you might expect that the function $\mathrm{sinc}\,x = \frac{\sin x}{x}$ should misbehave at zero, but plotting the function for values very near zero reveals no misbehavior at all, and it's possible to rewrite $\mathrm{sinc}\,x$ as a series that's even well-behaved exactly at $x=0$. What I'm arguing is that if the charges are smeared out in space (which is true for all physical charges) then the electric field near any particular blob of charge is finite. You might also read about the "shell theorem" which lets me ignore nearby-but-symmetric charges.
– rob
May 23 '15 at 4:36
For a charge distribution $\rho({\bf r'})$, the electric field at ${\bf r}$ is $${\bf E}({\bf r})=k\iiint\frac{\rho({\bf r'})}{|{\bf r}-{\bf r'}|^2}\hat{\bf R}d^3{\bf r'}$$ where ${\bf R}={\bf r}-{\bf r'}$.
I think the OP's claim is at positions where $\rho \ne 0$, the above integral is infinite because the integrand blows up at ${\bf R}={\bf 0}$.
I think the OP should first ask yourself whether you are asking a mathematics question or a physics question.
If you are asking a mathematics question, I think your question is similar to whether $$\int_0^1\frac{1}{\sqrt{x}}dx$$ is undefined (or infinite according to your argument), or equals $2$.
In my opinion, strictly speaking, the above integral is undefined, because the integrand is undefined at $x=0$.
However, since $$\lim_{\epsilon\rightarrow0^+}\int_\epsilon^1\frac{1}{\sqrt{x}}dx=2$$ we usually just say $$\int_0^1\frac{1}{\sqrt{x}}dx=2$$
So in my opinion, when we write $${\bf E}({\bf r})=k\iiint\frac{\rho({\bf r'})}{|{\bf r}-{\bf r'}|^2}\hat{\bf R}d^3{\bf r'}$$ we are actually saying $${\bf E}({\bf r})=\lim_{\epsilon\rightarrow0^+}k\iiint_{D_{\epsilon}}\frac{\rho({\bf r'})}{|{\bf r}-{\bf r'}|^2}\hat{\bf R}d^3{\bf r'}$$ where $D_\epsilon$ is the space minus a ball with radius $\epsilon$ centered at ${\bf r}$.
It can be shown that if $\rho({\bf r})$ is finite or does not go to infinity $\textit{too fast}$, then the limit exists.
If you accept that, then the physics question one should ask is whether this gives the correct answer in physics.
In classical EM, the $\textit{macroscopic}$ field ${\bf E}$ is in fact the average of the $\textit{microscopic}$ field ${\bf e}$. (Landau and Lifshitz, $\textit{Electrodynamics of Continuous Media}$, p.1.)
So you can consider ${\bf E}({\bf r})$ as the average of ${\bf e}$ over a ball with radius $\epsilon$ centered at ${\bf r}$.
Now, it can be argue that $${\bf E}({\bf r})=k\iiint_{D_{\epsilon}}\frac{\rho({\bf r'})}{|{\bf r}-{\bf r'}|^2}\hat{\bf R}d^3{\bf r'}$$ as follows.
It can be shown that (1) the average field over the ball due to charges outside the ball is the same as the total field due to all charges outside the ball at the center of the ball. , and (2) the average field over the ball due to charges inside the ball is $$-k\frac{{\bf p}}{\epsilon^3}$$ where ${\bf p}$ is the total dipole moment inside the ball. (Griffith, $\textit{Introduction to Electrodynamics}$, p. 156-157, problem 3.41).
So when $\epsilon$ is small macroscopically, ${\bf p}\rightarrow{\bf 0}$.
• Because $1/0$ is undefined in math. Apr 23 '16 at 12:59
• It's not $\infty$. It's undefined. Apr 23 '16 at 13:03
• @AnubhavGoel Explain what? Apr 23 '16 at 13:11
• When x→0, function is well defined and E should become ∞. Apr 23 '16 at 13:16
• Sorry? You mean the electric field AT the position of a point charge is well-defined and is infinity? Apr 23 '16 at 13:18
Your thinking is correct. At least, if the sphere is made up of small point charges, then the field will be infinite as you approach them.
There is a point you are missing when you say that this contradicts Gauss' law: Gauss' law only gives you the flux of the field. To get the field of the sphere from it in the textbook-way, you have to use symmetry. You argue then, that since a sphere is symmetric, the flux is simply field times area. But this is not true with your assumptions. A sphere consisting of discrete points is no sphere, it has at least no spherical symmetry. The contradiction is gone.
If we say, that the field equals a certain value, we mean the mean value. In your reasoning you are aiming exactly at a point charge. Then the field is infinite. But if you have points, you must also have gaps between them, and in those gaps the field is less than that which you get with the usual formula. If you say, that the points are dense (in the sense in which rational numbers are: between any two there is still one more), then they are infinitely many and have the charge of zero each.
You argue, that a value divided by zero is infinite ... but zero divided by zero is not! It can be, but it doesn't have to. And in this case you have the same zero-ness, if you divide them you get something finite.
In reality the electrons have a non-zero charge of course. But there are also finitely many of them. So there are real finite gaps.
But you are right - point charges are a strange thing. They are not consistent with classical electromagnetism. One could argue, that it's no problem that the potential gets infinite - just don't go there, than it's finite, one could say. But in the classical theory there is the concept of energy density of the field. And the energy of a point charge is infinite, if you do the calculation.
Well, this, too, would be no problem - as long as the number of charges is conserved, at least. But it isn't ... But the only thing that it proves is, that the world is not classical.
• I accept your answer. I had same feelings but couldn't explain it to myself. Apr 22 '16 at 15:20
• thank you :) ah, you asked me to edit something (I read it on mobile), so it worked like this already
– Ilja
Apr 23 '16 at 17:38 | 2022-01-21 19:03:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8682253360748291, "perplexity": 175.47666497500316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303512.46/warc/CC-MAIN-20220121162107-20220121192107-00208.warc.gz"} |
http://nuit-blanche.blogspot.com/2015/08/efficient-compressive-phase-retrieval.html | ## Monday, August 03, 2015
### Efficient Compressive Phase Retrieval with Constrained Sensing Vectors
Interesting that a random gaussian projection helps in the decoupling of two regularization terms in a nonlinear compressive sensing problem.
Efficient Compressive Phase Retrieval with Constrained Sensing Vectors by Sohail Bahmani, Justin Romberg
We propose a new approach to the problem of compressive phase retrieval in which the goal is to reconstruct a sparse vector from the magnitude of a number of its linear measurements. The proposed framework relies on constrained sensing vectors and a two-stage reconstruction method that consists of two standard convex optimization programs that are solved sequentially.
Various methods for compressive phase retrieval have been proposed in recent years, but none have come with strong efficiency and computational complexity guarantees. The main obstacle has been that there is no straightforward convex relaxations for the type of structure in the target. Given a set of underdetermined measurements, there is a standard framework for recovering a sparse matrix, and a standard framework for recovering a low-rank matrix. However, a general, efficient method for recovering a matrix which is jointly sparse and low-rank has remained elusive.
In this paper, we show that if the sensing vectors are chosen at random from an incoherent subspace, then the low-rank and sparse structures of the target signal can be effectively decoupled. We show that a recovery algorithm that consists of a low-rank recovery stage followed by a sparse recovery stage will produce an accurate estimate of the target when the number of measurements is $\mathsf O(k\,\log\frac{d}{k})$, where $k$ and $d$ denote the sparsity level and the dimension of the input signal. We also evaluate the algorithm through numerical simulation.
Join the CompressiveSensing subreddit or the Google+ Community or the Facebook page and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin. | 2017-04-29 01:38:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.806837797164917, "perplexity": 508.2627417776613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123172.42/warc/CC-MAIN-20170423031203-00152-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://plainmath.net/16826/insert-symbols-less-then-less-then-more-than-more-following-statements | # Insert one of the symbols << of >> to make the following statements true. a) If x >>, then x + 8 ___ 3. b) If x >> -3, then -5x + 18 ___ 3.
Insert one of the symbols << of >> to make the following statements true.
a) If $x⟩$, then x + 8 ___ 3.
b) If $x⟩-3$, then -5x + 18 ___ 3.
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unessodopunsep
We rewrite the inequality so that it matches the left side of the "then" part.
a)By adding 8 to each side, we have: x+8>5+8
x+8>13
So, insert>.
b)Multiply both sides by -5. Recall that when we multyply of divide both sides of an inequality by a negative number, we change the direction of the inequality: -5x<-15
Add 18 to both sides: -5x+18<3
So, insert <. | 2022-05-22 14:34:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 13, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8432484865188599, "perplexity": 1118.768904618878}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662545548.56/warc/CC-MAIN-20220522125835-20220522155835-00348.warc.gz"} |
https://de.zxc.wiki/wiki/Frequenzspektrum | Frequency spectrum
The frequency spectrum , usually simply spectrum , of a signal indicates its composition from different frequencies . In general, the frequency spectrum is a complex-valued function . Its amount is called the amplitude spectrum , its phase angle is called the phase (angle) spectrum . ${\ displaystyle {\ underline {X}}}$${\ displaystyle | {\ underline {X}} |}$${\ displaystyle \ arg {\ underline {X}}}$
The term frequency spectrum encompasses many different phenomena from all areas of physics such as optics, acoustics, electrodynamics or mechanics.
• Light is made up of waves of different frequencies. With the spectrum of light, its color usually changes, see color perception .
• The frequency of a tone determines its pitch . Among other things, the frequency spectrum of the sound characterizes the sound of a musical instrument or a human voice.
• The frequency mix of a broadcast signal contains the picture and sound information.
• The frequency of a mechanical oscillation determines how often the oscillation repeats itself in a certain time. A complicated mechanical vibration is, for example, the deflection of a seismograph during an earthquake . They are made up of vibrations of different frequencies.
The frequency spectrum of a signal can be calculated from the underlying signal by using the Fourier transform . The representation in the frequency domain is used in physics and technology to describe physical processes more simply than by functions of time or place.
Frequency spectrum of a time signal
Due to their frequent use, the class of so-called time signals is described first. The frequency spectrum of a time signal is based on the notion that a time-dependent signal x (t) can be composed as a sum or an integral of complex exponential functions of different frequencies using the transformation rules of Fourier series or Fourier transformation. The complex exponential functions are called "structure functions" in this context. The frequency spectrum describes the weighting (ie strength) with which the structure function associated with the respective frequency is included in the overall signal. The formulas for the inverse Fourier transformation are shown for the mathematical representation of the signal synthesis. To do this, it is necessary to differentiate which type of signal is present.
Periodic signal with a discrete spectrum
If the signal is a time-continuous periodic function with the period , the corresponding equation reads: ${\ displaystyle x}$${\ displaystyle T}$
${\ displaystyle \ displaystyle x (t) = \ sum _ {n = - \ infty} ^ {\ infty} {\ underline {X}} (n) \ mathrm {e} ^ {\ mathrm {i} 2 \ pi (n \ cdot f_ {0}) t}}$
The equation describes the signal x (t) as a sum of complex exponential oscillations of the frequencies . The function is called the spectrum of the signal x ${\ displaystyle e ^ {\ mathrm {i} 2 \ pi ft}}$${\ displaystyle f = 0, \ pm f_ {0}, \ pm 2 \ cdot f_ {0}, \ pm 3 \ cdot f_ {0} ...}$${\ displaystyle {\ underline {X}}}$
${\ displaystyle {\ underline {X}}: \ mathbb {Z} _ {0} \ rightarrow \ mathbb {C}, n \ rightarrow {\ underline {X}} (n) = {\ frac {1} {T }} \ int _ {0} ^ {T} x (t) \ mathrm {e} ^ {- \ mathrm {i} 2 \ pi (n \ cdot f_ {0}) t} dt,}$
with the fundamental frequency . The number is representative of n times the basic frequency. The complex exponential oscillation can be described by the equation . Since the spectrum is only defined for the discrete frequencies , one speaks of a discrete spectrum or a line spectrum . ${\ displaystyle f_ {0} = {\ frac {1} {T}}}$${\ displaystyle n \ in \ mathbb {Z}}$${\ displaystyle n \ cdot f_ {0}}$${\ displaystyle e ^ {\ mathrm {i} 2 \ pi ft}}$${\ displaystyle \ mathrm {e} ^ {\ mathrm {i} 2 \ pi ft} = \ cos (2 \ pi ft) + \ mathrm {i} \ sin (2 \ pi ft), ~ \ mathrm {i} = {\ sqrt {-1}}}$${\ displaystyle n \ cdot f}$
Non-periodic signal with a continuous spectrum
If the signal x (t) is a non-periodic time-continuous function with finite signal energy , the corresponding transformation equation reads:
${\ displaystyle x (t) = \ int _ {- \ infty} ^ {\ infty} {\ underline {X}} (f) {\ textrm {e}} ^ {\ mathrm {i} 2 \ pi ft} df}$
In this case, the function is called the spectrum of the signal${\ displaystyle {\ underline {X}}}$${\ displaystyle x}$
${\ displaystyle {\ underline {X}}: \ mathbb {R} \ rightarrow \ mathbb {C}, f \ rightarrow {\ underline {X}} (f) = \ int _ {- \ infty} ^ {\ infty } x (t) \ cdot \ mathrm {e} ^ {- \ mathrm {i} 2 \ pi ft} dt.}$
Since the spectrum is defined for all real-valued frequencies, one speaks of a so-called continuous spectrum. The spectrum of the continuous Fourier transformation can be represented as the limit case of the line spectrum of the Fourier series for the limit transition of an infinitely large signal period. ${\ displaystyle T \ to \ infty}$
Explanations and other signal classes
Both frequency spectra are defined for both positive and negative frequencies. For real-valued signals x (t), the spectra for positive and negative frequencies, however, are dependent on one another, and we have: . The asterisk indicates the complex conjugation . As a rule, the spectrum of negative frequencies is therefore only displayed for complex-valued signals. ${\ displaystyle {\ underline {X}} (f) = {\ underline {X}} (- f) ^ {*}}$${\ displaystyle ^ {*}}$
Within the framework of the theory of Fourier analysis , transformation formulas are also defined for other classes of signals, for example for time-discrete, value-continuous signals, i.e. H. sampled analog signals. The terms frequency spectrum , amplitude spectrum and phase spectrum are defined analogously as complex functions as well as their amounts and phases. The details are presented in the article about the Fourier transform and the links it contains.
In connection with non-periodic power signals such as noise signals, there is the term spectral power density , which, like the frequency spectrum, also describes the spectral composition of a signal. The special feature of non-periodic power signals is that they cannot be Fourier transformed. This can be seen from the fact that the associated transformation integrals diverge. Nevertheless, a connection with the term Fourier transformation can be established, which is important for metrological practice. If the signal is based on an ergodic development process, the spectral power density can approximately be determined by subjecting a partial signal of finite duration of the actually infinitely long signal to a Fourier transformation. The square of the Fourier transform is then approximately proportional to the spectral power density. ${\ displaystyle | {\ underline {X}} | ^ {2}}$
Examples
Elementary signals
The sound and spectral analysis clarifies z. B. the vowel formants as frequency ranges with increased intensity.
The spectra of elementary signals are contained in the descriptions of the associated signal transformations, see examples for the Fourier series and examples for the Fourier transformation . As an example, several spectra of simple signals are to be displayed. The fourth example shows the influence of the phase spectrum on a narrowband signal.
Amplitude spectrum of a sinusoidal signal. Amplitude spectrum of a square wave signal.
Amplitude spectrum of a square pulse. Amplitude spectrum of two burst signals with phase spectra.
Amplitude spectrum of an audio signal
As an example, consider the amplitude spectrum of the following violin tone
The spectrum of the violin tone depends on the time period chosen for analysis. If one looks at a signal excerpt that was recorded while the strings were being struck, one recognizes not only the fundamental frequency of f 0 = 294 Hz but also clear frequency components of the integer multiples . This can be explained by the fact that the string not only vibrates in its fundamental wave, in which the string experiences a deflection along its entire length, but also has additional nodes at 1/2, 1/3, 2/3, 1 / 4, 2/4, 3/4, ... the string length. The oscillation at a multiple of the fundamental is called overtone in musical parlance . The expression of the individual overtones is not only determined by the vibration of the string alone, but also by the overall arrangement of the instrument (string, resonance body, string pressure when bowing or deflection when plucking). In contrast to the signal excerpt during the bowing, the signal excerpt, which takes into account the decay of the tone, does not show any significant overtone components. ${\ displaystyle f_ {n} = nf_ {0}}$
Frequency spectrum of light
While in the radio range of the electromagnetic spectrum the frequency spectrum can still be determined from the temporal progression of the electric field strength , this is no longer possible in the spectral range of light , because the frequencies are over 100 terahertz . Usual plots of optical spectra (see spectroscopy ) often have the wavelength of the light or the energy of the light quanta as the x-axis. If, on the other hand, it is the frequency, one speaks of a frequency spectrum. Wavelength spectra are wider at the red end, frequency spectra at the blue end - wider and flatter if the spectrum is represented as spectral intensity per unit of the x-axis.
Spatial frequency spectra
If the underlying signal s does not depend on the time t but on the coordinates of the location, one speaks of a so-called spatial frequency spectrum. Spatial frequency spectra can be one, two or three dimensional, depending on whether one, two or three dimensional structures are being analyzed. They can have both a continuous and a discrete domain of definition.
Examples of structures with a continuous domain are
• the gray value gradient along a line (one-dimensional)
• the gray value curve of a black and white photograph (two-dimensional)
• the intensity distribution of a physical quantity in space (three-dimensional)
Examples of structures with a discrete domain are
• the gray value gradient on discrete points along a line (one-dimensional)
• the gray value curve on discrete points of a black and white photograph (two-dimensional), e.g. B. Pixel graphics
• the point distribution of a crystal lattice in space
As is analogous in the frequency spectrum of a time function to the spatial frequency spectrum based on the view is that the total signal s (x, y, z) by using the transformation rules of the Fourier series and the Fourier transform as a sum or an integral of complex exponentials of the spatial frequencies , and put together can be . ${\ displaystyle f_ {x}}$${\ displaystyle f_ {y}}$${\ displaystyle f_ {z}}$
Phase of the structural functions in the 2d Fourier transform
The exponential function can be illustrated by the location-dependent signal phase. In the case of a two-dimensional transformation, this is shown in the adjacent picture for different spatial frequencies. It can be seen that in general the vector indicates the direction of the maximum phase change. ${\ displaystyle {\ vec {f}} = (f_ {x}, f_ {y}, f_ {z})}$
Non-periodic signal with a continuous spectrum
If the signal s (x, y, z) is a non-periodic time-continuous function of the three position coordinates x, y and z, the corresponding transformation equation reads:
${\ displaystyle s (x, y, z) = \ int _ {- \ infty} ^ {\ infty} \ int _ {- \ infty} ^ {\ infty} \ int _ {- \ infty} ^ {\ infty } {\ underline {S}} (f_ {x}, f_ {y}, f_ {z}) {\ textrm {e}} ^ {i2 \ pi (f_ {x} x + f_ {y} y + f_ {z} z)} df_ {x} df_ {y} df_ {z}}$
The function is called the spatial frequency spectrum of the signal${\ displaystyle {\ underline {S}} (f_ {x}, f_ {y}, f_ {z})}$${\ displaystyle s}$
${\ displaystyle {\ underline {S}}: \ mathbb {R} ^ {3} \ rightarrow \ mathbb {C}, (f_ {x}, f_ {y}, f_ {z}) \ rightarrow {\ underline { S}} (f_ {x}, f_ {y}, f_ {z}) = \ int _ {- \ infty} ^ {\ infty} \ int _ {- \ infty} ^ {\ infty} \ int _ { - \ infty} ^ {\ infty} s (x, y, z) \ cdot \ mathrm {e} ^ {- \ mathrm {i} 2 \ pi (f_ {x} x + f_ {y} y + f_ { z} z)} dxdydz.}$
Measure the frequency spectrum
The frequency spectrum of an electrical signal can be measured with a spectrum analyzer or signal analyzer . The spectrum is then z. B. determined with the help of Fourier analysis (see also Fourier transformation ) or according to the principle of the heterodyne receiver from the time signal. The result of this transformation is the amplitudes of the respective frequency components A ( f ) as a function of the frequency f and, in the case of amplitude distributions that vary over time, a distribution A ( f, t ) as a function of the frequency f and the time t .
Characteristic spectra
Depending on the number and harmony of the frequencies it contains, the spectrum of a (one-dimensional) audio signal results in a sound (harmonic), a sound mixture (a few inharmonic frequencies), a noise (inharmonic) or a noise (all frequencies, occurring statistically).
Periodic signals usually have a line spectrum , while non-periodic signals such as pulses have a continuous frequency spectrum .
Frequency spectrum of a triangular voltage . Base frequency 220 Hz.
Other meanings
In a broader sense, frequency spectrum denotes a list of frequencies that have to be seen together in relation to a certain point of view, e.g. B. the frequency spectrum of radio and television channels ; see frequency band .
A response spectrum is used to design structures against the load from earthquakes .
Individual evidence
1. Rüdiger Hoffmann: Signal Analysis and Recognition: An Introduction for Information Technicians, Springer, 1998, p. 69. Quotation in connection with the complex Fourier series: “The series can be interpreted as an orthogonal development of the function x according to the system of structural functions, [... ] "${\ displaystyle \ phi _ {n} (t) = e ^ {jn \ omega _ {0} t}, n = - \ infty ... + \ infty}$
literature
• Curt Rint : Handbook for high frequency and electrical technicians, Volume 2. 13th edition, Hüthig and Pflaum Verlag GmbH, Heidelberg 1981, ISBN 3-7785-0699-4 .
• Gregor Häberle, Heinz Häberle, Thomas Kleiber: Expertise in radio, television and radio electronics. 3rd edition, Verlag Europa-Lehrmittel, Haan-Gruiten 1996, ISBN 3-8085-3263-7 .
• Horst Stöcker: Pocket book of physics. 4th edition, Verlag Harry Deutsch, Frankfurt am Main 2000, ISBN 3-8171-1628-4 .
• Thomas Görne: Sound engineering. 1st edition, Carl Hanser Verlag, Leipzig 2006, ISBN 3-446-40198-9 . | 2021-11-30 00:03:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 34, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7891634106636047, "perplexity": 1951.0217369109753}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358847.80/warc/CC-MAIN-20211129225145-20211130015145-00147.warc.gz"} |
https://mail.ergov.com/2rh9u/37c9c6-complex-conjugate-calculator-with-steps | Get the dividing complex numbers calculator available online for free only at BYJU'S Complex Conjugate,Absolute Calculator. Step 2: Click the blue arrow to submit and see your result! Just plug in the equation or numbers and the algebra solver will solve and show steps. The Math Calculator will evaluate your problem down to a final solution. 1: complex. Complex number calculator with steps. By using this website, you agree to our Cookie Policy. Complex Fractions Calculator. But to divide two complex numbers, say $$\dfrac{1+i}{2-i}$$, we multiply and divide this fraction by $$2+i$$.. An online simple and complex conjugate, Absolute calculation. Limit calculator counts a limit or border of a certain function. Well, since the conjugate of the product of two numbers is the product of their conjugates (that is, $(zw)^* = z^* w^*$), let's do it step by step. The final answer is simplified. 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The expression (2) was not written down for you to memorize, learn, or even use; the point was just for you to get some practice in seeing how a calculation like the one … The conjugate of a complex number requires you to change the sign of the bi term. The complex conjugate eigenvalue a − bi gives up to sign the same two solutions x 1 and x 2. In the example below, that’s “x” … This is a special property of conjugate complex numbers that will prove useful. Consider what happens when we multiply a complex number by its complex conjugate. Here is the complex conjugate calculator. Instructions. Ideal when solving equations in Algebra classes as well as classes such as Trigonometry, Calculus, Physics, Chemistry, Biology, Discrete Mathematics, Geometry, Complex Numbers. How to Use First, write the variable and the point at which taking the limit. i: Detailed Answer Simple Conjugate Function. It will find the eigenvalues of that matrix, and also outputs the corresponding eigenvectors.. For background on these concepts, see 7.Eigenvalues and Eigenvectors The complex conjugate has a very special property. If z = 3 – 4i, then z* = 3 + 4i. Polar to rectangular online calculator. First we need to conjugate $\sqrt{\frac{2}{a}}$, but since it's a real number, it is equal to its conjugate. Free Algebra Solver and Algebra Calculator showing step by step solutions. We know that to add or subtract complex numbers, we just add or subtract their real and imaginary parts.. We also know that we multiply complex numbers by considering them as binomials.. Casio fx-991es calculator tutorial #2. You can also add, subtraction, multiply, and divide and complete any arithmetic you need. For the conjugate complex number $$a-bi$$ schreibt man $$\overline{z}=a-bi$$. Fraction calculator that allows to do all types of calculations step by step on fractions. It also demonstrates elementary operations on complex numbers. Compute , Evaluate expressions involving Complex Numbers, Take the Square Root (Step by Step) , Find the Conjugate, Compute Arg(z), Modulus(z) Solve any Complex Equation Read Basics on Complex Analysis and Identities involving Trigonometric , Logarithmic, Exponential and Polynomial Functions, DeMoivre Theorem The calculator displays complex number and its conjugate on the complex plane, evaluate complex number absolute value and principal value of the argument . Using the graphing calculator with complex numbers. Complex numbers arithmetic This solver can performs basic operations with complex numbers i.e., addition, subtraction, multiplication and division . Complex conjugate - from wolfram mathworld. Eigenvalues and eigenvectors calculator. Simplifying Complex Numbers With Multiple Steps Multiplying By the Conjugate All Functions Operators + In order to show the steps, the calculator applies the same integration techniques that a human would apply. So in the example above $$\overline{5+3i}=5-3i$$ RedCrab Calculator Step-by-step explanations are provided. To make this possible, the first step is to understand the magnitude and structure of the essential number in the midst of this calculation: the conjugate number. Where’s the i?. Using the online integral calculator is very easy, just enter the equation you need to solve. : Step 3: Simplify the powers of i, specifically remember that i 2 = –1. In other words, it is the original complex number with the sign on the imaginary part changed. Conjugate of X+Y is X-Y, where X and Y are real numbers. Step 1: To divide complex numbers, you must multiply by the conjugate.To find the conjugate of a complex number all you have to do is change the sign between the two terms in the denominator. The conjugate of the complex number a + bi is a – bi.. The product of (a + bi)(a – bi) is a 2 + b 2.How does that happen? To check a complex number calculator program I wrote in C for a university course Comment/Request y''all need a design refresh on this website. Related Calculator: >> Complex Conjugate Calculator It is easy to find mistakes in your calculations when you can see every step of the process. Online calculator. The program that does this has been developed over several years and is written in Maxima's own programming language. Conjugate[z] or z\[Conjugate] gives the complex conjugate of the complex number z. Complex conjugates are indicated by a star (z*) or bar above the number — mathematicians love to argue about these notational conventions. Translate complete in context, with examples of … Dividing Complex Numbers To divide complex numbers, write the problem in fraction form first. Online calculator for 7th graders word problems, pde non homogeneous, a program that does algebra on ti 84 step by step, calculator for Rational exponents, logarithmic and exponential functions, free online fraction calculator simplest form, writing equations in vertex form, When adding integers with the same sign we can get the answer by. Here are some examples of complex numbers and their conjugates. One of the main purposes of the conjugate number is to make it possible to transform the complex denominator number (present in a fraction) into a real type number (an example we will see later). 2. Easy to use. Here, $$2+i$$ is the complex conjugate of $$2-i$$. Enter a + bi . Just type your formula into the top box. Step by Step - Complete the Square; Step by Step - Partial Fractions ... Exponents on a Scientific Calculator Ch 12. With the help of this calculator, one can easily add, subtract, multiply or divide any complex fractions. order of polynomial in the numerator greaters of equals to the order of polynomial in the denominator) calculator divides numerator to … Just in case you forgot how to determine the conjugate of a given complex number, see the table … Dividing Complex Numbers Read More » Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … The limit calculator helps to calculate limits at positive, negative and complex infinities. The complex number calculator allows to perform calculations with complex numbers (calculations with i). The product of the complex numbers and their conjugates is real. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step This website uses cookies to ensure you get the best experience. Fraction calculator : fraction_calculator. One-sided and two-sided being supported. Step 2: Distribute (or FOIL) in both the numerator and denominator to remove the parenthesis. Use the additional options on the calculator if you are not completely happy with the results. Alternatively, you can use the default button not to waste time. Complex number calculator: complex_number.
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https://hepdata.net/record/ins1220862 | • Browse all
Measurement of proton-proton elastic scattering and total cross-section at S**(1/2) = 7-TeV
The collaboration
Europhys.Lett. 101 (2013) 21002, 2013
Abstract (data abstract)
CERN-LHC. Measurement of the elastic differential cross section in proton-proton interactions at a centre-of-mass energy of 7 TeV. The data, which cover the |t| range 0.005-0.2 GeV**2, were collected using Roman Pot detectors very close to the outgoing beam in October 2011, allowing the precise extrapolation down to the optical point, t=0, and hence the derivation of the elastic as well as the total cross section via the optical theorem. An earler publication (CERN-PH-EP-2011-101,EPL 95(2011)41001) reports measurements in the higher |t| range from 0.36-2.5 GeV**2.
• #### Table 1
Data from T 4
10.17182/hepdata.66456.v1/t1
The measured differential elastic cross section.
• #### Table 2
Data from T 5
10.17182/hepdata.66456.v1/t2
The measured differential elastic cross section in the high |T| region. where it originally appeared as a plot, but was...
• #### Table 3
Data from T 6
10.17182/hepdata.66456.v1/t3
The fitted slope parameter for the elastic cross section fitted over 4 |T| ranges.
• #### Table 4
Data from T 7
10.17182/hepdata.66456.v1/t4
The fitted elastic cross section parameters. The quoted systematic (sys) errors are from the uncertainties in the T dependence, normalization...
• #### Table 5
Data from T 7
10.17182/hepdata.66456.v1/t5
The derived hadronic total and inelastic cross sections. The quoted systematic (sys) errors are from the uncertainty in the T... | 2017-11-25 09:20:45 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8727596998214722, "perplexity": 7683.3814937605775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934809746.91/warc/CC-MAIN-20171125090503-20171125110503-00274.warc.gz"} |
https://www.physicsforums.com/threads/find-the-laplace-of-t-3-e-5.25933/ | # Find the laplace of t^3/e^5
1. May 16, 2004
### angel
hi,
im trying to find the laplace of t^3/e^5
A casual glance seems to give the expression $$\frac{6}{s^{4}e^{5}}$$ for the Laplace transform. | 2017-08-21 16:35:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7238410115242004, "perplexity": 1185.581636344207}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109157.57/warc/CC-MAIN-20170821152953-20170821172953-00360.warc.gz"} |
http://math.stackexchange.com/questions/241798/compactness-across-induced-topological-spaces | Compactness Across Induced Topological Spaces
We have shown in class that if $E$ is a compact subset of a metric space $X$, then $\forall Y \subset X$ such that $E \subset Y$, E is compact in Y with the same metric as $X$. That is, compactness in a metric space is generalizable to any space with the same metric.
I'm trying to convince myself of this same notion in a general topological space. Take $T$ as a topological space, and $S \subset T$ a compact set. If we construct $Y \subset T$ with the induced topology from $T$, does it follow that if $S \subset Y$ then $S$ is compact in Y?
Any help would be greatly appreciated. Many thanks!
-
An open set in $Y$ is a subset of an open set in $T$. So if open sets in $Y$ cover $S$... – WimC Nov 21 '12 at 5:58
Yes, it does: the compactness of $S$ depends only on the topology on $S$.
Let $\langle T,\tau\rangle$ be a topological space and $S$ a compact subset of $T$. Let $\tau_S$ be the subspace topology on $S$. Let $\mathscr{U}\subseteq\tau_S$ be an $S$-open cover of $S$. For each $U\in\mathscr{U}$ there is a $V_U\in\tau$ such that $U=S\cap V_U$, so $\mathscr{V}=\{V_U:U\in\mathscr{U}\}$ is a $T$-open cover of $S$. $S$ is compact in $T$, so there is a finite $\mathscr{V}_0\subseteq\mathscr{V}$ that covers $S$. Let $\mathscr{U}_0=\{U\in\mathscr{U}:V_U\in\mathscr{V}_0\}$; then $\mathscr{U}_0$ is a finite subfamily of $\mathscr{U}$ covering $S$. In other words, the space $\langle S,\tau_S\rangle$ is a compact space in its own right, without any reference to $T$.
Now suppose that $S\subseteq Y\subseteq T$. Let $\tau_Y$ be the relative topology on $Y$ as a subspace of $T$ and $\tau_S'$ the relative topology on $S$ as a subspace of $\langle Y,\tau_Y\rangle$. Then $\tau_S'=\tau_S$, i.e., $S$ has the same topology as a subspace of $Y$ as it has as a subspace of $T$. (This is a straightforward exercise; if you get stuck, just ask.) Thus, if $\mathscr{U}$ is a $\tau_Y$-open cover of $S$ in $Y$, $\{U\cap S:U\in\mathscr{U}\}$ is a $\tau_S$-open cover of $S$ in $S$. $S$ is compact, so $\{U\cap S:U\in\mathscr{U}\}$ has a finite subcover, and from that it’s easy to extract a finite subfamily of $\mathscr{U}$ that covers $S$. | 2016-07-25 00:43:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9373111724853516, "perplexity": 31.824694994722556}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824201.28/warc/CC-MAIN-20160723071024-00065-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://hinduism.stackexchange.com/questions/2394/how-do-i-find-the-right-guru | # How do I find the right Guru?
How can we trust someone and accept him as Guru? Can we know or assess the spiritual level of a person? Are there any references for choosing one?
What should we do if we realize that the guru we chose wasn't the right one in terms of knowledge?
-
As per the scriptures, a right guru must be fulfilling two important qualities:
1. Srotriya : He must be well versed in all the sashtras, i.e. Vedas, Puranas, Philosophies, etc. (master in theory)
2. Brahmanistha: He must himself have realized God and stays fixed in Him. (master in practice)
So the scriptures say as below:
tad vijñānārthaṃ sa gurum evābhigacchet samit-pāṇiḥ śrotriyam brahma-niṣṭham [Mund. Up. - 1.2.12]
-To know That (God) he (the seeker) goes to a guru who is well versed in scriptures and situated in Brahman (God realized) .
tasmād guruṃ prapadyeta jijñāsuḥ śreya uttamam
śābde pare ca niṣṇātaṃ brahmaṇyupaśamāśrayam
[SB - 11.3.21]
Maning
Therefore, the seeker approaches a good and noble guru who is skillful in the knowledge of the scriptures and the supreme and who having taken the shelter of Brahma stays satisfied in Him.
After following a guru for few years if you find you are not improving or he is not the right person, then the only thing you should do is to stop following him and without any ill feelings towards him, just try to find someone else.
Technically one can do a lot of things after he finds himself cheated or a prey to the media hype, but he should focus upon continuing his development rather than getting revenge minded and focusing on someone else's ruin. Those negative feelings will only degrade him further instead of helping him move up.
Finding guru is actually a subjective thing. Even while one guru may work for a seeker, he may not be good for another one. And again, how would you know if he is God realized? Any one may show some cheap miracle or some spiritual power and can act as a guru. So it is generally said that it is not the student who finds the guru, it is the guru who finds the student. For example, Vivekananda first didn't accept Ramakrishna as his guru, even he mocked him. But later upon more interaction he felt his divinity and accepted as guru. So without properly analyzing it is very hard to find out who is a true guru and who is not. It is because true saints even act opposite to their nature sometimes. So from my experience I would say if one has faith in God and genuine interest, then eventually he will find the right guru even if he falls prey to others many times.
# Update
Adding more info regarding changing guru upon pbvamsi's suggestion. The thing about changing guru is that, it should be done only when its necessary and required. So I said one can change guru when he is not finding any improvement after following him.
Because not all gurus are perfect or of the same caliber and qualification, one may not find perfection by following only one guru. So the scripture says knowledge cannot be stable through one guru alone:
na hyekasmādgurorjñānaṃ susthiraṃ syātsupuṣkalam [SB -11.9.31]
- Not through one guru knowledge becomes steady and complete.
There are also real life examples when a guru sends his disciple to someone else for further improvements. Also the other thing is that, one guru may be perfect in yoga but doesn't know about devotion. So a student have to change his guru if he wants to learn devotion, knowledge or something else like it. So there is provision for changing guru. But only when it is necessary and required. If one finds a guru who is perfect and satisfies the above mentioned two conditions, there is no reason for someone to change guru. Moreover, in that case changing guru will break ananyata (complete surrender to only one) and the student cannot proceed further.
So your saying that a guru cannot be changed is right and true, but only when one has accepted the guru as one's true master and the guru also has accepted him as his dear disciple. In those cases, the guru and disciple become one soul in two bodies. Guru and shisya relationship is the most noble, divine and pure of all. It is one bond that never breaks. So before one has found his true guru, he can change many others. But once he has found him, there is no change.
-
request you to recheck your saying that we can change guru or request you to provide authentic reference on this. Only for that reason i've provided extended answer to your's accepted answer. Spiritual guru/sadguru cant be changed! – pbvamsi Sep 27 '14 at 6:47
thanks for the convincing edit. Worldly gurus who teaches worldly science/art can be changed any no of times but when it comes to spiritual guru, one has to wait till he finds the spiritually enlightened guru. If one is truly interested, guru will enter at right time! – pbvamsi Sep 27 '14 at 11:35
Are there any gurus who is of the cadre of 'Adi Sankaracharya'? – JavaTechnical Nov 11 '14 at 15:15
There is no privilege to change Guru in the disciple's lifetime. One has to take enough time in choosing a right Guru. Infact it is not the student who finds the guru, it is the guru who finds the student. Based on one's strong determination, he may get the right Guru.
After one makes stong determination, he will be privileged to meet many sadhus. With whom one's heart becomes peaceful and experiences sat-chit-ananda (true bliss eternal), in whose presence one gets courage and will power to solve all problems(a feeling that they will get solved by themselves), in whose presence all the worries & sorrows are forgotten, above all in whose presence one feels he is my savior, my relative, such a person can be approached as Guru. | 2015-01-27 14:21:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5092887282371521, "perplexity": 3507.3721765345886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422121981339.16/warc/CC-MAIN-20150124175301-00077-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://ncatlab.org/nlab/show/Maissam+Barkeshli | # nLab Maissam Barkeshli
Selected writings
## Selected writings
category: people
Created on May 11, 2022 at 07:47:29. See the history of this page for a list of all contributions to it. | 2022-12-04 15:26:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21831011772155762, "perplexity": 2745.7106190681375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710974.36/warc/CC-MAIN-20221204140455-20221204170455-00271.warc.gz"} |
https://gmatclub.com/forum/for-integers-a-and-b-if-a-3-a-2-b-1-2-7-what-is-the-105959.html | It is currently 19 Feb 2018, 04:02
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For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the
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For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
(1) a^2 - a = 12
(2) b^2 - b = 2
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aalriy wrote:
For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
(1) a^2 - a = 12
(2) b^2 - b = 2
Given: $$a$$ and $$b$$ are integers, also $$\sqrt{a^3-a^2-b}=7$$ --> $$a^3-a^2-b=49$$
(1) a^2 - a = 12 --> $$a=-3$$ or $$a=4$$. Now, both values of $$a$$ give an integer solution for $$b$$ ($$b=-85$$ or $$b=-1$$), so both values are valid. Not sufficient.
