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https://www.mathwarehouse.com/trigonometry/sine-cosine-tangent-practice2.php | Debug
# Sine, Cosine, Tangent Ratios
Practice writing the ratios
### Video Tutorial How to write sohcahtoa ratios, given side lengths
#### What do we mean by 'ratio' of sides?
Sine, cosine and tangent of an angle represent the ratios that are always true for given angles. Remember these ratios only apply to right triangles.
The 3 triangles pictured below illustrate this.
Diagram 1
Although the side lengths are different , each one has a 37° angle, and as you can see, the sine of 37 is always the same!
In other words, $$sin(\red { 37^{\circ} } )$$ is always $$\red {.6 }$$ .
(Note: I rounded to the nearest tenth) That's, of course, why we can use a calculator to find these sine, cosine and tangent ratios.
### Practice Problems
##### Problem 1
Step 1
In the diagram 2, what side is adjacent to $$\angle L$$?
$$\overline{ML}$$.
Step 2
What side is the hypotenuse?
$$\overline{ LN}$$
Step 3
Calculate $$cos(\angle L)$$.
Use sochcahtoa to help remember the ratio.
$cos(\angle \red L) = \frac{adjacent}{hypot} \\ cos(\angle \red L) = \frac{8}{10} \\ = .8$
Step 4
Calculate $$cos(\angle N)$$ (a different angle from prior question, look carefully at letters).
$cos(\angle \red N) = \frac{6}{10} \\ = .6$
##### Problem 2
What is the sine ratio of $$\angle C$$ ?
Remember the sine ratio
$sin(\angle C) =\frac{\text{opp } }{ hypot} \\ sin(\angle C) = \frac{6}{10} \\ sin(\angle C) = \frac{6}{10} \\ sin(\angle C) = .6$
##### Problem 3
What is the cosine ratio of $$\angle C$$ in $$\triangle ABC$$ ?
Remember the cosine ratio
$cos(\angle C) =\frac{\text{adj } }{ hypot} \\ cos(C) = \frac{4}{5} \\ cos(C) = .8$
##### Problem 4
What is the tangent ratio of $$\angle A$$ ?
Use sochcahtoa to help remember the ratio.
$tan(A) = \frac{opposite}{adjacent} \\ = \frac{24}{7} \\= 3.42857142$
##### Problem 5
Find the sine, cosine and tangent of $$\angle R$$.
Use sochcahtoa to help remember the ratios.
$$\text{ for } \angle \red R \\ \boxed { Sine } \\ sin(\red R )= \frac{opp}{hyp} \\ sin(\red R )= \frac{12}{13} \\ sin(\red R )= .923 \\ \boxed{ cosine } \\ cos(\red R)= \frac{adj}{hyp} \\ cos(\red R)= \frac{9}{13} = \\ cos(\red R).69 \\ \boxed{ tangent } \\ tan(\red R)= \frac{opp}{adj} \\ tan(\red R) = \frac{12}{9} \\ tan(\red R)= 1.3$$
##### Problem 6
What is $$sin(\angle X)$$?
$$sin(\red X ) = \frac{opp}{hyp} \\ sin(\red X ) = \frac{24}{25} \\ sin(\red X ) = .96$$
##### Problem 7
What is $$cos(\angle X)$$?
$$sin(\red X) = \frac{adj}{hyp} \\ sin(\red X) = \frac{7}{25} \\ sin(\red X) = .28$$
##### Problem 8
Calculate : $$\text{ a) } sin(\angle H) \\ \text{ b) }cos(\angle H) \\ \text{ c) } tan(\angle H)$$
$$sin(\angle \red H) = \frac{3}{5} \\ sin( \angle \red H) = .6 \\ cos(\angle \red H) =\frac{ 4}{5} \\ cos (\angle \red H)= .8 \\ tan(\angle \red H) = \frac{3}{4} \\ tan(\angle \red H)= .75$$
### Practice Problems II
##### Problem 9
Which angle below has a tangent ratio of $$\frac{3}{4}$$?
Diagram A
Diagram B
Use sochcahtoa to help remember the ratio for tangent.
$$tangent = \frac{opp}{adj}$$
So which angle has a tangent that is equivalent to $$\frac{3}{4}$$?
You only have 2 options. Either $$\angle L$$ or $$\angle K$$. So let's calculate the tangent for each angle.
$tan( \angle K) = \frac{12}{9} \\ \frac{12}{9} \red {\ne} \frac 3 4$
$tan( \angle L) = \frac{9}{12} \\ \frac{9}{12} = \frac 3 4$
##### Problem 10
Which angle below has a cosine of $$\frac{3}{5}$$?
Diagram A
Diagram B
Use sochcahtoa to help remember the ratio for tangent.
$$cosine = \frac{adj}{hyp}$$
So which angle has a cosine that is equivalent to $$\frac 3 5$$?
You only have 2 options. Either $$\angle L$$ or $$\angle K$$. So let's calculate the tangent for each angle.
$cos( \angle K) = \frac{9}{15} \\ \frac{9}{15} \red {\ne} \frac 3 4$
$cos(L) = \frac{12}{15} \\ \frac{12}{15} = \frac 3 5$
##### Problem 11
Which angle below has a tangent $$\approx$$ .29167?
This is a little trickier because you are given the ratio as a decimal; however, you only have two options. Either $$\angle A$$ or $$\angle C$$.
$$tan(\angle A) = \frac{48}{14} \\ tan(\angle A)= 3.42857$$
$$tan(\angle C) = \frac{14}{48} \\ tan(\angle C) = .29167$$
##### Problem 12
Which angle below has a tangent of 2.4?
Again, there are two options (angles R or P), but since your ratio is greater than 1 you might quickly be able to notice that it must be R.
$$tan(\angle P) = \frac{5}{12} \\ tan(\angle P) = 0. 41666$$
$$tan(\angle R) = \frac{12}{5} \\ tan(\angle R) = 2.4$$
### Word Problems
##### Problem 13
Step 1
It's easiest to do a word problem like this one, by first drawing the triangle and labelling the sides. We know the opposite side of $$\angle K$$ and we know the hypotenuse.
To get the tangent ratio we need to know the length of the adjacent side.
How can we find the length of the adjacent side?
Step 2
Use the Pythagorean theorem!
$$a^2 + b^2 = c^2 \\ 3^2 + b^2 = 5^2 \\ b^2 = 5^2- 3^2 = 25-9 = 16 \\ b = 4$$
Now, use the tangent ratio!
Step 3
Use the Pythagorean theorem!
$$tan(k) = \frac{opposite}{adjacent} \\ tan(k) = \frac{3}{4}$$
##### Problem 14
Step 1
Draw this triangle and label the sides:
Remember that the cosine ratio = $$\frac{adjacent}{hypotenuse}$$.
How can we find the length of the opposite side?
(Remember that sine involves the opposite so we need to find that somehow).
Step 2
Use the Pythagorean theorem!
$$a^2 + b^2 = c^2 \\ 7^2 + b^2 = 25^2 \\ b^2 = 25^2- 7^2 = 625 - 49 = 576 \\ b = \sqrt{576} =24$$
Now, use the sine ratio!
Step 3
$$sin(b) = \frac{opposite}{hypotenuse}$$
top | Practice I | Practice II | Word Problems | #Challenge Problem
##### Challenge Problem
Be careful!
In the triangle below, which angle has a sine ratio of 2.6?
There is no angle that has a sine of 2.6.
Remember that the maximum value of the sine ratio is 1.
The same, is true, by the way of cosine ratio (max value is 1). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9625828862190247, "perplexity": 1651.109426624075}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400198868.29/warc/CC-MAIN-20200920223634-20200921013634-00141.warc.gz"} |
https://blog.rishabh.co.in/blog/tag/power-quality-analyzer/ | ## Providing insights into inrush current monitoring and analysis using portable Power Quality Analyzer – NP45
Overview Several power quality issues occur in an electrical system like harmonics, voltage swells, dips and harmonics. Inrush current is also one such power quality issue that leads to voltage dips thereby affecting the power quality of the network in proximity to the source of causing it. These are present only for a short duration …
## Enhanced harmonic analysis with the help of Bar graph representation of harmonics in Power Quality Analyzer
Overview Nowadays, almost every industry undergoes issues related to harmonics due to the increase in usage of non-linear loads. As these loads are unavoidable because they perform important tasks like controlling or conversion, solutions to mitigate the ill effects of harmonics are to be devised. These solutions include passive filters consisting of detuned inductors and …
## Understanding cause of VFD failures using Power Quality Analyzer – Rish PQA
Overview Induction motors are widely used in industries due to their robust construction and operation. The speed of an induction motor is not required to be the same always and this depends on the requirement of the process, unnecessary operation of an induction motor at full speeds demands more energy and hence speed control is … | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8626822829246521, "perplexity": 1817.4109834379406}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335058.80/warc/CC-MAIN-20220927194248-20220927224248-00296.warc.gz"} |
https://www.physicsforums.com/threads/stress-buckling-question.842995/ | # Stress/buckling question
1. Nov 13, 2015
### Niall11
1. The problem statement, all variables and given/known
I have been asked to calculate the minimum length of this column (attatched) at which buckling is likely to occur
2. Relevant equations
E.S.R = sqrt(π^2*E / σ)
2nd moment of Area I = AK^2
E.S.R = L/K
I= π/32*(D^4-d^4)
3. The attempt at a solution
so E.S.R = sqrt(π^2*200*10^9 / 140*10^6) = 118.74
Since I = AK^2 and E.S.R = L/K
L= (E.S.R)*K = (E.S.R)*sqrt(I/A)
so I = π/32 * (0.08^4-0.06^4) = 2.75*10-6
Now L = 188.7 * sqrt(2.75*10-6 / Area)
ive got that far but don't know a suitable equation for area, I'm not looking for answers just for someone who can tell me if I'm on the right track, any help would be appreciated!!
#### Attached Files:
• ###### STRESS-BUCKLING COLUMN.PNG
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2. Nov 13, 2015
### SteamKing
Staff Emeritus
They're talking about the area of the cross section of the column. You are given the outer diameter and the inner diameter of the column.
What else do you need? Do you know how to calculate the area of a circle?
3. Nov 14, 2015
### Niall11
thanks for the fast reply, I was thinking I needed the area of the whole column for some reason.
yes area of circle = π*r^2 so area of outer circle - area of inner circle = cross sectional area of column;
π (0.04)^2 - π(0.03)^2 = 2.199*10^-3 m^2
meaning L = 118.7 * sqrt(2.75*10^-6 / 2.199*10^-3) = 4.19m
effective length is 1/2 of the column length for a column which both ends fixed therefore length = 4.19m * 2 = 8.38m
4. Nov 14, 2015
### SteamKing
Staff Emeritus
This formula for the second moment of area of the column is incorrect. You used the polar moment, which is Ix + Iy
You are supposed to use the least value of the second moment of area, which for a circle is I = (π/64)D4
Since this is a hollow cylinder, I = (π/64)*(Do4 - Di4)
You'll have to re-do your calculations of the length of the column accordingly.
5. Nov 14, 2015
### Niall11
oh right I have on my notes that least 2nd moment of area is π/32(D^4-d^4)!
so least 2nd moment of area = π/64 (D^4-d^4) = π/64 (0.08^4-0.06^4) = 1.374 * 10^-6
now effective L comes out at 2.94m and length 5.94m!
I'm hoping that is correct if i've used all the right formulae
6. Nov 14, 2015
### SteamKing
Staff Emeritus
Yes, your length calculation looks correct now.
FWIW, this same problem has cropped up on PF several times before.
7. Nov 14, 2015
### Niall11
Thanks a lot that was a big help, oh right I didn't realise ill look next time!
8. Nov 16, 2015
### Marv K
If the effective length is half of the actual length, are those figures not incorrect? I only ask cause I'm doing the same course and wanted to check. Is the effective length not 2.968m?
9. Nov 16, 2015
### SteamKing
Staff Emeritus
The effective length is approx. 2.94 m, but the OP made a slight error in doubling this figure to find the actual length, which should be approx. 5.88 m.
10. Nov 16, 2015
### Marv K
Ok, sorry to be annoying, but worried Ive done it wrong now.
If E.S.R. = 118.74
I = 1.374*10^-6
A = 2.199*10^-3
Then the equation is Le = 118.74*sqrt(1.374*10^-6)/(2.199*10^-3)
Which equals 2.968m?
Also, sorry if I've written that equation terribly.
11. Nov 16, 2015
### SteamKing
Staff Emeritus
I'm not sure what you point is here. Le = 2.968 m or Le = 2.94 m is the same number for all intents, given that this is a buckling problem.
Depending on how you round the intermediate computations, there will be some variation in the fractional portion of the final result. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8699429035186768, "perplexity": 2435.496996522156}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814300.52/warc/CC-MAIN-20180222235935-20180223015935-00667.warc.gz"} |
https://discrete.openmathbooks.org/dmoi3/sec_counting-combperm.html | ## Section1.3Combinations and Permutations
###### Investigate!
You have a bunch of chips which come in five different colors: red, blue, green, purple and yellow.
1. How many different two-chip stacks can you make if the bottom chip must be red or blue? Explain your answer using both the additive and multiplicative principles.
2. How many different three-chip stacks can you make if the bottom chip must be red or blue and the top chip must be green, purple or yellow? How does this problem relate to the previous one?
3. How many different three-chip stacks are there in which no color is repeated? What about four-chip stacks?
4. Suppose you wanted to take three different colored chips and put them in your pocket. How many different choices do you have? What if you wanted four different colored chips? How do these problems relate to the previous one?
A permutation is a (possible) rearrangement of objects. For example, there are 6 permutations of the letters a, b, c:
\begin{equation*} abc, ~~ acb, ~~ bac, ~~bca, ~~ cab, ~~ cba\text{.} \end{equation*}
We know that we have them all listed above —there are 3 choices for which letter we put first, then 2 choices for which letter comes next, which leaves only 1 choice for the last letter. The multiplicative principle says we multiply $$3\cdot 2 \cdot 1\text{.}$$
### Example1.3.1.
How many permutations are there of the letters a, b, c, d, e, f?
Solution.
We do NOT want to try to list all of these out. However, if we did, we would need to pick a letter to write down first. There are 6 choices for that letter. For each choice of first letter, there are 5 choices for the second letter (we cannot repeat the first letter; we are rearranging letters and only have one of each), and for each of those, there are 4 choices for the third, 3 choices for the fourth, 2 choices for the fifth and finally only 1 choice for the last letter. So there are $$6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$$ permutations of the 6 letters.
A piece of notation is helpful here: $$n!\text{,}$$ read “$$n$$ factorial”, is the product of all positive integers less than or equal to $$n$$ (for reasons of convenience, we also define 0! to be 1). So the number of permutation of 6 letters, as seen in the previous example is $$6! = 6\cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\text{.}$$ This generalizes:
### Permutations of $$n$$ elements.
There are $$n! = n\cdot (n-1)\cdot (n-2)\cdot \cdots \cdot 2\cdot 1$$ permutations of $$n$$ (distinct) elements.
### Example1.3.2.Counting Bijective Functions.
How many functions $$f:\{1,2,\ldots,8\} \to \{1,2,\ldots, 8\}$$ are bijective?
Solution.
Remember what it means for a function to be bijective: each element in the codomain must be the image of exactly one element of the domain. Using two-line notation, we could write one of these bijections as
\begin{equation*} f = \twoline{1 \amp 2 \amp 3 \amp 4 \amp 5 \amp 6 \amp 7 \amp 8} {3 \amp 1 \amp 5 \amp 8 \amp 7 \amp 6 \amp 2 \amp 4}\text{.} \end{equation*}
What we are really doing is just rearranging the elements of the codomain, so we are creating a permutation of 8 elements. In fact, “permutation” is another term used to describe bijective functions from a finite set to itself.
If you believe this, then you see the answer must be $$8! = 8 \cdot 7 \cdot\cdots\cdot 1 = 40320\text{.}$$ You can see this directly as well: for each element of the domain, we must pick a distinct element of the codomain to map to. There are 8 choices for where to send 1, then 7 choices for where to send 2, and so on. We multiply using the multiplicative principle.
Sometimes we do not want to permute all of the letters/numbers/elements we are given.
### Example1.3.3.
How many 4 letter “words” can you make from the letters a through f, with no repeated letters?
Solution.
This is just like the problem of permuting 4 letters, only now we have more choices for each letter. For the first letter, there are 6 choices. For each of those, there are 5 choices for the second letter. Then there are 4 choices for the third letter, and 3 choices for the last letter. The total number of words is $$6\cdot 5\cdot 4 \cdot 3 = 360\text{.}$$ This is not $$6!$$ because we never multiplied by 2 and 1. We could start with $$6!$$ and then cancel the 2 and 1, and thus write $$\frac{6!}{2!}\text{.}$$
In general, we can ask how many permutations exist of $$k$$ objects choosing those objects from a larger collection of $$n$$ objects. (In the example above, $$k = 4\text{,}$$ and $$n = 6\text{.}$$) We write this number $$P(n,k)$$ and sometimes call it a $$k$$-permutation of $$n$$ elements. From the example above, we see that to compute $$P(n,k)$$ we must apply the multiplicative principle to $$k$$ numbers, starting with $$n$$ and counting backwards. For example
\begin{equation*} P(10, 4) = 10\cdot 9 \cdot 8 \cdot 7\text{.} \end{equation*}
Notice again that $$P(10,4)$$ starts out looking like $$10!\text{,}$$ but we stop after 7. We can formally account for this “stopping” by dividing away the part of the factorial we do not want:
\begin{equation*} P(10,4) = \frac{10\cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = \frac{10!}{6!}\text{.} \end{equation*}
Careful: The factorial in the denominator is not $$4!$$ but rather $$(10-4)!\text{.}$$
### $$k$$-permutations of $$n$$ elements.
$$P(n,k)$$ is the number of $$k$$-permutations of $$n$$ elements, the number of ways to arrange $$k$$ objects chosen from $$n$$ distinct objects.
\begin{equation*} P(n,k) = \frac{n!}{(n-k)!} = n(n-1)(n-2)\cdots (n-(k-1))\text{.} \end{equation*}
Note that when $$n = k\text{,}$$ we have $$P(n,n) = \frac{n!}{(n-n)!} = n!$$ (since we defined $$0!$$ to be 1). This makes sense —we already know $$n!$$ gives the number of permutations of all $$n$$ objects.
### Example1.3.4.Counting injective functions.
How many functions $$f:\{1,2,3\} \to \{1,2,3,4,5,6,7,8\}$$ are injective?
Solution.
Note that it doesn't make sense to ask for the number of bijections here, as there are none (because the codomain is larger than the domain, there are no surjections). But for a function to be injective, we just can't use an element of the codomain more than once.
We need to pick an element from the codomain to be the image of 1. There are 8 choices. Then we need to pick one of the remaining 7 elements to be the image of 2. Finally, one of the remaining 6 elements must be the image of 3. So the total number of functions is $$8\cdot 7 \cdot 6 = P(8,3)\text{.}$$
What this demonstrates in general is that the number of injections $$f:A \to B\text{,}$$ where $$\card{A} = k$$ and $$\card{B} = n\text{,}$$ is $$P(n,k)\text{.}$$
Here is another way to find the number of $$k$$-permutations of $$n$$ elements: first select which $$k$$ elements will be in the permutation, then count how many ways there are to arrange them. Once you have selected the $$k$$ objects, we know there are $$k!$$ ways to arrange (permute) them. But how do you select $$k$$ objects from the $$n\text{?}$$ You have $$n$$ objects, and you need to choose $$k$$ of them. You can do that in $${n \choose k}$$ ways. Then for each choice of those $$k$$ elements, we can permute them in $$k!$$ ways. Using the multiplicative principle, we get another formula for $$P(n,k)\text{:}$$
\begin{equation*} P(n,k) = {n \choose k}\cdot k!\text{.} \end{equation*}
Now since we have a closed formula for $$P(n,k)$$ already, we can substitute that in:
\begin{equation*} \frac{n!}{(n-k)!} = {n \choose k} \cdot k!\text{.} \end{equation*}
If we divide both sides by $$k!$$ we get a closed formula for $${n \choose k}\text{.}$$
### Closed formula for $${n \choose k}$$.
\begin{equation*} {n \choose k} = \frac{n!}{(n-k)!k!} = \frac{n(n-1)(n-2)\cdots(n-(k-1))}{k(k-1)(k-2)\cdots 1}\text{.} \end{equation*}
We say $$P(n,k)$$ counts permutations, and $${n \choose k}$$ counts combinations. The formulas for each are very similar, there is just an extra $$k!$$ in the denominator of $${n \choose k}\text{.}$$ That extra $$k!$$ accounts for the fact that $${n \choose k}$$ does not distinguish between the different orders that the $$k$$ objects can appear in. We are just selecting (or choosing) the $$k$$ objects, not arranging them. Perhaps “combination” is a misleading label. We don't mean it like a combination lock (where the order would definitely matter). Perhaps a better metaphor is a combination of flavors — you just need to decide which flavors to combine, not the order in which to combine them.
To further illustrate the connection between combinations and permutations, we close with an example.
### Example1.3.5.
You decide to have a dinner party. Even though you are incredibly popular and have 14 different friends, you only have enough chairs to invite 6 of them.
1. How many choices do you have for which 6 friends to invite?
2. What if you need to decide not only which friends to invite but also where to seat them along your long table? How many choices do you have then?
Solution.
1. You must simply choose 6 friends from a group of 14. This can be done in $${14 \choose 6}$$ ways. We can find this number either by using Pascal's triangle or the closed formula: $$\frac{14!}{8!\cdot 6!} = 3003\text{.}$$
2. Here you must count all the ways you can permute 6 friends chosen from a group of 14. So the answer is $$P(14, 6)\text{,}$$ which can be calculated as $$\frac{14!}{8!} = 2162160\text{.}$$
Notice that we can think of this counting problem as a question about counting functions: how many injective functions are there from your set of 6 chairs to your set of 14 friends (the functions are injective because you can't have a single chair go to two of your friends).
How are these numbers related? Notice that $$P(14,6)$$ is much larger than $${14 \choose 6}\text{.}$$ This makes sense. $${14 \choose 6}$$ picks 6 friends, but $$P(14,6)$$ arranges the 6 friends as well as picks them. In fact, we can say exactly how much larger $$P(14,6)$$ is. In both counting problems we choose 6 out of 14 friends. For the first one, we stop there, at 3003 ways. But for the second counting problem, each of those 3003 choices of 6 friends can be arranged in exactly $$6!$$ ways. So now we have $$3003\cdot 6!$$ choices and that is exactly $$2162160\text{.}$$
Alternatively, look at the first problem another way. We want to select 6 out of 14 friends, but we do not care about the order they are selected in. To select 6 out of 14 friends, we might try this:
\begin{equation*} 14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9\text{.} \end{equation*}
This is a reasonable guess, since we have 14 choices for the first guest, then 13 for the second, and so on. But the guess is wrong (in fact, that product is exactly $$2162160 = P(14,6)$$). It distinguishes between the different orders in which we could invite the guests. To correct for this, we could divide by the number of different arrangements of the 6 guests (so that all of these would count as just one outcome). There are precisely $$6!$$ ways to arrange 6 guests, so the correct answer to the first question is
\begin{equation*} \frac{14 \cdot 13 \cdot 12 \cdot 11\cdot 10 \cdot 9}{6!}\text{.} \end{equation*}
Note that another way to write this is
\begin{equation*} \frac{14!}{8!\cdot 6!}\text{.} \end{equation*}
which is what we had originally.
### ExercisesExercises
#### 1.
A pizza parlor offers 10 toppings.
1. How many 3-topping pizzas could they put on their menu? Assume double toppings are not allowed.
2. How many total pizzas are possible, with between zero and ten toppings (but not double toppings) allowed?
3. The pizza parlor will list the 10 toppings in two equal-sized columns on their menu. How many ways can they arrange the toppings in the left column?
Solution.
1. $${10 \choose 3} = 120$$ pizzas. We must choose (in no particular order) 3 out of the 10 toppings.
2. $$2^{10} = 1024$$ pizzas. Say yes or no to each topping.
3. $$P(10,5) = 30240$$ ways. Assign each of the 5 spots in the left column to a unique pizza topping.
#### 2.
A combination lock consists of a dial with 40 numbers on it. To open the lock, you turn the dial to the right until you reach a first number, then to the left until you get to second number, then to the right again to the third number. The numbers must be distinct. How many different combinations are possible?
Solution.
Despite its name, we are not looking for a combination here. The order in which the three numbers appears matters. There are $$P(40,3) = 40\cdot 39 \cdot 38$$ different possibilities for the “combination”. This is assuming you cannot repeat any of the numbers (if you could, the answer would be $$40^3$$).
#### 3.
Using the digits 2 through 8, find the number of different 5-digit numbers such that:
1. Digits can be used more than once.
2. Digits cannot be repeated, but can come in any order.
3. Digits cannot be repeated and must be written in increasing order.
4. Which of the above counting questions is a combination and which is a permutation? Explain why this makes sense.
Solution.
1. This is just the multiplicative principle. There are 7 digits which we can select for each of the 5 positions, so we have $$7^5 = 16807$$ such numbers.
2. Now we have 7 choices for the first number, 6 for the second, etc. So there are $$7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 = P(7,5) = 2520$$ such numbers.
3. To build such a number we simply must select 5 different digits. After doing so, there will only be one way to arrange them into a 5-digit number. Thus there are $${7 \choose 5} = 21$$ such numbers.
4. The permutation is in part (b), while the combination is in part (c). At first this seems backwards, since usually we use combinations for when order does not matter. Here it looks like in part (c) that order does matter. The better way to distinguish between combinations and permutations is to ask whether we are counting different arrangements as different outcomes. In part (c), there is only one arrangement of any set of 5 digits, while in part (b) each set of 5 digits gives $$5!$$ different outcomes.
#### 4.
In an attempt to clean up your room, you have purchased a new floating shelf to put some of your 17 books you have stacked in a corner. These books are all by different authors. The new book shelf is large enough to hold 10 of the books.
1. How many ways can you select and arrange 10 of the 17 books on the shelf? Notice that here we will allow the books to end up in any order. Explain.
2. How many ways can you arrange 10 of the 17 books on the shelf if you insist they must be arranged alphabetically by author? Explain.
Hint.
Which question should have the larger answer? One of these is a combination, the other is a permutation.
#### 5.
Suppose you wanted to draw a quadrilateral using the dots below as vertices (corners). The dots are spaced one unit apart horizontally and two units apart vertically.
How many are squares?
How many are rectangles?
How many are parallelograms?
How many are trapezoids? (Here, as in calculus, a trapezoid is defined as a quadrilateral with at least one pair of parallel sides. In particular, parallelograms are trapezoids.)
How many are trapezoids that are not parallelograms?
Solution.
You can make $${7\choose 2}{7\choose 2} = 441$$ quadrilaterals.
There are 5 squares.
There are $${7 \choose 2}$$ rectangles.
There are $${7 \choose 2} + ({7 \choose 2}-1) + ({7 \choose 2} - 3) + ({7 \choose 2} - 6) + ({7 \choose 2} - 10) + ({7 \choose 2} - 15) = 91$$ parallelograms.
All of the quadrilaterals are trapezoids. To count the non-parallelogram trapezoids, compute $${7\choose 2}{7\choose 2} - \left[ {7 \choose 2} + ({7 \choose 2}-1) + ({7 \choose 2} - 3) + ({7 \choose 2} - 6) + ({7 \choose 2} - 10) + ({7 \choose 2} - 15) \right]\text{.}$$
#### 6.
How many triangles are there with vertices from the points shown below? Note, we are not allowing degenerate triangles - ones with all three vertices on the same line, but we do allow non-right triangles. Explain why your answer is correct.
Hint.
If you pick any three points, you can get a triangle, unless those three points are all on the $$x$$-axis or on the $$y$$-axis. There are other ways to start this as well, and any correct method should give the same answer.
Solution.
120.
#### 7.
An anagram of a word is just a rearrangement of its letters. How many different anagrams of “uncopyrightable” are there? (This happens to be the longest common English word without any repeated letters.)
Solution.
Since there are 15 different letters, we have 15 choices for the first letter, 14 for the next, and so on. Thus there are $$15!$$ anagrams.
#### 8.
How many anagrams are there of the word “assesses” that start with the letter “a”?
Hint.
We just need a string of 7 letters: 4 of one type, 3 of the other.
Solution.
There are $${7 \choose 2} = 21$$ anagrams starting with “a”.
#### 9.
How many anagrams are there of “anagram”?
Solution.
First, decide where to put the “a”s. There are 7 positions, and we must choose 3 of them to fill with an “a”. This can be done in $${7 \choose 3}$$ ways. The remaining 4 spots all get a different letter, so there are $$4!$$ ways to finish off the anagram. This gives a total of $${7 \choose 3}\cdot 4!$$ anagrams. Strangely enough, this is 840, which is also equal to $$P(7,4)\text{.}$$ To get the answer that way, start by picking one of the 7 positions to be filled by the “n”, one of the remaining 6 positions to be filled by the “g”, one of the remaining 5 positions to be filled by the “r”, one of the remaining 4 positions to be filled by the “m” and then put “a”s in the remaining 3 positions.
#### 10.
1. You need to divide up into foursomes (groups of 4 people): a first foursome, a second foursome, and so on. How many ways can you do this?
2. After all your hard work, you realize that in fact, you want each foursome to include one of the five Board members. How many ways can you do this?
Solution.
1. $${20 \choose 4}{16 \choose 4}{12 \choose 4}{8 \choose 4}{4 \choose 4}$$ ways.
2. $$5!{15 \choose 3}{12 \choose 3}{9 \choose 3}{6 \choose 3}{3 \choose 3}$$ ways.
#### 11.
How many different seating arrangements are possible for King Arthur and his 9 knights around their round table?
Hint.
There are 10 people seated around the table, but it does not matter where King Arthur sits, only who sits to his left, two seats to his left, and so on. So the answer is not $$10!\text{.}$$
Solution.
$$9!\text{.}$$
#### 12.
Consider sets $$A$$ and $$B$$ with $$|A| = 10$$ and $$|B| = 17\text{.}$$
1. How many functions $$f: A \to B$$ are there?
2. How many functions $$f: A \to B$$ are injective?
Solution.
1. $$17^{10}$$ functions. There are 17 choices for the image of each element in the domain.
2. $$P(17, 10)$$ injective functions. There are 17 choices for image of the first element of the domain, then only 16 choices for the second, and so on.
#### 13.
Consider functions $$f: \{1,2,3,4\} \to \{1,2,3,4,5,6\}\text{.}$$
1. How many functions are there total?
2. How many functions are injective?
3. How many of the injective functions are increasing? To be increasing means that if $$a \lt b$$ then $$f(a) \lt f(b)\text{,}$$ or in other words, the outputs get larger as the inputs get larger.
Solution.
1. $$6^4 = 1296$$ functions.
2. $$P(6, 4) = 6 \cdot 5 \cdot 4 \cdot 3 = 360$$ functions.
3. $${6 \choose 4} = 15$$ functions.
#### 14.
We have seen that the formula for $$P(n,k)$$ is $$\dfrac{n!}{(n-k)!}\text{.}$$ Your task here is to explain why this is the right formula.
1. Suppose you have 12 chips, each a different color. How many different stacks of 5 chips can you make? Explain your answer and why it is the same as using the formula for $$P(12,5)\text{.}$$
2. Using the scenario of the 12 chips again, what does $$12!$$ count? What does $$7!$$ count? Explain.
3. Explain why it makes sense to divide $$12!$$ by $$7!$$ when computing $$P(12,5)$$ (in terms of the chips).
4. Does your explanation work for numbers other than 12 and 5? Explain the formula $$P(n,k) = \frac{n!}{(n-k)!}$$ using the variables $$n$$ and $$k\text{.}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9096075892448425, "perplexity": 268.06875388967416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710473.38/warc/CC-MAIN-20221128034307-20221128064307-00145.warc.gz"} |
https://www.physicsforums.com/threads/are-a-vector-and-its-derivative-perpendicular-at-all-times.832192/ | # Are a vector and its derivative perpendicular at all times?
1. Sep 12, 2015
i'm dumb, sorry
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2. Sep 12, 2015
### Staff: Mentor
No. What if the vector is not changing direction? In general, the derivative of a vector has components both normal and tangent to the vector.
Chet
3. Sep 12, 2015
### Staff: Mentor
No. For an obvious example, consider a vector whose magnitude but not direction is increasing as a function of time: $\vec{F}(t+\Delta{t})-\vec{F}(t)$ points in the same direction as $\vec{F}(t)$. You're thinking of the case in which the magnitude of the vector is constant over time, in which case the derivative must indeed be perpendicular (as in acceleration in the case of uniform circular motion).
[Edit: Chet got there first but I used more Latex so I still win ]
Similar Discussions: Are a vector and its derivative perpendicular at all times? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9552405476570129, "perplexity": 1050.6845075480182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805023.14/warc/CC-MAIN-20171118190229-20171118210229-00295.warc.gz"} |
http://mathhelpforum.com/calculus/217772-relative-maxima-minima-closed-region.html | # Math Help - relative maxima and minima in a closed region
1. ## relative maxima and minima in a closed region
I'm working on a problem and it requires the relative maxima and minima of a closed region, I understand how to find the rel(max/min) of a function f(x,y) without the closed region(using fx, fy,fxy,fxx,fxy). Does anyone know how to do it with a closed region(ex circle x^2 + y^2 <= 1). Does it mean that all points outside of the circle I do not need to examine?
2. ## Re: relative maxima and minima in a closed region
One trick I've seen used for a circular boundary is to parametrize the boundary so that you have one variable with which to work.
In order to examine $f$ on the boudary of the region, we represent the circle $x^2+y^2=1$ by means of the parametric equations $x=\cos(t),\,y=\sin(t),\,0\le t\le2\pi$. Thus, on the boundary we can write $f$ as a function of a single variable $t$:
$f(t)=f(\cos(t),\sin(t))$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8888520002365112, "perplexity": 205.89470679136608}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223202457.0/warc/CC-MAIN-20140423032002-00160-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?s=71273066504b955af96aa7f2b5bee501&p=4534692 | # Coincidence events
by Manojg
Tags: coincidence, events
P: 45 Hello, I have a question about coincidence event. Let us take the decay of deuteron to proton and alpha particle, (d, pα). p and α goes in opposite direction. So, if we put two detectors in opposite directions, one of them will detect p and another will detect α simultaneously (within a small time window). If the solid angle covered by both detectors are same (say Ω) and if one of the detector detect p then it is sure that another one will detect α. If N0 be the total decay rate then number of coincidence event detected will be N0Ω. However, if the solid angle covered by the detectors are different, say Ω1 and Ω2 such that Ω1 > Ω2 then the number of coincidence event will be equal to the number of events detected by the second detector because other extra event detected by the larger detector won't be detected by the smaller detector. Are these reasoning right? Because in a book, I saw the number of coincidence event is N0Ω1Ω2. Thanks.
Mentor
P: 10,853
A deuteron is a stable nucleus with a single proton and a neutron. It cannot decay in the way you describe.
Lithium-5 quickly decays to proton+alpha particle - you have to produce it directly before the decay, and this usually will not happen without any momentum transfer.
If your decay is back to back:
If N0 be the total decay rate then number of coincidence event detected will be N0Ω.
Or 2 times this value, if both detectors are sensitive to both particles. And I guess you should divide it by the full solid angle of a sphere, or express Ω as ratio to that.
then the number of coincidence event will be equal to the number of events detected by the second detector because other extra event detected by the larger detector won't be detected by the smaller detector.
This is an upper limit.
Quote by Manojg Are these reasoning right? Because in a book, I saw the number of coincidence event is N0Ω1Ω2.
That would correspond to independently moving, isotropic decay products.
Related Discussions Set Theory, Logic, Probability, Statistics 9 Electrical Engineering 0 Introductory Physics Homework 16 Special & General Relativity 10 Set Theory, Logic, Probability, Statistics 4 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9336826801300049, "perplexity": 857.6519470600012}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00523-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.thefullwiki.org/Joules | # Joules: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
(Redirected to Joule article)
Joule Standard: SI derived unit Quantity: Energy Symbol: J Named after: James Prescott Joule Expressed in: 1 J = SI base units 1 kg·m2/s2 CGS units 1×107 erg
The joule (symbol J), named for James Prescott Joule, is the derived unit of energy in the International System of Units. It is the energy exerted by the force of one newton acting to move an object through a distance of one metre. In terms of dimensions:
$\rm 1\ J = 1\ N \cdot m = \left ( \frac{kg \cdot m}{s^2} \right ) \cdot m = \frac{kg \cdot m^2}{s^2}=Pa \cdot m^3= 1\ W \cdot s$
One joule is defined as the amount of work done by a force of one newton moving an object through a distance of one metre. Other relationships are:
## Contents
### Usage
This SI unit is named after James Prescott Joule. As with every SI unit whose name is derived from the proper name of a person, the first letter of its symbol is uppercase (J). When an SI unit is spelled out in English, it should always begin with a lowercase letter (joule), except where any word would be capitalized, such as at the beginning of a sentence or in capitalized material such as a title. Note that "degree Celsius" conforms to this rule because the "d" is lowercase.
—Based on The International System of Units, section 5.2.
## Confusion with newton metre
While it is dimensionally correct to express joules as newton metres or N·m, such use is discouraged[1] by the SI authority to avoid confusion with torque. Torque and energy are fundamentally different physical quantities. For example, adding 1 N·m of torque to 1 N·m of energy gives a dimensionally consistent result of 2 N·m, but this quantity is physically meaningless.
## Practical examples
One joule in everyday life is approximately:
• the energy required to lift a small apple one meter straight up.
• the energy released when that same apple falls one meter to the ground.
• the energy released as heat by a person at rest, every hundredth of a second.
• one hundredth of the energy a person can receive by drinking a drop of beer.
• the kinetic energy of an adult human moving at a speed of about a handspan every second.
• the kinetic energy of a tennis ball moving at 23 km/h (14 mph).[2]
## Multiples
For additional examples, see: Orders of magnitude (energy)
Submultiples Multiples Value Symbol Name Value 10–1 J dJ decijoule 101 J daJ decajoule 10–2 J cJ centijoule 102 J hJ hectojoule 10–3 J mJ millijoule 103 J kJ kilojoule 10–6 J µJ microjoule 106 J MJ megajoule 10–9 J nJ nanojoule 109 J GJ gigajoule 10–12 J pJ picojoule 1012 J TJ terajoule 10–15 J fJ femtojoule 1015 J PJ petajoule 10–18 J aJ attojoule 1018 J EJ exajoule 10–21 J zJ zeptojoule 1021 J ZJ zettajoule 10–24 J yJ yoctojoule 1024 J YJ yottajoule Common multiples are in bold face
### Nanojoule
The nanojoule (nJ) is equal to one billionth of one joule. One nanojoule is about 1/160 of the kinetic energy of a flying mosquito.[3]
### Microjoule
The microjoule (μJ) is equal to one millionth of one joule. The Large Hadron Collider (LHC) is expected to produce collisions on the order of 1 microjoule (7 TeV) per particle.
### Millijoule
The millijoule (mJ) is equal to one thousandth of one joule.
### Kilojoule
The kilojoule (kJ) is equal to one thousand joules. Food labels in some countries express food energy in kilojoules. One kilojoule is about the amount of solar radiation received by one square metre of the Earth in one second.[4]
### Megajoule
The megajoule (MJ) is equal to one million joules, or approximately the kinetic energy of a one-ton vehicle moving at 160 km/h (100 mph).
### Gigajoule
The gigajoule (GJ) is equal to one billion joules. Six gigajoules is about the amount of chemical energy in a barrel of oil.[5]
### Terajoule
The terajoule (TJ) is equal to one trillion joules. About 60 terajoules were released by the bomb that exploded over Hiroshima.[6]
## Conversions
1 joule is equal to:
Units defined exactly in terms of the joule include:
• 1 thermochemical calorie = 4.184 J[7]
• 1 International Table calorie = 4.1868 J[7]
• 1 watt hour = 3600 J
• 1 kilowatt hour = 3.6×106 J (or 3.6 MJ)
• 1 ton TNT = 4.184 GJ
## References
1. ^ From the official SI website: "A derived unit can often be expressed in different ways by combining base units with derived units having special names. Joule, for example, may formally be written newton metre, or kilogram metre squared per second squared. This, however, is an algebraic freedom to be governed by common sense physical considerations; in a given situation some forms may be more helpful than others. In practice, with certain quantities, preference is given to the use of certain special unit names, or combinations of unit names, to facilitate the distinction between different quantities having the same dimension."
2. ^ Ristinen, Robert A., and Jack J. Kraushaar. Energy and the Environment. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2006.
3. ^ CERN - Glossary
4. ^
5. ^ IRS publication
6. ^ Los Alamos National Laboratory report LA-8819, The yields of the Hiroshima and Nagasaki nuclear explosions by John Malik, September 1985. Available online at http://www.mbe.doe.gov/me70/manhattan/publications/LANLHiroshimaNagasakiYields.pdf
7. ^ a b The adoption of joules as units of energy, FAO/WHO Ad Hoc Committee of Experts on Energy and Protein, 1971. A report on the changeover from calories to joules in nutrition. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9213548302650452, "perplexity": 3547.4573357464883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105304.35/warc/CC-MAIN-20170819051034-20170819071034-00411.warc.gz"} |
https://www.itwissen.info/en/order-123043.html | # order
The term order is used in analog filter technology for their transfer function.
The transfer function of analog filters is largely determined by the ratio between the passband and the stopband. The transition from the passband to the stopband is given by the filter slope or the edge steep ness of the filter. The slope in turn depends on how many poles such a filter has. If such a filter is single-pole, then it is a 1st order filter and the slope is 6 dB/ octave.
A 1st order analog filter corresponds to an RC element, a combination of a resistor (R) and a capacitor (C). If two RC elements are connected in series, the slope increases to 12 dB/octave. It is then a 2nd order filter. With five RC elements, it is a 5th order filter whose slope is 30 dB/octave.
The higher the order of a filter, the steeper the slope.
Informations:
Englisch: order Updated at: 20.11.2012 #Words: 151 Links: analog, filter, decibel (dB), octave, ring control (RPR) (RC) Translations: DE Sharing: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.892402708530426, "perplexity": 2202.342196558265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103940327.51/warc/CC-MAIN-20220701095156-20220701125156-00759.warc.gz"} |
https://oneclass.com/class-notes/ca/utsc/math/matb24h3/163036-lecture10pdf.en.html | Class Notes (809,279)
Mathematics (837)
MATB24H3 (13)
Lecture
# Lecture10.pdf
8 Pages
181 Views
School
University of Toronto Scarborough
Department
Mathematics
Course
MATB24H3
Professor
Sophie Chrysostomou
Semester
Winter
Description
University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT B24S Fall 2011 Lecture 10 Cryptography Hill Ciphers This lecture will demonstrate a method of encoding and decoding mes- sages. Most of what was studied in this course so far will be used. This is the study of fields of the form Z where p is a prime, matrices, gaus- p sian elimination of matrices with entpies from Z , linear independence, linear transformations and change of basis matrices. The study of encoding and decoding secret messages is called cryptog- raphy. Codes are called ciphers. Uncoded messages are called plaintext and coded messages are ciphertext. The process of writing a plaintext into a ciphertext is called enciphering. The process of converting a ciphertext into plaintext is called deciphering. The process that we are going to study is called Hill Cipher after Lester S. Hill who introduced it. 1 Enciphering Using Hill n-cipher We will start by assigning the following numbers to the letters, period, comma and space as follows: A B C D E F G H I J K L M N O P Q R S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 T U V W X Y Z . , 20 21 22 23 24 25 26 27 28 0 1 Note that we have used the numbers from 0 to 28. These are the elements of Z29 Note also that 29 is a prime and from our previous observations this implies that Z is a field. (MATC15) Thus, every nonzero element has a 29 multiplicative inverse. Using the Hill n-cipher, to encipher we use the following steps: 1. We choose an invertible n × n matrix with entries from Z 29 . Call this matrix A. This matrix is known to the parties ciphering and deciphering text. 2. We divide the plaintext letters into groups of n letters each, adding a dummy letter to fill the last group if necessary. Next we replace each letter by its numerical equivalent using the above table. Thus forming groups of n numbers each. I.e. v i,1 ,i,2 , v i,nare the n numbers of the ith group. 3. Each group of numbers v , i,1··i,2 v i,nfrom above forms a column v i,1 v i, vector vi= . . We will call the vector v i plaintext vector and . v i,n Av i u iis corresponding ciphertext vector. 4. Convert each ciphertext vector into its alphabetic equivalent. 5. Send the ciphertext message. 2 Deciphering To decipher we use the following steps: 1. Receive the ciphetext message. 2. Divide the ciphertext letters into groups of n letters each, adding dummy letters to fill the last group if necessary . Next, replace each letter by its numerical equivalent using the above table. Thus forming groups of n numbers each. I.e. u , u ,··· , u are the n numbers of the ith i,1 i,2 i,n group. 2 3. Each group of n numbers u , u ,··· , u from above forms a column i,1 i,2 i,n ui,1 ui.2 vector ui= . . Then the plaintext vector ii v = Aui. . ui,n 4. We finally replace each of the numbers in the plaintext vectors with their alphabetic equivalents to get the plaintext. Originally Hill cipher was used with 26or Z27 Since 26 and 27 are not primes, they have nonzero elements that have no multiplicative inverses. This may pose some problems. That is why I use 29. In doing all the work in Z29 it is usefull to have the multiplicative inverses of all the nonzero elements. Here they are: a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 −1 a 1 15 10 22 6 5 25 11 13 3 8 17 9 27 2 20 12 21 26 20 21 22 23 24 25 26 27 28 16 18 4 24 23 7 19 14 28 ▯ ▯ 5 2 EXAMPLE: Use the Hill 2-cipher with A = 3 8 to encipher the plain- text ”I like linear algebra”. Solution: 1. First ensure that the matrix A is indeed invertible and fin. A ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ 5 2 ▯ 1 0 6R1 1 12 ▯6 0 R2+28R1 1 12 ▯ 6 0 3 8 ▯ 0 1 −10R 1 22 ▯0 10 −−−−−→ 0 10 ▯ 23 10 2 ▯ ▯ ▯ ▯ ▯ ▯ 3R2 1 12 ▯ 6 0 R1+17R2 1 0 ▯19 17 −−−→ 0 1 ▯11 1 −−−−− → 0 1 ▯11 1 ▯ ▯ Thus A is invertible and A1= 19 17 11 1 3 2. Then the plaintext is divided into groups of 2 letters each. This is done to their equivalent numerical values also: I L I K E L I N E A R 9 0 12 9 11 5 0 12 9 14 5 1 18 0 A L G E B R A 1 12 7 5 2 18 1 0 Since there are an odd number of numbers above, a dummy variable is placed at the end with numerical value 0. 3. The ▯o▯lowing▯ve▯tors a▯e ▯btaine▯: ▯ ▯ ▯ ▯ ▯ 9 12 11 0 9 5 v1= , v2= , v3= , v4= , 5 = , 6 = , 0 9 5 12 14 1 ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ v = 18 , v = 1 , v = 7 , v = 2 , and v = 1 . 7 0 8 12 9 5 10 18 11 0 4. Each of the above vectors is then multiplied with A to get Av = u i i where: ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ u1= 16 , u2= 20 , u3= 7 , u4= 24 , u5= 15 , u6= 27 , 27 21 15 9 23 23 ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ 3 0 13 17 5 u7= , u8= , u9= , u10= , and 11= . 25 12 3 5 3 5. These numbers are then placed next to each other along with their alphabetical equivalent: 16 17 20 21 7 15 24 9 15 23 27 23 3 25 0 12 13 3 17 5 5 3 P Q T U G O X I O W . W C Y L M C Q E E C Thus the ciphertext is : PQUVGOXIOW.WCY LMCQEEC 6. This is send to the party that we want to send the message to and who knows A and A−1. 4 Now the person receiving the ciphertext message must decipher it. This is how it is done: 1. Receive the ciphertext message. 2. Next we divide the ciphertext into grou
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https://www.physicsforums.com/threads/do-we-already-have-evidence-of-dark-energy-our-manifold.184720/ | # Do We Already Have Evidence Of Dark Energy - Our Manifold?
1. Sep 15, 2007
### zankaon
DO WE ALREADY HAVE EVIDENCE OF DARK ENERGY - OUR MANIFOLD?
