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kavyamanohar.com | 1,726,337,916,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651580.73/warc/CC-MAIN-20240914161327-20240914191327-00691.warc.gz | 313,795,810 | 12,491 | # Information, Entropy and Malayalam
It was during my undergraduate course, I was awestruck by the idea that Information is quantifiable. Until then information for me was an abstract term. It still would have been, if not in the realm of information storage, transmission and retreival.
As a layman we associate the term information to various contexts - text content of books and newspapers, videos from news rooms, speech from telephonic conversations, audio podcasts etc- all contains information. I should not forget to mention that we exchange information in our daily chit-chats too.
What attracts us to a source of information - a new book, a conversation or a video? It is either the novelty of the content or the way of presentation. The former is what I call the information content of the source. It surprises you with new ideas. The latter is an element of art that charms you to read a book, listen to a conversation or watch a video. But my concern in this writeup would of course be the first aspect, that helps you quantify or measure information. A news room debate may be heavily engaging and tempting to listen, but every participant would be just stating the obvious. Would you consider that debate informative? The answer is a big NO!!
## What is Information?
So far I have used the term information in a very loose sense. But when it comes to mathematical quantification, it has to be very binding. Even in the loose sense, it is meaningful to think a source is highly informative if it is giving out less probable or surprising outputs.
The text content of a newspaper article would just be a sequence of alphabets. But just imagine what diverse realms of knowledge can be depicted in English with just the permutations of the 26 alphabets!! The entire information content of that article can be considered to be built up of the information contributed by each alphabet. If that idea did surprise you, I am happy that I just transferred a hell lot of information to you.
A Simple Information Source
But it is difficult to explain how to measure the information contained in an abstract idea. So let me redefine the scope of the term source of information. Suppose there is a coin tossing machine. It tosses a fair coin in secret and lets you know the result by writing out a H and T for a Head and Tail respectively. This machine is a source of information. The result of the toss is unknown beforehand and there is an absolute uncertainity. The message written out H or T does carry some information to the outside world. If the machine continuously tosses and reports the result, there would be a sequence of Hs and Ts flowing out. Machine reporting an output is equivalent to someone reporting the result of a cricket match in a news article. The entire sequence of Hs and Ts contain information about the result of all tosses. On repeating the experiment many times you can expect approximately and equal number of Hs and Ts in your sequence. The information present in the sequence is equally distributed among the alphabets H and T.
Now let us expand the secret experiment of coin toss. Our new information source is a machine that randomly picks up a button in secret from a group of eight buttons - 4 blue, 2 green, 1 yellow and 1 red - and reports the colour. As in the previous case, instead of stating the color explicitly just share an agreed symbols B-blue, G-green, Y-yellow, R-red. If the secret pick experiment is repeated many number of times and the results are reported, we get a sequence, say BBYGBBRGGBBGYRBGBB.
In this particular case I can expect a half of the sequence would be Bs, a quarter would be Gs and one-eighth each would be Ys and Rs. That would hardly need any explanation. This is indicated in the folowing probability table.
P(B) = 1/2 =0.5
P(G) = 1/4 = 0.25
P(Y) = 1/8 = 0.125
P(R) = 1/8 = 0.125
Since B is a highly probable outcome, its occurance would not induce much surprise in you. In other words, among the symbols B, G, Y and R - B has the least information content in it. Y and R being the least probable outcomes, their occurance would be more surprising and hence contain more information. The information content in a source symbol is infact inversely realted to its probability of occurance.
### But what is Entropy?
Before quantifying this information let us agree on some unit to represent it. Let us choose binary digits (0 and 1) as the symbols to represent the outcome of any information source. That is instead of H and T use 0 and 1. To convey the outcome of the coin tossing machine, you need one bit. If you repeat the coin tossing experiment 32 times, you will need 32 x 1 = 32 bits to represent the sequence of outcomes.
Instead of B, G, Y and R let us use the binary words 00, 01, 10 and 11 respectively. That is to tell the outcome of the button picking machine you need two bits. Repeating the button picking experiment 32 times would require 32 x 2 = 64 bits to indicate the sequence of outcomes.
Now the question is, can it be represented in a more compact way? What if I use 1 for B, 01 for G, 000 for Y and 001 for R? So now I need 1 bit to say Blue was chosen while 3 bits to say Yellow. The sequence of 7 outcomes BBYGBBR can be now written as 1 1 000 01 1 1 001, using just 12 bits. If it were the earlier scheme we would have needed 14 bits.
That sounds cool!! But what did we just do?
Instead of assigning equal length binary word to each symbol B, G, Y and R , we assigned shortest word of 1 bit length to the symbol B which contains the least amount of information and the longest word of 3 bits to the the symbols Y and R with the highest quantity of information. When you represent a large sequence of outcomes like BBYGBBRGGBBGYRBGBB in bits, this would reduce the total number of bits as we have exemplified above.
Why does it reduce the total number of bits?
It is because you have used the shortest word for the most frequently occuring symbol B and the longest word for the least occuring symbols Y and R.
If you have not noticed already, the length of word assigned to each symbol was infact the base 2 logarithm of the inverse of probability of occurance of each symbol. This number(the base 2 logarithm of the inverse of probability) is the information content of that particular symbol in bits. You can read that once again!!
Probaility of each symbol
P(B) = 1/2 =0.5
P(G) = 1/4 = 0.25
P(Y) = 1/8 = 0.125
P(R) = 1/8 = 0.125
Code word and Information content of each symbol. Information is Quantified in this step!!
Code for B = 1. The Information content of B is given by:
$I (B)= log_2 {\frac{1}{0.5} }= 1 \ bit$
Code for G = 01. The Information content of G is given by:
$I (G)= log_2 {\frac{1}{0.25} }= 2 \ bits$
Code for Y = 000. The Information content of Y is given by:
$I (Y)= log_2 {\frac{1}{0.125} }= 3 \ bits$
Code for R = 001. The Information content of R is given by:
$I ( R)= log_2 {\frac{1}{0.125} }= 3 \ bits$
Initially we had used equal length code for each symbol. So the average code word length was 2 bits. Now our code word is variable length, the length corresponding to the information contained in each symbol.
The average information content per source symbol is $H(X)=\sum_{i=1}^n P(x_i) I(x_i) = 0.5 \times 1 bit + 0.25 \times 2bits + 0.125 \times 3bits + 0.125 \times 3 bits = 1.75 bits$ The term $H(X)$ is the entropy of the source.
On an average you need only 1.75 bits (unlike 2 bits we assumed initially) per source symbol to represent the information emitted by the source. The colours are predictable to some extend and that reduces the uncertainity and thus the information contained in it. If all the four colours were equally likely, you would have needed same number of bits to represent each letter and that number would have been 2, ie. ($log_2\frac{1}{0.25}$)
Higher the entropy of a source, on an average we need more number of bits to represent a source symbol. In the example above the codeword length was same as the information content of that symbol. This is possible because the information defined by $log_{2}(\frac{1}{P(x_i})$, happened to be integers. It need not always be the case. If it is not an integer, choose a code word length that is the next higher integer. Then the average codeword length would be higher than the entropy. A
Assigning a binary codeword to a symbol, that has a length (approximately) equal to the information contained in it is called entropy coding. A popular type of entropy coding is Huffman coding algorithm. Replacing every occurance of a symbol by its huffman code would essentially compress the content than it were represented by equal length codes.
This is infact an idea first proposed by Claude E.Shannon and the foundation for the theory of compression and communication. If you have observed the Morse code used in telegraphy, the same idea of assigning shortest code to most frequently occuring symbol can be seen.
Further, if you are really interested in the equation of entropy, check out the video in speech zone.
But the source need not always be the tossing machine!! Then how would I find the best code word length?
The source of information and its structure need not be known beforehand. You can always observe a large sequence of symbols emitted by a source and just count the number of times a particular symbol is repeated. Find the probaility of occurance of the symbol. Once you know the probability, you can find the information content associated with that symbol in bits. Averaging it over all symbols and their probabilities, would give you the entropy.
Shannon himself has tried to determine, the entropy of English language. If all the alphabets were equally likely, the entropy would have been $log_{2}(\frac{1}{\frac{1}{26}})=4.7 bits/ letter$. But on analysing English language, e is the most and q the least popular letters. Analysing the relative frequencies, he has reached the answer that the entropy is 4.14 bits/letter. The more we know about the statistical structure of a language, it becomes easily predictable and thus reduces the information content and thus the entropy.
The redundancy in a langauge can be further explored by considering the probability of a particular letter to follow another letter, and the entropy can be found to be still lower. From Shannon’s paper on Prediction and Entropy of Printed English
The redundancy,on the other hand, measures the amount of constraint imposed on a text in the language due to its statistical structure, e.g. in English the high frequency of the letter E, the strong tendency of H to follow T or of U to follow Q. It was estimated that when statistical effects extending over not more than eight letters are considered the entropy is roughly 2.3 bits per letter, the redundancy about 50 per cent
## Entropy of Malayalam Script
Every scripted language has an inherent structure that makes certain character sequences valid words. A probability modelling of the alphabets of the language with respect to the relative frequency of occurance of the alphabets can be done by analysing huge text corpora of that language. An experiment to determine the entropy of Malayalam script is elaborated next.
I extracted a subset of Malayalam sentence corpora maintained by Swathanthra Malayalam Computing. Regular Malayalam text content has English alphabets occuring ocassionally.
Then the following python script was used to filter out English and retain only those characters relevant to Malayalam. The relevant list of alphabets consist of all the Malaylam vowels(അ, ആ ..), consonants(ക, ച, ട…), vowel signs(ാ, ി…), Arabic numerals (0-9) and basic punctuations including space and newline characters .
A part of the sentance corpora consisting of curated content extracted from newspaper articles, blogposts etc. in textcorpus directory was filtered and the resulting content was placed in result directory as a single textfile malayalam.txt using the following shell script.
cat ./textcorpus/*.txt | python malayalam-filter.py > result/malayalam.txt
Now it is time for relative frequency analysis on malayalam.txt. Python libraries numpy, scipy and huffman helps in the statistical analysis and the code below is self explanatory.
On analysing 543865 characters, the resulting entropy is 4.944 bits/character. The result as given by the above script is as below:
The number of characters parsed 543865
Total number of unique characters: 82
writing content to character frequency table
The entropy is: 4.9443809924853666
writing content to huffman codebook
If you are curious to see the relative frequency of each characters, have a look at the character frequency table:
You can see for yourself ് is the most frequent character in Malayalam script and ഋ is the least occuring one.
Now if you apply Huffman encoding, the most frequent character will get the binary code-word of least length. See for yourself the huffman code for each of the above characters.
് has a 3 bit code word while ഋ has a 17 bit code word. You can experiment for youself by changing the corpora, and find out the corresponding entropies and binary codewords. On expanding the corpora by including the Malayalam wikipedia articles, I got an entropy of 5.024 bits/characters.
Infact there has been prior attempts in this regard for Malayalam and the details can be found in the following academic papers.
It reports an entropy of 4.8 bits per character. The difference might have been caused by the choice of corpora.
• “Malayalam text Compression”, Sajilal Divakaran, Biji C. L., Anjali C., Achuthsankar S. Nair, International Journal of Information System and Engineering, 2013
It reports an entropy of 5.47 bit per character. But the characters chosen are not the atomic characters, but mostly consonant conjuncts and that has made a difference.
What does that mean, if Malayalam has an entropy of 4.944 bits/letter?
In the simplest terms it indicate the limit to which you can compress Malayalam text content. If you have a text of 1000 characters in Malayalam, it would follow a probability distribution as indicated by the character frequency table. You can replace every character by the huffman code we derived above and the total number of bits you need to represent those 1000 characters would be approximately 4.944 bits/letter x 1000 = 4944 bits. But inorder to recover the original text you will have to look-up a huffman code table.
Representing a 1000 character text in Malayalam in Unicode encoding would have costed you 1000 x 24 bits = 24000 bits !! Remember every Malayalam unicode character takes 3 bytes of memory.
To wind up, there are still more possibilities to explore towards the statistical nature of the Malayalam text. There are restrictions like a vowel can never occur in a valid Malayalam word, unless it is at the start of the word. Such properties can be extracted from the N-gram character probalility modelling of Malayalam and the corresponding entropy would be smaller and can help you reach better compression.
That is all for now. Thanks for reading 😊 | 3,472 | 15,116 | {"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} | 2.96875 | 3 | CC-MAIN-2024-38 | latest | en | 0.951572 |
http://www.mywordsolution.com/question/work-out-the-expected-value-and-standard/920793 | 1,527,267,495,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794867140.87/warc/CC-MAIN-20180525160652-20180525180652-00632.warc.gz | 421,067,179 | 7,887 | +1-415-315-9853
info@mywordsolution.com
## Statistics
Find the expected value and standard deviation of the given frequency value.
Frequency Value 10 1 17 2 5 3 31 4
Work out the expected value and standard deviation of the value.
Statistics and Probability, Statistics
• Category:- Statistics and Probability
• Reference No.:- M920793
Have any Question?
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IF a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles ; and the three interior angles of every triangle are equal to two right angles.
The College Euclid: Comprising the First Six and the Parts of the Eleventh ... - Page 310
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Elias Loomis - Conic sections - 1849 - 252 pages
...PROPOSITION XXVII. THEOREM. If one side of a triangle is produced, the exterior angle is equal to the sum of the two interior and opposite, angles; and the three interior angles of every triangle are equal to two right angles. produced to D; then will the exterior angle ACD be equal to the sum of the...
## The English Journal of Education, Volume 4
Education - 1850 - 488 pages
...equal to two right angles: the two straight lines shall be parallel to one another. 2. If one side of a triangle be produced, the exterior angle is equal...every triangle are together equal to two right angles. 3. If the square described upon one of the sides of a triangle be equal to the squares described upon...
## A Collection of Problems and Examples Adapted to the "Elementary Course of ...
Harvey Goodwin - Mathematics - 1851 - 196 pages
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## The first two books of the Elements of Euclid, with additional figures ...
Euclides - 1852 - 152 pages
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## Five Years in an English University, Volume 2
Charles Astor Bristed - 1852 - 470 pages
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## The synoptical Euclid; being the first four books of Euclid's Elements of ...
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# Calculator
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The last calculation result is stored in Res memory; The
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## Assignment & Splitting Expression
The above expression contains two sub-expressions. The 1st one assign 5 to
X, the next one calculates X+5 = 10 (where X = 5).
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## Double Variables F(X,Y)
F2(X,Y) F3(X,Y,)
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F3(X,Y,Z)
Assign Fx Functions
Step 1:
2X+5 Input Based-X Equation
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Statistics
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2
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## Notice: Result will be display in appropriate data size
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## Base Calculation Examples
011 in Binary plus 0F in
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## 001 in Decimal (Default Base)
shift left two positions. The
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## 010 or 101 (In Binary). The
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Argument
## Argument of 2 + 2i, i.e The angle of 2 + 2i in Polar Form.
Fraction
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Sum, Product,) supports Matrices / Vectors.
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## Notice: Move the cursor (I Beam) into the target matrix
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## Press to convert the result to plain text form.
Matrix / Vector Example
## Calculate Eigen Values
M = V D V-1. 1 = 1 and 2 = 2
M=[ ]=[ ]x[ ]x[ ]
## Cramers Rule Solver
1X + 2Y + 3Z + 4T = 5
6X + 7Y + 8Z + 9T = 0
2X + 4Y + 1Z + 3T = 5
5X + 7Y + 8Z + 9T = 6
## Notice: Each element of the matrix must be matched to
each coefficient in the linear system.
In above example X=-6, Y=4, Z =1, T=0 is an only solution.
Common Equation Solvers
Mode Feature
Solver
## System of Linear Equations
with 2-Unknows Solver
## System of Linear Equations
with 3-Unkonws Solver
## Select the appropriate Mode for the target Equations.
Equation Solving Example
2X2-3X+5=0
## X+Y=2 & X-Y=3
Notice:
Input all of necessary coefficients (one by
one). Tap the coefficients or Press [Up] or
[Down] to select and change the value.
Regression
Key Function
Insert Current Result or
Expression to Data Set
XY, XY, XY
Open Regression Page
(Type, Equation, Error)
n Number of Pairs (X,Y)
Sum of all X
y Sum of all Y
y Sum of all XY
Sum of all XY divide by
y / y
Sum of all Y
xy / y would be considered as Sum of all
value x weight / total weight, where weight is y.
Data & Regression
Algebra Mode
Key Function
Open Streamline Scheme
Close Streamline Scheme
## Streamline is an advanced history section, which
displays the history of calculation. In addition, it
allows controlling, editing and operating with the
previous calculations.
Algebra Special Keys
Key Function
Simplify the Equation
Expand the Polynomial
Factor
An Integer or Polynomial
Evaluate the Approximate
Result
Plug In Variables to the
Equation
Save the Result to Variables or
Fx Functions
Notice:
+ All Operations appear once the streamline is opened,
The equation should be entered first before operating.
+ All Operations are applied to the latest (final) result.
+ Plug In Operation is applied to X Y Z Variables. The
other variables are plugged in automatically.
Algebra Operation Example
Simplify
## Note: Sin2(x) + Cos2(x) = 1
Expand
Factor
7! = 1234567, 6 = 32, 4 = 22
Eval
Key Function
Insert the Selected Equation to
Input Form
Copy the Equation as Plain Text
## Eval Evaluate the Approximate Value
Plug In Variables to the Selected
Plug (In)
Equation
Simpl
Simplify
Simplify the Selected Equation
Exp
Expand the Selected Polynomial
Expand
Fact Factor the
Factor Selected Number / Polynomial
Save the Selected Equation /
F1, F2, F3
Number to Fx Functions
Save the Selected Number to X,
X, Y, Z, M
Y, Z, M Variables
Integral, Derivative, Lim
Indefinite Integral
Derivative
## Leave the second argument empty to calculate derivate of an equation
Limit Calculation
## Notice: Only some common type of limit calculation
problem is supported.
Taylor Series Expansion
## TLor( F, Variable, Degree, Value)
Argument Description
F Function
Variable Respected Variable
Degree Maximum Degree
Value A Point
TLor returns the Taylor Series Expansion of [ F ]
respect to [ Variable ] at point [ Value ] to the
maximum of power expansion [ Degree ].
Examples | 1,934 | 7,467 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2021-17 | latest | en | 0.697314 |
https://www.indiabix.com/civil-engineering/theory-of-structures/discussion-4735-1-2 | 1,701,486,528,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100309.57/warc/CC-MAIN-20231202010506-20231202040506-00695.warc.gz | 901,859,845 | 7,000 | # Civil Engineering - Theory of Structures - Discussion
Discussion Forum : Theory of Structures - Section 1 (Q.No. 42)
42.
The ratio of the deflections of the free end of a cantilever due to an isolated load at l/3rd and 2/3rd of the span, is
Explanation:
No answer description is available. Let's discuss.
Discussion:
4 comments Page 1 of 1.
Bickey said: 7 years ago
For cantilever.
Deflection at free end=Wa^2*(3l-a)/6EI.
Where a is distance from fixed end.
Put a = l/3.
And a = 2l/3.
Then you will get the answer.
(3)
Binesh said: 4 years ago
Thanks @Bickey.
Atchaya said: 2 years ago
Thank you so much @Bickey.
Ni Kumar said: 2 years ago
Thank you for explaining the answer. | 208 | 693 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2023-50 | latest | en | 0.830819 |
https://gitlab.mpi-sws.org/iris/iris/-/commit/780f6b82266d8dd80f6471d805a2454a77b38dca.diff | 1,606,781,386,000,000,000 | text/plain | crawl-data/CC-MAIN-2020-50/segments/1606141515751.74/warc/CC-MAIN-20201130222609-20201201012609-00641.warc.gz | 308,753,588 | 15,632 | diff --git a/iris/agree.v b/iris/agree.v index 38aaf15f05ef102eab8d61a3fa0e6181a0cfdf3a..2525dff1f19c7bc7644ff4625a9a25a23aa55ea7 100644 --- a/iris/agree.v +++ b/iris/agree.v @@ -57,7 +57,6 @@ Next Obligation. by intros; simpl; split_ands; try apply agree_valid_0. Qed. Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. Global Instance agree_unit : Unit (agree A) := id. Global Instance agree_minus : Minus (agree A) := λ x y, x. -Global Instance agree_included : Included (agree A) := λ x y, y ≡ x ⋅ y. Instance: Commutative (≡) (@op (agree A) _). Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed. Definition agree_idempotent (x : agree A) : x ⋅ x ≡ x. @@ -80,6 +79,11 @@ Proof. repeat match goal with H : agree_is_valid _ _ |- _ => clear H end; by cofe_subst; rewrite !agree_idempotent. Qed. +Lemma agree_includedN (x y : agree A) n : x ≼{n} y ↔ y ={n}= x ⋅ y. +Proof. + split; [|by intros ?; exists y]. + by intros [z Hz]; rewrite Hz, (associative _), agree_idempotent. +Qed. Global Instance agree_cmra : CMRA (agree A). Proof. split; try (apply _ || done). @@ -87,23 +91,20 @@ Proof. rewrite <-(proj2 Hxy n'), (Hx n') by eauto using agree_valid_le. by apply dist_le with n; try apply Hxy. * by intros n x1 x2 Hx y1 y2 Hy. - * by intros x y1 y2 Hy ?; do 2 red; rewrite <-Hy. * intros x; split; [apply agree_valid_0|]. by intros n'; rewrite Nat.le_0_r; intros ->. * intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?]. rewrite (Hx n') by auto; symmetry; apply dist_le with n; try apply Hx; auto. * intros x; apply agree_idempotent. - * intros x y; change (x ⋅ y ≡ x ⋅ (x ⋅ y)). - by rewrite (associative _), agree_idempotent. * by intros x y n [(?&?&?) ?]. - * by intros x y; do 2 red; rewrite (associative _), agree_idempotent. + * by intros x y n; rewrite agree_includedN. Qed. Lemma agree_op_inv (x y1 y2 : agree A) n : validN n x → x ={n}= y1 ⋅ y2 → y1 ={n}= y2. Proof. by intros [??] Hxy; apply Hxy. Qed. Global Instance agree_extend : CMRAExtend (agree A). Proof. - intros x y1 y2 n ? Hx; exists (x,x); simpl; split. + intros n x y1 y2 ? Hx; exists (x,x); simpl; split. * by rewrite agree_idempotent. * by rewrite Hx, (agree_op_inv x y1 y2), agree_idempotent by done. Qed. @@ -131,15 +132,13 @@ Section agree_map. Global Instance agree_map_ne n : Proper (dist n ==> dist n) agree_map. Proof. by intros x1 x2 Hx; split; simpl; intros; [apply Hx|apply Hf, Hx]. Qed. Global Instance agree_map_proper: Proper ((≡)==>(≡)) agree_map := ne_proper _. - Global Instance agree_map_preserving : CMRAPreserving agree_map. + Global Instance agree_map_monotone : CMRAMonotone agree_map. Proof. split; [|by intros n x [? Hx]; split; simpl; [|by intros n' ?; rewrite Hx]]. - intros x y; unfold included, agree_included; intros Hy; rewrite Hy. - split; [split|done]. - * by intros (?&?&Hxy); repeat (intro || split); - try apply Hxy; try apply Hf; eauto using @agree_valid_le. - * by intros (?&(?&?&Hxy)&_); repeat split; - try apply Hxy; eauto using agree_valid_le. + intros x y n; rewrite !agree_includedN; intros Hy; rewrite Hy. + split; [split; simpl; try tauto|done]. + by intros (?&?&Hxy); repeat split; intros; + try apply Hxy; try apply Hf; eauto using @agree_valid_le. Qed. End agree_map. Lemma agree_map_id `{Cofe A} (x : agree A) : agree_map id x = x. diff --git a/iris/auth.v b/iris/auth.v index d204f361f63ba4a2e905fa031e2a531c4a93d201..23cfd6d983a09c4d2223de536963bfb4d380268a 100644 --- a/iris/auth.v +++ b/iris/auth.v @@ -1,27 +1,30 @@ Require Export iris.excl. Local Arguments disjoint _ _ !_ !_ /. -Local Arguments included _ _ !_ !_ /. -Record auth (A : Type) : Type := Auth { authorative : excl A ; own : A }. +Record auth (A : Type) : Type := Auth { authoritative : excl A ; own : A }. Arguments Auth {_} _ _. -Arguments authorative {_} _. +Arguments authoritative {_} _. Arguments own {_} _. Notation "∘ x" := (Auth ExclUnit x) (at level 20). Notation "∙ x" := (Auth (Excl x) ∅) (at level 20). Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅. -Instance auth_valid `{Valid A, Included A} : Valid (auth A) := λ x, - valid (authorative x) ∧ excl_above (own x) (authorative x). +Instance auth_valid `{Equiv A, Valid A, Op A} : Valid (auth A) := λ x, + valid (authoritative x) ∧ excl_above (own x) (authoritative x). Instance auth_equiv `{Equiv A} : Equiv (auth A) := λ x y, - authorative x ≡ authorative y ∧ own x ≡ own y. + authoritative x ≡ authoritative y ∧ own x ≡ own y. Instance auth_unit `{Unit A} : Unit (auth A) := λ x, - Auth (unit (authorative x)) (unit (own x)). + Auth (unit (authoritative x)) (unit (own x)). Instance auth_op `{Op A} : Op (auth A) := λ x y, - Auth (authorative x ⋅ authorative y) (own x ⋅ own y). + Auth (authoritative x ⋅ authoritative y) (own x ⋅ own y). Instance auth_minus `{Minus A} : Minus (auth A) := λ x y, - Auth (authorative x ⩪ authorative y) (own x ⩪ own y). -Instance auth_included `{Equiv A, Included A} : Included (auth A) := λ x y, - authorative x ≼ authorative y ∧ own x ≼ own y. + Auth (authoritative x ⩪ authoritative y) (own x ⩪ own y). +Lemma auth_included `{Equiv A, Op A} (x y : auth A) : + x ≼ y ↔ authoritative x ≼ authoritative y ∧ own x ≼ own y. +Proof. + split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|]. + intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto. +Qed. Instance auth_ra `{RA A} : RA (auth A). Proof. @@ -35,19 +38,19 @@ Proof. * by intros y1 y2 [Hy Hy'] [??]; split; simpl; rewrite <-?Hy, <-?Hy'. * by intros x1 x2 [Hx Hx'] y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy, ?Hy', ?Hx, ?Hx'. - * by intros x y1 y2 [Hy Hy'] [??]; split; simpl; rewrite <-?Hy, <-?Hy'. * by split; simpl; rewrite (associative _). * by split; simpl; rewrite (commutative _). * by split; simpl; rewrite ?(ra_unit_l _). * by split; simpl; rewrite ?(ra_unit_idempotent _). - * split; simpl; auto using ra_unit_weaken. - * intros ?? [??]; split; [by apply ra_valid_op_l with (authorative y)|]. + * intros ??; rewrite! auth_included; intros [??]. + by split; simpl; apply ra_unit_preserving. + * intros ?? [??]; split; [by apply ra_valid_op_l with (authoritative y)|]. by apply excl_above_weaken with (own x ⋅ own y) - (authorative x ⋅ authorative y); try apply ra_included_l. - * split; simpl; apply ra_included_l. - * by intros ?? [??]; split; simpl; apply ra_op_minus. + (authoritative x ⋅ authoritative y); try apply ra_included_l. + * by intros ??; rewrite auth_included; + intros [??]; split; simpl; apply ra_op_minus. Qed. Instance auth_ra_empty `{RA A, Empty A, !RAEmpty A} : RAEmpty (auth A). Proof. split. done. by intros x; constructor; simpl; rewrite (left_id _ _). Qed. Lemma auth_frag_op `{RA A} a b : ∘(a ⋅ b) ≡ ∘a ⋅ ∘b. -Proof. done. Qed. +Proof. done. Qed. \ No newline at end of file diff --git a/iris/cmra.v b/iris/cmra.v index e4c5589ae4eea4f2797c6c0f7f3d61348d1460fd..7b3a9d8b8ee2a4458c2d448430754022de352711 100644 --- a/iris/cmra.v +++ b/iris/cmra.v @@ -3,15 +3,18 @@ Require Export iris.ra iris.cofe. Class ValidN (A : Type) := validN : nat → A → Prop. Instance: Params (@validN) 3. -Class CMRA A `{Equiv A, Compl A, - Unit A, Op A, Valid A, ValidN A, Included A, Minus A} : Prop := { +Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ={n}= x ⋅ z. +Notation "x ≼{ n } y" := (includedN n x y) + (at level 70, format "x ≼{ n } y") : C_scope. +Instance: Params (@includedN) 4. + +Class CMRA A `{Equiv A, Compl A, Unit A, Op A, Valid A, ValidN A, Minus A} := { (* setoids *) cmra_cofe :> Cofe A; cmra_op_ne n x :> Proper (dist n ==> dist n) (op x); cmra_unit_ne n :> Proper (dist n ==> dist n) unit; cmra_valid_ne n :> Proper (dist n ==> impl) (validN n); cmra_minus_ne n :> Proper (dist n ==> dist n ==> dist n) minus; - cmra_included_proper x : Proper ((≡) ==> impl) (included x); (* valid *) cmra_valid_0 x : validN 0 x; cmra_valid_S n x : validN (S n) x → validN n x; @@ -21,18 +24,17 @@ Class CMRA A `{Equiv A, Compl A, cmra_commutative : Commutative (≡) (⋅); cmra_unit_l x : unit x ⋅ x ≡ x; cmra_unit_idempotent x : unit (unit x) ≡ unit x; - cmra_unit_weaken x y : unit x ≼ unit (x ⋅ y); + cmra_unit_preserving n x y : x ≼{n} y → unit x ≼{n} unit y; cmra_valid_op_l n x y : validN n (x ⋅ y) → validN n x; - cmra_included_l x y : x ≼ x ⋅ y; - cmra_op_minus x y : x ≼ y → x ⋅ y ⩪ x ≡ y + cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ={n}= y }. Class CMRAExtend A `{Equiv A, Dist A, Op A, ValidN A} := - cmra_extend_op x y1 y2 n : + cmra_extend_op n x y1 y2 : validN n x → x ={n}= y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z ={n}= (y1,y2) }. -Class CMRAPreserving - `{Included A, ValidN A, Included B, ValidN B} (f : A → B) := { - included_preserving x y : x ≼ y → f x ≼ f y; +Class CMRAMonotone + `{Dist A, Op A, ValidN A, Dist B, Op B, ValidN B} (f : A → B) := { + includedN_preserving n x y : x ≼{n} y → f x ≼{n} f y; validN_preserving n x : validN n x → validN n (f x) }. @@ -46,12 +48,11 @@ Structure cmraT := CMRAT { cmra_op : Op cmra_car; cmra_valid : Valid cmra_car; cmra_validN : ValidN cmra_car; - cmra_included : Included cmra_car; cmra_minus : Minus cmra_car; cmra_cmra : CMRA cmra_car; cmra_extend : CMRAExtend cmra_car }. -Arguments CMRAT _ {_ _ _ _ _ _ _ _ _ _ _}. +Arguments CMRAT _ {_ _ _ _ _ _ _ _ _ _}. Arguments cmra_car _ : simpl never. Arguments cmra_equiv _ _ _ : simpl never. Arguments cmra_dist _ _ _ _ : simpl never. @@ -60,12 +61,11 @@ Arguments cmra_unit _ _ : simpl never. Arguments cmra_op _ _ _ : simpl never. Arguments cmra_valid _ _ : simpl never. Arguments cmra_validN _ _ _ : simpl never. -Arguments cmra_included _ _ _ : simpl never. Arguments cmra_minus _ _ _ : simpl never. Arguments cmra_cmra _ : simpl never. Add Printing Constructor cmraT. Existing Instances cmra_equiv cmra_dist cmra_compl cmra_unit cmra_op - cmra_valid cmra_validN cmra_included cmra_minus cmra_cmra cmra_extend. + cmra_valid cmra_validN cmra_minus cmra_cmra cmra_extend. Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A. Canonical Structure cmra_cofeC. @@ -73,6 +73,13 @@ Section cmra. Context `{cmra : CMRA A}. Implicit Types x y z : A. +Lemma cmra_included_includedN x y : x ≼ y ↔ ∀ n, x ≼{n} y. +Proof. + split; [by intros [z Hz] n; exists z; rewrite Hz|]. + intros Hxy; exists (y ⩪ x); apply equiv_dist; intros n. + symmetry; apply cmra_op_minus, Hxy. +Qed. + Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (validN n). Proof. by split; apply cmra_valid_ne. Qed. Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (validN n). @@ -82,16 +89,15 @@ Proof. split; try by (destruct cmra; eauto using ne_proper, ne_proper_2 with typeclass_instances). * by intros x1 x2 Hx; rewrite !cmra_valid_validN; intros ? n; rewrite <-Hx. + * intros x y; rewrite !cmra_included_includedN. + eauto using cmra_unit_preserving. * intros x y; rewrite !cmra_valid_validN; intros ? n. by apply cmra_valid_op_l with y. + * intros x y [z Hz]; apply equiv_dist; intros n. + by apply cmra_op_minus; exists z; rewrite Hz. Qed. Lemma cmra_valid_op_r x y n : validN n (x ⋅ y) → validN n y. Proof. rewrite (commutative _ x); apply cmra_valid_op_l. Qed. -Lemma cmra_valid_included x y n : validN n y → x ≼ y → validN n x. -Proof. - rewrite ra_included_spec; intros Hvalid [z Hy]; rewrite Hy in Hvalid. - eauto using cmra_valid_op_l. -Qed. Lemma cmra_valid_le x n n' : validN n x → n' ≤ n → validN n' x. Proof. induction 2; eauto using cmra_valid_S. Qed. Global Instance ra_op_ne n : Proper (dist n ==> dist n ==> dist n) (⋅). @@ -101,24 +107,66 @@ Proof. Qed. Lemma cmra_unit_valid x n : validN n x → validN n (unit x). Proof. rewrite <-(cmra_unit_l x) at 1; apply cmra_valid_op_l. Qed. -Lemma cmra_included_dist_l x1 x2 x1' n : - x1 ≼ x2 → x1' ={n}= x1 → ∃ x2', x1' ≼ x2' ∧ x2' ={n}= x2. -Proof. - rewrite ra_included_spec; intros [z Hx2] Hx1; exists (x1' ⋅ z); split. - apply ra_included_l. by rewrite Hx1, Hx2. -Qed. Lemma cmra_op_timeless `{!CMRAExtend A} x1 x2 : validN 1 (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). Proof. intros ??? z Hz. - destruct (cmra_extend_op z x1 x2 1) as ([y1 y2]&Hz'&?&?); auto; simpl in *. + destruct (cmra_extend_op 1 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *. { by rewrite <-?Hz. } by rewrite Hz', (timeless x1 y1), (timeless x2 y2). Qed. + +(** * Included *) +Global Instance cmra_included_ne n : + Proper (dist n ==> dist n ==> iff) (includedN n). +Proof. + intros x x' Hx y y' Hy; unfold includedN. + by setoid_rewrite Hx; setoid_rewrite Hy. +Qed. +Global Instance cmra_included_proper:Proper ((≡) ==> (≡) ==> iff) (includedN n). +Proof. + intros n x x' Hx y y' Hy; unfold includedN. + by setoid_rewrite Hx; setoid_rewrite Hy. +Qed. +Lemma cmra_included_0 x y : x ≼{0} y. +Proof. by exists (unit x). Qed. +Lemma cmra_included_S x y n : x ≼{S n} y → x ≼{n} y. +Proof. by intros [z Hz]; exists z; apply dist_S. Qed. + +Lemma cmra_included_l n x y : x ≼{n} x ⋅ y. +Proof. by exists y. Qed. +Lemma cmra_included_r n x y : y ≼{n} x ⋅ y. +Proof. rewrite (commutative op); apply cmra_included_l. Qed. +Global Instance cmra_included_ord n : PreOrder (includedN n). +Proof. + split. + * by intros x; exists (unit x); rewrite ra_unit_r. + * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). + by rewrite (associative _), <-Hy, <-Hz. +Qed. +Lemma cmra_valid_includedN x y n : validN n y → x ≼{n} y → validN n x. +Proof. intros Hyv [z Hy]; rewrite Hy in Hyv; eauto using cmra_valid_op_l. Qed. +Lemma cmra_valid_included x y n : validN n y → x ≼ y → validN n x. +Proof. rewrite cmra_included_includedN; eauto using cmra_valid_includedN. Qed. +Lemma cmra_included_dist_l x1 x2 x1' n : + x1 ≼ x2 → x1' ={n}= x1 → ∃ x2', x1' ≼ x2' ∧ x2' ={n}= x2. +Proof. + intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using ra_included_l. + by rewrite Hx1, Hx2. +Qed. End cmra. -Instance cmra_preserving_id `{CMRA A} : CMRAPreserving (@id A). +Instance cmra_monotone_id `{CMRA A} : CMRAMonotone (@id A). Proof. by split. Qed. +Instance cmra_monotone_ra_monotone `{CMRA A, CMRA B} (f : A → B) : + CMRAMonotone f → RAMonotone f. +Proof. + split. + * intros x y; rewrite !cmra_included_includedN. + by intros ? n; apply includedN_preserving. + * intros x; rewrite !cmra_valid_validN. + by intros ? n; apply validN_preserving. +Qed. (* Also via [cmra_cofe; cofe_equivalence] *) Hint Cut [!*; ra_equivalence; cmra_ra] : typeclass_instances. @@ -139,15 +187,18 @@ Section discrete. * by intros [|n]; try apply ra_minus_proper. * by intros [|n]. * intros x; split; intros Hvalid. by intros [|n]. apply (Hvalid 1). + * by intros [|n]; try apply ra_unit_preserving. * by intros [|n]; try apply ra_valid_op_l. + * by intros [|n] x y; try apply ra_op_minus. Qed. Instance discrete_extend : CMRAExtend A. Proof. - intros x y1 y2 [|n] ??; [|by exists (y1,y2)]. + intros [|n] x y1 y2 ??; [|by exists (y1,y2)]. by exists (x,unit x); simpl; rewrite ra_unit_r. Qed. - Definition discreteC : cmraT := CMRAT A. + Definition discreteRA : cmraT := CMRAT A. End discrete. +Arguments discreteRA _ {_ _ _ _ _ _}. (** Product *) Instance prod_op `{Op A, Op B} : Op (A * B) := λ x y, (x.1 ⋅ y.1, x.2 ⋅ y.2). @@ -158,11 +209,20 @@ Instance prod_valid `{Valid A, Valid B} : Valid (A * B) := λ x, valid (x.1) ∧ valid (x.2). Instance prod_validN `{ValidN A, ValidN B} : ValidN (A * B) := λ n x, validN n (x.1) ∧ validN n (x.2). -Instance prod_included `{Included A, Included B} : Included (A * B) := - prod_relation (≼) (≼). Instance prod_minus `{Minus A, Minus B} : Minus (A * B) := λ x y, (x.1 ⩪ y.1, x.2 ⩪ y.2). - +Lemma prod_included `{Equiv A, Equiv B, Op A, Op B} (x y : A * B) : + x ≼ y ↔ x.1 ≼ y.1 ∧ x.2 ≼ y.2. +Proof. + split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|]. + intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto. +Qed. +Lemma prod_includedN `{Dist A, Dist B, Op A, Op B} (x y : A * B) n : + x ≼{n} y ↔ x.1 ≼{n} y.1 ∧ x.2 ≼{n} y.2. +Proof. + split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|]. + intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto. +Qed. Instance prod_cmra `{CMRA A, CMRA B} : CMRA (A * B). Proof. split; try apply _. @@ -171,7 +231,6 @@ Proof. * by intros n y1 y2 [Hy1 Hy2] [??]; split; simpl in *; rewrite <-?Hy1, <-?Hy2. * by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2]; split; simpl in *; rewrite ?Hx1, ?Hx2, ?Hy1, ?Hy2. - * by intros x y1 y2 [Hy1 Hy2] [??]; split; simpl in *; rewrite <-?Hy1, <-?Hy2. * split; apply cmra_valid_0. * by intros n x [??]; split; apply cmra_valid_S. * intros x; split; [by intros [??] n; split; apply cmra_valid_validN|]. @@ -180,10 +239,11 @@ Proof. * split; simpl; apply (commutative _). * split; simpl; apply ra_unit_l. * split; simpl; apply ra_unit_idempotent. - * split; apply ra_unit_weaken. + * intros n x y; rewrite !prod_includedN. + by intros [??]; split; apply cmra_unit_preserving. * intros n x y [??]; split; simpl in *; eapply cmra_valid_op_l; eauto. - * split; apply cmra_included_l. - * by intros x y [??]; split; apply cmra_op_minus. + * intros x y n; rewrite prod_includedN; intros [??]. + by split; apply cmra_op_minus. Qed. Instance prod_ra_empty `{RAEmpty A, RAEmpty B} : RAEmpty (A * B). Proof. @@ -191,18 +251,19 @@ Proof. Qed. Instance prod_cmra_extend `{CMRAExtend A, CMRAExtend B} : CMRAExtend (A * B). Proof. - intros x y1 y2 n [??] [??]; simpl in *. - destruct (cmra_extend_op (x.1) (y1.1) (y2.1) n) as (z1&?&?&?); auto. - destruct (cmra_extend_op (x.2) (y1.2) (y2.2) n) as (z2&?&?&?); auto. + intros n x y1 y2 [??] [??]; simpl in *. + destruct (cmra_extend_op n (x.1) (y1.1) (y2.1)) as (z1&?&?&?); auto. + destruct (cmra_extend_op n (x.2) (y1.2) (y2.2)) as (z2&?&?&?); auto. by exists ((z1.1,z2.1),(z1.2,z2.2)). Qed. Canonical Structure prodRA (A B : cmraT) : cmraT := CMRAT (A * B). -Instance prod_map_cmra_preserving `{CMRA A, CMRA A', CMRA B, CMRA B'} - (f : A → A') (g : B → B') `{!CMRAPreserving f, !CMRAPreserving g} : - CMRAPreserving (prod_map f g). +Instance prod_map_cmra_monotone `{CMRA A, CMRA A', CMRA B, CMRA B'} + (f : A → A') (g : B → B') `{!CMRAMonotone f, !CMRAMonotone g} : + CMRAMonotone (prod_map f g). Proof. split. - * by intros x1 x2 [??]; split; simpl; apply included_preserving. + * intros n x1 x2; rewrite !prod_includedN; intros [??]; simpl. + by split; apply includedN_preserving. * by intros n x [??]; split; simpl; apply validN_preserving. Qed. Definition prodRA_map {A A' B B' : cmraT} diff --git a/iris/cmra_maps.v b/iris/cmra_maps.v index 68591bece3b6bdabf4ba85c8938218d105dd461d..d4a20f725ba5d58e11b54654106d181a04f2fbee 100644 --- a/iris/cmra_maps.v +++ b/iris/cmra_maps.v @@ -3,18 +3,28 @@ Require Import prelude.pmap prelude.nmap prelude.zmap. Require Import prelude.stringmap prelude.natmap. (** option *) -Instance option_valid `{Valid A} : Valid (option A) := λ x, - match x with Some x => valid x | None => True end. -Instance option_validN `{ValidN A} : ValidN (option A) := λ n x, - match x with Some x => validN n x | None => True end. +Instance option_valid `{Valid A} : Valid (option A) := λ mx, + match mx with Some x => valid x | None => True end. +Instance option_validN `{ValidN A} : ValidN (option A) := λ n mx, + match mx with Some x => validN n x | None => True end. Instance option_unit `{Unit A} : Unit (option A) := fmap unit. Instance option_op `{Op A} : Op (option A) := union_with (λ x y, Some (x ⋅ y)). Instance option_minus `{Minus A} : Minus (option A) := difference_with (λ x y, Some (x ⩪ y)). -Inductive option_included `{Included A} : Included (option A) := - | Some_included x y : x ≼ y → Some x ≼ Some y - | None_included x : None ≼ x. -Existing Instance option_included. +Lemma option_includedN `{CMRA A} n mx my : + mx ≼{n} my ↔ n = 0 ∨ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼{n} y. +Proof. + split. + * intros [mz Hmz]; destruct n as [|n]; [by left|right]. + destruct mx as [x|]; [right|by left]. + destruct my as [y|]; [exists x, y|destruct mz; inversion_clear Hmz]. + destruct mz as [z|]; inversion_clear Hmz; split_ands; auto. + + by exists z. + + by cofe_subst. + * intros [->|[->|(x&y&->&->&z&Hz)]]; + try (by exists my; destruct my; constructor). + by exists (Some z); constructor. +Qed. Instance option_cmra `{CMRA A} : CMRA (option A). Proof. split. @@ -27,8 +37,6 @@ Proof. eapply (_ : Proper (dist _ ==> impl) (validN _)); eauto. * by destruct 1; inversion_clear 1; constructor; repeat apply (_ : Proper (dist _ ==> _ ==> _) _). - * intros [x|]; destruct 1; inversion_clear 1; constructor; - eapply (_ : Proper (equiv ==> impl) (included _)); eauto. * intros [x|]; unfold validN, option_validN; auto using cmra_valid_0. * intros n [x|]; unfold validN, option_validN; auto using cmra_valid_S. * by intros [x|]; unfold valid, validN, option_validN, option_valid; @@ -37,19 +45,21 @@ Proof. * intros [x|] [y|]; constructor; rewrite 1?(commutative _); auto. * by intros [x|]; constructor; rewrite cmra_unit_l. * by intros [x|]; constructor; rewrite cmra_unit_idempotent. - * intros [x|] [y|]; constructor; auto using cmra_unit_weaken. + * intros n mx my; rewrite !option_includedN;intros [|[->|(x&y&->&->&?)]];auto. + do 2 right; exists (unit x), (unit y); eauto using cmra_unit_preserving. * intros n [x|] [y|]; unfold validN, option_validN; simpl; eauto using cmra_valid_op_l. - * intros [x|] [y|]; constructor; auto using cmra_included_l. - * destruct 1 as [|[]]; constructor; eauto using cmra_op_minus. + * intros n mx my; rewrite option_includedN. + intros [->|[->|(x&y&->&->&?)]]; [done|by destruct my|]. + by constructor; apply cmra_op_minus. Qed. Instance option_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (option A). Proof. - intros mx my1 my2 n; destruct (decide (n = 0)) as [->|]. + intros n mx my1 my2; destruct (decide (n = 0)) as [->|]. { by exists (mx, None); repeat constructor; destruct mx; constructor. } destruct mx as [x|], my1 as [y1|], my2 as [y2|]; intros Hx Hx'; try (by exfalso; inversion Hx'; auto). - * destruct (cmra_extend_op x y1 y2 n) as ([z1 z2]&?&?&?); auto. + * destruct (cmra_extend_op n x y1 y2) as ([z1 z2]&?&?&?); auto. { by inversion_clear Hx'. } by exists (Some z1, Some z2); repeat constructor. * by exists (Some x,None); inversion Hx'; repeat constructor. @@ -57,10 +67,11 @@ Proof. * exists (None,None); repeat constructor. Qed. Instance option_fmap_cmra_preserving `{CMRA A, CMRA B} (f : A → B) - `{!CMRAPreserving f} : CMRAPreserving (fmap f : option A → option B). + `{!CMRAMonotone f} : CMRAMonotone (fmap f : option A → option B). Proof. split. - * by destruct 1 as [|[?|]]; constructor; apply included_preserving. + * intros n mx my; rewrite !option_includedN. + intros [->|[->|(x&y&->&->&?)]]; simpl; eauto 10 using @includedN_preserving. * by intros n [x|] ?; unfold validN, option_validN; simpl; try apply validN_preserving. Qed. @@ -74,14 +85,28 @@ Section map. Instance map_valid `{Valid A} : Valid (M A) := λ m, ∀ i, valid (m !! i). Instance map_validN `{ValidN A} : ValidN (M A) := λ n m, ∀ i, validN n (m!!i). Instance map_minus `{Minus A} : Minus (M A) := merge minus. - Instance map_included `{Included A} : Included (M A) := λ m1 m2, - ∀ i, m1 !! i ≼ m2 !! i. Lemma lookup_op `{Op A} m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i. Proof. by apply lookup_merge. Qed. Lemma lookup_minus `{Minus A} m1 m2 i : (m1 ⩪ m2) !! i = m1 !! i ⩪ m2 !! i. Proof. by apply lookup_merge. Qed. Lemma lookup_unit `{Unit A} m i : unit m !! i = unit (m !! i). Proof. by apply lookup_fmap. Qed. + Lemma map_included_spec `{CMRA A} (m1 m2 : M A) : + m1 ≼ m2 ↔ ∀ i, m1 !! i ≼ m2 !! i. + Proof. + split. + * intros [m Hm]; intros i; exists (m !! i). by rewrite <-lookup_op, Hm. + * intros Hm; exists (m2 ⩪ m1); intros i. + by rewrite lookup_op, lookup_minus, ra_op_minus. + Qed. + Lemma map_includedN_spec `{CMRA A} (m1 m2 : M A) n : + m1 ≼{n} m2 ↔ ∀ i, m1 !! i ≼{n} m2 !! i. + Proof. + split. + * intros [m Hm]; intros i; exists (m !! i). by rewrite <-lookup_op, Hm. + * intros Hm; exists (m2 ⩪ m1); intros i. + by rewrite lookup_op, lookup_minus, cmra_op_minus. + Qed. Instance map_cmra `{CMRA A} : CMRA (M A). Proof. split. @@ -91,7 +116,6 @@ Section map. * by intros n m1 m2 Hm ? i; rewrite <-(Hm i). * intros n m1 m1' Hm1 m2 m2' Hm2 i. by rewrite !lookup_minus, (Hm1 i), (Hm2 i). - * by intros m1 m2 m2' Hm2 ? i; rewrite <-(Hm2 i). * intros m i; apply cmra_valid_0. * intros n m Hm i; apply cmra_valid_S, Hm. * intros m; split; [by intros Hm n i; apply cmra_valid_validN|]. @@ -100,11 +124,12 @@ Section map. * by intros m1 m2 i; rewrite !lookup_op, (commutative _). * by intros m i; rewrite lookup_op, !lookup_unit, ra_unit_l. * by intros m i; rewrite !lookup_unit, ra_unit_idempotent. - * intros m1 m2 i; rewrite !lookup_unit, lookup_op; apply ra_unit_weaken. + * intros n x y; rewrite !map_includedN_spec; intros Hm i. + by rewrite !lookup_unit; apply cmra_unit_preserving. * intros n m1 m2 Hm i; apply cmra_valid_op_l with (m2 !! i). by rewrite <-lookup_op. - * intros m1 m2 i; rewrite lookup_op; apply ra_included_l. - * by intros m1 m2 Hm i; rewrite lookup_op, lookup_minus, ra_op_minus. + * intros x y n; rewrite map_includedN_spec; intros ? i. + by rewrite lookup_op, lookup_minus, cmra_op_minus by done. Qed. Instance map_ra_empty `{RA A} : RAEmpty (M A). Proof. @@ -114,10 +139,10 @@ Section map. Qed. Instance map_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (M A). Proof. - intros m m1 m2 n Hm Hm12. + intros n m m1 m2 Hm Hm12. assert (∀ i, m !! i ={n}= m1 !! i ⋅ m2 !! i) as Hm12' by (by intros i; rewrite <-lookup_op). - set (f i := cmra_extend_op (m !! i) (m1 !! i) (m2 !! i) n (Hm i) (Hm12' i)). + set (f i := cmra_extend_op n (m !! i) (m1 !! i) (m2 !! i) (Hm i) (Hm12' i)). set (f_proj i := proj1_sig (f i)). exists (map_imap (λ i _, (f_proj i).1) m, map_imap (λ i _, (f_proj i).2) m); repeat split; simpl; intros i; rewrite ?lookup_op, !lookup_imap. @@ -131,11 +156,12 @@ Section map. by destruct (m1 !! i), (m2 !! i); inversion_clear Hm12''. Qed. Definition mapRA (A : cmraT) : cmraT := CMRAT (M A). - Global Instance map_fmap_cmra_preserving `{CMRA A, CMRA B} (f : A → B) - `{!CMRAPreserving f} : CMRAPreserving (fmap f : M A → M B). + Global Instance map_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A → B) + `{!CMRAMonotone f} : CMRAMonotone (fmap f : M A → M B). Proof. split. - * by intros m1 m2 Hm i; rewrite !lookup_fmap; apply included_preserving. + * intros m1 m2 n; rewrite !map_includedN_spec; intros Hm i. + by rewrite !lookup_fmap; apply includedN_preserving. * by intros n m ? i; rewrite lookup_fmap; apply validN_preserving. Qed. Hint Resolve (map_fmap_ne (M:=M)) : typeclass_instances. @@ -143,8 +169,8 @@ Section map. CofeMor (fmap f : mapRA A → mapRA B). Global Instance mapRA_map_ne {A B} n : Proper (dist n ==> dist n) (@mapRA_map A B) := mapC_map_ne n. - Global Instance mapRA_mapcmra_preserving {A B : cmraT} (f : A -n> B) - `{!CMRAPreserving f} : CMRAPreserving (mapRA_map f) := _. + Global Instance mapRA_mapcmra_monotone {A B : cmraT} (f : A -n> B) + `{!CMRAMonotone f} : CMRAMonotone (mapRA_map f) := _. End map. Arguments mapRA {_} _ {_ _ _ _ _ _ _ _ _} _. diff --git a/iris/dra.v b/iris/dra.v index b7f573f0c406faf4fa30d5af6dd7958912dba615..aa3e707d88e0b91dec388ba41eea9a4a82276926 100644 --- a/iris/dra.v +++ b/iris/dra.v @@ -30,14 +30,19 @@ Instance validity_valid_proper `{Equiv A} (P : A → Prop) : Proof. intros ?? [??]; naive_solver. Qed. Local Notation V := valid. -Class DRA A `{Equiv A,Valid A,Unit A,Disjoint A,Op A,Included A, Minus A} := { + +Definition dra_included `{Equiv A, Valid A, Disjoint A, Op A} := λ x y, + ∃ z, y ≡ x ⋅ z ∧ V z ∧ x ⊥ z. +Instance: Params (@dra_included) 4. +Local Infix "≼" := dra_included. + +Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Minus A} := { (* setoids *) dra_equivalence :> Equivalence ((≡) : relation A); dra_op_proper :> Proper ((≡) ==> (≡) ==> (≡)) (⋅); dra_unit_proper :> Proper ((≡) ==> (≡)) unit; dra_disjoint_proper :> ∀ x, Proper ((≡) ==> impl) (disjoint x); dra_minus_proper :> Proper ((≡) ==> (≡) ==> (≡)) minus; - dra_included_proper :> Proper ((≡) ==> (≡) ==> impl) included; (* validity *) dra_op_valid x y : V x → V y → x ⊥ y → V (x ⋅ y); dra_unit_valid x : V x → V (unit x); @@ -51,20 +56,21 @@ Class DRA A `{Equiv A,Valid A,Unit A,Disjoint A,Op A,Included A, Minus A} := { dra_unit_disjoint_l x : V x → unit x ⊥ x; dra_unit_l x : V x → unit x ⋅ x ≡ x; dra_unit_idempotent x : V x → unit (unit x) ≡ unit x; - dra_unit_weaken x y : V x → V y → x ⊥ y → unit x ≼ unit (x ⋅ y); - dra_included_l x y : V x → V y → x ⊥ y → x ≼ x ⋅ y; - dra_disjoint_difference x y : V x → V y → x ≼ y → x ⊥ y ⩪ x; - dra_op_difference x y : V x → V y → x ≼ y → x ⋅ y ⩪ x ≡ y + dra_unit_preserving x y : V x → V y → x ≼ y → unit x ≼ unit y; + dra_disjoint_minus x y : V x → V y → x ≼ y → x ⊥ y ⩪ x; + dra_op_minus x y : V x → V y → x ≼ y → x ⋅ y ⩪ x ≡ y }. Section dra. Context A `{DRA A}. Arguments valid _ _ !_ /. +Hint Immediate dra_op_proper : typeclass_instances. -Instance: Proper ((≡) ==> (≡) ==> impl) (⊥). +Instance: Proper ((≡) ==> (≡) ==> iff) (⊥). Proof. - intros x1 x2 Hx y1 y2 Hy. - by rewrite Hy, (symmetry_iff (⊥) x1), (symmetry_iff (⊥) x2), Hx. + intros x1 x2 Hx y1 y2 Hy; split. + * by rewrite Hy, (symmetry_iff (⊥) x1), (symmetry_iff (⊥) x2), Hx. + * by rewrite <-Hy, (symmetry_iff (⊥) x2), (symmetry_iff (⊥) x1), <-Hx. Qed. Lemma dra_disjoint_rl x y z : V x → V y → V z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y. Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed. @@ -78,10 +84,8 @@ Proof. intros; symmetry; rewrite dra_commutative by eauto using dra_disjoint_rl. apply dra_disjoint_move_l; auto; by rewrite dra_commutative. Qed. -Hint Immediate dra_associative dra_commutative dra_unit_disjoint_l - dra_unit_l dra_unit_idempotent dra_unit_weaken dra_included_l - dra_op_difference dra_disjoint_difference dra_disjoint_rl - dra_disjoint_lr dra_disjoint_move_l dra_disjoint_move_r. +Hint Immediate dra_disjoint_move_l dra_disjoint_move_r. +Hint Unfold dra_included. Notation T := (validity (valid : A → Prop)). Program Instance validity_unit : Unit T := λ x, @@ -91,8 +95,6 @@ Program Instance validity_op : Op T := λ x y, Validity (validity_car x ⋅ validity_car y) (V x ∧ V y ∧ validity_car x ⊥ validity_car y) _. Next Obligation. by apply dra_op_valid; try apply validity_prf. Qed. -Instance validity_included : Included T := λ x y, - V y → V x ∧ validity_car x ≼ validity_car y. Program Instance validity_minus : Minus T := λ x y, Validity (validity_car x ⩪ validity_car y) (V x ∧ V y ∧ validity_car y ≼ validity_car x) _. @@ -108,21 +110,25 @@ Proof. * by intros ?? Heq ?; rewrite <-Heq. * intros x1 x2 [? Hx] y1 y2 [? Hy]; split; simpl; [|by intros (?&?&?); rewrite Hx, Hy]. - split; intros (?&?&?); split_ands'; try tauto. - + by rewrite <-Hx, <-Hy by done. - + by rewrite Hx, Hy by tauto. - * intros ??? [? Heq] Hle; split; [apply Hle; tauto|]. - rewrite <-Heq by tauto; apply Hle; tauto. - * intros [x px ?] [y py ?] [z pz ?]; - split; simpl; [naive_solver|intros; apply (associative _)]. - * intros [x px ?] [y py ?]; split; naive_solver. - * intros [x px ?]; split; naive_solver. - * intros [x px ?]; split; naive_solver. - * intros [x px ?] [y py ?]; split; naive_solver. + split; intros (?&?&z&?&?); split_ands'; try tauto. + + exists z. by rewrite <-Hy, <-Hx. + + exists z. by rewrite Hx, Hy by tauto. + * intros [x px ?] [y py ?] [z pz ?]; split; simpl; + [intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl + |intros; apply (associative _)]. + * intros [x px ?] [y py ?]; split; naive_solver eauto using dra_commutative. + * intros [x px ?]; split; + naive_solver eauto using dra_unit_l, dra_unit_disjoint_l. + * intros [x px ?]; split; naive_solver eauto using dra_unit_idempotent. + * intros x y Hxy; exists (unit y ⩪ unit x). + destruct x as [x px ?], y as [y py ?], Hxy as [[z pz ?] [??]]; simpl in *. + assert (py → unit x ≼ unit y) + by intuition eauto 10 using dra_unit_preserving. + constructor; [|symmetry]; simpl in *; + intuition eauto using dra_op_minus, dra_disjoint_minus, dra_unit_valid. * by intros [x px ?] [y py ?] (?&?&?). - * intros [x px ?] [y py ?]; split; naive_solver. - * intros [x px ?] [y py ?]; - unfold included, validity_included; split; naive_solver. + * intros [x px ?] [y py ?] [[z pz ?] [??]]; split; simpl in *; + intuition eauto 10 using dra_disjoint_minus, dra_op_minus. Qed. Definition dra_update (x y : T) : (∀ z, V x → V z → validity_car x ⊥ z → V y ∧ validity_car y ⊥ z) → x ⇝ y. diff --git a/iris/excl.v b/iris/excl.v index 698fc40a7ad23549b76e7b4da0e72d9e72585b22..7f4edafd2a3f8ef2fae745ee080aa30fdc5fc4a2 100644 --- a/iris/excl.v +++ b/iris/excl.v @@ -1,6 +1,5 @@ Require Export iris.cmra. Local Arguments disjoint _ _ !_ !_ /. -Local Arguments included _ _ !_ !_ /. Inductive excl (A : Type) := | Excl : A → excl A @@ -32,15 +31,6 @@ Instance excl_minus {A} : Minus (excl A) := λ x y, | Excl _, Excl _ => ExclUnit | _, _ => ExclBot end. -Instance excl_included `{Equiv A} : Included (excl A) := λ x y, - match x, y with - | Excl x, Excl y => x ≡ y - | ExclUnit, _ => True - | _, ExclBot => True - | _, _ => False - end. -Definition excl_above `{Included A} (x : A) (y : excl A) : Prop := - match y with Excl y' => x ≼ y' | ExclUnit => True | ExclBot => False end. Instance excl_ra `{Equiv A, !Equivalence (@equiv A _)} : RA (excl A). Proof. split. @@ -52,21 +42,23 @@ Proof. * constructor. * by destruct 1. * by do 2 destruct 1; constructor. - * by intros []; destruct 1; simpl; try (intros Hx; rewrite Hx). * by intros [?| |] [?| |] [?| |]; constructor. * by intros [?| |] [?| |]; constructor. * by intros [?| |]; constructor. * constructor. + * intros [?| |] [?| |]; exists ∅; constructor. * by intros [?| |] [?| |]. - * by intros [?| |] [?| |]. - * by intros [?| |] [?| |]; simpl; try constructor. - * by intros [?| |] [?| |] ?; try constructor. + * by intros [?| |] [?| |] [[?| |] Hz]; inversion_clear Hz; constructor. Qed. Instance excl_empty_ra `{Equiv A, !Equivalence (@equiv A _)} : RAEmpty (excl A). Proof. split. done. by intros []. Qed. Lemma excl_update {A} (x : A) y : valid y → Excl x ⇝ y. Proof. by destruct y; intros ? [?| |]. Qed. +Definition excl_above `{Equiv A, Op A} (x : A) (y : excl A) : Prop := + match y with Excl y' => x ≼ y' | ExclUnit => True | ExclBot => False end. +Instance: Params (@excl_above) 3. + Section excl_above. Context `{RA A}. Global Instance excl_above_proper : Proper ((≡) ==> (≡) ==> iff) excl_above. @@ -74,7 +66,8 @@ Section excl_above. Lemma excl_above_weaken (a b : A) x y : excl_above b y → a ≼ b → x ≼ y → excl_above a x. Proof. - destruct x as [a'| |], y as [b'| |]; simpl; intros ??; try done. - by intros Hab; rewrite Hab; transitivity b. + destruct x as [a'| |], y as [b'| |]; + intros ?? [[] Hz]; inversion_clear Hz; simpl in *; try done. + by setoid_subst; transitivity b. Qed. End excl_above. diff --git a/iris/language.v b/iris/language.v new file mode 100644 index 0000000000000000000000000000000000000000..c9d26dc74826b7494967edb91aa2161fb543ddb9 --- /dev/null +++ b/iris/language.v @@ -0,0 +1,31 @@ +Require Import prelude.prelude. + +Class language (E V S : Type) := Language { + to_expr : V → E; + of_expr : E → option V; + atomic : E → Prop; + prim_step : (E * S) → (E * S) → option E → Prop; + of_to_expr v : of_expr (to_expr v) = Some v; + to_of_expr e v : of_expr e = Some v → to_expr v = e; + values_stuck e σ s' ef : prim_step (e,σ) s' ef → of_expr e = None; + atomic_not_value e : atomic e → of_expr e = None; + atomic_step e1 σ1 e2 σ2 ef : + atomic e1 → + prim_step (e1,σ1) (e2,σ2) ef → + is_Some (of_expr e2) +}. + +Section foo. + Context `{language E V St}. + + Definition cfg : Type := (list E * St)%type. + Inductive step (ρ1 ρ2 : cfg) : Prop := + | step_atomic e1 σ1 e2 σ2 ef t1 t2 : + ρ1 = (t1 ++ e1 :: t2, σ1) → + ρ1 = (t1 ++ e2 :: t2 ++ option_list ef, σ2) → + prim_step (e1, σ1) (e2, σ2) ef → + step ρ1 ρ2. + + Definition steps := rtc step. + Definition stepn := nsteps step. +End foo. \ No newline at end of file diff --git a/iris/logic.v b/iris/logic.v index d3165e625a1ea8908d0a881fb0353a06ff4015ab..0e1ff71a0349369e55ae7a0fbb23fd6c888cdf33 100644 --- a/iris/logic.v +++ b/iris/logic.v @@ -48,7 +48,7 @@ Canonical Structure uPredC (M : cmraT) : cofeT := CofeT (uPred M). (** functor *) Program Definition uPred_map {M1 M2 : cmraT} (f : M2 → M1) - `{!∀ n, Proper (dist n ==> dist n) f, !CMRAPreserving f} + `{!∀ n, Proper (dist n ==> dist n) f, !CMRAMonotone f} (P : uPred M1) : uPred M2 := {| uPred_holds n x := P n (f x) |}. Next Obligation. by intros M1 M2 f ?? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed. Next Obligation. intros M1 M2 f _ _ P x; apply uPred_0. Qed. @@ -56,15 +56,15 @@ Next Obligation. naive_solver eauto using uPred_weaken, included_preserving, validN_preserving. Qed. Instance uPred_map_ne {M1 M2 : cmraT} (f : M2 → M1) - `{!∀ n, Proper (dist n ==> dist n) f, !CMRAPreserving f} : + `{!∀ n, Proper (dist n ==> dist n) f, !CMRAMonotone f} : Proper (dist n ==> dist n) (uPred_map f). Proof. by intros n x1 x2 Hx y n'; split; apply Hx; auto using validN_preserving. Qed. -Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAPreserving f} : +Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} : uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2). Lemma upredC_map_ne {M1 M2 : cmraT} (f g : M2 -n> M1) - `{!CMRAPreserving f, !CMRAPreserving g} n : + `{!CMRAMonotone f, !CMRAMonotone g} n : f ={n}= g → uPredC_map f ={n}= uPredC_map g. Proof. by intros Hfg P y n' ??; simpl; rewrite (dist_le _ _ _ _(Hfg y)) by lia. @@ -125,8 +125,7 @@ Next Obligation. by intros M P Q x; exists x, x. Qed. Next Obligation. intros M P Q x y n1 n2 (x1&x2&Hx&?&?) Hxy ? Hvalid. assert (∃ x2', y ={n2}= x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?). - { rewrite ra_included_spec in Hxy; destruct Hxy as [z Hy]. - exists (x2 ⋅ z); split; eauto using ra_included_l. + { destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using ra_included_l. apply dist_le with n1; auto. by rewrite (associative op), <-Hx, Hy. } rewrite Hy in Hvalid; exists x1, x2'; split_ands; auto. * apply uPred_weaken with x1 n1; eauto using cmra_valid_op_l. @@ -165,11 +164,11 @@ Next Obligation. Qed. Program Definition uPred_own {M : cmraT} (a : M) : uPred M := - {| uPred_holds n x := ∃ a', x ={n}= a ⋅ a' |}. -Next Obligation. by intros M a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed. + {| uPred_holds n x := a ≼{n} x |}. +Next Obligation. by intros M a x1 x2 n [a' ?] Hx; exists a'; rewrite <-Hx. Qed. Next Obligation. by intros M a x; exists x. Qed. Next Obligation. - intros M a x1 x n1 n2; rewrite ra_included_spec; intros [a' Hx1] [x2 Hx] ??. + intros M a x1 x n1 n2 [a' Hx1] [x2 Hx] ??. exists (a' ⋅ x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx. Qed. Program Definition uPred_valid {M : cmraT} (a : M) : uPred M := @@ -571,7 +570,7 @@ Lemma uPred_later_sep P Q : (▷ (P ★ Q))%I ≡ (▷ P ★ ▷ Q)%I. Proof. intros x n ?; split. * destruct n as [|n]; simpl; [by exists x, x|intros (x1&x2&Hx&?&?)]. - destruct (cmra_extend_op x x1 x2 n) + destruct (cmra_extend_op n x x1 x2) as ([y1 y2]&Hx'&Hy1&Hy2); auto using cmra_valid_S; simpl in *. exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1, Hy2]. * destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)]. @@ -682,10 +681,9 @@ Proof. Qed. Lemma uPred_own_unit (a : M) : uPred_own (unit a) ≡ (□ uPred_own (unit a))%I. Proof. - intros x n; split; [intros [a' Hx]|by apply uPred_always_elim]. - assert (∃ a'', unit (unit a ⋅ a') ≡ unit (unit a) ⋅ a'') as [a'' Ha] - by (rewrite <-ra_included_spec; auto using ra_unit_weaken). - by exists a''; rewrite Hx, Ha, ra_unit_idempotent. + intros x n; split; [intros [a' Hx]|by apply uPred_always_elim]. simpl. + rewrite <-(ra_unit_idempotent a), Hx. + apply cmra_unit_preserving, cmra_included_l. Qed. Lemma uPred_own_empty `{Empty M, !RAEmpty M} : True%I ⊆ uPred_own ∅. Proof. intros x [|n] ??; [done|]. by exists x; rewrite (left_id _ _). Qed. @@ -712,7 +710,7 @@ Global Instance uPred_sep_timeless P Q : TimelessP P → TimelessP Q → TimelessP (P ★ Q). Proof. intros ?? x [|n] Hvalid (x1&x2&Hx12&?&?); [done|]. - destruct (cmra_extend_op x x1 x2 1) as ([y1 y2]&Hx&Hy1&Hy2); auto; simpl in *. + destruct (cmra_extend_op 1 x x1 x2) as ([y1 y2]&Hx&Hy1&Hy2); auto; simpl in *. rewrite Hx12 in Hvalid; exists y1, y2; split_ands; [by rewrite Hx| |]. * apply timelessP; rewrite Hy1; eauto using cmra_valid_op_l. * apply timelessP; rewrite Hy2; eauto using cmra_valid_op_r. @@ -738,7 +736,7 @@ Global Instance uPred_own_timeless (a : M) : Timeless a → TimelessP (uPred_own a). Proof. intros ? x n ? [a' ?]. - destruct (cmra_extend_op x a a' 1) as ([b b']&Hx&Hb&Hb'); auto; simpl in *. + destruct (cmra_extend_op 1 x a a') as ([b b']&Hx&Hb&Hb'); auto; simpl in *. by exists b'; rewrite Hx, (timeless a b) by done. Qed. End logic. diff --git a/iris/ra.v b/iris/ra.v index a668bafe150a04978a1c19ee8b935e960ef946a2..001ffdb812a7d9318677419bb7c4f92df5ec2984 100644 --- a/iris/ra.v +++ b/iris/ra.v @@ -11,32 +11,30 @@ Instance: Params (@op) 2. Infix "⋅" := op (at level 50, left associativity) : C_scope. Notation "(⋅)" := op (only parsing) : C_scope. -Class Included (A : Type) := included : relation A. -Instance: Params (@included) 2. +Definition included `{Equiv A, Op A} (x y : A) := ∃ z, y ≡ x ⋅ z. Infix "≼" := included (at level 70) : C_scope. Notation "(≼)" := included (only parsing) : C_scope. Hint Extern 0 (?x ≼ ?x) => reflexivity. +Instance: Params (@included) 3. Class Minus (A : Type) := minus : A → A → A. Instance: Params (@minus) 2. Infix "⩪" := minus (at level 40) : C_scope. -Class RA A `{Equiv A, Valid A, Unit A, Op A, Included A, Minus A} : Prop := { +Class RA A `{Equiv A, Valid A, Unit A, Op A, Minus A} : Prop := { (* setoids *) ra_equivalence :> Equivalence ((≡) : relation A); ra_op_proper x :> Proper ((≡) ==> (≡)) (op x); ra_unit_proper :> Proper ((≡) ==> (≡)) unit; ra_valid_proper :> Proper ((≡) ==> impl) valid; ra_minus_proper :> Proper ((≡) ==> (≡) ==> (≡)) minus; - ra_included_proper x :> Proper ((≡) ==> impl) (included x); (* monoid *) ra_associative :> Associative (≡) (⋅); ra_commutative :> Commutative (≡) (⋅); ra_unit_l x : unit x ⋅ x ≡ x; ra_unit_idempotent x : unit (unit x) ≡ unit x; - ra_unit_weaken x y : unit x ≼ unit (x ⋅ y); + ra_unit_preserving x y : x ≼ y → unit x ≼ unit y; ra_valid_op_l x y : valid (x ⋅ y) → valid x; - ra_included_l x y : x ≼ x ⋅ y; ra_op_minus x y : x ≼ y → x ⋅ y ⩪ x ≡ y }. Class RAEmpty A `{Equiv A, Valid A, Op A, Empty A} : Prop := { @@ -44,6 +42,12 @@ Class RAEmpty A `{Equiv A, Valid A, Op A, Empty A} : Prop := { ra_empty_l :> LeftId (≡) ∅ (⋅) }. +Class RAMonotone + `{Equiv A, Op A, Valid A, Equiv B, Op B, Valid B} (f : A → B) := { + included_preserving x y : x ≼ y → f x ≼ f y; + valid_preserving x : valid x → valid (f x) +}. + (** Big ops *) Fixpoint big_op `{Op A, Empty A} (xs : list A) : A := match xs with [] => ∅ | x :: xs => x ⋅ big_op xs end. @@ -87,38 +91,27 @@ Lemma ra_unit_unit x : unit x ⋅ unit x ≡ unit x. Proof. by rewrite <-(ra_unit_idempotent x) at 2; rewrite ra_unit_r. Qed. (** ** Order *) -Lemma ra_included_spec x y : x ≼ y ↔ ∃ z, y ≡ x ⋅ z. -Proof. - split; [by exists (y ⩪ x); rewrite ra_op_minus|]. - intros [z Hz]; rewrite Hz; apply ra_included_l. -Qed. -Global Instance ra_included_proper' : Proper ((≡) ==> (≡) ==> iff) (≼). -Proof. - intros x1 x2 Hx y1 y2 Hy; rewrite !ra_included_spec. - by setoid_rewrite Hx; setoid_rewrite Hy. -Qed. +Instance ra_included_proper' : Proper ((≡) ==> (≡) ==> impl) (≼). +Proof. intros x1 x2 Hx y1 y2 Hy [z Hz]; exists z. by rewrite <-Hy, Hz, Hx. Qed. +Global Instance ra_included_proper : Proper ((≡) ==> (≡) ==> iff) (≼). +Proof. by split; apply ra_included_proper'. Qed. +Lemma ra_included_l x y : x ≼ x ⋅ y. +Proof. by exists y. Qed. +Lemma ra_included_r x y : y ≼ x ⋅ y. +Proof. rewrite (commutative op); apply ra_included_l. Qed. Global Instance: PreOrder included. Proof. split. - * by intros x; rewrite ra_included_spec; exists (unit x); rewrite ra_unit_r. - * intros x y z; rewrite ?ra_included_spec; intros [z1 Hy] [z2 Hz]. - exists (z1 ⋅ z2). by rewrite (associative _), <-Hy, <-Hz. + * by intros x; exists (unit x); rewrite ra_unit_r. + * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). + by rewrite (associative _), <-Hy, <-Hz. Qed. Lemma ra_included_unit x : unit x ≼ x. -Proof. by rewrite ra_included_spec; exists x; rewrite ra_unit_l. Qed. -Lemma ra_included_r x y : y ≼ x ⋅ y. -Proof. rewrite (commutative _); apply ra_included_l. Qed. +Proof. by exists x; rewrite ra_unit_l. Qed. Lemma ra_preserving_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. -Proof. - rewrite !ra_included_spec. - by intros (z1&Hz1); exists z1; rewrite Hz1, (associative (⋅)). -Qed. +Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1, (associative (⋅)). Qed. Lemma ra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. Proof. by intros; rewrite <-!(commutative _ z); apply ra_preserving_l. Qed. -Lemma ra_unit_preserving x y : x ≼ y → unit x ≼ unit y. -Proof. - rewrite ra_included_spec; intros [z Hy]; rewrite Hy; apply ra_unit_weaken. -Qed. (** ** Properties of [(⇝)] relation *) Global Instance ra_update_preorder : PreOrder ra_update. @@ -132,7 +125,7 @@ Proof. by intros x; rewrite (commutative op), (left_id _ _). Qed. Lemma ra_unit_empty x : unit ∅ ≡ ∅. Proof. by rewrite <-(ra_unit_l ∅) at 2; rewrite (right_id _ _). Qed. Lemma ra_empty_least x : ∅ ≼ x. -Proof. by rewrite ra_included_spec; exists x; rewrite (left_id _ _). Qed. +Proof. by exists x; rewrite (left_id _ _). Qed. (** * Big ops *) Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) big_op. diff --git a/iris/sts.v b/iris/sts.v index 71dbcdfd67eb3abc1fb1174c742292a767b3ac89..ceee5ea16c9e1cac54c2905e06d7bc5dac6678c5 100644 --- a/iris/sts.v +++ b/iris/sts.v @@ -1,34 +1,34 @@ Require Export iris.ra. -Require Import prelude.sets prelude.bsets iris.dra. +Require Import prelude.sets iris.dra. Local Arguments valid _ _ !_ /. Local Arguments op _ _ !_ !_ /. Local Arguments unit _ _ !_ /. Module sts. -Inductive t {A B} (R : relation A) (tok : A → bset B) := - | auth : A → bset B → t R tok - | frag : set A → bset B → t R tok. +Inductive t {A B} (R : relation A) (tok : A → set B) := + | auth : A → set B → t R tok + | frag : set A → set B → t R tok. Arguments auth {_ _ _ _} _ _. Arguments frag {_ _ _ _} _ _. Section sts_core. -Context {A B : Type} `{∀ x y : B, Decision (x = y)}. -Context (R : relation A) (tok : A → bset B). +Context {A B : Type} (R : relation A) (tok : A → set B). +Infix "≼" := dra_included. Inductive sts_equiv : Equiv (t R tok) := | auth_equiv s T1 T2 : T1 ≡ T2 → auth s T1 ≡ auth s T2 | frag_equiv S1 S2 T1 T2 : T1 ≡ T2 → S1 ≡ S2 → frag S1 T1 ≡ frag S2 T2. Global Existing Instance sts_equiv. -Inductive step : relation (A * bset B) := +Inductive step : relation (A * set B) := | Step s1 s2 T1 T2 : R s1 s2 → tok s1 ∩ T1 ≡ ∅ → tok s2 ∩ T2 ≡ ∅ → tok s1 ∪ T1 ≡ tok s2 ∪ T2 → step (s1,T1) (s2,T2). Hint Resolve Step. -Inductive frame_step (T : bset B) (s1 s2 : A) : Prop := +Inductive frame_step (T : set B) (s1 s2 : A) : Prop := | Frame_step T1 T2 : T1 ∩ (tok s1 ∪ T) ≡ ∅ → step (s1,T1) (s2,T2) → frame_step T s1 s2. Hint Resolve Frame_step. -Record closed (T : bset B) (S : set A) : Prop := Closed { +Record closed (T : set B) (S : set A) : Prop := Closed { closed_disjoint s : s ∈ S → tok s ∩ T ≡ ∅; closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }. @@ -37,8 +37,8 @@ Lemma closed_steps S T s1 s2 : Proof. induction 3; eauto using closed_step. Qed. Global Instance sts_valid : Valid (t R tok) := λ x, match x with auth s T => tok s ∩ T ≡ ∅ | frag S' T => closed T S' end. -Definition up (T : bset B) (s : A) : set A := mkSet (rtc (frame_step T) s). -Definition up_set (T : bset B) (S : set A) : set A := S ≫= up T. +Definition up (T : set B) (s : A) : set A := mkSet (rtc (frame_step T) s). +Definition up_set (T : set B) (S : set A) : set A := S ≫= up T. Global Instance sts_unit : Unit (t R tok) := λ x, match x with | frag S' _ => frag (up_set ∅ S') ∅ | auth s _ => frag (up ∅ s) ∅ @@ -55,12 +55,6 @@ Global Instance sts_op : Op (t R tok) := λ x1 x2, | frag _ T1, auth s T2 => auth s (T1 ∪ T2) | auth s T1, auth _ T2 => auth s (T1 ∪ T2) (* never happens *) end. -Inductive sts_included : Included (t R tok) := - | frag_frag_included S1 S2 T1 T2 S' : - T1 ⊆ T2 → S2 ≡ S1 ∩ S' → closed (T2 ∖ T1) S' → frag S1 T1 ≼ frag S2 T2 - | frag_auth_included s S T1 T2 : s ∈ S → T1 ⊆ T2 → frag S T1 ≼ auth s T2 - | auth_auth_included s T1 T2 : T1 ⊆ T2 → auth s T1 ≼ auth s T2. -Global Existing Instance sts_included. Global Instance sts_minus : Minus (t R tok) := λ x1 x2, match x1, x2 with | frag S1 T1, frag S2 T2 => frag (up_set (T1 ∖ T2) S1) (T1 ∖ T2) @@ -69,10 +63,9 @@ Global Instance sts_minus : Minus (t R tok) := λ x1 x2, | auth s T1, auth _ T2 => frag (up (T1 ∖ T2) s) (T1 ∖ T2) end. -Hint Extern 5 (equiv (A:=set _) _ _) => esolve_elem_of : sts. -Hint Extern 5 (equiv (A:=bset _) _ _) => esolve_elem_of : sts. -Hint Extern 5 (_ ∈ _) => esolve_elem_of : sts. -Hint Extern 5 (_ ⊆ _) => esolve_elem_of : sts. +Hint Extern 10 (equiv (A:=set _) _ _) => esolve_elem_of : sts. +Hint Extern 10 (_ ∈ _) => esolve_elem_of : sts. +Hint Extern 10 (_ ⊆ _) => esolve_elem_of : sts. Instance: Equivalence ((≡) : relation (t R tok)). Proof. split. @@ -112,7 +105,7 @@ Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. Proof. by intros T1 T2 HT S1 S2 HS; unfold up_set; rewrite HS, HT. Qed. Lemma elem_of_up s T : s ∈ up T s. Proof. constructor. Qed. -Lemma subseteq_up_set S T : S ⊆ up_set T S. +Lemma suseteq_up_set S T : S ⊆ up_set T S. Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed. Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ∩ T ≡ ∅) → closed T (up_set T S). Proof. @@ -135,16 +128,10 @@ Lemma closed_up_empty s : closed ∅ (up ∅ s). Proof. eauto using closed_up with sts. Qed. Lemma up_closed S T : closed T S → up_set T S ≡ S. Proof. - intros; split; auto using subseteq_up_set; intros s. + intros; split; auto using suseteq_up_set; intros s. unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?). induction Hstep; eauto using closed_step. Qed. -Lemma up_set_subseteq T1 T2 S1 S2 : - closed (T2 ∖ T1) S2 → up_set (T2 ∖ T1) (S1 ∩ S2) ⊆ S2. -Proof. - intros ? s2; unfold up_set; rewrite elem_of_bind; intros (s1&?&?). - apply closed_steps with (T2 ∖ T1) s1; auto with sts. -Qed. Global Instance sts_dra : DRA (t R tok). Proof. split. @@ -153,13 +140,13 @@ Proof. * by destruct 1; constructor; setoid_subst. * by intros ? [|]; destruct 1; inversion_clear 1; constructor; setoid_subst. * by do 2 destruct 1; constructor; setoid_subst. - * by do 2 destruct 1; inversion_clear 1; econstructor; setoid_subst. * assert (∀ T T' S s, closed T S → s ∈ S → tok s ∩ T' ≡ ∅ → tok s ∩ (T ∪ T') ≡ ∅). { intros S T T' s [??]; esolve_elem_of. } destruct 3; simpl in *; auto using closed_op with sts. * intros []; simpl; eauto using closed_up, closed_up_set with sts. - * destruct 3; simpl in *; setoid_subst; eauto using closed_up with sts. + * intros ???? (z&Hy&?&Hxz); destruct Hxz; inversion Hy;clear Hy; setoid_subst; + rewrite ?disjoint_union_difference; auto using closed_up with sts. eapply closed_up_set; eauto 2 using closed_disjoint with sts. * intros [] [] []; constructor; rewrite ?(associative _); auto with sts. * destruct 4; inversion_clear 1; constructor; auto with sts. @@ -168,22 +155,24 @@ Proof. * destruct 3; constructor; auto with sts. * intros []; constructor; auto using elem_of_up with sts. * intros [|S T]; constructor; auto with sts. - assert (S ⊆ up_set ∅ S); auto using subseteq_up_set with sts. + assert (S ⊆ up_set ∅ S); auto using suseteq_up_set with sts. * intros [s T|S T]; constructor; auto with sts. + by rewrite (up_closed (up _ _)) by auto using closed_up with sts. + by rewrite (up_closed (up_set _ _)) by auto using closed_up_set with sts. - * destruct 3 as [S1 S2| |]; simpl; - match goal with |- _ ≼ frag ?S _ => apply frag_frag_included with S end; - eauto using closed_up_empty, closed_up_set_empty; - unfold up_set; esolve_elem_of. - * destruct 3 as [S1 S2 T1 T2| |]; econstructor; eauto with sts. - by setoid_replace ((T1 ∪ T2) ∖ T1) with T2 by esolve_elem_of. - * destruct 3; constructor; eauto using elem_of_up with sts. - * destruct 3 as [S1 S2 T1 T2 S'| |]; constructor; - rewrite ?(commutative _ (_ ∖ _)), <-?union_difference; auto with sts. - assert (S2 ⊆ up_set (T2 ∖ T1) S2) by eauto using subseteq_up_set. - assert (up_set (T2 ∖ T1) (S1 ∩ S') ⊆ S') by eauto using up_set_subseteq. - esolve_elem_of. + * intros x y ?? (z&Hy&?&Hxz); exists (unit (x ⋅ y)). + destruct Hxz; inversion_clear Hy; simpl; split_ands; + auto using closed_up_set_empty, closed_up_empty; + constructor; unfold up_set; auto with sts. + * intros x y ?? (z&Hy&_&Hxz); destruct Hxz; inversion_clear Hy; + constructor; eauto using elem_of_up; auto with sts. + * intros x y ?? (z&Hy&?&Hxz); destruct Hxz as [S1 S2 T1 T2| |]; + inversion Hy; clear Hy; constructor; setoid_subst; + rewrite ?disjoint_union_difference by done; auto. + split; [|apply intersection_greatest; auto using suseteq_up_set with sts]. + apply intersection_greatest; [auto with sts|]. + intros s2; rewrite elem_of_intersection. + unfold up_set; rewrite elem_of_bind; intros (?&s1&?&?&?). + apply closed_steps with T2 s1; auto with sts. Qed. Lemma step_closed s1 s2 T1 T2 S Tf : step (s1,T1) (s2,T2) → closed Tf S → s1 ∈ S → T1 ∩ Tf ≡ ∅ → @@ -198,16 +187,15 @@ End sts. Section sts_ra. Context {A B : Type} `{∀ x y : B, Decision (x = y)}. -Context (R : relation A) (tok : A → bset B). +Context (R : relation A) (tok : A → set B). Definition sts := validity (valid : sts.t R tok → Prop). Global Instance sts_unit : Unit sts := validity_unit _. Global Instance sts_op : Op sts := validity_op _. -Global Instance sts_included : Included sts := validity_included _. Global Instance sts_minus : Minus sts := validity_minus _. Global Instance sts_ra : RA sts := validity_ra _. -Definition sts_auth (s : A) (T : bset B) : sts := to_validity (sts.auth s T). -Definition sts_frag (S : set A) (T : bset B) : sts := +Definition sts_auth (s : A) (T : set B) : sts := to_validity (sts.auth s T). +Definition sts_frag (S : set A) (T : set B) : sts := to_validity (sts.frag S T). Lemma sts_update s1 s2 T1 T2 : sts.step R tok (s1,T1) (s2,T2) → sts_auth s1 T1 ⇝ sts_auth s2 T2. diff --git a/prelude/collections.v b/prelude/collections.v index 619134db21a72ae5717e1a6659093b43f7b76dac..a0bbafe61a4e368eec1ba446070aa5e391823556 100644 --- a/prelude/collections.v +++ b/prelude/collections.v @@ -297,6 +297,8 @@ Section collection. Proof. esolve_elem_of. Qed. Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z. Proof. esolve_elem_of. Qed. + Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y. + Proof. esolve_elem_of. Qed. Section leibniz. Context `{!LeibnizEquiv C}. @@ -315,6 +317,8 @@ Section collection. Lemma difference_intersection_distr_l_L X Y Z : (X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z. Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed. + Lemma disjoint_union_difference_L X Y : X ∩ Y = ∅ → (X ∪ Y) ∖ X = Y. + Proof. unfold_leibniz. apply disjoint_union_difference. Qed. End leibniz. Section dec. | 20,029 | 54,916 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2020-50 | latest | en | 0.725391 |
http://www.ck12.org/algebra/Completing-the-Square/lesson/Completing-the-Square-When-the-Leading-Coefficient-Doesnt-Equal-1-ALG-II/ | 1,474,916,986,000,000,000 | text/html | crawl-data/CC-MAIN-2016-40/segments/1474738660882.14/warc/CC-MAIN-20160924173740-00139-ip-10-143-35-109.ec2.internal.warc.gz | 396,408,969 | 59,598 | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Completing the Square
## Create perfect square trinomials using the additive property of equality
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Practice Completing the Square
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Completing the Square When the Leading Coefficient Doesn't Equal 1
The area of a parallelogram is given by the equation 3x2+9x5=0\begin{align*}3x^2 + 9x - 5 = 0\end{align*}, where x is the length of the base. What is the length of this base?
### Completing the Square
When there is a number in front of x2\begin{align*}x^2\end{align*}, it will make completing the square a little more complicated.
Let's determine the number c that completes the square of 2x28x+c\begin{align*}2x^2 - 8x + c \end{align*}.
Previously, we just added (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}, but that was when a=1\begin{align*}a=1\end{align*}. Now that a1\begin{align*}a \neq 1\end{align*}, we have to take the value of a into consideration. Let's pull out the GCF of 2 and 8 first.
2(x24x)\begin{align*}2 \left(x^2 - 4x \right)\end{align*}
Now, there is no number in front of x2\begin{align*}x^2\end{align*}.
(b2)2=(42)2=4\begin{align*}\left(\frac{b}{2}\right)^2 = \left(\frac{4}{2}\right)^2 = 4\end{align*}.
Add this number inside the parenthesis and distribute the 2.
2(x24x+4)=2x24x+8\begin{align*}2 \left(x^2 - 4x +4 \right)=2x^2-4x+8\end{align*}
So, c=8\begin{align*}c=8\end{align*}.
Now, let's solve the following problems by completing the square.
1. Solve 3x29x+11=0\begin{align*}3x^2-9x+11=0\end{align*}.
Step 1: Write the polynomial so that x2\begin{align*}x^2\end{align*} and x\begin{align*}x\end{align*} are on the left side of the equation and the constants on the right.
3x29x=11\begin{align*}3x^2-9x=-11\end{align*}
Step 2: Pull out a\begin{align*}a\end{align*} from everything on the left side. Even if b\begin{align*}b\end{align*} is not divisible by a\begin{align*}a\end{align*}, the coefficient of x2\begin{align*}x^2\end{align*} needs to be 1 in order to complete the square.
3(x23x+)=11\begin{align*}3(x^2-3x+\underline{\;\;\;\;\;\;})=-11\end{align*}
Step 3: Now, complete the square. Determine what number would make a perfect square trinomial.
To do this, divide the x\begin{align*}x-\end{align*}term by 2 and square that number, or (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.
(b2)2=(32)2=94\begin{align*}\left(\frac{b}{2}\right)^2=\left(\frac{3}{2}\right)^2=\frac{9}{4}\end{align*}
Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add a(b2)2\begin{align*}{\color{red}a\cdot\left(\frac{b}{2}\right)^2}\end{align*} to keep the equation balanced.
3(x23x+94)=11+274\begin{align*}3\left(x^2-3x {\color{red}+\frac{9}{4}}\right)=-11 {\color{red}+\frac{27}{4}}\end{align*}
Step 5: Factor the left side and simplify the right.
3(x32)2=174\begin{align*}3\left(x-\frac{3}{2}\right)^2=-\frac{17}{4}\end{align*}
Step 6: Solve by using square roots.
(x32)2x32x=1712=±i172333=32±516i\begin{align*}\left(x-\frac{3}{2}\right)^2 &=-\frac{17}{12}\\ x-\frac{3}{2} &=\pm \frac{i\sqrt{17}}{2\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\\ x &=\frac{3}{2}\pm \frac{\sqrt{51}}{6}i\end{align*}
Be careful with the addition of Step 2 and the changes made to Step 4. A very common mistake is to add (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*} to both sides, without multiplying by a\begin{align*}a\end{align*} for the right side.
1. Solve 4x2+7x18=0\begin{align*}4x^2+7x-18=0\end{align*}.
Let’s follow the steps from problem #1 above.
Step 1: Write the polynomial so that x2\begin{align*}x^2\end{align*} and x\begin{align*}x\end{align*} are on the left side of the equation and the constants on the right.
4x27x=18\begin{align*}4x^2-7x=18\end{align*}
Step 2: Pull out a\begin{align*}a\end{align*} from everything on the left side.
4(x2+74x+)=18\begin{align*}4\left(x^2+\frac{7}{4}x+\underline{\;\;\;\;\;\;}\right)=18\end{align*}
Step 3: Now, complete the square. Find (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.
(b2)2=(78)2=4964\begin{align*}\left(\frac{b}{2}\right)^2=\left(\frac{7}{8}\right)^2=\frac{49}{64}\end{align*}
Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add a(b2)2\begin{align*}{\color{red}a\cdot\left(\frac{b}{2}\right)^2}\end{align*} to keep the equation balanced.
4(x2+74x+4964)=18+4916\begin{align*}4\left(x^2+\frac{7}{4}x{\color{red}+\frac{49}{64}}\right)=18{\color{red}+\frac{49}{16}}\end{align*}
Step 5: Factor the left side and simplify the right.
4(x+78)2=33716\begin{align*}4\left(x+\frac{7}{8}\right)^2=\frac{337}{16}\end{align*}
Step 6: Solve by using square roots.
(x+78)2x+78x=33764=±3378=78±3378\begin{align*}\left(x+\frac{7}{8}\right)^2 &=\frac{337}{64}\\ x+\frac{7}{8} &=\pm \frac{\sqrt{337}}{8}\\ x &=-\frac{7}{8} \pm \frac{\sqrt{337}}{8}\end{align*}
### Examples
#### Example 1
Earlier, you were asked to find the length of the base of the parallelogram.
We can't factor 3x2+9x5=0\begin{align*}3x^2 + 9x - 5 = 0\end{align*}, so let's follow the step-by-step process we learned in this lesson.
Step 1: Write the polynomial so that x2\begin{align*}x^2\end{align*} and x\begin{align*}x\end{align*} are on the left side of the equation and the constants on the right.
3x2+9x=5\begin{align*}3x^2 + 9x = 5\end{align*}
Step 2: Pull out a\begin{align*}a\end{align*} from everything on the left side.
3(x2+3x+)=5\begin{align*}3\left(x^2 + 3x+\underline{\;\;\;\;\;\;}\right) = 5\end{align*}
Step 3: Now, complete the square. Find (b2)2\begin{align*}\left(\frac{b}{2}\right)^2\end{align*}.
(b2)2=(32)2=94\begin{align*}\left(\frac{b}{2}\right)^2=\left(\frac{3}{2}\right)^2=\frac{9}{4}\end{align*}
Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add a(b2)2\begin{align*}{\color{red}a\cdot\left(\frac{b}{2}\right)^2}\end{align*} to keep the equation balanced.
3(x2+3x+94)=5+274\begin{align*}3\left(x^2+ 3x{\color{red}+\frac{9}{4}}\right)=5{\color{red}+\frac{27}{4}}\end{align*}
Step 5: Factor the left side and simplify the right.
3(x+32)2=474\begin{align*}3\left(x+\frac{3}{2}\right)^2=\frac{47}{4}\end{align*}
Step 6: Solve by using square roots.
(x+32)2x+32xx=4712=±4712=32±4723=32±1416\begin{align*}\left(x+\frac{3}{2}\right)^2 &=\frac{47}{12}\\ x+\frac{3}{2} &=\pm \frac{\sqrt{47}}{\sqrt{12}}\\ x &=-\frac{3}{2} \pm \frac{{\sqrt{47}}}{2\sqrt{3}}\\ x &=-\frac{3}{2} \pm \frac{{\sqrt{141}}}{6} \end{align*}
However, because x is the length of the parallelogram's base, it must have a positive value. Only x=32+1416\begin{align*}x =-\frac{3}{2} + \frac{{\sqrt{141}}}{6}\end{align*} results in a positive value, so the length of the base is x=32+1416\begin{align*}x =-\frac{3}{2} + \frac{{\sqrt{141}}}{6}\end{align*}.
#### Example 2
Solve the following quadratic equation by completing the square: 5x2+29x6=0\begin{align*}5x^2+29x-6=0\end{align*}.
5x2+29x65(x2+295x)5(x2+295x+841100)5(x+2910)2(x+2910)2x+2910xx=0=6=6+84120=96120=961100=±3110=2910±3110=6,15\begin{align*}5x^2+29x-6 &=0\\ 5\left(x^2+\frac{29}{5}x\right) &=6\\ 5\left(x^2+\frac{29}{5}x+\frac{841}{100}\right) &=6+\frac{841}{20}\\ 5\left(x+\frac{29}{10}\right)^2 &=\frac{961}{20}\\ \left(x+\frac{29}{10}\right)^2 &=\frac{961}{100}\\ x+\frac{29}{10} &=\pm \frac{31}{10}\\ x &=-\frac{29}{10} \pm \frac{31}{10}\\ x &=-6, \frac{1}{5}\end{align*}
#### Example 3
Solve the following quadratic by completing the square: 8x232x+4=0\begin{align*}8x^2-32x+4=0\end{align*}.
8x232x+48(x24x)8(x24x+4)8(x2)2(x2)2x2x=0=4=4+32=28=72=±7222=2±142\begin{align*}8x^2-32x+4 &=0\\ 8(x^2-4x) &=-4\\ 8(x^2-4x+4) &=-4+32\\ 8(x-2)^2 &=28\\ (x-2)^2 &=\frac{7}{2}\\ x-2 &=\pm \frac{\sqrt{7}}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}\\ x &=2\pm \frac{\sqrt{14}}{2}\end{align*}
### Review
Solve the quadratic equations by completing the square.
1. 6x212x7=0\begin{align*}6x^2-12x-7=0\end{align*}
2. 4x2+24x100=0\begin{align*}-4x^2+24x-100=0\end{align*}
3. 5x230x+55=0\begin{align*}5x^2-30x+55=0\end{align*}
4. 2x2x6=0\begin{align*}2x^2-x-6=0\end{align*}
5. 12x2+7x+8=0\begin{align*}\frac{1}{2}x^2+7x+8=0\end{align*}
6. 3x2+4x+15=0\begin{align*}-3x^2+4x+15=0\end{align*}
Solve the following equations by factoring, using square roots, or completing the square.