(2) b^2 - b = 2 --> $$b=-1$$ or $$b=2$$ --> if $$b=-1$$ then $$a^3-a^2=48$$ --> $$a^2(a-1)=48$$ --> $$a=4=integer$$ BUT if if $$b=2$$ then $$a^3-a^2=51$$ --> $$a^2(a-1)=51=3*17$$ --> this equation has no integer solution for $$a$$, hence only the first case is valid: $$b=-1$$ and $$a=4=integer$$. Sufficient.
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08 Dec 2010, 09:13
can you elaborate on how you solved $$a^2(a-1) = 48$$ to $$a = 4$$?
Similarly how did you conclude that $$a^2(a-1) = 51$$ will not have an interger as a solution?
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aalriy wrote:
can you elaborate on how you solved $$a^2(a-1) = 48$$ to $$a = 4$$?
Similarly how did you conclude that $$a^2(a-1) = 51$$ will not have an interger as a solution?
$$a^2(a-1)=48$$: as $$a$$ is an integer than we have that 48 is a product of a perfect square (a^2) and another positive integer (a-1): 48=1*48=4*12=16*3, so after some trial and error you'll get $$a=4$$;
$$a^2(a-1)=51=1*51$$: the same here. But 51 can be represented as a product of perfect square and another integer only in one way 51=1*51, which doesn't fit for a^2(a-1), so this equation doesn't have integer solution for $$a$$.
Hope it's clear.
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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03 Jul 2013, 01:15
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aalriy wrote:
For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
(1) a^2 - a = 12
(2) b^2 - b = 2
Given in the question stem that $$(a^3-a^2-b) = 49$$$$\to a^2(a-1) = 49+b$$
From F.S 1, we know that a(a-1) = 12, thus, $$a*12 = 49+b \to$$ a is an integer for both b = -1 or b = -13. Thus, we get two different values of a, Insufficient.
From F. S 2, we know upon solving for the quadratic $$b^2-b-2$$, the integral roots are -1 or 2.
Now,$$a^2(a-1) = 49+b$$
For b = 2, $$a^2(a-1) = 51$$. Assuming that a is odd/even, the given product is an arrangement like odd*odd*even = even, and 51 is not even.
Similarly, for a is even, the arrangement will be like even*even*odd = even, and just as above 51 is not even. Thus, $$b\neq{2}$$
For b = -1,$$a^2(a-1)$$ = 48. Now all the integral roots of this polynomial can only be factors of 48, including both negative and positive factors.
However, any negative factor will never satisfy the given polynomial as because (a-1) will become a negative expression, which can never equal 48.Hence, the given polynomial has no integral solutions in -48,-24,-12,-8,-6,-4,-3,-2,-1.
By the same logic, we know that any integral solution,if present will be one of the positive factors of 48.It fails for 1,2,3 and we find that a=4 is a root.If for a=4, the expression equals 48, then for a value above 4, the expression $$a^2(a-1)$$is bound to be greater than 48.
Thus, the only solution possible for the given polynomial is a=4, a unique value,Sufficient.
B.
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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03 Jul 2013, 08:54
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aalriy wrote:
For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
(1) a^2 - a = 12
(2) b^2 - b = 2
Tough one...Took more than 3 min 50 sec and ended up getting it wrong...
We can change the given equation to (a^3-a^2-b) = 49 (squaring both sides)
From St 1 we have a^2-a=12 ----> substituting in above given eqn we get
{a(a^2-a)- b} =49 ------> 12a-b=49 -----> a = (49+b)/12
Now a and b are integers therefore (49+b)/12 should be an integers
Possible values b=11, a=5
b=23, a =6, b=-1, a=4
So St 1 alone is not sufficient
St 2 says b^2-b =2 -----> b(b-1) =2 possible values of b are
b=2 or b=-1
Substituting in given expression we get
(a^3-a^2-b) = 49
a^3-a^2= 51 or------> a^2(a-1)= 51 (17*3) we see that 51 even after reducing to prime factors gives us no Integer value of a
a^3-a^2= 48 -----> a^2(a-1)= 48 (4^2)*3 and hence we get value of a= 4 or -4
Substituting values of b=-1 and a=4 or -4, we can see that only for a=4 the above given equation is proven.
Hence a =4
I did highlight the value of b=-1, a=4 from statement and can be taken as a hint without solving completely eqn 2
Ans B
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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03 Jul 2013, 09:00
mridulparashar1 wrote:
aalriy wrote:
For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
a^3-a^2= 48 -----> a^2(a-1)= 48 (4^2)*3 and hence we get value of a= 4 or -4
Substituting values of b=-1 and a=4 or -4, we can see that only for a=4 the above given equation is proven.
Hence a =4
I did highlight the value of b=-1, a=4 from statement and can be taken as a hint without solving completely eqn 2
Ans B
Minor mistake. The given polynomial will not yield a = -4 as a root.
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For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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10 Nov 2013, 08:31
Bunuel wrote:
aalriy wrote:
For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
(1) a^2 - a = 12
(2) b^2 - b = 2
Given: $$a$$ and $$b$$ are integers, also $$\sqrt{a^3-a^2-b}=7$$ --> $$a^3-a^2-b=49$$
(1) a^2 - a = 12 --> $$a=-3$$ or $$a=4$$. Now, both values of $$a$$ give an integer solution for $$b$$ ($$b=-85$$ or $$b=-1$$), so both values are valid. Not sufficient.
(2) b^2 - b = 2 --> $$b=-1$$ or $$b=2$$ --> if $$b=-1$$ then $$a^3-a^2=48$$ --> $$a^2(a-1)=48$$ --> $$a=4=integer$$ BUT if if $$b=2$$ then $$a^3-a^2=51$$ --> $$a^2(a-1)=51=3*17$$ --> this equation has no integer solution for $$a$$, hence only the first case is valid: $$b=-1$$ and $$a=4=integer$$. Sufficient.
if $$b=-1$$ then $$a^3-a^2=48$$ --> $$a^2(a-1)=48$$ --> $$a=4=integer$$
how do you solve something like that, is it just quick trial & error since you know a & b have to be integers and there's only a few values of a that would give results somewhat near 48?
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For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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10 Nov 2013, 11:10
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AccipiterQ wrote:
Bunuel wrote:
aalriy wrote:
For integers a and b, if (a^3 – a^2 – b)^1/2 = 7, what is the value of a?
(1) a^2 - a = 12
(2) b^2 - b = 2
Given: $$a$$ and $$b$$ are integers, also $$\sqrt{a^3-a^2-b}=7$$ --> $$a^3-a^2-b=49$$
(1) a^2 - a = 12 --> $$a=-3$$ or $$a=4$$. Now, both values of $$a$$ give an integer solution for $$b$$ ($$b=-85$$ or $$b=-1$$), so both values are valid. Not sufficient.
(2) b^2 - b = 2 --> $$b=-1$$ or $$b=2$$ --> if $$b=-1$$ then $$a^3-a^2=48$$ --> $$a^2(a-1)=48$$ --> $$a=4=integer$$ BUT if if $$b=2$$ then $$a^3-a^2=51$$ --> $$a^2(a-1)=51=3*17$$ --> this equation has no integer solution for $$a$$, hence only the first case is valid: $$b=-1$$ and $$a=4=integer$$. Sufficient.
if $$b=-1$$ then $$a^3-a^2=48$$ --> $$a^2(a-1)=48$$ --> $$a=4=integer$$
how do you solve something like that, is it just quick trial & error since you know a & b have to be integers and there's only a few values of a that would give results somewhat near 48?
Yes, that's correct. We know that $$a$$ is an integer, thus $$a^2(a-1)=(perfect \ square)*(positive \ integer)=48$$. From here you can use trial and error and find that $$a=4$$.
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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08 Sep 2014, 00:03
I have just one question, why isnt the absolute value used after the expression is squared?
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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08 Sep 2014, 03:10
I have just one question, why isnt the absolute value used after the expression is squared?
$$\sqrt{x}=2$$ --> $$x = 4$$. Where should there be an absolute value?
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink]
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Re: For integers a and b, if (a^3 a^2 b)^1/2 = 7, what is the [#permalink] 22 Oct 2017, 04:10
Display posts from previous: Sort by | 2018-02-19 12:02:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6939097046852112, "perplexity": 1783.6979421438996}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812584.40/warc/CC-MAIN-20180219111908-20180219131908-00637.warc.gz"} |
https://tex.stackexchange.com/questions/566388/latex-report-displays-page-number-is-different-places | # Latex report displays page number is different places
I am quite new to Latex and trying to make the report based on the template provided by my professor.
I am observing that latex is displaying the page number in different places such as for some page it displays the page number in the top-right corner and for some page, it displays in bottom-centre. Is this expected? Also, when the page number is displayed on the top-right corner then the chapter names appear on the top-left corner.
1. Is it possible for me to have the page number only on bottom-centre for all pages?
2. Also, why only on some pages the chapter heading is displayed on the top-left corner? Is this something to do with \documentclass[a4paper,12pt]{report} I am using?
3. Is it possible for me to have the roman numbers for contents, abstract, list of table and images? I know if I use book I can achieve this but if I use book then my images and text is getting messed up so I am using the report. Is there any possibility to achieve the same using the {report}.
The page number on bottom-centre and top-right: https://i.stack.imgur.com/hwJ1H.png
Following are the template which are provided by my professor:
\documentclass[a4paper,12pt]{report}
\usepackage{fontspec}
\usepackage{microtype}
\usepackage{polyglossia}
\setdefaultlanguage{english}
\usepackage[english]{selnolig}
\usepackage{geometry}
\usepackage[usegeometry,
margin=20mm,
twoside=semi]
\usepackage{acronym}
\renewcommand*{\acsfont}[1]{{\rmfamily #1}}
\usepackage[font=small,labelfont=bf]{caption}
\usepackage{csquotes}
\usepackage{graphicx}
\usepackage{icomma}
\usepackage{ltablex}
\usepackage{newfile}
\usepackage{overpic}
\usepackage[hyphens]{url}
\usepackage{url}
\parindent 2em
\parskip 0.5em
%THIS IS WHERE SOME COMMON conctent added
\usepackage{hyperref}
\usepackage[a-1b]{pdfx}
\usepackage{makeidx}
\makeindex
\setcounter{secnumdepth}{4}
\setcounter{tocdepth}{4}
\author{\dcauthorfirstname~\dcauthorlastname}
\title{\dctitle}
\date{\today}
\begin{document}
\frontmatter
\section*{\centering Acknowledgements}
%THIS IS WHERE I ADD THE Acknowledgements FROM ANOTHER FILE
\vfill
\thispagestyle{empty}
%-----------------------------
% Abstract
%-----------------------------
\section*{\centering Abstract}
%THIS IS WHERE I ADD THE ABSTRACT FROM ANOTHER FILE
\vfill
\clearpage{\pagestyle{empty}\cleardoublepage}
\tableofcontents \clearpage{\pagestyle{empty}\cleardoublepage}
\listoffigures \clearpage{\pagestyle{empty}\cleardoublepage}
\listoftables \clearpage{\pagestyle{empty}\cleardoublepage}
%THIS IS WHERE I ADD THE ACCRONYMS FROM ANOTHER FILE
\clearpage{\pagestyle{empty}\cleardoublepage}
\mainmatter
\acresetall
%---------------------------------------------------------------------------------------------------
\chapter{Introduction}
%THis is where I include the Introduction text from another file
\chapter{Conclusion}
%THis is where I include the Conclusion text from another file
\begin{appendix}
\chapter{Appendix}
\label{chap:Appendix}
%THis is where I include the Appendix text from another file
\end{appendix}
\backmatter
\printindex
\printbibliography
\end{document}
• we can not run your example code as it uses files not available, but avoid putting formatting into section headings like \chapter{\centering Introduction} as this will make \centering part of the text and will get used everywhere the text is used including page running heads and table of contents. Do you really use all those packages in this document? It seems a massively over-complicated preamble. Oct 11, 2020 at 19:04
• It is quite normal (and the default design in report) for chapter opening pages to have a different page style with no page head and the page number at the bottom. (\thispagestyle{plain}) Oct 11, 2020 at 19:05
• such templates make debugging anything harder as there are so many possibilities for conflict. If you just start from \documentclass[a4paper,12pt]{report} \begin{document} and only add packages that you need it is much easier to control. Oct 11, 2020 at 19:49
• @BATMAN_2008 I tried as you requested but it returns the error ! LaTeX Error: Unknown option margin=20mm' for package typearea'. Oct 11, 2020 at 20:15
• [H] disables the mechanism that latex uses to avoid bad white space at a page breaks (floating figures) so by design introduces ugly white space unless you manually position the figure at a good place in the source to get a good page break. It is opting for manual control and should only be used in exceptional cases. \pagenumbering{roman} then later \pagenumbering{arabic} will do your page numbers. Oct 12, 2020 at 7:09
In report class you can get a centred page number in the page foot, and no page head using
\pagestyle{plain}
(Which is the style already used on chapter openings)
You can use
\pagenumbering{roman}
and
\pagenumbering{arabic}
To use roman or arabic numbers, setting the page count to 1 in each case.
• Thanks a lot for the response. That worked !!!! Oct 12, 2020 at 12:22 | 2022-05-22 03:28:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6699986457824707, "perplexity": 1799.5724983975213}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662543797.61/warc/CC-MAIN-20220522032543-20220522062543-00408.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/480/C4sC4s7D15.html | Copied to
clipboard
## G = C4⋊C4⋊7D15order 480 = 25·3·5
### 1st semidirect product of C4⋊C4 and D15 acting through Inn(C4⋊C4)
Series: Derived Chief Lower central Upper central
Derived series C1 — C30 — C4⋊C4⋊7D15
Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×C4×D15 — C4⋊C4⋊7D15
Lower central C15 — C30 — C4⋊C4⋊7D15
Upper central C1 — C22 — C4⋊C4
Generators and relations for C4⋊C47D15
G = < a,b,c,d | a4=b4=c15=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 804 in 152 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C4⋊C47S3, C4×D15, C2×Dic15, C2×Dic15, C2×C60, C2×C60, C22×D15, C4⋊C47D5, C4×Dic15, C605C4, D303C4, C15×C4⋊C4, C2×C4×D15, C4⋊C47D15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, C4○D4, D10, C4×S3, C22×S3, D15, C42⋊C2, C4×D5, C22×D5, S3×C2×C4, D42S3, Q83S3, D30, C2×C4×D5, D42D5, Q82D5, C4⋊C47S3, C4×D15, C22×D15, C4⋊C47D5, C2×C4×D15, D42D15, Q83D15, C4⋊C47D15
Smallest permutation representation of C4⋊C47D15
On 240 points
Generators in S240
(1 174 20 155)(2 175 21 156)(3 176 22 157)(4 177 23 158)(5 178 24 159)(6 179 25 160)(7 180 26 161)(8 166 27 162)(9 167 28 163)(10 168 29 164)(11 169 30 165)(12 170 16 151)(13 171 17 152)(14 172 18 153)(15 173 19 154)(31 143 46 125)(32 144 47 126)(33 145 48 127)(34 146 49 128)(35 147 50 129)(36 148 51 130)(37 149 52 131)(38 150 53 132)(39 136 54 133)(40 137 55 134)(41 138 56 135)(42 139 57 121)(43 140 58 122)(44 141 59 123)(45 142 60 124)(61 237 76 222)(62 238 77 223)(63 239 78 224)(64 240 79 225)(65 226 80 211)(66 227 81 212)(67 228 82 213)(68 229 83 214)(69 230 84 215)(70 231 85 216)(71 232 86 217)(72 233 87 218)(73 234 88 219)(74 235 89 220)(75 236 90 221)(91 207 106 185)(92 208 107 186)(93 209 108 187)(94 210 109 188)(95 196 110 189)(96 197 111 190)(97 198 112 191)(98 199 113 192)(99 200 114 193)(100 201 115 194)(101 202 116 195)(102 203 117 181)(103 204 118 182)(104 205 119 183)(105 206 120 184)
(1 99 43 65)(2 100 44 66)(3 101 45 67)(4 102 31 68)(5 103 32 69)(6 104 33 70)(7 105 34 71)(8 91 35 72)(9 92 36 73)(10 93 37 74)(11 94 38 75)(12 95 39 61)(13 96 40 62)(14 97 41 63)(15 98 42 64)(16 110 54 76)(17 111 55 77)(18 112 56 78)(19 113 57 79)(20 114 58 80)(21 115 59 81)(22 116 60 82)(23 117 46 83)(24 118 47 84)(25 119 48 85)(26 120 49 86)(27 106 50 87)(28 107 51 88)(29 108 52 89)(30 109 53 90)(121 240 154 199)(122 226 155 200)(123 227 156 201)(124 228 157 202)(125 229 158 203)(126 230 159 204)(127 231 160 205)(128 232 161 206)(129 233 162 207)(130 234 163 208)(131 235 164 209)(132 236 165 210)(133 237 151 196)(134 238 152 197)(135 239 153 198)(136 222 170 189)(137 223 171 190)(138 224 172 191)(139 225 173 192)(140 211 174 193)(141 212 175 194)(142 213 176 195)(143 214 177 181)(144 215 178 182)(145 216 179 183)(146 217 180 184)(147 218 166 185)(148 219 167 186)(149 220 168 187)(150 221 169 188)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)(166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)(196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)(226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(121 122)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(136 143)(137 142)(138 141)(139 140)(144 150)(145 149)(146 148)(151 158)(152 157)(153 156)(154 155)(159 165)(160 164)(161 163)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(181 196)(182 210)(183 209)(184 208)(185 207)(186 206)(187 205)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)(211 240)(212 239)(213 238)(214 237)(215 236)(216 235)(217 234)(218 233)(219 232)(220 231)(221 230)(222 229)(223 228)(224 227)(225 226)
G:=sub<Sym(240)| (1,174,20,155)(2,175,21,156)(3,176,22,157)(4,177,23,158)(5,178,24,159)(6,179,25,160)(7,180,26,161)(8,166,27,162)(9,167,28,163)(10,168,29,164)(11,169,30,165)(12,170,16,151)(13,171,17,152)(14,172,18,153)(15,173,19,154)(31,143,46,125)(32,144,47,126)(33,145,48,127)(34,146,49,128)(35,147,50,129)(36,148,51,130)(37,149,52,131)(38,150,53,132)(39,136,54,133)(40,137,55,134)(41,138,56,135)(42,139,57,121)(43,140,58,122)(44,141,59,123)(45,142,60,124)(61,237,76,222)(62,238,77,223)(63,239,78,224)(64,240,79,225)(65,226,80,211)(66,227,81,212)(67,228,82,213)(68,229,83,214)(69,230,84,215)(70,231,85,216)(71,232,86,217)(72,233,87,218)(73,234,88,219)(74,235,89,220)(75,236,90,221)(91,207,106,185)(92,208,107,186)(93,209,108,187)(94,210,109,188)(95,196,110,189)(96,197,111,190)(97,198,112,191)(98,199,113,192)(99,200,114,193)(100,201,115,194)(101,202,116,195)(102,203,117,181)(103,204,118,182)(104,205,119,183)(105,206,120,184), (1,99,43,65)(2,100,44,66)(3,101,45,67)(4,102,31,68)(5,103,32,69)(6,104,33,70)(7,105,34,71)(8,91,35,72)(9,92,36,73)(10,93,37,74)(11,94,38,75)(12,95,39,61)(13,96,40,62)(14,97,41,63)(15,98,42,64)(16,110,54,76)(17,111,55,77)(18,112,56,78)(19,113,57,79)(20,114,58,80)(21,115,59,81)(22,116,60,82)(23,117,46,83)(24,118,47,84)(25,119,48,85)(26,120,49,86)(27,106,50,87)(28,107,51,88)(29,108,52,89)(30,109,53,90)(121,240,154,199)(122,226,155,200)(123,227,156,201)(124,228,157,202)(125,229,158,203)(126,230,159,204)(127,231,160,205)(128,232,161,206)(129,233,162,207)(130,234,163,208)(131,235,164,209)(132,236,165,210)(133,237,151,196)(134,238,152,197)(135,239,153,198)(136,222,170,189)(137,223,171,190)(138,224,172,191)(139,225,173,192)(140,211,174,193)(141,212,175,194)(142,213,176,195)(143,214,177,181)(144,215,178,182)(145,216,179,183)(146,217,180,184)(147,218,166,185)(148,219,167,186)(149,220,168,187)(150,221,169,188), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,143)(137,142)(138,141)(139,140)(144,150)(145,149)(146,148)(151,158)(152,157)(153,156)(154,155)(159,165)(160,164)(161,163)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,196)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226)>;
G:=Group( (1,174,20,155)(2,175,21,156)(3,176,22,157)(4,177,23,158)(5,178,24,159)(6,179,25,160)(7,180,26,161)(8,166,27,162)(9,167,28,163)(10,168,29,164)(11,169,30,165)(12,170,16,151)(13,171,17,152)(14,172,18,153)(15,173,19,154)(31,143,46,125)(32,144,47,126)(33,145,48,127)(34,146,49,128)(35,147,50,129)(36,148,51,130)(37,149,52,131)(38,150,53,132)(39,136,54,133)(40,137,55,134)(41,138,56,135)(42,139,57,121)(43,140,58,122)(44,141,59,123)(45,142,60,124)(61,237,76,222)(62,238,77,223)(63,239,78,224)(64,240,79,225)(65,226,80,211)(66,227,81,212)(67,228,82,213)(68,229,83,214)(69,230,84,215)(70,231,85,216)(71,232,86,217)(72,233,87,218)(73,234,88,219)(74,235,89,220)(75,236,90,221)(91,207,106,185)(92,208,107,186)(93,209,108,187)(94,210,109,188)(95,196,110,189)(96,197,111,190)(97,198,112,191)(98,199,113,192)(99,200,114,193)(100,201,115,194)(101,202,116,195)(102,203,117,181)(103,204,118,182)(104,205,119,183)(105,206,120,184), (1,99,43,65)(2,100,44,66)(3,101,45,67)(4,102,31,68)(5,103,32,69)(6,104,33,70)(7,105,34,71)(8,91,35,72)(9,92,36,73)(10,93,37,74)(11,94,38,75)(12,95,39,61)(13,96,40,62)(14,97,41,63)(15,98,42,64)(16,110,54,76)(17,111,55,77)(18,112,56,78)(19,113,57,79)(20,114,58,80)(21,115,59,81)(22,116,60,82)(23,117,46,83)(24,118,47,84)(25,119,48,85)(26,120,49,86)(27,106,50,87)(28,107,51,88)(29,108,52,89)(30,109,53,90)(121,240,154,199)(122,226,155,200)(123,227,156,201)(124,228,157,202)(125,229,158,203)(126,230,159,204)(127,231,160,205)(128,232,161,206)(129,233,162,207)(130,234,163,208)(131,235,164,209)(132,236,165,210)(133,237,151,196)(134,238,152,197)(135,239,153,198)(136,222,170,189)(137,223,171,190)(138,224,172,191)(139,225,173,192)(140,211,174,193)(141,212,175,194)(142,213,176,195)(143,214,177,181)(144,215,178,182)(145,216,179,183)(146,217,180,184)(147,218,166,185)(148,219,167,186)(149,220,168,187)(150,221,169,188), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,23)(17,22)(18,21)(19,20)(24,30)(25,29)(26,28)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(91,106)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(121,122)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(136,143)(137,142)(138,141)(139,140)(144,150)(145,149)(146,148)(151,158)(152,157)(153,156)(154,155)(159,165)(160,164)(161,163)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,196)(182,210)(183,209)(184,208)(185,207)(186,206)(187,205)(188,204)(189,203)(190,202)(191,201)(192,200)(193,199)(194,198)(195,197)(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(217,234)(218,233)(219,232)(220,231)(221,230)(222,229)(223,228)(224,227)(225,226) );
G=PermutationGroup([[(1,174,20,155),(2,175,21,156),(3,176,22,157),(4,177,23,158),(5,178,24,159),(6,179,25,160),(7,180,26,161),(8,166,27,162),(9,167,28,163),(10,168,29,164),(11,169,30,165),(12,170,16,151),(13,171,17,152),(14,172,18,153),(15,173,19,154),(31,143,46,125),(32,144,47,126),(33,145,48,127),(34,146,49,128),(35,147,50,129),(36,148,51,130),(37,149,52,131),(38,150,53,132),(39,136,54,133),(40,137,55,134),(41,138,56,135),(42,139,57,121),(43,140,58,122),(44,141,59,123),(45,142,60,124),(61,237,76,222),(62,238,77,223),(63,239,78,224),(64,240,79,225),(65,226,80,211),(66,227,81,212),(67,228,82,213),(68,229,83,214),(69,230,84,215),(70,231,85,216),(71,232,86,217),(72,233,87,218),(73,234,88,219),(74,235,89,220),(75,236,90,221),(91,207,106,185),(92,208,107,186),(93,209,108,187),(94,210,109,188),(95,196,110,189),(96,197,111,190),(97,198,112,191),(98,199,113,192),(99,200,114,193),(100,201,115,194),(101,202,116,195),(102,203,117,181),(103,204,118,182),(104,205,119,183),(105,206,120,184)], [(1,99,43,65),(2,100,44,66),(3,101,45,67),(4,102,31,68),(5,103,32,69),(6,104,33,70),(7,105,34,71),(8,91,35,72),(9,92,36,73),(10,93,37,74),(11,94,38,75),(12,95,39,61),(13,96,40,62),(14,97,41,63),(15,98,42,64),(16,110,54,76),(17,111,55,77),(18,112,56,78),(19,113,57,79),(20,114,58,80),(21,115,59,81),(22,116,60,82),(23,117,46,83),(24,118,47,84),(25,119,48,85),(26,120,49,86),(27,106,50,87),(28,107,51,88),(29,108,52,89),(30,109,53,90),(121,240,154,199),(122,226,155,200),(123,227,156,201),(124,228,157,202),(125,229,158,203),(126,230,159,204),(127,231,160,205),(128,232,161,206),(129,233,162,207),(130,234,163,208),(131,235,164,209),(132,236,165,210),(133,237,151,196),(134,238,152,197),(135,239,153,198),(136,222,170,189),(137,223,171,190),(138,224,172,191),(139,225,173,192),(140,211,174,193),(141,212,175,194),(142,213,176,195),(143,214,177,181),(144,215,178,182),(145,216,179,183),(146,217,180,184),(147,218,166,185),(148,219,167,186),(149,220,168,187),(150,221,169,188)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165),(166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195),(196,197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225),(226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,23),(17,22),(18,21),(19,20),(24,30),(25,29),(26,28),(31,39),(32,38),(33,37),(34,36),(40,45),(41,44),(42,43),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(91,106),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(121,122),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(136,143),(137,142),(138,141),(139,140),(144,150),(145,149),(146,148),(151,158),(152,157),(153,156),(154,155),(159,165),(160,164),(161,163),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(181,196),(182,210),(183,209),(184,208),(185,207),(186,206),(187,205),(188,204),(189,203),(190,202),(191,201),(192,200),(193,199),(194,198),(195,197),(211,240),(212,239),(213,238),(214,237),(215,236),(216,235),(217,234),(218,233),(219,232),(220,231),(221,230),(222,229),(223,228),(224,227),(225,226)]])
90 conjugacy classes
class 1 2A 2B 2C 2D 2E 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 6A 6B 6C 10A ··· 10F 12A ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 30 30 2 2 ··· 2 15 15 15 15 30 30 30 30 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4
90 irreducible representations
dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 D5 D6 C4○D4 D10 C4×S3 D15 C4×D5 D30 C4×D15 D4⋊2S3 Q8⋊3S3 D4⋊2D5 Q8⋊2D5 D4⋊2D15 Q8⋊3D15 kernel C4⋊C4⋊7D15 C4×Dic15 C60⋊5C4 D30⋊3C4 C15×C4⋊C4 C2×C4×D15 C4×D15 C5×C4⋊C4 C3×C4⋊C4 C2×C20 C30 C2×C12 C20 C4⋊C4 C12 C2×C4 C4 C10 C10 C6 C6 C2 C2 # reps 1 2 1 2 1 1 8 1 2 3 4 6 4 4 8 12 16 1 1 2 2 4 4
Matrix representation of C4⋊C47D15 in GL6(𝔽61)
60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 50 0 0 0 0 0 0 11
,
11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
43 17 0 0 0 0 43 0 0 0 0 0 0 0 47 30 0 0 0 0 31 25 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
1 60 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 60
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[43,43,0,0,0,0,17,0,0,0,0,0,0,0,47,31,0,0,0,0,30,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;
C4⋊C47D15 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7D_{15}
% in TeX
G:=Group("C4:C4:7D15");
// GroupNames label
G:=SmallGroup(480,857);
// by ID
G=gap.SmallGroup(480,857);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
×
𝔽 | 2022-12-05 15:47:57 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987773895263672, "perplexity": 4631.612961007967}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00418.warc.gz"} |
https://www.baryonbib.org/bib/75fc2360-a247-4742-b21e-7edee8f061c6 | PREPRINT
Deep Learning application for stellar parameters determination: II- Application to observed spectra of AFGK stars
Marwan Gebran, Frédéric Paletou, Ian Bentley, Rose Brienza, Kathleen Connick
Submitted on 31 October 2022
Abstract
In this follow-up paper, we investigate the use of Convolutional Neural Network for deriving stellar parameters from observed spectra. Using hyperparameters determined previously, we have constructed a Neural Network architecture suitable for the derivation of Teff, log g, [M/H], and vesini. The network was constrained by applying it to databases of AFGK synthetic spectra at different resolutions. Then, parameters of A stars from Polarbase, SOPHIE, and ELODIE databases are derived as well as FGK stars from the Spectroscopic Survey of Stars in the Solar Neighbourhood. The network model average accuracy on the stellar parameters are found to be as low as 80 K for Teff , 0.06 dex for log g, 0.08 dex for [M/H], and 3 km/s for vesini for AFGK stars.
Preprint
Comment: 13 pages, 7 figures. Accepted for publication in Open Astronomy, De Gruyter
Subjects: Astrophysics - Solar and Stellar Astrophysics; Astrophysics - Instrumentation and Methods for Astrophysics; Physics - Computational Physics | 2022-11-29 06:13:59 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8098635077476501, "perplexity": 6521.626851573403}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00471.warc.gz"} |
https://quantumai.google/reference/python/openfermion/ops/SymbolicOperator | # openfermion.ops.SymbolicOperator
Base class for FermionOperator and QubitOperator.
A SymbolicOperator stores an object which represents a weighted sum of terms; each term is a product of individual factors of the form (index, action), where index is a nonnegative integer and the possible values for action are determined by the subclass. For instance, for the subclass FermionOperator, action can be 1 or 0, indicating raising or lowering, and for QubitOperator, action is from the set {'X', 'Y', 'Z'}. The coefficients of the terms are stored in a dictionary whose keys are the terms. SymbolicOperators of the same type can be added or multiplied together.
Adding SymbolicOperators is faster using += (as this is done by in-place addition). Specifying the coefficient during initialization is faster than multiplying a SymbolicOperator with a scalar.
action_before_index Whether action comes before index in string representations.
Example: For QubitOperator, the actions are ('X', 'Y', 'Z') and the string representations look something like 'X0 Z2 Y3'. So the action comes before the index, and this function should return True. For FermionOperator, the string representations look like '0^ 1 2^ 3'. The action comes after the index, so this function should return False.
action_strings The string representations of the allowed actions.
Returns a tuple containing string representations of the possible actions, in the same order as the actions property.
actions The allowed actions.
Returns a tuple of objects representing the possible actions.
constant The value of the constant term.
different_indices_commute Whether factors acting on different indices commute.
## Methods
### accumulate
View source
Sums over SymbolicOperators.
### compress
View source
Eliminates all terms with coefficients close to zero and removes small imaginary and real parts.
Args
abs_tol(float): Absolute tolerance, must be at least 0.0
### get_operator_groups
View source
Gets a list of operators with a few terms.
Args
num_groups(int): How many operators to get in the end.
Returns
operators([self.class]): A list of operators summing up to self.
### get_operators
View source
Gets a list of operators with a single term.
Returns
operators([self.class]): A generator of the operators in self.
### identity
View source
Returns: multiplicative_identity (SymbolicOperator): A symbolic operator u with the property that ux = xu = x for all operators x of the same class.
### induced_norm
View source
Compute the induced p-norm of the operator.
If we represent an operator as $$\sum_{j} w_j H_j$$ where $$w_j$$ are scalar coefficients then this norm is $$\left(\sum_{j} \| w_j \|^p \right)^{\frac{1}{p} }$$ where $$p$$ is the order of the induced norm
Args
order(int): the order of the induced norm.
### isclose
View source
Check if other (SymbolicOperator) is close to self.
Comparison is done for each term individually. Return True if the difference between each term in self and other is less than EQ_TOLERANCE
Args
other(SymbolicOperator): SymbolicOperator to compare against.
### many_body_order
View source
Compute the many-body order of a SymbolicOperator.
The many-body order of a SymbolicOperator is the maximum length of a term with nonzero coefficient.
Returns
int
### zero
View source
Returns: additive_identity (SymbolicOperator): A symbolic operator o with the property that o+x = x+o = x for all operators x of the same class.
### __add__
View source
Returns
sum (SymbolicOperator)
### __div__
View source
For compatibility with Python 2.
### __eq__
View source
Approximate numerical equality (not true equality).
View source
### __mul__
View source
Return self * multiplier for a scalar, or a SymbolicOperator.
Args
multiplier A scalar, or a SymbolicOperator.
Returns
product (SymbolicOperator)
Raises
TypeError Invalid type cannot be multiply with SymbolicOperator.
### __ne__
View source
Return self!=value.
### __neg__
View source
Returns: negation (SymbolicOperator)
### __pow__
View source
Exponentiate the SymbolicOperator.
Args
exponent (int): The exponent with which to raise the operator.
Returns
exponentiated (SymbolicOperator)
Raises
ValueError Can only raise SymbolicOperator to non-negative integer powers.
### __radd__
View source
Returns
sum (SymbolicOperator)
### __rmul__
View source
Return multiplier * self for a scalar.
We only define rmul for scalars because the left multiply exist for SymbolicOperator and left multiply is also queried as the default behavior.
Args
multiplier A scalar to multiply by.
Returns
product A new instance of SymbolicOperator.
Raises
TypeError Object of invalid type cannot multiply SymbolicOperator.
### __rsub__
View source
Args: subtrahend (SymbolicOperator): The operator to subtract.
Returns
difference (SymbolicOperator)
### __sub__
View source
Args: subtrahend (SymbolicOperator): The operator to subtract.
Returns
difference (SymbolicOperator)
### __truediv__
View source
Return self / divisor for a scalar.
Note
This is always floating point division.
Args
divisor A scalar to divide by.
Returns
A new instance of SymbolicOperator.
Raises
TypeError Cannot divide local operator by non-scalar type.
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[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }] | 2022-10-04 08:05:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6704612970352173, "perplexity": 9881.212462326623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337480.10/warc/CC-MAIN-20221004054641-20221004084641-00242.warc.gz"} |
http://math.stackexchange.com/questions/58948/calculating-offset-from-planckian-locus | # Calculating offset from Planckian locus
I'm calculating the Correlated Color Temperature (CCT) from a chromacity pair, and I am trying to find how far from the Planckian Locus the coordinates are.
What I'm currently doing is I read RGB values off a sensor, I do a matrix transformation to get to XYZ. Then I calculate the xy values, and lastly I use McCamy’s formula to get CCT.
McCamy’s formula gives me the Correlated Color Temperature, the nearest point along the Planckian Locus for the xy coordinates (roughly a orange to blue axis). I am trying to find the green/magenta content of the light, so I need to find how far from the locus the coordinates are (roughly a green to magenta axis).
I guess McCamy’s formula finds the normal down to the locus, and that I need to calculate how long that normal is. I also have to do it fast (this is running on a small embedded processor). I'm using the CIE 1931 color space. (EDIT: The illustration is from CIE 1960 as pointed out below, but I could not find a chart showing isothermal lines for CIE 1931)
-
I'm a bit confused because you write that you calculate the CCT using $xy$ values using McCamy's formula (I'm assuming you mean this formula), but you included a diagram of the Planckian locus in $uv$, not in $xy$. Presumably when you say you're trying to find how far from the Planckian locus the coordinates are, you're referring to the Euclidean distance either in $xy$ space or in $uv$ space? In either case, this section has approximations where you can plug in the CCT to get the corresponding coordinates on the Planckian locus; then you just have to calculate the Euclidean distance of your point from that point using Pythagoras. I guess since you're already approximating by using McCamy's formula, this further approximation will be OK; if not, you can always do a full summation over the standard observer as shown further up in the article. | 2015-04-18 13:46:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8750021457672119, "perplexity": 177.81155747381985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246634333.17/warc/CC-MAIN-20150417045714-00193-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/gauss-law-in-dielectrics.841829/ | # Gauss' Law In Dielectrics
1. Nov 7, 2015
### BOAS
1. The problem statement, all variables and given/known data
A spherical capacitor consists of two concentric spherical conductors of radii $R_{1}$ and $R_2, (R_2 > R_1)$. The space between the two conductors is filled with a linear inhomogeneous dielectric whose relative permittivity varies with the distance $r$ from the centre of the spheres as $ε_r(r) = (c + r)/r$, with $c$ a constant. The inner sphere carries a total charge $q$ and the outer conductor is grounded.
Using Gauss’s law in dielectrics, compute the electric field (direction and magnitude) at a distance $R_1 < r < R_2$ from the centre of the spheres.
2. Relevant equations
3. The attempt at a solution
I think the charge on the inner sphere $q$, can be considered the free charge.
Gauss' law in dielectrics;
$\oint \vec D . d\vec a = q$
I don't know the polarisation vector.
$\vec D = \frac{q}{4 \pi R_1^2} \vec r = \epsilon_0 \vec E + \vec P$
$\vec D = \epsilon_r \epsilon_0 \vec E$
where $\epsilon_r = \frac{\epsilon}{\epsilon_0}$
$\vec E = \frac{\vec D}{\epsilon_r \epsilon_0}$
I think that this expression gives me the electric field inside the dielectric, but I am concerned that I have not considered the effect of the grounded outer shell.