If LIGO I with it’s 10^-21 sensitivity, VIRGO etc. don’t detect gravity waves, might this then be interpreted as indicating that C_R pseudo-Riemanian spacetime continuum (i.e. manifold’s) stiffness is not INsignificant; rather than the assumption that g.w.s propagate long distance, and that it just requires a more sensitive detector? Statistically LIGO I seems to have a large enough volume and sample size for inclusion of compact objects in NS and BH binary systems in tight orbits at least, even if not catching any coalescing events. However even for binary coalescence of BHs, might generated {g.w.} decay very rapidly? So resistance to deformation (normal stress: extension and compression, and even any shear stress) might not be insignificant. Might such stiffness (resistance to deformation/distortion) be considered as like inertia of C_R manifold? That is, {g.w.} have non-localized energy, but such energy is associated with deformation of manifold. Hence such {g.w.} energy might be considered as trying to overcome resistance to deformation (i.e. stiffness) of C_R manifold. Hence such inertia of manifold (resistance to deformation) would seem to represent a contribution to stress energy momentum tensor and it’s matrix representation; thus contributing not insignificantly to overall curvature? So if long range g.w.s are not detected, then might LIGO I actually be exploring a qualitative assessment (not limits) as to stiffness of C_R manifold? Thus might C_R manifold be quite robust to perturbation? Any such robustness would seem consistent with such manifold not breaking up (i.e. so no ‘foam’?) for near to, and at C_p Planck scale; hence also consistent with no quantization of manifold C_R? Also then less likely to have leakage of g.w.s propagating out of a manifold into another dimension i.e. brane? Also wouldn’t any such significant stiffness of C_R manifold be less consistent with deformations associated with superstrings? Also if the concept of inertia of manifold is descriptive, then any entertained recent new acceleration (i.e. resulting then in a strain or elasticity of manifold) of such C_R manifold would seem less likely. Might energy associated with resistance to deformation of manifold represent a significant portion of energy required to approach flatness? That is, rather than a quest for so-called DARK ENERGY, perhaps an additional significant contribution is right before us, in the form of ENERGY of manifold C_R; such stiffness of C_R manifold contributing to stress energy momentum tensor, and hence to curvature. How would one further explore such latter conjecture, other than any qualitative finding of no long range g.w. propagation? Perhaps one could consider all alternative possibilities of sources of energy sufficient for approach to flatness. Then to the extent that they can be found to be less probable and/or no supportive evidence, then the last standing definitive contributing source of such energy (i.e. energy of C_R manifold) might have to be in part (or in full) accepted. So have LIGO I, VIRGO already made a GREAT DISCOVERY - that is, the inertia of C_R manifold? So C_R manifold seems to have significant stiffness, and hence contributes a significant amount of energy to Tuv, and thus contributes significantly to curvature. SRM.
Last edited: Sep 15, 2007
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http://math.stackexchange.com/questions/149072/combinations-and-gaussian-function/149076 | # Combinations and Gaussian function
I notice that the function $\binom{C}{x}$, where $C$ is some constant, resembles a Gaussian function; for example, here is the plot for $\binom{20}{x}$:
This corresponds to the Gaussian function $a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }$, where $a$ is $\binom{20}{10}$, $b$ is 10, and $c$ (determined through curve fitting) is ~2.2689.
$\binom{20}{x}$ corresponds to $\frac{20!}{x!(20-x)!}$; how is this related to a Gaussian function?
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It's a bit hard to imagine that this question hasn't come up before, but I did not look very hard for it. – MJD May 24 '12 at 4:30
Agree, Mark. It is also covered in all the textbooks. Yet, it is possible to see this on you own without really studying the topic. I recall a certain schoolboy collecting data on the sum of the rolls of 5 dice... – Jyrki Lahtonen May 24 '12 at 4:37
Sure! I just meant that there might already be an answer on this site that is better than my answer, and we should link to it. – MJD May 24 '12 at 4:38
${20\choose x}*2^{-20}$ is the probability of getting exactly $x$ heads in 20 coin flips. Or put another way, it is the sum of 20 random variables, each of which has probability $\frac12$ of having the value 0 heads and probability $\frac12$ of having the value 1 head.
The central limit theorem says that the mean of a large number of independent, identical random variables is (subject to a few conditions) close to a Gaussian distribution. That is what we have here, the sum of 20 copies of the same distribution.
The central limit theorem is the reason for the ubiquity of the Gaussian distribution in the natural world, in things like people's heights. If you only know three things about probability, the central limit theorem should be one of them.
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Thanks for the explanation; it's very accessible. – Hypercube May 24 '12 at 4:49
Ignoring normalization, you have noticed an instance of the de Moivre-Laplace theorem (a special case of the central limit theorem) which shows how the binomial distribution can be approximated by a normal distribution for $n$ large. You have $n=20$ and $p=q =1/2$. The theorem tells us the mean for the relevant normal distribution is $n p = 10$ and the standard deviation $\sqrt{n p q} = \sqrt{5}$.
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That's interesting. I'll have to look more into it. – Hypercube May 24 '12 at 4:47
@Hypercube: It's worth it! – user26872 May 24 '12 at 4:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8815628290176392, "perplexity": 200.90086681740752}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860123023.37/warc/CC-MAIN-20160428161523-00029-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-the-following-system-x-2y-1-9x-y-1 | Algebra
Topics
# How do you solve the following system?: -x -2y =1, 9x -y = -1
Nov 14, 2015
We can solve this question by finding the value of x with respect to y from both equations. Since the value of x should be equal, the equations thus obtained can be solved for y. And then either of the initial equations can be used to find x.
#### Explanation:
1. $- x - 2 y$ = 1 => $- x = 1 + 2 y$ => $x = - \left(1 + 2 y\right)$ ----EQUATION 1
2. $9 x - y = - 1$ => $9 x = y - 1$ => $x = \frac{1}{9} \cdot \left(y - 1\right)$ ----EQUATION 2
Equating the values of x from EQUATION 1 and EQUATION 2;
we have;
$- \left(1 + 2 y\right) = \frac{1}{9} \cdot \left(y - 1\right)$
=> $- 9 - 18 y = y - 1$
=>$- 18 y - y = - 1 + 9$
=>$- 19 y = 8$
=>$y = - \frac{8}{19}$
Using EQUATION 1;
$x = - \left(1 + 2 y\right)$
=>$x = - \left(1 + 2 \cdot \left(- \frac{8}{19}\right)\right)$
=>$x = - \left(1 - \frac{16}{19}\right)$
=>$x = - \frac{3}{19}$
So there you have it!!
$x = - \frac{3}{19}$
and
$y = - \frac{8}{19}$
##### Impact of this question
317 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 17, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9035001993179321, "perplexity": 852.040300026897}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250607314.32/warc/CC-MAIN-20200122161553-20200122190553-00404.warc.gz"} |
https://worldwidescience.org/topicpages/s/strong+elastic+constant.html | #### Sample records for strong elastic constant
1. Elastic constants of calcite
Science.gov (United States)
Peselnick, L.; Robie, R.A.
1962-01-01
The recent measurements of the elastic constants of calcite by Reddy and Subrahmanyam (1960) disagree with the values obtained independently by Voigt (1910) and Bhimasenachar (1945). The present authors, using an ultrasonic pulse technique at 3 Mc and 25??C, determined the elastic constants of calcite using the exact equations governing the wave velocities in the single crystal. The results are C11=13.7, C33=8.11, C44=3.50, C12=4.82, C13=5.68, and C14=-2.00, in units of 1011 dyncm2. Independent checks of several of the elastic constants were made employing other directions and polarizations of the wave velocities. With the exception of C13, these values substantially agree with the data of Voigt and Bhimasenachar. ?? 1962 The American Institute of Physics.
2. Strain fluctuations and elastic constants
Energy Technology Data Exchange (ETDEWEB)
Parrinello, M.; Rahman, A.
1982-03-01
It is shown that the elastic strain fluctuations are a direct measure of elastic compliances in a general anisotropic medium; depending on the ensemble in which the fluctuation is measured either the isothermal or the adiabatic compliances are obtained. These fluctuations can now be calculated in a constant enthalpy and pressure, and hence, constant entropy, ensemble due to recent develpments in the molecular dynamics techniques. A calculation for a Ni single crystal under uniform uniaxial 100 tensile or compressive load is presented as an illustration of the relationships derived between various strain fluctuations and the elastic modulii. The Born stability criteria and the behavior of strain fluctuations are shown to be related.
3. Theory of the change of elastic constants by interstitials
International Nuclear Information System (INIS)
Breuer, N.; Dederichs, P.H.; Lehmann, C.; Leibfried, G.; Scholz, A.
1975-01-01
The theory of the change of elastic constants by point-defects, in particular by interstitials, is briefly summarized. The typical effects of spring changes in a defect lattice on the elastic data are discussed qualitatively. Numerical results for the change of elastic constants by self-interstitials and vacancies are given and compared with experimental data for Cu and Al
4. Elastic constants of diamond from molecular dynamics simulations
International Nuclear Information System (INIS)
Gao Guangtu; Van Workum, Kevin; Schall, J David; Harrison, Judith A
2006-01-01
The elastic constants of diamond between 100 and 1100 K have been calculated for the first time using molecular dynamics and the second-generation, reactive empirical bond-order potential (REBO). This version of the REBO potential was used because it was redesigned to be able to model the elastic properties of diamond and graphite at 0 K while maintaining its original capabilities. The independent elastic constants of diamond, C 11 , C 12 , and C 44 , and the bulk modulus were all calculated as a function of temperature, and the results from the three different methods are in excellent agreement. By extrapolating the elastic constant data to 0 K, it is clear that the values obtained here agree with the previously calculated 0 K elastic constants. Because the second-generation REBO potential was fit to obtain better solid-state force constants for diamond and graphite, the agreement with the 0 K elastic constants is not surprising. In addition, the functional form of the second-generation REBO potential is able to qualitatively model the functional dependence of the elastic constants and bulk modulus of diamond at non-zero temperatures. In contrast, reactive potentials based on other functional forms do not reproduce the correct temperature dependence of the elastic constants. The second-generation REBO potential also correctly predicts that diamond has a negative Cauchy pressure in the temperature range examined
5. Elastic constants and internal friction of fiber-reinforced composites
International Nuclear Information System (INIS)
Ledbetter, H.M.
1982-01-01
We review recent experimental studies at NBS on the anisotropic elastic constants and internal friction of fiber-reinforced composites. Materials that were studied include: boron-aluminum, boron-epoxy, graphite-epoxy, glass-epoxy, and aramid-epoxy. In all cases, elastic-constant direction dependence could be described by relationships developed for single crystals of homogeneous materials. Elastic stiffness and internal friction were found to vary inversely
6. Temperature dependence of grain boundary free energy and elastic constants
International Nuclear Information System (INIS)
Foiles, Stephen M.
2010-01-01
This work explores the suggestion that the temperature dependence of the grain boundary free energy can be estimated from the temperature dependence of the elastic constants. The temperature-dependent elastic constants and free energy of a symmetric Σ79 tilt boundary are computed for an embedded atom method model of Ni. The grain boundary free energy scales with the product of the shear modulus times the lattice constant for temperatures up to about 0.75 the melting temperature.
7. X-Ray Elastic Constants for Cubic Materials
Energy Technology Data Exchange (ETDEWEB)
Malen, K.
1974-10-15
The stress-strain relation to be used in X-ray stress measurements in anisotropic texture-free media is studied. The method for evaluation of appropriate elastic constants for a cubic medium is described. Some illustrative numerical examples have been worked out including line broadening due to elastic anisotropy. The elastic stress and strain compatibility at grain boundaries is taken into account using Kroner's method. These elastic constants obviously only apply when no internal stresses due to plastic deformation are present. The case of reorientation of free interstitials in the stress field can be taken into account
8. X-Ray Elastic Constants for Cubic Materials
International Nuclear Information System (INIS)
Malen, K.
1974-10-01
The stress-strain relation to be used in X-ray stress measurements in anisotropic texture-free media is studied. The method for evaluation of appropriate elastic constants for a cubic medium is described. Some illustrative numerical examples have been worked out including line broadening due to elastic anisotropy. The elastic stress and strain compatibility at grain boundaries is taken into account using Kroner's method. These elastic constants obviously only apply when no internal stresses due to plastic deformation are present. The case of reorientation of free interstitials in the stress field can be taken into account
9. Some Debye temperatures from single-crystal elastic constant data
Science.gov (United States)
Robie, R.A.; Edwards, J.L.
1966-01-01
The mean velocity of sound has been calculated for 14 crystalline solids by using the best recent values of their single-crystal elastic stiffness constants. These mean sound velocities have been used to obtain the elastic Debye temperatures ??De for these materials. Models of the three wave velocity surfaces for calcite are illustrated. ?? 1966 The American Institute of Physics.
10. A comparison of the elastic constants of laminated composite plates ...
African Journals Online (AJOL)
user
Symbols/Notations: C. = elastic stiffness of the composite e. = modified strains. Ef. = Young's modulus of the fibre material. Em. = Young's modulus of the matrix ..... Mechanics, Vol. 31, June 1964, pp. 223 – 231. 17 Sijian L. I. “Thre-Dimensional Elastic. Constants for Thick laminate” Journal of Composite Materials “ vol.
11. Elastic Constants of DyRhIn5
International Nuclear Information System (INIS)
Sanada, Naoyuki; Watanuki, Ryuta; Suzuki, Kazuya
2012-01-01
The components-separated magnetic transition in DyRhIn 5 was investigated by measuring the magnetic susceptibility and elastic constants. The magnetic susceptibility along the [001] direction indicates the antiferromagnetic ordering of c-component of the magnetic moments at T N = 28.1 K whereas the susceptibility along those of [100] and [110] direction show the paramagnetic behavior. The results of the elastic constant measurements suggest that the degeneracy of quadrupolar degrees of freedom exists in spite of the formation of magnetic order because the elastic softenings are observed below T N . The quadrupolar effect in DyRhIn 5 is discussed in terms of the symmetry classification.
12. Dynamic measurements of the elastic constants of glass wool
DEFF Research Database (Denmark)
Tarnow, Viggo
2005-01-01
, and this requires knowledge of the dynamic elastic constants of the fiber skeleton. The mechanical properties of glass wool are highly anisotropic. Previously only one of the elastic constants has been measured dynamically, but here all the elastic constants are reported. The measurement method is well known......The sound wave in the air between the fibers of glass wool exerts an oscillatory viscous drag on the fibers and excites a mechanical wave in the fiber skeleton. Accurate calculations of sound attenuation in glass wool must take the mechanical wave in the fiber skeleton into account...... formula. The elastic constants were measured in the frequency range 20–160 Hz for glass wool of mass density 30 kg/m3. The elastic constant C11 depended on the frequency; at 20 Hz it was 1.5+0.01i MPa, and at 160 Hz it was 2.6+0.06i MPa. The constant C33=12+0.6i kPa did not depend on frequency. The shear...
13. Calculation of elastic constants of BCC transition metals: tight-binding recursion method
International Nuclear Information System (INIS)
Masuda, K.; Hamada, N.; Terakura, K.
1984-01-01
The elastic constants of BCC transition metals (Fe, Nb, Mo and W) are calculated by using the tight-binding d band and the Born-Mayer repulsive potential. Introducing a small distortion characteristic to C 44 (or C') elastic deformation and calculating the energy change up to second order in the atomic displacement, the shear elastic constants C 44 and C' are determined. The elastic constants C 11 and C 12 are then calculated by using the relations B=1/3(C 11 + 2C 12 ) and C'=1/2(C 11 -C 12 ), where B is the bulk modulus. In general, the agreement between the present results and the experimental values is satisfactory. The characteristic elasticity behaviour, i.e. the strong Nsub(d) (number of d electrons) dependence of the observed anisotropy factor A=C 44 /C', will also be discussed. (author)
14. Calculation of Elastic Bond Constants in Atomistic Strain Analysis.
Science.gov (United States)
Chen, Haiyuan; Wang, Juanjuan; Ashalley, Eric; Li, Handong; Niu, Xiaobin
2015-12-01
Strain analysis has significance both for tailoring material properties and designing nanoscale devices. In particular, strain plays a vital role in engineering the growth thermodynamics and kinetics and is applicable for designing optoelectronic devices. In this paper, we present a methodology for establishing the relationship between elastic bond constants and measurable parameters, i.e., Poisson's ratio ν and systematic elastic constant K. At the atomistic level, this approach is within the framework of linear elastic theory and encompasses the neighbor interactions when an atom is introduced to stress. Departing from the force equilibrium equations, the relationships between ν, K, and spring constants are successfully established. Both the two-dimensional (2D) square lattice and common three-dimensional (3D) structures are taken into account in the procedure for facilitating, bridging the gap between structural complexity and numerical experiments. A new direction for understanding the physical phenomena in strain engineering is established.
15. Estimation of effective elastic constants for grid plate
International Nuclear Information System (INIS)
Shibanuma, Kiyoshi; Kuriyama, Masaaki; Okumura, Yoshikazu
1980-07-01
This article contains a method of estimation for the effective elastic constants of a grid plate, which is a flat perforated plate with pipes for cooling. The elastic constants of the grid plate are formulated for two symmetric axes. In the case of using OFCu(E 0 = 12500 kg/mm 2 , ν 0 = 0.34) as the material of the grid, the results are given as follows. E sub(L) = 3180 kg/mm 2 , E sub(T) = 3860 kg/mm 2 upsilon sub(LT) = 0.12, upsilon sub(TL) = 0.15 (author)
16. Asymptotics for Greeks under the constant elasticity of variance model
OpenAIRE
Kritski, Oleg L.; Zalmezh, Vladimir F.
2017-01-01
This paper is concerned with the asymptotics for Greeks of European-style options and the risk-neutral density function calculated under the constant elasticity of variance model. Formulae obtained help financial engineers to construct a perfect hedge with known behaviour and to price any options on financial assets.
17. Elastic constants of hard and soft nematic liquid crystals
NARCIS (Netherlands)
Tjipto-Margo, B.; Evans, G.T.; Allen, M.P.; Frenkel, D.
1992-01-01
The Frank elastic constants for a nematic liquid crystal have been calculated by computer simulations for a fluid of hard ellipsoids and by the Poniewierski-Stecki method for ellipsoids with and without an attractive square well. Required for the Poniewierski-Stecki method is the direct
18. Elastic constants in orthorhombic hen egg-white lysozyme crystals
Science.gov (United States)
Kitajima, N.; Tsukashima, S.; Fujii, D.; Tachibana, M.; Koizumi, H.; Wako, K.; Kojima, K.
2014-01-01
The ultrasonic sound velocities of cross-linked orthorhombic hen egg-white lysozyme (HEWL) crystals, including a large amount of water in the crystal, were measured using an ultrasonic pulse-echo method. As a result, seven elastic constants of orthorhombic crystals were observed to be C11 = 5.24 GPa, C22 = 4.87 GPa, C12 = 4.02 GPa, C33 = 5.23 GPa, C44 = 0.30 GPa, C55 = 0.40 GPa, and C66 = 0.43 GPa, respectively. However, C13 and C23 could not be observed because the suitable crystal planes could not be cut from bulk crystals. We conclude that the observed elastic constants of the cross-linked crystals are coincident with those of the intrinsic crystals without cross-linking. Moreover, the characteristics of the elastic constants in orthorhombic HEWL crystals are due to the fact that the shear elastic constants, C44, C55, and C66, are softer than in tetragonal crystals. That is, the shear components, C44, C55, and C66, are one half of those of the tetragonal crystals.
19. Single-crystal elastic constants of natural ettringite
KAUST Repository
Speziale, Sergio
2008-07-01
The single-crystal elastic constants of natural ettringite were determined by Brillouin spectroscopy at ambient conditions. The six non-zero elastic constants of this trigonal mineral are: C11 = 35.1 ± 0.1 GPa, C12 = 21.9 ±0.1 GPa, C13 = 20.0 ± 0.5 GPa, C14 = 0.6 ± 0.2 GPa, C33 = 55 ± 1 GPa, C44 = 11.0 ± 0.2 GPa. The Hill average of the aggregate bulk, shear modulus and the polycrystal Young\\'s modulus and Poisson\\'s ratio are 27.3 ± 0.9 GPa, 9.5 ± 0.8 GPa, 25 ± 2 GPa and 0.34 ± 0.02 respectively. The longitudinal and shear elastic anisotropy are C33/C11 = 0.64 ± 0.01 and C66/C44 =0.60 ± 0.01. The elastic anisotropy in ettringite is connected to its crystallographic structure. Stiff chains of [Al(OH)6]3- octahedra alternating with triplets of Ca2+ in eight-fold coordination run parallel to the c-axis leading to higher stiffness along this direction. The determination of the elastic stiffness tensor can help in the prediction of the early age properties of cement paste when ettringite crystals precipitate and in the modeling of both internal and external sulfate attack when secondary ettringite formation leads to expansion of concrete. © 2008 Elsevier Ltd. All rights reserved.
20. Hydrostatic pressure dependence of elastic constants for lead fluoride crystal
International Nuclear Information System (INIS)
Singh, R.K.; Rao, C.N.
1988-10-01
The variations of the second order elastic constants (SOEC) and longitudinal and shear moduli with hydrostatic pressure for the lead fluoride have been investigated theoretically, for the first time, by means of a three-body force potential (TBP) model. The significance of three-body interactions (TBI) has been clearly demonstrated in these investigations. The present TBP model has reproduced the pressure derivatives of the SOEC of PbF more satisfactorily than the shell model and other model calculations. (author). 24 refs, 3 figs, 2 tabs
1. High temperature elastic constant measurements: application to plutonium
International Nuclear Information System (INIS)
Bouchet, J.M.
1969-03-01
We present an apparatus with which we have measured the Young's modulus and the Poisson's ratio of several compounds from the resonance frequency of cylinders in the temperature range 0 deg. C-700 deg. C. We especially studied the elastic constants of plutonium and measured for the first time to our knowledge the Young's modulus of Pu δ and Pu ε . E δ 360 deg. C = 1.6 10 11 dy/cm 2 ; E ε 490 deg. C = 1.1 10 11 dy/cm 2 , σ ε = 0.25 ± 0.03 Using our results, we have calculated the compressibility, the Debye temperature, the Grueneisen constant and the electronic specific heat of Pu ε . (author) [fr
2. Stresses and elastic constants of crystalline sodium, from molecular dynamics
International Nuclear Information System (INIS)
Schiferl, S.K.
1985-02-01
The stresses and the elastic constants of bcc sodium are calculated by molecular dynamics (MD) for temperatures to T = 340K. The total adiabatic potential of a system of sodium atoms is represented by pseudopotential model. The resulting expression has two terms: a large, strictly volume-dependent potential, plus a sum over ion pairs of a small, volume-dependent two-body potential. The stresses and the elastic constants are given as strain derivatives of the Helmholtz free energy. The resulting expressions involve canonical ensemble averages (and fluctuation averages) of the position and volume derivatives of the potential. An ensemble correction relates the results to MD equilibrium averages. Evaluation of the potential and its derivatives requires the calculation of integrals with infinite upper limits of integration, and integrand singularities. Methods for calculating these integrals and estimating the effects of integration errors are developed. A method is given for choosing initial conditions that relax quickly to a desired equilibrium state. Statistical methods developed earlier for MD data are extended to evaluate uncertainties in fluctuation averages, and to test for symmetry. 45 refs., 10 figs., 4 tabs
3. Determination of the elastic constants of portlandite by Brillouin spectroscopy
KAUST Repository
Speziale, S.
2008-10-01
The single crystal elastic constants Cij and the shear and adiabatic bulk modulus of a natural portlandite (Ca(OH)2) crystal were determined by Brillouin spectroscopy at ambient conditions. The elastic constants, expressed in GPa, are: C11 = 102.0(± 2.0), C12 = 32.1(± 1.0), C13 = 8.4(± 0.4), C14 = 4.5(± 0.2), C33 = 33.6(± 0.7), C44 = 12.0(± 0.3), C66 = (C11-C12)/2 = 35.0(± 1.1), where the numbers in parentheses are 1σ standard deviations. The Reuss bounds of the adiabatic bulk and shear moduli are K0S = 26.0(± 0.3) GPa and G0 = 17.5(± 0.4) GPa, respectively, while the Voigt bounds of these moduli are K0S = 37.3(± 0.4) GPa and G0 = 24.4(± 0.3) GPa. The Reuss and Voigt bounds for the aggregate Young\\'s modulus are 42.8(± 1.0) GPa and 60.0(± 0.8) GPa respectively, while the aggregate Poisson\\'s ratio is equal to 0.23(± 0.01). Portlandite exhibits both large compressional elastic anisotropy with C11/C33 = 3.03(± 0.09) equivalent to that of the isostructural hydroxide brucite (Mg(OH)2), and large shear anisotropy with C66/C44 = 2.92(± 0.12) which is 11% larger than brucite. The comparison between the bulk modulus of portlandite and that of lime (CaO) confirms a systematic linear relationship between the bulk moduli of brucite-type simple hydroxides and the corresponding NaCl-type oxides. © 2008 Elsevier Ltd. All rights reserved.
4. Is the Armington Elasticity Really Constant across Importers?
OpenAIRE
Yilmazkuday, Hakan
2009-01-01
This paper shows that the Armington elasticity, which refers to both the elasticity of substitution across goods and the price elasticity of demand under the assumption of a large number of varieties, systematically changes from one importer country to another in an international trade context. Then a natural question to ask is "What determines the Armington elasticity?" The answer comes from the distinction between the elasticity of demand with respect to the destination price (i.e., the Arm...
5. Higher Order Elastic Constants, Gruneisen Parameters and Lattice Thermal Expansion of Lithium Niobate
Directory of Open Access Journals (Sweden)
Thresiamma Philip
2006-01-01
Full Text Available The second and third-order elastic constants and pressure derivatives of second- order elastic constants of trigonal LiNbO3 (lithium niobate have been obtained using the deformation theory. The strain energy density estimated using finite strain elasticity is compared with the strain dependent lattice energy density obtained from the elastic continuum model approximation. The second-order elastic constants and the non-vanishing third-order elastic constants along with the pressure derivatives of trigonal LiNbO3 are obtained in the present work. The second and third-order elastic constants are compared with available experimental values. The second-order elastic constant C11 which corresponds to the elastic stiffness along the basal plane of the crystal is less than C33 which corresponds to the elastic stiffness tensor component along the c-axis of the crystal. The pressure derivatives, dC'ij/dp obtained in the present work, indicate that trigonal LiNbO3 is compressible. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in LiNbO3. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with available reported values.
6. DFT calculation for elastic constants of orthorhombic structure within WIEN2K code: A new package (ortho-elastic)
International Nuclear Information System (INIS)
Reshak, Ali H.; Jamal, Morteza
2012-01-01
Highlights: ► A new package for calculating elastic constants of orthorhombic structure is released. ► The package called ortho-elastic. ► It is compatible with [FP-(L)APW+lo] method implemented in WIEN2k code. ► Several orthorhombic structure compounds were used to test the new package. ► Elastic constants calculated using this package show good agreement with experiment. - Abstract: A new package for calculating the elastic constants of orthorhombic structure is released. The package called ortho-elastic. The formalism of calculating the ortho-elastic constants is described in details. The package is compatible with the highly accurate all-electron full-potential (linearized) augmented plane-wave plus local orbital [FP-(L)APW+lo] method implemented in WIEN2k code. Several orthorhombic structure compounds were used to test the new package. We found that the calculated elastic constants using the new package show better agreement with the available experimental data than the previous theoretical results used different methods. In this package the second-order derivative E ″ (ε) of polynomial fit E=E(ε) of energy vs strains at zero strain (ε=0), used to calculate the orthorhombic elastic constants.
7. Evaluation of elastic constants of materials using the frequency spectrum
International Nuclear Information System (INIS)
Silva Neto, Ramiro J. da; Baroni, Douglas B.; Bittencourt, Marcelo de S.Q.
2015-01-01
The characterization of materials made with the support of non-destructive techniques has great importance in industrial applications. The ultrasonic techniques are distinguished by good resolution to measure small variations of wave velocities as a result of changes in the character suffered by a particular material. In general these ultrasonic techniques are studied in the time domain, which represents an experimental difficulties when thin materials are analyzed, as well as to attenuate the ultrasonic signal drastically. An ultrasonic technique that uses the frequency domain is used in this study aiming to provide good time measurements to calculate the elastic constants of the first order in an aluminum alloy 6351. With the aid of a statistical approach was possible to have good results of tests performed when compared by a time domain technique already well explored in Ultrasound works produced in the Nuclear Engineering Institute Laboratory (LABUS / IEN) and also presented in most of the package, in good agreement with the theoretical model established in literature and used to validate the experiment, which was found in the results with good approximation. The relevance of this work in the nuclear area is associated with the interest to know the mechanical properties of structural components of the nuclear industry, which is currently studied as a rule, resorting to the computer simulations or previously during the operation of the system. (author)
8. Measurement of the strong coupling constant using τ decays
Science.gov (United States)
Buskulic, D.; Decamp, D.; Goy, C.; Lees, J.-P.; Minard, M.-N.; Mours, B.; Pietrzyk, B.; Alemany, R.; Ariztizabal, F.; Comas, P.; Crespo, J. M.; Delfino, M.; Fernandez, E.; Fernandez-Bosman, M.; Gaitan, V.; Garrido, Ll.; Mattison, T.; Pacheco, A.; Padilla, C.; Pascual, A.; Creanza, D.; de Palma, M.; Farilla, A.; Iaselli, G.; Maggi, G.; Maggi, M.; Natali, S.; Nuzzo, S.; Quattromini, M.; Ranieri, A.; Raso, G.; Romano, F.; Ruggieri, F.; Selvaggi, G.; Silvestris, L.; Tempesta, P.; Zito, G.; Chai, Y.; Hu, H.; Huang, D.; Huang, X.; Lin, J.; Wang, T.; Xie, Y.; Xu, D.; Xu, R.; Zhang, J.; Zhang, L.; Zhao, W.; Bauerdick, L. A. T.; Blucher, E.; Bonvicini, G.; Boudreau, J.; Casper, D.; Drevermann, H.; Forty, R. W.; Ganis, G.; Gay, C.; Hagelberg, R.; Harvey, J.; Haywood, S.; Hilgart, J.; Jacobsen, R.; Jost, B.; Knobloch, J.; Lehraus, I.; Lohse, T.; Lusiani, A.; Martinez, M.; Mato, P.; Meinhard, H.; Minten, A.; Miotto, A.; Miquel, R.; Moser, H.-G.; Palazzi, P.; Perlas, J. A.; Pusztaszeri, J.-F.; Ranjard, F.; Redlinger, G.; Rolandi, L.; Rothberg, J.; Ruan, T.; Saich, M.; Schlatter, D.; Schmelling, M.; Sefkow, F.; Tejessy, W.; Wachsmuth, H.; Wiedenmann, W.; Wildish, T.; Witzeling, W.; Wotschack, J.; Ajaltouni, Z.; Badaud, F.; Bardadin-Otwinowska, M.; El Fellous, R.; Falvard, A.; Gay, P.; Guicheney, C.; Henrard, P.; Jousset, J.; Michel, B.; Montret, J.-C.; Pallin, D.; Perret, P.; Podlyski, F.; Proriol, J.; Prulhière, F.; Saadi, F.; Fearnley, T.; Hansen, J. D.; Hansen, J. R.; Hansen, P. H.; Møllerud, R.; Nilsson, B. S.; Efthymiopoulos, I.; Kyriakis, A.; Simopoulou, E.; Vayaki, A.; Zachariadou, K.; Badier, J.; Blondel, A.; Bonneaud, G.; Brient, J. C.; Fouque, G.; Orteu, S.; Rougé, A.; Rumpf, M.; Tanaka, R.; Verderi, M.; Videau, H.; Candlin, D. J.; Parsons, M. I.; Veitch, E.; Moneta, L.; Parrini, G.; Corden, M.; Georgiopoulos, C.; Ikeda, M.; Lannutti, J.; Levinthal, D.; Mermikides, M.; Sawyer, L.; Wasserbaech, S.; Antonelli, A.; Baldini, R.; Bencivenni, G.; Bologna, G.; Bossi, F.; Campana, P.; Capon, G.; Cerutti, F.; Chiarella, V.; D'Ettorre-Piazzoli, B.; Felici, G.; Laurelli, P.; Mannocchi, G.; Murtas, F.; Murtas, G. P.; Passalacqua, L.; Pepe-Altarelli, M.; Picchi, P.; Colrain, P.; Ten Have, I.; Lynch, J. G.; Maitland, W.; Morton, W. T.; Raine, C.; Reeves, P.; Scarr, J. M.; Smith, K.; Smith, M. G.; Thompson, A. S.; Turnbull, R. M.; Brandl, B.; Braun, O.; Geweniger, C.; Hanke, P.; Hepp, V.; Kluge, E. E.; Maumary, Y.; Putzer, A.; Rensch, B.; Stahl, A.; Tittel, K.; Wunsch, M.; Belk, A. T.; Beuselinck, R.; Binnie, D. M.; Cameron, W.; Cattaneo, M.; Colling, D. J.; Dornan, P. J.; Dugeay, S.; Greene, A. M.; Hassard, J. F.; Lieske, N. M.; Nash, J.; Payne, D. G.; Phillips, M. J.; Sedgbeer, J. K.; Tomalin, I. R.; Wright, A. G.; Girtler, P.; Kneringer, E.; Kuhn, D.; Rudolph, G.; Bowdery, C. K.; Brodbeck, T. J.; Finch, A. J.; Foster, F.; Hughes, G.; Jackson, D.; Keemer, N. R.; Nuttall, M.; Patel, A.; Sloan, T.; Snow, S. W.; Whelan, E. P.; Kleinknecht, K.; Raab, J.; Renk, B.; Sander, H.-G.; Schmidt, H.; Steeg, F.; Walther, S. M.; Wanke, R.; Wolf, B.; Aubert, J.-J.; Bencheikh, A. M.; Benchouk, C.; Bonissent, A.; Carr, J.; Coyle, P.; Drinkard, J.; Etienne, F.; Nicod, D.; Papalexiou, S.; Payre, P.; Roos, L.; Rousseau, D.; Schwemling, P.; Talby, M.; Adlung, S.; Assmann, R.; Bauer, C.; Blum, W.; Brown, D.; Cattaneo, P.; Dehning, B.; Dietl, H.; Dydak, F.; Frank, M.; Halley, A. W.; Lauber, J.; Lütjens, G.; Lutz, G.; Männer, W.; Richter, R.; Rotscheidt, H.; Schröder, J.; Schwarz, A. S.; Settles, R.; Seywerd, H.; Stierlin, U.; Stiegler, U.; Denis, R. St.; Wolf, G.; Boucrot, J.; Callot, O.; Cordier, A.; Davier, M.; Duflot, L.; Grivaz, J.-F.; Heusse, Ph.; Jaffe, D. E.; Janot, P.; Kim, D. W.; Le Diberder, F.; Lefrançois, J.; Lutz, A.-M.; Schune, M.-H.; Veillet, J.-J.; Videau, I.; Zhang, Z.; Abbaneo, D.; Bagliesi, G.; Batignani, G.; Bosisio, L.; Bottigli, U.; Bozzi, C.; Calderini, G.; Carpinelli, M.; Ciocci, M. A.; Dell'Orso, R.; Ferrante, I.; Fidecaro, F.; Foà, L.; Focardi, E.; Forti, F.; Giassi, A.; Giorgi, M. A.; Gregorio, A.; Ligabue, F.; Mannelli, E. B.; Marrocchesi, P. S.; Messineo, A.; Palla, F.; Rizzo, G.; Sanguinetti, G.; Spagnolo, P.; Steinberger, J.; Tenchini, R.; Tonelli, G.; Triggiani, G.; Vannini, C.; Venturi, A.; Verdini, P. G.; Walsh, J.; Betteridge, A. P.; Carter, J. M.; Green, M. G.; March, P. V.; Mir, Ll. M.; Medcalf, T.; Quazi, I. S.; Strong, J. A.; West, L. R.; Botterill, D. R.; Clifft, R. W.; Edgecock, T. R.; Edwards, M.; Fisher, S. M.; Jones, T. J.; Norton, P. R.; Salmon, D. P.; Thompson, J. C.; Bloch-Devaux, B.; Colas, P.; Duarte, H.; Kozanecki, W.; Lançon, E.; Lemaire, M. C.; Locci, E.; Perez, P.; Perrier, F.; Rander, J.; Renardy, J.-F.; Rosowsky, A.; Roussarie, A.; Schuller, J.-P.; Schwindling, J.; Si Mohand, D.; Vallage, B.; Johnson, R. P.; Litke, A. M.; Taylor, G.; Wear, J.; Ashman, J. G.; Babbage, W.; Booth, C. N.; Buttar, C.; Carney, R. E.; Cartwright, S.; Combley, F.; Hatfield, F.; Thompson, L. F.; Barberio, E.; Böhrer, A.; Brandt, S.; Cowan, G.; Grupen, C.; Lutters, G.; Rivera, F.; Schäfer, U.; Smolik, L.; Della Marina, R.; Giannini, G.; Gobbo, B.; Ragusa, F.; Bellantoni, L.; Chen, W.; Cinabro, D.; Conway, J. S.; Cowen, D. F.; Feng, Z.; Ferguson, D. P. S.; Gao, Y. S.; Grahl, J.; Harton, J. L.; Jared, R. C.; Leclaire, B. W.; Lishka, C.; Pan, Y. B.; Pater, J. R.; Saadi, Y.; Sharma, V.; Schmitt, M.; Shi, Z. H.; Walsh, A. M.; Weber, F. V.; Lan Wu, Sau; Wu, X.; Zheng, M.; Zobernig, G.; Aleph Collaboration
1993-06-01
The strong coupling constant is determined from the leptonic branching ratios, the lifetime, and the invariant mass distribution of the hadronic final state of the τ lepton, using data accumulated at LEP with the ALEPH detector. The strong coupling constant measurement, αs( mτ2) = 0.330±0.046, evolved to the Z mass yields αs( MZ2) = 0.188±0.005. The error includes experimental and theoretical uncertainties, the latter evaluated in the framework of the Shifman, Vainshtein and Zakharov (SVZ) approach. The method allows the non-perturbative contribution to the hadronic decay rate to be determined to be 0.3±0.5%.
9. Temperature and magnetic field dependence of the elastic constants in Nd3Se4
International Nuclear Information System (INIS)
Futterer, H.; Yohannes, T.; Bach, H.; Pelzl, J.; Nahm, K.; Kim, C.K.
1988-01-01
The elastic constants of Nd 3 Se 4 show an anomalous behaviour above and below T c =52 K. Band effects are assumed to be the origin of the softening of C' in the paramagnetic region. Below T c extra shifts arise from magneto-elastic coupling. An elastic hysteresis is observed which is directly related to the magnetic hysteresis loop
10. Elastic constants of a Laves phase compound: C15 NbCr2
International Nuclear Information System (INIS)
Ormeci, A.; Chu, F.; Wills, J.M.; Chen, S.P.; Albers, R.C.; Thoma, D.J.; Mitchell, T.E.
1997-01-01
The single-crystal elastic constants of C15 NbCr 2 have been computed by using a first-principles, self-consistent, full-potential total energy method. From these single-crystal elastic constants the isotropic elastic moduli are calculated using the Voigt and Reuss averages. The calculated values are in fair agreement with the experimental values. The implications of the results are discussed with regards to Poisson's ratio and the direction dependence of Young's modulus
11. Low-temperature monocrystal elastic constants of Fe-19Cr-10Ni
International Nuclear Information System (INIS)
Ledbetter, H.M.
1984-01-01
By a pulse-echo-overlap ultrasonic method, we determined the monocrystal elastic constants (C 11 , C 12 , C 44 ) of an Fe-19Cr-10Ni alloy between 295 and 4 K. In composition this laboratory alloy approximates a technological austenitic stainless steel: AISI 304. Many previous studies on polycrystalline steels found a low-temperature magnetic phase transition that affects physical properties, including elastic constants. At the transition, anomalies occur in all polycrystal elastic constants: Young's modulus, shear modulus, bulk modulus, and Poisson's ratio. The present study found that the transition, near 50 K, does not affect one monocrystal elastic constant: C 44 , the resistance to shear on a (100) plane in a [100]-type direction. We interpret this new observation from the viewpoint of a Born-type lattice model. Also, we comment about the relationship between the elastic-constant changes and the low-temperature magnetic state
12. The Elastic Constants Measurement of Metal Alloy by Using Ultrasonic Nondestructive Method at Different Temperature
Directory of Open Access Journals (Sweden)
Eryi Hu
2016-01-01
Full Text Available The ultrasonic nondestructive method is introduced into the elastic constants measurement of metal material. The extraction principle of Poisson’s ratio, elastic modulus, and shear modulus is deduced from the ultrasonic propagating equations with two kinds of vibration model of the elastic medium named ultrasonic longitudinal wave and transverse wave, respectively. The ultrasonic propagating velocity is measured by using the digital correlation technique between the ultrasonic original signal and the echo signal from the bottom surface, and then the elastic constants of the metal material are calculated. The feasibility of the correlation algorithm is verified by a simulation procedure. Finally, in order to obtain the stability of the elastic properties of different metal materials in a variable engineering application environment, the elastic constants of two kinds of metal materials in different temperature environment are measured by the proposed ultrasonic method.
13. Crystal structure and elastic constants of Dharwar cotton fibre using ...
Indian Academy of Sciences (India)
Wide-angle X-ray scattering (WAXS) recordings were carried out on raw Dharwar cotton fibres available in Karnataka. Using this data and employing linked atom least squares (LALS) method, we report here the molecular and crystal structure of these cotton fibres. Employing structural data, we have computed elastic ...
14. Elastic constants of an Fe-5Cr-26Mn austenitic steel, 76 to 400 K
International Nuclear Information System (INIS)
Ledbetter, H.M.; Austin, M.W.
1983-01-01
By measuring longitudinal-mode and transverse-mode sound velocities at frequencies near 10 MHz, we determined the complete engineering elastic constants - bulk modulus, shear modulus, Young modulus, Poisson ratio - for an Fe-5Cr-26Mn austenitic steel between 76 and 400 K. Due to a magnetic transition, all elastic constants behave anomalously below about 360 K. The bulk modulus begins to soften during cooling at some higher temperature. Except for Poisson's ratio, below the 360-K magnetic transition, all elastic constants resume an apparently normal temperature dependence. After increasing abruptly at the magnetic transition, Poisson's ratio increases with decreasing temperature
15. Defect-induced change of temperature-dependent elastic constants in BCC iron
Energy Technology Data Exchange (ETDEWEB)
Gao, N.; Setyawan, W.; Zhang, S. H.; Wang, Z. G.
2017-07-01
The effects of radiation-induced defects (randomly distributed vacancies, voids, and interstitial dislocation loops) on temperature-dependent elastic constants, C11, C12, and C44 in BCC iron, are studied with molecular dynamics method. The elastic constants are found to decrease with increasing temperatures for all cases containing different defects. The presence of vacancies, voids, or interstitial loops further decreases the elastic constants. For a given number of point defects, the randomly distributed vacancies show the strongest effect compared to voids or interstitial loops. All these results are expected to provide useful information to combine with experimental results for further understanding of radiation damage.
16. The strong coupling constant of QCD with four flavors
Energy Technology Data Exchange (ETDEWEB)
Tekin, Fatih
2010-11-01
In this thesis we study the theory of strong interaction Quantum Chromodynamics on a space-time lattice (lattice QCD) with four flavors of dynamical fermions by numerical simulations. In the early days of lattice QCD, only pure gauge field simulations were accessible to the computational facilities and the effects of quark polarization were neglected. The so-called fermion determinant in the path integral was set to one (quenched approximation). The reason for this approximation was mainly the limitation of computational power because the inclusion of the fermion determinant required an enormous numerical effort. However, for full QCD simulations the virtual quark loops had to be taken into account and the development of new machines and new algorithmic techniques made the so-called dynamical simulations with at least two flavors possible. In recent years, different collaborations studied lattice QCD with dynamical fermions. In our project we study lattice QCD with four degenerated flavors of O(a) improved Wilson quarks in the Schroedinger functional scheme and calculate the energy dependence of the strong coupling constant. For this purpose, we determine the O(a) improvement coefficient c{sub sw} with four flavors and use this result to calculate the step scaling function of QCD with four flavors which describes the scale evolution of the running coupling. Using a recursive finite-size technique, the {lambda} parameter is determined in units of a technical scale L{sub max} which is an unambiguously defined length in the hadronic regime. The coupling {alpha}{sub SF} of QCD in the so-called Schroedinger functional scheme is calculated over a wide range of energies non-perturbatively and compared with 2-loop and 3-loop perturbation theory as well as with the non-perturbative result for only two flavors. (orig.)
17. Ultrasonic measurement of elastic constants in fiber-reinforced polymer composites under influence of absorbed moisture
DEFF Research Database (Denmark)
Nielsen, S.A.; Toftegaard, H.
2000-01-01
This paper presents an attempt to quantify hygral aging in fiber-reinforced polymer composites by the elastic constants C-11 and C-33. Quantitative ultrasonic measurements of the elastic constants for three different unidirectional as well as three different cross-ply specimens were compared. The......, and typically moisture expansion coefficients are reported. Moreover, as the ultrasonic pulse form changed in the anisotropic materials, different broadband methods were used to calculate the elastic constants. (C) 2000 Published by Elsevier Science B.V. All rights reserved........ The specimens were manufactured with different moisture resistant surfaces and immersed in water for 24 h. By calculating the elastic constants, it was taken into account that hygral aging was accompanied by absorption of moisture in the polymer matrix. Moisture changed the laminate dimensions significantly...
18. Elastic constants of a Laves phase compound: C15 NbCr{sub 2}
Energy Technology Data Exchange (ETDEWEB)
Ormeci, A. [Koc Univ., Istanbul (Turkey)]|[Los Alamos National Lab., NM (United States); Chu, F.; Wills, J.M.; Chen, S.P.; Albers, R.C.; Thoma, D.J.; Mitchell, T.E. [Los Alamos National Lab., NM (United States)
1997-04-01
The single-crystal elastic constants of C15 NbCr{sub 2} have been computed by using a first-principles, self-consistent, full-potential total energy method. From these single-crystal elastic constants the isotropic elastic moduli are calculated using the Voigt and Reuss averages. The calculated values are in fair agreement with the experimental values. The implications of the results are discussed with regards to Poissons ratio and the direction dependence of Youngs modulus.
19. The relationship between elastic constants and structure of shock waves in a zinc single crystal
Science.gov (United States)
Krivosheina, M. N.; Kobenko, S. V.; Tuch, E. V.
2017-12-01
The paper provides a 3D finite element simulation of shock-loaded anisotropic single crystals on the example of a Zn plate under impact using a mathematical model, which allows for anisotropy in hydrostatic stress and wave velocities in elastic and plastic ranges. The simulation results agree with experimental data, showing the absence of shock wave splitting into an elastic precursor and a plastic wave in Zn single crystals impacted in the [0001] direction. It is assumed that the absence of an elastic precursor under impact loading of a zinc single crystal along the [0001] direction is determined by the anomalously large ratio of the c/a-axes and close values of the propagation velocities of longitudinal and bulk elastic waves. It is shown that an increase in only one elastic constant along the [0001] direction results in shock wave splitting into an elastic precursor and a shock wave of "plastic" compression.
20. Analysis of competitive power market with constant elasticity function
International Nuclear Information System (INIS)
Nguyen, D.H.M.; Wong, K.P.
2003-01-01
A solution method, for competitive power markets formulated as a Cournot game, that allows equilibrium to be determined without an explicit model of aggregated demand is presented. The method determines market equilibrium for all feasible demand conditions and thus provides a perspective on the market, independent of representative demand function, that reveals the inherent tendencies of producers in the market. Numerical solutions are determined by use of the new controlled genetic algorithm and constraint handling techniques. The solutions give production and demand elasticity distributions of the market at any feasible equilibrium price and volume. The solution distributions evaluated for the market with unspecified demand functions, were found to be consistent with previous results obtained from markets with specific demand functions. The ability of the new approach to all, and arbitrary, solutions allow specific markets to be examined, as well as very general observations to be made. Generally it was observed that: no inherent price constraint exists; price is more volatile for low volumes and high prices; market dominance and power are unaffected by price; and inelastic demand can give rise to equilibrium with lower price than responsive demand. (Author)
1. Elastic constants and Debye temperature of wz-AlN and wz-GaN ...
Indian Academy of Sciences (India)
First-principles calculations were performed to study the elastic stiffness constants ( C i j ) and Debye temperature ( D ) of wurzite (wz) AlN and GaN binary semiconductors at high pressure. The lattice constants were calculated from the optimized structure of these materials. The band gaps were calculated at point using ...
2. Exact result in strong wave turbulence of thin elastic plates
Science.gov (United States)
Düring, Gustavo; Krstulovic, Giorgio
2018-02-01
An exact result concerning the energy transfers between nonlinear waves of a thin elastic plate is derived. Following Kolmogorov's original ideas in hydrodynamical turbulence, but applied to the Föppl-von Kármán equation for thin plates, the corresponding Kármán-Howarth-Monin relation and an equivalent of the 4/5 -Kolmogorov's law is derived. A third-order structure function involving increments of the amplitude, velocity, and the Airy stress function of a plate, is proven to be equal to -ɛ ℓ , where ℓ is a length scale in the inertial range at which the increments are evaluated and ɛ the energy dissipation rate. Numerical data confirm this law. In addition, a useful definition of the energy fluxes in Fourier space is introduced and proven numerically to be flat in the inertial range. The exact results derived in this Rapid Communication are valid for both weak and strong wave turbulence. They could be used as a theoretical benchmark of new wave-turbulence theories and to develop further analogies with hydrodynamical turbulence.
3. Measurement of third-order elastic constants and applications to loaded structural materials.
Science.gov (United States)
Takahashi, Sennosuke; Motegi, Ryohei
2015-01-01
The objective of this study is to obtain the propagation velocity of an elastic wave in a loaded isotropic solid and to show the usefulness of the third-order elastic constant in determining properties of practical materials. As is well known, the infinitesimal elastic theory is unable to express the influence of stress on elastic wave propagating in loaded materials. To solve this problem, the authors derive an equation of motion for elastic wave in a finitely deformed state and use the Lagrangian description where the state before deformation is used as a reference, and Murnaghans finite deformation theory for the unidirectional deformed isotropic solid. Ordinary derivatives were used for the mathematical treatment and although the formulas are long the content is simple. The theory is applied to the measurement of the third-order elastic constants of common steels containing carbon of 0.22 and 0.32 wt%. Care is taken in preparing specimens to precise dimensions, in properly adhering of transducer to the surface of the specimen, and in having good temperature control during the measurements to obtain precise data. As a result, the stress at various sites in the structural materials could be estimated by measuring the elastic wave propagation times. The results obtained are graphed for illustration.