1. 4x24x8=0\begin{align*}4x^2-4x-8=0\end{align*}
2. 2x2+9x+7=0\begin{align*}2x^2+9x+7=0\end{align*}
3. 5(x+4)219=26\begin{align*}-5(x+4)^2-19=26\end{align*}
4. 3x2+30x5=0\begin{align*}3x^2+30x-5=0\end{align*}
5. 9x215x6=0\begin{align*}9x^2-15x-6=0\end{align*}
6. 10x2+40x+88=0\begin{align*}10x^2+40x+88=0\end{align*}
Problems 13-15 build off of each other.
1. Challenge Complete the square for ax2+bx+c=0\begin{align*}ax^2+bx+c=0\end{align*}. Follow the steps outlined in this lesson. Your final answer should be in terms of a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*}.
2. For the equation 8x2+6x5=0\begin{align*}8x^2+6x-5=0\end{align*}, use the formula you found in #13 to solve for x\begin{align*}x\end{align*}.
3. Is the equation in #14 factorable? If so, factor and solve it.
4. Error Analysis Examine the worked out problem below.
4x248x+114(x212x+)4(x212x+36)4(x6)2(x6)2x6x=0=11=11+36=25=254=±52=6±52172,72\begin{align*}4x^2-48x+11&=0\\ 4(x^2-12x+\underline{\;\;\;\;\;\;}) &=-11\\ 4(x^2-12x+36) &=-11+36\\ 4(x-6)^2 &=25\\ (x-6)^2 &=\frac{25}{4}\\ x-6 &=\pm \frac{5}{2}\\ x &=6\pm \frac{5}{2} \rightarrow \frac{17}{2},\frac{7}{2}\end{align*}
Plug the answers into the original equation to see if they work. If not, find the error and correct it.
To see the Review answers, open this PDF file and look for section 5.12.
### Notes/Highlights Having trouble? Report an issue.
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### Vocabulary Language: English
Perfect Square Trinomial
A perfect square trinomial is a quadratic expression of the form $a^2+2ab+b^2$ (which can be rewritten as $(a+b)^2$) or $a^2-2ab+b^2$ (which can be rewritten as $(a-b)^2$).
Square Root
The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9. | 4,207 | 10,285 | {"found_math": true, "script_math_tex": 77, "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": 4, "texerror": 0} | 4.875 | 5 | CC-MAIN-2016-40 | longest | en | 0.596377 |
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blue cobra
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### Re: Optics B/C
As I understand it, the energy of a photon is not continuous, but occurs only in discrete values. These values are described by Planck's constant in the equation E=hv. What I do not understand is how energy is quantized while frequency is not, or why frequency is quantized, and how exactly these relate to Planck's constant. Could someone please enlighten me?
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### Re: Optics B/C
Think of it this way. We know the universe is relative, right? The relationship between space and time has no static 0 point. That is, in space, you can never say what your absolute velocity is, only what it is relative to another object. So time cannot be universally quantized, because its value depends on how it is observed. The minimum frequency for light emitted from a hydrogen atom is about 24 Terahertz, but I can also wave this pencil around at 2.4 Hertz and the universe can't stop me. So time is not universally quantized, and frequency is not universally quantized, and by extension, energy is not universally quantized. But that doesn't mean that in certain situations and certain reference frames it can't be quantized. In fact, both frequency and energy are quantized when you get really really close to a single proton, also known as a hydrogen nucleus. in that case, only very specific frequencies and energies of photons can be created. But it is silly to try and find a way to universally quantize frequency or energy, because if I shine a 400 THz laser at the moon, and then I fly towards the moon really fast, the beam is no longer 400 THz. Does this answer your question?
blue cobra
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### Re: Optics B/C
Yes, thank you, I believe I understand now. I was erroneously assuming that E=hv as a mathematical relationship could only produce quantized values, whereas if we look at a hydrogen atom in an excited state moving to a lower energy state, it then becomes obvious how frequency becomes quantized within a certain reference frame. Thanks!
Last edited by blue cobra on August 24th, 2011, 11:44 am, edited 1 time in total.
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geekychic13
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### Re: Optics B/C
sooooooo, question. do we have to learn this for optics????
ichaelm wrote:Think of it this way. We know the universe is relative, right? The relationship between space and time has no static 0 point. That is, in space, you can never say what your absolute velocity is, only what it is relative to another object. So time cannot be universally quantized, because its value depends on how it is observed. The minimum frequency for light emitted from a hydrogen atom is about 24 Terahertz, but I can also wave this pencil around at 2.4 Hertz and the universe can't stop me. So time is not universally quantized, and frequency is not universally quantized, and by extension, energy is not universally quantized. But that doesn't mean that in certain situations and certain reference frames it can't be quantized. In fact, both frequency and energy are quantized when you get really really close to a single proton, also known as a hydrogen nucleus. in that case, only very specific frequencies and energies of photons can be created. But it is silly to try and find a way to universally quantize frequency or energy, because if I shine a 400 THz laser at the moon, and then I fly towards the moon really fast, the beam is no longer 400 THz. Does this answer your question?
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### Re: Optics B/C
geekychic13 wrote:sooooooo, question. do we have to learn this for optics????
ichaelm wrote:Think of it this way. We know the universe is relative, right? The relationship between space and time has no static 0 point. That is, in space, you can never say what your absolute velocity is, only what it is relative to another object. So time cannot be universally quantized, because its value depends on how it is observed. The minimum frequency for light emitted from a hydrogen atom is about 24 Terahertz, but I can also wave this pencil around at 2.4 Hertz and the universe can't stop me. So time is not universally quantized, and frequency is not universally quantized, and by extension, energy is not universally quantized. But that doesn't mean that in certain situations and certain reference frames it can't be quantized. In fact, both frequency and energy are quantized when you get really really close to a single proton, also known as a hydrogen nucleus. in that case, only very specific frequencies and energies of photons can be created. But it is silly to try and find a way to universally quantize frequency or energy, because if I shine a 400 THz laser at the moon, and then I fly towards the moon really fast, the beam is no longer 400 THz. Does this answer your question?
Not exactly. You should know that light exists as both an electromagnetic wave and a quantized particle, the photon. You should also know that one common source of light (i.e. photons) is when the electrons around an atom jump from a high-energy excited state to a low-energy state, releasing the extra energy as a photon.
In regards to ichaelm's post, he is essentially saying that in order for the frequency/wavelength of an electromagnetic wave to be quantized (that is, occur only in discrete quantities,) space-time itself would have to be quantized. This is because if space-time were smooth (which we currently believe it to be,) then the Doppler effect (i.e. red-shift) would allow the frequency of an electromagnetic wave to change smoothly. Because we currently think that space-time is smooth, then the frequency and wavelength of a photon do not occur only in discrete values.
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geekychic13
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### Re: Optics B/C
okay. i kinda understand, but not exactly. so we dont necessarily have to know that, but its related to something we have to know, right?
Infinity Flat wrote:
geekychic13 wrote:sooooooo, question. do we have to learn this for optics????
ichaelm wrote:Think of it this way. We know the universe is relative, right? The relationship between space and time has no static 0 point. That is, in space, you can never say what your absolute velocity is, only what it is relative to another object. So time cannot be universally quantized, because its value depends on how it is observed. The minimum frequency for light emitted from a hydrogen atom is about 24 Terahertz, but I can also wave this pencil around at 2.4 Hertz and the universe can't stop me. So time is not universally quantized, and frequency is not universally quantized, and by extension, energy is not universally quantized. But that doesn't mean that in certain situations and certain reference frames it can't be quantized. In fact, both frequency and energy are quantized when you get really really close to a single proton, also known as a hydrogen nucleus. in that case, only very specific frequencies and energies of photons can be created. But it is silly to try and find a way to universally quantize frequency or energy, because if I shine a 400 THz laser at the moon, and then I fly towards the moon really fast, the beam is no longer 400 THz. Does this answer your question?
Not exactly. You should know that light exists as both an electromagnetic wave and a quantized particle, the photon. You should also know that one common source of light (i.e. photons) is when the electrons around an atom jump from a high-energy excited state to a low-energy state, releasing the extra energy as a photon.
In regards to ichaelm's post, he is essentially saying that in order for the frequency/wavelength of an electromagnetic wave to be quantized (that is, occur only in discrete quantities,) space-time itself would have to be quantized. This is because if space-time were smooth (which we currently believe it to be,) then the Doppler effect (i.e. red-shift) would allow the frequency of an electromagnetic wave to change smoothly. Because we currently think that space-time is smooth, then the frequency and wavelength of a photon do not occur only in discrete values.
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### Re: Optics B/C
You might be expected to know about stuff like that, but most likely you won't. It's up to the person writing the test. I've seen tests that went into the resolving power of a diffraction-limited lens, but for most of the tests I took the hardest topic was double rainbows. So unless you go to a hard state competition or nationals, you probably don't need to go that deep.
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### Re: Optics B/C
Double Rainbows!?!?
Did you just need to know that it refracted once and reflected once for the primary rainbow and reflected again for the secondary rainbow? Or did you need to know more in depth?
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### Re: Optics B/C
We just needed to know how they work and what they look like in certain situations. But if you know how the refraction works, you can derive the rest of what you'd need to know. It was a question in a test that Bayard Rustin made for some invitationals.
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Unformatted text preview: v2 w3 u3 v1 w1 j u2w3 u1 v2 w2 u2 v1 w1 k w2 i u3v1 j u2v1 k u3w2 i u1w2 u2w1 k 59. u u u u1, u2, u3 ij u1 u 2 u1 u2 v u v v u2v3 u2v3 u2v3 u and u k u3 u3 u2u3 u3u2 i u1u3 u3u1 j u1u2 u2u1 k 0 61. u u u v v u3v2 i u3v2 u1 u3v2 v1 v v. u1v3 u3v1 u3v1 u3v1 j u1v3 u2 u1v3 v2 u1v2 u1v2 u1v2 u2v1 k u2v1 u3 u2v1 v3 0 0 Thus, u 63. u v u v sin If u and v are orthogonal, 2 and sin 1. Therefore, u v u v. Section 10.5 1. x (a) 1 3t, y z Lines and Planes in Space 2 t, z 2 5t (b) When t 0 we have P Q 10, 1, 17 . \ 1, 2, 2 . When t 3 we have PQ 9, 3, 15 \ The components of the vector and the coefficients of t are proportional since the line is parallel to PQ . x y (c) y 0 when t 2. Thus, x Point: 7, 0, 12 x z 0 when t 0 when t 1 . Point: 3 2 . Point: 5 7 and z 71 0, , 33 1 12 , ,0 55 12. 3. Point: (0, 0, 0 Direction vector: v (a) Parametric: x (b) Symmetric: x 1, 2, 3 t, y y 2 z 3 2t, z 3t Direction numbers: 1, 2, 3 5. Point: 2, 0, 3 2, 4, 2 2t, y y 4 z 4t, z 3 2 3 2t 2 2 2 2 Direction vector: v (a) Parametric: x (b) Symmetric: x Direction numbers: 2, 4, Section 10.5 Lines and Planes in Space 22 , ,1 33 17 i 3 11, 17t, y y 3 11 z 5 5 17 11 j 3 9 3 2 9 11t, z 2 3k 245 7. Point: 1, 0, 1 Direction vector: v Direction numbers: 3, (a) Parametric: x (b) Symmetric: x 3 1 1 3i 2j 2, 1 3t, y y 2 z 1 2t, z 1 1 t k 9. Points: 5, 3, 2, Direction vector: v Direction numbers: 17, (a) Parametric: x (b) Symmetric: x 9t 11. Points: 2, 3, 0 , 10, 8, 12 13. Point: 2, 3, 4 Direction vector: v Parametric: x 2, y k 3, z 4 t Direction numbers: 0, 0, 1 3 3 5 z 12 17. Li: v 3, 2, 4 6, 4, 6, 4, 8 6, 4, 6 8 6, 6, 6, 2, 5 on line 2, 5 on line 2, 5 not on line 5t, z 12t Direction vector: 8, 5, 12 Direction numbers: 8, 5, 12 (a) Parametric: x (b) Symmetric: x 8 2 2 8t, y y 15. Point: ( 2, 3, 1 Direction vector: v Parametric: x Symmetric: (a) On line (b) On line (c) Not on line y (d) Not on line 6 4 3 2 2 1 1 x 4 2 2 z 4i k 1 3, z 3 1 t 1 ,y 1 Direction numbers: 4, 0, L 2: v L 3: v L 4: v 4t, y not parallel to L1, L 2, nor L 3 Hence, L1 and L 2 are identical. L1 L 2 and...
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Ask a homework question - tutors are online | 1,188 | 2,352 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2017-04 | longest | en | 0.525922 |
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/4800/2/f/bc/3649/2/ | 1,670,020,326,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710916.70/warc/CC-MAIN-20221202215443-20221203005443-00130.warc.gz | 919,203,298 | 77,224 | # Properties
Label 4800.2.f.bc.3649.2 Level $4800$ Weight $2$ Character 4800.3649 Analytic conductor $38.328$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$4800 = 2^{6} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4800.f (of order $$2$$, degree $$1$$, not minimal)
## Newform invariants
Self dual: no Analytic conductor: $$38.3281929702$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$
## Embedding invariants
Embedding label 3649.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4800.3649 Dual form 4800.2.f.bc.3649.1
## $q$-expansion
$$f(q)$$ $$=$$ $$q+1.00000i q^{3} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -1.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} -6.00000i q^{17} -4.00000 q^{19} -1.00000i q^{27} -2.00000 q^{29} -8.00000 q^{31} +4.00000i q^{33} +2.00000i q^{37} -6.00000 q^{39} -6.00000 q^{41} +12.0000i q^{43} +8.00000i q^{47} +7.00000 q^{49} +6.00000 q^{51} +6.00000i q^{53} -4.00000i q^{57} +12.0000 q^{59} -14.0000 q^{61} -4.00000i q^{67} +8.00000 q^{71} +6.00000i q^{73} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{87} -10.0000 q^{89} -8.00000i q^{93} +2.00000i q^{97} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 8 q^{11} - 8 q^{19} - 4 q^{29} - 16 q^{31} - 12 q^{39} - 12 q^{41} + 14 q^{49} + 12 q^{51} + 24 q^{59} - 28 q^{61} + 16 q^{71} + 16 q^{79} + 2 q^{81} - 20 q^{89} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 8 * q^11 - 8 * q^19 - 4 * q^29 - 16 * q^31 - 12 * q^39 - 12 * q^41 + 14 * q^49 + 12 * q^51 + 24 * q^59 - 28 * q^61 + 16 * q^71 + 16 * q^79 + 2 * q^81 - 20 * q^89 - 8 * q^99
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$$.
$$n$$ $$577$$ $$901$$ $$1601$$ $$4351$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$
## Coefficient data
For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 4.00000i 0.696311i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 12.0000i 1.82998i 0.403473 + 0.914991i $$0.367803\pi$$
−0.403473 + 0.914991i $$0.632197\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 2.00000i − 0.214423i
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 0 0
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 6.00000i − 0.554700i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ − 6.00000i − 0.541002i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 0 0
$$129$$ −12.0000 −1.05654
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ 24.0000i 2.00698i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 7.00000i 0.577350i
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000i 0.901975i
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ − 14.0000i − 1.03491i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 24.0000i − 1.75505i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 10.0000i 0.712470i 0.934396 + 0.356235i $$0.115940\pi$$
−0.934396 + 0.356235i $$0.884060\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 0 0
$$213$$ 8.00000i 0.548151i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 36.0000 2.42162
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 28.0000i 1.85843i 0.369546 + 0.929213i $$0.379513\pi$$
−0.369546 + 0.929213i $$0.620487\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 2.00000i − 0.131024i −0.997852 0.0655122i $$-0.979132\pi$$
0.997852 0.0655122i $$-0.0208681\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 24.0000i − 1.52708i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 10.0000i 0.623783i 0.950118 + 0.311891i $$0.100963\pi$$
−0.950118 + 0.311891i $$0.899037\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ 0 0
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 0 0
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 0 0
$$293$$ 22.0000i 1.28525i 0.766179 + 0.642627i $$0.222155\pi$$
−0.766179 + 0.642627i $$0.777845\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 4.00000i − 0.232104i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 6.00000i − 0.344691i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 28.0000i 1.59804i 0.601302 + 0.799022i $$0.294649\pi$$
−0.601302 + 0.799022i $$0.705351\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 14.0000i − 0.786318i −0.919470 0.393159i $$-0.871382\pi$$
0.919470 0.393159i $$-0.128618\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 18.0000i − 0.995402i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ − 2.00000i − 0.109599i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 0 0
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ − 26.0000i − 1.38384i −0.721974 0.691920i $$-0.756765\pi$$
0.721974 0.691920i $$-0.243235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 2.00000i − 0.103556i −0.998659 0.0517780i $$-0.983511\pi$$
0.998659 0.0517780i $$-0.0164888\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 12.0000i − 0.609994i
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 4.00000i − 0.201773i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 38.0000i − 1.90717i −0.301131 0.953583i $$-0.597364\pi$$
0.301131 0.953583i $$-0.402636\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ − 48.0000i − 2.39105i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 8.00000i 0.396545i
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000i 0.195881i
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ − 8.00000i − 0.388973i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 0 0
$$433$$ 30.0000i 1.44171i 0.693087 + 0.720854i $$0.256250\pi$$
−0.693087 + 0.720854i $$0.743750\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 10.0000i − 0.472984i
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ 16.0000i 0.751746i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 0 0
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.0000i 1.29569i 0.761774 + 0.647843i $$0.224329\pi$$
−0.761774 + 0.647843i $$0.775671\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ 48.0000i 2.20704i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ 0 0
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 23.0000i − 1.02147i
$$508$$ 0 0
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 4.00000i 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 48.0000i 2.09091i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 12.0000i − 0.517838i
$$538$$ 0 0
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 10.0000i 0.429141i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 36.0000i − 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 0 0
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ −72.0000 −3.04528
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ 0 0
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 0 0
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 24.0000i 0.993978i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.00000i 0.165098i 0.996587 + 0.0825488i $$0.0263060\pi$$
−0.996587 + 0.0825488i $$0.973694\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ 0 0
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 0 0
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000i 0.0805170i 0.999189 + 0.0402585i $$0.0128181\pi$$
−0.999189 + 0.0402585i $$0.987182\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 16.0000i − 0.638978i
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 0 0
$$633$$ − 12.0000i − 0.476957i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 42.0000i 1.66410i
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 48.0000i − 1.88707i −0.331266 0.943537i $$-0.607476\pi$$
0.331266 0.943537i $$-0.392524\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 6.00000i − 0.234082i
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −6.00000 −0.233373 −0.116686 0.993169i $$-0.537227\pi$$
−0.116686 + 0.993169i $$0.537227\pi$$
$$662$$ 0 0
$$663$$ 36.0000i 1.39812i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ 14.0000i 0.539660i 0.962908 + 0.269830i $$0.0869676\pi$$
−0.962908 + 0.269830i $$0.913032\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −28.0000 −1.07296
$$682$$ 0 0
$$683$$ 28.0000i 1.07139i 0.844411 + 0.535695i $$0.179950\pi$$
−0.844411 + 0.535695i $$0.820050\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 10.0000i − 0.381524i
$$688$$ 0 0
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ 0 0
$$699$$ 2.00000 0.0756469
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ − 8.00000i − 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ − 14.0000i − 0.520666i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 32.0000i − 1.18681i −0.804902 0.593407i $$-0.797782\pi$$
0.804902 0.593407i $$-0.202218\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 72.0000 2.66302
$$732$$ 0 0
$$733$$ − 26.0000i − 0.960332i −0.877178 0.480166i $$-0.840576\pi$$
0.877178 0.480166i $$-0.159424\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 16.0000i − 0.589368i
$$738$$ 0 0
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 0 0
$$743$$ 16.0000i 0.586983i 0.955962 + 0.293492i $$0.0948173\pi$$
−0.955962 + 0.293492i $$0.905183\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ − 28.0000i − 1.02038i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 72.0000i 2.59977i
$$768$$ 0 0
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ −10.0000 −0.360141
$$772$$ 0 0
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 2.00000i 0.0714742i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 84.0000i − 2.98293i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 2.00000i 0.0708436i 0.999372 + 0.0354218i $$0.0112775\pi$$
−0.999372 + 0.0354218i $$0.988723\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 24.0000i 0.846942i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 18.0000i − 0.633630i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ − 24.0000i − 0.841717i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 48.0000i − 1.67931i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ − 48.0000i − 1.67317i −0.547833 0.836587i $$-0.684547\pi$$
0.547833 0.836587i $$-0.315453\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ − 42.0000i − 1.45521i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 0 0
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ − 22.0000i − 0.757720i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 2.00000i 0.0683187i 0.999416 + 0.0341593i $$0.0108754\pi$$
−0.999416 + 0.0341593i $$0.989125\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 8.00000i − 0.272323i −0.990687 0.136162i $$-0.956523\pi$$
0.990687 0.136162i $$-0.0434766\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 0 0
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 0 0
$$879$$ −22.0000 −0.742042
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 52.0000i 1.74994i 0.484178 + 0.874970i $$0.339119\pi$$
−0.484178 + 0.874970i $$0.660881\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 0 0
$$893$$ − 32.0000i − 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 16.0000 0.533630
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ − 48.0000i − 1.58857i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ 0 0
$$923$$ 48.0000i 1.57994i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ 0 0
$$933$$ 8.00000i 0.261908i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 42.0000i 1.37208i 0.727564 + 0.686040i $$0.240653\pi$$
−0.727564 + 0.686040i $$0.759347\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 60.0000i 1.94974i 0.222779 + 0.974869i $$0.428487\pi$$
−0.222779 + 0.974869i $$0.571513\pi$$
$$948$$ 0 0
$$949$$ −36.0000 −1.16861
$$950$$ 0 0
$$951$$ 14.0000 0.453981
$$952$$ 0 0
$$953$$ 14.0000i 0.453504i 0.973952 + 0.226752i $$0.0728108\pi$$
−0.973952 + 0.226752i $$0.927189\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 8.00000i − 0.258603i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ − 4.00000i − 0.128898i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 58.0000i 1.85558i 0.373097 + 0.927792i $$0.378296\pi$$
−0.373097 + 0.927792i $$0.621704\pi$$
$$978$$ 0 0
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ 18.0000 0.574696
$$982$$ 0 0
$$983$$ − 32.0000i − 1.02064i −0.859984 0.510321i $$-0.829527\pi$$
0.859984 0.510321i $$-0.170473\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ 28.0000i 0.888553i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18.0000i 0.570066i 0.958518 + 0.285033i $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ 0 0
$$999$$ 2.00000 0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
## Twists
By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4800.2.f.bc.3649.2 2
4.3 odd 2 4800.2.f.i.3649.1 2
5.2 odd 4 4800.2.a.cd.1.1 1
5.3 odd 4 960.2.a.c.1.1 1
5.4 even 2 inner 4800.2.f.bc.3649.1 2
8.3 odd 2 1200.2.f.g.49.2 2
8.5 even 2 600.2.f.b.49.1 2
15.8 even 4 2880.2.a.x.1.1 1
20.3 even 4 960.2.a.j.1.1 1
20.7 even 4 4800.2.a.r.1.1 1
20.19 odd 2 4800.2.f.i.3649.2 2
24.5 odd 2 1800.2.f.j.649.1 2
24.11 even 2 3600.2.f.c.2449.1 2
40.3 even 4 240.2.a.c.1.1 1
40.13 odd 4 120.2.a.b.1.1 1
40.19 odd 2 1200.2.f.g.49.1 2
40.27 even 4 1200.2.a.o.1.1 1
40.29 even 2 600.2.f.b.49.2 2
40.37 odd 4 600.2.a.c.1.1 1
60.23 odd 4 2880.2.a.bb.1.1 1
80.3 even 4 3840.2.k.j.1921.2 2
80.13 odd 4 3840.2.k.o.1921.1 2
80.43 even 4 3840.2.k.j.1921.1 2
80.53 odd 4 3840.2.k.o.1921.2 2
120.29 odd 2 1800.2.f.j.649.2 2
120.53 even 4 360.2.a.b.1.1 1
120.59 even 2 3600.2.f.c.2449.2 2
120.77 even 4 1800.2.a.n.1.1 1
120.83 odd 4 720.2.a.d.1.1 1
120.107 odd 4 3600.2.a.t.1.1 1
280.13 even 4 5880.2.a.a.1.1 1
360.13 odd 12 3240.2.q.g.2161.1 2
360.133 odd 12 3240.2.q.g.1081.1 2
360.173 even 12 3240.2.q.q.1081.1 2
360.293 even 12 3240.2.q.q.2161.1 2
By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.a.b.1.1 1 40.13 odd 4
240.2.a.c.1.1 1 40.3 even 4
360.2.a.b.1.1 1 120.53 even 4
600.2.a.c.1.1 1 40.37 odd 4
600.2.f.b.49.1 2 8.5 even 2
600.2.f.b.49.2 2 40.29 even 2
720.2.a.d.1.1 1 120.83 odd 4
960.2.a.c.1.1 1 5.3 odd 4
960.2.a.j.1.1 1 20.3 even 4
1200.2.a.o.1.1 1 40.27 even 4
1200.2.f.g.49.1 2 40.19 odd 2
1200.2.f.g.49.2 2 8.3 odd 2
1800.2.a.n.1.1 1 120.77 even 4
1800.2.f.j.649.1 2 24.5 odd 2
1800.2.f.j.649.2 2 120.29 odd 2
2880.2.a.x.1.1 1 15.8 even 4
2880.2.a.bb.1.1 1 60.23 odd 4
3240.2.q.g.1081.1 2 360.133 odd 12
3240.2.q.g.2161.1 2 360.13 odd 12
3240.2.q.q.1081.1 2 360.173 even 12
3240.2.q.q.2161.1 2 360.293 even 12
3600.2.a.t.1.1 1 120.107 odd 4
3600.2.f.c.2449.1 2 24.11 even 2
3600.2.f.c.2449.2 2 120.59 even 2
3840.2.k.j.1921.1 2 80.43 even 4
3840.2.k.j.1921.2 2 80.3 even 4
3840.2.k.o.1921.1 2 80.13 odd 4
3840.2.k.o.1921.2 2 80.53 odd 4
4800.2.a.r.1.1 1 20.7 even 4
4800.2.a.cd.1.1 1 5.2 odd 4
4800.2.f.i.3649.1 2 4.3 odd 2
4800.2.f.i.3649.2 2 20.19 odd 2
4800.2.f.bc.3649.1 2 5.4 even 2 inner
4800.2.f.bc.3649.2 2 1.1 even 1 trivial
5880.2.a.a.1.1 1 280.13 even 4 | 19,662 | 33,637 | {"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} | 2.5625 | 3 | CC-MAIN-2022-49 | latest | en | 0.286183 |
http://www.enotes.com/homework-help/im-greasy-little-maccas-worker-who-struggles-357698 | 1,477,121,150,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988718840.18/warc/CC-MAIN-20161020183838-00369-ip-10-171-6-4.ec2.internal.warc.gz | 432,047,451 | 10,242 | # What is 2*(7+4)/2
jeew-m | College Teacher | (Level 1) Educator Emeritus
Posted on
In maths there are several operations such as addition,subtraction,multiplication,division.
When we do maths operations there is a sequence we have to follow.
• First we need to solve the brackets if any.
• Then we have to do the multiplication and division
Now lets move on to your question.
2*(7+4)/2
Now first we have to solve the bracket.
(7+4) = 11
Now the question become;
2*11/2
Now we have to do the multiplication and division. you can do either one first.
I will do the multiplication first.
2*11 = 22
Now we have 22/2
22/2 = 11 | 177 | 639 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2016-44 | latest | en | 0.937734 |
http://www.ehow.com/how_8321036_calculate-total-pressure-container.html | 1,484,706,019,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560280133.2/warc/CC-MAIN-20170116095120-00154-ip-10-171-10-70.ec2.internal.warc.gz | 450,376,137 | 17,204 | # How to Calculate Total Pressure in a Container
Save
You can calculate the total pressure of a given amount of gas present in a container using the Ideal Gas Law. Because gases, unlike solids and liquids, have neither fixed shape nor fixed volume, you can take it that the volume of a gas equals the given or calculated volume of the container. Containers of gases should have no holes and should remain tightly corked to prevent the gas from diffusing through the holes. In standard SI units, pressure is measured in pascal (Pa). However, you can use other common units including the atmosphere (atm).
### Things You'll Need
• Calculator
• Define the parameters you will need. The Ideal Gas Law you will use states that the total pressure (P) in atmosphere equals the number of moles of the gas (n) in moles times molar gas constant (R) in liter atmospheres per mole per Kelvin times temperature (T) in Kelvin divided by volume (V) in liters. Therefore, you will need four parameters: "n", "R", "T", and "V".
• Calculate the number of moles. Some chemistry problems that involve calculating the total pressure of a gas in a container will require you to calculate the number of moles first. If you need to calculate the number of moles, then continue with this step, otherwise go to the next step. If you know the mass of the gas, divide it by the molar mass on a calculator and the result is "n". If you know the number of molecules, divide it by Avogadro's number on a calculator and the result is "n". If you know the volume of the gas at Standard Temperature and Pressure (STP), divide it by the molar volume at STP on a calculator and the result is "n".
• Calculate the volume of the gas. Some chemistry problems that involve calculating the total pressure of a gas in a container will require you to calculate the volume of the gas first. If you need to calculate the volume of gas, which is equivalent to the volume of the container, then continue with this step, otherwise go to the next step. If the container is a cylinder and you know the radius in meters and height in meters, calculate 3.1 times radius squared times height on a calculator and the result is "V" in meters cubed. Then multiply the result by 1000 on your calculator; you'll obtain the result in liters. If the container is a sphere and you know the radius in meters, calculate 1.3 times 3.1 times radius cubed on your calculator. The result is volume of the gas in meters cubed. To convert the result to liters, multiply the result by 1000 on your calculator. If the container is a rectangle and you know the length, width and height in meters, calculate length times width times height on your calculator. The result is volume of gas in meters cubed. To convert the result to liters, multiply the result by 1000 on your calculator.
• Perform the calculation. Calculate "n" times 0.082058 times "T" divided by "V" with the aid of a calculator to obtain "P" in atmospheres. The value for "P" is the total pressure of the gas in the container.
## Tips & Warnings
• Useful pressure converters include:
• 1 pascal (Pa) equals 1 N*m-2 equals 1 kg*m-1*s-2
• 1 atmosphere (atm) equals 1.01325*105 Pa
• 1 atmosphere (atm) equals 760 torr
• 1 bar equals 105 Pa
• The value for the molar gas constant (R) is 0.082058 L atm mol-1 K-1.
• Temperature equivalence in Kelvin (T) of t degree Celsius is T = 273 + t
• Number of moles of a gas is the ratio of given mass divided by molecular mass of the gas
• At standard temperature: 0°C = 273.15 K
• Standard volume of 1 mole of an ideal gas at STP equals 22.4 liters
• The Ideal Gas Law is used to solve problems regarding only gases of one substance, i.e., pure gases. When several substances are mixed in one container, use the Ideal Gas Law to determine the pressure of each substance, partial pressure, then use Dalton's law of partial pressures to determine the total pressure of the mixture in the container. Dalton's law of partial pressures states that the total pressure exerted on a container by several different gases equal the sum of the pressures exerted on the container by each gas.
## References
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### How to Build and Grow a Salad Garden On Your Balcony
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Submit Your Work! | 1,030 | 4,381 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2017-04 | latest | en | 0.87842 |
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1. Nov 10, 2013
Edira
1. The problem statement, all variables and given/known data
> Figure 1:
>
> A spool is hung on a spring. A part of the coil hangs in front of a
> homogeneous magnetic field. The coil is incorporated in an electrical
> circuit. See Figure 1.
>
> the coil KLMN has a rectangle shape, has a height of 10.0 cm, a width
> of 8.0 cm, and has 200 windings of which two are shown in Figure 1.
>
> The electrical circuit comprises a power supply, a sliding resistance
> and the coil, connected in the manner shown in Figure 1
>
>
> The distance between the bottom of the coil and the borders of the
> magnetic field is referred to as y. See Figure 1. If the power source
> is not yet connected, y is equal to 5.0 cm. The voltage source will
> now be connected, so that the spring will be stretched further and a
> Lorentz force will be exerted. The current in the coil can be changed
> by chaging the resistance. If there is a larger current, the spring
> will stretch more.
A.When calculating the LorentzForce : Fl=N*B*I*l Is the length (l) 8cm or 10cm? And why?
B. If the value of the current intensity is at a certain value, the coil length does not appear to increase any further with an increase in the current intensity.
Why is this the case when y=10 cm and how to apply the righthandrule in here?
2. Relevant equations
Fl=N*B*I*l
righthandrule
3. The attempt at a solution
A. I don't know why it should be 8cm...
I think 10cm, is because the height of the coil. But I don't see why it should be ten, because when there is no current y is already 5. And I don't see how you can use the righthandrule to show that the coil is going upward after y=10cm.
2. Nov 11, 2013
Basic_Physics
The current is up in the KL section of the coil, down in the MN section and from left to right in the NK section.
3. Nov 11, 2013
rude man
What is the direction of the force on the horizontal and vertical sections of the coil?
The coil does not go upward at y = 10 cm. It stays at 10 cm.
BTW it's not stated but from the description of the situation it's obvious that the B field points out of the page.
4. Nov 12, 2013
tiny-tim
Hi Edira! Welcome to PF!
it's Laplace force, not Lorentz force!
5. Nov 12, 2013
vanhees71
Why do you say it's Laplace Force? I've never heard about a Laplace force.
The Lorentz force, however, is indeed the well-established force acting on a charged particle in the electromagnetic field (written in Gaussian or Heaviside-Lorentz CGS units),
$$\vec{F}=q \left (\vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right ).$$
The continuum-mechanics version is the force density (force per volume fluid),
$$\vec{f}=\rho \left (\vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right ) = \rho \vec{E} + \frac{1}{c} \vec{j} \times \vec{B},$$
where $\rho$ is the charge density and $\vec{j}$ the current density of the charged fluid.
In your case you can get the total Lorentz force on the spool by
$$\vec{F}_{\text{mag}}=\frac{N I}{c} \int_{\text{spool}} \mathrm{d} \vec{r} \times \vec{B},$$
because for simplicity you can model the spool as an infinitely thin wire wound $N$ times over the rectangular frame.
I guess, what you are supposed to calculate is the situation where the spool is at rest (the fully dynamical problem is not so simple, particularly when you want to include the induced current and the backreaction to the motion of hte spool). For the static case you have to evaluate the force balance
$$\vec{F}_{\text{grav.}} +\vec{F}_{\text{spring}} + \vec{F}_{\text{mag}}=0$$
from which you can determine the value $y$ for a given current.
Note that there is a change as soon as the spool is fully within the region of the magnetic field!
6. Nov 12, 2013
rude man
??? It's the Lorentz force.
7. Nov 14, 2013
phyneach
Isn't Lorentz force 'Fnet= qE + q(V x B) = q{E + (V x B)}'?
8. Nov 14, 2013
rude man
Definitely!
9. Nov 15, 2013
tiny-tim
see eg http://en.wikipedia.org/wiki/Lorentz_force#Force_on_a_current-carrying_wire
the lorentz force is the name usually given to the force on a charged body …
it seems to me unhelpful to give the same name to the force on an uncharged body (a wire) in which the positive charges are stationary and the equal negative charges are moving
10. Nov 15, 2013
rude man
The force on a moving charge (aka "current") in a B field is the same whether the charge is moving in space or in a wire.
The force is not on the wire per se, it's on the moving charges. Strong binding forces convey the force on the charges to a force on the wire.
11. Nov 15, 2013
tiny-tim
but the lorentz force contains a E component
if the charge is not in a neutral wire, the lorentz force will not be NBIL
12. Nov 15, 2013
rude man
The Lorentz force is a general expression encompassing moving charge in a B field and static or moving charge in an E field.
Actually, there is also an E field and forces on static charges. The potential differences between different sections of the coil imply a corresponding E field between those sections and therefore free charges along the coil. So the entire Lorentz expression applies.
The E forces are in this case of negligible impact.
13. Nov 16, 2013
tiny-tim
but that's not an external E field
the lorentz force contains a E component from any external E field
if the charge is not in a neutral wire, the lorentz force will not be NBIL
14. Nov 16, 2013
rude man
A surface charge doesn't know that! All it sees is an E field and a force given by the Lorentz expression for an E field.
15. Nov 17, 2013
tiny-tim
not following you | 1,578 | 5,605 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.703125 | 4 | CC-MAIN-2018-13 | latest | en | 0.920651 |
https://codereview.stackexchange.com/questions/96395/simulating-gravitational-attraction-between-bodies/96396 | 1,718,325,688,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861517.98/warc/CC-MAIN-20240613221354-20240614011354-00516.warc.gz | 152,554,125 | 42,175 | Simulating gravitational attraction between bodies
I'm currently working on a simple simulator in Unity3D that involves planets orbiting each other. I wrote the following C# script to do that. It uses the classic gravitational formula,
$$F=G\frac{m_1m_2}{r^2}$$
using UnityEngine;
/// <summary>
/// Class used for calculating
/// how an object orbits another
/// object.
/// </summary>
/// <param name="parentObject">The parent gameObject to orbit.</param>
/// <param name="objectMass">The mass of the object.</param>
/// <param name="gravityConstant">The gravitational constant.</param>
[RequireComponent(typeof(Rigidbody))]
public class OrbitObject : MonoBehaviour
{
public OrbitObject parentObject;
public float objectMass;
public float gravityConstant;
private Vector3 parentPosition;
private float parentMass;
/// <summary>
/// Calculates the gravitational attraction.
/// </summary>
/// <returns>The force to apply</returns>
Vector3 CalculateGravitationalForce()
{
this.parentPosition = this.parentObject.transform.position;
Vector3 force = this.parentPosition - transform.position;
float distance = force.magnitude;
float strength = (this.gravityConstant * this.parentMass * this.objectMass) / (distance * distance);
force.Normalize();
force = force * strength;
return force;
}
/// <summary>
/// FixedUpdate is run once per frame.
/// FixedUpdate is used so that calculations
/// don't need to be multiplied by Time.deltaTime.
/// </summary>
void FixedUpdate()
{
}
/// <summary>
/// Start this instance.
/// </summary>
void Start()
{
this.parentPosition = this.parentObject.gameObject.transform.position;
this.parentMass = this.parentObject.objectMass;
}
}
For those who are curious, a gif of the script in action can be found here.
Just a note on your commenting:
Should you be breaking XML comments up across lines like that?
/// <summary>
/// Class used for calculating
/// how an object orbits another
/// object.
/// </summary>
I feel that should be three lines:
/// <summary>
/// Class used for calculating how an object orbits another object.
/// </summary>
Likewise, I feel that the following comment:
/// <summary>
/// FixedUpdate is run once per frame.
/// FixedUpdate is used so that calculations
/// don't need to be multiplied by Time.deltaTime.
/// </summary>
Could be rewritten as:
/// <summary>
/// FixedUpdate is run once per frame.
/// </summary>
/// <remarks>
/// FixedUpdate is used so that calculations don't need to be multiplied by Time.deltaTime.
/// </remarks>
Personally, I think breaking comments along multiple lines like that makes it harder to read. You're eyes have to jump to unnatural places.
Likewise, the <remarks> bit is nice because those comments aren't shown in Intellisense, but they can still be placed in the generated XML documentation files.
The only other bit I would mention, is the use of this. You should either use it consistently, or try to limit it. Sometimes (especially in larger cases) it helps to clear things up. But if you use it for some of the properties and not others, it can have a negative impact.
I'm referring to the following:
Vector3 CalculateGravitationalForce()
{
this.parentPosition = this.parentObject.transform.position;
Vector3 force = this.parentPosition - transform.position;
float distance = force.magnitude;
float strength = (this.gravityConstant * this.parentMass * this.objectMass) / (distance * distance);
force.Normalize();
force = force * strength;
return force;
}
Where does transform.position come from? The fact that you use this on all the other properties/fields of the class but not that one throws confusion into the mix. It's not a local variable, that much is true. But where exactly does it come from, is a little harder to notice right out of the box.
Again, these are all suggestions. Whether or not you like/use them is up to you, but this is how I would write it out.
Other answers have covered all of the points I'd make about the code already; I just wanted to get a bit physics-y.
Your implementation treats the "parentObject" as a fixed body and there is an implicit assumption about the relative masses of the two bodies.
That is to say, you are assuming that the parent object will have a mass very very much greater than the orbiting body and so will be relatively unaffected by the pull of the smaller body. That's fine for something like a satellite orbiting the earth but would be a bit odd if you have similar sized bodies orbiting each other in a different context.
For example, here's a two body system with bodies of similar masses in an elliptical orbit: https://en.wikipedia.org/wiki/Elliptic_orbit#/media/File:Orbit5.gif
Now, treating the real world as the real world is obviously much much more complicated. However, you could easily satisfy the physicist in me by renaming your object to CircularOrbit and ensure that the mass of the parent object is much greater.
This comes with the caveat that I don't actually know Unity so I might have misunderstood how the code works.
Your comments seem a little basic, at that, maybe write more than the basic reasoning of each function/execution, etc.
force = force * strength;
return force;
You can use the /=, *=, +=, -= operators for that:
return force *= strength;
I would also declare all your initial variables by type,
ParentObject
Float
Vector3 | 1,139 | 5,361 | {"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} | 3.140625 | 3 | CC-MAIN-2024-26 | latest | en | 0.683915 |
https://www.physicsforums.com/threads/what-is-the-correct-antiderivative-of-1-x-2.573496/ | 1,718,865,547,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861883.41/warc/CC-MAIN-20240620043158-20240620073158-00455.warc.gz | 812,958,773 | 17,470 | # What Is the Correct Antiderivative of 1/(x^2)?
• bobsmith76
In summary, an antiderivative is the reverse operation of a derivative and can be used to solve problems involving rates of change. The process for finding an antiderivative involves using integration rules and techniques, and not all functions have an antiderivative. The antiderivative is related to the definite integral through the Fundamental Theorem of Calculus.
bobsmith76
## Homework Statement
find the antiderivative of 1/(x^2)
## The Attempt at a Solution
I'm pretty sure you just find the antiderivatives of the numerator and the denominator.
the antiderivative of 1 is x.
the antiderivative of x^2 is (x^3)/3
mutliply the numerator by the inverse of the deominator and you get 3/(x^2)
The book says the answer is -1/x
bobsmith76 said:
## Homework Statement
find the antiderivative of 1/(x^2)
## The Attempt at a Solution
I'm pretty sure you just find the antiderivatives of the numerator and the denominator.
the antiderivative of 1 is x.
the antiderivative of x^2 is (x^3)/3
mutliply the numerator by the inverse of the deominator and you get 3/(x^2)
The book says the answer is -1/x
Have you ever seen the result ∫ x^n dx = x^(n+1)/(n+1) + C? It does not hold for n = -1 because that would involve division by zero on the right-hand-side, but otherwise n is unrestricted.
Please, please, please *get rid of the idea forever* that you can do the integral by integrating the numerator and denominator separately: that does not work! For example, integrate x, to get x^2/2. Now write x as x^2 / x and try your method of integrating the numerator and denominator separately---you will get the wrong answer.
RGV
Last edited:
... (posted an idea that was false, subsequently deleted it)
Then how do I find the antiderivative of a fraction?
Here's another problem
1. 2/3 sec^2 x/3
2. (2/3 tan x/3)/(x/3)
3. (2/3 tan x/3) * (3/x)
4. 2/3 * 3/x (tan x/3)
5. 2/x tan x/3
the book says the answer is
2 tan (x/3)
Ok, I saw from another website, that to find the antiderivative of a fraction you have to convert the fraction into a number with an exponent.
so -1/x = -x^-1
apply (x^n+1)/(n+1)
that comes to
-(x^0)/0
bobsmith76 said:
Then how do I find the antiderivative of a fraction?
Here's another problem
1. 2/3 sec^2 x/3
2. (2/3 tan x/3)/(x/3)
3. (2/3 tan x/3) * (3/x)
4. 2/3 * 3/x (tan x/3)
5. 2/x tan x/3
the book says the answer is
2 tan (x/3)
Assuming that you mean to find the anti-derivative of (2/3) sec2 (x/3) :
Rather than trying to follow those steps, which I can't make sense of ... Let's see if the book's answer is correct.
If 2 tan(x/3) is an anti-derivative of (2/3) sec2 (x/3), then the derivative of 2 tan(x/3), with respect to x, should be (2/3) sec2 (x/3) .
$\displaystyle \frac{d}{dx}\tan(x)=\sec^2(x)$
So that: $\displaystyle \frac{d}{dx}\left(2\,\tan\left(\frac{x}{3}\right) \right)=2\,\sec^2\left(\frac{x}{3}\right)\cdot \frac{1}{3}=\frac{2}{3}\, \sec^2\left(\frac{x}{3}\right)\,.$
Knowing the answer than finding the derivative can be done, but I still don't see how to find the answer starting from the antiderivative.
I got the answer to 1/x^-2 now.
bobsmith76 said:
Ok, I saw from another website, that to find the antiderivative of a fraction you have to convert the fraction into a number with an exponent.
so -1/x = -x^-1
apply (x^n+1)/(n+1)
that comes to
-(x^0)/0
The anti derivative of xn=xn+1/(n=1), except if n = -1.
Why that exception?
It's because x0=1, and the derivative of 1 is 0, not x-1.
The derivative of x-1 = ln(|x|)
bobsmith76 said:
Knowing the answer then finding the derivative can be done, but I still don't see how to find the answer starting from the antiderivative.
I got the answer to 1/x^-2 now.
The anti-derivative of 1/x-2 is easy, because 1/x-2 = x2.
Earlier, you wanted the anti-derivative of 1/x2, which you found to be -1/x . That's easy to get if you write 1/x2 as x-2 .
Details ... details.
Did you fail to read my first post, which specifically told you this result fails for n = -1?
RGV
## What is an antiderivative?
An antiderivative, also known as the indefinite integral, is the reverse operation of a derivative. It is a function that, when differentiated, produces a given function.
## Why is finding the antiderivative important?
Finding the antiderivative is important because it allows us to solve problems involving rates of change, such as calculating the area under a curve or finding the velocity of an object at a specific time.
## What is the process for finding the antiderivative?
The process for finding the antiderivative involves using integration rules and techniques, such as substitution and integration by parts, to manipulate the given function into a form that can be integrated. It may also involve using known antiderivatives for common functions.
## Can all functions have an antiderivative?
No, not all functions have an antiderivative. Functions that are continuous and have a finite derivative at every point in their domain have an antiderivative. However, some functions, such as the Dirichlet function, do not have an antiderivative.
## How is the antiderivative related to the definite integral?
The antiderivative and the definite integral are related through the Fundamental Theorem of Calculus, which states that the definite integral of a function can be calculated by finding its antiderivative and evaluating it at the limits of integration. In other words, the definite integral is the difference between the antiderivative evaluated at the upper and lower limits of integration.
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1K | 1,713 | 6,125 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2024-26 | latest | en | 0.926145 |
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Explore printable Irrational Numbers worksheets for 4th Grade
Irrational Numbers worksheets for Grade 4 are an essential tool for teachers looking to enhance their students' understanding of this important mathematical concept. By incorporating these worksheets into their lesson plans, educators can provide a solid foundation in number sense and help students develop a strong grasp of the different types of numbers, including rational and irrational numbers. These worksheets are specifically designed for Grade 4 students, ensuring that the content is age-appropriate and engaging. Teachers can use these worksheets to reinforce classroom instruction, as homework assignments, or even as assessment tools to gauge their students' progress in understanding irrational numbers.
Quizizz is an excellent platform that offers a wide range of educational resources, including Irrational Numbers worksheets for Grade 4, to support teachers in their quest to provide a comprehensive math education. In addition to worksheets, Quizizz also features interactive quizzes and games that can be used to supplement traditional teaching methods and keep students engaged in the learning process. Teachers can easily integrate Quizizz into their lesson plans, allowing them to track student progress and identify areas where additional support may be needed. By utilizing the various resources available on Quizizz, educators can create a dynamic and interactive learning environment that fosters a deep understanding of number sense and other essential math skills for their Grade 4 students. | 518 | 2,252 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2024-30 | latest | en | 0.953821 |
https://archive.softwareheritage.org/browse/content/sha1_git:56feeedcf1a1bccba01953516ebbfd2d23f2599b/?branch=refs/tags/R-2.9.0&path=man/exhaustive.search.Rd&snapshot_id=0f920b1e114986636ba2e45b5c1a83473fb6cf12 | 1,675,206,267,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00511.warc.gz | 119,321,846 | 10,378 | ##### swh:1:snp:0f920b1e114986636ba2e45b5c1a83473fb6cf12
Tip revision: 893e2d9
exhaustive.search.Rd
\name{exhaustive.search}
\alias{exhaustive.search}
\title{ Exhaustive search }
\description{
The algorithm for searching atrribute subset space.
}
\usage{
exhaustive.search(attributes, eval.fun)
}
\arguments{
\item{attributes}{ a character vector of all attributes to search in }
\item{eval.fun}{ a function taking as first parameter a character vector of all attributes and returning a numeric indicating how important a given subset is }
}
\details{
The algorithm searches the whole attribute subset space in breadth-first order.
}
\value{
A character vector of selected attributes.
}
\author{ Piotr Romanski }
\examples{
library(rpart)
data(iris)
evaluator <- function(subset) {
#k-fold cross validation
k <- 5
splits <- runif(nrow(iris))
results = sapply(1:k, function(i) {
test.idx <- (splits >= (i - 1) / k) & (splits < i / k)
train.idx <- !test.idx
test <- iris[test.idx, , drop=FALSE]
train <- iris[train.idx, , drop=FALSE]
tree <- rpart(as.simple.formula(subset, "Species"), train)
error.rate = sum(test\$Species != predict(tree, test, type="c")) / nrow(test)
return(1 - error.rate)
})
print(subset)
print(mean(results))
return(mean(results))
}
subset <- exhaustive.search(names(iris)[-5], evaluator)
f <- as.simple.formula(subset, "Species")
print(f)
} | 380 | 1,365 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2023-06 | latest | en | 0.441969 |
https://www.physicsforums.com/threads/tension-and-torque.197805/ | 1,653,424,971,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00606.warc.gz | 1,058,793,651 | 15,209 | # Tension! (and torque?)
1. 'A rope of negligible mass is stretched horizontally between two supports that are 3.44 meters apart. When an object of weight 3160 N is hung at the center of the rope, the is observed to sag by 35.0 cm. What is the tension in the rope?'
2. Does this have anything to do with torque?!?!
3. ATTEMPTS:
I divided 3.44 by 2 (1.72) to find the horizontal length from one support to the object. From that I found that the length of the rope from the object to one of the supports is 1.72^2 + .35^2 = X^2 (it's a triangle), x=1.75
I can assume because the object is in the center of the rope that the tension on both sides of the rope is equal. Can I also assume the forces sum to 0 because its stagnant? BAH I'm so confused, I don't know what to do!
The correct answer is 7920. Help!
it's not torque. draw a free-body diagram of the rope and label all the forces.
I drew a free-body diagram and I'm still confused. I really have no idea how to do this. How can I solve for tension!? I don't know the tension along the x axis or the y axis ? I know the angle - 11.5 degrees.
you DO know the downward force on the rope. you also know that the forces at either side should do what? is the rope moving? you have an angle and a side, you can get the other side, provided you draw a right triangle with the vectors.
ahhh i see, the rope is not moving so forces in the x and y direction are balanced.. i think i understand. so EFy = (<-- supposed to be sigma) = 0 = -W + 2Frope,y .. so Frope,y = 1580 on each side and the angle is 11 degrees .. so 1580*sin11=7925 .. about the right answer .. but that is the tension in the rope for only one side ? Oh dear, well thank you anyway !
remember that sin = opposite/hypotenuse
Last edited:
i think that sin = opposite / hypotenuse .. o/a = tan
oops, i'm tired :(
besides the original poster still used the sin funtion to calculate the hypotenuse by multiplying by the opposite if i follow correctly. | 520 | 1,972 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2022-21 | latest | en | 0.939553 |
https://studylib.net/doc/5871785/one-step-equation-foldable | 1,620,619,571,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243989030.87/warc/CC-MAIN-20210510033850-20210510063850-00357.warc.gz | 575,618,488 | 11,792 | # One-Step Equation Foldable
```Addition Equation
x+4 = 7
-4
-4
x+0 = 3
x = 3
Subtraction Equation
x–4 = 7
+4
+4
x + 0 = 11
x = 11
Multiplication Equation
Divide Line
3x = 24
3x = 24
3
3
x=8
Divide Line
Division Equation
Multiply
x = 7
7
(7)x = 2(7)
7
x = 14
Multiply | 130 | 268 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2021-21 | latest | en | 0.730086 |
https://www2.icp.uni-stuttgart.de/~icp/mediawiki/images/4/48/SS_2012_PC_blatt8_mc.py.txt | 1,596,735,051,000,000,000 | text/plain | crawl-data/CC-MAIN-2020-34/segments/1596439736972.79/warc/CC-MAIN-20200806151047-20200806181047-00253.warc.gz | 777,046,028 | 1,173 | from scipy import * from numpy.random import * import math import matplotlib.pyplot as pyplot import numpy as np from scipy.integrate import tplquad def expf3d(x1, x2, x3): return 1.0*(np.exp(x1**2+x2**2+x3**2)) def mc3d(f, N_list): res = [] for N in N_list: res.append(sum(f(uniform(0,1,N), uniform(0,1,N), uniform(0,1,N)))/N) return array(res) def avg(f, N): res = f() for i in range(1,N): res += f() return res/N ########################################## pyplot.figure() print mc3d(expf3d, [1000000]) N_range = logspace(0, 6, 10) #print N_range #print mc2d(f2d, N_range) #val, abserr = tplquad(ff, bounds_x[0], bounds_x[1], lambda y: bounds_y[0], lambda y: bounds_y[1], \ # lambda y, z: bounds_z[0], lambda y, z: bounds_z[1]) val3d, abserr = tplquad(lambda x,y,z: expf3d(x, y, z), 0, 1, lambda y: 0, lambda y: 1, lambda y, z: 0, lambda y, z: 1) #print mc3d(expf3d, N_range) pyplot.xscale("log") pyplot.yscale("log") pyplot.plot(N_range, avg(lambda: abs(mc3d(expf3d, N_range) - val3d), 20), "rx", markersize=3) #pyplot.plot(N_range, avg(lambda: abs(mc2d(f2d, N_range) - pi), 20), "bo", markersize=3) pyplot.show() #figure.savefig("pi.pdf") | 411 | 1,142 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2020-34 | latest | en | 0.215008 |
https://mathsgee.com/courses/master-linear-equations/ | 1,632,862,514,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780060882.17/warc/CC-MAIN-20210928184203-20210928214203-00252.warc.gz | 425,861,049 | 30,756 | # Master Linear Equations
Current Status
Not Enrolled
Price
\$5
Get Started
Course
Materials
Linear equations are written in the form $$y = mx + c$$ where $$y$$ is the dependent variable; $$m$$ is the gradient/slope; $$x$$ is the independent variable and $$c$$ is the $$y$$-intercept.
The biggest power of the independent variable, $$x$$ is 1, which implies that all linear equations are of order 1. These equations can be represented graphically with a straight line, hence the name linear equation. https://mathsgee.com/qna/31484/what-is-a-linear-equation | 141 | 560 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2021-39 | latest | en | 0.888836 |
https://www.popularmechanics.com/technology/a23928/solution-to-riddle-of-the-week-3/ | 1,670,006,799,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710916.40/warc/CC-MAIN-20221202183117-20221202213117-00132.warc.gz | 999,079,657 | 72,139 | You can get through this riddle with a little bit of trial and error, so if you haven't solved it yet, check out the problem again here, and then come back for validation if you like.
## Solution
The only way to be absolutely certain you will know which crates contain which fruits is if you ask your coworker to pull a fruit out of the crate that is labeled "A+O" to show to you. You know the crate is incorrectly labeled, so it doesn't have both apples and oranges in it. So if he pulls an apple out of the "A+O" crate, it has apples in it. If he pulls an orange out, it has oranges.
Let's say he pulls an apple out of the "A+O" crate, so you know it has apples in it. Now you know that the crate labeled "O" doesn't have just apples in it, because the "A+O" crate is the one with just apples in it, and you know the "O" crate doesn't have just oranges in it, because it is incorrectly labeled. So the "O" crate must have both apples and oranges in it, which leaves the "A" crate with just oranges.
Note: If your coworker pulls an orange out of the "A+O" crate at the beginning, the solution is inverted. | 278 | 1,110 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2022-49 | latest | en | 0.972745 |
https://www.answers.com/Q/What_is_more_proper_equivalent_to_or_equivalent_of | 1,553,473,764,000,000,000 | text/html | crawl-data/CC-MAIN-2019-13/segments/1552912203529.38/warc/CC-MAIN-20190324230359-20190325012359-00266.warc.gz | 668,461,199 | 49,796 | # What is more proper equivalent to or equivalent of?
Would you like to merge this question into it?
#### already exists as an alternate of this question.
Would you like to make it the primary and merge this question into it?
#### exists and is an alternate of .
It depends on the usage. If as a verb, "A is equivalent to B". If as a noun, "A is the equivalent of B".
7 people found this useful
# What is equivalence partitioning?
Answer A technique in black box testing is equivalence partitioning. Equivalence partitioning is designed to minimize the number of test cases by dividing tests in such a way that the system is expected to act the same way for all tests of each equivalence partition. Test inputs would be selected f ( Full Answer )
# What is an equivalent fraction?
An equivalent fraction is the same part of a whole. Example: would be two over three and EIGHT over twelve. it is equivalent(same as) to another fraction or whole number An equivalent fraction is one that is a multiple of the original. 1/2 is equivalent to 2/4. a equivalent fraction is when the va ( Full Answer )
# What is Equivalence class?
An equivalence relation ~ on A partitions into pairwise disjoint subsets called equivalence classes so that. 1. Within each class, every pair relates. 2. Between classes there is no relation. i.e. [x] = {a (element) A | a~x} and given two equivalence classes [a] and [b], either [a] = [b] or [a] i ( Full Answer )
# What are equivalent expressions?
Two or more expressions (equations are examples) that are equal to or equivalent to each other; for example: (X+2)(X+4) = X^2+6X+8.
# What is equivalent airspeed?
Equivalent airspeed is speed at sea level in which theincompressible dynamic pressure can be produces as if it were attrue airspeed. Equivalent airspeed is used to predict aircrafthandling.
# What is the equivalent of pi?
Pi can be written 2 ways, possibly more. The two ways I know of are these: 3.14 (rounded version) & 22 --- (the fraction form which is hard to write on a computer or this fraction form of 22/7) 7 The value of pi has never been determined because it is an irrational number but it's usually taken as 3 ( Full Answer )
# What is the equivalent for milliliter?
probably going to need some clarification for that question. Do you mean in the imperial system? If so then the answer is 0.0338140227 fluid ounces
# What is equivalent to a liter?
A liter is a metric unit of volume equal to approximately 1.056 liquid quarts, 0.908 dry quart, or 0.264 gallon.
# What is the equivalent of 0.05?
Put .05/1 move decimal places all the way right to whole numbers 5/100 factor out a five 1/20 ------------------the fractional equivalent of 0.05
# What is pi equivalent to?
3.14159 Pi is actually 3.141592654........... it goes on forever and that is all i can remember but u cant try using different websites as well. Some calculators say what pie is as well so just take a look around.
# How do you do equivalent decimals?
.60 if a number after the deciamal point is in the same decimal place as the same number you are trying to see if its equivalent to , then the zero behind it doesn't matter eg 3.600 is the same as 3.6 or 3.60 because the both of the 6's are in the tenths place. The zero behind the number d ( Full Answer )
# What is a equivalent triangle?
it is a triangle where you put the fractions on the bottom sides and the number your dividing it by on to. This cam also be known as a guess n check triangle
# What is the equivalence fallacy?
An Equivalence fallacy is the error of defining distinct and conflicting items in similar terms, thus equating tow items that are not, in fact, equal.. An author who suggests that one act of serious wrongdoing does not differ from a minor offence commits the fallacy of moral equivalence.. A differ ( Full Answer )
# What does equivalent means?
1. equal in value, measure, force, effect, significance, etc.: His silence is equivalent to an admission of guilt. 2. corresponding in position, function, etc.: In some ways their prime minister is equivalent to our president. 3. Geometry. having the same extent, as a triangle and a square of equal ( Full Answer )
# What is the equivalent to 6.0250?
Expressed as a mixed number in its simplest form, this is equal to 6 1/40 or six and one fortieth.
# What is an equivalent rate?
Equivalent Rates The Equivalent Rates calculation is used to find the nominal annual interest rate compounded n times a year equivalent to a given nominal rate compounded m times per year. Two nominal rates with different compounding frequencies are equivalent if they yield the same amount of i ( Full Answer )
# What are equivalent fracations?
Equivalent fractions are fractions whose results are equal to each other. For example, 3/4 (three quarters) and 6/8 (six eighths) are equivalent fractions, because the answer to both their sums is 0.75.\n \nAny two pairs of numbers expressed as a vulgar fraction whose sums are equal to each other c ( Full Answer )
# What is a foot equivalent to?
(Note that these numbers are rounded off) 1 foot - 305mm - 30cm - 0.305m - 0.000189 miles - 0.333334 Yards One sixth of a fathom.
# 0.4 and 0.40 is equivalent not equivalent?
The two values are equivalent in that they both represent around 4/10ths. However, 0.40 has one more significant figure, making it more precise than 0.4. Do not confuse "equivalent" with "equal", mathematically speaking. If something is equal, it is also equivalent. But if something is equivalent, ( Full Answer )
# What is 0.3 equivalent to?
It depends on what form you want. If you want a percentage, then 0.3 is equivalent to 30%. If you want a fraction, then 0.3 is equivalent to 3/10.
# How to get a equivalent?
First your score divide items second times it to 50 third add a 50 then that is the equivalent example 25 divide by 50 0.5 x 50 25 + 50 = 75 % .
# What is 0.04 equivalent to?
0.04 equivalent to = 1 /25; 4% 0.04* 100 /100 = 4 /100 or 1 /25 in fraction in lowest term 0.04 * 100% = 4%
# What is equivalent to 3tenths?
There are many answers to this: 1) 30/100 2) 300/1000 3) .30 or .3 4) 30%
# What is equivalent point?
In an acid-base titration for the reaction HCl + NaOH -> H 2 O + NaCl the equivalence point is when the hydrochloric acid and sodium hydroxide neutralise each other in equimolar proportions. When one mole of HCl is added to one mole of NaOH then equivalence point is reached. The 'end point' in a ti ( Full Answer )
# Which is equivalent to 4x3x2?
Which of what? You've given no options! But if your asking what is the equivalent of 4x3x2, the answer is 24. 4 multiplied by 3 is 12. 12 multiplied by 2 is 24. :)
# What is -log equivalent to?
The negative log of a number is the log of the number's reciprocal ('1' divided by the number).
# Is half a proper or equivalent fraction?
Sort of. It could be, but I think it is. But I'd go with...yes. Note: I'm just a kid, but it is.
# Why are American atheists more homophobic than their European equivalents?
American atheists are not homophobic as a class. In fact, the reader wonders at the source of your statistics. American atheists are not homophobic as a class. In fact, the reader wonders at the source of your statistics. American atheists are not homophobic as a class. In fact, the reader wonders ( Full Answer )
# What numbers are equivalent?
Pretty much any number has an equivalent, it is like saying hi and hello are both greetings, so you can have for example the number 1 has many equivalents 1/1=1, 3/3=1 100/100 =1 2^0=1...so on and so forth.... or 2 4/2=2 16/8=2 32/16 , 100/50=2 ...so on and so forth it is just another way to wri ( Full Answer )
# What is the equivalent of 1.8dl?
1.8dl = 0.18 liters 1.8dl = 180 ml 1.8dl = 6.087 fl. oz. There are 10 deciliters in a liter, and 100 milliliters in a deciliter. So there are 1.8 / 10 = .18 liters in 1.8 dl. Or there are 1.8 * 100ml = 180ml in 1.8dl. 1 deciliter is also equal to about 3.381 fluid ounces. That would mean that ( Full Answer )
# What are ratios that are equivalent?
any fractions that are equivalent are also equal ratios as a ratio and a fraction are pretty much the same thing.
# What is equivalent to fe490b?
If the expression in the question is a hexadecimal number, the equivalent decimal value is 16,664,843. If it was not a hexadecimal number, maybe you should have thought about stating what it was.
# What does equivalent ratios?
8 out of 10 boys play a sport , 25 out of 30 boys play a sport
# What is a equivalent factions?
equivalent fractions are two fractions that are equal to one another examples: 5/8=10/16, 1/9=10/90, 4/7=12/21, etc hope it helped!
# What is after 4.45 in equivalent?
There is no "after" in the context of fractions. They are infinitely dense: what that means that between any two fractions, there are infinitely many fractions. And between any two of them there are infinitely many, and so on.
# What is equivalent to 0.400?
.400 = .4 = 4/10 = 2/5 = 20/50 = 60/150 = 240/600 = an unlimited amount of things.
# What are equivalent representations?
They are different ways of writing the same number. For example, 0.04, 1/25, 0.2 2 , 5 -2 , 4% (and there are more) are all equivalent representations.
# How do you do equivalent fraction?
Example: . To get the equivalent fractions of 12/ 15: \n 1. Reduce the given fraction to its lowest term so as to get equivalent fractions of lesser value. \n 12 /15 = 4 /5 \n 2. Multiply 4 /5 by 2 /2, 3/ 3, 4 /4, 5 /5,... \n 4 /5 * 2 /2 = 8 /10 \n 4 /5 * 3 /3 = 12 /15 (this is the g ( Full Answer )
# What makes an equivalent fraction equivalent?
The thing that makes an equivalent fraction equivalent is theoverall value of the fraction, the ratio between the numerator andthe denominator.
# Which fraction is equivalent?
Every fraction has infinitely many fractions that are equivalent to it. So every fraction is equivalent. Every fraction has infinitely many fractions that are equivalent to it. So every fraction is equivalent. Every fraction has infinitely many fractions that are equivalent to it. So every frac ( Full Answer )
# How do you do an equivalent fraction?
Multiply the numerator (top) and the denominator (bottom) of the fraction by any non-zero integer. You will have an equivalent fraction. Multiply the numerator (top) and the denominator (bottom) of the fraction by any non-zero integer. You will have an equivalent fraction. Multiply the numerator ( ( Full Answer )
# What is the equivalent fractions of?
1/2 = 2/4 = 4/8 are three different looking fractions that all mean the same, and are equivalent fractions.
# How do you change proper fraction into equivalent fraction?
A proper fraction is when the numerator is smaller than thedenominator: 1 / 2, 3 / 4, and so on. An equivalent fraction is when different fractions have the samevalue: 1 / 2 = 2 / 4 = 5 / 10 all mean a half of something.
# What is equivalent to i63?
Nothing other than i63. Even 63i may not be the same unless it can be shown that multiplication is commutative.
# What is Btu equivalent to?
1 Btu is equal to ... 778.2 ft lb or 107.6 kg-m or 0.2520 Cal or 1055.056 J
# What equivalent set?
Two sets are equivalent if they have the same cardinality. For finite sets this means that they must have the same number of distinct elements. For infinite sets, equal cardinality means that there must be a one-to-one mapping from one set to the other. This can lead to some counter-intuitive resul ( Full Answer )
# How do you determine if more than two ratios are equivalent or not?
Cross-multiply them in pairs. The ratios are the same if and only if the answers are the same. Thus a/b = c/d iff a*d = b*c.
# What is an enharmonic equivalent?
A# = Bb B = Cb B# = C C# = Db D# = Eb E = Fb E# = F F# = Gb G# = Ab I would rather not go into double flats
# What is equivalent to7.5?
It is: 7.5 = 7 and a 1/2 as a mixed number or 15/2 as an improper fraction
# What are equivalent shapes?
The definition of equivalent is something that is ... having thesame valence; Geom. equal in area, volume, etc., but not of thesame shape. | 3,145 | 12,160 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2019-13 | latest | en | 0.937504 |
https://electronics.stackexchange.com/questions/557746/constant-gain-bandwidth-product-in-amplifiers | 1,722,714,452,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640377613.6/warc/CC-MAIN-20240803183820-20240803213820-00112.warc.gz | 184,540,399 | 41,538 | # Constant Gain-Bandwidth product in amplifiers
Can somebody explain to me how the gain-bandwidth product is constant in an amplifier or op-amp? I want to know the mathematical expression behind it if possible.
• What resources have you already looked at and what specific issue are you trying to understand within that context? (In a number of amplifiers, it is not a constant). Commented Apr 1, 2021 at 10:36
• Commented Apr 1, 2021 at 10:37
• You may have to elaborate the question (not in comments) with what you have understood/found out so far. Commented Apr 1, 2021 at 10:42
• Read the section AC imperfections: en.m.wikipedia.org/wiki/Operational_amplifier , if you are familiar with frequency response of "integrators", you should be able to figure out why. Commented Apr 1, 2021 at 10:51
• It is a direct consequence of the open-loop gain having the form of a single-pole or an integrator (doesn't matter which). I.e. A_OL=GBW/jω .. Here's my answer to a similar question with a quick walk-thru of the math Commented Apr 2, 2021 at 2:21
can somebody explain to me how the gain and bandwidth product is constant in an amplifier or OP AMP ?
Just look at the open loop gain of a typical op-amp: -
Mathematically this happens because an op-amp internally is equivalent to a DC gain stage with high gain followed by a single order low pass filter and, as we know with a single order low pass filter, the amplitude reduces above the 3 dB point at a rate proportional to frequency i.e. ten times the frequency means one-tenth the amplitude.
It's all embodied in a simple RC low-pass filter when we move beyond the cut-off frequency.
An RC low pass filter has a transfer function of: -
$$\dfrac{1}{1+j\omega RC}$$
And, when you get beyond the cut-off frequency the transfer function becomes asymptotic with $$\\dfrac{1}{j\omega RC}\$$ hence, gain is inversely proportional to frequency.
That slope has a constant gain-bandwidth product.
• that is fine, but it is based on a graphical method which may not rigorous all the time, is there any mathematical derivation of the statement? also is it fair to expect that all the amplifier posing band width product constant would have the same kind of (geometrically similar at least) frequency response curve as the one you have provided? Commented Apr 1, 2021 at 10:43
• The roll-off of amplitude against frequency is totally based on the maths of a single order low pass filter (as I said in my answer). It's not based on a graph method but I used a graph to make things easier to understand i.e. a picture speaks a thousand words. Most op-amps will adhere to the 20 dB/decade slope characteristic graph (and math) as shown. There will be some exceptions i.e. those there are not general purpose and those that are not unity gain stable but, in the main, all general purpose op-amps will comply @Sayan. Commented Apr 1, 2021 at 10:48 | 702 | 2,892 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.765625 | 4 | CC-MAIN-2024-33 | latest | en | 0.930255 |
https://www.convertit.com/Go/SmartPages/Measurement/Converter.ASP?From=yard&To=wavelength | 1,624,435,896,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488536512.90/warc/CC-MAIN-20210623073050-20210623103050-00244.warc.gz | 605,882,365 | 3,729 | New Online Book! Handbook of Mathematical Functions (AMS55)
Conversion & Calculation Home >> Measurement Conversion
Measurement Converter
Convert From: (required) Click here to Convert To: (optional) Examples: 5 kilometers, 12 feet/sec^2, 1/5 gallon, 9.5 Joules, or 0 dF. Help, Frequently Asked Questions, Use Currencies in Conversions, Measurements & Currencies Recognized Examples: miles, meters/s^2, liters, kilowatt*hours, or dC.
Conversion Result: ```yard = 0.9144 length (length) ``` Related Measurements: Try converting from "yard" to agate (typography agate), archin (Russian archin), astronomical unit, bolt (of cloth), cable length, chain (surveyors chain), digitus (Roman digitus), finger, foot, football field, Greek span, hand, marathon, parsec, Roman foot, sazhen (Russian sazhen), spindle, stadium (Roman stadium), UK mile (British mile), vara (Mexican vara), or any combination of units which equate to "length" and represent depth, fl head, height, length, wavelength, or width. Sample Conversions: yard = 9,144,000,000 angstrom, 1.29 archin (Russian archin), .00002065 arpentcan, .01563844 arpentlin, 1,440 bottom measure, .04545455 chain (surveyors chain), 49.32 digitus (Roman digitus), 2.38E-09 earth to moon (mean distance earth to moon), .5 fathom, 3 foot, 1.98 Greek cubit, 11.86 Greek palm, .00018939 league, 9.67E-17 light yr (light year), .00016458 nautical league, 12 palm, 216 pica (typography pica), 2,601.72 point (typography point), .00023286 ri (Japanese ri), .00085714 verst (Russian verst).
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Please read our Help Page and FAQ Page then post a message or send e-mail. Thanks! | 464 | 1,676 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2021-25 | latest | en | 0.687715 |
https://www.gamedev.net/forums/topic/550397-computing-the-bspline-kernel/ | 1,544,398,554,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376823228.36/warc/CC-MAIN-20181209232026-20181210013526-00176.warc.gz | 907,505,383 | 24,732 | # Computing the BSpline Kernel?
This topic is 3343 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
On wiki in the section: Uniform cubic B-splines http://en.wikipedia.org/wiki/B-spline the matrix form of a cubic BSpline is given. I have found the following implementation for this kernel: BSplineKernel( double u) { double absValue = vnl_math_abs( u ); double sqrValue = vnl_math_sqr( u ); if ( absValue < 1.0 ) { return ( 4.0 - 6.0 * sqrValue + 3.0 * sqrValue * absValue ) / 6.0; } else if ( absValue < 2.0 ) { return ( 8.0 - 12 * absValue + 6.0 * sqrValue - sqrValue * absValue ) / 6.0; } else { return 0.0; } } But the only resemblence I see is the division with 6. Out of curiosity does the above implementation relate to the formula on the wiki page or is it another way of expressing a cubic BSpline kernel??
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https://www.cyberdime.io/using-moroccan-tea-tray-mathematics-to-turn-robots-into-skilled-waiters/ | 1,686,089,569,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224653183.5/warc/CC-MAIN-20230606214755-20230607004755-00229.warc.gz | 799,100,883 | 34,169 | Select Page
# Using Moroccan tea tray mathematics to turn robots into skilled waiters
Cyberdime
Published: January 16, 2023
Researchers at the Munich Institute of Robotics and Machine intelligence (MIRMI) at the Technical University of Munich (TUM) have developed a model that enables a robot to serve tea and coffee faster and more safely than humans—with no sloshing. The mathematics behind the pendulum used in the concept is more than 300 years old.
Can a robot be a better waiter than a human being? To answer this question, Dr. Luis Figueredo, senior scientist in the team of Prof. Sami Haddadin, set up a robotic arm from the specialized robot maker Franka Emika and hooked up a computer. The robot’s hand grips a glass filled to the brim with water, raises it and rocks it back and forth without spilling a drop. “And it does it faster and more safely than a person,” says the scientist from the Munich Institute of Robotics and Machine Intelligence (MIRMI) of TUM.
## Copying the motion of the Moroccan tea tray
How does it work? The team simply fed the robot with algebraic formulas dating back several centuries. They based their mathematics on a Moroccan tea tray that applies the principle of a spherical pendulum.
With doctoral candidate Riddhiman Laha and masters student Rafael I. Cabral Muchacho, Figueredo embedded the dynamics of a spherical pendulum into the robot’s control software. That means that the robot’s movements are limited by the basic principles of geometry. With his team, he also integrated the correct angles, speeds and accelerations into the model. “When you understand how a pendulum moves and know how it works, it is suddenly quite simple,” says Figueredo.
## The mathematics of a pendulum: A simple solution to a complex problem
The science of slosh-free movement is a complex field. “Most approaches have focused mainly on limiting acceleration to keep the sloshing of liquids under control. Or they have worked with fluid dynamics to calculate how these substances behave in order to predict the trajectories,” explains Figueredo: “That takes at least a few minutes, if not hours, and the result is still uncertain.”
## Applications in health care and the transport of hazardous liquids
As a practical application, the scientists initially envision innovative robotic support for elderly people and those requiring nursing care. “But industries involved in the transport of materials posing biological and chemical hazards would probably be interested in a solution like this, too,” says Figueredo.
Safety remains a critical point: a robot should ideally be capable of recognizing dangerous situations. “For that we need better perception,” says Figueredo.
Sensors would then enable the machine to not only recognize people, but also predict their movements. That is the only way to completely rule out collisions with the robot. So far, the robot is working with tactile sensors as a safety mechanism. In the current slosh-free mode, the robot arm instantly retracts when it senses a collision, but also keeps the liquid safe. | 633 | 3,084 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2023-23 | longest | en | 0.92688 |
vu3rdd.me | 1,675,183,709,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499888.62/warc/CC-MAIN-20230131154832-20230131184832-00747.warc.gz | 644,327,719 | 3,534 | Motivation
Homebrewing in the 21st century is going to involve a lot of software. Most free software SDR programs out there today rely on the wdsp library written by Dr. Warren Pratt, NR0V. Dr. Warren has also written a fine manual for WDSP targeted at programmers who like to use wdsp in their SDR programs. This is mostly API documentation along with some details of the algorithms. However much of the math is missing in the docs and buried in the code.
My motivation is to understand how the "NR2" noise reduction algorithm works. I have mostly achieved it. Rather than describe it, I wanted to give references to the papers here so that curious minds can find them.
The big idea
WDSP has an "NR" algorithm. I haven't explored this yet, but this is the popular Weiner filtering algorithm that can be implemented using gradient descend (Least Mean Square algorithm is how it is implemented in practice).
The main idea in all noise reduction algorithm is this:
The "observable" is `y(n)` which has the speech signal with noise. Our motivation is to extract an approximation of the signal buried in noise. i.e. we need to estimate noise and subtract it. This is done either in time domain or frequency domain.
``````y(n) = s(n) + w(n)
``````
I believe NR using LMS is a time domain algorithm.
The other property is that noise is uncorrelated with speech signal, so the power spectrum of these two uncorrelated systems are additive as well. i.e.
``````|Y(f)² = |S(f)|² + |W(f)|²
``````
If we can "estimate" the noise spectral density, then we can get an estimate of signal power spectral density as follows:
``````|Ŝ(f)|² = |Y(f)|² - |Ŵ(f)|²
``````
Now, if we find the time domain values corresponding to `Ŝ(f)`, i.e. `ŝ(n)`, then we get back the speech signal with the noise removed. Well, I am grossly simplifying what "noise" is here. The estimate of speech that is reconstructed thus, is not without problems. There are things like musical noise artifacts that one can hear in such reconstructed speech. The quality also very much depends on the signal to noise ratio.
NR2 algorithm
NR2 uses a frequency domain method which has its origins in this famous paper by Ephraim and Malah called "Speech Enhancement using a minimum mean-square error short-time spectral amplitude estimator". Here we statistically model the speech signals and noise signals (or rather the probabilities of apriori SNR (clean signal to noise ratio) and aposteriori SNR ("observable signal" to noise ratio). A crucial step is to find an estimate of noise. This step is rather left out in the paper.
One popular way to track noise (in a time-frequency analysis) is with a method called "Optimal Smoothing Minimum Statistic". This method is detailed in the paper "Noise Power Spectral Density Estimation based on Optimal Smoothing and Minimum Statistics" by Rainer Martin.
WDSP Implementations
WDSP implements both the above steps for Noise Power Estimation (NPE in wdsp parlance). I could find clear mapping of equations in the OSMS method vs the equations in the paper) where as for the MMSE NPE, I could not fully correlate (pun intended) the equations in the code.
Any help welcome. Specifically, what I am looking for is a reference to the paper that details these steps in the mmse noise estimator. (Edit: I found some answers in section 4 of the paper "NOISE POWER ESTIMATION BASED ON THE PROBABILITY OF SPEECH PRESENCE" by Gerkmann and Hendriks. In the temporal smoothing step, it uses a smoothing factor β = 0.8. However, in the code, the variable `alpha_pow` is calculated differently. That is the next piece to dig into. It seem like wdsp author is trying to scale the fixed 0.8 value to work with a different overlap and sampling rate value.. The matlab code from the authors of the paper shows a fixed 0.8 value.)
Next steps
I want to understand how to introduce a new block into wdsp and start playing with variants of these algorithms. | 914 | 3,947 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2023-06 | longest | en | 0.953008 |
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E7_Lab05_Solutions_Fall_2010
# E7_Lab05_Solutions_Fall_2010 - Contents problem 1 Problem 2...
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Contents problem 1 Problem 2 Problem 3 problem 4 Problem 5 problem 6 problem 7 problem 8 problem 9 %Lab 05 Solutions problem 1 %1 a clear x = 10; y = -58; a = x<y %a is false (0) because x is not less than y b = x>=y %b is true (1) because x is greater than y c = x==y %c is false (0) because x is not equal to y d = (x - 68) >= y %d is true (1) because (x - 68) is equal to y e = x - 68 >= y %e is also true becuase math operations are carried out %before relational operations f = x - (68 >= y) %f is 9 because 10 minus 1 (true) equals 9 x=y; h = x~=y %h is false because x is equal to y i.e. x isn't not equal to y ii = ~x~=y %%ii is true because it is the opposite of h %j is true because 1 and 1 are both true k = 1|1 %k is true because 1 or 1 is true %l is false because 1 and 0 are not both true m = 1|0 %m is true because 1 OR 0 is true %n is true becasue doubles are considered to be true o = isnumeric(m) %o is false because m is logical, not numeric p = isnumeric( 'g' ) %p is false because 'g' is a character, not numeric q=isnumeric(6) %q is true because 6 is numeric r=isscalar([2 7]) %r is false because [2 7] is a vector, not scalar s=islogical(m) %s is true because m is a logical t=5 - m %t is 4 because a logical is converted to a value (1) when operated on u=class(t) %u is 'double' because t is a double (i.e. operating with logicals %returns a numeric result %1 b clear x = 10; y = -58; %58 - 10 is greater than 0 (true), y is less than x % (~true = false) true and false gives false. b = (x+y)/2+24==1 | x~=5+9
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Ask a homework question - tutors are online | 685 | 2,173 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2018-09 | latest | en | 0.843413 |
https://www.studypool.com/discuss/1122366/i-need-assistance-analyzing-a-two-way-anova | 1,477,403,065,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988720153.61/warc/CC-MAIN-20161020183840-00393-ip-10-171-6-4.ec2.internal.warc.gz | 1,001,172,830 | 751,796 | # I need assistance analyzing a two-way ANOVA
Aug 18th, 2015
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Question description
Submit your answers to the following questions using the ANOVA source table below. The table depicts a two-way ANOVA in which gender has two groups (male and female), marital status has three groups (married, single never married, divorced), and the means refer to happiness scores (n = 100):
1. What are the independent variables and their levels? What is the dependent variable?
2. State all null hypotheses associated with independent variables and their interaction? Also suggest alternate hypotheses?
3. What are the degrees of freedom for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance?
4. Calculate the mean square for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance.
5. Calculate the F ratio for 1) gender, 2) marital status, and 3) interaction between gender and marital status.
6. Identify the critical Fs at alpha = .05 for 1) gender, 2) marital status, and 3) interaction between gender and marital status.
7. If alpha is set at .05, what conclusions can you make?
Source Sum of Squares (degrees of freedom [df]) Mean Square Fobt. Fcrit. Gender 68.15 ? ? ? ? Marital Status 127.37 ? ? ? ? Gender * Marital Status (A x B) 41.90 ? ? ? ? Error (Within) 864.82 ? ? NA NA Total 1102.24 99 NA NA NA
Please Note: The table that you see in the assignment has been slightly modified from the one presented in the module notes since it is beyond the scope of this unit to have students calculate p values. Instead you are asked to calculate the F value and compare it to the critical F value to determine whether the test is significant or not.
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http://encyclopedia.kids.net.au/page/fi/Field_(algebra) | 1,716,398,520,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058560.36/warc/CC-MAIN-20240522163251-20240522193251-00758.warc.gz | 9,886,555 | 8,289 | ## Encyclopedia > Field (algebra)
Article Content
# Field
Redirected from Field (algebra)
In physics, a field is an assignment of a quantity to every point in space. We distinguish between scalar fields (such as the temperature at any given point) and vector fields (such as the electric or magnetic force at any given point).
A field, in abstract algebra, is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers.
Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or complex numbers. Fields used to be called rational domains.
The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other.
Definition: A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse.
Spelled out, this means that the following hold:
Closure of F under + and *
For all a,b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F);
Both + and * are associative
For all a,b,c in F, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c.
Both + and * are commutative
For all a,b belonging to F, a + b = b + a and a * b = b * a.
The operation * is distributive over the operation +
For all a,b,c, belonging to F, a * (b + c) = (a * b) + (a * c) and (b + c) * a = (b * a) + (c * a).
There exists an element 0 in F, such that for all a belonging to F, a + 0 = a and 0 + a= a .
Existence of a multiplicative identity
There exists an element 1 in F different from 0, such that for all a belonging to F, a * 1 = a and 1 * a = a.
For every a belonging to F, there exists an element -a in F, such that a + (-a) = 0 and (-a) + a = 0.
Existence of multiplicative inverses
For every a ≠ 0 belonging to F, there exists an element a-1 in F, such that a * a-1 = 1 and a-1 * a = 1.
The requirement 0 ≠ 1 ensures that the set which only contains a single zero is not a field. Directly from the axioms, one may show that (F, +) and (F - {0}, *) are commutative groups and that therefore (see elementary group theory) the additive inverse -a and the multiplicative inverse a-1 are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses:
(a*b)-1 = a-1 * b-1
provided both a and b are non-zero. Other useful rules include
-a = (-1) * a
and more generally
-(a * b) = (-a) * b = a * (-b)
as well as
a * 0 = 0,
all rules familiar from elementary arithmetic.
### Examples of Fields
• The smallest field has only two elements: 0 and 1. It is sometimes denoted by F2 or Z2 and can be defined by the two tables
``` + 0 1 * 0 1
0 0 1 0 0 0
1 1 0 1 0 1
```
It has important uses in computer science, especially in cryptography and coding theory[?].
• More generally: if q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements. No other finite fields exist. For instance, for a prime number p, the set of integers modulo p is a finite field with p elements: this is often written as Zp = {0,1,...,p-1} where the operations are defined by performing the operation in Z, dividing by p and taking the remainder, see modular arithmetic.
• The rational numbers can be extended to the fields of p-adic numbers for every prime number p.
• Let E and F be two fields with E a subfield of F (i.e., a subset of F containing 0 and 1, closed under the operations + and * of F and with its own operations defined by restriction). Let x be an element of F not in E. Then E(x) is defined to be the smallest subfield of F containing E and x. For instance, Q(i) is the subfield of the complex numbers C consisting of all numbers of the form a+bi where both a and b are rational numbers.
• For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
• If F is a field, and p(X) is an irreducible polynomial in the polynomial ring F[X], then the quotient F[X]/(p(X)) is a field with a subfield isomorphic to F. For instance, R[X]/(X2+1) is a field (in fact, it is isomorphic to the field of complex numbers).
• If V is an algebraic variety over F, then the rational functions VF form a field, the function field of V.
• The hyperreal numbers form a field containing the reals, plus infinitesimal and infinite numbers.
• The surreal numbers form a field containing the reals, except for the fact that they are a proper class, not a set. The set of all surreal numbers with birthday smaller than some inaccessible cardinal number form a field.
• The nimbers form a field, again except for the fact that they are a proper class. The set of nimbers with birthday smaller than 2^(2^n), the nimbers with birthday smaller than any infinite cardinal are all examples of fields.
### Further properties, definitions and facts
A field homomorphism between two fields E and F is a function f : E -> F such that f(x + y) = f(x) + f(y) and f(xy) = f(x) f(y) for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x-1) = f(x)-1 for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic if there exists a bijective homomorphism f : E -> F. The two fields are then identical for all practical purposes.
A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field. Such a subfield automatically has the same additive and multiplicative identities as F, and the additive and multiplicative inverses of an element of the subfield are the same as those of the same element in F. In order to check that a subset E of F is a subfield of F, one only has to check three properties:
1. E contains a non-zero element
2. For every x and y in E, x - y is in E
3. For every x and y in E with y ≠ 0, x / y is in E.
For example, Q is a subfield of R, which in turn is a subfield of C.
The set of non-zero elements of a field F is typically denoted by F×; it is an abelian group under multiplication. Every finite subgroup of F× is cyclic.
For every field F, there exists a (up to isomorphism) unique field G which contains F, is algebraic over F, and is algebraically closed. G is called the algebraic closure or F.
The characteristic of the field F is the smallest positive integer n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Zp has characteristic p.
If the characteristic of the field F is equal to the prime p, then p·x = 0 for every x in F, and (x + y) p = x p + y p for all x, y in F, a consequence of the binomial theorem. The map f(x) = x p is a field homomorphism F ->F, the "Frobenius homomorphism".
Every field has a unique smallest subfield, which is called the prime subfield and is contained in every other subfield. For fields of characteristic 0, the prime subfield is isomorphic to Q (the rationals). Fields of characteristic 0 are therefore always infinite. For fields of prime characteristic p, the prime subfield is isomorphic to Zp. Fields of prime characteristic can be either infinite or finite (see Finite field).
All the fields of importance in analysis (real numbers, complex numbers, p-adic numbers, nonstandard reals) carry a valuation[?] or an order, which turns them into topological spaces; addition, subtraction, multiplication and division are then continuous operations. All these fields have characteristic zero.
All Wikipedia text is available under the terms of the GNU Free Documentation License
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Lecture 8.docx - Angular Kinematics Measures Review When...
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Angular Kinematics Measures Review When the angle or angular velocity of interest is of a JOINT (between TWO different segments), then you are measuring a RELATIVE angle/angular velocity When the angle or angular velocity of interest is of a SEGMENT (a segment with respect to the space around it), then you are measuring an ABSOLUTE angle/angular velocity Tools for Measuring Angles Relative Angles o Goniometer used for direct measurement of relative joint angles protractor with arms centre placed over joint centre; arms aligned with long bones “elgon” is a goniometer with an electrical potentiometer at vertex Absolute Angles o Inclinometer is a gravitationally based instrument that identifies the absolute angle of orientation of a body segment challenge to accurately position the inclinometer Problem 8 (page 381 ) Measure and record the following angles for the drawing below: o a) The relative angle at the shoulder it is a joint angle shoulder is trunk to upper arm put protractor and it comes to 50 degrees o b) The relative angle at the elbow between upper arm and forearm 90 degrees o c) The absolute angle of the upper arm (use right horizontal as reference) not talking about a joint in the perspective of measuring a joint angle measuring a joint angle is taking 2 body segments and measuring angle between them need to have a consistent reference so when you compare 2 points in time, you know it’s a consistent reference absolute angle means referenced to something in the world o d) The absolute angle of the forearm (use right horizontal as reference) put a horizontal line and do it from the wrist to the right get 150 degrees Instantaneous Centre of Rotation when we have a joint angle, we talked about how angular motion occurs when you have that axis of rotation we are rotating around
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o in human motion, it is important to know whatever that imaginary line we are rotating around changes as we go through the motion In a joint, bones move (slide and spin) relative to one another o e.g., knee flexion: medial rotation & anterior slide of femur on tibia
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https://www.physicsforums.com/threads/this-is-strange-too.73343/ | 1,624,099,507,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487647232.60/warc/CC-MAIN-20210619081502-20210619111502-00392.warc.gz | 867,119,406 | 17,543 | # This is strange too
hey i need help with this as a high school grad ,and one people think happens 2 be a maths prodigy a junior student came once and asked me 2 help him integrate x! i could not give a solution ,now pls can anyone help with this puzzle?
matt grime
Homework Helper
What makes you think x! is even a function of a real variable as opposed to one of integers? There is a function Gamma, that agrees with the factorial at the integers, why don't you look it up? The Gamma Function, see, eg, mathworld.
Integrating the Gamma function isn’t going to be fun.
matt grime
Homework Helper
And integrating anything is ever fun?
The integral of (fu)dn is, such that f and u are constants
matt grime
Homework Helper
Hmm, not my idea of fun, but then all of analysis that isn't trying to be algebra is dull.
Zurtex
Homework Helper
matt grime said:
Hmm, not my idea of fun, but then all of analysis that isn't trying to be algebra is dull.
I love analysis, I used to love tricky integrals when I was first learning them as well
Give me any difficult maths challenge and I'll find some fun in it :!!)
matt grime
Homework Helper
Evidently you've never sat through a seminar on improving an upper bound from k^2 to k^{1.9999999999} on an eigenvalue of some operator defined on some probably oddly shaped domain, delivered in Russo-English for the benefit of the three other Russian analysts in the room with the poor sodding post grads (who were all algebraists/geometers, if they're different) like me who, at the time, didn't think they should miss these kinds of things.
Last edited:
shmoe
Homework Helper
Even better when you consider the k^2 bound probably takes 15 lines or less to prove while the k^{1.9999999999} takes 15 pages. For 30 pages you can improve this to k^{1.9999999995}.
If something interesting happens at k^{3/2} the next 50 years will see hundreds of impenetrably dense technical pages that improve the bound to k^{1.893453} before someone with a new bright idea comes by and hammers out k^{3/2} on the back of a napkin.
BobG
Homework Helper
JonF said:
The integral of (fu)dn is, such that f and u are constants
Not as fun as $$\frac{dx}{dn}=fu e^{-x}$$ such that f and u are constants. Although it seems more fun while you're doing it than when you're done.
HallsofIvy
Homework Helper
matt grime said:
Evidently you've never sat through a seminar on improving an upper bound from k^2 to k^{1.9999999999} on an eigenvalue of some operator defined on some probably oddly shaped domain, delivered in Russo-English for the benefit of the three other Russian analysts in the room with the poor sodding post grads (who were all algebraists/geometers, if they're different) like me who, at the time, didn't think they should miss these kinds of things.
Zurtex
Homework Helper
matt grime said:
Evidently you've never sat through a seminar on improving an upper bound from k^2 to k^{1.9999999999} on an eigenvalue of some operator defined on some probably oddly shaped domain, delivered in Russo-English for the benefit of the three other Russian analysts in the room with the poor sodding post grads (who were all algebraists/geometers, if they're different) like me who, at the time, didn't think they should miss these kinds of things.
:rofl:
Well, I assure you, I may not have had go through something quite so obscure but I've had my pains. I do remember sitting through 7 lectures on Euclid Algorithm, each one explaining no more than the last, in less than 2 weeks because the lecturers never bothered checking with each other what they were covering. I remember sitting through 5 lectures on the Taylor series without ever having ever come to a single example or how you generally derive them, after already previously encountering them thoroughly. The worst one at the moment is my "Advanced" Calculus lecture who knows clearly a lot less about the subject than I do and frequently makes mistakes that are on the level of a P.E high school teacher trying to teach it.
matt grime
Homework Helper
HallsofIvy said:
Oh, not really, I always think that of analysis; do I need a special reason?
u all not helping i am a high school grad and u all talk of analysis related problems those are not helping ,c'mon u are here 2 help and be helped
Zurtex
Homework Helper
abia ubong said:
u all not helping i am a high school grad and u all talk of analysis related problems those are not helping ,c'mon u are here 2 help and be helped
f(x)=x! is a function that goes from integers to integers. Generally for something to be integrated it needs to go from real numbers to real numbers, otherwise there is no area underneath it.
but i have been working on it thugh havenot gotten good result i was tryiong 2 find the general expansio of the factorial mean,but have not gotten it yuet if u can help give a general formula on how 2 expand generally ,i could get it.
matt grime | 1,194 | 4,907 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2021-25 | latest | en | 0.952104 |
https://mail.python.org/pipermail/python-list/2005-November/347673.html | 1,394,631,752,000,000,000 | text/html | crawl-data/CC-MAIN-2014-10/segments/1394021791079/warc/CC-MAIN-20140305121631-00076-ip-10-183-142-35.ec2.internal.warc.gz | 625,820,626 | 1,974 | Sorting dominoes
Yu-Xi Lim yuxi at ece.gatech.edu
Mon Nov 14 04:35:11 CET 2005
```DaveM wrote:
> Essentially, I'm trying to sort 12 dominoes. Each domino has two different
> numbers and there are two of every number. Identical dominoes are possible,
> but doubles are not.
>
> The problem is to place the dominoes in a line, side to side, so that two
> columns (or rows, depending how you orientate them) are formed, with each
> number appearing once in each column(row).
>
> I thought this would be the easy part of the program I'm writing, but so far
> some initial states have defeated every attempt I've made, short of
> re-shuffling the dominoes after an arbitrary number of attempts.
>
> I'm sure I'll work out an effective algorithm eventually, but I suspect I'm
> re-inventing the wheel, so any hints on the best way to tackle this would be
> appreciated!
>
> DaveM
Assuming your dominoes are expressed as a list of tuples, e.g. [ (1, 2),
(3, 2), (3, 1) ].
The pseudo-code would probably be as follows:
1) create a corresponding list of dominoes with each domino reversed
2) concatenate your original list of dominoes with the list of flipped
dominoes
3) sort the new concatenated list
4) take every other element in the sorted list, e.g. using [::2]
I hadn't actually thought about it enough to verify if it works all the
time.
``` | 365 | 1,347 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2014-10 | latest | en | 0.930429 |
http://creeksidelearning.com/category/learning-with-legos/ | 1,418,943,819,000,000,000 | text/html | crawl-data/CC-MAIN-2014-52/segments/1418802768044.102/warc/CC-MAIN-20141217075248-00138-ip-10-231-17-201.ec2.internal.warc.gz | 63,591,396 | 14,830 | ## LEGO Math Games: The Number Line Game
Step out of the workbook pages and into LEGO math games. If your child is learning to count, learning to add or subtract, working on negative and positive numbers, why not make it a game? The LEGO Number Line Game uses what you already have to make math fun.
Supplies
{affiliate link} Long sheet of paper from a roll of easel paper
Assorted LEGOs and a LEGO figure
This game can be played at any math level. We’ve used it in several ways.
1. Learning to count from 1-10, 1-30, 1-50, 1-100.
2. Adding and Subtracting. Call out equations to your child (or to the LEGO mini-figure) and have them move the LEGO figure along the number line. (“What’s 3+4, LEGO Man?”)
3. Working with Fact Families. “How many ways can LEGO Man get 10?” (5+5, 6+4, 8+2 and so on).
4. Working with negative numbers. Draw your number line to include negative and positive numbers. Give your child numbers to add and subtract that moves them along both the negative and positive parts of the number line.
What other ways could you use the LEGO Number Line? I’d love to hear your ideas. Post them in the comments below.
Follow Julie Kirkwood, Creekside Learning’s board Hands-On Math on Pinterest.
## Counting to 100
We are counting to 100 with objects found around the house.
Here’s our mason jar 100’s collection.
More information on how we made this collection and various ways to help your child learn how to count 100 can be found on Kiwi Crate DIY.
More hands-on math ideas on Pinterest.