Do I need to compute the electric displacement inside the outer shell due to the induced charge on it, and the field inside is the linear super position of the two?
2. Nov 8, 2015
### rude man
What does Gauss's law really say? Does it care about anything ouside the Gaussian surface?
3. Nov 8, 2015
### BOAS
Ah, I didn't think of that.
Gauss's law in a dielectric says that the flux through a closed surface is equal to the enclosed free charge. My argument seems reasonable in light of this. Although I notice my choice of gaussian surface (a sphere of radius $R_1$) should really be a sphere of radius $r , R_1 < r < R_2$
Last edited: Nov 8, 2015
4. Nov 8, 2015
### rude man
Good! Finish and show your work if you care to.
5. Nov 9, 2015
### BOAS
$\oint \vec D . d\vec a = Q_{fencl}$
I choose a spherical gaussian surface of radius $r, (R_{1} < r < R_{2})$.
$D(4 \pi r^2) = Q_{fencl}$
$\vec D = \frac{Q_{fencl}}{4 \pi r^{2}} \hat r$
*this step seems fishy to me* $Q_{fencl} = q$ Am I justified in saying that this is the case?
$\vec D = \frac{q}{4 \pi r^{2}} \hat r$
Since I do not know the polarisation vector $\vec D = \vec E \epsilon_0 + \vec P$, I can rearrange this to get $\vec D = \epsilon \vec E$ where $\epsilon = \epsilon_r \epsilon_0$
The electric field is therefore $\vec E = \frac{\vec D}{\epsilon_0 \epsilon_r}$
$\epsilon_r \epsilon_0 = \frac{\epsilon_0 (c + r)}{r}$
$\vec E = \frac{q}{4 \pi r^2} \frac{r}{\epsilon_0 (c + r)} \hat r = \frac{q}{4 \pi \epsilon_0 (cr + r^2)} \hat r$ for $r, (R_{1} < r < R_{2})$
6. Nov 9, 2015
### rude man
Why fishy? q is the total free charge within the gaussian surface, is it not?
The rest looks fine! You might, strictly for the sake of elegance, have moved the r out of the denominator parenthesis.
7. Nov 9, 2015
### BOAS
The question states that the conductor carries a total charge of q, which i'm imagining as having been added to an electrically neutral conductor. Why is it ok to ignore the electrons in the conduction band of the metal? Is it a case of free charge residing on the surface, and these electrons clearly don't meet that criteria?
Great, thank you.
8. Nov 9, 2015
### rude man
Because the conduction-band electrons are equal in number to the positive-charge "holes" left behind, leaving a net charge of q only inside the gaussian surface. It's the net charge that counts.
BTW the concept of "holes" is a matter for quantum mechnanics. Holes are not merely positively charged ions: https://en.wikipedia.org/wiki/Electron_hole
9. Nov 9, 2015
### BOAS
Ok, thanks.
I have met holes in the various solid state physics experiments i've done, but haven't taken a QM course yet. | 2018-02-21 04:03:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7701627612113953, "perplexity": 579.3080887265096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813322.19/warc/CC-MAIN-20180221024420-20180221044420-00205.warc.gz"} |
https://smellslike.ml/posts/efficient-transformers/ | Convolutional Neural Networks have been a boon to the computer vision community. Deep learning from high-bandwidth image/video datasets can be computationally and statistically much more efficient using the inductive bias of strong locality. This streamlines inference over big datasets or on resource-limited hardware.
To model sequential dependence in short sequences of low-dimensional data, we have often used LSTMs. However, researchers have recently found success adapting Transformer architectures to learn from image and video, both applications traditionally dominated by CNNs.
Vaswani et al’s pioneering work in machine translation introduced the Transformer, which utilizes attention mechanisms rather than recurrent or convolutional layers while encoding sequential structure through sinusoidal positional embeddings. Transformers would pave the way for many advances in NLP, most notably influencing the design of BERT.
Image and video data decoded into arrays admit sequential/grid representations. Video is generally recorded at frame rates sufficient to spatially resolve objects of interest, implying some degree of local spatial smoothness in image and video.
The space-time locality of convolutional kernels helps us to efficiently exploit this regularity to learn models with low parameter counts. Furthermore, sharing kernel weights over sliding windows combined with pooling helps to impose translation equivariance, a symmetry we expect to observe for many labeled datasets.
By comparison, self-attention in transformers is burdened by time & space complexity quadratic in the length of the input sequence. Despite this bottleneck, the model offers the capacity to learn from large-scale spatial interactions, spurring efforts to design more efficient transformers.
Google researchers describe a taxonomy of transformer variants, distinguished by the strategy used to sparsify attention.
Elementary reductions block or chunk the input sequences effectively quantizing the attention map. Similarly, strided or dilated attention helps to sample the input sequence. New schemes can be devised by combining these simple fixed patterns.
Advancing from handcrafted patterns, researchers considered learned attention maps. Some work reduces the token embedding space bucketing with LSH or using KMeans clustering.
Alex Graves suggests we consider “memory as attention through time” guiding research to introduce side-memory to limit the scope of a model’s attention.
Another conventient reduction is to assume a low-rank structure of the attention matrix to pass to a smaller N x k approximation (k « N). Ideas like kernel approximations and projection through Orthogonal Random Features offer this approach.
Vision Transformers (ViT) highlighted the potential for transformers in computer vision achieving SOTA performance comparable to models like Noisy Student and Big Transfer (BiT) across vision tasks after pretraining on larger (10M-100M) datasets.
The researchers reduced the compute bottleneck by tokenizing an image into patches while pointing out that specialized attention patterns suffer from a practical lack of hardware-accelerated implementations.
Transformers were further enhanced using Self-Supervised Learning (SSL) in Microsoft’s EsViT. Researchers borrow from BERT’s masked language modeling to incorporate local correlation information. This entails adding a term to the loss which encourages a student model to match a teacher’s soft label for a query patch, provided access only to distorted neighboring patches.
Facebook research into data-efficient Vision Transformers DeiT shows performant models trained on ImageNet-scale datasets. Researchers were interested in learning Transformer-based student models which benefit from the inductive bias of ConvNet-based teachers through knowledge distillation.
ConViT is another example hybridizing Transformers and ConvNets to endow the model with its inductive bias by initializing gated positional self-attention with convolutional priors using spatially-localized attention maps while relaxing any hard locality constraint.
The hard locality constraint enforced using convolutional layers helps mitigate the curse of dimensionality, but in the large data limit, may inhibit a model’s capacity to identify interactions occurring over larger spatial scales.
Comparing DeiT and ConViT with an aggregate metric of the attention-weighted distance between query and key patches, researchers find higher layers of ConViT attend to long-range interactions while promoting more diverse attention maps.
Considering the effect of initializing spatially-localized attention maps for training, we might ask whether attention maps concentrated in space-time could be useful in video object tracking. TrackFormer achieves tracking-by-attention after encoding frame level features extracted with a CNN backbone while dispensing with graph-based matching routines or appearance and motion models.
DeepMind’s Aloe applies self-supervised learning perform object tracking with transformers while characterizing the need to determine an appropriate level of resolution for input.
The Video Vision Transformer (ViViT) introduced a logical extension of spatial patches into the time dimension with tubelets:
Multiscale Vision Transformers introduced a hierarchical pooling attention which researchers contend helps the model to break permutation invariance and make better use of temporal information.
Even as Transformer research trends toward stronger data-driven priors, FNet shows the power of structured mixing. Researchers note a nominal reduction in accuracy on the GLUE benchmark by simply swapping the $$O(N^2)$$ self-attention in a BERT architecture for the highly-optimized $$O(N *log(N))$$ FFT. Perhaps this work has an extension to the image domain by applying FFT over image patches.
FNet authors suggest applications as a student model in knowledge distillation for deployment on resource-constrained environments.
Stanford researchers were motivated to introduce similar reductions to reduce computational bottlenecks of 1x1 depth-wise separable convolutions in MobileNet.
Others explored tiny Transformers for edge devices like the Arduino Nano BLE Sense, though limited by available tflite-micro ops. Making an addFNetMixer custom op for convolution by FFT could be an exciting contribution!
DeepMind researchers recently test the limits of Transformers over various data modalities including: audio, video, point cloud and text with Perceiver. This model utilizes cross-attention modules and latent factor projection to scale to high-dimensional input.
This success frames Transformers as the general purpose architecture and castes the utility of specialized architectures in doubt given the prevalence of big data. Indeed, researchers find Transformers generalizing well even to weakly-related tasks as training dataset scale increases.
After surveying the fontiers of research, we might conclude that specialized architectures like convolutional layers will remain en vogue for CV practitioners working in the small-medium data regime. But given sufficient volume of training data and/or aggressive augmentation and transfer learning, Transformers may reach higher performance using patterns learned over greater spatial scale.
We are learning to apply Transformers more generally and efficiently and expect to track increased adoption in ML systems.
Our first vision experiment using Transformers trains deit_base_patch16_224 over 100K theatrical posters labeled with one of 22 primary genre labels. After a few ours of fine-tuning with 2 Titan RTXs, our classifier reaches 80% accuracy on the approximately balanced dataset.
Consider these sample images and corresponding model logits for qualitative review.
Encouraged by the classifier’s performance, we decided to apply transformers for image similarity to compare against previous work. Adapting the DeiT approach with a keras implemention, we pair a ViT-based student with a ResNet-based teacher using knowledge distillation to fine-tune a genre-classifier before fine-tuning with the triplet loss.
While the ResNet50 teacher takes 224x224 input, we consider much lower resolutions for the ViT student:
We conjecture this extreme reduction can be justified in observing that theatrical posters are quite structured by conventional motifs designed to signal theme & genre. Further, we anticipate simple patterns like color palette and featured objects convey most of the signal, whereas edges and text or otherwise high-frequency information may offer lower-order improvements, hindered somewhat by sparse representation.
This extensive ViT comparison indicates that transfer learning can be quite effective for training ViT models and augmentation helps to match performance of models pretrained on much larger datasets. They also find larger patch sizes perform better than smaller model architectures.
A recent paper offers tips for successful knowledge distillation which guided our augmentation strategy. Starting with a genre classifier trained from scratch with logloss, we fine-tune with triplet-loss.
As recommended in the ViT comparison above, we can use transfer learning, selecting similarity model architectures by evaluating validation accuracy in the upstream classification task, which is much faster to train!
With an average of two genres labels per title, our dataset lends to multiple representations and augmentation strategies. For instance, we might consider each genre label to provide a new training sample. On the other hand, by framing a multi-label classification, we aim to utilize covariance in genre label distribution to enhance our distillation.
In another experiment, we apply the Multiscale Vision Transformer (MViT) to videos cropped down to faces for deepfake detection. With small changes to the config MVIT_B_16x4_CONV.yaml, we reach 75% validation accuracy on the balanced binary classification task, training the model on roughly 7K samples from scratch.
We also tried training this model on more of the raw deepfake detection challenge videos. However, focusing the model on faces with a detection pipeline proved to powerful an inductive bias to pass up for this small sample. | 2021-09-18 08:37:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3209240436553955, "perplexity": 4763.056714395883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056348.59/warc/CC-MAIN-20210918062845-20210918092845-00668.warc.gz"} |
https://www.allaboutcircuits.com/projects/c-biscuit-system-integration-and-testing/ | Finally, we have everything we need for a functional robot. Let’s put the pieces together and see what the C-BISCUIT demo bot can do.
### Supporting Information
• StackExchange thread on how to successfully use the UART on the Raspberry Pi 3 (it’s not as simple as you might think)
### Out with the Wand, In with the Pi
Electronics projects, like so many other things in life, don’t always work out the way we expect. This has definitely been the case with the C-BISCUIT demo bot, which has taken a few twists and turns on its way to becoming a functional system. And the biggest twist/turn of all happened at the end, when we decided to eliminate the Wandboard and replace it with a Raspberry Pi 3.
The dominant reason for this change was quite simple—the Wandboard had become an obstacle to our primary design goal, namely, transmitting live video from the robot to a computer via Wi-Fi. No one on the C-BISCUIT team knew exactly how to accomplish this after our initial attempt failed. It was one of those situations where we expected the third-party software environment to handle the details of the video transmission and, when it became clear that this was not exactly the case—well, let’s just say that the Wandboard quickly fell out of favor.
Fortunately, Patrick came up with the excellent idea of shelving the Wandboard and using a Raspberry Pi instead. This also meant using the Raspberry Pi Camera instead of the WandCam, because of course we cannot expect a Wandboard camera to seamlessly interface to a Raspberry Pi (and seamless is what we want here—maybe there’s some way to make a WandCam talk to a Pi, but we’d much rather pay another \$30 for a camera that just works).
### Why the Pi?
The Raspberry Pi has more fully-developed software support than the Wandboard, and it gives us access to a larger pool of experience (from both AAC contributors and the wider RPi community). It is also smaller and less expensive than the Wandboard. It’s possible that the Pi is not powerful enough for some of the image processing tasks that we want C-BISCUIT to support, but we can worry about that later. For now, we just need the wireless video feed and some basic serial communication, and the RPi 3 can certainly provide that functionality.
And we certainly have no regrets about using the Pi Camera (version 2). This module does indeed interface effortlessly to the RPi, and it provides high resolution (8 megapixels) along with a variety of useful features (customizable output resolution and frame rate, image rotation, brightness and contrast adjustment, etc.).
So now you know why C-BISCUIT ended up with a Pi instead of a Wand; here is the updated block diagram:
Fortunately, the RPi can use the same 5 V supply and UART connections that were originally intended for the Wandboard, so we didn’t need to make any modifications to the Robot Control Board (RCB). Hooray!
### The Hardware
Let’s take a look at how the robot is assembled, and then we’ll cover some details regarding the video and communication interfaces.
So the battery and RCB are on the upper level, and the RPi is down below. The camera is simply inserted into the dual-row header near the front edge of the RCB; I wrapped some electrical tape around the back row of pins so that they don’t short out anything on the back side of the camera PCB (the tape also ensures a snugger fit).
This arrangement wouldn’t be adequate for rough terrain, but it should be fine for initial testing and demonstration. Here are a few more assembly details:
• The bot is enabled whenever 12 V is delivered to the RCB, and 12 V is delivered to the RCB whenever the two power connectors are mated and the power switch is closed.
• The RPi is powered via its USB connector. I simply chopped off one end of a standard Micro USB cable and then connected the power and ground wires to the 5 V and GND screw terminals on the RCB. It turns out that the red wire is power and the black wire is ground, and I might dare to assume that all cable manufacturers follow this rather well-established color-code custom; nevertheless, always break out your multimeter and find a way to double-check the pinout before you plug everything in and flip the switch.
• Serial communication is enabled via two wires from the RPi’s UART Tx and Rx pins to the RCB’s UART Rx and Tx signals. A ground wire is not necessary because the power-supply connection ensures that the RCB and RPi are at the same ground potential. It would be good to include a ground specifically for the UART interface if we were worried about signal integrity, but in this case we’re not pushing any limits—we’re using 3.3 V logic at the astonishing rate of 9600 baud, and the UART lines have decent physical separation from the noisy motors.
• Notice how I have the robot resting on a book such that the tank treads are not in contact with the surface. This is a simple and very effective way to prevent the nightmare scenario in which your robot goes careening off your workbench after you experience a firmware malfunction or accidentally knock the power switch or what not.
### Best RPi Video Software in the World
Well, OK, I only tried two techniques; the first one didn’t work and the second one did. But after the difficulties with the Wandboard and the convoluted failure produced by the first RPi attempt, it was extremely impressive to see how well the second solution performed. The software package is called the RPi Cam Web Interface, and you can read all about it here.
I’m going to skip the details because the extensive wiki has everything you need to know to get started. The overall process is as follows: Connect the RPi to your network, either through a cable or Wi-Fi. Install the software on the RPi and run it. Then you simply open up your favorite browser and connect to the video stream by typing "http://RPi_IP_address:port/subfolder" into the address bar (see the wiki for details). What you see is this:
The low-latency video appears in the browser window, and you can also capture images or video and modify camera settings:
### This Is Not an Autonomous Robot
The robot won’t be very useful if all it can do is move around in circles or drive forward until it hits a wall. Thus, we need a convenient way to control its movement, and that is exactly what the following Python script provides:
import serial
ser = serial.Serial(
port='/dev/ttyS0',
baudrate = 9600,
parity=serial.PARITY_NONE,
stopbits=serial.STOPBITS_ONE,
bytesize=serial.EIGHTBITS,
timeout=1
)
while True:
response = raw_input("Enter command: ")
if response == "forward":
ser.write("forward\r".encode())
elif response =="reverse":
ser.write("reverse\r".encode())
elif response == "go":
ser.write("go\r".encode())
elif response == "stop":
ser.write("stop\r".encode())
elif response == "s":
ser.write("stop\r".encode())
elif not response:
ser.write("stop\r".encode())
elif response == "rotate r":
ser.write("rotate r\r".encode())
elif response == "rotate l":
ser.write("rotate l\r".encode())
elif response == "low speed":
ser.write("low speed\r".encode())
elif response == "high speed":
ser.write("high speed\r".encode())
elif response == "very high speed":
ser.write("very high speed\r".encode())
elif response == "battery voltage?":
ser.write("battery voltage?\r".encode())
elif response == "disable battery protection":
ser.write("disable battery protection\r".encode())
elif response == "enable battery protection":
ser.write("enable battery protection\r".encode())
elif response == "exit":
break
The script itself is self-explanatory. It asks for input and then looks at the string entered by the user in order to determine which (if any) ASCII message should be transmitted over the UART. The EFM8 microcontroller on the RCB then receives the message and controls the motors accordingly.
But how do we use the script if we have no direct access to the RPi? That’s where SSH comes in. (Click here for more information on controlling a Raspberry Pi via SSH.) We already have the Wi-Fi network connection, so we simply SSH into the RPi and then run the script via the console.
One thing you might notice in the Python script is that there are three commands for stop. In addition to “stop,” we also have “s” and an empty string. This is an important feature because it ensures that you will be able to stop the robot quickly, even under pressure.
Let’s say that you get distracted for a minute and the next thing you know, your bot is about to tumble down a staircase, or run into a door at high speed, or knock over your favorite houseplant. You might not have time to type “stop,” and that’s why the script is designed to interpret “s” as “stop”—surely you have time to hit one letter and then the enter key.
But then there is the worst-case scenario: your robot is in serious danger and you panic. Maybe you can’t remember the stop command, or maybe you’re in such a rush that you hit the wrong key when you’re trying to send an “s” command. Well, you’re covered here too, because all you have to do to stop the robot is press the enter key! If you press nothing more than enter, the return string from the raw_input() function will be empty, and the script interprets an empty string as a stop command.
This screenshot gives you the sequence of commands for the robot movements shown in the video that follows:
### The Firmware
The firmware is pretty straightforward. There is a simple UART interface for communication with the RPi, and the programmable counter array (PCA) is used to generate the PWM motor-drive signals. You can use the following link to download all the source and project files. The code uses descriptive identifiers and is thoroughly commented, so you shouldn’t have much trouble understanding the details.
One thing you will notice is additional files and code for USB communication. You can ignore this for now because USB functionality is not implemented in the demo-bot firmware.
### Conclusion
We now have a fully functional C-BISCUIT demonstration robot, which means that this series of articles is complete. However, remember that C-BISCUIT is not a single robot but rather a platform for robotics projects. In other words, this demo bot is only the beginning! | 2019-08-21 17:01:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25472185015678406, "perplexity": 1487.752859954266}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316075.15/warc/CC-MAIN-20190821152344-20190821174344-00488.warc.gz"} |
https://news.ycombinator.com/item?id=17075261 | I agree with this suggestion. It took me a year to slowly absorb the entire book of Statistics [0] including solving all exercises. It's just like walking to school but there is no external supervision. I made a rule to complete one chapter every evening including exercises and sticked to it.
Your story implies there are 365 chapters.
It implies he completed 365 chapters but says nothing about repetition of the chapters.
> sit down in a warm or hot roomWhile I agree with everything else, I'd have to vehemently disagree with this. Studies [1] have shown that warm temperatures severely diminish our performance on complex mental tasks.As some examples [2]:> Sales for scratch tickets, which require buyers to choose between many different options, fell by $594 with every 1° Fahrenheit increase in temperature. Sales for lotto tickets, which require fewer decisions on the part of the buyer, were not affected.> participants were asked to proofread an article while they were in either a warm (77°) or a cool (67°) room. Participants in warm rooms performed significantly worse than those in cool rooms, failing to identify almost half of the spelling and grammatical errors (those in cool rooms, on the hand, only missed a quarter of the mistakes). > You sit down in a warm or hot roomWhat is wrong with airconditioning? I am not sure about OP's reasoning, but I personally find it a bit 'motivating' to study in a slightly not-so-comfortable environment. I mean, it gives me sense that I am actually determined and am working hard. It also reminds me of my college days when even finding an air-conditioned room anywhere was just not possible. Could also give you the feeling of being uncomfortable. Then when you are struggling working through a problem you get so frustrated. And think "If only it weren't so damn hot in here." Then all you can think about is the heat, and you are so lost it cannot be returned. So then you give up for the day, and really haven't accomplished anything. Exactly, your mileage may vary, but my mindset has to be completely free from distractions to be productive.The library on my uni when I was in Math undergrad did not have AC at the beggining but was the only place where I could do any work, it was extremely difficult and I am sure impacted my progress. I find it impossible to think or stay focused in a hot or even warm environment. People are different I guess. I'm the same. During winter months, when I needed to cram a lot, I would open the windows wide, and sit with my jacket on. The cold would help me not fall asleep. A hot room sounds horrible, but the memories of college days does make sense to me. My college was freezing cold, and my search would be for a room where you didn't need to wear 2 sweaters to be comfortable.But yeah, the idea of studying in a really cold room "makes sense" to me, and this might be why. I think that's just building the idea that it's going to be a painful and uncomfortable process In cold parts of the world, warm has connotations of comfort, not cold... Is this part of the process? Visit a cold part of the world, set yourself up with a physics textbook in front of a fireplace...Actually, that sounds quite nice. my deduction: if you done it in a warm or hot room, you surely have enough will to do it. Seconded. I really believe there are no shortcuts to doing lots of problems. If you can afford it, getting a physics grad student to discuss problems that stumped you every now and then might also have quite good ROI, talking to physicists might also help convey some of the physics mindset(?).Reading this made me nostalgic for my days as a physics undergrad. I'll second this idea having survived a Physics BS doing just this. I'd also strongly recommend a series of books called Schaum's Outlines, they vary in quality but cover many advanced topics and have hundreds of solved problems in them. Schaum's Calculus was invaluable to refresh my memory of some of the details of "Calc 2" so I could be sure of passing a waiver exam (most schools would have waived it automatically on account of my AP credits but my school limited me to how many I could waive that way...) and get on with Calc 3. The book covered some Calc 3 too so continued being useful. I have a few others in the series, very handy."The reader who has read the book but cannot do the exercises has learned nothing." -- J.J. Sakurai(Incidentally, I tried reading Sakurai's Modern Quantum Mechanics on my own once and was immediately curb stomped. Lots of prep work required for that one...) Defs agree with op. I learned the more advanced maths I use daily in my thirties. It took about 3 years of exactly ops method. In my case, I found it motivating to take exams because it gives you a bit of skin in the game; forces you to prioritise your study at some point.A final thing: it's really worth doing. If you long for maths; it's likely it'll conceptually take you places you won't go without it. Do it! This is also what I did, going straight to the exercises except I used Calculus I by Apostol which covers some Linear Algebra. Perfect book if you need to redo math skills you've forgotten though plenty of times I had to Wikipedia, Khan Academy, and math.stackexchange in the beginning.There's also this free book, no answers though you could stackexchange if really stuck. I finished most of Apostol before starting it https://infinitedescent.xyz/ https://books.google.com/books/about/Physics.html?id=8MvvAAA...Is it that the text you’re referring to? We used the 5th edition in my physics course this year. It was a tough textbook to learn from but I feel like I learned a ton. I almost never went to class in university (Waterloo Engineering) and this is how I did it. The best is not letting them explain the concept to you first. Try to invent the math as you go along by covering the explanatory pages with pieces of paper and reading only one line at a time.It will stick with you forever. I second this, but, you will need some help initially. Follow the examples a few times - first with help, then without. Once you build your intuition, you will then be in a position to "invent" the maths as you go along. > Halliday and Resnick, early editions , printed in the late 60s and 70sAny particular reason to recommend the old editions over the latter ones? I actually had to look into this recently.The recent ones are less "textbook." The older ones are FILLED with information with graphics here and there but it's mostly text. The recent ones are very graphical so I would assume it has less total information. With that said, it's possible that there are techniques for learning that were not considered in the older texts.It is possible to look at samples online for you to compare if you want to see the difference. I do recommend getting the book if you decide to use it but that's just a personal preference. Textbooks have generally gotten less information dense over time. > If the textbook is in its 4th printing or so, the answers are correctIt's terrifying that it takes 4 printings before the answers should be considered trustworthy... Do you write bug free code? Is publishing a book the same thing as writing code? No, writing a math textbook involves (perhaps) thousands of things that might be wrong, none of which will have any impact on one another.Publishing a perfect book is difficult on par with writing code. Hell, Knuth is incredibly popular and crowdsources his error-checking, and TAOCP is still in its third edition. Very similar in some ways. There are a vast number of interconnected details that have the potential to be wrong and far fewer automated ways to catch any errors. Your "users" inevitably catch a lot of them at "runtime". Yes. You're programming a person instead of a computer but that's the only difference. I'm writing a book called "A Programmer's Introduction to Mathematics". Would you like to see a draft? Shoot me an email at mathintersectprogramming@gmail.comIt's an introduction to mathematics from a programmer's standpoint, with a big focus on taste and that second level of intuition beyond rote manipulation and memorization.Includes chapters on sets, graphs, calculus, linear algebra, and more! Each chapter has an application (a working Python implementation) of the ideas in the chapter. The applications range from physics to economics to machine learning and cryptography. One chapter even implements a Tensorflow-like neural network.There's also a mailing list: https://jeremykun.com/2016/04/25/book-mailing-list/ I can recommend anything Jeremy does, sight unseen. Take him up on his offer. Is this offer just for the OP, or for everyone? Disregard my email if it was meant only for the OP. Whats the progress / ETA for this? Sounds like the perfect thing for me right now Watch the 3Blue1Brown YouTube channel: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAwThese videos are frankly better explanation of college-level math concepts than most college classes.Also, now you probably care much more about the intuitions of mathematics over the raw mechanics of it. Once again, this channel perfectly exemplifies this concept. These videos are posted any time linear algebra is mentioned. I find it almost comical at this point. What good is this intuition? I struggle to understand the value that these videos bring, but I'm not saying there is no value. I'm just lost, and kind of jealous because I need a deep understanding of linear algebra for work. Those are good videos, and I also endorse them. However, you will not truly know what is going on until you solve some real problems with them.One interesting option that computer programmers have to understand linear algebra that most people do not is that you could go do some stuff with computer graphics. Grab three.js, and then once you've followed some tutorial somewhere to get a triangle on the screen, start doing things, but do them manually, including implementing matrix multiplication yourself. Modern graphics has moved so far up the stack nowadays that you probably shouldn't say that you "know 3D graphics" after that exercise, because you'll know 3D graphics circa 1995. But you will have a much better intuition for linear algebra, and those videos will either make sense, or be trivially obvious to you.(One of the reasons why the linear algebra videos can be so helpful is that it has historically been very easy to take an entire class on the topic and just grind numbers, without ever getting to that level of intuition. Differential equations, if you took physics that did not use them, can have a very similar problem, where you just grind through problems for a semester with no motivation.) It's not just linear algebra, 3blue1brown also has an entire series on undergraduate calculus, and series on Statistics, Linear Algebra II and Group Theory are in the works. Plus a large number of excellent videos on miscellaneous math topics.I think I understand what you're saying -- one needs more than just cool videos and cool intuition. You need to do exercises. This is a point made multiple times in those very videos.But, the intuition provided in those videos is absolutely excellent. As an example, look at the explanation of change-of-basis in the linear algebra I series. Ah, the videos come with pointers to exercises? I'm quite excited about https://www.edukera.com/ and possibilities for interactive learning through automated theorem proving. Thanks! Actually, there are not many pointers to exercises. I believe they may be planning to add some written materials, so maybe in the future, but not currently.The videos suggest pausing and trying to figure the next bit out yourself and a couple of the videos do end with a suggestion to prove something yourself. For me personally, understanding why it is done on a deeper level than is commonly taught helps me consolidate the concept more comprehensively and permanently. I think what stands out about those videos is that people who have no prior higher math education still get a shadow of intuition of what's actually going on, and people who already 'grokked' the concepts still got an alternative, simpler view on those concepts, resulting in at least a view 'a-ha!' moments for almost everyone. Did you actually watch the videos? Ha no. I can't watch videos, I read really fast and find videos painfully slow. Grant (the 3Blue1Brown guy) has an uncanny ability to explain difficult concepts and the fundamental intuition behind them. In many of his videos, he explains topics from the perspective of a person inventing that topic (such as in his first Calculus video [1]). I can't recommend his videos enough. It works for a lot of people, I'm just looking for something else, exercise-based computer-checked proofs to teach mathematics, something like that. Want it bad. I do x2 speed. But not all videos are slow. I second this. I was refreshing some of the linear algebra I learned in college recently and his videos gave far more insight into what linear algebra is actually about than I was taught in college. If everyone studying linear algebra in school watched his videos before taking the course they would have a much easier time learning it. His calculus series is of similarly high quality and I would imagine his other videos are too. It is important to really pause the videos at some points and do the "exercises". Doing the exercises is always a very important part of learning (I believe this holds in any field). Do them with pen and paper, not just in your head. I'm a web developer. In my 20's I used to love picking things up just for the of it. These days I'm more of a fan of JIT learning. Push the edges of my map as I go. I'm still constantly learning, but it's more iterative. Building on conquered territory and shifting the borders as needed (always outward, but the focus on which parts of the border to push changes.) Previously I was more like a crazed monkey and never holding any ground. I still feel the importance of occasionally sneaking outside my borders and going deep into enemy territory, but those are constrained efforts. Invade, gather booty, sift for intelligence value and then decide if it's worth a more serious invasion.Maybe figure out an actual destination and then devise a plan to get there. Deep diving into math and physics just for the sake of learning etc seems to be cargo-culting. Lawyers also sound smart until you realize they write like they do intentionally to keep people from figuring them out.> I follow a bunch of folks on the internet and idolize them for their multifaceted personalities - be it math, programming/problem solving, physics, music etc. And these people had a natural flair for math/physics which was nurtured by their environment which made them participate in IOI/ACPC etc.Sounds like they are good story-tellers along with whatever else they do. Have you tried putting anything out for others to consume? If you want to be like these people, then it would be good to start with writing / shipping things. If you have been doing that already, then post some links. ;) Anyone who tells you can "learn" math and pyhsics by just watching videos is lying to you. There is no substitute for actually doing lots of problems.Pick a book, pick a pace to work through it, and spend a few months going through it. Do the exercises in the back of each chapter, work through the solutions, and ask around if you still can't figure it out. Persistance and routine are key here.As for books, I like Stewart's Calculus, Lay's Linear Algebra, and Hammack's Book of Proof.For physics, I don't know what your background is. Giancoli is a popular undergrad freshman year book, where as griffith's electrodynamics is a bit more advanced. I second this but starting to watch actual undergrad lecture series is beneficial because sometimes the book your working through May just be missing that one piece of the puzzle you need to start getting enough of a grip to solve a problem.Also with physics, you need some tricks that are usually done in derivations or calculations (cause not everything can just be solved) and those do not necessarily appear in a book - but they do in lectures.Edit: the book gives you the fundamentals though on which you work. also, they will teach you the necessary abstraction - the first thing standing in my way of a degree in physics was my intuition and need to picture stuff. working a lot with differential geometry now, I've got some of that visualization back but with linear algebra even and quantum mechanics it can stand in your way. This!For the programmers in the house, it would be like claiming you can code python if you’ve never coded before but watched some videos of expert teachers explaining the structure of python programming.I often see people here on HN saying classroom lectures are more important than homework. But in my experience the real intuitive learning (at least for math and physics) doesn’t come until you’ve spent some time banging your head trying to figure things out solving problems (ie, applying the theory).One text (Wangsness E&M maybe) had a great student quote, roughly “I understand the principles but I can’t do the problems.” Giancoli's book doesn't use calculus. Halliday & Resnick (or one of its later updates from Crane) is a better bet in this regard. Take a look at brilliant.org. They teach mathematics in small chunks with practice. https://brilliant.org/courses/#math-advanced https://brilliant.org/courses/calculus-done-right/ https://brilliant.org/courses/linear-algebra/ People say you have to "do" math to learn it. Usually they make it sound like you need to do the exercises in the books. I think that doing just that can be boring and demotivating.I would suggest finding projects that can motivate you and help you exercise your math. Some suggestions of mathy things I regularly work on for fun:1. Make a video game. If it's a 3d game, you'll have to do your matrices, dot products, trigonometry, etc.2. shadertoy.com - This is a community site where people just program cool looking graphics for fun. All the code is open, so you can learn from it. Similar to game programming but without the mathless overhead. :)3. Machine learning projects - I love writing various machine learning things, but the project that has been a great ML playground has been my self driving toy car. It gives me plenty of opportunities to explore many aspects of machine learning and that helps drive my math knowledge. My car repo is here: https://github.com/otaviogood/carputer but a much easier project is donkeycar.com. ML will touch on linear algebra, calculus, probabilities/statistics, etc.The most important thing for learning is to be inspired and have fun with what you're learning. :) I can't speak to all of the things you want to learn, but I've learned some of them on my own. For calculus and linear algebra I'd go with Khan Academy, especially since it seems like all you need is a refresher for calculus. Graph theory and discrete math I did with MIT EdX courses. Their discrete course is pretty nice and I found it very easy to follow along with.Constraint solving and optimization problems aren't things I self studied, but you can find a variety of resources to help with those based on how you learn best. For me, I did them by taking a class and relying heavily on my textbooks. +1 to Khan Academy. Explanations are super clear. Their website allows you to work through practice problems too, which I think is the most important thing. For physics, I believe you fall in Leonard Susskind's target audience. You can get his book, or even better, watch his large amount of lectures: https://www.youtube.com/watch?v=iJfw6lDlTuASusskind is an eminence - he was Feynman's buddy back in the day. And he's entertaining as hell.Here's also Gerard 't Hooft's (Nobel laureate) list of concepts and books to master. If you finish that -in several years- you will be a qualified theoretical physicist. Whereas Susskind will give you more of an overview. http://www.goodtheorist.science/ Get a real pen and paper, get a real physical book, sit and solve problems with pen and paper for hours every day for a few months. Then you will pass the exams. Exactly. Every time I tutor someone in math, I tell them to use up at least a sheet of paper for every interesting question. When they do, their skills improve quickly. Saving paper is a false economy when it comes to math. I don't know why this is downvoted, but the process of writing and working problems on paper, at least for me, helps cement the knowledge. This is exactly right. I elaborate on the method in my response. While I agree with you, and love aj7's post, I'm going to push back slightly on the pen and paper.I used to do all my work (solutions to problems, notes) using pen and (plain! not lined) paper. However I realized a couple of years ago that becoming fluent in LaTeX was a better option for me. The reason is that, with the proof neatly typeset, and the ability to re-work and edit repeatedly without making a mess, I found that I think more precisely and systematically. I still do scratch work on paper, but writing up a clean copy as I go is very beneficial.In addition to those reasons, the other hugely important one is that my notes are now in git, I can grep them, and they don't add to the pile of objects that must be dealt with when moving to a new home.For best results you need to make a nice LaTeX set up. I use the Skim PDF reader so that it autorefreshes on file save, and set up a Makefile and make it so the PDF is recompiled on every file save. But whatever works for you, I'm sure there are easier setups. There's a lot to be said for using computer tools. If you're writing proofs, why not do it formally? [0] If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)? If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)The downside of course is that computers are very capable distraction vehicles, you need a bit of discipline to sit at one and study / do this sort of work at the same time for prolonged periods. Pulling out the ethernet cable can help but may not be sufficient depending on one's level of discipline and access to offline distractions.A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles. Certain modern enhancements are worth a qualified mention though. > If you're writing proofs, why not do it formally?Because that requires learning a formal proof-verification language. I'm certainly interested in that, but it is a distraction from learning undergraduate mathematics.> If you're working with graphical concepts, why not code them up, or use a drawing program (or hey, a graphing calculator) rather than pulling out a ruler and such (and maybe learning to draw at all if you don't know how)?> If you have sloppy handwriting (as I'm sure many of us here do), why not type in something you'll always be able to read later? (Along with whomever you show it to -- I did a lot of college homework using LaTeX. With macros I could do things way more efficiently, with comments I could go back and see what I was thinking at a misstep (if I wrote anything).)I'm confused; my post was advocating using software, so I'm unclear why you're suggesting I use software.> A lot of the old methods of learning actually work and so the advice is sound to strictly adhere to them when you're having struggles.What is that, a flat contradiction of my post?Very strange, maybe you meant to reply to a different post? My post was mainly adding agreement to yours with more specifics, "you" used is the "generic you".