4. Estimation of Single-Crystal Elastic Constants of Polycrystalline Materials from Back-Scattered Grain Noise
International Nuclear Information System (INIS)
Haldipur, P.; Margetan, F. J.; Thompson, R. B.
2006-01-01
Single-crystal elastic stiffness constants are important input parameters for many calculations in material science. There are well established methods to measure these constants using single-crystal specimens, but such specimens are not always readily available. The ultrasonic properties of metal polycrystals, such as velocity, attenuation, and backscattered grain noise characteristics, depend in part on the single-crystal elastic constants. In this work we consider the estimation of elastic constants from UT measurements and grain-sizing data. We confine ourselves to a class of particularly simple polycrystalline microstructures, found in some jet-engine Nickel alloys, which are single-phase, cubic, equiaxed, and untextured. In past work we described a method to estimate the single-crystal elastic constants from measured ultrasonic velocity and attenuation data accompanied by metallographic analysis of grain size. However, that methodology assumes that all attenuation is due to grain scattering, and thus is not valid if appreciable absorption is present. In this work we describe an alternative approach which uses backscattered grain noise data in place of attenuation data. Efforts to validate the method using a pure copper specimen are discussed, and new results for two jet-engine Nickel alloys are presented
5. Ultrasonic Determination of the Elastic Constants of Epoxy-natural Fiber Composites
Science.gov (United States)
Valencia, C. A. Meza; Pazos-Ospina, J. F.; Franco, E. E.; Ealo, Joao L.; Collazos-Burbano, D. A.; Garcia, G. F. Casanova
This paper shows the applications ultrasonic through-transmission technique to determine the elastic constants of two polymer-natural fiber composite materials with potential industrial application and economic and environmental advantages. The transversely isotropic coconut-epoxy and fique-epoxy samples were analyzed using an experimental setup which allows the sample to be rotated with respect to transducers faces and measures the time-of-flight at different angles of incidence. Then, the elastic properties of the material were obtained by fitting the experimental data to the Christoffel equation. Results show a good agreement between the measured elastic constants and the values predicted by an analytical model. The velocities as a function of the incidence angle are reported and the effect of the natural fiber on the stiffness of the composite is discussed.
6. Elastic constants and dimensions of imprinted polymeric nanolines determined from Brillouin light scattering
International Nuclear Information System (INIS)
Johnson, W L; Kim, S A; Geiss, R; Flannery, C M; Soles, C L; Wang, C; Stafford, C M; Wu, W-L; Torres, J M; Vogt, B D; Heyliger, P R
2010-01-01
Elastic constants and cross-sectional dimensions of imprinted nanolines of poly(methyl methacrylate) (PMMA) on silicon substrates are determined nondestructively from finite-element inversion analysis of dispersion curves of hypersonic acoustic modes of these nanolines measured with Brillouin light scattering. The results for the cross-sectional dimensions, under the simplifying assumption of vertical sides and a semicircular top, are found to be consistent with dimensions determined from critical-dimension small-angle x-ray scattering measurements. The elastic constants C 11 and C 44 are found to be, respectively, 11.6% and 3.1% lower than their corresponding values for bulk PMMA. This result is consistent with the dimensional dependence of the quasi-static Young's modulus determined from buckling measurements on PMMA films with lower molecular weights. This study provides the first evidence of size-dependent effects on hypersonic elastic properties of polymers.
7. First-principles elastic constants and phonons of delta-Pu
DEFF Research Database (Denmark)
Söderlind, P.; Landa, A.; Sadigh, B.
2004-01-01
Elastic constants and zone-boundary phonons of delta-plutonium have been calculated within the density-functional theory. The paramagnetic state of delta-Pu is modeled by disordered magnetism utilizing either the disordered local moment or the special quasirandom structure techniques...
8. Model for ultrasonic attenuation and elastic constant in chromium and its alloys
International Nuclear Information System (INIS)
Castro, E.P.; Marques, G.E.; Camargo, P.C. de.
1987-01-01
A theory based on the thermodynamics of a magnetic system under applied acoustic field is proposed. The calculated attenuation and longitudinal elastic constant for pure chromium and its alloys with diluted vanadium, show a good agreement with the experimental values. (Author) [pt
9. Econometric estimation of the “Constant Elasticity of Substitution" function in R
DEFF Research Database (Denmark)
Henningsen, Arne; Henningsen, Geraldine
The Constant Elasticity of Substitution (CES) function is popular in several areas of economics, but it is rarely used in econometric analysis because it cannot be estimated by standard linear regression techniques. We discuss several existing approaches and propose a new grid-search approach...
10. Temperature variation of higher-order elastic constants of MgO
Indian Academy of Sciences (India)
series of strains using Taylor's series expansion. The coefficients of quadratic, cu- ... as thermal expansion, specific heat at higher temperature, temperature variation of ultrasonic velocity and attenuation, .... such studies have an impression that linear variation of elastic constant is true. The experimental study shows that ...
11. Diffraction plane dependence of elastic constants in ferritic steel in neutron diffraction stress measurement
International Nuclear Information System (INIS)
Hayashi, Makoto; Ishiwata, Masayuki; Minakawa, Noriaki; Funahashi, Satoru; Root, J.H.
1995-01-01
Neutron diffraction measurement have been made to investigate the diffraction plane dependence of elastic constants in ferritic steel. The measured diffraction planes were 110, 220, 112, 222 and 200. In the measurement a small tensile specimen was loaded in the tensile test rig specially designed for a neutron diffractometer. The strains obtained for five diffraction planes increased almost in proportion to the applied stress up to 230 MPa nearly equivalent to the yield stress. The mean elastic constants obtained were E=243 GPa and ν=0.28 for 110, 220 and 112, 182 GPa and 0.31 for softest 200, and 268 GPa and 0.30 for stiffest 222, respectively. The bulk elastic constants, E=222 GPa and ν=0.29, measured by the strain gauges almost agreed with the mean values for 110, 220 and 112. The Kroner elastic model is found to account for the diffraction plane dependence of Young's modulus and Poisson's ratio of the ferritic steel. (author)
12. First-principles study of crystal structure, elastic stiffness constants, piezoelectric constants, and spontaneous polarization of orthorhombic Pna21-M2O3 (M = Al, Ga, In, Sc, Y)
Science.gov (United States)
2018-03-01
We perform first-principles calculations to investigate the crystal structure, elastic and piezoelectric properties, and spontaneous polarization of orthorhombic M2O3 (M = Al, Ga, In, Sc, Y) with Pna21 space group based on density functional theory. The lattice parameters, full elastic stiffness constants, piezoelectric stress and strain constants, and spontaneous polarization are successfully predicted. Comparison with available experimental and computational results indicates the validity of our computational results. Detailed analysis of the results clarifies the difference in the bonding character and the origin of the strong piezoelectric response and large spontaneous polarization.
13. Estimation of single crystal elastic constants using ultrasonic testing - a feasibility study
International Nuclear Information System (INIS)
Phani Kumar, K.K.; Rentala, Vamsi Krishna; Gautam, Jaiprakash; Mylavarapu, Phani
2015-01-01
Estimation of single crystal elastic constants (SCEC) of metallic materials is of paramount importance in the development of crystal plasticity based models as well as for studying the effect of microstructure on wave propagation. SCEC are usually determined destructively by tensile and shear loading a single crystal specimen. These constants can also be estimated non-destructively, using X-ray diffraction measurements on a polycrystalline specimen. However, the aforementioned procedures have a limitation of either the sample size (in case of X-ray diffraction) or, availability of single crystal (in case of destructive testing). Hence, in this study, an effort has been undertaken to estimate SCEC by subjecting polycrystalline specimens to ultrasonic testing. Ultrasonic longitudinal and shear velocities, longitudinal attenuation coefficient and ultrasonic backscattered grain noise will be measured on pure Cu specimen. Further, these parameters will also be calculated analytically using existing relationships involving, elastic constants, grain size probability level, ultrasonic longitudinal and shear wave velocities, attenuation coefficient and backscattered grain noise. By minimizing the difference between experimentally measured and analytically calculated ultrasonic parameters, an attempt will be made to estimate single crystal elastic constants. (author)
14. First-principles calculations on third-order elastic constants and internal relaxation for monolayer graphene
International Nuclear Information System (INIS)
Wang Rui; Wang Shaofeng; Wu Xiaozhi; Liang Xiao
2010-01-01
The method of homogeneous deformation is combined with first-principles total-energy calculations on determining third-order elastic constants and internal relaxation for monolayer graphene. We employ density functional theory (DFT) within generalized-gradient-approximation (GGA). The elastic constants are obtained from a polynomial fitted to the calculations of strain-energy and strain-stress relations. Our results agree well with recent calculations by DFT calculations, tight-binding atomistic simulations, and experiments with an atomic force microscope. The internal relaxation displacement has also been determined from ab initio calculations. The details of internal lattice relaxation by first principles are basically consistent with the previous molecular dynamics (MD) simulation. But for tiny deformation, there is an anomalous region in which the behavior of internal relaxation is backward action. In addition, we have also demonstrated that the symmetry of the relationship between the internal displacement and the infinitesimal stains can be satisfied.
15. A potential for Th from inversion of cohesive energy: Elastic constants
Energy Technology Data Exchange (ETDEWEB)
Jaroszewicz, S., E-mail: [email protected] [Gerencia de Investigacion y Aplicaciones, Comision Nacional de Energia Atomica (Argentina); Mosca, H.O. [Gerencia de Investigacion y Aplicaciones, Comision Nacional de Energia Atomica (Argentina); Garces, J.E. [DAEE, Centro Atomico Bariloche, Comision Nacional de Energia Atomica (Argentina)
2012-08-15
An interatomic pair potential for Th was derived by using the Chen-Mobius lattice inversion of cohesive energy for fcc Th as a starting point to develop a free-parameter potential suitable to be used in molecular dynamic calculations for predicting microstructure evolution and thermal properties in multicomponent nuclear fuel. The cohesive energy versus lattice parameter of Th was computed from first principles electronic structure calculations. The elastic constants for fcc Th were calculated by applying different types of strain to the starting crystal. Based on this information, the shear modulus, the Youngs modulus and the Poissons ratio were obtained. The computed elastic constants of fcc Th are found to be in a good agreement with experiments and previous theoretical results.
16. Peak effect in the magnetostriction of superconducting NbTi due to elastic constants
International Nuclear Information System (INIS)
Wyder, U.; Linden, P.J.E.M. van der; Meulen, H.P. van der; Gerber, A.; Duyn, V.H.M.; Perenboom, J.A.A.J.; Visser, A. de; Franse, J.J.M.
1995-01-01
The magnetostriction on polycrystalline superconducting NbTi perpendicular to the field is measured to study lattice deformation effects caused by fluxoids exerting forces on the lattice through pinning centers. A ''dip'' for increasing and a ''peak'' for decreasing fields in its length at 80% of the upper critical field (B c2 =12 T) was observed. Its temperature dependence is well explained by universal B c2 (T) scaling laws for pinning forces and elastic constants. Identical features are observed in the magnetization. In critical current no anomaly is found. An explanation in terms of an anomaly in elastic constants due to a field induced degeneracy of the ground state is discussed. (orig.)
17. Modelling and simulation of multi-phase effects on X-ray elasticity constants
CERN Document Server
Freour, S; Guillen, R; François, M X
2003-01-01
This paper deals with the calculation of X-ray Elasticity Constants (XEC) of phases embedded in multi-phase polycrystals. A three scales (macroscopic, pseudo-macroscopic, mesoscopic) model based on the classical self-consistent formalism is developed in order to analyse multi-phase effects on XEC values. Simulations are performed for cubic or hexagonal crystallographic structure phases embedded in several two-phases materials. In fact, it is demonstrated that XEC vary with the macroscopic stiffness of the whole polycrystal. In consequence, the constants of one particular phase depend on the elastic behaviour and the volume fraction of all the phases constituting the material. Now, XEC play a leading role in pseudo-macroscopic stresses determination by X-Ray Diffraction (XRD) methods. In this work, a quantitative analysis of the multi-phase effects on stresses determination by XRD methods was performed. Numerical results will be compared and discussed. (Abstract Copyright [2003], Wiley Periodicals, Inc.)
18. Magnetic susceptibility, electrical resistivity and elastic constants of antiferromagnetic UN single crystals
International Nuclear Information System (INIS)
Du Plessis, P. de V.; Doorn, C.F. van
1977-01-01
Susceptibility and electrical resistivity measurements on UN indicate Tsub(N) approximately 53 K. The spin-disorder resistivity is mainly proportional to 1-m 2 sub(n)(msub(n) is the reduced sublattice magnetization). The elastic constant C 44 shows a renormalization proportional to M 2 sub(n), whereas C 11 exhibits an anomalous softening of 10% well below Tsub(N) at 47 K. (Auth.)
19. Three-body interactions and the elastic constants of hcp solid 4He
Science.gov (United States)
Barnes, Ashleigh L.; Hinde, Robert J.
2017-09-01
The effect of three-body interactions on the elastic properties of hexagonal close packed solid 4He is investigated using variational path integral (VPI) Monte Carlo simulations. The solid's nonzero elastic constants are calculated, at T = 0 K and for a range of molar volumes from 7.88 cm3/mol to 20.78 cm3/mol, from the bulk modulus and the three pure shear constants C0, C66, and C44. Three-body interactions are accounted for using our recently reported perturbative treatment based on the nonadditive three-body potential of Cencek et al. Previous studies have attempted to account for the effect of three-body interactions on the elastic properties of solid 4He; however, these calculations have treated zero point motions using either the Einstein or Debye approximations, which are insufficient in the molar volume range where solid 4He is characterized as a quantum solid. Our VPI calculations allow for a more accurate treatment of the zero point motions which include atomic correlation. From these calculations, we find that agreement with the experimental bulk modulus is significantly improved when three-body interactions are considered. In addition, three-body interactions result in non-negligible differences in the calculated pure shear constants and nonzero elastic constants, particularly at higher densities, where differences of up to 26.5% are observed when three-body interactions are included. We compare to the available experimental data and find that our results are generally in as good or better agreement with experiment as previous theoretical investigations.
20. Phonon dispersions and elastic constants of disordered Pd-Ni alloys
Energy Technology Data Exchange (ETDEWEB)
Kart, S. Oezdemir [Department of Physics, Middle East Technical University, 06531 Ankara (Turkey)]. E-mail: [email protected]; Tomak, M. [Department of Physics, Middle East Technical University, 06531 Ankara (Turkey); Cagin, T. [Department of Chemical Engineering, Texas A and M University, College Station, TX 77845-3122 (United States)
2005-01-31
Phonon frequencies of Pd-Ni alloys are calculated by molecular dynamics (MD) simulation. Lattice dynamical properties computed from Sutton-Chen (SC) and quantum Sutton-Chen (Q-SC) potentials as a function of temperature are compared with each other. We present all interatomic force constants up to the 8th nearest-neighbor shell obtained by using the calculated potential. Elastic constants evaluated by two methods are consistent with each other. The transferability of the potential is also tested. The results are in good agreement with experimental data and other calculations.
1. Phonon dispersions and elastic constants of disordered Pd-Ni alloys
International Nuclear Information System (INIS)
Kart, S. Oezdemir; Tomak, M.; Cagin, T.
2005-01-01
Phonon frequencies of Pd-Ni alloys are calculated by molecular dynamics (MD) simulation. Lattice dynamical properties computed from Sutton-Chen (SC) and quantum Sutton-Chen (Q-SC) potentials as a function of temperature are compared with each other. We present all interatomic force constants up to the 8th nearest-neighbor shell obtained by using the calculated potential. Elastic constants evaluated by two methods are consistent with each other. The transferability of the potential is also tested. The results are in good agreement with experimental data and other calculations
2. Strong coupling constant extraction from high-multiplicity Z +jets observables
Science.gov (United States)
Johnson, Mark; Maître, Daniel
2018-03-01
We present a strong coupling constant extraction at next-to-leading order QCD accuracy using ATLAS Z +2 ,3,4 jets data. This is the first extraction using processes with a dependency on high powers of the coupling constant. We obtain values of the strong coupling constant at the Z mass compatible with the world average and with uncertainties commensurate with other next-to-leading order extractions at hadron colliders. Our most conservative result for the strong coupling constant is αS(MZ)=0.117 8-0.0043+0.0051 .
3. Magnetic field dependence of elastic constants in Pr1-xCaxMnO3
International Nuclear Information System (INIS)
Hazama, Hirofumi; Nemoto, Yuichi; Goto, Terutaka; Tomioka, Yasuhide; Tokura, Yoshinori; Asamitsu, Atsushi
2002-01-01
We have investigated charge and orbital (quadrupolar) ordering in perovskite manganites Pr 1-x Ca x MnO 3 (x=0.35, 0.40, 0.50) by ultrasonic measurements. Only transverse (C 11 - C 12 )/2 mode of the x=0.35, 0.40 and 0.50 compounds shows a softening from 400 K down to the charge ordering point T co =220 - 240 K. This softening is originated from the bilinear coupling of the quadrupole moment of the e g orbital in Mn 3+ ion to the elastic strain. Furthermore, the elastic softening in both (C 11 - C 12 )/2 and C 44 modes due to the coupling of the charge fluctuation mode to elastic strain is observed just above T co , especially for the x=0.50 compound. The considerable magnetic field dependence of the elastic constants is observed. In particular, the magnetic fields up to 12 T releases completely the elastic softening of (C 11 - C 12 )/2 mode due to the fact that the charge fluctuation is suppressed under 12 T in the x=0.40 compound. (author)
4. Estimation of Single-Crystal Elastic Constants from Ultrasonic Measurements on Polycrystalline Specimens
International Nuclear Information System (INIS)
Haldipur, P.; Margetan, F.J.; Thompson, R.B.
2004-01-01
In past work we reported on measurements of ultrasonic velocity, attenuation and backscattering in nickel-alloy materials used in the fabrication of rotating jet-engine components. Attenuation and backscattering were shown to be correlated to the average grain diameter, which varied with position in the billet specimens studied. The ultrasonic measurements and associated metallographic studies found the local microstructures to be approximately equiaxed and free of texture in these cubic-phase metals. In this paper we explore a method for deducing the single-crystal elastic constants of a metal using the combined ultrasonic and metallographic data for a polycrystalline specimen. We specifically consider the case seen in the jet-engine alloys: polycrystalline cubic microstructures having equiaxed, randomly oriented grains. We demonstrate how the three independent elastic constants {C11, C12, C44} can be deduced from the density, the mean grain diameter, the ultrasonic attenuation at one or more frequencies, and the longitudinal and shear wave speeds. The method makes use of the attenuation theory of Stanke and Kino, and the Hill averaging procedure for estimating the sonic velocity through a polycrystalline material. Elastic constant inputs to the velocity and attenuation models are adjusted to optimize the agreement with experiment. The method is demonstrated using several specimens of Inconel 718 and Waspaloy, and further tested using four specimens of pure Nickel
5. Applying a Bayesian Approach to Identification of Orthotropic Elastic Constants from Full Field Displacement Measurements
Directory of Open Access Journals (Sweden)
Le Riche R.
2010-06-01
Full Text Available A major challenge in the identification of material properties is handling different sources of uncertainty in the experiment and the modelling of the experiment for estimating the resulting uncertainty in the identified properties. Numerous improvements in identification methods have provided increasingly accurate estimates of various material properties. However, characterizing the uncertainty in the identified properties is still relatively crude. Different material properties obtained from a single test are not obtained with the same confidence. Typically the highest uncertainty is associated with respect to properties to which the experiment is the most insensitive. In addition, the uncertainty in different properties can be strongly correlated, so that obtaining only variance estimates may be misleading. A possible approach for handling the different sources of uncertainty and estimating the uncertainty in the identified properties is the Bayesian method. This method was introduced in the late 1970s in the context of identification [1] and has been applied since to different problems, notably identification of elastic constants from plate vibration experiments [2]-[4]. The applications of the method to these classical pointwise tests involved only a small number of measurements (typically ten natural frequencies in the previously cited vibration test which facilitated the application of the Bayesian approach. For identifying elastic constants, full field strain or displacement measurements provide a high number of measured quantities (one measurement per image pixel and hence a promise of smaller uncertainties in the properties. However, the high number of measurements represents also a major computational challenge in applying the Bayesian approach to full field measurements. To address this challenge we propose an approach based on the proper orthogonal decomposition (POD of the full fields in order to drastically reduce their
6. Estimation of parameters of constant elasticity of substitution production functional model
Science.gov (United States)
Mahaboob, B.; Venkateswarlu, B.; Sankar, J. Ravi
2017-11-01
Nonlinear model building has become an increasing important powerful tool in mathematical economics. In recent years the popularity of applications of nonlinear models has dramatically been rising up. Several researchers in econometrics are very often interested in the inferential aspects of nonlinear regression models [6]. The present research study gives a distinct method of estimation of more complicated and highly nonlinear model viz Constant Elasticity of Substitution (CES) production functional model. Henningen et.al [5] proposed three solutions to avoid serious problems when estimating CES functions in 2012 and they are i) removing discontinuities by using the limits of the CES function and its derivative. ii) Circumventing large rounding errors by local linear approximations iii) Handling ill-behaved objective functions by a multi-dimensional grid search. Joel Chongeh et.al [7] discussed the estimation of the impact of capital and labour inputs to the gris output agri-food products using constant elasticity of substitution production function in Tanzanian context. Pol Antras [8] presented new estimates of the elasticity of substitution between capital and labour using data from the private sector of the U.S. economy for the period 1948-1998.
7. Diffraction and single-crystal elastic constants of Inconel 625 at room and elevated temperatures determined by neutron diffraction
International Nuclear Information System (INIS)
Wang, Zhuqing; Stoica, Alexandru D.; Ma, Dong; Beese, Allison M.
2016-01-01
In this work, diffraction and single-crystal elastic constants of Inconel 625 have been determined by means of in situ loading at room and elevated temperatures using time-of-flight neutron diffraction. Theoretical models proposed by Voigt, Reuss, and Kroner were used to determine single-crystal elastic constants from measured diffraction elastic constants, with the Kroner model having the best ability to capture experimental data. The magnitude of single-crystal elastic moduli, computed from single-crystal elastic constants, decreases and the single crystal anisotropy increases as temperature increases, indicating the importance of texture in affecting macroscopic stress at elevated temperatures. The experimental data reported here are of great importance in understanding additive manufacturing of metallic components as: diffraction elastic constants are required for computing residual stresses from residual lattice strains measured using neutron diffraction, which can be used to validate thermomechanical models of additive manufacturing, while single-crystal elastic constants can be used in crystal plasticity modeling, for example, to understand mechanical deformation behavior of additively manufactured components.
8. Evaluating Bounds and Estimators for Constants of Random Polycrystals Composed of Orthotropic Elastic Materials
Energy Technology Data Exchange (ETDEWEB)
Berryman, J. G.
2012-03-01
While the well-known Voigt and Reuss (VR) bounds, and the Voigt-Reuss-Hill (VRH) elastic constant estimators for random polycrystals are all straightforwardly calculated once the elastic constants of anisotropic crystals are known, the Hashin-Shtrikman (HS) bounds and related self-consistent (SC) estimators for the same constants are, by comparison, more difficult to compute. Recent work has shown how to simplify (to some extent) these harder to compute HS bounds and SC estimators. An overview and analysis of a subsampling of these results is presented here with the main point being to show whether or not this extra work (i.e., in calculating both the HS bounds and the SC estimates) does provide added value since, in particular, the VRH estimators often do not fall within the HS bounds, while the SC estimators (for good reasons) have always been found to do so. The quantitative differences between the SC and the VRH estimators in the eight cases considered are often quite small however, being on the order of ±1%. These quantitative results hold true even though these polycrystal Voigt-Reuss-Hill estimators more typically (but not always) fall outside the Hashin-Shtrikman bounds, while the self-consistent estimators always fall inside (or on the boundaries of) these same bounds.
9. Elastic constants of neodymium single crystals in the temperature range 4.2--300 degreeK
International Nuclear Information System (INIS)
Greiner, J.D.; Schlader, D.M.; McMasters, O.D.; Gschneidner, K.A. Jr.; Smith, J.F.
1976-01-01
The elastic constants of a single crystal of the double hcp allotrope of neodymium have been measured over the temperature range 4.2--300 degreeK. The magnetic orderings which occur in neodymium near 7.5 and 19 degreeK are readily evident as cusps in the temperature dependences of some of the directly measured ultrasonic wave velocities, as well as in the associated elastic constants, and the character of the magnetic interactions is reflected in the differing effects on the various elastic constants. Comparison of the elastic constants of neodymium with those of seven other rare earths shows a trend with atomic number which is similar to trends which have been observed in other physical properties
10. Kelvin Notation for Stabilizing Elastic-Constant Inversion Notation Kelvin pour stabiliser l'inversion de constantes élastiques
Directory of Open Access Journals (Sweden)
Dellinger J.
2006-12-01
Full Text Available Inverting a set of core-sample traveltime measurements for a complete set of 21 elastic constants is a difficult problem. If the 21 elastic constants are directly used as the inversion parameters, a few bad measurements or an unfortunate starting guess may result in the inversion converging to a physically impossible solution . Even given perfect data, multiple solutions may exist that predict the observed traveltimes equally well. We desire the inversion algorithm to converge not just to a physically possible solution, but to the best(i. e. most physically likely solution of all those allowed. We present a new parameterization that attempts to solve these difficulties. The search space is limited to physically realizable media by making use of the Kelvin eigenstiffness-eigentensor representation of the 6 x 6 elastic stiffness matrix. Instead of 21 stiffnesses, there are 6 eigenstiffness parametersand 15 rotational parameters . The rotational parameters are defined using a Lie-algebra representation that avoids the artificial degeneracies and coordinate-system bias that can occur with standard polar representations. For any choice of these 21 real parameters, the corresponding stiffness matrix is guaranteed to be physically realizable. Furthermore, all physically realizable matrices can be represented in this way. This new parameterization still leaves considerable latitude as to which linear combinations of the Kelvin parameters to use, and how they should be ordered. We demonstrate that by careful choice and ordering of the parameters, the inversion can be relaxedfrom higher to lower symmetry simply by adding a few more parameters at a time. By starting from isotropy and relaxing to the general result in stages (isotropy, transverse isotropy, orthorhombic, general, we expect that the method should find the solution that is closest to isotropy of all those that fit the data. L'inversion d'un ensemble de mesures du temps de parcours d
11. Elastic Constants of Brucite (Mg(OH)2) and Diaspore (AlO(OH)) to 12 GPa by Brillouin Scattering
Science.gov (United States)
Jiang, F.; Speziale, S.; Majzlan, J.; Duffy, T. S.
2005-12-01
Hydroxides such as brucite, Mg(OH)2, and diaspore, AlO(OH), serve as analogs for the more complex hydrogen-bearing minerals found in subduction zones. The single-crystal elastic constants of these two minerals were determined by Brillouin scattering up to 12 GPa in a diamond anvil cell. A 16:3:1 methanol-ethanol-water and 4:1 methanol-ethanol mixture were used as pressure media and ruby as pressure calibrant. Two platelets of brucite and three platelets of diaspore were measured at room pressure and over nine elevated pressures. Brillouin spectra were recorded in 37 directions with a 5-degree step for each plane. All individual elastic stiffness constants were retrieved by fitting the velocity data to the Christoffel's equation. The individual elastic constants, aggregate bulk and shear moduli, were then fitted to the finite Eulerian stain equations to obtain their pressure derivatives. For brucite, aggregate moduli and their pressure derivatives are KT0=35.9(9) GPa, GO=31.3(2) GPa, (∂ Kt/∂ P)T0=8.9(3), (∂ G/∂ P)O=4.33(4) for the Reuss bound. Diaspore aggregate moduli and their pressure derivatives are KT0=147.0(7) GPa, GO=114.8(5) GPa, (∂ Ks/∂ P)T0=4.1(1), (∂ G/∂ P)O=1.6(1). For diaspore, both individual and aggregate elastic moduli define nearly linear modulus pressure trend within the measured pressure range. For Brucite, there are clear deviations from linear modulus - pressure trends. In comparison with diaspore, brucite exhibits strong anisotropy. The ratio of the linear compressibility of brucite along the c and a axes decreased from 4.7 to 1.3 over the examined pressure range. The shear anisotropy (C66/C44) decreased from 2.6 at ambient condition to 1.3 with increase of pressure to 12 GPa. Compression curves constructed from our Brillouin data for both brucite and diaspore are in good agreement with previous x-ray diffraction data.
12. The effects of rigid motions on elastic network model force constants.
Science.gov (United States)
Lezon, Timothy R
2012-04-01
Elastic network models provide an efficient way to quickly calculate protein global dynamics from experimentally determined structures. The model's single parameter, its force constant, determines the physical extent of equilibrium fluctuations. The values of force constants can be calculated by fitting to experimental data, but the results depend on the type of experimental data used. Here, we investigate the differences between calculated values of force constants and data from NMR and X-ray structures. We find that X-ray B factors carry the signature of rigid-body motions, to the extent that B factors can be almost entirely accounted for by rigid motions alone. When fitting to more refined anisotropic temperature factors, the contributions of rigid motions are significantly reduced, indicating that the large contribution of rigid motions to B factors is a result of over-fitting. No correlation is found between force constants fit to NMR data and those fit to X-ray data, possibly due to the inability of NMR data to accurately capture protein dynamics. Copyright © 2011 Wiley Periodicals, Inc.
13. Calculations of single crystal elastic constants for yttria partially stabilised zirconia from powder diffraction data
Science.gov (United States)
Lunt, A. J. G.; Xie, M. Y.; Baimpas, N.; Zhang, S. Y.; Kabra, S.; Kelleher, J.; Neo, T. K.; Korsunsky, A. M.
2014-08-01
Yttria Stabilised Zirconia (YSZ) is a tough, phase-transforming ceramic that finds use in a wide range of commercial applications from dental prostheses to thermal barrier coatings. Micromechanical modelling of phase transformation can deliver reliable predictions in terms of the influence of temperature and stress. However, models must rely on the accurate knowledge of single crystal elastic stiffness constants. Some techniques for elastic stiffness determination are well-established. The most popular of these involve exploiting frequency shifts and phase velocities of acoustic waves. However, the application of these techniques to YSZ can be problematic due to the micro-twinning observed in larger crystals. Here, we propose an alternative approach based on selective elastic strain sampling (e.g., by diffraction) of grain ensembles sharing certain orientation, and the prediction of the same quantities by polycrystalline modelling, for example, the Reuss or Voigt average. The inverse problem arises consisting of adjusting the single crystal stiffness matrix to match the polycrystal predictions to observations. In the present model-matching study, we sought to determine the single crystal stiffness matrix of tetragonal YSZ using the results of time-of-flight neutron diffraction obtained from an in situ compression experiment and Finite Element modelling of the deformation of polycrystalline tetragonal YSZ. The best match between the model predictions and observations was obtained for the optimized stiffness values of C11 = 451, C33 = 302, C44 = 39, C66 = 82, C12 = 240, and C13 = 50 (units: GPa). Considering the significant amount of scatter in the published literature data, our result appears reasonably consistent.
14. Frequency change and elastic constants of quartz irradiated by 1 MeV electrons
International Nuclear Information System (INIS)
Aoki, Takashi; Norisawa, Keizo; Sakisaka, Masakatsu.
1976-01-01
The fractions of frequency change Δf/f for quartz resonator plates exposed to 1 MeV electrons are measured as a function of electron fluence up to 2x10 17 electrons/cm 2 , from which the adiabatic elastic constants of c 44 , c 14 and c 66 are determined. The ratio of these fractional changes is found to have a correlation of Δc 44 /c 44 :Δ/c 14 ///c 14 /:Δc 66 /c 66 =1.3:3.1:1.0. The Δf/f values against fluence are calculated for various cut angles. The result indicates that the frequency in AT-cut resonators decreases a little and then increases ( -- 10 -5 ), but in BT- and Y-cut resonators a drastic increase and then a decrease (10 -3 -- 10 -2 ) are expected. (auth.)
15. Strong Nuclear Gravitational Constant and the Origin of Nuclear Planck Scale
Directory of Open Access Journals (Sweden)
Seshavatharam U. V. S.
2010-07-01
Full Text Available Whether it may be real or an equivalent, existence of strong nuclear gravitational con- stant G S is assumed. Its value is obtained from Fermi’s weak coupling constant as G S = 6 : 9427284 10 31 m 3 / kg sec 2 and thus “nuclear planck scale” is defined. For strong interaction existence of a new integral charged “confined fermion” of mass 105.383 MeV is assumed. Strong coupling constant is the ratio of nuclear planck energy = 11.97 MeV and assumed 105.383 MeV. 1 s = X s is defined as the strong interaction mass gen- erator. With 105.383 MeV fermion various nuclear unit radii are fitted. Fermi’s weak coupling constant, strong interaction upper limit and Bohr radius are fitted at funda- mental level. Considering Fermi’s weak coupling constant and nuclear planck length a new number X e = 294.8183 is defined for fitting the electron, muon and tau rest masses. Using X s , X e and 105 : 32 = 0 : 769 MeV as the Coulombic energy constant = E c , en- ergy coe cients of the semi-empirical mass formula are estimated as E v = 16 : 32 MeV ; E s = 19 : 37 MeV ; E a = 23 : 86 MeV and E p = 11 : 97 MeV where Coulombic energy term contains [ Z ] 2 : Starting from Z = 2 nuclear binding energy is fitted with two terms along with only one energy constant = 0.769 MeV. Finally nucleon mass and its excited levels are fitted.
16. Determination of elastic constants of fuels plates based on uranium by ultrasound testing
International Nuclear Information System (INIS)
Moreira Castro, Martin Ignacio
2015-01-01
Current nuclear reactors use as U-235 U-enriched compounds enriched with U-235, requiring U-alloys that increase the amount of atoms available for nuclear fission in a convenient way. This study was carried out on fuel plates manufactured in the Chilean Nuclear Energy Commission, whose cores are composed of a dispersed mixture Al-U 3 Si 2 and Al-U 7 Mo, with different densities of uranium, covered by a coating of Al6061. The objective was to characterize elastically and classify the fuel plates analyzed. Specifically, five Al-U 3 Si 2 fuel plates with 1.7 gU/cm 3 , eight A-U 3 Si 2 with 3.4 gU/cm 3 , five of A-l U 3 Si 2 with 4.8 gU/cm 3 were successfully studied. The apparent elastic constants (Young and Shear modules, and Poisson coefficient) were determined in the area where the fuel is located (MEAT) by means of an ultrasound sampling technique, thus being able to characterize them and classify them according to their composition. The behavior of the elastic constants generally shows a tendency to decrease as the amount of U 3 Si 2 particles dispersed in the MEAT zone of the fuel plates increases. In addition, the non-destructive test method used made it possible to detect several differences between the fuel plates analyzed, such as the amount of reduction in rolling, among others. Additionally, six experimental fuel miniplates were analyzed whose meat were formed by a dispersion of the Al-UMo type, specifically: two of Al-U 7 Mo with 6.0 gU/cm 3 , two of Al-U 7 Mo with 7.0 gU/ cm 3 and two of Al-U 7 Mo with 8.0 gU/cm 3 . The response of the U-Mo fuel miniplates against this technique was not good, so several ideas were proposed to improve this situation
17. Measurement of the Strong Coupling Constant from Inclusive Jet Production at the Tevatron Collider
Energy Technology Data Exchange (ETDEWEB)
Mesropian, Christina [Rockefeller Univ., New York, NY (United States)
2000-06-01
We present a measurement of the strong coupling constant from a single observable, the inclusive jet cross section. We use 86 pb-1 of data collected with the Collider Detector at Fermilab (CDF) from p$\\bar{p}$ collisions at √s = 1800 GeV. The data was analyzed and experimental systematic uncertainties estimated.
18. First-principles calculations of the elastic constants of the cubic, orthorhombic and hexagonal phases of BaF2
International Nuclear Information System (INIS)
Nyawere, P.W.O.; Makau, N.W.; Amolo, G.O.
2014-01-01
All the elastic constants of cubic, orthorhombic and hexagonal phases of BaF 2 have been calculated using first principles methods. We have employed density-functional theory within generalized gradient approximation (GGA) using a plane-wave pseudopotentials method and a plane-wave basis set. The calculated elastic constant values for a cubic phase compare well with recent theoretical and experimental calculations. The bulk modulus derived from the elastic constant calculations of orthorhombic phase of BaF 2 is 94.5 GPa and those of hexagonal phase is 161 GPa. These values are in good agreement with experimental data available. Stability of these phases of BaF 2 is also estimated in different crystallographic directions
19. Elastic removal self-shielding factors for light and medium nuclides with strong-resonance scattering
International Nuclear Information System (INIS)
Nakagawa, Masayuki; Ishiguro, Yukio; Tokuno, Yukio.
1978-01-01
The self-shielding factors for elastic removal cross sections of light and medium weight nuclides were calculated for the parameter, σ 0 within the conventional concept of the group constant sets. The numerical study were performed for obtaining a simple and accurate method. The present results were compared with the exact values and the conventional ones, and shown to be remarkably improved. It became apparent that the anisotropy of the elastic scattering did not affect to the self-shielding factors though it did to the infinite dilution cross sections. With use of the present revised set, the neutron flux were calculated in an iron medium and in a prototype FBR and compared with those by the fine spectrum calculations and the conventional set. The present set showed the considerable improvement in the vicinity of the large resonance regions of sodium, iron and oxygen. (auth.)
20. Using strong nonlinearity and high-frequency vibrations to control effective properties of discrete elastic waveguides
DEFF Research Database (Denmark)
Lazarov, Boyan Stefanov; Thomsen, Jon Juel; Snaeland, Sveinn Orri
2008-01-01
The aim of this article is to investigate how highfrequency (HF) excitation, combined with strong nonlinear elastic material behavior, influences the effective material or structural properties for low-frequency excitation and wave propagation. The HF effects are demonstrated on discrete linear...... spring-mass chains with non-linear inclusions. The presented analytical and numerical results suggest that the effective material properties can easily be altered by establishing finite amplitude HF standing waves in the non-linear regions of the chain....
1. Numerical investigation of influence of ionic space charge and flexoelectric polarization on measurements of elastic constants in nematic liquid crystals
Science.gov (United States)
Buczkowska, M.; Derfel, G.; Konowalski, M.
2009-06-01
Deformations of nematic layers caused by magnetic field allow determination of the elastic constants of liquid crystal. In this paper, we simulated numerically the deformations of planar and homeotropic nematic layers. The flexoelectric properties of the nematic and presence of ions were taken into account. Our aim was to show the influence of flexoelectricity on the results of the real measurement of the elastic constants k33 and k11. In these simulations, we calculated the optical phase difference ΔΦ between the ordinary and extraordinary rays of light passing through the layer placed between crossed polarizers as a function of the magnetic field induction B. One of the elastic constants can be calculated from the magnetic field threshold for deformation. The ratio k33/k11 can be found by means of fitting theoretical ΔΦ(B) dependence to the experimental results. The calculations reveal that the flexoelectric properties influence the deformations induced by the external magnetic field. In the case of highly pure samples, this may lead to false results of measurement of the elastic constants ratio k33/k11. This influence can be reduced if the nematic material contains ions of sufficiently high concentration. These results show that the flexoelectric properties may play an important role, especially in well purified samples.
2. Elastic Constants of Solids and Fluids with Initial Pressure via a Unified Approach Based on Equations-of-State
Science.gov (United States)
Cantrell, John H.
2014-01-01
The second and third-order Brugger elastic constants are obtained for liquids and ideal gases having an initial hydrostatic pressure p(sub 1). For liquids the second-order elastic constants are C(sub 11) = A + p(sub 1), C(sub 12) = A -- p(sub 1), and the third-order constants are C(sub 111) = --(B + 5A + 3p(sub 1)), C(sub 112) = --(B + A -- p(sub 1)), and C(sub 123) = A -- B -- p1, where A and B are the Beyer expansion coefficients in the liquid equation of state. For ideal gases the second order constants are C(sub 11) = p(sub 1)gamma + p9sub 1), C(sub 12) = p(sub 1)gamma -- p(sub 1), and the third-order constants are C(sub 111) = p(sub 1)(gamma(2) + 4gamma + 3), C(sub 112) = --p(sub 1)(gamma(2) -- 1), and C(sub 123) = --p(sub 1) (gamma(2) -- 2gamma + 1), where gamma is the ratio of specific heats. The inequality of C(sub 11) and C(sub 12) results in a nonzero shear constant C(sub 44) = (1/2)(C(sub 11) C(sub 12)) = p(sub 1) for both liquids and gases. For water at standard temperature and pressure the ratio of terms p1/A contributing to the second-order constants is approximately 4.3 x 10(-5). For atmospheric gases the ratio of corresponding terms is approximately 0.7. Analytical expressions that include initial stresses are derived for the material 'nonlinearity parameters' associated with harmonic generation and acoustoelasticity for fluids and solids of arbitrary crystal symmetry. The expressions are used to validate the relationships for the elastic constants of fluids.
3. Effect of Microstructure Constraints on the Homogenized Elastic Constants of Elastomeric Sylgard/GMB Syntactic Foam.
Energy Technology Data Exchange (ETDEWEB)
Brown, Judith Alice [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Steck, Daniel [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Brown, Judith Alice [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Long, Kevin Nicholas [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2017-08-01
Previous numerical studies of Sylgard filled with glass microballoons (GMB) have relied on various microstructure idealizations to achieve a large range of volume fractions with high mesh quality. This study investigates how different microstructure idealizations and constraints affect the apparent homogenized elastic constants in the virgin state of the material, in which all GMBs are intact and perfectly bonded to the Sylgard matrix, and in the fully damaged state of the material in which all GMBs are destroyed. In the latter state, the material behaves as an elastomeric foam. Four microstructure idealizations are considered relating to how GMBs are packed into a representative volume element (RVE): (1) no boundary penetration nor GMB-GMB overlap, (2) GMB-GMB overlap, (3) boundary penetration, and (4) boundary penetration and GMB-GMB overlap. First order computational homogenization with kinematically uniform displacement boundary conditions (KUBCs) was employed to determine the homogenized (apparent) bulk and shear moduli for the four microstructure idealizations in the intact and fully broken GMB material states. It was found that boundary penetration has a significant effect on the shear modulus for microstructures with intact GMBs, but that neither boundary penetration nor GMB overlap have a significant effect on homogenized properties for microstructures with fully broken GMBs. The primary conclusion of the study is that future investigations into Sylgard/GMB micromechanics should either force GMBs to stay within the RVE fully and/or use periodic BCs (PBCs) to eliminate the boundary penetration issues. The implementation of PBCs requires the improvement of existing tools in Sandia’s Sierra/SM code.
4. Thermal expansion and temperature variation of elastic constants of Li(H,D) and Na(H,D) systems
International Nuclear Information System (INIS)
Islam, A.K.M.A.; Hoque, M.T.
1994-11-01
An analysis of thermal expansion of Li(H,D) systems up to melting temperature has been performed using the theory of anharmonic lattice. The study has for the first time been extended to Na(H,D) systems where very little or no data are available. The calculated lattice constants of Li(H,D) systems show quite good agreement with experiment. The success of the present calculation with Li(H,D) and room temperature lattice constant data for Na(H,D) given an indication of the reliability of the computed lattice constants and thermal expansion coefficients for Na(H,D) systems. The study also allows us to predict the hitherto unknown lattice constants of Na(H,D) crystal at 0K. The temperature dependence of elastic constants for Li(H,D) systems has also been evaluated. Comparison with measurements shows the reliability of the present calculations. (author). 45 refs, 4 figs
5. Effective X-ray elastic constant measurement for in situ stress measurement of biaxially strained AA5754-O
International Nuclear Information System (INIS)
Iadicola, Mark A.; Gnäupel-Herold, Thomas H.
2012-01-01
Accurate measurement of stresses by X-ray diffraction requires accurate X-ray elastic constants. Calibration experiments are one method to determine these for a specific material in a specific condition. In this paper, uniaxial tension experiments are used to investigate the variation of these constants after uniaxial and equal-biaxial plastic deformation for an aluminum alloy (AA5754-O) of interest to the automotive industry. These data are critical for accurate measurement of the biaxial mechanical properties of the material using a recent experimental method combining specialized sheet metal forming equipment with portable X-ray diffraction equipment. The measured effective X-ray elastic constants show some minor variation with increased plastic deformation, and this behavior was found to be consistent for both uniaxially and equal-biaxially strained samples. The use of two average values for effective X-ray elastic constants, one in the rolling direction and one transverse to the rolling direction of the sheet material, is shown to be of sufficient accuracy for the combined tests of interest. Comparison of uniaxial data measured using X-ray diffraction and standard methods show good agreement, and biaxial stress–strain results show good repeatability. Additionally, the calibration data show some non-linear behavior, which is analyzed in regards to crystallographic texture and intergranular stress effects. The non-linear behavior is found to be the result of intergranular stresses based on comparison with additional measurements using other X-ray diffraction equipment and neutron diffraction.
6. Evaluation of single crystal elastic constants and stacking fault energy in high-nitrogen duplex stainless steel by in-situ neutron diffraction
International Nuclear Information System (INIS)
Kim, Yanghoo; Kim, Yong Min; Koh, Ji-Yeon; Lee, Tae-Ho; Woo, Wan Chuck; Han, Heung Nam
2016-01-01
Single crystal elastic constants of austenite and ferrite phases in high-nitrogen duplex stainless steel were evaluated by an elastic self-consistent model combined with an optimization process using in-situ neutron diffraction data. The optimized elastic constants were validated by the indentation moduli of each phase obtained by nanoindentation. In addition, the stacking fault energy of austenite was evaluated based on the neutron diffraction profile and the single crystal elastic constants and was subsequently correlated with the observed deformation microstructure.
7. The elastic constants and anisotropy of superconducting MgCNi3 and CdCNi3 under different pressure
KAUST Repository
Feng, Huifang
2013-11-23
The second-order elastic constants (SOECs) and third-order elastic constants (TOECs) of MgCNi3 and CdCNi3 are presented by using first-principles methods combined with homogeneous deformation theory. The Voigt-Reuss-Hill (VRH) approximation are used to calculate the bulk modulus B, shear modulus G, averaged Young\\'s modulus E and Poisson\\'s ratio ν for polycrystals and these effective modulus are consistent with the experiments. The SOECs under different pressure of MgCNi3 and CdCNi3 are also obtained based on the TOECs. Furthermore, the Zener anisotropy factor, Chung-Buessem anisotropy index, and the universal anisotropy index are used to describe the anisotropy of MgCNi3 and CdCNi3. The anisotropy of Young\\'s modulus of single-crystal under different pressure is also presented. © 2013 Springer Science+Business Media New York.
8. Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics
Science.gov (United States)
Motsepa, Tanki; Aziz, Taha; Fatima, Aeeman; Khalique, Chaudry Masood
2018-03-01
The optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.
9. Diffraction plane dependence of elastic constants in residual stress measurement by neutron diffraction
International Nuclear Information System (INIS)
Okido, Shinobu; Hayashi, Makoto; Morii, Yukio; Minakawa, Nobuaki; Tsuchiya, Yoshinori
1997-01-01
In a residual stress measurement by x-ray diffraction method and a neutron diffraction method, strictly speaking, the strain measurement of various diffracted surface was conducted and it is necessary to use its elastic modulus to convert from the strain to the stress. Then, in order to establish the residual stress measuring technique using neutron diffraction, it is an aim at first to make clear a diffraction surface dependency of elastic modulus for the stress conversion in various alloys. As a result of investigations the diffraction surface dependency of elastic module on SUS304 and STS410 steels by using RESA (Neutron diffractometer for residual stress analysis) installed at JRR-3M in Tokai Establishment of JAERI, following results are obtained. The elastic modulus of each diffraction surface considering till plastic region could be confirmed to be in a region of ±20% of that calculated by Kroner's model and to be useful for that used on conversion to the stress. And, error of this elastic modulus was thought to cause the transition and defect formed at inner portion of the materials due to a plastic deformation. (G.K.)
10. Ionization, photoelectron dynamics and elastic scattering in relativistic, ultra-strong field
Science.gov (United States)
Luo, Sui
Ultrastrong laser-matter interaction has direct bearing to next generation technologies including plasma acceleration, laser fusion and attosecond X-ray generation. The commonly known physics in strong field becomes different as one progress to ultrastrong field. The works presented in this dissertation theoretically study the influence of relativistic effect and magnetic component of the laser field on the ionization, photoelectron dynamics and elastic scattering processes. The influence of magnetic component (B laser) of circularly polarized (CP) ultrastrong fields (up to3 x 1022 W/cm2) on atomic bound state dynamics is investigated. The Poincare plots are used to find the changes in trajectory energies are on the order of a few percent for intensities up to1 x 1022 W/cm2. It is found that at intensities where ionization approaches 50% for the bound state, the small changes from Blaser of the circular polarized light can actually result in a several-fold decrease in ionization probability. The force on the bound electron exerted by the Lorentz force from B laser is perpendicular to the rotating plane of the circular polarized light, and this nature makes those trajectories which are aligned away from the minimum in the potential barrier stabilized against tunneling ionization. Our results provide a classical understanding for ionization in ultrastrong fields and indicate that relativistic effects in ultrastrong field ionization may most easily be seen with CP fields. The photoelectron energy spectra from elastic rescattering in ultrastrong laser fields (up to 2x1019 W/cm2) is studied by using a relativistic adaption of a semi-classical three-step recollision model. The Hartree-Fock scattering potentials are used in calculating the elastic rescattering for both hydrogenlike and noble gas species. It is found that there is a reduction in elastic rescattering for intensities beyond 6 x 1016 W/cm2 when the laser Lorentz deflection of the photoelectron exceeds its
11. Magnetoelastic contribution to the elastic constants of terbium, dysprosium and erbium
International Nuclear Information System (INIS)
Torikachvili, M.S.; Gama, S.; Kale, B.M.; Pinatti, D.G.; Donoho, P.L.