## Learning With Legos: Building an Iroquois Longhouse
Firefly, my 8 year old, is studying early American history this year, with the Joy Hakim series, A History of US. One of the fun projects he’s completed is an Iroquois Longhouse with Legos.
I thought it was important to emphasize that not all Native Americans lived in tipis when studying this time period. Iroquois and some other tribes lived in houses such as this one, along with many other families. This would be the equivalent of all the families on your block living in one house to keep warm throughout the winter and share resources.
I gave Firefly a printed picture of a longhouse and he set to work.
The Creek Dad assisted with construction of the roof. Here’s a look at the interior:
Linking up with Lego Fun Friday over at Frugal Fun For Boys. Stop by and see what other fun learning is happening with Legos.
For more ideas about what we are learning for early American history, follow our Pinterest boards:
Hands-On History
Early American History
Learning With American Girl
## Lego BrickFair Is Worth the Wait
We had a great time at the Lego BrickFair today in Virginia. There was some chatter on Twitter about the long lines to get in, which I hope will translate into a larger venue next year for this spectacular show. If you go tomorrow, or in the coming years, get there on the early or late side if you want to avoid the lines, but do go. It was well worth it. I’ll let the pictures speak for themselves. Here are some of our favorites.
The interactive building area was a big hit with kids. And some grown-ups, too. They could build and race their wheeled creations on the tracks or use the many tables provided to make things with what appeared to be thousands of bricks.
BrickFair runs through Sunday, August 5th, 2012, at the Dulles Expo Center in Chantilly, Virginia. It is a yearly event in this area. Go to www.BrickFair.com for more information.
I received four free tickets from BrickFair but was not required to write about it. All opinions expressed here are my own.
## BrickFair Virginia Lego Fan Festival with Ticket Give-Away
Show of hands: who has a Lego fan, or two or three, in their house? We love Legos in our house and use them frequently when learning about science, history, math and more.
So I was excited to learn about the BrickFair Lego Fan Festival of 2012, a Lego extravaganza featuring creations of all sizes, a stay-and-play area for kids to build with thousands of Lego bricks, and more than 1,000 exhibitors, including model train displays. All in all, over 100,000 square feet of awesome brick creations and activities.
BrickFair 2012 takes place on August 4th and 5th at the Dulles Expo Center in Chantilly, Virginia. Admission is \$10 per person (ages 3 and under are free; no strollers please). For more information, visit the BrickFair website.
And guess what? I have four free tickets to give away. That’s a \$40 value. All you have to do to enter is leave a comment on THIS POST here on THIS BLOG using the word BrickFair in your comment. Also, leave me a way to contact you if you win. Last day to enter is Wednesday, July 18th.
Photos courtesy of BrickFair.
Disclosure: I received 8 free tickets (4 for myself, and 4 to give away) from BrickFair in exchange for posting about BrickFair on my blog.
This contest is now closed. | 1,178 | 4,901 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2014-52 | latest | en | 0.913828 |
http://www.siue.edu/text/its/bb/fac_staff/bb_step4g.shtml | 1,481,212,867,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541864.24/warc/CC-MAIN-20161202170901-00073-ip-10-31-129-80.ec2.internal.warc.gz | 704,402,080 | 10,255 | Information Technology Services
# Blackboard - Faculty & Staff
## Step 3: Add Materials and Set up Course
The Grade Center affords the opportunity to post not only grades associated with work completed within Blackboard, but also the addition of grades for work completed outside of Blackboard. Students can access their grades by clicking on the "My Grades" link in the Course Menu.
### Setting up the Grade Center
There are a number of settings you must review and/or configure before you use the Grade Center. It is recommended that you approach them in the following order:
#### Review and/or Create New Grading Schema
Some items are automatically added to the Grade Center when they are added to the course (e.g. Blackboard assignments, blogs, discussion boards, journals, surveys, Turnitin assignments, and tests). Assessments not delivered within Blackboard must be manually added directly into the Grade Center by adding a Grade Column. See How to Create Grade Columns.
#### Extra Credit
The Total Points Possible for an Extra Credit or Attendance Demerit column must be set to zero. There is no equitable way to weight Extra Credit or Attendance Demerits as columns in the Weighted Total. In order for Extra Credit or an Attendance Demerit to be applied in a Weighted Total, they must first be associated with a category that is being weighted. See How to Create an Extra Credit Column.
Calculated Columns afford the opportunity to combine the scores of a number of individual standard columns to create a new calculated column. Perhaps the most frequent use for a Calculated Column would be to take the scores of a number of quizzes, drop the lowest quiz score, and create a new Calculated Column for quizzes where the lowest quiz score has been dropped. See How to Create Total Column or How to Create Weighted Columns.
#### Review and/or Configure the Total & Weighted Total Column Settings
• The Total column should be used when the final grade is to be calculated on a straight point basis: total points earned divided by total points possible.
• The Weighted Total column should be used when the final grade is to be calculated by averaging the grade for items and/or categories and then multiplying that by a percentage for weighting. See Configuring the Weighted Total in the Grade Center.
Take a few minutes to review and configure either the Total or the Weighted Total column settings. It is recommended that you not show the total you are not using to your students. By default, both of these columns are set up to include All Grade Columns in their calculations and to calculate as a running total (including only those items with grades or attempts). If everything in your Grade Center is supposed to be included in the total, then everything should be in order for the Total column. In addition, if you have included manually added practice quizzes, for instance, that are not supposed to be included in the overall grade for the term, you will want to make certain that those columns are not included in the Total and/or Weighted Total column settings.
Note: If you are using the Weighted Total Column you must select either Percentage or Letter Grade to display the grade. If you display Score, the value will not appear to add up and will lead to confusion.
Note: When the Grade Center is set to calculate the Total or Weighted Total as a "running total," you must manually enter a grade of "zero" for any assessment a student does not complete, otherwise, that student's grade will be calculated against points possible, reduced by the points possible for the item not completed.
Once your Grade Center includes the columns you need with the grading preferences you have specified, you will want to organize it in order to view it in a way that meets your needs and your preferences. See About Organizing the Grade Center Data.
#### Organize Smart Views
• If you are teaching multiple sections of the same course in a single Blackboard class, you could create separate views of the Grade Center for each section of the class.
• If you have a student coming to discuss his or her grades, you could create a view that shows just that student's grades.
• If you wish to identify those students whose performance on an assessment was less than stellar, you could create a view of those students who scored beneath a certain benchmark
• Learn how to apply Smart Views in the Grade Center.
The "Needs Grading" screen provides quick access to all items in need of grading, rather than going to the Full Grade Center and having to hunt for items that need your attention. Attempts are placed in a queue for easy navigation among items.
There are a variety of ways that grades can be entered into the Grade Center. For electronic assessments delivered through Blackboard (tests, surveys, discussion boards, journals, blogs, wikis, assignments, Turnitin assignments), please refer to resources for those specific assessment tools for additional information. For manually created columns, users will simply manually enter grades.
There are several options available for dropping the lowest "x" grades for an assessment.
• Dropping a Grade by Exempting - Exempting removes the grade from the Running Total and the Running Weighted Totals columns but does not delete the grade information. Exempting grades requires you to manually search for the lowest grade for each students and manually exempt that one. An Exempted grade can be reversed. This option can be done directly in the cells of the Grade Center but can be time consuming with a large class. Exempting a grade is most useful when a student enrolls late in a course and you wish to exempt them from a graded assessment completed prior to the time they enrolled in the course. The total points possible for that student will be reduced by the points possible for the exempted assessment.
• Dropping a Grade using the Running Weighted Totals Column - If you are using the Weighted Total column to weight all the grades in your Grade Center, this is the option to use. Generally it is used when weighting by Category. When you select the Category to weight, you will see a setting allowing you to drop the lowest "x" grades.
• Dropping a Grade using a Calculated Column - If you are not using the Weighted Total, but using the Total (points) column, this is the option to use. You will be creating a Calculated Averaged Column to average the grades and drop the lowest. Then you will need to modify the Total column to include the Calculated Average column and not the individual columns that feed into that Calculated Average. It is up to your discretion as to whether or not to show this Calculated Average column to your students. | 1,342 | 6,729 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2016-50 | longest | en | 0.906291 |
https://www.media4math.com/NY-4.MD.5a | 1,685,435,236,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224645417.33/warc/CC-MAIN-20230530063958-20230530093958-00420.warc.gz | 982,830,846 | 13,224 | ## NY-4.MD.5a: An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
There are 49 resources.
Title Description Thumbnail Image Curriculum Topics
## Definition--Circle Concepts--Arc Length
Definition--Circle Concepts--Arc Length
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Area of a Circle
Definition | Circle Concepts | Area of a Circle
This is part of a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Center of a Circle
Definition--Circle Concepts--Center of a Circle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Chord
Definition--Circle Concepts--Chord
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circle
Definition | Circle Concepts | Circle
This is part of a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circle Inscribed in a Square
Definition--Circle Concepts--Circle Inscribed in a Square
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circle Inscribed in a Triangle
Definition--Circle Concepts--Circle Inscribed in a Triangle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circles on a Coordinate Plane
Definition--Circle Concepts--Circles on a Coordinate Plane
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circular Arc
Definition--Circle Concepts--Circular Arc
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circular Composite Figures
Definition--Circle Concepts--Circular Composite Figures
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circular Cross-Sections
Definition--Circle Concepts--Circular Cross-Sections
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circular Functions
Definition--Circle Concepts--Circular Functions
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circular Models
Definition--Circle Concepts--Circular Models
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circular Numerical Models
Definition--Circle Concepts--Circular Numerical Models
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circumcenter of Triangle
Definition--Circle Concepts--Circumcenter of Triangle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circumference
Definition--Circle Concepts--Circumference
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Circumscribed Angle
Definition--Circle Concepts--Circumscribed Angle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Concentric Circles
Definition--Circle Concepts--Concentric Circles
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Diameter
Definition--Circle Concepts--Diameter
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Equation of a Circles
Definition--Circle Concepts--Equation of a Circles
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Hexagon Inscribed in a Circle
Definition--Circle Concepts--Hexagon Inscribed in a Circle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Incenter of a Triangle
Definition--Circle Concepts--Incenter of a Triangle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Inscribed Angle
Definition--Circle Concepts--Inscribed Angle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Inscribed Angle Theorem
Definition--Circle Concepts--Inscribed Angle Theorem
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle
## Definition--Circle Concepts--Point of Tangency to a Circle
Definition--Circle Concepts--Point of Tangency to a Circle
This is a collection of definitions related to circles and related topics. This includes general definitions for parts of a circle, constructions involving circles, and other topics.
Definition of a Circle | 1,525 | 8,221 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2023-23 | longest | en | 0.862264 |
https://spidersolitaireaddict.blog/2019/05/03/my-first-meaningful-post-hooray/ | 1,675,208,334,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00209.warc.gz | 543,456,964 | 22,799 | # My first meaningful post, hooray! :)
## Why Spider Solitaire?
Welcome to my blog on Spider Solitaire.
In an age dominated by Netflix, Fortnite, Randy Rainbow songs, internet memes, griping about the NBN, checking Twitter feeds every 5 minutes, Tetris99 (okay, even I have to admit the concept is kinda cool) and what-not, it seems Spider Solitaire (indeed most abstract mathematical games in general) is becoming a lost art. If you’ve never heard of the game you’re not alone. I’ve even come across one dude at work who didn’t even know what Sudoku was one year after it became a thing. But I digress.
Like most armchair critics, I consider myself to be pretty good at Spider Solitaire. But I also have some concrete data: I can win about half the time on four-suit level without undoing moves. I know very few people who have claimed (let alone proved) to be expert at Spider Solitaire. Certainly one can google basic strategies easily enough, but I find most of it to be superficial. For instance, Joe Bloggs might show a complete game from start to finish, but the game turned out to be ridiculously easy. Jane Citizen might say that empty columns are better than smoking, it’s always better to build sequences in the same suit and bears defecate in the woods. But that fails the Duh Test since every man, dog and millipede on the planet could have figured that out by themselves. I believe Spider Solitaire is a lot deeper than the trash advice I found on the internet. Hence this blog was born.
## Okay, so what is Spider Solitaire?
You have made it to the final table of the World Series of Spider Solitaire. You look at the left-most column. The Seven of Hearts, then the Deuce of Clubs, followed by five face-down cards. After studying the rest of the layout you announce that you are all-in! The audience gasps with horror: is this guy on tilt? Unperturbed, you shift the Seven and Deuce out of the way and expose the hidden cards one by one. The last card gives you a straight flush in diamonds, which you immediately move onto the foundations. You take a large bite of your last Oreo. Crunch. Victory is mathematically certain and the rest is a formality. You take home an \$8 million pay cheque and a gold bracelet and the good guys live happily ever after … or something like that.
Okay that’s probably not the most accurate description of how the game is played, but I do believe Spider is the Cadillac of Solitaire games.
To be continued … | 550 | 2,464 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2023-06 | latest | en | 0.947246 |
https://matlabprojects.org/simulation-projects-in-matlab/ | 1,721,337,749,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514859.56/warc/CC-MAIN-20240718191743-20240718221743-00801.warc.gz | 345,097,980 | 14,936 | • Matlab
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## SIMULATION PROJECTS IN MATLAB
The linear algebra programming is very simple,so we move on to the matrix programming language as Simulation projects in Matlab. It work under the batch job as well as the interactive sessions.The sensor image can be measured based on the size of each photosite. The.The pixels provide photons,which converted to electric charges and the ones & zeroes.the numbers like any numbers that run via head which have no physical size.the difficult task of image processing is segmentation of images.in the images,extract objects by segmentation.
### 2015 IEEE SIMULATION PROJECTS IN MATLAB
#### Characteristics of simulation projects in Matlab:
• Interactive environment are generated for problem solving,design and iterative exploration.
• It is the great development tool for increase code quality, performance maximization, maintability using the programming interface.
• Library for mathematical functions for statistics,fourier analysis ,filtering,linear algebra.
• Ordinary differential equations,numerical integration,optimization for visualizing data &tools for custom plots.
The data with more information provide a small value for discrimination .the original measurements is inefficient in pattern recognition which obsure interpretation.
• Redundant information:
In spectral data more information are repeated from image to image which complicates analysis and unnecessarily classification.
• Spatial vs spectral information:
The small portion of information content of more images represent gray values.
• Information vs useful information:
Variability in image data is little or given classification problem with no value which declares the random or systematic variability as the target of interest or changes in the background targets not of immediate interest.
Content based images & video retrieval are the interested application in reduction methods.the feature vector with great dimensional generate problem in construction of data structures for retrieval & search.
##### Simulation projects in matlab using classification procedure:
• Selection of feature.
• The training stage.
• Appropriate classification algorithm procedure.
• Post classification smothering.
• Accuracy assessment.
Optical character recognition (OCR) is a technology which allow user to transform variety of documents like PDF files,scanned paper documents,image captured by digital camera as searchable & editable data.
You can Watch all Subjects Matlab & Simulink latest Innovative Project Results
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• Unique Work | 840 | 4,450 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2024-30 | latest | en | 0.828928 |
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답변 | 1,040 | 3,245 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2024-26 | latest | en | 0.769722 |
https://www.netexplanations.com/telangana-scert-class-9-math-solution-chapter-15-proofs-in-mathematics-exercise-15-2/ | 1,675,638,819,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500294.64/warc/CC-MAIN-20230205224620-20230206014620-00697.warc.gz | 924,215,946 | 13,102 | # Telangana SCERT Class 9 Math Solution Chapter 15 Proofs in Mathematics Exercise 15.2
## Telangana SCERT Solution Class IX (9) Math Chapter 15 Proofs in Mathematics Exercise 15.2
1.)
(i) Based on the two statements we can conclude that – Jeevan is mortal.
(ii) No, x could be from any other state of India other than Andhra Pradesh and Telangana.
(iii) Based on the two statements we can conclude that Guly has a red tongue
(iv) All smart people are not presidents. There are a lot of smart people in different – fields or different line of work all of them need not be presidents.
2.)
Given,
3 3 U 8
“If a cord has consonant on one side. There is a odd number on the other side”.
Clearly, we have to check B & 8.
If B has an even number on the other sides or if 8 has consonant – on the other side then the rule is broken.
3.) From the given clues we can conclude that the answer is 35.
(a) Statement – a does not help because the number clearly has more than one digit.
(b) Statement – b doesn’t help because the only multiply 7
7 is 70, which is divided by 10.
(c) Statement – c help because it opts out lots of number that are not multiple of 7.
(d) Helps because being odd number a lot of number.
(e) Statement – e does not help because the multiple of 7 and 11 is 77 and both the digits are not equal.
(f) Statement – f does not help.
(g) Statement – g for the opts out some possibilities.
(h) Using statement h we can conclude that 35 is the answers so it obviously helps.
Therefore, only c, d, g & h can be used to get the number rest are not required.
Updated: October 5, 2021 — 4:30 pm | 418 | 1,623 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2023-06 | latest | en | 0.922411 |
https://www.growingiq.com/dlt-collection/Ms.-Raejean/level2/saturday/63fad91b-b02b-4967-a364-5ba0fab41586 | 1,638,025,788,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964358189.36/warc/CC-MAIN-20211127133237-20211127163237-00080.warc.gz | 897,900,101 | 182,569 | ## Ms. Raejean
### Target 1
###### Lesson Type:
Continuation
Measurment
:
Comparison
Measure and compare length in nonstandard units.
###### 1:
Measure and compare length in nonstandard units.
###### 2:
Understand that different measurements will be obtained for the same object with non-standard units, based on the unit selected.
###### 3:
Explain how two people can measure the same object and get different lengths.
###### 4:
Understand that a non-standard unit must be placed end-to-end (with no gaps or overlap).
2nd
###### Vocabulary:
Measure, Lengths, Non-Standard Units
Activities:
We continued to measure the lengths of different objects using non-standard units. Students used cubes and paper clips. We reviewed how non-standard units had to be placed one after the other (no gaps or overlaps). We also discussed the different measurements that students found for the same objects. It could be due to how students placed the paper clips, but some students used a combination of big and small paper clips.
## Absent Students:
### Target 2
:
###### 1:
Write an addition equation to express the sum of equal addends.
###### 2:
Set the foundation for later exploration of multiplication (i.e. 2+2+2+2 =? How many 2s did you add? When you add 4 number 2s together, what do you get?)
###### 3:
Use addition to find the total number of objects in an array.
2nd
###### Vocabulary:
Activities:
Students continued to count objects using repeated addition. We discussed how many times a number was added. Students wrote an addition equation, and we connected it to a multiplication equation. Ex. How many times did you add the number 5? 5 times: 5 + 5 + 5 + 5 + 5 = 5 x 5. This week, we used a hundreds chart to find out how many groups are in a number. Ex. How many groups of 3 are in 12? Students counted by 3s until they landed on 12. 3 + 3 + 3 + 3 = 12. 3 was added 4 times, so 3 x 4 = 12.
### Target 3
:
###### 1:
Relate a reflection to a fold line, where an image will leave its impression on the other side, and that impression will be backwards (the fold line will create an image that has matching parts).
###### 2:
Identify the reflection of a given image.
2nd
###### Vocabulary:
Symmetry, Reflection, Mirror Image
Activities:
We did a quick review of symmetry using pattern blocks. Symmetry is when you make the mirror image of an object. Students created a symmetrical turkey by repeating the pattern block on the opposite side. Whatever pattern block was used on the left, must also be used on the right. | 608 | 2,562 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.71875 | 5 | CC-MAIN-2021-49 | latest | en | 0.93272 |
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waeting on Jod
Registered: 15.02.2005
From:
Posted: Friday 29th of Dec 19:18 hi People out there I really hope some math expert reads this. I am stuck on this test that I have to turn in in the next couple of days and I can’t seem to find a way to finish it. You see, my professor has given us this assignment on math practice on gcf and lcm worksheets, graphing and radical equations and I just can’t seem to get the hang of it. I am thinking of going to some private tutor to help me solve it. If someone can lend me a hand, I will highly grateful.
nxu
Registered: 25.10.2006
From: Siberia, Russian Federation
Posted: Saturday 30th of Dec 08:32 Student can’t seem to think of anything beyond tutoring. Why don’t you try something yourself? There are numerous tools for math practice on gcf and lcm worksheets which are a lot better than tutoring. Try Algebrator, and you will never need a tuition .
alhatec16
Registered: 10.03.2002
From: Notts, UK.
Posted: Monday 01st of Jan 09:35 It’s true, even I’ve been using this tool since sometime now and it really helped me in solving on math practice on gcf and lcm worksheets and math practice on gcf and lcm worksheets. I also used it to clear my concepts in topics such as graphing circles and equation properties. If you are don’t have much time, then I would highly recommend this software, and well even if you are not , I still would!
EvenettB
Registered: 01.08.2003
From: North America
Posted: Wednesday 03rd of Jan 10:41 Sounds interesting. Where can I find this software ?
Registered: 10.07.2002
From: NW AR, USA
Posted: Thursday 04th of Jan 10:45 I'm sorry . I should have included the link our first time around : http://www.solve-variable.com/solving-quadratic-equations.html. I don't know about a test copy , but the legitimate providers of Algebrator, as opposed to some store fronts of imitation software , offer a complete money back guarantee . Hence, you can acquire the official version , examine the program out and send it back should one not be content by the execution and functionality . Although I believe you are gonna love this software , I'm very interested in discovering from you or anyone should there be something where the program does not function . I do not wish to recommend Algebrator for a function it cannot do . However the next problem found will probably be the first one!
Svizes
Registered: 10.03.2003
From: Slovenia
Posted: Friday 05th of Jan 20:28 Algebrator is a very incredible product and is surely worth a try. You will also find several interesting stuff there. I use it as reference software for my math problems and can swear that it has made learning math more fun .
All Right Reserved. Copyright 2005-2018 | 1,230 | 5,119 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2018-30 | latest | en | 0.834807 |
https://socratic.org/questions/how-do-you-solve-log-2-x-log-4-x-log-8-x-log-16-x-25#235012 | 1,638,486,466,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964362297.22/warc/CC-MAIN-20211202205828-20211202235828-00570.warc.gz | 574,199,549 | 6,378 | # How do you solve log_2 x + log_4 x + log_8 x + log_16 x = 25?
Mar 4, 2016
Solution $\implies x = {2}^{12}$
#### Explanation:
${\log}_{2} x + {\log}_{4} x + {\log}_{8} x + {\log}_{16} x = 25$
$\implies \frac{1}{\log} _ x 2 + \frac{1}{\log} _ x {2}^{2} + \frac{1}{\log} _ x {2}^{3} + \frac{1}{\log} _ x {2}^{4} = 25$
$\implies \frac{1}{\log} _ x 2 + \frac{1}{2 {\log}_{x} 2} + \frac{1}{3 {\log}_{x} 2} + \frac{1}{4 {\log}_{x} 2} = 25$
$\implies \frac{1}{\log} _ x 2 \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\right) = 25$
$\implies {\log}_{2} x \frac{12 + 6 + 4 + 3}{12} = 25$
$\implies {\log}_{2} x = 25 \cdot \frac{12}{25} = 12$
$\implies x = {2}^{12}$
Mar 4, 2016
We can start by converting all of the $\log$ terms to the same base using the rule;
${\log}_{c} b = \log \frac{b}{\log} c$
$\log \frac{x}{\log} 2 + \log \frac{x}{\log} 4 + \log \frac{x}{\log} 8 + \log \frac{x}{\log} 16 = 25$
Now we can pull $\log x$ out of the numerators.
$\log x \left(\frac{1}{\log} 2 + \frac{1}{\log} 4 + \frac{1}{\log} 8 + \frac{1}{\log} 16\right) = 25$
Notice that all of the denominators are powers of 2, so we can rewrite as;
$\log x \left(\frac{1}{\log} 2 + \frac{1}{\log} \left({2}^{2}\right) + \frac{1}{\log} \left({2}^{3}\right) + \frac{1}{\log} \left({2}^{4}\right)\right) = 25$
Using the identity, $\log \left({x}^{y}\right) = y \cdot \log x$, we can rewrite the denominators as multiples of $\log 2$.
$\log x \left(\frac{1}{\log} 2 + \frac{1}{2 \log 2} + \frac{1}{3 \log 2} + \frac{1}{4 \log 2}\right) = 25$
Now we can pull $\log 2$ out of the denominators.
$\log \frac{x}{\log} 2 \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\right) = 25$
We can rewrite all of the fractions in term of the lowest denominator.
$\log \frac{x}{\log} 2 \left(\frac{12}{12} + \frac{6}{12} + \frac{4}{12} + \frac{3}{12}\right) = 25$
Sum the terms in parenthesis.
$\log \frac{x}{\log} 2 \left(\frac{25}{12}\right) = 25$
We can combine all of the constant terms by multiplying both sides by the inverse of the fraction on the left hand side.
$\log \frac{x}{\log} 2 \left(\frac{25}{12}\right) \left(\frac{12}{25}\right) = 25 \left(\frac{12}{25}\right)$
Combine the $\log$ terms by reversing the process we did in step one and eliminate all of the constant terms that cancel.
${\log}_{2} x = 12$
Using the definition of a logarithm, we write this statement in terms of $x$.
$x = {2}^{12}$
Solve.
$x = 4096$ | 996 | 2,418 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 27, "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} | 4.65625 | 5 | CC-MAIN-2021-49 | latest | en | 0.584746 |
https://a1calculator.com/health/body-fat-calculator/ | 1,721,864,844,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763518532.61/warc/CC-MAIN-20240724232540-20240725022540-00679.warc.gz | 62,586,885 | 19,131 | ## Introduction to Body Fat Percentage Calculation:
This body fat calculator is designed to estimate the percentage of body fat in relation to total body weight. While it provides a rough estimate, it is essential to note that it should not replace professional medical advice. If health concerns arise, consulting with a doctor is recommended.
## Human Body Composition:
Over 98% of the human body comprises six essential elements: oxygen, carbon, hydrogen, nitrogen, calcium, and phosphorus. Additionally, molecules such as water, proteins, lipids, hydroxyapatite, carbohydrates, and nucleic acids play crucial roles.
## Understanding Body Fat:
Body fat, or adipose tissue, serves the dual purpose of storing energy and providing insulation. There are two types of body fat: essential, necessary for life and reproductive functions (higher in women); and storage, accumulated for protection, particularly for organs.
## How to Calculate Body Fat:
To estimate body fat percentage, input age, gender, weight, and height into the calculator. The formula involves BMI and gender values. While this method provides an estimate, it cannot replace professional assessment.
## Normal Body Fat Percentage:
Comparing your calculated body fat percentage with recommended values is crucial. The American Council on Exercise provides average ranges for different groups:
• Essential fat: 10–13% (women), 2–5% (men)
• Athletes: 14–20% (women), 6–13% (men)
• Fitness: 21–24% (women), 14-17% (men)
• Average: 25–31% (women), 18–24% (men)
• Obese: 32%+ (women), 25%+ (men)
## Importance of Controlling Body Fat:
Maintaining a healthy body fat percentage is vital for hormonal balance. Excessive body fat can lead to health issues such as heart diseases, hormonal imbalances, diabetes, and complications during pregnancy.
## Alternative Body Fat Measurement Methods:
Various methods exist, including underwater weighing, skinfold measurements, BMI-based calculations, and ultrasound. Each method has its advantages and limitations, catering to diverse preferences and needs.
## Tips for Reducing Body Fat:
• Plan for long-term, gradual fat loss with moderate exercises.
• Combine weight lifting and cardio exercises for varied impact.
• Follow a balanced diet with less saturated fats and opt for oils.
• Stay hydrated with water and incorporate proper snacks.
• Avoid extreme diets and plan occasional “cheat days.”
• Regularly monitor your progress for motivation.
## Is Fat Bad?
While excess fat poses health risks, it is essential for energy, cell growth, hormone production, and organ protection. Examining extreme cases emphasizes the importance of maintaining a balanced approach.
## Frequently Asked Questions (FAQs):
#### 1. What is a good body fat percentage?
For women, a body fat range of 13-31% is considered healthy, while for men, the range is 5-24%. Body fat is not the most accurate measure of health; it becomes more acceptable to have more body fat as you age, due to metabolism slowing.
#### 2. How does fat leave the body?
Fat primarily leaves the body through the lungs when exhaling. Additionally, it can be released via sweat, urine, and feces.
#### 3. How does the body burn fat?
The body burns fat when there is insufficient energy in the bloodstream to meet current needs. Fatty acids are released into the blood, transported to muscles, broken down for energy, and the byproduct, CO2, is expelled through the lungs or urine.
#### 4. What does fat do for your body?
Fats are essential for providing energy, contributing to cell growth, forming essential hormones, protecting organs from physical damage, and providing insulation. However, excessive fatty tissues can act as pathways for diseases.
#### 5. Are body fat scales accurate?
Body fat scales can be accurate, but results may vary by about 5%. To maximize accuracy, follow the instructions closely.
#### 6. How do you check your body fat?
Various methods include skinfold calipers, body circumference measurements, dual-energy X-ray absorptiometry (DXA), hydrostatic (underwater) weighing, bioelectrical impedance analysis (BIA), and estimation from BMI.
#### 7. What is the average percentage of body fat?
In the USA, the average body fat percentage varies by age and gender. For example, for women aged 16-19, it was 35%, for those 20-39, it was 38%, and for those 40-59, it was 41%.
#### 8. How much body fat can I lose in a month?
Losing 1-3% of body fat per month is considered healthy and achievable. However, fat loss depends on various factors, so consulting with a doctor or dietitian before significant weight loss is advisable.
#### 9. When does your body burn fat?
The body burns fat when there is not enough energy in the bloodstream to meet current needs. If you’ve recently eaten, the body uses that energy first before tapping into fat stores. If there’s no fat to burn, the body may burn muscle instead. | 1,066 | 4,926 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2024-30 | latest | en | 0.876982 |
https://www.abcteach.com/directory/subjects-sports-physical-education-soccer-world-cup-9806-6-0 | 1,511,186,212,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934806066.5/warc/CC-MAIN-20171120130647-20171120150647-00208.warc.gz | 751,023,063 | 25,028 | You are an abcteach Member, but you are logged in to the Free Site. To access all member features, log into the Member Site.
# Physical Education World Cup Worksheets
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• Perfect for the World Cup. Use these patterns to design uniforms, balls and flags for an imaginary soccer team.
• [created with abctools] Unscramble the words, from FIFA to score.
• [created with abctools] Find the words, from FIFA to score, in this soccer-player shaped word search.
• Two writing prompts, Soccer Story and Competition, with a World Cup theme.
• Match countries to flags, countries to locations, and teams to regions. Fill in the pie chart to identify teams by regions, and answer questions based on the information you find. This geography unit combines research skills and FUN! for the World Cup. Wide range of activities for all abilities.
• Answer the targeted questions to help organize thoughts for a report on a World Cup participating country. Includes directions.
• Practice completing and reading graphs. Use your research skills to discover the scores for the 2002 World Cups, and then fill in the graphs to understand the scoring system. This is fun math, and a great way to prepare for the next World Cup! Answers and directions are included.
• Practice completing and reading a graph. Use your research skills to discover the scores for one group from the 2010 World Cup. Complete the graph and answer the questions. This is an easier version for younger students. Directions are included.
• Practice completing and understanding graphs with information on the 2006 World Cup. Color in graph and answer math comprehension questions. This is fun math, and a great way to teach about the current World Cup! Directions and answers included.
• Practice completing and reading graphs. Use your research skills to discover the scores for the 2003 World Cup, and then fill in the graphs to understand the scoring system. Finally, complete the fractions charts based on the information. This is a good project for practicing reading a chart and "translating" the information. Answers and directions are included.
• Practice completing and understanding graphs. Record the scores for the 2007 Women's World Cup, and then fill in the graphs. This is fun math, and a great way to enjoy the Women's World Cup! Directions are included.
• Match the participating country to the continent; match the country to the region; answer the questions about the participating regions; label the pie graph.
• Answer the targeted questions to help organize thoughts for a report on a World Cup participating country. Includes directions.
• Match the country participating in the World Cup to its continent or region. Color in the pie graph.
• Match the countries participating in the 2010 World Cup soccer games with the correct continent or region. Color in the pie graph.
• Practice completing and understanding graphs. Record the scores for the 2010 World Cup games, and then fill in the graphs. This is fun math, and a great way to enjoy the 2010 World Cup. 10 pages.
• Match countries to flags, countries to locations, and teams to regions. Fill in the pie chart to identify teams by regions, and answer questions based on the information you find. This geography unit combines research skills and fun for the World Cup. Wide range of activities for all abilities.
• Each full color page features the flags of the World Cup participants in the their assigned groups. All groups in one easy download.
• A reading comprehension tracing the history of World Cup soccer from its beginnings in 1904.
• This World Cup Center: Flag 2 B&W clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This World Cup Center: Flag 2 B&W clipart is provided in jpeg format.
• This World Cup Center: Soccer Ball B&W clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This World Cup Center: Soccer Ball B&W clipart is provided in jpeg format.
• This World Cup Center: Soccer Shirt Back B&W clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This World Cup Center: Soccer Shirt Back B&W clipart is provided in jpeg format.
• This World Cup Center: Soccer Shirt Front B&W clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This World Cup Center: Soccer Shirt Front B&W clipart is provided in jpeg format.
• Get ready for the World Cup with these three soccer-themed bookmarks featuring English and German phrases ("I love soccer"). More World Cup soccer bookmarks and other materials are available on our member site.
• 3 pages, 3 bookmarks each, featuring English and German phrases related to the World Cup ("Ich mag Fussball")
• 3 pages, 3 bookmarks each, featuring English and German phrases related to the World Cup ("Wer ist der beste Spieler?")
• 1 page with 3 horizontal bookmarks, featuring English and German phrases related to the World Cup ("Fussball ist mein Lieblingssport")
• 1 page with 3 bookmarks, featuring negative English and German phrases related to the World Cup for the less excited students ("Fussball ist langweilig")
• 3 pages, 3 bookmarks each, featuring U.S. English and German phrases related to the World Cup ("Ich mag Fussball")
• 3 pages, 3 bookmarks each, featuring U.S. English and German phrases related to the World Cup ("Wer ist der beste Spieler?")
• 1 page with 3 horizontal bookmarks, featuring U.S. English and German phrases related to the World Cup ("Fussball ist mein Lieblingssport")
• 1 page with 3 bookmarks, featuring negative U.S. English and German phrases related to the World Cup for the less excited students ("Fussball ist langweilig")
• [created with abctools] 1 ACROSS: Covers the middle of the field.
• Seventy terms related to the World Cup, including players, plays, penalties, and more, to match to their definitions. These soccer ball cards make a great memory game!
• [created with abctools] Thirty-five vocabulary words, from FIFA to YELLOW CARD.
• Our World Cup word search, crossword puzzle, word unscramble, and vocabulary matching game are all combined here, with Learning Center directions for use.
• Each full color page features the flags of the World Cup participants in their assigned groups. All groups in one easy download.
• abcteach has a multitude of World Cup materials available! Here are some suggestions for using our materials as Learning Center activities - language arts, math, social studies, and more. Find inspiration and direction here.
• Use this handy chart to track the winners of each group match.
• Practice completing and reading graphs. Use your research skills to discover the scores for the 2002 World Cup, and then fill in the graphs to understand the scoring system. Finally, complete the fractions charts based on the information. This is a good project for practicing reading a chart and "translating" the information. Answers and directions are included.
• Scalloped edge (Full color)
• Scalloped edge (full color)
• This World Cup Center: Flag 2 (coloring page) clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This World Cup Center: Flag 2 (coloring page) clipart is provided in jpeg format.
• This World Cup Center: Soccer Ball (coloring page) clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This World Cup Center: Soccer Ball (coloring page) clipart is provided in jpeg format.
• Use this 'Word Wall: Soccer' printable worksheet in the classroom or at home. Your students will love this 'Word Wall: Soccer'. From "ball" to "World Cup". These vocabulary building word strips are great for word walls.
• Use this 'ABC Order: Soccer Vocabulary' printable worksheet in the classroom or at home. Your students will love this 'ABC Order: Soccer Vocabulary'. 30 words, from "ball" to "World Cup".
• Use this 'Missing Letters: Soccer Vocabulary' printable worksheet in the classroom or at home. Your students will love this 'Missing Letters: Soccer Vocabulary'. Fill in the missing letters for these 30 soccer-related words, from "ball" to "World Cup".
• Use this 'Spelling List: Soccer Vocabulary' printable worksheet in the classroom or at home. Your students will love this 'Spelling List: Soccer Vocabulary'. A list of 30 soccer-related words, from BALL to WORLD CUP, to use as a spelling lesson.
• [Created with abctools] Unscramble the 30 soccer-related words, from "ball" to "World Cup".
• This 2010 WC: South Africa World Cup Flag B&W clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This 2010 WC: South Africa World Cup Flag B&W clipart is provided in jpeg format.
• This 2010 WC: South Africa World Cup Flag Color clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This 2010 WC: South Africa World Cup Flag Color clipart is provided in jpeg format.
• This 2006 WC: Germany World Cup Venues Map B&W clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This 2006 WC: Germany World Cup Venues Map B&W clipart is provided in jpeg format.
• This 2006 WC: Germany World Cup Venues Map Color clipart is great to illustrate your teaching materials. As an abcteach member you have unlimited access to our 22,000+ clipart illustrations and can use them for commercial use. This 2006 WC: Germany World Cup Venues Map Color clipart is provided in jpeg format.
• This Math - 2010 Graphs (upper el/middle) World Cup Soccer Center is perfect to practice graphing skills. Your elementary grade students will love this Math - 2010 Graphs (upper el/middle) World Cup Soccer Center. Practice completeing and understanding graphs with information on the 2010 World Cup. Color in graphs for goals made and against. Answers included. Could be used to compare statistics for 2014 World Cup. | 2,323 | 10,731 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2017-47 | latest | en | 0.923862 |
https://www.netlib.org/lapack/explore-html-3.6.1/d5/d99/group__single__eig_ga4c87fd3a41eaf68bcb5349b3935eb7a6.html | 1,632,753,037,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780058450.44/warc/CC-MAIN-20210927120736-20210927150736-00159.warc.gz | 888,065,468 | 6,243 | LAPACK 3.6.1 LAPACK: Linear Algebra PACKage
subroutine sget51 ( integer ITYPE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( * ) WORK, real RESULT )
SGET51
Purpose:
``` SGET51 generally checks a decomposition of the form
A = U B V'
where ' means transpose and U and V are orthogonal.
Specifically, if ITYPE=1
RESULT = | A - U B V' | / ( |A| n ulp )
If ITYPE=2, then:
RESULT = | A - B | / ( |A| n ulp )
If ITYPE=3, then:
RESULT = | I - UU' | / ( n ulp )```
Parameters
[in] ITYPE ``` ITYPE is INTEGER Specifies the type of tests to be performed. =1: RESULT = | A - U B V' | / ( |A| n ulp ) =2: RESULT = | A - B | / ( |A| n ulp ) =3: RESULT = | I - UU' | / ( n ulp )``` [in] N ``` N is INTEGER The size of the matrix. If it is zero, SGET51 does nothing. It must be at least zero.``` [in] A ``` A is REAL array, dimension (LDA, N) The original (unfactored) matrix.``` [in] LDA ``` LDA is INTEGER The leading dimension of A. It must be at least 1 and at least N.``` [in] B ``` B is REAL array, dimension (LDB, N) The factored matrix.``` [in] LDB ``` LDB is INTEGER The leading dimension of B. It must be at least 1 and at least N.``` [in] U ``` U is REAL array, dimension (LDU, N) The orthogonal matrix on the left-hand side in the decomposition. Not referenced if ITYPE=2``` [in] LDU ``` LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.``` [in] V ``` V is REAL array, dimension (LDV, N) The orthogonal matrix on the left-hand side in the decomposition. Not referenced if ITYPE=2``` [in] LDV ``` LDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1.``` [out] WORK ` WORK is REAL array, dimension (2*N**2)` [out] RESULT ``` RESULT is REAL The values computed by the test specified by ITYPE. The value is currently limited to 1/ulp, to avoid overflow. Errors are flagged by RESULT=10/ulp.```
Date
November 2011
Definition at line 151 of file sget51.f.
151 *
152 * -- LAPACK test routine (version 3.4.0) --
153 * -- LAPACK is a software package provided by Univ. of Tennessee, --
154 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155 * November 2011
156 *
157 * .. Scalar Arguments ..
158 INTEGER itype, lda, ldb, ldu, ldv, n
159 REAL result
160 * ..
161 * .. Array Arguments ..
162 REAL a( lda, * ), b( ldb, * ), u( ldu, * ),
163 \$ v( ldv, * ), work( * )
164 * ..
165 *
166 * =====================================================================
167 *
168 * .. Parameters ..
169 REAL zero, one, ten
170 parameter ( zero = 0.0, one = 1.0e0, ten = 10.0e0 )
171 * ..
172 * .. Local Scalars ..
173 INTEGER jcol, jdiag, jrow
174 REAL anorm, ulp, unfl, wnorm
175 * ..
176 * .. External Functions ..
177 REAL slamch, slange
178 EXTERNAL slamch, slange
179 * ..
180 * .. External Subroutines ..
181 EXTERNAL sgemm, slacpy
182 * ..
183 * .. Intrinsic Functions ..
184 INTRINSIC max, min, real
185 * ..
186 * .. Executable Statements ..
187 *
188 result = zero
189 IF( n.LE.0 )
190 \$ RETURN
191 *
192 * Constants
193 *
194 unfl = slamch( 'Safe minimum' )
195 ulp = slamch( 'Epsilon' )*slamch( 'Base' )
196 *
197 * Some Error Checks
198 *
199 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
200 result = ten / ulp
201 RETURN
202 END IF
203 *
204 IF( itype.LE.2 ) THEN
205 *
206 * Tests scaled by the norm(A)
207 *
208 anorm = max( slange( '1', n, n, a, lda, work ), unfl )
209 *
210 IF( itype.EQ.1 ) THEN
211 *
212 * ITYPE=1: Compute W = A - UBV'
213 *
214 CALL slacpy( ' ', n, n, a, lda, work, n )
215 CALL sgemm( 'N', 'N', n, n, n, one, u, ldu, b, ldb, zero,
216 \$ work( n**2+1 ), n )
217 *
218 CALL sgemm( 'N', 'C', n, n, n, -one, work( n**2+1 ), n, v,
219 \$ ldv, one, work, n )
220 *
221 ELSE
222 *
223 * ITYPE=2: Compute W = A - B
224 *
225 CALL slacpy( ' ', n, n, b, ldb, work, n )
226 *
227 DO 20 jcol = 1, n
228 DO 10 jrow = 1, n
229 work( jrow+n*( jcol-1 ) ) = work( jrow+n*( jcol-1 ) )
230 \$ - a( jrow, jcol )
231 10 CONTINUE
232 20 CONTINUE
233 END IF
234 *
235 * Compute norm(W)/ ( ulp*norm(A) )
236 *
237 wnorm = slange( '1', n, n, work, n, work( n**2+1 ) )
238 *
239 IF( anorm.GT.wnorm ) THEN
240 result = ( wnorm / anorm ) / ( n*ulp )
241 ELSE
242 IF( anorm.LT.one ) THEN
243 result = ( min( wnorm, n*anorm ) / anorm ) / ( n*ulp )
244 ELSE
245 result = min( wnorm / anorm, REAL( N ) ) / ( n*ulp )
246 END IF
247 END IF
248 *
249 ELSE
250 *
251 * Tests not scaled by norm(A)
252 *
253 * ITYPE=3: Compute UU' - I
254 *
255 CALL sgemm( 'N', 'C', n, n, n, one, u, ldu, u, ldu, zero, work,
256 \$ n )
257 *
258 DO 30 jdiag = 1, n
259 work( ( n+1 )*( jdiag-1 )+1 ) = work( ( n+1 )*( jdiag-1 )+
260 \$ 1 ) - one
261 30 CONTINUE
262 *
263 result = min( slange( '1', n, n, work, n, work( n**2+1 ) ),
264 \$ REAL( N ) ) / ( n*ulp )
265 END IF
266 *
267 RETURN
268 *
269 * End of SGET51
270 *
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:116
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
Here is the call graph for this function:
Here is the caller graph for this function: | 2,017 | 5,565 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2021-39 | latest | en | 0.783929 |
https://discuss.leetcode.com/topic/47776/can-someone-help-to-check-my-solution-output-limit-exceed | 1,513,156,835,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948522343.41/warc/CC-MAIN-20171213084839-20171213104839-00727.warc.gz | 525,198,541 | 12,046 | # Can someone help to check my solution output limit exceed
• /**
• Definition for a binary tree node.
• public class TreeNode {
• public int val;
• public TreeNode left;
• public TreeNode right;
• public TreeNode(int x) { val = x; }
• }
*/
public class Solution {
//use memery to save the contain root and target, so no need to check again
Dictionary<TreeNode,Dictionary<TreeNode,bool>> memarizeSearchResult = new Dictionary<TreeNode,Dictionary<TreeNode,bool>>();
public TreeNode LowestCommonAncestor(TreeNode root, TreeNode p, TreeNode q) {
//check null
if(root==null)
return root;
//check if anyone is the root
if(p==root)
return root;
if(q==root)
return root;
//check if anyone in the left
if(IsContainReal(root.left, p) && IsContainReal(root.left, q))
return LowestCommonAncestor(root.left, p, q);
//check if anyone in the right
if(IsContainReal(root.right, p) && IsContainReal(root.right, q))
return LowestCommonAncestor(root.right, p, q);
return root;
}
public bool IsContainReal(TreeNode root, TreeNode target)
{
if(root==null)
return false;
if(root==target)
return true;
return IsContain(root.left,target) || IsContain(root.right,target);
}
public bool IsContain(TreeNode root, TreeNode target)
{
if(root==null)
return false;
if(!memarizeSearchResult.ContainsKey(root))
{
var newDic=new Dictionary<TreeNode,bool>();
memarizeSearchResult[root][target] = IsContainReal(root,target);
}
if(!memarizeSearchResult[root].ContainsKey(target))
{ | 349 | 1,471 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2017-51 | latest | en | 0.250275 |
http://www.ck12.org/concept/Function-Notation/?difficulty=at%20grade | 1,435,620,966,000,000,000 | text/html | crawl-data/CC-MAIN-2015-27/segments/1435375090887.26/warc/CC-MAIN-20150627031810-00239-ip-10-179-60-89.ec2.internal.warc.gz | 405,306,356 | 15,195 | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# Function Notation
## Explore f(x) notation for functions
Levels are CK-12's student achievement levels.
Basic Students matched to this level have a partial mastery of prerequisite knowledge and skills fundamental for proficient work.
At Grade (Proficient) Students matched to this level have demonstrated competency over challenging subject matter, including subject matter knowledge, application of such knowledge to real-world situations, and analytical skills appropriate to subject matter.
Advanced Students matched to this level are ready for material that requires superior performance and mastery.
## Function Notation
Learn how to use function notation.
4
• Practice
0%
## Function Notation Practice
4
• Critical Thinking
## Function Notation Discussion Questions
A list of student-submitted discussion questions for Function Notation.
0
## Function Notation Pre and Post Read
To organize ideas, increase comprehension, synthesize learning, demonstrate understanding of key concepts, and reinforce vocabulary using a Quickwrite.
0
• Real World Application
## Nike
Students represent the relationship of Nike's profit from producing shoes as functions.
3
• Study Guide | 235 | 1,254 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2015-27 | latest | en | 0.822741 |
https://cityoftehran.com/dr-martens-sneakers-best-salebrboys-dr-martens-rozarya-sneakers-best-sale-black-t-canvas-dr-martens-boys-sneakers-best-sale-dr-martens-uk-store-p-53.html | 1,675,160,752,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499857.57/warc/CC-MAIN-20230131091122-20230131121122-00211.warc.gz | 184,925,220 | 5,020 | # App that solves math problems
There are a lot of App that solves math problems that are available online. Our website will give you answers to homework.
## The Best App that solves math problems
In this blog post, we will be discussing about App that solves math problems. Solving domain and range can be tricky, but there are a few helpful tips that can make the process easier. First, it is important to remember that the domain is the set of all values for which a function produces a result, while the range is the set of all values that the function can produce. In other words, the domain is the inputs and the range is the outputs. To solve for either the domain or range, begin by identifying all of the possible values that could be inputted or outputted. Then, use this information to determine which values are not possible given the constraints of the function. For example, if a function can only produce positive values, then any negative values in the input would be excluded from the domain. Solving domain and range can be challenging, but with a little practice it will become easier and more intuitive.
Composition of functions solver is a mathematical tool that allows two or more functions to be composed into a single function. The process of composition is relatively simple: the output of one function is used as the input for the next function in the sequence. Composition of functions solver can be used to solve problems in a variety of fields, including physics, engineering, and economics. In each case, the goal is to find a way to simplify a complex problem by breaking it down into smaller, more manageable pieces. Composition of functions solver is an essential tool for anyone working with complex systems.
There are a few different ways to solve equations with e. The first way is to use the definition of e. e is equal to the limit as n goes to infinity of (1+1/n)^n. This can be rewritten as (1/n)((1+1/n)^n). So, if you have an equation that has e in it, you can divide both sides by 1/n and then take the nth root of both sides. This will usually give you a numerical answer that is very close to the actual value of e. Another way to solve equations with e is to use a graphing calculator. If you graph the equation, the point where the graph crosses the x-axis will be the value of e that solves the equation. You can also use online calculators or software programs to solve equations with e. These methods will usually give you a more accurate answer than using the definition of e.
When it comes to solving math problems, there is no one-size-fits-all solution. The best approach depends on the nature of the problem, as well as the skills and knowledge of the person solving it. However, there are a few general tips that can help make solving math problems easier. First, it is important to take the time to understand the problem. What is being asked for? What information is given? Once you have a clear understanding of the problem, you can begin to consider different approaches. Sometimes, visual aids such as charts or diagrams can be helpful in solving math problems. Other times, it may be helpful to break the problem down into smaller steps. And sometimes, simply taking a step back and looking at the problem from a different perspective can make all the difference. There is no single right way to solve math problems. However, by taking the time to understand the problem and trying different approaches, it is usually possible to find a solution that works.
## Math checker you can trust
I was pleasantly surprised at its handwriting recognition. Has solved some hard quadratic, simultaneous and linear equations. Didn't like complex solution to quadratic (I.e., finding a root of a negative number) Tested it with a simple yet viral puzzle which sets math camps afire because it is solved 2 ways both CORRECT and WRONG simultaneously depending on how x(a) is interpreted. Otherwise, this is a fantastic app which students should find extremely helpful. Well done!
Kayleigh Griffin
Great app! It can solve most math questions you scan and the answers are almost always correct. It also gives great step by step tutorials on how to solve the question. It's a really great app and I highly recommend it if you are struggling on a math question.
Gisselle Martin
Algebra solver inequalities Perfect square solver Mathematics app for iphone Difference of quotient solver Free math solving apps | 902 | 4,458 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.4375 | 4 | CC-MAIN-2023-06 | latest | en | 0.936022 |
http://bendwavy.org/klitzing/dimensions/polyhedra-neu.htm | 1,369,430,686,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368705069221/warc/CC-MAIN-20130516115109-00081-ip-10-60-113-184.ec2.internal.warc.gz | 24,119,478 | 7,224 | ```
----
3D
----
```
This page is available sorted by point-group symmetry (below) or by complexity or by similarity.
This dimension is accessible for pictures. Thus most of the following uniform polyhedra pages provide such. Further all pictures bear links to VRML models.
For most of those, which are derivable as any kind of snubs, pictures (and VRMLs) on their derivation are provided in addition. There the color coding is: red are the elements to be alternated, yellow are the faceting faces underneath (sefa( . )), the starting figure is given as wire frame. Those figures in general do not show uniform representants, it is the starting figure which is chosen to be uniform.
Especially the Grünbaumians can be best understood, if the (abstract) incidence matrices of those degenerate polyhedra with (geometrical) complete coincidence of some elements are not investigated individually, but independently of the special symmetry, i.e. by considering simultanuously the general Schwarz triangle o-p-o-q-o-r-*a and deriving therefrom the individual cases. (For the notation of virtual nodes like *a see here.)
Just as for the Grünbaumians, especially the holosnubs with ...β3... elements are better understood from the consideration of ...βn... with general odd n.
*) Some of the partial snubs which contain both s and x nodes, respectively partial holosnubs which contain both β and x nodes, do not have a uniform representation. Those are only possible with different edge lengths, e.g. as mere alternated faceting from their uniform starting figure. This latter representation surely is always possible.
### Tetrahedral Symmetries (up)
o3o3o (convex) o3/2o3o3*a (µ=2) o3/2o3o (µ=3) quasiregulars ```x3o3o - tet o3x3o - oct ``` ```x3/2o3o3*a - 2tet (?) o3/2o3x3*a - 2tet (?) ``` ```x3/2o3o - tet o3/2x3o - oct o3/2o3x - tet ``` otherWythoffians ```x3x3o - tut x3o3x - co x3x3x - toe a3b3c - (general variant) ``` ```x3/2x3o3*a - 2oct (?) x3/2o3x3*a - oho x3/2x3x3*a - 2tut (?) ``` ```x3/2x3o - 3tet (?) x3/2o3x - 2thah (?) o3/2x3x - tut x3/2x3x - cho+4{6/2} (?) ``` (partial)snubs andholosnubs ```β3o3o - 2tet (?) o3β3o - oct+6{4} (?) β3x3o - 2oct (?) x3β3o - (?) *) β3β3o - 2oct+8{3} (?) β3o3x - oho β3o3β - (?) *) β3x3x - 2tut (?) x3β3x - 2co (?) β3β3x - 2co (?) β3x3β - (?) *) s3s3s - ike ``` ```β3/2o3o3*a - 4tet (?) o3/2o3β3*a - 4tet (?) β3/2x3o3*a - 2oct (?) β3/2β3o3*a - 2oct+12{4} (?) β3/2o3x3*a - oho+8{3} (?) x3/2o3β3*a - 6tet (?) β3/2o3β3*a - 2oct+12{4} (?) β3/2x3x3*a - oho+8{3} (?) x3/2x3β3*a - 4oct (?) β3/2β3x3*a - (?) *) β3/2x3β3*a - 4oct (?) s3/2s3s3*a - 2oct+8{3} (?) ``` ```β3/2o3o - 2tet (?) o3/2β3o - oct+6{4} (?) o3/2o3β - 2tet (?) β3/2x3o - 2tet (?) x3/2β3o - 2tet (?) β3/2β3o - 6tet (?) β3/2o3x - 2oct (?) x3/2o3β - 2oct (?) β3/2o3β - (?) *) o3/2β3x - (?) *) o3/2x3β - 2oct (?) o3/2β3β - 2oct+8{3} (?) β3/2x3x - oho x3/2β3x - (?) *) x3/2x3β - 6tet (?) β3/2β3x - 4thah (?) β3/2x3β - 6tet (?) ... ``` o3/2o3/2o (µ=5) o3/2o3/2o3/2*a (µ=6) quasiregulars ```x3/2o3/2o - tet o3/2x3/2o - oct ``` ```x3/2o3/2o3/2*a - 2tet (?) ``` otherWythoffians ```x3/2x3/2o - 3tet (?) x3/2o3/2x - co x3/2x3/2x - 2oct+6{4} (?) ``` ```x3/2x3/2o3/2*a - 2oct (?) x3/2x3/2x3/2*a - 6tet (?) ``` (partial)snubs andholosnubs ```β3/2o3/2o - 2tet (?) o3/2β3/2o - oct+6{4} (?) s3/2s3/2s - gike ... ``` ```β3/2o3/2o3/2*a - 4tet (?) β3/2x3/2o3/2*a - 2oct (?) β3/2β3/2o3/2*a - 2oct+12{4} (?) ... ```
### Octahedral Symmetries (up)
o3o4o (convex) o3/2o4o4*a (µ=2) o4/3o3o4*a (µ=4) quasiregulars ```x3o4o - oct o3x4o - co o3o4x - cube ``` ```x3/2o4o4*a - oct+6{4} (?) o3/2o4x4*a - 2cube (?) ``` ```x4/3o3o4*a - 2cube (?) o4/3x3o4*a - oct+6{4} (?) o4/3o3x4*a - oct+6{4} (?) ``` otherWythoffians ```x3x4o - toe x3o4x - sirco o3x4x - tic x3x4x - girco a3b4c - (general variant) ``` ```x3/2x4o4*a - 2co (?) x3/2o4x4*a - socco x3/2x4x4*a - 2tic (?) ``` ```x4/3x3o4*a - gocco x4/3o3x4*a - socco o4/3x3x4*a - 2cho (?) x4/3x3x4*a - cotco ``` (partial)snubs andholosnubs ```β3o4o - oct+6{4} (?) o3β4o - (?) *) o3o4s - tet β3x4o - 2co (?) x3β4o - (?) *) s3s4o - ike β3o4x - socco x3o4s - tut β3o4β - (?) *) o3β4x - (?) *) o3x4s - co o3β4β - 2co+16{3} (?) β3x4x - 2tic (?) x3β4x - 2sirco (?) x3x4s - toe s3s4x - sirco β3x4β - (?) *) x3β4β - 2sirco (?) s3s4s - snic ``` ```... ``` ```... ``` o3/2o4o (µ=5) o4/3o3o (µ=7) o4/3o3/2o (µ=11) quasiregulars ```x3/2o4o - oct o3/2x4o - co o3/2o4x - cube ``` ```x4/3o3o - cube o4/3x3o - co o4/3o3x - oct ``` ```x4/3o3/2o - cube o4/3x3/2o - co o4/3o3/2x - oct ``` otherWythoffians ```x3/2x4o - 2oct+6{4} (?) x3/2o4x - querco o3/2x4x - tic x3/2x4x - sroh+8{6/2} (?) ``` ```x4/3x3o - quith x4/3o3x - querco o4/3x3x - toe x4/3x3x - quitco ``` ```x4/3x3/2o - quith x4/3o3/2x - sirco o4/3x3/2x - 2oct+6{4} (?) x4/3x3/2x - groh+8{6/2} (?) ``` (partial)snubs andholosnubs ```β3/2o4o - oct+6{4} (?) s3/2s4o - gike ... ``` ```o4/3o3β - oct+6{4} (?) o4/3s3s - ike ... ``` ```o4/3o3/2β - oct+6{4} (?) o4/3s3/2s - gike ... ``` o4/3o4/3o3/2*a (µ=14) quasiregulars ```x4/3o4/3o3/2*a - oct+6{4} (?) o4/3x4/3o3/2*a - 2cube (?) ``` otherWythoffians ```x4/3x4/3o3/2*a - gocco x4/3o4/3x3/2*a - 2co (?) x4/3x4/3x3/2*a - 2quith (?) ``` (partial)snubs andholosnubs ```... ```
### Icosahedral Symmetries (up)
o3o5o (convex) o5/2o3o3*a (µ=2) o3/2o5o5*a (µ=2) quasiregulars ```x3o5o - ike o3x5o - id o3o5x - doe ``` ```x5/2o3o3*a - sidtid o5/2o3x3*a - 2ike (?) ``` ```x3/2o5o5*a - cid o3/2o5x5*a - 2doe (?) ``` otherWythoffians ```x3x5o - ti x3o5x - srid o3x5x - tid x3x5x - grid a3b5c - (general variant) ``` ```x5/2x3o3*a - 2id (?) x5/2o3x3*a - siid x5/2x3x3*a - 2ti (?) ``` ```x3/2x5o5*a - 2id (?) x3/2o5x5*a - saddid x3/2x5x5*a - 2tid (?) ``` (partial)snubs andholosnubs ```β3o5o - cid o3β5o - (?) *) o3o5β - sidtid β3x5o - 2id (?) x3β5o - (?) *) β3β5o - seside β3o5x - saddid x3o5β - siid β3o5β - (?) *) o3β5x - (?) *) o3x5β - 2id (?) o3β5β - 2id+40{3} (?) β3x5x - 2tid (?) x3β5x - 2srid (?) x3x5β - 2ti (?) β3β5x - 2srid (?) β3x5β - (?) *) x3β5β - 2srid (?) s3s5s - snid ``` ```... s5/2s3s3*a - seside ``` ```... ``` o5/2o5o (µ=3) o5/3o3o5*a (µ=4) o5/2o5/2o5/2*a (µ=6) quasiregulars ```x5/2o5o - sissid o5/2x5o - did o5/2o5x - gad ``` ```x5/3o3o5*a - ditdid o5/3x3o5*a - gacid o5/3o3x5*a - cid ``` ```x5/2o5/2o5/2*a - 2sissid (?) ``` otherWythoffians ```x5/2x5o - 3doe (?) x5/2o5x - raded o5/2x5x - tigid x5/2x5x - sird+12{10/2} (?) ``` ```x5/3x3o5*a - gidditdid x5/3o3x5*a - sidditdid o5/3x3x5*a - ided x5/3x3x5*a - idtid ``` ```x5/2x5/2o5/2*a - 2did (?) x5/2x5/2x5/2*a - 6doe (?) ``` (partial)snubs andholosnubs ```... s5/2s5s - siddid ``` ```... s5/3s3s5*a - sided ``` ```... ``` o3/2o3o5*a (µ=6) o5/4o5o5*a (µ=6) o5/2o3o (µ=7) quasiregulars ```x3/2o3o5*a - gidtid o3/2x3o5*a - 2gike (?) o3/2o3x5*a - gidtid ``` ```x5/4o5o5*a - 2gad (?) o5/4o5x5*a - 2gad (?) ``` ```x5/2o3o - gissid o5/2x3o - gid o5/2o3x - gike ``` otherWythoffians ```x3/2x3o5*a - 3ike+gad (?) x3/2o3x5*a - 2seihid (?) o3/2x3x5*a - giid x3/2x3x5*a - siddy+20{6/2} (?) ``` ```x5/4x5o5*a - 2did (?) x5/4o5x5*a - 2sidhid (?) x5/4x5x5*a - 2tigid (?) ``` ```x5/2x3o - 2gad+ike (?) x5/2o3x - sicdatrid o5/2x3x - tiggy x5/2x3x - ri+12{10/2} (?) ``` (partial)snubs andholosnubs ```... s3/2s3s5*a - 5ike+gad (?) ``` ```... ``` ```... s5/2s3s - gosid ``` o3/2o5/2o5*a (µ=8) o5/3o5o (µ=9) o5/4o3o5*a (µ=10) quasiregulars ```x3/2o5/2o5*a - cid o3/2x5/2o5*a - gacid o3/2o5/2x5*a - ditdid ``` ```x5/3o5o - sissid o5/3x5o - did o5/3o5x - gad ``` ```x5/4o3o5*a - 2doe (?) o5/4x3o5*a - cid o5/4o3x5*a - cid ``` otherWythoffians ```x3/2x5/2o5*a - sidtid+gidtid (?) x3/2o5/2x5*a - sidditdid o3/2x5/2x5*a - ike+3gad (?) x3/2x5/2x5*a - id+seihid+sidhid (?) ``` ```x5/3x5o - quit sissid x5/3o5x - cadditradid o5/3x5x - tigid x5/3x5x - quitdid ``` ```x5/4x3o5*a - sidtid+ditdid (?) x5/4o3x5*a - saddid o5/4x3x5*a - 2gidhei (?) x5/4x3x5*a - siddy+12{10/4} (?) ``` (partial)snubs andholosnubs ```... ``` ```... s5/3s5s - isdid ``` ```... ``` o5/3o5/2o3*a (µ=10) o3/2o5o (µ=11) o5/3o3o (µ=13) quasiregulars ```x5/3o5/2o3*a - gacid o5/3x5/2o3*a - 2gissid (?) o5/3o5/2x3*a - gacid ``` ```x3/2o5o - ike o3/2x5o - id o3/2o5x - doe ``` ```x5/3o3o - gissid o5/3x3o - gid o5/3o3x - gike ``` otherWythoffians ```x5/3x5/2o3*a - gaddid x5/3o5/2x3*a - 2sidhei (?) o5/3x5/2x3*a - ditdid+gidtid (?) x5/3x5/2x3*a - giddy+12{10/2} (?) ``` ```x3/2x5o - 2ike+gad (?) x3/2o5x - gicdatrid o3/2x5x - tid x3/2x5x - sird+20{6/2} (?) ``` ```x5/3x3o - quit gissid x5/3o3x - qrid o5/3x3x - tiggy x5/3x3x - gaquatid ``` (partial)snubs andholosnubs ```... s5/3s5/2s3*a - gisdid ``` ```... β3/2β5o - sirsid s3/2s5s - 4ike+gad (?) ``` ```... s5/3s3s - gisid ``` o5/4o3o3*a (µ=14) o3/2o5/2o5/2*a (µ=14) o5/4o5/2o3*a (µ=16) quasiregulars ```x5/4o3o3*a - gidtid o5/4o3x3*a - 2gike (?) ``` ```x3/2o5/2o5/2*a - gacid o3/2o5/2x5/2*a - 2gissid (?) ``` ```x5/4o5/2o3*a - cid o5/4x5/2o3*a - ditdid o5/4o5/2x3*a - gacid ``` otherWythoffians ```x5/4x3o3*a - 2gid (?) x5/4o3x3*a - giid x5/4x3x3*a - 2tiggy (?) ``` ```x3/2x5/2o5/2*a - 2gid (?) x3/2o5/2x5/2*a - ditdid+gidtid (?) x3/2x5/2x5/2*a - 2ike+4gad (?) ``` ```x5/4x5/2o3*a - 3sissid+gike (?) x5/4o5/2x3*a - ided o5/4x5/2x3*a - ike+3gad (?) x5/4x5/2x3*a - did+sidhei+gidhei (?) ``` (partial)snubs andholosnubs ```... ``` ```... ``` ```... ``` o3/2o5/2o (µ=17) o3/2o5/3o3*a (µ=18) o5/3o5/3o5/2*a (µ=18) quasiregulars ```x3/2o5/2o - gike o3/2x5/2o - gid o3/2o5/2x - gissid ``` ```x3/2o5/3o3*a - 2ike (?) o3/2x5/3o3*a - sidtid o3/2o5/3x3*a - sidtid ``` ```x5/3o5/3o5/2*a - 2sissid (?) o5/3x5/3o5/2*a - 2sissid (?) ``` otherWythoffians ```x3/2x5/2o - 2gike+sissid (?) x3/2o5/2x - qrid o3/2x5/2x - 2gad+ike (?) x3/2x5/2x - 2gidtid+5cube (?) ``` ```x3/2x5/3o3*a - sissid+3gike (?) x3/2o5/3x3*a - siid o3/2x5/3x3*a - 2geihid (?) x3/2x5/3x3*a - giddy+20{6/2} (?) ``` ```x5/3x5/3o5/2*a - 2gidhid (?) x5/3o5/3x5/2*a - 2did (?) x5/3x5/3x5/2*a - 2quitsissid (?) ``` (partial)snubs andholosnubs ```... ``` ```... ``` ```... ``` o5/4o3o (µ=19) o5/4o5/2o (µ=21) o3/2o3/2o5/2*a (µ=22) quasiregulars ```x5/4o3o - doe o5/4x3o - id o5/4o3x - ike ``` ```x5/4o5/2o - gad o5/4x5/2o - did o5/4o5/2x - sissid ``` ```x3/2o3/2o5/2*a - sidtid o3/2x3/2o5/2*a - 2ike (?) ``` otherWythoffians ```x5/4x3o - 2sissid+gike (?) x5/4o3x - gicdatrid o5/4x3x - ti x5/4x3x - ri+12{10/4} (?) ``` ```x5/4x5/2o - 3gissid (?) x5/4o5/2x - cadditradid o5/4x5/2x - 3doe (?) x5/4x5/2x - 2ditdid+5cube (?) ``` ```x3/2x3/2o5/2*a - sissid+3gike (?) x3/2o3/2x5/2*a - 2id (?) x3/2x3/2x5/2*a - 4ike+2gad (?) ``` (partial)snubs andholosnubs ```... ``` ```... ``` ```... s3/2s3/2s5/2*a - sirsid ``` o3/2o5/3o (µ=23) o3/2o5/3o5/3*a (µ=26) o5/4o5/3o (µ=27) quasiregulars ```x3/2o5/3o - gike o3/2x5/3o - gid o3/2o5/3x - gissid ``` ```x3/2o5/3o5/3*a - gacid o3/2o5/3x5/3*a - 2gissid (?) ``` ```x5/4o5/3o - gad o5/4x5/3o - did o5/4o5/3x - sissid ``` otherWythoffians ```x3/2x5/3o - 2gike+sissid (?) x3/2o5/3x - sicdatrid o3/2x5/3x - quit gissid x3/2x5/3x - gird+20{6/2} (?) ``` ```x3/2x5/3o5/3*a - 2gid (?) x3/2o5/3x5/3*a - gaddid x3/2x5/3x5/3*a - 2quitgissid ``` ```x5/4x5/3o - 3gissid (?) x5/4o5/3x - raded o5/4x5/3x - quit sissid x5/4x5/3x - gird+12{10/4} (?) ``` (partial)snubs andholosnubs ```... s3/2s5/3s - girsid ``` ```... ``` ```... ``` o5/4o3/2o (µ=29) o5/4o3/2o5/3*a (µ=32) o5/4o3/2o3/2*a (µ=34) quasiregulars ```x5/4o3/2o - doe o5/4x3/2o - id o5/4o3/2x - ike ``` ```x5/4o3/2o5/3*a - ditdid o5/4x3/2o5/3*a - cid o5/4o3/2x5/3*a - gacid ``` ```x5/4o3/2o3/2*a - gidtid o5/4o3/2x3/2*a - 2gike (?) ``` otherWythoffians ```x5/4x3/2o - 2sissid+gike (?) x5/4o3/2x - srid o5/4x3/2x - 2ike+gad (?) x5/4x3/2x - 2sidtid+5cube (?) ``` ```x5/4x3/2o5/3*a - 3sissid+gike (?) x5/4o3/2x5/3*a - gidditdid o5/4x3/2x5/3*a - sidtid+gidtid (?) x5/4x3/2x5/3*a - gid+geihid+gidhid (?) ``` ```x5/4x3/2o3/2*a - 2gid (?) x5/4o3/2x3/2*a - 3ike+gad (?) x5/4x3/2x3/2*a - 2sissid+4gike (?) ``` (partial)snubs andholosnubs ```... ``` ```... ``` ```... ``` o5/4o5/4o3/2*a (µ=38) o5/4o5/4o5/4*a (µ=42) quasiregulars ```x5/4o5/4o3/2*a - cid o5/4x5/4o3/2*a - 2doe (?) ``` ```x5/4o5/4o5/4*a - 2gad (?) ``` otherWythoffians ```x5/4x5/4o3/2*a - sidtid+ditdid (?) x5/4o5/4x3/2*a - 2id (?) x5/4x5/4x3/2*a - 4sissid+2gike (?) ``` ```x5/4x5/4o5/4*a - 2did (?) x5/4x5/4x5/4*a - 6gissid (?) ``` (partial)snubs andholosnubs ```... s5/4s5/4s3/2*a - 4ike+2gad (?) ``` ```... ```
### Prismatic Symmetries (up)
o ono (convex) o on/do (µ=d) o o o (convex) products ofquasiregulars ```x x3o - trip x x4o - cube x x5o - pip x x6o - hip x x8o - op x x10o - dip x x12o - twip x xno - n-p ``` ```x x5/2o - stip x x8/3o - stop x x10/3o - stiddip x xn/do - n/d-p x xn/2o - n/2-p ``` ```x x x - cube ``` otherWythoffians ```x x3x - hip x x4x - op x x5x - dip x x6x - twip x xnx - 2n-p ``` ```x x4/3x - stop x x5/3x - stiddip x xn/dx - 2n/d-p x xn/2x - (2n)/2-p ``` (partial)snubs andholosnubs ```s2s3s - oct s2s4s - squap s2xno - {n} s2onx - {n} s2xnx - {2n} x2βnx - 2n/2-p β2βno - n/2-ap β2βnx - 2n/2-p x2sns - n-p s2sns - n-ap x2s2no - n-p x2s2nx - 2n-p s2s2no - n-ap s2s2nx - 2n-p ``` ```s2sn/ds - n/d-ap s2s2n/do - n/d-ap s2s2n/dx - 2n/d-p ``` ```s2s2s - tet ```
### other non-kaleidoscopical uniform polyhedra (up)
hemi reduced others ```hemi( x3/2o3x ) - thah hemi( o4/3x3x4*a ) - cho hemi( ? ) - oho hemi( x3/2o3x5*a ) - seihid hemi( x5/4o5x5*a ) - sidhid hemi( o5/4x3x5*a ) - gidhei hemi( x5/3o5/2x3*a ) - sidhei hemi( o3/2x5/3x3*a ) - geihid hemi( x5/3x5/3o5/2*a ) - gidhid ``` ```reduced( x3/2x3x , by 4{6/2} ) - cho reduced( x3/2x4x , by 8{6/2} ) - sroh reduced( x4/3x3/2x , by 8{6/2} ) - groh reduced( x5/2x5x , by 12{10/2} ) - sird reduced( x3/2x3x5*a , by 20{6/2} ) - siddy reduced( x5/2x3x , by 12{10/2} ) - ri reduced( x3/2x5/2x5*a , by id ) - seihid & sidhid reduced( x5/4x3x5*a , by 12{10/4} ) - siddy reduced( x5/3x5/2x3*a , by 12{10/2} ) - giddy reduced( x3/2x5x , by 20{6/2} ) - sird reduced( x5/4x5/2x3*a , by did ) - gidhei & sidhei reduced( x3/2x5/3x3*a , by 20{6/2} ) - giddy reduced( x5/4x3x , by 12{10/4} ) - ri reduced( x3/2x5/3x , by 20{6/2} ) - gird reduced( x5/4x5/3x , by 12{10/4} ) - gird reduced( x5/4x3/2x5/3*a , by gid ) - geihid & gidhid reduced( xx3/2ox&#x , by {6/2} ) - thah ``` ```gidrid gidisdrid ```
© 2004-2013 top of page | 7,621 | 14,152 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2013-20 | latest | en | 0.881649 |
http://mrorr-isageek.com/ | 1,419,189,596,000,000,000 | text/html | crawl-data/CC-MAIN-2014-52/segments/1418802772265.125/warc/CC-MAIN-20141217075252-00005-ip-10-231-17-201.ec2.internal.warc.gz | 195,273,275 | 26,685 | # Logarithmic Warm Up
Our goal in Advanced Functions today was to graph y = log(x) and transformations of y = log(x). Here was what we did as a warm up/minds-on. Everyone started with a whiteboard and a device of some sort (Smartphones, iPads, tablets—I have access to a set of iPads for anyone without a device. This class being a grade 12 class….just about everyone has their own device).
I gave everyone in my class a number. Some got integers 1-20 and some got fractions 1/4 through 1.
My instructions:
1. Take your number, n, and find log(n). Write your number and log(n) as an ordered pair! (n, log(n))
I started the PearDeck presentation which showed them this slide……and gave them all a movable point.
2. Move your point to the location of the ordered pair you have! So my students started moving their points around and on the projector screen we can see everyone’s points all at once in real time! So we are basically watching the creation of y = log(x).
Sorry for the poor quality of pic…it was on the fly
You could see the looks on their faces as the graph was being created….pretty cool
From here we took a note on the properties of the function….then kept using PearDeck to analyze transformations of y = log(x). We saw Graphs then –> wrote equations and then saw equations –> drew graphs. We could do everything right in PearDeck so we could see all of our answers all of the time! PearDeck works through your Google account…..give it a try!
I thought it was pr
# Projects from 2013-2014, iTunesU, iBooks
I feel like since I started using Twitter for Pd and following the #MTBos my learning curve has been steep! As a result, when I look back at resources I have created I find myself wishing that I had done it “this way” or “that way” or used a different task here or there.
This is no problem for creating lessons, activities, tasks to use with MY students because I can always modify, change, manipulate!
My regrets show up when I’m involved in a project that gets published or shared out beyond my reach and afterwards my thinking has changed by seeing a great blog post, or a new activity, or by just having a discussion with a colleague. I wish I had all the time in the world to keep everything “up to date”
Our blogs, Google Drive Folders, Dropboxes are easily update-able and where are current lessons are!
Here are a few of those projects that I hit publish and find it hard to keep up to date: Each of these has lots of stuff I’m proud of, and some stuff I wish I could find the time to “update”
ITunesU Course – MPM1D (created April 2014)
Last spring I created an iTunesU course for my school board (Lambton-Kent).
Purpose: To share some digital resources I have used, or, are using in my grade 9 Principles of mathematics course.
https://itunes.apple.com/ca/course/principles-mathematics-public/id946920145
iBook – Measurement (created July 2014)
Whatcha Thinking – independent use of 3 Act Math Tasks for MFM1P
iBook – Linear Relations (created July 2013)
# Popcorn Pandemonium
My afternoon grade 9 applied class (as a group) is very outspoken, loud, and restless (maybe it’s because it’s the afternoon and they have been sitting at desks all day). They have been a challenge to keep on task. So….I am trying to find opportunities for them to be outspoken, loud, and restless.
A few weeks ago I came across this post by John Berray. Using/eating marshmallows to compare rates of change. I loved his idea of “experiencing rate of change” I decided to re-purpose his lesson to meet our goal of—> “I can solve a linear systems of equations by graphing.” I also took his recommendation of using popcorn instead of marshmallows…..and it paid off!!
Here is the low down…. we start the “Math Dial” off low.
ME: OK you are going to have a good o’ fashion popcorn eating contest!
Here are a few from math tweeps
here are a few questions we can address with this problem.
• When will Tim and Don eat the same amount as Jon?
• Who will eat the most when the minute is up?
• Will there be a time when Tim and Don eat the exact same amount?
• When would Don eat more than Tim?
ME: Ok lets figure out who will eat the most in the 1 minute. But I want to recreate the video with you guys.
So I made a giant bowl of popcorn. (Don’t have time to make enough popcorn? — have kids give high fives to a timer instead)
Arrange groups of 2 or 3 and everybody grabbed some popcorn to start!
Round 1:
In each group kids are to choose who to mimic, Jon, Tim, or Don. They are to eat just like them! Allow them to ask about how fast each person is eating….or how much did each start with, etc.
Show Act 2 to answer those questions:
Tell them to get their timers ready….because they will eat just like one of those guys. Ready…..all you Tims and Jons eat your starting amount … Set….Go!
Start the timers and eat!
Question 1:
After they are finished, have them work out on their whiteboards who would eat the most in a minute.
Question 2:
When would Tim & Don eat the same as Jon if ever? (Great potential here for integer solutions talk).
Question 3:
During the minute, at anytime did Tim and Don eat the same?
If there was no time limit find when Tim & Don would eat the same?
Used this handout so they could create tables of values. Had them graph in Desmos!
The awesome thing was that my students were desperately trying to find the equations to match their graphs….they didn’t want to plot all the points. I visited each group helping them find the equations if needed. Once the equations were in desmos they knew where to look.
Act 3 – The reveal of who ate the most in a minute
Round 2: Do it all over again with new eating patterns!
Here are two possible eating pattern cards to give out:
Students who finished early worked on our Crazy Taxi vs. a new Insane Cab
(@mathletepearce has a nice write up on using the Crazy taxi problem in class.)
Next day! Solving Multi-step equations…..will solve this systems of equations algebraically.
Here’s how I taught students how to solve trigonometric equations in our grade 12 advanced functions class.
Started with this Ferris wheel problem
What has been working well is starting our “math” at a very low level…..like on a dial…..then we slowly turn the dial up….adding more “math” in. Read more about the Math Dial from a comment on Dan Meyer’s blog here.
Starting with this video the math on the “math dial” is very low.
I asked: What questions do you have after seeing this….
How fast is it spinning?
What’s the period?
Where will the red dot be after 3 min?
And that last one is the question we studied.
Act 2:
From 101qs.com
Almost all kids solved this problem using proportions! They kept the dial in the low position still! They realized that it takes 5 seconds to travel from dot to dot. Therefore it takes 40 seconds to go all the way around. They divide 3 minutes up into 40 second sections and get 4.5 rotations. The dot will end at the top of the Wheel!! But the Trigonometry in me was screaming to get out……I asked, “Did anyone create a trig equation to model the height?” — cue crickets!
So we cranked the math dial up a tad!
I said:
When I go on a ferris wheel I always look for my house.” We talked about how high that might be in relation to Dan’s problem….we settled on about maybe 40 feet.
My question: How long will it take to get to that height?
Guesses? Will it be a nice number? No? Why not?
Crank it up a bit more …
Let’s create an equation for the height in terms of time (we had already learned how to do this and it was no problem for the class) .
Now, to solve our question we have to solve this equation!
Student: That looks super hard!
Me: It does doesn’t it!
Let’s make that our goal!
We don’t want the math dial going up too quickly!
Gotta keep the math dial low for a bit more…
We solve this as a class, then another, and another, slowly building up our skills; slowly bringing the dial up. We stop at the end of the class. I assign a few more like the ones above. “Let’s get good at these so we can do the super hard one… Practice these for homework….”
Next day:
We take up the assigned questions then get back on track! We then solve these:
We have a discussion on how many solutions there are here… and plop down a graphical solution in Desmos
The math dial is getting up there…
Me: “Are you ready to try the big one?”
We do it! And everyone is into it….they have been waiting two days to see the answer! And the dial is pretty far up there!
After:
One student says: “That was pretty awesome! ”
That was my highlight of the day! Best compliment for a teacher!
We then show the graphical solution in Desmos.
Oh…..and we started class playing Pictionary (It’s our Wednesday thing) there was a tie and we have a good o’l match of Rock, Paper, Scissors to declare the winner. It was Intense!!!
# Many Many Volumes
In our senior math classes (advanced functions & calculus) we come across a problem like this….
I really like these problems, they have great potential but not really in this form. Let’s jazz it up and spend an entire class with this
Them:
What size is that rectangle?
Why are the corners cut?
Is volume always the same?
Etc,
My question:
What size of that square do we cut out so the box has the biggest volume?
Play the video again and have them yell out when they think the box has the largest volume.
Have them guess
What is too small?
What is too large?
Have them take their guess for the size of the corner and find the volume of the box
Draw a picture of the “card board” label the dimensions.
Draw the squares to cut out. Optional (Cut them out) make the boxes.
What’s the new length?
What’s the new width?
What’s the height?
What’s the volume?
Is this the max?
How can we check?
Have them do another? And another.
Have them come up to your computer and enter their height and volume in the Desmos page for each box.
Now, let’s generalize!
This time let your guess be x and find an expression for the volume.
What’s the new width? Take 8 and subtract twice your guess. (8-2x) Now the length? What is the height now??
Put that expression into Desmos and let them see the function, let them point to the maximum.
For calculus: have them find the maximum using derivatives!
Show them this video to check their guesses.
From here we can solve problems like our original textbook question. The kids are invested now and they are ready to use the equation to find the value of x where the volume is say 24 cubic units.
Further reading: Jonathan Newman’s volume of a box Activity
Credits: Algebra in motion for the Geometers Sketchpad file. Dan Meyer – this lesson mimics his Circle Square lesson.
UPDATE [Nov. 27, 2014]
Luke Walsh created a Desmos Sketch that seems super useful!
# Filling it up!
In our grade 9 applied class we are finishing off linear relations and moving into solving equations. I want an activity that is hands-on, engaging, and shows a purpose to solving linear equations.
Here is some thoughts on an activity I want to try. Let me know what you think. Any feedback would be greatly appreciated.
Here it is: Filling it up!!
Show them this picture….
Let them wonder, let them ask what that thing in the pitcher is.
ME:
“How many would be too much?”
“How many would be not enough?”
“How many is just right?”
Have them record the guess. “We’ll compare our answer to our guess”
Next,
ME: Let’s find out how many.
Organize them into groups of 3.
ME: What are we going to need?
We’ll need volume of the pitcher, volume of the cup, and volume of the weight.
Have discussion on:
What shape is the cup? …..is it more like a cylinder or a cone? Which is it closest to? What formula for volume will you use? Will you be right?
What shape is the pitcher? What shape is the weight?
Choose 1 member of you group to find the volume of the cup; choose 1 member for volume of the pitcher; choose 1 member to find volume of the weight.
Have the items around the room like stations:
Each member will find the volume of their object and bring it back to the group.
Allow the students to work
Here are some scaffolding questions I can use (Please feel free to give me some more)
• What’s changing as you fill up the pitcher?
• What volume is left after the weight?
Here is a possible solution….
My idea is this could be great context for introducing solving equations using opposite operations! Use their technique and show how the volume grows as the cups increase. Use Desmos and relate it to y = mx + b.
show them how their strategy is the same as solving 5562 = 1511 + 335x. Boom! Context for solving equations!
My ideas for extensions would be to put objects like….
in the pitcher. Count how many cups to fill the pitcher now. Use our equation to solve for the volume of the car. [Corresponding Grade 9 Academic learning goal: Find the y-intercept (initial value) of a linear equation given the slope (rate of change) and a point. ]
What do you think?? Think it would work? I would love some feedback!
# The Best Estimates
So Dan sent out this tweet.
Wasn’t sure if he was asking Andrew to make a blog post or anyone, but I decided to share my thoughts!
If you’re not familiar with Estimation180…..become familiar quick!! The challenges/estimates have been great conversation starters, warm ups, and intros to math concepts in my classes for the last couple years!!!
So, to answer Dan’s question…..My favourite Estimates have been the ones that make the students do a double take! They make us say No Way!!! or How is that right?
Here is my favourite…
Day 52
It’s awesome because of the controversy! Very few kids guess that there are actually 12 ounces/355ml in that glass! Most think it must be more than the can! In my class we have had great discussion on reasons. Most say the camera angle in the picture is deceiving. They get angry because they think I tricked them. From this point on they are skeptical about all given information!! Awesome! Love it!
I love this whole line up of estimates. Great discussion come out of why the tall vase has the same volume as the Dessert Dish on day 54….
….and the glass on day 57
I think the kids get a kick out of watching the video answers too!
I’m a huge fan of these types of estimates too …. ”
## How many small vases will it take to fill the large vase?
by these types I mean “How many of these fit in there?” These have worked wonders for some of our problem solving skills. After we reveal the the answer we take, for example the total ounces in the large container and try to work backwards and see if we can figure out how many small containers fit. (by dividing). By using these estimates as warm ups it has been an easy transition to solve problems like…
In the past my grade 9 applied students have struggled with this type of problem. After using the “how many fit” estimates my students’ ability on this type have dramatically improved!
These are a few of my favourite things…
Again ….check this site out now…..Estimation180. Thanks Andrew Stadel!
# Amazing Race Review Activity
We should not attach speed as a factor in our math learning but I love the intensity my students show when we do the Amazing Race Review.
I first saw this activity from a friend of mine Brian McBain. He created a review game where students travelled around the school completing challenges.The first to make it back to the room after completing all challenges was the winner…..just like in the show The Amazing Race.
Today was a review day on trigonometric expressions. I just grabbed some “Knowledgey” questions from the review section of the text. My goal here was to get them to practice the basics. I wanted to provide them some feedback on the application type questions….so i didn’t include them in the race.
and like…
where they are to complete the review question and then use their answer to figure out where to go next.
I also threw in some like….
where they had to come back to my room and complete a challenge.
Each clue was placed around or in the room indicated. With permission from the teachers of the room the students had to actually go in a classroom and look around for the clue.
When found each clue looked like…
They had to scan the QR code which revealed the clue.
Making the QR codes is pretty easy…..
1. Take a picture of your clues and put them in a Google Drive folder.
2. For each clue, grab the shareable link and paste it into the QR code generator here: http://www.qrstuff.com/
3. Download that QR code and paste it into a sheet like above. And you’ve got a clue!!!
I staggered the start so each group didn’t just follow each other from room to room. I gave them a recording sheet so they could keep track of their clues and work. I set them off and said “Complete all clues in the correct order and you will be eligible to crack the code for the prize.”
Here is the code to crack….
This years class was pretty intense. The kids were racing each other down the halls and blocking each other from looking at their work. When all groups made it back to the room it was a heated match of “who can crack the code” first!! When finally the group opened the lock …..the class erupted! Some in cheers and some not so much!! You could put anything in that box for the prize and they would be happy!!! Stickers is usually my go to choice!!! Kids have a fun time practicing some skills!!
# Stacking Cups!
So we did Dan’s Meyer’s stacking cups lesson in class today!!! I first saw this activity from Andrew Stadel in his 3-Act math collection. Not sure who first came up with it though. But thanks to both of you!
I started class by stacking the cups up in front of them…..allowed them wonder what was going on. They had questions like
“What are you doing?
“Are we having Hot Chocolate?”
“Are we going to use them to drink something?”
“How many cups do you have?”
and “How tall are you in cups?”
and bingo there we go!
I told them that is our task for today…To discover how tall I am in cups! I then had them estimate how many cups it would be! They were uncomfortable to start. They wanted to guess perfectly so they wanted to know how tall I was. They tried to put cups next to me as I walked around. They wanted me to lie down! I said just make an estimate to start off! I wanted them to guess so we had something to compare their final answers to. I wanted them to continually checking their work against their initial guess.
After a few minutes of estimating one group asked: “Are we stacking them like this…..
or like this…..
Awesome!!!! I said “Does it matter?” and they all yelled yes!!! So we then agreed that we had TWO problems to solve. So we put up two sets of estimates!!! We decided to stack them like the second picture first!
Estimates
“Did you need anything from me?”
they asked for: Rulers, my height, and Desmos!
I gave them all of those things…….everyone wrote frantically when I said I was 183 cm tall!!!
They worked! I saw groups stacking cups, recording values in Desmos, and measuring!
Almost all groups realized that the stack height was only changing by the lip amount and I saw a lot of this…
which had me excited!!! It gave me a chance to say: “Tell me about this, why do you think this is correct?” It was so interesting to hear their responses…..they were convinced they were right so I said let’s plot this in desmos and see if the equation matches the table
Oh!!
They knew they were wrong…..but what was awesome is that they knew how to fix it!!!! Desmos is awesome for this. It’s like a visual self correction machine! We discussed that the start of the line didn’t seem to match up with our points. Then the ahaa! happened.
“We didn’t use the zero row for our start value.” They fixed it and were visually rewarded with a correct answer.
After our equations were in desmos, the kids dragged their finger along the line until they reached a height of 183 cm and read off the number of cups! For the kids who seemed ahead of the game this was my chance to introduce solving equations by using opposite operations!
Finally we stacked the cups to verify.
Round 2: Stack the cups end to end.
Most groups divided my height with the height of 1 cup…..21 cups….give or take….So great! It gave us context when we discussed opposite operations when solving equations.
I found it was great that we had two problems in one! We are discussing how to distinguish between partial variation problems and direct variation problems. And here is one scenario where we got to look at each!!! Such a valuable activity!
Oh……did you want to know my height in cups (overlapped)??? —–> 128!
## Below are the list of Ontario Curriculum Expectations covered in this activity—-> Look at them all!!!!
• pose problems, identify variables, and formulate hypotheses associated with relationships between two variables
• carry out an investigation or experiment involving relationships between two variables, including the collection and organization of data, using appropriate methods, equipment, and/or technology (e.g., surveying; using measuring tools, scientific probes, the Internet) and techniques
• describe trends and relationships observed in data, make inferences from data, com- pare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses
• compare the properties of direct variation and partial variation in applications, and identify the initial value
• express a linear relation as an equation in two variables, using the rate of change and the initial value
• describe the meaning of the rate of change and the initial value for a linear relation arising from a realistic situation
• determine values of a linear relation by using a table of values, by using the equa- tion of the relation, and by interpolating or extrapolating from the graph of the relation.
# Wouldn’t it be awesome…
Desmos,
I can’t stop thinking about the great stuff from Penny Circle, Waterline, Central Park, Desman, and Function Carnival. Specifically the collaboration; the crowd sourcing of data and responses!
In Penny Circle, I love the fact that the student gets to do a few instances of selecting a circle and filling it with pennies. Then the data is grouped with the rest of the class….and voila!! we have a scatterplot!
I would love for this option of crowd sourcing content as a regular option. Wouldn’t it be awesome for when we complete the Vroom Vroom activity or the Barbie Bungee activity that we could ask students to record a few pieces of data in their table….like this,
but up on the projector the class sees this?
Wouldn’t it also be awesome if I asked the class to draw me a line with slope -2 ….the student would see theirs….
but we would all see this?
I’ve been using PearDeck for some lessons lately and we’ve been able to crowd source some stuff like..
Put the moveable point on A solution to the inequality f(x) > 40
Day 23 – Pear Deck!!
We’ve also been able to crowd source by the old fashion way…….everyone write their points up on the board then we can all graph. This is still great don’t get me wrong……i’m just wishing!
Wouldn’t it also be awesome when we go to make Math Art with our Function Art project…..we all work together to make a picture like…
I know that Texas Instruments has TI Navigator which tries to link students up ……but in my opinion it’s not as nice or easy as Desmos is to use!!!! Maybe this is already possible in Desmos and I just don’t know it. Or maybe there is something else out there……but I doubt it.
I would love it if my class could all work together…..keep our technology social! These are just some wishes! Love Desmos no matter what! | 5,404 | 23,734 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2014-52 | longest | en | 0.959841 |
http://www.chegg.com/homework-help/questions-and-answers/imagine-mike-lives-earth-rest-sara-traveling-past-theearth-velocity-8900m-s-meteor-moving--q144103 | 1,475,066,797,000,000,000 | text/html | crawl-data/CC-MAIN-2016-40/segments/1474738661367.29/warc/CC-MAIN-20160924173741-00163-ip-10-143-35-109.ec2.internal.warc.gz | 383,977,212 | 13,851 | Imagine Mike lives on the earth at rest. Sara is traveling past theearth with a velocity of 8900m/s. If a meteor moving in the samedirection passes Sara, Mike sees the meteor velocity to be13600m/s. Suppose that a photon of light is passing Sara instead ofa meteor. Mike measures the photon speed to be c, thespeed of light. What speed does Sara measure? | 84 | 354 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2016-40 | latest | en | 0.908245 |
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# Take Home Lab: Ocean Circulation This lab covers several aspects of Ocean Circulation, a process that governs much of the biology of the oceans. NOTE...
Take Home Lab: Ocean Circulation This lab covers several aspects of Ocean Circulation, a process that governs much of the biology of the oceans. It is a computer exercise that will help you understand the types of data that oceanographers collect and how that data can be used. NOTE This is a lab that we you will do at home because there are no labs the week of Thanksgiving. This lab is due anytime prior to Mid-Term 1. This will make the material most useful to your understanding of the lecture material. Instructions: Go to http://funnel.sfsu.edu/courses/geol103/labs/currents/currents.home.html. Follow along the lab exercise and answer the questions below. Part 1 Section 1 A) Plot the 1995 data from 3 buoys in the North Pacific (Table 1) on the Eastern and Western Pacific maps provided below. Use longitude and latitude data to plot the position of each buoy location during the year; then connect the locations with lines and draw an arrow to show the direction of motion. Note that a negative longitude is the same as a west longitude, e.g. -130=130W, +130=130E. Aternatively, you can use the website http://www.aquarius.geomar.de/make_map.html . If you use the website (recommended), use the following settings: North 60, South -20 East -80, West -240 Click: Grid On Map, b/w map, plot user defined locations (from the exercise). When plotting the bouys, you might find it easier to shorten their names (or name the start and end points) so that the map doesn’t become too crowded. Map can be printing using the Create Map button. B) What are the names of the surface currents that moved the buoys whose courses you plotted (refer to linked map)? Buoy 12410: Buoy 15022: Buoy 22217: C) What is the name of the current that moves water past the coast of California?
D) Do you think it carries warm or cold water past the coast of California?
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# 20 Introduction to Animals Worksheet
Animal Science Uintah High School FFA Chapter introduction to animals worksheet chapter 32, chapter 24 introduction to animals worksheet answers, chapter 25 introduction to animals worksheet answers, introduction to animals skills worksheet answers, introduction to animals worksheet, via: uintahffa.net
Numbering Worksheets for Kids. Kids are usually introduced to this topic matter during their math education. The main reason behind this is that learning math can be done with the worksheets. With an organized worksheet, kids will be able to describe and explain the correct answer to any mathematical problem. But before we talk about how to create a math worksheet for kids, let’s have a look at how children learn math.
In elementary school, children are exposed to a number of different ways of teaching them how to do a number of different subjects. Learning these subjects is important because it would help them develop logical reasoning skills. It is also an advantage for them to understand the concept behind all mathematical concepts.
To make the learning process easy for children, the educational methods used in their learning should be easy. For example, if the method is to simply count, it is not advisable to use only numbers for the students. Instead, the learning process should also be based on counting and dividing numbers in a meaningful way.
The main purpose of using a worksheet for kids is to provide a systematic way of teaching them how to count and multiply. Children would love to learn in a systematic manner. In addition, there are a few benefits associated with creating a worksheet. Here are some of them:
Children have a clear idea about the number of objects that they are going to add up. A good worksheet is one which shows the addition of different objects. This helps to give children a clear picture about the actual process. This helps children to easily identify the objects and the quantities that are associated with it.
This worksheet helps the child’s learning. It also provides children a platform to learn about the subject matter. They can easily compare and contrast the values of various objects. They can easily identify the objects and compare it with each other. By comparing and contrasting, children will be able to come out with a clearer idea.
He or she will also be able to solve a number of problems by simply using a few cells. He or she will learn to organize a worksheet and manipulate the cells. to arrive at the right answer to any question.
This worksheet is a vital part of a child’s development. When he or she comes across an incorrect answer, he or she can easily find the right solution by using the help of the worksheets. He or she will also be able to work on a problem without having to refer to the teacher. And most importantly, he or she will be taught the proper way of doing the mathematical problem.
Math skills are the most important part of learning and developing. Using the worksheet for kids will improve his or her math skills.
Many teachers are not very impressed when they see the number of worksheets that are being used by their children. This is actually very much true in the case of elementary schools. In this age group, the teachers often feel that the child’s performance is not good enough and they cannot just give out worksheets.
However, what most parents and educators do not realize is that there are several ways through which you can improve the child’s performance. You just need to make use of a worksheet for kids. elementary schools.
As a matter of fact, there is a very good option for your children to improve their performance in math. You just need to look into it.
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| Decipher
| Ron Dubren | Peter Meekin
| 暂缺
|抽象 |数学
"The Ultimate Number Game" Number Quest plays sort of like Boggle crossed with Othello/Reversi; this game is essentially a number version of its cousin, Overturn. The playing board is a plastic tray into which nine tiles (from a set of eighteen) are placed in a random order. Each tile has four raised pegs on it, each with a different digit (from 0-9) printed on it. On their turn, each player must construct an equation by selecting at least three adjacent digits (digits may be adjacent both orthogonally and diagonally) that form a valid equation when the proper operational symbols are added (i.e. 347 becomes 3+4=7, 9752149 becomes 97+52=149, 678234444 becomes 678-234=444, etc.) Only addition, subtraction, multiplication, and division are allowed. No fractions or decimals are permitted. Equations must follow normal sequence (i.e. 97+52=149 is valid; 149=97+52 isn't)and may not contain more than one operation (i.e. 4-1-2=2 is not valid.) A given digit may not be used more than once in the same equation, and equations must make use of the digits on at least two separate tiles. 0 may never appear by itself in an equation (i.e. 4+0=4 and 5-5=0 are invalid.) The equation is claimed by placing plastic rings of the player's chosen color (either green or gold) around the pegs of the digits forming the equation. After the first player's first turn, all subsequent turns must use at least one previously used number. If necessary, the player forming the equation flips over the ring so used (rings are gold on one side and green on the other), a la Othello/Reversi. The game ends when all 36 rings have been used, or when no more equations are possible. Players then score one point for each ring of their color. Player with the most points is the winner. Although Number Quest is essentially a two player game, more can be accommodated by forming teams. | 480 | 1,934 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2024-38 | latest | en | 0.932678 |
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• Isotropicaf
In summary, the conversation suggests using the equation |r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta} for simplifying the integration process. The conversation also confirms that the integral of the sum is equal to the sum of the integrals.