> it is a distraction from learning undergraduate mathematicsArguably so is LaTeX. But it's desirable that students (or just people learning the same material, later) spend some of their undergraduate time learning new things, right? And not just because it's new, but hopefully because it's better. Learning new/different things is just a small step further beyond learning old things with new/different assistants. And maybe some things will have to be cut out, like 17th century prose-proofs (edit: and even just moving to structured proofs without full formal tools is an improvement...), or square roots by hand (http://www.theodoregray.com/BrainRot/) One thing that I can add, is that the process of neatly recording something really helps cement the process. My professor for dynamics and mechanics of materials required homework to include diagrams of the problem, neatly drawn, on unlined paper. Often I would find that each problem would take three sheets of paper (I'm a horrible draftsman), but I am horribly glad after the fact that I invested all that time.It is painful, but I don't think there is any easy way of actually learning without just sitting down and doing problems. Have you considered auditing a course at a community college? Very few people (myself included) are motivated enough to work enough problems without the threat of assigned homework. You need to do enough problems on a topic that you are no longer struggling, then do 4-6 more. Those last problems are, IMO, the most important, they actually cement the concepts in long term memory.As far as books, I can recommend Schaums Outlines for good examples of worked-through example problems.Edit:fixed typos To quickly review a broad range of math up until 1st or 2nd year of university, I really recommend Khan Academy https://www.khanacademy.org/math . I am currently using it to brush up my math skills for machine learning.Before going to Khan Academy, I started reading a rigorous math textbook, but my motivation didn't last long. You really need high motivation to complete a rigorous textbook, but Khan Academy is different and I am finally able to continuously improve my math skills.The best thing I like about Khan Academy is the large amount and instant feedback of exercises that you don't get from regular textbooks. I really wished that Khan Academy was there when I was a kid.To get deep knowledge of math, I think that rigorous textbooks are the way to go, but before those and to prepare for them, I really recommend Khan Academy. Being in your thirties has little to do with learning. How you learn is much more important than your age.If you learn best in a classroom, you may have a local college that teaches math in the evenings. (I got my Master's in Statistics that way.)If you learn best in small chunks, Khan Academy has differential and integral calculus and linear algebra, to start you out.If you learn best from books... there are hundreds of great textbooks.Best wishes to you. Keep up a lifetime of learning! Very true.My learning actually accelerated in my 30s because knowledge pays compound interest -- the more knowledge you have, the faster it is to acquire new knowledge. Assuming one has continued to pursue learning, someone in their 30s would have built up a significant enough semantic tree to pin new knowledge to.Most people find it hard to learn in their 30s because they lack the energy, environment (+kids, +spouse, etc.) or internal drive that provides them the impetus. Others find it hard to learn because of bad habits and a poor foundation (their semantic tree wasn't that well built up in their youth). But their actual abilities (even memory) haven't actually degraded all that much.And of course, there are some who find it hard because they have reached the limits of their cognitive abilities (un-PC as it sounds, this is a real thing). You have to know if this is the case. Most of the time it is not.I would start by building up a good foundation. Learn the basics well but don't get hung up on understanding every little detail.Chunk your learning and use your little victories to drive you (brain hack: humans are a sucker for little victories). Use the Feynman method (learn by teaching).Drill yourself with exercises rather than trying to understand everything -- math is one of those things where it is easier to learn hands-on by working on problems BEFORE understanding the definitions fully... understanding comes later (the patterns will emerge once your semantic tree is solid). It's a process of cognitive dissonance where you actively wrestle with problems rather than passively work through them.People who try to understand math by reading alone (or by watching videos) tend to fail in real life -- they tend to be able to recite definitions but their ability to execute on their knowledge is weak.This is a standard rookie mistake, and the reason why so many American kids are weaker at math compared to their Asian counterparts. Drilling--even if mindless at frst--really does help, especially when you're starting out on a new subject. It helps you develop muscle memory which in turn gives you confidence to move to the next level. I could say I'm in the same boat. Always wanted to learn such things, but never found the motivation to do so in an effective way.I picked up various books and different learning strategies along the years but couldn't move forward cause I could not see any practical use for what I was trying to learn.Fast forward a few years and now I'm learning both physics and mathematics.What changed is I started working with 3D development for the furniture industry and a while later I got interested in woodworking.Started doing some woodworking projects and had to learn some basic geometry and trigonometry to calculate cuts.Now I'm interested in mechanical machines and electrical machines. To be able to build my own machines I have to learn some physics and other branches of mathematics and that's what I have been doing for the past months.I probably cannot work with formal physics or mathematics but I was able to learn a lot of the concepts behind the formulas and calculations and I believe that is much more important, at least, at first.The bottom line is you need to find something that motivates you and make you want to learn. That's how it worked for me. Honestly, despite all the crap universities get, taking an undergraduate degree with a double major in physics and maths is an awesome way to do this. You'll meet people who are similarly passionate, be naturally competitive with them which is a motivating force not to be underestimated, and you'll meet a diverse set of teachers who each will have some awesome insights into these fields and you'll get to see first-hand how they think about solving problems.Physics, and to a lesser extent maths[1], are topics where the top 1% of ability are actually concentrated at universities. My advice would be find a cheap university nearby and start enrolling in courses. If you're bright, motivated and take ownership of your own learning, the faculty will love interacting with you. If you're doing it to learn, don't sweat about the prestige of the place. There are people everywhere who will be much better than you at this stuff, and in some ways it's extremely motivating if you feel like with some hard work you can surpass some of your teachers, and it's extremely motivating when the best teachers recognize you as having more potential than the average student. You're never gonna feel either of these things at MIT.[1] The problem with maths in academia is that it's massively biased toward proofs of mathematics and not use of mathematics. I've met very few PhD mathematicians who are even as close to as good at applying appropriate mathematics to problems then someone with a PhD in physics who consider themselves >50% theorist. PhD mathematicians are wonderfully knowledgeable if you say "tell me about this field of mathematics" and it's a field they know. But there is a certain extent to which they like to work by building things on a frictionless ice world, and get uncomfortable if asked to build something on the rough ground of the real world. Hi,I'm 30 and trying to relearn the math courses I did in college (Computer Science degree) and more. I am currently using Standford & MIT's open couseware. I feel like I am moving slower than I would if I were in a course but able to grasp the material better at this rate... I made good grades in my math courses but like you, I didn't have to use them in software engineering that much. I would like to get into a field that requires a stronger grasp of mathematics but also has a need for programming and computation (maybe machine learning or computational biology). I feel like I'm getting tired of being a software engineer (defense contractor) at a small company and looking for something higher levelCalculus (with a pdf version of the text book): https://ocw.mit.edu/resources/res-18-001-calculus-online-tex...Linear Algebra (text book link: https://www.amazon.com/exec/obidos/ASIN/0980232716/ref=as_at...) https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...Optimization course & book link (Stanford) https://web.stanford.edu/~boyd/cvxbook/Statistics: http://greenteapress.com/wp/think-stats-2e/ I'm taking you at your word and assuming you truly want to reach the cutting edge of knowledge and learn things like QFT, Gauge Theory, String Theory, etc. alongside the math needed for it.This is a long-term project, so I'd recommend by starting a bit with "learning about learning". There is a great, fairly short Coursera course called "learning how to learn".Things not covered in the above course: - Your learning ability is not actually much lower in your 30s than it was in your 20s. You have a relatively benign rate of learning decline, until your late 50s / early 60s, when it drops quite a bit. You can still learn a lot, but it's meaningfully harder. (learning rate != thinking rate / creation rate!!)You'll likely be able to write good papers into your late 60s, and perhaps 70s. There are exceptions and people who do significant work even later, but that's more unlikely.When I started learning stuff again in my late 20s, I felt frustrated because I'd take a couple courses over a year, and by the time the year's over, I'd forget the first one. We all know what this feels like - we've forgotten most of what we've learned in college that we don't use in our profession.When I was a teenager, I had great recall for things I've learned only once or twice. I didn't realize this was unusual and thought I got very bad at learning. In fact, the vast majority of people, including high IQ people, will need spaced repetition and study to retain things for a long period.I'd recommend the following "schedule" to absorb things into y memory permanently. (By "learn" I mean read, do problems, write summaries.. it's a wide range):Days to repeat: 0 (initial learning), 1, 6, 15, 37, 93, 234, 586, 1464, 3662, 9155This would suggest interleaving classes instead of learning things sequentially for optimal time management. It's also a bigger time investment than people usually think of upfront, but pays dividends later on as the material builds-up like a cathedral of knowledge.Like other commenters I'll also repeat: Do problems, problems, problems. The struggle is where the learning happens.On the other hand, I wouldn't worry too much about super-high IQ etc. I don't think it's a strict requirement to have an extraordinary IQ to learn grad school physics and math. Great post. Rather thought provoking. Though I dropped out of college in my late teens, I started taking classes again 17 years later later am doing much better than I did before. As a sysadmin, I was always reading all sorts of subjects and pursuing different hobbies that further expanded my knowledge. The one subject I have been having issues is with Math, but that is due to lack of effort and stretching myself too thin. That is a fascinating number series. Is it taught in the Coursera course you mentioned? Hi, I calculated it basing it on the super-memo algorithm. That algorithm is more sophisticated and geared towards cards / smaller pieces of knowledge; but I think it works equally well as "re-study" / "re-learn" reminders for larger chunks of material. The overall idea is to capture https://en.wikipedia.org/wiki/Forgetting_curve fairly well with the repetition frequency. Taste, you say? The Princeton Companion is your friend: https://press.princeton.edu/titles/8350.html https://press.princeton.edu/titles/10592.html I took a Masters of Mathematics with the Open University in my thirties.The (my) short answer is grind. Get a good textbook on subject of interest, start reading, start scribbling, start answering the questions. That's how I did it. Three or four days a week, two to six hours a day, grind grind grind GRIND GRIND GRIND GRIND. It's geology; time and pressure.Now and then, when really stuck, finding someone who can illuminate a point for you is worth it, but that helped me a lot, lot less than one might think.Anticipate not understanding large chunks of it. Anticipate pressing on anyway. Anticipate not being able to answer many questions. Anticipate having to read three or four different treatments of the same thing in order to get a real understanding. Anticipate that some of it you will never understand. Anticipate that watching youtube is not a substitute. Anticipate that the sheer information density of well-written text means you might spend an hour on a single page. Anticipate questions taking you six hours to solve, leaving your table and floor strewn with the history of your consciousness. If you're prepared for all that, and it's a price you're willing to pay, there is no reason to not simply start now. Pick up the first good textbook, start grinding now. Time and pressure. It's so easy to waste time preparing to start learning; beyond making sure it's a decent textbook and getting some pencils and paper in a quiet room, the only preparation is accepting that this is going to be a long grind, and embracing it. I'm doing my OU Masters in Maths now, in my 40s. It is definitely hard, but I'm enjoying it. Personally I struggle to learning things thoroughly unless I'm working in the subject, or I have exams to do. My own learning workflow is to flip through a chapter to get an overview, then read/re-read it thoroughly, then go through the exercises on the chapters quickly looking at the answers. Read the chapter again, then try doing the exercises without help. I've found YouTube pretty good for getting the intuition behind some ideas. Read some books, practice exercises, and find an area of interest.Start with some liberal-arts introduction to a particular topic of interest and delve in.I often find myself recommending Introduction to Graph Theory [0]. It is primarily aimed at liberal arts people who are math curious but may have been damaged or put off by the typical pedagogy of western mathematics. It will start you off by introducing some basic material and have you writing proofs in a simplistic style early on. I find the idea of convincing yourself it works is a better approach to teaching than to simply memorize formulas.Another thing to ask yourself is, what will I gain from this? Mathematics requires a sustained focus and long-term practice. Part of it is rote memorization. It helps to maintain your motivation if you have a reason, a driving reason, to continue this practice. Even if it's simply a love of mathematics itself.For me it was graphics at first... and today it's formal proofs and type theory.Mathematics is beautiful. I'm glad we have it.Update: I also recommend keeping a journal of your progress. It will be helpful to revisit later when you begin to forget older topics and will help you to create a system for keeping your knowledge fresh as you progress to more advanced topics. Here's how I would do it, my 2 cents:1) Find a good source of information --- typically, this is either very good lectures (like on youtube), a good textbook, or good lecture notes.2) Do problems. There is a fairly large gap between those that just watch the lectures and those that have sat down and try to go through each and every step of the logic, and that's what everyone here (on HN) is pointing out when they similarly mention doing problems.2b) Have solutions to those problems. I make this a separate point because it's important to spend quality time on a problem yourself before looking at the solutions. At the end of the day, if you read the problems and then the solution right away, that's much closer to reading the textbook itself instead of the more rigorous learning one goes through when trying things themselves.If you were to ask me what textbooks or lectures I recommend, I think that's a more personal question than many here might guess. What topics are you most interested in? Are you really just solely interested in a solid background? How patient are you when doing problems?Regardless, I'll give my two cents for textbooks anyway. In no particular order:1) Griffiths E&M: https://www.amazon.com/Introduction-Electrodynamics-David-J-...2) Axler, Linear Algebra Done Right: https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Ma...Good luck! First, you can't go back to your twenties and you shouldn't try.When you are still in a school environment, there's an environment that for doing problems for problem's sake. For all you have trained so far, be able to solve problems is how you were measured and feels like a life's purpose.By thirty you probably get the hint that life is not about solving fake problems, and most of the knowledge you learn at school is useless and pointless. If you try the suggestion to sit down and do exercises, I doubt you would be able to keep at it long enough for any gain.But at thirty, you have the luxury of not worry about midterms and finals and you probably can afford multiple books. So first, find your love of physics. Second, collect old text books. Third, read them. Fourth, criticize them. Fourth, throw away the book that you've digested or is bad. When you can throw away all the books (the knowledges are all online anyway), you are learned. I'm learning Japanese at 35 with the goal of becoming business-fluent in five years. I decided to be very systematic about it.Anki is a very good memorization tool, I would use it aggressively.I study for one hour every day before work. Math probably requires more time to grind through hard problems.I would hire a tutor or find a partner.A few things I'd recommend:write down why you're doing what you're doing.write down the gap in your abilities you'd like to fill so you can track your progressdon't compare yourself to others, compare yourself to yourself.It is very possible to develop engineering math chops late in life (I did it!) but outside of a fulltime education context it requires organization and sacrifice.edit: on the bright side, I'm so pleased with my current language learning pace that I intend to double down on it and make it a lifelong commitment. It really feels good to learn something new and interesting. I'm eyeballing self-learning an EE degree next, so I'm curious how it goes for you Oh Anki, how I love it. 2199 cards and counting Math is something that needs to be learned by rigorous practiceI find it helpful to first learn the theory via 3blue1brownhttps://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAwI'm currently working through the bookofproof math problems found herehttps://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472...It goes into how to actually write mathmatical proofs / discrete math which I believe is extremely important to any math branches and computer-science in general.I was lucky to have a great foundational math background in highschool, my calc 1/2 teacher was considered one of the best in my state. I practiced at least 30 problems every night in that class for 2 years If you are located in Los Angeles, I would recommend looking into the Michael Miller math class series taught thru UCLA extension. This is an 11 session 7-10pm math graduate level math course.I've taken courses on Algebraic Number Theory, Lie Groups and Lie Algebras, and Measure Theory. It's hard, but definitely doable. It is definitely more on the abstract math side, which I enjoy a lot. It is very different than the practical engineering-focused math I learned in school.One of the students has an introduction into what you can expect. http://boffosocko.com/2015/09/22/dr-michael-miller-math-clas... The epitome of what you want is to find a mentor, a chalkboard, and 3-4 hour chunks of time you can dedicate to learning. Repeat about 2x a week for a year, and do independent study with a book on one side of the table and a notepad on the other between classes. I enjoyed reading the No Bullshit Guide to Math and Physics: https://gumroad.com/l/noBSmath It's a brief overview but covers a lot of topics. The author's writing style and the book's layout worked well for me.The same person wrote a book for linear algebra as well but I have not read it: https://gumroad.com/l/noBSLA Hi atxhx, thx for the plug!Here is a link to an extended PDF previews of the books: https://minireference.com/static/excerpts/noBSguide_v5_previ... and https://minireference.com/static/excerpts/noBSguide2LA_previ... May I ask what your goal is? Do you want to be able to say read papers? Do research? It's easier to offer suggestions from there. Are you more interested in math or physics? I think at your stage, having a problem that interests you and then seeking out knowledge to understand that problem might be a good way to go...What is your current area of work? For example, if you are a programmer, then collaborating with a physicist or mathematician on a problem could be a worthwhile exchange. Regardless of whether you will actually put a maths degree to use on the job market—which no doubt you could, once you attain it—pursuing one can bring great rewards. I am in my thirties too (mid-thirties now) and recently took up maths again. Lucky for me, I come from the German-speaking realm, where there is a distance learning university that offers a solid BSc programme in mathematics at roughly one eighth of what someone would pay for tuition in the UK. I do not know what options you have in that regard, but if being enrolled in a programme does not seem off-putting to you, it might be worth checking out. I am mostly on my own, so I guess if I were studying entirely independently, I would not be doing much different. There are benefits though that I would not want to miss out on—I get my (bi-weekly) assignments reviewed, there are platforms where your teachers and fellow students are communicating, and, last but not least, there are exams that provide for the 'hard facts' as to whether you have been studying your stuff right. Not to mention the degree you are awarded if you succeed. To me it is beyond question that distance education is the right way for me to do this. Like you, I am working in business software development, and I simply cannot attend a brick and mortar university because I do not have the time, or would have to accept the severe cut-back in income resulting from a reduction of my work hours. (Besides, I have been to brick and mortar before, and not being very social, I do not think I am missing out on the social aspects of it at all.) So if there is any distance education option that suits your needs, it might greatly augment your self-directed learning. For motivation, if there is a nearby university, start attending the relevant departments' colloquia. They are generally open to all, and will start to get you up to date on what's new across all of math/physics.My favorite undergraduate students when I was TA'ing were all students who had returned to school after spending time in the world. They knew why they were there, knew that the material was worth learning, and asked lots of questions.Go get it -- start small, don't stop. This will expose you to cutting edge research going on. But I disagree entirely on this approach to learning fundamentals of math and physics.The colloquia are usually very subject specific. When I did my PhD in condensed matter physics, depending on the speaker, sometimes it could be 10 minutes into the lecture when it delves into narrow field-specific material I don’t understand (eg, a speaker talking about particle physics or astrophysics). Good speakers and even non-physicists can follow the whole talk regardless of subject matter. But good speakers are rare. And colloquia are typically for benefit of the department (students + faculty), so will quickly gloss over fundamentals into the real meat.So you might learn about super cool research happening, which is great. But you aren’t likely to learn key fundamentals. I went back to school in my late 20s for this. Ultimately what everyone says is true, you learn the stuff by doing problems and at the end of the day lectures are of marginal use and really the learning happens when it's you and the textbook(s).I personally went back to school because it was a way of putting pressure on myself. There's a lot of boring drills before it gets particularly interesting. The biggest value of school/coursework for me was the fact that it was a declaration of "I am going to do this now, and there are external motivators (grades, exams) to ensure that I stay on track."That said, if I had it to do over again, for the money I spent, I wonder if hiring a graduate students/postdocs or even professors as tutors would have been better. I suppose that even then, when the material gets boring or time consuming it's easy to walk away, whereas if you invest time and money into a class you basically are in good shape to shame yourself into succeeding. (plus being in the academic environment helps to get a better sense of the broader landscape of material)Maybe you could find a buddy to work with, like people do with the gym or whatever to shame each other into staying on task. Also, I should mention, one big lesson learned... Maths build on each other. Don't be embarrassed to start with a pre-calculus text/course.You probably don't remember much from high school, and most low-div calculus problems are some simple calculus rules combined with a bunch of high school algebra/arithmetic manipulation that if you don't have it all ready at your fingertips, you _will_struggle. (this means memorizing all those trig identities again) The hardest part of low-div calculus is this stuff... The actual rules of calculus are simple, easy to grasp and dare I say it, intuitive.... if you can do the pre-calc. What works for me is purchasing and reading textbooks (look for online college syllabuses for good ones). Probably the best way to read maths texts is to work the problems, but what I do is read it through once or twice. Then switch to a different text on the same subject. Things will slowly start to click, although of course you don't understand something until you can explain it (i.e. write a blog post about it) to someone else. > purchasing and reading textbooksPlus, you can get really, REALLY good deals on used college textbooks (some of which are still in pristine - as in never even been opened - condition). https://www.3blue1brown.com has you covered for linear algebra. Now as far as the math needed for physics, I highly suggest Roger Penrose's 'Road to Reality' (https://www.amazon.com/Road-Reality-Complete-Guide-Universe/...). As the reviews say it's not an easy read but what it does provide you with is all the mathematics you're going to need to learn to understand today's physics. The book provides a high-level overview of the mathematics - which is technically complete but so concise that it's difficult to learn from. So use that to take a deeper dive into a mathematical subject. What the book is really providing is a roadmap: you need to understand these concepts from these mathematical disciplines to understand this area of physics and then proceeds with the high-level description of those concepts. Take the deep dive as needed and you'll be amply rewarded. Covered for linear algebra? Yes those videos have some nice visuals but the material is just scratching the surface. A lot of resources on the awesome github: https://github.com/rossant/awesome-math If you want to be serious about math you should get a feel for what mathematics means to mathematicians.A common view is that mathematicians prove theorems. While true, a skill that is not emphasized enough is learning (by heart) and understanding definitions. Learning by heart here also means something slightly different than simply being able to recite definitions and theorems. You have to be able to compare the objects that you define and get a feel for how a definition is really a manipulation of a basic intuition.A great book to start with is Rudin's Principles of mathematical analysis. You will get a much deeper appreciation of what calculus is. If you want any chance of understanding the mathematical tools used in theoretical physics (operators, Hilbert spaces, Fourier decompositions to get solutions to differential equations etc.) you will have to understand analysis. I'm in a similar situation myself. My plan is to buy a high school textbook and work my way through it, chapter by chapter. I assume that the selection of topics in such a curriculum is reasonable and if the presentation is deficient I'll supplement with YouTube etc until I understand. Same here. I've been eyeing few books on Amazon myself. Most come with answer sheet at the back to check your work.It's all about doing it like we did in high school. Pen, graph paper, and maybe a calculator. Drilling down the practice will help with theory. You want personal sessions with a mathematics professor to help plan your curriculum and direct your learning! Here's where you can make that happen:Check out https://edeeu.education/. The founder, Alex Coward, is an ex-Berkeley math professor. To apply to have him as your director of studies, fill out the form at the bottom of https://edeeu.education/alexander-coward. This is affordable (currently$330/month).I have the same aims as you, also learning math/physics in my 30s, and it's been a huge boost to move from purely self-directed online learning to having sessions with a math professor and a planned curriculum.
You might also find https://physicstravelguide.com/ useful. It's an expository wiki that sorts physics and related math explanations/books etc. by the required level of sophistication.This means, especially as a beginner when you are stuck you can easily find an explanation that you can understand.I would recommend that you start with physics and only learn math on a "just-in-time learning" basis. Mathematics is infinitely large and it's too easy to get lost. Physics, in comparison, is relatively constrained. When you learn the basics of modern physics you will automatically get a good understanding of lots of important math topics plus you will always automatically know why they are important.
I did the same as you, but in my 40’s. It’s better to start with books that are “too easy” than those that are “too hard” (for your level, whatever that is). I started with Ken stroud’s ‘engineering math’, then did calculus 1,2,3 and linear algebra at community college (online, with proctored exams), then used chartrand’s “mathematical proofs” to learn proofs, which you must know in order to learn, understand and enjoy upper div math (and Physics). I’m now doing an MS in math and stats and loving it, but you need to pace yourself in a way that you can understand and enjoy the math and physics you are doing - and as many people have commented, that means doing lots of problems . Feel free to message me if you want to chat
Take refresher classes at community college. This is what I did. I did all the calculus and linear algebra classes on offer. For me this was very valuable. My goal was to be able to read mathematics in research papers.
A phenomenal resource for physics is the text for the Feynman lectures. And the great part - they're all online, excellently organized, and free: http://www.feynmanlectures.caltech.edu/ The lectures alone will not provide mastery, but they will provide a very solid foundation for understanding the breadth of most of all of physics. The one thing those lectures are desperately missing is a similarly well organized and presented problem set.
Was a physics student. IMO the best book you can drill questions from is Boas' Mathematical Methods in the Physical Sciences. It offers fairly succinct yet comprehensive overviews of various fields of math. I still have my copy sitting at home. It is considered an essential textbook for any physics student.
I am not sure where you got the impression that it's essential for any physics student, but I have never used it and don't think it's mattered much in my upbringing as a physicist.
Buy a book and take a course. It won’t be enjoyable but being examined on what you learn will provide focus. Find a small group of people doing the same or similar learning, so you can discuss the different problems each of you will have. Persevere and plan carefully. Otherwise you’ll waste time and energy. Don’t get sidetracked by great presentations and new tech from internet-based resources; remember what the objective is...
Why? What purpose do you have for textbook physics? Think of what you're going to do with that knowledge. Are you going to become a physics teacher? Do you just want to impress "a bunch of folks on the internet"?The time in your life for grinding on textbook knowledge is over. That was age 0 to {insert age at end of college/academic career} non-inclusive.Think very carefully where you want to spend your motivation and discipline.
Understanding the "Why" important. However, it seems clear that he/she has already answered that question, and now they are trying to figure out the "how". Knowledge is power no matter where it comes from - a textbook, the internet, a master or simply studying the natural world. Seems odd to discourage someone from expanding their understanding of the world.
I fell in love with physics by reading this guy: Paul G. Hewitt https://www.amazon.com/Conceptual-Physics-High-School-Progra...
This is a bit of an evergreen so just searching HN will net you piles of threads with lots of advice and references to resources.
Pick a related topic that makes use of what you want to learn, and learn what you want to learn as side effect from practice in the related topic. Another poster already mentioned linear algebra for simple computer graphics.From there you could branch out into more dynamic stuff, like realtime 3D rendering or particle simulations, where you'd need calculus.
Getting the right material to study is only small part of it. The real difficult part is finding all the time it takes and also finding company that has similar interest and is willing to invest time seriously with you. I struggled with the latter part. It is very frustrating to not be able to ask someone if what I am doing is right or not.
You might find my site interesting. It attempts to cover basic algebra in a more formal, proof oriented style. I designed it with adults revisiting mathematics and wanting to move on to higher mathematics in mind.
I've posted a full undergraduate curriculum based on free resources here:https://edeeu.education/undergraduate-mathematics-curriculumI'm also accepting students. See the website for how that works. It's a lot of fun!
If you don't have the patience for doing random math problems and just want to understand things abstractly, try this channel:
But how do I learn math/physics in 2018?Jokes aside, try https://www.coursera.org/ or https://www.khanacademy.org/
I learned Calculus myself through a book called "Calculus - an intuitive and physical approach", by Morris Kline. It is cheap and very didatic.I never really studied physics, but I found the first books from the "Feynman lectures on Physics" to be very good.
I forgot to mention that the Feynman lectures on Physics are available online for free: http://www.feynmanlectures.caltech.edu/
Pick a problem that interests you and involves lots of physics and math. This worked for me.
I've had the same thoughts about (re-)learning some math. I ran across this a while back, anyone know if these are good?
You should go here: https://betterexplained.com/ I found this site explains basic math concepts intuitively and very easy to understand.
Youtube, EDX , brilliant.org and Khan academy are all good resources. I'm taking hybrid online Math classes at my local community college; trying to get through all the Math requirements.
I am doing this now. I just took a Discrete Math class and I am taking a calc refresher in the Fall. I have been reading several text books as well for practice and reinforcement.
Start with simple books to warm up those grey cells. For maths, I recommend Mathematical Circles: Full of fun discrete math problems.
I would suggest getting text books with loads of homework problems with solutions and actually sit down to work through the problems.
YouTube is my preferred method of learning. It's easy and can quickly expose you to a variety of teaching styles.
get spivaks calculus. Took me months to get though chapter 1 :D, but gave me through understanding of how to think about maths and how to prove stuff and that proof are the real fun of math. If you can't prove it, you don't understand it.
I have no idea why someone would have down-voted you - Spivak is brilliant. It's not really about calculus, it's really about Real Analysis, and it's excellent.Highly recommended.
I think your goal to "learn math/physics" is too broad. Steven Hawkins was still trying to learn about physics before his death.Define what "understanding physics" means to you and then figure out how to get to your goal.
There are two approaches that work well. The first is to embark on the standard, formative curriculum. The second is to start with a handful of problems that interest you and go pick up stuff piecemeal on the way to solving those.Approach 1. Decide if you want to learn physics or applied mathematics. They're not the same. The math you describe is more on the operations research side of applied mathematics, and not terribly relevant to physics. You say your goals are physics. In that case you're in luck. The curriculum is utterly standard and consists of four passes through the material.The first pass is one or two years long and is roughly what's in Halliday and Resnick or Tipler's physics books: Newtonian mechanics, some wave motion, some thermodynamics and statistical mechanics, electromagnetism, and a little "modern physics" (special relativity and a bit of quantum theory). Meanwhile you study calculus of a single variable, multivariable and vector calculus, and a little bit of ordinary differential equations, and do a year of laboratories.The second pass is a semester of classical mechanics covering Lagrangian and Hamiltonian mechanics, a semester of statistical mechanics and thermodynamics, a year of electromagnetism, and a year of quantum mechanics, paired with a year of mathematical methods (linear algebra, special functions, curvilinear coordinates, a little tensor calculus, some linear partial differential equations, and a lot of Fourier analysis) and a year of more advanced laboratories. Here ends the undergraduate curriculum. At this point astrophysicists tend to separate off and start learning the knowledge for that domain instead of the second semesters of electromagnetism and quantum mechanics. Their labs are also different.The third pass is the first couple years of graduate school, and goes through the same subjects again in more depth. No labs this time. A mathematical methods course only if a student needs more help. Quantum field theory for those going that direction. General relativity for those going another. Advanced statistical mechanics or other special topics for those going into condensed matter.The fourth path is the student by themselves, integrating it all in preparation for doctoral qualifying exams.If this approach sounds like what you're after, start by getting a 1960's edition of Halliday and Resnick's physics (which is better than the present editions and quite cheap used), a Schaum's outline of calculus.Approach 2. Pick a handful of problems. What actually interests you? Not what sounds fascinating or what seems prestigious. What's interesting? Cloud shapes? Bird lifecycles? What are you actually curious enough about to spend some time poking at?
I'm a bit wary of some of the suggestions here. I'm a similar position to you (at least, when it comes to maths). The difference for me is that I'm fairly certain that I'm one of the least accomplished people responding to your post. Whereas most of the others responding appear to be of the same ilk as the people you idolise. I know I'm cut from a different piece of cloth as those people (and I don't mean that resentfully - I just want to be realistic about myself). So some of the suggestions may be perfect for them. For people like me (and I'm not suggesting that you are, but your circumstances sound similar to mine), something with more of a safety net may be more realistic. I know I don't have the fortitude to sit and persevere with a text book for hours on end. Sometimes, I need some of the groundwork to be laid down for me - at least, when it comes to things like maths and scientific ideas.I can't say I've found a successful way of learning and retaining maths knowledge over the past few years. That retaining bit is important - I can pick up something, give it a go and get it right, but if I don't do it again, I forget what I've learnt. I had some early success with the OU book series 'Countdown to Mathematics' (https://www.amazon.co.uk/Countdown-Mathematics-1-v/dp/020113...), but the problem ultimately is that it covers materials in a block format; once you've covered a topic and solved problems, you move on to the next topic. Khan Academy has the same issue and is arguably worse because, so far as I've found, it requires learners to know what they don't know, rather than structures a learning programme where each topic follows sequentially (maybe there is a way to do this but I've found the KA UX to be complicated and confusing).I've tried more 'sophisticated' maths learning solutions that claim to account for learners' knowledge and weaknesses, but there are various shortfalls with them and none is aimed at learners older than schoolchildren. It's not so much the provision of learning materials that I'm frustrated with but the process of testing and retaining what I learn. I would dearly, dearly love to find a tool that can assess what I know, tell me what to learn next, test me on the topic BUT continues to do so as I progress, making use of spaced repetition and interleaving. It seems like the perfect use of both techniques - like Anki - but so far as I've found, nobody has done this for maths learners. Anyway, I've digressed...
Here's how I go from dumb to less dumb:- on Saturday, I wanted to learn how to write a Kubernetes configuration file from scratch, so I decided I'd deploy a static web page- for the static web page, I decided it should return pictures of teapots w/ a 418 status code, and initially tried to return responses using netcat, which I got working on my local machine, but not in a container- instead of using netcat, I decided that nginx is for people that don't like over-engineering their weekend hack projects, so clearly I needed to write my own web server and hacked out a janky Elixir server that serves up a poop emoji teapot image [1][2]- then I started working on an overly over-engineered HTTP server, which so far only has date headers [3]- then last night I randomly wondered how HTTP 2 works, looked for the RFC [4]- then I remembered working on the date header, and I wondered how headers work in HTTP2, and I learned they use Huffman encoding, and so my next side tangent is to read up on Huffman encoding and add HTTP 2 header support to my HTTP server! [5]TL;DR -- I made a poop joke and turned it into a learning opportunity
This is awesomely overengineered and not-invented-here, and I mean that in the best possible way. Well done! | 2021-01-20 10:23:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3658488988876343, "perplexity": 821.2765652378096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519984.9/warc/CC-MAIN-20210120085204-20210120115204-00045.warc.gz"} |
http://math.eretrandre.org/tetrationforum/showthread.php?tid=3&pid=3890&mode=threaded | • 2 Vote(s) - 3 Average
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Andrew Robbins' Tetration Extension andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 09/04/2009, 06:42 AM (08/26/2009, 04:01 PM)tommy1729 Wrote: The idea is to arrive at the same superlogaritms by choosing an appropriate ( of possibly many ) solution for development points$x_0$ and $x_1$ . if that is possible for all the points on the (real) interval [$x_0$,$x_1$] then i assume THAT PARTICULAR SLOG has a radius at least $x_1 - x_0$ when developped at $x_0$. I have thought about this, and I have a discussion about this too. I'd like to write up the discussion here, but I'd rather just upload my notes. Mind you these notes are not final at all, but rather unsolicited ramblings of a curious mind. The section I'm talking about is draft/Robbins_paper3.pdf (pages 10-16), which is also what I was referring to here, when I said a "detailed investigation of the coefficients". I noticed that if you tabulate the coefficients of z (Taylor coefficients of the super-logarithm) with the coefficients of ln(a) in the numerator and denominator of each coefficient, then the diagonals of this tabulation make interesting patterns. So I started finding expressions for the diagonals of this table, and got stuck on the 5th diagonal. That is why I haven't really developed this idea much farther. If any of the expressions can be proven with induction or the like, then it would form the basis of a proof for the base-1 slog and base-$\infty$ slog (which only require the first row and the last row). I thought that maybe if we had expressions for all coefficients as functions of n, then perhaps we could find a closed form for $\text{slog}_e'(0)$ or something. Andrew Robbins Attached Files Robbins_final.zip (Size: 926.38 KB / Downloads: 279) Robbins_draft.zip (Size: 246.16 KB / Downloads: 293) « Next Oldest | Next Newest »
Messages In This Thread Andrew Robbins' Tetration Extension - by bo198214 - 08/07/2007, 04:38 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/18/2007, 08:20 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/19/2007, 09:50 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/20/2007, 02:22 PM RE: Andrew Robbins' Tetration Extension - by andydude - 11/12/2007, 08:43 AM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 06/26/2009, 10:51 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 06/27/2009, 09:39 AM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 06/28/2009, 12:08 AM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/06/2007, 04:17 AM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/06/2007, 04:27 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/06/2007, 10:57 AM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/06/2007, 01:58 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/06/2007, 03:58 PM RE: Andrew Robbins' Tetration Extension - by jaydfox - 11/12/2007, 09:14 AM RE: Andrew Robbins' Tetration Extension - by andydude - 11/12/2007, 09:56 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/12/2007, 08:05 PM RE: Andrew Robbins' Tetration Extension - by andydude - 11/13/2007, 12:16 AM RE: Andrew Robbins' Tetration Extension - by bo198214 - 11/13/2007, 10:21 AM RE: Andrew Robbins' Tetration Extension - by andydude - 11/13/2007, 05:45 PM RE: Andrew Robbins' Tetration Extension - by Gottfried - 03/17/2008, 07:52 AM RE: Andrew Robbins' Tetration Extension - by Gottfried - 03/17/2008, 06:09 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 06/29/2009, 08:20 PM RE: Andrew Robbins' Tetration Extension - by andydude - 07/27/2009, 08:10 AM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/11/2009, 12:18 PM RE: Andrew Robbins' Tetration Extension - by jaydfox - 08/11/2009, 07:06 PM RE: Andrew Robbins' Tetration Extension - by jaydfox - 08/11/2009, 07:12 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/23/2009, 02:45 PM RE: Andrew Robbins' Tetration Extension - by bo198214 - 08/23/2009, 03:23 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/26/2009, 04:01 PM RE: Andrew Robbins' Tetration Extension - by andydude - 09/04/2009, 06:42 AM RE: Andrew Robbins' Tetration Extension - by Gottfried - 12/28/2009, 05:21 PM RE: Andrew Robbins' Tetration Extension - by tommy1729 - 08/18/2016, 12:29 PM RE: Andrew Robbins' Tetration Extension - by Gottfried - 08/22/2016, 04:19 PM
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Users browsing this thread: 1 Guest(s) | 2020-01-28 06:30:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 8, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8116722702980042, "perplexity": 13492.479101931063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251776516.99/warc/CC-MAIN-20200128060946-20200128090946-00326.warc.gz"} |
http://vdumoulin.github.io/articles/timit-part-6 | Lately I’ve been working on enabling Pylearn2 to iterate over variable-length sequences. In this post, I’ll discuss my progress so far.
### The problem
Some types of models (such as convolutional or recurrent neural nets) naturally deal with variable-length inputs. Unfortunately, for the moment, this type of input is not well supported in Pylearn2: all Space subclasses expect the data to be a tensor whose first dimension is the batch axis and whose other dimensions are of fixed size. This means a sequence of fixed-sized elements cannot be stored in those spaces, because all time steps of the sequence would be considered as separate examples.
Even more fundamentally, there is no straightforward way to represent data structures containing variable-length elements in Theano. This means even if we solve the Space problem in Pylearn2, we’re limited to batches of size 1 unless some TypedList data structure is implemented in Theano.
### New spaces
I wrote two new Space subclasses (VectorSequenceSpace and IndexSequenceSpace) to deal with variable-length sequences. They’re very similar to the corresponding VectorSpace and IndexSpace, with few key differences:
• Because of Theano restrictions, an object in living in a *SequenceSpace is considered to represent a single example, unlike e.g. VectorSpace, which considers objects as batches of examples.
• A *SequenceSpace expects objects living in its space to be matrices whose first dimension is time and whose second dimension represent a fixed-sized state, e.g. a features vector.
• In order to enforce the fact that we’re dealing with a single example, it is impossible to convert a *SequenceSpace into a *Space. Doing otherwise would give rise to confusing behaviour: by going from a VectorSequenceSpace to a VectorSpace, suddenly every time step of the sequence is considered as a separate example. The only conversion allowed is from an IndexSequenceSpace to a VectorSequenceSpace.
• Some methods such as get_total_dimension() don’t make sense when dealing with variable-length sequences and are not implemented.
### New TIMIT wrapper
I also wrote a new TIMIT wrapper called TIMITSequences, which uses VectorSequenceSpace and IndexSequenceSpace to represent its data. Iterating over this dataset returns whole sequences. These sequences are segmented in frames of frame_length and form matrices whose first dimension is time and whose second dimension is what a sliding window of this length sees as it’s passing through the sequence.
As a proof-of-concept, I also wrote a toy RNN model (which you can find here) to train on this dataset. I haven’t had time to play with it a lot, but I hope to find time to do so this week and next week and present some results in another blog post. | 2017-04-25 12:22:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5218762159347534, "perplexity": 1187.986479729544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120349.46/warc/CC-MAIN-20170423031200-00183-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://biolearnr.blogspot.com/2017/05/project-organisation.html | ## Tuesday, 16 May 2017
### Project organisation
How do you organise your projects? I'm sure no-one but me cares how I organise mine. Here, I'm trying to put down my rules and my reasons for them, so that when I start breaking my rules I can send myself to the corner of naughty bioinformaticians. Have a look at this if you want the same kind of info, but from someone with a lot more clout.