1979-01-01
We have measured the dependence of the longitudinal and shear elastic waves velocity of Tb, Dy and Er single-crystals on magnetization and temperature. Data were taken in magnetic fields up to 75 kOe, in the temperatue range of 4.2-300 K, which included all the ordered phases of these materials. We found qualitatively that to explain the observed behavior, the magnetoelastic interaction have to be treated up to 2nd order in the strains and that the 3rd order elastic energy can not be neglected
12. High Magnetic Field Study of Elastic Constants of the Cage-structure Compound SmBe13
International Nuclear Information System (INIS)
Mombetsu, S; Murazumi, T; Hiura, K; Yamazaki, S; Hidaka, H; Yanagisawa, T; Amitsuka, H; Shimizu, Y; Yasin, S; Zherlitsyn, S; Wosnitza, J
2016-01-01
Ultrasonic measurements were performed on the cage-structure compound SmBe 13 . We have investigated the magnetic field-temperature phase diagram of this material by using pulsed magnetic fields. We found that the low-temperature magnetic order is suppressed by a magnetic field of 43 T for H ∥ [001], which is smaller than the estimated value from mean-field approximation assuming the Γ 8 quartet crystal-electric-field ground state and simple antiferromagnetic order. We found that the elastic constant C 44 shows softening below the ordering temperature and has a local minimum below 7 T. These facts suggest that the low- temperature state is not a simple antiferromagnetically ordered state. In addition, no elastic anomaly due to rattling modes was found in the present measurements. (paper)
13. Experimental study and finite element analysis based on equivalent load method for laser ultrasonic measurement of elastic constants.
Science.gov (United States)
Zhan, Yu; Liu, Changsheng; Zhang, Fengpeng; Qiu, Zhaoguo
2016-07-01
The laser ultrasonic generation of Rayleigh surface wave and longitudinal wave in an elastic plate is studied by experiment and finite element method. In order to eliminate the measurement error and the time delay of the experimental system, the linear fitting method of experimental data is applied. The finite element analysis software ABAQUS is used to simulate the propagation of Rayleigh surface wave and longitudinal wave caused by laser excitation on a sheet metal sample surface. The equivalent load method is proposed and applied. The pulsed laser is equivalent to the surface load in time and space domain to meet the Gaussian profile. The relationship between the physical parameters of the laser and the load is established by the correction factor. The numerical solution is in good agreement with the experimental result. The simple and effective numerical and experimental methods for laser ultrasonic measurement of the elastic constants are demonstrated. Copyright © 2016. Published by Elsevier B.V.
14. Precision determination of the strong coupling constant within a global PDF analysis arXiv
CERN Document Server
Ball, Richard D.; Del Debbio, Luigi; Forte, Stefano; Kassabov, Zahari; Rojo, Juan; Slade, Emma; Ubiali, Maria
We present a determination of the strong coupling constant $\\alpha_s(m_Z)$ based on the NNPDF3.1 determination of parton distributions, which for the first time includes constraints from jet production, top-quark pair differential distributions, and the $Z$ $p_T$ distributions using exact NNLO theory. Our result is based on a novel extension of the NNPDF methodology - the correlated replica method - which allows for a simultaneous determination of $\\alpha_s$ and the PDFs with all correlations between them fully taken into account. We study in detail all relevant sources of experimental, methodological and theoretical uncertainty. At NNLO we find $\\alpha_s(m_Z) = 0.1185 \\pm 0.0005^\\text{(exp)}\\pm 0.0001^\\text{(meth)}$, showing that methodological uncertainties are negligible. We conservatively estimate the theoretical uncertainty due to missing higher order QCD corrections (N$^3$LO and beyond) from half the shift between the NLO and NNLO $\\alpha_s$ values, finding $\\Delta\\alpha^{\\rm th}_s =0.0011$.
15. Measurement of jet production with the ATLAS detector and extraction of the strong coupling constant
CERN Document Server
Sawyer, Lee; The ATLAS collaboration
2017-01-01
The production of jets at hadron colliders provides a stringent test of perturbative QCD at the highest energies. The process can also be used to probe the gluon density function of the proton. Specific topologies can be used to extract the strong coupling constant. The ATLAS collaboration has recently measured the inclusive jet production cross section in data collected at a center-of-mass energy of 8TeV and 13TeV. The measurements have been performed differentially in jet rapidity and transverse momentum. The collaboration also presents a first measurement of the di-jet cross section at a center-of-mass energy of 13TeV as a function of the di-jet mass and rapidity. The results have been compared with state-of-the-art theory predictions at NLO in pQCD, interfaced with different parton distribution functions and can be used to constrain the proton structure. We also present new measurements of transverse energy-energy correlations (TEEC) and their associated asymmetries (ATEEC) in multi-jet events at a center...
16. The ATLAS Measurements of Jet Production and the Strong Coupling Constant
CERN Document Server
Sawyer, Lee; The ATLAS collaboration
2017-01-01
The production of jets at hadron colliders provides a stringent test of perturbative QCD at the highest energies. The process can also be used to probe the gluon density in the parton distribution function of the proton. Specific topologies can be used to extract the strong coupling constant. The ATLAS collaboration has recently measured the inclusive jet production cross section in data collected at a center-of-mass energy of 8 TeV and 13 TeV. The measurements have been performed differentially in jet rapidity and transverse momentum. The collaboration also presents a first measurement of the dijet cross section at a center-of-mass energy of 13 TeV as a function of the dijet invariant mass and rapidity. The results have been compared with state-of-the-art theory predictions at NLO in pQCD, interfaced with different parton distribution functions and can be used to constrain the proton structure. We also present new measurements of transverse energy-energy correlations (TEEC) and their associated asymmetries (...
17. How strong is it? The interpretation of force and compliance constants as bond strength descriptors.
Science.gov (United States)
Brandhorst, Kai; Grunenberg, Jörg
2008-08-01
Knowledge about individual covalent or non-covalent bond strengths is the Holy Grail of many modern molecular sciences. Recent developments of new descriptors for such interaction strengths based on potential constants are summarised in this tutorial review. Several publications for and against the use of compliance matrices (inverse force constants matrix) have appeared in the literature in the last few years. However the mathematical basis for understanding, and therefore interpreting, compliance constants is still not well developed. We therefore summarise the theoretical foundations and point to the advantages and disadvantages of the use of force constants versus compliance constants for the description of both non-covalent and covalent interactions.
18. Elastic constants and the structural phase transition in La2-xSrxCuO4
International Nuclear Information System (INIS)
Sarrao, J.L.; Lei, Ming; Stekel, A.; Bell, T.M.; Leisure, R.G.; Sham, L.J.; Visscher, W.M.; Migliori, A.; Bussmann-Holder, A.; Tanaka, I.; Kojima, H.
1991-01-01
Resonant ultrasound spectroscopy is used to measure the temperature dependence of all six elastic moduli of La 2-x Sr x CuO 4 . A giant softening (> 50% reduction) in the in-plane shear modulus, c 66 , is observed and is attributed to the tetragonal-orthorhombic structural phase transition in this material. This phase transition and the corresponding softening is examined with a simple anharmonic mechanical model and a macroscopic Ginsburg-Landau formalism exploiting the full symmetry of the crystal. 16 refs., 5 figs
19. Non-linear optical measurement of the twist elastic constant in thermotropic and DNA lyotropic chiral nematics.
Science.gov (United States)
Lucchetti, Liana; Fraccia, Tommaso P; Ciciulla, Fabrizio; Bellini, Tommaso
2017-07-10
Throughout the whole history of liquid crystals science, the balancing of intrinsic elasticity with coupling to external forces has been the key strategy for most application and investigation. While the coupling of the optical field to the nematic director is at the base of a wealth of thoroughly described optical effects, a significant variety of geometries and materials have not been considered yet. Here we show that by adopting a simple cell geometry and measuring the optically induced birefringence, we can readily extract the twist elastic coefficient K 22 of thermotropic and lyotropic chiral nematics (N*). The value of K 22 we obtain for chiral doped 5CB thermotropic N* well matches those reported in the literature. With this same strategy, we could determine for the first time K 22 of the N* phase of concentrated aqueous solutions of DNA oligomers, bypassing the limitations that so far prevented measuring the elastic constants of this class of liquid crystalline materials. The present study also enlightens the significant nonlinear optical response of DNA liquid crystals.
20. Bounds and self-consistent estimates for elastic constants of polycrystals composed of orthorhombics or crystals with higher symmetries.
Science.gov (United States)
Berryman, James G
2011-04-01
Methods for computing Hashin-Shtrikman bounds and related self-consistent estimates of elastic constants for polycrystals composed of crystals having orthorhombic symmetry have been known for about three decades. However, these methods are underutilized, perhaps because of some perceived difficulties with implementing the necessary computational procedures. Several simplifications of these techniques are introduced, thereby reducing the overall computational burden, as well as the complications inherent in mapping out the Hashin-Shtrikman bounding curves. The self-consistent estimates of the effective elastic constants are very robust, involving a quickly converging iteration procedure. Once these self-consistent values are known, they may then be used to speed up the computations of the Hashin-Shtrikman bounds themselves. It is shown furthermore that the resulting orthorhombic polycrystal code can be used as well to compute both bounds and self-consistent estimates for polycrystals of higher-symmetry tetragonal, hexagonal, and cubic (but not trigonal) materials. The self-consistent results found this way are shown to be the same as those obtained using the earlier methods, specifically those methods designed specially for each individual symmetry type. But the Hashin-Shtrikman bounds found using the orthorhombic code are either the same or (more typically) tighter than those found previously for these special cases (i.e., tetragonal, hexagonal, and cubic). The improvement in the Hashin-Shtrikman bounds is presumably due to the additional degrees of freedom introduced into the available search space.
1. Bounds and self-consistent estimates for elastic constants of granular polycrystals composed of orthorhombics or crystal with higher symmetries
Energy Technology Data Exchange (ETDEWEB)
Berryman, J. G.
2011-02-01
Methods for computing Hashin-Shtrikman bounds and related self-consistent estimates of elastic constants for polycrystals composed of crystals having orthorhombic symmetry have been known for about three decades. However, these methods are underutilized, perhaps because of some perceived difficulties with implementing the necessary computational procedures. Several simplifications of these techniques are introduced, thereby reducing the overall computational burden, as well as the complications inherent in mapping out the Hashin-Shtrikman bounding curves. The self-consistent estimates of the effective elastic constants are very robust, involving a quickly converging iteration procedure. Once these self-consistent values are known, they may then be used to speed up the computations of the Hashin-Shtrikman bounds themselves. It is shown furthermore that the resulting orthorhombic polycrystal code can be used as well to compute both bounds and self-consistent estimates for polycrystals of higher-symmetry tetragonal, hexagonal, and cubic (but not trigonal) materials. The self-consistent results found this way are shown to be the same as those obtained using the earlier methods, specifically those methods designed specially for each individual symmetry type. But the Hashin-Shtrikman bounds found using the orthorhombic code are either the same or (more typically) tighter than those found previously for these special cases (i.e., tetragonal, hexagonal, and cubic). The improvement in the Hashin-Shtrikman bounds is presumably due to the additional degrees of freedom introduced into the available search space.
2. Elastic constants of non-modulated Ni-Mn-Ga martensite
Czech Academy of Sciences Publication Activity Database
Sedlák, Petr; Seiner, Hanuš; Bodnárová, Lucie; Heczko, Oleg; Landa, Michal
2017-01-01
Roč. 136, July (2017), s. 20-23 ISSN 1359-6462 R&D Projects: GA ČR GA17-00062S Institutional support: RVO:61388998 ; RVO:68378271 Keywords : acoustic methods * elastic behavior * ferromagnetic shape memory alloys * martensitic phase transformation Subject RIV: BM - Solid Matter Physics ; Magnetism; BM - Solid Matter Physics ; Magnetism (FZU-D) OBOR OECD: Condensed matter physics (including formerly solid state physics, supercond.); Condensed matter physics (including formerly solid state physics, supercond.) (FZU-D) Impact factor: 3.747, year: 2016 http://ac.els-cdn.com/S1359646217301768/1-s2.0-S1359646217301768-main.pdf?_tid=9b99b306-4a83-11e7-8ec6-00000aacb35e&acdnat=1496731657_35d3b5f3132e926d5bc8c6043961bb6d
3. Molecular dynamics simulations to calculate glass transition temperature and elastic constants of novel polyethers.
Science.gov (United States)
Sarangapani, Radhakrishnan; Reddy, Sreekantha T; Sikder, Arun K
2015-04-01
Molecular dynamics simulations studies are carried out on hydroxyl terminated polyethers that are useful in energetic polymeric binder applications. Energetic polymers derived from oxetanes with heterocyclic side chains with different energetic substituents are designed and simulated under the ensembles of constant particle number, pressure, temperature (NPT) and constant particle number, volume, temperature (NVT). Specific volume of different amorphous polymeric models is predicted using NPT-MD simulations as a function of temperature. Plots of specific volume versus temperature exhibited a characteristic change in slope when amorphous systems change from glassy to rubbery state. Several material properties such as Young's, shear, and bulk modulus, Poisson's ratio, etc. are predicted from equilibrated structures and established the structure-property relations among designed polymers. Energetic performance parameters of these polymers are calculated and results reveal that the performance of the designed polymers is comparable to the benchmark energetic polymers like polyNIMMO, polyAMMO and polyBAMO. Overall, it is worthy remark that this molecular simulations study on novel energetic polyethers provides a good guidance on mastering the design principles and allows us to design novel polymers of tailored properties. Copyright © 2015 Elsevier Inc. All rights reserved.
4. Application of resonant ultrasound spectroscopy to determine elastic constants of plasma-sprayed coatings with high internal friction
Czech Academy of Sciences Publication Activity Database
Sedmák, P.; Seiner, Hanuš; Sedlák, Petr; Landa, Michal; Mušálek, Radek; Matějíček, Jiří
2013-01-01
Roč. 232, October (2013), s. 747-757 ISSN 0257-8972 R&D Projects: GA ČR(CZ) GA101/09/0702; GA ČR GA13-13616S; GA ČR(CZ) GPP108/12/P552; GA ČR(CZ) GAP108/12/1872 Grant - others:Rada Programu interní podpory projektů mezinárodní spolupráce AV ČR(CZ) M100761203 Program:M Institutional support: RVO:61388998 ; RVO:61389021 Keywords : plasma-sprayed coatings * elastic constants * resonant ultrasound spectroscopy * internal friction * anisotropy Subject RIV: BI - Acoustics; BI - Acoustics (UFP-V) Impact factor: 2.199, year: 2013 http://www.sciencedirect.com/science/article/pii/S0257897213006063
5. Non-perturbative computation of the strong coupling constant on the lattice
International Nuclear Information System (INIS)
Sommer, Rainer; Humboldt-Universitaet, Berlin; Wolff, Ulli
2015-01-01
We review the long term project of the ALPHA collaboration to compute in QCD the running coupling constant and quark masses at high energy scales in terms of low energy hadronic quantities. The adapted techniques required to numerically carry out the required multiscale non-perturbative calculation with our special emphasis on the control of systematic errors are summarized. The complete results in the two dynamical flavor approximation are reviewed and an outlook is given on the ongoing three flavor extension of the programme with improved target precision.
6. Sub-Micrometer Zeolite Films on Gold-Coated Silicon Wafers with Single-Crystal-Like Dielectric Constant and Elastic Modulus
Energy Technology Data Exchange (ETDEWEB)
Tiriolo, Raffaele [Department of Medical and Surgical Sciences, University Magna Graecia of Catanzaro, Viale Europa 88100 Catanzaro Italy; Rangnekar, Neel [Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE Minneapolis MN 55455 USA; Zhang, Han [Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE Minneapolis MN 55455 USA; Shete, Meera [Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE Minneapolis MN 55455 USA; Bai, Peng [Department of Chemistry and Chemistry Theory Center, University of Minnesota, 207 Pleasant St SE Minneapolis MN 55455 USA; Nelson, John [Characterization Facility, University of Minnesota, 12 Shepherd Labs, 100 Union St. S.E. Minneapolis MN 55455 USA; Karapetrova, Evguenia [Surface Scattering and Microdiffraction, X-ray Science Division, Argonne National Laboratory, 9700 S. Cass Ave, Building 438-D002 Argonne IL 60439 USA; Macosko, Christopher W. [Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE Minneapolis MN 55455 USA; Siepmann, Joern Ilja [Department of Chemistry and Chemistry Theory Center, University of Minnesota, 207 Pleasant St SE Minneapolis MN 55455 USA; Lamanna, Ernesto [Department of Health Sciences, University Magna Graecia of Catanzaro, Viale Europa 88100 Catanzaro Italy; Lavano, Angelo [Department of Medical and Surgical Sciences, University Magna Graecia of Catanzaro, Viale Europa 88100 Catanzaro Italy; Tsapatsis, Michael [Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE Minneapolis MN 55455 USA
2017-05-08
A low-temperature synthesis coupled with mild activation produces zeolite films exhibiting low dielectric constant (low-k) matching the theoretically predicted and experimentally measured values for single crystals. This synthesis and activation method allows for the fabrication of a device consisting of a b-oriented film of the pure-silica zeolite MFI (silicalite-1) supported on a gold-coated silicon wafer. The zeolite seeds are assembled by a manual assembly process and subjected to optimized secondary growth conditions that do not cause corrosion of the gold underlayer, while strongly promoting in-plane growth. The traditional calcination process is replaced with a non-thermal photochemical activation to ensure preservation of an intact gold layer. The dielectric constant (k), obtained through measurement of electrical capacitance in a metal-insulator-metal configuration, highlights the ultralow k approximate to 1.7 of the synthetized films, which is among the lowest values reported for an MFI film. There is large improvement in elastic modulus of the film (E approximate to 54 GPa) over previous reports, potentially allowing for integration into silicon wafer processing technology.
7. The maximum penalty criterion for ridge regression: application to the calibration of the force constant in elastic network models.
Science.gov (United States)
Dehouck, Yves; Bastolla, Ugo
2017-07-17
Tikhonov regularization, or ridge regression, is a popular technique to deal with collinearity in multivariate regression. We unveil a formal analogy between ridge regression and statistical mechanics, where the objective function is comparable to a free energy, and the ridge parameter plays the role of temperature. This analogy suggests two novel criteria for selecting a suitable ridge parameter: specific-heat (C v ) and maximum penalty (MP). We apply these fits to evaluate the relative contributions of rigid-body and internal fluctuations, which are typically highly collinear, to crystallographic B-factors. This issue is particularly important for computational models of protein dynamics, such as the elastic network model (ENM), since the amplitude of the predicted internal motion is commonly calibrated using B-factor data. After validation on simulated datasets, our results indicate that rigid-body motions account on average for more than 80% of the amplitude of B-factors. Furthermore, we evaluate the ability of different fits to reproduce the amplitudes of internal fluctuations in X-ray ensembles from the B-factors in the corresponding single X-ray structures. The new ridge criteria are shown to be markedly superior to the commonly used two-parameter fit that neglects rigid-body rotations and to the full fits regularized under generalized cross-validation. In conclusion, the proposed fits ensure a more robust calibration of the ENM force constant and should prove valuable in other applications.
8. Elastic wave propagation and stop-band generation in strongly damaged solids
Directory of Open Access Journals (Sweden)
G. Carta
2014-07-01
Full Text Available In this work, we study the propagation of elastic waves in elongated solids with an array of equallyspaced deep transverse cracks, focusing in particular on the determination of stop-bands. We consider solids with different types of boundary conditions and different lengths, and we show that the eigenfrequencies associated with non-localized modes lie within the pass-bands of the corresponding infinite periodic system, provided that the solids are long enough. In the stop-bands, instead, eigenfrequencies relative to localized modes may be found. Furthermore, we use an asymptotic reduced model, whereby the cracked solid is approximated by a beam with elastic connections. This model allows to derive the dynamic properties of damaged solids through analytical methods. By comparing the theoretical dispersion curves yielded by the asymptotic reduced model with the numerical outcomes obtained from finite element computations, we observe that the asymptotic reduced model provides a better fit to the numerical data as the slenderness ratio increases. Finally, we illustrate how the limits of the stop-bands vary with the depth of the cracks.
9. Effects of Ni vacancy, Ni antisite, Cr and Pt on the third-order elastic constants and mechanical properties of NiAl
KAUST Repository
Wu, Shaohua
2014-12-01
Effects of Ni vacancy, Ni antisite in Al sublattice, Cr in Al sublattice, Pt in Ni sublattice on the second-order elastic constants (SOECs) and third-order elastic constants (TOECs) of the B2 NiAl have been investigated using the first-principles methods. Lattice constant and the SOECs of NiAl are in good agreement with the previous results. The brittle/ductile transition map based on Pugh ratio G/B and Cauchy pressure Pc shows that Ni antisite, Cr, Pt and pressure can improve the ductility of NiAl, respectively. Ni vacancy and lower pressure can enhance the Vickers hardness Hv of NiAl. The density of states (DOS) and the charge density difference are also used to analysis the effects of vacancy, Ni antisite, Cr and Pt on the mechanical properties of NiAl, and the results are in consistent with the transition map. © 2014 Elsevier Ltd. All rights reserved.
10. Evaluation of the strong coupling constant {alpha}{sub s} using the ATLAS inclusive jet cross-section data
Energy Technology Data Exchange (ETDEWEB)
Malaescu, B. [European Organization for Nuclear Research (CERN), Geneva (Switzerland); Starovoitov, P. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-03-15
We perform a determination of the strong coupling constant using the latest ATLAS inclusive jet cross section data, from proton-proton collisions at {radical}(s)=7 TeV, and their full information on the bin-to-bin correlations. Several procedures for combining the statistical information from the different data inputs are studied and compared. The theoretical prediction is obtained using NLO QCD, and it also includes non-perturbative corrections. Our determination uses inputs with transverse momenta between 45 and 600 GeV, the running of the strong coupling being also tested in this range. Good agreement is observed when comparing our result with the world average at the Z-boson scale, as well as with the most recent results from the Tevatron. (orig.)
11. Measurements of the elastic stiffness constants of single-crystal SmCo5 and of liquid-phase sintered SmCo5 permanent magnet material
International Nuclear Information System (INIS)
Doane, D.A.
1977-01-01
The five elastic stiffness constants were determined for both single-crystal SmCo 5 and for the commercially processed liquid-phase sintered (LPS) SmCo 5 permanent magnet material. The LPS material is an aligned polycrystalline aggregate of SmCo 5 crystallites oriented so that their magnetically easy c axes are approximately parallel. The elastic constants were obtained from the velocities of propagation of ultrasound in various directions in samples of known thickness and density. For the single crystal, the room-temperature values of the constants (in units of 10 12 dyn/cm 2 ) are c 11 =1.968 +- 2%, c 12 =1.032 +- 4%, c 13 =1.049 +- 4%, c 33 =2.398 +- 2%, and c 44 =0.483 +- 2%, and for the LPS permanent magnet material, c 11 =1.330 +- 2%, c 12 =0.616 +- 5%, c 13 =0.485 +- 5%, c 33 =1.659 +- 2%, and c 44 =0.419 +- 2%. The decrease in elastic constants in SmCo 5 relative to cobalt can be related qualitatively to a corresponding decrease in the number of nearest-neighbor cobalt bonds in SmCo 5
12. Strong coupling constant extraction from high-multiplicity $Z+\text{jets}$ observables
OpenAIRE
Johnson, Mark; Maître, Daniel
2018-01-01
We present a strong coupling constant extraction at next-to-leading order QCD accuracy using ATLAS Z+2,3,4 jets data. This is the first extraction using processes with a dependency on high powers of the coupling constant. We obtain values of the strong coupling constant at the Z mass compatible with the world average and with uncertainties commensurate with other next-to-leading order extractions at hadron colliders. Our most conservative result for the strong coupling constant is αS(MZ)=0.11...
13. Determination of the strong coupling constant from the inclusive jet cross section in pp- collisions at sqrt s=1.96 TeV
NARCIS (Netherlands)
Abazov, V.M.; et al., [Unknown; Ancu, L.S.; de Jong, S.J.; Filthaut, F.; Galea, C.F.; Hegeman, J.G.; Houben, P.; Meijer, M.M.; Svoisky, P.; van den Berg, P.J.; van Leeuwen, W.M.
2009-01-01
We determine the strong coupling constant alpha(s) and its energy dependence from the p(T) dependence of the inclusive jet cross section in pp collisions at s=1.96 TeV. The strong coupling constant is determined over the transverse momentum range 50 < p(T)< 145 GeV. Using perturbative QCD
14. Surface Brillouin scattering measurement of the elastic constants of single crystal InAs{sub 0.91}Sb{sub 0.09}
Energy Technology Data Exchange (ETDEWEB)
Kotane, L M; Comins, J D; Every, A G [Materials Physics Research Institute, School of Physics, University of the Witwatersrand, Johannesburg, Wits 2050 (South Africa); Botha, J R, E-mail: [email protected] [Department of Physics, Nelson Mandela Metropolitan University, Port Elizabeth (South Africa)
2011-01-01
Surface Brillouin scattering of light has been used to measure the angular dependence of the Rayleigh surface acoustic wave (SAW), pseudo surface acoustic wave (PSAW) and longitudinal lateral wave (LLW) speeds in a (100)-oriented single crystal of the ternary semiconductor alloy InAs{sub 0.91}Sb{sub 0.09}. The wave speed measurements have been used to determine the room temperature values of the elastic constants C{sub 11}, C{sub 12} and C{sub 44} of the alloy. A simple and robust fitting procedure has been implemented for recovering the elastic constants, in which the merit function is constructed from explicit secular functions that determine the surface and lateral wave speeds in the [001] and [011] crystallographic directions. In the fitting, relatively larger weighting factors have been assigned to the SAW and PSAW data because of the greater precision with which the surface modes can be measured as compared with the lateral wave.
15. Surface Brillouin scattering measurement of the elastic constants of single crystal InAs0.91Sb0.09
International Nuclear Information System (INIS)
Kotane, L M; Comins, J D; Every, A G; Botha, J R
2011-01-01
Surface Brillouin scattering of light has been used to measure the angular dependence of the Rayleigh surface acoustic wave (SAW), pseudo surface acoustic wave (PSAW) and longitudinal lateral wave (LLW) speeds in a (100)-oriented single crystal of the ternary semiconductor alloy InAs 0.91 Sb 0.09 . The wave speed measurements have been used to determine the room temperature values of the elastic constants C 11 , C 12 and C 44 of the alloy. A simple and robust fitting procedure has been implemented for recovering the elastic constants, in which the merit function is constructed from explicit secular functions that determine the surface and lateral wave speeds in the [001] and [011] crystallographic directions. In the fitting, relatively larger weighting factors have been assigned to the SAW and PSAW data because of the greater precision with which the surface modes can be measured as compared with the lateral wave.
16. Study of dynamic properties for NaK binary liquid alloy using first principle and theoretical predictions of isothermal bulk modulus using elastic constants
Energy Technology Data Exchange (ETDEWEB)
Thakur, Anil, E-mail: [email protected]; Kashyap, Rajinder [Department of Physics, Govt. P. G. College Solan-173212, Himachal Pradesh (India); Sharma, Nalini; Ahluwalia, P. K. [Department of Physics, Himachal Pradesh University Shimla-171005, Himachal Pradesh (India)
2014-04-24
Study of atomic motions in the binary liquid alloys have been studied in terms of dynamical variables like velocity auto correlation, power spectrum and mean square displacement. Elastic constants and isothermal bulk modulus have been calculated to see the effeectiveness of ab-initio pseudopotentials which has been used in this paper. This appraoch is free from the fitting parameters and results obtained using this appraoch have been found very close to the average values.
17. Determination of the strong coupling constant $\\alpha_s$ in multijet production with the ATLAS detector at the LHC.
CERN Document Server
Llorente Merino, Javier; The ATLAS collaboration
2018-01-01
A measurement of transverse energy--energy correlations and its asymmetry in $pp$ collisions recorded by the ATLAS detector at the LHC at $\\sqrt{s} = 8$ TeV is presented. The results are intepreted as a precision test of Quantum Chromodynamics, used to determine the strong coupling constant $\\alpha_s(m_Z)$ and to test asymptotic freedom up to scales close to 1 TeV. A global fit to the transverse energy--energy correlation distributions yields $\\alpha_{\\mathrm{s}}(m_Z) = 0.1162 \\pm 0.0011 \\mbox{ (exp.)}^{+0.0084}_{-0.0070} \\mbox{ (theo.)}$, while a global fit to the asymmetry distributions yields a value of $\\alpha_{\\mathrm{s}}(m_Z) = 0.1196 \\pm 0.0013 \\mbox{ (exp.)}^{+0.0075}_{-0.0045} \\mbox{ (theo.)}$.
18. The influence of fragmentation models on the determination of the strong coupling constant in e+e- annihilation into hadrons
International Nuclear Information System (INIS)
Behrend, H.J.; Chen, C.; Fenner, H.; Schachter, M.J.; Schroeder, V.; Sindt, H.; D'Agostini, G.; Apel, W.D.; Banerjee, S.; Bodenkamp, J.; Chrobaczek, D.; Engler, J.; Fluegge, G.; Fries, D.C.; Fues, W.; Gamerdinger, K.; Hopp, G.; Kuester, H.; Mueller, H.; Randoll, H.; Schmidt, G.; Schneider, H.; Boer, W. de; Buschhorn, G.; Grindhammer, G.; Grosse-Wiesmann, P.; Gunderson, B.; Kiesling, C.; Kotthaus, R.; Kruse, U.; Lierl, H.; Lueers, D.; Oberlack, H.; Schacht, P.; Colas, P.; Cordier, A.; Davier, M.; Fournier, D.; Grivaz, J.F.; Haissinski, J.; Journe, V.; Klarsfeld, A.; Laplanche, F.; Le Diberder, F.; Mallik, U.; Veillet, J.J.; Field, J.H.; George, R.; Goldberg, M.; Grossetete, B.; Hamon, O.; Kapusta, F.; Kovacs, F.; London, G.; Poggioli, L.; Rivoal, M.; Aleksan, R.; Bouchez, J.; Carnesecchi, G.; Cozzika, G.; Ducros, Y.; Gaidot, A.; Jadach, S.; Lavagne, Y.; Pamela, J.; Pansart, J.P.; Pierre, F.
1983-01-01
Hadronic events obtained with the CELLO detector at PETRA were compared with first-order QCD predictions using two different models for the fragmentation of quarks and gluons, the Hoyer model and the Lund model. Both models are in reasonable agreement with the data, although they do not completely reproduce the details of many distributions. Several methods have been applied to determine the strong coupling constant αsub(s). Although within one model the value of αsub(s) varies by 20% among the different methods, the values determined using the Lund model are 30% or more larger (depending on the method used) than the values determined with the Hoyer model. Our results using the Hoyer model are in agreement with previous results based on this approach. (orig.)
19. Elastic properties
International Nuclear Information System (INIS)
Ledbetter, H.M.
1983-01-01
This chapter investigates the following five aspects of engineering-material solid-state elastic constants: general properties, interrelationships, relationships to other physical properties, changes during cooling from ambient to near-zero temperature, and near-zero-temperature behavior. Topics considered include compressibility, bulk modulus, Young's modulus, shear modulus, Poisson's ratio, Hooke's law, elastic-constant measuring methods, thermodynamic potentials, higher-order energy terms, specific heat, thermal expansivity, magnetic materials, structural phase transitions, polymers, composites, textured aggregates, and other-phenomena correlations. Some of the conclusions concerning polycrystalline elastic properties and their temperature dependence are: elastic constants are physical, not mechanical, properties which relate thermodynamically to other physical properties such as specific heat and thermal expansivity; elastic constants at low temperatures are nearly temperature independent, as required by the third law of thermodynamics; and elastic constants can be used to study directional properties of materials, such as textured aggregates and composites
20. PDF constraints and extraction of the strong coupling constant from the inclusive jet cross section at 7 TeV
CERN Document Server
CMS Collaboration
2013-01-01
The recent CMS measurement of the inclusive jet cross section at 7~TeV extends the accessible phase space in jet transverse momentum up to 2 TeV and ranges up to 2.5 in absolute jet rapidity. At the same time the experimental uncertainties are smaller than in previous publications such that these data constrain the parton distribution functions of the proton, notably for the gluon at high fractions of the proton momentum, and provide valuable input to determine the strong coupling at high momentum scales. The impact on the extraction of the parton distribution functions is investigated. Using predictions from theory at next-to-leading order, complemented with electroweak corrections, the strong coupling constant is determined from the inclusive jet cross section to be $\\alpha_S(M_Z) = 0.1185 \\pm 0.0019\\,\\mathrm{(exp.)} \\pm 0.0028\\,\\mathrm{(\\mathrm{PDF})} \\pm 0.0004\\,\\mathrm{(\\mathrm{NP})} ^{+0.0055}_{-0.0022}\\,\\mathrm{(\\mathrm{scale})}$, which is in agreement with the world average.
1. Measurement of the strong interaction coupling constant αs by jet study in the H1 experiment
International Nuclear Information System (INIS)
Squinabol, F.
1997-01-01
The H1 experiment allows to study hadronic jets produced in deep inelastic lepton (27.5 GeV) scattering off protons (820 GeV). The coupling constant of the strong interaction α s can be extracted from the measurement of the 2-jets rate in the final state. The use of the JADE algorithm is optimal for events with high energy transfer (100-4,000 GeV 2 ), corresponding to the 1994 and 1995 data. The error on α s (M Z 0 2 ) is dominated by the uncertainty from the hadronic energy measurement and the experimental resolution effects on jets. The theoretical error is dominated by the renormalization scale dependence. The final result is (M Z 0 2 ) 0.118 -0.008 +0.008 . This analysis is extended to smaller momentum transfers (25-100 GeV 2 ) using the factorizable K t algorithm, taking the transferred momentum as energy scale of the particle re-clustering. The result α s (M Z 0 2 ) 0.117 -0.008 +0.009 is compatible with the previous one. The precision of the measurement performed in this thesis is 7%. A precision of 4% could be achieved after progresses in the theoretical framework and/or after a significant increase of the luminosity. (author)
2. Non-linear optical measurement of the twist elastic constant in thermotropic and DNA lyotropic chiral nematics
OpenAIRE
Lucchetti, Liana; Fraccia, Tommaso P.; Ciciulla, Fabrizio; Bellini, Tommaso
2017-01-01
Throughout the whole history of liquid crystals science, the balancing of intrinsic elasticity with coupling to external forces has been the key strategy for most application and investigation. While the coupling of the optical field to the nematic director is at the base of a wealth of thoroughly described optical effects, a significant variety of geometries and materials have not been considered yet. Here we show that by adopting a simple cell geometry and measuring the optically induced bi...
3. Empirical correlation among the dynamic elastic constants and the waves P and S velocities in rocks; Correlaciones empiricas entre las constantes elasticas dinamicas y las velocidades de las ondas P y S de las rocas
Energy Technology Data Exchange (ETDEWEB)
Contreras Lopez, Enrique [Instituto de Investigaciones Electricas, Cuernavaca (Mexico)
1995-12-31
Departing from the analysis of a data base on the velocities of the compression waves (V{sub p}) and the transverse waves (V{sub s}) in a group of 97 specimens of sedimentary, igneous and metamorphic rocks, the existence of four types of empirical correlation very well entailed between the dynamic elastic constants and the velocities V{sub p} and V{sub s}. These correlation allow the estimation with a very close approximation the elastic dynamic constants without the need of having available of the complete set of data (V{sub p}, V{sub s} and total density) that is normally required for its determination. The identified correlation is mathematically expressed by means of adjustment equations that reproduce in all of the cases the experimental values with a standard error of estimation within 10%, for the universe of rocks studied and with much less error for different specific lithological groups. The application methodologies of the correlation found for different cases of practical interest, are described. [Espanol] A partir del analisis de una base de datos experimentales sobre la velocidad de las ondas compresionales (V{sub p}) y de las ondas transversales (V{sub s}) de un conjunto de 97 especimenes de rocas sedimentarias, igneas y metamorficas, se identifica la existencia de cuatro tipos de correlaciones empiricas muy bien comportadas entre las constantes elasticas dinamicas y las velocidades V{sub p} y V{sub s}. Estas correlaciones permiten estimar con muy buena aproximacion las constantes elasticas dinamicas de las rocas sin tener que disponer del conjunto completo de datos (V{sub p}, V{sub s} y densidad total) que normalmente se requieren para su determinacion. Las correlaciones identificadas se expresan matematicamente mediante ecuaciones de ajuste que reproducen en todos los casos los valores experimentales con un error estandar de estimacion dentro de 10% para el universo de las rocas estudiadas, y con mucho menor error para diferentes grupos litologicos
4. Constant, cycling, hot and cold thermal environments: strong effects on mean viability but not on genetic estimates
DEFF Research Database (Denmark)
Ketola, Tarmo; Kellermann, Vanessa; Kristensen, Torsten Nygård
2012-01-01
and their fluctuations. How species will respond to these changes is uncertain, particularly as there is a lack of studies which compare genetic performances in constant vs. fluctuating environments. In this study, we used a nested full-sib/half-sib breeding design to examine how the genetic variances and heritabilities...
5. On the model dependence of the determination of the strong coupling constant in second order QCD from e+e--annihilation into hadrons
International Nuclear Information System (INIS)
Achterberg, O.; D'Agostini, G.; Apel, W.D.; Engler, J.; Fluegge, G.; Forstbauer, B.; Fries, D.C.; Fues, W.; Gamerdinger, K.; Henkes, T.; Hopp, G.; Krueger, M.; Kuester, H.; Mueller, H.; Randoll, H.; Schmidt, G.; Schneider, H.; Boer, W. de; Buschhorn, G.; Grindhammer, G.; Grosse-Wiesmann, P.; Gunderson, B.; Kiesling, C.; Kotthaus, R.; Kruse, U.; Lierl, H.; Lueers, D.; Oberlack, H.; Schacht, P.; Bonneaud, G.; Colas, P.; Cordier, A.; Davier, M.; Fournier, D.; Grivaz, J.F.; Haissinski, J.; Journe, V.; Laplanche, F.; Le Diberder, F.; Mallik, U.; Ros, E.; Veillet, J.J.; Behrend, H.J.; Fenner, H.; Schachter, M.J.; Schroeder, V.; Sindt, H.
1983-12-01
Hadronic events obtained with the CELLO detector at PETRA are compared with second order QCD predictions using different models for the fragmentation of quarks and gluons into hadrons. We find that the model dependence in the determination of the strong coupling constant persists when going from first to second order QCD calculations. (orig.)
6. Strong enhancement of piezoelectric constants in ScxAl1−xN: First-principles calculations
Directory of Open Access Journals (Sweden)
Hiroyoshi Momida
2016-06-01
Full Text Available We theoretically investigate the piezoelectricity of ScxAl1−xN in the entire range of x by first-principles calculations. We find that the piezoelectric constants of wurtzite-type ScxAl1−xN significantly enhance as x increases from 0 to 0.75. However, the energy stability analyses between structure phases show that the cubic-type phases become more stable than the wurtzite-type phases at x of approximately 0.5 and higher, interfering with the ability of wurtzite-type ScxAl1−xN to realize the maximum piezoelectricity. Moreover, our study on element combination dependences on piezoelectricity in A0.5B0.5N (A = Sc, Y, La and B = Al, Ga, In indicates that Sc, Y, and La have the strongest effect on the enhancement of piezoelectric constants in AlN, GaN, and InN, respectively.
7. Strategy Iteration Is Strongly Polynomial for 2-Player Turn-Based Stochastic Games with a Constant Discount Factor
DEFF Research Database (Denmark)
Hansen, Thomas Dueholm; Miltersen, Peter Bro; Zwick, Uri
2013-01-01
-based stochastic games with discounted zero-sum rewards. This provides the first strongly polynomial algorithm for solving these games, solving a long standing open problem. Combined with other recent results, this provides a complete characterization of the complexity the standard strategy iteration algorithm...... terminates after at most O(m1−γ log n1−γ) iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the strategy iteration (or strategy improvement) algorithm, a generalization of Howard’s policy iteration algorithm used for solving 2-player turn...... for 2-player turn-based stochastic games; it is strongly polynomial for a fixed discount factor, and exponential otherwise....
8. Strategy iteration is strongly polynomial for 2-player turn-based stochastic games with a constant discount factor
DEFF Research Database (Denmark)
Hansen, Thomas Dueholm; Miltersen, Peter Bro; Zwick, Uri
2011-01-01
iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the strategy iteration (or strategy improvement) algorithm, a generalization of Howard's policy iteration algorithm used for solving 2-player turn-based stochastic games with discounted zero......-sum rewards. This provides the first strongly polynomial algorithm for solving these games, resolving a long standing open problem....
9. Determination of the strong coupling constant from the inclusive jet cross section in ppbar collisions at sqrt(s)=1.96 TeV
Energy Technology Data Exchange (ETDEWEB)
Abazov, V.M.; /Dubna, JINR; Abbott, B.; /Oklahoma U.; Abolins, M.; /Michigan State U.; Acharya, B.S.; /Tata Inst.; Adams, M.; /Illinois U., Chicago; Adams, T.; /Florida State U.; Aguilo, E.; /Alberta U. /Simon Fraser U. /York U., Canada /McGill U.; Ahsan, M.; /Kansas State U.; Alexeev, G.D.; /Dubna, JINR; Alkhazov, G.; /St. Petersburg, INP; Alton, A.; /Michigan U. /Northeastern U.
2009-11-01
We determine the strong coupling constant {alpha}{sub s} and its energy dependence from the p{sub T} dependence of the inclusive jet cross section in p{bar p} collisions at {radical}s = 1.96 TeV. The strong coupling constant is determined over the transverse momentum range 50 < p{sub T} < 145 GeV. Using perturbative QCD calculations to order {Omicron}({alpha}{sub s}{sup 3}) combined with {Omicron}({alpha}{sub s}{sup 4}) contributions from threshold corrections, we obtain {alpha}{sub s}(M{sub Z}) = 0.1173{sub -0.0049}{sup +0.0041}. This is the most precise result obtained at a hadron-hadron collider.
10. Characterization for elastic constants of fused deposition modelling-fabricated materials based on the virtual fields method and digital image correlation
Science.gov (United States)
Cao, Quankun; Xie, Huimin
2017-12-01
Fused deposition modelling (FDM), a widely used rapid prototyping process, is a promising technique in manufacturing engineering. In this work, a method for characterizing elastic constants of FDM-fabricated materials is proposed. First of all, according to the manufacturing process of FDM, orthotropic constitutive model is used to describe the mechanical behavior. Then the virtual fields method (VFM) is applied to characterize all the mechanical parameters (Q_{11}, Q_{22}, Q_{12}, Q_{66}) using the full-field strain, which is measured by digital image correlation (DIC). Since the principal axis of the FDM-fabricated structure is sometimes unknown due to the complexity of the manufacturing process, a disk in diametrical compression is used as the load configuration so that the loading angle can be changed conveniently. To verify the feasibility of the proposed method, finite element method (FEM) simulation is conducted to obtain the strain field of the disk. The simulation results show that higher accuracy can be achieved when the loading angle is close to 30°. Finally, a disk fabricated by FDM was used for the experiment. By rotating the disk, several tests with different loading angles were conducted. To determine the position of the principal axis in each test, two groups of parameters (Q_{11}, Q_{22}, Q_{12}, Q_{66}) are calculated by two different groups of virtual fields. Then the corresponding loading angle can be determined by minimizing the deviation between two groups of the parameters. After that, the four constants (Q_{11}, Q_{22}, Q_{12}, Q_{66}) were determined from the test with an angle of 27°.
11. Measurement of the strong coupling constant {alpha}{sub s} with hadronic jets in deep inelastic scattering; Mesure de la constante de couplage forte {alpha}{sub s} avec les jets hadroniques en diffusion inelastique profonde
Energy Technology Data Exchange (ETDEWEB)
Gouzevitch, Maxime
2008-12-15
In this analysis we have used the production of hard jets in neutral-current DIS for the extraction of the strong coupling constant {alpha}{sub s}. The jets have been selected in the NC DIS events at large momentum transvers 1505. Three jet observables normalized to the total NC DIS cross section have been used: Inclusive jet multiplicity as well as the production rates of 2-jet and 3-jet events. The prediction of the renormalization-group equation for the evolution of the strong coupling constant has been successfully tested for two orders of magnitude between Q=2 QeV to Q=122 GeV. The better precision on {alpha}{sub s}(m{sub Z}) has been obtained with the combination ob the three observables at Q{sup 2}>150 GeV{sup 2}: {alpha}{sub s}(m{sub Z})=0.1180{+-}0.0007(exp.){sub -0.0034}{sup +0.0050}(th.){+-}0.0017(pdf.).
12. A "theory of relativity" for cognitive elasticity of time and modality dimensions supporting constant working memory capacity: involvement of harmonics among ultradian clocks?
Science.gov (United States)
Glassman, R B
2000-02-01
13. Development of ACROSS (Accurately Controlled, Routinely Operated, Signal System) to realize constant monitoring the invisible earth's interiors by means of stationary coherent elastic and electromagnetic waves
International Nuclear Information System (INIS)
Kumazawa, Mineo; Kunitomo, Takahiro; Nakajima, Takahiro; Fujii, Naoyuki; Shigeta, Naotaka; Tsuruga, Kayoko; Hasada, Yoko; Nagao, Hiromichi; Matsumoto, Hiroshi; Kasahara, Junzo
2007-03-01
The developmental study made at Tono Geoscience Center under the Earthquake Frontier Research Project since 1996 is reported for a brand new technology system called ACROSS (Accurately Controlled, Routinely Operated, Signal System invented at Kagoya University in 1994). Various technology elements have been combined together under a specific theoretical framework for the underground exploration and monitoring of structures and physical states. The ACROSS is essentially a spectroscopy of the underground space consisted of complex media subjected to environmental noise. The robustness against noise is devised by utilizing coherent elastic and electromagnetic waves with phase controlled very accurately. Demanded hardware technology has been developed successfully and know how has been accumulated for practical applications. Accurate synchronization of transmission and observation systems has provided us with reliable data on the tensor transfer function between the source and receiver, which is equivalent to Green function within a limited frequency range. Several examples of the field application are demonstrated by the test experiments at Tono Mine site. After the developmental works of 10 years, the ACROSS is brought to be a practical method applied to the remote monitoring of temporal variation of underground states at the Horonobe Underground Research Laboratory and also it is being applied to the expected focal region of the coming Tokai earthquake near Hamaoka in Shizuoka prefecture. Whereas ACROSS technology is not mature enough yet, it is shown to be a potential and versatile methodology applied even for the health monitoring of the construction such as building strongly coupled with the ground in addition to the underground study. (author)
14. Traveltime dispersion in an isotropic elastic mantle: strong lower-mantle signal in differential-frequency residuals
Science.gov (United States)
Schuberth, Bernhard S. A.; Zaroli, Christophe; Nolet, Guust
2015-12-01
We study wavefield effects of direct P- and S-waves in elastic and isotropic 3-D seismic structures derived from the temperature field of a high-resolution mantle circulation model. More specifically, we quantify the dispersion of traveltime residuals caused by diffraction in structures with dynamically constrained length scales and magnitudes of the lateral variations in seismic velocities and density. 3-D global wave propagation is simulated using a spectral element method. Intrinsic attenuation (i.e. dissipation of seismic energy) is deliberately neglected, so that any variation of traveltimes with frequency can be attributed to structural effects. Traveltime residuals are measured at 15, 22.5, 34 and 51 s dominant periods by cross-correlation of 3-D and 1-D synthetic waveforms. Additional simulations are performed for a model in which 3-D structure is removed in the upper 800 km to isolate the dispersion signal of the lower mantle. We find that the structural length scales inherent to a vigorously convecting mantle give rise to significant diffraction-induced body-wave traveltime dispersion. For both P- and S-waves, the difference between long-period and short-period residuals for a given source-receiver pair can reach up to several seconds for the period bands considered here. In general, these differential-frequency' residuals tend to increase in magnitude with increasing short-period delay. Furthermore, the long-period signal typically is smaller in magnitude than the short-period one; that is, wave-front healing is efficient independent of the sign of the residuals. Unlike the single-frequency residuals, the differential-frequency residuals are surprisingly similar between the lower-mantle' and the whole-mantle' model for corresponding source-receiver pairs. The similarity is more pronounced in case of S-waves and varies between different combinations of period bands. The traveltime delay acquired in the upper mantle seems to cancel in these differential
15. Determination of the strong coupling constant α{sub s} (m{sub Z}) from measurements of the total cross section for top-antitop-quark production
Energy Technology Data Exchange (ETDEWEB)
Klijnsma, Thomas; Dissertori, Guenther [ETH Zurich, Institute for Particle Physics, Zurich (Switzerland); Bethke, Siegfried [Max-Planck-Institute of Physics, Munich (Germany); Salam, Gavin P. [CERN, Theoretical Physics Department, Geneva (Switzerland); CNRS, UMR 7589, LPTHE, Paris (France)
2017-11-15
We present a determination of the strong coupling constant α{sub s} (m{sub Z}) using inclusive top-quark pair production cross section measurements performed at the LHC and at the Tevatron. Following a procedure first applied by the CMS Collaboration, we extract individual values of α{sub s} (m{sub Z}) from measurements by different experiments at several centre-of-mass energies, using QCD predictions complete in NNLO perturbation theory, supplemented with NNLL approximations to all orders, and suitable sets of parton distribution functions. The determinations are then combined using a likelihood-based approach, where special emphasis is put on a consistent treatment of theoretical uncertainties and of correlations between various sources of systematic uncertainties. Our final combined result is α{sub s} (m{sub Z}) = 0.1177{sup +0.0034}{sub -0.0036}. (orig.)