Isotropicaf
Homework Statement
Hello,
Imagine the Hamiltonian of a two-atom molecule, you have the kinetic energy of one+the other and then you have a relative distance potential ( V(|r2-r1|^2), ri is 3 dimensional).
How to change the variable to solve the following integral with infinite limits?
Relevant Equations
Intg( exp( |r2-r1|^2) dr1dr2)
I think its going to be intg(dr2)intg(exp(r^2) dr) or something like that.
Try putting one variable on the axis
Abhishek11235 said:
Try putting one variable on the axis
Im sorry, i don't know what you mean by that, you mean i should assume r1 as a constant and analyse how |r2-r1|^2 behaves in this condition ?
Isotropicaf said:
you mean i should assume r1 as a constant and analyse how |r2-r1|^2 behaves in this condition ?
No!
##|r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta}##
Now the integrations are easy!
Abhishek11235 said:
No!
##|r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta}##
Now the integrations are easy!
Oh thanks that totally solved my problem, seems obvious now ahah just to check, the integral of the sum is the sum of the integrals?
## 1. What is the concept of "Integral of relative distance-dependent potential"?
The integral of relative distance-dependent potential is a mathematical concept used in physics and chemistry to calculate the energy associated with the interaction between two particles at a given distance. It takes into account the distance between the particles and the strength of their interaction.
## 2. How is the integral of relative distance-dependent potential calculated?
The integral is calculated by integrating the relative distance-dependent potential function over the entire distance range between the two particles. This involves finding the anti-derivative of the potential function and evaluating it at the given distance limits.
## 3. What is the significance of the integral of relative distance-dependent potential?
The integral provides a measure of the total energy associated with the interaction between two particles. It can be used to understand the stability of a system and predict the behavior of particles at different distances.
## 4. How does the integral of relative distance-dependent potential relate to other mathematical concepts?
The integral of relative distance-dependent potential is closely related to the concept of potential energy, which is the energy associated with the position of particles in a system. It is also related to the concept of force, as the derivative of the potential function gives the force between the particles.
## 5. Can the integral of relative distance-dependent potential be applied to all types of interactions?
Yes, the concept can be applied to any type of interaction between two particles, such as gravitational, electrostatic, or van der Waals interactions. However, the specific form of the potential function may vary depending on the type of interaction being studied.
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# Use the following to answer the questions below:Breakfast is often considered to be the most ...
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Use the following to answer the questions below:
Breakfast is often considered to be the most important meal of the day. Data on the amount of sugar (g) per serving for randomly selected cereals from three different brands (General Mills, Kellogg's, and Kashi) are summarized in the provided plot and table.
Brand n s General Mills 26 8.538 4.492 Kashi 16 8.500 3.183 Kellogg's 33 10.636 3.516 Overall 75 9.453 3.916
Construct the ANOVA table and test, at the 5% significance level, for a difference in mean amount of sugar among the three brands. Use two decimal places in all decimal values. Is there enough evidence to conclude that the average amount of sugar per serving differs significantly among the three brands.
▸ Yes
▸ No
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## Statistics: Unlocking the Power of Data
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Crack Origin 8-6 Practice Factoring Ax2 Bx C http://urlin.us/5aitc
Krantz Book.pdf - Central Washington University www.cwu.edu/~lundin/Courses/math260/Krantz%20Book.pdf Mar 2, 2015 practice a sensible statement in mathematics is either true or false, .. claims that for every x, the number x C 1 is less than x. 8x 8y; x2 C y2. 0 The second claims that it is not the case that B.x/ fails for .8 3/ .8 6/ D 8 8 C 8 .6/ C .3/ 8 C . 3/ .6/ : origin; but if R is large then the image of CR under p is a. 6 Factoring Polynomials and Solving Equations - Ace www.learningace.com/doc//6-factoring-polynomials-and-solving-equations Jan 3, 2012 Write the equation asax2 + bx + c ? 0. 2. ? Because 2(4) - ? ? 0, the value 4 is called a(n) of the polynomial 2 x - ??. ab ? 0 implies that a ?. to download the study guide. - Conwell-Egan Catholic High School www.conwell-egan.org/wp-content//Parent-and-Student-Study-Guide.pdf Standardized Test Practice Evaluate the expression 2 (3 4)2 6 5(2). .. The ordered pair (0, 0) corresponds to the origin. . Use the secret code in the box at the right to crack these problems. 1. 2. set of ordered pairs: {(3, 2), (4, 3), (8, 6 )} Practice. 9-3. To factor a trinomial of the form x2 bx c, find two numbers, m and n, . PDF (Entire book) authors.library.caltech.edu/25031/1/Calc1studentw.pdf Factoring. This is a technique that is learned best through practice. A good starting point is to tions of the form Ax + Bx + C = 0 by solving for y = x2 and taking. 8-6 Practice B ans - Math Men mathmenalg1.weebly.com/uploads/2/4/8//8-6_practice_b_ans.pdf Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 55. 8 -6. Practice. Form K. Factoring ax2 + bx + c. Factor each expression. 1. 3n. Algebra 1 Parent & Student Workbook - Azinga Cartoons www.azingacartoons.com/Algebra%201%20Parent%20&<wbr>%20Student%20Workbook_GR.pdf A 14. B 23. C 141. D 77. Examples. Practice. Examples. Verbal Expression .. The ordered pair (0, 0) corresponds to the origin. . Use the secret code in the box at the right to crack these problems. 1. 2. This is called factoring by grouping. .. Axis of symmetry for graph of y ax2 bx c, where a. 0, is x . b. 2a. Answers:1–9. Algebra 2 - Scribd https://www.scribd.com/doc/137186258/Algebra-2 The diagram shows some important subsets of the real numbers. ,i> Ê Õ“LiÀà ,> ̈œ˜> Ê Õ“LiÀà ύ. ä°x x ä. ˜Ìi}iÀà ϖ. Ç ÓÚÚ ™. ÀÀ>̈œ˜> Ê Õ“LiÀÃ. е x ȖÎ û. chapter 1 - ResearchGate https://www.researchgate.net/Ben//0f67be622fd10f9cafe5c88c It is quite probable that surveying had its origin in the ancient Egypt. . It c omprise 100 million parcels of land, triangle survey, measurable survey, . Here, the surveyor is issued with a professional number to practice as either a By comparing the equation x 2 – 2 x -4 = 0 with ax2 bx +c = 0, we have a = 1, b =-2, c = -1. a8336db058
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# Homework 1 - ECON 398 HOMEWORK 1 Professor Ozdenoren 1 For...
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ECON 398 HOMEWORK 1 Professor Ozdenoren 1) For the following games. List the strategies available to all players. What is the equilibrium outcome? What are the equilibrium strategies? a) 0,4 -1,-1 2,2 1 2 Out In Accomodate Price War b) U D L R L R l r 1 1 2 2 1,4 5,2 3,3 2,0 6,2 2) Consider the following football example: The Offense can choose to run or pass, and the defense can play run defense, play pass defense, or blitz. The game is sequential: the offense goes first, then the defense, and finally, IF (AND ONLY IF) the defense plays Blitz, the offense can choose to change their play from run to pass. The payoffs are: (2,-2) if offense plays run and defense plays run; (5,-5) if offense plays run, and defense plays pass; (15, -15) if the offense plays run, the defense plays blitz, and the offense stays with the run; (-5, 5) if the offense plays run, the defense plays blitz, and the offense changes to pass; (8, -8) if the offense plays pass, and the defense plays
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1. Illustrating the Central Limit Theorem
ERIC Educational Resources Information Center
Corcoran, Mimi
2016-01-01
Statistics is enjoying some well-deserved limelight across mathematics curricula of late. Some statistical concepts, however, are not especially intuitive, and students struggle to comprehend and apply them. As an AP Statistics teacher, the author appreciates the central limit theorem as a foundational concept that plays a crucial role in…
2. Visualizing the Central Limit Theorem through Simulation
ERIC Educational Resources Information Center
Ruggieri, Eric
2016-01-01
The Central Limit Theorem is one of the most important concepts taught in an introductory statistics course, however, it may be the least understood by students. Sure, students can plug numbers into a formula and solve problems, but conceptually, do they really understand what the Central Limit Theorem is saying? This paper describes a simulation…
3. Central limit theorems under special relativity.
PubMed
McKeague, Ian W
2015-04-01
Several relativistic extensions of the Maxwell-Boltzmann distribution have been proposed, but they do not explain observed lognormal tail-behavior in the flux distribution of various astrophysical sources. Motivated by this question, extensions of classical central limit theorems are developed under the conditions of special relativity. The results are related to CLTs on locally compact Lie groups developed by Wehn, Stroock and Varadhan, but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior.
4. "Dealing" with the Central Limit Theorem
ERIC Educational Resources Information Center
Matz, David C.; Hause, Emily L.
2008-01-01
We describe an easy-to-employ, hands-on demonstration using playing cards to illustrate the central limit theorem. This activity allows students to see how a collection of sample means drawn from a nonnormally distributed population will be normally distributed. Students who took part in the demonstration reported it to be helpful in understanding…
5. Temporal Distributional Limit Theorems for Dynamical Systems
Dolgopyat, Dmitry; Sarig, Omri
2017-02-01
Suppose {T^t} is a Borel flow on a complete separable metric space X, f:X→ R is Borel, and xin X. A temporal distributional limit theorem is a scaling limit for the distributions of the random variables X_T:=int _0^t f(T^s x)ds, where t is chosen randomly uniformly from [0, T], x is fixed, and T→ ∞. We discuss such laws for irrational rotations, Anosov flows, and horocycle flows.
6. Central limit theorem: the cornerstone of modern statistics
PubMed Central
2017-01-01
According to the central limit theorem, the means of a random sample of size, n, from a population with mean, µ, and variance, σ2, distribute normally with mean, µ, and variance, σ2n. Using the central limit theorem, a variety of parametric tests have been developed under assumptions about the parameters that determine the population probability distribution. Compared to non-parametric tests, which do not require any assumptions about the population probability distribution, parametric tests produce more accurate and precise estimates with higher statistical powers. However, many medical researchers use parametric tests to present their data without knowledge of the contribution of the central limit theorem to the development of such tests. Thus, this review presents the basic concepts of the central limit theorem and its role in binomial distributions and the Student's t-test, and provides an example of the sampling distributions of small populations. A proof of the central limit theorem is also described with the mathematical concepts required for its near-complete understanding. PMID:28367284
7. Central limit theorems for percolation models
Cox, J. Theodore; Grimmett, Geoffrey
1981-06-01
Let p ≠ 1/2 be the open-bond probability in Broadbent and Hammersley's percolation model on the square lattice. Let W x be the cluster of sites connected to x by open paths, and let γ(n) be any sequence of circuits with interiors|γ limits^ circ (n)| to infty . It is shown that for certain sequences of functions { f n },S_n = sum _{x in γ limits^ circ (n)} f_n (W_x ) converges in distribution to the standard normal law when properly normalized. This result answers a problem posed by Kunz and Souillard, proving that the number S n of sites inside γ(n) which are connected by open paths to γ(n) is approximately normal for large circuits γ(n).
8. Improving Conceptions in Analytical Chemistry: The Central Limit Theorem
ERIC Educational Resources Information Center
Rodriguez-Lopez, Margarita; Carrasquillo, Arnaldo, Jr.
2006-01-01
This article describes the central limit theorem (CLT) and its relation to analytical chemistry. The pedagogic rational, which argues for teaching the CLT in the analytical chemistry classroom, is discussed. Some analytical chemistry concepts that could be improved through an understanding of the CLT are also described. (Contains 2 figures.)
9. Understanding the Sampling Distribution and the Central Limit Theorem.
ERIC Educational Resources Information Center
Lewis, Charla P.
The sampling distribution is a common source of misuse and misunderstanding in the study of statistics. The sampling distribution, underlying distribution, and the Central Limit Theorem are all interconnected in defining and explaining the proper use of the sampling distribution of various statistics. The sampling distribution of a statistic is…
10. Nonequilibrium fluctuation theorems in the presence of local heating
Pradhan, Punyabrata; Kafri, Yariv; Levine, Dov
2008-04-01
We study two nonequilibrium work fluctuation theorems, the Crooks theorem and the Jarzynski equality, for a test system coupled to a spatially extended heat reservoir whose degrees of freedom are explicitly modeled. The sufficient conditions for the validity of the theorems are discussed in detail and compared to the case of classical Hamiltonian dynamics. When the conditions are met the fluctuation theorems are shown to hold despite the fact that the immediate vicinity of the test system goes out of equilibrium during an irreversible process. We also study the effect of the coupling to the heat reservoir on the convergence of ⟨exp(-βW)⟩ to its theoretical mean value, where W is the work done on the test system and β is the inverse temperature. It is shown that the larger the local heating, the slower the convergence.
11. On local-hidden-variable no-go theorems
Methot, A. A.
2006-06-01
The strongest attack against quantum mechanics came in 1935 in the form of a paper by Einstein, Podolsky, and Rosen. It was argued that the theory of quantum mechanics could not be called a complete theory of Nature, for every element of reality is not represented in the formalism as such. The authors then put forth a proposition: we must search for a theory where, upon knowing everything about the system, including possible hidden variables, one could make precise predictions concerning elements of reality. This project was ultimately doomed in 1964 with the work of Bell, who showed that the most general local hidden variable theory could not reproduce correlations that arise in quantum mechanics. There exist mainly three forms of no-go theorems for local hidden variable theories. Although almost every physicist knows the consequences of these no-go theorems, not every physicist is aware of the distinctions between the three or even their exact definitions. Thus, we will discuss here the three principal forms of no-go theorems for local hidden variable theories of Nature. We will define Bell theorems, Bell theorems without inequalities, and pseudo-telepathy. A discussion of the similarities and differences will follow.
12. Entropy Inequalities for Stable Densities and Strengthened Central Limit Theorems
Toscani, Giuseppe
2016-10-01
We consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function. By showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a Lévy centered density of parameter λ >1 . This introduces a new information-theoretic approach to the central limit theorem for stable laws, in which the main argument is shown to be the relative fractional Fisher information, recently introduced in Toscani (Ricerche Mat 65(1):71-91, 2016). In particular, it is proven that, with respect to the relative fractional Fisher information, the Lévy density satisfies an analogous of the logarithmic Sobolev inequality, which allows to pass from the monotonicity and decay to zero of the relative fractional Fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate.
13. Zero-Bounded Limits as a Special Case of the Squeeze Theorem for Evaluating Single-Variable and Multivariable Limits
ERIC Educational Resources Information Center
Gkioulekas, Eleftherios
2013-01-01
Many limits, typically taught as examples of applying the "squeeze" theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful…
14. Mixing rates and limit theorems for random intermittent maps
Bahsoun, Wael; Bose, Christopher
2016-04-01
We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps {{T}α} using the full parameter range 0<α <∞ , in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e. smallest α) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (theorem 1.1) establishes sharp estimates on the position of return time intervals for the quenched dynamics. The main applications of this estimate are to limit laws (in particular, CLT and stable laws, depending on the parameters chosen in the range 0<α <1 ) for the associated skew product; these are detailed in theorem 3.2. Since our estimates in theorem 1.1 also hold for 1≤slant α <∞ we study a second class of random transformations derived from piecewise affine Gaspard-Wang maps, prove existence of an infinite (σ-finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.
15. A pointwise limit theorem for filtered backprojection in computed tomography.
PubMed
Ye, Yangbo; Zhu, Jiehua; Wang, Ge
2003-05-01
Computed tomography (CT) is one of the most important areas in the modern science and technology. The most popular approach for image reconstruction is filtered backprojection. It is essential to understand the limit behavior of the filtered backprojection algorithms. The classic results on the limit of image reconstruction are typically done in the norm sense. In this paper, we use the method of limited bandwidth to handle filtered backprojection-based image reconstruction when the spectrum of an underlying image is not absolutely integrable. Our main contribution is, assuming the method of limited bandwidth, to prove a pointwise limit theorem for a class of functions practically relevant and quite general. Further work is underway to extend the theory and explore its practical applications.
16. No-local-broadcasting theorem for multipartite quantum correlations.
PubMed
Piani, Marco; Horodecki, Paweł; Horodecki, Ryszard
2008-03-07
We prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled, as long as it does not simply encode a multipartite classical probability distribution. Said quantumness is revealed by the new task of local broadcasting, i.e., of locally sharing preestablished correlations, which is feasible if and only if correlations are stricly classical. Our operational approach leads to natural definitions of measures for quantumness of correlations. It also reproduces the standard no-broadcasting theorem as a special case.
17. Using Computers To Teach the Concepts of the Central Limit Theorem.
ERIC Educational Resources Information Center
Mittag, Kathleen Cage
A pivotal theorem which is of critical importance to statistical inference in probability and statistics is the Central Limit Theorem (CLT). The theorem concerns the sampling distribution of random samples taken from a population, including population distributions that do not have to be normal distributions. This paper contains a brief history of…
18. Local Paley Wiener theorems for functions analytic on unit spheres
Damelin, S. B.; Devaney, A. J.
2007-04-01
The purpose of this paper is to provide new and simplified statements of local Paley-Wiener theorems on the (n - 1)-dimensional unit sphere realized as a subset of n = 2, 3 Euclidean space. More precisely, given a function f:{\\bb C}^n\\to {\\bb C}, n=2,3 , whose restriction to an n - 1 sphere is analytic, we establish necessary and sufficient conditions determining whether f is the Fourier transform of a compactly supported, bounded function F:{\\bb R}^n\\to{\\bb C} . The essence of this investigation is that, because of the local nature of the problem, the mapping f → F is not in general invertible and so the problem cannot be studied via a Fourier integral. Our proofs are new.
19. Planetary Accretion, Oxygen Isotopes and the Central Limit Theorem
NASA Technical Reports Server (NTRS)
Nuth, Joseph A., III; Hill, Hugh G. M.; Vondrak, Richard R. (Technical Monitor)
2001-01-01
The accumulation of presolar dust into increasingly larger aggregates (CAIs and Chondrules, Asteroids, Planets) should result in a very drastic reduction in the numerical spread in oxygen isotopic composition between bodies of similar size, in accord with the Central Limit Theorem. Observed variations in oxygen isotopic composition are many orders of magnitude larger than would be predicted by a simple, random accumulation model that begins in a well-mixed nebula - no matter which size-scale objects are used as the beginning or end points of the calculation. This discrepancy implies either that some as yet unspecified process acted on the solids in the Solar Nebula to increase the spread in oxygen isotopic composition during each and every stage of accumulation or that the nebula was heterogeneous and maintained this heterogeneity throughout most of nebular history. Large-scale nebular heterogeneity would have significant consequences for many areas of cosmochemistry, including the application of some well-known isotopic systems to the dating of nebular events or the prediction of bulk compositions of planetary bodies on the basis of a uniform cosmic abundance.
20. Finiteness theorems for limit cycles: a digest of the revised proof
Ilyashenko, Yu S.
2016-02-01
This is the first paper in a series of two presenting a digest of the proof of the finiteness theorem for limit cycles of a planar polynomial vector field. At the same time we sketch the proof of the following two theorems: an analogous result for analytic vector fields, and a description of the asymptotics of the monodromy transformation for polycycles of such fields.
1. Central limit theorems and suppression of anomalous diffusion for systems with symmetry
Gottwald, Georg A.; Melbourne, Ian
2016-10-01
We give general conditions for the central limit theorem and weak convergence to Brownian motion (the weak invariance principle/functional central limit theorem) to hold for observables of compact group extensions of nonuniformly expanding maps. In particular, our results include situations where the central limit theorem would fail, and anomalous behaviour would prevail, if the compact group were not present. This has important consequences for systems with noncompact Euclidean symmetry and provides the rigorous proof for a conjecture made in our paper: a Huygens principle for diffusion and anomalous diffusion in spatially extended systems. Gottwald and Melbourne (2013 Proc. Natl Acad. Sci. USA 110 8411-6).
2. On a new proof of the Lindeberg-Feller classical limit theorem
2015-09-01
In recent papers researchers describe some of the new types of properties characterization of the normal distribution. This paper gives a new one based on the characterization of these properties, the proof of the classical limit theorem Lindeberg-Feller.
3. Uniform Limit Theorems for Synchronous Processes with Applications to Queues
DTIC Science & Technology
1989-10-01
first moment. In the present paper we investi gate conditions under which the Cesaro averaged functionals ;(f) 1- jfoE((6,X))dj converge uniformly...Proposition 3.1 both apply to positive HRMP’s. So, for example, given any initial state Z 0 = :, it follows that the Cesaro averaged measures (A) 1! 7foEIA o...collection of measures (see Theorem 2.1 of [41)). Continuing in the spirit of Cesaro convergence we have Proposition 4.1. If Z is a positive HRMP with
4. The Implicit Function Theorem and Non-Existence of Limit of Functions of Several Variables
ERIC Educational Resources Information Center
dos Santos, A. L. C.; da Silva, P. N.
2008-01-01
We use the Implicit Function Theorem to establish a result of non-existence of limit to a certain class of functions of several variables. We consider functions given by quotients such that both the numerator and denominator functions are null at the limit point. We show that the non-existence of the limit of such function is related with the…
5. Signal-locality, uncertainty, and the subquantum H-theorem. II
Valentini, Antony
1991-08-01
In the pilot-wave formulation, signal-locality and the uncertainty principle are shown to be valid only for the equilibrium distribution P=| Ψ| 2 (which arises from the subquantum H-theorem proved earlier). The H-theorem then explains the emergence of effective locality and uncertainty from a deeper nonlocal and deterministic theory. In order to explain the present uneasy “peaceful coexistence” (or “conspiracy”) between relativity and quantum theory, we suggest that a subquantum analogue of Boltzmann's heat death has actually happened in the real universe.
6. Metric rigidity theorems on Hermitian locally symmetric spaces
PubMed Central
Mok, Ngaiming
1986-01-01
Let X = Ω/Γ be a compact quotient of an irreducible bounded symmetric domain Ω of rank ≥2 by a discrete group ω of automorphisms without fixed points. It is well known that the Kähler-Einstein metric g on X carries seminegative curvature (in the sense of Griffiths). I show that any Hermitian metric h on X carrying seminegative curvature must be a constant multiple of g. This can be applied to prove rigidity theorems of holomorphic maps from X into Hermitian manifolds (Y, k) carrying seminegative curvature. These results are also generalized to the case of quotients of finite volume. On the other hand, let (Xc, gc) be an irreducible compact Hermitian symmetric manifold of rank ≥2. Then gc is Kähler and carries semipositive holomorphic bisectional curvature. I prove that any Kähler h on Xc carrying semipositive holomorphic bisectional curvature must be equal to gc up to a constant multiple and up to a biholomorphic transformation of Xc. PMID:16593680
7. A Central Limit Theorem for the Effective Conductance: Linear Boundary Data and Small Ellipticity Contrasts
Biskup, M.; Salvi, M.; Wolff, T.
2014-06-01
Given a resistor network on with nearest-neighbor conductances, the effective conductance in a finite set with a given boundary condition is the minimum of the Dirichlet energy over functions with the prescribed boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box converges to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and ellipticity contrasts will be addressed in a subsequent paper.
8. The optical theorem for local source excitation of a particle near a plane interface
Eremin, Yuri; Wriedt, Thomas
2015-11-01
Based on classic Maxwell's theory and the Gauss Theorem we extended the Optical Theorem to the case of a penetrable particle excited by a local source deposited near a plane interface. We demonstrate that the derived Extinction Cross-Section involves the total point source radiating cross-section and some definite integrals responsible for the scattering by the interface. The derived extinction cross-section can be employed to estimate the quantum yield and the optical antenna efficiency without computation of the absorption cross-section.
9. Computability, Gödel's incompleteness theorem, and an inherent limit on the predictability of evolution.
PubMed
Day, Troy
2012-04-07
The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century, there have been remarkable advances in the development of theory for this diversification, and the theory's success rests, in part, on the scope of its applicability. A great deal of this theory focuses on a relatively small subset of the space of potential genotypes, chosen largely based on historical or contemporary patterns, and then predicts the evolutionary dynamics within this pre-defined set. To what extent can such an approach be pushed to a broader perspective that accounts for the potential open-endedness of evolutionary diversification? There have been a number of significant theoretical developments along these lines but the question of how far such theory can be pushed has not been addressed. Here a theorem is proven demonstrating that, because of the digital nature of inheritance, there are inherent limits on the kinds of questions that can be answered using such an approach. In particular, even in extremely simple evolutionary systems, a complete theory accounting for the potential open-endedness of evolution is unattainable unless evolution is progressive. The theorem is closely related to Gödel's incompleteness theorem, and to the halting problem from computability theory.
10. Sanov and central limit theorems for output statistics of quantum Markov chains
SciTech Connect
2015-02-15
In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.
11. Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations
Fouxon, Itzhak; Denisov, Sergey; Zaburdaev, Vasily; Barkai, Eli
2017-04-01
We consider super-diffusive Lévy walks in d≥slant 2 dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called ‘infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d > 1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.
12. Limitations of sensitivity, specificity, likelihood ratio, and bayes' theorem in assessing diagnostic probabilities: a clinical example.
PubMed
Moons, K G; van Es, G A; Deckers, J W; Habbema, J D; Grobbee, D E
1997-01-01
We evaluated the extent to which the sensitivity, specificity, and likelihood ratio of the exercise test to diagnose coronary artery disease vary across subgroups of a certain patient population. Among 295 patients suspected of coronary artery disease, as independently determined by coronary angiography, we assessed variation in sensitivity and specificity according to patient history, physical examination, exercise test results, and disease severity in 207 patients with and 88 patients without coronary artery disease, respectively. The sensitivity varied substantially according to sex (women 30% and men 64%), systolic blood pressure at baseline (53% to 65%), expected workload (50% to 64%), systolic blood pressure at peak exercise (50% to 67%), relative workload (33% to 68%), and number of diseased vessels (39% to 77%). The specificity varied across subgroups of sex (men 89% and women 97%) and relative workload (85% to 98%). The likelihood ratio varied (3.8 to 17.0) across the same patient subgroups, as did the sensitivity. As each population tends to be heterogeneous with respect to patient characteristics, no single level of these parameters can be given that is adequate for all subgroups. Use of these parameters as a basis for calculating diagnostic probabilities in individual patients using Bayes' theorem has serious limitations.
13. A Microsoft® Excel Simulation Illustrating the Central Limit Theorem's Appropriateness for Comparing the Difference between the Means of Any Two Populations
ERIC Educational Resources Information Center
Moen, David H.; Powell, John E.
2008-01-01
Using Microsoft® Excel, several interactive, computerized learning modules are developed to illustrate the Central Limit Theorem's appropriateness for comparing the difference between the means of any two populations. These modules are used in the classroom to enhance the comprehension of this theorem as well as the concepts that provide the…
14. Fluctuation limits of a locally regulated population and generalized Langevin equations
2015-06-01
We consider a locally regulated spatial population model introduced by Bolker and Pacala. Based on the deterministic approximation studied by Fournier and Méléard, we prove that the fluctuation theorem holds under some mild moment conditions. The limiting process is shown to be an infinite-dimensional Gaussian process solving a generalized Langevin equation. In particular, we further consider its properties in one dimension case, which is characterized as a time-inhomogeneous Ornstein-Uhlenbeck process.
15. Resolution limits of ultrafast ultrasound localization microscopy
Desailly, Yann; Pierre, Juliette; Couture, Olivier; Tanter, Mickael
2015-11-01
As in other imaging methods based on waves, the resolution of ultrasound imaging is limited by the wavelength. However, the diffraction-limit can be overcome by super-localizing single events from isolated sources. In recent years, we developed plane-wave ultrasound allowing frame rates up to 20 000 fps. Ultrafast processes such as rapid movement or disruption of ultrasound contrast agents (UCA) can thus be monitored, providing us with distinct punctual sources that could be localized beyond the diffraction limit. We previously showed experimentally that resolutions beyond λ/10 can be reached in ultrafast ultrasound localization microscopy (uULM) using a 128 transducer matrix in reception. Higher resolutions are theoretically achievable and the aim of this study is to predict the maximum resolution in uULM with respect to acquisition parameters (frequency, transducer geometry, sampling electronics). The accuracy of uULM is the error on the localization of a bubble, considered a point-source in a homogeneous medium. The proposed model consists in two steps: determining the timing accuracy of the microbubble echo in radiofrequency data, then transferring this time accuracy into spatial accuracy. The simplified model predicts a maximum resolution of 40 μm for a 1.75 MHz transducer matrix composed of two rows of 64 elements. Experimental confirmation of the model was performed by flowing microbubbles within a 60 μm microfluidic channel and localizing their blinking under ultrafast imaging (500 Hz frame rate). The experimental resolution, determined as the standard deviation in the positioning of the microbubbles, was predicted within 6 μm (13%) of the theoretical values and followed the analytical relationship with respect to the number of elements and depth. Understanding the underlying physical principles determining the resolution of superlocalization will allow the optimization of the imaging setup for each organ. Ultimately, accuracies better than the size
16. On the limit theorem for life time distribution connected with some reliability systems and their validation by means of the Monte Carlo method
Gheorghe, Munteanu Bogdan; Alexei, Leahu; Sergiu, Cataranciuc
2013-09-01
We prove the limit theorem for life time distribution connected with reliability systems when their life time is a Pascal Convolution of independent and identically distributed random variables. We show that, in some conditions, such distributions may be approximated by means of Erlang distributions. As a consequnce, survival functions for such systems may be, respectively, approximated by Erlang survival functions. By using Monte Carlo method we experimantally confirm the theoretical results of our theorem.
17. Beyond Gisin's Theorem and its Applications: Violation of Local Realism by Two-Party Einstein-Podolsky-Rosen Steering.
PubMed
Chen, Jing-Ling; Su, Hong-Yi; Xu, Zhen-Peng; Wu, Yu-Chun; Wu, Chunfeng; Ye, Xiang-Jun; Żukowski, Marek; Kwek, L C
2015-06-25
We demonstrate here that for a given mixed multi-qubit state if there are at least two observers for whom mutual Einstein-Podolsky-Rosen steering is possible, i.e. each observer is able to steer the other qubits into two different pure states by spontaneous collapses due to von Neumann type measurements on his/her qubit, then nonexistence of local realistic models is fully equivalent to quantum entanglement (this is not so without this condition). This result leads to an enhanced version of Gisin's theorem (originally: all pure entangled states violate local realism). Local realism is violated by all mixed states with the above steering property. The new class of states allows one e.g. to perform three party secret sharing with just pairs of entangled qubits, instead of three qubit entanglements (which are currently available with low fidelity). This significantly increases the feasibility of having high performance versions of such protocols. Finally, we discuss some possible applications.
18. Restructuring local distribution services: Possibilities and limitations
SciTech Connect
Duann, D.J.
1994-08-01
The restructuring of local distribution services is now the focus of the natural gas industry. It is the last major step in the reconstitution of the natural gas industry and a critical clement in realizing the full benefits of regulatory and market reforms that already have taken place in the wellhead and interstate markets. It could also be the most important regulatory initiative for most end-use customers because they are affected directly by the costs and reliability of distribution services. Several factors contribute to the current emphasis on distribution service restructuring. They include the unbundling and restructuring of upstream markets, a realization of the limitations of supply-side options (such as gas procurement oversight), and the increased diversity and volatility of gas demand facing local distribution companies. Local distribution service is not one but a series of activities that start with commodity gas procurement and extend to transportation, load balancing, storage, and metering and billing of services provided. There are also considerable differences in the economies of scale and scope associated with these various activities. Thus, a mixture of supply arrangements (such as a competitive market or a monopoly) is required for the most efficient delivery of local distribution services. A distinction must be made between the supply of commodity gas and the provision of a bundled distribution service. This distinction and identification of the best supply arrangements for various distribution service components are the most critical factors in developing appropriate restructuring policies. For most state public utility commissions the criteria for service restructuring should include pursuing the economies of scale and scope in gas distribution, differentiating and matching gas service reliability and quality with customer requirements, and controlling costs associated with the search, negotiation, and contracting of gas services.
19. Central limit theorem for the solution to the heat equation with moving time
Liu, Junfeng; Tudor, Ciprian A.
2016-03-01
We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.
20. Computer-Enriched Instruction (CEI) Is Better for Preview Material Instead of Review Material: An Example of a Biostatistics Chapter, the Central Limit Theorem
ERIC Educational Resources Information Center
See, Lai-Chu; Huang, Yu-Hsun; Chang, Yi-Hu; Chiu, Yeo-Ju; Chen, Yi-Fen; Napper, Vicki S.
2010-01-01
This study examines the timing using computer-enriched instruction (CEI), before or after a traditional lecture to determine cross-over effect, period effect, and learning effect arising from sequencing of instruction. A 2 x 2 cross-over design was used with CEI to teach central limit theorem (CLT). Two sequences of graduate students in nursing…
1. Classroom Research: Assessment of Student Understanding of Sampling Distributions of Means and the Central Limit Theorem in Post-Calculus Probability and Statistics Classes
ERIC Educational Resources Information Center
Lunsford, M. Leigh; Rowell, Ginger Holmes; Goodson-Espy, Tracy
2006-01-01
We applied a classroom research model to investigate student understanding of sampling distributions of sample means and the Central Limit Theorem in post-calculus introductory probability and statistics courses. Using a quantitative assessment tool developed by previous researchers and a qualitative assessment tool developed by the authors, we…
2. Nonparametric Functional Central Limit Theorem for Time Series Regression with Application to Self-normalized Confidence Interval.
PubMed
Kim, Seonjin; Zhao, Zhibiao; Shao, Xiaofeng
2015-01-01
This paper is concerned with the inference of nonparametric mean function in a time series context. The commonly used kernel smoothing estimate is asymptotically normal and the traditional inference procedure then consistently estimates the asymptotic variance function and relies upon normal approximation. Consistent estimation of the asymptotic variance function involves another level of nonparametric smoothing. In practice, the choice of the extra bandwidth parameter can be difficult, the inference results can be sensitive to bandwidth selection and the normal approximation can be quite unsatisfactory in small samples leading to poor coverage. To alleviate the problem, we propose to extend the recently developed self-normalized approach, which is a bandwidth free inference procedure developed for parametric inference, to construct point-wise confidence interval for nonparametric mean function. To justify asymptotic validity of the self-normalized approach, we establish a functional central limit theorem for recursive nonparametric mean regression function estimates under primitive conditions and show that the limiting process is a Gaussian process with non-stationary and dependent increments. The superior finite sample performance of the new approach is demonstrated through simulation studies.
3. Generalised Central Limit Theorems for Growth Rate Distribution of Complex Systems
Takayasu, Misako; Watanabe, Hayafumi; Takayasu, Hideki
2014-04-01
We introduce a solvable model of randomly growing systems consisting of many independent subunits. Scaling relations and growth rate distributions in the limit of infinite subunits are analysed theoretically. Various types of scaling properties and distributions reported for growth rates of complex systems in a variety of fields can be derived from this basic physical model. Statistical data of growth rates for about 1 million business firms are analysed as a real-world example of randomly growing systems. Not only are the scaling relations consistent with the theoretical solution, but the entire functional form of the growth rate distribution is fitted with a theoretical distribution that has a power-law tail.
4. Quasi-local approximation of non-local exchange-correlation kernels in the adiabatic-connection fluctuation-dissipation theorem
Lu, Deyu
The adiabatic-connection fluctuation-dissipation theorem (ACFDT) is a formal theoretical framework to treat van der Waals (vdW) dispersion interactions. Under the random phase approximation (RPA), it yields the correct asymptotic behavior at large distances, but the short-range correlation is overestimated. It has been demonstrated that non-local exchange-correlation kernels can systematically correct the errors of RPA for homogenous electron gas. However, direct extension of non-local kernels derived from the electron gas model to inhomogeneous systems raises several issues. In addition to the high computational expense, the non-local kernels worsen the rare gas dimer binding curve as compared to RPA. In this study, we propose a quasi-local approximation of the non-local kernel in order to address these issues. This research used resources of the Center for Functional Nanomaterials, which is a U.S. DOE Office of Science Facility, at Brookhaven National Laboratory under Contract No. DE-SC0012704.
5. Limits on the local dark matter density
Garbari, Silvia; Read, Justin I.; Lake, George
2011-09-01
We revisit systematics in determining the local dark matter density ρdm from the vertical motion of stars in the solar neighbourhood. Using a simulation of a Milky Way like galaxy, we determine the data quality required to detect ρdm at its expected local value. We introduce a new method for recovering ρdm that uses moments of the Jeans equations, combined with a Markov chain Monte Carlo technique, to marginalize over the unknown parameters. Given sufficiently good data, we show that our method can recover the correct local dark matter density even in the face of disc inhomogeneities, non-isothermal tracers and a non-separable distribution function. We illustrate the power of our technique by applying it to Hipparcos data. We first make the assumption that the A- and F-star tracer populations are isothermal. This recovers ρdm= 0.003+0.009- 0.007 M⊙ pc-3 (ρdm= 0.11+0.34- 0.27 GeV cm-3, with 90 per cent confidence), consistent with previous determinations. However, the vertical dispersion profile of these tracers is poorly known. If we assume instead a non-isothermal profile similar to that of the blue disc stars from SDSS DR-7 recently measured, we obtain a fit with a very similar χ2 value, but with ρdm= 0.033+0.008- 0.009 M⊙ pc-3 (ρdm= 1.25+0.30- 0.34 GeV cm-3 with 90 per cent confidence). This highlights that it is vital to measure the vertical dispersion profile of the tracers to recover an unbiased estimate of ρdm.
6. Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics: II. Noether's theorems and Casimirs
Webb, G. M.; Dasgupta, B.; McKenzie, J. F.; Hu, Q.; Zank, G. P.
2014-03-01
Conservation laws in ideal gas dynamics and magnetohydrodynamics (MHD) associated with fluid relabeling symmetries are derived using Noether's first and second theorems. Lie dragged invariants are discussed in terms of the MHD Casimirs. A nonlocal conservation law for fluid helicity applicable for a non-barotropic fluid involving Clebsch variables is derived using Noether's theorem, in conjunction with a fluid relabeling symmetry and a gauge transformation. A nonlocal cross helicity conservation law involving Clebsch potentials, and the MHD energy conservation law are derived by the same method. An Euler-Poincaré variational approach is also used to derive conservation laws associated with fluid relabeling symmetries using Noether's second theorem.
7. Visual Theorems.
ERIC Educational Resources Information Center
Davis, Philip J.
1993-01-01
Argues for a mathematics education that interprets the word "theorem" in a sense that is wide enough to include the visual aspects of mathematical intuition and reasoning. Defines the term "visual theorems" and illustrates the concept using the Marigold of Theodorus. (Author/MDH)
8. Wold-Cramer Concordance Theorems for Interpolation of q-Variate Stationary Process over Locally Compact Abelian Groups
ERIC Educational Resources Information Center
Makagon, A.; Weron, A.
1976-01-01
Salehi and Scheidt have derived several Wold-Cramer concordance theorems for q-variate stationary processes over discrete groups. This paper characterizes the concordance of the Wold decomposition with respect to families arising in the interpolation problem and the Cramer decomposition for non-full-rank q-variate stationary processes over certain…
9. Dulac's memoir "On limit cycles" and related problems of the local theory of differential equations
Il'yashenko, Yu S.
1985-12-01
CONTENTSIntroductionChapter I. Dulac's theorem and its generalization § 1. Definitions § 2. Reduction of the finiteness problem to the study of a neighbourhood of a compound cycle with elementary singular points § 3. Correspondence maps § 4. Composition of correspondence maps § 5. Remarks on Dulac's theory § 6. Two finiteness theoremsChapter II. Smooth orbital classification of elementary singular points of plane vector fields § 1. Survey of known results and sketch of a proof of the classification theorem § 2. Formal normal forms § 3. Proof of the classification theorem for degenerate elementary singular pointsConclusionAppendix. Example of a flat quadratic system having four limit cycles (after Shi Sonling)References
10. Bell's theorem and Bayes' theorem
Garrett, A. J. M.
1990-12-01
Bell's theorem is expounded as an analysis in Bayesian probabilistic inference. Assume that the result of a spin measurement on a spin- 1/2 particle is governed by a variable internal to the particle (local, “hidden”), and examine pairs of particles having zero combined angular momentum so that their internal variables are correlated: knowing something about the internal variable of one tells us something about that of the other. By measuring the spin of one particle, we infer something about its internal variable; through the correlation, about the internal variable of the second particle, which may be arbitrarily distant and is by hypothesis unchanged by this measurement (locality); and make (probabilistic) prediction of spin observations on the second particle. Each link in this chain has a counterpart in the Bayesian analysis of the situation. Irrespective of the details of the internal variable description, such prediction is violated by measurements on many particle pairs, so that locality—effectively the only physics invoked—fails. The time ordering of the two measurements is not Lorentz-invariant, implying acausality. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of the statistics of the results imply it has a nonlocal—acausal interpretation; one such, the “transactional” interpretation, is presented to demonstrable advantage, and some misconceptions about quantum theory are pursued. The “unobservability” loophole in photonic Bell experiments is proven to be closed. It is shown that this mechanism cannot be used for signalling; signalling would become possible only if the hidden variables, which we insist must underlie the statistical character of the observations (the alternative is to give up), are uncovered in deviations from quantum predictions. Their reticence is understood as a consequence of their nonlocality: it is not easy to isolate and measure something nonlocal. Once the hidden variables
11. Studies on Bell's theorem
Guney, Veli Ugur
In this work we look for novel classes of Bell's inequalities and methods to produce them. We also find their quantum violations including, if possible, the maximum one. The Jordan bases method that we explain in Chapter 2 is about using a pair of certain type of orthonormal bases whose spans are subspaces related to measurement outcomes of incompatible quantities on the same physical system. Jordan vectors are the briefest way of expressing the relative orientation of any two subspaces. This feature helps us to reduce the dimensionality of the parameter space on which we do searches for optimization. The work is published in [24]. In Chapter 3, we attempt to find a connection between group theory and Bell's theorem. We devise a way of generating terms of a Bell's inequality that are related to elements of an algebraic group. The same group generates both the terms of the Bell's inequality and the observables that are used to calculate the quantum value of the Bell expression. Our results are published in [25][26]. In brief, Bell's theorem is the main tool of a research program that was started by Einstein, Podolsky, Rosen [19] and Bohr [8] in the early days of quantum mechanics in their discussions about the core nature of physical systems. These debates were about a novel type of physical states called superposition states, which are introduced by quantum mechanics and manifested in the apparent inevitable randomness in measurement outcomes of identically prepared systems. Bell's huge contribution was to find a means of quantifying the problem and hence of opening the way to experimental verification by rephrasing the questions as limits on certain combinations of correlations between measurement results of spatially separate systems [7]. Thanks to Bell, the fundamental questions related to the nature of quantum mechanical systems became quantifiable [6]. According to Bell's theorem, some correlations between quantum entangled systems that involve incompatible
12. A rapid-pressure correlation representation consistent with the Taylor-Proudman theorem materially-frame-indifferent in the 2D limit
NASA Technical Reports Server (NTRS)
Ristorcelli, J. R.; Lumley, J. L.; Abid, R.
1994-01-01
A nonlinear representation for the rapid-pressure correlation appearing in the Reynolds stress equations, consistent with the Taylor-Proudman theorem, is presented. The representation insures that the modeled second-order equations are frame-invariant with respect to rotation when the flow is two-dimensional in planes perpendicular to the axis of rotation. The representation satisfies realizability in a new way: a special ansatz is used to obtain analytically, the values of coefficients valid away from the realizability limit: the model coefficients are functions of the state of the turbulence that are valid for all states of the mechanical turbulence attaining their constant limiting values only when the limit state is achieved. Utilization of all the mathematical constraints are not enough to specify all the coefficients in the model. The unspecified coefficients appear as free parameters which are used to insure that the representation is asymptotically consistent with the known equilibrium states of a homogeneous sheared turbulence. This is done by insuring that the modeled evolution equations have the same fixed points as those obtained from computer and laboratory experiments for the homogeneous shear. Results of computations of the homogeneous shear, with and without rotation, and with stabilizing and destabilizing curvature, are shown. Results are consistently better, in a wide class of flows which the model not been calibrated, than those obtained with other nonlinear models.
13. Accounting for Limited Detection Efficiency and Localization Precision in Cluster Analysis in Single Molecule Localization Microscopy
PubMed Central
Shivanandan, Arun; Unnikrishnan, Jayakrishnan; Radenovic, Aleksandra
2015-01-01
Single Molecule Localization Microscopy techniques like PhotoActivated Localization Microscopy, with their sub-diffraction limit spatial resolution, have been popularly used to characterize the spatial organization of membrane proteins, by means of quantitative cluster analysis. However, such quantitative studies remain challenged by the techniques’ inherent sources of errors such as a limited detection efficiency of less than 60%, due to incomplete photo-conversion, and a limited localization precision in the range of 10 – 30nm, varying across the detected molecules, mainly depending on the number of photons collected from each. We provide analytical methods to estimate the effect of these errors in cluster analysis and to correct for them. These methods, based on the Ripley’s L(r) – r or Pair Correlation Function popularly used by the community, can facilitate potentially breakthrough results in quantitative biology by providing a more accurate and precise quantification of protein spatial organization. PMID:25794150
14. Local versus basin-scale limitation of marine nitrogen fixation.
PubMed
Weber, Thomas; Deutsch, Curtis
2014-06-17
Nitrogen (N) fixation by diazotrophic plankton is the primary source of this crucial nutrient to the ocean, but the factors limiting its rate and distribution are controversial. According to one view, the ecological niche of diazotrophs is primarily controlled by the ocean through internally generated N deficits that suppress the growth of their competitors. A second view posits an overriding limit from the atmosphere, which restricts diazotrophs to regions where dust deposition satisfies their high iron (Fe) requirement, thus separating N sources from sinks at a global scale. Here we use multiple geochemical signatures of N2 fixation to show that the Fe limitation of diazotrophs is strong enough to modulate the regional distribution of N2 fixation within ocean basins--particularly the Fe-poor Pacific--but not strong enough to influence its partition between basins, which is instead governed by rates of N loss. This scale-dependent limitation of N2 fixation reconciles local observations of Fe stress in diazotroph communities with an inferred spatial coupling of N sources and sinks. Within this regime of intermediate Fe control, the oceanic N reservoir would respond only weakly to enhanced dust fluxes during glacial climates, but strongly to the reduced fluxes hypothesized under anthropogenic climate warming.
15. Limits of localized control in extended nonlinear systems
Handel, Andreas
We investigate the limits of localized linear control in spatially extended, nonlinear systems. Spatially extended, nonlinear systems can be found in virtually every field of engineering and science. An important category of such systems are fluid flows. Fluid flows play an important role in many commercial applications, for instance in the chemical, pharmaceutical and food-processing industries. Other important fluid flows include air- or water flows around cars, planes or ships. In all these systems, it is highly desirable to control the flow of the respective fluid. For instance control of the air flow around an airplane or car leads to better fuel-economy and reduced noise production. Usually, it is impossible to apply control everywhere. Consider an airplane: It would not be feasibly to cover the whole body of the plane with control units. Instead, one can place the control units at localized regions, such as points along the edge of the wings, spaced as far apart from each other as possible. These considerations lead to an important question: For a given system, what is the minimum number of localized controllers that still ensures successful control? Too few controllers will not achieve control, while using too many leads to unnecessary expenses and wastes resources. To answer this question, we study localized control in a class of model equations. These model equations are good representations of many real fluid flows. Using these equations, we show how one can design localized control that renders the system stable. We study the properties of the control and derive several expressions that allow us to determine the limits of successful control. We show how the number of controllers that are needed for successful control depends on the size and type of the system, as well as the way control is implemented. We find that especially the nonlinearities and the amount of noise present in the system play a crucial role. This analysis allows us to determine under
16. Exponential Localization of Photons
Bialynicki-Birula, Iwo
1998-06-01
It is shown that photons can be localized in space with an exponential falloff of the energy density and photodetection rates. The limits of localization are determined by the fundamental Paley-Wiener theorem. A direct mathematical connection between the spatial localization of photons and the decay in time of quantum mechanical systems is established.
17. Application of the Central Limit Theorem in microbial risk assessment: high number of servings reduces the Coefficient of Variation of food-borne burden-of-illness.
PubMed
Pérez-Rodríguez, Fernando; Zwietering, Marcel H
2012-02-15
The Central Limit Theorem (CLT) is proposed as a means of understanding microbial risk in foods from a Public Health perspective. One variant of the CLT states that as the number of random variables, each with a finite mean and variance, increases (→∞), the distribution of the sum (or mean) of those variables approximates a normal distribution. On the basis of the CLT, the hypothesis introduced by this paper states that the Coefficient of Variation (CV) of the annual number of food-borne illness cases decreases as a result of a larger number of exposures (or servings) (n). Second-order Monte-Carlo analysis and classical statistics were used to support the hypothesis, based on existing risk models on Listeria monocytogenes in deli meat products focused on elderly people in the United States. Likewise, the hypothesis was tested on epidemiological data of annual incidence of salmonellosis and listeriosis in different countries (i.e. different n). Although different sources of error affected the accuracy of the results, both the Monte-Carlo analysis (in silico) and epidemiological data (in vivo), especially for salmonellosis, demonstrated that the CV of the annual number of cases decreased as n increased as stated by the CLT. Furthermore, results from this work showed that classical statistical methods can be helpful to provide reliable risk estimates based on simple and well-established statistical principles.
18. Local and Global Limits on Visual Processing in Schizophrenia
PubMed Central
Tibber, Marc S.; Anderson, Elaine J.; Bobin, Tracy; Carlin, Patricia; Shergill, Sukhwinder S.; Dakin, Steven C.
2015-01-01
Schizophrenia has been linked to impaired performance on a range of visual processing tasks (e.g. detection of coherent motion and contour detection). It has been proposed that this is due to a general inability to integrate visual information at a global level. To test this theory, we assessed the performance of people with schizophrenia on a battery of tasks designed to probe voluntary averaging in different visual domains. Twenty-three outpatients with schizophrenia (mean age: 40±8 years; 3 female) and 20 age-matched control participants (mean age 39±9 years; 3 female) performed a motion coherence task and three equivalent noise (averaging) tasks, the latter allowing independent quantification of local and global limits on visual processing of motion, orientation and size. All performance measures were indistinguishable between the two groups (ps>0.05, one-way ANCOVAs), with one exception: participants with schizophrenia pooled fewer estimates of local orientation than controls when estimating average orientation (p = 0.01, one-way ANCOVA). These data do not support the notion of a generalised visual integration deficit in schizophrenia. Instead, they suggest that distinct visual dimensions are differentially affected in schizophrenia, with a specific impairment in the integration of visual orientation information. PMID:25689281
19. Local and global limits on visual processing in schizophrenia.
PubMed
Tibber, Marc S; Anderson, Elaine J; Bobin, Tracy; Carlin, Patricia; Shergill, Sukhwinder S; Dakin, Steven C
2015-01-01
Schizophrenia has been linked to impaired performance on a range of visual processing tasks (e.g. detection of coherent motion and contour detection). It has been proposed that this is due to a general inability to integrate visual information at a global level. To test this theory, we assessed the performance of people with schizophrenia on a battery of tasks designed to probe voluntary averaging in different visual domains. Twenty-three outpatients with schizophrenia (mean age: 40±8 years; 3 female) and 20 age-matched control participants (mean age 39±9 years; 3 female) performed a motion coherence task and three equivalent noise (averaging) tasks, the latter allowing independent quantification of local and global limits on visual processing of motion, orientation and size. All performance measures were indistinguishable between the two groups (ps>0.05, one-way ANCOVAs), with one exception: participants with schizophrenia pooled fewer estimates of local orientation than controls when estimating average orientation (p = 0.01, one-way ANCOVA). These data do not support the notion of a generalised visual integration deficit in schizophrenia. Instead, they suggest that distinct visual dimensions are differentially affected in schizophrenia, with a specific impairment in the integration of visual orientation information.
20. Vorticity, Stokes' Theorem and the Gauss's Theorem
Narayanan, M.
2004-12-01
Vorticity is a property of the flow of any fluid and moving fluids acquire properties that allow an engineer to describe that particular flow in greater detail. It is important to recognize that mere motion alone does not guarantee that the air or any fluid has vorticity. Vorticity is one of four important quantities that define the kinematic properties of any fluid flow. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. However, the divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into, or away from the region through its boundary. This is also known as Gauss's Theorem. It should also be noted that there are many useful extensions of Gauss's Theorem, including the extension to include surfaces of discontinuity in V. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. Integral (Surface) [(DEL X V)] . dS = Integral (Contour) [V . dx] In this paper, the author outlines and stresses the importance of studying and teaching these mathematical techniques while developing a course in Hydrology and Fluid Mechanics. References Arfken, G. "Gauss's Theorem." 1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985. Morse, P. M. and Feshbach, H. "Gauss's Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 37-38, 1953. Eric W. Weisstein. "Divergence Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DivergenceTheorem.html
1. Structure theorem for Vaisman completely solvable solvmanifolds
Sawai, Hiroshi
2017-04-01
Locally conformal Kähler manifold is said to be a Vaisman manifold if the Lee form is parallel with respect to the Riemannian metric. In this paper, we have the structure theorem for Vaisman completely solvable solvmanifolds.
2. Local Longitudinal Microwave Instability Limits During Bunch Rotation
SciTech Connect
Ng, K. Y.
2012-10-23
Bunch width compression can be accomplished by rf rotating an elongated bunch with minimal energy spread. The formation of tails at the two bunch ends produces disconnected regions in many time-advance slices. The stability limits of longitudinal microwave growth for each time-advance slice of the beam will be affected. This effect is studied and the Keil-Schnell stability limits for such time-advance slices are derived. Application is made to the bunch-width compression in the Fermilab Compressor Ring, destined for pion and subsequently muon production
3. Local Revenues for Schools: Limits and Options in California
ERIC Educational Resources Information Center
Perry, Mary; Edwards, Brian
2009-01-01
With K-12 schools representing the single largest expenditure in the state budget, education funding has been a central issue throughout the chaos that has recently characterized California's budget process. School districts throughout California have endured deep cuts in state funding, and more cuts are looming. But state law severely limits the…
4. Quantum-limited estimation of continuous spontaneous localization
McMillen, S.; Brunelli, M.; Carlesso, M.; Bassi, A.; Ulbricht, H.; Paris, M. G. A.; Paternostro, M.
2017-01-01
We apply the formalism of quantum estimation theory to extract information about potential collapse mechanisms of the continuous spontaneous localization (CSL) form. In order to estimate the strength with which the field responsible for the CSL mechanism couples to massive systems, we consider the optomechanical interaction between a mechanical resonator and a cavity field. Our estimation strategy passes through the probing of either the state of the oscillator or that of the electromagnetic field that drives its motion. In particular, we concentrate on all-optical measurements, such as homodyne and heterodyne measurements. We also compare the performances of such strategies with those of a spin-assisted optomechanical system, where the estimation of the CSL parameter is performed through time-gated spinlike measurements.
5. Understanding Rolle's Theorem
ERIC Educational Resources Information Center
Parameswaran, Revathy
2009-01-01
This paper reports on an experiment studying twelfth grade students' understanding of Rolle's Theorem. In particular, we study the influence of different concept images that students employ when solving reasoning tasks related to Rolle's Theorem. We argue that students' "container schema" and "motion schema" allow for rich…
6. The Parity Theorem Shuffle
ERIC Educational Resources Information Center
Smith, Michael D.
2016-01-01
The Parity Theorem states that any permutation can be written as a product of transpositions, but no permutation can be written as a product of both an even number and an odd number of transpositions. Most proofs of the Parity Theorem take several pages of mathematical formalism to complete. This article presents an alternative but equivalent…
7. An extension theorem for conformal gauge singularities
Lübbe, Christian; Tod, Paul
2009-11-01
We analyze conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmological singularity and prove a local extension theorem along a congruence of timelike conformal geodesics.
8. The Floquet Adiabatic Theorem revisited
Weinberg, Phillip; Bukov, Marin; D'Alessio, Luca; Kolodrubetz, Michael; Davidson, Shainen; Polkovnikov, Anatoli
2015-03-01
The existance of the adiabatic theorem for Floquet systems has been the subject of an active debate with different articles reaching opposite conclusions over the years. In this talk we clarify the situation by deriving a systematic expansion in the time-derivatives of a slow parameter for the occupation probabilities of the Floque states. Our analysis shows that the in a certain limit the transition between Floquet eigenstates are suppressed and it is possible to define an adiabatic theorem for Floquet systems. Crucially we observe however that the conditions for adiabaticity in ordinary and Floquet systems are different and that this difference can become important when the amplitude of the periodic driving is large. We illustrate our results with specific examples of a periodically driven harmonic oscillator and cold atoms in optical lattices which are relevant in current experiments.
9. Local superconducting coupling in the strong-localization limit of ultrathin granular metal films
SciTech Connect
Haviland, D. B.; Jaeger, H. M.; Orr, B. G.; Goldman, A. M.
1989-07-01
The onset of fluctuations into the superconducting state, as identified bythe appearance of a local minimum in the resistance versus temperature, isfound to be coincident with the characteristic activation energy ofnormal-state conduction falling to a value of the order of the superconductingenergy gap of the material at /ital T/=0. The model of conduction used isvariable-range hopping with a Coulomb gap, appropriate to granular metals.
10. Cooperation Among Theorem Provers
NASA Technical Reports Server (NTRS)
Waldinger, Richard J.
1998-01-01
In many years of research, a number of powerful theorem-proving systems have arisen with differing capabilities and strengths. Resolution theorem provers (such as Kestrel's KITP or SRI's SNARK) deal with first-order logic with equality but not the principle of mathematical induction. The Boyer-Moore theorem prover excels at proof by induction but cannot deal with full first-order logic. Both are highly automated but cannot accept user guidance easily. The purpose of this project, and the companion project at Kestrel, has been to use the category-theoretic notion of logic morphism to combine systems with different logics and languages.
11. Local intelligent electronic device (IED) rendering templates over limited bandwidth communication link to manage remote IED
DOEpatents
Bradetich, Ryan; Dearien, Jason A; Grussling, Barry Jakob; Remaley, Gavin
2013-11-05
The present disclosure provides systems and methods for remote device management. According to various embodiments, a local intelligent electronic device (IED) may be in communication with a remote IED via a limited bandwidth communication link, such as a serial link. The limited bandwidth communication link may not support traditional remote management interfaces. According to one embodiment, a local IED may present an operator with a management interface for a remote IED by rendering locally stored templates. The local IED may render the locally stored templates using sparse data obtained from the remote IED. According to various embodiments, the management interface may be a web client interface and/or an HTML interface. The bandwidth required to present a remote management interface may be significantly reduced by rendering locally stored templates rather than requesting an entire management interface from the remote IED. According to various embodiments, an IED may comprise an encryption transceiver.
12. Anti-Bell - Refutation of Bell's theorem
Barukčić, Ilija
2012-12-01
In general, Albert Einstein as one of "the founding fathers of quantum mechanics" had some problems to accept especially the Copenhagen dominated interpretation of quantum mechanics. Einstein's dissatisfaction with Copenhagen's interpretation of quantum mechanics, the absence of locality and causality within the Copenhagen dominated quantum mechanics lead to the well known Einstein, Podolsky and Rosen thought experiment. According to Einstein et al., the Copenhagen dominated quantum mechanics cannot be regarded as a complete physical theory. The Einstein, Podolsky and Rosen thought experiment was the origin of J. S. Bell's publication in 1964; known as Bell's theorem. Meanwhile, some dramatic violations of Bell's inequality (by so called Bell test experiments) have been reported which is taken as an empirical evidence against local realism and causality at quantum level and as positive evidence in favor of the Copenhagen dominated quantum mechanics. Thus far, Quantum mechanics is still regarded as a "strictly" non-local theory. The purpose of this publication is to refute Bell's original theorem. Thus far, if we accept Bell's theorem as correct, we must accept that +0> = +1. We can derive a logical contradiction out of Bell's theorem, Bell's theorem is refuted.
13. Trigonometry, Including Snell's Theorem.
ERIC Educational Resources Information Center
Kent, David
1980-01-01
Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)
14. Pompeiu's Theorem Revisited
ERIC Educational Resources Information Center
2009-01-01
Pompeiu's theorem states that if ABC is an "equilateral" triangle and M a point in its plane, then MA, MB, and MC form a new triangle. In this article, we have a new look at this theorem in the realm of arbitrary triangles. We discover what we call Pompeiu's Area Formula, a neat equality relating areas of triangles determined by the points A, B,…
15. Reaching the ultimate performance limit given by non-local effects in BOTDA sensors
Dominguez-Lopez, Alejandro; Yang, Zhisheng; Soto, Marcelo A.; Angulo-Vinuesa, Xabier; Martin-Lopez, S.; Thevenaz, Luc; Gonzalez-Herraez, Miguel
2015-09-01
Non-local effects have been traditionally identified as one of the most limiting factors of the performance of Brillouin optical time-domain analyzers. These phenomena, directly linked with the energy gained/lost by the pump pulse, limit the probe power and ultimately the SNR of the system. Several solutions have been proposed, although none offers the possibility to increase the probe power until its limit, the onset of amplified spontaneous Brillouin scattering. In this work, we propose a technique that avoids non-local effects and permits to set the probe power at its maximum, reaching a 100 km sensing distance with 2 meter resolution.
16. Influence of localized surface plasmons on Pauli blocking and optical limiting in graphene under femtosecond pumping
Bongu, Sudhakara Reddy; Bisht, Prem B.; Namboodiri, Raman C. K.; Nayak, Pranati; Ramaprabhu, Sundara; Kelly, Thomas J.; Fallon, Colm; Costello, John T.
2014-08-01
The Pauli blocking limit and optical limiting threshold have been found to be modified following silver-nanoparticle decoration of functionalized hydrogen induced exfoliated graphene. Femtosecond Z-scan experiments have been used to measure the Pauli blocking range, optical limiting threshold, and the third order nonlinear susceptibility (χ(3)) values. The observed results have been explained by modified band structure of graphene in the presence of silver nanoparticles and their localized surface plasmon resonances.
17. Fluctuation theorem in dynamical systems with quenched disorder
Drocco, Jeffrey; Olson Reichhardt, Cynthia; Reichhardt, Charles
2010-03-01
We demonstrate that the fluctuation theorem of Gallavotti and Cohen can be used to characterize far from equilibrium dynamical nonthermal systems in the presence of quenched disorder where strong fluctuations or crackling noise occur. By observing the frequency of entropy-destroying trajectories, we show that the theorem holds in specific dynamical regimes near the threshold for motion, indicating that these systems might be ideal candidates for understanding what types of nonthermal fluctuations could be used in constructing generalized fluctuation theorems. We also discuss how the theorem could be tested with global or local probes in systems such as superconducting vortices, magnetic domain walls, stripe phases, Coulomb glasses and earthquake models.
18. Navier Stokes Theorem in Hydrology
Narayanan, M.
2005-12-01
In a paper presented at the 2004 AGU International Conference, the author outlined and stressed the importance of studying and teaching certain important mathematical techniques while developing a course in Hydrology and Fluid Mechanics. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. In an article entitled "Corrections to Fluid Dynamics" R. F. Streater, (Open Systems and Information Dynamics, 10, 3-30, 2003.) proposes a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit he obtains a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations. In 2001, David Hoff published an article in Journees Equations aux derivees partielles. (Art. No. 7, 9 p.). His paper is entitled : Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. In his paper, David Hoff proves the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time
19. Plasmonic localized heating beyond the diffraction limit via magnetic polariton excitation
Alshehri, Hassan; Ying, Xiaoyan; Wang, Hao; Wang, Liping
2016-09-01
Optical localized heating in the nanoscale has recently attracted great attention due to its unique small hot spot size with high energy. However, the hot spot size is conventionally constrained by the diffraction limit. Plasmonic localized heating can provide solutions to this limitation in nanoscale patterning, cancer treatment, and data storage. Plasmonic approaches to overcome the diffraction limit in hot spot size have mainly utilized the excitation of surface plasmon or localized surface plasmon resonance. However, achieving plasmonic localized heating by the excitation of magnetic polariton has not been researched extensively yet. In this work, we numerically investigated the optical response of a nanoscale metamaterial composed of a gold nanowire array and a gold film separated by an ultrathin polymer spacer using ANSYS High Frequency Structural Simulator. A strong absorption peak at the wavelength of 760 nm was exhibited, and the underlying physical mechanism for the strong absorption was verified via the local electromagnetic field distribution to be magnetic resonance excitation. An inductor-capacitor circuit model was used to predict the magnetic resonance wavelength and compare with the numerical results for varied geometrical parameters. Volume loss density due to the strong local optical energy confinement was transferred as heat generation to an ANSYS thermal solver to obtain the local temperature profile. The steady state temperature profile shows an average temperature of 145 °C confined in a local area as small as 33 nm within the spacer, with a full-width at half-maximum of 50 nm along the x-direction. Moreover, the temperature rise from ambient drops to half its maximum value at a distance of 5 nm from the top of the spacer along the z-direction. This clearly demonstrates plasmonic localized heating beyond the diffraction limit via magnetic polariton excitation. Furthermore, the transient temperature profile shows that the system reached
20. A generalization of Nekhoroshev's theorem
Bates, Larry; Cushman, Richard
2016-11-01
Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville-Arnol'd on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev's theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev's theorem does in the abelian case.
1. Adaptation of Medicago truncatula to nitrogen limitation is modulated via local and systemic nodule developmental responses.
PubMed
Jeudy, Christian; Ruffel, Sandrine; Freixes, Sandra; Tillard, Pascal; Santoni, Anne Lise; Morel, Sylvain; Journet, Etienne-Pascal; Duc, Gérard; Gojon, Alain; Lepetit, Marc; Salon, Christophe
2010-02-01
Adaptation of Medicago truncatula to local nitrogen (N) limitation was investigated to provide new insights into local and systemic N signaling. The split-root technique allowed a characterization of the local and systemic responses of NO(3)(-) or N(2)-fed plants to localized N limitation. (15)N and (13)C labeling were used to monitor plant nutrition. Plants expressing pMtENOD11-GUS and the sunn-2 hypernodulating mutant were used to unravel mechanisms involved in these responses. Unlike NO(3)(-)-fed plants, N(2)-fixing plants lacked the ability to compensate rapidly for a localized N limitation by up-regulating the N(2)-fixation activity of roots supplied elsewhere with N. However they displayed a long-term response via a growth stimulation of pre-existing nodules, and the generation of new nodules, likely through a decreased abortion rate of early nodulation events. Both these responses involve systemic signaling. The latter response is abolished in the sunn mutant, but the mutation does not prevent the first response. Local but also systemic regulatory mechanisms related to plant N status regulate de novo nodule development in Mt, and SUNN is required for this systemic regulation. By contrast, the stimulation of nodule growth triggered by systemic N signaling does not involve SUNN, indicating SUNN-independent signaling.
2. Immigration, local dispersal limitation, and the repeatability of community composition under neutral and niche dynamics.
PubMed
Ai, Dexiecuo; Desjardins-Proulx, Philippe; Chu, Chengjin; Wang, Gang
2012-01-01
Repeatability of community composition has been a critical aspect for community structure, which is closely associated with community stability, predictability, conservation biology and ecological restoration. It has been shown that both immigration and local dispersal limitation can affect the community composition in both neutral and niche model. Hence, we use a spatially explicit individual-based model to investigate the potential influence of immigration rate and strength of local dispersal limitation on repeatability in both neutral and niche models. Similarity measures are used to quantify repeatability. We examine the repeatability of community composition among replicate communities (which means the same community repeats many times), and between niche and neutral replicate communities. We find the correlation between repeatability and immigration rate is positive in the neutral model and an inverted unimodal in the niche model. The correlation between repeatability and local dispersal distance is positive in the niche model and negative in the neutral model. High repeatability between niche communities and neutral communities is observed with high immigration rates or when high local dispersal distance appears in the niche model or low local dispersal distance in the neutral model. Our results show that repeatability of community composition is not only dependent on the types of community models (niche vs. neutrality) but also strongly determined by immigration rates and local dispersal limitation.
3. New limits on the violation of local position invariance of gravity
Shao, Lijing; Wex, Norbert
2013-08-01
Within the parameterized post-Newtonian (PPN) formalism, there could be an anisotropy of local gravity induced by an external matter distribution, even for a fully conservative metric theory of gravity. It reflects the breakdown of the local position invariance of gravity and, within the PPN formalism, is characterized by the Whitehead parameter ξ. We present three different kinds of observation, from the Solar system and radio pulsars, to constrain it. The most stringent limit comes from recent results on the extremely stable pulse profiles of solitary millisecond pulsars, that gives |\\hat{\\xi }| < 3.9 \\times 10^{-9} (95% CL), where the hat denotes the strong-field generalization of ξ. This limit is six orders of magnitude more constraining than the current best limit from superconducting gravimeter experiments. It can be converted into an upper limit of ˜4 × 10-16 on the spatial anisotropy of the gravitational constant. Communicated by C M Will
4. Spatial fluctuation theorem
Pérez-Espigares, Carlos; Redig, Frank; Giardinà, Cristian
2015-08-01
For non-equilibrium systems of interacting particles and for interacting diffusions in d-dimensions, a novel fluctuation relation is derived. The theorem establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti-Cohen fluctuation theorem related to the time-reversal symmetry. This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.
5. Plasmonic local heating beyond diffraction limit by the excitation of magnetic polariton
Alshehri, Hassan; Wang, Hao; Ma, Yanchao; Wang, Liping
2015-08-01
In recent years, optical local heating in the nanoscale has attracted great attention due to its unique features of small hot spot size and high energy density. Plasmonic local heating can provide solutions to several challenges in data storage and cancer treatment. Research conducted in this field to achieve plasmonic local heating has mainly utilized the excitation of localized surface plasmon (LSP) or surface plasmon resonance (SPR). However, achieving plasmonic local heating by the excitation of magnetic polariton (MP) has not been researched extensively yet. We numerically investigate the optical response of a nanostructure composed of a gold nanowire on a gold surface separated by a polymer spacer using the ANSYS High Frequency Structural Simulator (HFSS). The structure exhibits a strong absorption peak at the wavelength of 750 nm, and the underlying physical mechanism is verified by the local electromagnetic field distribution to be the magnetic resonance excitation. By incorporating the volume loss density due to the strong local optical energy confinement as the heat generation, nanoscale temperature distribution within the structure is numerically obtained with a thermal solver after assigning proper boundary conditions. The results show a maximum temperature of 158.5°C confined in a local area on the order of 35 nm within the ultrathin polymer layer, which clearly demonstrates the plasmonic local heating effect beyond diffraction limit by excitation of MP.
6. Strong genetic differentiation but not local adaptation toward the range limit of a coastal dune plant.
PubMed
Samis, Karen E; López-Villalobos, Adriana; Eckert, Christopher G
2016-11-01
All species have limited geographic distributions; but the ecological and evolutionary mechanisms causing range limits are largely unknown. That many species' geographic range limits are coincident with niche limits suggests limited evolutionary potential of marginal populations to adapt to conditions experienced beyond the range. We provide a test of range limit theory by combining population genetic analysis of microsatellite polymorphisms with a transplant experiment within, at the edge of, and 60 km beyond the northern range of a coastal dune plant. Contrary to expectations, lifetime fitness increased toward the range limit with highest fitness achieved by most populations at and beyond the range edge. Genetic differentiation among populations was strong, with very low, nondirectional gene flow suggesting range limitation via constraints to dispersal. In contrast, however, local adaptation was negligible, and a distance-dependent decline in fitness only occurred for those populations furthest from home when planted beyond the range limit. These results challenge a commonly held assumption that stable range limits match niche limits, but also raise questions about the unique value of peripheral populations in expanding species' geographical ranges.
7. Virial Theorem and Scale Transformations.
ERIC Educational Resources Information Center
Kleban, Peter
1979-01-01
Discussed is the virial theorem, which is useful in classical, quantum, and statistical mechanics. Two types of derivations of this theorem are presented and the relationship between the two is explored. (BT)
8. Rediscovering Schreinemakers' Theorem.
ERIC Educational Resources Information Center
Bathurst, Bruce
1983-01-01
Schreinemakers' theorem (arrangement of curves around an invariant point), derived from La Chatelier's principle, can be rediscovered by students asked to use the principle when solving a natural problem such as "How does diluting a mineral/fluid alter shape of a pressure/temperature diagram?" Background information and instructional…
9. Limits of slow sound propagation and transparency in lossy, locally resonant periodic structures
Theocharis, G.; Richoux, O.; Romero García, V.; Merkel, A.; Tournat, V.
2014-09-01
We investigate sound propagation in lossy, locally resonant periodic structures by studying an air-filled tube periodically loaded with Helmholtz resonators and taking into account the intrinsic viscothermal losses. In particular, by tuning the resonator with the Bragg gap in this prototypical locally resonant structure, we study the limits and various characteristics of slow sound propagation. While in the lossless case the overlapping of the gaps results in slow-sound-induced transparency of a narrow frequency band surrounded by a strong and broadband gap, the inclusion of the unavoidable losses imposes limits to the slowdown factor and the maximum transmission. Experiments, theory, and finite element simulations have been used for the characterization of acoustic wave propagation by tuning the Helmholtz/Bragg frequencies and the total amount of loss both for infinite and finite lattices. This study contributes to the field of locally resonant acoustic metamaterials and slow sound applications.
10. The Fluctuation Theorem
Evans, Denis J.; Searles, Debra J.
2002-11-01
The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in thermostatted dissipative non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy 'thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been in the last few of years that the generality of the Theorem has been appreciated. We now know that the Second Law of Thermodynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quantifies the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observation that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversed 'anti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of
11. Cooperation Among Theorem Provers
NASA Technical Reports Server (NTRS)
Waldinger, Richard J.
1998-01-01
This is a final report, which supports NASA's PECSEE (Persistent Cognizant Software Engineering Environment) effort and complements the Kestrel Institute project "Inference System Integration via Logic Morphism". The ultimate purpose of the project is to develop a superior logical inference mechanism by combining the diverse abilities of multiple cooperating theorem provers. In many years of research, a number of powerful theorem-proving systems have arisen with differing capabilities and strengths. Resolution theorem provers (such as Kestrel's KITP or SRI's, SNARK) deal with first-order logic with equality but not the principle of mathematical induction. The Boyer-Moore theorem prover excels at proof by induction but cannot deal with full first-order logic. Both are highly automated but cannot accept user guidance easily. The PVS system (from SRI) in only automatic within decidable theories, but it has well-designed interactive capabilities: furthermore, it includes higher-order logic, not just first-order logic. The NuPRL system from Cornell University and the STeP system from Stanford University have facilities for constructive logic and temporal logic, respectively - both are interactive. It is often suggested - for example, in the anonymous "QED Manifesto"-that we should pool the resources of all these theorem provers into a single system, so that the strengths of one can compensate for the weaknesses of others, and so that effort will not be duplicated. However, there is no straightforward way of doing this, because each system relies on its own language and logic for its success. Thus. SNARK uses ordinary first-order logic with equality, PVS uses higher-order logic. and NuPRL uses constructive logic. The purpose of this project, and the companion project at Kestrel, has been to use the category-theoretic notion of logic morphism to combine systems with different logics and languages. Kestrel's SPECWARE system has been the vehicle for the implementation.
12. Ferromagnetism beyond Lieb's theorem
Costa, Natanael C.; Mendes-Santos, Tiago; Paiva, Thereza; Santos, Raimundo R. dos; Scalettar, Richard T.
2016-10-01
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction U is turned on, Lieb proved rigorously that at half-filling (ρ =1 ) the ground state has a nonzero spin. In this paper we consider a "CuO2 lattice" (also known as "Lieb lattice," or as a decorated square lattice), in which "d orbitals" occupy the vertices of the squares, while "p orbitals" lie halfway between two d orbitals; both d and p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up , originally considered by Lieb, and the inhomogeneous (IH) case, Ud≠Up . For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U . For the IH system at half-filling, we argue that the case Up≠Ud falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0 ,Ud=U and Up=U ,Ud=0 . We found that the different environments of d and p sites lead to a ferromagnetic insulator when Ud=0 ; by contrast, Up=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density ρ =1 /3 , strong antiferromagnetic correlations set in, caused by the presence of one fermion on each
Wares, Arsalan
2015-02-01
The purpose of this paper is to provide examples of 'non-traditional' proof-related activities or theorems that can be explored through paper folding by university and high-school students. These theorems were encountered through playful acts of paper folding by the author. The author used these activities successfully with preservice teachers. The paper contains proof outlines for each theorem.
PubMed
Barnum, Howard; Barrett, Jonathan; Leifer, Matthew; Wilce, Alexander
2007-12-14
We prove a generalized version of the no-broadcasting theorem, applicable to essentially any nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with "superquantum" correlations. A strengthened version of the quantum no-broadcasting theorem follows, and its proof is significantly simpler than existing proofs of the no-broadcasting theorem.
Barnum, Howard; Barrett, Jonathan; Leifer, Matthew; Wilce, Alexander
2007-12-01
We prove a generalized version of the no-broadcasting theorem, applicable to essentially any nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with “superquantum” correlations. A strengthened version of the quantum no-broadcasting theorem follows, and its proof is significantly simpler than existing proofs of the no-broadcasting theorem.
16. Pick's Theorem: What a Lemon!
ERIC Educational Resources Information Center
Russell, Alan R.
2004-01-01
Pick's theorem can be used in various ways just like a lemon. This theorem generally finds its way in the syllabus approximately at the middle school level and in fact at times students have even calculated the area of a state considering its outline with the help of the above theorem.
17. Intragrain charge transport in kesterite thin films—Limits arising from carrier localization
Hempel, Hannes; Redinger, Alex; Repins, Ingrid; Moisan, Camille; Larramona, Gerardo; Dennler, Gilles; Handwerg, Martin; Fischer, Saskia F.; Eichberger, Rainer; Unold, Thomas
2016-11-01
Intragrain charge carrier mobilities measured by time-resolved terahertz spectroscopy in state of the art Cu2ZnSn(S,Se)4 kesterite thin films are found to increase from 32 to 140 cm2 V-1 s-1 with increasing Se content. The mobilities are limited by carrier localization on the nanometer-scale, which takes place within the first 2 ps after carrier excitation. The localization strength obtained from the Drude-Smith model is found to be independent of the excited photocarrier density. This is in accordance with bandgap fluctuations as a cause of the localized transport. Charge carrier localization is a general issue in the probed kesterite thin films, which were deposited by coevaporation, colloidal inks, and sputtering followed by annealing with varying Se/S contents and yield 4.9%-10.0% efficiency in the completed device.
18. Intragrain charge transport in kesterite thin films—Limits arising from carrier localization
SciTech Connect
Hempel, Hannes; Redinger, Alex; Repins, Ingrid; Moisan, Camille; Larramona, Gerardo; Dennler, Gilles; Handwerg, Martin; Fischer, Saskia F.; Eichberger, Rainer; Unold, Thomas
2016-11-03
Intragrain charge carrier mobilities measured by time-resolved terahertz spectroscopy in state of the art Cu2ZnSn(S,Se)4 kesterite thin films are found to increase from 32 to 140 cm2V-1s-1 with increasing Se content. The mobilities are limited by carrier localization on the nanometer-scale, which takes place within the first 2 ps after carrier excitation. The localization strength obtained from the Drude-Smith model is found to be independent of the excited photocarrier density. This is in accordance with bandgap fluctuations as a cause of the localized transport. Lastly, charge carrier localization is a general issue in the probed kesterite thin films, which were deposited by coevaporation, colloidal inks, and sputtering followed by annealing with varying Se/S contents and yield 4.9%-10.0% efficiency in the completed device.
19. Intragrain charge transport in kesterite thin films—Limits arising from carrier localization
DOE PAGES
Hempel, Hannes; Redinger, Alex; Repins, Ingrid; ...
2016-11-03
Intragrain charge carrier mobilities measured by time-resolved terahertz spectroscopy in state of the art Cu2ZnSn(S,Se)4 kesterite thin films are found to increase from 32 to 140 cm2V-1s-1 with increasing Se content. The mobilities are limited by carrier localization on the nanometer-scale, which takes place within the first 2 ps after carrier excitation. The localization strength obtained from the Drude-Smith model is found to be independent of the excited photocarrier density. This is in accordance with bandgap fluctuations as a cause of the localized transport. Lastly, charge carrier localization is a general issue in the probed kesterite thin films, which weremore » deposited by coevaporation, colloidal inks, and sputtering followed by annealing with varying Se/S contents and yield 4.9%-10.0% efficiency in the completed device.« less
20. Building local human resources to implement SLMTA with limited donor funding: The Ghana experience
PubMed Central
Nkrumah, Bernard; van der Puije, Beatrice; Bekoe, Veronica; Adukpo, Rowland; Kotey, Nii A.; Yao, Katy; Fonjungo, Peter N.; Luman, Elizabeth T.; Duh, Samuel; Njukeng, Patrick A.; Addo, Nii A.; Khan, Fazle N.; Woodfill, Celia J.I.