There are a few motivating factors behind my choice of approach:
1) It should be quick to set up a new project;
2) Each project should have a consistent structure;
3) A build tool should be used to track dependencies amongst code / data / results, so that updates to a given file trickle down to any dependent files (without forcing nondependent files to be remade)
[hence a Snakefile for use in snakemake is used at the head of each project];
4) I should be able to git-clone one of my projects from my personal bitbucket ...
[all configs, docs, scripts should be present and under version control];
5) ... and be able to setup and run it on most linux boxes (given the availability of the input data and modulo some configurations, eg, to link to that data) ...
[filepaths etc. should be hardware-independent within the project, non-standard shells and the like should be avoided];
6) ... and generate all results / reports for that project automatically ...
[related: any results quoted in reports should be dynamically updateable];
7) ... and it should only take a few lines of command line code to go from step [4] to step [6];
8) Each project should run in a self-specified, version-explicit, environment
[so we use conda and related projects anaconda & bioconda to manage all external libraries / packages / programs that are used];
9) If multiple projects need to use a particular script / library that I've written, each project should keep an internal copy of that script
[so that project-generic code can be updated without breaking a working project];
10) Big projects should be allowed to contain loosely-coupled subprojects to keep everything neat
[but the subprojects should respect the principles above];
[TODO: add in what you forgot: projects shouldn't tread on each other's toes; the value of packaging; whither exploratory analysis; the volatility of external data; how to limit code duplication; issues with data used by multiple projects?]
A typical project template looks like this:
<JOBNAME>/
- conf/
- check_these_dirs.txt
- copy_these_files.txt
- job_specific_vars.sh
- make_these_subdirs.txt
- touch_these_files.txt
- <JOB_SPECIFIC_CONFIG_FILES>
- data/
- ext/
- int/
- job/
- doc/
- figure/
- notebook.[Rmd|lyx]
- <various_throwaway_ideas>.ipynb
- lib/
- <PKGNAME>/ # R package DESCRIPTION/R/tests etc
- <PKGNAME>.tar.gz
- Makefile
- conf/
- include_into_rpackage.txt
- global_rfuncs/
- R/
- <COPIES_OF_MULTIPROJECT_RFILES>
- tests/testthat/
- <... AND TESTS>
- local_rfuncs/
- R/
- <PROJECT_SPECIFIC_RFILES>
- tests/testthat/
- <... AND TESTS>
- setup.DESCRIPTION.R
- requirements.txt # OR environment.yaml
- results/
- scripts/
- check_env.sh
- setup_dirs.sh
- setup_libs.sh
- setup.sh
- <VARIOUS_SCRIPTS>
- Snakefile
- <OPTIONAL:subprojects/>
- temp/
- TODO.txt
Initialisation:
The default contents of ./conf/ and ./scripts are added by a script new_project.sh
The script setup.sh defines all the rest of the project-specific directory structure and builds a project-specific R package based on the contents of the files in ./conf/ and the scripts setup_dirs.sh and setup_libs.sh. I modify the config scripts/files to specify the names of the job, the conda-environment in which the job is to run, and the optional job-specific R package and I specify other things, like whether an R kernel is required, which files should be copied/linked/touched before running the project.
Environment set up:
While adding code to a project you'll need to import external libraries / programs. For the sake of reproducibility you should keep track of the versions of the resources you are using. The easiest, and most portable, way to do this is by using conda to install any external dependencies of your code. I make a new environment for each main project that I'm working on
(conda create --name some_project)
and use
(conda install -c <repos> <package>)
to add extra stuff to it. Having done this, always output a requirements.txt file (or environment.yaml, see here) containing the currently-installed conda stuff for the project
(conda list --explicit --name some_project > requirements.txt)
and keep this under version control. Then if you need to run the same project on a different computer, you should be able to create an identical environment using
(conda create --name some_project --file-spec requirements.txt).
Code:
Code for the project gets put into either scripts/ or lib/.
./scripts mostly contains python / shell / R scripts and snakemake-recipe-scripts that are either imported into, or called as subworkflows, by the main project Snakefile.
./lib contains any R function files, class definitions or pipelines that are used in the project. Some of these R files are used by several projects - these are copied (not linked) from a separate repository into ./lib/global_rfuncs: I have to ensure that in the future, any given project will still run - if the global R files for a project were linked to the repository versions, any update to the repository versions may break the future compilation of this project. R functions (etc) that are written specifically for this project are put into ./lib/local_rfuncs. I use a makefile to build and install the project-specific R package. Having an R package for each project means the R-scripts I put in ./scripts are more lightweight than if they contained loads of function definitions.
I don't need an equivalent package / library for python functions because I write so few python scripts and the script vs function isolation provided by if __name__ == '__main__' means I can easily use functions from one python script inside another.
Unlike in the Noble paper above, I don't put separate src/ and bin/ directories in projects because I don't compile jack.
Data:
There are two main ways to pass project-specific data into a project: via filepaths in ./conf or in ./data. Minor routes exist: through hard-coding information into project-specific scripts, notebooks or the R package for the project, or (for non-project-specific data) by importing bioconductor packages.
If I want to be able to modify a datafile by hand without hard-coding the info in a script/library, the file has to go in ./conf - not in ./data. For example, I recently did a meta-analysis of a lot of GEO/ArrayExpress datasets - having checked the relevant papers/abstracts, not all of the datasets identified by my automated searches were of value to the meta-analysis, I encoded my inclusion/exclusion criteria for these datasets in a yaml file in ./conf. (because the info may have been used by multiple docs/scripts, it had to be somewhere other than ./scripts or ./lib).
All datasets generated by the current project (and most data used by the current project) is accessed via ./data, although ./data may contain links to files/dirs elsewhere on the file system. This means I can use relative subdir filepaths throughout the project, and indeed, I never use full paths within the scripts/docs/libs for the project (except during setup.sh if a full-path is mentioned in one of the config files). That is, any script/makefile/notebook used in a project uses filepaths relative to the main working directory for the project (ie, the dir where Snakefile is sitting) and none of them use filepaths that ../../../..... off into the wilderness (nonetheless, I do allow subproject Snakefiles to have dependencies on other parallel subprojects). The only way to use files that are external to the current working directory and it's subdirs, is to either copy them into one of the project directories or link to them from ./data/ext (for datasets downloaded from elsewhere) or ./data/int (for datasets generated by our lab, or for results generated by another one of my projects). Any datasets generated during running a project are put into ./data/job - so a project shouldn't write to ./data/int, ./data/ext or ./conf.
Docs:
I used to write project notebooks in lyx (with knitr and the like) but I've started writing it all in Rmarkdown and might move over to bookdown eventually (cos it looks like I can stitch several subproject notebooks together more easily in bookdown, but I (or someone) need to put bookdown on anaconda first). Reasons for switching: Rmarkdown and .Rmd files seem better at these things (some may be due to my own ignorance however):
Including dynamic results within paragraphs;
Automated building of pdfs (and other formats);
Working directory bewilderments
Reproducibility and portability (lyx is a bit bloated for conda)
Running the whole damn thing:
- Clone the project
- Move into the project's main directory
- # Possibly modify config files in ./conf
- Create conda env
- Activate conda env
- Run ./scripts/setup.sh to setup all dir structure and data links and to build/install the R package into the conda envcompilation
- Run snakemake
Now I change my mind and rewrite the whole thing... | 2022-12-04 04:26:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43319782614707947, "perplexity": 5037.348866013617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710962.65/warc/CC-MAIN-20221204040114-20221204070114-00028.warc.gz"} |
https://solvedlib.com/in-a-certain-mens-track-and-field-event-the,272717 | In a certain men's track and field event, the shotput has a mass of 7.25 kg...
Question:
In a certain men's track and field event, the shotput has a mass of 7.25 kg and is released eastward with a speed of 17.2 m/s at 41.0
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https://tex.stackexchange.com/questions/124465/prevent-latex-from-putting-big-figure-alone-on-the-page?noredirect=1 | # Prevent latex from putting big figure alone on the page [duplicate]
I have a figure which occupies around 75% of the page and Latex keeps putting this figure alone on a page (in the middle of the page). What I want is to have some of the lines of the following paragraph right after the figure, even if there is space for only 4 or 5 lines. I tried with \nopagebreak and minipage, but nothing seems to work. Thanks!
EDIT: I fixed it by setting the [ht] parameters for the figure.
• You can do \begin{figure*} which will put the figure where LaTeX see fits and text should follow above or below it as much as possible. – dustin Jul 17 '13 at 19:37
• nope, not working – Vlad Jul 17 '13 at 19:39
• Can you post a picture of what it looks on the page? My guess would be with the margins and the size there is no feasible room to have any text. You could adjust the margins, check the bounding box of the figure, or scale down the figure. – dustin Jul 17 '13 at 19:43
• @dustin that is not what figure* does. – David Carlisle Jul 17 '13 at 19:47
• LaTeX does not allow floats on test pages to exceed a specified fraction of the page. see tex.stackexchange.com/questions/39017/… for the parameters you can set. – David Carlisle Jul 17 '13 at 19:49 | 2021-01-20 08:02:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8263981342315674, "perplexity": 773.7685979848122}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519923.26/warc/CC-MAIN-20210120054203-20210120084203-00359.warc.gz"} |
https://starkware.co/starkex/docs/escapeverifier.sol.html | Escape Verifier¶
contract EscapeVerifier is IFactRegistry
function verifyEscape(uint256[] calldata escapeProof)
external
Verifies that the contents of a vault belong to a certain Merkle commitment (root).
The Merkle commitment uses the Pedersen hash variation described next:
• Hash constants: A sequence $$p_i$$ of 504 points on an elliptic curve and an additional $$ec_{shift}$$ point
• Input: A vector of 504 bits $$b_i$$
• Output: The 252 bits x coordinate of $$(ec_{shift} + \sum_i b_i*p_i)$$
The following table describes the expected escapeProof format. Note that unlike a standard Merkle proof, the escapeProof contains both the nodes along the Merkle path and their siblings. The proof ends with the expected root and the ID of the vault for which the proof is submitted (which implies the location of the nodes within the Merkle tree).
starkKey (252) tokenId (252) zeros (8) hash(starkKey, tokenId) (252) quantizedAmount (252) zeros (8) left_node_0 (252) right_node_0 (252) zeros (8) … left_node_n (252) right_node_n (252) zeros (8) root (252) zeros (4) vaultId (248) zeros (8)
If the proof is accepted, this is registered under the following claim hash that may later be queried for validity:
claimHash = keccak256(starkKey, tokenId, quantizedAmount, vaultRoot, treeHeight, vaultId)
For information about when this module is to be used, see Escapes.
function isValid(bytes32 hash)
external
view
returns (bool val) | 2020-09-25 09:40:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3636661171913147, "perplexity": 8552.356661734271}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400223922.43/warc/CC-MAIN-20200925084428-20200925114428-00010.warc.gz"} |
http://www.physicspages.com/2015/07/12/heat-capacity-and-specific-heat-capacity/ | # Heat capacity and specific heat capacity
Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.41.
The heat capacity ${C}$ of a sample of matter is the amount of heat that must be added to raise the temperature of the sample by 1 K. As a rule, the heat capacity depends on all the state quantities (pressure, volume and temperature) as well as on the method by which it is measured (is pressure or volume held constant). The general definition is
$\displaystyle C\equiv\frac{Q}{\Delta T} \ \ \ \ \ (1)$
where ${Q}$ is the heat added and ${\Delta T}$ is the resulting temperature change. It’s important to note that ${C}$ is defined by considering only the heat which flows into the substance (recall that heat is defined to be energy flow due to temperature difference only), so we can’t use the change in thermal energy ${\Delta U}$ on its own to calculate ${C}$; we must also know how much work ${W}$ is done on the substance in order to isolate ${Q}$:
$\displaystyle C=\frac{\Delta U-W}{\Delta T} \ \ \ \ \ (2)$
The total heat capacity of a sample therefore depends on how much matter is in the sample, so it’s not a property of a particular substance in general (such as melting point). We can, however, define the specific heat capacity, which is the heat capacity per unit mass:
$\displaystyle c\equiv\frac{C}{m} \ \ \ \ \ (3)$
[Note that ${c}$ here is not the speed of light! As I’ve been posting a lot of relativity and electrodynamics recently, this post requires a bit of a phase change in your way of thinking.]
The calorie is the amount of heat required to raise the temperature of 1 gram of water by 1 K. This fact suggests a method of measuring the heat capacity of some substances, at least in the temperature range of liquid water. For example, if we take a chunk of metal with a mass of 100 grams, we can drop it in boiling water long enough for its temperature to become ${100^{\circ}\mbox{ C}}$. We can then transfer it quickly (so as not to lose any heat in the transfer) to a styrofoam cup (styrofoam has a very low heat capacity itself so serves as a good thermal insulator) containing 250 g of water at ${20^{\circ}\mbox{ C}}$, and seal the cup with a styrofoam lid. We can wait a few minutes to let the metal come into thermal equilibrium with the water, and then measure the temperature of the water. Suppose this is measured as ${24^{\circ}\mbox{ C}}$.
The whole process involves heat transfer only, as the only transfer of energy that occurs is due to temperature differences between the metal and, first, the boiling water and, second, the water in the cup. Thus all temperature changes are due to ${Q}$ on its own.
The water in the cup absorbs heat ${Q}$ from the hot metal:
$\displaystyle Q$ $\displaystyle =$ $\displaystyle C_{w}\Delta T\ \ \ \ \ (4)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 250\mbox{ g}\times1\mbox{ cal g}^{-1}\mbox{K}^{-1}\times4\mbox{ K}\ \ \ \ \ (5)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 1000\mbox{ cal} \ \ \ \ \ (6)$
Therefore the heat lost by the metal is ${-1000\mbox{ cal}}$. The heat capacity ${C_{m}}$ of the metal is found by using ${\Delta T=24-100=-76\mbox{ K}}$:
$\displaystyle C_{m}=\frac{Q}{\Delta T}=\frac{-1000}{-76}=13.16\mbox{ cal K}^{-1} \ \ \ \ \ (7)$
The specific heat capacity of the metal is
$\displaystyle c_{m}=\frac{C_{m}}{m}=\frac{13.16}{100}=0.1316\mbox{ cal g}^{-1}\mbox{K}^{-1}=0.55\mbox{ J g}^{-1}\mbox{K}^{-1} \ \ \ \ \ (8)$
## 8 thoughts on “Heat capacity and specific heat capacity”
1. Pingback: Latent heat | Physics pages
2. Pingback: Heat equation | Physics pages | 2017-02-27 22:49:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 31, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7659906148910522, "perplexity": 206.1089098312113}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501173866.98/warc/CC-MAIN-20170219104613-00259-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://mathhelpforum.com/math-software/85297-help-me-plot-equation-matlab.html | # Thread: help me to plot this equation in Matlab....
1. ## help me to plot this equation in Matlab....
this is a differential equation...and taking the inverse laplace transform it became a step function....
this is the function...
y(t) = 1/2[(sin2(t-1)u(t-1)/2)+cos(2t)]
2. Originally Posted by thegeniusguy
this is a differential equation...and taking the inverse laplace transform it became a step function....
this is the function...
y(t) = 1/2[(sin2(t-1)u(t-1)/2)+cos(2t)]
> t=0:0.1:10;
> plot(t,y) --> | 2017-06-26 21:36:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8611545562744141, "perplexity": 3706.4877917768335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320865.14/warc/CC-MAIN-20170626203042-20170626223042-00444.warc.gz"} |
https://par.nsf.gov/biblio/10131283 | Quantifying thermal refugia connectivity by combining temperature modeling, distributed temperature sensing, and thermal infrared imaging
Abstract. Watershed-scale stream temperature models are often one-dimensional because they require fewer data and are more computationally efficient than two- or three-dimensional models. However, one-dimensional models assume completely mixed reaches and ignore small-scale spatial temperature variability, which may create temperature barriers or refugia for cold-water aquatic species. Fine spatial- and temporal-resolution stream temperature monitoring provides information to identify river features with increased thermal variability. We used distributed temperature sensing (DTS) to observe small-scale stream temperature variability, measured as a temperature range through space and time, within two 400 m reaches in summer 2015 in Nevada's East Walker and main stem Walker rivers. Thermal infrared (TIR) aerial imagery collected in summer 2012 quantified the spatial temperature variability throughout the Walker Basin. We coupled both types of high-resolution measured data with simulated stream temperatures to corroborate model results and estimate the spatial distribution of thermal refugia for Lahontan cutthroat trout and other cold-water species. Temperature model estimates were within the DTS-measured temperature ranges 21 % and 70 % of the time for the East Walker River and main stem Walker River, respectively, and within TIR-measured temperatures 17 %, 5 %, and 5 % of the time for the East Walker, West Walker, and main stem Walker rivers, respectively. DTS, TIR, and modeled more »
Authors:
; ;
Award ID(s):
Publication Date:
NSF-PAR ID:
10131283
Journal Name:
Hydrology and Earth System Sciences
Volume:
23
Issue:
7
Page Range or eLocation-ID:
2965 to 2982
ISSN:
1607-7938
National Science Foundation
##### More Like this
1. Managed aquifer recharge (MAR) is typically used to enhance the agricultural water supply but may also be promising to maintain summer streamflows and temperatures for cold-water fish. An existing aquifer model, water temperature data, and analysis of water administration were used to assess potential benefits of MAR to cold-water fisheries in Idaho’s Snake River. This highly-regulated river supports irrigated agriculture worth US $10 billion and recreational trout fisheries worth$100 million. The assessment focused on the Henry’s Fork Snake River, which receives groundwater from recharge incidental to irrigation and from MAR operations 8 km from the river, addressing (1) the quantity and timing of MAR-produced streamflow response, (2) the mechanism through which MAR increases streamflow, (3) whether groundwater inputs decrease the local stream temperature, and (4) the legal and administrative hurdles to using MAR for cold-water fisheries conservation in Idaho. The model estimated a long-term 4%–7% increase in summertime streamflow from annual MAR similar to that conducted in 2019. Water temperature observations confirmed that recharge increased streamflow via aquifer discharge rather than reduction in river losses to the aquifer. In addition, groundwater seeps created summer thermal refugia. Measured summer stream temperature at seeps was within the optimal temperature range formore »
2. Abstract. Landslides are the main source of sediment in most mountain ranges. Rivers then act as conveyor belts, evacuating landslide-derived sediment. Sediment dynamics are known to influence landscape evolution through interactions among landslide sediment delivery, fluvial transport and river incision into bedrock. Sediment delivery and its interaction with river incision therefore control the pace of landscape evolution and mediate relationships among tectonics, climate and erosion. Numerical landscape evolution models (LEMs) are well suited to study the interactions among these surface processes. They enable evaluation of a range of hypotheses at varying temporal and spatial scales. While many models have been used to study the dynamic interplay between tectonics, erosion and climate, the role of interactions between landslide-derived sediment and river incision has received much less attention. Here, we present HyLands, a hybrid landscape evolution model integrated within the TopoToolbox Landscape Evolution Model (TTLEM) framework. The hybrid nature of the model lies in its capacity to simulate both erosion and deposition at any place in the landscape due to fluvial bedrock incision, sediment transport, and rapid, stochastic mass wasting through landsliding. Fluvial sediment transport and bedrock incision are calculated using the recently developed Stream Power with Alluvium Conservation and Entrainment (SPACE)more »
3. We examined the patterns of propagule recruitment to assess the timescale and trajectory of succession and the possible roles of physical factors in controlling benthic community structure in a shallow High Arctic kelp bed in the Beaufort Sea, Alaska. Spatial differences in established epilithic assemblages were evaluated against static habitat attributes (depth, distance from river inputs) and environmental factors (temperature, salinity, current speed, underwater light) collected continuously over 2–6 years. Our measurements revealed that bottom waters remained below freezing (mean winter temperatures ∼−1.8°C) and saline (33–36) with negligible light levels for 8–9 months. In contrast, the summer open water period was characterized by variable salinities (22–36), higher temperatures (up to 8–9°C) and measurable irradiance (1–8 mol photons m –2 day –1 ). An inshore, near-river site experienced strong, acute, springtime drops in salinity to nearly 0 in some years. The epilithic community was dominated by foliose red algae (47–79%), prostrate kelps (2–19%), and crustose coralline algae (0–19%). Strong spatial distinctions among sites included a positive correlation between cover by crustose coralline algae and distance to river inputs, but we found no significant relationships between multi-year means of physical factors and functional groups. Low rates of colonization and the very slowmore »
4. Abstract. Ecohydrological models are powerful tools to quantify the effects that independent fluxes may have on catchment storage dynamics. Here, we adapted the tracer-aided ecohydrological model, EcH2O-iso, for cold regions with the explicit conceptualization of dynamic soil freeze–thaw processes. We tested the model at the data-rich Krycklan site in northern Sweden with multi-criterion calibration using discharge, stream isotopes and soil moisture in three nested catchments. We utilized the model's incorporation of ecohydrological partitioning to evaluate the effect of soil frost on evaporation and transpiration water ages, and thereby the age of source waters. The simulation of stream discharge, isotopes, and soil moisture variability captured the seasonal dynamics at all three stream sites and both soil sites, with notable reductions in discharge and soil moisture during the winter months due to the development of the frost front. Stream isotope simulations reproduced the response to the isotopically depleted pulse of spring snowmelt. The soil frost dynamics adequately captured the spatial differences in the freezing front throughout the winter period, despite no direct calibration of soil frost to measured soil temperature. The simulated soil frost indicated a maximum freeze depth of 0.25 m below forest vegetation. Water ages of evaporation and transpiration reflect themore »
5. Food availability is a primary factor limiting the abundance of wild populations, but quantifying it requires an understanding of when and where prey are vulnerable to predators. Salmonid fishes in streams are commonly thought to forage on drifting aquatic invertebrates during daylight hours. However, past studies also report benthic and nocturnal foraging despite the predominant view of salmonids as diurnal drift-feeding predators. We used instream videography to assess foraging mode and energy intake for stream-dwelling Yellowstone Cutthroat Trout Oncorhynchus clarkii bouvieri. We recorded the foraging behavior of wild fish with a waterproof video camera and estimated energy intake based on fish size, foraging rate, retention rate, and caloric values of prey. Fish captured prey primarily from the water column and surface, targeting drifting invertebrates during daytime hours; however, they also foraged from the stream benthos and during nighttime. Yellowstone Cutthroat Trout foraging rate was most strongly related to foraging location in the stream, diel period, and month. Energy intake was highest from daytime drift-foraging behavior and exceeded a modeled metabolic limit of food intake during October and November. Nocturnal and benthic foraging contributed the smallest proportion of total foraging attempts but was observed over all months of our study andmore » | 2022-12-07 20:10:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3234865963459015, "perplexity": 10842.659589638482}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00618.warc.gz"} |
https://hal.inria.fr/hal-01194682 | # Error bounds in stochastic-geometric normal approximation
Abstract : We provide normal approximation error bounds for sums of the form $\sum_x \xi_x$, indexed by the points $x$ of a Poisson process (not necessarily homogeneous) in the unit $d$-cube, with each term $\xi_x$ determined by the configuration of Poisson points near to $x$ in some sense. We consider geometric graphs and coverage processes as examples of our general results.
Keywords :
Document type :
Conference papers
Domain :
Cited literature [22 references]
https://hal.inria.fr/hal-01194682
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Monday, September 7, 2015 - 12:51:05 PM
Last modification on : Wednesday, May 10, 2017 - 5:41:04 PM
Long-term archiving on: : Tuesday, December 8, 2015 - 12:59:57 PM
### File
dmAI0104.pdf
Publisher files allowed on an open archive
### Citation
Mathew D. Penrose, Tom Rosoman. Error bounds in stochastic-geometric normal approximation. Fifth Colloquium on Mathematics and Computer Science, 2008, Kiel, Germany. pp.71-94, ⟨10.46298/dmtcs.3557⟩. ⟨hal-01194682⟩
Record views | 2022-08-10 05:10:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5098406076431274, "perplexity": 4923.639454792626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571147.84/warc/CC-MAIN-20220810040253-20220810070253-00750.warc.gz"} |
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# The Impact of Wind Scalings on Stellar Growth and the Baryon Cycle in Cosmological Simulations
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### Abstract
Many phenomenologically successful cosmological galaxy formation simulations employ kinetic winds to model galactic outflows, a crucial ingredient in obtaining predictions that agree with various observations. Yet systematic studies of how variations in kinetic wind scalings might alter observable galaxy properties are rare. Here we employ GADGET-3 simulations to study how the baryon cycle, stellar mass function, and other galaxy and CGM predictions vary as a function of the assumed outflow speed $$v_w$$ and the scaling of the mass loading factor $$\eta$$ with velocity dispersion $$\sigma$$. We design our fiducial model to reproduce the measured wind properties at 25% of the virial radius from the Feedback In Realistic Environments (FIRE) simulations. We find that a strong dependence of $$\eta \sim \sigma^5$$ in low mass haloes with $$\sigma < 106\ \mathrm{km\ s^{-1}}$$ is required to match the faint end of the stellar mass functions at $$z > 1$$. The wind speed also has a major impact, with faster winds significantly reducing wind recycling and heating more halo gas. Both effects result in less stellar mass growth in massive haloes and impact high ionization absorption in halo gas. We cannot simultaneously match the stellar content at $$z=2$$ and $$z=0$$ within a single model, suggesting that an additional feedback source such as AGN might be required in massive galaxies at lower redshifts, but the amount needed depends strongly on assumptions regarding the outflow properties. We run a 50 $$\mathrm{Mpc/h}$$, $$2\times576^3$$ simulation with our fiducial parameters and show that it matches a range of star-forming galaxy properties at $$z\sim0-2$$. In closing, the results from simulations of galaxy formation are much more sensitive to small changes in the feedback implementation than to the hydrodynamic technique.
### Author and article information
###### Journal
17 October 2019
###### Article
1910.08199
http://arxiv.org/licenses/nonexclusive-distrib/1.0/
###### Custom metadata
31 pages, 19 figures, submitted to MNRAS
astro-ph.GA
Galaxy astrophysics | 2020-07-14 12:51:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5558586120605469, "perplexity": 2615.1874572555485}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655880665.3/warc/CC-MAIN-20200714114524-20200714144524-00346.warc.gz"} |
http://mathhelpforum.com/calculus/39907-quantum-mechanics.html | # Math Help - Quantum Mechanics
1. ## Quantum Mechanics
Hi, can someone let me know what kind of maths I need to learn to be able to understand and solve the Schrodinger Equation for a hydrogen atom?
According to my textbook the Schrodinger Equation tells you the principal quantum number, the azimuthal quantum number and the magnetic quantum number, can someone let me know HOW these are derived from the solution?
Lastly, can I determine the shapes of the s, p, d and f orbitals of a hydrogen atom by solving the Schrodinger Equation?
(I posted this on the physics help forum but decided it has more to do with maths, how do I delete my other post?
2. The solution to the Hydrogen Schrodinger Equation is rigorous but is managed by separation of variables:
$\psi(r, \theta, \phi) = R(r)P(\theta)F(\phi)$
Where
$R(r) = n = \text{Principal Quantum Number}$
$P(\theta) = l = \text{Orbital Quantum Number}$
$F(\phi) = m_l = \text{Magnetic Quantum Number}$
The separation leads to three separate equations. Solving these spatial equations gives the three quantum numbers.
Here's a picture:
And here's a website:
Hydrogen Schrodinger Equation
I think this is what you're looking for. | 2014-07-25 00:24:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7814421057701111, "perplexity": 313.1937896679476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997892557.70/warc/CC-MAIN-20140722025812-00239-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://gitlab.math.tu-dresden.de/spraetor/dune-amdis/-/blame/5d5e8c7d1409c0e8e2d65e08dc6a81f3a3f0768b/doc/doxygen/Doxylocal | Doxylocal 2.12 KB
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Praetorius, Simon committed Mar 13, 2016 1 2 3 ``````# This file contains local changes to the doxygen configuration # please us '+=' to add file/directories to the lists `````` Praetorius, Simon committed May 07, 2016 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 ``````FILE_PATTERNS += *.hpp \ *.cpp HIDE_SCOPE_NAMES = YES HIDE_UNDOC_CLASSES = NO INTERNAL_DOCS = NO MARKDOWN_SUPPORT = YES EXCLUDE_SYMBOLS = AMDiS::Impl \ AMDiS::traits::Impl \ AMDiS::detail \ itl::details PREDEFINED += HAVE_UMFPACK \ HAVE_ALBERTA \ HAVE_UG \ AMDIS_BACKEND_MTL `````` Praetorius, Simon committed Mar 13, 2016 22 23 24 25 26 ``````# The INPUT tag can be used to specify the files and/or directories that contain # documented source files. You may enter file names like "myfile.cpp" or # directories like "/usr/src/myproject". Separate the files or directories # with spaces. `````` Praetorius, Simon committed May 07, 2016 27 28 29 30 31 ``````INPUT += @top_srcdir@/dune/amdis \ @top_srcdir@/dune/amdis/common \ @top_srcdir@/dune/amdis/utility \ @top_srcdir@/dune/amdis/linear_algebra \ @top_srcdir@/dune/amdis/linear_algebra/mtl `````` Praetorius, Simon committed Mar 13, 2016 32 ``````# see e.g. dune-grid for the examples of mainpage and modules `````` Praetorius, Simon committed May 07, 2016 33 34 ``````#INPUT += @srcdir@/mainpage \ # @srcdir@/modules `````` Praetorius, Simon committed Mar 13, 2016 35 36 37 38 39 `````` # The EXCLUDE tag can be used to specify files and/or directories that should # excluded from the INPUT source files. This way you can easily exclude a # subdirectory from a directory tree whose root is specified with the INPUT tag. `````` Praetorius, Simon committed May 07, 2016 40 41 ``````EXCLUDE += @top_srcdir@/dune/amdis/test \ @top_srcdir@/dune/amdis/linear_algebra/istl `````` Praetorius, Simon committed Mar 13, 2016 42 43 44 45 46 `````` # The EXAMPLE_PATH tag can be used to specify one or more files or # directories that contain example code fragments that are included (see # the \include command). `````` Praetorius, Simon committed May 07, 2016 47 ``````EXAMPLE_PATH += @top_srcdir@/src `````` Praetorius, Simon committed Mar 13, 2016 48 49 50 51 52 53 `````` # The IMAGE_PATH tag can be used to specify one or more files or # directories that contain image that are included in the documentation (see # the \image command). # IMAGE_PATH += @top_srcdir@/dune/amdis/pics`````` | 2021-06-15 21:59:07 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8859568238258362, "perplexity": 6902.0181434296555}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487621627.41/warc/CC-MAIN-20210615211046-20210616001046-00104.warc.gz"} |
https://nhigham.com/2022/04/19/whats-new-in-matlab-r2022a/ | # What’s New in MATLAB R2022a?
In this post I discuss some of the new features in MATLAB R2022a, focusing on ones that relate to my particular interests. See the release notes for a detailed list of the many changes in MATLAB and its toolboxes. For my articles about new features in earlier releases, see here.
## Themes
MATLAB Online now has themes, including a dark theme (which is my preference). We will have to wait for a future release for themes to be supported on desktop MATLAB.
## Economy Factorizations
One can now write `qr(A,'econ')` instead of `qr(A,0)` and `gsvd(A,B,'econ')` instead of `gsvd(A,B)` for the “economy size” decompositions. This is useful as the `'econ'` form is more descriptive. The `svd` function already supported the `'econ'` argument. The economy-size QR factorization is sometimes called the thin QR factorization.
## Tie Breaking in the round Function
The `round` function, which rounds to the nearest integer, now breaks ties by rounding away from zero by default and has several other tie-breaking options (albeit not stochastic rounding). See a sequence of four blog posts on this topic by Cleve Moler starting with this one from February 2021.
## Tolerances for null and orth
The `null` (nullspace) and `orth` (orthonormal basis for the range) functions now accept a tolerance as a second argument, and any singular values less than that tolerance are treated as zero. The default tolerance is `max(size(A)) * eps(norm(A))`. This change brings the two functions into line with `rank`, which already accepted the tolerance. If you are working in double precision (the MATLAB default) and your matrix has inherent errors of order $10^{-8}$ (for example), you might set the tolerance to $10^{-8}$, since singular values smaller than this are indistinguishable from zero.
## Unit Testing Reports
The unit testing framework can now generate docx, html, and pdf reports after test execution, by using the function `generatePDFReport` in the latter case. This is useful for keeping a record of test results and for printing them. We use unit testing in Anymatrix and have now added an option to return the results in a variable so that the user can call one of these new functions.
## Checking Arrays for Special Values
Previously, if you wanted to check whether a matrix had all finite values you would need to use a construction such as `all(all(isfinite(A)))` or `all(isfinite(A),'all')`. The new `allfinite` function does this in one go: `allfinite(A)` returns true or false according as all the elements of `A` are finite or not, and it works for arrays of any dimension.
Similarly, `anynan` and `anymissing` check for NaNs or missing values. A missing value is a NaN for numerical arrays, but is indicated in other ways for other data types.
## Linear Algebra on Multidimensional Arrays
The new `pagemldivide`, `pagemrdivide`, and `pageinv` functions solve linear equations and calculate matrix inverses using pages of $d$-dimensional arrays, while `tensorprod` calculates tensor products (inner products, outer products, or a combination of the two) between two $d$-dimensional arrays.
## Animated GIFs
The append option of the `exportgraphics` function now supports the GIF format, enabling one to create animated GIFs (previously only multipage PDF files were supported). The key command is `exportgraphics(gca,file_name,"Append",true)`. There are other ways of creating animated GIFs in MATLAB, but this one is particularly easy. Here is an example M-file (based on cheb3plot in MATLAB Guide) with its output below.
```%CHEB_GIF Animated GIF of Chebyshev polynomials.
% Based on cheb3plot in MATLAB Guide.
x = linspace(-1,1,1500)';
p = 49
Y = ones(length(x),p);
Y(:,2) = x;
for k = 3:p
Y(:,k) = 2*x.*Y(:,k-1) - Y(:,k-2);
end
delete cheby_animated.gif
a = get(groot,'defaultAxesColorOrder'); m = length(a);
for j = 1:p-1 % length(k)
plot(x,Y(:,j),'LineWidth',1.5,'color',a(1+mod(j-1,m),:));
xlim([-1 1]), ylim([-1 1]) % Must freeze axes.
title(sprintf('%2.0f', j),'FontWeight','normal')
exportgraphics(gca,"cheby_animated.gif","Append",true)
end
``` | 2022-10-06 04:39:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4329860210418701, "perplexity": 1047.110174600693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337723.23/warc/CC-MAIN-20221006025949-20221006055949-00542.warc.gz"} |
http://www.nag.com/numeric/MB/manual64_24_1/html/E04/e04nsf.html | Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int
Chapter Contents
Chapter Introduction
NAG Toolbox
# NAG Toolbox: nag_opt_qpconvex2_sparse_option_string (e04ns)
## Purpose
nag_opt_qpconvex2_sparse_option_string (e04ns) may be used to supply individual optional parameters to nag_opt_qpconvex2_sparse_solve (e04nq). The initialization function nag_opt_qpconvex2_sparse_init (e04np) must have been called before calling nag_opt_qpconvex2_sparse_option_string (e04ns).
## Syntax
[cw, iw, rw, ifail] = e04ns(string, cw, iw, rw)
[cw, iw, rw, ifail] = nag_opt_qpconvex2_sparse_option_string(string, cw, iw, rw)
## Description
nag_opt_qpconvex2_sparse_option_string (e04ns) may be used to supply values for optional parameters to nag_opt_qpconvex2_sparse_solve (e04nq). It is only necessary to call nag_opt_qpconvex2_sparse_option_string (e04ns) for those parameters whose values are to be different from their default values. One call to nag_opt_qpconvex2_sparse_option_string (e04ns) sets one parameter value.
Each optional parameter is defined by a single character string, of up to 72$72$ characters, consisting of one or more items. The items associated with a given option must be separated by spaces, or equals signs [ = ]$\left[=\right]$. Alphabetic characters may be upper or lower case. The string
`Print Level = 1`
is an example of a string used to set an optional parameter. For each option the string contains one or more of the following items:
– a mandatory keyword; – a phrase that qualifies the keyword; – a number that specifies an integer or double value. Such numbers may be up to 16$16$ contiguous characters in Fortran's I, F, E or D formats, terminated by a space if this is not the last item on the line.
For nag_opt_qpconvex2_sparse_option_string (e04ns), each user-specified option is not normally printed as it is defined, but this printing may be turned on using the keyword List. Thus the statement
```[cw, iw, rw, ifail] = e04ns('List', cw, iw, rw);
```
turns on printing of this and subsequent options. Printing may be turned off again using the keyword Nolist.
Optional parameter settings are preserved following a call to nag_opt_qpconvex2_sparse_solve (e04nq) and so the keyword Defaults is provided to allow you to reset all the optional parameters to their default values before a subsequent call to nag_opt_qpconvex2_sparse_solve (e04nq).
A complete list of optional parameters, their abbreviations, synonyms and default values is given in Section [Optional Parameters] in (e04nq).
None.
## Parameters
### Compulsory Input Parameters
1: string – string
A single valid option string (see Section [Description] in nag_opt_qpconvex2_sparse_option_string (e04ns) and Section [Optional Parameters] in (e04nq)).
2: cw( : $:$) – cell array of strings
Note: the dimension of the array cw must be at least lencw${\mathbf{lencw}}$ (see nag_opt_qpconvex2_sparse_init (e04np)).
3: iw( : $:$) – int64int32nag_int array
Note: the dimension of the array iw must be at least leniw${\mathbf{leniw}}$ (see nag_opt_qpconvex2_sparse_init (e04np)).
4: rw( : $:$) – double array
Note: the dimension of the array rw must be at least lenrw${\mathbf{lenrw}}$ (see nag_opt_qpconvex2_sparse_init (e04np)).
None.
None.
### Output Parameters
1: cw( : $:$) – cell array of strings
Note: the dimension of the array cw must be at least lencw${\mathbf{lencw}}$ (see nag_opt_qpconvex2_sparse_init (e04np)).
2: iw( : $:$) – int64int32nag_int array
Note: the dimension of the array iw must be at least leniw${\mathbf{leniw}}$ (see nag_opt_qpconvex2_sparse_init (e04np)).
3: rw( : $:$) – double array
Note: the dimension of the array rw must be at least lenrw${\mathbf{lenrw}}$ (see nag_opt_qpconvex2_sparse_init (e04np)).
4: ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).
## Error Indicators and Warnings
Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
The initialization function nag_opt_qpconvex2_sparse_init (e04np) has not been called.
ifail = 2${\mathbf{ifail}}=2$
The supplied option is invalid. Check that the keywords are neither ambiguous nor misspelt.