16. Computation with Inverse States in a Finite Field FPα: The Muon Neutrino Mass, the Unified Strong-Electroweak Coupling Constant, and the Higgs Mass
International Nuclear Information System (INIS)
Dai, Yang; Borisov, Alexey B.; Boyer, Keith; Rhodes, Charles K.
2000-01-01
The construction of inverse states in a finite field F P α enables the organization of the mass scale with fundamental octets in an eight-dimensional index space that identifies particle states with residue class designations. Conformance with both CPT invariance and the concept of supersymmetry follows as a direct consequence of this formulation. Based on two parameters (P α and g α ) that are anchored on a concordance of physical data, this treatment leads to (1) a prospective mass for the muon neutrino of approximately27.68 meV, (2) a value of the unified strong-electroweak coupling constant α* = (34.26) -1 that is physically defined by the ratio of the electron neutrino and muon neutrino masses, and (3) a see-saw congruence connecting the Higgs, the electron neutrino, and the muon neutrino masses. Specific evaluation of the masses of the corresponding supersymmetric Higgs pair reveals that both particles are superheavy (> 10 18 GeV). No renormalization of the Higgs masses is introduced, since the calculational procedure yielding their magnitudes is intrinsically divergence-free. Further, the Higgs fulfills its conjectured role through the see-saw relation as the particle defining the origin of all particle masses, since the electron and muon neutrino systems, together with their supersymmetric partners, are the generators of the mass scale and establish the corresponding index space. Finally, since the computation of the Higgs masses is entirely determined by the modulus of the field P α , which is fully defined by the large-scale parameters of the universe through the value of the universal gravitational constant G and the requirement for perfect flatness (Omega = 1.0), the see-saw congruence fuses the concepts of mass and space and creates a new unified archetype
17. On the evaluation of temperature dependence of elastic constants of martensitic phases in shape memory alloys from resonant ultrasound spectroscopy studies
Czech Academy of Sciences Publication Activity Database
Landa, Michal; Sedlák, Petr; Šittner, Petr; Seiner, Hanuš; Heller, Luděk
481-482, - (2008), s. 567-573 ISSN 0921-5093 R&D Projects: GA ČR GA101/06/0768 Institutional research plan: CEZ:AV0Z20760514; CEZ:AV0Z10100520 Keywords : modal resonant ultrasound spectroscopy * elastic properties * shape memory alloy s Subject RIV: BI - Acoustics Impact factor: 1.806, year: 2008
18. A calorimetric measurement of the strong coupling constant in electron-positron annihilation at a center-of-mass energy of 91.6 GeV
International Nuclear Information System (INIS)
Martirena, S.G.
1994-04-01
In this work, a measurement of the strong coupling constant α s in e + e - annihilation at a center-of-mass energy of 91.6 GeV is presented. The measurement was performed with the SLD at the Stanford Linear Collider facility located at the Stanford Linear Accelerator Center in California. The procedure used consisted of measuring the rate of hard gluon radiation from the primary quarks in a sample of 9,878 hadronic events. After defining the asymptotic manifestation of partons as 'jets', various phenomenological models were used to correct for the hadronization process. A value for the QCD scale parameter Λ bar MS , defined in the bar MS renormalization convention with 5 active quark flavors, was then obtained by a direct fit to O(α s 2 ) calculations. The value of α s obtained was α s (M z0 ) = 0.122 ± 0.004 -0.007 +0.008 where the uncertainties are experimental (combined statistical and systematic) and theoretical (systematic) respectively. Equivalently, Λ bar MS = 0.28 -0.10 +0.16 GeV where the experimental and theoretical uncertainties have been combined
19. Determination of the strong coupling constant from the measurement of inclusive multijet event cross sections in pp collisions at $\\sqrt{s} = 8~\\mathrm{TeV}$
CERN Document Server
CMS Collaboration
2017-01-01
A measurement of inclusive multijet event cross sections is presented from proton-proton collisions recorded at $\\sqrt{s} = 8\\,$TeV with the CMS detector and corresponding to an integrated luminosity of $19.7\\,\\mathrm{fb}^{-1}$. Jets are reconstructed with the anti-k$_t$ clustering algorithm for a jet size parameter $R=0.7$ in a phase space region ranging up to jet transverse momenta $p_\\mathrm{T}$ of $2.0\\,$TeV and an absolute rapidity of $|y|=2.5$. The inclusive 2-jet and 3-jet event cross sections are measured as a function of the average $p_\\mathrm{T}$ of the two leading jets. The data are well described by predictions at next-to-leading order in perturbative quantum chromodynamics and additionally are compared to several Monte Carlo event generators. The strong coupling constant at the scale of the Z boson mass is inferred from a fit of the ratio of the 3-jet over 2-jet event cross section giving \\alpha_s(M_Z) = 0.1150\\,\\pm0.0010\\,\\textrm{(exp)}\\,\\pm0.0013\\,\\textrm{(PDF)}\\, \\pm0.0015\\,\\textrm{(NP)}\\,^{+... 20. Regularized unfolding of jet cross sections in deep-inelastic ep scattering at HERA and determination of the strong coupling constant Energy Technology Data Exchange (ETDEWEB) Britzger, Daniel Andreas 2013-10-15 In this thesis double-differential cross sections for jet production in neutral current deep-inelastic e{sup {+-}}p scattering (DIS) are presented at the center-of-mass energy of {radical}(s)=319 GeV, and in the kinematic range of the squared four-momentum transfer 150< Q{sup 2}<15 000 GeV{sup 2} and the inelasticity 0.2strong coupling constant {alpha}{sub s}(M{sub Z}) at the scale of the mass of the Z{sup 0} boson in the framework of perturbative quantum chromodynamics in next-to-leading order. Values are derived separately for the absolute 1. Elastic interaction of a crack with a microcrack array. I - Formulation of the problem and general form of the solution. II - Elastic solution for two crack configurations (piecewise constant and linear approximations) Science.gov (United States) Chudnovsky, A.; Dolgopolsky, A.; Kachanov, M. 1987-01-01 The elastic interactions of a two-dimensional configuration consisting of a crack with an array of microcracks located near the tip are studied. The general form of the solution is based on the potential representations and approximations of tractions on the microcracks by polynomials. In the second part, the technique is applied to two simple two-dimensional configurations involving one and two microcracks. The problems of stress shielding and stress amplification (the reduction or increase of the effective stress intensity factor due to the presence of microcracks) are discussed, and the refinements introduced by higher order polynomial approximations are illustrated. 2. Microstructural evolution in inhomogeneous elastic media International Nuclear Information System (INIS) Jou, H.J.; Leo, P.H.; Lowengrub, J.S. 1997-01-01 We simulate the diffusional evolution of microstructures produced by solid state diffusional transformations in elastically stressed binary alloys in two dimensions. The microstructure consists of arbitrarily shaped precipitates embedded coherently in an infinite matrix. The precipitate and matrix are taken to be elastically isotropic, although they may have different elastic constants (elastically inhomogeneous). Both far-field applied strains and mismatch strains between the phases are considered. The diffusion and elastic fields are calculated using the boundary integral method, together with a small scale preconditioner to remove ill-conditioning. The precipitate-matrix interfaces are tracked using a nonstiff time updating method. The numerical method is spectrally accurate and efficient. Simulations of a single precipitate indicate that precipitate shapes depend strongly on the mass flux into the system as well as on the elastic fields. Growing shapes (positive mass flux) are dendritic while equilibrium shapes (zero mass flux) are squarish. Simulations of multiparticle systems show complicated interactions between precipitate morphology and the overall development of microstructure (i.e., precipitate alignment, translation, merging, and coarsening). In both single and multiple particle simulations, the details of the microstructural evolution depend strongly o the elastic inhomogeneity, misfit strain, and applied fields. 57 refs., 24 figs 3. Other paradigms: growth rate constants and tumor burden determined using computed tomography data correlate strongly with the overall survival of patients with renal cell carcinoma. Science.gov (United States) Stein, Wilfred D; Huang, Hui; Menefee, Michael; Edgerly, Maureen; Kotz, Herb; Dwyer, Andrew; Yang, James; Bates, Susan E 2009-01-01 In solid tumors, where curative therapies still elude oncologists, novel paradigms are needed to assess the efficacy of new therapies and those already approved. We used radiologic measurements obtained in patients with metastatic renal cell carcinoma enrolled in a phase II study of the epothilone B analog, ixabepilone (Ixempra), to address this issue. Using a novel 2-phase mathematical equation, we used the radiologic measurements to estimate the concomitant rates of tumor regression and growth (regression and growth rate constants). Eighty-one patients were enrolled on the ixabepilone trial at the time of this analysis. Growth rate constants were determined using computed tomography measurements obtained exclusively while a patient was enrolled on study. The growth rate constants of renal cell carcinomas treated with ixabepilone were significantly reduced compared with those of tumors in patients who received placebo in a previous trial. Furthermore, a correlation with overall survival was found for both the growth rate constant and the initial tumor burden; and this correlation was even stronger when both the growth rate constant and the initial tumor burden were combined. The readily amenable mathematical model described herein has potential applications to many tumor types that can be assessed with imaging modalities. Because the growth rate constant seems to be a surrogate for survival, assessment could aid in the evaluation of relative efficacies of different therapies and perhaps in assessing the potential individual benefit of an experimental therapy. 4. Possibilités actuelles du calcul des constantes élastiques de polymères par des méthodes de simulation atomistique Current Possibilities of the Computation of Elastic Constants of Polymers Using Atomistic Simulations Directory of Open Access Journals (Sweden) Dal Maso F. 2006-12-01 Full Text Available Les propriétés élastiques des phases amorphe et cristalline pures de polymères semi-cristallins ne sont en général pas mesurables directement avec les moyens physiques habituels. Il est donc nécessaire de recourir à des méthodes de calcul numérique. Cet article décrit certaines de ces méthodes, fondées sur des modélisations atomistiques, ainsi qu'une évaluation des implémentations actuelles. Il est montré que la méthode proposée par Zehnder et al. (1996 fournit les meilleurs résultats, au prix d'un temps long de calcul, dû à la dynamique moléculaire. Néanmoins, aucune de ces méthodes n'est vraiment utilisable simplement au jour le jour, car elles requièrent des moyens importants de calcul. Elastic properties of pure crystalline and amorphous phases of a semicrystalline polymer are usually not directly measurable by usual physical means. It therefore is necessary to resort to numerical computing methods. This paper describes some of these methods, based on atomistic simulations, as well as an assessment of current implementations. It is shown that the method proposed by Zehnder et al. (1996 gives the best results, at the expense of long computing time, due to molecular dynamic simulation. Nevertheless none of these methods are really usable on a daily basis, since there are demanding important computing capabilities. 5. Bounds and self-consistent estimates for elastic constants of polycrystals of hcp solid He $4$ Energy Technology Data Exchange (ETDEWEB) Berryman, James G. 2012-03-01 Recent advances in methods for computing both Hashin-Shtrikman bounds and related selfconsistent (or CPA) estimates of elastic constants for polycrystals composed of randomly oriented crystals can be applied successfully to hexagonal close packed solid He{sup 4}. In particular, since the shear modulus C{sub 44} of hexagonal close-packed solid He is known to undergo large temperature variations when 20 mK {<=} T {<=} 200 mK, bounds and estimates computed with this class of effective medium methods, while using C{sub 44} {r_arrow} 0 as a proxy for melting, are found to be both qualitatively and quantitatively very similar to prior results obtained using Monte Carlo methods. Hashin- Shtrikman bounds provide significantly tighter constraints on the polycrystal behavior than do the traditional Voigt and Reuss bounds. 6. Cosmological constant as integration constant Science.gov (United States) Treder, H.-J. 1994-08-01 Einstein's field theory of elementary particles (Einstein 1919) yields black holes with a mass M approximately G-1 Lambda-1/2 c2 and a charge Q approximately G-1/2 Lambda-1/2 c2, their curvature radius is Lambda-1/2. Here Lambda is an integration constant of Einstein's 'trace-less' gravitation equations. The choice Lambda = G-1 h-1 c3 for this constant defines Planck ions and implies 'strong-gravity'. The choice Lambda = lambda = 3Hinf exp 2 c-2 (where Hinf means the Hubble parameter of a final de Sitter cosmos) involves 'weak-gravity' and describes an electro-vac spherical universe. 7. Theory of Spacetime Elasticity Science.gov (United States) Gusev, Andrei A.; Lurie, Sergey A. We present the theory of spacetime elasticity and demonstrate that it involves traditional thermoelasticity. Assuming linear-elastic constitutive behavior and using spacetime transversely-isotropic elastic constants, we derive all principal thermodynamic relations of classical thermoelasticity. We introduce the spacetime principle of virtual work, and use it to derive the equations of motion for both reversible and dissipative thermoelastic dynamics. We show that spacetime elasticity directly implies the Fourier and the Maxwell-Cattaneo laws of heat conduction. However, spacetime elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity, complemented by the spectrum of boundary and interface conditions. We argue that the presented framework of spacetime elasticity should prove adequate for describing the thermoelastic phenomena occurring at low temperatures, for interpreting the results of molecular simulations of heat conduction in solids, and also for the optimal heat and stress management in the microelectronic components and the thermoelectric devices. 8. Temperature dependence of elastic properties of paratellurite International Nuclear Information System (INIS) Silvestrova, I.M.; Pisarevskii, Y.V.; Senyushenkov, P.A.; Krupny, A.I. 1987-01-01 New data are presented on the temperature dependence of the elastic wave velocities, elastic stiffness constants, and thermal expansion of paratellurite. It is shown that the external pressure appreciably influences the elastic properties of TeO 2 , especially the temperature dependence of the elastic modulus connected with the crystal soft mode. (author) 9. Determination of the strong coupling constant α{sub s}(m{sub Z}) in next-to-next-to-leading order QCD using H1 jet cross section measurements Energy Technology Data Exchange (ETDEWEB) Andreev, V.; Belousov, A.; Fomenko, A.; Gogitidze, N.; Lebedev, A.; Malinovski, E.; Soloviev, Y.; Vazdik, Y. [Lebedev Physical Institute, Moscow (Russian Federation); Baghdasaryan, A.; Zohrabyan, H. [Yerevan Physics Institute, Yerevan (Armenia); Begzsuren, K.; Ravdandorj, T. [Institute of Physics and Technology of the Mongolian Academy of Sciences, Ulaanbaatar (Mongolia); Bertone, V. [Vrije University, Department of Physics and Astronomy, Amsterdam (Netherlands); National Institute for Subatomic Physics (NIKHEF), Amsterdam (Netherlands); Bolz, A.; Britzger, D.; Huber, F.; Sauter, M.; Schoening, A. [Universitaet Heidelberg, Physikalisches Institut, Heidelberg (Germany); Boudry, V.; Specka, A. [LLR, Ecole Polytechnique, CNRS/IN2P3, Palaiseau (France); Brandt, G. [Universitaet Goettingen, II. Physikalisches Institut, Goettingen (Germany); Brisson, V.; Jacquet, M.; Pascaud, C.; Zhang, Z.; Zomer, F. [LAL, Universite Paris-Sud, CNRS/IN2P3, Orsay (France); Buniatyan, A.; Newman, P.R.; Thompson, P.D. [University of Birmingham, School of Physics and Astronomy, Birmingham (United Kingdom); Bylinkin, A. [Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region (Russian Federation); Bystritskaya, L.; Fedotov, A. [Institute for Theoretical and Experimental Physics, Moscow (Russian Federation); Campbell, A.J.; Dodonov, V.; Eckerlin, G.; Elsen, E.; Fleischer, M.; Gayler, J.; Ghazaryan, S.; Haidt, D.; Jung, H.; Katzy, J.; Kleinwort, C.; Kruecker, D.; Krueger, K.; Levonian, S.; Lipka, K.; List, B.; List, J.; Meyer, A.B.; Meyer, J.; Niebuhr, C.; Olsson, J.E.; Pirumov, H.; Pitzl, D.; Placakyte, R.; Schmitt, S.; Sefkow, F.; South, D.; Steder, M.; Wuensch, E.; Zlebcik, R. [DESY, Hamburg (Germany); Cantun Avila, K.B.; Contreras, J.G. [CINVESTAV, Departamento de Fisica Aplicada, Merida, Yucatan (Mexico); Cerny, K.; Salek, D.; Valkarova, A.; Zacek, J. [Charles University, Faculty of Mathematics and Physics, Prague (Czech Republic); Chekelian, V.; Grindhammer, G.; Kiesling, C.; Lobodzinski, B. [Max-Planck-Institut fuer Physik, Munich (Germany); Cvach, J.; Hladky, J.; Reimer, P. [Academy of Sciences of the Czech Republic, Institute of Physics, Prague (Czech Republic); Currie, J. [Durham University, Institute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Durham (United Kingdom); Dainton, J.B.; Gabathuler, E.; Greenshaw, T.; Klein, M.; Kostka, P.; Kretzschmar, J.; Laycock, P.; Maxfield, S.J.; Mehta, A.; Patel, G.D. [University of Liverpool, Department of Physics, Liverpool (United Kingdom); Daum, K.; Meyer, H. [Fachbereich C, Universitaet Wuppertal, Wuppertal (Germany); Diaconu, C.; Hoffmann, D.; Vallee, C. [Aix Marseille Universite, CNRS/IN2P3, CPPM UMR 7346, Marseille (France); Dobre, M.; Rotaru, M. [Horia Hulubei National Institute for R and D in Physics and Nuclear Engineering (IFIN-HH), Bucharest (Romania); Egli, S.; Horisberger, R.; Ozerov, D. [Paul Scherrer Institute, Villigen (Switzerland); Favart, L.; Grebenyuk, A.; Hreus, T.; Janssen, X.; Roosen, R.; Mechelen, P.Van [Brussels and Universiteit Antwerpen, Inter-University Institute for High Energies ULB-VUB, Antwerp (Belgium); Feltesse, J.; Schoeffel, L. [Irfu/SPP, CE Saclay, Gif-sur-Yvette (France); Gehrmann, T.; Mueller, K.; Niehues, J.; Robmann, P.; Straumann, U.; Truoel, P. [Physik-Institut der Universitaet Zuerich, Zurich (Switzerland); Goerlich, L.; Mikocki, S.; Nowak, G.; Sopicki, P. [Polish Academy of Sciences, Institute of Nuclear Physics, Krakow (Poland); Gouzevitch, M.; Petrukhin, A. [IPNL, Universite Claude Bernard Lyon 1, CNRS/IN2P3, Villeurbanne (France); Grab, C.; Huss, A. [ETH Zuerich, Institut fuer Teilchenphysik, Zurich (Switzerland); Gwenlan, C.; Radescu, V. [Oxford University, Department of Physics, Oxford (United Kingdom); Henderson, R.C.W. [University of Lancaster, Department of Physics, Lancaster (United Kingdom); Jung, A.W. [Purdue University, Department of Physics and Astronomy, West Lafayette, IN (United States); Kapichine, M.; Morozov, A.; Spaskov, V. [Joint Institute for Nuclear Research, Dubna (Russian Federation); Kogler, R. [Universitaet Hamburg, Institut fuer Experimentalphysik, Hamburg (Germany); Landon, M.P.J.; Rizvi, E.; Traynor, D. [Queen Mary University of London, School of Physics and Astronomy, London (United Kingdom); Lange, W.; Naumann, T. [DESY, Zeuthen (Germany); Martyn, H.U. [I. Physikalisches Institut der RWTH, Aachen (Germany); Perez, E. [CERN, Geneva (Switzerland); Picuric, I.; Raicevic, N. [University of Montenegro, Faculty of Science, Podgorica (Montenegro); Polifka, R. [Charles University, Faculty of Mathematics and Physics, Prague (Czech Republic); University of Toronto, Department of Physics, Toronto, ON (Canada); Rabbertz, K. [Karlsruher Institut fuer Technologie (KIT), Institut fuer Experimentelle Teilchenphysik (ETP), Karlsruhe (Germany); Rostovtsev, A. [Institute for Information Transmission Problems RAS, Moscow (Russian Federation); Sankey, D.P.C. [STFC, Rutherford Appleton Laboratory, Didcot, Oxfordshire (United Kingdom); Sauvan, E. [Aix Marseille Universite, CNRS/IN2P3, CPPM UMR 7346, Marseille (France); Universite de Savoie, CNRS/IN2P3, LAPP, Annecy-le-Vieux (France); Shushkevich, S. [Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics, Moscow (Russian Federation); Stella, B. [Universita di Roma Tre, Dipartimento di Fisica, Rome (Italy); INFN Roma 3 (Italy); Sutton, M.R. [University of Sussex, Department of Physics and Astronomy, Brighton (United Kingdom); Sykora, T. [Brussels and Universiteit Antwerpen, Inter-University Institute for High Energies ULB-VUB, Antwerp (Belgium); Charles University, Faculty of Mathematics and Physics, Prague (Czech Republic); Tsakov, I. [Institute for Nuclear Research and Nuclear Energy, Sofia (Bulgaria); Tseepeldorj, B. [Institute of Physics and Technology of the Mongolian Academy of Sciences, Ulaanbaatar (MN); Ulaanbaatar University, Ulaanbaatar (MN); Wegener, D. [TU Dortmund, Institut fuer Physik, Dortmund (DE); Collaboration: H1 Collaboration 2017-11-15 The strong coupling constant α{sub s} is determined from inclusive jet and dijet cross sections in neutral-current deep-inelastic ep scattering (DIS) measured at HERA by the H1 collaboration using next-to-next-to-leading order (NNLO) QCD predictions. The dependence of the NNLO predictions and of the resulting value of α{sub s}(m{sub Z}) at the Z-boson mass m{sub Z} are studied as a function of the choice of the renormalisation and factorisation scales. Using inclusive jet and dijet data together, the strong coupling constant is determined to be α{sub s}(m{sub Z}) = 0.1157(20){sub exp}(29){sub th}. Complementary, α{sub s}(m{sub Z}) is determined together with parton distribution functions of the proton (PDFs) from jet and inclusive DIS data measured by the H1 experiment. The value α{sub s}(m{sub Z}) = 0.1142(28){sub tot} obtained is consistent with the determination from jet data alone. The impact of the jet data on the PDFs is studied. The running of the strong coupling is tested at different values of the renormalisation scale and the results are found to be in agreement with expectations. (orig.) 10. Calculation of the total plasma concentration of nonvolatile weak acids and the effective dissociation constant of nonvolatile buffers in plasma for use in the strong ion approach to acid-base balance in cats. Science.gov (United States) McCullough, Sheila M; Constable, Peter D 2003-08-01 To determine values for the total concentration of nonvolatile weak acids (Atot) and effective dissociation constant of nonvolatile weak acids (Ka) in plasma of cats. Convenience plasma samples of 5 male and 5 female healthy adult cats. Cats were sedated, and 20 mL of blood was obtained from the jugular vein. Plasma was tonometered at 37 degrees C to systematically vary PCO2 from 8 to 156 mm Hg, thereby altering plasma pH from 6.90 to 7.97. Plasma pH, PCO2, and concentrations of quantitatively important strong cations (Na+, K+, and Ca2+), strong anions (Cl-, lactate), and buffer ions (total protein, albumin, and phosphate) were determined. Strong ion difference was estimated from the measured strong ion concentrations and nonlinear regression used to calculate Atot and Ka from the measured pH and PCO2 and estimated strong ion difference. Mean (+/- SD) values were as follows: Atot = 24.3 +/- 4.6 mmol/L (equivalent to 0.35 mmol/g of protein or 0.76 mmol/g of albumin); Ka = 0.67 +/- 0.40 x 10(-7); and the negative logarithm (base 10) of Ka (pKa) = 7.17. At 37 degrees C, pH of 7.35, and a partial pressure of CO2 (PCO2) of 30 mm Hg, the calculated venous strong ion difference was 30 mEq/L. These results indicate that at a plasma pH of 7.35, a 1 mEq/L decrease in strong ion difference will decrease pH by 0.020, a 1 mm Hg decrease in PCO2 will increase plasma pH by 0.011, and a 1 g/dL decrease in albumin concentration will increase plasma pH by 0.093. 11. Temperature dependence of the dielectric, piezoelectric, and elastic constants for Pb(Mg1/3Nb2/3)O3-PbZrO3-PbTiO3 piezocrystals Science.gov (United States) Zhang, Shujun; Lee, Sung-Min; Kim, Dong-Ho; Lee, Ho-Yong; Shrout, Thomas R. 2007-12-01 Pb(Mg1/3Nb2/3)O3-PbZrO3-PbTiO3 (PMN-PZT) ferroelectric single crystals were grown successfully using the solid state crystal growth method. The crystals were found to exhibit higher Curie temperatures TCs and ferroelectric phase transition temperatures TR-Ts when compared to the binary Pb(Mg1/3Nb2/3)O3-PbTiO3 system. Specifically the composition 0.4Pb(Mg1/3Nb2/3)O3-0.25PbZrO3-0.35PbTiO3 which lies close to the rhombohedral-tetragonal morphotropic phase boundary, was investigated. The full set of materials constants, including elastic (c and s), piezoelectric (d, g, e, and h), and dielectric permittivity (ɛ /ɛ0), were determined using the IEEE standards as a function of temperature ranging between 25 and 100°C. The electromechanical coupling factors k33 and k32 were found to be 93.3% and 93.5%, respectively, with corresponding piezoelectric coefficients d33 and d32 on the order of 1530 and -1440pC/N. Together with high phase transition temperatures (TC of 216°C and TR-T on the order of 144°C) and high coercive field EC ˜4.6kV /cm, make PMN-PZT single crystals promising candidates for high temperature actuator and transducer applications. 12. Constraints on parton distribution functions and extraction of the strong coupling constant from the inclusive jet cross section in pp collisions at\\sqrt{s}$= 7 TeV CERN Document Server Khachatryan, Vardan; Tumasyan, Armen; Adam, Wolfgang; Bergauer, Thomas; Dragicevic, Marko; Erö, Janos; Friedl, Markus; Fruehwirth, Rudolf; Ghete, Vasile Mihai; Hartl, Christian; Hörmann, Natascha; Hrubec, Josef; Jeitler, Manfred; Kiesenhofer, Wolfgang; Knünz, Valentin; Krammer, Manfred; Krätschmer, Ilse; Liko, Dietrich; Mikulec, Ivan; Rabady, Dinyar; Rahbaran, Babak; Rohringer, Herbert; Schöfbeck, Robert; Strauss, Josef; Treberer-Treberspurg, Wolfgang; Waltenberger, Wolfgang; Wulz, Claudia-Elisabeth; Mossolov, Vladimir; Shumeiko, Nikolai; Suarez Gonzalez, Juan; Alderweireldt, Sara; Bansal, Monika; Bansal, Sunil; Cornelis, Tom; De Wolf, Eddi A; Janssen, Xavier; Knutsson, Albert; Luyckx, Sten; Ochesanu, Silvia; Rougny, Romain; Van De Klundert, Merijn; Van Haevermaet, Hans; Van Mechelen, Pierre; Van Remortel, Nick; Van Spilbeeck, Alex; Blekman, Freya; Blyweert, Stijn; D'Hondt, Jorgen; Daci, Nadir; Heracleous, Natalie; Keaveney, James; Lowette, Steven; Maes, Michael; Olbrechts, Annik; Python, Quentin; Strom, Derek; Tavernier, Stefaan; Van Doninck, Walter; Van Mulders, Petra; Van Onsem, Gerrit Patrick; Villella, Ilaria; Caillol, Cécile; Clerbaux, Barbara; De Lentdecker, Gilles; Dobur, Didar; Favart, Laurent; Gay, Arnaud; Grebenyuk, Anastasia; Léonard, Alexandre; Mohammadi, Abdollah; Perniè, Luca; Reis, Thomas; Seva, Tomislav; Thomas, Laurent; Vander Velde, Catherine; Vanlaer, Pascal; Wang, Jian; Zenoni, Florian; Adler, Volker; Beernaert, Kelly; Benucci, Leonardo; Cimmino, Anna; Costantini, Silvia; Crucy, Shannon; Dildick, Sven; Fagot, Alexis; Garcia, Guillaume; Mccartin, Joseph; Ocampo Rios, Alberto Andres; Ryckbosch, Dirk; Salva Diblen, Sinem; Sigamani, Michael; Strobbe, Nadja; Thyssen, Filip; Tytgat, Michael; Yazgan, Efe; Zaganidis, Nicolas; Basegmez, Suzan; Beluffi, Camille; Bruno, Giacomo; Castello, Roberto; Caudron, Adrien; Ceard, Ludivine; Da Silveira, Gustavo Gil; Delaere, Christophe; Du Pree, Tristan; Favart, Denis; Forthomme, Laurent; Giammanco, Andrea; Hollar, Jonathan; Jafari, Abideh; Jez, Pavel; Komm, Matthias; Lemaitre, Vincent; Nuttens, Claude; Pagano, Davide; Perrini, Lucia; Pin, Arnaud; Piotrzkowski, Krzysztof; Popov, Andrey; Quertenmont, Loic; Selvaggi, Michele; Vidal Marono, Miguel; Vizan Garcia, Jesus Manuel; Beliy, Nikita; Caebergs, Thierry; Daubie, Evelyne; Hammad, Gregory Habib; Aldá Júnior, Walter Luiz; Alves, Gilvan; Brito, Lucas; Correa Martins Junior, Marcos; Dos Reis Martins, Thiago; Mora Herrera, Clemencia; Pol, Maria Elena; Carvalho, Wagner; Chinellato, Jose; Custódio, Analu; Melo Da Costa, Eliza; De Jesus Damiao, Dilson; De Oliveira Martins, Carley; Fonseca De Souza, Sandro; Malbouisson, Helena; Matos Figueiredo, Diego; Mundim, Luiz; Nogima, Helio; Prado Da Silva, Wanda Lucia; Santaolalla, Javier; Santoro, Alberto; Sznajder, Andre; Tonelli Manganote, Edmilson José; Vilela Pereira, Antonio; Bernardes, Cesar Augusto; Dogra, Sunil; Tomei, Thiago; De Moraes Gregores, Eduardo; Mercadante, Pedro G; Novaes, Sergio F; Padula, Sandra; Aleksandrov, Aleksandar; Genchev, Vladimir; Iaydjiev, Plamen; Marinov, Andrey; Piperov, Stefan; Rodozov, Mircho; Stoykova, Stefka; Sultanov, Georgi; Vutova, Mariana; Dimitrov, Anton; Glushkov, Ivan; Hadjiiska, Roumyana; Kozhuharov, Venelin; Litov, Leander; Pavlov, Borislav; Petkov, Peicho; Bian, Jian-Guo; Chen, Guo-Ming; Chen, He-Sheng; Chen, Mingshui; Du, Ran; Jiang, Chun-Hua; Plestina, Roko; Romeo, Francesco; Tao, Junquan; Wang, Zheng; Asawatangtrakuldee, Chayanit; Ban, Yong; Li, Qiang; Liu, Shuai; Mao, Yajun; Qian, Si-Jin; Wang, Dayong; Zou, Wei; Avila, Carlos; Chaparro Sierra, Luisa Fernanda; Florez, Carlos; Gomez, Juan Pablo; Gomez Moreno, Bernardo; Sanabria, Juan Carlos; Godinovic, Nikola; Lelas, Damir; Polic, Dunja; Puljak, Ivica; Antunovic, Zeljko; Kovac, Marko; Brigljevic, Vuko; Kadija, Kreso; Luetic, Jelena; Mekterovic, Darko; Sudic, Lucija; Attikis, Alexandros; Mavromanolakis, Georgios; Mousa, Jehad; Nicolaou, Charalambos; Ptochos, Fotios; Razis, Panos A; Bodlak, Martin; Finger, Miroslav; Finger Jr, Michael; Assran, Yasser; Ellithi Kamel, Ali; Mahmoud, Mohammed; Radi, Amr; Kadastik, Mario; Murumaa, Marion; Raidal, Martti; Tiko, Andres; Eerola, Paula; Fedi, Giacomo; Voutilainen, Mikko; Härkönen, Jaakko; Karimäki, Veikko; Kinnunen, Ritva; Kortelainen, Matti J; Lampén, Tapio; Lassila-Perini, Kati; Lehti, Sami; Lindén, Tomas; Luukka, Panja-Riina; Mäenpää, Teppo; Peltola, Timo; Tuominen, Eija; Tuominiemi, Jorma; Tuovinen, Esa; Wendland, Lauri; Talvitie, Joonas; Tuuva, Tuure; Besancon, Marc; Couderc, Fabrice; Dejardin, Marc; Denegri, Daniel; Fabbro, Bernard; Faure, Jean-Louis; Favaro, Carlotta; Ferri, Federico; Ganjour, Serguei; Givernaud, Alain; Gras, Philippe; Hamel de Monchenault, Gautier; Jarry, Patrick; Locci, Elizabeth; Malcles, Julie; Rander, John; Rosowsky, André; Titov, Maksym; Baffioni, Stephanie; Beaudette, Florian; Busson, Philippe; Charlot, Claude; Dahms, Torsten; Dalchenko, Mykhailo; Dobrzynski, Ludwik; Filipovic, Nicolas; Florent, Alice; Granier de Cassagnac, Raphael; Mastrolorenzo, Luca; Miné, Philippe; Mironov, Camelia; Naranjo, Ivo Nicolas; Nguyen, Matthew; Ochando, Christophe; Paganini, Pascal; Regnard, Simon; Salerno, Roberto; Sauvan, Jean-Baptiste; Sirois, Yves; Veelken, Christian; Yilmaz, Yetkin; Zabi, Alexandre; Agram, Jean-Laurent; Andrea, Jeremy; Aubin, Alexandre; Bloch, Daniel; Brom, Jean-Marie; Chabert, Eric Christian; Collard, Caroline; Conte, Eric; Fontaine, Jean-Charles; Gelé, Denis; Goerlach, Ulrich; Goetzmann, Christophe; Le Bihan, Anne-Catherine; Van Hove, Pierre; Gadrat, Sébastien; Beauceron, Stephanie; Beaupere, Nicolas; Boudoul, Gaelle; Bouvier, Elvire; Brochet, Sébastien; Carrillo Montoya, Camilo Andres; Chasserat, Julien; Chierici, Roberto; Contardo, Didier; Depasse, Pierre; El Mamouni, Houmani; Fan, Jiawei; Fay, Jean; Gascon, Susan; Gouzevitch, Maxime; Ille, Bernard; Kurca, Tibor; Lethuillier, Morgan; Mirabito, Laurent; Perries, Stephane; Ruiz Alvarez, José David; Sabes, David; Sgandurra, Louis; Sordini, Viola; Vander Donckt, Muriel; Verdier, Patrice; Viret, Sébastien; Xiao, Hong; Tsamalaidze, Zviad; Autermann, Christian; Beranek, Sarah; Bontenackels, Michael; Edelhoff, Matthias; Feld, Lutz; Hindrichs, Otto; Klein, Katja; Ostapchuk, Andrey; Perieanu, Adrian; Raupach, Frank; Sammet, Jan; Schael, Stefan; Weber, Hendrik; Wittmer, Bruno; Zhukov, Valery; Ata, Metin; Brodski, Michael; Dietz-Laursonn, Erik; Duchardt, Deborah; Erdmann, Martin; Fischer, Robert; Güth, Andreas; Hebbeker, Thomas; Heidemann, Carsten; Hoepfner, Kerstin; Klingebiel, Dennis; Knutzen, Simon; Kreuzer, Peter; Merschmeyer, Markus; Meyer, Arnd; Millet, Philipp; Olschewski, Mark; Padeken, Klaas; Papacz, Paul; Reithler, Hans; Schmitz, Stefan Antonius; Sonnenschein, Lars; Teyssier, Daniel; Thüer, Sebastian; Weber, Martin; Cherepanov, Vladimir; Erdogan, Yusuf; Flügge, Günter; Geenen, Heiko; Geisler, Matthias; Haj Ahmad, Wael; Heister, Arno; Hoehle, Felix; Kargoll, Bastian; Kress, Thomas; Kuessel, Yvonne; Künsken, Andreas; Lingemann, Joschka; Nowack, Andreas; Nugent, Ian Michael; Perchalla, Lars; Pooth, Oliver; Stahl, Achim; Asin, Ivan; Bartosik, Nazar; Behr, Joerg; Behrenhoff, Wolf; Behrens, Ulf; Bell, Alan James; Bergholz, Matthias; Bethani, Agni; Borras, Kerstin; Burgmeier, Armin; Cakir, Altan; Calligaris, Luigi; Campbell, Alan; Choudhury, Somnath; Costanza, Francesco; Diez Pardos, Carmen; Dooling, Samantha; Dorland, Tyler; Eckerlin, Guenter; Eckstein, Doris; Eichhorn, Thomas; Flucke, Gero; Garay Garcia, Jasone; Geiser, Achim; Gunnellini, Paolo; Hauk, Johannes; Hempel, Maria; Horton, Dean; Jung, Hannes; Kalogeropoulos, Alexis; Kasemann, Matthias; Katsas, Panagiotis; Kieseler, Jan; Kleinwort, Claus; Krücker, Dirk; Lange, Wolfgang; Leonard, Jessica; Lipka, Katerina; Lobanov, Artur; Lohmann, Wolfgang; Lutz, Benjamin; Mankel, Rainer; Marfin, Ihar; Melzer-Pellmann, Isabell-Alissandra; Meyer, Andreas Bernhard; Mittag, Gregor; Mnich, Joachim; Mussgiller, Andreas; Naumann-Emme, Sebastian; Nayak, Aruna; Novgorodova, Olga; Ntomari, Eleni; Perrey, Hanno; Pitzl, Daniel; Placakyte, Ringaile; Raspereza, Alexei; Ribeiro Cipriano, Pedro M; Roland, Benoit; Ron, Elias; Sahin, Mehmet Özgür; Salfeld-Nebgen, Jakob; Saxena, Pooja; Schmidt, Ringo; Schoerner-Sadenius, Thomas; Schröder, Matthias; Seitz, Claudia; Spannagel, Simon; Vargas Trevino, Andrea Del Rocio; Walsh, Roberval; Wissing, Christoph; Aldaya Martin, Maria; Blobel, Volker; Centis Vignali, Matteo; Draeger, Arne-Rasmus; Erfle, Joachim; Garutti, Erika; Goebel, Kristin; Görner, Martin; Haller, Johannes; Hoffmann, Malte; Höing, Rebekka Sophie; Kirschenmann, Henning; Klanner, Robert; Kogler, Roman; Lange, Jörn; Lapsien, Tobias; Lenz, Teresa; Marchesini, Ivan; Ott, Jochen; Peiffer, Thomas; Pietsch, Niklas; Poehlsen, Jennifer; Pöhlsen, Thomas; Rathjens, Denis; Sander, Christian; Schettler, Hannes; Schleper, Peter; Schlieckau, Eike; Schmidt, Alexander; Seidel, Markus; Sola, Valentina; Stadie, Hartmut; Steinbrück, Georg; Troendle, Daniel; Usai, Emanuele; Vanelderen, Lukas; Vanhoefer, Annika; Barth, Christian; Baus, Colin; Berger, Joram; Böser, Christian; Butz, Erik; Chwalek, Thorsten; De Boer, Wim; Descroix, Alexis; Dierlamm, Alexander; Feindt, Michael; Frensch, Felix; Giffels, Manuel; Hartmann, Frank; Hauth, Thomas; Husemann, Ulrich; Katkov, Igor; Kornmayer, Andreas; Kuznetsova, Ekaterina; Lobelle Pardo, Patricia; Mozer, Matthias Ulrich; Müller, Thomas; Nürnberg, Andreas; Quast, Gunter; Rabbertz, Klaus; Ratnikov, Fedor; Röcker, Steffen; Sieber, Georg; Simonis, Hans-Jürgen; Stober, Fred-Markus Helmut; Ulrich, Ralf; Wagner-Kuhr, Jeannine; Wayand, Stefan; Weiler, Thomas; Wolf, Roger; Anagnostou, Georgios; Daskalakis, Georgios; Geralis, Theodoros; Giakoumopoulou, Viktoria Athina; Kyriakis, Aristotelis; Loukas, Demetrios; Markou, Athanasios; Markou, Christos; Psallidas, Andreas; Topsis-Giotis, Iasonas; Agapitos, Antonis; Kesisoglou, Stilianos; Panagiotou, Apostolos; Saoulidou, Niki; Stiliaris, Efstathios; Aslanoglou, Xenofon; Evangelou, Ioannis; Flouris, Giannis; Foudas, Costas; Kokkas, Panagiotis; Manthos, Nikolaos; Papadopoulos, Ioannis; Paradas, Evangelos; Bencze, Gyorgy; Hajdu, Csaba; Hidas, Pàl; Horvath, Dezso; Sikler, Ferenc; Veszpremi, Viktor; Vesztergombi, Gyorgy; Zsigmond, Anna Julia; Beni, Noemi; Czellar, Sandor; Karancsi, János; Molnar, Jozsef; Palinkas, Jozsef; Szillasi, Zoltan; Makovec, Alajos; Raics, Peter; Trocsanyi, Zoltan Laszlo; Ujvari, Balazs; Swain, Sanjay Kumar; Beri, Suman Bala; Bhatnagar, Vipin; Gupta, Ruchi; Bhawandeep, Bhawandeep; Kalsi, Amandeep Kaur; Kaur, Manjit; Kumar, Ramandeep; Mittal, Monika; Nishu, Nishu; Singh, Jasbir; Kumar, Ashok; Kumar, Arun; Ahuja, Sudha; Bhardwaj, Ashutosh; Choudhary, Brajesh C; Kumar, Ajay; Malhotra, Shivali; Naimuddin, Md; Ranjan, Kirti; Sharma, Varun; Banerjee, Sunanda; Bhattacharya, Satyaki; Chatterjee, Kalyanmoy; Dutta, Suchandra; Gomber, Bhawna; Jain, Sandhya; Jain, Shilpi; Khurana, Raman; Modak, Atanu; Mukherjee, Swagata; Roy, Debarati; Sarkar, Subir; Sharan, Manoj; Abdulsalam, Abdulla; Dutta, Dipanwita; Kailas, Swaminathan; Kumar, Vineet; Mohanty, Ajit Kumar; Pant, Lalit Mohan; Shukla, Prashant; Topkar, Anita; Aziz, Tariq; Banerjee, Sudeshna; Bhowmik, Sandeep; Chatterjee, Rajdeep Mohan; Dewanjee, Ram Krishna; Dugad, Shashikant; Ganguly, Sanmay; Ghosh, Saranya; Guchait, Monoranjan; Gurtu, Atul; Kole, Gouranga; Kumar, Sanjeev; Maity, Manas; Majumder, Gobinda; Mazumdar, Kajari; Mohanty, Gagan Bihari; Parida, Bibhuti; Sudhakar, Katta; Wickramage, Nadeesha; Bakhshiansohi, Hamed; Behnamian, Hadi; Etesami, Seyed Mohsen; Fahim, Ali; Goldouzian, Reza; Khakzad, Mohsen; Mohammadi Najafabadi, Mojtaba; Naseri, Mohsen; Paktinat Mehdiabadi, Saeid; Rezaei Hosseinabadi, Ferdos; Safarzadeh, Batool; Zeinali, Maryam; Felcini, Marta; Grunewald, Martin; Abbrescia, Marcello; Calabria, Cesare; Chhibra, Simranjit Singh; Colaleo, Anna; Creanza, Donato; De Filippis, Nicola; De Palma, Mauro; Fiore, Luigi; Iaselli, Giuseppe; Maggi, Giorgio; Maggi, Marcello; My, Salvatore; Nuzzo, Salvatore; Pompili, Alexis; Pugliese, Gabriella; Radogna, Raffaella; Selvaggi, Giovanna; Sharma, Archana; Silvestris, Lucia; Venditti, Rosamaria; Abbiendi, Giovanni; Benvenuti, Alberto; Bonacorsi, Daniele; Braibant-Giacomelli, Sylvie; Brigliadori, Luca; Campanini, Renato; Capiluppi, Paolo; Castro, Andrea; Cavallo, Francesca Romana; Codispoti, Giuseppe; Cuffiani, Marco; Dallavalle, Gaetano-Marco; Fabbri, Fabrizio; Fanfani, Alessandra; Fasanella, Daniele; Giacomelli, Paolo; Grandi, Claudio; Guiducci, Luigi; Marcellini, Stefano; Masetti, Gianni; Montanari, Alessandro; Navarria, Francesco; Perrotta, Andrea; Primavera, Federica; Rossi, Antonio; Rovelli, Tiziano; Siroli, Gian Piero; Tosi, Nicolò; Travaglini, Riccardo; Albergo, Sebastiano; Cappello, Gigi; Chiorboli, Massimiliano; Costa, Salvatore; Giordano, Ferdinando; Potenza, Renato; Tricomi, Alessia; Tuve, Cristina; Barbagli, Giuseppe; Ciulli, Vitaliano; Civinini, Carlo; D'Alessandro, Raffaello; Focardi, Ettore; Gallo, Elisabetta; Gonzi, Sandro; Gori, Valentina; Lenzi, Piergiulio; Meschini, Marco; Paoletti, Simone; Sguazzoni, Giacomo; Tropiano, Antonio; Benussi, Luigi; Bianco, Stefano; Fabbri, Franco; Piccolo, Davide; Ferretti, Roberta; Ferro, Fabrizio; Lo Vetere, Maurizio; Robutti, Enrico; Tosi, Silvano; Dinardo, Mauro Emanuele; Fiorendi, Sara; Gennai, Simone; Gerosa, Raffaele; Ghezzi, Alessio; Govoni, Pietro; Lucchini, Marco Toliman; Malvezzi, Sandra; Manzoni, Riccardo Andrea; Martelli, Arabella; Marzocchi, Badder; Menasce, Dario; Moroni, Luigi; Paganoni, Marco; Pedrini, Daniele; Ragazzi, Stefano; Redaelli, Nicola; Tabarelli de Fatis, Tommaso; Buontempo, Salvatore; Cavallo, Nicola; Di Guida, Salvatore; Fabozzi, Francesco; Iorio, Alberto Orso Maria; Lista, Luca; Meola, Sabino; Merola, Mario; Paolucci, Pierluigi; Azzi, Patrizia; Bacchetta, Nicola; Biasotto, Massimo; Bisello, Dario; Branca, Antonio; Carlin, Roberto; Checchia, Paolo; Dall'Osso, Martino; Dorigo, Tommaso; Dosselli, Umberto; Galanti, Mario; Gasparini, Fabrizio; Gasparini, Ugo; Giubilato, Piero; Gonella, Franco; Gozzelino, Andrea; Kanishchev, Konstantin; Lacaprara, Stefano; Margoni, Martino; Montecassiano, Fabio; Pazzini, Jacopo; Pozzobon, Nicola; Ronchese, Paolo; Tosi, Mia; Vanini, Sara; Ventura, Sandro; Zucchetta, Alberto; Gabusi, Michele; Ratti, Sergio P; Re, Valerio; Riccardi, Cristina; Salvini, Paola; Vitulo, Paolo; Biasini, Maurizio; Bilei, Gian Mario; Ciangottini, Diego; Fanò, Livio; Lariccia, Paolo; Mantovani, Giancarlo; Menichelli, Mauro; Saha, Anirban; Santocchia, Attilio; Spiezia, Aniello; Androsov, Konstantin; Azzurri, Paolo; Bagliesi, Giuseppe; Bernardini, Jacopo; Boccali, Tommaso; Broccolo, Giuseppe; Castaldi, Rino; Ciocci, Maria Agnese; Dell'Orso, Roberto; Donato, Silvio; Fiori, Francesco; Foà, Lorenzo; Giassi, Alessandro; Grippo, Maria