2016-01-01
Background In 2009, Ghana adopted the Strengthening Laboratory Management Toward Accreditation (SLMTA) programme in order to improve laboratory quality. The programme was implemented successfully with limited donor funding and local human resources. Objectives To demonstrate how Ghana, which received very limited PEPFAR funding, was able to achieve marked quality improvement using local human resources. Method Local partners led the SLMTA implementation and local mentors were embedded in each laboratory. An in-country training-of-trainers workshop was conducted in order to increase the pool of local SLMTA implementers. Three laboratory cohorts were enrolled in SLMTA in 2011, 2012 and 2013. Participants from each cohort attended in a series of three workshops interspersed with improvement projects and mentorship. Supplemental training on internal audit was provided. Baseline, exit and follow-up audits were conducted using the Stepwise Laboratory Quality Improvement Process Towards Accreditation (SLIPTA) checklist. In November 2013, four laboratories underwent official SLIPTA audits by the African Society for Laboratory Medicine (ASLM). Results The local SLMTA team successfully implemented three cohorts of SLMTA in 15 laboratories. Seven out of the nine laboratories that underwent follow-up audits have reached at least one star. Three out of the four laboratories that underwent official ASLM audits were awarded four stars. Patient satisfaction increased from 25% to 70% and sample rejection rates decreased from 32% to 10%. On average, $40 000 was spent per laboratory to cover mentors' salaries, SLMTA training and improvement project support. Conclusion Building in-country capacity through local partners is a sustainable model for improving service quality in resource-constrained countries such as Ghana. Such models promote country ownership, capacity building and the use of local human resources for the expansion of SLMTA. PMID:26937417 1. Evaluation of Forming Limit by the 3 Dimensional Local Bifurcation Theory SciTech Connect Nishimura, Ryuichi; Nakazawa, Yoshiaki; Ito, Koichi; Uemura, Gen; Mori, Naomichi 2007-05-17 A theoretical prediction and evaluation method for the sheet metal formability is developed on the basis of the three-dimensional local bifurcation theory previously proposed by authors. The forming limit diagram represented on the plane defined by the ratio of stress component to work-hardening rate is perfectly independent of plastic strain history. The upper and the lower limit of the sheet formability are indicated by the 3D critical line and the Stoeren-Rice's critical line on this plane, respectively. In order to verify the above mentioned behavior of the proposed forming limit diagram, the experimental research is also conducted. From the standpoint of the mechanical instability theory, a new concept called instability factor is introduced. It represents a degree of acceleration by current stress for developing the local bifurcation mode toward a fracture. The instability factor provides a method to evaluate a forming allowance which is useful to appropriate identification for a forming limit and to optimize the forming condition. The proposed criterion provides not only the moment to initiate the necking but also the local bifurcation mode vector and the direction of necking line. 2. Silhouette-Slice Theorems DTIC Science & Technology 1987-03-20 with standard expressions of spherical trigonometry is sinr)0 = cos0 sini//0 (4.37) which is consistent with the results obtained previously with...theorems for discrete transforms. However, sampling questions inlroduce difficult obstacles in the develop- ment of a discrete theory. First, sampling...additional obstacle to discrete represen- tations of the CT. An example of qualitative predication of the shape of silhouettes with the Silhouette-Slice 3. The Steep Nekhoroshev's Theorem NASA Astrophysics Data System (ADS) Guzzo, M.; Chierchia, L.; Benettin, G. 2016-03-01 Revising Nekhoroshev's geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev's theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be {1/(2nα_1\\cdotsα_{n-2}}) ({α_i}'s being Nekhoroshev's steepness indices and {n ≥ 3} the number of degrees of freedom). On the base of a heuristic argument, we conjecture that the new stability exponent is optimal. 4. Effects of hole localization on limiting p-type conductivity in oxide and nitride semiconductors SciTech Connect Lyons, J. L.; Janotti, A.; Van de Walle, C. G. 2014-01-07 We examine how hole localization limits the effectiveness of substitutional acceptors in oxide and nitride semiconductors and explain why p-type doping of these materials has proven so difficult. Using hybrid density functional calculations, we find that anion-site substitutional impurities in AlN, GaN, InN, and ZnO lead to atomic-like states that localize on the impurity atom itself. Substitution with cation-site impurities, on the other hand, triggers the formation of polarons that become trapped on nearest-neighbor anions, generally leading to large ionization energies for these acceptors. Unlike shallow effective-mass acceptors, these two types of deep acceptors couple strongly with the lattice, significantly affecting the optical properties and severely limiting prospects for achieving p-type conductivity in these wide-band-gap materials. 5. 7 CFR 2201.9 - Limitation on the applicability of the definition of Local Television Broadcast Signals. Code of Federal Regulations, 2011 CFR 2011-01-01 ... Local Television Broadcast Signals. 2201.9 Section 2201.9 Agriculture Regulations of the Department of Agriculture (Continued) LOCAL TELEVISION LOAN GUARANTEE BOARD LOCAL TELEVISION LOAN GUARANTEE PROGRAM-PROGRAM REGULATIONS General § 2201.9 Limitation on the applicability of the definition of Local Television... 6. 7 CFR 2201.9 - Limitation on the applicability of the definition of Local Television Broadcast Signals. Code of Federal Regulations, 2013 CFR 2013-01-01 ... Local Television Broadcast Signals. 2201.9 Section 2201.9 Agriculture Regulations of the Department of Agriculture (Continued) LOCAL TELEVISION LOAN GUARANTEE BOARD LOCAL TELEVISION LOAN GUARANTEE PROGRAM-PROGRAM REGULATIONS General § 2201.9 Limitation on the applicability of the definition of Local Television... 7. 7 CFR 2201.9 - Limitation on the applicability of the definition of Local Television Broadcast Signals. Code of Federal Regulations, 2012 CFR 2012-01-01 ... Local Television Broadcast Signals. 2201.9 Section 2201.9 Agriculture Regulations of the Department of Agriculture (Continued) LOCAL TELEVISION LOAN GUARANTEE BOARD LOCAL TELEVISION LOAN GUARANTEE PROGRAM-PROGRAM REGULATIONS General § 2201.9 Limitation on the applicability of the definition of Local Television... 8. 7 CFR 2201.9 - Limitation on the applicability of the definition of Local Television Broadcast Signals. Code of Federal Regulations, 2010 CFR 2010-01-01 ... Local Television Broadcast Signals. 2201.9 Section 2201.9 Agriculture Regulations of the Department of Agriculture (Continued) LOCAL TELEVISION LOAN GUARANTEE BOARD LOCAL TELEVISION LOAN GUARANTEE PROGRAM-PROGRAM REGULATIONS General § 2201.9 Limitation on the applicability of the definition of Local Television... 9. 7 CFR 2201.9 - Limitation on the applicability of the definition of Local Television Broadcast Signals. Code of Federal Regulations, 2014 CFR 2014-01-01 ... Local Television Broadcast Signals. 2201.9 Section 2201.9 Agriculture Regulations of the Department of Agriculture (Continued) LOCAL TELEVISION LOAN GUARANTEE BOARD LOCAL TELEVISION LOAN GUARANTEE PROGRAM-PROGRAM REGULATIONS General § 2201.9 Limitation on the applicability of the definition of Local Television... 10. How colorful are fruits? Limited color diversity in fleshy fruits on local and global scales. PubMed Stournaras, Kalliope E; Lo, Eugenia; Böhning-Gaese, Katrin; Cazetta, Eliana; Dehling, D Matthias; Schleuning, Matthias; Stoddard, Mary Caswell; Donoghue, Michael J; Prum, Richard O; Schaefer, H Martin 2013-04-01 The colors of fleshy fruits are considered to be a signal to seed-dispersing animals, but their diversity remains poorly understood. Using an avian color space to derive a sensory morphospace for fruit color, we tested four hypotheses of fruit color diversity: fruit colors occupy a limited area of the color space; they are less diverse than flower colors; fruit colors within localities are similar to each other; and fruit color diversity reflects phylogeny. The global fruit color diversity of 948 primarily bird-dispersed plant species and the color diversity of localities were compared with null models of random, unconstrained evolution of fruit color. Fruit color diversity was further compared with the diversity of 1300 flower colors. Tests of phylogenetic effects on fruit color were used to assess the degree of correspondence with phylogeny. Global and local fruit color diversity was limited compared with null models and fruits have achieved only half the color diversity of flowers. Interestingly, we found little indication of phylogenetic conservatism. Constraints resulting from the chemical properties of pigments probably limit global fruit and flower color diversity. Different types of selection on fruits and flowers may further explain the smaller color diversity of fruits. 11. Upper limit of applicability of the local similarity theory in the stable atmospheric boundary layer NASA Astrophysics Data System (ADS) Grachev, A. A.; Andreas, E. L.; Fairall, C. W.; Guest, P. S.; Persson, P. O. G. 2012-04-01 The applicability of the classical Monin-Obukhov similarity theory (1954) has been limited by constant flux assumption, which is valid in a narrow range z/L < 0.1 in the stable boundary layer (SBL). Nieuwstadt (1984) extended the range of applicability of the original theory using the local scaling (height-dependent) in place of the surface scaling, but the limits of applicability of the local similarity theory in the SBL have been blurred. Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to clarify this issue. Based on spectral analysis of wind velocity and temperature fluctuations, it is shown that when both gradient Richardson number, Ri, and flux Richardson number, Rf, exceed a "critical value" about 0.2-0.25, inertial subrange associated with a Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in the supercritical regime but this is non-Kolmogorov turbulence and it decays rapidly with further increasing stability. The similarity theory is based on the turbulent fluxes in the high frequency part of the spectra associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band collapse along with the Kolmogorov energy cascade. Therefore, applicability of the local Monin-Obukhov similarity theory in the SBL is limited by inequalities Ri < Ri_cr and Rf < Rf_cr (however, Rf_cr = 0.2-0.25 is a primary threshold). Application of this prerequisite shows that both the flux-profile and flux-variances relationships follow to the classical Monin-Obukhov local z-less predictions after the irrelevant cases have been filtered out. 12. A Geometrical Approach to Bell's Theorem NASA Technical Reports Server (NTRS) Rubincam, David Parry 2000-01-01 Bell's theorem can be proved through simple geometrical reasoning, without the need for the Psi function, probability distributions, or calculus. The proof is based on N. David Mermin's explication of the Einstein-Podolsky-Rosen-Bohm experiment, which involves Stern-Gerlach detectors which flash red or green lights when detecting spin-up or spin-down. The statistics of local hidden variable theories for this experiment can be arranged in colored strips from which simple inequalities can be deduced. These inequalities lead to a demonstration of Bell's theorem. Moreover, all local hidden variable theories can be graphed in such a way as to enclose their statistics in a pyramid, with the quantum-mechanical result lying a finite distance beneath the base of the pyramid. 13. Rotationally invariant proof of Bell's theorem without inequalities SciTech Connect Cabello, Adan 2003-03-01 The singlet state of two spin-(3/2) particles allows a proof of Bell's theorem without inequalities with two distinguishing features: any local observable can be regarded as an Einstein-Podolsky-Rosen element of reality, and the contradiction with local realism occurs not only for some specific local observables but for any rotation whereof. 14. Tests of the lattice index theorem SciTech Connect Jordan, Gerald; Hoellwieser, Roman; Faber, Manfried; Heller, Urs M. 2008-01-01 We investigate the lattice index theorem and the localization of the zero modes for thick classical center vortices. For nonorientable spherical vortices, the index of the overlap Dirac operator differs from the topological charge although the traces of the plaquettes deviate only by a maximum of 1.5% from trivial plaquettes. This may be related to the fact that even in Landau gauge some links of these configuration are close to the nontrivial center elements. 15. Bayes' theorem in paleopathological diagnosis. PubMed Byers, Steven N; Roberts, Charlotte A 2003-05-01 The utility of Bayes' theorem in paleopathological diagnoses is explored. Since this theorem has been used heavily by modern clinical medicine, its usefulness in that field is described first. Next, the mechanics of the theorem are discussed, along with methods for deriving the prior probabilities needed for its application. Following this, the sources of these prior probabilities and their accompanying problems in paleopathology are considered. Finally, an application using prehistoric rib lesions is presented to demonstrate the utility of this method to paleopathology. 16. Recurrence theorems: A unified account SciTech Connect Wallace, David 2015-02-15 I discuss classical and quantum recurrence theorems in a unified manner, treating both as generalisations of the fact that a system with a finite state space only has so many places to go. Along the way, I prove versions of the recurrence theorem applicable to dynamics on linear and metric spaces and make some comments about applications of the classical recurrence theorem in the foundations of statistical mechanics. 17. Nanoplasmonics simulations at the basis set limit through completeness-optimized, local numerical basis sets SciTech Connect Rossi, Tuomas P. Sakko, Arto; Puska, Martti J.; Lehtola, Susi; Nieminen, Risto M. 2015-03-07 We present an approach for generating local numerical basis sets of improving accuracy for first-principles nanoplasmonics simulations within time-dependent density functional theory. The method is demonstrated for copper, silver, and gold nanoparticles that are of experimental interest but computationally demanding due to the semi-core d-electrons that affect their plasmonic response. The basis sets are constructed by augmenting numerical atomic orbital basis sets by truncated Gaussian-type orbitals generated by the completeness-optimization scheme, which is applied to the photoabsorption spectra of homoatomic metal atom dimers. We obtain basis sets of improving accuracy up to the complete basis set limit and demonstrate that the performance of the basis sets transfers to simulations of larger nanoparticles and nanoalloys as well as to calculations with various exchange-correlation functionals. This work promotes the use of the local basis set approach of controllable accuracy in first-principles nanoplasmonics simulations and beyond. 18. Nanoplasmonics simulations at the basis set limit through completeness-optimized, local numerical basis sets. PubMed Rossi, Tuomas P; Lehtola, Susi; Sakko, Arto; Puska, Martti J; Nieminen, Risto M 2015-03-07 We present an approach for generating local numerical basis sets of improving accuracy for first-principles nanoplasmonics simulations within time-dependent density functional theory. The method is demonstrated for copper, silver, and gold nanoparticles that are of experimental interest but computationally demanding due to the semi-core d-electrons that affect their plasmonic response. The basis sets are constructed by augmenting numerical atomic orbital basis sets by truncated Gaussian-type orbitals generated by the completeness-optimization scheme, which is applied to the photoabsorption spectra of homoatomic metal atom dimers. We obtain basis sets of improving accuracy up to the complete basis set limit and demonstrate that the performance of the basis sets transfers to simulations of larger nanoparticles and nanoalloys as well as to calculations with various exchange-correlation functionals. This work promotes the use of the local basis set approach of controllable accuracy in first-principles nanoplasmonics simulations and beyond. 19. Limitations in the spectral method for graph partitioning: Detectability threshold and localization of eigenvectors. PubMed Kawamoto, Tatsuro; Kabashima, Yoshiyuki 2015-06-01 Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both un-normalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit. 20. Limitations in the spectral method for graph partitioning: Detectability threshold and localization of eigenvectors NASA Astrophysics Data System (ADS) Kawamoto, Tatsuro; Kabashima, Yoshiyuki 2015-06-01 Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both un-normalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit. 1. Multidimensional Tauberian theorems for generalized functions NASA Astrophysics Data System (ADS) Drozhzhinov, Yu N. 2016-12-01 This is a brief survey of multidimensional Tauberian theorems for generalized functions. Included are theorems of Hardy-Littlewood type, Tauberian and Abelian comparison theorems of Keldysh type, theorems of Wiener type, and Tauberian theorems for generalized functions with values in Banach spaces. Bibliography: 58 titles. 2. Phase space localization for anti-de Sitter quantum mechanics and its zero curvature limit NASA Technical Reports Server (NTRS) Elgradechi, Amine M. 1993-01-01 Using techniques of geometric quantization and SO(sub 0)(3,2)-coherent states, a notion of optimal localization on phase space is defined for the quantum theory of a massive and spinning particle in anti-de Sitter space time. It is shown that this notion disappears in the zero curvature limit, providing one with a concrete example of the regularizing character of the constant (nonzero) curvature of the anti-de Sitter space time. As a byproduct a geometric characterization of masslessness is obtained. 3. Theorems on positive data: on the uniqueness of NMF. PubMed Laurberg, Hans; Christensen, Mads Graesbøll; Plumbley, Mark D; Hansen, Lars Kai; Jensen, Søren Holdt 2008-01-01 We investigate the conditions for which nonnegative matrix factorization (NMF) is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors. 4. Theorems on Positive Data: On the Uniqueness of NMF PubMed Central Laurberg, Hans; Christensen, Mads Græsbøll; Plumbley, Mark D.; Hansen, Lars Kai; Jensen, Søren Holdt 2008-01-01 We investigate the conditions for which nonnegative matrix factorization (NMF) is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors. PMID:18497868 5. Wigner-Araki-Yanase theorem beyond conservation laws NASA Astrophysics Data System (ADS) Tukiainen, Mikko 2017-01-01 The ability to measure every quantum observable is ensured by a fundamental result in quantum measurement theory. Nevertheless, additive conservation laws associated with physical symmetries, such as the angular momentum conservation, may lead to restrictions on the measurability of the observables. Such limitations are imposed by the theorem of Wigner, Araki, and Yanase (WAY). In this paper a formulation of the WAY theorem is presented rephrasing the measurability limitations in terms of quantum incompatibility. This broader mathematical basis enables us to both capture and generalize the WAY theorem by allowing us to drop the assumptions of additivity and even conservation of the involved quantities. Moreover, we extend the WAY theorem to the general level of positive operator-valued measures. 6. Limiter DOEpatents Cohen, S.A.; Hosea, J.C.; Timberlake, J.R. 1984-10-19 A limiter with a specially contoured front face is provided. The front face of the limiter (the plasma-side face) is flat with a central indentation. In addition, the limiter shape is cylindrically symmetric so that the limiter can be rotated for greater heat distribution. This limiter shape accommodates the various power scrape-off distances lambda p, which depend on the parallel velocity, V/sub parallel/, of the impacting particles. 7. Crop response to localized organic amendment in soils with limiting physical properties NASA Astrophysics Data System (ADS) Lordan, Joan; Pascual, Miquel; Fonseca, Francisco; Villar, Josep Maria; Montilla, Victor; Papió, Josep; Rufat, Josep 2013-04-01 This 2-year study evaluated the use of rice husk as a localized organic amendment in a soil with limiting physical properties. The research was conducted in a commercial peach orchard planted in 2011 using a ridge planting system. Six soil and water management treatments were evaluated in 18 experimental units, which were set up in the field using a randomized complete block design. The treatments were compared both in terms of soil physical properties and crop response. Soil amendment with rice husk was the most effective technique. It improved soil conditions (soil infiltration and soil porosity), providing a better soil environment for root activity and thereby resulted in better crop performance. Concerning growth parameters, the amended treatment presented the highest overall values without negatively affecting crop water status. These techniques were suitable for mitigating the effects of soils with limiting physical conditions. Localized applications of amendments, as proposed in this work, imply an important reduction in application rates. It is important to consider an efficient use of by-products since there is a growing interest in industrial and agronomical exploitations. 8. A locally-blazed ant trail achieves efficient collective navigation despite limited information PubMed Central Fonio, Ehud; Heyman, Yael; Boczkowski, Lucas; Gelblum, Aviram; Kosowski, Adrian; Korman, Amos; Feinerman, Ofer 2016-01-01 Any organism faces sensory and cognitive limitations which may result in maladaptive decisions. Such limitations are prominent in the context of groups where the relevant information at the individual level may not coincide with collective requirements. Here, we study the navigational decisions exhibited by Paratrechina longicornis ants as they cooperatively transport a large food item. These decisions hinge on the perception of individuals which often restricts them from providing the group with reliable directional information. We find that, to achieve efficient navigation despite partial and even misleading information, these ants employ a locally-blazed trail. This trail significantly deviates from the classical notion of an ant trail: First, instead of systematically marking the full path, ants mark short segments originating at the load. Second, the carrying team constantly loses the guiding trail. We experimentally and theoretically show that the locally-blazed trail optimally and robustly exploits useful knowledge while avoiding the pitfalls of misleading information. DOI: http://dx.doi.org/10.7554/eLife.20185.001 PMID:27815944 9. Critical properties of the Anderson localization transition and the high-dimensional limit NASA Astrophysics Data System (ADS) Tarquini, E.; Biroli, G.; Tarzia, M. 2017-03-01 In this paper we present a thorough study of transport, spectral, and wave-function properties at the Anderson localization critical point in spatial dimensions d =3 , 4, 5, 6. Our aim is to analyze the dimensional dependence and to assess the role of the d →∞ limit provided by Bethe lattices and treelike structures. Our results strongly suggest that the upper critical dimension of Anderson localization is infinite. Furthermore, we find that dU=∞ is a much better starting point compared to dL=2 to describe even three-dimensional systems. We find that critical properties and finite-size scaling behavior approach by increasing d those found for Bethe lattices: the critical state becomes an insulator characterized by Poisson statistics and corrections to the thermodynamics limit become logarithmic in the number N of lattice sites. In the conclusion, we present physical consequences of our results, propose connections with the nonergodic delocalized phase suggested for the Anderson model on infinite-dimensional lattices, and discuss perspectives for future research studies. 10. Propagule Limitation, Disparate Habitat Quality, and Variation in Phenotypic Selection at a Local Species Range Boundary PubMed Central Moore, Kara A.; Stanton, Maureen L. 2014-01-01 Adaptation to novel conditions beyond current range boundaries requires the presence of suitable sites within dispersal range, but may be impeded when emigrants encounter poor habitat and sharply different selection pressures. We investigated fine-scale spatial heterogeneity in ecological dynamics and selection at a local population boundary of the annual plant Gilia tricolor. In two years, we planted G. tricolor seeds in core habitat, margin habitat at the edge of the local range, and exterior habitat in order to measure spatial and temporal variation in habitat quality, opportunity for selection, and selection on phenotypic traits. We found a striking decline in average habitat quality with distance from the population core, yet some migrant seeds were successful in suitable, unoccupied microsites at and beyond the range boundary. Total and direct selection on four out of five measured phenotypic traits varied across habitat zones, as well as between years. Moreover, the margin habitat often exerted unique selection pressures that were not intermediate between core and exterior habitats. This study reveals that a combination of ecological and evolutionary forces, including propagule limitation, variation in habitat quality and spatial heterogeneity in phenotypic selection may reduce opportunities for adaptive range expansion, even across a very local population boundary. PMID:24717472 11. Propagule limitation, disparate habitat quality, and variation in phenotypic selection at a local species range boundary. PubMed Moore, Kara A; Stanton, Maureen L 2014-01-01 Adaptation to novel conditions beyond current range boundaries requires the presence of suitable sites within dispersal range, but may be impeded when emigrants encounter poor habitat and sharply different selection pressures. We investigated fine-scale spatial heterogeneity in ecological dynamics and selection at a local population boundary of the annual plant Gilia tricolor. In two years, we planted G. tricolor seeds in core habitat, margin habitat at the edge of the local range, and exterior habitat in order to measure spatial and temporal variation in habitat quality, opportunity for selection, and selection on phenotypic traits. We found a striking decline in average habitat quality with distance from the population core, yet some migrant seeds were successful in suitable, unoccupied microsites at and beyond the range boundary. Total and direct selection on four out of five measured phenotypic traits varied across habitat zones, as well as between years. Moreover, the margin habitat often exerted unique selection pressures that were not intermediate between core and exterior habitats. This study reveals that a combination of ecological and evolutionary forces, including propagule limitation, variation in habitat quality and spatial heterogeneity in phenotypic selection may reduce opportunities for adaptive range expansion, even across a very local population boundary. 12. Limited availability of ZBP1 restricts axonal mRNA localization and nerve regeneration capacity. PubMed Donnelly, Christopher J; Willis, Dianna E; Xu, Mei; Tep, Chhavy; Jiang, Chunsu; Yoo, Soonmoon; Schanen, N Carolyn; Kirn-Safran, Catherine B; van Minnen, Jan; English, Arthur; Yoon, Sung Ok; Bassell, Gary J; Twiss, Jeffery L 2011-09-30 Subcellular localization of mRNAs is regulated by RNA-protein interactions. Here, we show that introduction of a reporter mRNA with the 3'UTR of β-actin mRNA competes with endogenous mRNAs for binding to ZBP1 in adult sensory neurons. ZBP1 is needed for axonal localization of β-actin mRNA, and introducing GFP with the 3'UTR of β-actin mRNA depletes axons of endogenous β-actin and GAP-43 mRNAs and attenuates both in vitro and in vivo regrowth of severed axons. Consistent with limited levels of ZBP1 protein in adult neurons, mice heterozygous for the ZBP1 gene are haploinsufficient for axonal transport of β-actin and GAP-43 mRNAs and for regeneration of peripheral nerve. Exogenous ZBP1 can rescue the RNA transport deficits, but the axonal growth deficit is only rescued if the transported mRNAs are locally translated. These data support a direct role for ZBP1 in transport and translation of mRNA cargos in axonal regeneration in vitro and in vivo. 13. Roo: A parallel theorem prover SciTech Connect Lusk, E.L.; McCune, W.W.; Slaney, J.K. 1991-11-01 We describe a parallel theorem prover based on the Argonne theorem-proving system OTTER. The parallel system, called Roo, runs on shared-memory multiprocessors such as the Sequent Symmetry. We explain the parallel algorithm used and give performance results that demonstrate near-linear speedups on large problems. 14. Theorem of Mystery: Part 1 ERIC Educational Resources Information Center Lopez-Real, Francis 2008-01-01 While the author was searching the web, he came across an article by Michael Keyton of IMSA (Illinois Mathematics and Science Academy) called "Theorems of mystery". The phrase is Keyton's own, and he defines such a theorem as "a result that has considerable structure with minimal hypotheses." The simplest of his 10 examples is one that many… 15. The 1965 Penrose singularity theorem NASA Astrophysics Data System (ADS) Senovilla, José M. M.; Garfinkle, David 2015-06-01 We review the first modern singularity theorem, published by Penrose in 1965. This is the first genuine post-Einsteinian result in general relativity, where the fundamental and fruitful concept of the closed trapped surface was introduced. We include historical remarks, an appraisal of the theorem's impact, and relevant current and future work that belongs to its legacy. 16. Geometry of the Adiabatic Theorem ERIC Educational Resources Information Center Lobo, Augusto Cesar; Ribeiro, Rafael Antunes; Ribeiro, Clyffe de Assis; Dieguez, Pedro Ruas 2012-01-01 We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.… 17. Equivalence theorem in effective theories NASA Astrophysics Data System (ADS) Chicherin, D.; Gorbenko, V.; Vereshagin, V. 2011-11-01 The famous equivalence theorem is reexamined in order to make it applicable to the case of effective theories. We slightly modify the formulation of this theorem and prove it based on the notion of the generating functional for Green functions. This allows one to trace (directly in terms of graphs) the mutual cancellation of different groups of contributions. 18. A Decomposition Theorem for Finite Automata. ERIC Educational Resources Information Center Santa Coloma, Teresa L.; Tucci, Ralph P. 1990-01-01 Described is automata theory which is a branch of theoretical computer science. A decomposition theorem is presented that is easier than the Krohn-Rhodes theorem. Included are the definitions, the theorem, and a proof. (KR) 19. Limiter DOEpatents Cohen, Samuel A.; Hosea, Joel C.; Timberlake, John R. 1986-01-01 A limiter with a specially contoured front face accommodates the various power scrape-off distances .lambda..sub.p, which depend on the parallel velocity, V.sub..parallel., of the impacting particles. The front face of the limiter (the plasma-side face) is flat with a central indentation. In addition, the limiter shape is cylindrically symmetric so that the limiter can be rotated for greater heat distribution. 20. Localized reversal of the perpendicular velocity in Tore Supra ohmic, L-mode, limited plasmas NASA Astrophysics Data System (ADS) Trier, E.; Hennequin, P.; Gürcan, Ö. D.; Sabot, R.; Bucalossi, J.; Guimarães-Filho, Z. O.; Bourdelle, C.; Clairet, F.; Falchetto, G.; Fenzi, C.; Garbet, X.; Maget, P.; Vermare, L.; The Tore Supra Team 2017-04-01 In Tore Supra plasmas, the perpendicular velocity measured by Doppler reflectometry was observed to reverse in a localized zone close to a normalized radius ∼0.5–0.6, changing from a negative value (corresponding to a negative radial electric field E r) to a positive value ({{E}\\text{r}}>0 ). This occurs in L-mode, ohmic plasmas with a negligible external momentum input, a non-circular limited cross-section, and an edge safety factor close to 3. This reversal is favoured by a decrease in the magnetic field, or an increase in density. It is accompanied by a characteristic behaviour of the MHD activity signal, whose amplitude decrease during a ramp-down of the edge safety factor as it approaches {{q}a}∼ 3.1 –3.2. A m/n = 2/1 mode is involved in the mechanism causing these observations. 1. Spatial resolution limits for the localization of noise sources using direct sound mapping NASA Astrophysics Data System (ADS) Fernandez Comesaña, D.; Holland, K. R.; Fernandez-Grande, E. 2016-08-01 One of the main challenges arising from noise and vibration problems is how to identify the areas of a device, machine or structure that produce significant acoustic excitation, i.e. the localization of main noise sources. The direct visualization of sound, in particular sound intensity, has extensively been used for many years to locate sound sources. However, it is not yet well defined when two sources should be regarded as resolved by means of direct sound mapping. This paper derives the limits of the direct representation of sound pressure, particle velocity and sound intensity by exploring the relationship between spatial resolution, noise level and geometry. The proposed expressions are validated via simulations and experiments. It is shown that particle velocity mapping yields better results for identifying closely spaced sound sources than sound pressure or sound intensity, especially in the acoustic near-field. 2. Non-local meta-conformal invariance in diffusion-limited erosion NASA Astrophysics Data System (ADS) Henkel, Malte 2016-12-01 The non-stationary relaxation and physical ageing in the diffusion-limited erosion process (dle) is studied through the exact solution of its Langevin equation, in d spatial dimensions. The dynamical exponent z = 1, the growth exponent β =\\max (0,(1-d)/2) and the ageing exponents a=b=d-1 and {λ }C={λ }R=d are found. In d = 1 spatial dimension, a new representation of the meta-conformal Lie algebra, isomorphic to {sl}(2,{{R}})\\oplus {sl}(2,{{R}}), acts as a dynamical symmetry of the noise-averaged dle Langevin equation. Its infinitesimal generators are non-local in space. The exact form of the full time-space dependence of the two-time response function of dle is reproduced for d = 1 from this symmetry. The relationship to the terrace-step-kink model of vicinal surfaces is discussed. 3. Avoiding the local-minimum problem in multi-agent systems with limited sensing and communication NASA Astrophysics Data System (ADS) Okamoto, Makiko; Akella, Maruthi R. 2016-06-01 In this paper, we consider a control problem for nonholonomic multi-agent systems in which agents and obstacles operate within a circular-shaped work area. We assume that agents only have limited sensing and communication ranges. We propose a novel control scheme using potential functions that drives agents from the initial to the goal configuration while avoiding collision with other agents, obstacles, and the boundary of the work area. The control scheme employs an avoidance strategy that ensures that the agents are never trapped at local minima that are typically encountered with most potential function-based approaches. A numerical simulation is presented to demonstrate the validity and effectiveness of the proposed control scheme. 4. Generalized Sampling Theorem for Bandpass Signals NASA Astrophysics Data System (ADS) Prokes, Ales 2006-12-01 The reconstruction of an unknown continuously defined function[InlineEquation not available: see fulltext.] from the samples of the responses of[InlineEquation not available: see fulltext.] linear time-invariant (LTI) systems sampled by the[InlineEquation not available: see fulltext.]th Nyquist rate is the aim of the generalized sampling. Papoulis (1977) provided an elegant solution for the case where[InlineEquation not available: see fulltext.] is a band-limited function with finite energy and the sampling rate is equal to[InlineEquation not available: see fulltext.] times cutoff frequency. In this paper, the scope of the Papoulis theory is extended to the case of bandpass signals. In the first part, a generalized sampling theorem (GST) for bandpass signals is presented. The second part deals with utilizing this theorem for signal recovery from nonuniform samples, and an efficient way of computing images of reconstructing functions for signal recovery is discussed. 5. Temporomandibular joint injection with corticosteroid and local anesthetic for limited mouth opening. PubMed Samiee, Aveed; Sabzerou, Daniel; Edalatpajouh, Faraz; Clark, Glenn T; Ram, Saravanan 2011-09-01 It is unclear whether temporomandibular joint (TMJ) injections with local anesthetic and corticosteroid are an effective first-line management modality for patients with limited mouth opening. The purpose of this study was to determine the effectiveness of TMJ injections in patients with disc displacement without reduction (DDWOR), i.e. closed lock, at the University of Southern California Orofacial Pain and Oral Medicine Center. A retrospective chart review was conducted using a database of over 4000 patient records from 2003-2010. We identified 17 patients (16 female; 1 male) between the ages of 16 and 70 years who had been diagnosed with DDWOR and received a TMJ injection. Active mouth opening before injection ranged between 15 and 40 mm (average 29 mm), and active mouth opening after injection and manual mobilization ranged between 25 and 50 mm (average 39 mm). The average increase in mouth opening after injection and manual mobilization was 10 mm (P = 0.0004). TMJ injection with corticosteroid and local anesthetic is suitable as an alternative first-line management modality for DDWOR. 6. Metapopulation dynamics override local limits on long-term parasite persistence. PubMed Ram, Karthik; Preisser, Evan L; Gruner, Daniel S; Strong, Donald R 2008-12-01 A simple null model, particularly germane to small and vulnerable organisms such as parasites, is that local conditions set a stage upon which larger-scale dynamics play out. Soil moisture strongly influences survival of entomopathogenic nematodes (EPN), which in turn drive trophic cascades by protecting vegetation from root-feeding herbivores. In this study, we examine the mechanisms responsible for patchy occurrence of an entomopathogenic nematode, Heterorhabditis marelatus, in a California coastal prairie. One hypothesis proposes that biotic factors such as competition and natural enemies could regulate occurrence of EPN populations. We found that fungi and other enemies of EPN, although locally potent, did not explain the patterns of incidence across sites. Abiotic factors also have strong effects on EPN persistence, especially for vulnerable free-living stages. Thus, we tested the hypothesis that patchy occurrence of EPN on a large landscape was driven by differences in soil moisture. Our research uses long-term data on nematode incidence in combination with a landscape-level experiment to demonstrate the lack of a correlation between soil moisture and long-term persistence. A year-long experiment showed EPN mortality was weakly correlated with soil moisture among our study sites. Thirteen years of data, however, showed that colonization rates were highly correlated with long-term persistence. Sites with highest long-term persistence experienced the highest rates of rhizosphere colonization, extinction, and turnover. As a result, we concluded that metapopulation dynamics override limitations set by local and short-term abiotic conditions to determine long-term persistence in this parasite-driven trophic cascade. 7. Analogues of Chernoff's theorem and the Lie-Trotter theorem SciTech Connect Neklyudov, Alexander Yu 2009-10-31 This paper is concerned with the abstract Cauchy problem .x=Ax, x(0)=x{sub 0} element of D(A), where A is a densely defined linear operator on a Banach space X. It is proved that a solution x( {center_dot} ) of this problem can be represented as the weak limit lim {sub n{yields}}{sub {infinity}}{l_brace}F(t/n){sup n}x{sub 0}{r_brace}, where the function F:[0,{infinity}){yields}L(X) satisfies the equality F'(0)y=Ay, y element of D(A), for a natural class of operators. As distinct from Chernoff's theorem, the existence of a global solution to the Cauchy problem is not assumed. Based on this result, necessary and sufficient conditions are found for the linear operator C to be closable and for its closure to be the generator of a C{sub 0}-semigroup. Also, we obtain new criteria for the sum of two generators of C{sub 0}-semigroups to be the generator of a C{sub 0}-semigroup and for the Lie-Trotter formula to hold. Bibliography: 13 titles. 8. Multi Sensor Fusion Framework for Indoor-Outdoor Localization of Limited Resource Mobile Robots PubMed Central Marín, Leonardo; Vallés, Marina; Soriano, Ángel; Valera, Ángel; Albertos, Pedro 2013-01-01 This paper presents a sensor fusion framework that improves the localization of mobile robots with limited computational resources. It employs an event based Kalman Filter to combine the measurements of a global sensor and an inertial measurement unit (IMU) on an event based schedule, using fewer resources (execution time and bandwidth) but with similar performance when compared to the traditional methods. The event is defined to reflect the necessity of the global information, when the estimation error covariance exceeds a predefined limit. The proposed experimental platforms are based on the LEGO Mindstorm NXT, and consist of a differential wheel mobile robot navigating indoors with a zenithal camera as global sensor, and an Ackermann steering mobile robot navigating outdoors with a SBG Systems GPS accessed through an IGEP board that also serves as datalogger. The IMU in both robots is built using the NXT motor encoders along with one gyroscope, one compass and two accelerometers from Hitecnic, placed according to a particle based dynamic model of the robots. The tests performed reflect the correct performance and low execution time of the proposed framework. The robustness and stability is observed during a long walk test in both indoors and outdoors environments. PMID:24152933 9. Multi sensor fusion framework for indoor-outdoor localization of limited resource mobile robots. PubMed Marín, Leonardo; Vallés, Marina; Soriano, Ángel; Valera, Ángel; Albertos, Pedro 2013-10-21 This paper presents a sensor fusion framework that improves the localization of mobile robots with limited computational resources. It employs an event based Kalman Filter to combine the measurements of a global sensor and an inertial measurement unit (IMU) on an event based schedule, using fewer resources (execution time and bandwidth) but with similar performance when compared to the traditional methods. The event is defined to reflect the necessity of the global information, when the estimation error covariance exceeds a predefined limit. The proposed experimental platforms are based on the LEGO Mindstorm NXT, and consist of a differential wheel mobile robot navigating indoors with a zenithal camera as global sensor, and an Ackermann steering mobile robot navigating outdoors with a SBG Systems GPS accessed through an IGEP board that also serves as datalogger. The IMU in both robots is built using the NXT motor encoders along with one gyroscope, one compass and two accelerometers from Hitecnic, placed according to a particle based dynamic model of the robots. The tests performed reflect the correct performance and low execution time of the proposed framework. The robustness and stability is observed during a long walk test in both indoors and outdoors environments. 10. Upper limits on the probability of an interstellar civilization arising in the local Solar neighbourhood NASA Astrophysics Data System (ADS) Cartin, Daniel 2015-10-01 At this point in time, there is very little empirical evidence on the likelihood of a space-faring species originating in the biosphere of a habitable world. However, there is a tension between the expectation that such a probability is relatively high (given our own origins on Earth), and the lack of any basis for believing the Solar System has ever been visited by an extraterrestrial colonization effort. From the latter observational fact, this paper seeks to place upper limits on the probability of an interstellar civilization arising on a habitable planet in its stellar system, using a percolation model to simulate the progress of such a hypothetical civilization's colonization efforts in the local Solar neighbourhood. To be as realistic as possible, the actual physical positions and characteristics of all stars within 40 parsecs of the Solar System are used as possible colony sites in the percolation process. If an interstellar civilization is very likely to have such colonization programmes, and they can travel over large distances, then the upper bound on the likelihood of such a species arising per habitable world is of the order of 10-3 on the other hand, if civilizations are not prone to colonize their neighbours, or do not travel very far, then the upper limiting probability is much larger, even of order one. 11. Turbulence, flows and edge localized mode (ELM) dynamics in limiter H-mode plasmas in TEXTOR NASA Astrophysics Data System (ADS) Soldatov, S.; Krämer-Flecken, A.; Kantor, M.; Unterberg, B.; Sun, Y.; Van Oost, G.; Reiter, D.; TEXTOR Team 2010-08-01 The turbulence, plasma flow and edge localized mode (ELM) dynamics in the limiter H-mode TEXTOR plasmas are investigated. Properties of both ambient turbulence within 0 < k⊥ < 4.2 cm-1 and coherent modes are studied on the ELM time scale in detail. The turbulence level near the pedestal is shown to evolve several times with the period of ELMs. Within the inter-ELM period the 'silent stage' is found which is characterized by an extremely low (below that for Ohmic plasmas) turbulence level and a phase growth in the reflectometry signal. The silent stage is associated with the quasi-steady state when the pedestal is formed and confinement is improved between two successive ELMs. Quasi-coherent density oscillations near the pedestal region with m ≈ 3, 5, 16 and 38 are measured with correlation reflectometry. Low-m modes are found to reveal the signatures of precursor mode. At first, the radial structure of the rotation shear and radial electric field Er in limiter H-mode in TEXTOR is presented. The characteristic negative electric field well with the sharp gradient ∇Er ≈ 250 V cm-2 at ≈2 cm inside separatrix is resolved. The Er × B rotation profile defines both the resulting plasma rotation in the electron diamagnetic drift direction and a significant rotation shear near the separatrix which exceeds the decorrelation rate of ambient turbulence by several times. 12. Tau leaping of stiff stochastic chemical systems via local central limit approximation NASA Astrophysics Data System (ADS) Yang, Yushu; Rathinam, Muruhan 2013-06-01 Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system S1↔S2. The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the LCLA-τ. The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solve these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that LCLA-τ performs favorably for stiff systems and that the LCLA-τ is also able to capture bimodal distributions unlike the CLA itself. The proposed LCLA-τ method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation. 13. 20 CFR 661.310 - Under what limited conditions may a Local Board directly be a provider of core services... Code of Federal Regulations, 2011 CFR 2011-04-01 ... 20 Employees' Benefits 3 2011-04-01 2011-04-01 false Under what limited conditions may a Local...-Stop Operator? 661.310 Section 661.310 Employees' Benefits EMPLOYMENT AND TRAINING ADMINISTRATION, DEPARTMENT OF LABOR STATEWIDE AND LOCAL GOVERNANCE OF THE WORKFORCE INVESTMENT SYSTEM UNDER TITLE I OF... 14. 20 CFR 661.310 - Under what limited conditions may a Local Board directly be a provider of core services... Code of Federal Regulations, 2012 CFR 2012-04-01 ... 20 Employees' Benefits 4 2012-04-01 2012-04-01 false Under what limited conditions may a Local...-Stop Operator? 661.310 Section 661.310 Employees' Benefits EMPLOYMENT AND TRAINING ADMINISTRATION, DEPARTMENT OF LABOR (CONTINUED) STATEWIDE AND LOCAL GOVERNANCE OF THE WORKFORCE INVESTMENT SYSTEM UNDER... 15. Comparison theorems for causal diamonds NASA Astrophysics Data System (ADS) Berthiere, Clément; Gibbons, Gary; Solodukhin, Sergey N. 2015-09-01 We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the redshift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction, and it takes its maximal value at the center. As a by-product of our discussion we rederive Bishop's inequality without assuming the positivity of the spatial Ricci tensor. We then generalize our considerations to arbitrary, static and not necessarily spherically symmetric, asymptotically flat spacetimes. In the case of spacetimes with a horizon our generalization involves the so-called domain of dependence. The respective volume, expressed in terms of the duration measured by a distant observer compared with the volume of the domain in Minkowski spacetime, exhibits behaviors which differ if d =4 or d >4 . This peculiarity of four dimensions is due to the logarithmic subleading term in the asymptotic expansion of the metric near infinity. In terms of the invariant duration measured by a comoving observer associated with the diamond we establish an inequality which is universal for all d . We suggest some possible applications of our results including comparison theorems for entanglement entropy, causal set theory, and fundamental limits on computation. 16. The Digital Morphological Sampling Theorem NASA Astrophysics Data System (ADS) Haralick, Robert M.; Zhuang, Xinhua; Lin, Charlotte; Lee, James 1988-02-01 There are potential industrial applications for any methodology which inherently reduces processing time and cost and yet produces results sufficiently close to the result of full processing. It is for this reason that a morphological sampling theorem is important. The morphological sampling theorem described in this paper states: (1) how a digital image must be morphologically filtered before sampling in order to preserve the relevant information after sampling; (2) to what precision an appropriately morphologically filtered image can be reconstructed after sampling; and (3) the relationship between morphologically operating before sampling and the more computationally efficient scheme of morphologically operating on the sampled image with a sampled structuring element. The digital sampling theorem is developed first for the case of binary morphology and then it is extended to gray scale morphology through the use of the umbra homomorphism theorems. 17. Racial Isolation, Poverty, and the Limits of Local Control in Oakland ERIC Educational Resources Information Center Noguera, Pedro A. 2004-01-01 Historically, schools in the United States have been governed at the local level by elected school boards, and finances have been raised primarily through local property taxes. While local control theoretically allows for greater responsiveness to local concerns, it does not take into account the vast inequality among and between communities in… 18. Factor and Remainder Theorems: An Appreciation ERIC Educational Resources Information Center Weiss, Michael 2016-01-01 The high school curriculum sometimes seems like a disconnected collection of topics and techniques. Theorems like the factor theorem and the remainder theorem can play an important role as a conceptual "glue" that holds the curriculum together. These two theorems establish the connection between the factors of a polynomial, the solutions… 19. Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes SciTech Connect Woolgar, Eric; Wylie, William 2016-02-15 We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the “pure Bakry-Émery” N = ∞ case with f uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite N-values N ∈ (n, ∞) and N ∈ (−∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (−∞, 1] and N = ∞, we are able to replace the boundedness of f by a weaker condition on the integral of f along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when N = 1, the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in a prior work on the N-Bakry-Émery curvature in Riemannian signature when N = 1 and appears to be a general feature. 20. Basis set limit and systematic errors in local-orbital based all-electron DFT NASA Astrophysics Data System (ADS) Blum, Volker; Behler, Jörg; Gehrke, Ralf; Reuter, Karsten; Scheffler, Matthias 2006-03-01 With the advent of efficient integration schemes,^1,2 numeric atom-centered orbitals (NAO's) are an attractive basis choice in practical density functional theory (DFT) calculations of nanostructured systems (surfaces, clusters, molecules). Though all-electron, the efficiency of practical implementations promises to be on par with the best plane-wave pseudopotential codes, while having a noticeably higher accuracy if required: Minimal-sized effective tight-binding like calculations and chemically accurate all-electron calculations are both possible within the same framework; non-periodic and periodic systems can be treated on equal footing; and the localized nature of the basis allows in principle for O(N)-like scaling. However, converging an observable with respect to the basis set is less straightforward than with competing systematic basis choices (e.g., plane waves). We here investigate the basis set limit of optimized NAO basis sets in all-electron calculations, using as examples small molecules and clusters (N2, Cu2, Cu4, Cu10). meV-level total energy convergence is possible using <=50 basis functions per atom in all cases. We also find a clear correlation between the errors which arise from underconverged basis sets, and the system geometry (interatomic distance). ^1 B. Delley, J. Chem. Phys. 92, 508 (1990), ^2 J.M. Soler et al., J. Phys.: Condens. Matter 14, 2745 (2002). 1. Limits of the measurability of the local quantum electromagnetic-field amplitude NASA Astrophysics Data System (ADS) Compagno, G.; Persico, F. 1998-03-01 The precision with which the amplitude of the free electromagnetic field can be measured locally in QED is evaluated by analyzing a well-known gedanken experiment originally proposed by Bohr and Rosenfeld (BR). The analysis is performed by applying standard theoretical techniques familiar in quantum optics. The main result obtained for the precision is significantly different from the generally accepted Bohr-Rosenfeld result. This leads to questioning the widely accepted notion of the compensating field, fostered by these authors. A misconception at the origin of this notion is pointed out by a careful investigation of the self-force acting on the apparatus designed to measure the field. The correct expression for this self-force is found to be at variance with that proposed by Bohr and Rosenfeld and generally accepted. It is argued that, as a consequence of this new expression and in contrast with the generally accepted view, no compensating force of nonelectromagnetic nature is required in order to perform measurements of the quantum field amplitude with any desired accuracy. It is shown that the only limitations to the precision of the measurement, in the BR gedanken experiment, arise from the time-energy uncertainty principle, as well as from the finite dimensions of the measuring apparatus. 2. Systemic protein delivery by muscle-gene transfer is limited by a local immune response PubMed Central Wang, Lixin; Dobrzynski, Eric; Schlachterman, Alexander; Cao, Ou; Herzog, Roland W. 2005-01-01 Adeno-associated viral (AAV) vectors have been successfully used for therapeutic expression of systemic transgene products (such as factor IX or erythropoietin) following in vivo administration to skeletal muscle of animal models of inherited hematologic disorders. However, an immune response may be initiated if the transgene product represents a neoantigen. Here, we use ovalbumin (OVA) as a model antigen and demonstrate immune-mediated elimination of expression on muscle-directed AAV-2 gene transfer. Administration to immune competent mice resulted in transient systemic OVA expression. Within 10 days, OVA-specific T-helper cells had been activated in draining lymph nodes, an inflammatory immune response ensued, and OVA-expressing muscle fibers were destroyed by a cytotoxic CD8+ T-cell response. Use of a muscle-specific promoter did not prevent this immune response. Adoptively transferred CD4+ cells transgenic for a T-cell receptor specific to OVA peptide-major histocompatibility complex class II showed antigen-specific, vector dose-dependent proliferation confined to the draining lymph nodes of AAV-OVA–transduced muscle within 5 days after gene transfer and subsequently participated in lymphocytic infiltration of transduced muscle. This study documents that a local immune response limits sustained expression of a secreted protein in muscle gene transfer, a finding that may have consequences for design of clinical protocols. PMID:15713796 3. Freezing of tissue-limits for the autoradiographic localization of diffusible substances. PubMed Frederik, P M; Busing, W M 1979-11-01 Frozen thin sections and sections from freeze-dried and embedded tissue are used for the autoradiographic localization of diffusible substances at the electron microscope level. The presence of ice crystals in such sections may limit the autoradiographic resolution. Ice crystals are formed during freezing and may grow during subsequent processing of tissue. The contribution of ice crystal growth to the final image was estimated by measuring the distribution of the ice crystal sizes in freeze-etch replicas and in sections from freeze-dried and embedded tissues. A surface layer (10-15 mu) without visible ice crystals was present in both preparations. Beneath this surface layer the diameter of ice crystals increased towards the interior with the same relationship between crystal size and distance from the surface in the freeze-etch preparation as in the freeze-dry preparation. Ice crystal growth occurring during a much longer time during freeze-drying compared to freeze-etching does not significantly contribute to the final image in the electron microscope. The formation of ice crystals during freezing determines to a large extent the image (and therefore the autoradiographic resolution) of freeze-dry preparations and this probably holds also for thin cryosections of which examples are given. 4. Spin groups of super metrics and a theorem of Rogers NASA Astrophysics Data System (ADS) Fulp, Ronald 2017-01-01 We derive the canonical forms of super Riemannian metrics and the local isometry groups of such metrics. For certain super metrics we also compute the simply connected covering groups of the local isometry groups and interpret these as local spin groups of the super metric. Super metrics define reductions OSg of the relevant frame bundle. When principal bundles S˜g exist with structure group the simply connected covering group G ˜ of the structure group of OSg , representations of G ˜ define vector bundles associated to S˜g whose sections are "spinor fields" associated with the super metric g . Using a generalization of a Theorem of Rogers, which is itself one of the main results of this paper, we show that for super metrics we call body reducible, each such simply connected covering group G ˜ is a super Lie group with a conventional super Lie algebra as its corresponding super Lie algebra. Some of our results were known to DeWitt (1984) using formal Grassmann series and others were known by Rogers using finitely many Grassmann generators and passing to a direct limit. We work exclusively in the category of G∞ supermanifolds with G∞ mappings. Our supernumbers are infinite series of products of Grassmann generators subject to convergence in the ℓ1 norm introduced by Rogers (1980, 2007). 5. Enhancing Local Climate Projections of Precipitation: Assets and Limitations of Quantile Mapping Techniques for Statistical Downscaling NASA Astrophysics Data System (ADS) Ivanov, Martin; Kotlarski, Sven; Schär, Christoph 2015-04-01 The Swiss CH2011 scenarios provide a portfolio of climate change scenarios for the region of Switzerland, specifically tailored for use in climate impact research. Although widely applied by a variety of end-users, these scenarios are subject to several limitations related to the underlying delta change methodology. Examples are difficulties to appropriately account for changes in the spatio-temporal variability of meteorological fields and for changes in extreme events. The recently launched ELAPSE project (Enhancing local and regional climate change projections for Switzerland) is connected to the EU COST Action VALUE (www.value-cost.eu) and aims at complementing CH2011 by further scenario products, including a bias-corrected version of daily scenarios at the site scale. For this purpose the well-established empirical quantile mapping (QM) methodology is employed. Here, daily temperature and precipitation output of 15 GCM-RCM model chains of the ENSEMBLES project is downscaled and bias-corrected to match observations at weather stations in Switzerland. We consider established QM techniques based on all empirical quantiles or linear interpolation between the empirical percentiles. In an attempt to improve the downscaling of extreme precipitation events, we also apply a parametric approximation of the daily precipitation distribution by a dynamically weighted mixture of a Gamma distribution for the bulk and a Pareto distribution for the right tail for the first time in the context of QM. All techniques are evaluated and intercompared in a cross-validation framework. The statistical downscaling substantially improves virtually all considered distributional and temporal characteristics as well as their spatial distribution. The empirical methods have in general very similar performances. The parametric method does not show an improvement over the empirical ones. Critical sites and seasons are highlighted and discussed. Special emphasis is placed on investigating the 6. Nambu-Goldstone theorem and spin-statistics theorem NASA Astrophysics Data System (ADS) Fujikawa, Kazuo 2016-05-01 On December 19-21 in 2001, we organized a yearly workshop at Yukawa Institute for Theoretical Physics in Kyoto on the subject of “Fundamental Problems in Field Theory and their Implications”. Prof. Yoichiro Nambu attended this workshop and explained a necessary modification of the Nambu-Goldstone theorem when applied to non-relativistic systems. At the same workshop, I talked on a path integral formulation of the spin-statistics theorem. The present essay is on this memorable workshop, where I really enjoyed the discussions with Nambu, together with a short comment on the color freedom of quarks. 7. Four theorems on the psychometric function. PubMed May, Keith A; Solomon, Joshua A 2013-01-01 In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, Δx. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull "slope" parameter, β, can be approximated by β(Noise) x β(Transducer), where β(Noise) is the β of the Weibull function that fits best to the cumulative noise distribution, and β(Transducer) depends on the transducer. We derive general expressions for β(Noise) and β(Transducer), from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when d' ∝ (Δx)(b), β ≈ β(Noise) x b. We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4-0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull β reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of β for contrast discrimination suggests that, if internal noise is stimulus 8. A categorical account of the Hofmann-Mislove theorem NASA Astrophysics Data System (ADS) Townsend, Christopher F. 2005-11-01 A categorical account is given of the Hofmann-Mislove theorem, describing the Scott open filters on a frame. The account is stable under an order duality and so is shown to also cover Bunge and Funk's constructive description of the points of the lower power locale. 9. Equipartition theorem and the dynamics of liquids SciTech Connect Levashov, Valentin A.; Egami, Takeshi; Aga, Rachel S; Morris, James R 2008-01-01 In liquids, phonons have a very short lifetime and the total potential energy does not depend linearly on temperature. Thus it may appear that atomic vibrations in liquids cannot be described by the harmonic-oscillator model and that the equipartition theorem for the potential energy is not upheld. In this paper we show that the description of the local atomic dynamics in terms of the atomic-level stresses provides such a description, satisfying the equipartition theorem. To prove this point we carried out molecular-dynamics simulations with several pairwise potentials, including the Lennard-Jones potential, the modified Johnson potential, and the repulsive part of the Johnson potential, at various particle number densities. In all cases studied the total self-energy of the atomic-level stresses followed the (3/2)kBT law. From these results we suggest that the concept of local atomic stresses can provide description of thermodynamic properties of glasses and liquids on the basis of harmonic atomistic excitations. An example of application of this approach to the description of the glass transition temperature in metallic glasses is discussed. 10. New double soft emission theorems NASA Astrophysics Data System (ADS) Cachazo, Freddy; He, Song; Yuan, Ellis Ye 2015-09-01 We study the behavior of the tree-level S-matrix of a variety of theories as two particles become soft. By analogy with the recently found subleading soft theorems for gravitons and gluons, we explore subleading terms in double soft emissions. We first consider double soft scalar emissions and find subleading terms that are controlled by the angular momentum operator acting on hard particles. The order of the subleading theorems depends on the presence or not of color structures. Next we obtain a compact formula for the leading term in a double soft photon emission. The theories studied are a special Galileon, Dirac-Born-Infeld, Einstein-Maxwell-Scalar, nonlinear sigma model and Yang-Mills-Scalar. We use the recently found Cachazo-He-Yuan representation of these theories in order to give a simple proof of the leading order part of all these theorems. 11. The Variation Theorem Applied to H-2+: A Simple Quantum Chemistry Computer Project ERIC Educational Resources Information Center Robiette, Alan G. 1975-01-01 Describes a student project which requires limited knowledge of Fortran and only minimal computing resources. The results illustrate such important principles of quantum mechanics as the variation theorem and the virial theorem. Presents sample calculations and the subprogram for energy calculations. (GS) 12. Quantum cryptography without Bell's theorem NASA Astrophysics Data System (ADS) Bennett, Charles H.; Brassard, Gilles; Mermin, N. David 1992-02-01 Ekert has described a cryptographic scheme in which Einstein-Podolsky-Rosen (EPR) pairs of particles are used to generate identical random numbers in remote places, while Bell's theorem certifies that the particles have not been measured in transit by an eavesdropper. We describe a related but simpler EPR scheme and, without invoking Bell's theorem, prove it secure against more general attacks, including substitution of a fake EPR source. Finally we show our scheme is equivalent to the original 1984 key distribution scheme of Bennett and Brassard, which uses single particles instead of EPR pairs. 13. Advantages and limitations of navigation-based multicriteria optimization (MCO) for localized prostate cancer IMRT planning SciTech Connect McGarry, Conor K.; Bokrantz, Rasmus; O’Sullivan, Joe M.; Hounsell, Alan R. 2014-10-01 Efficacy of inverse planning is becoming increasingly important for advanced radiotherapy techniques. This study’s aims were to validate multicriteria optimization (MCO) in RayStation (v2.4, RaySearch Laboratories, Sweden) against standard intensity-modulated radiation therapy (IMRT) optimization in Oncentra (v4.1, Nucletron BV, the Netherlands) and characterize dose differences due to conversion of navigated MCO plans into deliverable multileaf collimator apertures. Step-and-shoot IMRT plans were created for 10 patients with localized prostate cancer using both standard optimization and MCO. Acceptable standard IMRT plans with minimal average rectal dose were chosen for comparison with deliverable MCO plans. The trade-off was, for the MCO plans, managed through a user interface that permits continuous navigation between fluence-based plans. Navigated MCO plans were made deliverable at incremental steps along a trajectory between maximal target homogeneity and maximal rectal sparing. Dosimetric differences between navigated and deliverable MCO plans were also quantified. MCO plans, chosen as acceptable under navigated and deliverable conditions resulted in similar rectal sparing compared with standard optimization (33.7 ± 1.8 Gy vs 35.5 ± 4.2 Gy, p = 0.117). The dose differences between navigated and deliverable MCO plans increased as higher priority was placed on rectal avoidance. If the best possible deliverable MCO was chosen, a significant reduction in rectal dose was observed in comparison with standard optimization (30.6 ± 1.4 Gy vs 35.5 ± 4.2 Gy, p = 0.047). Improvements were, however, to some extent, at the expense of less conformal dose distributions, which resulted in significantly higher doses to the bladder for 2 of the 3 tolerance levels. In conclusion, similar IMRT plans can be created for patients with prostate cancer using MCO compared with standard optimization. Limitations exist within MCO regarding conversion of navigated plans to 14. Towards a novel no-hair theorem for black holes SciTech Connect Hertog, Thomas 2006-10-15 We provide strong numerical evidence for a new no-scalar-hair theorem for black holes in general relativity, which rules out spherical scalar hair of static four-dimensional black holes if the scalar field theory, when coupled to gravity, satisfies the Positive Energy Theorem. This sheds light on the no-scalar-hair conjecture for Calabi-Yau compactifications of string theory, where the effective potential typically has negative regions but where supersymmetry ensures the total energy is always positive. In theories where the scalar tends to a negative local maximum of the potential at infinity, we find the no-scalar-hair theorem holds provided the asymptotic conditions are invariant under the full anti-de Sitter symmetry group. 15. On the role of sharp chains in the transport theorem NASA Astrophysics Data System (ADS) Falach, L.; Segev, R. 2016-03-01 A generalized transport theorem for convecting irregular domains is presented in the setting of Federer's geometric measure theory. A prototypical r-dimensional domain is viewed as a flat r-chain of finite mass in an open set of an n-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to follow a continuous succession of Lipschitz embedding so that the spatial gradient may be nonexistent in a subset of the domain with zero measure. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in Rn. A time-dependent property is naturally assigned to the evolving region via the action of an r-cochain on the current associated with the domain. Applying a representation theorem for cochains, the properties are shown to be locally represented by an r-form. Using these notions, a generalized transport theorem is presented. 16. Stochastic thermodynamics, fluctuation theorems and molecular machines. PubMed Seifert, Udo 2012-12-01 Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production. 17. Stochastic thermodynamics, fluctuation theorems and molecular machines NASA Astrophysics Data System (ADS) Seifert, Udo 2012-12-01 Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production. 18. 20 CFR 661.310 - Under what limited conditions may a Local Board directly be a provider of core services... Code of Federal Regulations, 2010 CFR 2010-04-01 ... 20 Employees' Benefits 3 2010-04-01 2010-04-01 false Under what limited conditions may a Local Board directly be a provider of core services, intensive services, or training services, or act as a One-Stop Operator? 661.310 Section 661.310 Employees' Benefits EMPLOYMENT AND TRAINING ADMINISTRATION, DEPARTMENT OF LABOR STATEWIDE AND... 19. Angle Defect and Descartes' Theorem ERIC Educational Resources Information Center Scott, Paul 2006-01-01 Rene Descartes lived from 1596 to 1650. His contributions to geometry are still remembered today in the terminology "Descartes' plane". This paper discusses a simple theorem of Descartes, which enables students to easily determine the number of vertices of almost every polyhedron. (Contains 1 table and 2 figures.) 20. Discovering the Inscribed Angle Theorem ERIC Educational Resources Information Center Roscoe, Matt B. 2012-01-01 Learning to play tennis is difficult. It takes practice, but it also helps to have a coach--someone who gives tips and pointers but allows the freedom to play the game on one's own. Learning to act like a mathematician is a similar process. Students report that the process of proving the inscribed angle theorem is challenging and, at times,… 1. Generalized Pump-restriction Theorem SciTech Connect Sinitsyn, Nikolai A; Chernyak, Vladimir Y 2008-01-01 We formulate conditions under which periodic modulations of parameters on a finite graph with stochastic transitions among its nodes do not lead to overall pump currents through any given link. Our theorem unifies previously known results with the new ones and provides a universal approach to explore futher restrictions on stochastic pump effect in non-adiabatically driven systems with detailed balance. 2. Expanding the Interaction Equivalency Theorem ERIC Educational Resources Information Center Rodriguez, Brenda Cecilia Padilla; Armellini, Alejandro 2015-01-01 Although interaction is recognised as a key element for learning, its incorporation in online courses can be challenging. The interaction equivalency theorem provides guidelines: Meaningful learning can be supported as long as one of three types of interactions (learner-content, learner-teacher and learner-learner) is present at a high level. This… 3. Arriving at the Pythagorean Theorem. ERIC Educational Resources Information Center Jaramillo, James; Brown, Jonathan Caius This lesson plan uses group activity and manipulative materials to teach English-speaking students (ages 15-16) of diverse ethnic backgrounds an operatonal understanding of the Pythagorean Theorem. It is based on theories of constructivism and holism and includes teacher instructions, discussion questions, a retrospective vision, and an ancillary… 4. Interpretation of the quantum formalism and Bell's theorem SciTech Connect Santos, E. ) 1991-02-01 It is argued that quantum mechanics must be interpreted according to the Copenhagen interpretation. Consequently the formalism must be used in a purely operational way. The relation between realism, hidden variables, and the Bell inequalities is discussed. The proof of impossibility of local hidden-variables theories (Bell theorem) is criticized on the basis that the quantum mechanical states violating local realism are not physically realizable states. 5. Analytical proof of Gisin's theorem for three qubits SciTech Connect Choudhary, Sujit K.; Ghosh, Sibasish; Kar, Guruprasad; Rahaman, Ramij 2010-04-15 Gisin's theorem assures that for any pure bipartite entangled state, there is violation of the inequality of Bell and of Clauser, Horne, Shimony, and Holt, revealing its contradiction with local realistic model. Whether a similar result holds for three-qubit pure entangled states remained unresolved. We show analytically that all three-qubit pure entangled states violate a Bell-type inequality, derived on the basis of local realism, by exploiting the Hardy's nonlocality argument. 6. Optical theorem for multipole sources in wave diffraction theory NASA Astrophysics Data System (ADS) Eremin, Yu. A.; Sveshnikov, A. G. 2016-05-01 The optical theorem is generalized to the case of local body excitation by multipole sources. It is found that, to calculate the extinction cross section, it is sufficient to calculate the scattered field derivatives at a single point. It is shown that the Purcell factor, which is a rather important parameter, can be represented in analytic form. The result is generalized to the case of a local scatterer incorporated in a homogeneous halfspace. 7. Generalized Fourier slice theorem for cone-beam image reconstruction. PubMed Zhao, Shuang-Ren; Jiang, Dazong; Yang, Kevin; Yang, Kang 2015-01-01 The cone-beam reconstruction theory has been proposed by Kirillov in 1961, Tuy in 1983, Feldkamp in 1984, Smith in 1985, Pierre Grangeat in 1990. The Fourier slice theorem is proposed by Bracewell 1956, which leads to the Fourier image reconstruction method for parallel-beam geometry. The Fourier slice theorem is extended to fan-beam geometry by Zhao in 1993 and 1995. By combining the above mentioned cone-beam image reconstruction theory and the above mentioned Fourier slice theory of fan-beam geometry, the Fourier slice theorem in cone-beam geometry is proposed by Zhao 1995 in short conference publication. This article offers the details of the derivation and implementation of this Fourier slice theorem for cone-beam geometry. Especially the problem of the reconstruction from Fourier domain has been overcome, which is that the value of in the origin of Fourier space is 0/0. The 0/0 type of limit is proper handled. As examples, the implementation results for the single circle and two perpendicular circle source orbits are shown. In the cone-beam reconstruction if a interpolation process is considered, the number of the calculations for the generalized Fourier slice theorem algorithm is O(N^4), which is close to the filtered back-projection method, here N is the image size of 1-dimension. However the interpolation process can be avoid, in that case the number of the calculations is O(N5). 8. Pythagorean Theorem Proofs: Connecting Interactive Websites ERIC Educational Resources Information Center Lin, Cheng-Yao 2007-01-01 There are over 400 proofs of the Pythagorean Theorem. Some are visual proofs, others are algebraic. This paper features several proofs of the Pythagorean Theorem in different cultures--Greek, Chinese, Hindu and American. Several interactive websites are introduced to explore ways to prove this beautiful theorem. (Contains 8 figures.) 9. A Fundamental Theorem on Particle Acceleration SciTech Connect Xie, Ming 2003-05-01 A fundamental theorem on particle acceleration is derived from the reciprocity principle of electromagnetism and a rigorous proof of the theorem is presented. The theorem establishes a relation between acceleration and radiation, which is particularly useful for insightful understanding of and practical calculation about the first order acceleration in which energy gain of the accelerated particle is linearly proportional to the accelerating field. 10. A note on generalized Weyl's theorem NASA Astrophysics Data System (ADS) Zguitti, H. 2006-04-01 We prove that if either T or T* has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every . An application is given for algebraically paranormal operators. 11. Generalizations of Ptolemy and Brahmagupta Theorems ERIC Educational Resources Information Center Ayoub, Ayoub B. 2007-01-01 The Greek astronomer Ptolemy of Alexandria (second century) and the Indian mathematician Brahmagupta (sixth century) each have a significant theorem named after them. Both theorems have to do with cyclic quadrilaterals. Ptolemy's theorem states that: In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of two… 12. Khalfin's Theorem and Neutral Mesons Subsystem NASA Astrophysics Data System (ADS) Urbanowski, Krzysztof 2009-01-01 The consequences of Khalfin's Theorem are discussed. we find, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. Within a given model we examine numerically the Khalfin's Theorem and show in a graphic form how the Khalfin's Theorem works. 13. Saoithín: A Theorem Prover for UTP NASA Astrophysics Data System (ADS) Butterfield, Andrew Saoithín is a theorem prover developed to support the Unifying Theories of Programming (UTP) framework. Its primary design goal was to support the higher-order logic, alphabets, equational reasoning and "programs as predicates" style that is prevalent in much of the UTP literature, from the seminal work by Hoare & He [HH98] onwards. This paper describes the key features of the theorem prover, with an emphasis on the underlying foundations, and how these affect the design and implementation choices. These key features include: a formalisation of a UTP Theory; support for common proof strategies; sophisticated goal/law matching ; and user-defined language constructs. A simple theory of designs with some proof extracts is used to illustrate the above features. The theorem prover has been used with undergraduate students and we discuss some of those experiences. The paper then concludes with a discussion of current limitations and planned improvements to the tool. 14. Numerous strategies but limited implementation guidance in US local adaptation plans NASA Astrophysics Data System (ADS) Woodruff, Sierra C.; Stults, Missy 2016-08-01 Adaptation planning offers a promising approach for identifying and devising solutions to address local climate change impacts. Yet there is little empirical understanding of the content and quality of these plans. We use content analysis to evaluate 44 local adaptation plans in the United States and multivariate regression to examine how plan quality varies across communities. We find that plans draw on multiple data sources to analyse future climate impacts and include a breadth of strategies. Most plans, however, fail to prioritize impacts and strategies or provide detailed implementation processes, raising concerns about whether adaptation plans will translate into on-the-ground reductions in vulnerability. Our analysis also finds that plans authored by the planning department and those that engaged elected officials in the planning process were of higher quality. The results provide important insights for practitioners, policymakers and scientists wanting to improve local climate adaptation planning and action. 15. The sensitive period for auditory localization in barn owls is limited by age, not by experience. PubMed Knudsen, E I; Knudsen, P F 1986-07-01 Early in life, the barn owl passes through a sensitive period during which it can interpret and make use of abnormal auditory cues for accurate sound localization. This capacity is lost at about 8 weeks of age, just after the head and ears reach adult size (knudsen et al. 1984a). The end of the sensitive period could be triggered either by an age-dependent process or by the exposure of the auditory system to stable or adult-like cues. To distinguish between these alternatives, we subjected baby owls to constant abnormal cues (chronic monaural occlusion) or to frequently changing abnormal cues (alternating monaural occlusion) throughout the sensitive period. In the first group of animals (n = 2), one ear was plugged continuously until 73 or 79 d of age, respectively, and then the earplug was switched to the opposite ear. Although these animals adjusted sound localization accuracy during the initial chronic monaural occlusion, they could not localize sounds at all after the earplug was switched to the opposite ear, and they remained unable to localize sounds as long as the opposite ear remained occluded (7 and 27 weeks, respectively). When the second monaural occlusion was finally removed, both birds localized sounds with errors that were similar to the errors they exhibited immediately after removal of the first monaural occlusion. One bird that was 127-d-old at the time the second earplug was removed corrected its localization error; the other bird, 250-d-old when the second earplug was removed, did not.(ABSTRACT TRUNCATED AT 250 WORDS) 16. Hijacked organic, limited local, faulty fair trade: what's a radical to eat? PubMed Engler, Mark 2012-01-01 Organic farming has been hijacked by big business. Local food can have a larger carbon footprint than products shipped in from overseas. Fair trade doesn't address the real concerns of farmers in the global South. As the food movement has moved from the countercultural fringe to become a mainstream phenomenon, organic, local, and fair trade advocates have been beset by criticism from overt foes and erstwhile allies alike. Now that Starbucks advertises fair trade coffee and Kraft owns Boca soy burgers, it's fair to ask, "What's a radical to eat?" 17. Bell's theorem, inference, and quantum transactions NASA Astrophysics Data System (ADS) Garrett, A. J. M. 1990-04-01 Bell's theorem is expounded as an analysis in Bayesian inference. Assuming the result of a spin measurement on a particle is governed by a causal variable internal (hidden, “local”) to the particle, one learns about it by making a spin measurement; thence about the internal variable of a second particle correlated with the first; and from there predicts the probabilistic result of spin measurements on the second particle. Such predictions are violated by experiment: locality/causality fails. The statistical nature of the observations rules out signalling; acausal, superluminal, or otherwise. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of experiment imply that it has a nonlocal/acausal interpretation. Cramer's new transactional interpretation, which incorporates this feature by adapting the Wheeler-Feynman idea of advanced and retarded processes to the quantum laws, is advocated. It leads to an invaluable way of envisaging quantum processes. The usual paradoxes melt before this, and one, the “delayed choice” experiment, is chosen for detailed inspection. Nonlocality implies practical difficulties in influencing hidden variables, which provides a very plausible explanation for why they have not yet been found; from this standpoint, Bell's theorem reinforces arguments in favor of hidden variables. 18. De Finetti Theorem on the CAR Algebra NASA Astrophysics Data System (ADS) Crismale, Vitonofrio; Fidaleo, Francesco 2012-10-01 The symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present paper we extend the De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, after showing that a symmetric state is automatically even under the natural action of the parity automorphism, we prove that the compact convex set of such states is a Choquet simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of permutations previously described) are precisely the product states in the sense of Araki-Moriya. In order to do that, we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self-containing interest. 19. Complex virial theorem and complex scaling SciTech Connect Junker, B.R. 1983-06-01 We present the simple generalization to complex energies of the normal global real scaling used for bound-state calculations to produce a variational energy which satisfies the virial theorem. We show that in two limiting cases, one or the other of which is almost always p satisfied in all calculations, the virially stabilized complex energy is sensitive to only the real part or the imaginary part of the complex virial expression. We then compute the virial expression for a number of wave functions for the 1s2s/sup 2/ /sup 2/S He/sup -/, 1s2s2p /sup 2/P/sup o/ He/sup -/, and 1s/sup 2/2s/sup 2/kp /sup 2/P/sup o/ Be/sup -/ resonances and the corresponding virially stabilized resonance energies. In all calculations one of the limiting cases was applicable. 20. Limitations of a localized surface plasmon resonance sensor on Salmonella detection Technology Transfer Automated Retrieval System (TEKTRAN) We have designed a localized surface plasmon resonance (LSPR) biosensor to perform the whole cell detection of Salmonella using gold nanoparticls fabricated by oblique angle deposition technique. The LSPR sensor showed a plasmon peak shift due to the Salmonella antigen and anti-Salmonella antibody r... 1. Taking account of local culture: limits to the development of a professional ethos. PubMed Goopy, Suzanne E 2005-06-01 The need to extend the discussion of culture in the study of nursing, combined with an enthusiasm for the possibility of viewing nursing from a new perspective, provides the impetus for this study. Based on fieldwork undertaken in the intensive care unit (RICU) of a major public hospital in Rome (Italy), this paper explores some of the key aspects of the social relations and local staff culture of one particular group of Italian nurses. In a climate of globalization, where the deployment of dominant Anglo-American ideas is difficult to counter, the RICU presents as a setting which challenges the widespread assumptions of universal standards of nursing practice. By building a picture of the working world of these particular nurses, we are assisted in our understanding of nursing practice as a local cultural activity. In exploring the significance of local culture this paper brings into question the universality of the current paradigm of professionalism and professional identity, and emphasizes the value of acknowledging local culture. 2. Localization of type I interferon receptor limits interferon-induced TLR-3 in epithelial cells EPA Science Inventory This study aimed to expand on the role of type I IFNs in the influenza-induced upregulation of TLR3 and determine whether and how the localization of the IFN-alpha/beta receptor (IFNAR) in respiratory epithelial cells could modify IFN-induced responses. Using differentiated prima... 3. 40 CFR 1400.11 - Limitation on dissemination to State and local government officials. Code of Federal Regulations, 2010 CFR 2010-07-01 ... and local government officials. 1400.11 Section 1400.11 Protection of Environment ENVIRONMENTAL PROTECTION AGENCY AND DEPARTMENT OF JUSTICE ACCIDENTAL RELEASE PREVENTION REQUIREMENTS; RISK MANAGEMENT PROGRAMS UNDER THE CLEAN AIR ACT SECTION 112(r)(7); DISTRIBUTION OF OFF-SITE CONSEQUENCE... 4. Distant views and local realities: The limits of global assessments to restore the fragmented phosphorus cycle Technology Transfer Automated Retrieval System (TEKTRAN) With more sophisticated data compilation and analytical capabilities, the evolution of “big data” analysis has occurred rapidly. We examine the meta-analysis of “big data” representing phosphorus (P) flows and stocks in global agriculture and address the need to consider local nuances of farm operat... 5. Implementation of a workplace smoking ban in bars: The limits of local discretion PubMed Central Montini, Theresa; Bero, Lisa A 2008-01-01 Background In January 1998, the California state legislature extended a workplace smoking ban to bars. The purpose of this study was to explore the conditions that facilitate or hinder compliance with a smoking ban in bars. Methods We studied the implementation of the smoking ban in bars by interviewing three sets of policy participants: bar employers responsible for complying with the law; local government officials responsible for enforcing the law; and tobacco control activists who facilitated implementation. We transcribed the interviews and did a qualitative analysis of the text. Results The conditions that facilitated bar owners' compliance with a smoking ban in bars included: if the cost to comply was minimal; if the bars with which they were in competition were in compliance with the smoking ban; and if there was authoritative, consistent, coordinated, and uniform enforcement. Conversely, the conditions that hindered compliance included: if the law had minimal sanctions; if competing bars in the area allowed smoking; and if enforcement was delayed or inadequate. Conclusion Many local enforcers wished to forfeit their local discretion and believed the workplace smoking ban in bars would be best implemented by a state agency. The potential implication of this study is that, given the complex nature of local politics, smoking bans in bars are best implemented at a broader provincial or national level. PMID:19063716 6. The Spin-S Einstein - Podolsky - Rosen Experiment and the Recovery of Local Realism in the Classical Limit. NASA Astrophysics Data System (ADS) Garg, Anupam Kumar This thesis consists of a study of the general spin Einstein-Podolsky-Rosen experiment, with a view to seeing how classical behavior is recovered in the infinite -spin limit. In Bohm's version of the experiment, two spin -s particles fly apart in the singlet state. Quantum theory predicts that if the spin of one particle is measured to be m along a direction a, the other will necessarily have a sping of -m along a. Since the spins of the two far apart particles can be measured in an infinitesimal time interval, the requirement of locality suggests that the particles' spins are predetermined along all directions, in defiance of quantum theoretic precepts. It was shown by Bell and by Clauser and Horne that an alternative point of view called local realism which attempts to explain the spin correlations by introducing hidden variables mut be incompatible with the quantitative predictions of quantum theory for spin- 1/2. Chapters II and III of this thesis contain extensions of their arguments to arbitrary spin. It is found that no matter how large the spin gets, quantum mechanics and local realism do not become more compatible, and classical mechanics does not emerge as a smooth infinite-spin limit. Chapter IV gives a general procedure (with several examples) for deciding whether or not a given set of joint distributions for the spins of the two particles is compatible with local realism. In Chapter V it is argued that local realism can emerge in the infinite-spin limit only if limitations in detector resolution (quite unrelated to those imposed by the uncertainty principle) are explicitly included. Some questions stemming from this point of view are answered for spin- 1/2, and a specific model for including detector error is studied for higher spin. 7. Generalized Bloch theorem and chiral transport phenomena NASA Astrophysics Data System (ADS) Yamamoto, Naoki 2015-10-01 Bloch theorem states the impossibility of persistent electric currents in the ground state of nonrelativistic fermion systems. We extend this theorem to generic systems based on the gauged particle number symmetry and study its consequences on the example of chiral transport phenomena. We show that the chiral magnetic effect can be understood as a generalization of the Bloch theorem to a nonequilibrium steady state, similarly to the integer quantum Hall effect. On the other hand, persistent axial currents are not prohibited by the Bloch theorem and they can be regarded as Pauli paramagnetism of relativistic matter. An application of the generalized Bloch theorem to quantum time crystals is also discussed. 8. The de Finetti theorem for test spaces NASA Astrophysics Data System (ADS) Barrett, Jonathan; Leifer, Matthew 2009-03-01 We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem. 9. Local oscillator phase noise limitation on the resolution of acoustic delay line wireless passive sensor measurement NASA Astrophysics Data System (ADS) Chrétien, N.; Friedt, J.-M.; Martin, G. 2014-06-01 The role of the phase noise of a local oscillator driving a pulsed-mode RADAR used for probing surface acoustic wave sensors is investigated. The echo delay, representative of the acoustic velocity, and hence the physical quantity probed by the sensor, is finely measured as a phase. Considering that the intrinsic oscillator phase fluctuation defines the phase noise measurement resolution, we experimentally and theoretically assess the relation between phase noise, measurement range, and measurand resolution. 10. The Limits of the Czar’s Ukase: Drug Policy at the Local Level DTIC Science & Technology 1990-06-01 for which post-arrest data are available, approximately seven times as many persons were sen- tenced to incarceration in state prisons and local jails...referred to treatment programs than incarcerated . No doubt the cocaine epidemic would have put a great strain on big city treatment agencies anyway, bitt... recidivism . cooperation among treat ment and correct ions agencies wil haet-buh more efl’ ctive than it was during pre- vious drug abuse epidemics. 2 11. Genetic shift in local rice populations during rice breeding programs in the northern limit of rice cultivation in the world. PubMed Fujino, Kenji; Obara, Mari; Ikegaya, Tomohito; Tamura, Kenichi 2015-09-01 The rapid accumulation of pre-existing mutations may play major roles in the establishment and shaping of adaptability for local regions in current rice breeding programs. The cultivated rice, Oryza sativa L., which originated from tropical regions, is now grown worldwide due to the concerted efforts of breeding programs. However, the process of establishing local populations and their origins remain unclear. In the present study, we characterized DNA polymorphisms in the rice variety KITAAKE from Hokkaido, one of the northern limits of rice cultivation in the world. Indel polymorphisms were attributed to transposable element-like insertions, tandem duplications, and non-TE deletions as the original mutation events in the NIPPONBARE and KITAAKE genomes. The allele frequencies of the KITAAKE alleles markedly shifted to the current variety types among the local population from Hokkaido in the last two decades. The KITAAKE alleles widely distributed throughout wild rice and cultivated rice over the world. These have accumulated in the local population from Hokkaido via Japanese landraces as the ancestral population of Hokkaido. These results strongly suggested that combinations of pre-existing mutations played a role in the establishment of adaptability. This approach using the re-sequencing of local varieties in unique environmental conditions will be useful as a genetic resource in plant breeding programs in local regions. 12. Quantum mechanics vs local realism near the classical limit:A Bell inequality for spin s SciTech Connect Mermin, N.D. 1980-07-15 The quantitative quantum-mechanical analysis of the Einstein-Podolsky-Rosen experiment for correlated particles of arbitrary spin s is shown to contradict a generalized form of Bell's inequality, for suitable orientations of the detectors. As the classical (s ..-->.. infinity ) limit is approached, the range of angles for which the contradiction arises vanishes as 1/s. 13. Equivalence theorem of uncertainty relations NASA Astrophysics Data System (ADS) Li, Jun-Li; Qiao, Cong-Feng 2017-01-01 We present an equivalence theorem to unify the two classes of uncertainty relations, i.e. the variance-based ones and the entropic forms, showing that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators. This means that an uncertainty relation in the language of entropy may be mapped onto a variance-based one, and vice versa. Employing the equivalence theorem, alternative formulations of entropic uncertainty relations are obtained for the qubit system that are stronger than the existing ones in the literature, and variance-based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations. 14. Uniqueness Theorem for Black Objects SciTech Connect Rogatko, Marek 2010-06-23 We shall review the current status of uniqueness theorem for black objects in higher dimensional spacetime. At the beginning we consider static charged asymptotically flat spacelike hypersurface with compact interior with both degenerate and non-degenerate components of the event horizon in n-dimensional spacetime. We gave some remarks concerning partial results in proving uniqueness of stationary axisymmetric multidimensional solutions and winding numbers which can uniquely characterize the topology and symmetry structure of black objects. 15. Splitting theorem for Z2n -supermanifolds NASA Astrophysics Data System (ADS) Covolo, Tiffany; Grabowski, Janusz; Poncin, Norbert 2016-12-01 Smooth Z2n -supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the 'scalar product' of the involved Z2n -degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. The classical Batchelor-Gawȩdzki theorem says that any smooth supermanifold is diffeomorphic to the 'superization' ΠE of a vector bundle E. It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of Z2n -supermanifolds with its local model made of formal power series. We give a positive answer to this question. 16. A global conformal extension theorem for perfect fluid Bianchi space-times SciTech Connect Luebbe, Christian Tod, Paul 2008-12-15 A global extension theorem is established for isotropic singularities in polytropic perfect fluid Bianchi space-times. When an extension is possible, the limiting behaviour of the physical space-time near the singularity is analysed. 17. Non-local rheological properties of granular flows near a jamming limit. SciTech Connect Aranson, I. S.; Tsimring, L. S.; Malloggi, F.; Clement, E.; Materials Science Division; Univ. of California at San Diego; CNRS-ESPCI Univ. 2008-01-01 We study the rheology of sheared granular flows close to a jamming transition. We use the approach of partially fluidized theory (PFT) with a full set of equations extending the thin layer approximation derived previously for the description of the granular avalanches phenomenology. This theory provides a picture compatible with a local rheology at large shear rates [G. D. R. Midi, Eur. Phys. J. E 14, 341 (2004)] and it works in the vicinity of the jamming transition, where a description in terms of a simple local rheology comes short. We investigate two situations displaying important deviations from local rheology. The first one is based on a set of numerical simulations of sheared soft two-dimensional circular grains. The next case describes previous experimental results obtained on avalanches of sandy material flowing down an incline. Both cases display, close to jamming, significant deviations from the now standard Pouliquen's flow rule [O. Pouliquen, Phys. Fluids 11, 542 (1999); 11, 1956 (1999)]. This discrepancy is the hallmark of a strongly nonlocal rheology and in both cases, we relate the empirical results and the outcomes of PFT. The numerical simulations show a characteristic constitutive structure for the fluid part of the stress involving the confining pressure and the material stiffness that appear in the form of an additional dimensionless parameter. This constitutive relation is then used to describe the case of sandy flows. We show a quantitative agreement as far as the effective flow rules are concerned. A fundamental feature is identified in PFT as the existence of a jammed layer developing in the vicinity of the flow arrest that corroborates the experimental findings. Finally, we study the case of solitary erosive granular avalanches and relate the outcome with the PFT analysis. 18. Overcoming non-local effects and Brillouin threshold limitations in Brillouin distributed sensors NASA Astrophysics Data System (ADS) Urricelqui, Javier; Ruiz-Lombera, Rubén.; Sagues, Mikel; Mirapeix, Jesús; López-Higuera, José M.; Loayssa, Alayn 2015-09-01 We demonstrate, for the first time to our knowledge, a Brillouin optical time domain analysis sensor that is able to operate with a probe power larger than the Brillouin threshold of the deployed sensing fiber and that is free from detrimental non-local effects. The technique is based on a dual-probe-sideband setup in which a frequency modulation of the probes waves along the fiber is introduced. This makes the frequency of maximum interaction between pump and probes to vary along the fiber, thus mitigating the pump pulse depletion and making it possible to use very large probe power, which brings an improved signal-to-noise ratio in detection. 19. Radially dependent large-scale dynamos in global cylindrical shear flows and the local cartesian limit NASA Astrophysics Data System (ADS) Ebrahimi, F.; Blackman, E. G. 2016-06-01 For cylindrical differentially rotating plasmas, we study large-scale magnetic field generation from finite amplitude non-axisymmetric perturbations by comparing numerical simulations with quasi-linear analytic theory. When initiated with a vertical magnetic field of either zero or finite net flux, our global cylindrical simulations exhibit the magnetorotational instability (MRI) and large-scale dynamo growth of radially alternating mean fields, averaged over height and azimuth. This dynamo growth is explained by our analytic calculations of a non-axisymmetric fluctuation-induced electromotive force that is sustained by azimuthal shear of the fluctuating fields. The standard Ω effect' (shear of the mean field by differential rotation) is unimportant. For the MRI case, we express the large-scale dynamo field as a function of differential rotation. The resulting radially alternating large-scale fields may have implications for angular momentum transport in discs and corona. To connect with previous work on large-scale dynamos with local linear shear and identify the minimum conditions needed for large-scale field growth, we also solve our equations in local Cartesian coordinates. We find that large-scale dynamo growth in a linear shear flow without rotation can be sustained by shear plus non-axisymmetric fluctuations - even if not helical, a seemingly previously unidentified distinction. The linear shear flow dynamo emerges as a more restricted version of our more general new global cylindrical calculations. 20. Experiments with central-limit properties of spatial samples from locally covariant random fields USGS Publications Warehouse Barringer, T.H.; Smith, T.E. 1992-01-01 When spatial samples are statistically dependent, the classical estimator of sample-mean standard deviation is well known to be inconsistent. For locally dependent samples, however, consistent estimators of sample-mean standard deviation can be constructed. The present paper investigates the sampling properties of one such estimator, designated as the tau estimator of sample-mean standard deviation. In particular, the asymptotic normality properties of standardized sample means based on tau estimators are studied in terms of computer experiments with simulated sample-mean distributions. The effects of both sample size and dependency levels among samples are examined for various value of tau (denoting the size of the spatial kernel for the estimator). The results suggest that even for small degrees of spatial dependency, the tau estimator exhibits significantly stronger normality properties than does the classical estimator of standardized sample means. ?? 1992. 1. COMPUTATIONAL MODELING OF CATHODIC LIMITATIONS ON LOCALIZED CORROSION OF WETTED SS 316L, AT ROOM TEMPERATURE SciTech Connect F. Cui; F.J. Presuel-Moreno; R.G. Kelly 2005-10-13 The ability of a SS316L surface wetted with a thin electrolyte layer to serve as an effective cathode for an active localized corrosion site was studied computationally. The dependence of the total net cathodic current, I{sub net}, supplied at the repassivation potential E{sub rp} (of the anodic crevice) on relevant physical parameters including water layer thickness (WL), chloride concentration ([Cl{sup -}]) and length of cathode (Lc) were investigated using a three-level, full factorial design. The effects of kinetic parameters including the exchange current density (i{sub o,c}) and Tafel slope ({beta}{sub c}) of oxygen reduction, the anodic passive current density (i{sub p}) (on the cathodic surface), and E{sub rp} were studied as well using three-level full factorial designs of [Cl{sup -}] and Lc with a fixed WL of 25 {micro}m. The study found that all the three parameters WL, [Cl{sup -}] and Lc as well as the interactions of Lc x WL and Lc x [Cl{sup -}] had significant impact on I{sub net}. A five-factor regression equation was obtained which fits the computation results reasonably well, but demonstrated that interactions are more complicated than can be explained with a simple linear model. Significant effects on I{sub net} were found upon varying either i{sub o,c}, {beta}{sub c}, or E{sub rp}, whereas i{sub p} in the studied range was found to have little impact. It was observed that I{sub net} asymptotically approached maximum values (I{sub max}) when Lc increased to critical minimum values. I{sub max} can be used to determine the stability of coupled localized corrosion and the critical Lc provides important information for experimental design and corrosion protection. 2. Variation of Pollinator Assemblages and Pollen Limitation in a Locally Specialized System: The Oil-producing Nierembergia linariifolia (Solanaceae) PubMed Central Cosacov, Andrea; Nattero, Julieta; Cocucci, Andrea A. 2008-01-01 Background and Aims Few studies have examined the dynamics of specialist plant–pollinator interactions at a geographical scale. This knowledge is crucial for a more general evolutionary and ecological understanding of specialized plant–pollinator systems. In the present study, variations in pollinator activity, assemblage composition and pollen limitation were explored in the oil-producing species Nierembergia linariifolia (Solanaceae). Methods Pollen limitation in fruit and seed production was analysed by supplementary hand pollination in five wild populations. Pollinator activity and identity were recorded while carrying out supplementary pollination to assess the effect of pollinators on the degree of pollen limitation. In two populations, pollen limitation was discriminated into quantitative and qualitative components by comparing supplementation and hand cross-pollination in fruit set and seed set. The effect of flower number per plant on the number of flowers pollinated per visitor per visit to a plant was examined in one of these populations as a possible cause of low-quality pollination by increasing geitonogamy. Results and Conclusions Although pollen limitation was evident along time and space, differences in magnitude were detected among populations and years that were greatly explained by pollinator activity, which was significantly different across populations. Floral display size had a significant effect on the visitation rate per flower. Limitation by quality clearly affected one population presumably due to a high proportion of geitonogamous pollen. The great inter-population variation in plant–pollinator interaction (both in pollinator assemblages composition and pollinator activity) and fitness consequences, suggests that this system should be viewed as a mosaic of locally selective processes and locally specialized interactions. PMID:18765440 3. Spatial distribution of limited resources and local density regulation in juvenile Atlantic salmon. PubMed Finstad, Anders G; Einum, Sigurd; Ugedal, Ola; Forseth, Torbjørn 2009-01-01 1. Spatial heterogeneity of resources may influence competition among individuals and thus have a fundamental role in shaping population dynamics and carrying capacity. In the present study, we identify shelter opportunities as a limiting resource for juvenile Atlantic salmon (Salmo salar L.). Experimental and field studies are combined in order to demonstrate how the spatial distribution of shelters may influence population dynamics on both within and among population scales. 2. In closed experimental streams, fish performance scaled negatively with decreasing shelter availability and increasing densities. In contrast, the fish in open stream channels dispersed according to shelter availability and performance of fish remaining in the streams did not depend on initial density or shelters. 3. The field study confirmed that spatial variation in densities of 1-year-old juveniles was governed both by initial recruit density and shelter availability. Strength of density-dependent population regulation, measured as carrying capacity, increased with decreasing number of shelters. 4. Nine rivers were surveyed for spatial variation in shelter availability and increased shelter heterogeneity tended to decrease maximum observed population size (measured using catch statistics of adult salmon as a proxy). 5. Our studies highlight the importance of small-scale within-population spatial structure in population dynamics and demonstrate that not only the absolute amount of limiting resources but also their spatial arrangement can be an important factor influencing population carrying capacity. 4. The Critical Richardson Number and Limits of Applicability of Local Similarity Theory in the Stable Boundary Layer NASA Astrophysics Data System (ADS) Grachev, Andrey A.; Andreas, Edgar L.; Fairall, Christopher W.; Guest, Peter S.; Persson, P. Ola G. 2013-04-01 Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin-Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson number, Ri, and the flux Richardson number, Rf, exceed a critical value' of about 0.20-0.25, the inertial subrange associated with the Richardson-Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson-Kolmogorov energy cascade weakens; therefore, the applicability of local Monin-Obukhov similarity theory in stable conditions is limited by the inequalities Ri < Ri cr and Rf < Rf cr. However, it is found that Rf cr = 0.20-0.25 is a primary threshold for applicability. Applying this prerequisite shows that the data follow classical Monin-Obukhov local z-less predictions after the irrelevant cases (turbulence without the Richardson-Kolmogorov cascade) have been filtered out. 5. Theoretical limit of localized surface plasmon resonance sensitivity to local refractive index change and its comparison to conventional surface plasmon resonance sensor. PubMed Zalyubovskiy, Sergiy J; Bogdanova, Maria; Deinega, Alexei; Lozovik, Yurii; Pris, Andrew D; An, Kwang Hyup; Hall, W Paige; Potyrailo, Radislav A 2012-06-01 In this paper, the theoretical sensitivity limit of the localized surface plasmon resonance (LSPR) to the surrounding dielectric environment is discussed. The presented theoretical analysis of the LSPR phenomenon is based on perturbation theory. Derived results can be further simplified assuming quasistatic limit. The developed theory shows that LSPR has a detection capability limit independent of the particle shape or arrangement. For a given structure, sensitivity is directly proportional to the resonance wavelength and depends on the fraction of the electromagnetic energy confined within the sensing volume. This fraction is always less than unity; therefore, one should not expect to find an optimized nanofeature geometry with a dramatic increase in sensitivity at a given wavelength. All theoretical results are supported by finite-difference time-domain calculations for gold nanoparticles of different geometries (rings, split rings, paired rings, and ring sandwiches). Numerical sensitivity calculations based on the shift of the extinction peak are in good agreement with values estimated by perturbation theory. Numerical analysis shows that, for thin (≤10 nm) analyte layers, sensitivity of the LSPR is comparable with a traditional surface plasmon resonance sensor and LSPR has the potential to be significantly less sensitive to temperature fluctuations. 6. Improved optical limiting performance of laser-ablation-generated metal nanoparticles due to silica-microsphere-induced local field enhancement. PubMed Du, Zheren; Chen, Lianwei; Kao, Tsung-Sheng; Wu, Mengxue; Hong, Minghui 2015-01-01 For practical application, optical limiting materials must exhibit a fast response and a low threshold in order to be used for the protection of the human eye and electro-optical sensors against intense light. Many nanomaterials have been found to exhibit optical limiting properties. Laser ablation offers the possibility of fabricating nanoparticles from a wide range of target materials. For practical use of these materials, their optical limiting performance, including optical limiting threshold and the ability to efficiently attenuate high intensity light, needs to be improved. In this paper, we fabricate nanoparticles of different metals by laser ablation in liquid. We study the optical nonlinear properties of the laser-generated nanoparticle dispersion. Silica microspheres are used to enhance the optical limiting performance of the nanoparticle dispersion. The change in the optical nonlinear properties of the laser-generated nanoparticle dispersion caused by silica microspheres is studied. It is found that the incident laser beam is locally focused by the microspheres, leading to an increased optical nonlinearity of the nanoparticle dispersion. 7. Limited and localized outbreak of newly emergent type 2 vaccine-derived poliovirus in Sichuan, China. PubMed Yan, Dongmei; Zhang, Yong; Zhu, Shuangli; Chen, Na; Li, Xiaolei; Wang, Dongyan; Ma, Xiaozhen; Zhu, Hui; Tong, Wenbin; Xu, Wenbo 2014-07-01 From August 2011 to February 2012, an outbreak caused by type 2 circulating vaccine-derived poliovirus (cVDPV) occurred in Aba County, Sichuan, China. During the outbreak, four type 2 VDPVs (≥0.6% nucleotide divergence in the VP1 region relative to the Sabin 2 strain) were isolated from 3 patients with acute flaccid paralysis (AFP) and one close contact. In addition, a type 2 pre-VDPV (0.3% to 0.5% divergence from Sabin 2) that was genetically related to these type 2 VDPVs was isolated from another AFP patient. These 4 patients were all unimmunized children 0.7 to 1.1 years old. Nucleotide sequencing revealed that the 4 VDPV isolates differed from Sabin 2 by 0.7% to 1.2% in nucleotides in the VP1 region and shared 5 nucleotide substitutions with the pre-VDPV. All 5 isolates were closely related, and all were S2/S3/S2/S3 recombinants sharing common recombination crossover sites. Although the two major determinants of attenuation and temperature sensitivity phenotype of Sabin 2 (A481 in the 5' untranslated region and Ile143 in the VP1 protein) had reverted in all 5 isolates, one VDPV (strain CHN16017) still retained the temperature sensitivity phenotype. Phylogenetic analysis of the third coding position of the complete P1 coding region suggested that the cVDPVs circulated locally for about 7 months following the initiating oral poliovirus vaccine (OPV) dose. Our findings reinforce the point that cVDPVs can emerge and spread in isolated communities with immunity gaps and highlight the emergence risks of type 2 cVDPVs accompanying the trivalent OPV used. To solve this issue, it is recommended that type 2 OPV be removed from the trivalent OPV or that inactivated polio vaccine (IPV) be used instead. 8. PKA antagonizes CLASP-dependent microtubule stabilization to re-localize Pom1 and buffer cell size upon glucose limitation PubMed Central Kelkar, Manasi; Martin, Sophie G. 2015-01-01 Cells couple growth with division and regulate size in response to nutrient availability. In rod-shaped fission yeast, cell-size control occurs at mitotic commitment. An important regulator is the DYRK-family kinase Pom1, which forms gradients from cell poles and inhibits the mitotic activator Cdr2, itself localized at the medial cortex. Where and when Pom1 modulates Cdr2 activity is unclear as Pom1 medial cortical levels remain constant during cell elongation. Here we show that Pom1 re-localizes to cell sides upon environmental glucose limitation, where it strongly delays mitosis. This re-localization is caused by severe microtubule destabilization upon glucose starvation, with microtubules undergoing catastrophe and depositing the Pom1 gradient nucleator Tea4 at cell sides. Microtubule destabilization requires PKA/Pka1 activity, which negatively regulates the microtubule rescue factor CLASP/Cls1/Peg1, reducing CLASP's ability to stabilize microtubules. Thus, PKA signalling tunes CLASP's activity to promote Pom1 cell side localization and buffer cell size upon glucose starvation. PMID:26443240 9. On Liouville's theorem in fluid mechanics NASA Astrophysics Data System (ADS) Morrison, P. J.; Bouchet, F.; Thalabard, S.; Zaboronski, O. V. 2011-11-01 Since the early work of Burgers it has been known that discretizations of fluid models possess a version of Liouville's theorem on conservation of phase space volume. In fact, spectral representations of two-dimensional turbulence are known to have a detailed version of this theorem. The existence of such Liouville theorems led many (e.g. Burgers, Lee, Kraichnan and Montgomery) to consider various statistical mechanical approaches to turbulence. We show how this theorem arises naturally from the Hamiltonian structure of inviscid fluid equations. 10. Cosmological perturbations and the Weinberg theorem SciTech Connect Akhshik, Mohammad; Firouzjahi, Hassan; Jazayeri, Sadra E-mail: firouz@ipm.ir 2015-12-01 The celebrated Weinberg theorem in cosmological perturbation theory states that there always exist two adiabatic scalar modes in which the comoving curvature perturbation is conserved on super-horizon scales. In particular, when the perturbations are generated from a single source, such as in single field models of inflation, both of the two allowed independent solutions are adiabatic and conserved on super-horizon scales. There are few known examples in literature which violate this theorem. We revisit the theorem and specify the loopholes in some technical assumptions which violate the theorem in models of non-attractor inflation, fluid inflation, solid inflation and in the model of pseudo conformal universe. 11. Fluctuation theorem for partially masked nonequilibrium dynamics NASA Astrophysics Data System (ADS) Shiraishi, Naoto; Sagawa, Takahiro 2015-01-01 We establish a generalization of the fluctuation theorem for partially masked nonequilibrium dynamics. We introduce a partial entropy production with a subset of all possible transitions, and show that the partial entropy production satisfies the integral fluctuation theorem. Our result reveals the fundamental properties of a broad class of autonomous as well as nonautonomous nanomachines. In particular, our result gives a unified fluctuation theorem for both autonomous and nonautonomous Maxwell's demons, where mutual information plays a crucial role. Furthermore, we derive a fluctuation-dissipation theorem that relates nonequilibrium stationary current to two kinds of equilibrium fluctuations. 12. Fluctuation theorem for the renormalized entropy change in the strongly nonlinear nonequilibrium regime. PubMed Sughiyama, Yuki; Abe, Sumiyoshi 2008-08-01 A nonlinear relaxation process is considered for a macroscopic thermodynamic quantity, generalizing recent work by Taniguchi and Cohen [J. Stat. Phys. 126, 1 (2006)] that was based on the Onsager-Machlup theory. It is found that the fluctuation theorem holds in the nonlinear nonequilibrium regime if the change of entropy characterized by local equilibria is appropriately renormalized. The fluctuation theorem for the ordinary entropy change is recovered in the linear near-equilibrium case. 13. The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces NASA Astrophysics Data System (ADS) Huang, Jianhua 2005-12-01 In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679-690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results. 14. INTERPOLATION THEOREMS FOR THE SPACES L_{p,q} NASA Astrophysics Data System (ADS) Ovchinnikov, V. I. 1989-02-01 A sharp or optimal interpolation theorem is proved for the Lorentz spaces L_{p,q}, generalizing the Marcinkiewicz theorem and refining the Riesz-Thorin theorem and the Stein-Weiss theorem. This theorem extends to the spaces \\overline{X}_{\\theta,p} of the real method constructed from any Banach pair; thus it extends also to Besov spaces.Bibliography: 12 titles. 15. 26 CFR 48.6427-6 - Limitation on credit or refund of tax paid on fuel used in intercity, local or school buses after... Code of Federal Regulations, 2012 CFR 2012-04-01 ... fuel used in intercity, local or school buses after July 31, 1984. 48.6427-6 Section 48.6427-6 Internal... intercity, local or school buses after July 31, 1984. (a) Limitation on amount of credit or refund—(1) In... except where fuel is used in a bus while such bus is being operated as a “qualified local bus” in... 16. 26 CFR 48.6427-6 - Limitation on credit or refund of tax paid on fuel used in intercity, local or school buses after... Code of Federal Regulations, 2013 CFR 2013-04-01 ... fuel used in intercity, local or school buses after July 31, 1984. 48.6427-6 Section 48.6427-6 Internal... intercity, local or school buses after July 31, 1984. (a) Limitation on amount of credit or refund—(1) In... except where fuel is used in a bus while such bus is being operated as a “qualified local bus” in... 17. 26 CFR 48.6427-6 - Limitation on credit or refund of tax paid on fuel used in intercity, local or school buses after... Code of Federal Regulations, 2011 CFR 2011-04-01 ... fuel used in intercity, local or school buses after July 31, 1984. 48.6427-6 Section 48.6427-6 Internal... intercity, local or school buses after July 31, 1984. (a) Limitation on amount of credit or refund—(1) In... except where fuel is used in a bus while such bus is being operated as a “qualified local bus” in... 18. Uniqueness theorems in bioluminescence tomography. PubMed Wang, Ge; Li, Yi; Jiang, Ming 2004-08-01 Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed in the University of Iowa. While the forward imaging model is described by the well-known diffusion equation, the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy data. Our primary goal in this paper is to establish the solution uniqueness for BLT under practical constraints despite the ill-posedness of the inverse problem in the general case. After a review on the inverse source literature, we demonstrate that in the general case the BLT solution is not unique by constructing the set of all the solutions to this inverse problem. Then, we show the uniqueness of the solution in the case of impulse sources. Finally, we present our main theorem that solid/hollow ball sources can be uniquely determined up to nonradiating sources. For better readability, the exact conditions for and rigorous proofs of the theorems are given in the Appendices. Further research directions are also discussed. 