## Accuracy
Not applicable.
nag_opt_qpconvex2_sparse_option_integer_set (e04nt) or nag_opt_qpconvex2_sparse_option_double_set (e04nu) may also be used to supply optional parameters to nag_opt_qpconvex2_sparse_solve (e04nq).
## Example
```function nag_opt_qpconvex2_sparse_option_string_example
string = 'Iterations limit 50';
[cw, iw, rw, ifail] = nag_opt_qpconvex2_sparse_init();
[cw, iw, rw, ifail] = nag_opt_qpconvex2_sparse_option_string(string, cw, iw, rw);
[ivalue, cw, iw, rw, ifail] = nag_opt_qpconvex2_sparse_option_integer_get(string, cw, iw, rw);
ivalue, ifail
```
```
ivalue =
50
ifail =
0
```
```function e04ns_example
string = 'Iterations limit 50';
[cw, iw, rw, ifail] = e04np;
[cw, iw, rw, ifail] = e04ns(string, cw, iw, rw);
[ivalue, cw, iw, rw, ifail] = e04nx(string, cw, iw, rw);
ivalue, ifail
```
```
ivalue =
50
ifail =
0
``` | 2015-09-04 16:34:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 18, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9146857261657715, "perplexity": 7743.435888465483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645356369.70/warc/CC-MAIN-20150827031556-00275-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mail.scipy.org/pipermail/numpy-discussion/2010-June/050968.html | # [Numpy-discussion] Tensor contraction
Friedrich Romstedt friedrichromstedt@gmail....
Sun Jun 13 14:37:50 CDT 2010
2010/6/13 Alan Bromborsky <abrombo@verizon.net>:
> I am writing symbolic tensor package for general relativity. In making
> symbolic tensors concrete
> I generate numpy arrays stuffed with sympy functions and symbols.
That sound's interesting.
> The
> operations are tensor product
> (numpy.multiply.outer), permutation of indices (swapaxes), partial and
> covariant (both vector operators that
> increase array dimensions by one) differentiation, and contraction.
I would like to know more precisely what this differentiations do, and
how it comes that they add an index to the tensor.
> I think I need to do the contraction last
> to make sure everything comes out correctly. Thus in many cases I would
> be performing multiple contractions
> on the tensor resulting from all the other operations.
Hm, ok, so I guess I shall give my 1 cent now.
Ok.
# First attempt (FYI, failed):
# The general procedure is, to extract a multi-dimensional diagonal array.
# The sum \sum_{ij = 0}^{M} \sum_{kl = 0}^{N} is actually the sum over a
# 2D array with indices I \equiv i \equiv j and K \equiv k \equiv
l. Meaning:
# \sum_{(I, K) = (0, 0)}^{(M, N)}.
# Thus, if we extract the indices with 2D arrays [[0], [1], ...,
[N - 1]] for I and
# [[0, 1, ..., M - 1]] on the other side for K, then numpy's broadcasting
# mechanism will broadcast them to the same shape, yielding (N, M) arrays.
# Then finally we sum over this X last dimensions when there were X
# contractions, and we're done.
# Hmmm, when looking closer at the problem, it seems that this isn't
# adequate. Because we would have to insert open slices, but cannot
# create them outside of the [] operator ...
# So now follows second attemt:
def contract(arr, *contractions):
"""*CONTRACTIONS is e.g.:
(0, 1), (2, 3)
meaning two contractions, one of 0 & 1, and one of 2 & 2,
but also:
(0, 1, 2),
is allowed, meaning contract 0 & 1 & 2."""
# First, we check if we can contract using the *contractions* given ...
for contraction in contractions:
# Extract the dimensions used.
dimensions = numpy.asarray(arr.shape)[list(contraction)]
# Check if they are all the same.
dimensionsdiff = dimensions - dimensions[0]
if numpy.abs(dimensionsdiff).sum() != 0:
raise ValueError('Contracted indices must be of same dimension.')
# So now, we can contract.
#
# First, pull the contracted dimensions all to the front ...
# The names of the indices.
names = range(arr.ndim)
# Pull all of the contractions.
names_pulled = []
for contraction in contractions:
names_pulled = names_pulled + list(contraction)
# Remove the pulled index names from the pool:
for used_index in contraction:
# Some more sanity check
if used_index not in names:
raise ValueError('Each index can only be used in one
contraction.')
names.remove(used_index)
# Concatenate the pulled indices and the left-over indices.
names_final = names_pulled + names
# Perform the swap.
arr = arr.transpose(names_final)
# Perform the contractions ...
for contraction in contractions:
# The respective indices are now, since we pulled them, the
frontmost indices:
ncontraction = len(contraction)
# The index array:
# shape[0] = shape[1] = ... = shape[ncontraction - 1]
I = numpy.arange(0, arr.shape[0])
# Perform contraction:
index = [I] * ncontraction
arr = arr[tuple(index)].sum(axis=0)
# If I made no mistake, we should be done now.
return arr
Ok, it didn't get much shorter than Pauli's solution, so you decide ...
> One question to
> ask would be considering that I am stuffing
> the arrays with symbolic objects and all the operations on the objects
> would be done using the sympy modules,
> would using numpy operations to perform the contractions really save any
> time over just doing the contraction in
> python code with a numpy array.
I don't know anything about sympy. I think there's some typo around:
I guess you mean creating some /sympy/ array and doing the operations
using that instead of using a numpy array having sympy dtype=object
content?
Friedrich
More information about the NumPy-Discussion mailing list | 2015-03-27 00:00:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.63665771484375, "perplexity": 6702.591547804741}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131293283.10/warc/CC-MAIN-20150323172133-00240-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://im.kendallhunt.com/HS/students/1/1/13/index.html | # Lesson 13
More Standard Deviation
• Let’s continue to interpret standard deviation.
### 13.1: Math Talk: Outlier Math
Evaluate mentally.
$$0.5 \boldcdot 30$$
$$1.5 \boldcdot 30$$
$$100- 1.5 \boldcdot 30$$
$$100-1.5 \boldcdot 18$$
### 13.2: Info Gap: African and Asian Elephants
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
1. Silently read the information on your card.
2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
4. Read the problem card, and solve the problem independently.
5. Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
1. Silently read your card and think about what information you need to answer the question.
2. Ask your partner for the specific information that you need.
3. Explain to your partner how you are using the information to solve the problem.
4. When you have enough information, share the problem card with your partner, and solve the problem independently.
5. Read the data card, and discuss your reasoning.
### 13.3: Interpreting Measures of Center and Variability
For each situation, you are given two graphs of data, a measure of center for each, and a measure of variability for each.
• Interpret the measure of center in terms of the situation.
• Interpret the measure of variability in terms of the situation.
• Compare the two data sets.
1. The heights of the 40 trees in each of two forests are collected.
2. The number of minutes it takes Lin and Noah to finish their tests in German class is collected for the year.
3. The number of raisins in a cereal with a name brand and the generic version of the same cereal are collected for several boxes.
One use of standard deviation is it gives a natural scale as to how far above or below the mean a data point is. This is incredibly useful for comparing points from two different distributions.
For example, they say you cannot compare apples and oranges, but here is a way. The average weight of a granny smith weighs 128 grams with a standard deviation of about 10 grams. The average weight of a navel orange is 140 grams with a standard deviation of about 14 grams. If we have a 148 gram granny smith apple and a 161 gram navel orange, we might wonder which is larger for its species even though they are both about 20 grams above their respective mean. We could say that the apple, which is 2 standard deviations above its mean, is larger for its species than the orange, which is only 1.5 standard deviations above its mean.
1. How many standard deviations above the mean height of a tree in forest A is its tallest tree?
2. How many standard deviations above the mean height of a tree in forest B is its tallest tree?
3. Which tree is taller in its forest?
### Summary
The more variation a distribution has, the greater the standard deviation. A more compact distribution will have a lesser standard deviation.
The first dot plot shows the number of points that a player on a basketball team made during each of 15 games. The second dot plot shows the number of points scored by another player during the same 15 games.
The data in the first plot has a mean of approximately 3.87 points and standard deviation of about 2.33 points. The data in the second plot has a mean of approximately 7.73 points and a standard deviation of approximately 4.67 points. The second distribution has greater variability than first distribution because the data is more spread out. This is shown in the standard deviation for the second distribution being greater than the standard deviation for the first distribution.
Standard deviation is calculated using the mean, so it makes sense to use it as a measure of variability when the mean is appropriate to use for the measure of center. In cases where the median is a more appropriate measure of center, the interquartile range is still a better measure of variability than standard deviation.
### Glossary Entries
• standard deviation
A measure of the variability, or spread, of a distribution, calculated by a method similar to the method for calculating the MAD (mean absolute deviation). The exact method is studied in more advanced courses.
• statistic
A quantity that is calculated from sample data, such as mean, median, or MAD (mean absolute deviation). | 2020-09-23 00:40:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5879424214363098, "perplexity": 703.0853127727468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400208095.31/warc/CC-MAIN-20200922224013-20200923014013-00787.warc.gz"} |
http://www.oalib.com/relative/970768 | Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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Sylvain Arlot Mathematics , 2007, Abstract: We present a new family of model selection algorithms based on the resampling heuristics. It can be used in several frameworks, do not require any knowledge about the unknown law of the data, and may be seen as a generalization of local Rademacher complexities and $V$-fold cross-validation. In the case example of least-square regression on histograms, we prove oracle inequalities, and that these algorithms are naturally adaptive to both the smoothness of the regression function and the variability of the noise level. Then, interpretating $V$-fold cross-validation in terms of penalization, we enlighten the question of choosing $V$. Finally, a simulation study illustrates the strength of resampling penalization algorithms against some classical ones, in particular with heteroscedastic data.
Sylvain Arlot Mathematics , 2009, DOI: 10.1214/08-EJS196 Abstract: In this paper, a new family of resampling-based penalization procedures for model selection is defined in a general framework. It generalizes several methods, including Efron's bootstrap penalization and the leave-one-out penalization recently proposed by Arlot (2008), to any exchangeable weighted bootstrap resampling scheme. In the heteroscedastic regression framework, assuming the models to have a particular structure, these resampling penalties are proved to satisfy a non-asymptotic oracle inequality with leading constant close to 1. In particular, they are asympotically optimal. Resampling penalties are used for defining an estimator adapting simultaneously to the smoothness of the regression function and to the heteroscedasticity of the noise. This is remarkable because resampling penalties are general-purpose devices, which have not been built specifically to handle heteroscedastic data. Hence, resampling penalties naturally adapt to heteroscedasticity. A simulation study shows that resampling penalties improve on V-fold cross-validation in terms of final prediction error, in particular when the signal-to-noise ratio is not large.
Mathematics , 2006, Abstract: We consider the problem of estimating the density $g$ of identically distributed variables $X\_i$, from a sample $Z\_1, ..., Z\_n$ where $Z\_i=X\_i+\sigma\epsilon\_i$, $i=1, ..., n$ and $\sigma \epsilon\_i$ is a noise independent of $X\_i$ with known density $\sigma^{-1}f\_\epsilon(./\sigma)$. We generalize adaptive estimators, constructed by a model selection procedure, described in Comte et al. (2005). We study numerically their properties in various contexts and we test their robustness. Comparisons are made with respect to deconvolution kernel estimators, misspecification of errors, dependency,... It appears that our estimation algorithm, based on a fast procedure, performs very well in all contexts.
Statistics , 2012, Abstract: Adaptive nuclear-norm penalization is proposed for low-rank matrix approximation, by which we develop a new reduced-rank estimation method for the general high-dimensional multivariate regression problems. The adaptive nuclear norm of a matrix is defined as the weighted sum of the singular values of the matrix. For example, the pre-specified weights may be some negative power of the singular values of the data matrix (or its projection in regression setting). The adaptive nuclear norm is generally non-convex under the natural restriction that the weight decreases with the singular value. However, we show that the proposed non-convex penalized regression method has a global optimal solution obtained from an adaptively soft-thresholded singular value decomposition. This new reduced-rank estimator is computationally efficient, has continuous solution path and possesses better bias-variance property than its classical counterpart. The rank consistency and prediction/estimation performance bounds of the proposed estimator are established under high-dimensional asymptotic regime. Simulation studies and an application in genetics demonstrate that the proposed estimator has superior performance to several existing methods. The adaptive nuclear-norm penalization can also serve as a building block to study a broad class of singular value penalties.
Statistics , 2013, Abstract: In the causal adjustment setting, variable selection techniques based on either the outcome or treatment allocation model can result in the omission of confounders or the inclusion of spurious variables in the propensity score. We propose a variable selection method based on a penalized likelihood which considers the response and treatment assignment models simultaneously. The proposed method facilitates confounder selection in high-dimensional settings. We show that under some conditions our method attains the oracle property. The selected variables are used to form a double robust regression estimator of the treatment effect. Simulation results are presented and economic growth data are analyzed.
Quantitative Biology , 2014, DOI: 10.1073/pnas.1409572111 Abstract: The efficient recognition of pathogens by the adaptive immune system relies on the diversity of receptors displayed at the surface of immune cells. T-cell receptor diversity results from an initial random DNA editing process, called VDJ recombination, followed by functional selection of cells according to the interaction of their surface receptors with self and foreign antigenic peptides. To quantify the effect of selection on the highly variable elements of the receptor, we apply a probabilistic maximum likelihood approach to the analysis of high-throughput sequence data from the $\beta$-chain of human T-cell receptors. We quantify selection factors for V and J gene choice, and for the length and amino-acid composition of the variable region. Our approach is necessary to disentangle the effects of selection from biases inherent in the recombination process. Inferred selection factors differ little between donors, or between naive and memory repertoires. The number of sequences shared between donors is well-predicted by the model, indicating a purely stochastic origin of such "public" sequences. We find a significant correlation between biases induced by VDJ recombination and our inferred selection factors, together with a reduction of diversity during selection. Both effects suggest that natural selection acting on the recombination process has anticipated the selection pressures experienced during somatic evolution.
PLOS Biology , 2014, DOI: 10.1371/journal.pbio.1001868 Abstract: Gene amplification of specific loci has been described in all kingdoms of life. In the protozoan parasite Leishmania, the product of amplification is usually part of extrachromosomal circular or linear amplicons that are formed at the level of direct or inverted repeated sequences. A bioinformatics screen revealed that repeated sequences are widely distributed in the Leishmania genome and the repeats are chromosome-specific, conserved among species, and generally present in low copy number. Using sensitive PCR assays, we provide evidence that the Leishmania genome is continuously being rearranged at the level of these repeated sequences, which serve as a functional platform for constitutive and stochastic amplification (and deletion) of genomic segments in the population. This process is adaptive as the copy number of advantageous extrachromosomal circular or linear elements increases upon selective pressure and is reversible when selection is removed. We also provide mechanistic insights on the formation of circular and linear amplicons through RAD51 recombinase-dependent and -independent mechanisms, respectively. The whole genome of Leishmania is thus stochastically rearranged at the level of repeated sequences, and the selection of parasite subpopulations with changes in the copy number of specific loci is used as a strategy to respond to a changing environment.
Chinese Science Bulletin , 2013, DOI: 10.1007/s11434-012-5635-8 Abstract: Leptin is an adiposity-secreted hormone that is pivotal in regulating feeding behavior, energy metabolism and body mass. The study of leptin has been of crucial importance for public health and pharmaceutical intervention given its role in obesity. Generally, leptin is highly conserved due to its functional importance. However, episodes of rapid sequence evolution and positive selection have been observed in some mammalian species, indicating that the leptin functions in these animals may have undergone adaptive modification to their environments. In this article, we review the adaptive evolution of leptin and its potential functional consequences. This review is expected to guide future research of molecular evolution and functional assays of this gene, and also to provide a theoretical foundation for the use of leptin in therapeutic applications.
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/176419 Abstract: This paper studies the adaptive fault estimation problems for stochastic Markovian jump systems (MJSs) with time delays. With the aid of the selected Lyapunov-Krasovskii functional, the adaptive fault estimation algorithm based on adaptive observer is proposed to enhance the rapidity and accuracy performance of fault estimation. A sufficient condition on the existence of adaptive observer is presented and proved by means of linear matrix inequalities techniques. The presented results are extended to multiple time-delayed MJSs. Simulation results illustrate that the validity of the proposed adaptive faults estimation algorithms.
电子与信息学报 , 2008, Abstract: A novel support vector regression method AEPSVR is proposed in this paper. First, an approximate regression function is obtained using -SVR method, and then a new adaptive error penalization function is introduced to enhance the robust performance of SVR such that a robust support vector regression is derived. Because the proposed AEPSVR here is based on -SVR, so various optimization methods for SVR can be used. Experimental results show that the proposed AEPSVR can reduce the affect of outliers, and have the very good generalization capability.
Page 1 /100 Display every page 5 10 20 Item | 2019-11-13 01:34:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5560498237609863, "perplexity": 1170.5579011843965}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665976.26/warc/CC-MAIN-20191113012959-20191113040959-00313.warc.gz"} |
https://codegolf.stackexchange.com/questions/13152/shortest-code-to-produce-infinite-output/92483#92483 | # Shortest code to produce infinite output
Write the shortest code you can that produces an infinite output.
That's all. You code will only be disqualified if it stops producing output at some point. As always in code golf, the shortest code wins.
Here's a list of answers that I think are really clever, so they can get credit:
var QUESTION_ID=13152,OVERRIDE_USER=8611;function answersUrl(e){return"https://api.stackexchange.com/2.2/questions/"+QUESTION_ID+"/answers?page="+e+"&pagesize=100&order=desc&sort=creation&site=codegolf&filter="+ANSWER_FILTER}function commentUrl(e,s){return"https://api.stackexchange.com/2.2/answers/"+s.join(";")+"/comments?page="+e+"&pagesize=100&order=desc&sort=creation&site=codegolf&filter="+COMMENT_FILTER}function getAnswers(){jQuery.ajax({url:answersUrl(answer_page++),method:"get",dataType:"jsonp",crossDomain:!0,success:function(e){answers.push.apply(answers,e.items),answers_hash=[],answer_ids=[],e.items.forEach(function(e){e.comments=[];var s=+e.share_link.match(/\d+/);answer_ids.push(s),answers_hash[s]=e}),e.has_more||(more_answers=!1),comment_page=1,getComments()}})}function getComments(){jQuery.ajax({url:commentUrl(comment_page++,answer_ids),method:"get",dataType:"jsonp",crossDomain:!0,success:function(e){e.items.forEach(function(e){e.owner.user_id===OVERRIDE_USER&&answers_hash[e.post_id].comments.push(e)}),e.has_more?getComments():more_answers?getAnswers():process()}})}function getAuthorName(e){return e.owner.display_name}function process(){var e=[];answers.forEach(function(s){var r=s.body;s.comments.forEach(function(e){OVERRIDE_REG.test(e.body)&&(r="<h1>"+e.body.replace(OVERRIDE_REG,"")+"</h1>")});var a=r.match(SCORE_REG);a&&e.push({user:getAuthorName(s),size:+a[2],language:a[1],link:s.share_link})}),e.sort(function(e,s){var r=e.size,a=s.size;return r-a});var s={},r=1,a=null,n=1;e.forEach(function(e){e.size!=a&&(n=r),a=e.size,++r;var t=jQuery("#answer-template").html();t=t.replace("{{PLACE}}",n+".").replace("{{NAME}}",e.user).replace("{{LANGUAGE}}",e.language).replace("{{SIZE}}",e.size).replace("{{LINK}}",e.link),t=jQuery(t),jQuery("#answers").append(t);var o=e.language;/<a/.test(o)&&(o=jQuery(o).text()),s[o]=s[o]||{lang:e.language,user:e.user,size:e.size,link:e.link}});var t=[];for(var o in s)s.hasOwnProperty(o)&&t.push(s[o]);t.sort(function(e,s){return e.lang>s.lang?1:e.lang<s.lang?-1:0});for(var c=0;c<t.length;++c){var i=jQuery("#language-template").html(),o=t[c];i=i.replace("{{LANGUAGE}}",o.lang).replace("{{NAME}}",o.user).replace("{{SIZE}}",o.size).replace("{{LINK}}",o.link),i=jQuery(i),jQuery("#languages").append(i)}}var ANSWER_FILTER="!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe",COMMENT_FILTER="!)Q2B_A2kjfAiU78X(md6BoYk",answers=[],answers_hash,answer_ids,answer_page=1,more_answers=!0,comment_page;getAnswers();var SCORE_REG=/<h\d>\s*([^\n,]*[^\s,]),.*?(\d+)(?=[^\n\d<>]*(?:<(?:s>[^\n<>]*<\/s>|[^\n<>]+>)[^\n\d<>]*)*<\/h\d>)/,OVERRIDE_REG=/^Override\s*header:\s*/i;
body{text-align:left!important}#answer-list,#language-list{padding:10px;width:290px;float:left}table thead{font-weight:700}table td{padding:5px}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <link rel="stylesheet" type="text/css" href="//cdn.sstatic.net/codegolf/all.css?v=83c949450c8b"> <div id="answer-list"> <h2>Leaderboard</h2> <table class="answer-list"> <thead> <tr><td></td><td>Author</td><td>Language</td><td>Size</td></tr></thead> <tbody id="answers"> </tbody> </table> </div><div id="language-list"> <h2>Winners by Language</h2> <table class="language-list"> <thead> <tr><td>Language</td><td>User</td><td>Score</td></tr></thead> <tbody id="languages"> </tbody> </table> </div><table style="display: none"> <tbody id="answer-template"> <tr><td>{{PLACE}}</td><td>{{NAME}}</td><td>{{LANGUAGE}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr></tbody> </table> <table style="display: none"> <tbody id="language-template"> <tr><td>{{LANGUAGE}}</td><td>{{NAME}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr></tbody> </table>
• All answers disqualified because at some point the Earth will be swallowed by the sun, and at some point the universe will die :P
– Doorknob
Nov 9 '13 at 20:00
• Does "infinite until your computer crashes" count? <_< Nov 10 '13 at 1:39
• If I write mine in Piet, can I count the pixels of the text the other programs used? I believe the smallest possible repeating Piet program would be 6 pixels. That beats Befunge if "off" pixels still count. Nov 12 '13 at 20:27
• @Izkata So any answer that crashes your computer is also allowed :D Jul 11 '14 at 20:11
• @Doorknob So really, the challenge is to produce infinite output in a finite amount of time. Sounds easy enough. Apr 10 '15 at 21:15
## J 11
echo@-^:_]1
Outputs 0's because inversion (-) does not converge when repeated ad infinitum (^:_).
A couple of attempts for AVR, starting with one of the older MEGA/TINY models:
inc r1
out DIRC, r0
out DIRC, r1
rjmp .-2
And for XMEGA:
inc r0
sts PORTC_DIR, r0
rjmp .-2
Both of these rely on registers being cleared at reset. The output is toggling the direction of PORTC pin 0 (input/output). You need to connect a pull-up resistor to this pin to create a square wave.
# Coffeescript 15
alert 0 while 1
• Or: loop alert() Apr 11 '15 at 15:57
# Bash - 13
cat /dev/zero
OK, not the shortest, but I'm surprised this isn't here already.
• Even shorter: cat /dev/zero. Jun 22 '14 at 22:05
• Ah, thanks! I knew I was forgetting something :) Jun 22 '14 at 22:07
# Bash - 7 Characters
echo
$0 Prints endless newlines, at least until your computer crashes. ## k4/q (11) {x}{x""}/-1 The second function, {x""}, sends the empty string to x. If x is an integer, it's interpreted as a file descriptor. If x is 1 or 2, this means to print to stdout or stderr. If x is -1 or -2, this means to additionally print a newline. The result of sending a string to a file descriptor is that file descriptor. The first function {x} is an idempotent function. (k has an actual idempotent function, ::, but in this context, it would have to be written (::), so this is shorter.) The construct g f/x, where g and f are functions, is a variation on functional "fold": f x is called, then g f x is tested as a boolean; if it is false, execution stops; if not, f is called on the result of the prior call. (The return value is the result of the last call to f.) Every integer but 0 is truthy, so the -1 returned by {x""} -1 allows the execution to continue. As a bonus, and at no extra character cost, if you swap the / for a \, you'll waste infinite amounts of RAM too: g f\x does the same thing, but it saves all the intermediate results as it runs--its return value is the full list of intermediate returns of f. Just for fun, here's another alternate version. It's the same length, but only valid in k (not q): {x}{x@$x}/1
This one prints "1"s forever, through more or less the same method, but a couple details are slightly different.
# CJam, 6 bytes
1{1p}h
or
{1p1}g
which prints an infinite number of 1\n without risk of stack overflow (as there's no recursion).
I think CJam might be younger than the question, but this answer isn't a winner, so I don't see any harm.
# QBasic, 6 (5?) characters
?1:RUN
? is short for PRINT, and RUN without arguments runs the current file from the top. This is the shortest (and most interesting) way to get an infinite loop in QBasic.
If, as in the accepted answer, infinite newlines count, then I present this 5-character version:
?:RUN
# LOOP, 1
.
Outputs 000000000...
# This Programming Language, 2
In the spirit of the Befunge answer,
i'
Outputs 105 indefinitely.
# Turing Machine Code (9)
(This is the dialect of turing machine code used by many online turing machine simulators)
0 * 1 r *
It will endlessly ouput ones while moving rightward along the tape.
# Mathematica, 34 33 bytes
\$IterationLimit=∞;#0@Print@1&[]
If there were no limitation on the maximum length of evaluation chain in Mathematica, #0@Print@1&[] is only 13 bytes, 4 bytes shorter than For[,0<1,Print@1].
Qt 5.4 QMake pro file, 42 chars
defineReplace(A){log(A)return($$A())}$$A()
Is an infinite loop, but sadly gets caught out by qmake interpreter before it can get very far.
AAAA[...]Ran into infinite recursion (depth > 100).
Not sure if that counts under disqualification.
## Self-modifying Brainfuck, 4 bytes
Prints ] forever. Note that the traditional BF programs work as well.
<[.]
The pointer starts one to the right of the source code on the tape, so moving left once puts the pointer at ].
## Simplefunge, 3 chars
Does it count if, you know, it is your own language?
>o<
Prints 0 over and over again, as the tape is initialized as a tape of infinite zeros.
# PlatyPar, 3 Bytes
1WA
Explanation:
1 ## push 1 to the stack
W ## while(stack[-1])
Works the same as my truth machine, except manually inputting the 1 instead of asking for it.
# Detour, 1 bytes
,
Try it online!
, outputs its contents, and passes it on to the next cell (,). In the event of not being given input, a 0 will be passed in.
## BASTARD 30 Bytes
{(., <> {fi out {t 0}(.)})(.)}
This makes use of a few things that are still in question with the spec, but I'm the authour, so what the hell.
Note: BASTARD is still under development, and I haven't finished the spec, or even a rough REPL, so it probably should be disqualified for everything.
### Explanation:
1. Everything has to sit in a {} block.
2. We define a function called . that takes no arguments.
3. The function prints an undefined variable from the stack, which equates to 'nil'.
4. The function also calls itself, thus an infinite loop.
5. We call the infinite function.
Note: Anonymous functions may or may not be introduced in the future. The current syntax proposed would reduce the above to 24 bytes:
{(<> {fi out {t 0}(@)})}
The only new concept here is @, which is a self-reference to the anonymous function.
• I suggest deleting this answer temporarily until you have released the language Mar 1 '16 at 18:03
# Reng v.1.2, 1 byte
Try it here!
n
Pops N and prints N. N is always zero, popping off the empty stack.
• If Reng is newer than this challenge, you should state that in your answer. Mar 21 '16 at 3:39
# Fuzzy Octo Guacamole, 3 bytes (Non-Competing)
(non-competing, FOG is newer than the challenge)
(1)
Infinitely outputs 1.
Due to weird implicit output and recursive loops (it executes the code inside the loop using recursion), 1 is outputted by the implicit output.
• Since FOG is newer than this challenge, you should state that in your answer. Mar 21 '16 at 3:39
• @Dennis thanks for catching that. Mar 21 '16 at 16:06
# Scratch, 3 bytes
(scoring used)
Adds a blank item to list = repeatedly and indefinitely.
• I can't see your script because of a 503 error. I think that using the button to upload an image is better, since it uploads to i.stack.imgur.com. Oct 6 '16 at 18:57
• That's because CubeUpload is down at the moment. Oct 8 '16 at 18:05
• Well, can you posts the ScratchBlocks version of the script at least (golfed)? Oct 8 '16 at 18:11
## Burlesque - 2 Bytes
bc
basically just creates an infinite list and outputting infinite lists will obviously produce infinite output.
Don't try this in the online shell because some browsers freeze when trying to render the result because it's huge :) (at some point either the browser stops rendering or the webserver closes the stream).
# Quetzalcoatl (non-competing), 14 chars
while 1: ::' '
This is for an older version of Quetzalcoatl.
• The latest revision of your answers makes it identical to this answer. May 6 '16 at 15:41
## Hexagony, 1 byte
!
Try it online!
Prints an infinite amount of 0s by printing the value of the initial memory edge (which happens to be 0) over and over. The ! is executed in a loop because the source code is toroidal.
# Woefully, 49 bytes (newer than challenge)
||| |
|| |
| |
|| |
||| |
|||| |
||||| |
|||||| |
Explanation
v v represents char pointer, instruction pointer finds first space after the char
|||A| pointed at by char pointer
||A| A- Push zero
|A|
||B|
|||B| B- Pop and print (number)
||||B|
|||||B|
||||||B|
[end] End- go back to character char pointer is pointing at. Char pointer has not moved
so it will just execute the same again
## Sonic Pi, 19 Bytes
loop do puts "" end
Sonic Pi is a sound language, but it's also fully a programming language.
• What is sonic pi, for those who don't know? May 6 '16 at 15:33
# zsh (8 chars)
</dev/z*
Analogous to cat /dev/zero.
Note: This does depend on there not being any other files in /dev starting with z, other than /dev/zero.
• Given that this particular configuration of /dev is the default on many UNIX-like operating systems (e.g. Ubuntu, which I had easily available to test), I'm pretty sure there are "implementations" where this works.
– user62131
Nov 26 '16 at 21:36
TI-83 Hex Assembly, 6 bytes
PROGRAM:I
:AsmPrgm
:EF0A45
:C3959D
Run with Asm(prgmI). Prints garbage over and over again. The only way to stop the printing is to physically remove the batteries from the calculator and re-insert them, at which point the calculator's RAM will be cleared. I count each hex digit pair as one byte.
# Casio FX-7000G, 6 bytes
Lbl 0◢
Goto 0
This uses the calculator's own encoding, where each token is stored as a byte.
Lbl 0 sets the label of the first line to 0. The triangle means "print last value", which is in this case 0. The next line is your standard Goto statement, jumping back to the top so the value can be printed again.
Due to the calculator's limitations, the user must press EXE after each printed value before the next can be displayed.
• Very cool. The FX-7000G was cutting-edge tech in the mid 80s. Reminds me of my old FX-3900P (RIP) Jul 29 '19 at 10:20
# Labyrinth, 1 byte
!
Labyrinth's stack has implicit zeroes at the bottom and the ! character outputs the integer representation of top of the stack. Since the program never finds anywhere else to go besides this single character, it keeps repeating that instruction. | 2021-10-15 20:53:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.28578469157218933, "perplexity": 5579.362545748126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323583083.92/warc/CC-MAIN-20211015192439-20211015222439-00183.warc.gz"} |
https://math.stackexchange.com/questions/1995563/algorithm-theta-notation-how-constant-c-2-geq-1-2-is-derived-from-the-inequ | # Algorithm Theta Notation : How constant $c_2 \geq 1/2$ is derived from the inequality $c_1 \leq 1/2 - 3/n \leq c_2$
I was learning ϴ(n) notation in my course "Asymptotic Analysis for Algorithms" when I encountered the following example:
For any non-negative constants $c_1\geq 0,c_2\geq 0,n\geq n_0$ we have the following inequality:
$$c_1\leq\frac{1}{2}-\frac{3}{n}\leq c_2$$
For these constants to satisfy this inequality, will be $c_2\geq\frac{1}{2}$ when $n\geq 1$. This is the part I'm having issues with.
I tried to derive it assuming $n \geq 1$ and I found:
\begin{align*} c_2 &\geq \frac{1}{2} - \frac{3}{n} \\ \implies c_2 &\geq \frac{1}{2} - \frac{3}{1} \\ \implies c_2 &\geq -\frac{5}{2}` \end{align*}
I managed to show the left inequality holds when $n\geq 7$ and $c_1\leq \frac{1}{14}$.
• $3/n$ is positive, so $1/2-3/n$ will be smaller than $1/2$. In other words, $1/2-3/n \leq 1/2$. Thus you can choose $c_2$ to be anything bigger than this upper bound of $1/2$ if you want it to be an upper bound too. So $1/2-3/n \leq c_2$ if $c_2 \geq 1/2$. – Antonio Vargas Nov 2 '16 at 7:49
The point is $-3/n$ is increasing to zero, so for large $n$ $1/2-3/n$ is approximately (but less than) $1/2$
In other words $1/2-3/n \ge 1/2-3/1$ is wrong, so you cannot derive the second inequality from the first one. | 2019-09-16 20:09:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9996923208236694, "perplexity": 134.0774440419341}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572934.73/warc/CC-MAIN-20190916200355-20190916222355-00410.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/201969-help-groups.html | 1. ## Help with groups
Hi
My wife is trying to do the question below. I cannot help out and wondered if someone could take me through it.
Thanks
2. ## Re: Help with groups
Originally Posted by Roger12
Hi
My wife is trying to do the question below. I cannot help out and wondered if someone could take me through it.
Because $a+b+1=b+a+1$ we see $a*b=b*a$ so it is commutative.
$(a*b)*c=(a*b)+c+1=(a+b+1)+c+1=a+(b+c+1)+1=a*(b*c)$ so it is associative.
You wife needs to show that $-1$ is the operational identity.
You wife needs to show that $-a-2$ is the operational inverse of $a$.
3. ## Re: Help with groups
verifying that -1 is the identity is rather easy one you "know" -1 is the identity. but how did Plato know it was?
it involves a bit of "detective work". let's find out what the identity would have to be, if there actually is one.
by definition, an identity for * is an element e of Z, with:
a*e = e*a = a, for any element a of Z.
since we know * is commutative (because of Plato's post above), it suffices to find e with a*e = a, for all a (in Z).
so we write down an equation in Z, and "solve for e".
if a*e = a, what this means in terms of the operations we're used to is:
a+e+1 = a.
we can, once we're "back in the ordinary integers", subtract a from both sides, to get:
e+1 = 0, which makes it clear e = -1 (since that's the only integer e with e+1 = 0, or: if you like, subtract 1 from both sides).
we can use "the same trick" to find out what a-1 has to be.
by definition, a-1 is an element b of Z with a*b = b*a = e.
now, we know what "e" is (see above), so we do pretty much what we did above: write a*b = e in terms of regular operations in Z, and "solve for b".
a*b = e translates into:
a+b+1 = -1
subtract 1 from both sides:
a+b = -2
subtract a from both sides:
b = -2-a (since this formula only depends on "a", we're good).
my point being, it's often not profitable to just "guess" what the identity and/or inverse might be. often, with just a little bit of reasoning, we can figure out what they HAVE to be. | 2017-05-23 22:57:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7612906098365784, "perplexity": 1140.4124882459237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607704.68/warc/CC-MAIN-20170523221821-20170524001821-00344.warc.gz"} |
http://lt-jds.jinr.ru/record/50170?ln=ru | / Experiment-HEP arXiv:1011.4193
Prompt and non-prompt J/psi production in pp collisions at sqrt(s) = 7 TeV
Abstract: The production of J/psi mesons is studied in pp collisions at sqrt(s)=7 TeV with the CMS experiment at the LHC. The measurement is based on a dimuon sample corresponding to an integrated luminosity of 314 inverse nanobarns. The J/psi differential cross section is determined, as a function of the J/psi transverse momentum, in three rapidity ranges. A fit to the decay length distribution is used to separate the prompt from the non-prompt (b hadron to J/psi) component. Integrated over J/psi transverse momentum from 6.5 to 30 GeV/c and over rapidity in the range |y| < 2.4, the measured cross sections, times the dimuon decay branching fraction, are 70.9 \pm 2.1 (stat.) \pm 3.0 (syst.) \pm 7.8(luminosity) nb for prompt J/psi mesons assuming unpolarized production and 26.0 \pm 1.4 (stat.) \pm 1.6 (syst.) \pm 2.9 (luminosity) nb for J/psi mesons from b-hadron decays.