Teresa; Ligabue, Franco; Lomtadze, Teimuraz; Martini, Luca; Messineo, Alberto; Moon, Chang-Seong; Palla, Fabrizio; Rizzi, Andrea; Savoy-Navarro, Aurore; Serban, Alin Titus; Spagnolo, Paolo; Squillacioti, Paola; Tenchini, Roberto; Tonelli, Guido; Venturi, Andrea; Verdini, Piero Giorgio; Vernieri, Caterina; Barone, Luciano; Cavallari, Francesca; D'imperio, Giulia; Del Re, Daniele; Diemoz, Marcella; Jorda, Clara; Longo, Egidio; Margaroli, Fabrizio; Meridiani, Paolo; Micheli, Francesco; Nourbakhsh, Shervin; Organtini, Giovanni; Paramatti, Riccardo; Rahatlou, Shahram; Rovelli, Chiara; Santanastasio, Francesco; Soffi, Livia; Traczyk, Piotr; Amapane, Nicola; Arcidiacono, Roberta; Argiro, Stefano; Arneodo, Michele; Bellan, Riccardo; Biino, Cristina; Cartiglia, Nicolo; Casasso, Stefano; Costa, Marco; Degano, Alessandro; Demaria, Natale; Finco, Linda; Mariotti, Chiara; Maselli, Silvia; Migliore, Ernesto; Monaco, Vincenzo; Musich, Marco; Obertino, Maria Margherita; Ortona, Giacomo; Pacher, Luca; Pastrone, Nadia; Pelliccioni, Mario; Pinna Angioni, Gian Luca; Potenza, Alberto; Romero, Alessandra; Ruspa, Marta; Sacchi, Roberto; Solano, Ada; Staiano, Amedeo; Tamponi, Umberto; Belforte, Stefano; Candelise, Vieri; Casarsa, Massimo; Cossutti, Fabio; Della Ricca, Giuseppe; Gobbo, Benigno; La Licata, Chiara; Marone, Matteo; Schizzi, Andrea; Umer, Tomo; Zanetti, Anna; Chang, Sunghyun; Kropivnitskaya, Anna; Nam, Soon-Kwon; Kim, Dong Hee; Kim, Gui Nyun; Kim, Min Suk; Kong, Dae Jung; Lee, Sangeun; Oh, Young Do; Park, Hyangkyu; Sakharov, Alexandre; Son, Dong-Chul; Kim, Tae Jeong; Kim, Jae Yool; Song, Sanghyeon; Choi, Suyong; Gyun, Dooyeon; Hong, Byung-Sik; Jo, Mihee; Kim, Hyunchul; Kim, Yongsun; Lee, Byounghoon; Lee, Kyong Sei; Park, Sung Keun; Roh, Youn; Choi, Minkyoo; Kim, Ji Hyun; Park, Inkyu; Ryu, Geonmo; Ryu, Min Sang; Choi, Young-Il; Choi, Young Kyu; Goh, Junghwan; Kim, Donghyun; Kwon, Eunhyang; Lee, Jongseok; Seo, Hyunkwan; Yu, Intae; Juodagalvis, Andrius; Komaragiri, Jyothsna Rani; Md Ali, Mohd Adli Bin; Casimiro Linares, Edgar; Castilla-Valdez, Heriberto; De La Cruz-Burelo, Eduard; Heredia-de La Cruz, Ivan; Hernandez-Almada, Alberto; Lopez-Fernandez, Ricardo; Sánchez Hernández, Alberto; Carrillo Moreno, Salvador; Vazquez Valencia, Fabiola; Pedraza, Isabel; Salazar Ibarguen, Humberto Antonio; Morelos Pineda, Antonio; Krofcheck, David; Butler, Philip H; Reucroft, Steve; Ahmad, Ashfaq; Ahmad, Muhammad; Hassan, Qamar; Hoorani, Hafeez R; Khan, Wajid Ali; Khurshid, Taimoor; Shoaib, Muhammad; Bialkowska, Helena; Bluj, Michal; Boimska, Bożena; Frueboes, Tomasz; Górski, Maciej; Kazana, Malgorzata; Nawrocki, Krzysztof; Romanowska-Rybinska, Katarzyna; Szleper, Michal; Zalewski, Piotr; Brona, Grzegorz; Bunkowski, Karol; Cwiok, Mikolaj; Dominik, Wojciech; Doroba, Krzysztof; Kalinowski, Artur; Konecki, Marcin; Krolikowski, Jan; Misiura, Maciej; Olszewski, Michał; Wolszczak, Weronika; Bargassa, Pedrame; Beirão Da Cruz E Silva, Cristóvão; Faccioli, Pietro; Ferreira Parracho, Pedro Guilherme; Gallinaro, Michele; Lloret Iglesias, Lara; Nguyen, Federico; Rodrigues Antunes, Joao; Seixas, Joao; Varela, Joao; Vischia, Pietro; Afanasiev, Serguei; Bunin, Pavel; Gavrilenko, Mikhail; Golutvin, Igor; Gorbunov, Ilya; Kamenev, Alexey; Karjavin, Vladimir; Konoplyanikov, Viktor; Lanev, Alexander; Malakhov, Alexander; Matveev, Viktor; Moisenz, Petr; Palichik, Vladimir; Perelygin, Victor; Shmatov, Sergey; Skatchkov, Nikolai; Smirnov, Vitaly; Zarubin, Anatoli; Golovtsov, Victor; Ivanov, Yury; Kim, Victor; Levchenko, Petr; Murzin, Victor; Oreshkin, Vadim; Smirnov, Igor; Sulimov, Valentin; Uvarov, Lev; Vavilov, Sergey; Vorobyev, Alexey; Vorobyev, Andrey; Andreev, Yuri; Dermenev, Alexander; Gninenko, Sergei; Golubev, Nikolai; Kirsanov, Mikhail; Krasnikov, Nikolai; Pashenkov, Anatoli; Tlisov, Danila; Toropin, Alexander; Epshteyn, Vladimir; Gavrilov, Vladimir; Lychkovskaya, Natalia; Popov, Vladimir; Pozdnyakov, Ivan; Safronov, Grigory; Semenov, Sergey; Spiridonov, Alexander; Stolin, Viatcheslav; Vlasov, Evgueni; Zhokin, Alexander; Andreev, Vladimir; Azarkin, Maksim; Dremin, Igor; Kirakosyan, Martin; Leonidov, Andrey; Mesyats, Gennady; Rusakov, Sergey V; Vinogradov, Alexey; Belyaev, Andrey; Boos, Edouard; Dubinin, Mikhail; Dudko, Lev; Ershov, Alexander; Gribushin, Andrey; Klyukhin, Vyacheslav; Kodolova, Olga; Lokhtin, Igor; Obraztsov, Stepan; Petrushanko, Sergey; Savrin, Viktor; Snigirev, Alexander; Azhgirey, Igor; Bayshev, Igor; Bitioukov, Sergei; Kachanov, Vassili; Kalinin, Alexey; Konstantinov, Dmitri; Krychkine, Victor; Petrov, Vladimir; Ryutin, Roman; Sobol, Andrei; Tourtchanovitch, Leonid; Troshin, Sergey; Tyurin, Nikolay; Uzunian, Andrey; Volkov, Alexey; Adzic, Petar; Ekmedzic, Marko; Milosevic, Jovan; Rekovic, Vladimir; Alcaraz Maestre, Juan; Battilana, Carlo; Calvo, Enrique; Cerrada, Marcos; Chamizo Llatas, Maria; Colino, Nicanor; De La Cruz, Begona; Delgado Peris, Antonio; Domínguez Vázquez, Daniel; Escalante Del Valle, Alberto; Fernandez Bedoya, Cristina; Fernández Ramos, Juan Pablo; Flix, Jose; Fouz, Maria Cruz; Garcia-Abia, Pablo; Gonzalez Lopez, Oscar; Goy Lopez, Silvia; Hernandez, Jose M; Josa, Maria Isabel; Navarro De Martino, Eduardo; Pérez-Calero Yzquierdo, Antonio María; Puerta Pelayo, Jesus; Quintario Olmeda, Adrián; Redondo, Ignacio; Romero, Luciano; Senghi Soares, Mara; Albajar, Carmen; de Trocóniz, Jorge F; Missiroli, Marino; Moran, Dermot; Brun, Hugues; Cuevas, Javier; Fernandez Menendez, Javier; Folgueras, Santiago; Gonzalez Caballero, Isidro; Brochero Cifuentes, Javier Andres; Cabrillo, Iban Jose; Calderon, Alicia; Duarte Campderros, Jordi; Fernandez, Marcos; Gomez, Gervasio; Graziano, Alberto; Lopez Virto, Amparo; Marco, Jesus; Marco, Rafael; Martinez Rivero, Celso; Matorras, Francisco; Munoz Sanchez, Francisca Javiela; Piedra Gomez, Jonatan; Rodrigo, Teresa; Rodríguez-Marrero, Ana Yaiza; Ruiz-Jimeno, Alberto; Scodellaro, Luca; Vila, Ivan; Vilar Cortabitarte, Rocio; Abbaneo, Duccio; Auffray, Etiennette; Auzinger, Georg; Bachtis, Michail; Baillon, Paul; Ball, Austin; Barney, David; Benaglia, Andrea; Bendavid, Joshua; Benhabib, Lamia; Benitez, Jose F; Bernet, Colin; Bloch, Philippe; Bocci, Andrea; Bonato, Alessio; Bondu, Olivier; Botta, Cristina; Breuker, Horst; Camporesi, Tiziano; Cerminara, Gianluca; Colafranceschi, Stefano; D'Alfonso, Mariarosaria; D'Enterria, David; Dabrowski, Anne; David Tinoco Mendes, Andre; De Guio, Federico; De Roeck, Albert; De Visscher, Simon; Di Marco, Emanuele; Dobson, Marc; Dordevic, Milos; Dupont-Sagorin, Niels; Elliott-Peisert, Anna; Eugster, Jürg; Franzoni, Giovanni; Funk, Wolfgang; Gigi, Dominique; Gill, Karl; Giordano, Domenico; Girone, Maria; Glege, Frank; Guida, Roberto; Gundacker, Stefan; Guthoff, Moritz; Hammer, Josef; Hansen, Magnus; Harris, Philip; Hegeman, Jeroen; Innocente, Vincenzo; Janot, Patrick; Kousouris, Konstantinos; Krajczar, Krisztian; Lecoq, Paul; Lourenco, Carlos; Magini, Nicolo; Malgeri, Luca; Mannelli, Marcello; Marrouche, Jad; Masetti, Lorenzo; Meijers, Frans; Mersi, Stefano; Meschi, Emilio; Moortgat, Filip; Morovic, Srecko; Mulders, Martijn; Musella, Pasquale; Orsini, Luciano; Pape, Luc; Perez, Emmanuelle; Perrozzi, Luca; Petrilli, Achille; Petrucciani, Giovanni; Pfeiffer, Andreas; Pierini, Maurizio; Pimiä, Martti; Piparo, Danilo; Plagge, Michael; Racz, Attila; Rolandi, Gigi; Rovere, Marco; Sakulin, Hannes; Schäfer, Christoph; Schwick, Christoph; Sharma, Archana; Siegrist, Patrice; Silva, Pedro; Simon, Michal; Sphicas, Paraskevas; Spiga, Daniele; Steggemann, Jan; Stieger, Benjamin; Stoye, Markus; Takahashi, Yuta; Treille, Daniel; Tsirou, Andromachi; Veres, Gabor Istvan; Wardle, Nicholas; Wöhri, Hermine Katharina; Wollny, Heiner; Zeuner, Wolfram Dietrich; Bertl, Willi; Deiters, Konrad; Erdmann, Wolfram; Horisberger, Roland; Ingram, Quentin; Kaestli, Hans-Christian; Kotlinski, Danek; Langenegger, Urs; Renker, Dieter; Rohe, Tilman; Bachmair, Felix; Bäni, Lukas; Bianchini, Lorenzo; Buchmann, Marco-Andrea; Casal, Bruno; Chanon, Nicolas; Dissertori, Günther; Dittmar, Michael; Donegà, Mauro; Dünser, Marc; Eller, Philipp; Grab, Christoph; Hits, Dmitry; Hoss, Jan; Lustermann, Werner; Mangano, Boris; Marini, Andrea Carlo; Martinez Ruiz del Arbol, Pablo; Masciovecchio, Mario; Meister, Daniel; Mohr, Niklas; Nägeli, Christoph; Nessi-Tedaldi, Francesca; Pandolfi, Francesco; Pauss, Felicitas; Peruzzi, Marco; Quittnat, Milena; Rebane, Liis; Rossini, Marco; Starodumov, Andrei; Takahashi, Maiko; Theofilatos, Konstantinos; Wallny, Rainer; Weber, Hannsjoerg Artur; Amsler, Claude; Canelli, Maria Florencia; Chiochia, Vincenzo; De Cosa, Annapaola; Hinzmann, Andreas; Hreus, Tomas; Kilminster, Benjamin; Lange, Clemens; Millan Mejias, Barbara; Ngadiuba, Jennifer; Robmann, Peter; Ronga, Frederic Jean; Taroni, Silvia; Verzetti, Mauro; Yang, Yong; Cardaci, Marco; Chen, Kuan-Hsin; Ferro, Cristina; Kuo, Chia-Ming; Lin, Willis; Lu, Yun-Ju; Volpe, Roberta; Yu, Shin-Shan; Chang, Paoti; Chang, You-Hao; Chang, Yu-Wei; Chao, Yuan; Chen, Kai-Feng; Chen, Po-Hsun; Dietz, Charles; Grundler, Ulysses; Hou, George Wei-Shu; Kao, Kai-Yi; Lei, Yeong-Jyi; Liu, Yueh-Feng; Lu, Rong-Shyang; Majumder, Devdatta; Petrakou, Eleni; Tzeng, Yeng-Ming; Wilken, Rachel; Asavapibhop, Burin; Singh, Gurpreet; Srimanobhas, Norraphat; Suwonjandee, Narumon; Adiguzel, Aytul; Bakirci, Mustafa Numan; Cerci, Salim; Dozen, Candan; Dumanoglu, Isa; Eskut, Eda; Girgis, Semiray; Gokbulut, Gul; Gurpinar, Emine; Hos, Ilknur; Kangal, Evrim Ersin; Kayis Topaksu, Aysel; Onengut, Gulsen; Ozdemir, Kadri; Ozturk, Sertac; Polatoz, Ayse; Sunar Cerci, Deniz; Tali, Bayram; Topakli, Huseyin; Vergili, Mehmet; Akin, Ilina Vasileva; Bilin, Bugra; Bilmis, Selcuk; Gamsizkan, Halil; Karapinar, Guler; Ocalan, Kadir; Sekmen, Sezen; Surat, Ugur Emrah; Yalvac, Metin; Zeyrek, Mehmet; Albayrak, Elif Asli; Gülmez, Erhan; Isildak, Bora; Kaya, Mithat; Kaya, Ozlem; Yetkin, Taylan; Cankocak, Kerem; Vardarli, Fuat Ilkehan; Levchuk, Leonid; Sorokin, Pavel; Brooke, James John; Clement, Emyr; Cussans, David; Flacher, Henning; Goldstein, Joel; Grimes, Mark; Heath, Greg P; Heath, Helen F; Jacob, Jeson; Kreczko, Lukasz; Lucas, Chris; Meng, Zhaoxia; Newbold, Dave M; Paramesvaran, Sudarshan; Poll, Anthony; Senkin, Sergey; Smith, Vincent J; Williams, Thomas; Bell, Ken W; Belyaev, Alexander; Brew, Christopher; Brown, Robert M; Cockerill, David JA; Coughlan, John A; Harder, Kristian; Harper, Sam; Olaiya, Emmanuel; Petyt, David; Shepherd-Themistocleous, Claire; Thea, Alessandro; Tomalin, Ian R; Womersley, William John; Worm, Steven; Baber, Mark; Bainbridge, Robert; Buchmuller, Oliver; Burton, Darren; Colling, David; Cripps, Nicholas; Cutajar, Michael; Dauncey, Paul; Davies, Gavin; Della Negra, Michel; Dunne, Patrick; Ferguson, William; Fulcher, Jonathan; Futyan, David; Gilbert, Andrew; Hall, Geoffrey; Iles, Gregory; Jarvis, Martyn; Karapostoli, Georgia; Kenzie, Matthew; Lane, Rebecca; Lucas, Robyn; Lyons, Louis; Magnan, Anne-Marie; Malik, Sarah; Mathias, Bryn; Nash, Jordan; Nikitenko, Alexander; Pela, Joao; Pesaresi, Mark; Petridis, Konstantinos; Raymond, David Mark; Rogerson, Samuel; Rose, Andrew; Seez, Christopher; Sharp, Peter; Tapper, Alexander; Vazquez Acosta, Monica; Virdee, Tejinder; Zenz, Seth Conrad; Cole, Joanne; Hobson, Peter R; Khan, Akram; Kyberd, Paul; Leggat, Duncan; Leslie, Dawn; Martin, William; Reid, Ivan; Symonds, Philip; Teodorescu, Liliana; Turner, Mark; Dittmann, Jay; Hatakeyama, Kenichi; Kasmi, Azeddine; Liu, Hongxuan; Scarborough, Tara; Charaf, Otman; Cooper, Seth; Henderson, Conor; Rumerio, Paolo; Avetisyan, Aram; Bose, Tulika; Fantasia, Cory; Lawson, Philip; Richardson, Clint; Rohlf, James; St John, Jason; Sulak, Lawrence; Alimena, Juliette; Berry, Edmund; Bhattacharya, Saptaparna; Christopher, Grant; Cutts, David; Demiragli, Zeynep; Dhingra, Nitish; Ferapontov, Alexey; Garabedian, Alex; Heintz, Ulrich; Kukartsev, Gennadiy; Laird, Edward; Landsberg, Greg; Luk, Michael; Narain, Meenakshi; Segala, Michael; Sinthuprasith, Tutanon; Speer, Thomas; Swanson, Joshua; Breedon, Richard; Breto, Guillermo; Calderon De La Barca Sanchez, Manuel; Chauhan, Sushil; Chertok, Maxwell; Conway, John; Conway, Rylan; Cox, Peter Timothy; Erbacher, Robin; Gardner, Michael; Ko, Winston; Lander, Richard; Miceli, Tia; Mulhearn, Michael; Pellett, Dave; Pilot, Justin; Ricci-Tam, Francesca; Searle, Matthew; Shalhout, Shalhout; Smith, John; Squires, Michael; Stolp, Dustin; Tripathi, Mani; Wilbur, Scott; Yohay, Rachel; Cousins, Robert; Everaerts, Pieter; Farrell, Chris; Hauser, Jay; Ignatenko, Mikhail; Rakness, Gregory; Takasugi, Eric; Valuev, Vyacheslav; Weber, Matthias; Burt, Kira; Clare, Robert; Ellison, John Anthony; Gary, J William; Hanson, Gail; Heilman, Jesse; Ivova Rikova, Mirena; Jandir, Pawandeep; Kennedy, Elizabeth; Lacroix, Florent; Long, Owen Rosser; Luthra, Arun; Malberti, Martina; Olmedo Negrete, Manuel; Shrinivas, Amithabh; Sumowidagdo, Suharyo; Wimpenny, Stephen; Branson, James G; Cerati, Giuseppe Benedetto; Cittolin, Sergio; D'Agnolo, Raffaele Tito; Holzner, André; Kelley, Ryan; Klein, Daniel; Letts, James; Macneill, Ian; Olivito, Dominick; Padhi, Sanjay; Palmer, Christopher; Pieri, Marco; Sani, Matteo; Sharma, Vivek; Simon, Sean; Sudano, Elizabeth; Tadel, Matevz; Tu, Yanjun; Vartak, Adish; Welke, Charles; Würthwein, Frank; Yagil, Avraham; Barge, Derek; Bradmiller-Feld, John; Campagnari, Claudio; Danielson, Thomas; Dishaw, Adam; Dutta, Valentina; Flowers, Kristen; Franco Sevilla, Manuel; Geffert, Paul; George, Christopher; Golf, Frank; Gouskos, Loukas; Incandela, Joe; Justus, Christopher; Mccoll, Nickolas; Richman, Jeffrey; Stuart, David; To, Wing; West, Christopher; Yoo, Jaehyeok; Apresyan, Artur; Bornheim, Adolf; Bunn, Julian; Chen, Yi; Duarte, Javier; Mott, Alexander; Newman, Harvey B; Pena, Cristian; Rogan, Christopher; Spiropulu, Maria; Timciuc, Vladlen; Vlimant, Jean-Roch; Wilkinson, Richard; Xie, Si; Zhu, Ren-Yuan; Azzolini, Virginia; Calamba, Aristotle; Carlson, Benjamin; Ferguson, Thomas; Iiyama, Yutaro; Paulini, Manfred; Russ, James; Vogel, Helmut; Vorobiev, Igor; Cumalat, John Perry; Ford, William T; Gaz, Alessandro; Krohn, Michael; Luiggi Lopez, Eduardo; Nauenberg, Uriel; Smith, James; Stenson, Kevin; Ulmer, Keith; Wagner, Stephen Robert; Alexander, James; Chatterjee, Avishek; Chaves, Jorge; Chu, Jennifer; Dittmer, Susan; Eggert, Nicholas; Mirman, Nathan; Nicolas Kaufman, Gala; Patterson, Juliet Ritchie; Ryd, Anders; Salvati, Emmanuele; Skinnari, Louise; Sun, Werner; Teo, Wee Don; Thom, Julia; Thompson, Joshua; Tucker, Jordan; Weng, Yao; Winstrom, Lucas; Wittich, Peter; Winn, Dave; Abdullin, Salavat; Albrow, Michael; Anderson, Jacob; Apollinari, Giorgio; Bauerdick, Lothar AT; Beretvas, Andrew; Berryhill, Jeffrey; Bhat, Pushpalatha C; Bolla, Gino; Burkett, Kevin; Butler, Joel Nathan; Cheung, Harry; Chlebana, Frank; Cihangir, Selcuk; Elvira, Victor Daniel; Fisk, Ian; Freeman, Jim; Gao, Yanyan; Gottschalk, Erik; Gray, Lindsey; Green, Dan; Grünendahl, Stefan; Gutsche, Oliver; Hanlon, Jim; Hare, Daryl; Harris, Robert M; Hirschauer, James; Hooberman, Benjamin; Jindariani, Sergo; Johnson, Marvin; Joshi, Umesh; Kaadze, Ketino; Klima, Boaz; Kreis, Benjamin; Kwan, Simon; Linacre, Jacob; Lincoln, Don; Lipton, Ron; Liu, Tiehui; Lykken, Joseph; Maeshima, Kaori; Marraffino, John Michael; Martinez Outschoorn, Verena Ingrid; Maruyama, Sho; Mason, David; McBride, Patricia; Merkel, Petra; Mishra, Kalanand; Mrenna, Stephen; Musienko, Yuri; Nahn, Steve; Newman-Holmes, Catherine; O'Dell, Vivian; Prokofyev, Oleg; Sexton-Kennedy, Elizabeth; Sharma, Seema; Soha, Aron; Spalding, William J; Spiegel, Leonard; Taylor, Lucas; Tkaczyk, Slawek; Tran, Nhan Viet; Uplegger, Lorenzo; Vaandering, Eric Wayne; Vidal, Richard; Whitbeck, Andrew; Whitmore, Juliana; Yang, Fan; Acosta, Darin; Avery, Paul; Bortignon, Pierluigi; Bourilkov, Dimitri; Carver, Matthew; Cheng, Tongguang; Curry, David; Das, Souvik; De Gruttola, Michele; Di Giovanni, Gian Piero; Field, Richard D; Fisher, Matthew; Furic, Ivan-Kresimir; Hugon, Justin; Konigsberg, Jacobo; Korytov, Andrey; Kypreos, Theodore; Low, Jia Fu; Matchev, Konstantin; Milenovic, Predrag; Mitselmakher, Guenakh; Muniz, Lana; Rinkevicius, Aurelijus; Shchutska, Lesya; Snowball, Matthew; Sperka, David; Yelton, John; Zakaria, Mohammed; Hewamanage, Samantha; Linn, Stephan; 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Sarangi, Tapas; Savin, Alexander; Smith, Wesley H; Taylor, Devin; Verwilligen, Piet; Vuosalo, Carl; Woods, Nathaniel 2015-06-26 The inclusive jet cross section for proton-proton collisions at a centre-of-mass energy of 7$~\\mathrm{TeV}$was measured by the CMS Collaboration at the LHC with data corresponding to an integrated luminosity of 5.0$~\\mathrm{fb}^{-1}$. The measurement covers a phase space up to 2$~\\mathrm{TeV}$in jet transverse momentum and 2.5 in absolute jet rapidity. The statistical precision of these data leads to stringent constraints on the parton distribution functions of the proton. The data provide important input for the gluon density at high fractions of the proton momentum and for the strong coupling constant at large energy scales. Using predictions from perturbative quantum chromodynamics at next-to-leading order, complemented with electroweak corrections, the constraining power of these data is investigated and the strong coupling constant at the Z boson mass$M_{\\mathrm{Z}}$is determined to be$\\alpha_S(M_{\\mathrm{Z}}) = 0.1185 \\pm 0.0019\\,(\\mathrm{exp})\\,^{+0.0060}_{-0.0037}\\,(\\mathrm{theo})$, which is in a... 13. Measurement of the inclusive 3-jet production differential cross section in proton-proton collisions at 7 TeV and determination of the strong coupling constant in the TeV range CERN Document Server Khachatryan, Vardan; Tumasyan, Armen; Adam, Wolfgang; Bergauer, Thomas; Dragicevic, Marko; Erö, Janos; Fabjan, Christian; Friedl, Markus; Fruehwirth, Rudolf; Ghete, Vasile Mihai; Hartl, Christian; Hörmann, Natascha; Hrubec, Josef; Jeitler, Manfred; Kiesenhofer, Wolfgang; Knünz, Valentin; Krammer, Manfred; Krätschmer, Ilse; Liko, Dietrich; Mikulec, Ivan; Rabady, Dinyar; Rahbaran, Babak; Rohringer, Herbert; Schöfbeck, Robert; Strauss, Josef; Taurok, Anton; Treberer-Treberspurg, Wolfgang; Waltenberger, Wolfgang; Wulz, Claudia-Elisabeth; Mossolov, Vladimir; Shumeiko, Nikolai; Suarez Gonzalez, Juan; Alderweireldt, Sara; Bansal, Monika; Bansal, Sunil; Cornelis, Tom; De Wolf, Eddi A; Janssen, Xavier; Knutsson, Albert; Luyckx, Sten; Ochesanu, Silvia; Rougny, Romain; Van De Klundert, Merijn; Van Haevermaet, Hans; 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Schettler, Hannes; Schleper, Peter; Schlieckau, Eike; Schmidt, Alexander; Seidel, Markus; Sola, Valentina; Stadie, Hartmut; Steinbrück, Georg; Troendle, Daniel; Usai, Emanuele; Vanelderen, Lukas; Vanhoefer, Annika; Barth, Christian; Baus, Colin; Berger, Joram; Böser, Christian; Butz, Erik; Chwalek, Thorsten; De Boer, Wim; Descroix, Alexis; Dierlamm, Alexander; Feindt, Michael; Frensch, Felix; Giffels, Manuel; Hartmann, Frank; Hauth, Thomas; Husemann, Ulrich; Katkov, Igor; Kornmayer, Andreas; Kuznetsova, Ekaterina; Lobelle Pardo, Patricia; Mozer, Matthias Ulrich; Müller, Thomas; Nürnberg, Andreas; Quast, Gunter; Rabbertz, Klaus; Ratnikov, Fedor; Röcker, Steffen; Sieber, Georg; Simonis, Hans-Jürgen; Stober, Fred-Markus Helmut; Ulrich, Ralf; Wagner-Kuhr, Jeannine; Wayand, Stefan; Weiler, Thomas; Wolf, Roger; Anagnostou, Georgios; Daskalakis, Georgios; Geralis, Theodoros; Giakoumopoulou, Viktoria Athina; Kyriakis, Aristotelis; Loukas, Demetrios; Markou, Athanasios; Markou, Christos; Psallidas, Andreas; Topsis-Giotis, Iasonas; Agapitos, Antonis; Kesisoglou, Stilianos; Panagiotou, Apostolos; Saoulidou, Niki; Stiliaris, Efstathios; Aslanoglou, Xenofon; Evangelou, Ioannis; Flouris, Giannis; Foudas, Costas; Kokkas, Panagiotis; Manthos, Nikolaos; Papadopoulos, Ioannis; Paradas, Evangelos; Bencze, Gyorgy; Hajdu, Csaba; Hidas, Pàl; Horvath, Dezso; Sikler, Ferenc; Veszpremi, Viktor; Vesztergombi, Gyorgy; Zsigmond, Anna Julia; Beni, Noemi; Czellar, Sandor; Karancsi, János; Molnar, Jozsef; Palinkas, Jozsef; Szillasi, Zoltan; Raics, Peter; Trocsanyi, Zoltan Laszlo; Ujvari, Balazs; Swain, Sanjay Kumar; Beri, Suman Bala; Bhatnagar, Vipin; Gupta, Ruchi; Bhawandeep, Bhawandeep; Kalsi, Amandeep Kaur; Kaur, Manjit; Kumar, Ramandeep; Mittal, Monika; Nishu, Nishu; Singh, Jasbir; Kumar, Ashok; Kumar, Arun; Ahuja, Sudha; Bhardwaj, Ashutosh; Choudhary, Brajesh C; Kumar, Ajay; Malhotra, Shivali; Naimuddin, Md; Ranjan, Kirti; Sharma, Varun; Banerjee, Sunanda; Bhattacharya, Satyaki; Chatterjee, Kalyanmoy; Dutta, Suchandra; Gomber, Bhawna; Jain, Sandhya; Jain, Shilpi; Khurana, Raman; Modak, Atanu; Mukherjee, Swagata; Roy, Debarati; Sarkar, Subir; Sharan, Manoj; Abdulsalam, Abdulla; Dutta, Dipanwita; Kailas, Swaminathan; Kumar, Vineet; Mohanty, Ajit Kumar; Pant, Lalit Mohan; Shukla, Prashant; Topkar, Anita; Aziz, Tariq; Banerjee, Sudeshna; Bhowmik, Sandeep; Chatterjee, Rajdeep Mohan; Dewanjee, Ram Krishna; Dugad, Shashikant; Ganguly, Sanmay; Ghosh, Saranya; Guchait, Monoranjan; Gurtu, Atul; Kole, Gouranga; Kumar, Sanjeev; Maity, Manas; Majumder, Gobinda; Mazumdar, Kajari; Mohanty, Gagan Bihari; Parida, Bibhuti; Sudhakar, Katta; Wickramage, Nadeesha; Bakhshiansohi, Hamed; Behnamian, Hadi; Etesami, Seyed Mohsen; Fahim, Ali; Goldouzian, Reza; Khakzad, Mohsen; Mohammadi Najafabadi, Mojtaba; Naseri, Mohsen; Paktinat Mehdiabadi, Saeid; Rezaei Hosseinabadi, Ferdos; Safarzadeh, Batool; Zeinali, Maryam; Felcini, Marta; Grunewald, Martin; Abbrescia, Marcello; Barbone, Lucia; Calabria, Cesare; Chhibra, Simranjit Singh; Colaleo, Anna; Creanza, Donato; De Filippis, Nicola; De Palma, Mauro; Fiore, Luigi; Iaselli, Giuseppe; Maggi, Giorgio; Maggi, Marcello; My, Salvatore; Nuzzo, Salvatore; Pompili, Alexis; Pugliese, Gabriella; Radogna, Raffaella; Selvaggi, Giovanna; Silvestris, Lucia; Venditti, Rosamaria; Zito, Giuseppe; Abbiendi, Giovanni; Benvenuti, Alberto; Bonacorsi, Daniele; Braibant-Giacomelli, Sylvie; Brigliadori, Luca; Campanini, Renato; Capiluppi, Paolo; Castro, Andrea; Cavallo, Francesca Romana; Codispoti, Giuseppe; Cuffiani, Marco; Dallavalle, Gaetano-Marco; Fabbri, Fabrizio; Fanfani, Alessandra; Fasanella, Daniele; Giacomelli, Paolo; Grandi, Claudio; Guiducci, Luigi; Marcellini, Stefano; Masetti, Gianni; Montanari, Alessandro; Navarria, Francesco; Perrotta, Andrea; Primavera, Federica; Rossi, Antonio; Rovelli, Tiziano; Siroli, Gian Piero; Tosi, Nicolò; Travaglini, Riccardo; Albergo, Sebastiano; Cappello, Gigi; Chiorboli, Massimiliano; Costa, Salvatore; Giordano, Ferdinando; Potenza, Renato; Tricomi, Alessia; Tuve, Cristina; Barbagli, Giuseppe; Ciulli, Vitaliano; Civinini, Carlo; D'Alessandro, Raffaello; Focardi, Ettore; Gallo, Elisabetta; Gonzi, Sandro; Gori, Valentina; Lenzi, Piergiulio; Meschini, Marco; Paoletti, Simone; Sguazzoni, Giacomo; Tropiano, Antonio; Benussi, Luigi; Bianco, Stefano; Fabbri, Franco; Piccolo, Davide; Ferretti, Roberta; Ferro, Fabrizio; Lo Vetere, Maurizio; Robutti, Enrico; Tosi, Silvano; Dinardo, Mauro Emanuele; Fiorendi, Sara; Gennai, Simone; Gerosa, Raffaele; Ghezzi, Alessio; Govoni, Pietro; Lucchini, Marco Toliman; Malvezzi, Sandra; Manzoni, Riccardo Andrea; Martelli, Arabella; Marzocchi, Badder; Menasce, Dario; Moroni, Luigi; Paganoni, Marco; Pedrini, Daniele; Ragazzi, Stefano; Redaelli, Nicola; Tabarelli de Fatis, Tommaso; Buontempo, Salvatore; Cavallo, Nicola; Di Guida, Salvatore; Fabozzi, Francesco; Iorio, Alberto Orso Maria; Lista, Luca; Meola, Sabino; Merola, Mario; Paolucci, Pierluigi; Azzi, Patrizia; Bacchetta, Nicola; Bisello, Dario; Branca, Antonio; Carlin, Roberto; Checchia, Paolo; Dall'Osso, Martino; Dorigo, Tommaso; Galanti, Mario; Gasparini, Fabrizio; Gasparini, Ugo; Giubilato, Piero; Gozzelino, Andrea; Kanishchev, Konstantin; Lacaprara, Stefano; Margoni, Martino; Meneguzzo, Anna Teresa; Pazzini, Jacopo; Pozzobon, Nicola; Ronchese, Paolo; Simonetto, Franco; Torassa, Ezio; Tosi, Mia; Vanini, Sara; Ventura, Sandro; Zotto, Pierluigi; Zucchetta, Alberto; Gabusi, Michele; Ratti, Sergio P; Re, Valerio; Riccardi, Cristina; Salvini, Paola; Vitulo, Paolo; Biasini, Maurizio; Bilei, Gian Mario; Ciangottini, Diego; Fanò, Livio; Lariccia, Paolo; Mantovani, Giancarlo; Menichelli, Mauro; Saha, Anirban; Santocchia, Attilio; Spiezia, Aniello; Androsov, Konstantin; Azzurri, Paolo; Bagliesi, Giuseppe; Bernardini, Jacopo; Boccali, Tommaso; Broccolo, Giuseppe; Castaldi, Rino; Ciocci, Maria Agnese; Dell'Orso, Roberto; Donato, Silvio; Fiori, Francesco; Foà, Lorenzo; Giassi, Alessandro; Grippo, Maria Teresa; Ligabue, Franco; Lomtadze, Teimuraz; Martini, Luca; Messineo, Alberto; Moon, Chang-Seong; Palla, Fabrizio; Rizzi, Andrea; Savoy-Navarro, Aurore; Serban, Alin Titus; Spagnolo, Paolo; Squillacioti, Paola; Tenchini, Roberto; Tonelli, Guido; Venturi, Andrea; Verdini, Piero Giorgio; Vernieri, Caterina; Barone, Luciano; Cavallari, Francesca; D'imperio, Giulia; Del Re, Daniele; Diemoz, Marcella; Grassi, Marco; Jorda, Clara; Longo, Egidio; Margaroli, Fabrizio; Meridiani, Paolo; Micheli, Francesco; Nourbakhsh, Shervin; Organtini, Giovanni; Paramatti, Riccardo; Rahatlou, Shahram; Rovelli, Chiara; Santanastasio, Francesco; Soffi, Livia; Traczyk, Piotr; Amapane, Nicola; Arcidiacono, Roberta; Argiro, Stefano; Arneodo, Michele; Bellan, Riccardo; Biino, Cristina; Cartiglia, Nicolo; Casasso, Stefano; Costa, Marco; Degano, Alessandro; Demaria, Natale; Finco, Linda; Mariotti, Chiara; Maselli, Silvia; Migliore, Ernesto; Monaco, Vincenzo; Musich, Marco; Obertino, Maria Margherita; Ortona, Giacomo; Pacher, Luca; Pastrone, Nadia; Pelliccioni, Mario; Pinna Angioni, Gian Luca; Potenza, Alberto; Romero, Alessandra; Ruspa, Marta; Sacchi, Roberto; Solano, Ada; Staiano, Amedeo; Tamponi, Umberto; Belforte, Stefano; Candelise, Vieri; Casarsa, Massimo; Cossutti, Fabio; Della Ricca, Giuseppe; Gobbo, Benigno; La Licata, Chiara; Marone, Matteo; Schizzi, Andrea; Umer, Tomo; Zanetti, Anna; Chang, Sunghyun; Kropivnitskaya, Anna; Nam, Soon-Kwon; Kim, Dong Hee; Kim, Gui Nyun; Kim, Min Suk; Kong, Dae Jung; Lee, Sangeun; Oh, Young Do; Park, Hyangkyu; Sakharov, Alexandre; Son, Dong-Chul; Kim, Tae Jeong; Kim, Jae Yool; Song, Sanghyeon; Choi, Suyong; Gyun, Dooyeon; Hong, Byung-Sik; Jo, Mihee; Kim, Hyunchul; Kim, Yongsun; Lee, Byounghoon; Lee, Kyong Sei; Park, Sung Keun; Roh, Youn; Choi, Minkyoo; Kim, Ji Hyun; Park, Inkyu; Ryu, Geonmo; Ryu, Min Sang; Choi, Young-Il; Choi, Young Kyu; Goh, Junghwan; Kim, Donghyun; Kwon, Eunhyang; Lee, Jongseok; Seo, Hyunkwan; Yu, Intae; Juodagalvis, Andrius; Komaragiri, Jyothsna Rani; Md Ali, Mohd Adli Bin; Castilla-Valdez, Heriberto; De La Cruz-Burelo, Eduard; Heredia-de La Cruz, Ivan; Hernandez-Almada, Alberto; Lopez-Fernandez, Ricardo; Sánchez Hernández, Alberto; Carrillo Moreno, Salvador; Vazquez Valencia, Fabiola; Pedraza, Isabel; Salazar Ibarguen, Humberto Antonio; Casimiro Linares, Edgar; Morelos Pineda, Antonio; Krofcheck, David; Butler, Philip H; Reucroft, Steve; Ahmad, Ashfaq; Ahmad, Muhammad; Hassan, Qamar; Hoorani, Hafeez R; Khalid, Shoaib; Khan, Wajid Ali; Khurshid, Taimoor; Shah, Mehar Ali; Shoaib, Muhammad; Bialkowska, Helena; Bluj, Michal; Boimska, Bożena; Frueboes, Tomasz; Górski, Maciej; Kazana, Malgorzata; Nawrocki, Krzysztof; Romanowska-Rybinska, Katarzyna; Szleper, Michal; Zalewski, Piotr; Brona, Grzegorz; Bunkowski, Karol; Cwiok, Mikolaj; Dominik, Wojciech; Doroba, Krzysztof; Kalinowski, Artur; Konecki, Marcin; Krolikowski, Jan; Misiura, Maciej; Olszewski, Michał; Wolszczak, Weronika; Bargassa, Pedrame; Beirão Da Cruz E Silva, Cristóvão; Faccioli, Pietro; Ferreira Parracho, Pedro Guilherme; Gallinaro, Michele; Lloret Iglesias, Lara; Nguyen, Federico; Rodrigues Antunes, Joao; Seixas, Joao; Varela, Joao; Vischia, Pietro; Afanasiev, Serguei; Bunin, Pavel; Gavrilenko, Mikhail; Golutvin, Igor; Gorbunov, Ilya; Kamenev, Alexey; Karjavin, Vladimir; Konoplyanikov, Viktor; Lanev, Alexander; Malakhov, Alexander; Matveev, Viktor; Moisenz, Petr; Palichik, Vladimir; Perelygin, Victor; Shmatov, Sergey; Skatchkov, Nikolai; Smirnov, Vitaly; Zarubin, Anatoli; Golovtsov, Victor; Ivanov, Yury; Kim, Victor; Levchenko, Petr; Murzin, Victor; Oreshkin, Vadim; Smirnov, Igor; Sulimov, Valentin; Uvarov, Lev; Vavilov, Sergey; Vorobyev, Alexey; Vorobyev, Andrey; Andreev, Yuri; Dermenev, Alexander; Gninenko, Sergei; Golubev, Nikolai; Kirsanov, Mikhail; Krasnikov, Nikolai; Pashenkov, Anatoli; Tlisov, Danila; Toropin, Alexander; Epshteyn, Vladimir; Gavrilov, Vladimir; Lychkovskaya, Natalia; Popov, Vladimir; Safronov, Grigory; Semenov, Sergey; Spiridonov, Alexander; Stolin, Viatcheslav; Vlasov, Evgueni; Zhokin, Alexander; Andreev, Vladimir; Azarkin, Maksim; Dremin, Igor; Kirakosyan, Martin; Leonidov, Andrey; Mesyats, Gennady; Rusakov, Sergey V; Vinogradov, Alexey; Belyaev, Andrey; Boos, Edouard; Dubinin, Mikhail; Dudko, Lev; Ershov, Alexander; Gribushin, Andrey; Klyukhin, Vyacheslav; Kodolova, Olga; Lokhtin, Igor; Obraztsov, Stepan; Petrushanko, Sergey; Savrin, Viktor; Snigirev, Alexander; Azhgirey, Igor; Bayshev, Igor; Bitioukov, Sergei; Kachanov, Vassili; Kalinin, Alexey; Konstantinov, Dmitri; Krychkine, Victor; Petrov, Vladimir; Ryutin, Roman; Sobol, Andrei; Tourtchanovitch, Leonid; Troshin, Sergey; Tyurin, Nikolay; Uzunian, Andrey; Volkov, Alexey; Adzic, Petar; Ekmedzic, Marko; Milosevic, Jovan; Rekovic, Vladimir; Alcaraz Maestre, Juan; Battilana, Carlo; Calvo, Enrique; Cerrada, Marcos; Chamizo Llatas, Maria; Colino, Nicanor; De La Cruz, Begona; Delgado Peris, Antonio; Domínguez Vázquez, Daniel; Escalante Del Valle, Alberto; Fernandez Bedoya, Cristina; Fernández Ramos, Juan Pablo; Flix, Jose; Fouz, Maria Cruz; Garcia-Abia, Pablo; Gonzalez Lopez, Oscar; Goy Lopez, Silvia; Hernandez, Jose M; Josa, Maria Isabel; Navarro De Martino, Eduardo; Pérez-Calero Yzquierdo, Antonio María; Puerta Pelayo, Jesus; Quintario Olmeda, Adrián; Redondo, Ignacio; Romero, Luciano; Senghi Soares, Mara; Albajar, Carmen; de Trocóniz, Jorge F; Missiroli, Marino; Moran, Dermot; Brun, Hugues; Cuevas, Javier; Fernandez Menendez, Javier; Folgueras, Santiago; Gonzalez Caballero, Isidro; Brochero Cifuentes, Javier Andres; Cabrillo, Iban Jose; Calderon, Alicia; Duarte Campderros, Jordi; Fernandez, Marcos; Gomez, Gervasio; Graziano, Alberto; Lopez Virto, Amparo; Marco, Jesus; Marco, Rafael; Martinez Rivero, Celso; Matorras, Francisco; Munoz Sanchez, Francisca Javiela; Piedra Gomez, Jonatan; Rodrigo, Teresa; Rodríguez-Marrero, Ana Yaiza; Ruiz-Jimeno, Alberto; Scodellaro, Luca; Vila, Ivan; Vilar Cortabitarte, Rocio; Abbaneo, Duccio; Auffray, Etiennette; Auzinger, Georg; Bachtis, Michail; Baillon, Paul; Ball, Austin; Barney, David; Benaglia, Andrea; Bendavid, Joshua; Benhabib, Lamia; Benitez, Jose F; Bernet, Colin; Bianchi, Giovanni; Bloch, Philippe; Bocci, Andrea; Bonato, Alessio; Bondu, Olivier; Botta, Cristina; Breuker, Horst; Camporesi, Tiziano; Cerminara, Gianluca; Colafranceschi, Stefano; D'Alfonso, Mariarosaria; D'Enterria, David; Dabrowski, Anne; David Tinoco Mendes, Andre; De Guio, Federico; De Roeck, Albert; De Visscher, Simon; Di Marco, Emanuele; Dobson, Marc; Dordevic, Milos; Dorney, Brian; Dupont-Sagorin, Niels; Elliott-Peisert, Anna; Eugster, Jürg; Franzoni, Giovanni; Funk, Wolfgang; Gigi, Dominique; Gill, Karl; Giordano, Domenico; Girone, Maria; Glege, Frank; Guida, Roberto; Gundacker, Stefan; Guthoff, Moritz; Hammer, Josef; Hansen, Magnus; Harris, Philip; Hegeman, Jeroen; Innocente, Vincenzo; Janot, Patrick; Kousouris, Konstantinos; Krajczar, Krisztian; Lecoq, Paul; Lourenco, Carlos; Magini, Nicolo; Malgeri, Luca; Mannelli, Marcello; Marrouche, Jad; Masetti, Lorenzo; Meijers, Frans; Mersi, Stefano; Meschi, Emilio; Moortgat, Filip; Morovic, Srecko; Mulders, Martijn; Musella, Pasquale; Orsini, Luciano; Pape, Luc; Perez, Emmanuelle; Perrozzi, Luca; Petrilli, Achille; Petrucciani, Giovanni; Pfeiffer, Andreas; Pierini, Maurizio; Pimiä, Martti; Piparo, Danilo; Plagge, Michael; Racz, Attila; Rolandi, Gigi; Rovere, Marco; Sakulin, Hannes; Schäfer, Christoph; Schwick, Christoph; Sharma, Archana; Siegrist, Patrice; Silva, Pedro; Simon, Michal; Sphicas, Paraskevas; Spiga, Daniele; Steggemann, Jan; Stieger, Benjamin; Stoye, Markus; Takahashi, Yuta; Treille, Daniel; Tsirou, Andromachi; Veres, Gabor Istvan; Wardle, Nicholas; Wöhri, Hermine Katharina; Wollny, Heiner; Zeuner, Wolfram Dietrich; Bertl, Willi; Deiters, Konrad; Erdmann, Wolfram; Horisberger, Roland; Ingram, Quentin; Kaestli, Hans-Christian; Kotlinski, Danek; Langenegger, Urs; Renker, Dieter; Rohe, Tilman; Bachmair, Felix; Bäni, Lukas; Bianchini, Lorenzo; Buchmann, Marco-Andrea; Casal, Bruno; Chanon, Nicolas; Dissertori, Günther; Dittmar, Michael; Donegà, Mauro; Dünser, Marc; Eller, Philipp; Grab, Christoph; Hits, Dmitry; Hoss, Jan; Lustermann, Werner; Mangano, Boris; Marini, Andrea Carlo; Martinez Ruiz del Arbol, Pablo; Masciovecchio, Mario; Meister, Daniel; Mohr, Niklas; Nägeli, Christoph; Nessi-Tedaldi, Francesca; Pandolfi, Francesco; Pauss, Felicitas; Peruzzi, Marco; Quittnat, Milena; Rebane, Liis; Rossini, Marco; Starodumov, Andrei; Takahashi, Maiko; Theofilatos, Konstantinos; Wallny, Rainer; Weber, Hannsjoerg Artur; Amsler, Claude; Canelli, Maria Florencia; Chiochia, Vincenzo; De Cosa, Annapaola; Hinzmann, Andreas; Hreus, Tomas; Kilminster, Benjamin; Lange, Clemens; Millan Mejias, Barbara; Ngadiuba, Jennifer; Robmann, Peter; Ronga, Frederic Jean; Taroni, Silvia; Verzetti, Mauro; Yang, Yong; Cardaci, Marco; Chen, Kuan-Hsin; Ferro, Cristina; Kuo, Chia-Ming; Lin, Willis; Lu, Yun-Ju; Volpe, Roberta; Yu, Shin-Shan; Chang, Paoti; Chang, You-Hao; Chang, Yu-Wei; Chao, Yuan; Chen, Kai-Feng; Chen, Po-Hsun; Dietz, Charles; Grundler, Ulysses; Hou, George Wei-Shu; Kao, Kai-Yi; Lei, Yeong-Jyi; Liu, Yueh-Feng; Lu, Rong-Shyang; Majumder, Devdatta; Petrakou, Eleni; Tzeng, Yeng-Ming; Wilken, Rachel; Asavapibhop, Burin; Singh, Gurpreet; Srimanobhas, Norraphat; Suwonjandee, Narumon; Adiguzel, Aytul; Bakirci, Mustafa Numan; Cerci, Salim; Dozen, Candan; Dumanoglu, Isa; Eskut, Eda; Girgis, Semiray; Gokbulut, Gul; Gurpinar, Emine; Hos, Ilknur; Kangal, Evrim Ersin; Kayis Topaksu, Aysel; Onengut, Gulsen; Ozdemir, Kadri; Ozturk, Sertac; Polatoz, Ayse; Sunar Cerci, Deniz; Tali, Bayram; Topakli, Huseyin; Vergili, Mehmet; Akin, Ilina Vasileva; Bilin, Bugra; Bilmis, Selcuk; Gamsizkan, Halil; Karapinar, Guler; Ocalan, Kadir; Sekmen, Sezen; Surat, Ugur Emrah; Yalvac, Metin; Zeyrek, Mehmet; Gülmez, Erhan; Isildak, Bora; Kaya, Mithat; Kaya, Ozlem; Cankocak, Kerem; Vardarlı, Fuat Ilkehan; Levchuk, Leonid; Sorokin, Pavel; Brooke, James John; Clement, Emyr; Cussans, David; Flacher, Henning; Goldstein, Joel; Grimes, Mark; Heath, Greg P; Heath, Helen F; Jacob, Jeson; Kreczko, Lukasz; Lucas, Chris; Meng, Zhaoxia; Newbold, Dave M; Paramesvaran, Sudarshan; Poll, Anthony; Senkin, Sergey; Smith, Vincent J; Williams, Thomas; Bell, Ken W; Belyaev, Alexander; Brew, Christopher; Brown, Robert M; Cockerill, David JA; Coughlan, John A; Harder, Kristian; Harper, Sam; Olaiya, Emmanuel; Petyt, David; Shepherd-Themistocleous, Claire; Thea, Alessandro; Tomalin, Ian R; Womersley, William John; Worm, Steven; Baber, Mark; Bainbridge, Robert; Buchmuller, Oliver; Burton, Darren; Colling, David; Cripps, Nicholas; Cutajar, Michael; Dauncey, Paul; Davies, Gavin; Della Negra, Michel; Dunne, Patrick; Ferguson, William; Fulcher, Jonathan; Futyan, David; Gilbert, Andrew; Hall, Geoffrey; Iles, Gregory; Jarvis, Martyn; Karapostoli, Georgia; Kenzie, Matthew; Lane, Rebecca; Lucas, Robyn; Lyons, Louis; Magnan, Anne-Marie; Malik, Sarah; Mathias, Bryn; Nash, Jordan; Nikitenko, Alexander; Pela, Joao; Pesaresi, Mark; Petridis, Konstantinos; Raymond, David Mark; Rogerson, Samuel; Rose, Andrew; Seez, Christopher; Sharp, Peter; Tapper, Alexander; Vazquez Acosta, Monica; Virdee, Tejinder; Zenz, Seth Conrad; Cole, Joanne; Hobson, Peter R; Khan, Akram; Kyberd, Paul; Leggat, Duncan; Leslie, Dawn; Martin, William; Reid, Ivan; Symonds, Philip; Teodorescu, Liliana; Turner, Mark; Dittmann, Jay; Hatakeyama, Kenichi; Kasmi, Azeddine; Liu, Hongxuan; Scarborough, Tara; Charaf, Otman; Cooper, Seth; Henderson, Conor; Rumerio, Paolo; Avetisyan, Aram; Bose, Tulika; Fantasia, Cory; Lawson, Philip; Richardson, Clint; Rohlf, James; St John, Jason; Sulak, Lawrence; Alimena, Juliette; Berry, Edmund; Bhattacharya, Saptaparna; Christopher, Grant; Cutts, David; Demiragli, Zeynep; Dhingra, Nitish; Ferapontov, Alexey; Garabedian, Alex; Heintz, Ulrich; Kukartsev, Gennadiy; Laird, Edward; Landsberg, Greg; Luk, Michael; Narain, Meenakshi; Segala, Michael; Sinthuprasith, Tutanon; Speer, Thomas; Swanson, Joshua; Breedon, Richard; Breto, Guillermo; Calderon De La Barca Sanchez, Manuel; Chauhan, Sushil; Chertok, Maxwell; Conway, John; Conway, Rylan; Cox, Peter Timothy; Erbacher, Robin; Gardner, Michael; Ko, Winston; Lander, Richard; Miceli, Tia; Mulhearn, Michael; Pellett, Dave; Pilot, Justin; Ricci-Tam, Francesca; Searle, Matthew; Shalhout, Shalhout; Smith, John; Squires, Michael; Stolp, Dustin; Tripathi, Mani; Wilbur, Scott; Yohay, Rachel; Cousins, Robert; Everaerts, Pieter; Farrell, Chris; Hauser, Jay; Ignatenko, Mikhail; Rakness, Gregory; Takasugi, Eric; Valuev, Vyacheslav; Weber, Matthias; Burt, Kira; Clare, Robert; Ellison, John Anthony; Gary, J William; Hanson, Gail; Heilman, Jesse; Ivova Rikova, Mirena; Jandir, Pawandeep; Kennedy, Elizabeth; Lacroix, Florent; Long, Owen Rosser; Luthra, Arun; Malberti, Martina; Nguyen, Harold; Olmedo Negrete, Manuel; Shrinivas, Amithabh; Sumowidagdo, Suharyo; Wimpenny, Stephen; Andrews, Warren; Branson, James G; Cerati, Giuseppe Benedetto; Cittolin, Sergio; D'Agnolo, Raffaele Tito; Evans, David; Holzner, André; Kelley, Ryan; Klein, Daniel; Lebourgeois, Matthew; Letts, James; Macneill, Ian; Olivito, Dominick; Padhi, Sanjay; Palmer, Christopher; Pieri, Marco; Sani, Matteo; Sharma, Vivek; Simon, Sean; Sudano, Elizabeth; Tadel, Matevz; Tu, Yanjun; Vartak, Adish; Welke, Charles; Würthwein, Frank; Yagil, Avraham; Barge, Derek; Bradmiller-Feld, John; Campagnari, Claudio; Danielson, Thomas; Dishaw, Adam; Flowers, Kristen; Franco Sevilla, Manuel; Geffert, Paul; George, Christopher; Golf, Frank; Gouskos, Loukas; Incandela, Joe; Justus, Christopher; Mccoll, Nickolas; Richman, Jeffrey; Stuart, David; To, Wing; West, Christopher; Yoo, Jaehyeok; Apresyan, Artur; Bornheim, Adolf; Bunn, Julian; Chen, Yi; Duarte, Javier; Mott, Alexander; Newman, Harvey B; Pena, Cristian; Rogan, Christopher; Spiropulu, Maria; Timciuc, Vladlen; Vlimant, Jean-Roch; Wilkinson, Richard; Xie, Si; Zhu, Ren-Yuan; Azzolini, Virginia; Calamba, Aristotle; Carlson, Benjamin; Ferguson, Thomas; Iiyama, Yutaro; Paulini, Manfred; Russ, James; Vogel, Helmut; Vorobiev, Igor; Cumalat, John Perry; Ford, William T; Gaz, Alessandro; Luiggi Lopez, Eduardo; Nauenberg, Uriel; Smith, James; Stenson, Kevin; Ulmer, Keith; Wagner, Stephen Robert; Alexander, James; Chatterjee, Avishek; Chu, Jennifer; Dittmer, Susan; Eggert, Nicholas; Mirman, Nathan; Nicolas Kaufman, Gala; Patterson, Juliet Ritchie; Ryd, Anders; Salvati, Emmanuele; Skinnari, Louise; Sun, Werner; Teo, Wee Don; Thom, Julia; Thompson, Joshua; Tucker, Jordan; Weng, Yao; Winstrom, Lucas; Wittich, Peter; Winn, Dave; Abdullin, Salavat; Albrow, Michael; Anderson, Jacob; Apollinari, Giorgio; Bauerdick, Lothar AT; Beretvas, Andrew; Berryhill, Jeffrey; Bhat, Pushpalatha C; Bolla, Gino; Burkett, Kevin; Butler, Joel Nathan; Cheung, Harry; Chlebana, Frank; Cihangir, Selcuk; Elvira, Victor Daniel; Fisk, Ian; Freeman, Jim; Gao, Yanyan; Gottschalk, Erik; Gray, Lindsey; Green, Dan; Grünendahl, Stefan; Gutsche, Oliver; Hanlon, Jim; Hare, Daryl; Harris, Robert M; Hirschauer, James; Hooberman, Benjamin; Jindariani, Sergo; Johnson, Marvin; Joshi, Umesh; Kaadze, Ketino; Klima, Boaz; Kreis, Benjamin; Kwan, Simon; Linacre, Jacob; Lincoln, Don; Lipton, Ron; Liu, Tiehui; Lykken, Joseph; Maeshima, Kaori; Marraffino, John Michael; Martinez Outschoorn, Verena Ingrid; Maruyama, Sho; Mason, David; McBride, Patricia; Merkel, Petra; Mishra, Kalanand; Mrenna, Stephen; Musienko, Yuri; Nahn, Steve; Newman-Holmes, Catherine; O'Dell, Vivian; Prokofyev, Oleg; Sexton-Kennedy, Elizabeth; Sharma, Seema; Soha, Aron; Spalding, William J; Spiegel, Leonard; Taylor, Lucas; Tkaczyk, Slawek; Tran, Nhan Viet; Uplegger, Lorenzo; Vaandering, Eric Wayne; Vidal, Richard; Whitbeck, Andrew; Whitmore, Juliana; Yang, Fan; Acosta, Darin; Avery, Paul; Bortignon, Pierluigi; Bourilkov, Dimitri; Carver, Matthew; Cheng, Tongguang; Curry, David; Das, Souvik; De Gruttola, Michele; Di Giovanni, Gian Piero; Field, Richard D; Fisher, Matthew; Furic, Ivan-Kresimir; Hugon, Justin; Konigsberg, Jacobo; Korytov, Andrey; Kypreos, Theodore; Low, Jia Fu; Matchev, Konstantin; Milenovic, Predrag; Mitselmakher, Guenakh; Muniz, Lana; Rinkevicius, Aurelijus; Shchutska, Lesya; Snowball, Matthew; Sperka, David; Yelton, John; Zakaria, Mohammed; Hewamanage, Samantha; Linn, Stephan; Markowitz, Pete; Martinez, German; Rodriguez, Jorge Luis; Adams, Todd; Askew, Andrew; Bochenek, Joseph; Diamond, Brendan; Haas, Jeff; Hagopian, Sharon; Hagopian, Vasken; Johnson, Kurtis