19. Impurity-limited resistance and phase interference of localized impurities under quasi-one dimensional nano-structures NASA Astrophysics Data System (ADS) Sano, Nobuyuki 2015-12-01 The impurity-limited resistance and the effect of the phase interference among localized multiple impurities in the quasi-one dimensional (quasi-1D) nanowire structures are systematically investigated under the framework of the scattering theory. We derive theoretical expressions of the impurity-limited resistance in the nanowire under the linear response regime from the Landauer formula and from the Boltzmann transport equation (BTE) with the relaxation time approximation. We show that the formula from the BTE exactly coincides with that from the Landauer approach with the weak-scattering limit when the energy spectrum of the in-coming electrons from the reservoirs is narrow and, thus, point out a possibility that the distinction of the impurity-limited resistances derived from the Landauer formula and that of the BTE could be made clear. The derived formulas are applied to the quasi-1D nanowires doped with multiple localized impurities with short-range scattering potential and the validity of various approximations on the resistance are discussed. It is shown that impurity scattering becomes so strong under the nanowire structures that the weak-scattering limit breaks down in most cases. Thus, both phase interference and phase randomization simultaneously play a crucial role in determining the impurity-limited resistance even under the fully coherent framework. When the impurity separation along the wire axis direction is small, the constructive phase interference dominates and the resistance is much greater than the average resistance. As the separation becomes larger, however, it approaches the series resistance of the single-impurity resistance due to the phase randomization. Furthermore, under the uniform configuration of impurities, the space-average resistance of multiple impurities at room temperature is very close to the series resistance of the single-impurity resistance, and thus, each impurity could be regarded as an independent scattering center. The 20. Impurity-limited resistance and phase interference of localized impurities under quasi-one dimensional nano-structures SciTech Connect Sano, Nobuyuki 2015-12-28 The impurity-limited resistance and the effect of the phase interference among localized multiple impurities in the quasi-one dimensional (quasi-1D) nanowire structures are systematically investigated under the framework of the scattering theory. We derive theoretical expressions of the impurity-limited resistance in the nanowire under the linear response regime from the Landauer formula and from the Boltzmann transport equation (BTE) with the relaxation time approximation. We show that the formula from the BTE exactly coincides with that from the Landauer approach with the weak-scattering limit when the energy spectrum of the in-coming electrons from the reservoirs is narrow and, thus, point out a possibility that the distinction of the impurity-limited resistances derived from the Landauer formula and that of the BTE could be made clear. The derived formulas are applied to the quasi-1D nanowires doped with multiple localized impurities with short-range scattering potential and the validity of various approximations on the resistance are discussed. It is shown that impurity scattering becomes so strong under the nanowire structures that the weak-scattering limit breaks down in most cases. Thus, both phase interference and phase randomization simultaneously play a crucial role in determining the impurity-limited resistance even under the fully coherent framework. When the impurity separation along the wire axis direction is small, the constructive phase interference dominates and the resistance is much greater than the average resistance. As the separation becomes larger, however, it approaches the series resistance of the single-impurity resistance due to the phase randomization. Furthermore, under the uniform configuration of impurities, the space-average resistance of multiple impurities at room temperature is very close to the series resistance of the single-impurity resistance, and thus, each impurity could be regarded as an independent scattering center. The 1. Initiation of Combustion of a Gel-Like Condensed Substance by a Local Source of Limited Power NASA Astrophysics Data System (ADS) Glushkov, D. O.; Kuznetsov, G. V.; Strizhak, P. A. 2017-01-01 A physical and a mathematical model of the gas-phase ignition of a gel-like condensed substance, containing a combustible (hydrazine) and an oxidizer (liquefied oxygen) in its composition, at a cryogenic initial temperature have been developed. A numerical investigation of the integral characteristics of the interrelated physicochemical processes occurring in the initiation of combustion of such a substance by a typical energy source of limited heat content (an individual small-size particle heated to a high temperature) has been perfumed. The dependence of the delay time of ignition of the indicated substance on the heat content of a local heat source was determined. The minimum values of the main parameters of hot particles (their temperature and sizes), at which the ignition conditions are fulfilled, were estimated. It is shown that the delay time of ignition of a gel-like condensed substance depends mainly on the initial temperature of an energy source. The characteristic features of the conditions and regimes of initiation of combustion of condensed substances found in different aggregate states (solid, liquid, gel-like) under conditions of their local heating by a heat source of limiting power were analyzed. 2. High thermal tolerance of a rainbow trout population near its southern range limit suggests local thermal adjustment PubMed Central Verhille, Christine E.; English, Karl K.; Cocherell, Dennis E.; Farrell, Anthony P.; Fangue, Nann A. 2016-01-01 Transformation of earth's ecosystems by anthropogenic climate change is predicted for the 21st century. In many regions, the associated increase in environmental temperatures and reduced precipitation will have direct effects on the physiological performance of terrestrial and aquatic ectotherms and have already threatened fish biodiversity and important fisheries. The threat of elevated environmental temperatures is particularly salient for members of the Oncorhynchus genus living in California, which is the southern limit of their range. Here, we report the first assessments of the aerobic capacity of a Californian population of wild Oncorhynchus mykiss Walbaum in relationship to water temperature. Our field measurements revealed that wild O. mykiss from the lower Tuolumne River, California maintained 95% of their peak aerobic scope across an impressive temperature range (17.8–24.6°C). The thermal range for peak performance corresponds to local high river temperatures, but represents an unusually high temperature tolerance compared with conspecifics and congeneric species from northern latitudes. This high thermal tolerance suggests that O. mykiss at the southern limit of their indigenous distribution may be locally adjusted relative to more northern populations. From fisheries management and conservation perspectives, these findings challenge the use of a single thermal criterion to regulate the habitat of the O. mykiss species along the entirety of its distribution range. PMID:27957333 3. Local and Regional Diversity Reveals Dispersal Limitation and Drift as Drivers for Groundwater Bacterial Communities from a Fractured Granite Formation. PubMed Beaton, E D; Stevenson, Bradley S; King-Sharp, Karen J; Stamps, Blake W; Nunn, Heather S; Stuart, Marilyne 2016-01-01 Microorganisms found in terrestrial subsurface environments make up a large proportion of the Earth's biomass. Biogeochemical cycles catalyzed by subsurface microbes have the potential to influence the speciation and transport of radionuclides managed in geological repositories. To gain insight on factors that constrain microbial processes within a formation with restricted groundwater flow we performed a meta-community analysis on groundwater collected from multiple discrete fractures underlying the Chalk River Laboratories site (located in Ontario, Canada). Bacterial taxa were numerically dominant in the groundwater. Although these were mainly uncultured, the closest cultivated representatives were from the phenotypically diverse Betaproteobacteria, Deltaproteobacteria, Bacteroidetes, Actinobacteria, Nitrospirae, and Firmicutes. Hundreds of taxa were identified but only a few were found in abundance (>1%) across all assemblages. The remainder of the taxa were low abundance. Within an ecological framework of selection, dispersal and drift, the local and regional diversity revealed fewer taxa within each assemblage relative to the meta-community, but the taxa that were present were more related than predicted by chance. The combination of dispersion at one phylogenetic depth and clustering at another phylogenetic depth suggest both niche (dispersion) and filtering (clustering) as drivers of local assembly. Distance decay of similarity reveals apparent biogeography of 1.5 km. Beta diversity revealed greater influence of selection at shallow sampling locations while the influences of dispersal limitation and randomness were greater at deeper sampling locations. Although selection has shaped each assemblage, the spatial scale of groundwater sampling favored detection of neutral processes over selective processes. Dispersal limitation between assemblages combined with local selection means the meta-community is subject to drift, and therefore, likely reflects the 4. Local and Regional Diversity Reveals Dispersal Limitation and Drift as Drivers for Groundwater Bacterial Communities from a Fractured Granite Formation PubMed Central Beaton, E. D.; Stevenson, Bradley S.; King-Sharp, Karen J.; Stamps, Blake W.; Nunn, Heather S.; Stuart, Marilyne 2016-01-01 Microorganisms found in terrestrial subsurface environments make up a large proportion of the Earth’s biomass. Biogeochemical cycles catalyzed by subsurface microbes have the potential to influence the speciation and transport of radionuclides managed in geological repositories. To gain insight on factors that constrain microbial processes within a formation with restricted groundwater flow we performed a meta-community analysis on groundwater collected from multiple discrete fractures underlying the Chalk River Laboratories site (located in Ontario, Canada). Bacterial taxa were numerically dominant in the groundwater. Although these were mainly uncultured, the closest cultivated representatives were from the phenotypically diverse Betaproteobacteria, Deltaproteobacteria, Bacteroidetes, Actinobacteria, Nitrospirae, and Firmicutes. Hundreds of taxa were identified but only a few were found in abundance (>1%) across all assemblages. The remainder of the taxa were low abundance. Within an ecological framework of selection, dispersal and drift, the local and regional diversity revealed fewer taxa within each assemblage relative to the meta-community, but the taxa that were present were more related than predicted by chance. The combination of dispersion at one phylogenetic depth and clustering at another phylogenetic depth suggest both niche (dispersion) and filtering (clustering) as drivers of local assembly. Distance decay of similarity reveals apparent biogeography of 1.5 km. Beta diversity revealed greater influence of selection at shallow sampling locations while the influences of dispersal limitation and randomness were greater at deeper sampling locations. Although selection has shaped each assemblage, the spatial scale of groundwater sampling favored detection of neutral processes over selective processes. Dispersal limitation between assemblages combined with local selection means the meta-community is subject to drift, and therefore, likely reflects the 5. Bring the Pythagorean Theorem "Full Circle" ERIC Educational Resources Information Center Benson, Christine C.; Malm, Cheryl G. 2011-01-01 Middle school mathematics generally explores applications of the Pythagorean theorem and lays the foundation for working with linear equations. The Grade 8 Curriculum Focal Points recommend that students "apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and… 6. TAUBERIAN THEOREMS FOR MATRIX REGULAR VARIATION PubMed Central MEERSCHAERT, M. M.; SCHEFFLER, H.-P. 2013-01-01 Karamata’s Tauberian theorem relates the asymptotics of a nondecreasing right-continuous function to that of its Laplace-Stieltjes transform, using regular variation. This paper establishes the analogous Tauberian theorem for matrix-valued functions. Some applications to time series analysis are indicated. PMID:24644367 7. The Pythagorean Theorem: I. The finite case PubMed Central Kadison, Richard V. 2002-01-01 The Pythagorean Theorem and variants of it are studied. The variations evolve to a formulation in terms of noncommutative, conditional expectations on von Neumann algebras that displays the theorem as the basic result of noncommutative, metric, Euclidean Geometry. The emphasis in the present article is finite dimensionality, both “discrete” and “continuous.” PMID:11929992 8. A Note on Morley's Triangle Theorem ERIC Educational Resources Information Center Mueller, Nancy; Tikoo, Mohan; Wang, Haohao 2012-01-01 In this note, we offer a proof of a variant of Morley's triangle theorem, when the exterior angles of a triangle are trisected. We also offer a generalization of Morley's theorem when angles of an "n"-gon are "n"-sected. (Contains 9 figures.) 9. The Classical Version of Stokes' Theorem Revisited ERIC Educational Resources Information Center Markvorsen, Steen 2008-01-01 Using only fairly simple and elementary considerations--essentially from first year undergraduate mathematics--we show how the classical Stokes' theorem for any given surface and vector field in R[superscript 3] follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the… 10. General Theorems about Homogeneous Ellipsoidal Inclusions ERIC Educational Resources Information Center Korringa, J.; And Others 1978-01-01 Mathematical theorems about the properties of ellipsoids are developed. Included are Poisson's theorem concerning the magnetization of a homogeneous body of ellipsoidal shape, the polarization of a dielectric, the transport of heat or electricity through an ellipsoid, and other problems. (BB) 11. Fragile X Mental Retardation Protein Is Required to Maintain Visual Conditioning-Induced Behavioral Plasticity by Limiting Local Protein Synthesis PubMed Central Liu, Han-Hsuan 2016-01-01 Fragile X mental retardation protein (FMRP) is thought to regulate neuronal plasticity by limiting dendritic protein synthesis, but direct demonstration of a requirement for FMRP control of local protein synthesis during behavioral plasticity is lacking. Here we tested whether FMRP knockdown in Xenopus optic tectum affects local protein synthesis in vivo and whether FMRP knockdown affects protein synthesis-dependent visual avoidance behavioral plasticity. We tagged newly synthesized proteins by incorporation of the noncanonical amino acid azidohomoalanine and visualized them with fluorescent noncanonical amino acid tagging (FUNCAT). Visual conditioning and FMRP knockdown produce similar increases in FUNCAT in tectal neuropil. Induction of visual conditioning-dependent behavioral plasticity occurs normally in FMRP knockdown animals, but plasticity degrades over 24 h. These results indicate that FMRP affects visual conditioning-induced local protein synthesis and is required to maintain the visual conditioning-induced behavioral plasticity. SIGNIFICANCE STATEMENT Fragile X syndrome (FXS) is the most common form of inherited intellectual disability. Exaggerated dendritic protein synthesis resulting from loss of fragile X mental retardation protein (FMRP) is thought to underlie cognitive deficits in FXS, but no direct evidence has demonstrated that FMRP-regulated dendritic protein synthesis affects behavioral plasticity in intact animals. Xenopus tadpoles exhibit a visual avoidance behavior that improves with visual conditioning in a protein synthesis-dependent manner. We showed that FMRP knockdown and visual conditioning dramatically increase protein synthesis in neuronal processes. Furthermore, induction of visual conditioning-dependent behavioral plasticity occurs normally after FMRP knockdown, but performance rapidly deteriorated in the absence of FMRP. These studies show that FMRP negatively regulates local protein synthesis and is required to maintain visual 12. PET Cell Tracking Using 18F-FLT is Not Limited by Local Reuptake of Free Radiotracer PubMed Central MacAskill, Mark G.; Tavares, Adriana S.; Wu, Junxi; Lucatelli, Christophe; Mountford, Joanne C.; Baker, Andrew H.; Newby, David E.; Hadoke, Patrick W. F. 2017-01-01 Assessing the retention of cell therapies following implantation is vital and often achieved by labelling cells with 2′-[18F]-fluoro-2′-deoxy-D-glucose (18F-FDG). However, this approach is limited by local retention of cell-effluxed radiotracer. Here, in a preclinical model of critical limb ischemia, we assessed a novel method of cell tracking using 3′-deoxy-3′-L-[18F]-fluorothymidine (18F-FLT); a clinically available radiotracer which we hypothesise will result in minimal local radiotracer reuptake and allow a more accurate estimation of cell retention. Human endothelial cells (HUVECs) were incubated with 18F-FDG or 18F-FLT and cell characteristics were evaluated. Dynamic positron emission tomography (PET) images were acquired post-injection of free 18F-FDG/18F-FLT or 18F-FDG/18F-FLT-labelled HUVECs, following the surgical induction of mouse hind-limb ischemia. In vitro, radiotracer incorporation and efflux was similar with no effect on cell viability, function or proliferation under optimised conditions (5 MBq/mL, 60 min). Injection of free radiotracer demonstrated a faster clearance of 18F-FLT from the injection site vs. 18F-FDG (p ≤ 0.001), indicating local cellular uptake. Using 18F-FLT-labelling, estimation of HUVEC retention within the engraftment site 4 hr post-administration was 24.5 ± 3.2%. PET cell tracking using 18F-FLT labelling is an improved approach vs. 18F-FDG as it is not susceptible to local host cell reuptake, resulting in a more accurate estimation of cell retention. PMID:28287126 13. Limited Pollen Dispersal Contributes to Population Genetic Structure but Not Local Adaptation in Quercus oleoides Forests of Costa Rica PubMed Central Deacon, Nicholas John; Cavender-Bares, Jeannine 2015-01-01 Background Quercus oleoides Cham. and Schlect., tropical live oak, is a species of conservation importance in its southern range limit of northwestern Costa Rica. It occurs in high-density stands across a fragmented landscape spanning a contrasting elevation and precipitation gradient. We examined genetic diversity and spatial genetic structure in this geographically isolated and genetically distinct population. We characterized population genetic diversity at 11 nuclear microsatellite loci in 260 individuals from 13 sites. We monitored flowering time at 10 sites, and characterized the local environment in order to compare observed spatial genetic structure to hypotheses of isolation-by-distance and isolation-by-environment. Finally, we quantified pollen dispersal distances and tested for local adaptation through a reciprocal transplant experiment in order to experimentally address these hypotheses. Results High genetic diversity is maintained in the population and the genetic variation is significantly structured among sampled sites. We identified 5 distinct genetic clusters and average pollen dispersal predominately occurred over short distances. Differences among sites in flowering phenology and environmental factors, however, were not strictly associated with genetic differentiation. Growth and survival of upland and lowland progeny in their native and foreign environments was expected to exhibit evidence of local adaptation due to the more extreme dry season in the lowlands. Seedlings planted in the lowland garden experienced much higher mortality than seedlings in the upland garden, but we did not identify evidence for local adaptation. Conclusion Overall, this study indicates that the Costa Rican Q. oleoides population has a rich population genetic history. Despite environmental heterogeneity and habitat fragmentation, isolation-by-distance and isolation-by-environment alone do not explain spatial genetic structure. These results add to studies of genetic 14. Using Pictures to Enhance Students' Understanding of Bayes' Theorem ERIC Educational Resources Information Center Trafimow, David 2011-01-01 Students often have difficulty understanding algebraic proofs of statistics theorems. However, it sometimes is possible to prove statistical theorems with pictures in which case students can gain understanding more easily. I provide examples for two versions of Bayes' theorem. 15. On the Theorem of Correspondence. PubMed Krøjgaard, Peter 2017-03-01 In a recent paper, Mammen (Integrative Psychological and Behavioral Science, 50, 196-233, 2016a) brought novel arguments into the discussion concerning the importance of being able to single out and track objects through space and time. Mammen offered a formal account of two basic, yet distinct, ways in which we as human beings encounter objects in the real world, that is, sense and choice categories. In this paper I discuss aspects of his theory and in particular the Theorem of Correspondence. I shall attempt to argue that Mammen's formal account is indeed a novel and powerful analytical generic tool allowing us to see the important relevance in different domains of being able to establish choice categories. Meanwhile, I will attempt to show that evidence from the so-called multiple object tracking studies -- even though these use highly artificial stimuli -- provide compelling evidence in support of Mammen's formal account. 16. Singlet and triplet instability theorems SciTech Connect Yamada, Tomonori; Hirata, So 2015-09-21 A useful definition of orbital degeneracy—form-degeneracy—is introduced, which is distinct from the usual energy-degeneracy: Two canonical spatial orbitals are form-degenerate when the energy expectation value in the restricted Hartree–Fock (RHF) wave function is unaltered upon a two-electron excitation from one of these orbitals to the other. Form-degenerate orbitals tend to have isomorphic electron densities and occur in the highest-occupied and lowest-unoccupied molecular orbitals (HOMOs and LUMOs) of strongly correlated systems. Here, we present a mathematical proof of the existence of a triplet instability in a real or complex RHF wave function of a finite system in the space of real or complex unrestricted Hartree–Fock wave functions when HOMO and LUMO are energy- or form-degenerate. We also show that a singlet instability always exists in a real RHF wave function of a finite system in the space of complex RHF wave functions, when HOMO and LUMO are form-degenerate, but have nonidentical electron densities, or are energy-degenerate. These theorems provide Hartree–Fock-theory-based explanations of Hund’s rule, a singlet instability in Jahn–Teller systems, biradicaloid electronic structures, and a triplet instability during some covalent bond breaking. They also suggest (but not guarantee) the spontaneous formation of a spin density wave (SDW) in a metallic solid. The stability theory underlying these theorems extended to a continuous orbital-energy spectrum proves the existence of an oscillating (nonspiral) SDW instability in one- and three-dimensional homogeneous electron gases, but only at low densities or for strong interactions. 17. Posterior Probability and Fluctuation Theorem in Stochastic Processes NASA Astrophysics Data System (ADS) Ohkubo, Jun 2009-12-01 A generalization of fluctuation theorems in stochastic processes is proposed. The new theorem is written in terms of posterior probabilities, which are introduced via Bayes’ theorem. In conventional fluctuation theorems, a forward path and its time reversal play an important role, so that a microscopically reversible condition is essential. In contrast, the microscopically reversible condition is not necessary in the new theorem. It is shown that the new theorem recovers various theorems and relations previously known, such as the Gallavotti-Cohen-type fluctuation theorem, the Jarzynski equality, and the Hatano-Sasa relation, when suitable assumptions are employed. 18. Localized tip enhanced Raman spectroscopic study of impurity incorporated single GaN nanowire in the sub-diffraction limit SciTech Connect Patsha, Avinash E-mail: dhara@igcar.gov.in; Dhara, Sandip; Tyagi, A. K. 2015-09-21 The localized effect of impurities in single GaN nanowires in the sub-diffraction limit is reported using the study of lattice vibrational modes in the evanescent field of Au nanoparticle assisted tip enhanced Raman spectroscopy (TERS). GaN nanowires with the O impurity and the Mg dopants were grown by the chemical vapor deposition technique in the catalyst assisted vapor-liquid-solid process. Symmetry allowed Raman modes of wurtzite GaN are observed for undoped and doped nanowires. Unusually very strong intensity of the non-zone center zone boundary mode is observed for the TERS studies of both the undoped and the Mg doped GaN single nanowires. Surface optical mode of A{sub 1} symmetry is also observed for both the undoped and the Mg doped GaN samples. A strong coupling of longitudinal optical (LO) phonons with free electrons, however, is reported only in the O rich single nanowires with the asymmetric A{sub 1}(LO) mode. Study of the local vibration mode shows the presence of Mg as dopant in the single GaN nanowires. 19. Local density of electromagnetic states in plasmonic nanotapers: spatial resolution limits with nitrogen-vacancy centers in diamond nanospheres. PubMed Salas-Montiel, Rafael; Berthel, Martin; Beltrán-Madrigal, Josslyn; Huant, Serge; Drezet, Aurélien; Blaize, Sylvain 2017-03-21 One of the most explored single quantum emitters for the development of nanoscale fluorescence lifetime imaging is the nitrogen-vacancy (NV) color center in diamond. Indeed, a NV center does not experience fluorescence bleaching and blinking at room temperature. Furthermore, its optical properties are preserved when embedded into nanodiamond hosts. This letter focuses on the modeling of the local density of states (LDOS) in a plasmonic nanofocusing structure with NV center acting as local illumination sources. Numerical calculations of the LDOS near such nanostructure were done with a classical electric dipole radiation placed inside a diamond sphere as well as near-field optical fluorescence lifetime imaging of the structure. We found that Purcell factors higher than 10 can be reached with diamond nanospheres of radius less than 5 nm and at a distance of less than 20 nm from the surface of the structure. Although the spatial resolution of the experiment is limited by the size of the nanodiamond, our work supports the analysis and interpretation of single NV color center in nanodiamond as a probe for scanning near-field optical microscopy. 20. Precision atomic spectroscopy for improved limits on variation of the fine structure constant and local position invariance. PubMed Fortier, T M; Ashby, N; Bergquist, J C; Delaney, M J; Diddams, S A; Heavner, T P; Hollberg, L; Itano, W M; Jefferts, S R; Kim, K; Levi, F; Lorini, L; Oskay, W H; Parker, T E; Shirley, J; Stalnaker, J E 2007-02-16 We report tests of local position invariance and the variation of fundamental constants from measurements of the frequency ratio of the 282-nm 199Hg+ optical clock transition to the ground state hyperfine splitting in 133Cs. Analysis of the frequency ratio of the two clocks, extending over 6 yr at NIST, is used to place a limit on its fractional variation of <5.8x10(-6) per change in normalized solar gravitational potential. The same frequency ratio is also used to obtain 20-fold improvement over previous limits on the fractional variation of the fine structure constant of |alpha/alpha|<1.3x10(-16) yr-1, assuming invariance of other fundamental constants. Comparisons of our results with those previously reported for the absolute optical frequency measurements in H and 171Yb+ vs other 133Cs standards yield a coupled constraint of -1.5x10(-15) 1. Kato type operators and Weyl's theorem NASA Astrophysics Data System (ADS) Duggal, B. P.; Djordjevic, S. V.; Kubrusly, Carlos 2005-09-01 A Banach space operator T satisfies Weyl's theorem if and only if T or T* has SVEP at all complex numbers [lambda] in the complement of the Weyl spectrum of T and T is Kato type at all [lambda] which are isolated eigenvalues of T of finite algebraic multiplicity. If T* (respectively, T) has SVEP and T is Kato type at all [lambda] which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all [lambda][set membership, variant]iso[sigma](T)), then T satisfies a-Weyl's theorem (respectively, T* satisfies a-Weyl's theorem). 2. The Lax-Onsager regression theorem' revisited NASA Astrophysics Data System (ADS) Lax, Melvin 2000-05-01 It is stated by Ford and O'Connell in this festschrift issue and elsewhere that "there is no quantum regression theorem" although Lax "obtained a formula for correlation in a driven quantum system that has come to be called the quantum regression theorem". This produces a puzzle: "How can it be that a non-existent theorem gives correct results?" Clarification will be provided in this paper by a description of the Lax procedure, with a quantitative estimate of the error for a damped harmonic oscillator based on expressions published in the 1960's. 3. 26 CFR 48.6427-6 - Limitation on credit or refund of tax paid on fuel used in intercity, local or school buses after... Code of Federal Regulations, 2010 CFR 2010-04-01 ... fuel used in intercity, local or school buses after July 31, 1984. 48.6427-6 Section 48.6427-6 Internal... intercity, local or school buses after July 31, 1984. (a) Limitation on amount of credit or refund—(1) In general. In the case of fuel sold or used after July 31, 1984, on which tax was imposed under section... 4. Imbedding Locally Euclidean and Conformally Euclidean Metrics NASA Astrophysics Data System (ADS) Aleksandrov, V. A. 1992-02-01 The possibility of imbedding n-dimensional locally Euclidean metrics in the large in Rn is studied by means of the global inverse function theorem in the forms suggested by Hadamard, John, Levy and Plastock. The imbeddability of conformally Euclidean metrics is studied by means of a theorem of Zorich on the removability of an isolated singularity of a locally quasiconformal mapping. 5. Duality Theorems in Ergodic Transport NASA Astrophysics Data System (ADS) Lopes, Artur O.; Mengue, Jairo K. 2012-11-01 We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose σ is the shift acting on Bernoulli space X={1,2,…, d}ℕ, and, consider a fixed continuous cost function c: X× X→ℝ. Denote by Π the set of all Borel probabilities π on X× X, such that, both its x and y marginals are σ-invariant probabilities. We are interested in the optimal plan π which minimizes ∫ c dπ among the probabilities in Π. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on c. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs c the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan π by convex combinations of plans such that the support projects in periodic orbits. 6. ALGEBRAIC DEPENDENCE THEOREMS ON COMPLEX PSEUDOCONCAVE SPACES DTIC Science & Technology The notion of pseudoconcave space is introduced and classical theorems on algebraic dependence of meromorphic functions are extended for this new class of spaces and for sections in a coherent sheaf. (Author) 7. Strain localization in brittle-ductile shear zones: fluid abundant vs fluid limited conditions (an example from Wyangala area, Australia) NASA Astrophysics Data System (ADS) Spruzeniece, L.; Piazolo, S. 2015-04-01 This study focuses on physiochemical processes occurring in a brittle-ductile shear zone at both fluid-present and fluid-limited conditions. In the studied shear zone (Wyangala, SE Australia), a coarse-grained two feldspar-quartz-biotite granite is transformed into a medium grained orthogneiss at the shear zone margins and a fine-grained quartz-muscovite phyllonite in the central parts. The orthogneiss displays cataclasis of feldspar and crystal-plastic deformation of quartz. Quartz accommodates most of the deformation and is extensively recrystallized showing distinct crystallographic preferred orientation (CPO). Feldspar-to-muscovite, biotite-to-muscovite and albitization reactions occur locally at porphyroclasts' fracture surfaces and margins. However, the bulk rock composition shows very little change in respect to the wall rock composition. In contrast, in the shear zone centre quartz occurs as large, weakly deformed porphyroclasts, in sizes similar to that in the wall rock, suggesting that it has undergone little deformation. Feldspars and biotite are almost completely reacted to muscovite, which is arranged in a fine-grained interconnected matrix. Muscovite-rich layers contain significant amounts of fine-grained intermixed quartz with random CPO. These domains are interpreted to have accommodated most of the strain. Bulk rock chemistry data shows a significant increase in SiO2 and depletion in NaO content compared to the wall rock composition. We suggest that the high and low strain fabrics represent markedly different scenarios and cannot be interpreted as a simple sequential development with respect to strain. We suggest that the fabrics and mineralogical changes in the shear zone centre have formed due to fluid influx probably along an initially brittle fracture. Here, hydration reactions dramatically changed the rheological properties of the rock. In the newly produced muscovite-quartz layers creep cavitation associated with grain boundary sliding and 8. Sahoo- and Wayment-Type Integral Mean Value Theorems ERIC Educational Resources Information Center Tiryaki, Aydin; Cakmak, Devrim 2010-01-01 In this article, by using Rolle's theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment ["An integral mean value theorem", Math. Gazette 54 (1970), pp. 300-301] and Sahoo ["Some results related to the integral mean value… 9. The Great Emch Closure Theorem and a combinatorial proof of Poncelet's Theorem NASA Astrophysics Data System (ADS) Avksentyev, E. A. 2015-11-01 The relations between the classical closure theorems (Poncelet's, Steiner's, Emch's, and the zigzag theorems) and some of their generalizations are discussed. It is known that Emch's Theorem is the most general of these, while the others follow as special cases. A generalization of Emch's Theorem to pencils of circles is proved, which (by analogy with the Great Poncelet Theorem) can be called the Great Emch Theorem. It is shown that the Great Emch and Great Poncelet Theorems are equivalent and can be derived one from the other using elementary geometry, and also that both hold in the Lobachevsky plane as well. A new closure theorem is also obtained, in which the construction of closure is slightly more involved: closure occurs on a variable circle which is tangent to a fixed pair of circles. In conclusion, a combinatorial proof of Poncelet's Theorem is given, which deduces the closure principle for an arbitrary number of steps from the principle for three steps using combinatorics and number theory. Bibliography: 20 titles. 10. A Converse of Fermat's Little Theorem ERIC Educational Resources Information Center Bruckman, P. S. 2007-01-01 As the name of the paper implies, a converse of Fermat's Little Theorem (FLT) is stated and proved. FLT states the following: if p is any prime, and x any integer, then x[superscript p] [equivalent to] x (mod p). There is already a well-known converse of FLT, known as Lehmer's Theorem, which is as follows: if x is an integer coprime with m, such… 11. A Physical Proof of the Pythagorean Theorem NASA Astrophysics Data System (ADS) Treeby, David 2017-02-01 What proof of the Pythagorean theorem might appeal to a physics teacher? A proof that involved the notion of mass would surely be of interest. While various proofs of the Pythagorean theorem employ the circumcenter and incenter of a right-angled triangle, we are not aware of any proof that uses the triangle's center of mass. This note details one such proof. Though far from the most elegant approach, we believe it to be novel. 12. Littlewood-Paley Theorem for Schrodinger Operators DTIC Science & Technology 2006-07-26 26 JUL 2006 2. REPORT TYPE 3. DATES COVERED 00-00-2006 to 00-00-2006 4. TITLE AND SUBTITLE Littlewood -Paley theorem for Schrodinger operators...associated with H are well defined. We further give a Littlewood -Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes...unclassified c THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 LITTLEWOOD -PALEY THEOREM FOR SCHRÖDINGER 13. Localization and interaction effects of epitaxial Bi2Se3 bulk states in two-dimensional limit NASA Astrophysics Data System (ADS) Dey, Rik; Roy, Anupam; Pramanik, Tanmoy; Guchhait, Samaresh; Sonde, Sushant; Rai, Amritesh; Register, Leonard F.; Banerjee, Sanjay K. 2016-10-01 Quantum interference effects and electron-electron interactions are found to play an important role in two-dimensional (2D) bulk transport of topological insulator (TI) thin films, which were previously considered as 2D electron gas (2DEG) and explained on basis of Hikami-Larkin-Nagaoka formula and Lee-Ramakrishnan theory. The distinct massive Dirac-type band structure of the TI bulk state gives rise to quantum corrections to conductivity due to interference and interaction effects, which are quite different from that of a 2DEG. We interpret the experimental findings employing Lu-Shen theory particularly derived for the TI system in the 2D limit. The surface and the bulk conductions are identified based on slopes of logarithmic temperature-dependent conductivities with magnetic fields. The perpendicular field magnetoresistance is analyzed considering suppression of weak antilocalization/localization of the surface/bulk electrons by the applied field. We propose corresponding theoretical models to explain the parallel and tilted field magnetoresistance. The effect of the band structure is found to be crucial for an accurate explanation of the magnetotransport results in the TI thin film. 14. Assessing Landscape Constraints on Species Abundance: Does the Neighborhood Limit Species Response to Local Habitat Conservation Programs? PubMed Central Jorgensen, Christopher F.; Powell, Larkin A.; Lusk, Jeffery J.; Bishop, Andrew A.; Fontaine, Joseph J. 2014-01-01 Landscapes in agricultural systems continue to undergo significant change, and the loss of biodiversity is an ever-increasing threat. Although habitat restoration is beneficial, management actions do not always result in the desired outcome. Managers must understand why management actions fail; yet, past studies have focused on assessing habitat attributes at a single spatial scale, and often fail to consider the importance of ecological mechanisms that act across spatial scales. We located survey sites across southern Nebraska, USA and conducted point counts to estimate Ring-necked Pheasant abundance, an economically important species to the region, while simultaneously quantifying landscape effects using a geographic information system. To identify suitable areas for allocating limited management resources, we assessed land cover relationships to our counts using a Bayesian binomial-Poisson hierarchical model to construct predictive Species Distribution Models of relative abundance. Our results indicated that landscape scale land cover variables severely constrained or, alternatively, facilitated the positive effects of local land management for Ring-necked Pheasants. PMID:24918779 15. The Nekhoroshev Theorem and Long-Term Stabilities in the Solar System NASA Astrophysics Data System (ADS) Guzzo, M. 2015-06-01 The Nekhoroshev theorem has been often indicated in the last decades as the reference theorem for explaining the dynamics of several systems which are stable in the long-term. The Solar System dynamics provides a wide range of possible and useful applications. In fact, despite the complicated models which are used to numerically integrate realistic Solar System dynamics as accurately as possible, when the integrated solutions are chaotic the reliability of the numerical integrations is limited, and a theoretical long-term stability analysis is required. After the first formulation of Nekhoroshev's theorem in 1977, many theoretical improvements have been achieved. On the one hand, alternative proofs of the theorem itself led to consistent improvements of the stability estimates; on the other hand, the extensions which were necessary to apply the theorem to the systems of interest for Solar System Dynamics, in particular concerning the removal of degeneracies and the implementation of computer assisted proofs, have been developed. In this review paper we discuss some of the motivations and the results which have made Nekhoroshev's theorem a reference stability result for many applications in the Solar System dynamics. 16. Conditioned Limit Theorems for Some Null Recurrent Markov Processes DTIC Science & Technology 1976-08-01 this conlus ion is the lolloing Suppos I in Pt > t 0 for all (t - nd (iv) hold X10 I J -or each is arN > mtv ,x is an inureas tip function of St hen (v...Diffusion Processes and Their Sample Paths, Springer-Verlag, second printing, (1973). 39. Jacobsen , M., Splitting times for Markov processes and a 17. Central Limit Theorem: New SOCR Applet and Demonstration Activity ERIC Educational Resources Information Center Dinov, Ivo D.; Christou, Nicholas; Sanchez, Juana 2008-01-01 Modern approaches for information technology based blended education utilize a variety of novel instructional, computational and network resources. Such attempts employ technology to deliver integrated, dynamically linked, interactive content and multi-faceted learning environments, which may facilitate student comprehension and information… 18. Pedagogical Simulation of Sampling Distributions and the Central Limit Theorem ERIC Educational Resources Information Center Hagtvedt, Reidar; Jones, Gregory Todd; Jones, Kari 2007-01-01 Students often find the fact that a sample statistic is a random variable very hard to grasp. Even more mysterious is why a sample mean should become ever more Normal as the sample size increases. This simulation tool is meant to illustrate the process, thereby giving students some intuitive grasp of the relationship between a parent population… 19. Determination of forming limit diagrams of AA6013-T6 aluminum alloy sheet using a time and position dependent localized necking criterion NASA Astrophysics Data System (ADS) Dicecco, S.; Butcher, C.; Worswick, M.; Boettcher, E.; Chu, E.; Shi, C. 2016-11-01 The forming limit behaviour of AA6013-T6 aluminium alloy sheet was characterized under isothermal conditions at room temperature (RT) and 250°C using limiting dome height (LDH) tests. Full field strain measurements were acquired throughout testing using in situ stereoscopic digital image correlation (DIC) techniques. Limit strain data was generated from the resulting full field strain measurements using two localized necking criteria: ISO12004- 2:2008 and a time and position dependent criterion, termed the “Necking Zone” (NZ) approach in this paper, introduced by Martinez-Donaire et al. (2014). The limit strains resulting from the two localization detection schemes were compared. It was found that the ISO and NZ limit strains at RT are similar on the draw-side of the FLD, while the NZ approach yields a biaxial major limit strain 14.8% greater than the ISO generated major limit strain. At 250°C, the NZ generated major limit strains are 31-34% greater than the ISO generated major limit strains for near uniaxial, plane strain and biaxial loading conditions, respectively. The significant variance in limit strains between the two methodologies at 250°C highlights the need for a validation study regarding warm FLC determination. 20. Limited clonal heterogeneity of antigen-specific T cells localizing in the pleural space during mycobacterial infection. PubMed Central Manca, F; Rossi, G; Valle, M T; Lantero, S; Li Pira, G; Fenoglio, D; De Bruin, J; Costantini, M; Damiani, G; Balbi, B 1991-01-01 To detect possible differences in phenotype and fine specificity for mycobacterial antigens between CD4-positive T cells from peripheral blood (PB) and from inflammatory sites, we identified four patients presenting with a mycobacterial pleural exudate (PE) rich in PPD-specific lymphocytes and with a negative skin test to tuberculin purified protein derivative (PPD) and a negative proliferative response of PB lymphocytes to PPD at the same time. Several weeks after chemotherapy, these patients converted to PPD responsiveness in the periphery, and PPD-specific clones could be generated from PB at this stage. The phenotypic comparison of PE lymphocytes and concomitant PB lymphocytes obtained before treatment showed an increase of CD8 cells and a high frequency of HLA-DR-positive activated T cells in PE. The frequency of tetanus toxoid-specific and Candida albicans-specific proliferating T cells was lower than that of PPD-specific cells in PE but not in PB. PPD-specific clones were derived initially from PE and from PB once the patients had converted to PPD responsiveness. The two sets of clones from each patient were compared for proliferative response to mycobacterial antigen clusters of defined molecular weight ranges. A large number of PE-derived clones (36%) responded to a fraction of 27 to 35 kDa, whereas only one clone from PB responded to the same fraction. The purified antigen P32 (32 kDa), a soluble mycobacterial protein, stimulated PE-derived clones that were responsive to the 37- to 27-kDa fraction but did not stimulate PB-derived clones. The data demonstrate that PE- and PB-derived lymphocytes differ both in phenotype and in fine specificity, suggesting a limited clonal heterogeneity of T cells localizing at the inflammatory site in tuberculous patients without a PPD response in the periphery. Therefore T cells compartmentalized at inflammatory sites provide information that is different from that provided by T cells in the periphery. PMID:1898906 1. A Program Certification Assistant Based on Fully Automated Theorem Provers NASA Technical Reports Server (NTRS) Denney, Ewen; Fischer, Bernd 2005-01-01 We describe a certification assistant to support formal safety proofs for programs. It is based on a graphical user interface that hides the low-level details of first-order automated theorem provers while supporting limited interactivity: it allows users to customize and control the proof process on a high level, manages the auxiliary artifacts produced during this process, and provides traceability between the proof obligations and the relevant parts of the program. The certification assistant is part of a larger program synthesis system and is intended to support the deployment of automatically generated code in safety-critical applications. 2. Generalized parametric down conversion, many particle interferometry, and Bell's theorem NASA Technical Reports Server (NTRS) Choi, Hyung Sup 1992-01-01 A new field of multi-particle interferometry is introduced using a nonlinear optical spontaneous parametric down conversion (SPDC) of a photon into more than two photons. The study of SPDC using a realistic Hamiltonian in a multi-mode shows that at least a low conversion rate limit is possible. The down converted field exhibits many stronger nonclassical phenomena than the usual two photon parametric down conversion. Application of the multi-particle interferometry to a recently proposed many particle Bell's theorem on the Einstein-Podolsky-Rosen problem is given. 3. Ergodic theorem, ergodic theory, and statistical mechanics PubMed Central Moore, Calvin C. 2015-01-01 This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics. PMID:25691697 4. Republication of: A theorem on Petrov types NASA Astrophysics Data System (ADS) Goldberg, J. N.; Sachs, R. K. 2009-02-01 This is a republication of the paper “A Theorem on Petrov Types” by Goldberg and Sachs, Acta Phys. Pol. 22 (supplement), 13 (1962), in which they proved the Goldberg-Sachs theorem. The article has been selected for publication in the Golden Oldies series of General Relativity and Gravitation. Typographical errors of the original publication were corrected by the editor. The paper is accompanied by a Golden Oldie Editorial containing an editorial note written by Andrzej Krasiński and Maciej Przanowski and Goldberg’s brief autobiography. The editorial note explains some difficult parts of the proof of the theorem and discusses the influence of results of the paper on later research. 5. Fate of the Bose insulator in the limit of strong localization and low Cooper-pair density in ultrathin films NASA Astrophysics Data System (ADS) Hollen, S. M.; Fernandes, G. E.; Xu, J. M.; Valles, J. M. 2014-10-01 A Bose insulator composed of a low density of strongly localized Cooper pairs develops at the two-dimensional superconductor to insulator transition (SIT) in a number of thin film systems. Investigations of ultrathin amorphous PbBi films far from the SIT described here provide evidence that the Bose insulator gives way to a second insulating phase with decreasing film thickness. At a critical film thickness dc the magnetoresistance changes sign from positive, as expected for boson transport, to negative, as expected for fermion transport, signs of local Cooper-pair phase coherence effects on transport vanish, and the transport activation energy exhibits a kink. Below dc pairing fluctuation effects remain visible in the high-temperature transport while the activation energy continues to rise. These features show that Cooper pairing persists and suggest that the localized unpaired electron states involved in transport are interspersed among regions of strongly localized Cooper pairs in this strongly localized, low Cooper-pair density phase. 6. Finite de Finetti theorem for conditional probability distributions describing physical theories NASA Astrophysics Data System (ADS) Christandl, Matthias; Toner, Ben 2009-04-01 We work in a general framework where the state of a physical system is defined by its behavior under measurement and the global state is constrained by no-signaling conditions. We show that the marginals of symmetric states in such theories can be approximated by convex combinations of independent and identical conditional probability distributions, generalizing the classical finite de Finetti theorem of Diaconis and Freedman. Our results apply to correlations obtained from quantum states even when there is no bound on the local dimension, so that known quantum de Finetti theorems cannot be used. 7. An invariance theorem in acoustic scattering theory NASA Astrophysics Data System (ADS) Ha-Duong, T. 1996-10-01 Karp's theorem states that if the far-field pattern corresponding to the scattering of a time-harmonic acoustic plane wave by a sound-soft obstacle is invariant under the group of orthogonal transformations in 0266-5611/12/5/007/img1 (rotations in 0266-5611/12/5/007/img2), then the scatterer is a sphere (circle). The theorem is generalized to the case where the invariant group of the far field pattern is only a subgroup of the orthogonal group, and for a class of mixed boundary conditions. 8. At math meetings, enormous theorem eclipses fermat. PubMed Cipra, B 1995-02-10 Hardly a word was said about Fermat's Last Theorem at the joint meetings of the American Mathematical Society and the Mathematical Association of America, held this year from 4 to 7 January in San Francisco. For Andrew Wiles's proof, no news is good news: There are no reports of mistakes. But mathematicians found plenty of other topics to discuss. Among them: a computational breakthrough in the study of turbulent diffusion and progress in slimming down the proof of an important result in group theory, whose original size makes checking the proof of Fermat's Last Theorem look like an afternoon's pastime. 9. Local fields in conductor surface electromigration: A first-principles study in the low-bias ballistic limit SciTech Connect Bevan, Kirk H; Zhu, Wenguang; Stocks, George Malcolm; Guo, Hong; Zhang, Zhenyu 2012-01-01 Utilizing first-principles quantum transport calculations, we investigate the role of local fields in conductor surface electromigration. A nanometer-thick Ag(100) thin film is adopted as our prototypical conductor, where we demonstrate the existence of intense local electric fields at atomic surface defects under an external bias. It is shown that such local fields can play an important role in driving surface electromigration and electrical breakdown. The intense fields originate from the relatively short (atomic-scale) screening lengths common to most elemental metals. This general short-range screening trend is established self-consistently within an intuitive picture of linear response electrostatics. The findings shed new light on the underlying physical origins of surface electromigration and point to the possibility of harnessing local fields to engineer electromigration at the nanoscale. 10. Fixed-point theorems for families of weakly non-expansive maps NASA Astrophysics Data System (ADS) Mai, Jie-Hua; Liu, Xin-He 2007-10-01 In this paper, we present some fixed-point theorems for families of weakly non-expansive maps under some relatively weaker and more general conditions. Our results generalize and improve several results due to Jungck [G. Jungck, Fixed points via a generalized local commutativity, Int. J. Math. Math. Sci. 25 (8) (2001) 497-507], Jachymski [J. Jachymski, A generalization of the theorem by Rhoades and Watson for contractive type mappings, Math. Japon. 38 (6) (1993) 1095-1102], Guo [C. Guo, An extension of fixed point theorem of Krasnoselski, Chinese J. Math. (P.O.C.) 21 (1) (1993) 13-20], Rhoades [B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257-290], and others. 11. Kochen-Specker theorem as a precondition for secure quantum key distribution SciTech Connect Nagata, Koji 2005-07-15 We show that (1) the violation of the Ekert 1991 inequality is a sufficient condition for certification of the Kochen-Specker (KS) theorem, and (2) the violation of the Bennett-Brassard-Mermin 1992 (BBM92) inequality is, also, a sufficient condition for certification of the KS theorem. Therefore the success in each quantum key distribution protocol reveals the nonclassical feature of quantum theory, in the sense that the KS realism is violated. Further, it turned out that the Ekert inequality and the BBM inequality are depictured by distillable entanglement witness inequalities. Here, we connect the success in these two key distribution processes into the no-hidden-variables theorem and into witness on distillable entanglement. We also discuss the explicit difference between the KS realism and Bell's local realism in the Hilbert space formalism of quantum theory. 12. 20 CFR 668.825 - Does the WIA administrative cost limit for States and local areas apply to section 166 grants? Code of Federal Regulations, 2012 CFR 2012-04-01 ... 20 Employees' Benefits 4 2012-04-01 2012-04-01 false Does the WIA administrative cost limit for States and local areas apply to section 166 grants? 668.825 Section 668.825 Employees' Benefits EMPLOYMENT AND TRAINING ADMINISTRATION, DEPARTMENT OF LABOR (CONTINUED) INDIAN AND NATIVE AMERICAN... 13. 20 CFR 668.825 - Does the WIA administrative cost limit for States and local areas apply to section 166 grants? Code of Federal Regulations, 2014 CFR 2014-04-01 ... 20 Employees' Benefits 4 2014-04-01 2014-04-01 false Does the WIA administrative cost limit for States and local areas apply to section 166 grants? 668.825 Section 668.825 Employees' Benefits EMPLOYMENT AND TRAINING ADMINISTRATION, DEPARTMENT OF LABOR (CONTINUED) INDIAN AND NATIVE AMERICAN... 14. 20 CFR 668.825 - Does the WIA administrative cost limit for States and local areas apply to section 166 grants? Code of Federal Regulations, 2013 CFR 2013-04-01 ... 20 Employees' Benefits 4 2013-04-01 2013-04-01 false Does the WIA administrative cost limit for States and local areas apply to section 166 grants? 668.825 Section 668.825 Employees' Benefits EMPLOYMENT AND TRAINING ADMINISTRATION, DEPARTMENT OF LABOR (CONTINUED) INDIAN AND NATIVE AMERICAN... 15. General proof of the Greenberger-Horne-Zeilinger theorem SciTech Connect Chen Zeqian 2004-09-01 It is proved that all states of three spin-(1/2) particles exhibiting an 'all versus nothing' contradiction between quantum mechanics and the local realism of Einstein, Podolsky, and Rosen are exactly the Greenberger-Horne-Zeilinger (GHZ) states and the states obtained from them by local unitary transformations. The proof is obtained by showing that there are at most four elements (except for a different sign) in a set of mutually commuting nonlocal spin observables in the three-qubit system and using the certain algebraic properties that Pauli's matrices satisfy. We show that only does such a set of four nonlocal spin observables present a Greenberger-Horne-Zeilinger-Mermin-like argument. This also reveals the equivalence between the GHZ theorem and maximal violation of the Bell inequality. 16. Bell's theorem and quantum mechanics NASA Astrophysics Data System (ADS) Rosen, Nathan 1994-02-01 Bell showed that assuming locality leads to a disagreement with quantum mechanics. Here the nature of the nonlocality that follows from quantum mechanics is investigated. Note by the Editor—Readers will recognize Professor Rosen, author of this paper, as one of the co-authors of the famous EPR paper, Albert Einstein, Boris Podolsky, and Nathan Rosen, Can Quantum-Mechanical Description of Physical Reality be considered Complete?'', Phys. Rev. 47, 770-780 (1935). Robert H. Romer, Editor 17. Note on the theorems of Bjerknes and Crocco NASA Technical Reports Server (NTRS) Theodorsen, Theodore 1946-01-01 The theorems of Bjerknes and Crocco are of great interest in the theory of flow around airfoils at Mach numbers near and above unity. A brief note shows how both theorems are developed by short vector transformations. 18. Fluctuation theorem applied to the Nosé-Hoover thermostated Lorentz gas. PubMed Gilbert, Thomas 2006-03-01 We present numerical evidence supporting the validity of the Gallavotti-Cohen fluctuation theorem applied to the driven Lorentz gas with Nosé-Hoover thermostating. It is moreover argued that the asymptotic form of the fluctuation formula is independent of the amplitude of the driving force in the limit where it is small. 19. Fluctuation theorem applied to the Nosé-Hoover thermostated Lorentz gas NASA Astrophysics Data System (ADS) Gilbert, Thomas 2006-03-01 We present numerical evidence supporting the validity of the Gallavotti-Cohen fluctuation theorem applied to the driven Lorentz gas with Nosé-Hoover thermostating. It is moreover argued that the asymptotic form of the fluctuation formula is independent of the amplitude of the driving force in the limit where it is small. 20. Student Research Project: Goursat's Other Theorem ERIC Educational Resources Information Center Petrillo, Joseph 2009-01-01 In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Edouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of… 1. The Pythagorean Theorem and the Solid State ERIC Educational Resources Information Center Kelly, Brenda S.; Splittgerber, Allan G. 2005-01-01 Packing efficiency and crystal density can be calculated from basic geometric principles employing the Pythagorean theorem, if the unit-cell structure is known. The procedures illustrated have applicability in courses such as general chemistry, intermediate and advanced inorganic, materials science, and solid-state physics. 2. Type Theory, Computation and Interactive Theorem Proving DTIC Science & Technology 2015-09-01 Springer, Heidelberg, 61-76, 2014. [9] Jeremy Avigad and John Harrison , “Formally verified mathematics,” Communications of the ACM, 57(4):66-75, 2014. [10...inequalities," in Gerwin Klein and Ruben Gamboa, eds., Interactive Theorem Proving 2014, Springer, Heidelberg, 61-76, 2014. 9) Jeremy Avigad and John Harrison 3. Generalized Friedland's theorem for C0-semigroups NASA Astrophysics Data System (ADS) Cichon, Dariusz; Jung, Il Bong; Stochel, Jan 2008-07-01 Friedland's characterization of bounded normal operators is shown to hold for infinitesimal generators of C0-semigroups. New criteria for normality of bounded operators are furnished in terms of Hamburger moment problem. All this is achieved with the help of the celebrated Ando's theorem on paranormal operators. 4. On Viviani's Theorem and Its Extensions ERIC Educational Resources Information Center Abboud, Elias 2010-01-01 Viviani's theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. Here, in an extension of this result, we show, using linear programming, that any convex polygon can be divided into parallel line segments on which the sum of the distances to the sides of the polygon is constant. Let us say… 5. Abel's Theorem Simplifies Reduction of Order ERIC Educational Resources Information Center Green, William R. 2011-01-01 We give an alternative to the standard method of reduction or order, in which one uses one solution of a homogeneous, linear, second order differential equation to find a second, linearly independent solution. Our method, based on Abel's Theorem, is shorter, less complex and extends to higher order equations. 6. Codimension- p Paley-Wiener theorems NASA Astrophysics Data System (ADS) Yang, Yan; Qian, Tao; Sommen, Frank 2007-04-01 We obtain the generalized codimension- p Cauchy-Kovalevsky extension of the exponential function e^{i1, y,tinmathbf{R}q, and prove the corresponding codimension- p Paley-Wiener theorems. 7. Tennis Rackets and the Parallel Axis Theorem ERIC Educational Resources Information Center Christie, Derek 2014-01-01 This simple experiment uses an unusual graph straightening exercise to confirm the parallel axis theorem for an irregular object. Along the way, it estimates experimental values for g and the moment of inertia of a tennis racket. We use Excel to find a 95% confidence interval for the true values. 8. Fundamental Theorems of Algebra for the Perplexes ERIC Educational Resources Information Center Poodiak, Robert; LeClair, Kevin 2009-01-01 The fundamental theorem of algebra for the complex numbers states that a polynomial of degree n has n roots, counting multiplicity. This paper explores the "perplex number system" (also called the "hyperbolic number system" and the "spacetime number system") In this system (which has extra roots of +1 besides the usual [plus or minus]1 of the… 9. The soft photon theorem for bremsstrahlung SciTech Connect Heller, L. 1990-01-01 We review this theorem and discuss the possible importance of the second term in the expansion of the cross section in powers of the photon momentum, especially for radiation from particle coming from the decay of resonances. 10 refs., 4 figs. 10. A non-differentiable Noether's theorem NASA Astrophysics Data System (ADS) Cresson, Jacky; Greff, Isabelle 2011-02-01 In the framework of the nondifferentiable embedding of Lagrangian systems, defined by Cresson and Greff [non-dierentiable embedding of lagrangian systems and partial dierential equations. Preprint Max-Plank-Institut für Mathematik in den Naturwissenschaften, Leipzig 16, 26 (2010)], we prove a Noether's theorem based on the lifting of one-parameter groups of diffeomorphisms. 11. Reflection theorem for Lorentz-Minkowski spaces NASA Astrophysics Data System (ADS) Lee, Nam-Hoon 2016-07-01 We generalize the reflection theorem of the Lorentz-Minkowski plane to that of the Lorentz-Minkowski spaces of higher dimensions. As a result, we show that an isometry of the Lorentz-Minkowski spacetime is a composition of at most 5 reflections. 12. Ptolemy's Theorem and Familiar Trigonometric Identities. ERIC Educational Resources Information Center Bidwell, James K. 1993-01-01 Integrates the sum, difference, and multiple angle identities into an examination of Ptolemy's Theorem, which states that the sum of the products of the lengths of the opposite sides of a quadrilateral inscribed in a circle is equal to the product of the lengths of the diagonals. (MDH) 13. Canonical perturbation expansions to large order from classical hypervirial and Hellmann-Feynman theorems. NASA Astrophysics Data System (ADS) McRae, S. M.; Vrscay, E. R. 1992-09-01 The classical hypervirial and Hellmann-Feynman theorems are used to formulate a "perturbation theory without Fourier series" that can be used to generate canonical series expansions for the energies of perturbed periodic orbits for separable classical Hamiltonians. Here, the method is applied to one-dimensional anharmonic oscillators and radial Kepler problems. In all cases, the classical series for energies and expectation values are seen to correspond to the expansions associated with their quantum mechanical counterparts through an appropriate action preserving classical limit. This "action fixing" is inherent in the classical Hellmann-Feynman theorem applied to periodic orbits. 14. The electric dipole moment of the electron: An intuitive explanation for the evasion of Schiff's theorem NASA Astrophysics Data System (ADS) Commins, Eugene D.; Jackson, J. David; Demille, David P. 2007-06-01 In most experimental searches for the electron electric dipole moment, one searches for a linear Stark effect in a paramagnetic atom or molecule and interprets the result in terms of the electric dipole moment of the unpaired valence electron(s). Schiff's theorem states that in the limit of nonrelativistic quantum mechanics, there can be no linear Stark effect to first order in the electric dipole moment. Sandars has shown that Schiff's theorem is not applicable when special relativity is taken into account. We give a heuristic explanation for this relativistic effect, which corrects a widespread misconception in the literature. 15. On a Theorem of Weinstein. DTIC Science & Technology 1986-02-01 literture - see e.g. [6 -81 so we will be rather sketchy. Suppose first that H 2 *Then n is determined as the solution of an ordinary differential equation ...finding periodic solutions of differential equations , Nonlinear Evolution Equations (M. G. Crandall, editor), Academic Press, New York, (1978), 225-251...in X.of the form: =-tw(r) !A() - n’’n (2.19) d nI(0,z) - z where )dn) =A (ri) ’’’()14~- *In (2.19), w is a locally Lipschitz continuous function 16. Applications of square-related theorems NASA Astrophysics Data System (ADS) Srinivasan, V. K. 2014-04-01 The square centre of a given square is the point of intersection of its two diagonals. When two squares of different side lengths share the same square centre, there are in general four diagonals that go through the same square centre. The Two Squares Theorem developed in this paper summarizes some nice theoretical conclusions that can be obtained when two squares of different side lengths share the same square centre. These results provide the theoretical basis for two of the constructions given in the book of H.S. Hall and F.H. Stevens , 'A Shorter School Geometry, Part 1, Metric Edition'. In page 134 of this book, the authors present, in exercise 4, a practical construction which leads to a verification of the Pythagorean theorem. Subsequently in Theorems 29 and 30, the authors present the standard proofs of the Pythagorean theorem and its converse. In page 140, the authors present, in exercise 15, what amounts to a geometric construction, whose verification involves a simple algebraic identity. Both the constructions are of great importance and can be replicated by using the standard equipment provided in a 'geometry toolbox' carried by students in high schools. The author hopes that the results proved in this paper, in conjunction with the two constructions from the above-mentioned book, would provide high school students an appreciation of the celebrated theorem of Pythagoras. The diagrams that accompany this document are based on the free software GeoGebra. The author formally acknowledges his indebtedness to the creators of this free software at the end of this document. 17. A Simple Geometrical Derivation of the Spatial Averaging Theorem. ERIC Educational Resources Information Center Whitaker, Stephen 1985-01-01 The connection between single phase transport phenomena and multiphase transport phenomena is easily accomplished by means of the spatial averaging theorem. Although different routes to the theorem have been used, this paper provides a route to the averaging theorem that can be used in undergraduate classes. (JN) 18. Extending the Principal Axis Theorem to Fields Other than R. ERIC Educational Resources Information Center Friedberg, Stephen H. 1990-01-01 That the principal axis theorem does not extend to any finite field is demonstrated. Presented are four examples that illustrate the difficulty in extending the principal axis theorem to fields other than the field of real numbers. Included are a theorem and proof that uses only a simple counting argument. (KR) 19. Using Dynamic Geometry to Explore Non-Traditional Theorems ERIC Educational Resources Information Center Wares, Arsalan 2010-01-01 The purpose of this article is to provide examples of "non-traditional" theorems that can be explored in a dynamic geometry environment by university and high school students. These theorems were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these theorems. The… 20. A Health Impact Assessment of a Proposed Bill to Decrease Speed Limits on Local Roads in Massachusetts (U.S.A.) PubMed Central James, Peter; Ito, Kate; Banay, Rachel F.; Buonocore, Jonathan J.; Wood, Benjamin; Arcaya, Mariana C. 2014-01-01 Decreasing traffic speeds increases the amount of time drivers have to react to road hazards, potentially averting collisions, and makes crashes that do happen less severe. Boston’s regional planning agency, the Metropolitan Area Planning Council (MAPC), in partnership with the Massachusetts Department of Public Health (MDPH), conducted a Health Impact Assessment (HIA) that examined the potential health impacts of a proposed bill in the state legislature to lower the default speed limits on local roads from 30 miles per hour (mph) to 25 mph. The aim was to reduce vehicle speeds on local roads to a limit that is safer for pedestrians, cyclists, and children. The passage of this proposed legislation could have had far-reaching and potentially important public health impacts. Lower default speed limits may prevent around 18 fatalities and 1200 serious injuries to motorists, cyclists and pedestrians each year, as well as promote active transportation by making local roads feel more hospitable to cyclists and pedestrians. While a lower speed limit would increase congestion and slightly worsen air quality, the benefits outweigh the costs from both a health and economic perspective and would save the state approximately$62 million annually from prevented fatalities and injuries. PMID:25279544
1. Quantum Stratonovich calculus and the quantum Wong-Zakai theorem
SciTech Connect
Gough, John
2006-11-15
We extend the Ito(bar sign)-to-Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes. Working within the framework of quantum stochastic calculus, we define Stratonovich calculus as an algebraic modification of the Ito(bar sign) one and give conditions for the existence of Stratonovich time-ordered exponentials. We show that conversion formula for the coefficients has a striking resemblance to Green's function formulas from standard perturbation theory. We show that the calculus conveniently describes the Markov limit of regular open quantum dynamical systems in much the same way as in the Wong-Zakai approximation theorems of classical stochastic analysis. We extend previous limit results to multiple-dimensions with a proof that makes use of diagrammatic conventions.
2. Revisiting MHD stability comparison theorems: Some surprising new results
Cerfon, Antoine; Freidberg, Jeffrey
2009-05-01
The classic MHD stability comparison theorems (Kruskal-Oberman, Rosenbluth-Rostoker) show that ideal MHD yields the most stringent stability limits according to the hierarchy δWCGL>δWKIN>δWMHD. This has long justified the use of ideal MHD for conservative predictions of MHD stability boundaries. We reexamine these theorems, with the following conclusions:(1) It is crucial to distinguish between ergodic and closed field line systems.(2) It is essential to account for resonant particles in the kinetic MHD model.(3) For ergodic systems the original kinetic MHD analysis over-estimates stability: δWKIN>δWMHD. Our new result predicts δWKIN=δWMHD.(4) For closed line systems plasma compressibility effects become important, and resonant particle effects vanish. Both the original and new analysis predict δWKIN>δWMHD. However, using a Vlasov-Fluid model with Vlasov ions and fluid electrons we show that both δWKIN and δWMHD, while mathematically correct, yield the wrong physical result. The V-F model shows that at marginal stability the compressibility stabilization term vanishes identically! For ergodic systems, marginal stability is always incompressible, so δWKIN=δWMHD=δWVF. For compressible modes in closed line systems, however, perpendicular resonant particle effects cancel the stabilizing effect of plasma compressibility predicted by ideal and kinetic MHD: δWKIN>δWMHD>δWVF.
3. Generalized virial theorem in Palatini f(R) gravity
SciTech Connect
Sefiedgar, A. S.; Atazadeh, K.; Sepangi, H. R.
2009-09-15
We use the collision-free Boltzmann equation in Palatini f(R) gravity to derive the virial theorem within the context of the Palatini approach. It is shown that the virial mass is proportional to certain geometrical terms appearing in the Einstein field equations which contributes to gravitational energy and that such geometric mass can be attributed to the virial mass discrepancy in a cluster of galaxies. We then derive the velocity dispersion relation for clusters, followed by the metric tensor components inside the cluster as well as the f(R) Lagrangian in terms of the observational parameters. Since these quantities may also be obtained experimentally, the f(R) virial theorem is a convenient tool to test the viability of f(R) theories in different models. Finally, we discuss the limitations of our approach in light of the cosmological averaging used and questions that have been raised in the literature against such averaging procedures in the context of the present work.
4. Bosonization Theorem and a Model of High-Tc Superconductor.
Ren, Hai-Cang
1996-03-01
For a purely fermionic system on a lattice, there exists a different, but well defined system on the same lattice, consisting both of bona fide fermions and bosons with an interaction depending on a parameter G characterizing on-site repulsion between particles(R. Friedberg, T. D. Lee and H. C. Ren, Phys. Rev. B50, 10190 (1994).). The energy spectrum and the scattering matrix of the former are identical to those in the finite-energy sector of the latter in the hard-core limit, G→∞. This theorem is particularly useful for the description of a fermionic system whose low-lying spectrum consists of bosonic resonances. We argue that the high-Tc superconductors belong to this category and the long-range order in the superphase can be identified with the condensation of resonance bosons. A short coherence length, results from μSR experiments, measurements of the Hall number and the anomalous behavior of H_c2 near T=0 can be understood in terms of this resonance-boson model(R. Friedberg, T. D. Lee and H. C. Ren, Phys. Rev. B42, 4122 (1990).). We have also examined the possibility of a bosonic d-wave resonance(O.Tchernyshyov, A.S.Blaer and H.Ren, in the current Proceedings.). In this case, the bosonization theorem predicts coexistence of an s-wave bosonic condensate and a d-wave gap parameter for fermions.
5. Gradient estimates for u=ΔF(u) on manifolds and some Liouville-type theorems
Xu, Xiangjin
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: u=ΔF(u), with F(u)>0, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): u=Δ(u), p>0, and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson-Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yau's celebrated Liouville theorem for positive harmonic functions.
6. The Need for, and Limits of, Scientific Information in Local Decision Making on Unconventional Oil and Gas Development
Phartiyal, P.; Field, P.; Kansal, T.
2014-12-01
Scientific information on, and regulatory oversight of, the U.S. oil and gas extraction have been outpaced by the scale and extent of development, particularly in states like Pennsylvania. Through recent convenings and focus groups with local officials from municipalities and counties facing such development, we asked how scientific information can be gathered and communicated to help policymakers make decisions on whether to proceed with development and, if so, what regulatory and non-regulatory approaches to consider to manage the risks from such activity. We found that the highly technical nature of unconventional oil and gas development can make conveying information difficult and public conversations harder. And, although there is scientific agreement on areas of greater risk, such as air, water, and socioeconomics effects, communities vary widely in their perceptions and concerns about these. Local leaders expressed concerns about the availability and accessibility of information: much of it is scattered, sourced from a variety of sources and viewpoints, and is viewed with confusion, skepticism, or disbelief among various stakeholders. In order to generate independent and trusted information, baseline testing, monitoring and enforcement, and data sharing are needed - but the specifics of who would do the studies, who would fund them, and how much data one would need before decisions can be made remain largely unclear. One reason for this uncertainty is the patchwork and contested nature of regulation between local, state, tribal, and federal authorities. Another is the fragmented operations disbursed across the landscape, numerous kinds and scales of operators, and the host of actors involved in land access, well development, production, and piping, lead to disjointed sources of studies, data, and communication. Another reason is that the impacts of oil and gas development activities are nested and complex, each affecting the other at varied levels, local to
7. Limited segregation of different types of sound localization information among classes of units in the inferior colliculus.
PubMed
Chase, Steven M; Young, Eric D
2005-08-17
The auditory system uses three cues to decode sound location: interaural time differences (ITDs), interaural level differences (ILDs), and spectral notches (SNs). Initial processing of these cues is done in separate brainstem nuclei, with ITDs in the medial superior olive, ILDs in the lateral superior olive, and SNs in the dorsal cochlear nucleus. This work addresses the nature of the convergence of localization information in the central nucleus of the inferior colliculus (ICC). Ramachandran et al. (1999) argued that ICC neurons of types V, I, and O, respectively, receive their predominant inputs from ITD-, ILD-, and SN-sensitive brainstem nuclei, suggesting that these ICC response types should be differentially sensitive to localization cues. Here, single-unit responses to simultaneous manipulation of pairs of localization cues were recorded, and the mutual information between discharge rate and individual cues was quantified. Although rate responses to cue variation were generally consistent with those expected from the hypothesized anatomical connections, the differences in information were not as large as expected. Type I units provide the most information, especially about SNs in the physiologically useful range. Type I and O units provide information about ILDs, even at low frequencies at which actual ILDs are very small. ITD information is provided by a subset of all low-frequency neurons. Type V neurons provide information mainly about ITDs and the average binaural intensity. These results are the first to quantify the relative representation of cues in terms of information and suggest a variety of degrees of cue integration in the ICC.
8. A torus bifurcation theorem with symmetry
NASA Technical Reports Server (NTRS)
Vangils, S. A.; Golubitsky, M.
1989-01-01
Hopf bifurcation in the presence of symmetry, in situations where the normal form equations decouple into phase/amplitude equations is described. A theorem showing that in general such degeneracies are expected to lead to secondary torus bifurcations is proved. By applying this theorem to the case of degenerate Hopf bifurcation with triangular symmetry it is proved that in codimension two there exist regions of parameter space where two branches of asymptotically stable two-tori coexist but where no stable periodic solutions are present. Although a theory was not derived for degenerate Hopf bifurcations in the presence of symmetry, examples are presented that would have to be accounted for by any such general theory.
9. H-theorem in quantum physics
PubMed Central
Lesovik, G. B.; Lebedev, A. V.; Sadovskyy, I. A.; Suslov, M. V.; Vinokur, V. M.
2016-01-01
Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy. PMID:27616571
10. Aging and nonergodicity beyond the Khinchin theorem
PubMed Central
Burov, S.; Metzler, R.; Barkai, E.
2010-01-01
The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin’s theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics. PMID:20624984
11. Fluctuation theorem for constrained equilibrium systems.
PubMed
Gilbert, Thomas; Dorfman, J Robert
2006-02-01
We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as isokinetic and Nosé-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless, finite-time averages of the phase-space contraction rate have nontrivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for nonequilibrium stationary states and appropriate to constrained equilibrium states. Moreover, we show that these fluctuations are distributed according to a Gaussian curve for long enough times. Three different systems are considered here: namely, (i) a fluid composed of particles interacting with Lennard-Jones potentials, (ii) a harmonic oscillator with Nosé-Hoover thermostatting, and (iii) a simple hyperbolic two-dimensional map.
12. Fluctuation theorem for constrained equilibrium systems
Gilbert, Thomas; Dorfman, J. Robert
2006-02-01
We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as isokinetic and Nosé-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless, finite-time averages of the phase-space contraction rate have nontrivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for nonequilibrium stationary states and appropriate to constrained equilibrium states. Moreover, we show that these fluctuations are distributed according to a Gaussian curve for long enough times. Three different systems are considered here: namely, (i) a fluid composed of particles interacting with Lennard-Jones potentials, (ii) a harmonic oscillator with Nosé-Hoover thermostatting, and (iii) a simple hyperbolic two-dimensional map.
13. H-theorem in quantum physics.
PubMed
Lesovik, G B; Lebedev, A V; Sadovskyy, I A; Suslov, M V; Vinokur, V M
2016-09-12
Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy.
14. About the Stokes decomposition theorem of waves
Lacaze, B.
2011-06-01
The Stokes decomposition theorem deals with the electrical field E→=X,Y of a light beam. The theorem asserts that a beam can be viewed as the sum of two differently polarized parts. This result was recently discussed for light in the frame of the unified theory of coherence. We study the general case of an electromagnetic wave which can be in radio, radar, communications, or light. We assume stationary components with any power spectrum and finite or infinite bandwidth. We show that an accurate definition of polarization and unpolarization is a key parameter which rules the set of solutions of the problem. When dealing with a "strong definition" of unpolarization, the problem is treated in the frame of stationary processes and linear invariant filters. When dealing with a "weak definition", solutions are given by elementary properties of bidimensional random variables.
15. Construction of momentum theorem using cross moments
Hahm, T. S.; Wang, Lu; Diamond, P. H.
2009-11-01
Charney-Drazin theorem has been extended to Hasegawa Wakatani system for zonal flow problem in magnetic fusion [P.H. Diamond, et al., Plasma Phys. Control. Fusion 50, 124018 (2008)]. For this model, the guiding center density is the potential vorticity and zonal flow is influenced by the particle flux. In this work we construct momentum theorems in terms of a hierarchy of cross moments , , and . Then we show that the particle flux, momentum flux, and heat flux influence the zonal flow for each system respectively. This work was supported by U. S. Department of Energy Contract No. DE--AC02--09CH11466 (TSH, LW), China Scholarship Council (LW), U. S. DOE SciDAC center for Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas, and the U. S. DOE SciDAC-FSP Center for Plasma Edge Simulation (TSH).
16. Approaching the bulk limit with finite cluster calculations using local increments: the case of LiH.
PubMed
Stoll, Hermann; Doll, Klaus
2012-02-21
Finite-cluster calculations employing high-level wavefunction-based ab initio methods and extended atomic-orbital basis sets are used to determine local energy increments for bulk LiH. It is shown that these increments can be converged with respect to cluster size and point-charge embedding so as to yield bulk cohesive energies with an accuracy of better than 1 mE(h), both at the Hartree-Fock and at correlated levels. Instrumental for the efficiency of the scheme is the introduction of non-orthogonal orbitals, at an intermediate stage.
17. The Ultrathin Limit and Dead-layer Effects in Local Polarization Switching of BiFeO3
SciTech Connect
Maksymovych, Petro; Huijben, Mark; Pan, Minghu; Jesse, Stephen; Balke, Nina; Chang, Hye Jung; Borisevich, Albina Y; Baddorf, Arthur P; Rijnders, Guus; Blank, Dave H. A.; Ramesh, R.; Kalinin, Sergei V
2012-01-01
Using piezoresponse force microscopy in ultra-high vacuum, polarization switching has been detected and quantified in epitaxial BiFeO3 films from 200 down to ~ 4 unit cells. Local remnant piezoresponse was used to infer the applied electric field inside the ferroelectric volume, and account for the elusive effect of dead-layers in ultrathin films. The dead-layer manifested itself in the slower than anticipated decrease of the switching bias with film thickness, yielding apparent Kay-Dunn scaling of the switching field, while the statistical analysis of hysteresis loops revealed lateral variation of the dead-layer with sub-10 nm resolution.
18. Characterizing Companions to Low-Mass Stars: A Large-Scale, Volume-Limited Survey of Local M-dwarfs
Ward-Duong, Kimberly; Patience, J.; De Rosa, R.; Rajan, A.
2013-01-01
M-dwarfs constitute the major fraction of stars within both the solar neighborhood and nearby star-forming regions. However, key M-dwarf companion characteristics - including multiplicity fraction, mass ratios, and separation distributions - are less certain for field stars, due to limited sample sizes and non-uniform selection criteria. Studies of star-forming regions often compare results to solar-type field stars due to the extensive population statistics available for G-dwarfs, but field M-dwarfs represent a more analogous population for comparison due to their prevalence. We present results on a stellar and substellar companion study covering separations from ~1 - 10,000 AU, based on a volume-limited survey of ~300 M-dwarfs within 15 pc. Our study constrains the frequency of binary companions and the shape of the companion separation and mass ratio distributions. Diffraction-limited, mid-to-near infrared archival data were obtained from the Very Large Telescope, Hubble Space Telescope, and Canada-France-Hawaii Telescope, to detect nearby companions to M-dwarfs from ~1 to 100 AU. To supplement the high-resolution data, wide-field archival plates were searched for companions with separations of 100 to 10,000 AU. The all-sky survey data include multiple epochs, and follow up observations at higher resolution will allow us to confirm or reject the new companion candidates detected during our analysis. These multi-epoch observations provide confirmation of common proper motions, thereby minimizing background contamination and providing comprehensive statistics for M-star binaries. Preliminary analysis of an initial subset of the sample suggests a lower limit to the multiplicity of 23 ± 7% within the restricted separation range. Characterizations of the binary frequency for M-dwarfs provide crucial insights into the low-mass star formation environment, and hold additional implications for the frequency and evolutionary histories of their associated disks and
19. [Objectivity of BSE symptoms using Bayes theorem].
PubMed
Hässig, M; Urech Hässig, B; Knubben-Schweizer, G
2011-12-01
In clinical epidemiology the Bayes theorem finds ever more use to render clinical acting more objective. It is shown that unusual examinations of BSE (bovine spongiform encephalopathy) as noise producing with ladle covers may quite objectively be evaluated. With the help of the likelihood ratio computed thereby, also a ranking of importance (clinical utility) of symptoms can be provided. The single most important symptom for BSE is photosensibility.
20. Volume integral theorem for exotic matter
SciTech Connect
Nandi, Kamal Kanti; Zhang Yuanzhong; Kumar, K.B. Vijaya
2004-12-15
We answer an important question in general relativity about the volume integral theorem for exotic matter by suggesting an exact integral quantifier for matter violating Averaged Null Energy Condition (ANEC). It is checked against some well-known static, spherically symmetric traversable wormhole solutions of general relativity with a sign reversed kinetic term minimally coupled scalar field. The improved quantifier is consistent with the principle that traversable wormholes can be supported by arbitrarily small quantities of exotic matter.
1. Spontaneously broken spacetime symmetries and Goldstone's theorem.
PubMed
Low, Ian; Manohar, Aneesh V
2002-03-11
Goldstone's theorem states that there is a massless mode for each broken symmetry generator. It has been known for a long time that the naive generalization of this counting fails to give the correct number of massless modes for spontaneously broken spacetime symmetries. We explain how to get the right count of massless modes in the general case, and discuss examples involving spontaneously broken Poincaré and conformal invariance.
2. Infinite flag varieties and conjugacy theorems
PubMed Central
Peterson, Dale H.; Kac, Victor G.
1983-01-01
We study the orbit of a highest-weight vector in an integrable highest-weight module of the group G associated to a Kac-Moody algebra [unk](A). We obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of [unk](A). In particular, we prove conjugacy theorems for Cartan and Borel subalgebras of [unk](A), so that the Cartan matrix A is an invariant of [unk](A). PMID:16593298
3. Haag's theorem in noncommutative quantum field theory
SciTech Connect
Antipin, K. V.; Mnatsakanova, M. N.; Vernov, Yu. S.
2013-08-15
Haag's theorem was extended to the general case of noncommutative quantum field theory when time does not commute with spatial variables. It was proven that if S matrix is equal to unity in one of two theories related by unitary transformation, then the corresponding one in the other theory is equal to unity as well. In fact, this result is valid in any SO(1, 1)-invariant quantum field theory, an important example of which is noncommutative quantum field theory.
4. Asynchronous networks: modularization of dynamics theorem
Bick, Christian; Field, Michael
2017-02-01
Building on the first part of this paper, we develop the theory of functional asynchronous networks. We show that a large class of functional asynchronous networks can be (uniquely) represented as feedforward networks connecting events or dynamical modules. For these networks we can give a complete description of the network function in terms of the function of the events comprising the network: the modularization of dynamics theorem. We give examples to illustrate the main results.
5. A stability theorem for energy-balance climate models
NASA Technical Reports Server (NTRS)
Cahalan, R. F.; North, G. R.
1979-01-01
The paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feedback, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a 'slope stability' theorem, i.e., if the local slope of the steady-state iceline latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed.
6. From Loops to Trees By-passing Feynman's Theorem
SciTech Connect
Catani, Stefano; Gleisberg, Tanju; Krauss, Frank; Rodrigo, German; Winter, Jan-Christopher
2008-04-22
We derive a duality relation between one-loop integrals and phase-space integrals emerging from them through single cuts. The duality relation is realized by a modification of the customary + i0 prescription of the Feynman propagators. The new prescription regularizing the propagators, which we write in a Lorentz covariant form, compensates for the absence of multiple cut contributions that appear in the Feynman Tree Theorem. The duality relation can be applied to generic one-loop quantities in any relativistic, local and unitary field theories. It is suitable for applications to the analytical calculation of one-loop scattering amplitudes, and to the numerical evaluation of cross-sections at next-to-leading order.
7. Boundary trace embedding theorems for variable exponent Sobolev spaces
Fan, Xianling
2008-03-01
We study boundary trace embedding theorems for variable exponent Sobolev space W1,p([dot operator])([Omega]). Let [Omega] be an open (bounded or unbounded)[thin space]domain in satisfying strong local Lipschitz condition. Under the hypotheses that p[set membership, variant]L[infinity]([Omega]), 1[less-than-or-equals, slant]infp(x)[less-than-or-equals, slant]supp(x)N, we prove that there is a continuous boundary trace embedding W1,p([dot operator])([Omega])-->Lq([dot operator])([not partial differential][Omega]) provided q([dot operator]), a measurable function on [not partial differential][Omega], satisfies condition for x[set membership, variant][not partial differential][Omega].
8. Sanitary landfill local-scale flow and transport modeling in support of alternative concentrations limit demonstrations, Savannah River Site
SciTech Connect
Kelly, V.A.; Beach, J.A.; Statham, W.H.; Pickens, J.F.
1993-02-19
The Savannah River Site (SRS) is a Department of Energy (DOE) facility located near Aiken, South Carolina which is currently operated and managed by Westinghouse Savannah River Company (WSRC). The Sanitary Landfill (Sanitary Landfill) at the SRS is located approximately 2,000 feet Northwest of Upper Three Runs Creek (UTRC) on an approximately 70 acre site located south of Road C between the SRS B-Area and UTRC. The Sanitary Landfill has been receiving wastes since 1974 and operates as an unlined trench and fill operation. The original landfill site was 32 acres. This area reached its capacity around 1987 and a Northern Expansion of 16 acres and a Southern Expansion of 22 acres were added in 1987. The Northern Expansion has not been used for waste disposal to date and the Southern Expansion is expected to reach capacity in 1992 or 1993. The waste received at the Sanitary Landfill is predominantly paper, plastics, rubber, wood, metal, cardboard, rags saturated with degreasing solvents, pesticide bags, empty cans, and asbestos in bags. The landfill is not supposed to receive any radioactive wastes. However, tritium has been detected in the groundwater at the site. Gross alpha and gross beta are also evaluated at the landfill. The objectives of this modeling study are twofold: (1) to create a local scale Sanitary Landfill flow model to study hydraulic effects resulting from capping the Sanitary Landfill; and (2) to create a Sanitary Landfill local scale transport model to support ACL Demonstrations for a RCRA Part B Permit Renewal.
9. Delayed plastic relaxation limit in SiGe islands grown by Ge diffusion from a local source
SciTech Connect
Vanacore, G. M.; Zani, M.; Tagliaferri, A.; Nicotra, G.; Bollani, M.; Bonera, E.; Montalenti, F.; Picco, A.; Boioli, F.; Capellini, G.; Isella, G.; Osmond, J.
2015-03-14
The hetero-epitaxial strain relaxation in nano-scale systems plays a fundamental role in shaping their properties. Here, the elastic and plastic relaxation of self-assembled SiGe islands grown by surface-thermal-diffusion from a local Ge solid source on Si(100) are studied by atomic force and transmission electron microscopies, enabling the simultaneous investigation of the strain relaxation in different dynamical regimes. Islands grown by this technique remain dislocation-free and preserve a structural coherence with the substrate for a base width as large as 350 nm. The results indicate that a delay of the plastic relaxation is promoted by an enhanced Si-Ge intermixing, induced by the surface-thermal-diffusion, which takes place already in the SiGe overlayer before the formation of a critical nucleus. The local entropy of mixing dominates, leading the system toward a thermodynamic equilibrium, where non-dislocated, shallow islands with a low residual stress are energetically stable. These findings elucidate the role of the interface dynamics in modulating the lattice distortion at the nano-scale, and highlight the potential use of our growth strategy to create composition and strain-controlled nano-structures for new-generation devices.
10. Theorem Proving In Higher Order Logics
NASA Technical Reports Server (NTRS)
Carreno, Victor A. (Editor); Munoz, Cesar A.; Tahar, Sofiene
2002-01-01
The TPHOLs International Conference serves as a venue for the presentation of work in theorem proving in higher-order logics and related areas in deduction, formal specification, software and hardware verification, and other applications. Fourteen papers were submitted to Track B (Work in Progress), which are included in this volume. Authors of Track B papers gave short introductory talks that were followed by an open poster session. The FCM 2002 Workshop aimed to bring together researchers working on the formalisation of continuous mathematics in theorem proving systems with those needing such libraries for their applications. Many of the major higher order theorem proving systems now have a formalisation of the real numbers and various levels of real analysis support. This work is of interest in a number of application areas, such as formal methods development for hardware and software application and computer supported mathematics. The FCM 2002 consisted of three papers, presented by their authors at the workshop venue, and one invited talk.
11. Haag's Theorem and Parameterized Quantum Field Theory
Seidewitz, Edwin
2017-01-01
Haag's theorem is very inconvenient; it means that the interaction picture exists only if there is no interaction''. In traditional quantum field theory (QFT), Haag's theorem states that any field unitarily equivalent to a free field must itself be a free field. But the derivation of the Dyson series perturbation expansion relies on the use of the interaction picture, in which the interacting field is unitarily equivalent to the free field, but which must still account for interactions. So, the usual derivation of the scattering matrix in QFT is mathematically ill defined. Nevertheless, perturbative QFT is currently the only practical approach for addressing realistic scattering, and it has been very successful in making empirical predictions. This success can be understood through an alternative derivation of the Dyson series in a covariant formulation of QFT using an invariant, fifth path parameter in addition to the usual four position parameters. The parameterization provides an additional degree of freedom that allows Haag's Theorem to be avoided, permitting the consistent use of a form of interaction picture in deriving the Dyson expansion. The extra symmetry so introduced is then broken by the choice of an interacting vacuum.
12. 3-D Ultrasound Localization Microscopy for Identifying Microvascular Morphology Features of Tumor Angiogenesis at a Resolution Beyond the Diffraction Limit of Conventional Ultrasound
PubMed Central
Lin, Fanglue; Shelton, Sarah E.; Espíndola, David; Rojas, Juan D.; Pinton, Gianmarco; Dayton, Paul A.
2017-01-01
Angiogenesis has been known as a hallmark of solid tumor cancers for decades, yet ultrasound has been limited in its ability to detect the microvascular changes associated with malignancy. Here, we demonstrate the potential of 'ultrasound localization microscopy' applied volumetrically in combination with quantitative analysis of microvascular morphology, as an approach to overcome this limitation. This pilot study demonstrates our ability to image complex microvascular patterns associated with tumor angiogenesis in-vivo at a resolution of tens of microns - substantially better than the diffraction limit of traditional clinical ultrasound, yet using an 8 MHz clinical ultrasound probe. Furthermore, it is observed that data from healthy and tumor-bearing tissue exhibit significant differences in microvascular pattern and density. Results suggests that with continued development of these novel technologies, ultrasound has the potential to detect biomarkers of cancer based on the microvascular 'fingerprint' of malignant angiogenesis rather than through imaging of blood flow dynamics or the tumor mass itself. PMID:28042327
13. Localized epigenetic silencing of a damage-activated WNT enhancer limits regeneration in mature Drosophila imaginal discs.
PubMed
Harris, Robin E; Setiawan, Linda; Saul, Josh; Hariharan, Iswar K
2016-02-03
Many organisms lose the capacity to regenerate damaged tissues as they mature. Damaged Drosophila imaginal discs regenerate efficiently early in the third larval instar (L3) but progressively lose this ability. This correlates with reduced damage-responsive expression of multiple genes, including the WNT genes wingless (wg) and Wnt6. We demonstrate that damage-responsive expression of both genes requires a bipartite enhancer whose activity declines during L3. Within this enhancer, a damage-responsive module stays active throughout L3, while an adjacent silencing element nucleates increasing levels of epigenetic silencing restricted to this enhancer. Cas9-mediated deletion of the silencing element alleviates WNT repression, but is, in itself, insufficient to promote regeneration. However, directing Myc expression to the blastema overcomes repression of multiple genes, including wg, and restores cellular responses necessary for regeneration. Localized epigenetic silencing of damage-responsive enhancers can therefore restrict regenerative capacity in maturing organisms without compromising gene functions regulated by developmental signals.
14. IgE Antibodies, FcεRIαand IgE-mediated Local Anaphylaxis Can Limit Snake Venom Toxicity
PubMed Central
Gaudenzio, Nicolas; Reber, Laurent Lionel; Sibilano, Riccardo; Tsai, Mindy; Galli, Stephen Joseph
2015-01-01
Background Type 2 cytokine-related (i.e., type 2) immune responses associated with development of antigen-specific Immunoglobulin E antibodies (IgE) can contribute to pathology in allergic diseases and to fatal anaphylaxis. However, recent findings in mice indicate that IgE also can enhance defense against honeybee venom. Objective We tested whether IgE antibodies, IgE-dependent effector mechanisms, and a local anaphylactic reaction to an unrelated antigen can enhance defense against Russell's viper venom (RVV) and determined whether such responses can be influenced by immunization protocol or mouse strain. Methods We compared the resistance of RVV-immunized wild-type, IgE-deficient, and Fcer1a-deficient mice following injection of a potentially lethal dose of RVV. Results A single prior exposure to RVV enhanced the ability of wild-type mice, but not mice lacking IgE or functional FcεRI, to survive challenge with a potentially lethal amount of RVV. Moreover, IgE-dependent local passive cutaneous anaphylaxis in response to challenge with an antigen not naturally present in RVV significantly enhanced resistance to the venom. Finally, we observed different effects on resistance to RVV or honeybee venoms in BALB/c versus C57BL/6 mice which had received a second exposure to that venom prior to challenge with a high dose of that venom. Conclusion These observations illustrate the potential benefit of IgE-dependent effector mechanisms in acquired host defense against venoms. The extent to which type 2 immune responses against venoms can decrease pathology associated with envenomation seems to be influenced by the type of venom, the frequency of venom exposure, and the genetic background of the host. PMID:26410782
15. Sea lamprey carcasses exert local and variable food web effects in a nutrient-limited Atlantic coastal stream
USGS Publications Warehouse
Weaver, Daniel M.; Coghlan Jr., Stephen M.; Zydlewski, Joseph
2016-01-01
Resource flows from adjacent ecosystems are critical in maintaining structure and function of freshwater food webs. Migrating sea lamprey (Petromyzon marinus) deliver a pulsed marine-derived nutrient subsidy to rivers in spring when the metabolic demand of producers and consumers are increasing. However, the spatial and temporal dynamics of these nutrient subsidies are not well characterized. We used sea lamprey carcass additions in a small stream to examine changes in nutrients, primary productivity, and nutrient assimilation among consumers. Algal biomass increased 57%–71% immediately adjacent to carcasses; however, broader spatial changes from multiple-site carcass addition may have been influenced by canopy cover. We detected assimilation of nutrients (via δ13C and δ15N) among several macroinvertebrate families including Heptageniidae, Hydropsychidae, and Perlidae. Our research suggests that subsidies may evoke localized patch-scale effects on food webs, and the pathways of assimilation in streams are likely coupled to adjacent terrestrial systems. This research underscores the importance of connectivity in streams, which may influence sea lamprey spawning and elicit varying food web responses from carcass subsidies due to fine-scale habitat variables.
16. Localized epigenetic silencing of a damage-activated WNT enhancer limits regeneration in mature Drosophila imaginal discs
PubMed Central
Harris, Robin E; Setiawan, Linda; Saul, Josh; Hariharan, Iswar K
2016-01-01
Many organisms lose the capacity to regenerate damaged tissues as they mature. Damaged Drosophila imaginal discs regenerate efficiently early in the third larval instar (L3) but progressively lose this ability. This correlates with reduced damage-responsive expression of multiple genes, including the WNT genes wingless (wg) and Wnt6. We demonstrate that damage-responsive expression of both genes requires a bipartite enhancer whose activity declines during L3. Within this enhancer, a damage-responsive module stays active throughout L3, while an adjacent silencing element nucleates increasing levels of epigenetic silencing restricted to this enhancer. Cas9-mediated deletion of the silencing element alleviates WNT repression, but is, in itself, insufficient to promote regeneration. However, directing Myc expression to the blastema overcomes repression of multiple genes, including wg, and restores cellular responses necessary for regeneration. Localized epigenetic silencing of damage-responsive enhancers can therefore restrict regenerative capacity in maturing organisms without compromising gene functions regulated by developmental signals. DOI: http://dx.doi.org/10.7554/eLife.11588.001 PMID:26840050
17. Uniqueness theorem for black hole space-times with multiple disconnected horizons
Armas, Jay; Harmark, Troels
2010-05-01
We show uniqueness of stationary and asymptotically flat black hole spacetimes with multiple disconnected horizons and with two rotational Killing vector fields in the context of five-dimensional minimal supergravity (Einstein-Maxwell-Chern-Simons gravity). The novelty in this work is the introduction in the uniqueness theorem of intrinsic local charges measured near each horizon as well as the measurement of local fluxes besides the asymptotic charges that characterize a particular solution. A systematic method of defining the boundary conditions on the fields that specify a black hole space-time is given based on the study of its rod structure (domain structure). Also, an analysis of known solutions with disconnected horizons is carried out as an example of an application of this theorem. ”But the perfect scientist is also a gardener: he believes that beauty is knowledge.” Gonçalo M. Tavares in Brief Notes on Science
18. Limiting the risk of cardiac toxicity with esophageal-sparing intensity modulated radiotherapy for locally advanced lung cancers
PubMed Central
Panettieri, Vanessa; Ruben, Jeremy D.; Senthi, Sashendra
2016-01-01
Background Intensity modulated radiotherapy (IMRT) is routinely utilized in the treatment of locally advanced non-small cell lung cancer (NSCLC). RTOG 0617 found that overall survival was impacted by increased low (5 Gy) and intermediate (30 Gy) cardiac doses. We evaluated the impact of esophageal-sparing IMRT on cardiac doses with and without the heart considered in the planning process and predicted toxicity compared to 3D-conventional radiotherapy (3DCRT). Methods Ten consecutive patients with N2 Stage III NSCLC treated to 60 Gy in 30 fractions, between February 2012 and September 2014, were evaluated. For each patient, 3DCRT and esophageal-sparing IMRT plans were generated. IMRT plans were then created with and without the heart considered in the optimization process. To compare plans, the dose delivered to 95% and 99% of the target (D95% and D99%), and doses to the esophagus, lung and heart were compared by determining the volume receiving X dose (VXGy) and the normal tissue complication probability (NTCP) calculated. Results IMRT reduced maximum esophagus dose to below 60 Gy in all patients and produced significant reductions to V50Gy, V40Gy and esophageal NTCP. The cost of this reduction was a non-statistically, non-clinically significant increase in low dose (5 Gy) lung exposure that did not worsen lung NTCP. IMRT plans produced significant cardiac sparing, with the amount of improvement correlating to the amount of heart overlapping with the target. When included in plan optimization, for selected patients further sparing of the heart and improvement in heart NTCP was possible. Conclusions Esophageal-sparing IMRT can significantly spare the heart even if it is not considered in the optimization process. Further sparing can be achieved if plan optimization constrains low and intermediate heart doses, without compromising lung doses. PMID:27162670
19. Keeping It Local: Dispersal Limitations of Coral Larvae to the High Latitude Coral Reefs of the Houtman Abrolhos Islands.
PubMed
Markey, Kathryn L; Abdo, Dave A; Evans, Scott N; Bosserelle, Cyprien
2016-01-01
In 2011 the first recorded bleaching event for the high latitude Houtman Abrolhos Islands (HAI) coral communities was documented. This bleaching event highlighted the question of whether a supply of 'heat tolerant' coral recruits from the tropical north would be sufficient to provide a level of resistance for these reefs to future warming events. Using Lagrangian modelling we showed that due to its regional isolation, large-scale larval input from potential tropical northern source populations to the HAI is unlikely, despite the southward flowing Leeuwin current. Successful recruitment to artificial substrates was recorded following the bleaching event. However, this was negligible (0.4 ± 0.1 recruits per tile) compared to 2013 post impact recruitment (128.8 ± 15.8 recruits per tile). Our data therefore provides preliminary evidence suggesting that the connectivity of the HAI with coral communities in the north is limited, and population maintenance and recovery is likely driven primarily by self-recruitment. Given the low thermal tolerance of the HAI coral communities, the dominance of Acropora, and the apparent reliance on self-recruitment, an increased frequency of thermally anomalous conditions at the HAI (such as experienced in 2011) has the potential to reduce the long-term stability of the HAI coral populations and species that depend upon them.
20. Keeping It Local: Dispersal Limitations of Coral Larvae to the High Latitude Coral Reefs of the Houtman Abrolhos Islands
PubMed Central
Markey, Kathryn L.; Abdo, Dave A.; Evans, Scott N.; Bosserelle, Cyprien
2016-01-01
In 2011 the first recorded bleaching event for the high latitude Houtman Abrolhos Islands (HAI) coral communities was documented. This bleaching event highlighted the question of whether a supply of ‘heat tolerant’ coral recruits from the tropical north would be sufficient to provide a level of resistance for these reefs to future warming events. Using Lagrangian modelling we showed that due to its regional isolation, large-scale larval input from potential tropical northern source populations to the HAI is unlikely, despite the southward flowing Leeuwin current. Successful recruitment to artificial substrates was recorded following the bleaching event. However, this was negligible (0.4 ± 0.1 recruits per tile) compared to 2013 post impact recruitment (128.8 ± 15.8 recruits per tile). Our data therefore provides preliminary evidence suggesting that the connectivity of the HAI with coral communities in the north is limited, and population maintenance and recovery is likely driven primarily by self-recruitment. Given the low thermal tolerance of the HAI coral communities, the dominance of Acropora, and the apparent reliance on self-recruitment, an increased frequency of thermally anomalous conditions at the HAI (such as experienced in 2011) has the potential to reduce the long-term stability of the HAI coral populations and species that depend upon them. PMID:26812259
1. Limited proteolysis of human leukocyte interferon-. cap alpha. 2 and localization of the monoclonal antibody-binding antigenic determinant
SciTech Connect
Kostrov, S.V.; Chernovskaya, T.V.; Khodova, O.M.; Borukhov, S.I.; Ryzhavskaya, A.S.; Izotova, L.S.; Strongin, A.Ya.
1986-05-20
Large peptide fragments of human leukocyte interferon-..cap alpha..2 (INF-..cap alpha..2) were produced by limited proteolysis with trypsin, pepsin, thermolysin, and Bacillus amyloliquefaciens serine proteinase, and the ability of the fragments to react with murine monoclonal antibodies NK2, directed toward INF-..cap alpha..2, was studied by the immunoblotting technique. The region of the sequence 110-149 is the most sensitive to proteinase attack and evidently is exposed on the surface of the INF-..cap alpha..2 molecule. The INF-..cap alpha..2 fragments 1-139, 1-147, and 1-149 react with antibodies, whereas the fragments 1-109 and 1-112 do not bind NK2 antibodies. A comparison of the primary structure of the families of human leukocyte and murine leukocyte INF in the region of the sequence 110-139 and an analysis of the ability of human INF differing in amino acid sequence to interact with NK2 antibodies suggested that the antigenic determinant that binds monoclonal antibodies NK2 is the sequence Glu/sub 114/-Asp/sub 115/-Ser/sub 116/-He/sub 117/ of the INF-..cap alpha..2 molecule.
2. Near distance approximation in astrodynamical applications of Lambert's theorem
Rauh, Alexander; Parisi, Jürgen
2014-01-01
The smallness parameter of the approximation method is defined in terms of the non-dimensional initial distance between target and chaser satellite. In the case of a circular target orbit, compact analytical expressions are obtained for the interception travel time up to third order. For eccentric target orbits, an explicit result is worked out to first order, and the tools are prepared for numerical evaluation of higher order contributions. The possible transfer orbits are examined within Lambert's theorem. For an eventual rendezvous it is assumed that the directions of the angular momenta of the two orbits enclose an acute angle. This assumption, together with the property that the travel time should vanish with vanishing initial distance, leads to a condition on the admissible initial positions of the chaser satellite. The condition is worked out explicitly in the general case of an eccentric target orbit and a non-coplanar transfer orbit. The condition is local. However, since during a rendezvous maneuver, the chaser eventually passes through the local space, the condition propagates to non-local initial distances. As to quantitative accuracy, the third order approximation reproduces the elements of Mars, in the historical problem treated by Gauss, to seven decimals accuracy, and in the case of the International Space Station, the method predicts an encounter error of about 12 m for an initial distance of 70 km.
3. On the Chern-Gauss-Bonnet theorem for the noncommutative 4-sphere
Arnlind, Joakim; Wilson, Mitsuru
2017-01-01
We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Chern-Gauss-Bonnet type theorem for the noncommutative 4-sphere.
4. Index Theorem for Topological Excitations on R^3 \\times S^1 and Chern-Simons Theory
SciTech Connect
Poppitz, Erich; Unsal, Mithat
2008-12-12
We derive an index theorem for the Dirac operator in the background of various topological excitations on an R{sup 3} x S{sup 1} geometry. The index theorem provides more refined data than the APS index for an instanton on R{sup 4} and reproduces it in decompactification limit. In the R{sup 3} limit, it reduces to the Callias index theorem. The index is expressed in terms of topological charge and the {eta}-invariant associated with the boundary Dirac operator. Neither topological charge nor {eta}-invariant is typically an integer, however, the non-integer parts cancel to give an integer-valued index. Our derivation is based on axial current non-conservation--an exact operator identity valid on any four-manifold--and on the existence of a center symmetric, or approximately center symmetric, boundary holonomy (Wilson line). We expect the index theorem to usefully apply to many physical systems of interest, such as low temperature (large S{sup 1}, confined) phases of gauge theories, center stabilized Yang-Mills theories with vector-like or chiral matter (at S{sup 1} of any size), and supersymmetric gauge theories with supersymmetry-preserving boundary conditions (also at any S{sup 1}). In QCD-like and chiral gauge theories, the index theorem should shed light into the nature of topological excitations responsible for chiral symmetry breaking and the generation of mass gap in the gauge sector. We also show that imposing chirally-twisted boundary condition in gauge theories with fermions induces a Chern-Simons term in the infrared. This suggests that some QCD-like gauge theories should possess components with a topological Chern-Simons phase in the small S{sup 1} regime.
5. Index theorem for topological excitations on R3 × S1 and Chern-Simons theory
Poppitz, Erich; Ünsal, Mithat
2009-03-01
We derive an index theorem for the Dirac operator in the background of various topological excitations on an R3 × S1 geometry. The index theorem provides more refined data than the APS index for an instanton on R4 and reproduces it in decompactification limit. In the R3 limit, it reduces to the Callias index theorem. The index is expressed in terms of topological charge and the η-invariant associated with the boundary Dirac operator. Neither topological charge nor η-invariant is typically an integer, however, the non-integer parts cancel to give an integer-valued index. Our derivation is based on axial current non-conservation — an exact operator identity valid on any four-manifold — and on the existence of a center symmetric, or approximately center symmetric, boundary holonomy (Wilson line). We expect the index theorem to usefully apply to many physical systems of interest, such as low temperature (large S1, confined) phases of gauge theories, center stabilized Yang-Mills theories with vector-like or chiral matter (at S1 of any size), and supersymmetric gauge theories with supersymmetry-preserving boundary conditions (also at any S1). In QCD-like and chiral gauge theories, the index theorem should shed light into the nature of topological excitations responsible for chiral symmetry breaking and the generation of mass gap in the gauge sector. We also show that imposing chirally-twisted boundary condition in gauge theories with fermions induces a Chern-Simons term in the infrared. This suggests that some QCD-like gauge theories should possess components with a topological Chern-Simons phase in the small S1 regime.
6. The van Cittert-Zernike theorem for electromagnetic fields.
PubMed
Ostrovsky, Andrey S; Martínez-Niconoff, Gabriel; Martínez-Vara, Patricia; Olvera-Santamaría, Miguel A
2009-02-02
The van Cittert-Zernike theorem, well known for the scalar optical fields, is generalized for the case of vector electromagnetic fields. The deduced theorem shows that the degree of coherence of the electromagnetic field produced by the completely incoherent vector source increases on propagation whereas the degree of polarization remains unchanged. The possible application of the deduced theorem is illustrated by an example of optical simulation of partially coherent and partially polarized secondary source with the controlled statistical properties.
7. Borsuk-Ulam theorem in infinite-dimensional Banach spaces
Gel'man, B. D.
2002-02-01
The well-known classical Borsuk-Ulam theorem has a broad range of applications to various problems. Its generalization to infinite-dimensional spaces runs across substantial difficulties because its statement is essentially finite-dimensional. A result established in the paper is a natural generalization of the Borsuk-Ulam theorem to infinite-dimensional Banach spaces. Applications of this theorem to various problems are discussed.
8. A Converse of the Mean Value Theorem Made Easy
ERIC Educational Resources Information Center
Mortici, Cristinel
2011-01-01
The aim of this article is to discuss some results about the converse mean value theorem stated by Tong and Braza [J. Tong and P. Braza, "A converse of the mean value theorem", Amer. Math. Monthly 104(10), (1997), pp. 939-942] and Almeida [R. Almeida, "An elementary proof of a converse mean-value theorem", Internat. J. Math. Ed. Sci. Tech. 39(8)…
9. A qualitative approach to Bayes' theorem.
PubMed
Medow, Mitchell A; Lucey, Catherine R
2011-12-01
While decisions made according to Bayes' theorem are the academic normative standard, the theorem is rarely used explicitly in clinical practice. Yet the principles can be followed without intimidating mathematics. To do so, one can first categorise the prior-probability of the disease being tested for as very unlikely (less likely than 10%), unlikely (10-33%), uncertain (34-66%), likely (67-90%) or very likely (more likely than 90%). Usually, for disorders that are very unlikely or very likely, no further testing is needed. If the prior probability is unlikely, uncertain or likely, a test and a Bayesian-inspired update process incorporating the result can help. A positive result of a good test increases the probability of the disorder by one likelihood category (eg, from uncertain to likely) and a negative test decreases the probability by one category. If testing is needed to escape the extremes of likelihood (eg, a very unlikely but particularly dangerous condition or in the circumstance of population screening, or a very likely condition with a particularly noxious treatment), two tests may be needed to achieve. Negative results of tests with sensitivity ≥99% are sufficient to rule-out a diagnosis; positive results of tests with specificity ≥99% are sufficient to rule-in a diagnosis. This method overcomes some common heuristic errors: ignoring the base rate, probability adjustment errors and order effects. The simplicity of the method, while still adhering to the basic principles of Bayes' theorem, has the potential to increase its application in clinical practice.
10. Generating Test Templates via Automated Theorem Proving
NASA Technical Reports Server (NTRS)
1997-01-01
Testing can be used during the software development process to maintain fidelity between evolving specifications, program designs, and code implementations. We use a form of specification-based testing that employs the use of an automated theorem prover to generate test templates. A similar approach was developed using a model checker on state-intensive systems. This method applies to systems with functional rather than state-based behaviors. This approach allows for the use of incomplete specifications to aid in generation of tests for potential failure cases. We illustrate the technique on the cannonical triangle testing problem and discuss its use on analysis of a spacecraft scheduling system.
11. Penrose's singularity theorem in a Finsler spacetime
Babak Aazami, Amir; Javaloyes, Miguel Angel
2016-01-01
We translate Penrose's singularity theorem to a Finsler spacetime. To that end, causal concepts in Lorentzian geometry are extended, including definitions and properties of focal points and trapped surfaces, with careful attention paid to the differences that arise in the Finslerian setting. This activity is supported by the programme 'Young leaders in research' 18942/JLI/13 by Fundación Séneca, Regional Agency for Science and Technology from the Region of Murcia, and by the World Premier International Research Center Initiative (WPI), MEXT, Japan.
12. Generalizations of Brandl's theorem on Engel length
Quek, S. G.; Wong, K. B.; Wong, P. C.
2013-04-01
Let n < m be positive integers such that [g,nh] = [g,mh] and assume that n and m are chosen minimal with respect to this property. Let gi = [g,n+ih] where i = 1,2,…,m-n. Then π(g,h) = (g1,…,gm-n) is called the Engel cycle generated by g and h. The length of the Engel cycle is m-n. A group G is said to have Engel length r, if all the length of the Engel cycles in G divides r. In this paper we discuss the Brandl's theorem on Engel length and give some of its generalizations.
13. No-cloning theorem on quantum logics
SciTech Connect
2009-10-15
This paper discusses the no-cloning theorem in a logicoalgebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning operation if and only if it is a Boolean algebra. That is, only classical theory admits the cloning of states. If unsharp propositions are to be included in the theory, then a notion of effect algebra is considered. We proved that an atomic Archimedean effect algebra admitting cloning operation is a Boolean algebra. This paper also presents a partial result, indicating a relation between the cloning on effect algebras and hidden variables.
14. On the inversion of Fueter's theorem
Dong, Baohua; Kou, Kit Ian; Qian, Tao; Sabadini, Irene
2016-10-01
The well known Fueter theorem allows to construct quaternionic regular functions or monogenic functions with values in a Clifford algebra defined on open sets of Euclidean space R n + 1, starting from a holomorphic function in one complex variable or, more in general, from a slice hyperholomorphic function. Recently, the inversion of this theorem has been obtained for odd values of the dimension n. The present work extends the result to all dimensions n by using the Fourier multiplier method. More precisely, we show that for any axially monogenic function f defined in a suitable open set in R n + 1, where n is a positive integer, we can find a slice hyperholomorphic function f → such that f =Δ (n - 1) / 2 f →. Both the even and the odd dimensions are treated with the same, viz., the Fourier multiplier, method. For the odd dimensional cases the result obtained by the Fourier multiplier method coincides with the existing result obtained through the pointwise differential method.