Note: * Temporary entry *
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(Еще не рецензированная) | 2020-07-02 05:57:22 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9517218470573425, "perplexity": 11958.544339483193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655878519.27/warc/CC-MAIN-20200702045758-20200702075758-00482.warc.gz"} |
http://mwhf.mpl-bauen.de/vector-in-r.html | R is a system for statistical computation and graphics. Since the vector is perpendicular to the normal (which is itself perpendicular to the plane) and we are in $\mathbb{R^3}$ it must be parallel to the plane. R Data Types. table is an extension of data. Vector implements List Interface. Als Antwort wird TRUE/FALSE ausgegeben. The set of all polynomials with coefficients in R and having degree less than or equal to n, denoted Pn, is a vector space. The vector is a very important tool in R programming. Download free Vectors and Graphic Vectors. Find the velocity of the motorboat that would be required to achieve this resultant velocity. And here is a vector of logical values. 0 in steps of 0. Cointegration was introduced in a series. A vector is a sequence of data elements of the same basic type. There are many ways of doing the same thing in R. Nevertheless, we will just call them members in this site. I simple want to append to a vector. R makes this very easy using indexing. Vector Magic always traces your bitmap, carefully teasing out the underlying shapes in it, and provides you with a real vector image with all of its benefits. using System; using System. Following table shows the logical operators supported by R language. A vector ~v2R3 is a 3-tuple of real numbers (v. org [mailto:r-help-bounces at r-project. Here is a vector containing three numeric values 2, 3 and 5 :. If the index is negative, it would strip the member whose position has the same absolute value as the negative index. R makes this very easy using indexing. On the graph, u is the unit vector (in black) pointing in the same direction as vector OA, and i, j, and k (the unit vectors in the x-, y-and z-directions respectively) are marked in green. I am not sure whether this is really Rstudio's issue. We run into high dimensional vectors even in fields like neuroscience. e character vector is similar as creating numeric vector. Right away, you can see we'll need a 100-dimensional vector to describe Fred's neuronal activity at any point in time. The R manual defines a vector as “a single entity consisting of a collection of things. Generic; // Simple business object. In order to express R in terms of the chosen coordinate system, we use the three unit vectors i, j and k. In a previous article Vector Operation In R , we have already seen how the addition of two vectors works in R. 3:1 does not include the value 1 as the last value since the increment does not line up with the endpoint. A unit vector is a vector that has a magnitude of 1 with no units. Vectors can be broken into i j and k, representing the x y and z axes, respectively. In each iteration, val takes on the value of corresponding element of x. Anyway: The atomic vector is the simplest R data type. , the addition of two vectors in R. In this tutorial, you will familiarize yourself with basic operations on vectors and get yourself accustomed to basic rules like subsetting and functions like name() and sum(). Recall in the section above, to reference values within a vector we simply supply it a vector of either index numbers, or TRUE / FALSE values indicating which values to keep. The c function (mnemonic for concatenate or combine) allows you to quickly enter data into R. The syntax for data. Norm of tangent vector is the norm of derivative. The best selection of Royalty Free R Vector Art, Graphics and Stock Illustrations. The replacement form can be used to reset the length of a vector. You can use a data structure to hold a vector. As a consequence, R is R = R x i + R y j + R z k º x i + y j + z k, As the object moves through space, the position vector traces a path. A data frame, a matrix-like structure whose columns may be of differing types (numeric, logical, factor and character and so on). It provides, among other things, a pro-gramming language, high level graphics, interfaces to other languages and debugging facilities. TRUE and FALSE are part of the R language, where T and F are global variables set to these. We can, however, create a pseudo vector in C in a couple of different ways. Every week we add new premium graphics by the thousands. Suppose that we had an attribute vector stored in a UCINET file (other vectors could also be in the file, but this algorithm operates on a single vector), that measured whether each of 100 large donors had given funds in support of (+1) or in opposition to (-1) a given ballot proposition. If we imagine moving the vector so it's tail is at the origin then the endpoint Pdetermines. Returns the number of elements in the vector. Logical Data Type. There are also separate programs available to vectorize bitmap images. The operation to concatenate vectors or matrices under MATLAB is defined as a combination of the variables in a single vector or matrix. The zero vector is also a linear combination of v 1 and v 2, since 0 = 0 v 1 + 0 v 2. 25),0), (1,1), (0,sqrt(5)). Vectors are the most basic R data objects and there are six types of atomic vectors. Note: By default, R displays the elements of an array as a series of horizontal values. Creating Vectors: age <- c(20,22,18,45) Here “age” is the name of vector and “c” is the shorthand of combine. Vector in R : Vectors are collection of data items of similar data type. Negative Index. Support Vector Machines in R In this tutorial, you'll try to gain a high-level understanding of how SVMs work and then implement them using R. The rules for determining the attributes of the result are rather complicated. Given some vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, we denote the distance between those two points in the following manner. 0k, All In Meters. You will want to create your own data type (vector type) by using the typedef keyword:. The only other vector this works for is b) which isn't a unit vector. Given a vector of data one common task is to isolate particular entries or censor items that meet some criteria. A vector's type can be checked with the typeof() function. A unit vector in ℝ 3 was called a right versor by W. A tutorial to perform basic operations with spatial data in R, such as importing and exporting data (both vectorial and raster), plotting, analysing and making maps. We shall see how it works in R studio. Don't have an account? Sign up now. A vector can be "scaled" off the unit vector. An important aspect of working with R objects is knowing how to “index” them Indexing means selecting a subset of the elements in order to use them in further analysis or possibly change them Here we focus just on three kinds of vector indexing: positional, named reference, and logical Any of these indexing techniques works the same for all classes of vectors. The scalar product of two vectors can be constructed by taking the component of one vector in the direction of the other and multiplying it times the magnitude of the other vector. Any number of vector quantities of the same type (i. These functions allow crossing the data in a number of ways and avoid explicit use of loop constructs. Simply put, a vector is a sequence of data elements of the same basic type. "paste" in Unix) diff(x) # Returns. If we handed the plot function only one vector, the x-axis would consist of sequential integers. In the example, you want the third element, so you would type "yourelement <- yourvector[3]". , geometric distributions. A vector can be "scaled" off the unit vector. An AR model explains one variable linearly with its own previous values, while a VAR explains a vector of variables with the vector's previous values. 9779939 # Get a vector of 3 numbers from 0 to 100 runif. using System; using System. So, how can we set up and use a vector in R? We can construct a vector from a series of individual elements, using the c() function, as follows:. Because of this, vector images are considered resolution independent. bootstrap _b: regress mpg weight length. In the previous assignment, you created 3 vectors: a numeric vector, a character vector and a boolean vector. to v until the total is 20000 or more. Replicating a Vector in C. Unit vectors can be used in 2 dimensions: Here we show that the vector a is made up of 2 "x" unit vectors and 1. Definition S Subspace Suppose that V and W are two vector spaces that have identical definitions of vector addition and scalar multiplication, and that W is a subset of V, W⊆V. The equation for making a prediction for a new input using the dot product between the input (x) and each support vector (xi) is calculated as follows: f(x) = B0 + sum(ai * (x,xi)) This is an equation that involves calculating the inner products of a new input vector (x) with all support vectors in training data. Multiplying a Matrix by Another Matrix. If this step is possible, can one use "mutate" to split each name so that for example, the first three characters from the file name are added to a second vector?. vector , a generic, attempts to coerce its argument into a vector of mode mode (the default is to coerce to whichever vector mode is most convenient): if the result is atomic all attributes are removed. A tutorial to perform basic operations with spatial data in R, such as importing and exporting data (both vectorial and raster), plotting, analysing and making maps. No matter how it's written, the de nition of a vector space looks like abstract nonsense the rst time you see it. Negates this vector. Vector-R is designed to launch a 60 kg (130 lb) payload to a 1,000 km (620 mi) Sun-synchronous orbit, suitable for CubeSats and other small satellites. Clicking on the end of a vector will also reveal its individual components. In the following article, I'm going to provide you with 3 examples for the application of the rbind function in R. One can regard a column vector of length r as an r × 1 matrix and a row vector of length c as a 1×c matrix. We are going to discuss R vector functions in this tutorial. Suppose x is a vector. Vector in R : Vectors are collection of data items of similar data type. We shall see how it works in R studio. Parameters x Another vector container of the same type (i. Suppose there is an integer vector OBS containing the elements 2, 3, 7, i. The Time Derivative of a Vector in a Rotating Coordinate System. Matrices and other arrays are produced in LaTeX using the \textbf{array} environment. Find a vector parameterization for the line passing through P = (10, 8, 7) with direction vector v=< 4,-3,-2 > (Use t for the paramterized variable. frame columns are: But if you are writing library code (like vtreat) you end up working with. A vector is a sequence of data elements of the same basic type. Overview: ZNB Vector Network Analyzer - Vector network analyzer features a wide dynamic range, low trace noise and high output power necessary in electronic goods and mobile radio industry. Three-Dimensional Rotation Matrices 1. This distinction has a visual impact but the nature of a variable is usually readily apparent from the context in which it is used. A vector can be "scaled" off the unit vector. I need to estimate the quantile of a new number. Support Vector Machine With R In this article I will show how to use R to perform a Support Vector Regression. If the corresponding members of the original vector are included in the slice, then vector members are TRUE and otherwise FALSE. Arrays might store vertex data for complex shapes, recent keystrokes from the keyboard, or data read from a file. In machine learning, support vector machines are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis. Three Styles for LaTeX Vector Notation filed in LaTeX , Math on Jun. working with vector data, and GDAL for working with raster data. For example, if you have the vector {A,B,A,A,C}, it will output the following data frame: value count A 3 B 1 C…. Again, this is stating that is a result of combining the vectors using scalar multiplication (the c’s) and addition. I would like to make a partial differential equation by using the following notation: dQ/dt (without / but with a real numerator and denomenator). Researchers have developed a new and improved viral vector -- a virus-based vehicle that delivers therapeutic genes -- for use in gene therapy for sickle cell disease. Accessing Elements of Vector in R An element of the vector is accessed by their position. Download free Vectors and Graphic Vectors. “mcs-ftl” — 2010/9/8 — 0:40 — page 3 — #9 Introduction This text explains how to use mathematical models and methods to analyze prob-lems that arise in computer science. If a vector field F with zero divergence is defined on a ball in R 3, then there exists some vector field G on the ball with F = curl G. The equation for making a prediction for a new input using the dot product between the input (x) and each support vector (xi) is calculated as follows: f(x) = B0 + sum(ai * (x,xi)) This is an equation that involves calculating the inner products of a new input vector (x) with all support vectors in training data. A vector's type can be checked with the typeof() function. Describe the characteristics of 3 key vector data structures: points, lines and polygons. Anyway, it may be of value to you to try using memory. for any scalar. That is to say, ANOVA tests for the. vector returns TRUE if x is a vector of the specified mode having no attributes. R Data Types. Vector in R : Vectors are collection of data items of similar data type. I'm trying to. Hopefully the reader can well imagine the definition of a vector in. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C). We can also use for-loops to create or extend vectors, as R will automatically make a vector larger to accommodate values we assign to it. For example, lets create a numeric vector: Additionally, you can create a. org Subject: [R] how to count number of elements in a vector that are not NA ? Hi, is there a simpler way to count the number of elements in a vector that are not NA than this:. 1×n(C) or Mn×1(C) is a vector space with its field of scalars being either R or C. In this post, I want to show how to run a vector autoregression (VAR) in R. The gradient vector <8x,2y> is plotted at the 3 points (sqrt(1. In this case, two classes are red and blue balls. Distance is a scalar ("3 km") Displacement is a vector ("3 km Southeast") You can walk a long distance, but your displacement may be small (or zero if you return to the start). It is ideal for beginners and lays a strong foundation in learning of R. But to multiply a matrix by another matrix we need to do the "dot product" of rows and columns what does that mean?. Any help is appreciated. It provides, among other things, a pro-gramming language, high level graphics, interfaces to other languages and debugging facilities. Show that each of these is a vector space over the complex numbers. Collections. dimnames = NULL, useNames = FALSE) x: logical vector or array. In the previous assignment, you created 3 vectors: a numeric vector, a character vector and a boolean vector. result that can be a matrix, a vector, or a scalar, depending on the computations involved. All logos work with EPS, AI, PSD and Adobe PDF. Cointegration was introduced in a series. So the first element of ivec would be ivec[0], the second element would be ivec[1], and so on. Subsetting a list works in the same way as subsetting an atomic vector. Vectors are the most basic R data objects and there are six types of atomic vectors. Let me make the vector. Each element of the first vector is compared with the corresponding element of the second vector. However, C is not an object oriented language, so creating a true vector is virtually impossible. Suppose x is a vector. The gradient vector <8x,2y> is plotted at the 3 points (sqrt(1. I usually use any and all to check logical statements applied across a vector, check for NA values, or to examine a vector of logical values. The former is a data model, the latter is an R class just like data. rep Cookbook for R. In machine learning, support vector machines are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis. Unit vectors can be used in 2 dimensions: Here we show that the vector a is made up of 2 "x" unit vectors and 1. For example, suppose we have a vector of 10 values, but the fourth one is missing. exp(x) function compute the exponential value of a number or number vector, e x. ind = FALSE, useNames = TRUE) arrayInd(ind,. I guess since it's in QGIS there would be a solution in Python, but I'd like to integrate it into a R workflow if possible. You want to generate random numbers. Another important property of a vector is its length. Multivariate analysis of variance (MANOVA) is simply an ANOVA with several dependent variables. Although a vector has magnitude and direction, it does not have position. Find an orthonormal basis of the three-dimensional vector space R^3 containing a given vector as one basis vector. In the end my vector (for each variable) wıll be a single column by 800 rows filled with the data that I need Please help. Vector-merge for merging vector-like objects. The Vector is the most basic Data structure in R programming. In this article. probability vector. In the introduction to support vector machine classifier article, we learned about the key aspects as well as the mathematical foundation behind SVM classifier. In my case, more than 100. Support Vector Machine In R: With the exponential growth in AI, Machine Learning is becoming one of the most sort after fields. Subsetting a list works in the same way as subsetting an atomic vector. Support Vector Regression (SVR). Should the sort be increasing or decreasing? For the "radix" method, this can be a vector of length equal to the number of arguments in For the other methods. Vector images solve this issue of resizing images. result that can be a matrix, a vector, or a scalar, depending on the computations involved. Answer to: Consider the following vector function. Initialize a vector in C++ (5 different ways) Following are different ways to create and initialize a vector in C++ STL. We run into high dimensional vectors even in fields like neuroscience. Returns TRUE or FALSE Use as. When rotating about the z -axis, only coordinates of x and y will change and the z -coordinate will be the same. Vectors are the most basic form of data in R. Returns NA if no cases. There are two types of concatenation operation - horizontal and vertical concatenation. Nevertheless, we will just call them members in this site. For completeness: In big vectors, you can use the indices to speed things up (we do that often in simulations, where functions typically run 1000 to 10000 times). Earlier today I got help from this page on how. No characters other than real numbers are accepted by the calculator. And since the transpose of a column vector is a row vector, we use lower-case, boldface letters plus a prime to represent row vectors. Thus, vector b would be a column vector, and vector b' would be a row vector. With vector r in meters and time(t) in seconds. “mcs-ftl” — 2010/9/8 — 0:40 — page 3 — #9 Introduction This text explains how to use mathematical models and methods to analyze prob-lems that arise in computer science. Feel free to browse through our site and learn about encephalitis, Lyme disease, Africanized honeybees, and a variety of other vector. Vector, in physics, a quantity that has both magnitude and direction. Clicking on the end of a vector will also reveal its individual components. R Vector Data Type. Vector fields along a curve. Multiplying a Matrix by Another Matrix. , the addition of two vectors in R. So if the populations of the city and the suburbs are given by the vector , after one year the proportions remain the same (though the people may move between the city and the suburbs). Even when you write just one value in R, it becomes a vector of length 1 and belongs to one of the above vector types. It even works as one would hope when test is a vector. 0k, All In Meters. Vector Addition; Resultants; Vector Components; Vector Resolution; Component Addition; Relative Velocity and Riverboat Problems; Independence of Perpendicular Components of Motion; Lesson 2 - Projectile Motion; What is a Projectile? Characteristics of a Projectile's Trajectory; Horizontal and Vertical Components of Velocity; Horizontal and. Vector-comparison for comparing, ordering, and tabulating vector-like objects. As you see, the way the equations are displayed depends on the delimiter, in this case and . In the first example we keep appending the square of an N(0. which(x, arr. There is a part 2 coming that will look at density plots with ggplot , but first I thought I would go on a tangent to give some examples of the apply family, as they. Libraries can implement different strategies for growth to balance between memory usage and reallocations, but in any case, reallocations should only happen at logarithmically growing intervals of size so that the insertion of individual elements at the end of the vector can be provided with amortized constant time complexity (see push_back). If data is a vector, a single value used for replacement Additional arguments for methods. The c function (mnemonic for concatenate or combine) allows you to quickly enter data into R. We can use a logical vector, of the same length as your data, as an index and R will pull out the elements of the data vector for which the corresponding indices are TRUE. R wouldn’t be R if it didn’t have some kind of vectorized version of an if…else statement. TRUE and FALSE are part of the R language, where T and F are global variables set to these. Another important property of a vector is its length. The data types can be logical, integer, double, character, complex or raw. In my continued playing around with plyr’s ddply function I was trying to group a data frame by one of its columns and return a count of the number of rows with. To refer to individual elements of the vector you use the element's subscript inside square brackets. Scalar Product of Vectors. In vectors, data is inserted at the end. With a matrix. , geometric distributions. In the code snippet above, notice how the numeric values are being coerced into character strings when the two vectors are combined. Type "yourelement <- yourvector[number]" where "number" is the position of the element you want. R - Factors - Factors are the data objects which are used to categorize the data and store it as levels. These are the basic building blocks that all R objects are built from. Syntax for unlist function in R: unlist(x, recursive = TRUE, use. It deals with the restructuring of data: what it is and how to perform it using base R functions and the {reshape} package. The former is a data model, the latter is an R class just like data. Just an adaptation of the stats:sd function to return the functionality found in R < 2. since I am writing blog post that hosted by Github with Editor Atom, and use plugin markdown-preview-plus and mathjax-wra. The bool specialization of vector provides an additional overload for this function (see vector::swap). The significance of div 6. In machine learning, support vector machines are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis. Returns NA if no cases. And since the transpose of a column vector is a row vector, we use lower-case, boldface letters plus a prime to represent row vectors. I need to write a function that receives 2 arguments in R: first a number = X second a vector = V I need that this function would return the max number of the identical straight occurrences of X for example: f(6, c(7,6,6,3,7,9,3,6,6,6,8,9) should return 3. for sort an R object with a class or a numeric, complex, character or logical vector. These numbers (the 0 and the 1) are called subscripts. A vector is a basic data structure which plays an important role in R programming. load ("vect") loads this package, and demo ("vect") displays a demonstration. Gabor Grothendieck I thought it would be interesting to actually time this. The vector is a very important tool in R programming. Vectors are the most basic R data objects and there are six types of atomic vectors. ” A collection of numbers, for example, is a numeric vector — the first five integer numbers form a numeric vector of length 5. Accessing Elements of Vector in R An element of the vector is accessed by their position. 8 Some special matrices - An n×n matrix is a square matrix - A matrix A is symmetric if A = A>. To refer to individual elements of the vector you use the element's subscript inside square brackets. The main idea of support vector machine is to find the optimal hyperplane (line in 2D, plane in 3D and hyperplane in more than 3 dimensions) which maximizes the margin between two classes. The ts() function will convert a numeric vector into an R time series object. I guess since it's in QGIS there would be a solution in Python, but I'd like to integrate it into a R workflow if possible. vector(mode = "logical", length = 0). For examples, the following two vectors n and s are combined into a new vector containing elements from both vectors. First, let's create some. Recall in the section above, to reference values within a vector we simply supply it a vector of either index numbers, or TRUE / FALSE values indicating which values to keep. In the following article, I'm going to provide you with 3 examples for the application of the rbind function in R. The R programming syntax is extremely easy to learn, even for users with no previous programming experience. (ii) Intersection between line and plane: For a line with equation r=a+λm and a plane with equation r•n=k. 0] or in sparse format as (3, [0, 2], [1. Given some vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, we denote the distance between those two points in the following manner. The distinction between row vectors and column vectors is essential. And since the transpose of a column vector is a row vector, we use lower-case, boldface letters plus a prime to represent row vectors. You will find a summary of the most popular approaches in the following. Because of this it is convenient to define the moment as the vector M = r x F. rm=TRUE) Arguments. Above, you can find the basic code for rbind in R. In the previous assignment, you created 3 vectors: a numeric vector, a character vector and a boolean vector. Since the vector is perpendicular to the normal (which is itself perpendicular to the plane) and we are in $\mathbb{R^3}$ it must be parallel to the plane. The result is presented for both the settings of the Simplify parameter. In R, a string is considered to be a character vector, but an R character vector would be an array of strings in any other programming language. These purple, these are all bolded, just because those are vectors, but sometimes it's kind of onerous to keep bolding things. and when you pass it normally, it just makes a copy of the vector and operates on that correct? so the original vector is never changed. If you make a longer vector — say, with the numbers from 1 to 30 — you see more indices. There are also separate programs available to vectorize bitmap images. The basic equation is: A x = λ x ; we say that λ is an eigenvalue of A. Multivariate analysis of variance (MANOVA) is simply an ANOVA with several dependent variables. In R, Vector is a basic data structure that contains element of similar type (logical, integer, double, character, complex or raw). Researchers have developed a new and improved viral vector -- a virus-based vehicle that delivers therapeutic genes -- for use in gene therapy for sickle cell disease. Thus, vector b would be a column vector, and vector b' would be a row vector. Because of this it is convenient to define the moment as the vector M = r x F. Look at the end of the vector, and you will find the point specified! P(r) r. Download free Vectors and Graphic Vectors. Ce petit lanceur d'environ 6 tonnes peut placer sur une orbite basse un ou plusieurs nano-satellites ayant une masse totale d'environ 60 kg [1]. A value in R can be a numerical value (e. The fact that there is more than one way to express the vector v in R 2 as a linear combination of the vectors in C provides another indication that C cannot be a basis for R 2. This course will introduce a powerful classifier, the support vector machine (SVM) using an intuitive, visual approach. k is the last value in the vector only when the increment lines up to exactly land on k. match returns a vector of the positions of (first) matches of its first argument in its second. A vector is a set of numbers that are arranged in a single row or single column. You can give a name to the elements of a vector with the names() function. Is there a way to replicate the Split Vector Layer function from QGIS in R? I have a Shapefile containing around 20 polygons that I'd like to split into 20 separate files. The post-fixed 3 simply refers to the vector being in 3-dimensions (x,y,z). Unit vectors can be used in 2 dimensions: Here we show that the vector a is made up of 2 "x" unit vectors and 1. To find the magnitude of a vector, first determine its horizontal and vertical components on their respective number lines around the origin. Introduction. Logical Operators. Parameters x Another vector container of the same type (i. The simplest such structure is the numeric vector, which is a single entity consisting of an ordered collection of numbers. Note that a vector can be empty and still have a mode. R uses this information for internal handling of strings and character vectors. Quickly create a vector of ones and zeros. You place the vector elements separated by a comma between the parentheses. The set of all linear combinations of a collection of vectors v 1, v 2,…, v r from R n is called the span of { v 1. The grep function takes your regex as the first argument, and the input vector as the second argument. The replacement form can be used to reset the length of a vector. Vector objects can be placed over other objects, and the object below will show through. A vector is the simplest type of data structure in R. The R programming syntax is extremely easy to learn, even for users with no previous programming experience. Vector, in physics, a quantity that has both magnitude and direction. If a vector is shortened, extra values are discarded and when a vector is lengthened, it is padded out to its new length with NAs. The first source of confusion for me is the R type system. A vector in R is also an object that contains elements having the same data type. In the current lesson, we'll see what can be done when the two worlds of vector data and raster data intersect. By using a logical index vector in R, we can form a new vector from a given vector, which has the same length as the original vector. On the graph, u is the unit vector (in black) pointing in the same direction as vector OA, and i, j, and k (the unit vectors in the x-, y-and z-directions respectively) are marked in green. NAs are allowed and omitted (treated as if FALSE). | 2019-11-13 16:02:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.567776620388031, "perplexity": 582.8724369375561}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496667262.54/warc/CC-MAIN-20191113140725-20191113164725-00363.warc.gz"} |
https://math.stackexchange.com/questions/520246/fundamental-solutions-of-wave-equations-in-terms-of-heaviside-functions | # Fundamental solutions of wave equations in terms of Heaviside functions
I know that for the canonical one dimensional wave equations $u_{\mu\eta}=0$, the fundamental solution $F$ satisfies $\frac{\partial}{\partial\mu\partial\eta}F=\delta(\mu,\eta)=\delta(\mu)\delta(\eta)$. Thus one solution would be $F=H(\mu)H(\eta)$. However, my question is if I introduce change of variables by $\mu=x-ct, \eta=x+ct$, what would the fundamental solution be? The one dimensional wave operator corresponds to $\partial^{2}_t-c^2\partial^{2}_x=-\frac{c^2}{4}\partial^2_{\mu\eta}$, what would the fundamental solution $F$ correspond to?
• I know d'Alembert solution, but what I am trying to do is to find fundamental solution which is a distribution such that $(\partial^2_t-c^2\partial^2_x)F=\delta$, where $\delta$ is the Dirac function. – Xuxu Oct 10 '13 at 0:05 | 2019-10-15 12:17:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9513831734657288, "perplexity": 95.46968933207552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986658566.9/warc/CC-MAIN-20191015104838-20191015132338-00352.warc.gz"} |
https://brilliant.org/discussions/thread/arithmetic-mean-geometric-mean-inequality/ | # Arithmetic Mean-Geometric Mean Inequality
This week, we learn about the Arithmetic Mean-Geometric Mean Inequality, which relates the arithmetic mean and geometric mean of non-negative real values.
How would you use AM-GM to solve the following?
1. Given that $$a, b$$ and $$c$$ are positive numbers such that $$abc=1$$, show that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a+b+c.$
2. Share a problem which uses the AM-GM technique.
Note by Calvin Lin
4 years, 6 months ago
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\boxed{123} $$\boxed{123}$$ | 2018-06-18 17:42:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9996728897094727, "perplexity": 6601.726981976086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267860684.14/warc/CC-MAIN-20180618164208-20180618184208-00008.warc.gz"} |
https://toc.ui.ac.ir/?_action=article&sb=326&_sb=05A+Combinatorics%3A+Enumerative+combinatorics | Main Subjects = 05A Combinatorics: Enumerative combinatorics
Number of Articles: 11
##### 1. Statistics on restricted Fibonacci words
Articles in Press, Corrected Proof, Available Online from 14 August 2020
Omer Egecloglu
##### 2. Determinant identities for toeplitz-hessenberg matrices with tribonacci entries
Volume 9, Issue 2, June 2020, Pages 89-109
Taras Goy; Mark Shattuck
##### 3. Refinements of the Bell and Stirling numbers
Volume 7, Issue 4, December 2018, Pages 25-42
Tanay Wakhare
##### 4. Combinatorial parameters on bargraphs of permutations
Volume 7, Issue 2, June 2018, Pages 1-16
Toufik Mansour; Mark Shattuck
##### 5. The log-convexity of the fubini numbers
Volume 7, Issue 2, June 2018, Pages 17-23
Qing Zou
##### 6. Majorization and the number of bipartite graphs for given vertex degrees
Volume 7, Issue 1, March 2018, Pages 19-30
Annabell Berger
##### 7. The site-perimeter of words
Volume 6, Issue 2, June 2017, Pages 37-48
Aubrey Blecher; Charlotte Brennan; Arnold Knopfmacher; Toufik Mansour
##### 8. Congruences from $q$-Catalan Identities
Volume 5, Issue 4, December 2016, Pages 57-67
Qing Zou
##### 9. Note on the skew energy of oriented graphs
Volume 4, Issue 1, March 2015, Pages 57-61
Jun He; Ting-Zhu Huang
##### 10. Complete solution to a conjecture of Zhang-Liu-Zhou
Volume 3, Issue 4, December 2014, Pages 55-58
Mostafa Tavakoli; F. Rahbarnia; M. Mirzavaziri; A. R. Ashrafi
##### 11. On the number of cliques and cycles in graphs
Volume 2, Issue 2, June 2013, Pages 27-33 | 2021-01-26 22:51:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19777563214302063, "perplexity": 6989.327329255495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704803737.78/warc/CC-MAIN-20210126202017-20210126232017-00246.warc.gz"} |
https://math.stackexchange.com/questions/706528/let-a-in-operatornamemat-n-mathbb-r-ai-j-langle-v-i-v-j-rangle-wh/706535#706535 | # Let $A \in \operatorname{Mat}_n(\mathbb R), A(i,j) := \langle v_i,v_j\rangle$ where $v_1,\dotsc,v_n$ is a basis. Show $A$ is invertible.
Let $V$ be a real inner product space with basis $U=(v_1, \ldots, v_n)$ and $A \in Mat_{n,n}(\mathbb R)$ with $(i,j)$-entry equal to $\langle v_i v_j \rangle$.
I've shown the identity $\langle x, y \rangle = [x]_U^T \cdot A \cdot [y]_U \ \forall x,y \in V$ by writing $x= c_1 v_1 + \ldots c_n x_n$ and $y= d_1 v_1 + \ldots d_n v_n$, and then expanding inner product using axioms and afterwards computing the matrix product.
However I must show $A$ is invertible.
I try to show this by writing $Ax = 0$ and show $x = 0$.
$Ax = 0 \iff x_1 \langle v_i, v_1 \rangle + \ldots + x_j \langle v_i, v_j \rangle + \ldots \langle v_i, v_n \rangle = 0 \iff \langle v_i, x_1 v_1 + \ldots + x_j v_j + \ldots + x_n v_n \rangle = 0$
I want to conclude $x_1 v_1 + \ldots + x_j v_j + \ldots + x_n v_n = 0$ and then since $v_i : 1 \le i \le n$ is a basis we have $x_j = 0 : 1 \le j \le n$. However, how can I do this ?
By the way, the matrix $A$ is called the Gram matrix for $v_1, \ldots, v_n$.
• Do you know that you can orthogonalize bases in finite dimensional real inner product spaces? Mar 10, 2014 at 9:37
What happens is that your vector $v=a_1x_1+\ldots +a_nx_n$ is orthogonal to all the $v_i$. So $v$ is orthogonal to any linear combination of the $v_i$, so $v$ is orthogonal to itself. Since $<,>$ is an inner product, this is only possible if $v=0$.
• Depends on whether $V = ℝ^n$ and “$〈–,–〉$” denotes the euclidean inner product. Mar 10, 2014 at 9:44
• I think $[y]_U$ should denote the coordinates of $y$ in base $U$. This identity holds for abstract inner product spaces as well. Mar 10, 2014 at 9:48
• @k.stm please see my revised version. Mar 10, 2014 at 9:54
• Ewan why is it only possible if $v = 0$ that $v$ is ortogonal to every $v_i$ ? Mar 10, 2014 at 9:55
• Maybe this is yet to prove! Mar 10, 2014 at 9:57
There is a base change $S$ such that $US$ is a orthogonal base. You can prove this by induction on the dimension $n$. Let $v_1', …, v_n'$ be the transformed orthogonal base $US$. Calculate that $S^TAS = (〈v_i',v_j'〉)_{i,j = 1, …,n}$. To do this, you can use the identity you already have proven.
Here $US$ denotes $(s_{1,1}v_1 + …, s_{n,1}v_n, …, s_{1,n}v_1 + … s_{n,n}v_n)$, i.e. the system obtained by using the columns of $S$ to linearly combine out of $(v_1, …, v_n)$.
Because $A$ is invertible if and only if $S^TAS$ is invertible, you can assume without loss of generality your base to be an orthogonal system. Then your argument will work.
Let $(a_1,...,a_n)$ be the vector which makes it 0. Then in the first row $v_1(a_1 v_1+.... a_n v_n)=0$ implies $a_1=0$. Similarly you can show all $a_n$ are 0.
• Depends on whether $V = ℝ^n$ and “〈–,–〉” denotes the euclidean inner product. Mar 10, 2014 at 9:47 | 2022-07-04 07:29:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9625285267829895, "perplexity": 133.27443949746535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104354651.73/warc/CC-MAIN-20220704050055-20220704080055-00251.warc.gz"} |
https://byjus.com/question-answer/an-otherwise-infinite-straight-wire-has-two-concentric-loops-of-radii-a-and-b-carrying/ | Question
An otherwise infinite, straight wire has two concentric loops of radii $$a$$ and $$b$$ carrying equal currents in opposite directions as shown in figure. The magnetic field at the common center is zero for:
A
ab=π1π
B
ab=ππ+1
C
ab=π1π+1
D
ab=π+1π1
Solution
The correct option is B $$\cfrac { a }{ b } =\cfrac { \pi } { \pi +1 }$$$${ B }_{ centre }=0$$$$\cfrac { { \mu }_{ 0 }I }{ 4\pi b } \bigodot +\cfrac { { \mu }_{ 0 }I }{ 4\pi b } \bigodot +\cfrac { { \mu }_{ 0 }I }{ 2b } \bigodot +\cfrac { { \mu }_{ 0 }I }{ 2a } \bigotimes =0$$$$\Rightarrow \cfrac { { \mu }_{ 0 }I }{ 2\pi b } +\cfrac { { \mu }_{ 0 }I }{ 2b } -\cfrac { { \mu }_{ 0 }I }{ 2a } \Rightarrow \cfrac { 1 }{ 2\pi b } +\cfrac { 1 }{ 2b } =\cfrac { 1 }{ 2a }$$$$\Rightarrow \cfrac { a }{ b } =\cfrac { \pi }{ \pi +1 }$$PhysicsNCERTStandard XII
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View More | 2022-01-29 04:35:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32735323905944824, "perplexity": 4814.006960365726}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320299927.25/warc/CC-MAIN-20220129032406-20220129062406-00168.warc.gz"} |
https://www.physicsforums.com/threads/question-on-newtons-second-law-of-motion.257505/ | # Homework Help: Question on Newton's Second Law of Motion
1. Sep 18, 2008
### Nivlac2425
1. The problem statement, all variables and given/known data
Here's the question:
An arrow, starting from rest, leaves the bow with a speed of 25.0 m/s. If the average force exerted by the bow were doubled, all else remaining the same, with what speed would the arrow leave the bow?
2. Relevant equations
F = ma
3. The attempt at a solution
I know I need to find a new initial velocity for the arrow for which the force is doubled. And the question states that all else remains constant. I may need to find the acceleration using F = ma but I'm just not sure where to start at all. I just need a small kick so I know where I'm going
2. Sep 18, 2008
### Dschumanji
If a force F gives the arrow an acceleration of a, what would a force of 2F accelerate the arrow by? Use Newton's second law, it's simple algebra.
3. Sep 19, 2008
### Nivlac2425
It's double the initial accceleration right? But how would I find the speed of that?
4. Sep 19, 2008
### Dschumanji
Yup.
You can assume the force is applied for the same amount of time in each case, so in each case the acceleration lasts that long. Set up some equations that relate the velocity of the arrow after t seconds for the first and second case.
5. Oct 5, 2008
### kealh
Umm i dont get what you are telling him? Would you mind explain? please.
6. Oct 5, 2008
### Dschumanji
I really think that I made an error on this problem. If the acceleration is applied for the same duration on the arrow in each case then the velocity in the second case would just be double the velocity in the first case. If the acceleration is applied over the same distance then the velocity of the of the arrow in the second case is $$\sqrt{2}$$ times more than the velocity in the first case. I don't know which one is right.
The question is very vague when it says "All else remaining the same." In reality, to get double the average force out of the bow, the distance over which the average force is applied (equal to how far back you pull the string) is not the same in both cases; you could assume that this distance can't be changed. You could keep the distance constant in both cases but that would require more tension in the bow string, which you could also assume can't be changed too. It follows that the amount of time the average force is applied would be different in each case.
Anyone else have some insight?
7. Oct 5, 2008
### Dschumanji
Really? Nobody has any input?
8. Oct 6, 2008
### kealh
Eh yea i'm still confuse. but thank though
9. Oct 6, 2008
### nasu
The confusion comes from the word "average" in the problem. You can have time average or distance average. The result depends on which one you use.
I suppose the problem may belong to a textbook chapter about momentum where you have the equation
Fave= delta (mv).
In this case the average is over time.