F; Prosper, Harrison; Veeraraghavan, Venkatesh; Weinberg, Marc; Baarmand, Marc M; Hohlmann, Marcus; Kalakhety, Himali; Yumiceva, Francisco; Adams, Mark Raymond; Apanasevich, Leonard; Bazterra, Victor Eduardo; Berry, Douglas; Betts, Russell Richard; Bucinskaite, Inga; Cavanaugh, Richard; Evdokimov, Olga; Gauthier, Lucie; Gerber, Cecilia Elena; Hofman, David Jonathan; Khalatyan, Samvel; Kurt, Pelin; Moon, Dong Ho; O'Brien, Christine; Silkworth, Christopher; Turner, Paul; Varelas, Nikos; Albayrak, Elif Asli; Bilki, Burak; Clarida, Warren; Dilsiz, Kamuran; Duru, Firdevs; Haytmyradov, Maksat; Merlo, Jean-Pierre; Mermerkaya, Hamit; Mestvirishvili, Alexi; Moeller, Anthony; Nachtman, Jane; Ogul, Hasan; Onel, Yasar; Ozok, Ferhat; Penzo, Aldo; Rahmat, Rahmat; Sen, Sercan; Tan, Ping; Tiras, Emrah; Wetzel, James; Yetkin, Taylan; Yi, Kai; Barnett, Bruce Arnold; Blumenfeld, Barry; Bolognesi, Sara; Fehling, David; Gritsan, Andrei; Maksimovic, Petar; Martin, Christopher; Swartz, Morris; Baringer, Philip; Bean, Alice; Benelli, Gabriele; Bruner, Christopher; Kenny III, Raymond Patrick; Malek, Magdalena; Murray, Michael; Noonan, Daniel; Sanders, Stephen; Sekaric, Jadranka; Stringer, Robert; Wang, Quan; Wood, Jeffrey Scott; Barfuss, Anne-Fleur; Chakaberia, Irakli; Ivanov, Andrew; Khalil, Sadia; Makouski, Mikhail; Maravin, Yurii; Saini, Lovedeep Kaur; Shrestha, Shruti; Skhirtladze, Nikoloz; Svintradze, Irakli; Gronberg, Jeffrey; Lange, David; Rebassoo, Finn; Wright, Douglas; Baden, Drew; Belloni, Alberto; Calvert, Brian; Eno, Sarah Catherine; Gomez, Jaime; Hadley, Nicholas John; Kellogg, Richard G; Kolberg, Ted; Lu, Ying; Marionneau, Matthieu; Mignerey, Alice; Pedro, Kevin; Skuja, Andris; Tonjes, Marguerite; Tonwar, Suresh C; Apyan, Aram; Barbieri, Richard; Bauer, Gerry; Busza, Wit; Cali, Ivan Amos; Chan, Matthew; Di Matteo, Leonardo; Dutta, Valentina; Gomez Ceballos, Guillelmo; Goncharov, Maxim; Gulhan, Doga; Klute, Markus; Lai, Yue Shi; Lee, Yen-Jie; Levin, Andrew; Luckey, Paul David; Ma, Teng; Paus, Christoph; Ralph, Duncan; Roland, Christof; Roland, Gunther; Stephans, George; Stöckli, Fabian; Sumorok, Konstanty; Velicanu, Dragos; Veverka, Jan; Wyslouch, Bolek; Yang, Mingming; Zanetti, Marco; Zhukova, Victoria; Dahmes, Bryan; Gude, Alexander; Kao, Shih-Chuan; Klapoetke, Kevin; Kubota, Yuichi; Mans, Jeremy; Pastika, Nathaniel; Rusack, Roger; Singovsky, Alexander; Tambe, Norbert; Turkewitz, Jared; Acosta, John Gabriel; Oliveros, Sandra; Avdeeva, Ekaterina; Bloom, Kenneth; Bose, Suvadeep; Claes, Daniel R; Dominguez, Aaron; Gonzalez Suarez, Rebeca; Keller, Jason; Knowlton, Dan; Kravchenko, Ilya; Lazo-Flores, Jose; Malik, Sudhir; Meier, Frank; Snow, Gregory R; Zvada, Marian; Dolen, James; Godshalk, Andrew; Iashvili, Ia; Kharchilava, Avto; Kumar, Ashish; Rappoccio, Salvatore; Alverson, George; Barberis, Emanuela; Baumgartel, Darin; Chasco, Matthew; Haley, Joseph; Massironi, Andrea; Morse, David Michael; Nash, David; Orimoto, Toyoko; Trocino, Daniele; Wang, Ren-Jie; Wood, Darien; Zhang, Jinzhong; Hahn, Kristan Allan; Kubik, Andrew; Mucia, Nicholas; Odell, Nathaniel; Pollack, Brian; Pozdnyakov, Andrey; Schmitt, Michael Henry; Stoynev, Stoyan; Sung, Kevin; Velasco, Mayda; Won, Steven; Brinkerhoff, Andrew; Chan, Kwok Ming; Drozdetskiy, Alexey; Hildreth, Michael; Jessop, Colin; Karmgard, Daniel John; Kellams, Nathan; Lannon, Kevin; Luo, Wuming; Lynch, Sean; Marinelli, Nancy; Pearson, Tessa; Planer, Michael; Ruchti, Randy; Valls, Nil; Wayne, Mitchell; Wolf, Matthias; Woodard, Anna; Antonelli, Louis; Brinson, Jessica; Bylsma, Ben; Durkin, Lloyd Stanley; Flowers, Sean; Hill, Christopher; Hughes, Richard; Kotov, Khristian; Ling, Ta-Yung; Puigh, Darren; Rodenburg, Marissa; Smith, Geoffrey; Winer, Brian L; Wolfe, Homer; Wulsin, Howard Wells; Driga, Olga; Elmer, Peter; Hebda, Philip; Hunt, Adam; Koay, Sue Ann; Lujan, Paul; Marlow, Daniel; Medvedeva, Tatiana; Mooney, Michael; Olsen, James; Piroué, Pierre; Quan, Xiaohang; Saka, Halil; Stickland, David; Tully, Christopher; Werner, Jeremy Scott; Zuranski, Andrzej; Brownson, Eric; Mendez, Hector; Ramirez Vargas, Juan Eduardo; Barnes, Virgil E; Benedetti, Daniele; Bortoletto, Daniela; De Mattia, Marco; Gutay, Laszlo; Hu, Zhen; Jha, Manoj; Jones, Matthew; Jung, Kurt; Kress, Matthew; Leonardo, Nuno; Lopes Pegna, David; Maroussov, Vassili; Miller, David Harry; Neumeister, Norbert; Radburn-Smith, Benjamin Charles; Shi, Xin; Shipsey, Ian; Silvers, David; Svyatkovskiy, Alexey; Wang, Fuqiang; Xie, Wei; Xu, Lingshan; Yoo, Hwi Dong; Zablocki, Jakub; Zheng, Yu; Parashar, Neeti; Stupak, John; Adair, Antony; Akgun, Bora; Ecklund, Karl Matthew; Geurts, Frank JM; Li, Wei; Michlin, Benjamin; Padley, Brian Paul; Redjimi, Radia; Roberts, Jay; Zabel, James; Betchart, Burton; Bodek, Arie; Covarelli, Roberto; de Barbaro, Pawel; Demina, Regina; Eshaq, Yossof; Ferbel, Thomas; Garcia-Bellido, Aran; Goldenzweig, Pablo; Han, Jiyeon; Harel, Amnon; Khukhunaishvili, Aleko; Petrillo, Gianluca; Vishnevskiy, Dmitry; Ciesielski, Robert; Demortier, Luc; Goulianos, Konstantin; Lungu, Gheorghe; Mesropian, Christina; Arora, Sanjay; Barker, Anthony; Chou, John Paul; Contreras-Campana, Christian; Contreras-Campana, Emmanuel; Duggan, Daniel; Ferencek, Dinko; Gershtein, Yuri; Gray, Richard; Halkiadakis, Eva; Hidas, Dean; Kaplan, Steven; Lath, Amitabh; Panwalkar, Shruti; Park, Michael; Patel, Rishi; Salur, Sevil; Schnetzer, Steve; Somalwar, Sunil; Stone, Robert; Thomas, Scott; Thomassen, Peter; Walker, Matthew; Rose, Keith; Spanier, Stefan; York, Andrew; Bouhali, Othmane; Castaneda Hernandez, Alfredo; Eusebi, Ricardo; Flanagan, Will; Gilmore, Jason; Kamon, Teruki; Khotilovich, Vadim; Krutelyov, Vyacheslav; Montalvo, Roy; Osipenkov, Ilya; Pakhotin, Yuriy; Perloff, Alexx; Roe, Jeffrey; Rose, Anthony; Safonov, Alexei; Sakuma, Tai; Suarez, Indara; Tatarinov, Aysen; Akchurin, Nural; Cowden, Christopher; Damgov, Jordan; Dragoiu, Cosmin; Dudero, Phillip Russell; Faulkner, James; Kovitanggoon, Kittikul; Kunori, Shuichi; Lee, Sung Won; Libeiro, Terence; Volobouev, Igor; Appelt, Eric; Delannoy, Andrés G; Greene, Senta; Gurrola, Alfredo; Johns, Willard; Maguire, Charles; Mao, Yaxian; Melo, Andrew; Sharma, Monika; Sheldon, Paul; Snook, Benjamin; Tuo, Shengquan; Velkovska, Julia; Arenton, Michael Wayne; Boutle, Sarah; Cox, Bradley; Francis, Brian; Goodell, Joseph; Hirosky, Robert; Ledovskoy, Alexander; Li, Hengne; Lin, Chuanzhe; Neu, Christopher; Wood, John; Clarke, Christopher; Harr, Robert; Karchin, Paul Edmund; Kottachchi Kankanamge Don, Chamath; Lamichhane, Pramod; Sturdy, Jared; Belknap, Donald; Carlsmith, Duncan; Cepeda, Maria; Dasu, Sridhara; Dodd, Laura; Duric, Senka; Friis, Evan; Hall-Wilton, Richard; Herndon, Matthew; Hervé, Alain; Klabbers, Pamela; Lanaro, Armando; Lazaridis, Christos; Levine, Aaron; Loveless, Richard; Mohapatra, Ajit; Ojalvo, Isabel; Perry, Thomas; Pierro, Giuseppe Antonio; Polese, Giovanni; Ross, Ian; Sarangi, Tapas; Savin, Alexander; Smith, Wesley H; Taylor, Devin; Verwilligen, Piet; Vuosalo, Carl; Woods, Nathaniel 2015-05-01 This paper presents a measurement of the inclusive 3-jet production differential cross section at a proton-proton centre-of-mass energy of 7 TeV using data corresponding to an integrated luminosity of 5 fb$^{-1}$collected with the CMS detector. The analysis is based on the three jets with the highest transverse momenta. The cross section is measured as a function of the invariant mass of the three jets in a range of 445-3270 GeV and in two bins of the maximum rapidity of the jets up to a value of 2. A comparison between the measurement and the prediction from perturbative QCD at next-to-leading order is performed. Within uncertainties, data and theory are in agreement. The sensitivity of the observable to parameters of the theory such as the parton distribution functions of the proton and the strong coupling constant$\\alpha_S$is studied. A fit to all data points with 3-jet masses larger than 664 GeV gives a value of the strong coupling constant of$\\alpha_S(M_\\mathrm{Z})$= 0.1171$\\pm$0.0013 (exp)$^{+0...
14. Tiny cause with huge impact: polar instability through strong magneto-electric-elastic coupling in bulk EuTiO3.
Science.gov (United States)
Reuvekamp, Patrick; Caslin, Kevin; Guguchia, Zurab; Keller, Hugo; Kremer, Reinhard K; Simon, Arndt; Köhler, Jürgen; Bussmann-Holder, Annette
2015-07-08
EuTiO3 exhibits strong magneto-electric coupling at the onset of antiferromagnetic order below TN = 5.7 K. The dielectric permittivity drops at TN by 7% and recovers to normal values with increasing magnetic field. This effect is shown to stem from tiny lattice effects as seen in magnetostriction data which directly affect the soft optic mode and its polarizability coordinate. By combining experimental results with theory we show that marginal changes in the lattice parameter of the order of 0.01% have a more than 1000% effect on the transverse optic soft mode of ETO and thus easily induce a ferroelectric instability.
15. On the impact of the elastic-plastic flow upon the process of destruction of the solenoid in a super strong pulsed magnetic field
Science.gov (United States)
Krivosheev, S. I.; Magazinov, S. G.; Alekseev, D. I.
2018-01-01
At interaction of super strong magnetic fields with a solenoid material, a specific mode of the material flow forms. To describe this process, magnetohydrodynamic approximation is traditionally used. The formation of plastic shock-waves in material in a rapidly increasing pressure of 100 GPa/μs, can significantly alter the distribution of the physical parameters in the medium and affect the flow modes. In this paper, an analysis of supporting results of numerical simulations in comparison with available experimental data is presented.
16. Temperature-dependent elastic properties of Ti1−xAlxN alloys
International Nuclear Information System (INIS)
Shulumba, Nina; Hellman, Olle; Rogström, Lina; Raza, Zamaan; Tasnádi, Ferenc; Odén, Magnus; Abrikosov, Igor A.
2015-01-01
Ti 1−x Al x N is a technologically important alloy that undergoes a process of high temperature age-hardening that is strongly influenced by its elastic properties. We have performed first principles calculations of the elastic constants and anisotropy using the symmetry imposed force constant temperature dependent effective potential method, which include lattice vibrations and therefore the effects of temperature, including thermal expansion and intrinsic anharmonicity. These are compared with in situ high temperature x-ray diffraction measurements of the lattice parameter. We show that anharmonic effects are crucial to the recovery of finite temperature elasticity. The effects of thermal expansion and intrinsic anharmonicity on the elastic constants are of the same order, and cannot be considered separately. Furthermore, the effect of thermal expansion on elastic constants is such that the volume change induced by zero point motion has a significant effect. For TiAlN, the elastic constants soften non-uniformly with temperature: C 11 decreases substantially when the temperature increases for all compositions, resulting in an increased anisotropy. These findings suggest that an increased Al content and annealing at higher temperatures will result in a harder alloy
17. Extraction of the strong coupling constant from the measurement of inclusive multijet event cross-sections in pp collisions at center of mass energy of 8 TeV
CERN Document Server
Kaur, Anterpreet
2017-01-01
A measurement of inclusive multijet event cross sections is presented from proton-proton collisions recorded at 8 TeV with the CMS detector and corresponding to an integrated luminosity of 19.7/fb. Jets are reconstructed with the anti-kt clustering algorithm for a jet size parameter R=0.7 in a phase space region ranging up to jet transverse momenta pT of 2.0 TeV and rapidity of IyI lt 2.5. The inclusive 2-jet and 3-jet event cross sections are measured as a function of the average pT of the two leading jets. The results are compared to fixed-order predictions of perturbative QCD and to simulations using various Monte Carlo event generators including parton showers, hadronisation, and multiparton interactions. A fit of the strong coupling constant is performed with the ratio of the 3-jet over 2-jet event cross section.
18. Nuclear constants
International Nuclear Information System (INIS)
Foos, J.
1999-01-01
This paper is written in two tables. The first one describes the different particles (bosons and fermions). The second one gives the isotopes nuclear constants of the different elements, for Z = 1 to 56. (A.L.B.)
19. Nuclear constants
International Nuclear Information System (INIS)
Foos, J.
2000-01-01
This paper is written in two tables. The first one describes the different particles (bosons and fermions). The second one gives the isotopes nuclear constants of the different elements, for Z = 56 to 68. (A.L.B.)
20. Nuclear constants
International Nuclear Information System (INIS)
Foos, J.
1998-01-01
This paper is made of two tables. The first table describes the different particles (bosons and fermions) while the second one gives the nuclear constants of isotopes from the different elements with Z = 1 to 25. (J.S.)
1. Nuclear constants
International Nuclear Information System (INIS)
Foos, J.
1999-01-01
This paper is written in two tables. The first one describes the different particles (bosons and fermions). The second one gives the isotopes nuclear constants of the different elements, for Z = 56 to 68. (A.L.B.)
2. Lagrange multipliers in elastic-plastic torsion problem for nonlinear monotone operators
Science.gov (United States)
Giuffrè, S.; Maugeri, A.; Puglisi, D.
2015-08-01
The existence of Lagrange multipliers as a Radon measure is ensured for an elastic-plastic torsion problem associated to a nonlinear strictly monotone operator. A regularization of this result, namely the existence of Lp Lagrange multipliers, is obtained under strong monotonicity assumption on the operator. Moreover, the relationships between elastic-plastic torsion problem and the obstacle problem are investigated. Finally, an example of the so-called "Von Mises functions" is provided, namely of solutions of the elastic-plastic torsion problem, associated to nonlinear monotone operators, which are not obtained by means of the obstacle problem in the case f =constant.
3. Elasticity of semiflexible polymers in two dimensions
Science.gov (United States)
Prasad, Ashok; Hori, Yuko; Kondev, Jané
2005-10-01
We study theoretically the entropic elasticity of a semiflexible polymer, such as DNA, confined to two dimensions. Using the worm-like-chain model we obtain an exact analytical expression for the partition function of the polymer pulled at one end with a constant force. The force-extension relation for the polymer is computed in the long chain limit in terms of Mathieu characteristic functions. We also present applications to the interaction between a semiflexible polymer and a nematic field, and derive the nematic order parameter and average extension of the polymer in a strong field.
4. Elasticity of a quantum monolayer solid
DEFF Research Database (Denmark)
Bruch, Ludwig Walter
1992-01-01
A perturbation-theory formulation of the zero-temperature elastic constants is used to verify symmetry relations for a (monolayer) triangluar lattice. A generalization of the Cauchy relation between the two elastic constants of the triangular lattice with central-pair-potential interactions...
5. Determination of the top-quark pole mass and strong coupling constant from the $t\\bar{t}$ production cross section in pp collisions at $\\sqrt{s}$ = 7 TeV
CERN Document Server
2014-01-20
The inclusive cross section for top-quark pair production measured by the CMS experiment in proton-proton collisions at a center-of-mass energy of 7 TeV is compared to the QCD prediction at next-to-next-to-leading order with various parton distribution functions to determine the top-quark pole mass, $m_t^{pole}$, or the strong coupling constant, $\\alpha_S$. With the parton distribution function set NNPDF2.3, a pole mass of 176.7$^{+3.0}_{-2.8}$ GeV is obtained when constraining $\\alpha_S$ at the scale of the Z boson mass, $m_Z$, to the current world average. Alternatively, by constraining $m_t^{pole}$ to the latest average from direct mass measurements, a value of $\\alpha_S(m_Z)$ = 0.1151$^{+0.0028}_{-0.0027}$ is extracted. This is the first determination of $\\alpha_S$ using events from top-quark production.
6. Measurement of transverse energy-energy correlations in multi-jet events in $pp$ collisions at $\\sqrt{s} = 7$ TeV using the ATLAS detector and determination of the strong coupling constant $\\alpha_{\\mathrm{s}}(m_Z)$
CERN Document Server
2015-09-24
High transverse momentum jets produced in pp collisions at a centre of mass energy of 7 TeV are used to measure the transverse energy--energy correlation function and its associated azimuthal asymmetry. The data were recorded with the ATLAS detector at the LHC in the year 2011 and correspond to an integrated luminosity of 158 $\\mathrm{pb}^{-1}$. The selection criteria demand the average transverse momentum of the two leading jets in an event to be larger than 250 GeV. The data at detector level are well described by Monte Carlo event generators. They are unfolded to the particle level and compared with theoretical calculations at next-to-leading-order accuracy. The agreement between data and theory is good and provides a precision test of perturbative Quantum Chromodynamics at large momentum transfers. From this comparison, the strong coupling constant given at the $Z$ boson mass is determined to be $\\alpha_{\\mathrm{s}}(m_Z) = 0.1173 \\pm 0.0010 \\mbox{ (exp.) }^{+0.0065}_{-0.0026} \\mbox{ (theo.)}$.
7. Measurement of transverse energy–energy correlations in multi-jet events in pp collisions at s=7 TeV using the ATLAS detector and determination of the strong coupling constant αs(mZ
Directory of Open Access Journals (Sweden)
2015-11-01
Full Text Available High transverse momentum jets produced in pp collisions at a centre of mass energy of 7 TeV are used to measure the transverse energy–energy correlation function and its associated azimuthal asymmetry. The data were recorded with the ATLAS detector at the LHC in the year 2011 and correspond to an integrated luminosity of 158 pb−1. The selection criteria demand the average transverse momentum of the two leading jets in an event to be larger than 250 GeV. The data at detector level are well described by Monte Carlo event generators. They are unfolded to the particle level and compared with theoretical calculations at next-to-leading-order accuracy. The agreement between data and theory is good and provides a precision test of perturbative Quantum Chromodynamics at large momentum transfers. From this comparison, the strong coupling constant given at the Z boson mass is determined to be αs(mZ=0.1173±0.0010 (exp. −0.0026+0.0065 (theo..
8. Determination of the strong coupling constant α{sub s} from transverse energy-energy correlations in multijet events at √(s) = 8 TeV using the ATLAS detector
Energy Technology Data Exchange (ETDEWEB)
Aaboud, M. [Univ. Mohamed Premier et LPTPM, Oujda (Morocco). Faculte des Sciences; Aad, G. [CPPM, Aix-Marseille Univ. et CNRS/IN2P3, Marseille (France); Abbott, B. [Oklahoma Univ., Norman, OK (United States). Homer L. Dodge Dept. of Physics and Astronomy; Collaboration: ATLAS Collaboration; and others
2017-12-15
Measurements of transverse energy-energy correlations and their associated asymmetries in multi-jet events using the ATLAS detector at the LHC are presented. The data used correspond to √(s) = 8 TeV proton-proton collisions with an integrated luminosity of 20.2 fb{sup -1}. The results are presented in bins of the scalar sum of the transverse momenta of the two leading jets, unfolded to the particle level and compared to the predictions from Monte Carlo simulations. A comparison with next-to-leading-order perturbative QCD is also performed, showing excellent agreement within the uncertainties. From this comparison, the value of the strong coupling constant is extracted for different energy regimes, thus testing the running of α{sub s}(μ) predicted in QCD up to scales over 1 TeV. A global fit to the transverse energy-energy correlation distributions yields α{sub s}(m{sub Z}) = 0.1162 ± 0.0011 (exp.){sup +0.0084}{sub -0.0070} (theo.), while a global fit to the asymmetry distributions yields a value of α{sub s}(m{sub Z}) = 0.1196 ± 0.0013 (exp.){sup +0.0075}{sub -0.0045} (theo.). (orig.)
9. Temperature-dependent elastic properties of Ti{sub 1−x}Al{sub x}N alloys
Energy Technology Data Exchange (ETDEWEB)
Shulumba, Nina [Department of Physics, Chemistry, and Biology (IFM), Linköping University, SE-581 83 Linköping (Sweden); Functional Materials, Saarland University, D-66123 Saarbrücken (Germany); Hellman, Olle [Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125 (United States); Department of Physics, Chemistry, and Biology (IFM), Linköping University, SE-581 83 Linköping (Sweden); Rogström, Lina; Raza, Zamaan; Tasnádi, Ferenc; Odén, Magnus [Department of Physics, Chemistry, and Biology (IFM), Linköping University, SE-581 83 Linköping (Sweden); Abrikosov, Igor A. [Department of Physics, Chemistry, and Biology (IFM), Linköping University, SE-581 83 Linköping (Sweden); Materials Modeling and Development Laboratory, NUST “MISIS,” 119049 Moscow (Russian Federation); LACOMAS Laboratory, Tomsk State University, 634050 Tomsk (Russian Federation)
2015-12-07
Ti{sub 1−x}Al{sub x}N is a technologically important alloy that undergoes a process of high temperature age-hardening that is strongly influenced by its elastic properties. We have performed first principles calculations of the elastic constants and anisotropy using the symmetry imposed force constant temperature dependent effective potential method, which include lattice vibrations and therefore the effects of temperature, including thermal expansion and intrinsic anharmonicity. These are compared with in situ high temperature x-ray diffraction measurements of the lattice parameter. We show that anharmonic effects are crucial to the recovery of finite temperature elasticity. The effects of thermal expansion and intrinsic anharmonicity on the elastic constants are of the same order, and cannot be considered separately. Furthermore, the effect of thermal expansion on elastic constants is such that the volume change induced by zero point motion has a significant effect. For TiAlN, the elastic constants soften non-uniformly with temperature: C{sub 11} decreases substantially when the temperature increases for all compositions, resulting in an increased anisotropy. These findings suggest that an increased Al content and annealing at higher temperatures will result in a harder alloy.
10. Elastic strips
OpenAIRE
Chubelaschwili, David; Pinkall, Ulrich
2010-01-01
Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Moebius strip, but did not give an integrable solution. We derive two conservation laws, which describe the equilibrium equations of elastic strips. In applying these laws we find...
11. Orthodontic elastic materials.
Science.gov (United States)
Wong, A K
1976-04-01
Latex elastics and synthetic elastomers have certain similarities and differences. In the fracture tests the latex elastics showed a greater amount of loss in strength than plastic elastomers when stretched over a 21 day period. There is a great variability, as much as 50%, in the tensile strength of the plastic materials taken from the same batch and stretched under the same conditions. The Ormco Power Chain was more resilient than the Unitek AlastiK chain. The Unitek AlastiKs had more force and stretched less. The force decay of synthetic elastomers, stretched over a specific length and time, exhibited a great loss in force. This loss could be as great as 73% during the first day. The decay of force continued at a slower rate during the rest of the 21 day period. Unitek AlastiK C2 double links, when stretched 17 millimeters, had a higher initial force averaging 641 grams (22.5 ounces) than the Ormco Power Chain which averages 342 grams (12.0 ounces). In one day the force was reduced to 171 grams (6.0 ounces) for both materials. The elastic materials within the same batch showed a great variation in the modulus of elasticity under different test conditions. The approximate force generated when stretched dry, within the elastic limit, was 22 grams per millimeter for 3/16 inches heavy latex elastics. The Unitek AlastiK C2 gave a force of 89 grams per millimeter, while the Ormco Power Chain had a value of 46 grams per millimeter. The modulus of elasticity of all of the materials was much lower after immersion in the water bath. The force decay under constant force application to latex, elastic, polymer chains, and tied loops showed that the greatest amount of force decay occurred during the first three hours in the water bath. The forces remained relatively the same throughout the rest of the test period. The elastic materials undergo permanent deformation in shape. The synthetic elastomers exhibited plastic deformation when the elastomers were stretched 17
12. Determination of the strong coupling constant αs(MZ2) under regardment of completely resummed leading and next-to-leading logarithms. Analysis of global event variables measured in hadronic Z decays
International Nuclear Information System (INIS)
Wehr, A.
1994-06-01
The value of the strong coupling constant α s is determined from a combined analysis of the global event shape variables thrust, heavy jet mass and total and wide jet broadening. The extraction of α s includes the full calculation of O(α s 2 ) terms and leading and next-to-leading logarithms resummed to all orders of α s . The analysis is based on data taken with the DELPHI detector at LEP during 1991 and 1992. The dependence of the result on the detailed matching of the resummed and fixed order terms is studied. The result from the combined theory is compared with values coming from a pure NLLA analysis and as pure O(α s 2 ) analysis, respectively. It is found that the inclusion of the resummed logarithms allows the description of the data in the two jet range and reduces the scale dependence of α s (M Z 2 ) compared to pure O(α s 2 ) theory. The value using the combined NLLA+O(α s 2 ) theory at the scale μ 2 =M Z 2 is α S (M Z 2 )=0.118±0.007. The running of α s is measured from the 1991 data in an energy range from 88.5 to 93.7 GeV. The slope of α s obtained at the Z peak is dα s /dQ/ Q=Mz =-(2.9±2.8)x10 -4 GeV -1 . This value is compatible with QCD and exludes an abelian gluon model with more than two standard deviations. (orig.)
13. Elastic anisotropy of crystals
Directory of Open Access Journals (Sweden)
Christopher M. Kube
2016-09-01
Full Text Available An anisotropy index seeks to quantify how directionally dependent the properties of a system are. In this article, the focus is on quantifying the elastic anisotropy of crystalline materials. Previous elastic anisotropy indices are reviewed and their shortcomings discussed. A new scalar log-Euclidean anisotropy measure AL is proposed, which overcomes these deficiencies. It is based on a distance measure in a log-Euclidean space applied to fourth-rank elastic tensors. AL is an absolute measure of anisotropy where the limiting case of perfect isotropy yields zero. It is a universal measure of anisotropy applicable to all crystalline materials. Specific examples of strong anisotropy are highlighted. A supplementary material provides an anisotropy table giving the values of AL for 2,176 crystallite compounds.
14. Graphene nanoribbon as an elastic damper
Science.gov (United States)
Evazzade, Iman; Lobzenko, Ivan P.; Saadatmand, Danial; Korznikova, Elena A.; Zhou, Kun; Liu, Bo; Dmitriev, Sergey V.
2018-05-01
Heterostructures composed of dissimilar two-dimensional nanomaterials can have nontrivial physical and mechanical properties which are potentially useful in many applications. Interestingly, in some cases, it is possible to create heterostructures composed of weakly and strongly stretched domains with the same chemical composition, as has been demonstrated for some polymer chains, DNA, and intermetallic nanowires supporting this effect of two-phase stretching. These materials, at relatively strong tension forces, split into domains with smaller and larger tensile strains. Within this region, average strain increases at constant tensile force due to the growth of the domain with the larger strain, at the expense of the domain with smaller strain. Here, the two-phase stretching phenomenon is described for graphene nanoribbons with the help of molecular dynamics simulations. This unprecedented feature of graphene that is revealed in our study is related to the peculiarities of nucleation and the motion of the domain walls separating the domains of different elastic strain. It turns out that the loading–unloading curves exhibit a hysteresis-like behavior due to the energy dissipation during the domain wall nucleation and motion. Here, we put forward the idea of implementing graphene nanoribbons as elastic dampers, efficiently converting mechanical strain energy into heat during cyclic loading–unloading through elastic extension where domains with larger and smaller strains coexist. Furthermore, in the regime of two-phase stretching, graphene nanoribbon is a heterostructure for which the fraction of domains with larger and smaller strain, and consequently its physical and mechanical properties, can be tuned in a controllable manner by applying elastic strain and/or heat.
15. Elastic Beanstalk
CERN Document Server
Vliet, Jurg; Wel, Steven; Dowd, Dara
2011-01-01
While it's always been possible to run Java applications on Amazon EC2, Amazon's Elastic Beanstalk makes the process easier-especially if you understand how it works beneath the surface. This concise, hands-on book not only walks you through Beanstalk for deploying and managing web applications in the cloud, you'll also learn how to use this AWS tool in other phases of development. Ideal if you're a developer familiar with Java applications or AWS, Elastic Beanstalk provides step-by-step instructions and numerous code samples for building cloud applications on Beanstalk that can handle lots
16. Pressure dependence of the elastic constants and vibrational anharmonicity of Pd sub 3 sub 9 Ni sub 1 sub 0 Cu sub 3 sub 0 P sub 2 sub 1 bulk metallic glass
CERN Document Server
Wang Li; Sun, L L; Wang, W H; Wang, W K
2003-01-01
The pressure dependence of the acoustic velocities of a Pd sub 3 sub 9 Ni sub 1 sub 0 Cu sub 3 sub 0 P sub 2 sub 1 bulk metallic glass have been investigated up to 0.5 GPa at room temperature with the pulse echo overlap method. Two independent second-order elastic coefficients C sub 1 sub 1 and C sub 4 sub 4 and their pressure derivatives are yielded. The vibrational anharmonicity is shown by calculating both the acoustic mode Grueneisen parameters in the long-wavelength limit and the thermal Grueneisen parameter, and this result is compared with that for the Pd sub 4 sub 0 Ni sub 4 sub 0 P sub 2 sub 0 bulk glass.
17. Effect of Pressure on Elastic Constants, Generalized Stacking Fault Energy, and Dislocation Properties in Antiperovskite-Type Ni-Rich Nitrides ZnNNi3 and CdNNi3
KAUST Repository
Liu, Lili
2014-07-31
The elastic properties and generalized stacking fault energy curves of antiperovskite-type Ni-rich nitrides MNNi3 (M = Zn, Cd) under different pressure have been obtained from the first-principles calculations. By using the variational method, the core width and Peierls stresses of (Formula presented.) edge dislocation and screw dislocation in ZnNNi3 and CdNNi3 within the improved Peierls-Nabarro (P-N) model in which the lattice discrete effect is taken into account have been investigated. Whatever the material or the pressure range, the Peierls stress of edge dislocation is smaller than that of screw dislocation. This also demonstrates that the edge dislocation is considered to be the dominant factor in determining the plastic behavior of MNNi3 (M = Zn, Cd) in the pressure range of 0–30 GPa.
18. Elastic properties of magnetostrictive rare-earth-iron alloys
International Nuclear Information System (INIS)
Cullen, J.R.; Blessing, G.; Rinaldi, S.
1978-01-01
The elastic properties of certain magnetostrictive rare-earth-iron alloys, namely polycrystalline Tbsub(0.3)Dysub(0.7)Fesub(2), Smsub(0.88)Dysub(0.12)Fesub(2)and amorphous TbFesub(2), were investigated ultrasonically. In all cases two shear waves were observed propagating simultaneously when a magnetic field was applied perpendicular to the direction of propagation. A model to explain this behaviour, based on magnetic-elastic coupling within local regions of these disordered materials, is developed and discussed in two limiting cases: (i) strongly coupled regions for which an effective isotropic magneto-elastic coupling is appropriate, and (ii) materials for which the elastic properties of the conglomerate are determined by averaging over those of independent regions. Experimental results up to fields of 25 kOe on the alloys mentioned above are exhibited and compared with the limiting cases (i) and (ii). In the case of polycrystalline Tbsub(0.3)Dysub(0.7)Fesub(2) further comparison is made between the determination of the magneto-elastic coupling constants using this model and the determination by using the results of a previous single-crystal study. (author)
19. Rotational elasticity
Science.gov (United States)
Vassiliev, Dmitri
2017-04-01
We consider an infinite three-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis that gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory, we choose the coframe and a density. We write down the general dynamic variational functional for our rotational theory of elasticity, assuming our material to be physically linear but the kinematic model geometrically nonlinear. Allowing geometric nonlinearity is natural when dealing with rotations because rotations in dimension three are inherently nonlinear (rotations about different axes do not commute) and because there is no reason to exclude from our study large rotations such as full turns. The main result of the talk is an explicit construction of a class of time-dependent solutions that we call plane wave solutions; these are travelling waves of rotations. The existence of such explicit closed-form solutions is a non-trivial fact given that our system of Euler-Lagrange equations is highly nonlinear. We also consider a special case of our rotational theory of elasticity which in the stationary setting (harmonic time dependence and arbitrary dependence on spatial coordinates) turns out to be equivalent to a pair of massless Dirac equations. The talk is based on the paper [1]. [1] C.G.Boehmer, R.J.Downes and D.Vassiliev, Rotational elasticity, Quarterly Journal of Mechanics and Applied Mathematics, 2011, vol. 64, p. 415-439. The paper is a heavily revised version of preprint https://arxiv.org/abs/1008.3833
20. Elastic properties of graphite and interstitial defects
International Nuclear Information System (INIS)
Ayasse, J.-B.
1977-01-01
The graphite elastic constants C 33 and C 44 , reflecting the interaction of the graphitic planes, were experimentally measured as a function of irradiation and temperature. A model of non-central strength atomic interaction was established to explain the experimental results obtained. This model is valid at zero temperature. The temperature dependence of the elastic properties was analyzed. The influence of the elastic property variations on the specific heat of the lattice at very low temperature was investigated [fr
1. Second order method for solving 3D elasticity equations with complex interfaces
Science.gov (United States)
Wang, Bao; Xia, Kelin; Wei, Guo-Wei
2015-08-01
Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three-dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equations. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques have been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuities of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both L∞ and L2 error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins.
2. Parity Non-Conservation in Proton-Proton Elastic Scattering
International Nuclear Information System (INIS)
Brown, V.R.; B.F. Gibson; J.A. Carlson; R. Schiavilla
2002-01-01
The parity non-conserving longitudinal asymmetry in proton-proton (pp) elastic scattering is calculated in the lab-energy range 0-350 MeV using contemporary, realistic strong-interaction potentials combined with a weak-interaction potential comprised of rho- and omega-meson exchanges as exemplified by the DDH model. Values for the rho- and omega-meson coupling constants, h rho rho rho and h rho rho omega , are determined from comparison with the measured asymmetries at 13.6 MeV, 45 MeV, and 221 MeV
3. Constant time INEPT CT-HSQC (CTi-CT-HSQC) – A new NMR method to measure accurate one-bond J and RDCs with strong 1H–1H couplings in natural abundance
NARCIS (Netherlands)
Yu, B.; van Ingen, H.|info:eu-repo/dai/nl/297054651; Freedberg, D.I.
2013-01-01
Strong (1)H-(1)H coupling can significantly reduce the accuracy of (1)J(CH) measured from frequency differences in coupled HSQC spectra. Although accurate (1)J(CH) values can be extracted from spectral simulation, it would be more convenient if the same accurate (1)J(CH) values can be obtained
4. Quantification of local and global elastic anisotropy in ultrafine grained gradient microstructures, produced by linear flow splitting
DEFF Research Database (Denmark)
Niehuesbernd, Jörn; Müller, Clemens; Pantleon, Wolfgang
2013-01-01
Severely deformed materials often show strong plastic strain gradients, which can lead to a variety of gradients in microstructure and texture. Since the elastic behavior of a material is in most cases linked to its crystallographic texture, gradients in the elastic properties are also possible....... The local grain orientations determined by EBSD measurements were used to calculate the elastic tensors at several positions along the strain gradient. Based on the geometric mean, the calculated local elastic constants were transferred into global ones by appropriate weighting. Ultrasonic measurements were...... carried out to determine the macroscopic stiffness tensor of the severely deformed parts, showing a good agreement with the global stiffness tensor calculated from orientation data....
5. Influence of temperature and atmosphere on the strength and elastic modulus of solid oxide fuel cell anode supports
DEFF Research Database (Denmark)
Ni, De Wei; Charlas, Benoit; Kwok, Kawai
2016-01-01
need to be characterized to ensure reliable operation. In this study, the effect of reduction temperature on microstructural stability, high temperature strength and elastic modulus of Ni-YSZ anode supports were investigated. The statistical distribution of strength was determined from a large number...... of samples (∼30) at each condition to ensure high statistical validity. It is revealed that the microstructure and mechanical properties of the Ni-YSZ strongly depend on the reduction temperature. Further studies were conducted to investigate the temperature dependence of the strength and elastic modulus...... for both the unreduced and reduced Ni(O)-YSZ anode supports. With increasing temperature, the strength and elastic modulus of the reduced Ni-YSZ specimens drop almost linearly. In contrast, the strength and elastic modulus of the unreduced NiO-YSZ remain almost constant over the investigated temperature...
6. The first constant-domain (CH1) exon of human IGHG2 is polymorphic and in strong linkage disequilibrium with the CH2 exon polymorphism encoding the G2m(n+) allotype in Caucasians
DEFF Research Database (Denmark)
Hougs, L; Svejgaard, A; Barington, T
2001-01-01
, this amino acid position is expected to be surface exposed in IgG2. Besides this structural difference, we identified two silent nucleotide polymorphisms in the CH1 region and seven in the introns. Finally, we developed a sequence-specific PCR typing system detecting the polymorphisms in the CH1 and CH2......Here we describe a hitherto unknown proline/threonine polymorphism at residue 72 of the human IgG2 CH1 domain (EU numbering 189) and show that it is linked to the known valine/methionine polymorphism at residue 52 of CH2 (EU numbering 282) defining the G2m(n+)/G2m(n-) allotypes. We sequenced...... the entire constant region of the heavy-chain gene for secreted IgG2 in five IGHG2*02 homozygous individuals covering CH1, hinge, CH2, and CH3 regions (approximately 2 kb). Proline 72 in CH1 of G2m(n-) is changed to threonine in the G2m(n+) [G2m(23)] allotype. Based on the crystal structure of human IgG1...