15. Random numbers certified by Bell's theorem.
PubMed
Pironio, S; Acín, A; Massar, S; de la Giroday, A Boyer; Matsukevich, D N; Maunz, P; Olmschenk, S; Hayes, D; Luo, L; Manning, T A; Monroe, C
2010-04-15
Randomness is a fundamental feature of nature and a valuable resource for applications ranging from cryptography and gambling to numerical simulation of physical and biological systems. Random numbers, however, are difficult to characterize mathematically, and their generation must rely on an unpredictable physical process. Inaccuracies in the theoretical modelling of such processes or failures of the devices, possibly due to adversarial attacks, limit the reliability of random number generators in ways that are difficult to control and detect. Here, inspired by earlier work on non-locality-based and device-independent quantum information processing, we show that the non-local correlations of entangled quantum particles can be used to certify the presence of genuine randomness. It is thereby possible to design a cryptographically secure random number generator that does not require any assumption about the internal working of the device. Such a strong form of randomness generation is impossible classically and possible in quantum systems only if certified by a Bell inequality violation. We carry out a proof-of-concept demonstration of this proposal in a system of two entangled atoms separated by approximately one metre. The observed Bell inequality violation, featuring near perfect detection efficiency, guarantees that 42 new random numbers are generated with 99 per cent confidence. Our results lay the groundwork for future device-independent quantum information experiments and for addressing fundamental issues raised by the intrinsic randomness of quantum theory.
16. Group Theoretical Interpretation of von Neumann's Theorem on Composite Systems.
ERIC Educational Resources Information Center
Bergia, S.; And Others
1979-01-01
Shows that von Neumann's mathematical theorem on composite systems acquires a transparent physical meaning with reference to a suitable physical example; a composite system in a state of definite angular momentum. Gives an outline of the theorem, and the results are restated in Dirac's notation, thus generalizing von Neumann's results which were…
17. Generalizations of Karp's theorem to elastic scattering theory
Tuong, Ha-Duong
Karp's theorem states that if the far field pattern corresponding to the scattering of a time-harmonic acoustic plane wave by a sound-soft obstacle in R2 is invariant under the group of rotations, then the scatterer is a circle. The theorem is generalized to the elastic scattering problems and the axisymmetric scatterers in R3.
18. When 95% Accurate Isn't: Exploring Bayes's Theorem
ERIC Educational Resources Information Center
2011-01-01
Bayes's theorem is notorious for being a difficult topic to learn and to teach. Problems involving Bayes's theorem (either implicitly or explicitly) generally involve calculations based on two or more given probabilities and their complements. Further, a correct solution depends on students' ability to interpret the problem correctly. Most people…
19. Unique Factorization and the Fundamental Theorem of Arithmetic
ERIC Educational Resources Information Center
Sprows, David
2017-01-01
The fundamental theorem of arithmetic is one of those topics in mathematics that somehow "falls through the cracks" in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like "every natural number can be broken down into a…
20. On Euler's Theorem for Homogeneous Functions and Proofs Thereof.
ERIC Educational Resources Information Center
Tykodi, R. J.
1982-01-01
Euler's theorem for homogenous functions is useful when developing thermodynamic distinction between extensive and intensive variables of state and when deriving the Gibbs-Duhem relation. Discusses Euler's theorem and thermodynamic applications. Includes six-step instructional strategy for introducing the material to students. (Author/JN)
1. Solving boundary-value electrostatics problems using Green's reciprocity theorem
Hu, Ben Yu-Kuang
2001-12-01
Formal solutions to electrostatics boundary-value problems are derived using Green's reciprocity theorem. This method provides a more transparent interpretation of the solutions than the standard Green's function derivation. An energy-based argument for the reciprocity theorem is also presented.
2. Estimating Filtering Errors Using the Peano Kernel Theorem
SciTech Connect
Jerome Blair
2009-02-20
The Peano Kernel Theorem is introduced and a frequency domain derivation is given. It is demonstrated that the application of this theorem yields simple and accurate formulas for estimating the error introduced into a signal by filtering it to reduce noise.
3. Leaning on Socrates to Derive the Pythagorean Theorem
ERIC Educational Resources Information Center
Percy, Andrew; Carr, Alistair
2010-01-01
The one theorem just about every student remembers from school is the theorem about the side lengths of a right angled triangle which Euclid attributed to Pythagoras when writing Proposition 47 of "The Elements". Usually first met in middle school, the student will be continually exposed throughout their mathematical education to the…
4. On the Weighted Mean Value Theorem for Integrals
ERIC Educational Resources Information Center
Polezzi, M.
2006-01-01
The Mean Value Theorem for Integrals is a powerful tool, which can be used to prove the Fundamental Theorem of Calculus, and to obtain the average value of a function on an interval. On the other hand, its weighted version is very useful for evaluating inequalities for definite integrals. This article shows the solutions on applying the weighted…
5. Interactive Theorem Finding through Continuous Variation of Geometric Configurations.
ERIC Educational Resources Information Center
Schumann, Heinz
1991-01-01
Described and evaluated are microcomputers as a tool for construction in geometry education and heuristic theorem finding through interactive continuous variation of geometric configurations. Numerous examples of theorem finding processes are provided using the prototype graphics system CABRI-Geometer. (MDH)
6. Level reduction and the quantum threshold theorem
Aliferis, Panagiotis (Panos)
Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow.Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology.The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise.One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the
7. The virial theorem for the polarizable continuum model
SciTech Connect
Cammi, R.
2014-02-28
The electronic virial theorem is extended to molecular systems within the framework of the Polarizable Continuum Model (PCM) to describe solvation effects. The theorem is given in the form of a relation involving the components of the energy (kinetic and potential) of a molecular solute and its electrostatic properties (potential and field) at the boundary of the cavity in the continuum medium. The virial theorem is also derived in the presence of the Pauli repulsion component of the solute-solvent interaction. Furthermore, it is shown that these forms of the PCM virial theorem may be related to the virial theorem of more simple systems as a molecule in the presence of fixed point charges, and as an atom in a spherical box with confining potential.
8. Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements.
PubMed
Li, Ke; Smith, Graeme
2015-04-24
We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multifold product states. The approximation is measured by distinguishability under measurements that are implementable by fully-one-way local operations and classical communication (LOCC). Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially-one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.
9. Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements
Li, Ke; Smith, Graeme
2015-04-01
We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multifold product states. The approximation is measured by distinguishability under measurements that are implementable by fully-one-way local operations and classical communication (LOCC). Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially-one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.
10. Test of the steady-state fluctuation theorem in turbulent Rayleigh-B'enard convection
Tong, Penger; Shang, Xiaodong
2005-03-01
Local convective heat flux in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in an aspect-ratio-one cell filled with water. It is found that large positive fluctuations of the vertical heat flux occurs more often in the plume-dominated sidewall region and their histograms are highly asymmetric. The statistical properties of the time-averaged local flux fluctuations are analyzed and the results are compared with the predictions of the steady state fluctuation theorem of Gallavotti and Cohen. Work supported by the Research Grants Council of Hong Kong SAR under Grant Nos. HKUST603003 (P.T.) and CUHK403003 (K.Q.X.).
11. Test of steady-state fluctuation theorem in turbulent Rayleigh-Bénard convection
Shang, X.-D.; Tong, P.; Xia, K.-Q.
2005-07-01
The local entropy production rate σ(r,t) in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in an aspect-ratio-one cell filled with water. The statistical properties of the time-averaged σ(r,t) are analyzed and the results are compared with the predictions of the steady-state fluctuation theorem (SSFT) of Gallavotti and Cohen. The experiment reveals that the SSFT can indeed be extended to the local variables, but further development is needed in order to incorporate the common dynamic complexities of far-from-equilibrium systems into the theory.
12. Test of steady-state fluctuation theorem in turbulent Rayleigh-Bénard convection.
PubMed
Shang, X-D; Tong, P; Xia, K-Q
2005-07-01
The local entropy production rate sigma (r,t) in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in an aspect-ratio-one cell filled with water. The statistical properties of the time-averaged sigma (r,t) are analyzed and the results are compared with the predictions of the steady-state fluctuation theorem (SSFT) of Gallavotti and Cohen. The experiment reveals that the SSFT can indeed be extended to the local variables, but further development is needed in order to incorporate the common dynamic complexities of far-from-equilibrium systems into the theory.
13. Differential diagnosis in immunohistochemistry with Bayes theorem.
PubMed
Vollmer, Robin T
2009-05-01
When immunohistochemical stains that are specific for specific tumor diagnoses do not yield diagnostic results, we often turn to less specific immunohistochemical stains and consider the resulting lists of possible tumor types. Typically, such lists are ordered according to tumor sensitivities for the stains. In probability terminology, sensitivity is the conditional probability of a positive stain given a specific tumor. Yet, the most useful probability to know is the probability of a specific tumor diagnosis, given a set of staining results. Bayes theorem provides this probability. To illustrate its use for differential diagnosis, I apply it here to the situation of carcinomas of uncertain primary site and use the information provided by stains for cytokeratin 7 and cytokeratin 20.
14. Elementary theorems regarding blue isocurvature perturbations
Chung, Daniel J. H.; Yoo, Hojin
2015-04-01
Blue CDM-photon isocurvature perturbations are attractive in terms of observability and may be typical from the perspective of generic mass relations in supergravity. We present and apply three theorems useful for blue isocurvature perturbations arising from linear spectator scalar fields. In the process, we give a more precise formula for the blue spectrum associated with the axion model of Kasuya and Kawasaki [Axion Isocurvature Fluctuations with Extremely Blue Spectrum, Phys. Rev. D 80, 023516 (2009).], which can in a parametric corner give a factor of O (10 ) correction. We explain how a conserved current associated with Peccei-Quinn symmetry plays a crucial role and explicitly plot several example spectra including the breaks in the spectra. We also resolve a little puzzle arising from a naive multiplication of isocurvature expression that sheds light on the gravitational imprint of the adiabatic perturbations on the fields responsible for blue isocurvature fluctuations.
15. A Stochastic Tikhonov Theorem in Infinite Dimensions
SciTech Connect
Buckdahn, Rainer Guatteri, Giuseppina
2006-03-15
The present paper studies the problem of singular perturbation in the infinite-dimensional framework and gives a Hilbert-space-valued stochastic version of the Tikhonov theorem. We consider a nonlinear system of Hilbert-space-valued equations for a 'slow' and a 'fast' variable; the system is strongly coupled and driven by linear unbounded operators generating a C{sub 0}-semigroup and independent cylindrical Brownian motions. Under well-established assumptions to guarantee the existence and uniqueness of mild solutions, we deduce the required stability of the system from a dissipativity condition on the drift of the fast variable. We avoid differentiability assumptions on the coefficients which would be unnatural in the infinite-dimensional framework.
16. Walking Through the Impulse-Momentum Theorem
Haugland, Ole Anton
2013-02-01
Modern force platforms are handy tools for investigating forces during human motion. Earlier they were very expensive and were mostly used in research laboratories. But now even platforms that can measure in two directions are quite affordable. In this work we used the PASCO 2-Axis Force Platform. The analysis of the data can serve as a nice illustration of qualitative or quantitative use of the impulse-momentum theorem p - p0 = ∫t0t Fdt = I. The most common use of force platforms is to study the force from the base during the push-off period of a vertical jump. I think this is an activity of great value, and I would recommend it. The use of force platforms in teaching is well documented in research literature.1-4
17. Extended Ehrenfest theorem with radiative corrections
de la Peña, L.; Cetto, A. M.; Valdés-Hernández, A.
2015-10-01
A set of basic evolution equations for the mean values of dynamical variables is obtained from the Fokker-Planck equation applied to the general problem of a particle subject to a random force. The specific case of stochastic electrodynamics is then considered, in which the random force is due to the zero-point radiation field. Elsewhere it has been shown that when this system reaches a state of energy balance, it becomes controlled by an equation identical to Schrödinger’s, if the radiationless approximation is made. The Fokker-Planck equation was shown to lead to the Ehrenfest theorem under such an approximation. Here we show that when the radiative terms are not neglected, an extended form of the Ehrenfest equation is obtained, from which follow, among others, the correct formulas for the atomic lifetimes and the (nonrelativistic) Lamb shift.
18. Quantum violation of fluctuation-dissipation theorem
Shimizu, Akira; Fujikura, Kyota
2017-02-01
We study quantum measurements of temporal equilibrium fluctuations in macroscopic quantum systems. It is shown that the fluctuation-dissipation theorem, as a relation between observed quantities, is partially violated in quantum systems, even if measurements are made in an ideal way that emulates classical ideal measurements as closely as possible. This is a genuine quantum effect that survives on a macroscopic scale. We also show that the state realized during measurements of temporal equilibrium fluctuations is a ‘squeezed equilibrium state’, which is macroscopically identical to the pre-measurement equilibrium state but is squeezed by the measurement. It is a time-evolving state, in which macrovariables fluctuate and relax. We also explain some of subtle but important points, careless treatments of which often lead to unphysical results, of the linear response theory.
19. Jarzynski equality, Crooks fluctuation theorem, and the fluctuation theorems of heat for arbitrary initial states
Gong, Zongping; Quan, H. T.
2015-07-01
By taking full advantage of the dynamic property imposed by the detailed balance condition, we derive a new refined unified fluctuation theorem (FT) for general stochastic thermodynamic systems. This FT involves the joint probability distribution functions of the final phase-space point and a thermodynamic variable. Jarzynski equality, Crooks fluctuation theorem, and the FTs of heat as well as the trajectory entropy production can be regarded as special cases of this refined unified FT, and all of them are generalized to arbitrary initial distributions. We also find that the refined unified FT can easily reproduce the FTs for processes with the feedback control, due to its unconventional structure that separates the thermodynamic variable from the choices of initial distributions. Our result is heuristic for further understanding of the relations and distinctions between all kinds of FTs and might be valuable for studying thermodynamic processes with information exchange.
20. Non-targeted profiling of semi-polar metabolites in Arabidopsis root exudates uncovers a role for coumarin secretion and lignification during the local response to phosphate limitation
PubMed Central
Ziegler, Jörg; Schmidt, Stephan; Chutia, Ranju; Müller, Jens; Böttcher, Christoph; Strehmel, Nadine; Scheel, Dierk; Abel, Steffen
2016-01-01
Plants have evolved two major strategies to cope with phosphate (Pi) limitation. The systemic response, mainly comprising increased Pi uptake and metabolic adjustments for more efficient Pi use, and the local response, enabling plants to explore Pi-rich soil patches by reorganization of the root system architecture. Unlike previous reports, this study focused on root exudation controlled by the local response to Pi deficiency. To approach this, a hydroponic system separating the local and systemic responses was developed. Arabidopsis thaliana genotypes exhibiting distinct sensitivities to Pi deficiency could be clearly distinguished by their root exudate composition as determined by non-targeted reversed-phase ultraperformance liquid chromatography electrospray ionization quadrupole-time-of-flight mass spectrometry metabolite profiling. Compared with wild-type plants or insensitive low phosphate root 1 and 2 (lpr1 lpr2) double mutant plants, the hypersensitive phosphate deficiency response 2 (pdr2) mutant exhibited a reduced number of differential features in root exudates after Pi starvation, suggesting the involvement of PDR2-encoded P5-type ATPase in root exudation. Identification and analysis of coumarins revealed common and antagonistic regulatory pathways between Pi and Fe deficiency-induced coumarin secretion. The accumulation of oligolignols in root exudates after Pi deficiency was inversely correlated with Pi starvation-induced lignification at the root tips. The strongest oligolignol accumulation in root exudates was observed for the insensitive lpr1 lpr2 double mutant, which was accompanied by the absence of Pi deficiency-induced lignin deposition, suggesting a role of LPR ferroxidases in lignin polymerization during Pi starvation. PMID:26685189
1. On soft limits of inflationary correlation functions
Assassi, Valentin; Baumann, Daniel; Green, Daniel
2012-11-01
Soft limits of inflationary correlation functions are both observationally relevant and theoretically robust. Various theorems can be proven about them that are insensitive to detailed model-building assumptions. In this paper, we re-derive several of these theorems in a universal way. Our method makes manifest why soft limits are such an interesting probe of the spectrum of additional light fields during inflation. We illustrate these abstract results with a detailed case study of the soft limits of quasi-single-field inflation.
2. STABILITY OF GAS CLOUDS IN GALACTIC NUCLEI: AN EXTENDED VIRIAL THEOREM
SciTech Connect
2016-03-10
Cold gas entering the central 1–10{sup 2} pc of a galaxy fragments and condenses into clouds. The stability of the clouds determines whether they will be turned into stars or can be delivered to the central supermassive black hole (SMBH) to turn on an active galactic nucleus (AGN). The conventional criteria to assess the stability of these clouds, such as the Jeans criterion and Roche (or tidal) limit, are insufficient here, because they assume the dominance of self-gravity in binding a cloud, and neglect external agents, such as pressure and tidal forces, which are common in galactic nuclei. We formulate a new scheme for judging this stability. We first revisit the conventional Virial theorem, taking into account an external pressure, to identify the correct range of masses that lead to stable clouds. We then extend the theorem to further include an external tidal field, which is equally crucial for the stability in the region of our interest—in dense star clusters, around SMBHs. We apply our extended Virial theorem to find new solutions to controversial problems, namely, the stability of the gas clumps in AGN tori, the circum-nuclear disk in the Galactic Center, and the central molecular zone of the Milky Way. The masses we derive for these structures are orders of magnitude smaller than the commonly used Virial masses (equivalent to the Jeans mass). Moreover, we prove that these clumps are stable, contrary to what one would naively deduce from the Roche (tidal) limit.
3. Modeling state transition of hippocampal local field potential between theta rhythm and large irregular amplitude activity by bifurcation between a limit cycle and chaotic dynamics
Tokuda, K.; Katori, Y.; Aihara, K.
2013-01-01
Here we propose a possible mathematical structure of the state transition of the hippocampal local field potential (LFP) between theta rhythm and large irregular amplitude activity (LIA) in terms of nonlinear dynamics. The basic idea is that the alternation of the state between theta rhythm and LIA can be interpreted as a bifurcation of the attractor between a limit cycle and chaotic dynamics. Tsuda et al. reported that a network composed of simple class 1 model neurons connected with gap junctions shows both synchronous periodic behavior and asynchronous chaotic behavior [1]. Here we model the network of hippocampal interneurons extending their model. The network is composed of electrically coupled simple 2-dimensional neurons with natural resonant frequency in the theta frequency. We incorporate a periodic external force representing the medial septal afferent. The system converges on a limit cycle under this external force, but shows chaotic dynamics without this external force. Furthermore, the external noise realized rapid alteration of the state obeying the change of the amplitude of the septal input.
4. Generalized Optical Theorem Detection in Random and Complex Media
Tu, Jing
The problem of detecting changes of a medium or environment based on active, transmit-plus-receive wave sensor data is at the heart of many important applications including radar, surveillance, remote sensing, nondestructive testing, and cancer detection. This is a challenging problem because both the change or target and the surrounding background medium are in general unknown and can be quite complex. This Ph.D. dissertation presents a new wave physics-based approach for the detection of targets or changes in rather arbitrary backgrounds. The proposed methodology is rooted on a fundamental result of wave theory called the optical theorem, which gives real physical energy meaning to the statistics used for detection. This dissertation is composed of two main parts. The first part significantly expands the theory and understanding of the optical theorem for arbitrary probing fields and arbitrary media including nonreciprocal media, active media, as well as time-varying and nonlinear scatterers. The proposed formalism addresses both scalar and full vector electromagnetic fields. The second contribution of this dissertation is the application of the optical theorem to change detection with particular emphasis on random, complex, and active media, including single frequency probing fields and broadband probing fields. The first part of this work focuses on the generalization of the existing theoretical repertoire and interpretation of the scalar and electromagnetic optical theorem. Several fundamental generalizations of the optical theorem are developed. A new theory is developed for the optical theorem for scalar fields in nonhomogeneous media which can be bounded or unbounded. The bounded media context is essential for applications such as intrusion detection and surveillance in enclosed environments such as indoor facilities, caves, tunnels, as well as for nondestructive testing and communication systems based on wave-guiding structures. The developed scalar
5. Global stabilization using LSS-Theorem: Applications to Robotics and Aerospace Vehicles
Selman, AbdulRazzak
Underactuated mechanical systems are gaining interest as they can sometimes provide the desired motion or functionality at reduced cost due to their using fewer expensive actuators. The term "underactuated" refers to the fact that such mechanical systems have fewer actuators than degrees of freedom, which makes them very difficult to control. Moreover, underactuated robots have nonlinear dynamics which must be tackled with nonlinear control techniques. Furthermore, control theory for underactuated mechanical systems has been an active area of research for the past 15-20 years. Most of the research has focused on local and global asymptotic stabilization by feedback. Underactuated systems can either possess nonminimum phase or minimum phase characteristics. For minimum phase underactuated systems, the stabilization problem is rather simple and many existing control design methodologies have been proved powerful in providing a solution to this problem. For nonminimum phase underactuated systems, asymptotic stabilization problem has been, and still is, an attractive subject to the researchers in the field of nonlinear control system and theory. In particular, global asymptotic stabilization (GAS) at a desired equilibrium point of such systems by means of a single smooth static or dynamic state feedback law is still largely an open problem in the literature. In this thesis, the problem of GAS via a smooth static state feedback law is addressed for a class of an underactuated nonlinear system that is affine (possibly non affine) in the control, partially feedback linearizable, nonminimum phase and (possibly) has a non-integrable acceleration constraint. The core result of the thesis is formulated through a theorem that the author refers to through this thesis as the Legend of Salah Salman (LSS) Theorem. LSS theorem states the existence of a smooth static state feedback law that globally asymptotically stabilizes the origin of the nonlinear underactuated system that is
6. Entropy for quantum pure states and quantum H theorem
Han, Xizhi; Wu, Biao
2015-06-01
We construct a complete set of Wannier functions that are localized at both given positions and momenta. This allows us to introduce the quantum phase space, onto which a quantum pure state can be mapped unitarily. Using its probability distribution in quantum phase space, we define an entropy for a quantum pure state. We prove an inequality regarding the long-time behavior of our entropy's fluctuation. For a typical initial state, this inequality indicates that our entropy can relax dynamically to a maximized value and stay there most of time with small fluctuations. This result echoes the quantum H theorem proved by von Neumann [Zeitschrift für Physik 57, 30 (1929), 10.1007/BF01339852]. Our entropy is different from the standard von Neumann entropy, which is always zero for quantum pure states. According to our definition, a system always has bigger entropy than its subsystem even when the system is described by a pure state. As the construction of the Wannier basis can be implemented numerically, the dynamical evolution of our entropy is illustrated with an example.
7. An Almost Sure Ergodic Theorem for Quasistatic Dynamical Systems
Stenlund, Mikko
2016-09-01
We prove an almost sure ergodic theorem for abstract quasistatic dynamical systems, as an attempt of taking steps toward an ergodic theory of such systems. The result at issue is meant to serve as a working counterpart of Birkhoff's ergodic theorem which fails in the quasistatic setup. It is formulated so that the conditions, which essentially require sufficiently good memory-loss properties, could be verified in a straightforward way in physical applications. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then illustrate the use of the theorem by examples.
8. Fluctuation theorem for Hamiltonian systems: Le Chatelier's principle.
PubMed
Evans, D J; Searles, D J; Mittag, E
2001-05-01
For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields.
9. Fluctuation theorem for Hamiltonian Systems: Le Chatelier's principle
Evans, Denis J.; Searles, Debra J.; Mittag, Emil
2001-05-01
For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields.
10. Localization and Dualities in Three-dimensional Superconformal Field Theories
Willett, Brian
In this thesis we apply the technique of localization to three-dimensional N = 2 superconformal field theories. We consider both theories which are exactly superconformal, and those which are believed to flow to nontrivial superconformal fixed points, for which we consider implicitly these fixed points. We find that in such theories, the partition function and certain supersymmetric observables, such as Wilson loops, can be computed exactly by a matrix model. This matrix model consists of an integral over g , the Lie algebra of the gauge group of the theory, of a certain product of 1-loop factors and classical contributions. One can also consider a space of supersymmetric deformations of the partition function corresponding to the set of abelian global symmetries. In the second part of the thesis we apply these results to test dualities. We start with the case of ABJM theory, which is dual to M-theory on an asymptotically AdS4 x S7 background. We extract strong coupling results in the field theory, which can be compared to semiclassical, weak coupling results in the gravity theory, and a nontrivial agreement is found. We also consider several classes of dualities between two three-dimensional field theories, namely, 3D mirror symmetry, Aharony duality, and Giveon-Kutasov duality. Here the dualities are typically between the IR limits of two Yang-Mills theories, which are strongly coupled in three dimensions since Yang-Mills theory is asymptotically free here. Thus the comparison is again very nontrivial, and relies on the exactness of the localization computation. We also compare the deformed partition functions, which tests the mapping of global symmetries of the dual theories. Finally, we discuss some recent progress in the understanding of general three-dimensional theories in the form of the F-theorem, a conjectured analogy to the a-theorem in four dimensions and c-theorem in two dimensions, which is closely related to the localization computation.
11. Local source identification of trace metals in urban/industrial mixed land-use areas with daily PM10 limit value exceedances
Fernández-Olmo, Ignacio; Andecochea, Carlos; Ruiz, Sara; Fernández-Ferreras, José Antonio; Irabien, Angel
2016-05-01
This study presents the analysis of the concentration levels, inter-site variation and source identification of trace metals at three urban/industrial mixed land-use sites of the Cantabria region (northern Spain), where local air quality plans were recently approved because the number of exceedances of the daily PM10 limit value according to the Directive 2008/50/EC had been relatively high in the last decade (more than 35 instances per year). PM10 samples were collected for over three years at the Torrelavega (TORR) and Los Corrales (CORR) sites and for over two years at the Camargo (GUAR) site and analysed for the presence of arsenic (As), cadmium (Cd), chromium (Cr), copper (Cu), lead (Pb), nickel (Ni), titanium (Ti), vanadium (V), molybdenum (Mo), manganese (Mn), iron (Fe), antimony (Sb) and zinc (Zn). Analysis of enrichment factors revealed an anthropogenic origin of most of the studied elements; Zn, Cd, Mo, Pb and Cu were the most enriched elements at the three sites, with Fe and V as the least enriched elements. Positive Matrix Factorisation (PMF) and pollutant roses (Cu at TORR, Zn at CORR and Mn at GUAR) were used to identify the local sources of the studied metals. Analysis of PMF results revealed the main sources of trace metals at each site as road traffic at the TORR site, iron foundry and casting industry at the CORR site and a ferro-manganese alloy industry at the GUAR site. Other sources were also identified at these sites, but with much lower contributions, such as minor industrial sources, combustion and traffic mixed with the previous sources.
12. Strain localization in brittle-ductile shear zones: fluid-abundant vs. fluid-limited conditions (an example from Wyangala area, Australia)
Spruzeniece, L.; Piazolo, S.
2015-07-01
This study focuses on physiochemical processes occurring in a brittle-ductile shear zone at both fluid-present and fluid-limited conditions. In the studied shear zone (Wyangala, SE Australia), a coarse-grained two-feldspar-quartz-biotite granite is transformed into a medium-grained orthogneiss at the shear zone margins and a fine-grained quartz-muscovite phyllonite in the central parts. The orthogneiss displays cataclasis of feldspar and crystal-plastic deformation of quartz. Quartz accommodates most of the deformation and is extensively recrystallized, showing distinct crystallographic preferred orientation (CPO). Feldspar-to-muscovite, biotite-to-muscovite and albitization reactions occur locally at porphyroclasts' fracture surfaces and margins. However, the bulk rock composition shows very little change in respect to the wall rock composition. In contrast, in the shear zone centre quartz occurs as large, weakly deformed porphyroclasts in sizes similar to that in the wall rock, suggesting that it has undergone little deformation. Feldspars and biotite are almost completely reacted to muscovite, which is arranged in a fine-grained interconnected matrix. Muscovite-rich layers contain significant amounts of fine-grained intermixed quartz with random CPO. These domains are interpreted to have accommodated most of the strain. Bulk rock chemistry data show a significant increase in SiO2 and depletion in NaO content compared to the wall rock composition. We suggest that the high- and low-strain microstructures in the shear zone represent markedly different scenarios and cannot be interpreted as a simple sequential development with respect to strain. Instead, we propose that the microstructural and mineralogical changes in the shear zone centre arise from a local metasomatic alteration around a brittle precursor. When the weaker fine-grained microstructure is established, the further flow is controlled by transient porosity created at (i) grain boundaries in fine
13. The Pythagorean Theorem and the Solid State
Kelly, Brenda S.; Splittgerber, Allen G.
2005-05-01
Solid-state parameters such as radius ratios, packing efficiencies, and crystal densities may be calculated for various crystal structures from basic Euclidean geometry relating to the Pythagorean theorem of right triangles. Because simpler cases are often discussed in the standard inorganic chemistry texts, this article only presents calculations for closest-packed A-type lattices (one type of particle) and several compound AB lattices (A and B particles) including sodium chloride, cesium chloride, zinc blende (sphalerite), wurtzite, and fluorite. For A-type metallic crystals, the use of recommended values of atomic radii results in calculated densities within 1% of observed values. For AB lattices, assuming ionic crystals, the use of recommended values of ionic radii results in density determinations that are usually but not always close to observed values. When there is covalent character to the bonding, the use of covalent radii results in calculated densities that correlate well with observed values. If interionic or interatomic spacings are used, the calculated densities are always close to the observed values. As indicated by a survey of the standard inorganic texts, these calculations are generally not presented. However, as an illustration of the application of simple mathematical principles to the study of chemistry, discussion of the methods presented in this manuscript may be of value in classroom presentations pertaining to the solid state.
14. Digital superresolution and the generalized sampling theorem
2007-02-01
The technique of reconstructing a higher-resolution (HR) image of size ML×ML by digitally processing L×L subpixel-shifted lower-resolution (LR) copies of it, each of size M×M, has now become well established. This particular digital superresolution problem is analyzed from the standpoint of the generalized sampling theorem. It is shown both theoretically and by computer simulation that the choice of regularly spaced subpixel shifts for the LR images tends to maximize the robustness and minimize the error of reconstruction of the HR image. In practice, since subpixel-level control of LR image shifts may be nearly impossible to achieve, however, a more likely scenario, which is also discussed, is one involving random subpixel shifts. It is shown that without reasonably tight bounds on the range of random shifts, the reconstruction is likely to fail in the presence of even small amounts of noise unless either reliable prior information or additional data are available.
15. On the Spin-Statistics Theorem
Peshkin, Murray
2002-05-01
M.V. Berry and J.M. Robbins* (B) have explained the spin-statistics theorem (SST) within nonrelativistic quantum mechanics (QM), without using relativity or field theory. For two identical spinless particles, their starting point is a coordinate space which consists of unordered pairs r,r' where r and r' represent two points in space, not particle labels. The point r,r' is the point r',r\\. That has topological consequences for the 6D configuration space and for the wave functions |r,r'>. More generally, spin variables are appended and there are N vectors. B gave a beautiful mathematical analysis to go from there to the usual SST under stated assumptions of QM. They also explored alternative assumptions that give unusual results but that may not be physical. I seek additional insight by recasting B's analysis into a form that emphasizes the relative orbital angular momenta of pairs of particles. I report here on the spinless case, where boson statistics emerges in a transparent way. This approach appears to exclude unusual possibilities. Work supported by U.S. DOE contract W-31-109-ENG-38. *Proc. R. Soc. Lond. A 453, 1771 (1997).
16. Generalized Browder's and Weyl's theorems for Banach space operators
Curto, Raúl E.; Han, Young Min
2007-12-01
We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of [sigma](T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f[set membership, variant]H((T)), the space of functions analytic on an open neighborhood of [sigma](T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f[set membership, variant]H([sigma](T)).
17. Gibbs Paradox Revisited from the Fluctuation Theorem with Absolute Irreversibility
Murashita, Yûto; Ueda, Masahito
2017-02-01
The inclusion of the factor ln (1 /N !) in the thermodynamic entropy proposed by Gibbs is shown to be equivalent to the validity of the fluctuation theorem with absolute irreversibility for gas mixing.
18. The Pythagorean Theorem: II. The infinite discrete case
PubMed Central
2002-01-01
The study of the Pythagorean Theorem and variants of it as the basic result of noncommutative, metric, Euclidean Geometry is continued. The emphasis in the present article is the case of infinite discrete dimensionality. PMID:16578869
19. Comparison theorems for neutral stochastic functional differential equations
Bai, Xiaoming; Jiang, Jifa
2016-05-01
The comparison theorems under Wu and Freedman's order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same.
20. Forest Carbon Uptake and the Fundamental Theorem of Calculus
ERIC Educational Resources Information Center
Zobitz, John
2013-01-01
Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.
1. Fluctuation theorem in driven nonthermal systems with quenched disorder
SciTech Connect
Reichhardt, Charles; Reichhardt, C J; Drocco, J A
2009-01-01
We demonstrate that the fluctuation theorem of Evans and Searles can be used to characterize the class of dynamics that arises in nonthermal systems of collectively interacting particles driven over random quenched disorder. By observing the frequency of entropy-destroying trajectories, we show that there are specific dynamical regimes near depinning in which this theorem holds. Hence the fluctuation theorem can be used to characterize a significantly wider class of non-equilibrium systems than previously considered. We discuss how the fluctuation theorem could be tested in specific systems where noisy dynamics appear at the transition from a pinned to a moving phase such as in vortices in type-II superconductors, magnetic domain walls, and dislocation dynamics.
2. A Computer Science Version of Goedel’s Theorem.
DTIC Science & Technology
1983-08-01
The author presents a simplified proof of Godel’s theorem by appealing to well-known programming concepts. The significance of Goedel’s result to computer science , mathematics and logic is discussed. (Author)
3. Two time physics and Hamiltonian Noether theorem for gauge systems
SciTech Connect
Nieto, J. A.; Ruiz, L.; Silvas, J.; Villanueva, V. M.
2006-09-25
Motivated by two time physics theory we revisited the Noether theorem for Hamiltonian constrained systems. Our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints.
4. Conformal frames and the validity of Birkhoff's theorem
Capozziello, S.; Sáez-Gómez, D.
2012-07-01
Birkhoff's theorem is one of the most important statements of Einstein's general relativity, which generally can not be extended to modified theories of gravity. Here we study the validity of the theorem in scalar-tensor theories using a perturbative approach, and compare the results in the so-called Einstein and Jordan frames. The implications of the results question the physical equivalence between both frames, at least in perturbations.
5. No-broadcasting theorem and its classical counterpart.
PubMed
Kalev, Amir; Hen, Itay
2008-05-30
Although it is widely accepted that "no-broadcasting"-the nonclonability of quantum information-is a fundamental principle of quantum mechanics, an impossibility theorem for the broadcasting of general density matrices has not yet been formulated. In this Letter, we present a general proof for the no-broadcasting theorem, which applies to arbitrary density matrices. The proof relies on entropic considerations, and as such can also be directly linked to its classical counterpart, which applies to probabilistic distributions of statistical ensembles.
6. Levinson theorem for Aharonov-Bohm scattering in two dimensions
SciTech Connect
Sheka, Denis D.; Mertens, Franz G.
2006-11-15
We apply the recently generalized Levinson theorem for potentials with inverse-square singularities [Sheka et al., Phys. Rev. A 68, 012707 (2003)] to Aharonov-Bohm systems in two dimensions (2D). By this theorem, the number of bound states in a given mth partial wave is related to the phase shift and the magnetic flux. The results are applied to 2D soliton-magnon scattering.
7. Non-linear energy conservation theorem in the framework of special relativity
Pérez Teruel, Ginés R.
2015-07-01
In this work we revisit the study of the gravitational interaction in the context of the special theory of relativity. It is found that, as long as the equivalence principle is respected, a relativistic nonlinear energy conservation theorem arises in a natural way. We interpret that this nonlinear conservation law stresses the nonlinear character of the gravitational interaction. The theorem reproduces the energy conservation theorem of Newtonian mechanics in the corresponding low energy limit, but also allows to derive some standard results of post-Newtonian gravity, such as the formula of the gravitational redshift. Guided by this conservation law, we develop a Lagrangian formalism for a particle in a gravitational field. We realize that the Lagrangian can be written in an explicit covariant fashion, and turns out to be the geodesic Lagrangian of a curved Lorentzian manifold. Therefore, any attempt to describe gravity within the special theory, leads outside their own domains towards a curved space-time. Thus, the pedagogical content of the paper may be useful as a starting point to discuss the problem of gravitation in the context of the special theory, as a preliminary step before introducing general relativity.
8. Nyquist-Shannon sampling theorem applied to refinements of the atomic pair distribution function
Farrow, Christopher L.; Shaw, Margaret; Kim, Hyunjeong; Juhás, Pavol; Billinge, Simon J. L.
2011-10-01
We have systematically studied the optimal real-space sampling of atomic pair distribution (PDF) data by comparing refinement results from oversampled and resampled data. Based on nickel and a complex perovskite system, we show that not only is the optimal sampling bounded by the Nyquist interval described by the Nyquist-Shannon (NS) sampling theorem as expected, but near this sampling interval, the data points in the PDF are minimally correlated, which results in more reliable uncertainty estimates in the modeling. Surprisingly, we find that PDF refinements quickly become unstable for data on coarser grids. Although the Nyquist-Shannon sampling theorem is well known, it has not been applied to PDF refinements, despite the growing popularity of the PDF method and its adoption in a growing number of communities. Here, we give explicit expressions for the application of NS sampling theorem to the PDF case, and establish through modeling that it is working in practice, which lays the groundwork for this to become more widely adopted. This has implications for the speed and complexity of possible refinements that can be carried out many times faster than currently with no loss of information, and it establishes a theoretically sound limit on the amount of information contained in the PDF that will prevent over-parametrization during modeling.
9. Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains
Shargel, Benjamin Hertz; Chou, Tom
2009-10-01
Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry in the rate function of either the time-averaged entropy production or heat dissipation of a process. Such theorems have been proved for various general classes of continuous-time deterministic and stochastic processes, but always under the assumption that the forces driving the system are time independent, and often relying on the existence of a limiting ergodic distribution. In this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous-time processes without a stationary distribution, considering specifically a finite state Markov chain driven by periodic transition rates. We find that for both entropy production and heat dissipation, the usual Gallavotti-Cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process, in which the trajectory and the driving protocol have been time-reversed. The effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process, and vice-versa, revealing that the distributions of fluctuations in universes in which time moves forward and backward are related. As an additional result, the asymptotic time-averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics.
10. Exploring the limit of accuracy for density functionals based on the generalized gradient approximation: Local, global hybrid, and range-separated hybrid functionals with and without dispersion corrections
SciTech Connect
2014-05-14
The limit of accuracy for semi-empirical generalized gradient approximation (GGA) density functionals is explored by parameterizing a variety of local, global hybrid, and range-separated hybrid functionals. The training methodology employed differs from conventional approaches in 2 main ways: (1) Instead of uniformly truncating the exchange, same-spin correlation, and opposite-spin correlation functional inhomogeneity correction factors, all possible fits up to fourth order are considered, and (2) Instead of selecting the optimal functionals based solely on their training set performance, the fits are validated on an independent test set and ranked based on their overall performance on the training and test sets. The 3 different methods of accounting for exchange are trained both with and without dispersion corrections (DFT-D2 and VV10), resulting in a total of 491 508 candidate functionals. For each of the 9 functional classes considered, the results illustrate the trade-off between improved training set performance and diminished transferability. Since all 491 508 functionals are uniformly trained and tested, this methodology allows the relative strengths of each type of functional to be consistently compared and contrasted. The range-separated hybrid GGA functional paired with the VV10 nonlocal correlation functional emerges as the most accurate form for the present training and test sets, which span thermochemical energy differences, reaction barriers, and intermolecular interactions involving lighter main group elements.
11. Exploring the limit of accuracy for density functionals based on the generalized gradient approximation: Local, global hybrid, and range-separated hybrid functionals with and without dispersion corrections
2014-05-01
The limit of accuracy for semi-empirical generalized gradient approximation (GGA) density functionals is explored by parameterizing a variety of local, global hybrid, and range-separated hybrid functionals. The training methodology employed differs from conventional approaches in 2 main ways: (1) Instead of uniformly truncating the exchange, same-spin correlation, and opposite-spin correlation functional inhomogeneity correction factors, all possible fits up to fourth order are considered, and (2) Instead of selecting the optimal functionals based solely on their training set performance, the fits are validated on an independent test set and ranked based on their overall performance on the training and test sets. The 3 different methods of accounting for exchange are trained both with and without dispersion corrections (DFT-D2 and VV10), resulting in a total of 491 508 candidate functionals. For each of the 9 functional classes considered, the results illustrate the trade-off between improved training set performance and diminished transferability. Since all 491 508 functionals are uniformly trained and tested, this methodology allows the relative strengths of each type of functional to be consistently compared and contrasted. The range-separated hybrid GGA functional paired with the VV10 nonlocal correlation functional emerges as the most accurate form for the present training and test sets, which span thermochemical energy differences, reaction barriers, and intermolecular interactions involving lighter main group elements.
12. The PBR theorem: Whose side is it on?
Ben-Menahem, Yemima
2017-02-01
This paper examines the implications of the PBR theorem for the debate on the reality of the quantum state. The theorem seeks to undermine epistemic interpretations of the quantum state and support realist interpretations thereof, but there remains ambiguity about the precise nature of epistemic interpretations, and thus ambiguity about the implications of the theorem. The aim of this paper is to examine a radical epistemic interpretation that is not undermined by the theorem and is, arguably, strengthened by it. It is this radical interpretation, rather than the one assumed by the PBR theorem, that many epistemic theorists subscribe to. In order to distinguish the radical epistemic interpretation from alternative interpretations of quantum states-in particular, to distinguish it from instrumentalism-a historical comparison of different approaches to the meaning of quantum probabilities is provided. The comparison highlights, in particular, Schrödinger's work on the nature of quantum probabilities as distinct from probabilities in statistical mechanics, and the implications of this distinction for an epistemic interpretation of probability in the two areas. Schrödinger's work also helps to identify the difficulties in the PBR definition of an epistemic interpretation and is shown to anticipate the radical alternative that is not undermined by the theorem.
13. Inelastic light and electron scattering in parabolic quantum dots in magnetic field: Implications of generalized Kohn's theorem
Kushwaha, Manvir S.
2016-03-01
We investigate a one-component, quasi-zero-dimensional, quantum plasma exposed to a parabolic potential and an applied magnetic field in the symmetric gauge. If the size of such a system as can be realized in the semiconducting quantum dots is on the order of the de Broglie wavelength, the electronic and optical properties become highly tunable. Then the quantum size effects challenge the observation of many-particle phenomena such as the magneto-optical absorption, Raman intensity, and electron energy loss spectrum. An exact analytical solution of the problem leads us to infer that these many-particle phenomena are, in fact, dictated by the generalized Kohn's theorem in the long-wavelength limit. Maneuvering the confinement and/or the magnetic field furnishes the resonance energy capable of being explored with the FIR, Raman, or electron energy loss spectroscopy. This implies that either of these probes should be competent in observing the localized magnetoplasmons in the system. A deeper insight into the physics of quantum dots is paving the way for their implementation in diverse fields such as quantum computing and medical imaging.
14. Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons
Giustina, Marissa; Versteegh, Marijn A. M.; Wengerowsky, Sören; Handsteiner, Johannes; Hochrainer, Armin; Phelan, Kevin; Steinlechner, Fabian; Kofler, Johannes; Larsson, Jan-Åke; Abellán, Carlos; Amaya, Waldimar; Pruneri, Valerio; Mitchell, Morgan W.; Beyer, Jörn; Gerrits, Thomas; Lita, Adriana E.; Shalm, Lynden K.; Nam, Sae Woo; Scheidl, Thomas; Ursin, Rupert; Wittmann, Bernhard; Zeilinger, Anton
2015-12-01
Local realism is the worldview in which physical properties of objects exist independently of measurement and where physical influences cannot travel faster than the speed of light. Bell's theorem states that this worldview is incompatible with the predictions of quantum mechanics, as is expressed in Bell's inequalities. Previous experiments convincingly supported the quantum predictions. Yet, every experiment requires assumptions that provide loopholes for a local realist explanation. Here, we report a Bell test that closes the most significant of these loopholes simultaneously. Using a well-optimized source of entangled photons, rapid setting generation, and highly efficient superconducting detectors, we observe a violation of a Bell inequality with high statistical significance. The purely statistical probability of our results to occur under local realism does not exceed 3.74 ×10-31 , corresponding to an 11.5 standard deviation effect.
15. Use of Lambert's theorem for the n-dimensional Coulomb problem
SciTech Connect
Kanellopoulos, Vassiliki; Kleber, Manfred; Kramer, Tobias
2009-07-15
We present the analytical solution in closed form for the semiclassical limit of the quantum-mechanical Coulomb Green's function in position space in n dimensions. We utilize a projection method which has its roots in Lambert's theorem and which allows us to treat the system as an essentially one-dimensional problem. The semiclassical result assumes a simple analytical form and is well suited for a numerical evaluation. The method can also be extended to classically forbidden space regions. Already for moderately large principal quantum numbers {nu}{>=}5, the semiclassical Green's function is found to be an excellent approximation to the quantum-mechanical Green's function.
16. Dynamical control of quantum systems in the context of mean ergodic theorems
Bernád, J. Z.
2017-02-01
Equidistant and non-equidistant single pulse ‘bang-bang’ dynamical controls are investigated in the context of mean ergodic theorems. We show the requirements in which the limit of infinite pulse control for both the equidistant and the non-equidistant dynamical control converges to the same unitary evolution. It is demonstrated that the generator of this evolution can be obtained by projecting the generator of the free evolution onto the commutant of the unitary operator representing the pulse. Inequalities are derived to prove this statement and in the case of non-equidistant approach these inequalities are optimised as a function of the time intervals.
17. Asymptotic behavior and Denjoy-Wolff theorems for Hilbert metric nonexpansive maps
Lins, Brian C.
We study the asymptotic behavior of fixed point free Hilbert metric nonexpansive maps on bounded convex domains. For such maps, we prove that the omega limit sets are contained in a convex subset of the boundary when the domain is either polyhedral or two dimensional. Similar results are obtained for several classes of positive operators defined on closed cones, including linear maps, affine linear maps, max-min operators, and reproduction-decimation operators. We discuss the relationship between these results and other Denjoy-Wolff type theorems. In particular, we investigate the interaction of nonexpansive maps with the horofunction boundary in the Hilbert geometry and in finite dimensional normed spaces.
18. The solution to the phase retrieval problem using the sampling theorem
Arsenault, H. H.; Chalasinska-Macukow, K.
1983-10-01
The 2D phase-retrieval problem is investigated analytically, and the solution is applied to an optics example. In the case considered, only the Fourier-transform modulus and the support of the object function are known. The approach taken is based on the Whittaker-Shannon sampling theorem (Goodman, 1968), using the sin c function as interpolator. The algorithm developed is shown to converge rapidly and give an accurate representation of band-limited objects if the two sampling grids are carefully chosen, as illustrated in a test calculation using a 5 x 5-pixel object.
19. Formalization of the Integral Calculus in the PVS Theorem Prover
NASA Technical Reports Server (NTRS)
Butler, Ricky W.
2004-01-01
The PVS Theorem prover is a widely used formal verification tool used for the analysis of safety-critical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht's classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
20. Limit laws for Zipf's law
Eliazar, Iddo
2011-01-01
In this communication we establish stochastic limit laws leading from Zipf's law to Pareto's and Heaps' laws. We consider finite ensembles governed by Zipf's law and study their asymptotic statistics as the ensemble size tends to infinity. A Lorenz-curve analysis establishes three types of limit laws for the ensembles' statistical structure: 'communist', 'monarchic', and Paretian. Further considering a dynamic setting in which the ensembles grow stochastically in time, a functional central limit theorem analysis establishes a Gaussian approximation for the ensembles' stochastic growth. The Gaussian approximation provides a generalized and corrected formulation of Heaps' law.
1. Attractive Hubbard model with disorder and the generalized Anderson theorem
SciTech Connect
Kuchinskii, E. Z. Kuleeva, N. A. Sadovskii, M. V.
2015-06-15
Using the generalized DMFT+Σ approach, we study the influence of disorder on single-particle properties of the normal phase and the superconducting transition temperature in the attractive Hubbard model. A wide range of attractive potentials U is studied, from the weak coupling region, where both the instability of the normal phase and superconductivity are well described by the BCS model, to the strong-coupling region, where the superconducting transition is due to Bose-Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures much higher than the superconducting transition temperature. We study two typical models of the conduction band with semi-elliptic and flat densities of states, respectively appropriate for three-dimensional and two-dimensional systems. For the semi-elliptic density of states, the disorder influence on all single-particle properties (e.g., density of states) is universal for an arbitrary strength of electronic correlations and disorder and is due to only the general disorder widening of the conduction band. In the case of a flat density of states, universality is absent in the general case, but still the disorder influence is mainly due to band widening, and the universal behavior is restored for large enough disorder. Using the combination of DMFT+Σ and Nozieres-Schmitt-Rink approximations, we study the disorder influence on the superconducting transition temperature T{sub c} for a range of characteristic values of U and disorder, including the BCS-BEC crossover region and the limit of strong-coupling. Disorder can either suppress T{sub c} (in the weak-coupling region) or significantly increase T{sub c} (in the strong-coupling region). However, in all cases, the generalized Anderson theorem is valid and all changes of the superconducting critical temperature are essentially due to only the general disorder widening of the conduction band.
2. Remarks on asymptotic symmetries and the subleading soft photon theorem
Conde, Eduardo; Mao, Pujian
2017-01-01
A deep connection has been recently established between soft theorems and symmetries at null infinity in gravity and gauge theories, recasting the former as Ward identities of the latter. In particular, different orders (in the frequency of the soft particle) in the soft theorems are believed to be controlled by different asymptotic symmetries. In this paper we argue that this need not be the case by focusing on the soft photon theorem. We argue that the subleading soft factor follows from the same symmetry responsible for the leading one, namely certain residual (large) gauge transformations of the gauge theory. In particular, expanding the associated charge in inverse powers of the radial coordinate, the (sub)leading charge yields the (sub)leading soft factor.
3. Noncommutative topology and the world’s simplest index theorem
PubMed Central
van Erp, Erik
2010-01-01
In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool for proving results in classical analysis and geometry. PMID:20418506
4. Time dependent electromagnetic fields and 4-dimensional Stokes' theorem
Andosca, Ryan; Singleton, Douglas
2016-11-01
Stokes' theorem is central to many aspects of physics—electromagnetism, the Aharonov-Bohm effect, and Wilson loops to name a few. However, the pedagogical examples and research work almost exclusively focus on situations where the fields are time-independent so that one need only deal with purely spatial line integrals (e.g., ∮ A . d x ) and purely spatial area integrals (e.g., ∫ ( ∇ × A ) . d a = ∫ B . d a ). Here, we address this gap by giving some explicit examples of how Stokes' theorem plays out with time-dependent fields in a full 4-dimensional spacetime context. We also discuss some unusual features of Stokes' theorem with time-dependent fields related to gauge transformations and non-simply connected topology.
5. Generalized Bezout's Theorem and its applications in coding theory
NASA Technical Reports Server (NTRS)
Berg, Gene A.; Feng, Gui-Liang; Rao, T. R. N.
1996-01-01
This paper presents a generalized Bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more curves for a large class of plane curves. A new approach to determine a lower bound on the minimum distance (and also the generalized Hamming weights) for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bezout theorem. Examples of more efficient linear codes are constructed using the generalized Bezout theorem and the new approach. For d = 4, the linear codes constructed by the new construction are better than or equal to the known linear codes. For d greater than 5, these new codes are better than the known codes. The Klein code over GF(2(sup 3)) is also constructed.
6. Formulation of Liouville's theorem for grand ensemble molecular simulations
Delle Site, Luigi
2016-02-01
Liouville's theorem in a grand ensemble, that is for situations where a system is in equilibrium with a reservoir of energy and particles, is a subject that, to our knowledge, has not been explicitly treated in literature related to molecular simulation. Instead, Liouville's theorem, a central concept for the correct employment of molecular simulation techniques, is implicitly considered only within the framework of systems where the total number of particles is fixed. However, the pressing demand of applied science in treating open systems leads to the question of the existence and possible exact formulation of Liouville's theorem when the number of particles changes during the dynamical evolution of the system. The intention of this paper is to stimulate a debate about this crucial issue for molecular simulation.
7. Quantum de Finetti theorem in phase-space representation
Leverrier, Anthony; Cerf, Nicolas J.
2009-07-01
The quantum versions of de Finetti’s theorem derived so far express the convergence of n -partite symmetric states, i.e., states that are invariant under permutations of their n parties, toward probabilistic mixtures of independent and identically distributed (IID) states of the form σ⊗n . Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a quantum de Finetti theorem particularly relevant to continuous-variable systems. Specifically, an n -mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge toward a probabilistic mixture of IID Gaussian states (actually, n identical thermal states).
8. Strong Kochen-Specker theorem and incomputability of quantum randomness
Abbott, Alastair A.; Calude, Cristian S.; Conder, Jonathan; Svozil, Karl
2012-12-01
The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is noncontextual and consistent with quantum mechanics. If we require noncontextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefinite. Various technical and subtle aspects relating to this formal proof and its connection to quantum mechanics are discussed. This result is then utilized for the proposal and certification of a dichotomic quantum random number generator operating in a three-dimensional Hilbert space.
9. Model Checking Failed Conjectures in Theorem Proving: A Case Study
NASA Technical Reports Server (NTRS)
Pike, Lee; Miner, Paul; Torres-Pomales, Wilfredo
2004-01-01
Interactive mechanical theorem proving can provide high assurance of correct design, but it can also be a slow iterative process. Much time is spent determining why a proof of a conjecture is not forthcoming. In some cases, the conjecture is false and in others, the attempted proof is insufficient. In this case study, we use the SAL family of model checkers to generate a concrete counterexample to an unproven conjecture specified in the mechanical theorem prover, PVS. The focus of our case study is the ROBUS Interactive Consistency Protocol. We combine the use of a mechanical theorem prover and a model checker to expose a subtle flaw in the protocol that occurs under a particular scenario of faults and processor states. Uncovering the flaw allows us to mend the protocol and complete its general verification in PVS.
10. Heat Capacity and the Equipartition Theorem
ERIC Educational Resources Information Center
Dence, Joseph B.
1972-01-01
Limitations of classical mechanics in understanding molecular properties are discussed. Modifications introduced by quantum mechanics enable the instructor to include and integrate important concepts from thermodynamics, quantum mechanics, spectroscopy, and statistics. (DF)
11. Muonium Spectrum Beyond the Nonrelativistic Limit
SciTech Connect
Weber, Axel
2008-07-02
A generalization of the Gell-Mann-Low theorem is applied to the antimuon-electron system. The bound state spectrum is extracted numerically. As a result, fine and hyperfine structure are reproduced correctly near the nonrelativistic limit (and for arbitrary masses). We compare the spectrum for the relativistic value {alpha} = 0.3 with corresponding calculations in light-front quantization.
12. Reasoning by analogy as an aid to heuristic theorem proving.
NASA Technical Reports Server (NTRS)
Kling, R. E.
1972-01-01
When heuristic problem-solving programs are faced with large data bases that contain numbers of facts far in excess of those needed to solve any particular problem, their performance rapidly deteriorates. In this paper, the correspondence between a new unsolved problem and a previously solved analogous problem is computed and invoked to tailor large data bases to manageable sizes. This paper outlines the design of an algorithm for generating and exploiting analogies between theorems posed to a resolution-logic system. These algorithms are believed to be the first computationally feasible development of reasoning by analogy to be applied to heuristic theorem proving.
13. Nonlinear Dynamic Maximum Power Theorem, with Numerical Method
DTIC Science & Technology
1983-09-01
Desoer , "The Maximum Power Transfer Theorem for n-Ports," IEEE Trans. Circuit Theory , vol. CT-20, no. 3, pp. 328-330, May 1973. [2] J.L.Wyatt, Jr. and L.O...327-330, May 1974. [10] H. Flanders, "On the Maximal Power Transfer Theorem for n-Ports," Int. J. Circuit Theory and Applications, vol. 4, pp. 319-344...conditions in section 3.1), then the (noncausal) matched load has the form shown in Fig. 2. 3.3) Circuit Example Suppose the source takes the specific
14. Quantum Theory of Jaynes' Principle, Bayes' Theorem, and Information
Haken, Hermann
2014-12-01
After a reminder of Jaynes' maximum entropy principle and of my quantum theoretical extension, I consider two coupled quantum systems A,B and formulate a quantum version of Bayes' theorem. The application of Feynman's disentangling theorem allows me to calculate the conditional density matrix ρ (A|B) , if system A is an oscillator (or a set of them), linearly coupled to an arbitrary quantum system B. Expectation values can simply be calculated by means of the normalization factor of ρ (A|B) that is derived.
15. Fluidized Granular Medium as an Instance of the Fluctuation Theorem
Feitosa, Klebert; Menon, Narayanan
2004-04-01
We study the statistics of the power flux into a collection of inelastic beads maintained in a fluidized steady state by external mechanical driving. The power shows large fluctuations, including frequent large negative fluctuations, about its average value. The relative probabilities of positive and negative fluctuations in the power flux are in close accord with the fluctuation theorem of Gallavotti and Cohen, even at time scales shorter than those required by the theorem. We also compare an effective temperature that emerges from this analysis to the kinetic granular temperature.
16. Fluidized granular medium as an instance of the fluctuation theorem.
PubMed
Feitosa, Klebert; Menon, Narayanan
2004-04-23
We study the statistics of the power flux into a collection of inelastic beads maintained in a fluidized steady state by external mechanical driving. The power shows large fluctuations, including frequent large negative fluctuations, about its average value. The relative probabilities of positive and negative fluctuations in the power flux are in close accord with the fluctuation theorem of Gallavotti and Cohen, even at time scales shorter than those required by the theorem. We also compare an effective temperature that emerges from this analysis to the kinetic granular temperature.
17. Fluctuation theorems for total entropy production in generalized Langevin systems
Ghosh, Bappa; Chaudhury, Srabanti
2017-01-01
The validity of the fluctuation theorems for total entropy production of a colloidal particle embedded in a non-Markovian heat bath driven by a time-dependent force in a harmonic potential is probed here. The dynamics of the system is modeled by the generalized Langevin equation with colored noise. The distribution function of the total entropy production is calculated and the detailed fluctuation theorem contains a renormalized temperature term which arises due to the non-Markovian characteristics of the thermal bath.
18. Extensions of the Feynman-Hellman theorem and applications
Singh, S. Brajamani; Singh, C. A.
1989-10-01
Epstein's [Am. J. Phys. 22, 613 (1954)] off-diagonal and higher-order extensions of the Feynman-Hellmann theorem, obtained by using the basic technique of parameter differentiation under the integral sign, are further pursued. Epstein's rederivation of the Rayleigh-Schrödinger perturbation expansion is also extended to include the degenerate case. The same approach is also used to obtain the Lennard-Jones-Brillouin-Wigner perturbation theory. The quantum virial theorem and its off-diagonal generalization is deduced and its application is illustrated by taking the example of the linear harmonic oscillator. The semiclassical expression for the kinetic energy is obtained directly from the quantization condition.
19. Generalization of Carey's equality and a theorem on stationary population.
PubMed
Srinivasa Rao, Arni S R; Carey, James R
2015-09-01
Carey's Equality pertaining to stationary models is well known. In this paper, we have stated and proved a fundamental theorem related to the formation of this Equality. This theorem will provide an in-depth understanding of the role of each captive subject, and their corresponding follow-up duration in a stationary population. We have demonstrated a numerical example of a captive cohort and the survival pattern of medfly populations. These results can be adopted to understand age-structure and aging process in stationary and non-stationary population models.
20. General self-tuning solutions and no-go theorem
SciTech Connect
Förste, Stefan; Kim, Jihn E.; Lee, Hyun Min E-mail: jihnekim@gmail.com
2013-03-01
We consider brane world models with one extra dimension. In the bulk there is in addition to gravity a three form gauge potential or equivalently a scalar (by generalisation of electric magnetic duality). We find classical solutions for which the 4d effective cosmological constant is adjusted by choice of integration constants. No go theorems for such self-tuning mechanism are circumvented by unorthodox Lagrangians for the three form respectively the scalar. It is argued that the corresponding effective 4d theory always includes tachyonic Kaluza-Klein excitations or ghosts. Known no go theorems are extended to a general class of models with unorthodox Lagrangians.
1. Finite de Finetti theorem for infinite-dimensional systems.
PubMed
D'Cruz, Christian; Osborne, Tobias J; Schack, Rüdiger
2007-04-20
We formulate and prove a de Finetti representation theorem for finitely exchangeable states of a quantum system consisting of k infinite-dimensional subsystems. The theorem is valid for states that can be written as the partial trace of a pure state |Psi/Psi| chosen from a family of subsets {Cn} of the full symmetric subspace for n subsystems. We show that such states become arbitrarily close to mixtures of pure power states as n increases. We give a second equivalent characterization of the family {Cn}.
2. Distributed Online Judge System for Interactive Theorem Provers
Mizuno, Takahisa; Nishizaki, Shin-ya
2014-03-01
In this paper, we propose a new software design of an online judge system for interactive theorem proving. The distinctive feature of this architecture is that our online judge system is distributed on the network and especially involves volunteer computing. In volunteers' computers, network bots (software robots) are executed and donate computational resources to the central host of the online judge system. Our proposed design improves fault tolerance and security. We gave an implementation to two different styles of interactive theorem prover, Coq and ACL2, and evaluated our proposed architecture. From the experiment on the implementation, we concluded that our architecture is efficient enough to be used practically.
3. Quantum Locality?
SciTech Connect
Stapp, Henry
2011-11-10
vagaries that he cites do not upset the proof in question. It is show here in detail why the precise statement of this theorem justifies the specified application of CQT. It is also shown, in response to his challenge, why a putative proof of locality that he has proposed is not valid.
4. Test of the steady-state fluctuation theorem in turbulent Rayleigh-B'enard convection
Tong, Penger; Shang, Xiaodong
2005-11-01
Local entropy production rate σ(r,t) in turbulent thermal convection is obtained from simultaneous velocity and temperature measurements in an aspect-ratio-one cell filled with water. The statistical properties of the time-averaged σ(r,t) are analyzed and the results are compared with the predictions of the steady state fluctuation theorem (SSFT) of Gallavotti and Cohen. The experiment reveals that the SSFT can indeed be extended to the local variables, but further development is needed in order to incorporate the common dynamic complexities of far-from-equilibrium systems into the theory. *Work supported by the Research Grants Council of Hong Kong SAR under Grant Nos. HKUST603504 (P.T.) and CUHK403003 (K.Q.X.).
5. Unifying Framework for Relaxations of the Causal Assumptions in Bell's Theorem
Chaves, R.; Kueng, R.; Brask, J. B.; Gross, D.
2015-04-01
Bell's theorem shows that quantum mechanical correlations can violate the constraints that the causal structure of certain experiments impose on any classical explanation. It is thus natural to ask to which degree the causal assumptions—e.g., locality or measurement independence—have to be relaxed in order to allow for a classical description of such experiments. Here we develop a conceptual and computational framework for treating this problem. We employ the language of Bayesian networks to systematically construct alternative causal structures and bound the degree of relaxation using quantitative measures that originate from the mathematical theory of causality. The main technical insight is that the resulting problems can often be expressed as computationally tractable linear programs. We demonstrate the versatility of the framework by applying it to a variety of scenarios, ranging from relaxations of the measurement independence, locality, and bilocality assumptions, to a novel causal interpretation of Clauser-Horne-Shimony-Holt inequality violations.
6. Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations
Belolipetskii, A. A.; Ter-Krikorov, A. M.
2016-11-01
The functional equation f( x,ɛ) = 0 containing a small parameter ɛ and admitting regular and singular degeneracy as ɛ → 0 is considered. By the methods of small parameter, a function x n 0(ɛ) satisfying this equation within a residual error of O(ɛ n+1) is found. A modified Newton's sequence starting from the element x n 0(ɛ) is constructed. The existence of the limit of Newton's sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton's iterative sequence). The deviation of the limit of Newton's sequence from the initial approximation x n 0(ɛ) has the order of O(ɛ n+1), which proves the asymptotic character of the approximation x n 0(ɛ). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.
7. Index theorem and Majorana zero modes along a non-Abelian vortex in a color superconductor
SciTech Connect
Fujiwara, Takanori; Fukui, Takahiro; Nitta, Muneto; Yasui, Shigehiro
2011-10-01
Color superconductivity in high-density QCD exhibits the color-flavor-locked phase. To explore zero modes in the color-flavor-locked phase in the presence of a non-Abelian vortex with an SU(2) symmetry in the vortex core, we apply the index theorem to the Bogoliubov-de Gennes (BdG) Hamiltonian. From the calculation of the topological index, we find that triplet, doublet and singlet sectors of SU(2) have certain number of chiral Majorana zero modes in the limit of vanishing chemical potential. We also solve the BdG equation by the use of the series expansion to show that the number of zero modes and their chirality match the result of the index theorem. From particle-hole symmetry of the BdG Hamiltonian, we conclude that if and only if the index of a given sector is odd, one zero mode survives generically for a finite chemical potential. We argue that this result should hold nonperturbatively even in the high-density limit.
8. Elliptical billiard systems and the full Poncelet's theorem in n dimensions
Chang, Shau-Jin; Crespi, Bruno; Shi, Kang-Jie
1993-06-01
In this work is presented a generalization of Poncelet's theorem to n dimensions which is refered to as the full Poncelet's theorem. The theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon, then there exists an (n-1)-parameter family of polygons having the same property. A physical realization and a projective geometrical proof of this theorem are given. If all the reflecting quadrics coincide, the above theorem reduces to the n-dimensional Poncelet's theorem presented by Chang and Friedberg. The geometrical proof is a finite construction based on a preliminary theorem which extends Hart's lemma. The full Poncelet's theorem may thus be extended to projective geometries over most fields, including discrete ones.
9. No-scalar-hair theorem for spherically symmetric reflecting stars
Hod, Shahar
2016-11-01
It is proved that spherically symmetric compact reflecting objects cannot support static bound-state configurations made of scalar fields whose self-interaction potential V (ψ2) is a monotonically increasing function of its argument. Our theorem rules out, in particular, the existence of massive scalar hair outside the surface of a spherically symmetric compact reflecting star.
10. An Elementary Proof of a Converse Mean-Value Theorem
ERIC Educational Resources Information Center
Almeida, Ricardo
2008-01-01
We present a new converse mean value theorem, with a rather elementary proof. [The work was supported by Centre for Research on Optimization and Control (CEOC) from the "Fundacaopara a Ciencia e a Tecnologia" FCT, co-financed by the European Community Fund FEDER/POCTI.
11. Kochen-Specker Theorem as a Precondition for Quantum Computing
2016-12-01
We study the relation between the Kochen-Specker theorem (the KS theorem) and quantum computing. The KS theorem rules out a realistic theory of the KS type. We consider the realistic theory of the KS type that the results of measurements are either +1 or -1. We discuss an inconsistency between the realistic theory of the KS type and the controllability of quantum computing. We have to give up the controllability if we accept the realistic theory of the KS type. We discuss an inconsistency between the realistic theory of the KS type and the observability of quantum computing. We discuss the inconsistency by using the double-slit experiment as the most basic experiment in quantum mechanics. This experiment can be for an easy detector to a Pauli observable. We cannot accept the realistic theory of the KS type to simulate the double-slit experiment in a significant specific case. The realistic theory of the KS type can not depicture quantum detector. In short, we have to give up both the observability and the controllability if we accept the realistic theory of the KS type. Therefore, the KS theorem is a precondition for quantum computing, i.e., the realistic theory of the KS type should be ruled out.
12. Fermat's Last Theorem for Factional and Irrational Exponents
ERIC Educational Resources Information Center
Morgan, Frank
2010-01-01
Fermat's Last Theorem says that for integers n greater than 2, there are no solutions to x[superscript n] + y[superscript n] = z[superscript n] among positive integers. What about rational exponents? Irrational n? Negative n? See what an undergraduate senior seminar discovered.
13. An Experiment on a Physical Pendulum and Steiner's Theorem
ERIC Educational Resources Information Center
Russeva, G. B.; Tsutsumanova, G. G.; Russev, S. C.
2010-01-01
Introductory physics laboratory curricula usually include experiments on the moment of inertia, the centre of gravity, the harmonic motion of a physical pendulum, and Steiner's theorem. We present a simple experiment using very low cost equipment for investigating these subjects in the general case of an asymmetrical test body. (Contains 3 figures…
14. Boltzmann's "H"-Theorem and the Assumption of Molecular Chaos
ERIC Educational Resources Information Center
Boozer, A. D.
2011-01-01
We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann "H"-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric.…
Marinho, Rubens M., Jr.
2009-09-01
This reply answers the issues raised in the comment on my paper (Marinho Jr 2007 Eur. J. Phys. 28 37-43), obtains the Laplace-Runge-Lenz vector (Goldstein 2002 Classical Mechanics 3rd edn (Reading, MA: Addison-Wesley)) using Noether's theorem and includes a Maple program used to derive the results.
16. Establishing Appropriate Conditions: Students Learning to Apply a Theorem
ERIC Educational Resources Information Center
Scataglini-Belghitar, Giovanna; Mason, John
2012-01-01
During a sequence of tutorials conducted by the first author, it became evident that students were not seeing how to apply the theorem concerning a continuous function on a closed and bounded interval attaining its extreme values in situations in which it is necessary first to construct the closed and bounded interval by reasoning about properties…
17. An Extension of the Mean Value Theorem for Integrals
ERIC Educational Resources Information Center
Khalili, Parviz; Vasiliu, Daniel
2010-01-01
In this note we present an extension of the mean value theorem for integrals. The extension we consider is motivated by an older result (here referred as Corollary 2), which is quite classical for the literature of Mathematical Analysis or Calculus. We also show an interesting application for computing the sum of a harmonic series.
18. Null conformal Killing-Yano tensors and Birkhoff theorem
Ferrando, Joan Josep; Sáez, Juan Antonio
2016-04-01
We study the space-times admitting a null conformal Killing-Yano tensor whose divergence defines a Killing vector. We analyze the similarities and differences with the recently studied non null case (Ferrando and Sáez in Gen Relativ Gravit 47:1911, 2015). The results by Barnes concerning the Birkhoff theorem for the case of null orbits are analyzed and generalized.
19. The Unforgettable Experience of a Workshop on Pythagoras Theorem
ERIC Educational Resources Information Center
2011-01-01
The author conducted a workshop with colleagues in which awareness of Pythagoras' theorem was raised. This workshop was an unforgettable event in the author's life because it was the first time that she had interacted with teachers from a different school system, and it allowed her to develop presentation skills and confidence in her own…
20. Hamiltonian Noether theorem for gauge systems and two time physics
Villanueva, V. M.; Nieto, J. A.; Ruiz, L.; Silvas, J.
2005-08-01
The Noether theorem for Hamiltonian constrained systems is revisited. In particular, our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints. We apply our results to the relativistic point particle, to the Friedberg et al model and, with special emphasis, to two time physics.
1. Geometric Demonstration of the Fundamental Theorems of the Calculus
ERIC Educational Resources Information Center
Sauerheber, Richard D.
2010-01-01
After the monumental discovery of the fundamental theorems of the calculus nearly 350 years ago, it became possible to answer extremely complex questions regarding the natural world. Here, a straightforward yet profound demonstration, employing geometrically symmetric functions, describes the validity of the general power rules for integration and…
2. Weak convergence theorems for a countable family of Lipschitzian mappings
Nilsrakoo, Weerayuth; Saejung, Satit
2009-08-01
This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37-43], Tada-Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359-370], and Plubtieng-Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.
3. A fixed point theorem for certain operator valued maps
NASA Technical Reports Server (NTRS)
Brown, D. R.; Omalley, M. J.
1978-01-01
In this paper, we develop a family of Neuberger-like results to find points z epsilon H satisfying L(z)z = z and P(z) = z. This family includes Neuberger's theorem and has the additional property that most of the sequences q sub n converge to idempotent elements of B sub 1(H).
4. Thermodynamic laws and equipartition theorem in relativistic Brownian motion.
PubMed
Koide, T; Kodama, T
2011-06-01
We extend the stochastic energetics to a relativistic system. The thermodynamic laws and equipartition theorem are discussed for a relativistic Brownian particle and the first and the second law of thermodynamics in this formalism are derived. The relation between the relativistic equipartition relation and the rate of heat transfer is discussed in the relativistic case together with the nature of the noise term.
5. A Computer Algorithm from DeMoivre's Theorem.
ERIC Educational Resources Information Center
Boyd, James N.
1982-01-01
Details are given of a simple computer program written in BASIC which calculates the sine of an angle through an application of DeMoivre's Theorem. The program is included in the material, and the program's success is discussed in terms of why the approximation works. (MP)
6. On Feynman's Triangle Problem and the Routh Theorem
ERIC Educational Resources Information Center
Man, Yiu-Kwong
2009-01-01
In this article, we give a brief history of the Feynman's Triangle problem and describe a simple method to solve a general version of this problem, which is called the Routh Theorem. This method could be found useful to school teachers, instructors or lecturers who are involved in teaching geometry.
7. A shape theorem for Riemannian first-passage percolation
LaGatta, T.; Wehr, J.
2010-05-01
Riemannian first-passage percolation is a continuum model, with a distance function arising from a random Riemannian metric in Rd. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability of 1.
8. Four Proofs of the Converse of the Chinese Remainder Theorem
ERIC Educational Resources Information Center
Dobbs, D. E.
2008-01-01
Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [double-struck z][subscript m] [direct sum] [double-struck…
9. Two Theorems on Dissipative Energy Losses in Capacitor Systems
ERIC Educational Resources Information Center
Newburgh, Ronald
2005-01-01
This article examines energy losses in charge motion in two capacitor systems. In the first charge is transferred from a charged capacitor to an uncharged one through a resistor. In the second a battery charges an originally uncharged capacitor through a resistance. Analysis leads to two surprising general theorems. In the first case the fraction…
10. A Theorem and its Application to Finite Tampers
DOE R&D Accomplishments Database
Feynman, R. P.
1946-08-15
A theorem is derived which is useful in the analysis of neutron problems in which all neutrons have the same velocity. It is applied to determine extrapolated end-points, the asymptotic amplitude from a point source, and the neutron density at the surface of a medium. Formulas fro the effect of finite tampers are derived by its aid, and their accuracy discussed.
11. Fixed point theorems for generalized contractions in ordered metric spaces
2008-05-01
The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, MEC. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505-2517], A. Petrusel, I.A. Rus [A. Petrusel, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411-418] and R.P. Agarwal, M.A. El-Gebeily, D. O'Regan [R.P. Agarwal, M.A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given.
12. Two Familiar Theorems: Modified So Their Proofs Are Comprehensible to First Semester Calculus Students.
ERIC Educational Resources Information Center
Young, Anne Ludington
1996-01-01
Error estimates for tangent line approximations and for numerical integration are found using special cases of the error formulas for Taylor's Theorem and the Trapezoidal Rule, respectively. Proofs of these theorems rely on a modification of Rolle's Theorem. (Author/MKR)
13. A variational theorem for creep with applications to plates and columns
NASA Technical Reports Server (NTRS)
Sanders, J Lyell, Jr; Mccomb, Harvey G , Jr; Schlechte, Floyd R
1958-01-01
A variational theorem is presented for a body undergoing creep. Solutions to problems of the creep behavior of plates, columns, beams, and shells can be obtained by means of the direct methods of the calculus of variations in conjunction with the stated theorem. The application of the theorem is illustrated for plates and columns by the solution of two sample problems.
14. Limitations of a Research, Development and Diffusion (RD and D) Strategy in Diffusion: A Case Study of Nine Local Implementations of a State-Adopted Curriculum.
ERIC Educational Resources Information Center
Schumacher, Sally
This study of the first year of an inter-organizational diffusion effort by a national educational laboratory, a state department of education, and nine local school districts focuses on the implementation phase within the Research, Development and Diffusion (RD and D) strategy of an aesthetic education program which uses the arts as the…
15. On the logical structure of Bell theorems.
Walgate, Jonathan; Broadbent, Anne; Carteret, Hilary; Methot, Andre
2007-03-01
Some specific predictions of quantum mechanics are inconsistent with local realism, a phenomenon known as nonlocality. Despite overwhelming evidence for quantum mechanics, the practical difficulties of detector efficiency and coordinating space-like separated measurements have provided loopholes for a classical worldview. New experiments have been proposed to meet these challenges, based around a new kind of nonlocality proof called an `EPR Bell inequality''. Much investment is now being made to realize these proofs experimentally. We show all these proposals are fundamentally flawed. We focus on a series of designs that have appeared in PRL and PRA for loophole-free Bell experiments. These experiments use hyperentangled two-photon quantum states to generate experimental data supposedly at odds with local realism. We show how to produce identical results using a single coin, and explain where the logical flaw can be found. To understand our nonclassical world, we must understand precisely the experimental evidence for nonlocality. This tempting shortcut is a logical, theoretical and experimental dead end. http://www.arxiv.org/abs/quant-ph/0512201.
16. Serre duality, Abel's theorem, and Jacobi inversion for supercurves over a thick superpoint
Rothstein, Mitchell J.; Rabin, Jeffrey M.
2015-04-01
The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1 | q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincaré duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic groups and class fields, combined with an invariance result for the topology on the dual of the space of répartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.
17. From Multileg Loops to Trees (by-passing Feynman's Tree Theorem)
SciTech Connect
Rodrigo, German; Catani, Stefano; Gleisberg, Tanju; Krauss, Frank; Winter, Jan-Christopher; /Fermilab
2011-10-14
We illustrate a duality relation between one-loop integrals and single-cut phase-space integrals. The duality relation is realised by a modification of the customary +i0 prescription of the Feynman propagators. The new prescription regularizing the propagators, which we write in a Lorentz covariant form, compensates for the absence of multiple-cut contributions that appear in the Feynman Tree Theorem. The duality relation can be extended to generic one-loop quantities, such as Green's functions, in any relativistic, local and unitary field theories. The physics program of LHC requires the evaluation of multi-leg signal and background processes at next-to-leading order (NLO). In the recent years, important efforts have been devoted to the calculation of many 2 {yields} 3 processes and some 2 {yields} 4 processes. We have recently proposed a method to compute multi-leg one-loop cross sections in perturbative field theories. The method uses combined analytical and numerical techniques. The starting point of the method is a duality relation between one-loop integrals and phase-space integrals. In this respect, the duality relation has analogies with the Feynman's Tree Theorem (FTT). The key difference with the FTT is that the duality relation involves only single cuts of the one-loop Feynman diagrams. In this talk, we illustrate the duality relation, and discuss its correspondence, similarities, and differences with the FTT.
18. On the notion of free will in the Free Will Theorem
Landsman, Klaas
2017-02-01
The (Strong) Free Will Theorem (FWT) of Conway and Kochen (2009) on the one hand follows from uncontroversial parts of modern physics and elementary mathematical and logical reasoning, but on the other hand seems predicated on an undefined notion of free will (allowing physicists to ;freely choose; the settings of their experiments). This makes the theorem philosophically vulnerable, especially if it is construed as a proof of indeterminism or even of libertarian free will (as Conway & Kochen suggest). However, Cator and Landsman (Foundations of Physics 44, 781-791, 2014) previously gave a reformulation of the FWT that does not presuppose indeterminism, but rather assumes a mathematically specific form of such ;free choices; even in a deterministic world (based on a non-probabilistic independence assumption). In the present paper, which is a philosophical sequel to the one just mentioned, I argue that the concept of free will used in the latter version of the FWT is essentially the one proposed by Lewis (1981), also known as 'local miracle compatibilism' (of which I give a mathematical interpretation that might be of some independent interest also beyond its application to the FWT). As such, the (reformulated) FWT in my view challenges compatibilist free will à la Lewis (albeit in a contrived way via bipartite EPR-type experiments), falling short of supporting libertarian free will.
19. Quantization of conductance minimum and index theorem
Ikegaya, Satoshi; Suzuki, Shu-Ichiro; Tanaka, Yukio; Asano, Yasuhiro
2016-08-01
We discuss the minimum value of the zero-bias differential conductance Gmin in a junction consisting of a normal metal and a nodal superconductor preserving time-reversal symmetry. Using the quasiclassical Green function method, we show that Gmin is quantized at (4 e2/h ) NZES in the limit of strong impurity scatterings in the normal metal at the zero temperature. The integer NZES represents the number of perfect transmission channels through the junction. An analysis of the chiral symmetry of the Hamiltonian indicates that NZES corresponds to the Atiyah-Singer index in mathematics.
20. The physical origins of the uncertainty theorem | 104,842 | 493,600 | {"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} | 2.828125 | 3 | CC-MAIN-2017-39 | longest | en | 0.900533 |
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b) What is LIBOR and what are the advantages and limitations of the LIBOR?
c) What drives interbank rates? In answering the questions for this project, you will be required to use a variety of references and the international finance concepts learnt in this unit.
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# Symbolic Logic Outline - PowerPoint PPT Presentation
Symbolic Logic Outline. Another Boolean Operation Joining the Premises Together More on OR What If a Premise is False? #1 What If a Premise is False? #2 What If Both Premises are False? The OR Operation Truth Table for OR Operation Boolean OR is Inclusive What is Exclusive OR?
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Symbolic Logic Outline
• Another Boolean Operation
• Joining the Premises Together
• More on OR
• What If a Premise is False? #1
• What If a Premise is False? #2
• What If Both Premises are False?
• The OR Operation
• Truth Table for OR Operation
• Boolean OR is Inclusive
• What is Exclusive OR?
• The NOT Operation
• Truth Table for NOT Operation
• Symbolic Logic Outline
• What is Logic?
• How Do We Use Logic?
• Logical Inferences #1
• Logical Inferences #2
• Symbolic Logic #1
• Symbolic Logic #2
• What If a Premise is False? #1
• What If a Premise is False? #2
• What If a Premise is False? #3
• What If Both Premises are False?
• Boolean Values #1
• Boolean Values #2
• Boolean Values #2
• The AND Operation
• Truth Table for AND Operation
Symbolic Logic Lesson
CS1313 Spring 2009
What is Logic?
“Logic is the study of the methods and principles used to distinguish good (correct) from bad (incorrect) reasoning.”
Irving M. Copi, Introduction to Logic, 6th ed., Macmillan Publishing Co., New York, 1982, p. 3.
Symbolic Logic Lesson
CS1313 Spring 2009
How Do We Use Logic?
Every day, we put logic to work in making decisions about our lives, such as:
• how to dress (for example, Will it be hot or cold?);
• what to eat and drink (for example, Will we need caffeine to stay up studying?);
• where to go (for example, Is it a Monday, in which case I need to go to CS1313?).
Symbolic Logic Lesson
CS1313 Spring 2009
Logical Inferences #1
We make logical inferencesto reason about the decisions we need to make:
• It’s cold this morning, so I need to wear a sweatshirt and jeans, not just a t-shirt and shorts.
• I’ve got a big exam tomorrow that I haven’t studied for, so I’d better drink a couple pots of coffee.
• It’s Monday, so I’d better be on time for CS1313 or I’ll be late for the quiz.
Symbolic Logic Lesson
CS1313 Spring 2009
Logical Inferences #2
We can even construct more complicated chains of logic:
• I have a programming project due soon.
• I have been putting off working on it.
• Therefore, I must start working on it today.
Symbolic Logic Lesson
CS1313 Spring 2009
Symbolic Logic #1
In logic as in many topics, it sometimes can be easier to manage the various pieces of a task if we represent them symbolically.
• Let D be the statement “I have a programming project due soon.”
• Let L be the statement “I have been putting off working on my programming project.”
• Let W be the statement “I must start working on my programming project today.”
We can then represent the chain of logic like so:
D and L => W
Symbolic Logic Lesson
CS1313 Spring 2009
Symbolic Logic #2
D and L => W
This can be read in two ways:
• “D and L impliesW.”
• “IfD is true andL is true, thenW is true.”
Symbolic Logic Lesson
CS1313 Spring 2009
What If a Premise is False? #1
D and L => W
What if L is nottrue?
What if I’ve already started working on my programming project?
In that case, the statement
“I have been putting off working on my programming project”
is not true; it is false.
So then the statement
D and L
is also false. Why?
Symbolic Logic Lesson
CS1313 Spring 2009
What If a Premise is False? #2
D and L => W
If the statement L is false, then why is the statement “D and L” also false?
Well, in this example, L is the statement “I have been putting off working on my programming project.” If this statement is false, then the following statement is true: “I haven’t been putting off working on my programming project.”
In that case, the statement W — “I must start working on my programming project today” — cannot be true, because I’ve alreadystarted working on it, so I can’t startworking on it now.
Symbolic Logic Lesson
CS1313 Spring 2009
What If a Premise is False? #3
D and L => W
What if D is false?
What if I don’thave a programming project due soon?
Well, statement D is “I have a programming project due soon.” So if I don’t have a programming project due soon, then statement D is false.
In that case, statement W — “I must start working on my programming project today”—is also false, because I don’t have a programming project due soon, so I don’t need to start working on it today.
Symbolic Logic Lesson
CS1313 Spring 2009
What If Both Premises are False?
D and L => W
What if both D and L are false?
In that case, I don’t have a programming project due soon, and I’ve already gotten started on the one that’s due in, say, a month, so I definitely don’t need to start working on it today.
Symbolic Logic Lesson
CS1313 Spring 2009
Boolean Values #1
A Booleanvalue is a value that is either true or false.
The name Boolean comes from George Boole, one of the 19th century mathematicians most responsible for formalizing the rules of symbolic logic.
So, in our example, statements D, L and W all are Boolean statements, because each of them is either true or false — that is, the valueof each statement is either true or false.
http://thefilter.blogs.com/photos/uncategorized/boole.jpg
Symbolic Logic Lesson
CS1313 Spring 2009
Boolean Values #2
D and L => W
We can express this idea symbolically; for example:
D = true
L = false
W = false
Note that
L = false
is read as “The statement L is false.”
Symbolic Logic Lesson
CS1313 Spring 2009
Boolean Values #2
L = false
is read as “The statement L is false.”
In our programming project example, this means that the statement “I have been putting off working on my programming project” is false, which means that the statement “It is not the case that I have been putting off working on my programming project” is true, which in turn means that the statement “I haven’tbeen putting off working on my programming project” is true.
So, in this case, “L = false” means that I already have started working on my programming project.
Symbolic Logic Lesson
CS1313 Spring 2009
The AND Operation
From this example, we can draw some general conclusions about the statement
“S1 and S2”
for anystatement S1 and anystatement S2:
• If S1 is trueand S2 is true, then “S1 and S2” is true.
• If S1 is false and S2 is true, then “S1 and S2” is false.
• If S1 is trueand S2 is false, then “S1 and S2” is false.
• If S1 is false and S2 is false, then “S1 and S2” is false.
Symbolic Logic Lesson
CS1313 Spring 2009
Truth Table for AND Operation
“S1 and S2”
We can represent this statement with a truth table:
To read this, put your left index finger on the value of statement S1 (that is, either true or false) at the left side of a row, and put your right index finger on the value of statement S2 at the top of a column. Slide your left index finger rightward, and slide your right index finger downward, until they meet. The value under the two fingers is the value of the statement “S1 and S2.”
Symbolic Logic Lesson
CS1313 Spring 2009
Another Boolean Operation
Suppose you want to know whether today is a good day to wear a jacket. You might want to come up with rules to help you make this decision:
• If it’s raining in the morning, then I’ll wear a jacket today.
• If it’s cold in the morning, then I’ll wear a jacket today.
So, for example, if you wake up one morning and it’s cold, then you wear a jacket that day.
Likewise, if you wake up one morning and it’s raining, then you wear a jacket that day.
Symbolic Logic Lesson
CS1313 Spring 2009
Joining the Premises Together
We can construct a generalrule by joining these two rules together:
If it’s raining in the morning
OR
it’s cold in the morning,
then I’ll wear a jacket today.
Symbolic Logic Lesson
CS1313 Spring 2009
More on OR
We can apply symbolic logic to this set of statements, like so:
• Let R be the statement “It’s raining in the morning.”
• Let C be the statement “It’s cold in the morning.”
• Let J be the statement “I’ll wear a jacket today.”
We can then represent the chain of logic like so:
R orC =>J
This can be read in two ways:
• “R or C implies J.”
• “If R is true or C is true, then J is true.”
Symbolic Logic Lesson
CS1313 Spring 2009
What If a Premise is False? #1
What if C is nottrue? For example, what if it’s hot in the morning?
In that case, the statement “It’s cold in the morning” is not true; it is false.
So then what about the statement “R or C”?
Well, even if it’s hot in the morning, if it’s raining you want your jacket anyway.
In other words, if R is true, then even though
C is false, still “R or C” is true.
Symbolic Logic Lesson
CS1313 Spring 2009
What If a Premise is False? #2
Suppose that it’s not raining in the morning, but it is cold.
Then the statement “It’s raining in the morning” is false, and the statement “It’s cold in the morning” is true – and so the statement “I’ll wear a jacket today” is true.
In other words, if R is false and C is true, then
“R or C” is also true.
Symbolic Logic Lesson
CS1313 Spring 2009
What If Both Premises are False?
What if both R and C are false?
In that case, it’s neither raining nor cold in the morning, so I won’t wear my jacket.
In other words, if R is false and C is false,
then “R or C” is false.
Symbolic Logic Lesson
CS1313 Spring 2009
The OR Operation
From this example, we can draw some general conclusions about the statement
“S1 or S2”
for anystatement S1 and anystatement S2:
• If S1 is trueand S2 is true, then “S1 or S2” is true.
• If S1 is false and S2 is true, then “S1 or S2” is true.
• If S1 is trueand S2 is false, then “S1 or S2” is true.
• If S1 is false and S2 is false, then “S1 or S2” is false.
Symbolic Logic Lesson
CS1313 Spring 2009
Truth Table for OR Operation
“S1 or S2”
We can represent this statement with a truth table:
To read this, put your left index finger on the value of statement S1 (that is, either true or false) at the left side of a row, and put your right index finger on the value of statement S2 at the top of a column. Slide your left index finger rightward, and slide your right index finger downward, until they meet. The value under the two fingers is the value of the statement “S1 or S2.”
Symbolic Logic Lesson
CS1313 Spring 2009
Boolean OR is Inclusive
In symbolic logic, the Boolean operation OR is inclusive, meaning that it can be the case that both statements are true.
In the jacket example, if it’s raining and it’s cold, then you’ll take your jacket.
So Boolean OR is equivalent to “and/or” in normal colloquial speaking.
Symbolic Logic Lesson
CS1313 Spring 2009
What is Exclusive OR?
We know that the Boolean OR operation is inclusive.
But, there’s also such a thing as exclusive OR, denoted XOR.
XOR is like OR, except that if both statements are true, then the result is false.
We WON’T be worrying about XOR in this course.
Symbolic Logic Lesson
CS1313 Spring 2009
The NOT Operation
Boolean logic has another very important operation: NOT, which changes a true value to false and a false value to true.
In real life, you’ve probably said something like this:
“I care what you think – NOT!”
Notice that the NOT exactly negates the meaning of the sentence: the sentence means “I don’t care what you think.”
From this example, we can draw some conclusions about the statement “not S,” for any statement S:
• If S is true, then “not S” is false.
• If S is false, then “not S” is true.
Symbolic Logic Lesson
CS1313 Spring 2009
Truth Table for NOT Operation
“NOTS”
We can represent this statement with a truth table:
Symbolic Logic Lesson
CS1313 Spring 2009 | 3,219 | 12,425 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.25 | 4 | CC-MAIN-2018-43 | latest | en | 0.808872 |
https://fresherbell.com/quizdiscuss/python/what-will-be-the-output-of-the-following-python-co903 | 1,702,205,773,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679101779.95/warc/CC-MAIN-20231210092457-20231210122457-00644.warc.gz | 301,835,315 | 9,263 | # Quiz Discussion
What will be the output of the following Python code snippet if x=1?
x<<2
Course Name: Python
• 1]
4
• 2]
2
• 3]
1
• 4]
8
##### Solution
No Solution Present Yet
#### Top 5 Similar Quiz - Based On AI&ML
Quiz Recommendation System API Link - https://fresherbell-quiz-api.herokuapp.com/fresherbell_quiz_api
# Quiz
1
Discuss
Which among the following list of operators has the highest precedence?
+, -, **, %, /, <<, >>, |
• 1]
%
• 2]
|
• 3]
**
• 4]
<<, >>
##### Solution
2
Discuss
What will be the output of the following Python code snippet?
bool(‘False’)
bool()
• 1]
True
True
• 2]
False
True
• 3]
True
False
• 4]
False
False
##### Solution
3
Discuss
To find the decimal value of 1111, that is 15, we can use the function:
• 1]
int(‘1111’,2)
• 2]
int(1111,2)
• 3]
int(‘1111’,10)
• 4]
int(1111,10)
##### Solution
4
Discuss
What will be the output of the following Python code?
if (9 < 0) and (0 < -9):
print("hello")
elif (9 > 0) or False:
print("good")
else:
print("bad")
• 1]
• 2]
good
• 3]
hello
• 4]
error
##### Solution
5
Discuss
What will be the output of the following Python expression?
bin(29)
• 1]
‘0b11011’
• 2]
‘0b11111’
• 3]
‘0b11101’
• 4]
‘0b10111’
##### Solution
6
Discuss
What will be the output of the following Python code snippet?
not(3>4)
not(1&1)
• 1]
True
True
• 2]
True
False
• 3]
False
True
• 4]
False
False
##### Solution
7
Discuss
What will be the output of the following Python expression?
4^12
• 1]
2
• 2]
4
• 3]
8
• 4]
12
##### Solution
8
Discuss
What is the two’s complement of -44?
• 1]
10110011
• 2]
11101011
• 3]
11010100
• 4]
1011011
##### Solution
9
Discuss
What will be the output of the following Python code if a=10 and b =20?
a=10
b=20
a=a^b
b=a^b
a=a^b
print(a,b)
• 1]
20 20
• 2]
20 10
• 3]
10 10
• 4]
10 20
##### Solution
10
Discuss
Which of the following expressions results in an error?
• 1]
int(’10.8’)
• 2]
float(’10.8’)
• 3]
int(‘10’)
• 4]
float(‘10’)
# Quiz | 730 | 2,032 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2023-50 | longest | en | 0.550276 |
http://mathhelpforum.com/algebra/5491-i-need-more-help-print.html | 1,527,472,994,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794870604.65/warc/CC-MAIN-20180528004814-20180528024814-00441.warc.gz | 187,648,121 | 2,667 | # I need more help!!!
• Sep 13th 2006, 06:19 PM
Suwanee
I need more help!!!
This is the problem.
A number is 42 less than its square. Find all such numbers.
Can you please tell me how to do this? what steps to follow. Thanks.
• Sep 13th 2006, 06:23 PM
ThePerfectHacker
Quote:
Originally Posted by Suwanee
This is the problem.
A number is 42 less than its square. Find all such numbers.
Can you please tell me how to do this? what steps to follow. Thanks.
x=x^2-42
Thus,
x^2-x-42=0
Thus,
(x-7)(x+6)=0
• Sep 13th 2006, 06:46 PM
Suwanee
Thanks!!!
thank you.
• Sep 14th 2006, 04:21 AM
topsquark
Quote:
Originally Posted by Suwanee
thank you.
Keep this up and you'll have more posts than Quick before too long. :)
-Dan | 237 | 721 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2018-22 | latest | en | 0.942873 |
https://community.smartsheet.com/discussion/77391/changing-status-symbols-if-a-task-is-not-completed-before-due-date | 1,723,550,299,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641076695.81/warc/CC-MAIN-20240813110333-20240813140333-00849.warc.gz | 140,407,021 | 107,414 | # Changing status symbols if a task is not completed before due date
Options
Hello,
I recently found someone with a similar issue after doing a search in the Community and hoped it would help me but I haven't been able to succeed even though it got me close.
I need to do the following:
• If project has not started = "Grey"
• If project is in progress (percentage is increasing 1%-99% to show in progress) = Green
• If project reaches 100% at or before end date = "Green"
• If project is at 75% or below and we're 2 days away from the end date, project is not complete = "Yellow"
• If project/task is at end date and the percentage complete is not at 100% = "Red" (I guess based on the bullet above, 76% and up = "Red")
Like the person who had the same issue above, I have "Start Date", "End Date", "% Complete" columns.
I hope this is clear. Let me know if you have any questions.
Thanks,
Judy
Tags:
• ✭✭✭✭✭✭
Options
Try something like this...
=IF([% Complete]@row = 1, "Green", IF([End Date]@row < TODAY(), "Red", IF(AND([% Complete]@row<= 0.75, [End Date]@row <= TODAY(2)), "Yellow", IF([% Complete]@row = 0, "Grey", "Green"))))
• ✭✭✭✭✭✭
Options
Try something like this...
=IF([% Complete]@row = 1, "Green", IF([End Date]@row < TODAY(), "Red", IF(AND([% Complete]@row<= 0.75, [End Date]@row <= TODAY(2)), "Yellow", IF([% Complete]@row = 0, "Grey", "Green"))))
• edited 03/17/21
Options
It appears to be working for all scenarios but "Grey". When I change a % Complete to 0%, there is no status symbol, instead it's blank. When I was playing around with your original formula from the last forum, I had: =IF(AND([% Complete]@row = 0, [Start Date]@row <= TODAY()), "Gray"... and that worked but not sure how to apply it here.
• Options
Hi Paul,
I found the error! It was just the spelling of "Grey", it needed to be "Gray". Thanks again for all your help.
• ✭✭✭✭✭✭
Options
Sorry about that. I always mix that one up even in my own sheets. You'd think that I would remember it by now.
Glad you were able to figure it out and get it working.
• Options
I do the same and that's how I was able to figure it out hahaha. Thanks again and have a great weekend!
## Help Article Resources
Want to practice working with formulas directly in Smartsheet?
Check out the Formula Handbook template! | 663 | 2,314 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2024-33 | latest | en | 0.939009 |
https://www.riddlesforkids.net/an-apprentice-baker-made-40-liters-of-60-sugar-syrup-instead-of-40-liters-of-45-sugar-syrup/ | 1,628,118,603,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046155188.79/warc/CC-MAIN-20210804205700-20210804235700-00500.warc.gz | 998,964,194 | 24,623 | # An apprentice baker made 40 liters of 60% sugar syrup instead of 40 liters of 45% sugar syrup.
An apprentice baker made 40 liters of 60% sugar syrup instead of 40 liters of 45% sugar syrup.
How much syrup should she drain and replace with distilled water so that the resulting solution is 40 liters of 45% syrup?
Solution:
Firstly, lets find the only syrup. 40 liters of 60% sugar syrup == 26 liters syrup and 16 litres water.
For the 45% syrup, we must add some woter. We have 26 liters syrup.
If we have a 45% sugar syrup, we need 31.7 liters water for 26 liters syrup.
At the begining, we had 16 liters water. So 31.7 – 16 = 15.6 liters added water.