And is the only equation in this chapter involving an average so there is no confusion... | 2018-05-20 10:49:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7774767279624939, "perplexity": 482.64048021402004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794863277.18/warc/CC-MAIN-20180520092830-20180520112830-00231.warc.gz"} |
https://github.com/dhermes/bossylobster-blog/blob/master/templated_content/2012-05-15-reverse-calculating-interest-rate.template | # dhermes/bossylobster-blog
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--- title: Reverse Calculating An Interest Rate date: 2012-05-15 author: Danny Hermes (dhermes@bossylobster.com) tags: Finance, Interest Rate, Mortgage, Newton-Raphson Method, Numerical Analysis, Python slug: reverse-calculating-interest-rate comments: true github_slug: templated_content/2012-05-15-reverse-calculating-interest-rate.template --- I was recently playing around with some loan data and only happened to have the term (or length, or duration) of the loan, the amount of the recurring payment (in this case monthly) and the remaining principal owed on the loan. I figured there was an easy way to get at the interest rate, but wasn't sure how. After some badgering from my coworker [+Paul](https://plus.google.com/104679465567407024302), I searched the web and found a [tool](http://www.calcamo.net/loancalculator/quickcalculations/loan-rate.php5) from [CALCAmo](http://www.calcamo.net/) (a site just for calculating amortizations). Problem solved, right? Wrong. I wanted to know why; I had to [go deeper](http://knowyourmeme.com/memes/we-need-to-go-deeper). So I did a bit of math and a bit of programming and I was where I needed to be. I'll break the following down into parts before going on full steam. - Break down the amortization schedule in terms of the variables we have and the one we want - Determine a function we want to find zeros of - Write some code to implement the Newton-Raphson method - Utilize the Newton-Raphson code to find an interest rate - Bonus: Analyze the function to make sure we are right Step I: Break Down the [Amortization Schedule][amort-sched] ----------------------------------------------------------- We can do this using the series {{ get_katex("\\left\\{P_i\\right\\}_i") }} of principal owed, which varies over time and will go to zero once paid off. In this series, {{ get_katex("P_0") }} is the principal owed currently and {{ get_katex("P_i") }} is the principal owed after {{ get_katex("i") }} payments have been made. (Assuming monthly payments, this will be after {{ get_katex("i") }} months.) If the term is {{ get_katex("T") }} periods, then we have {{ get_katex("P_T = 0") }}. We have already introduced the term {{ get_katex("T") }}; we also need the value of the recurring (again, usually monthly) payment {{ get_katex("R,") }} the interest rate {{ get_katex("r") }} and the initial principal owed {{ get_katex("P_0 = P") }}. #### Time-Relationship between Principal Values If after {{ get_katex("i") }} periods, {{ get_katex("P_i") }} is owed, then after one period has elapsed, we will owe {{ get_katex("P_i \\cdot m") }} where {{ get_katex("m = m(r)") }} is some multiplier based on the length of the term. For example if each period is one month, then we divide our rate by {{ get_katex("12") }} for the interest and add {{ get_katex("1") }} to note that we are adding to existing principal: {{ get_katex("m(r) = 1 + \\frac{r}{12}.", blockquote=True) }} In addition to the interest, we will have paid off {{ get_katex("R") }} hence {{ get_katex("P_{i + 1} = P_i \\cdot m - R.", blockquote=True) }} #### Formula for {{ get_katex("P_i") }} Using this, we can actually determine {{ get_katex("P_i") }} strictly in terms of {{ get_katex("m, R") }} and {{ get_katex("P") }}. First, note that {{ get_katex("\\begin{aligned} P_2 = P_1 \\cdot m - R &= (P_0 \\cdot m - R) \\cdot m - R \\\\ &= P \\cdot m^2 - R(m + 1) \\end{aligned}", blockquote=True) }} since {{ get_katex("P_0 = P") }}. We can show inductively that {{ get_katex("\\displaystyle P_i = P \\cdot m^i - R \\cdot \\sum_{j = 0}^{i - 1} m^j.", blockquote=True) }} We already have the base case {{ get_katex("i = 1,") }} by definition. Assuming it holds for {{ get_katex("i,") }} we see that {{ get_katex("\\begin{aligned} P_{i + 1} = P_i \\cdot m - R &= m \\cdot \\left(P \\cdot m^i - R \\cdot \\sum_{j = 0}^{i - 1} m^j\\right) - R \\\\ &= P \\cdot m^{i + 1} - R \\cdot \\sum_{j = 1}^{i} m^j - R, \\end{aligned}", blockquote=True) }} and our induction is complete. (We bump the index {{ get_katex("j") }} since we are multiplying each {{ get_katex("m^j") }} by {{ get_katex("m") }}.) Each term in the series is related to the previous one (except {{ get_katex("P_0,") }} since time can't be negative in this case). Step II: Determine a Function we want to find Zeros of ------------------------------------------------------ Since we know {{ get_katex("P_T = 0") }} and {{ get_katex("\\displaystyle P_T = P \\cdot m^T - R \\cdot \\sum_{j = 0}^{T - 1} m^j,") }} we actually have a polynomial in place that will let us solve for {{ get_katex("m") }} and in so doing, solve for {{ get_katex("r") }}. To make our lives a tad easier, we'll do some rearranging. First, note that {{ get_katex("\\displaystyle \\sum_{j = 0}^{T - 1} m^j =m^{T - 1} + \\cdots + m + 1 = \\frac{m^T - 1}{m - 1}.", blockquote=True) }} We calculate this sum of a geometric series here, but I'll just refer you to the [Wikipedia page](http://en.wikipedia.org/wiki/Geometric_series) instead. With this reduction we want to solve {{ get_katex("\\begin{aligned} & 0 = P \\cdot m^T - R \\cdot \\frac{m^T - 1}{m - 1} \\\\ & \\Longleftrightarrow P \\cdot m^T \\cdot (m - 1) =R \\cdot(m^T - 1). \\end{aligned}", blockquote=True) }} With that, we have accomplished Step II, we have found a function (parameterized by {{ get_katex("P, T") }} and {{ get_katex("R") }} which we can use zeros from to find our interest rate: {{ get_katex("\\begin{aligned} f_{P, T, R}(m) &= P \\cdot m^T \\cdot (m - 1) -R \\cdot(m^T - 1) \\\\ &= P \\cdot m^{T + 1} - (P + R) \\cdot m^T + R. \\end{aligned}", blockquote=True) }} Step III: Write some code to implement the [Newton-Raphson method][newton-raph] --------------------------------------------------------------------- We use the Newton-Raphson method to get super-duper-close to a zero of the function.For in-depth coverage, see the Wikipedia page on the Newton-Raphson method, but I'll give some cursory coverage below. The methods used to show that a fixed point is found are not necessary for the intuition behind the method. #### Intuition behind the method For the intuition, assume we know (and can compute) a function {{ get_katex("f,") }} its derivative {{ get_katex("f'") }} at a value {{ get_katex("x") }}. Assume there is some zero {{ get_katex("y") }} nearby {{ get_katex("x") }}. Since they are close, we can approximate the slope of the line between the points {{ get_katex("(x, f(x))") }} and {{ get_katex("(y, f(y))") }} with the derivative nearby. Since we know {{ get_katex("x,") }} we use {{ get_katex("f'(x)") }} and intuit that {{ get_katex("f'(x) = \\text{slope} = \\frac{f(y) - f(x)}{y - x} \\Rightarrow y - x = \\frac{f(y) - f(x)}{f'(x)}.", blockquote=True) }} But, since we know that {{ get_katex("y") }} is a zero, {{ get_katex("f(y) - f(x) = -f(x)") }} hence {{ get_katex("y - x = \\frac{-f(x)}{f'(x)} \\Rightarrow y = x - \\frac{f(x)}{f'(x)}.", blockquote=True) }} Using this method, one can start with a given value {{ get_katex("x_0 = x") }} and compute better and better approximations of a zero via the iteration above that determines {{ get_katex("y") }}. We use a sequence to do so: {{ get_katex("x_{i + 1} = x_i - \\frac{f(x_i)}{f'(x_i)}", blockquote=True) }} and stop calculating the {{ get_katex("x_i") }} either after {{ get_katex("f(x_i)") }} is below a preset threshold or after the fineness of the approximation {{ get_katex("\\left|x_i - x_{i + 1}\\right|") }} goes below a (likely different) preset threshold. Again, there is much that can be said about these approximations, but we are trying to accomplish things today, not theorize. **Programming Newton-Raphson** To perform Newton-Raphson, we'll implement a Python function that takes the initial guess {{ get_katex("x_0") }} and the functions {{ get_katex("f") }} and {{ get_katex("f'") }}. We'll also (arbitrarily) stop after the value {{ get_katex("f(x_i)") }} drops below {{ get_katex("10^{-8}") }} in absolute value. python def newton_raphson_method(guess, f, f_prime): def next_value(value): return value - f(value)*1.0/f_prime(value) current = guess while abs(f(current)) > 10**(-8): current = next_value(current) return current As you can see, once we have f and f_prime, everything else is easy because all the work in calculating the next value (via next_value) is done by the functions. Step IV: Utilize the Newton-Raphson code to find an Interest Rate ----------------------------------------------------------------- We first need to implement {{ get_katex("f_{P, T, R}(m) = P \\cdot m^{T + 1} - (P + R) \\cdot m^T + R") }} and {{ get_katex("f'_{P, T, R}") }} in Python. Before doing so, we do a simple derivative calculation: {{ get_katex("f_{P, T, R}'(m) = P \\cdot (T + 1) \\cdot m^T - (P + R) \\cdot T \\cdot m^{T - 1}.", blockquote=True) }} With these [formulae](http://dictionary.reference.com/browse/formulae) in hand, we write a function which will spit out the corresponding f and f_prime given the parameters {{ get_katex("P") }} (principal), {{ get_katex("T") }} (term) and {{ get_katex("R") }} (payment): python def generate_polynomials(principal, term, payment): def f(m): return (principal*(m**(term + 1)) - (principal + payment)*(m**term) + payment) def f_prime(m): return (principal*(term + 1)*(m**term) - (principal + payment)*term*(m**(term - 1))) return (f, f_prime) Note that these functions only take a single argument (m), but we are able to use the other parameters from the parent scope beyond the life of the call to generate_polynomials due to [closure](http://en.wikipedia.org/wiki/Closure_(computer_science)) in Python. In order to solve, we need an initial guess, but we need to know the relationship between {{ get_katex("m") }} and {{ get_katex("r") }} before we can determine what sort ofguessmakes sense. In addition, once a value for {{ get_katex("m") }} is returned from Newton-Raphson, we need to be able to turn it into an {{ get_katex("r") }} value so functions m and m_inverse should be implemented. For our dummy case here, we'll assume monthly payments (and compounding): python def m(r): return 1 + r/12.0 def m_inverse(m_value): return 12.0*(m_value - 1) Using these, and assuming that an interest rate of **10%** is a good guess, we can put all the pieces together: python def solve_for_interest_rate(principal, term, payment, m, m_inverse): f, f_prime = generate_polynomials(principal, term, payment) guess_m = m(0.10) # ten percent as a decimal m_value = newton_raphson_method(guess_m, f, f_prime) return m_inverse(m_value) To check that this makes sense, let's plug in some values. Using the [bankrate.com loan calculator](http://www.bankrate.com/calculators/mortgages/mortgage-calculator.aspx), if we have a 30-year loan (with {{ get_katex("12 \\cdot 30 = 360") }} months of payments) of $100,000 with an interest rate of 7%, the monthly payment would be$665.30. Plugging this into our pipeline: python >>> principal = 100000 >>> term = 360 >>> payment = 665.30 >>> solve_for_interest_rate(principal, term, payment, m, m_inverse) 0.0699996284703 And we see the rate of 7% is approximated quite well! Bonus: Analyze the function to make sure we are right ----------------------------------------------------- Coming soon. We will analyze the derivative and concavity to make sure that our guess yield the correct (and unique) zero. [amort-sched]: http://en.wikipedia.org/wiki/Amortization_schedule [newton-raph]: http://en.wikipedia.org/wiki/Newton's_method | 2018-12-17 11:57:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6204420924186707, "perplexity": 5034.234574323704}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376828507.57/warc/CC-MAIN-20181217113255-20181217135255-00404.warc.gz"} |
https://deustotech.github.io/DyCon-Blog/tutorial/wp03/P0009 | # Control of reaction-diffusion under state constraints - Numerical exploration of controls
Author: - 04 April 2020
##### Control of reaction-diffusion
This is a post in a collection of several on control of reaction-diffusion under state constraints
In this tutorial we will observe different phenomena that arise in the control with state-constraints and moreover we will observe different ways to approach a desired target.
# Observation of several phenomena
In the simulations below, we will consider the following optimal control problem:
In Figure 1, the initial state was $u_0=1$ and we can observe how the solution has been successfully driven to the target $u_1=0.33$. For this length of the domain there is no barrier (see Barriers)), for this reason the optimal solution can achieve the target.
The trajectory of the control can also be understood from the theoretical understanding of the controllability under state constraints.
In the first phase, the control takes values that are below the target $u_1=0.33$. This behavior is necessary because for this length of the domain there exist nontrivial solutions with boundary value $0.33$ that are between $0$ and $1$.
These nontrivial solutions do not constitute an intrinsic obstruction since we know that the equation is controllable for this length of the domain (see [1,3] and Staircase post and Barriers post). However, the strategy of setting boundary equal to $0.33$ for a long time and then doing a local control will not work.
The second oscillatory phase has been already observed in the unconstrained case (see [4]).
Figure 1. At the left, the controlled state, at the right, the control function. $u_0=1$ and $u_1=0.33$.
Another important feature is the existence of a minimal controllability time. In Figure 2, $u_0=0$ and the target is $u_1=0.33$. From the theoretical perspective, if the time horizon is large enough we are always able to go from the initial state to the target. This is due to the existence of an admissible continuous path of steady states (Staircase post).
However, in Figure 2, one observes that the target $u_1=0.33$ has been not achieved. The reason is that $T$ is not large enough for the equation to be controllable.
Figure 2. At the left, the controlled state displayed in space-time, at the right, different snapshots of the state, darker curves are associated with larger times. $u_0=0$ and $u_1=0.33$ .
The other phenomenology already mentioned above and in the blog post Barriers is the lack of controllability due to the emergence of nontrivial solutions. In Figure 3, the initial state is $u_0=1$ and the target is $u_1=0$. We can observe how the state gets stuck to the barrier.
Figure 3. At the left, the controlled state displayed in space-time, at the right, different snapshots of the state, darker curves are associated with larger times. $u_0=1$ and $u_1=0$ .
## Exploration
Now we turn our attention to the exploration of different control strategies.
### Quasistatic
In Figure 4, we seek to find a control that is quasistatic, for doing so we penalize the time derivative of the control in the discrete level and we add the restriction $u(x,T)=0.33$.
Figure 4. At the left, the controlled state displayed in space-time, at the right, the control function. $u_0=0$ and $u_1=0.33$ .
# Minimal control time
In Figure 5, we compute the control in minimal time by minimizing the functional:
Figure 5. At the left, the controlled state displayed in space-time, at the right, the control function. $u_0=0$ and $u_1=0.33$ .
Figure 5 points out that the control in minimal time has the bang-bang property.
### Minimal flow control
In Figure 6, we compute the control with minimal flow by minimizing the functional:
Figure 6.
## References:
[1] D. Ruiz-Balet and E. Zuazua. Control of certain parabolic models from biology and social sciences. Preprint available at https://cmc.deusto.eus/domenec-ruiz-balet/.
[2] J.-M. Coron, E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equa- tions, SIAM J. Control. Optim. 43 (2) (2004) 549–569.
[3] C. Pouchol, E. Trélat, E. Zuazua, Phase portrait control for 1d monostable and bistable reac tion–diffusion equations, Nonlinearity 32 (3) (2019) 884–909.
[4] R. Glowinski, J. L. Lions and J. He. “Exact and approximate controllability for distributed parameter systems: a numerical approach.” Encyclopedia of Mathematics and its Applications (2008). | 2021-04-20 13:03:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7883497476577759, "perplexity": 588.7242124444017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039398307.76/warc/CC-MAIN-20210420122023-20210420152023-00598.warc.gz"} |
https://learn.careers360.com/ncert/ncert-solutions-class-7-science-chapter-10-respiration-in-organisms/ | # NCERT Solutions for Class 7 Science Chapter 10 Respiration in Organisms
NCERT solutions for class 7 science chapter 10 respiration in organisms: This chapter is very important, as in this you will learn about respiration. It is essential for the survival of living organisms and it helps in the release of energy from food. In the CBSE NCERT solutions for class 7 science chapter 10 respiration in organisms, you will get questions from different types of respiration and its processes. You will study that oxygen you inhale is used to breakdown glucose into carbon dioxide along with water and in this process energy releases. This breakdown of glucose occurs in the cells of an organism, that's why it is also called cellular respiration and if the food is broken down with the help of oxygen, then it will be called aerobic respiration and without the use of oxygen, it will become anaerobic respiration. In solutions of NCERT for Class 7 Science Chapter 10 Respiration in Organisms, you have to study breathing, as it is a part of the process of respiration during which an organism takes in the oxygen-rich air and gives out air rich in carbon dioxide, as based on this you will get questions in NCERT solutions for class 7 science chapter 10 respiration in organisms. If you need help in finding solutions of NCERT book to any other class, then don't worry NCERT solutions will help you out.
## Here are the important topics of CBSE NCERT solutions for class 7 science chapter 10 respiration in organisms mentioned below:
10.1 Why Do We Respire?
10.2 Breathing
10.3 How Do We Breathe?
10.4 What Do We Breathe Out?
10.5 Breathing in Other Animals
10.6 Breathing Under Water
10.7 Do Plants Also Respire?
After going through the solutions for NCERT class 7 science chapter 10 respiration in organisms, you must be able to understand all the answers the following questions:
NCERT solutions for class 7 science chapter 10 respiration in organisms: Solved Exercise Questions:
The muscles of an athlete undergo anaerobic respiration which means the oxygen is used up faster than being supplied by the blood.
Thus, to inhale more oxygen, an athlete breathes faster and deeper than usual after finishing the race.
The similarities between aerobic and anaerobic respiration are:
(i) Carbon dioxide is produced in both reactions.
(ii) Energy is released in both reactions.
The differences between aerobic and anaerobic respiration are:
Aerobic Anaerobic Takes place in the presence of oxygen Takes place without oxygen Complete break down of food Partial break down of food Carbon dioxide and water as the end product Ethanol, carbon dioxide as the end product(in yeast) Produces more energy Produces less energy
Sneezing removes the foreign particle like dust particles, smoke from the air inhaled which ensures good air quality inside our body.
When we inhale, unwanted particles get trapped in the hair present in our nasal cavity. However, sometimes these particles may get past the hair causing irritation in the lining of the cavity. This causes sneezing.
We know,
Snail will inhale oxygen and exhale carbon dioxide whereas the plant will inhale carbon dioxide for respiration.
The test with snail and without plant will have maximum carbon dioxide concentration. Therefore, the order of test tube containing $CO_{2}$ is $A>C>B$.
Hence, test-tube A will have the highest concentration of $CO_{2}$.
(a) In cockroaches, air enters the body through
(i) lungs
(ii) gills
(iii) spiracles
(iv) skin
A cockroach has small openings called spiracles on the sides of its body.
Therefore, (a) In cockroaches, air enters the body through (iii) spiracles
(i) carbon dioxide
(ii) lactic acid
(iii) alcohol
(iv) water
(b) During heavy exercise, we get cramps in the legs due to the accumulation of (ii) lactic acid
During heavy exercise, muscle cells respire anaerobically because oxygen is used faster as compared to its supply by the blood. The partial breakdown of glucose produces lactic acid whose accumulation causes muscle cramps.
(c) Normal range of breathing rate per minute in an average adult person at rest is:
(i) 9–12
(ii) 15–18
(iii) 21–24
(iv) 30–33
Normal range of breathing rate per minute in an average adult person at rest is (ii) 15–18
This means on an average, an adult human being breathes in and out 15–18 times in a minute.
(d) During exhalation, the ribs
(i) move outwards
(ii) move downwards
(iii) move upwards
(iv) do not move at all
During exhalation, the ribs (ii) move downwards
During exhalation, ribs move down and inwards reducing the size of the chest cavity causing the air to be pushed out of the lungs.
## Q6. Match the items in Column I with those in Column II:
Column I Column II
(a) Yeast (i) Earthworm
(b) Diaphragm (ii) Gills
(c) Skin (iii) Alcohol
(d) Leaves (iv) Chest cavity
(e) Fish (v) Stomata
(f) Frog (vi) Lungs and skin
(vii) Tracheae
Column I Column II (a) Yeast (iii) Alcohol (b) Diaphragm (iv) Chest cavity (c) Skin (i) Earthworm (d) Leaves (v) Stomata (e) Fish (ii) Gills (f) Frog (vi) Lungs and skin
(i) During heavy exercise the breathing rate of a person slows down. (T/F)
(ii) Plants carry out photosynthesis only during the day and respiration only at night. (T/F)
(iii) Frogs breathe through their skins as well as their lungs. (T/F)
(iv) The fishes have lungs for respiration. (T/F)
(v) The size of the chest cavity increases during inhalation. (T/F)
(i) During heavy exercise the breathing rate of a person slows down. False
- During heavy exercise, we breathe faster and deeper than usual
(ii) Plants carry out photosynthesis only during the day and respiration only at night. False
- Respiration is an essential process required for the plants to live. It occurs day and night. Whereas, photosynthesis takes place only during the day.
(iii) Frogs breathe through their skins as well as their lungs. True
(iv) The fishes have lungs for respiration. False
- Fishes do not have lungs. They breathe through gills.
(v) The size of the chest cavity increases during inhalation. True
- During inhalation, ribs move up and outwards and the diaphragm moves down. This movement increases space in our chest cavity and air rushes into the lungs.
(i) The air tubes of insects
(ii) Skeletal structures surrounding chest cavity
(iii) Muscular floor of chest cavity
(iv) Tiny pores on the surface of leaf
(v) Small openings on the sides of the body of an insect
(vi) The respiratory organs of human beings
(vii) The openings through which we inhale
(viii) An anaerobic organism
(ix) An organism with tracheal system
(i) The air tubes of insects - TRACHEA
(ii) Skeletal structures surrounding chest cavity - RIBS
(iii) Muscular floor of chest cavity - DIAPHRAGM
(iv) Tiny pores on the surface of leaf - STOMATA
(v) Small openings on the sides of the body of an insect - SPIRACLES
(vi) The respiratory organs of human beings - LUNGS
(vii) The openings through which we inhale - NOSTRILS
(viii) An anaerobic organism - YEAST
(ix) An organism with tracheal system - ANT
## Q9. The mountaineers carry oxygen with them because:
(a) At an altitude of more than 5 km there is no air.
(b) The amount of air available to a person is less than that available on the ground.
(c) The temperature of air is higher than that on the ground.
(d) The pressure of air is higher than that on the ground.
The mountaineers carry oxygen with them because the amount of oxygen availability to a person is less than that available on the ground. (b)
As we move higher in altitude, the density of the atmosphere lessens.
If you are facing any issue in understanding the answers of the above questions, then don't worry and again go through that NCERT Book and try to understand all the concepts of every topic and then write the answers of all these questions by your own and then compare them with NCERT solutions for class 7 science chapter 10 respiration in organisms. You will see that your knowledge, as well as the understanding of these concepts, will increase and you will be able to explain the answers in a better way.
## NCERT Solutions for Class 7 Science – Chapter-wise
Chapter 1 Solutions of NCERT for class 7 science chapter 1 Nutrition in Plants Chapter 2 CBSE NCERT solutions for class 7 science chapter 2 Nutrition in Animals Chapter 3 NCERT solutions for class 7 science chapter 3 Fibre to Fabric Chapter 4 Solutions of NCERT for class 7 science chapter 4 Heat Chapter 5 CBSE NCERT solutions for class 7 science chapter 5 Acids, Bases and Salts Chapter 6 NCERT solutions for class 7 science chapter 6 Physical and Chemical Changes Chapter 7 Solutions of NCERT for class 7 science chapter 7 Weather, Climate and Adaptations of Animals to Climate Chapter 8 CBSE NCERT solutions for class 7 science chapter 8 Winds, storms and cyclones Chapter 9 NCERT solutions for class 7 science chapter 9 Soil Chapter 10 NCERT solutions for class 7 science chapter 10 Respiration in Organisms Chapter 11 CBSE NCERT solutions for class 7 science chapter 11 Transportation in Animals and Plants Chapter 12 NCERT solutions for class 7 science chapter 12 Reproduction in Plants Chapter 13 Solutions of NCERT for class 7 science chapter 13 Motion and Time Chapter 14 CBSE NCERT solutions for class 7 science chapter 14 Electric Current and its Effects Chapter 15 NCERT solutions for class 7 science chapter 15 Light Chapter 16 Solutions of NCERT for class 7 science chapter 16 Water: A Precious Resource Chapter 17 CBSE NCERT solutions for class 7 science chapter 17 Forests: Our Lifeline Chapter 18 NCERT Solutions for Class 7 Science Chapter 18 Wastewater Story
## NCERT Solutions for Class 7- Subject Wise
NCERT Solutions for Class 7 Maths NCERT Solutions for Class 7 Science
## What are the benefits of NCERT solutions for class 7 science chapter 10 respiration in organisms:
• Solutions for NCERT class 7 science chapter 10 respiration in organisms will also boost your knowledge.
• You will get all the answers to this chapter and it will help you to score good marks in your school exam.
• NCERT is the base of your learning.
• NCERT solutions for class 7 science chapter 10 respiration in organisms will also help you to understand all the concepts easily. | 2020-02-28 16:36:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33469074964523315, "perplexity": 5189.324068623337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147234.52/warc/CC-MAIN-20200228135132-20200228165132-00304.warc.gz"} |
https://www.cut-the-knot.org/arithmetic/algebra/HungNguyenViet.shtml | An Inequality in Reciprocals
Problem
Leo Giugiuc has kindly communicated to me the following problem, along with an elegant solution. The problem is due to Hung Nguyen Viet, Hanoi, Vietnam.
Preliminaries
Introduce function $f:\,[0, \infty)^3\rightarrow\mathbb{R},\;$ with
\begin{align} f(u, v, w) = 4(u^3+v^3+w^2)&+12(u^2w+v^2u+w^2v)\\ &-15(u^2v+v^2w+w^2u)-3uvw. \end{align}
Since $f\;$ is cyclic and homogeneous polynomial of degree three, then, according to the Pham Kim Hung's theorem, $f(u,v,w)\ge 0\;$ for all $u,v,w\ge 0\;$ if and only if $f(1,1,1)\ge 0;$ and, for all $u\ge 0,\;$ $f(u, 1,0)\ge 0.\;$ This is indeed so in our case: $f(1,1,1)=0\ge 0\;$ and $f(u,1,0)=(u-2)^2(4u+1)\ge 0,\;$ for $u\ge 0.$
Thus, we conclude that
$\displaystyle \frac{4}{3}(u^3+v^3+w^2)+4(u^2w+v^2u+w^2v)\ge 5(u^2v+v^2w+w^2u)+uvw,$
for all $u,v,w\ge 0.$
Solution
So, for $t\in (0,1],$
\displaystyle\begin{align} \frac{4}{3}(t^{3a}+t^{3b}+t^{3c})&+4(t^{a+2b}+t^{b+2c}+t^{c+2a})\\ &\ge 5(t^{2a+b}+t^{2b+c}+t^{2c+a})+t^{a+b+c}. \end{align}
Next,
$\displaystyle\int^1_0\frac{1}{t}\left[\frac{4}{3}(t^{3a}+t^{3b}+t^{3c})+4(t^{a+2b}+t^{b+2c}+t^{c+2a})\right]\\ \displaystyle\qquad\;\space\;\;\ge \int^1_0\frac{1}{t}\left[5(t^{2a+b}+t^{2b+c}+t^{2c+a})+t^{a+b+c}\right].$
computing which,
\displaystyle\begin{align}\frac{4}{9}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)&+4\left(\frac{1}{a+2b}+\frac{1}{b+2c}+\frac{1}{c+2a}\right)\\ &\ge 5\left(\frac{1}{2a+b}+\frac{1}{2b+c}+\frac{1}{2c+a}\right)+\frac{1}{a+b+c}. \end{align}
Illustration
Nassim Nicholas Taleb has kindly provided the following amazing illustration:
[an error occurred while processing this directive] | 2018-08-15 10:21:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6444053053855896, "perplexity": 1516.3361210394353}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210040.24/warc/CC-MAIN-20180815083101-20180815103101-00631.warc.gz"} |
https://www.gradesaver.com/textbooks/business/accounting/accounting-tools-for-business-decision-making-5th-edition/chapter-1-introduction-to-financial-statements-problems-set-b-page-37/p1-4b | ## Accounting: Tools for Business Decision Making, 5th Edition
a) Preacher Corporation Statement of Cash Flows December 31, 2014 Cash flows from operating activities $\ \ \ \ \$ Cash receipts from operating activities $\ \ \ \ \ \ \ \ \ \$ $\$162,000\ \ \ \ \ $Cash payments for operating activities$\ \ \ \ \ \ \ \,\ \ \ \underline{(154,000)}\ \ \ \ \ \ \ \ \ \ $Net cash provided by operating activities$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$\ \ \ \ 8,000$ Cash flows from investing activities $\ \ \ \ \$ Cash paid to purchase equipment $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ \underline{(20,000)}$ $\ \ \ \ \$ $\ \ \ \ \$ Net cash used by investing activities $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ (20,000)$ Cash flows from financing activities $\ \ \ \ \$ Issuance of common stock $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ \ \ \ \ \ \ \ \ \ \ 0$ $\ \ \ \ \$ Issuance of bonds payable $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\ \ \ 40,000$ $\ \ \ \ \$ Dividends paid $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\underline{\ \ \ \ \ (2,000)}$ $\ \ \ \ \$ $\ \ \ \ \$ Net cash provided by financing activities $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\underline{38,000}$ Net increase in cash $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $26,000$ Cash at beginning of period $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\underline{11,000}$ Cash at end of period $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $\underline{\underline{37,000}}$ b) Cash on hand increased, but only because the company incurred debt in the form of bonds. | 2018-07-20 09:14:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6540283560752869, "perplexity": 931.6489169553754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591575.49/warc/CC-MAIN-20180720080634-20180720100634-00079.warc.gz"} |
https://scicomp.stackexchange.com/tags/fluid-dynamics/new | # Tag Info
Generally, the viscosity we speak about (which is linked to the relax. time) is the kinematic viscosity $\nu$, not $\mu$ as you write. So, by replacing, you will find the right expression. | 2020-01-21 06:00:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6570085883140564, "perplexity": 610.2157550041176}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601615.66/warc/CC-MAIN-20200121044233-20200121073233-00423.warc.gz"} |
https://www.r-bloggers.com/2013/12/setting-axis-limits-on-ggplot-charts/ | Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
I’ve been doodling some chart in R/ggplot using geom_text() to generate a labelled scatterplot.
The chart actually builds up several layers using different datasets, so it’s not obvious how to set the ranges cleanly: I know the lower bound I want for the y-axis (y=0), but I want to let the upper bound float.
There’s also an issue with the labels overflowing the edges left and right.
So here are a couple of lines to make everything better (chart is in g):
#Find the current ymax value for upper bound
#(via http://stackoverflow.com/questions/7705345/how-can-i-extract-plot-axes-ranges-for-a-ggplot2-object#comment24184444_8167461 )
gy=ggplot_build(g)$panel$ranges[[1]]\$y.range[2]
g=g+ylim(0,gy)
#Handle the overflow by expanding the x-axis
g=g+scale_x_continuous(expand=c(0.1,0)) | 2021-05-13 08:47:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4408608078956604, "perplexity": 2014.6229137370224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243990584.33/warc/CC-MAIN-20210513080742-20210513110742-00382.warc.gz"} |
https://bodheeprep.com/cat-2017-question-solution-36 | Bodhee Prep-CAT Online Preparation
| Best Online CAT PreparationFor Enquiry CALL @ +91-95189-40261
# CAT 2017 [slot 2] Question with solution 02
Question 36:
In a 10 km race, A, B, and C, each running at uniform speed, get the gold, silver, and bronze medals, respectively. I f A beats B by 1 km and B beats C by 1 km, then by how many metres does A beat C?
Explanation:
By the time A traveled 10 KM, B traveled 9 KM
Hence $Speed_A : Speed_B = 10:9$
Similarly $Speed_B : Speed_C = 10:9$
Hence $Speed_A : Speed_B : Speed_C = 100:90:81$
Hence by the time A traveled 10 KMs , C should have traveled 8.1 KMs
So A beat C by 1.9 KMs = 1900 Mts
Previous QuestionNext Question
### CAT 2023Classroom Course
We are starting classroom course for CAT 2023 in Gurugram from the month of December. | 2022-12-05 14:39:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2901119589805603, "perplexity": 8331.702293918888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00724.warc.gz"} |
https://code.bioconductor.org/browse/GMRP/blob/master/man/snpPositAnnot.Rd | \name{snpPositAnnot}
\alias{snpPositAnnot}
\title{
\emph{SNP} Position Annotation
}
\description{
This function is used to perform position annotation analysis of \verb{SNP}s chosen from \verb{GWAS}.
}
\usage{
snpPositAnnot(SNPdata, SNP_hg19="chr", main)
}
\arguments{
\item{SNPdata}{
\verb{SNPdata} may be hg19 that is a string vector(chr##.########) or two numeric vectors (chromosome number and SNP position).
}
\item{SNP_hg19}{
a string parameter. It may be "hg19" or "chr". If SNP_hg19="hg19",then \verb{SNPdata} contain a string vector of hg19 or if \verb{SNP_hg19}="chr", then \verb{SNPdata} consist of at lest two columns: \verb{chr} and \verb{posit}. \verb{chr} is chromosome number and posit is \verb{SNP} physical position on chromosomes. If data sheet has chromosome X, then character "X" should be changed to 23 in chr vector or chr23.######## in hg19 vector.
}
\item{main}{
a string which is title of graph. If no title is given, then main="".
}
}
\value{
Return a set of numbers of SNPs between which interval length > \bold{LG} on 23 chromosomes. This function also creates a histogram for averaged distances between SNPs and SNP numbers on chromosomes.
}
\author{
Yuan-De Tan
\email{tanyuande@gmail.com}
}
\note{
This function can also be applied to hg18 data with \code{SNP_hg19}="hg18".
}
\seealso{ | 2023-03-25 07:55:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43343493342399597, "perplexity": 12912.570659665646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945317.85/warc/CC-MAIN-20230325064253-20230325094253-00099.warc.gz"} |
https://indico.cern.ch/event/299180/contributions/1659568/ | # TWEPP 2014 - Topical Workshop on Electronics for Particle Physics
Sep 22 – 26, 2014
Centre des Congrès - Aix en Provence, France
Europe/Zurich timezone
## Development of Front-end Electronics for LumiCal Detector in CMOS 130 nm Technology
Sep 24, 2014, 2:00 PM
25m
Centre des Congrès - Aix en Provence, France
#### Centre des Congrès - Aix en Provence, France
14 boulevard Carnot 13100
Oral ASICs
### Speaker
Jakub Moron (AGH University of Science and Technology (PL))
### Description
The design and preliminary measurement results of a multichannel, variable gain front-end electronics for LumiCal detector at future Linear Collider are presented. The 8-channel prototype was designed and fabricated in a 130 nm CMOS technology. Each channel comprises a charge sensitive preamplifier and CR-RC shaper with pole-zero cancellation circuit. Measurement results confirm full functionality of the prototype and compliance with all requirements imposed by the detector specification. Power consumption of the front-end is around 1.5 mW per channel with 50 ns peaking time and noise ENC below 900 e$^{-}$.
### Summary
In the very forward region of a future Linear Collider a special kind of calorimeter is foreseen - the Luminosity Calorimeter (LumiCal). The LumiCal will measure the luminosity with a precision of better than 0.001 at 500 GeV centre-of-mass energy and 0.003 at 1 TeV centre-of-mass energy. The sandwich type calorimeter consist of tungsten absorber disks of 3.5 mm thickness, corresponding to one radiation length, interspersed with sensor layers. Each sensor layer is segmented radially and azimuthally into pads. Due to the high occupancy originating from beamstrahlung and two-photon processes, the LumiCal needs a fast readout.
The variable gain, 8-channel front-end electronics was developed for the LumiCal. The ASIC was designed, simulated and fabricated in 130 nm CMOS technology. Each channel comprises a charge sensitive preamplifier and a CR-RC shaper with pole-zero cancellation circuit. The shaper peaking time of about 50 ns was chosen to fulfill the timings requirements. Two charge gain modes were implemented - the physics mode and the calibration mode. In the physics mode (low gain) the detector should be sensitive to electromagnetic showers resulting in a high energy deposition and the front-end should process signals up to about 6 pC per channel. In the calibration mode (high gain) it should detect signals from relativistic muons, i.e. should be able to register the minimum ionizing particles (MIPs). The proposed sensor geometry results in a wide range of capacitive load (5 pF - 35 pF) connected to a single front-end channel. Because of expected high occupancy per channel, the front-end should be fast enough to resolve signals from subsequent beam bunches separated in time of about 350 ns. Severe requirements on power dissipation of readout electronics may be strongly relaxed if the power is switched off between bunch trains, i.e. the power pulsing is applied. This is feasible since in the ILC experiment 200 ms pause is foreseen after each 1 ms length bunch train. The power pulsing was implemented in the designed prototype.
Measurements show that the ASIC is functional and fulfils the requirements. The measured charge gain in the calibration mode is around 4.23 mV/fC, with dynamic range up to 100 fC, and 0.1 mV/fC in the physics mode, with dynamic range up to 6 pC. The peaking time of around 52 ns was obtained in both modes. The designed front-end works with detector capacitance in the range of 0 to 50 pF. The measured noise ENC at 10 pF detector capacitance is below 900 e$^{-}$ which gives the signal to noise ratio (SNR) above 20. The measured power consumption of the prototype is around 1.5 mW per channel at typical biasing. Using the power pulsing, with a duty factor of 1/200, the average power consumption is reduced to around 10 $\mu$W per channel. After power cycling the ASIC recovers completely within 1 us.
### Primary authors
Jakub Moron (AGH University of Science and Technology (PL)) Krzysztof Piotr Swientek (AGH University of Science and Technology (PL)) Marek Idzik (AGH University of Science and Technology (PL)) Miroslaw Firlej (AGH University of Science and Technology (PL)) Tomasz Fiutowski (AGH University of Science and Technology)
### Presentation materials
Paper Slides | 2021-10-18 04:50:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5434709191322327, "perplexity": 4578.765863650786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585196.73/warc/CC-MAIN-20211018031901-20211018061901-00558.warc.gz"} |
https://ve7xen.com/blog/2012/02/07/antennas-ordered--future-projects/ | # VE7XEN's electronics blog
## Antennas Ordered & Future Projects
Shortly after getting my ham license I started fooling around with antennas. Built myself a very shoddy 2m ground plane 1/4 wave just to get an antenna outside for use with my HT, as the whips can't get anywhere from inside. Then I decided to upgrade to a proper antenna - a coaxial dipole design by W6NBC published in the July 2009 QST. This worked fantastically well for a couple of months, then a solder joint broke between the antenna connector and the radiator element. I've repaired it, waterproofed it better and it just doesn't work anymore - I'm scared to transmit into it as the receive signal is so bad. I've taken it apart several times and reflowed the problematic solder joint to no avail - it seems something inside the lamp-tube coax has gone wrong. Given how much grief this design has given me I decided to just pay for a commercial design that will work properly. I considered building myself one of the popular copper pipe J-poles, but without equipment to properly tune it, and with the price of copper it's as expensive as buying one and not nearly as foolproof. I want something a little more substantial than the ladder-line J-pole design.
After a look around at some of the inexpensive designs, I thought I'd stick with the J-pole design and go for an Arrow OSJ 146/440 dual-band J-pole. Based on the reviews this seems to be a well-built, well-performing antenna that should be small enough not to be obtrusive. The only other serious options at a similar price were poorly-constructed ground planes. Anything else decent was about double the price. So I'll give a try, it was only $40 after all. I just have to remember to take care to waterproof the coax connection... Since I was making an order with a fairly substantial shipping cost, I figured I'd also pick up something else that I want for an upcoming project - a cheap, small, easy to hide VHF antenna that doesn't need great performance. I'm working on a from-scratch APRS tracker design (and making good progress!), and would like my own install to be somewhat 'stealth'. I've selected a small through-the-glass design by Workman, the WEP2000. This antenna probably sucks, but with the 25W radio I plan to use, and just for APRS tracking, it should do the job and fit the criteria of being simple to install and hide-able enough to be less-than-obvious. Perhaps I will swap it for a simple HT antenna used internally to the car later, and use this one for a real radio. We'll see how it performs... it can't be worse than the HT+whip I'm current using while mobile! This little guy was only$25, including a whack of coax pre-attached.
I've also begun construction on the UHFSDR by WB6DHW, which is a 1.75MHz - 700MHz RX/TX SDR based around the Si570. I'll need to add some filtering and a poweramp for serious use, but it should be fun to play with even on its own. Stay tuned for more on that project, as well as my APRS tracker once I get far enough to begin real testing on that. | 2020-02-22 19:27:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3666468560695648, "perplexity": 2020.4355658322204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145713.39/warc/CC-MAIN-20200222180557-20200222210557-00108.warc.gz"} |
https://brilliant.org/problems/a-b-c-6/ | # A, B, C
Geometry Level 3
In right angled triangle $$ABC$$, $$\angle ACB=90^\circ$$ . If $$2\tan A = \sin B$$ , what does $$\sin A$$ equal?
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https://cdl-prep.com/oregon-air-brake-practice-test | # Free Oregon CDL Air Brakes Practice Test 2023
Do you need an Oregon Air Brakes endorsement or an L endorsement for your commercial driving license? The Oregon CDL Air Brake has some differences from other endorsements because your license will receive a mark of restriction if you fail the test. So having good preparation before exam day is very necessary. To ensure that our questions are relevant, all of our CDL practice test packs are based on the Oregon CDL manual. Each question has a detailed explanation for you to thoroughly learn the format and the topic. Don't be afraid of having a restriction on your license. Let’s try our Oregon CDL Practice Test to get ready to pass the Oregon CDL Air Brake Test now.
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OR CDL Air Brakes Test format:
25 questions
80% passing score
List of questions
1.
Air brakes consist of ________.
2.
When inspecting air brakes:
3.
The safety valve discharges automatically at the pressure of:
4.
An air brake system is fully charged at what psi?
5.
The spring brakes used on the chambers in a straight truck will bring you to a stop when air pressure drops below ___ psi.
6.
The supply pressure gauge shows how much pressure:
7.
The main factor in causing brake fade is:
8.
9.
To make sure that the spring brakes come on automatically, do all of the following except:
10.
When testing air leakage rates for single vehicles, initial loss rate should be less than:
11.
When the safety valve releases air this indicates ___________.
12.
Another name for the brake pedal is the "foot valve", or:
13.
What should a driver do when the low air pressure warning signal comes on?
14.
Some vehicles are equipped with a control allowing you to apply the spring brakes gradually, called a:
15.
If your truck has a dual air brake system, and one of the systems is low on pressure:
16.
A device that limits the amount of air provided to the front brakes is a:
17.
Brake drums (or discs) must not have cracks longer than ____ the width of the friction area.
18.
If one air system is very low on pressure _______.
19.
If your vehicle has an alcohol evaporator, the evaporator's purpose is to _____.
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