## One Reply to “An apprentice baker made 40 liters of 60% sugar syrup instead of 40 liters of 45% sugar syrup.”
1. Anonymous says:
How does one obtain 26 liters of syrup from 40 gallons of 60% concentration? | 237 | 873 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2021-31 | latest | en | 0.91828 |
https://discusstest.codechef.com/t/hackerrank-ad-infinitum-13-math-programming-contest/11990 | 1,718,440,279,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861584.65/warc/CC-MAIN-20240615062230-20240615092230-00676.warc.gz | 179,295,608 | 6,817 | # HackerRank Ad Infinitum 13 - Math Programming Contest
Probleam Here
I was trying to solve this problem and I am not able to pass the test cases please help me.
My approach for this problem:
``````1.Number N get the frequency of all prime factors [a:i, b:j, c:k…..] where a,b,c.. is prime factor of N and i,j,k.. are frequencies of a,b,c.. respectively
2.Now sum of all factor of N will be [a^0+a^1+…a^i]*[b^0+b^1…a^j]*[c^0+c^1+..c^k]
To solve this problem I took two array nFact[] and nFactP[]
nFact[] Holds frequencies of each prime factor of any number on its index(like count sort)
nFactP[] Holds the sum of powers of any number on its index like for example 3 is index and nFact[A] holds 2 then nFactP[A] will hold [3^0+3^1+3^2]
Now when a number X is multiplied to N then do prime factorization of X
if F is a factor of X then update these:
1.ans=ans/nFactP[F] (clear the previous value of nFactP[F] from the answer)
2.nFact[F]=nFact[F]+1 (Add 1 to frequency of F)
3.nFactP[F]=nFactP[F]+(F^nFact[F]) (Add one more power of F (Highest Power))
4.asn=ans*nFactP[F]
Print ans for given X
``````
Is there anything wrong with my logic or I implemented it wrong ?
Brute Force Here
Solution Here
I found something in your implementation (might be considered wrong in point 1) in your algo too, but it’s not obvious).
You must not use integer division for for dividing in modular arithmetic. The correct implementation of “division” in this case is multiplication with the modular inverse:
x/y \operatorname{mod} p := x* y^{-1} \operatorname{mod} p
which can be computed either with the extended euclidean algorithm or by using the identity
y^{-1}=y^{p-2} \operatorname{mod} p
There are a few things that seem wrong to me,
1. (a/b)%MOD != (a%MOD)/(b%MOD) // take the inverse module here instead.
2. Also take care of negative values while doing a modulo , say // if(nFactP[i]<0)nFactP[i]+=MOD;//same for all
3. Also assign all your arrays as long long int.
Rest the logic seems fine ,but why add 1 to nFact[F] and keep updating the ans?? If F was factor of previous n just take the module inverse of the ans due to previous nFact[F] and after calculating power of F in x, Do it once !
OK admit that is a bug but I also try to do it in python without any modular arithmetic and I was printing only (ans mod p) and guess what that was also wrong answer even for small integers.
I don’t get what you are trying to say in last two lines
I was telling you an optimisation as you were using power function at each occurrence of a factor F of x, so an extra O(log(n)) at each step. But never mind, i ran your code, No issue of time limit. | 732 | 2,658 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2024-26 | latest | en | 0.827772 |
http://blog.aarp.org/2013/02/12/aarp-experience-corps-mentoring-volunteering-academic-community/ | 1,481,325,375,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542851.96/warc/CC-MAIN-20161202170902-00331-ip-10-31-129-80.ec2.internal.warc.gz | 31,230,574 | 32,068 | # Celebrating 100 Days of School on the Wall
Today my class of Experience Corps students celebrated their 100th day of school! When I got to school the teacher had placed “100th day of school” signs all over the walls. To celebrate, the children made crowns with a “100” featured prominently on the front, and decorated their headband with 100 hearts, numbers, triangles, dots, letters-whatever they were moved to create. And even if they didn’t get 100 “somethings” down on the paper, when the adults stapled the bands to fit the kids’ heads, they all looked stunning-especially as a group.
100 days of Learning!
The class then read The 100th Day of School as a shared reading. That means the teacher reads aloud and the children follow along, pointing to the sequence of words as they are spoken. I sat with my kids and pointed to the words while they took turns pointing with me. The exercise builds their recognition of “words” in sentence form, even if the word is not in their vocabulary. So, while they may not be capable of recognizing the word “because,” they are learning that it is the 4th word in the sentence and seeing it in a sentence frame.
After the shared reading, the teacher gathered all the boys and girls back together to work on a numbers exercise. All year long they have been learning to count to 100, with a particular emphasis on counting by 10’s as a guide.
Here’s what happened: She placed 100 chocolate kisses throughout the classroom-in some easy and some not so easy places to find-and then the kids went on a hunt. Each kiss had a number written on the bottom, so each child took their random “finds” back to their seats. Then they placed each kiss with its number on a corresponding sheet labeled 1-100. As the teacher called out the numbers, each kid brought up their kiss until there were 100 kisses lined up neatly in rows on her master sheet.
The morning was full of fun and learning, and that’s the kind of engagement kids need on celebratory occasions like this. It’s the oldest formula in the world. Horace, Aristotle and others said it a long, long time ago: the best way to teach is to both “delight and instruct.” | 496 | 2,176 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2016-50 | longest | en | 0.966262 |
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Beginner
## Calculating Switch Power Requirements for VoIP
I am planning for a VoIP install and I need to make sure we have enough power on the 3850 switch to support the phones. The phones consume 13 watts max, switch is 48 ports, there will be a phone plugged into each port, available watts per switch module in 5 switch stack is 420 watts, and each switch has one 715 watt power supply. How do I calculate how much power I will need to support all of the phones? Is it as simple as multiplying the phone wattage by number of ports and comparing to available watts for the switch module?
1 ACCEPTED SOLUTION
Accepted Solutions
Hall of Fame Expert
## Re: Calculating Switch Power Requirements for VoIP
Hello Mario3,
you would need at least 13W *48 for the phones and you need also to provide power to make the switch to stay alive and function correctly.
13W*48 = 620 W you would need this power avaiable for POE
if you power supply is only 715W I am afraid you will not be able to put a phone on each port.
I think you would need a 900W power supply as minimum.
To support also more demanding POE devices 1100W power supplies are used.
see the datasheet
https://www.cisco.com/c/en/us/products/collateral/switches/catalyst-3850-series-switches/datasheet_c78-720918.html
for 48 ports POE+ or UPOE you need a 1100 W PS with 800 W available for POE.
Also what happens in your stack if one power supply fails ?
you should deploy redundant power supplies on each switch
Dual redundant modular power supplies
The Cisco Catalyst 3850 Series Switches support dual redundant power supplies.[6] The switch ships with one power supply by default, and the second power supply can be purchased at the time of ordering the switch or at a later time. If only one power supply is installed, it should always be in power supply bay 1. The switch also ships with three field-replaceable fans
or you should use 1100W PS and stack power cables
>>>Cisco StackPower® technology provides power stacking among stack members for power redundancy
Hope to help
Giuseppe
2 REPLIES 2
Hall of Fame Expert
## Re: Calculating Switch Power Requirements for VoIP
Hello Mario3,
you would need at least 13W *48 for the phones and you need also to provide power to make the switch to stay alive and function correctly.
13W*48 = 620 W you would need this power avaiable for POE
if you power supply is only 715W I am afraid you will not be able to put a phone on each port.
I think you would need a 900W power supply as minimum.
To support also more demanding POE devices 1100W power supplies are used.
see the datasheet
https://www.cisco.com/c/en/us/products/collateral/switches/catalyst-3850-series-switches/datasheet_c78-720918.html
for 48 ports POE+ or UPOE you need a 1100 W PS with 800 W available for POE.
Also what happens in your stack if one power supply fails ?
you should deploy redundant power supplies on each switch
Dual redundant modular power supplies
The Cisco Catalyst 3850 Series Switches support dual redundant power supplies.[6] The switch ships with one power supply by default, and the second power supply can be purchased at the time of ordering the switch or at a later time. If only one power supply is installed, it should always be in power supply bay 1. The switch also ships with three field-replaceable fans
or you should use 1100W PS and stack power cables
>>>Cisco StackPower® technology provides power stacking among stack members for power redundancy
Hope to help
Giuseppe
Beginner
## Re: Calculating Switch Power Requirements for VoIP
Giuseppe,
Thanks! This is what I was trying to confirm. | 887 | 3,736 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2019-43 | latest | en | 0.903441 |
https://www.convert-measurement-units.com/convert+fC+m+to+yC+m.php | 1,596,917,629,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738351.71/warc/CC-MAIN-20200808194923-20200808224923-00129.warc.gz | 645,076,797 | 15,422 | Convert fC·m to yC·m (Electric dipole moment)
## fC·m into yC·m
Measurement Categorie:
Original value: Original unit: aC·mC·mDebye [D]fC·mFr·cmKilodebye [kD]mC·mMegadebye [MD]nC·mpC·myC·mzC·mµC·m Target unit: aC·mC·mDebye [D]fC·mFr·cmKilodebye [kD]mC·mMegadebye [MD]nC·mpC·myC·mzC·mµC·m numbers in scientific notation
https://www.convert-measurement-units.com/convert+fC+m+to+yC+m.php
# Convert fC·m to yC·m:
1. Choose the right category from the selection list, in this case 'Electric dipole moment'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'fC·m'.
4. Finally choose the unit you want the value to be converted to, in this case 'yC·m'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '381 fC·m'. In so doing, either the full name of the unit or its abbreviation can be used. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Electric dipole moment'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '92 fC·m to yC·m' or '91 fC·m into yC·m' or '1 fC·m -> yC·m' or '33 fC·m = yC·m'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(76 * 19) fC·m'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '381 fC·m + 1143 yC·m' or '78mm x 18cm x 63dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 8.731 864 118 070 9×1029. For this form of presentation, the number will be segmented into an exponent, here 29, and the actual number, here 8.731 864 118 070 9. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 8.731 864 118 070 9E+29. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 873 186 411 807 090 000 000 000 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.
## How many yC·m make 1 fC·m?
1 fC·m = 1 000 000 000 yC·m - Measurement calculator that can be used to convert fC·m to yC·m, among others. | 1,008 | 3,776 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2020-34 | latest | en | 0.831167 |
http://lbartman.com/worksheet/fraction-worksheets-grade-4.php | 1,553,135,208,000,000,000 | text/html | crawl-data/CC-MAIN-2019-13/segments/1552912202476.48/warc/CC-MAIN-20190321010720-20190321032720-00124.warc.gz | 120,997,446 | 12,132 | ## lbartman.com - the pro math teacher
• Subtraction
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a - b - c
a x b x c
a : b : c
Public on 01 Oct, 2016 by Cyun Lee
### equivalent fractions worksheet education
Name : __________________
Seat Num. : __________________
Date : __________________
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84 + 12 = ...
45 + 81 = ...
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17 + 91 = ...
74 + 88 = ...
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67 + 84 = ...
58 + 21 = ...
17 + 29 = ...
79 + 39 = ...
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36 + 91 = ...
show printable version !!!hide the show
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integer multiplication and division worksheet | 1,366 | 3,725 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2019-13 | longest | en | 0.175152 |
http://physicstasks.eu/1292/small-cork-boat | 1,719,237,400,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198865383.8/warc/CC-MAIN-20240624115542-20240624145542-00638.warc.gz | 25,247,378 | 8,873 | ## Small cork boat
A 50 g weighing cork boat floats on the water surface. The boat has a shape as shown in the picture. A drop of detergent is added to the spot A. The surface tension of the mixture of water and detergent has a quarter value compared to pure water.
a) Explain what happens to the boat and why.
b) What is the magnitude of the force affecting the boat and what acceleration would the boat move with, if we neglected the friction force of the water?
Consider the following dimensions of the boat:
a = 20 cm, b = 5 cm, α = 90° and β = 60°.
• #### Hint
The water acts on boat edges with surface force due to the surface tension. This force points horizontally and perpendicularly to the boat. Its magnitude is proportional to the surface tension and the length of the boat edge.
Draw the forces acting on each of the boat edges and think over where their resultant points.
• #### Notation
a = 20 cm, b = 5 cm, α = 90°, β = 60° boat dimensions m = 50 g = 0.05 kg boat weight σm = σv relation between surface tensions of water with the detergent and pure water From tables: σv = 73 mN m-1 pure water surface tension
• #### Analysis and solution of part a)
Due to the surface tension, the surface of liquid acts at its border on a container or an object standing out of the liquid by the surface force. This force is always perpendicular to the edge of the surface. We can also imagine that the liquid tries to “reduce its surface”. The acting force is proportional to the surface tension (constant characterizing the fluid) and the length at which it acts.
In the vertical direction, the boat is affected by gravitational force and buoyancy force. Because the boat floats on the surface, these forces equal in magnitude and point to the opposite directions. The water surface acts on the boat with surface forces in the horizontal direction (see the figure below).
If the boat is surrounded by water on all sides, the resultant of all these forces is zero and the boat does not move. This will be discussed in detail in the numerical solution section.
If we place the drop of the detergent behind the boat, the local surface tension is reduced and thus also the forces acting on the rear side of the boat. Therefore the forces acting on the front side of the boat prevail and cause a motion in the forward direction. The surface forces acting on the boat are summarized in the following figure and the resultant force is found so that we can calculate the acceleration caused by this force according to the boat weight.
• #### Solution of part b)
It is evident from the figures shown in the analysis section that the surface forces acting on the “inclined boat parts” are crucial so that we can calculate the resultant surface force acting on the boat. The forces are summarized into the detailed figure below.
The resultant of the “blue” forces $$\vec{F}_1$$ and $$\vec{F}_2$$ affecting the front side of the boat is marked $$\vec{F}_{12}$$. Analogously, the resultant of the “red” forces $$\vec{F}_3$$ and $$\vec{F}_4$$ affecting the back side of the boat will be referred as $$\vec{F}_{34}$$.
The task is to calculate the resultant of these forces, i.e. the force F:
$\vec{F}\,=\,\vec{F}_1\,+\vec{F}_2\,+\,+\vec{F}_3\,+\,\vec{F}_4\,=\,\vec{F}_{12}\,+\,\vec{F}_{34}$
The forces $$\vec{F}_1$$ and $$\vec{F}_2$$ act on the side of the boat with the following length: $\frac{\frac{b}{2}}{\cos \frac{\beta}{2}}\,.$ Therefore for their sizes:
$F_1\,=\,F_2\,=\, \sigma_v \, \frac{b}{ 2 \cos \frac{\beta}{2}}$ $F_{12}\,=\, 2F_1 \,\cos \frac{\beta}{2} \,=\, 2\sigma_v \, \frac{b}{ 2 \cos \frac{\beta}{2}}\,\cos \frac{\beta}{2}\,=\, 2\sigma_v b$
Similarly, the magnitude of the forces $$\vec{F}_3$$ and $$\vec{F}_4$$ and their resultant $$\vec{F}_{34}$$ can be expressed. One can not forget, that apart from a different angle, the surface tension must be substituted, because there is the detergent solution behind the boat.
$F_3\,=\,F_4\,=\, \sigma_m \, \frac{b}{ 2 \cos \frac{\alpha}{2}}$ $F_{34}\,=\, 2F_3 \,\cos \frac{\alpha}{2} \,=\, 2\sigma_m \, \frac{b}{ 2 \cos \frac{\alpha}{2}}\,\cos \frac{\alpha}{2}\,=\, 2\sigma_m b$
The forces $$\vec{F}_{12}$$ and $$\vec{F}_{34}$$ act in opposite directions, therefore the resultant force magnitude is given by the subtraction of their magnitudes:
$F \,=\, F_{12}\,-\, F_{34}\, =\, \sigma_v b \,-\, \sigma_m b \,=\, (\sigma_v - \sigma_m ) b$
It is clearly evident from this relation, that the resultant of the acting forces is zero before the detergent drop is added (because in this moment the surface tensions behind and in front of the boat are equal). Therefore the boat remains static.
When we recall the specified values:
$F \,=\, (\sigma_v - \frac{1}{4}\sigma_v ) b\,=\, \frac{3}{4}\sigma_v b \,=\, \frac{3}{4}\,\cdot \, 0.073 \,\cdot\, 0.05 \, \mathrm{N}$ $F \,\dot=\, 2.7\, \mathrm{mN}$
A value of the boat acceleration caused by this force is determined by the Newton’s second law:
$a \,=\, \frac{F}{m}\,=\,\frac{3 \sigma_v b}{4m}\,=\,\frac{3\,\cdot\,0.073\,\cdot\,0.05}{4\,\cdot \,0.05}\, \mathrm{m\,s}^{-2}$ $a \,\dot=\, 0.05\,\mathrm{m\,s}^{-2}$ | 1,485 | 5,141 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5 | 4 | CC-MAIN-2024-26 | latest | en | 0.899077 |
http://math.drhuang.com/science/mathematics/sirmodel.htm | 1,619,126,232,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618039604430.92/warc/CC-MAIN-20210422191215-20210422221215-00246.warc.gz | 57,599,913 | 4,384 | SIR Model Fractional Calculus Computer Algebra System math software
+ + =
# SIR model
Beta:Beta controls the rate of infection spreads, per contact between infected and non-infected, per unit time.
Gamma: Gamma controls the rate infected individuals move into the recovered phase.
Initial
Susceptible: Infected: Recovered:
Days:
Time (days)
Overview:
The SIR models the flows of people between three states: susceptible (S), infected (I), and resistant (R). Each of those variables represents the number of people in those groups. The parameters alpha and beta partially control how fast people move from being susceptible to infected (alpha), and from infected to resistant (beta).
The SIR model is used where individuals infect each other directly (rather than through a disease vector such as a mosquito). An individual who recovers from the illness is also modeled to have perfect immunity to the disease thereafter. Contact between people is also modeled to be random.
The rate that people become infected is proportional to the number of people who are infected, and the number of people who are susceptible. If there are lots of people infected, the chances of a susceptible coming into contact with someone who is infected is high. Likewise, if there are very few people who are susceptible, the chances of a susceptible coming into contact with an infected is lower (since most of the contact would be between the non-susceptible people--either infected or resistant).
Instructions:
The boxes on the right side of the page control the parameters of the model. The page should load with some parameters already in the box. Click "submit" to run the model. The parameters can all be modified and the model re-run. The parameters are
Beta The parameter controlling how often a susceptible-infected contact results in a new infection. Gamma The rate an infected recovers and moves into the resistant phase. Initial susceptible The number of susceptible individuals at the beginning of the model run. Initial infected The number of infected individuals at the beginning of the model run. Initial recovered The number of recovered individuals at the beginning of the model run. Iterations Controls how long the model will run (each iteration is .01 units of time).
This program runs on your computer, so some computers may run faster than others. It is probably a good idea not to set the number iterations or the initial populations too high, since it will take longer to run. Note that cookies must be enabled for the algorithm to function.
Details:
This is an ordinary differential equation model, described by the following equation:
The simulation uses the fourth-order Runge-Kutta algorithm to solve it numerically, with a step size fixed at 0.01, written in JavaScript. The plotting methods are from the flot module. Both the ode simulation and the script in this page calling it are new, so there may still be some unanticipated bugs (I am also fairly new to the language, so my code may be inefficient or bizarre in places). Internet Explorer may have a different look than other browsers, since it has not yet adopted the canvas element, which is used in plotting. | 633 | 3,195 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2021-17 | latest | en | 0.916237 |
https://share.cocalc.com/share/0596bb90a3ac140e1b29a590466db3d0fa60b437/Programming%20Exercise%201.sagews?viewer=share | 1,563,510,247,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195525974.74/warc/CC-MAIN-20190719032721-20190719054721-00288.warc.gz | 543,142,982 | 6,259 | SharedProgramming Exercise 1.sagewsOpen in CoCalc
Author: Dylan Samuelian
Description: Galois Theory Programming Exercise 1
# There is three parts to this Programming Exercise. The first two parts consist of an introduction to Python and Sage Maths syntax.
# You can hide each part by using the small v arrow on your left at line 1 of each part.
# When using the Run button, only the code in the part that you are currently editing is executed. Each part is independent from the other.
##################################################################
# Python Basics (You can skip this if you're already familiar with the syntax and you can hide this part of the code using the small v arrow on your left at line 1) #
##################################################################
# We will mostly work with lists and functions, so you should be a bit comfortable with them.
######### Basics #########
# In Python, you don't need to declare what type a variable is when assigning it.
myvar = 3
mystring = "Galois"
myvar
mystring
# Basic maths operations on numbers are done as you would expect.
myvar + 5
myvar*7
myvar/2
myvar//2 # Integer division
myvar%2 # Modulo
myvar += 2
myvar
mystring += " Theory"
print(mystring)
# You can swap the variable names using the syntax "a,b = b,a"
myvar, mystring = mystring, myvar
######### Lists #########
# List are not restricted to elements of one type
mylist = [3, "Hello", 7.5]
# Lists are indexed from 0 to len(mylist)-1
mylist[0]
# mylist[4] # This gives an error
# You can add or remove stuff at the end of your list as you would expect using ".pop()" and ".append()"
mylist.pop() # Note that this returns the element removed, so that you could write "x = mylist.pop()"
mylist.append("World")
mylist.append(130)
mylist.append(178)
# You can extract slices of your list using [i:j] to get [mylist[i], mylist[i+1], ..., mylist[j-1]]. Omitting i means you want to start from 0 and omitting j means you go until the end of the list.
mylist
mylist[1:3]
mylist[1:]
mylist[:2]
# You can check whether an element is in your list using the key word "in".
3 in mylist # => True
"EPFL" in mylist # => False
# If an element is in your list, you can remove its first ocurrence using ".remove()" and get the index of first occurrence using ".index()"
# If the element is not in the list, ".remove()" will return an error.
mylist.index("World")
mylist.remove("World")
mylist
# You can delete the i-th element using "del"
del mylist[1]
mylist
# You can add an element at index i using ".insert()" and you can concatenate two lists using ".extend(other_list)"
mylist.insert(2,"Hello World")
mylist
######### For Loops, While Loops, If-Else statements #########
# You can get the integers in [a,b) using range(a,b).
rangelist = range(5,10)
# You can iterate over the elements of a list (without them being numbers) :
for number in rangelist:
if number in [3, 4]:
# "Break" terminates a for loop without executing the "else" clause.
break
elif number == 8:
print(number)
elif number%2 == 0:
continue # "Continue" starts the next iteration of the loop without executing the rest of the code.
print(number)
else:
pass # Does nothing
i = 0
while i < 3:
print(i)
i = i+1
######### Functions #########
# The syntax of a function is as follows. You can return a value using the key word "return"
def myfunction(var1,var2,var3):
pass
##################################################################
# Sage Math Introduction #
##################################################################
# SageMath Basics Reference : https://doc.sagemath.org/html/fr/tutorial/
# Commands for polynomials : http://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_gf2x.html
# Commands for finite fields : http://doc.sagemath.org/html/en/reference/finite_rings/sage/rings/finite_rings/finite_field_constructor.html
# Our goal wil be to implement the Extended Euclidean Algorithm.
# First, we familiarize with the syntax of rings.
# The polynomial ring QQ[x] is defined as follow :
R.<x> = PolynomialRing(QQ)
R
# Note that we can replace QQ by RR; or CC; or a finite field F with F = GF(p) for a prime p; or a finite field F with F.<a> = GF(q), where q = p^n for a prime p and 'a' is a name for a primitive (p^n-1)-th root of unity; or any other fields that SageMath implements.
# We can define elements of QQ[x] and do basic operations with polynomials in a natural way :
f = x^2 + 1
g = 3*x^3 + 2*x - 5
# We can also define a polynomial using a vector of coefficients [a0, a1, ..., an] :
h = R([0, 1, 1, 0, 8])
h
# We can check in which ring they belong using ".parent()"
f.parent()
f*g
g%f
f/g # This works and gives automatically a rational element in the fraction field "FractionField(R)" of R "=" QQ[x].
(f/g).parent()
f+g
2*f
# You can get the list of coefficients [a0, a1, ..., an] using ".list()"
f.list()
g.list()
# We can check irreducibility using ".is_irreducible()"
f.is_irreducible() # => True
g.is_irreducible() # => False
# You can also use ".degree()", ".gcd(other_polynomial)" in a straightfoward way.
# You can get the factorization of a polynomial into irreducible factors of R using ".factor()" and get the list of factors with multiplicity using "list()"
g.factor()
list(g.factor())
# We can consider an ideal in R generated by some polynomials [f_1, ..., f_k]
I1 = Ideal(R, [f,g])
I1
I2 = Ideal(R, [f])
I2
# And quotient the ring R by this ideal.
S.<y> = R.quotient_ring(I2) # We change the variable to avoid any ambiguity
S
# You can consider a polynomial p as an element of the quotient ring S using "S(p)" or "S([vector of coefficients])"
S(y^3 + 1) == S(-y+1) # => True
##################################################################
# Sage Math Exercises #
##################################################################
R.<x> = PolynomialRing(QQ)
##################################################################
# Exercise 1 #
##################################################################
# The goal of this exercise is to implement the standard Euclidean Division for non-zero elements f, g of PolyRing, where PolyRing is a ring of polynomials over some field (for example, PolyRing = R "=" QQ[x] as above)
# The function must return a list [h, r] of polynomials such that f = gh + r, deg(r) < deg(g) (except when g = 0).
# This is the standard recursive algorithm, which follows from the proof (by induction) that h and r do exist.
# Two lines are missing. The exercise is to fill them.
def euclideanDivision(f,g, PolyRing):
degree_f = f.degree()
degree_g = g.degree()
f_coef = f.list()
g_coef = g.list()
if g == PolyRing([0]):
return (PolyRing([0]), f)
if degree_g > degree_f:
pass # return .... # MISSING
elif degree_g == degree_f:
return (PolyRing([h]), f-g*h)
else:
pass # h1 = ... # MISSING
f_ = f - g*h1
h2, r = euclideanDivision(f_,g,PolyRing)
return [h1+h2,r]
# Test your function on a few examples.
##################################################################
# Exercise 2 #
##################################################################
# The goal of this exercise is to implement the Extended Euclidean Algorithm for non-zero elements f, g of PolyRing, where PolyRing is a ring of polynomials over some field (for example, PolyRing = R "=" QQ[x] as above).
# The function must return a triple [h, a, b], such that af + bg = h = gcd(f, g).
# Three lines are missing. The exercise is to fill them.
def extEuclidean(f,g, PolyRing):
r0, s0, t0 = f, 1, 0 # Multiple assignments
r1, s1, t1 = g, 0, 1
while r1 != PolyRing([0]):
pass # q = ... # MISSING
tmp_r, tmp_s, tmp_t = r1, s1, t1
pass # r1 = ... # MISSING
s1 = s0 - q*s1
t1 = t0 - q*t1
r0,s0,t0 = tmp_r, tmp_s, tmp_t
pass # return ... # MISSING
# Compare the output of "extEuclidean(f,g,R)" with the output of "f.xgcd(g)". What's the difference between both ?
range(10)
[x if x%2 == 0 else 1 for x in range(1,11)]
[[(x,y) if x%2 == 0 and y%2 == 0 else (1,1) for x in range(1,11)] for y in range(1,11)]
[1,2]+[3,4]
t = [1,2]
t.append(3)
2 in t | 2,147 | 8,060 | {"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} | 3.5 | 4 | CC-MAIN-2019-30 | latest | en | 0.810543 |
http://stackoverflow.com/questions/4394716/please-help-me-with-this-simple-elementary-pseudocode | 1,394,718,597,000,000,000 | text/html | crawl-data/CC-MAIN-2014-10/segments/1394678674071/warc/CC-MAIN-20140313024434-00067-ip-10-183-142-35.ec2.internal.warc.gz | 137,858,275 | 16,955 | # Please help me with this simple, elementary pseudocode
This should be very easy for you guys, but I still can't get it visualized in my mind. Here's what I'm trying to do:
``````if (flag)
for (i = 0; i < 10; i++)
{LARGE_BLOCK_OF_CODE (that visits an array in order)}
else
for (i = 9; i >= 0; i--)
{LARGE_BLOCK_OF_CODE (that visits an array in REVERSE order)}
``````
The problem is that this block of code cannot be placed in a function, and I don't want to copy/paste it twice. Is there any way I can do that in this manner:
``````if (flag)
loop = for (i = 0; i < 10; i++);
else
loop = for (i = 9; i >= 0; i--);
loop
{LARGE_BLOCK_OF_CODE (that visits an array in the order specified in 'loop')}
``````
??
-
"this block of code cannot be placed in a function" Why not? – Lightness Races in Orbit Jul 22 '11 at 22:59
add comment
## 4 Answers
Create variables for the start and end of the loop, and create a third variable to determine the 'direction' for the iteration.
Then just let the loop run through your ONE block of code. Like this...
``````int iStart = 0;
int iEnd = 9;
int iIncrement = +1;
if (reverse)
{
iStart = 9;
iEnd = 0;
iIncrement = -1;
}
for (i = iStart; i != iEnd; i += iIncrement)
{
//...large block of code here...
}
``````
-
Looks good. Thanks! – SoftwareGuy Dec 9 '10 at 18:41
My pleasure, "user486800". :) – FlipScript Dec 10 '10 at 4:02
add comment
Put that LARGE_BLOCK_OF_CODE in a function, receiving order as a parameter. Then reuse it in any number of loops you need.
-
`The problem is that this block of code cannot be placed in a function` – pinichi Dec 9 '10 at 4:16
If it's C or C++ anything can be placed in a function. The only question is how much work is involved in putting it in a function. Another alternative: Put that LARGE_BLOCK_OF_CODE in a macro. – Android Eve Dec 9 '10 at 5:09
That's what I felt like doing at first, but I didn't write that code initially, so I don't want to mess with it unless within the limits they assign me. Not to mention the fact that I don't understand most of it... >_> – SoftwareGuy Dec 9 '10 at 18:45
@user486800 +1 for explaining the constraint. – Android Eve Dec 9 '10 at 18:54
add comment
I don't know how big your array is. Maybe it is possible to generate an additional array that holds the required indexes in the required order:
``````indexes = []
for (i = 0; i < 10; i++)
indexes.add(i)
if (!flag)
indexes.reverse()
for each (i in indexes)
{LARGE_BLOCK_OF_CODE}
``````
-
I like it. FlipScript's answer seems a little bit easier for me, but this is a nice backup plan if the other one does not work for me. Thank you for the insight! – SoftwareGuy Dec 9 '10 at 18:43
True. My first approach was like FlipScript's too. But he beat me by 2 minutes so I changed it to this. As you can see in his solution, it's a bit difficult to get the values for the end conditions right. (I guess both are off by one.) But in the end it's more efficient and intuitive. On the other hand the solution in my post is ready for some really funky loops. You could even shuffle the indexes array and thus iterate in a random order. Maybe I'll just link to this post whenever someone wants to do this. – Robert Dec 13 '10 at 0:23
add comment
``````for (i = flag?0:9; i != flag?10:0; i += flag?1:-1)
{
//LARGE_BLOCK_OF_CODE
}
``````
Well I know...:p thats too many if else.
And yes, FlipScript's solution is better though it might not look neat. :)
-
`for (i = 0; i >= 9; i += 1)`??? – pinichi Dec 9 '10 at 4:29
that's the result when flag = true ^^. You must use `!=` not `>=` and replace `9` by `10` – pinichi Dec 9 '10 at 4:39
Hehe....sorry my bad. Thanx for pointing out. – loxxy Dec 9 '10 at 4:47
I like this compact version, but FlipScript's elaborate version explains it better. Thanks. – SoftwareGuy Dec 9 '10 at 18:41
I like this, too... but it would probably give me a headache a year or two from now when I have to revisit the code and didn't remember how it worked. I hate to have to study my own code! – FlipScript Dec 10 '10 at 4:03
add comment | 1,186 | 4,052 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2014-10 | latest | en | 0.883802 |
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Stress and deflections are primarily function of diameter. Buckling could be said combination of both diameter and thickness. Stiffness (for vibration) primarily function of diameter. Mass (and material costs) primarily function of thickness.
4. Tackles...
Also, adding more pulley means that the vertical movement reduced for each additional pulley. using 1 pulley, F = W, you move F 1 m, you get W lifted for 1 m. using 2 pulley, F = W/2, you move F 1 m, you get W lifted for only 0.5 m. using 3 pulley, F = W/3, you move F 1 m, you get W lifted for only 0.333 m. But mostly, what DrD said.... friction
5. Why all news graduate engineer have to suffer for job
Wait until you work for a while and realize that most the "mechanical engineers" can't grasp the basic concept of mechanics, fluid mechanics and any basic subjects of mechanical engineering. They won't bother about anything that don't involve money. But they positioned higher than you and obviously better salaried than you.
6. Want to know about Go kart Analysis
Vi= 45 km/h = 12.5 m/s Vf= 0 km/h = m/s t = 0.2 sec (industrial standards) m = 5 x m_original = ... (5g thingy) Impulse F x dt = m x (Vi - Vf) Applied load F to front end of the chassis.
7. is this formula ok?
It should be E not E^2. Other than that it looks correct. Established books make mistakes (albeit rarely). Here's for other potentially conflicting reference https://www.machinemfg.com/load-analysis-driven-power-calculation-symmetrical-3-roll-plate-bending-machine/ I've seen very rare occasion where constants do have dimensional units. (I recall a book by Niemann on Design of Machine Elements, the original German edition) Niemann & Hirt, Maschinenelemente Volume 1: Konstruktion und Berechnung von Verbindungen, Lagern, Wellen Volume 2: Getriebe allgemein, Zahnradgetriebe - Grundlagen, Stirnradgetriebe Volume 3: Schraubrad-, Kegelrad-, Schnecken-, Ketten-, Riemen-, Reibradgetriebe, Kupplungen, Bremsen, Freiläufe Never seen it in American books or European books using SI units. There are occasions that constants have dimensional units in practical standards/codes as well. But that's a different story.
8. can pump run with AIR/Gas
Can you use ur feet to write? with some practice you can. It may never be better than your hand writing, but you can have a decent feet writing.
9. Force/Torque on a cylindrical system
I think this problem goes to T = J x alpha = J x dw/dt But yeah, DrD is correct. Without braking torque, since dw/dt =0, this equation is lacking something.
10. How can I manufacture this spiral slide?
I think that was made by 3D printing.
11. Flow visualization on a flat plate
Google these terms: boundary layer laminar flow point of separation
This is a professional engineering report on its own. You can’t expect this for a free.
13. How do I overcome self doubt in obtaining my engineering degree?
Give you perspective from other way. Unlike a PhD program, both for undergraduate and Master degree program, you can’t be failed by your professor (unless you are very unlucky or very dumb). The only reason of failing is you giving up. In PhD program, you could be failed by your professor during your thesis defense.
14. linear actuator rod extension
What your purpose? Additional length means additional applied force and bending moment. You need to check the working stress on the screwed connection. | 989 | 4,048 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2020-05 | longest | en | 0.908994 |
http://www.primidi.com/bernoulli_equation | 1,537,834,057,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267160842.79/warc/CC-MAIN-20180924224739-20180925005139-00115.warc.gz | 382,523,849 | 3,234 | # Bernoulli Equation
Bernoulli equation may refer to:
• Bernoulli differential equation
• Bernoulli's equation, in fluid dynamics
• Euler-Bernoulli beam equation, in solid mechanics
### Other articles related to "bernoulli, equation, bernoulli equation, equations":
Bernoulli's Principle - Incompressible Flow Equation
... Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow ... A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is (A) where is the fluid flow speed at a point on a streamline, is the acceleration due to gravity, is the ... For conservative force fields, Bernoulli's equation can be generalized as where Ψ is the force potential at the point considered on the streamline ...
Bernoulli's Principle - Derivations of Bernoulli Equation
... Bernoulli equation for incompressible fluids The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation ... x(t) With density ρ constant, the equation of motion can be written as by integrating with respect to x where C is a constant, sometimes referred to as the Bernoulli constant ... Rather, Bernoulli's principle was inherently derived by a simple manipulation of the momentum equation ...
### Famous quotes containing the word equation:
A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.
Norman Mailer (b. 1923) | 350 | 1,640 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2018-39 | latest | en | 0.946557 |
http://roguebasin.roguelikedevelopment.org/index.php?title=Bresenham%27s_Line_Algorithm&diff=next&oldid=29232 | 1,596,948,433,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738425.43/warc/CC-MAIN-20200809043422-20200809073422-00500.warc.gz | 101,463,305 | 12,976 | # Bresenham's Line Algorithm
(Difference between revisions)
Bresenham's Line Algorithm is a way of drawing a line segment onto a square grid. It is especially useful for roguelikes due to their cellular nature.
In libtcod it is accessible using `line(x1, y1, x2, y2, callback)`. Below are several hand-coded implementations in various languages.
## C#
Here is a simple way of using the algorithm in C# with delegates.
```// Author: Jason Morley (Source: http://www.morleydev.co.uk/blog/2010/11/18/generic-bresenhams-line-algorithm-in-visual-basic-net/)
using System;
namespace Bresenhams
{
/// <summary>
/// The Bresenham algorithm collection
/// </summary>
public static class Algorithms
{
private static void Swap<T>(ref T lhs, ref T rhs) { T temp; temp = lhs; lhs = rhs; rhs = temp; }
/// <summary>
/// The plot function delegate
/// </summary>
/// <param name="x">The x co-ord being plotted</param>
/// <param name="y">The y co-ord being plotted</param>
/// <returns>True to continue, false to stop the algorithm</returns>
public delegate bool PlotFunction(int x, int y);
/// <summary>
/// Plot the line from (x0, y0) to (x1, y10
/// </summary>
/// <param name="x0">The start x</param>
/// <param name="y0">The start y</param>
/// <param name="x1">The end x</param>
/// <param name="y1">The end y</param>
/// <param name="plot">The plotting function (if this returns false, the algorithm stops early)</param>
public static void Line(int x0, int y0, int x1, int y1, PlotFunction plot)
{
bool steep = Math.Abs(y1 - y0) > Math.Abs(x1 - x0);
if (steep) { Swap<int>(ref x0, ref y0); Swap<int>(ref x1, ref y1); }
if (x0 > x1) { Swap<int>(ref x0, ref x1); Swap<int>(ref y0, ref y1); }
int dX = (x1 - x0), dY = Math.Abs(y1 - y0), err = (dX / 2), ystep = (y0 < y1 ? 1 : -1), y = y0;
for (int x = x0; x <= x1; ++x)
{
if (!(steep ? plot(y, x) : plot(x, y))) return;
err = err - dY;
if (err < 0) { y += ystep; err += dX; }
}
}
}
}```
## C++
Here's a C++ version; plot() draws a "dot" at (x, y):
```#include <cstdlib>
////////////////////////////////////////////////////////////////////////////////
void Bresenham(int x1,
int y1,
int const x2,
int const y2)
{
int delta_x(x2 - x1);
// if x1 == x2, then it does not matter what we set here
signed char const ix((delta_x > 0) - (delta_x < 0));
delta_x = std::abs(delta_x) << 1;
int delta_y(y2 - y1);
// if y1 == y2, then it does not matter what we set here
signed char const iy((delta_y > 0) - (delta_y < 0));
delta_y = std::abs(delta_y) << 1;
plot(x1, y1);
if (delta_x >= delta_y)
{
// error may go below zero
int error(delta_y - (delta_x >> 1));
while (x1 != x2)
{
if ((error >= 0) && (error || (ix > 0)))
{
error -= delta_x;
y1 += iy;
}
// else do nothing
error += delta_y;
x1 += ix;
plot(x1, y1);
}
}
else
{
// error may go below zero
int error(delta_x - (delta_y >> 1));
while (y1 != y2)
{
if ((error >= 0) && (error || (iy > 0)))
{
error -= delta_y;
x1 += ix;
}
// else do nothing
error += delta_x;
y1 += iy;
plot(x1, y1);
}
}
}```
A template metaprogram implementation (requires the Boost.MPL library):
```#include "boost/mpl/bool.hpp"
#include "boost/mpl/char.hpp"
#include "boost/mpl/for_each.hpp"
#include "boost/mpl/bitwise.hpp"
#include "boost/mpl/shift_left.hpp"
#include "boost/mpl/list.hpp"
#include "boost/mpl/push_front.hpp"
#include "boost/mpl/max.hpp"
#include "boost/mpl/minus.hpp"
#include "boost/mpl/arithmetic.hpp"
#include "boost/mpl/pair.hpp"
namespace mpl = boost::mpl;
template <std::size_t N,
typename x1, typename y1,
typename delta_x, typename delta_y,
typename ix, typename iy,
typename error, typename swap>
struct bresenham_line_pair :
mpl::push_front<typename bresenham_line_pair<N - 1,
typename mpl::plus<x1, ix>::type,
typename mpl::if_c<(error::value >= 0)
&& (error::value || (ix::value > 0)),
typename mpl::plus<y1, iy>::type, y1>::type,
delta_x, delta_y,
ix, iy,
typename mpl::if_c<(error::value >= 0)
&& (error::value || (ix::value > 0)),
typename mpl::minus<typename mpl::plus<error,
delta_y>::type, delta_x>::type,
typename mpl::plus<error, delta_y>::type>::type,
swap>::type,
typename mpl::if_<swap, mpl::pair<y1, x1>, mpl::pair<x1, y1> >::type
>
{
};
template <typename x1, typename y1,
typename delta_x, typename delta_y,
typename ix, typename iy,
typename error, typename swap>
struct bresenham_line_pair<0, x1, y1, delta_x, delta_y, ix, iy, error, swap> :
mpl::list<typename mpl::if_<swap, mpl::pair<y1, x1>,
mpl::pair<x1, y1> >::type>
{
};
template <int x1, int y1, int x2, int y2>
class bresenham_line
{
public:
typedef typename mpl::minus<mpl::int_<x2>, mpl::int_<x1> >::type dx;
typedef typename mpl::minus<mpl::int_<y2>, mpl::int_<y1> >::type dy;
typedef typename mpl::if_<mpl::less<dx, mpl::int_<0> >,
mpl::int_<-1>, mpl::int_<1> >::type ix;
typedef typename mpl::if_<mpl::less<dy, mpl::int_<0> >,
mpl::int_<-1>, mpl::int_<1> >::type iy;
typedef typename mpl::max<dx, mpl::negate<dx> >::type abs_dx;
typedef typename mpl::max<dy, mpl::negate<dy> >::type abs_dy;
typedef typename mpl::shift_left<abs_dx, mpl::char_<1> >::type delta_x;
typedef typename mpl::shift_left<abs_dy, mpl::char_<1> >::type delta_y;
typedef typename mpl::if_<mpl::less<delta_x, delta_y>, mpl::bool_<true>,
mpl::bool_<false> >::type swap;
typedef typename mpl::if_<swap,
typename mpl::minus<delta_x, abs_dy>::type,
typename mpl::minus<delta_y, abs_dx>::type>::type error;
typedef typename mpl::max<abs_dx, abs_dy>::type N;
typedef typename mpl::if_<swap,
bresenham_line_pair<N::value, mpl::int_<y1>, mpl::int_<x1>,
delta_y, delta_x, iy, ix, error, swap>,
bresenham_line_pair<N::value, mpl::int_<x1>, mpl::int_<y1>,
delta_x, delta_y, ix, iy, error, swap> >::type::type sequence_type;
};
struct plotter
{
template<typename T>
inline void operator()(T)
{
plot(T::first::type::value, T::second::type::value);
}
};
int main(int, char*[])
{
mpl::for_each<bresenham_line<0, 0, 20, 10>::sequence_type>(plotter());
return 0;
}```
## Python
This Python version returns a list of (x, y) tuples. It was converted from the Ruby version below, but also reverses the list to begin with the first coordinates.
```def get_line(x1, y1, x2, y2):
points = []
issteep = abs(y2-y1) > abs(x2-x1)
if issteep:
x1, y1 = y1, x1
x2, y2 = y2, x2
rev = False
if x1 > x2:
x1, x2 = x2, x1
y1, y2 = y2, y1
rev = True
deltax = x2 - x1
deltay = abs(y2-y1)
error = int(deltax / 2)
y = y1
ystep = None
if y1 < y2:
ystep = 1
else:
ystep = -1
for x in range(x1, x2 + 1):
if issteep:
points.append((y, x))
else:
points.append((x, y))
error -= deltay
if error < 0:
y += ystep
error += deltax
# Reverse the list if the coordinates were reversed
if rev:
points.reverse()
return points```
## Ruby
Here's a Ruby version, it returns an array of points, each being a hash with 2 elements (x and y).
```def get_line(x0,x1,y0,y1)
points = []
steep = ((y1-y0).abs) > ((x1-x0).abs)
if steep
x0,y0 = y0,x0
x1,y1 = y1,x1
end
if x0 > x1
x0,x1 = x1,x0
y0,y1 = y1,y0
end
deltax = x1-x0
deltay = (y1-y0).abs
error = (deltax / 2).to_i
y = y0
ystep = nil
if y0 < y1
ystep = 1
else
ystep = -1
end
for x in x0..x1
if steep
points << {:x => y, :y => x}
else
points << {:x => x, :y => y}
end
error -= deltay
if error < 0
y += ystep
error += deltax
end
end
return points
end```
## VB.NET
Here is a generic way of using the algorithm in VB.NET using delegates.
```' Author: Jason Morley (Source: http://www.morleydev.co.uk/blog/2010/11/18/generic-bresenhams-line-algorithm-in-visual-basic-net/)
Module BresenhamsLineAlgorithm
Sub Swap(ByRef X As Long, ByRef Y As Long)
Dim t As Long = X
X = Y
Y = t
End Sub
' If the plot function returns true, the bresenham's line algorithm continues.
' if the plot function returns false, the algorithm stops
Delegate Function PlotFunction(ByVal x As Long, ByVal y As Long) As Boolean
Sub Bresenham(ByVal x1 As Long, ByVal y1 As Long, ByVal x2 As Long, ByVal y2 As Long, ByVal plot As PlotFunction)
Dim steep As Boolean = (Math.Abs(y2 - y1) > Math.Abs(x2 - x1))
If (steep) Then
Swap(x1, y1)
Swap(x2, y2)
End If
If (x1 > x2) Then
Swap(x1, x2)
Swap(y1, y2)
End If
Dim deltaX As Long = x2 - x1
Dim deltaY As Long = y2 - y1
Dim err As Long = deltaX / 2
Dim ystep As Long
Dim y As Long = y1
If (y1 < y2) Then
ystep = 1
Else
ystep = -1
End If
For x As Long = x1 To x2
Dim result As Boolean
If (steep) Then result = plot(y, x) Else result = plot(x, y)
If (Not result) Then Exit Sub
err = err - deltaY
If (err < 0) Then
y = y + ystep
err = err + deltaX
End If
Next
End Sub
Function plot(ByVal x As Long, ByVal y As Long) As Boolean
Console.WriteLine(x.ToString() + " " + y.ToString())
Return True 'This just prints each co-ord
End Function
Sub Main()
' example
Bresenham(1, 1, 10, 15, New PlotFunction(AddressOf plot))
End Sub
End Module```
A slightly verbose version in Haskell. See the discussion page for a variant one line shorter, but IMHO less readable. I bet other version, more readable and more succinct, can be written.
```-- | See <http://roguebasin.roguelikedevelopment.org/index.php/Digital_lines>.
balancedWord :: Int -> Int -> Int -> [Int]
balancedWord p q eps | eps + p < q = 0 : balancedWord p q (eps + p)
balancedWord p q eps = 1 : balancedWord p q (eps + p - q)
-- | Bresenham's line algorithm.
-- Includes the first point and goes through the second to infinity.
bla :: (Int, Int) -> (Int, Int) -> [(Int, Int)]
bla (x0, y0) (x1, y1) =
let (dx, dy) = (x1 - x0, y1 - y0)
xyStep b (x, y) = (x + signum dx, y + signum dy * b)
yxStep b (x, y) = (x + signum dx * b, y + signum dy)
(p, q, step) | abs dx > abs dy = (abs dy, abs dx, xyStep)
| otherwise = (abs dx, abs dy, yxStep)
walk w xy = xy : walk (tail w) (step (head w) xy)
in walk (balancedWord p q 0) (x0, y0)``` | 3,246 | 9,693 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2020-34 | latest | en | 0.545306 |
https://www.jiskha.com/questions/261445/I-was-given-gross-accounts-receivable-15-200-000-and-17-600-000-respectively-allowance | 1,561,011,442,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999141.54/warc/CC-MAIN-20190620044948-20190620070948-00553.warc.gz | 793,383,956 | 5,067 | # accounting
I was given gross accounts receivable \$15,200,000 and \$17,600,000 respectively; allowance for uncollectible accounts at the begining and end of the year \$1,400,000 and \$1,550,000; expense for estimated uncollectible accouts \$750,00 which was 1% of sales; no recoveries of accounts written off in previous years. I need to reconstruct the journal entries using th allowance method
1. 👍 0
2. 👎 0
3. 👁 41
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More Similar Questions | 866 | 3,323 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2019-26 | latest | en | 0.925142 |
https://techcommunity.microsoft.com/t5/excel/excel-dilemma/m-p/3872292 | 1,723,702,075,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641151918.94/warc/CC-MAIN-20240815044119-20240815074119-00541.warc.gz | 420,827,970 | 58,759 | # Excel dilemma
Copper Contributor
# Excel dilemma
Hello,
I currently use Microsoft 365 on the Microsoft 10 platform and I need to verify the total cost for our water discharge to the POTW. This cost is broken down into the number of gallons that are discharged. I thought I could use the IF function but I have been unsuccessful. We normally discharge between 3,000,000 to 4,000,000 gallons a month and this data is sent to the local Municipality to calculate our total discharge fees. I am wanting to verify the fees. The cost is broken down in gallons of discharge as follows.
The first 100,000 gallons cost \$10.30 per 1000 gallons.
The second 650,000 gallons cost \$7.66 per 1000 gallons or >100,000 gallons but <=750,000 gallons.
The third 1,250,000 gallons cost \$6.22 per 1000 gallons or >750,000 gallons but <= 1,250,000 gallons.
The fourth 30,000,000 gallons cost \$ 5.25 per 1000 gallons or >1,250,000 gallons but <= 30,000,000 gallons.
As I stated above, we discharge between 3,000,000 to 4,000,000 gallons a month would I use the IF function to calculate the total cost per month or is there a better way to calculate these charges?
4 Replies
# Re: Excel dilemma
Your description is inconsistent, for example
"The third 1,250,000 gallons cost \$6.22 per 1000 gallons or >750,000 gallons but <= 1,250,000 gallons."
From 750,000 to 1,250,000 is 500,000 and not 1,250,000.
So you may have to change the values shown below.
I'd use a lookup table:
You can then use a SUMPRODUCT formula:
The formula in C2 is
=SUMPRODUCT((B2>\$E\$2:\$E\$5)*(B2-\$E\$2:\$E\$5),\$G\$2:\$G\$5)/1000
# Re: Excel dilemma
Thank you, Hans, I understand your point but the municipality charges us a cumulative fee. What I mean is, as I stated the first 100,000 is charged then the next 650,000, and finally the next 1,250,000. These three flows are added together or 2,000,000 and anything above the 2,000,000 is charged \$5.25. Does that now make sense? Yes, I know it's confusing but that is how they make their money.
# Re: Excel dilemma
That was very unclear (to me) from your first post.
Change the lookup range to
Threshold Cost per 1000 Difference 0 \$10.30 \$10.30 100,000 \$7.66 -\$2.64 750,000 \$6.22 -\$1.44 2,000,000 \$5.25 -\$0.97
Formulas:
Threshold Cost per 1000 Difference 0 10.3 =F2 100000 7.66 =F3-F2 750000 6.22 =F4-F3 2000000 5.25 =F5-F4
# Re: Excel dilemma
Since you have 365, this is overkill for the problem in hand.
``= MAP(dischargeVolume, TieredCostλ(threshold, costPer1000))``
The list 'dischargeVolume' corresponds to the the different months, threshold and costPer1000 are names given to the lookup data. The point of the lambda function is that it conceals the complexity.
``````= LAMBDA(dischargeVol,
LET(
limit, VSTACK(DROP(threshold, 1), ∞),
volume, MAP(threshold, limit,
LAMBDA(th, lt, MAX(MIN(lt, +dischargeVol) - th, 0))
),
SUM(volume * costPer1000) / 1000
)
)`````` | 870 | 2,916 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2024-33 | latest | en | 0.885856 |
http://archive.ambermd.org/200412/0033.html | 1,726,888,727,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725701427996.97/warc/CC-MAIN-20240921015054-20240921045054-00681.warc.gz | 2,986,848 | 3,419 | # AMBER: one question about dielectric dependent model
From: Xiaowei (David) Li <"Xiaowei>
Date: Sun, 05 Dec 2004 12:42:56 -0500
Dear AMBER users:
It is widely used technique to mimic the the influence of solvent using
implicit solvent(Generally, water). In AMBER, my understanding is that
there are two methods to use implicit solvent. One is Generalized-Born
(GB) model , and the other is to use distance-dependent dielectric
model. Beause there is no GB model implemented in NMODE, so I use
distantce-dependent dielectic model when I perform normal mode analysis
of nucleic acids. My question is concering about the distance-dependent
model in AMBER.
Based on the Debye-Hückel model, the screened Coulomb potential (phi) is
phi=(Z*q)/(epsilon *r) exp(-kr), here the 1/k is the D-H length, epsilon
is the dielectric constant in vacumm.
Therefore, the effective dielectric constant is (epsilon * exp(kr)). If
|kr|<1, then exp(kr) is approximated as (1+kr). So the effective
dielectric constant is (epsilon*(1+kr)).
In NMODE, we can set dielc=1-4 and idiel=0 to get the effective
dielectric constant as (1-4)*r, which means kr>>1 and the dielectric
constant is distant dependent. However, if the amount of water is very
small, then k is very close to 0. Thus, we should set the dielectric
constant as (epsilon*(1+kr)), instead of (epsilon*kr). But it seems to
me that there is no options in AMBER to set the dielectric constant as
(epsilon*(1+kr)). In other words, we can set constant dielectric or
distant-dependent dielectric constant in AMBER. But there is
discontinuity between these two cases. This is my understanding about
this question. If I am wrong, please feel free to correct me. Thank you
very much.
Best,
-----------------------------------------------------------------------
The AMBER Mail Reflector
To post, send mail to amber.scripps.edu
To unsubscribe, send "unsubscribe amber" to majordomo.scripps.edu
Received on Sun Dec 05 2004 - 17:53:00 PST
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# How do you teleport a part to a character?
I am just starting to learn how to use position and I am confused. I believe the answer is in the lines of:
local hit = Workspace.WiseDumbledore
Workspace.Part.Position = (hit.x,hit.y,hit.z)
Is that correct? Please correct me if I am wrong.
Thanks -WiseDumbledore
0
It's close to being correct, but in roblox, you use vectors for positions and forces.
local hit = game.Workspace.WiseDumbledore.Torso.Position
game.Workspace.Part.Position = hit --Since 'hit' is already an vector, you don't need to use 'Vector.new()'
If you want the part to get moved to that exact position, ignoring the fact that there are other parts in the way, you use coordinate frame variable(CFrame):
local hit = game.Workspace.WiseDumbledore.Torso.Position
game.Workspace.Part.CFrame= CFrame.new(hit)
1
I think it would display an Error. You haven't specified the player, instead, your hit variable is in Workspace, Players are not in Workspace. Line 2 looks correct.
Now it depends really, if you want this to be repeated, you need to add a loop. There are many loops. The while true do loop makes it go forever. I am not so familiar with other loops, since I need to revise them. If you wish to do the while true do loop, the following code should be correct:
hit = AAA -- First of all, Find the player's TORSO... Replace "AAA" With the Torso's Path (eg: game.Players. etc...
Part = Workspace.Part --The Part that you want to teleport
while true do --Keeps this going on forever
Part.Position = Vector3.new(hit.Position.x, hit.Position.y, hit.Position.z) --You forgot the Vector3.new().
wait() --Add how many second it should wait before moving again. Or leave it, so that it doesn't crash.
end
--[[ NOTE:
Always leave a "wait() " in the loop, or else Studio WILL Crash.]]
0
duckwit 1399
7 years ago
No, it is not correct. Position is a property of BasePart, but WiseDumbledore is a Model, a Model does not have x, y, or z as Properties.
What you should do is get the position (a Vector3) of the Torso (a specific part of the Character) and then you can assign it to the Part that you want:
positionOfTorso = workspace.WiseDumbledore.Torso.Position
workspace.Part.Position = positionOfTorso
Alternatively, you can set the CFrame (CoordinateFrame) of the Part and this will allow you to put parts inside each other (i.e. instead of placing the Part on top of the player)
positionOfTorso = workspace.WiseDumbledore.Torso.Position
workspace.Part.CFrame = CFrame.new(positionOfTorso)
... or you could just do it in one line by referring directly to the CFrame of the Torso
workspace.Part.CFrame = workspace.WiseDumbledore.Torso.CFrame
Hope that helps! | 667 | 2,701 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2021-39 | latest | en | 0.864934 |
http://at.metamath.org/ileuni/nlimsucg.html | 1,580,006,132,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251684146.65/warc/CC-MAIN-20200126013015-20200126043015-00546.warc.gz | 14,378,718 | 5,019 | Intuitionistic Logic Explorer < Previous Next > Nearby theorems Mirrors > Home > ILE Home > Th. List > nlimsucg Structured version GIF version
Theorem nlimsucg 4226
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
nlimsucg (A 𝑉 → ¬ Lim suc A)
Proof of Theorem nlimsucg
StepHypRef Expression
1 limord 4081 . . . . . 6 (Lim suc A → Ord suc A)
2 ordsuc 4225 . . . . . 6 (Ord A ↔ Ord suc A)
31, 2sylibr 137 . . . . 5 (Lim suc A → Ord A)
4 limuni 4082 . . . . 5 (Lim suc A → suc A = suc A)
53, 4jca 290 . . . 4 (Lim suc A → (Ord A suc A = suc A))
6 ordtr 4064 . . . . . . . 8 (Ord A → Tr A)
7 unisucg 4100 . . . . . . . . 9 (A 𝑉 → (Tr A suc A = A))
87biimpa 280 . . . . . . . 8 ((A 𝑉 Tr A) → suc A = A)
96, 8sylan2 270 . . . . . . 7 ((A 𝑉 Ord A) → suc A = A)
109eqeq2d 2033 . . . . . 6 ((A 𝑉 Ord A) → (suc A = suc A ↔ suc A = A))
11 ordirr 4209 . . . . . . . . 9 (Ord A → ¬ A A)
12 eleq2 2083 . . . . . . . . . 10 (suc A = A → (A suc AA A))
1312notbid 579 . . . . . . . . 9 (suc A = A → (¬ A suc A ↔ ¬ A A))
1411, 13syl5ibrcom 146 . . . . . . . 8 (Ord A → (suc A = A → ¬ A suc A))
15 sucidg 4102 . . . . . . . . 9 (A 𝑉A suc A)
1615con3i 549 . . . . . . . 8 A suc A → ¬ A 𝑉)
1714, 16syl6 29 . . . . . . 7 (Ord A → (suc A = A → ¬ A 𝑉))
1817adantl 262 . . . . . 6 ((A 𝑉 Ord A) → (suc A = A → ¬ A 𝑉))
1910, 18sylbid 139 . . . . 5 ((A 𝑉 Ord A) → (suc A = suc A → ¬ A 𝑉))
2019expimpd 345 . . . 4 (A 𝑉 → ((Ord A suc A = suc A) → ¬ A 𝑉))
215, 20syl5 28 . . 3 (A 𝑉 → (Lim suc A → ¬ A 𝑉))
2221con2d 542 . 2 (A 𝑉 → (A 𝑉 → ¬ Lim suc A))
2322pm2.43i 43 1 (A 𝑉 → ¬ Lim suc A)
Colors of variables: wff set class Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 = wceq 1228 ∈ wcel 1374 ∪ cuni 3554 Tr wtr 3828 Ord word 4048 Lim wlim 4050 suc csuc 4051 This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 532 ax-in2 533 ax-io 617 ax-5 1316 ax-7 1317 ax-gen 1318 ax-ie1 1363 ax-ie2 1364 ax-8 1376 ax-10 1377 ax-11 1378 ax-i12 1379 ax-bnd 1380 ax-4 1381 ax-17 1400 ax-i9 1404 ax-ial 1409 ax-i5r 1410 ax-ext 2004 ax-setind 4204 This theorem depends on definitions: df-bi 110 df-3an 875 df-tru 1231 df-nf 1330 df-sb 1628 df-clab 2009 df-cleq 2015 df-clel 2018 df-nfc 2149 df-ne 2188 df-ral 2289 df-rex 2290 df-v 2537 df-dif 2897 df-un 2899 df-in 2901 df-ss 2908 df-sn 3356 df-pr 3357 df-uni 3555 df-tr 3829 df-iord 4052 df-ilim 4055 df-suc 4057 This theorem is referenced by: (None)
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4.7
별점
10,701개의 평가
2,126개의 리뷰
## 강좌 소개
In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning....
## 최상위 리뷰
EC
2019년 9월 9일
Excellent review of Linear Algebra even for those who have taken it at school. Handwriting of the first instructor wasn't always legible, but wasn't too bad. Second instructor's handwriting is better.
PL
2018년 8월 25일
Great way to learn about applied Linear Algebra. Should be fairly easy if you have any background with linear algebra, but looks at concepts through the scope of geometric application, which is fresh.
필터링 기준:
## Mathematics for Machine Learning: Linear Algebra의 2,137개 리뷰 중 226~250
교육 기관: Deleted A
2020년 10월 23일
It feels a bit intimidating at first!
Then you realize that it was a while ago since you needed this part of the brain.
Things might seem simple in some videos, but trust me it pays off in the end!
The last part of this specialisation requires you to be on your toes!
교육 기관: Diogo P
2019년 7월 22일
This is an awesome course! You probably were like me, with a foundation in maths shaky due to poor understanding of the underlying principles. This course re-centers math around intuition, making it much easier to understand and apply the concepts with confidence.
교육 기관: Andi S R
2019년 12월 23일
I really like the approach of this course: build the intuition of the core concepts with an easy language and loads of examples. This has helped me a lot to understand finally the eigenvector and eigenvalues, for example. I strongly recommend to take this course.
교육 기관: Maksim S
2021년 11월 21일
Fantastic course. Sam presents the main idea in the summary: course gives understanding how mathematical skills can be applied to the real problems providing nice examples instead of blind drilling with number only. Thank you, I believe in magic of math again :)
교육 기관: Prateek S
2020년 6월 25일
This was one of the best courses I have ever had. The courses structure was awesome and the instructors were very clear with what they were teaching. The assignments were good. Anyone with a fair understanding of high school algebra should be able to understand.
교육 기관: AKSHAY K
2021년 5월 17일
Although we have learnt these topics way back in our high schools but here I get the real graphical and applicational understanding of the linear algebra. Now I can visualise thing that how it gonna work so it will definately open new doors to research for me.
2020년 9월 5일
It's been really fun, It broadens my view about linear algebra and its relation with machine learning. It also helped me a lot to understand the topic which was in my college course. My favorite part was the Gaussian elimination and the Gram-Schmidt Process.
교육 기관: SINGH S
2020년 5월 24일
I would like to say that this was one of the best courses that I've learned online during these difficult times of COVID-19 Pandemic. the teachers professor David Dye and Professor Samuel J Cooper were very friendly in teaching , all my concepts got cleared.
교육 기관: Sol S
2021년 4월 17일
A really great course for building foundational intuition around linear algebra. Great for data folks who are used to applying libraries (e.g. scikit-learn, numpy, scipy), but want to gain a deeper understanding of what the methods are doing under the hood.
교육 기관: Juan M E
2020년 9월 6일
Superb course! One of the best I have taken. I already knew linear algebra, but you always find rich tips and different ways of understanding the main topics. Congratulations to both professors. It is quite visible the passion they have and try to share it.
교육 기관: Anna U
2020년 1월 14일
An excellently simple explanation of concepts of linear algebra. Applause for lector. I really liked this course and found it very useful for those newbies in machine learning like myself. I recommend this course to all my friends and others interested in.
2020년 7월 1일
This course has an amazing way of teaching. So u understand the concepts of mathematics that was seeming harder to me before i applied for this course. If you are beginner at Machine learning and worried about mathematics you must go through this course.
교육 기관: Rishabh T
2020년 8월 6일
This is one of the best courses I have seen in coursera. The material was good and instructors were excellent. The subject and topics were explained in a very simple and interesting manner making it very easy to understand and also fun at the same time.
교육 기관: Rahul R
2020년 6월 13일
I highly recommend this course to anyone who wants to build a general understanding of linear algebra and its real-world application. Both the instructors are highly capable of communicating the intuition behind every steps and algorithm to the viewers.
교육 기관: Kamile D
2021년 6월 30일
I really enjoyed this course. Although the video material was quite dificult to comprehend at times, it was sensibly put together in short video batches and also with many tests in between that really heped to digest and fully understand the material.
교육 기관: DASGAONKAR Y N
2020년 7월 12일
This was one of the best courses for linear algebra for a working professionals, students and researchers who want to brush up their skills as this course is very different from the regular book stuff.It was more practical with real world assignments.
교육 기관: Loc N
2020년 1월 9일
Awesome course! Entertaining and digestible, with great assignments despite some hiccups in file organization and a slight lack of response from admins (understandable because the course is old). It was so awesome that I had to go and tweet the profs.
교육 기관: Lorenzo
2019년 9월 27일
it's a very well structured and well taught course. The lecturers have the ability to keep the students interested in the subject and the various exercises at the end of each session are a very good way to find out where extra work/research is needed.
교육 기관: Volodymyr C
2019년 1월 27일
Clearly explained and key equations are derived with good step sizes. Quizzes and assignments are challenging (which is good!) and have high expectations for learners (which is really good for my motivation). Overall, I am really enjoying this course.
교육 기관: Ion S
2021년 6월 3일
This course was perfect except for the last assignment. It took on average over double the time to complete (including for me), and I died multiple times :). Other than that the course helped me a lot with my understanding about matrices and vectors!
교육 기관: Agamjyot C
2020년 6월 3일
A really nice course, I had already done a Linear Algebra module in the university. But that was mostly mugging up and not knowing what this is used for. This course's geometric interpretation of all topics, helped me a lot and give a lot of insight.
교육 기관: Nut P
2020년 3월 25일
The presentation an way of teaching is excellence; however, the course should add more reference or additional source or materials for more in dept detail for the person who feel that the simplified explanation in the course are still not sufficient.
교육 기관: Steven J R
2021년 2월 25일
So exhausted, but amazed how the computer managed to process this kind of things for us everyday!
The lecturer were so nice and the explanation was so clear and funny too, although there are several assignments (which on my opinion) is sooo hard :((
교육 기관: Jonathan M
2020년 4월 10일
Extremely helpful. I haven't taken a linear algebra class in almost 5 years and by going through these videos it helped me regain an intuition towards the subject. The videos do a good job of tying the material back to machine learning as a whole.
교육 기관: Cyprien P
2020년 7월 3일
Great maths refresher content, with very useful 2D geometrics examples helping to build the intuition rather than just explaining the maths. I feel like I can understand this part of linear algebra now, and I know what to search for when I won't. | 2,085 | 8,831 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2022-21 | longest | en | 0.928993 |
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0
440
2
A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are 4, \$A\$, and 7. If the integer is a multiple of \$17_{10}\$, what is the units digit?
Guest Mar 16, 2017
#1
0
Can you clarify your question without those \$ signs?
Guest Mar 16, 2017
#2
+5
If the number is multiple of 17 to base 10, then the Hex. number is:
4A71 =19,057 / 17 =1,121.
Guest Mar 16, 2017 | 150 | 445 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2018-30 | latest | en | 0.802957 |
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Get Full Access to College Physics - 9 Edition - Chapter 23 - Problem 5
Get Full Access to College Physics - 9 Edition - Chapter 23 - Problem 5
×
# Solved: An object is placed 16.0 cm away from a convex
ISBN: 9780840062062 220
## Solution for problem 5 Chapter 23
College Physics | 9th Edition
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1
Problem 5
An object is placed 16.0 cm away from a convex mirror with a focal length of magnitude 6.00 cm. What is the location of the image? (a) 9.60 cm in front of the mirror (b) 4.36 cm in front of the mirror (c) 9.60 cm behind the mirror (d) 4.36 cm behind the mirror (e) 10.0 cm in front of the mirror
Step-by-Step Solution:
Step 1 of 3
Projectile motion can be separated into its components like so: The equations for projectile motion are: g is always negative because it points towards the ground which is in the negative direction to find the range of a projectile the range is the maximum when the angle of the projectile is at a 45 degree angle therefore, to find the time it takes to fly the range use equation 4, which, when rearranged gets you , one answer will be zero and it corresponds to before the projectile started moving to find the maximum height reached the trajectory, or path the projectile makes can be found using for circular motion: o even though an object can move in orbit ar
Step 2 of 3
Step 3 of 3
#### Related chapters
Unlock Textbook Solution | 430 | 1,612 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.328125 | 3 | CC-MAIN-2021-17 | latest | en | 0.839875 |
https://wiki-helper.com/question/9-the-product-of-two-integers-is-396-if-one-integer-is-12-find-the-other-integer-kitu-40392691-6/ | 1,632,284,957,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057329.74/warc/CC-MAIN-20210922041825-20210922071825-00592.warc.gz | 660,190,997 | 15,458 | ## 9. The product of two integers is 396. If one integer is (-12). Find the other Integer?
Question
9. The product of two integers is 396. If one integer is (-12). Find the other Integer?
in progress 0
2 months 2021-07-27T05:12:20+00:00 2 Answers 0 views 0
## Answers ( )
1. Step-by-step explanation:
hope it is helpful mark me as brainiliest
2. Answer:
x ×-12=396
x=396/-12
x= -33
Other integer is -33 | 134 | 414 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.359375 | 3 | CC-MAIN-2021-39 | latest | en | 0.733607 |
http://mathhelpforum.com/calculus/201867-definiteness-constrained-quadratics-help.html | 1,527,083,166,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794865651.2/warc/CC-MAIN-20180523121803-20180523141803-00268.warc.gz | 182,495,287 | 10,581 | 1. ## Definiteness of Constrained Quadratics - HELP
I am so stuck with these questions. Great appreciation for any help! Thanks
Determine the definiteness of the following constrained quadratic:
Q(x1,x2) = x12 +2x1x2 - x22 subject to x1 + x2 = 0
2. ## Re: Definiteness of Constrained Quadratics - HELP
Don't know what "definiteness" is (in a mathematical sense) but if you replace $\displaystyle x_2$ with $\displaystyle -x_1$ you obtain
$\displaystyle Q(x_1, x_2) = x_1^2 + 2x_1(-x_1) - (-x_1)^2$
$\displaystyle Q(x_1, x_2) = -2x_1^2$.
3. ## Re: Definiteness of Constrained Quadratics - HELP
If it's any use, you get a 'no, not positive definite' from testing by plugging the coefficients into the matrix shown in the result at the top of this pdf:
Also,
Positive definiteness - Wikipedia, the free encyclopedia
4. ## Re: Definiteness of Constrained Quadratics - HELP
Originally Posted by tom@ballooncalculus
You get a 'no, not positive definite' from testing by plugging the coefficients into the matrix shown in the result at the top of this pdf: | 307 | 1,061 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2018-22 | latest | en | 0.748731 |
https://atcoder.jp/contests/abc104/submissions/4433569?lang=en | 1,590,795,651,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347406785.66/warc/CC-MAIN-20200529214634-20200530004634-00424.warc.gz | 260,690,491 | 5,161 | Contest Duration: ~ (local time) (100 minutes) Back to Home
Submission #4433569
Source Code Expand
Copy
```import math
D,G = map(int,input().split())
L = []
for i in range(D):
L.append(list(map(int,input().split())))
n = 2
ans = float('inf')
def Base_10_to_n(X, n):
if (int(X/n)):
return Base_10_to_n(int(X/n),n)+str(X%n)
return str(X%n)
for i in range(2**D):
K = Base_10_to_n(i, n).zfill(D)
S = 0
sub = 0
for j in range(D):
if K[j] == '1':
S += L[j][0]*100*(j+1)+L[j][1]
sub += L[j][0]
if S >= G:
if sub <= ans:
ans = sub
else:
cnt = 0
for l in range(D):
if cnt == 0:
if K[D-l-1] == '0':
cnt = 1
if G-S <= L[D-l-1][0]*100*(D-l)+L[D-l-1][1]:
if G-S > L[D-l-1][0]*100*(D-l):
if ans >= sub+L[D-l-1][0]:
ans = sub+L[D-l-1][0]
else:
if ans >= sub+math.ceil((G-S)/(100*(D-l))):
ans = sub+math.ceil((G-S)/(100*(D-l)))
print(ans)```
#### Submission Info
Submission Time 2019-03-02 23:23:11+0900 C - All Green Syuko4omi Python3 (3.4.3) 300 928 Byte AC 29 ms 3064 KB
#### Test Cases
Set Name Score / Max Score Test Cases
Sample 0 / 0 a01, a02, a03, a04
All 300 / 300 a01, a02, a03, a04, b05, b06, b07, b08, b09, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24
Case Name Status Exec Time Memory
a01 20 ms 3064 KB
a02 18 ms 3064 KB
a03 18 ms 3064 KB
a04 18 ms 3064 KB
b05 18 ms 3064 KB
b06 29 ms 3064 KB
b07 28 ms 3064 KB
b08 18 ms 3064 KB
b09 18 ms 3064 KB
b10 18 ms 3064 KB
b11 19 ms 3064 KB
b12 21 ms 3064 KB
b13 23 ms 3064 KB
b14 28 ms 3064 KB
b15 29 ms 3064 KB
b16 29 ms 3064 KB
b17 28 ms 3064 KB
b18 28 ms 3064 KB
b19 28 ms 3064 KB
b20 20 ms 3064 KB
b21 23 ms 3064 KB
b22 28 ms 3064 KB
b23 28 ms 3064 KB
b24 28 ms 3064 KB | 744 | 1,649 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2020-24 | latest | en | 0.197614 |
http://comscigate.com/java/course/KjellJavaIntro/Notes/chap22/progExercises22.html | 1,519,005,931,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891812306.16/warc/CC-MAIN-20180219012716-20180219032716-00457.warc.gz | 84,693,016 | 1,933 | created 08/13/99
# Chapter 22 Programming Exercises
Many of your programs from previous chapters can be used with file input and output without modification. Pick a few of them and play. Perhaps make a few modifications, such as removing prompts.
## Exercise 1 --- Maximum
Write a program that reads 5 integers from a file, computes their and their maximum and prints these values to the screen. Do this by modifying the summing program from the chapter. Insert a new `int` variable called `max` which you should initialize to the first value in the file. This will call for an extra set of input statements before the loop starts. To compute the maximum you will need an `if` statement nested inside the loop.
## Exercise 2 --- Floating Point Input
Write a program that adds up five floating point numbers input from a file (or from the keyboard.) As always with input from text files, use `readLine()` to get a string of characters. This string must contain only characters that designate a floating point number, like -0.184 or 1.23E+12. Characters that designate an integer, like 412 will also work for floating point input. Recall how to convert a string into a floating point number:
```String lineIn;
double val;
. . .
val = ( Double.valueOf( lineIn ) ).doubleValue();
```
You will need to use the `equals()` method of `String`. Set a boolean variable to true or false depending on the user's first input string. There will be an `if` statement inside the loop that determines if a prompt is written. | 334 | 1,517 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2018-09 | latest | en | 0.861857 |
https://discuss.leetcode.com/topic/29428/for-those-who-is-not-so-clear-about-inorder-successors | 1,513,622,841,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948619804.88/warc/CC-MAIN-20171218180731-20171218202731-00509.warc.gz | 545,120,837 | 10,181 | # For those who is not so clear about inorder successors.
• Here is a good explaination I found:
http://stackoverflow.com/questions/5471731/in-order-successor-in-binary-search-tree
If you have right child, do this approach (case 1 above):
If you don't have a right child, do this approach (case 2 above):
• Python solution of iterative and recursive:
#method 1: iterative :
``````def inorderSuccessor(self,root,p):
suc=None
while root:
# if root.val>p.val: #if root.val=p.val we still move right, cause we need find a value greater than him.
#greater value is successor candidate:
suc=root
root=root.left
else:
root=root.right
return suc
``````
#method 2: recursive:
``````def inorderSuccessor(self,root,p):
if root == None:return None
if root.val>p.val:#means root is at right side of p, then move left:
suc=self.inorderSuccessor(root.left,p)
#suc is none means this is a dead end, and since we are moving left, so successor is the previous one, which is root.
if suc !=None:
return suc
else:
return root
else:
#means root is at right side of p, it could be the successor,if it's left value
# Or it could be far right from P, then move left:
return self.inorderSuccessor(root.right,p)``````
• I method 1, p is not used???
• http://www.geeksforgeeks.org/?p=9999 | 342 | 1,276 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2017-51 | latest | en | 0.745961 |
https://testbook.com/question-answer/get-the-value-of-xrm-sqrt140-xsqrt1--63a9e70d91409ae0c07be4c4 | 1,723,749,604,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641311225.98/warc/CC-MAIN-20240815173031-20240815203031-00385.warc.gz | 438,796,971 | 46,119 | # Get the value of X:$$\rm \sqrt{(140-x)}=\sqrt{(12+\sqrt{169})}$$
This question was previously asked in
RPF Constable (2018) Official Paper (Held On: 03 Feb, 2019 Shift 2)
View all RPF Constable Papers >
1. 245
2. 425
3. 115
4. 135
Option 3 : 115
Free
RPF Constable Full Test 1
1.9 Lakh Users
120 Questions 120 Marks 90 Mins
## Detailed Solution
Calculation:
$$\rm \sqrt{(140-x)}=\sqrt{(12+\sqrt{169})}$$
$$\rm \sqrt{(140-x)}=\sqrt{(12+13)}$$
$$\rm \sqrt{(140-x)}=\sqrt{(25)}$$
$${(140-x)}=25$$
x = 140 - 25
x = 115
∴ Option 3 is the correct answer.
Last updated on Jul 25, 2024
-> RPF Constable 2024 Notification has been released. A total of 4208 Vacancies have been announced for the post of Constable. Interested candidates had applied online from the 15th April to 14th May 2024.
-> Earlier, Candidates had paid the application fee online from 18th to 20th May 2024.
-> The application modification window was active from 15th to 24th May 2024.
-> A total of 4208 Vacancies have been announced for the post of Constable.
-> The RPF Constable vacancy information is released for postings all across India.
-> Candidates planning to appear for the exam can refer to the RPF Constable Previous Year Papers to analyze the level of the exam and increase the chances of selection. | 398 | 1,299 | {"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} | 4.09375 | 4 | CC-MAIN-2024-33 | latest | en | 0.891931 |
https://www.physicsforums.com/threads/electromagnetism-problem.896522/ | 1,527,254,910,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794867092.48/warc/CC-MAIN-20180525121739-20180525141739-00624.warc.gz | 807,903,749 | 15,981 | # Homework Help: Electromagnetism problem
Tags:
1. Dec 9, 2016
### nafuturo
https://postimg.org/image/lg27z8ngn/
1. The problem statement, all variables and given/known data
in the first one. I need to find the electrical force
in the second one. Lorentz force which exerts the wire on the other element
third. Magnetization (M) for Nickel
quarter. Electromotive force induced on the dipstick
Translate:
1.)If the power of a dipolar bond between two water molecules is represented by equation: (IMAGE) What is the electric force of attraction between the two molecules 2.)If a current (I) I2 = 3 A flows through the rectangle of the figure and through the infinite straight wire flows a current (I) I1 = 2, find the Lorentz force which exerts the wire on the other element 3.)Calculate the theoretical value of magnetization M for nickel (Z = 28), if two unpaired 3d electrons (d ^ 34s ^ 3) contribute to the magnetization. The nickel is FCC with a = 0.352 um as seen in the figure (IMAGE) 4.)A duct rod of length l = 20 cm moves parallel to a straight cable with a constant speed = 3 m / s, as in the figure. If a current I = 15 A flows through the cable, fiend the Electromotive force induced on the dipstick
2. Dec 9, 2016 | 333 | 1,236 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2018-22 | longest | en | 0.902784 |
https://www.physicsforums.com/threads/arc-length-of-curve-find-solution-2-2-to-2-4.438183/ | 1,723,055,030,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640707024.37/warc/CC-MAIN-20240807174317-20240807204317-00379.warc.gz | 719,161,930 | 18,824 | # Arc Length of Curve: Find Solution (-2,2) to (2,4)
• sepah50
In summary: A1=&A2=&A3=&A4=&A5=&A6=&A7=&A8=&A9=&output=tableIn summary, the homework statement is to find the arclength of the section y=x2 between points (-2,2) and (2,4). The Attempt at a Solution found that y'=2x and then used the integral table to find the arclength to be 1.4. The other intergral 4x^2=(2x)^2 can be integrated out by parts (and using some elementary substitutions). Thanks for the advice.
sepah50
## Homework Statement
Find the arclength of the section y=x2 between points (-2,2) and (2,4)
## Homework Equations
L = $$\int\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}$$
## The Attempt at a Solution
So what I did first is find the derivative of y=x2 which is y'=2x
So I put that into the formula and get $$\int\sqrt{1+4x^2}$$ from limit point 1 to 4
From this point on I attempted to do u-substitution but it didn;t seem to work.. Anyone have any suggestions o how I can get it done?
Last edited:
Trig substitution works well here.
Char. Limit said:
Trig substitution works well here.
Yes, some people have told me to use Trigonometric substitution but the thing is that my teacher hasn't taught us that yet!
Represent it as:
$$\sqrt{1 + 4 x^{2}} = \frac{1 + 4 x^{2}}{\sqrt{1 + 4 x^{2}}} = \frac{1}{\sqrt{1 + 4 x^{2}}} + \frac{4 x^{2}}{\sqrt{1 + 4 x^{2}}}$$
The integral:
$$\int{\frac{dt}{\sqrt{1 + t^{2}}}} = \sinh^{-1}{(t)} = \ln{(t + \sqrt{1 + t^{2}})}$$
is sometimes quoted as a table integral. To my knowledge, there is no other way to prove except by those substitutions mentioned earlier, or hyperbolic ones.
The other intergral:
$$4 \, \int{\frac{x^{2} \, dx}{\sqrt{1 + 4 x^{2}}}}$$
can be integrated out by parts (and using some elementary substitutions). Good luck!
Well, basically here's how it works. Let's say that...
$$2x=tan(\theta)$$
Then this is also true.
$$2 dx = sec^2(\theta) d\theta$$
Just plug those values in for dx and 4x^2=(2x)^2 and then you have a new integral in theta.
But don't forget to change your bounds too.
Char. Limit said:
Well, basically here's how it works. Let's say that...
$$2x=tan(\theta)$$
Then this is also true.
$$2 dx = sec^2(\theta) d\theta$$
Just plug those values in for dx and 4x^2=(2x)^2 and then you have a new integral in theta.
But don't forget to change your bounds too.
Thanks for the advice, it is much appreciated! What I found out is that I can use the integral table and use this formula: http://www.sosmath.com/tables/integral/integ11/integ11.html" It is #8 of this list. Would that work?
Last edited by a moderator:
## 1. What is the formula for finding the arc length of a curve?
The formula for finding the arc length of a curve is L = ∫ √(1+(dy/dx)^2) dx, where dy/dx represents the derivative of the curve.
## 2. How is the starting and ending point of the curve determined?
The starting and ending point of the curve are determined by the given coordinates, (-2,2) and (2,4) in this case. These points represent the beginning and end of the curve.
## 3. Can you explain the steps for finding the arc length of a curve?
First, find the derivative of the curve. Then, plug the derivative into the formula L = ∫ √(1+(dy/dx)^2) dx. Integrate the expression and substitute in the coordinates of the starting and ending points. Finally, solve for L, which represents the arc length of the curve.
## 4. Is there a shortcut for finding the arc length of a straight line?
Yes, the arc length of a straight line can be found using the Pythagorean Theorem, where L = √((x2-x1)^2 + (y2-y1)^2), where (x1,y1) and (x2,y2) represent the starting and ending points of the line.
## 5. Can the arc length of a curve be negative?
No, the arc length of a curve cannot be negative. It represents the distance along the curve and therefore, is always a positive value.
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895 | 1,297 | 4,396 | {"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} | 4.46875 | 4 | CC-MAIN-2024-33 | latest | en | 0.871369 |
https://mrorr-isageek.com/tag/imovie/ | 1,709,252,542,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474893.90/warc/CC-MAIN-20240229234355-20240301024355-00587.warc.gz | 414,847,880 | 29,424 | # Animated Patterns Gallery Walk
A major expectation for our grade 9 applied class is to “connect various representations of a linear relation, and solve problems using the representations.” Early in the spiralled grade 9 course I bring in Fawn’s Visual Patterns website as warm ups. We routinely continue the patterns, create tables, equations, and graphs to show the representations. Students also create their own patterns.
More and more I notice that grade 9 applied students don’t see what I see when looking at patterns (which is definitely not a bad thing). I love hearing all about how students see the patterns. However, I always see the patterns as growing/shrinking…..what I mean is that I see that one shape morphing into a bigger/smaller version. What I’ve heard from some students though is that they see each figure as a separate object, separate things that looks slightly different. I wanted to explore if students seeing the patterns morph instead of seeing them as separate objects could help them with seeing connections among the different forms of the relation.
To start the class I showed this video:
I asked: What do you notice?
Students described the pattern to each other while sitting in pairs. We decided that if the first set of shapes represented figure 1….then every figure after that showed two more shapes being added in. I asked them to go ahead and find out how many shapes were in figure 108.
I gave out the following set of instructions:
• Create your own animated pattern video
• Create a tough pattern for your classmates to discover. Ex: Show how the pattern changes in other ways than figure 1 then figure 2 then figure 3. Maybe show how your pattern changes from figure 1 to figure 3 then figure 5.
• Display your video around the room for a gallery walk. In your display hide the table and equation and answer to your question.
They went to work on building & shooting their patterns. Having them skip figure numbers made them really think about how to create their patterns and how the equations related. Since they were invested in their own patterns they worked hard at creating the tables and equations.
After they created their video they were to create a display for a gallery walk. The gallery walk gave us a purpose to practice finding rates of change, determining equations, generating equations and solving problems. We wanted to see the creative patterns from our classmates and see if we could solve each others problems. Like a challenge! Each display showed the video and then under flap of paper was an answer to a problem with a table and equation. Students left their display and visited each others displays with a recording sheet.
We spent two class days working on building the videos/patterns and the gallery walk. There are a variety of stop animation apps on the app store. My students used various different ones. Some students used iMovie.
I felt students were stronger on knowing why we need to find the rate of change for our equations and not just take the first difference value. The one-two combo of actually building the patterns and then making them move through animation built a deeper understanding of the representations than just completing a worksheet!! | 634 | 3,244 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2024-10 | longest | en | 0.963394 |
https://dmoj.ca/problem/cco13p1 | 1,604,196,566,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107922746.99/warc/CC-MAIN-20201101001251-20201101031251-00059.warc.gz | 283,064,831 | 8,876 | ## CCO '13 P1 - All Your Base Belong to Palindromes
View as PDF
Points: 10 (partial)
Time limit: 1.0s
Memory limit: 512M
Problem type
Allowed languages
Ada, Assembly, Awk, Brain****, C, C#, C++, COBOL, CommonLisp, D, Dart, F#, Forth, Fortran, Go, Groovy, Haskell, Intercal, Java, JS, Kotlin, Lisp, Lua, Nim, ObjC, OCaml, Octave, Pascal, Perl, PHP, Pike, Prolog, Python, Racket, Ruby, Rust, Scala, Scheme, Sed, Swift, TCL, Text, Turing, VB, Zig
##### Canadian Computing Competition: 2013 Stage 2, Day 1, Problem 1
Most of the time, humans have fingers. This fact is the main reason that our numbering system is base-: the number really means . Notice that each digit in base- is in the range from .
Of course, there are other bases we can use: binary (base-), octal (base-) and hexadecimal (base-) are common bases that really cool people use when trying to impress others. In base-, the digits are in the range from , with each digit (when read from right to left) being the multiplier of next larger power of .
So, for example (in base-) is:
• in base-
• in base- ()
• in base- ()
Noticing the above, you can see that is a palindrome in these three different bases. A palindrome is a sequence which is the same even if it is written in reverse order: English words such as dad, mom, and racecar are palindromes, and numbers like , , and are also palindromes.
Given a particular number (in base-), for what bases () is the representation of in base- a palindrome?
#### Input Specification
There will be one line, containing the integer ().
For test cases worth of the points, you may assume .
#### Output Specification
The output should consist of a sequence of increasing integers, each on its own line, indicating which bases have the property that written in that base is a palindrome. Note that we will only concern ourselves with bases which are less than , and that the first possible valid base is .
#### Sample Input
9
#### Output for Sample Input
2
8
#### Explanation of Output for Sample Input
The number was shown to be a palindrome in base- and in base- in the problem description. The other bases do not lead to palindromes. For example, in base-, is expressed as , and in base-, is expressed as .
• commented on Nov. 2, 2017, 8:09 a.m. edit 3
(fixed by Kirito)
• commented on Nov. 2, 2017, 12:56 p.m.
Fixed. Thanks for pointing that out!
• commented on March 15, 2015, 9:17 p.m.
How can I express something higher than base 62?
• commented on March 15, 2015, 9:53 p.m.
Instead of representing the digits with numbers and/or letters, you can use just a vector of numbers.
Base 62 :
• commented on March 15, 2015, 11:17 p.m.
Thanks! | 711 | 2,675 | {"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} | 3.328125 | 3 | CC-MAIN-2020-45 | longest | en | 0.89271 |
https://www.coursehero.com/file/6468256/h7color/ | 1,529,571,452,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864110.40/warc/CC-MAIN-20180621075105-20180621095105-00202.warc.gz | 802,225,002 | 62,106 | h7color - ECE 580 Math 587 SPRING 2011 Correspondence 18...
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ECE 580 / Math 587 SPRING 2011 Correspondence # 18 April 23, 2011 ASSIGNMENT 7 Reading Assignment: Text: Chapters 7 (7.1 - 7.8), 8 , and 9 (9.1-9.5). Recommended Reading: Curtain & Pritchard: Chapter 12; Balakrishnan: Chapter 2. Problems (to be handed in): Due Date: Tuesday, May 3 . 48. Let f be a functional on C [0 , 1], defined by f ( x ) = x := max 0 t 1 | x ( t ) | Determine a class of functions, D C [0 , 1], so that if x D the Gateaux differential δf ( x ; h ) exists for all h , and is linear in h . Hint : First show that if x C [0 , 1] has a unique maximum at a point t o (0 , 1), which is also the unique maximum of | x ( t ) | , and h is an arbitrary element of C [0 , 1], then lim α 0 1 α {∥ x + αh ∥ − | x ( t o ) + αh ( t o ) |} = 0 . 49. i) Repeat Problem 48 above for the functional f ( x ) = max 0 t 1 x ( t ) ii) Do the same for the functional f ( x ) = 1 0 | x ( t ) | d t 50. Obtain the Gateaux differentials and Gateaux derivatives of the following transformations, and indicate in each case whether the Gateaux derivative is a linear operator or not.
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http://danaernst.com/archive/fall2011/ma4140/homework.html | 1,566,241,801,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027314904.26/warc/CC-MAIN-20190819180710-20190819202710-00213.warc.gz | 56,745,938 | 4,114 | # Homework
When doing your homework, I encourage you to use the Elements of Style of Proofs as a reference.
## Daily Homework
The following assignments are to be turned in at the end of the indicated class period. I reserve the right to modify the assignment if the need arises. These exercises will form the basis of the student-led presentations each day. Daily assignments will be graded on a $\checkmark$-system. Exercises marked with * are eligible for the formal Weekly Write-ups.
Note: On each Daily Homework, please write (i) your name, (ii) name of course, and (iii) Daily Homework number.
• Daily Homework 1: Read all of Chapter 1 and complete Problem 1 (Due Friday, September 9). Note: Those of you that haven't had linear algebra shouldn't panic when you look at the sixth example of a binary operation.
• Daily Homework 2: Re-read all of Chapter 1 and complete Problems 2, 3, 4, 5, 6* (Due Monday, September 12).
• Daily Homework 3: Read Chapter 2 up to and including the hint immediately following Theorem 10. In addition, complete Theorem 7, Theorem 8*, Corollary 9*, and Theorem 10* (Due Wednesday, September 14).
• Daily Homework 4: Read Chapter 2 up to and including Theorem 12. In addition, re-do Theorem 10* (unless you are confident in your original write-up) and complete Corollary 11 and Theorem 12* (Due Friday, September 16).
• Daily Homework 5: Read Chapter 2 up to and including Theorem 14. In addition, complete Problem 13, Theorem 14* (Due Monday, September 19).
• Daily Homework 6: Read Chapter 2 up to and including Theorem 16. In addition, complete Theorem 15* and Theorem 16* (Due Wednesday, September 21).
• Daily Homework 7: Read the rest of Chapter 2. In addition, complete Lemma 18* and Theorem 19* (Due Friday, September 23). Note: I have not asked you to prove Theorem 17, but you should read and understand it.
• Daily Homework 8: Read Chapter 3 up to and including Problem 23. In addition, complete Lemma 20*, Lemma 21*, Lemma 22, Problem 23 (Due Monday, September 26).
• Daily Homework 9: Read Chapter 3 up to and including Problem 27. In addition, complete Problem 24, Problem 25, Lemma 26*, Problem 27 (Due Friday, September 30).
• Daily Homework 10: Read Chapter 3 up to and including Corollary 29. In addition, complete Theorem 28* and Corollary 29* (Due Monday, October 3).
• Daily Homework 11: Read Chapter 3 up to and including Theorem 31. In addition, complete Lemma 30* and Theorem 31* (Due Wednesday, October 5).
• Daily Homework 12: Finish reading Chapter 3. In addition, complete Theorem 32*, Lemma 33*, Theorem 34* (Due Friday, October 7).
• Daily Homework 13: Read Chapter 4 up to and including Theorem 38. In addition, complete Theorem 38* (Due Monday, October 17).
• Daily Homework 14: Read Chapter 4 up to and including Lemma 42. In addition, complete Problem 39, Problem 40, Lemma 41*, Lemma 42* (Due Wednesday, October 19).
• Daily Homework 15: Read Chapter 4 up to and including Lagrange's Theorem. In addition, complete Lemma 43* and Theorem 44* (Due Friday, October 21).
• Daily Homework 16: Finish reading Chapter 4. In addition, complete Corollary 45*, Corollary 46*, and Theorem 47* (Due Monday, October 24).
• Daily Homework 17: Read Chapter 5 starting with paragraph before Theorem 56. In addition, complete Theorem 56* and Lemma 57* (Due Wednesday, October 26).
• Daily Homework 18: Finish reading Chapter 5. In addition, complete Theorem 58* (you may assume that function composition is associative) (Due Friday, October 28).
• Daily Homework 19: Start reading Chapter 6 up to and including Lemma 61. In addition, complete Lemma 60* and Lemma 61* (Due Monday, October 31).
• Daily Homework 20: Complete Theorem 63* (Due Wednesday, November 2).
• Daily Homework 21: Complete Lemma 64* and Theorem 65* (Due Friday, November 4).
• Daily Homework 22: Complete Theorem 68* and Lemma 69* (Due Monday, November 7).
• Daily Homework 23: Complete Theorem 70* (Due Wednesday, November 9).
• Daily Homework 24: Read Chapter 6 up to and including Theorem 72. In addition, complete Theorem 71* and Theorem 72* (Due Monday, November 14).
• Daily Homework 25: Read Chapter 6 up to and including Lemma 73. In addition, finish up Theorem 71* and Theorem 72*, and then complete Theorem 73* (Due Wednesday, November 16).
• Daily Homework 26: Read Chapter 6 up to and including Lemma 77. In addition, complete Lemma 75*, Lemma 76*, and Lemma 77* (Due Wednesday, November 30).
• Daily Homework 27: Read the rest of Chapter 6. In addition, finish up Lemma 77* if you haven't already and complete Lemma 78* and Lemma 79* (Due Friday, December 2).
• Daily Homework 28: Complete Theorem 80* and any outstanding problems. (Due Monday, December 5).
• Daily Homework 29: Complete Theorem "86" (just (i) iff (ii)), Problem 85, Problem 83, and Corollary 88 (Due Wednesday, December 7).
• Daily Homework 30: Read and understand Corollary 90 and Theorem 95. Also, complete Lemma 96 and Theorem 97. (Due Friday, December 9).
## Weekly Write-ups
In addition to the Daily Homework, you will also be required to submit two formally written proofs each week. By 5PM on Tuesday of week $n$, you should submit the proofs of any two theorems that were turned in for Daily Homework during week $n-1$. Beginning with the second Weekly Write-up, you will be required to type your submission. You can either email me your Weekly Write-up as a PDF file, share them with me via PSU Google Docs, or submit a hard copy in person.
Note: On each Weekly Write-up, please write (i) your name, (ii) name of course, and (iii) Weekly Write-up number. You should type your Weekly Write-ups using $\LaTeX$, MS Word, OpenOffice/LibreOffice, or Google Docs. If you plan to email me your file, then you should send me a PDF (if you need help with this, please let me know). Furthermore, you should name your file using the following convention:
WeeklyXLast-Name.pdf
where you replace "X" with the assignment number and replace "Last-Name" with your actual last name. Please don't put any spaces in your file name.
• Weekly Write-up 1: Choose any two *-problems from Daily Homework 2, 3, 4 (Due Tuesday, September 20 by 5PM). Note: This assignment does not need to be typed.
• Weekly Write-up 2: Choose any two *-problems from Daily Homework 5, 6, 7 (Due Friday, September 30 by 5PM). Note: Beginning with this assignment, your weekly write-ups should be typed (using $\LaTeX$, MS Word, Google Docs, etc.). Regardless of what you use to type up your assignment, if you email me your assignment, please email me a PDF file following the naming convention discussed above.
• Weekly Write-up 3: Choose any two *-problems from Daily Homework 8 and 9 (Due Tuesday, October 4 by 5PM).
• Weekly Write-up 4: Choose any two *-problems from Daily Homework 10, 11, 12 (Due Thursday, October 18 by 5PM).
• Weekly Write-up 5: Choose any two *-problems from Daily Homework 13, 14, 15 (Due Thursday, October 27 by 5PM).
• Weekly Write-up 6: Choose any two *-problems from Daily Homework 16, 17, 18 (Due Thursday, November 3 by 5PM).
• Weekly Write-up 7: Choose any two *-problems from Daily Homework 19, 20, 21, 22, 23 (Due Tuesday, November 15 by 5PM).
• Weekly Write-up 8: Choose any two *-problems from Daily Homework 24, 25, 26, 27, 28 (Due Thursday, December 8 by 5PM).
## LaTeX Template for Homework
You can find a $\LaTeX$ template for typing up your homework in my public ScribTeX folder. In this directory you will also find a help file for using $\LaTeX$ with ScribTeX. You can download the entire directory and then if you are using ScribTeX, you can upload the files you are interested in using. Alternatively, you can copy the content of the file and paste it into a new tex file.
For a rough overview of some of the basics of $\LaTeX$, you might find the Quick LaTeX Guide useful. | 2,091 | 7,803 | {"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} | 2.71875 | 3 | CC-MAIN-2019-35 | latest | en | 0.934598 |
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