7. Constant Proportion Portfolio Insurance
DEFF Research Database (Denmark)
Jessen, Cathrine
2014-01-01
Portfolio insurance, as practiced in 1987, consisted of trading between an underlying stock portfolio and cash, using option theory to place a floor on the value of the position, as if it included a protective put. Constant Proportion Portfolio Insurance (CPPI) is an option-free variation...... on the theme, originally proposed by Fischer Black. In CPPI, a financial institution guarantees a floor value for the “insured” portfolio and adjusts the stock/bond mix to produce a leveraged exposure to the risky assets, which depends on how far the portfolio value is above the floor. Plain-vanilla portfolio...... insurance largely died with the crash of 1987, but CPPI is still going strong. In the frictionless markets of finance theory, the issuer’s strategy to hedge its liability under the contract is clear, but in the real world with transactions costs and stochastic jump risk, the optimal strategy is less obvious...
8. Crystal structure and elastic constants of Dharwar cotton fibre using ...
Indian Academy of Sciences (India)
WINTEC
iPi i n Pi i. D D. ⎡. ⎤. = −. −. −. ⎢. ⎥. ⎢. ⎥. ⎣. ⎦. ∫. ∫. (3). In cotton fibres, it is rare to obtain multiple reflections and hence we cannot use Warren and Averbach multiple order method. We have used single order method (Hall and. Somashekar 1991; Pope and Balzar 2002; Balzar 2004) to obtain crystal size and lattice strain, ...
9. Geometric approach to the pressure tensor and the elastic constants
NARCIS (Netherlands)
den Otter, Wouter K.; Kröhn, M.; Clarke, J.H.R.
2002-01-01
Expressions are obtained for the pressure tensor in the canonical and the microcanonical ensemble for both isolated and periodic systems, using the same geometric approach to thermodynamic derivatives as has been used previously to define the configurational temperature. The inherent freedom of the
10. Elastic constants of nanoporous III-V semiconductors
Czech Academy of Sciences Publication Activity Database
Janovská, Michaela; Sedlák, Petr; Kruisová, Alena; Seiner, Hanuš; Landa, Michal; Grym, Jan
2015-01-01
Roč. 48, č. 24 (2015) ISSN 0022-3727 R&D Projects: GA ČR GB14-36566G Institutional support: RVO:61388998 ; RVO:67985882 Keywords : nanoporous semiconductors * resonant ultrasound spectroscopy * finite elements modelling Subject RIV: BM - Solid Matter Physics ; Magnetism; BM - Solid Matter Physics ; Magnetism (URE-Y) Impact factor: 2.772, year: 2015 http://iopscience.iop.org/0022-3727/48/24/245102/article
11. A Comparative Analysis of the Effective Elastic Constants of ...
African Journals Online (AJOL)
Results of various finite element and closed form models developed in the attempt to evaluate and establish accurate values of the Young's modulus, E; the shear modulus G; and the Poisson's ratio,< for laminated composite plates having soft matrix and high fibre volume fraction are discussed in this paper. Their merits and ...
12. Crystal structure and elastic constants of Dharwar cotton fibre using ...
Indian Academy of Sciences (India)
WINTEC
Abstract. Wide-angle X-ray scattering (WAXS) recordings were carried out on raw Dharwar cotton fibres available in Karnataka. Using this data and employing linked atom least squares (LALS) method, we report here the molecular and crystal structure of these cotton fibres. Employing structural data, we have computed.
13. Computational Elastic Knots
KAUST Repository
Zhao, Xin
2013-05-01
Elastic rods have been studied intensively since the 18th century. Even now the theory of elastic rods is still developing and enjoying popularity in computer graphics and physical-based simulation. Elastic rods also draw attention from architects. Architectural structures, NODUS, were constructed by elastic rods as a new method of form-finding. We study discrete models of elastic rods and NODUS structures. We also develop computational tools to find the equilibria of elastic rods and the shape of NODUS. Applications of elastic rods in forming torus knot and closing Bishop frame are included in this thesis.
14. Elastic and optical behaviour of some europium monochalcogenides
International Nuclear Information System (INIS)
Islam, A.K.M.A.; Shahdatullah, M.S.
1994-11-01
A study of the elastic and optical properties of some Eu-monochalcogenides with NaCl structure has been carried out in this paper. Various anharmonic properties e.g. thermal expansion, third order elastic constants, Grueneisen parameter, and the pressure and temperature derivatives of second order elastic constants of EuS and EuO are also studied. A comparison of the calculated elastic and dielectric properties with the available experimental results and other theoretical estimates gives an indication of the applicability of the methods applied. (author). 49 refs, 3 figs, 3 tabs
15. Marangoni elasticity of flowing soap films
Science.gov (United States)
Kim, Ildoo; Mandre, Shreyas
2017-08-01
We measure the Marangoni elasticity of a flowing soap film to be 22 mN/m irrespective of its width, thickness, flow speed, or the bulk soap concentration. We perform this measurement by generating an oblique shock in the soap film and measuring the shock angle, flow speed, and thickness. We postulate that the elasticity is constant because the film surface is crowded with soap molecules. Our method allows nondestructive measurement of flowing soap film elasticity and the value 22 mN/m is likely applicable to other similarly constructed flowing soap films.
16. Elasticity of some mantle crystal structures. II.
Science.gov (United States)
Wang, H.; Simmons, G.
1973-01-01
The single-crystal elastic constants are determined as a function of pressure and temperature for rutile structure germanium dioxide (GeO2). The data are qualitatively similar to those of rutile TiO2 measured by Manghnani (1969). The compressibility in the c direction is less than one-half that in the a direction, the pressure derivative of the shear constant is negative, and the pressure derivative of the bulk modulus has a relatively high value of about 6.2. According to an elastic strain energy theory, the negative shear modulus derivative implies that the kinetic barrier to diffusion decreases with increasing pressure.
17. Tissue elasticity and the ageing elastic fibre
OpenAIRE
Sherratt, Michael J.
2009-01-01
The ability of elastic tissues to deform under physiological forces and to subsequently release stored energy to drive passive recoil is vital to the function of many dynamic tissues. Within vertebrates, elastic fibres allow arteries and lungs to expand and contract, thus controlling variations in blood pressure and returning the pulmonary system to a resting state. Elastic fibres are composite structures composed of a cross-linked elastin core and an outer layer of fibrillin microfibrils. Th...
18. Pricing perpetual American options under multiscale stochastic elasticity of variance
International Nuclear Information System (INIS)
Yoon, Ji-Hun
2015-01-01
Highlights: • We study the effects of the stochastic elasticity of variance on perpetual American option. • Our SEV model consists of a fast mean-reverting factor and a slow mean-revering factor. • A slow scale factor has a very significant impact on the option price. • We analyze option price structures through the market prices of elasticity risk. - Abstract: This paper studies pricing the perpetual American options under a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a fast mean-reverting Ornstein–Ulenbeck process and a slowly varying diffusion process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices with respect to model parameters. Our results enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk
19. Tissue elasticity and the ageing elastic fibre.
Science.gov (United States)
Sherratt, Michael J
2009-12-01
The ability of elastic tissues to deform under physiological forces and to subsequently release stored energy to drive passive recoil is vital to the function of many dynamic tissues. Within vertebrates, elastic fibres allow arteries and lungs to expand and contract, thus controlling variations in blood pressure and returning the pulmonary system to a resting state. Elastic fibres are composite structures composed of a cross-linked elastin core and an outer layer of fibrillin microfibrils. These two components perform distinct roles; elastin stores energy and drives passive recoil, whilst fibrillin microfibrils direct elastogenesis, mediate cell signalling, maintain tissue homeostasis via TGFβ sequestration and potentially act to reinforce the elastic fibre. In many tissues reduced elasticity, as a result of compromised elastic fibre function, becomes increasingly prevalent with age and contributes significantly to the burden of human morbidity and mortality. This review considers how the unique molecular structure, tissue distribution and longevity of elastic fibres pre-disposes these abundant extracellular matrix structures to the accumulation of damage in ageing dermal, pulmonary and vascular tissues. As compromised elasticity is a common feature of ageing dynamic tissues, the development of strategies to prevent, limit or reverse this loss of function will play a key role in reducing age-related morbidity and mortality.
20. Strong solutions to a Navier–Stokes–Lamé system on a domain with a non-flat boundary
International Nuclear Information System (INIS)
Kukavica, Igor; Ziane, Mohammed; Tuffaha, Amjad
2011-01-01
In this paper, we consider a Navier–Stokes–Lamé system modeling a fluid–structure interaction. For a general domain, we establish local well-posedness for strong solutions in which initial velocity u 0 belongs to H 1 while the initial data (w 0 , w 1 ) for the elasticity equation belongs to (H 3/2+k , H 1/2+k ) for any k in (0, k 0 ) where k 0 is an explicit positive constant
1. Elastic, mechanical, and thermodynamic properties of Bi-Sb binaries: Effect of spin-orbit coupling
Science.gov (United States)
Singh, Sobhit; Valencia-Jaime, Irais; Pavlic, Olivia; Romero, Aldo H.
2018-02-01
Using first-principles calculations, we systematically study the elastic stiffness constants, mechanical properties, elastic wave velocities, Debye temperature, melting temperature, and specific heat of several thermodynamically stable crystal structures of BixSb1 -x (0 strong spin-orbit coupling (SOC) effects, and topological features in the electronic band structure. We analyze the bulk modulus (B ), Young's modulus (E ), shear modulus (G ), B /G ratio, and Poisson's ratio (ν ) as a function of the Bi concentration in BixSb1 -x . The effect of SOC on the above-mentioned properties is further investigated. In general, we observe that the SOC effects cause elastic softening in most of the studied structures. Three monoclinic structures of Bi-Sb binaries are found to exhibit significantly large auxetic behavior due to the hingelike geometric structure of bonds. The Debye temperature and the magnitude of the elastic wave velocities monotonically increase with increasing Sb concentration. However, anomalies were observed at very low Sb concentration. We also discuss the specific-heat capacity versus temperature data for all studied binaries. Our theoretical results are in excellent agreement with the existing experimental and theoretical data. The comprehensive understanding of the material properties such as hardness, mechanical strength, melting temperature, propagation of the elastic waves, auxeticity, and heat capacity is vital for practical applications of the studied binaries.
2. Ion exchange equilibrium constants
CERN Document Server
Marcus, Y
2013-01-01
Ion Exchange Equilibrium Constants focuses on the test-compilation of equilibrium constants for ion exchange reactions. The book first underscores the scope of the compilation, equilibrium constants, symbols used, and arrangement of the table. The manuscript then presents the table of equilibrium constants, including polystyrene sulfonate cation exchanger, polyacrylate cation exchanger, polymethacrylate cation exchanger, polysterene phosphate cation exchanger, and zirconium phosphate cation exchanger. The text highlights zirconium oxide anion exchanger, zeolite type 13Y cation exchanger, and
3. Measurement of the ratio of the inclusive 3-jet cross section to the inclusive 2-jet cross section in pp collisions at $\\sqrt{s}$ = 7 TeV and first determination of the strong coupling constant in the TeV range
CERN Document Server
2013-10-19
A measurement is presented of the ratio of the inclusive 3-jet cross section to the inclusive 2-jet cross section as a function of the average transverse momentum, , of the two leading jets in the event. The data sample was collected during 2011 at a proton-proton centre-of-mass energy of 7 TeV with the CMS detector at the LHC, corresponding to an integrated luminosity of 5.0 inverse femtobarns. The strong coupling constant at the scale of the Z boson mass is determined to be alphaS[MZ] = 0.1148 +/- 0.0014 (exp.) +/- 0.0018(PDF) +0.0050/-0.0000 (scale), by comparing the ratio in the range 0.42 < 1.39 TeV to the predictions of perturbative QCD at next-to-leading order. This is the first determination of alphaS[MZ] from measurements at momentum scales beyond 0.6 TeV. The predicted ratio depends only indirectly on the evolution of the parton distribution functions of the proton such that this measurement also serves as a test of the evolution of the strong coupling constant beyond 0.42 TeV. No deviation from...
4. Influence of temperature on elastic properties of caesium cyanide
International Nuclear Information System (INIS)
Singh, Preeti; Gaur, N.K.; Singh, R.K.
2007-01-01
An extended three body force shell model (ETSM), which incorporates the effects of translational-rotational (TR) coupling, three body interactions (TBI) and anharmonicity, has been applied to investigate the temperature dependence of the second order elastic constants (c ij , i,j=1,2) of CsCN. The elastic constant c 44 obtained by us shows an anomalous behaviour with the variation of temperature. The variations of elastic constants (c 11 , c 12 , c 44 ) with temperature are almost in excellent agreement with Brillouin scattering measured data. We have also evaluated the temperature variations of the third order elastic constants (c ijk ) and the pressure derivatives of the c ij in the CsCN material. However, their values could not be compared due to lack of experimental data. (copyright 2007 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)
5. Boron nitride elastic and thermal properties. Irradiation effects
International Nuclear Information System (INIS)
Jager, Bernard.
1977-01-01
The anisotropy of boron nitride (BN) and especially thermal and elastic properties were studied. Specific heat and thermal conductivity between 1.2 and 300K, thermal conductivity between 4 and 350K and elastic constants C 33 and C 44 were measured. BN was irradiated with electrons at 77K and with neutrons at 27K to determine properties after irradiation [fr
6. Elastic Appearance Models
DEFF Research Database (Denmark)
Hansen, Mads Fogtmann; Fagertun, Jens; Larsen, Rasmus
2011-01-01
This paper presents a fusion of the active appearance model (AAM) and the Riemannian elasticity framework which yields a non-linear shape model and a linear texture model – the active elastic appearance model (EAM). The non-linear elasticity shape model is more flexible than the usual linear subs...
7. In Silico Measurement of Elastic Moduli of Nematic Liquid Crystals
Science.gov (United States)
Sidky, Hythem; de Pablo, Juan J.; Whitmer, Jonathan K.
2018-03-01
Experiments on confined droplets of the nematic liquid crystal 5CB have questioned long-established bounds imposed on the elastic free energy of nematic systems. This elasticity, which derives from molecular alignment within nematic systems, is quantified through a set of moduli which can be difficult to measure experimentally and, in some cases, can only be probed indirectly. This is particularly true of the surfacelike saddle-splay elastic term, for which the available experimental data indicate values on the cusp of stability, often with large uncertainties. Here, we demonstrate that all nematic elastic moduli, including the saddle-splay elastic constant k24, may be calculated directly from atomistic molecular simulations. Importantly, results obtained through in silico measurements of the 5CB elastic properties demonstrate unambiguously that saddle-splay elasticity alone is unable to describe the observed confined morphologies.
8. ELASTIC CHARACTERIZATION OF Eucalyptus citriodora WOOD
Directory of Open Access Journals (Sweden)
2003-01-01
Full Text Available This paper contributed to the elastic characterization of Eucalyptus citriodora grown inBrazil, considering an orthotropic model and evaluating its most important elastic constants.Considering this as a reference work to establish basic elastic ratios — several important elasticconstants of Brazilian woods were not determined yet - the experimental set-up utilized one tree of 65years old from plantations of “Horto Florestal Navarro de Andrade”, at Rio Claro-SP, Brazil. All theexperimental procedures attended NBR 7190/97 – Brazilian Code for wooden structures –withconventional tension and compression tests. Results showed statistical identity between compressionand tension modulus of elasticity. The relation observed between longitudinal and radial modulus ofelasticity was 10 (EL/ER ≈ 10 and same relation, considering shear modulus (modulus of rigidity was20 (EL/GLR ≈ 20. These results, associated with Poisson’s ratios herein determined, allow theoreticalmodeling of wood mechanical behavior in structures.
9. Dielectric, elastic, anelastic and conductivity behaviour of ...
Indian Academy of Sciences (India)
The presence of two phases was confirmed by X-ray diffraction. The temperature variation of dielectric constant, ', dielectric loss, tan , d.c. conductivity, a.c. conductivity, elastic and anelastic behaviour of ferrite–ferroelectric composites were studied in the temperature range 30–350°C. The a.c. conductivity measurements ...
10. Elastic least-squares reverse time migration
KAUST Repository
Feng, Zongcai
2017-03-08
We use elastic least-squares reverse time migration (LSRTM) to invert for the reflectivity images of P- and S-wave impedances. Elastic LSRTMsolves the linearized elastic-wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration. Numerical tests on synthetic data and field data reveal the advantages of elastic LSRTM over elastic reverse time migration (RTM) and acoustic LSRTM. For our examples, the elastic LSRTM images have better resolution and amplitude balancing, fewer artifacts, and less crosstalk compared with the elastic RTM images. The images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic LSRTM images. Similar to conventional leastsquares migration, elastic LSRTM also requires an accurate estimation of the P- and S-wave migration velocity models. However, the problem remains that, when there are moderate errors in the velocity model and strong multiples, LSRTMwill produce migration noise stronger than that seen in the RTM images.
11. Elasticity of Substitution and Antidumping Measures
DEFF Research Database (Denmark)
Drud Hansen, Jørgen; Meinen, Philipp; Nielsen, Jørgen Ulff-Møller
therefore also vary inversely with the elasticity of substitution at least for countries which have a strong focus on prices in the determination of their anti-dumping measures. We test this for ten countries from 1990 to 2009 using data on anti-dumping from Chad Bown (2010) and US-data at 8-digit level......Abstract This paper analyzes the role of the elasticity of substitution for anti-dumping decisions across countries. In monopolistic competition models with cost heterogeneous firms across countries, price differences vary inversely with the elasticity of substitution. Anti-dumping duties should...
12. Elasticity of a Filament with Kinks
Science.gov (United States)
2017-12-01
Using the wormlike chain model, we analytically study the elasticity of a filament with kinks. We calculate the position probability density function and the force constant of a kinked filament with a general kink angle. Then, using the mathematical induction, we obtain the positional-orientational probability density function of a filament with regular kinks. For this filament, we compute the force constant in two different directions. In longitudinal direction of the filament, the force constant is proportional to the inverse of the number of the segments, i.e., 1 / m, while in transverse direction, it is proportional to 1/m^3.
13. Nonlinear Elasticity of Borocarbide Superconductor YNi2B2C: A First-Principles Study
Directory of Open Access Journals (Sweden)
Lili Liu
2017-01-01
Full Text Available First-principles calculations combined with homogeneous deformation methods are used to investigate the second- and third-order elastic constants of YNi2B2C with tetragonal structure. The predicted lattice constants and second-order elastic constants of YNi2B2C agree well with the available data. The effective second-order elastic constants are obtained from the second- and third-order elastic constants for YNi2B2C. Based on the effective second-order elastic constants, Pugh’s modulus ratio, Poisson’s ratio, and Vickers hardness of YNi2B2C under high pressure are further investigated. It is shown that the ductility of YNi2B2C increases with increasing pressure.
14. Integrated magnetic and elastic force systems.
Science.gov (United States)
Bourauel, Christoph; Köklü, Saduman O; Vardimon, Alexander D
2002-08-01
Magnetic force increases as the distance (d) of the force- generating elements (F approximately 1/d(2)) decreases, whereas elastic force decreases as the distance decreases (F approximately kd). These opposing characteristics suggest that combining both force systems will establish an integrated system with a long-range working ability. The objective of this study was to determine the vertical closure force (F(X)) and the transverse axis moment (M(Y)) of an integrated force system, ie, attracting magnets with elastics (vertical or Classes II and III). F(X) and M(Y) were examined on the orthodontic measurement and simulation system. It was found that the integrated force system had a positive closure force (+F(X)) that never declined to 0 and a long working range. Three regions characterized the force-deflection curve of F(X): the magnetic region (0-3 mm, for magnets with 3/16-in medium elastics), in which the decline in magnetic force was larger than the increase in elastic force (6.3-2.5 N); the constant region (3-7 mm), in which the decline in magnetic force equaled the increase in elastic force (2.5-2.9 N); and the elastic region (7-10 mm), in which there was only an increase in elastic force (2.9-3.5 N). The transverse axis moment (+M(Y)), which tends to close the bite, developed especially in magnets with a single vertical elastic. Clinically, inactivation of vertical elastics by closing the mouth can be overruled by the integrated force system because it exerts adequate force level at both short and long distances.
15. Elastic scattering at the LHC
CERN Document Server
Kaspar, Jan; Deile, M
The seemingly simple elastic scattering of protons still presents a challenge for the theory. In this thesis we discuss the elastic scattering from theoretical as well as experimental point of view. In the theory part, we present several models and their predictions for the LHC. We also discuss the Coulomb-hadronic interference, where we present a new eikonal calculation to all orders of alpha, the fine-structure constant. In the experimental part we introduce the TOTEM experiment which is dedicated, among other subjects, to the measurement of the elastic scattering at the LHC. This measurement is performed primarily with the Roman Pot (RP) detectors - movable beam-pipe insertions hundreds of meters from the interaction point, that can detect protons scattered to very small angles. We discuss some aspects of the RP simulation and reconstruction software. A central point is devoted to the techniques of RP alignment - determining the RP sensor positions relative to each other and to the beam. At the end we pres...
16. The Fine Structure Constant
Indian Academy of Sciences (India)
The article discusses the importance of the fine structure constant in quantum mechanics, along with the brief history of how it emerged. Al- though Sommerfelds idea of elliptical orbits has been replaced by wave mechanics, the fine struc- ture constant he introduced has remained as an important parameter in the field of ...
17. The Cosmological Constant
Directory of Open Access Journals (Sweden)
Carroll Sean M.
2001-01-01
Full Text Available This is a review of the physics and cosmology of the cosmological constant. Focusing on recent developments, I present a pedagogical overview of cosmology in the presence of a cosmological constant, observational constraints on its magnitude, and the physics of a small (and potentially nonzero vacuum energy.
18. On Aryabhata's Planetary Constants
OpenAIRE
Kak, Subhash
2001-01-01
This paper examines the theory of a Babylonian origin of Aryabhata's planetary constants. It shows that Aryabhata's basic constant is closer to the Indian counterpart than to the Babylonian one. Sketching connections between Aryabhata's framework and earlier Indic astronomical ideas on yugas and cyclic calendar systems, it is argued that Aryabhata's system is an outgrowth of an earlier Indic tradition.
19. ElasticSearch cookbook
CERN Document Server
Paro, Alberto
2013-01-01
Written in an engaging, easy-to-follow style, the recipes will help you to extend the capabilities of ElasticSearch to manage your data effectively.If you are a developer who implements ElasticSearch in your web applications, manage data, or have decided to start using ElasticSearch, this book is ideal for you. This book assumes that you've got working knowledge of JSON and Java
20. On elastic waves in an thinly-layered laminated medium with stress couples under initial stress
Directory of Open Access Journals (Sweden)
P. Pal Roy
1988-01-01
Full Text Available The present work is concerned with a simple transformation rule in finding out the composite elastic coefficients of a thinly layered laminated medium whose bulk properties are strongly anisotropic with a microelastic bending rigidity. These elastic coefficients which were not known completely for a layered laminated structure, are obtained suitably in terms of initial stress components and Lame's constants λi, μi of initially isotropic solids. The explicit solutions of the dynamical equations for a prestressed thinly layered laminated medium under horizontal compression in a gravity field are derived. The results are discussed specifying the effects of hydrostatic, deviatoric and couple stresses upon the characteristic propagation velocities of shear and compression wave modes.
1. Clay behaviour under thermal gradients elastic and plastic strains
International Nuclear Information System (INIS)
Pintado, Xavier; Autio, Jorma; Punkkinen, Olli
2010-01-01
Document available in extended abstract form only. The nuclear waste repositories will generate strong temperature gradients at the clay barrier. The heat and water transport generate volume change in the clay. An experimental work is proposed here. The clay reference is the MX-80. The test device imposes a fixed heat flow in one side of the sample and maintains constant the temperature on the other side. Two samples are tested for symmetry. The samples are unconfined and the total mass of water remains constant. This situation creates a strong thermal gradient in the samples. The final radial strains in some places of the sample, the total vertical strain and the water content distribution will be measured just at the end of the test and some weeks later in order to distinguish the elastic strains from the plastic strains. The test period mustn't be longer than two weeks because a large quantity of water loses through the rubber membrane and the heads of the sample. The maximum temperature reached in the cooper is 90 degrees because with higher temperature, the rubber membrane is damaged. This test is already simulated by a numerical code. Thermal, thermo-hydraulic and thermo-hydro-mechanical analyses are being done. These analyses allow studying the different fluxes inside the sample and its quantification. Water content distribution is compared with the water content calculated from the reference parameters in the clay. The water distribution and the change of diameter after the test will also be studied. This experimental work will allow to know what is the percentage of the strains elastic or plastic and check the mechanical model. The experimental diameter change is compared with the diameter change calculated from the reference parameters of the clay. (authors)
2. Anisotropy in elastic properties of lithium sodium sulphate ...
Indian Academy of Sciences (India)
Figure 1. Photograph of lithium sodium sulphate hexahydrate. (308 K) and lithium sodium sulphate crystals grown at 313 K,. 315 K, 317 K, 319 K and 323 K. velocities in these crystals in the specified directions are tabulated in table 1. The values of the elastic constants, compliance constants and Poisson's ratios of LSSW.
3. Elastic anisotropy and low-temperature thermal expansion in the shape memory alloy Cu-Al-Zn.
Science.gov (United States)
Kuruvilla, Santhosh Potharay; Menon, C S
2008-04-01
Cu-based shape memory alloys are known for their technologically important pseudo-elastic and shapememory properties, which are intimately associated with the martensitic transformation. A combination of deformation theory and finite-strain elasticity theory has been employed to arrive at the expressions for higher order elastic constants of Cu-Al-Zn based on Keating's approach. The second- and third-order elastic constants are in good agreement with the measurements. The aggregate elastic properties like bulk modulus, pressure derivatives, mode Grüneisen parameters of the elastic waves, low temperature limit of thermal expansion, and the Anderson-Grüneisen parameter are also presented.
4. The cosmological constant problem
International Nuclear Information System (INIS)
Dolgov, A.D.
1989-05-01
A review of the cosmological term problem is presented. Baby universe model and the compensating field model are discussed. The importance of more accurate data on the Hubble constant and the Universe age is stressed. 18 refs
5. Deconstructing the Cosmological Constant
CERN Document Server
Jejjala, V; Minic, D; Jejjala, Vishnu; Leigh, Robert G.; Minic, Djordje
2003-01-01
Deconstruction provides a novel way of dealing with the notoriously difficult ultraviolet problems of four-dimensional gravity. This approach also naturally leads to a new perspective on the holographic principle, tying it to the fundamental requirements of unitarity and diffeomorphism invariance, as well as to a new viewpoint on the cosmological constant problem. The numerical smallness of the cosmological constant is implied by a unique combination of holography and supersymmetry, opening a new window into the fundamental physics of the vacuum.
6. ElasticSearch cookbook
CERN Document Server
Paro, Alberto
2015-01-01
If you are a developer who implements ElasticSearch in your web applications and want to sharpen your understanding of the core elements and applications, this is the book for you. It is assumed that you've got working knowledge of JSON and, if you want to extend ElasticSearch, of Java and related technologies.
7. Measuring global gasoline and diesel price and income elasticities
International Nuclear Information System (INIS)
Dahl, Carol A.
2012-01-01
Price and income elasticities of transport fuel demand have numerous applications. They help forecast increases in fuel consumption as countries get richer, they help develop appropriate tax policies to curtail consumption, help determine how the transport fuel mix might evolve, and show the price response to a fuel disruption. Given their usefulness, it is understandable why hundreds of studies have focused on measuring such elasticities for gasoline and diesel fuel consumption. In this paper, I focus my attention on price and income elasticities in the existing studies to see what can be learned from them. I summarize the elasticities from these historical studies. I use statistical analysis to investigate whether income and price elasticities seem to be constant across countries with different incomes and prices. Although income and price elasticities for gasoline and diesel fuel are not found to be the same at high and low incomes and at high and low prices, patterns emerge that allow me to develop suggested price and income elasticities for gasoline and diesel demand for over one hundred countries. I adjust these elasticities for recent fuel mix policies, and suggest an agenda of future research topics. - Research highlights: ► Surveyed econometric studies of transport fuel demand. ► Developed price elasticities of demand for gasoline and diesel fuel for 120 countries. ► Developed income elasticities of demand for gasoline and diesel fuel for 120 countries. ► Suggested a research agenda for future work.
8. First-principles calculations of structural, elastic, thermodynamic, and electronic properties of anti-perovskites A III CNi3 (A III = Al, Ga, In)
Science.gov (United States)
Saadaoui, Fatiha; Driss Khodja, Fatima Zohra; Kadoun, Abd-Ed-Daïm; Driss Khodja, Mohammed; Elias, Abdelkader; Boudali, Abdelkader
2015-12-01
We have performed first-principles calculations of structural, elastic, thermodynamic, and electronic properties of anti-perovskites AIIICNi3 (AIII = Al, Ga, In), by using the full-potential linearized augmented plane wave (FP-LAPW) method combined with the quasi-harmonic Debye model. We carried out our calculations within the local density approximation (LDA) and the generalized gradient approximation (GGA-PBE and GGA-PBEsol functionals). Our results constitute interesting first predictions in the case of many elastic parameters of the anti-perovskites AIIICNi3, among them elastic parameters of AlCNi3 and GaCNi3 and some polycrystalline elastic parameters of InCNi3. We also report for the first time calculated values, at ambient conditions, of Grüneisen parameter, thermal expansion coefficient, specific heat at constant pressure, specific heat at constant volume, isothermal bulk modulus, and adiabatic bulk modulus for AlCNi3, GaCNi3, and InCNi3. Band structure, total and partial densities of states, and charge density have been obtained and analyzed. Electronic structure results show metallic behavior for the three compounds. Ni 3 d states play dominant role near the Fermi level and there is a strong hybridization between Ni 3 d and C 2 p states. In addition, as AIIICNi3 synthesized samples are expected to be carbon-deficient, we calculated structural, elastic, and thermodynamic properties of sub-stoichiometric AlC x Ni3 materials.
9. Fundamental physics constants
International Nuclear Information System (INIS)
Cohen, E.R.; Taylor, B.N.
1995-01-01
Present technological applications require the values used for the fundamental physical and chemical constants to be more and more precise and at the same time coherent. Great importance is then attached to the task of coordinating and comparing the most recent experimental data, extracting from them as a whole, by means of a least square fit, a set of values for the fundamental constants as precise and coherent as possible. The set of values which is at present in usage, derives from a fit performed in 1986, but new experimental results already promise a large reduction in the uncertainties of various constants. A new global fit that will implement such reductions is scheduled for completion in 1995 or 1996
10. Unconventional emergence of elastic softening induced by magnetic fields in the unusual heavy-fermion compound PrFe sub 4 P sub 1 sub 2
CERN Document Server
Nakanishi, Y; Yamaguchi, T; Hazama, H; Nemoto, Y; Goto, T; Matsuda, T D; Sugawara, H; Sato, H
2002-01-01
Ultrasonic measurement on the filled skutterudite compound PrFe sub 4 P sub 1 sub 2 exhibits a mysterious temperature dependence of the elastic constant (C sub 1 sub 1 - C sub 1 sub 2)/2. Pronounced elastic softening at low temperatures is revived by applying a magnetic field. This fact strongly suggests the 4f-multiplet ground state of the Pr ion split by the crystalline electric field (CEF) to be a GAMMA sub 3 non-Kramers doublet. The expectation value of a quadrupole moment with GAMMA sub 3 symmetry in the CEF ground state, which leads to elastic softening at low temperature, was evaluated by theoretical fitting to the present results. This may imply that suppression of the electric quadrupole Kondo effect occurs in PrFe sub 4 P sub 1 sub 2 and the quadrupole moment becomes steady due to the application of a magnetic field. (letter to the editor)
11. Elastic scattering phenomenology
Energy Technology Data Exchange (ETDEWEB)
Mackintosh, R.S. [The Open University, School of Physical Sciences, Milton Keynes (United Kingdom)
2017-04-15
We argue that, in many situations, fits to elastic scattering data that were historically, and frequently still are, considered ''good'', are not justifiably so describable. Information about the dynamics of nucleon-nucleus and nucleus-nucleus scattering is lost when elastic scattering phenomenology is insufficiently ambitious. It is argued that in many situations, an alternative approach is appropriate for the phenomenology of nuclear elastic scattering of nucleons and other light nuclei. The approach affords an appropriate means of evaluating folding models, one that fully exploits available empirical data. It is particularly applicable for nucleons and other light ions. (orig.)
12. Radiographic constant exposure technique
DEFF Research Database (Denmark)
Domanus, Joseph Czeslaw
1985-01-01
The constant exposure technique has been applied to assess various industrial radiographic systems. Different X-ray films and radiographic papers of two producers were compared. Special attention was given to fast film and paper used with fluorometallic screens. Radiographic image quality was tes...... was tested by the use of ISO wire IQI's and ASTM penetrameters used on Al and Fe test plates. Relative speed and reduction of kilovoltage obtained with the constant exposure technique were calculated. The advantages of fast radiographic systems are pointed out...
13. Radiographic constant exposure technique
DEFF Research Database (Denmark)
Domanus, Joseph Czeslaw
1985-01-01
The constant exposure technique has been applied to assess various industrial radiographic systems. Different X-ray films and radiographic papers of two producers were compared. Special attention was given to fast film and paper used with fluorometallic screens. Radiographic image quality...... was tested by the use of ISO wire IQI's and ASTM penetrameters used on Al and Fe test plates. Relative speed and reduction of kilovoltage obtained with the constant exposure technique were calculated. The advantages of fast radiographic systems are pointed out...
14. Vibrations of a pipe on elastic foundations
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
This new term is opposed to the centrifugal term U2w,xx and has a strong effect on the stability. Lilkova-Markova & Lolov (2003) investigated the influence of the transverse force at the free end of the dynamic stability of a cantilevered pipe placed on a Winkler elastic foundation. In this paper, two cases of cantilevered pipes ...
15. Constant conditional entropy and related hypotheses
International Nuclear Information System (INIS)
Ferrer-i-Cancho, Ramon; Dębowski, Łukasz; Moscoso del Prado Martín, Fermín
2013-01-01
Constant entropy rate (conditional entropies must remain constant as the sequence length increases) and uniform information density (conditional probabilities must remain constant as the sequence length increases) are two information theoretic principles that are argued to underlie a wide range of linguistic phenomena. Here we revise the predictions of these principles in the light of Hilberg’s law on the scaling of conditional entropy in language and related laws. We show that constant entropy rate (CER) and two interpretations for uniform information density (UID), full UID and strong UID, are inconsistent with these laws. Strong UID implies CER but the reverse is not true. Full UID, a particular case of UID, leads to costly uncorrelated sequences that are totally unrealistic. We conclude that CER and its particular cases are incomplete hypotheses about the scaling of conditional entropies. (letter)
16. Calculation of liquid-crystal Frank constants by computer simulation
NARCIS (Netherlands)
Allen, M.P.; Frenkel, D.
1988-01-01
We present the first calculations, by computer simulation, of the Frank elastic constants of a liquid crystal composed of freely rotating and translating molecules. Extensive calculations are performed for hard prolate ellipsoids at a single density, and for hard spherocylinders at three densities.
17. The Yamabe constant
International Nuclear Information System (INIS)
1991-01-01
The set of riemannian three-metrics with positive Yamabe constant defines the space of independent data for the gravitational field. The boundary of this set is investigated, and it is shown that metrics close to the boundary satisfy the positive-energy theorem. (Author) 18 refs
18. FORMATION CONSTANTS AND THERMODYNAMIC ...
African Journals Online (AJOL)
, Ni(II), Cu(II) and Zn(II) ions has been ... A good deal of work has been reported on the preparation and structural investigation of. Schiff base ... Formation constants and thermodynamic parameters of Co, Ni, Cu and Zn complexes. Bull. Chem.
19. Elastic magnetic electron scattering
International Nuclear Information System (INIS)
Sick, I.
1985-01-01
The paper surveys the field of elastic magnetic electron scattering. Magnetic scattering as a configuration analyzer; magnetic form factors of high multipole order; absolute spectroscopic factors; and non-nucleonic constituents; are all discussed. (U.K.)
20. Statistical mechanics of elasticity
CERN Document Server
Weiner, JH
2012-01-01
Advanced, self-contained treatment illustrates general principles and elastic behavior of solids. Topics include thermoelastic behavior of crystalline and polymeric solids, interatomic force laws, behavior of solids, and thermally activated processes. 1983 edition.
1. Graviton fluctuations erase the cosmological constant
Science.gov (United States)
Wetterich, C.
2017-10-01
Graviton fluctuations induce strong non-perturbative infrared renormalization effects for the cosmological constant. The functional renormalization flow drives a positive cosmological constant towards zero, solving the cosmological constant problem without the need to tune parameters. We propose a simple computation of the graviton contribution to the flow of the effective potential for scalar fields. Within variable gravity, with effective Planck mass proportional to the scalar field, we find that the potential increases asymptotically at most quadratically with the scalar field. The solutions of the derived cosmological equations lead to an asymptotically vanishing cosmological ;constant; in the infinite future, providing for dynamical dark energy in the present cosmological epoch. Beyond a solution of the cosmological constant problem, our simplified computation also entails a sizeable positive graviton-induced anomalous dimension for the quartic Higgs coupling in the ultraviolet regime, substantiating the successful prediction of the Higgs boson mass within the asymptotic safety scenario for quantum gravity.
2. Dumbbell formation for elastic capsules in nonlinear extensional Stokes flows
Science.gov (United States)
Dimitrakopoulos, P.
2017-06-01
Cross-slot and four-roll-mill microdevices are commonly used for particle manipulation and characterization owing to the stagnation-point flow at the device center. Because of the solid boundaries, these devices may generate extensional Stokes flows where the velocity is a nonlinear function of position associated with a decreased pressure at the particle edges and an increased pressure at the particle middle. Our computational investigation shows that in this class of Stokes flows, an elastic capsule made of a strain-hardening membrane develops two distinct steady-state conformations at strong flows, i.e., an elongated weak dumbbell shape with rounded edges at low flow nonlinearity and a laterally extended dumbbell shape at high flow nonlinearity. These effects are more pronounced for the less strain-hardening capsules which develop a flat extended middle where the two sides of the membrane approach each other. The strong stability properties of the strain-hardening capsules (owing to the development of strong membrane tensions) contrast significantly with the behavior of droplets in these nonlinear flows which are unable to achieve highly deformed steady-state dumbbell shapes owing to their constant surface tension.
3. Mastering ElasticSearch
CERN Document Server
Kuc, Rafal
2013-01-01
A practical tutorial that covers the difficult design, implementation, and management of search solutions.Mastering ElasticSearch is aimed at to intermediate users who want to extend their knowledge about ElasticSearch. The topics that are described in the book are detailed, but we assume that you already know the basics, like the query DSL or data indexing. Advanced users will also find this book useful, as the examples are getting deep into the internals where it is needed.
4. Deflation of elastic surfaces
Science.gov (United States)
Quilliet, Catherine; Quemeneur, François; Marmottant, Philippe; Imhof, Arnout; Pépin-Donat, Brigitte; van Blaaderen, Alfons
2010-03-01
The deflation of elastic spherical surfaces has been numerically investigated, and show very different types of deformations according the range of elastic parameters, some of them being quantitatively explained through simple calculations. This allows to retrieve various shapes observed on hollow shells (from colloidal to centimeter scale), on lipid vesicles, or on some biological objects. The extension of this process to other geometries allows to modelize vegetal objects such as the ultrafast trap of carnivorous plants.
5. Translation of selected papers published in Nuclear Constants, No. 1, Moscow 1988
International Nuclear Information System (INIS)
1988-12-01
The document contains the English translation from Russian of the following two papers published in Nuclear Constants No. 1, Moscow 1988: 239 Pu Neutron Cross-Sections in the Resolved-Resonance Region; Elastic and Quasi-Elastic Nucleon Scattering on Vanadium. A separate abstract was prepared for each of these two papers. Refs, figs and tabs
6. Stressed-deformed state of mountain rocks in elastic stage and between elasticity
Directory of Open Access Journals (Sweden)
Samedov A.M.
2017-12-01
Full Text Available The problems of the stress-strain state of rocks in the elastic stage and beyond the elastic limits, and the ways of schematizing the tension and compression diagrams were reviewed in the article. To simplify calculations outside the elastic range, the tension (compression diagrams are usually schematized, i.e. are replaced by curved smooth lines having a fairly simple mathematical expression and at the same time well coinciding with the experimentally obtained diagrams. When diagram is to be schematized, it is necessary to take a constant temperature of superheated water steam if a rock test is planned in a relaxed form. Note that when the diagram is schematizing, the difference between the limits of proportionality and fluidity is erased. This allows the limit of proportionality to be considered the limit of fluidity. Schematicization can be carried out in the area where the tensile strength (compression is planned to be destroyed with the established weakening of rocks by exposure to water steam or chemical reagents. Samples of rocks in natural form were tested and weakened by means of superheated water steam (220 °C and more and chemical reagents for tension and compression. The data are obtained, the diagrams of deformation are constructed and schematized in the elastic stage and beyond the elastic limit. Based on the schematic diagrams of deformation, the components of stress and strain were composed in the elastic stage and beyond the elastic limit. It is established in the publication that rocks under compression and stretching deform, both within the elastic stage, and beyond the limits of elasticity. This could be seen when the samples, both in natural and in weakened state, with superheated water steam (more than 220 °C or chemical reagents were tested. In their natural form, they are mainly deformed within the elastic stage and are destroyed as a brittle material, and in a weakened form they can deform beyond the elastic stage and
7. Superpropulsion of Droplets and Soft Elastic Solids
Science.gov (United States)
Raufaste, Christophe; Chagas, Gabriela Ramos; Darmanin, Thierry; Claudet, Cyrille; Guittard, Frédéric; Celestini, Franck
2017-09-01
We investigate the behavior of droplets and soft elastic objects propelled with a catapult. Experiments show that the ejection velocity depends on both the projectile deformation and the catapult acceleration dynamics. With a subtle matching given by a peculiar value of the projectile/catapult frequency ratio, a 250% kinetic energy gain is obtained as compared to the propulsion of a rigid projectile with the same engine. This superpropulsion has strong potentialities: actuation of droplets, sorting of objects according to their elastic properties, and energy saving for propulsion engines.
8. Renormalization of Newton's constant
Science.gov (United States)
Falls, Kevin
2015-12-01
The problem of obtaining a gauge independent beta function for Newton's constant is addressed. By a specific parametrization of metric fluctuations a gauge independent functional integral is constructed for the semiclassical theory around an arbitrary Einstein space. The effective action then has the property that only physical polarizations of the graviton contribute, while all other modes cancel with the functional measure. We are then able to compute a gauge independent beta function for Newton's constant in d dimensions to one-loop order. No Landau pole is present provided Ng<18 , where Ng=d (d -3 )/2 is the number of polarizations of the graviton. While adding a large number of matter fields can change this picture, the absence of a pole persists for the particle content of the standard model in four spacetime dimensions.
9. Production in constant evolution
International Nuclear Information System (INIS)
Lozano, T.
2009-01-01
The Cofrentes Nuclear Power Plant now has 25 years of operation behind it: a quarter century adding value and demonstrating the reasons why it is one of the most important energy producing facilities in the Spanish power market. Particularly noteworthy is the enterprising spirit of the plant, which has strived to continuously improve with the large number of modernization projects that it has undertaken over the past 25 years. The plant has constantly evolved thanks to the amount of investments made to improve safety and reliability and the perseverance to stay technologically up to date. Efficiency, training and teamwork have been key to the success of the plant over these 25 years of constant change and progress. (Author)
10. Elasticity of methane hydrate phases at high pressure
Energy Technology Data Exchange (ETDEWEB)
Beam, Jennifer; Yang, Jing; Liu, Jin [Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712 (United States); Liu, Chujie [Laboratory of Seismology and Physics of Earth’s Interior, School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui 230026 (China); Lin, Jung-Fu, E-mail: [email protected] [Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712 (United States); Center for High Pressure Science and Advanced Technology Research (HPSTAR), Shanghai 201203 (China)
2016-04-21
Determination of the full elastic constants (c{sub ij}) of methane hydrates (MHs) at extreme pressure-temperature environments is essential to our understanding of the elastic, thermodynamic, and mechanical properties of methane in MH reservoirs on Earth and icy satellites in the solar system. Here, we have investigated the elastic properties of singe-crystal cubic MH-sI, hexagonal MH-II, and orthorhombic MH-III phases at high pressures in a diamond anvil cell. Brillouin light scattering measurements, together with complimentary equation of state (pressure-density) results from X-ray diffraction and methane site occupancies in MH from Raman spectroscopy, were used to derive elastic constants of MH-sI, MH-II, and MH-III phases at high pressures. Analysis of the elastic constants for MH-sI and MH-II showed intriguing similarities and differences between the phases′ compressional wave velocity anisotropy and shear wave velocity anisotropy. Our results show that these high-pressure MH phases can exhibit distinct elastic, thermodynamic, and mechanical properties at relevant environments of their respective natural reservoirs. These results provide new insight into the determination of how much methane exists in MH reservoirs on Earth and on icy satellites elsewhere in the solar system and put constraints on the pressure and temperature conditions of their environment.
11. The Fine Structure Constant
Indian Academy of Sciences (India)
important parameter in the field of atomic struc- ture. The values of the constants of ... tions in their core that produce carbon. As a result, .... atom in 1913. In other words, the size of a hydrogen atom is a factor α−2 ≈ 20000 times the size of an elec- tron. Another way of looking at α is to consider the ratio of the orbital speed of ...
12. The cosmological constant
International Nuclear Information System (INIS)
Mellor, F.
1989-01-01
Astronomical observations predict to an extremely accurate degree that the cosmological term in Einstein's equations should be zero. This conflicts with the predictions from particle theories of a non-zero cosmological term. Attempts to resolve this paradox range from arguments based on the anthropic principle to supersymmetric theories to quantum cosmological proposals. These approaches are discussed here and the history of the cosmological constant is reviewed. (author)
13. Connecting Fundamental Constants
International Nuclear Information System (INIS)
Di Mario, D.
2008-01-01
A model for a black hole electron is built from three basic constants only: h, c and G. The result is a description of the electron with its mass and charge. The nature of this black hole seems to fit the properties of the Planck particle and new relationships among basic constants are possible. The time dilation factor in a black hole associated with a variable gravitational field would appear to us as a charge; on the other hand the Planck time is acting as a time gap drastically limiting what we are able to measure and its dimension will appear in some quantities. This is why the Planck time is numerically very close to the gravitational/electric force ratio in an electron: its difference, disregarding a π√(2) factor, is only 0.2%. This is not a coincidence, it is always the same particle and the small difference is between a rotating and a non-rotating particle. The determination of its rotational speed yields accurate numbers for many quantities, including the fine structure constant and the electron magnetic moment
14. The Hubble Constant.
Science.gov (United States)
Jackson, Neal
2015-01-01
I review the current state of determinations of the Hubble constant, which gives the length scale of the Universe by relating the expansion velocity of objects to their distance. There are two broad categories of measurements. The first uses individual astrophysical objects which have some property that allows their intrinsic luminosity or size to be determined, or allows the determination of their distance by geometric means. The second category comprises the use of all-sky cosmic microwave background, or correlations between large samples of galaxies, to determine information about the geometry of the Universe and hence the Hubble constant, typically in a combination with other cosmological parameters. Many, but not all, object-based measurements give H 0 values of around 72-74 km s -1 Mpc -1 , with typical errors of 2-3 km s -1 Mpc -1 . This is in mild discrepancy with CMB-based measurements, in particular those from the Planck satellite, which give values of 67-68 km s -1 Mpc -1 and typical errors of 1-2 km s -1 Mpc -1 . The size of the remaining systematics indicate that accuracy rather than precision is the remaining problem in a good determination of the Hubble constant. Whether a discrepancy exists, and whether new physics is needed to resolve it, depends on details of the systematics of the object-based methods, and also on the assumptions about other cosmological parameters and which datasets are combined in the case of the all-sky methods.
15. Universe of constant
Science.gov (United States)
Yongquan, Han
2016-10-01
The ideal gas state equation is not applicable to ordinary gas, it should be applied to the Electromagnetic gas'' that is applied to the radiation, the radiation should be the ultimate state of matter changes or initial state, the universe is filled with radiation. That is, the ideal gas equation of state is suitable for the Singular point and the universe. Maybe someone consider that, there is no vessel can accommodate radiation, it is because the Ordinary container is too small to accommodate, if the radius of your container is the distance that Light through an hour, would you still think it can't accommodates radiation? Modern scientific determinate that the radius of the universe now is about 1027 m, assuming that the universe is a sphere whose volume is approximately: V = 4.19 × 1081 cubic meters, the temperature radiation of the universe (cosmic microwave background radiation temperature of the universe, should be the closest the average temperature of the universe) T = 3.15k, radiation pressure P = 5 × 10-6 N / m 2, according to the law of ideal gas state equation, PV / T = constant = 6 × 1075, the value of this constant is the universe, The singular point should also equal to the constant Author: hanyongquan
16. Nonlinear elastic waves in materials
CERN Document Server
Rushchitsky, Jeremiah J
2014-01-01
The main goal of the book is a coherent treatment of the theory of propagation in materials of nonlinearly elastic waves of displacements, which corresponds to one modern line of development of the nonlinear theory of elastic waves. The book is divided on five basic parts: the necessary information on waves and materials; the necessary information on nonlinear theory of elasticity and elastic materials; analysis of one-dimensional nonlinear elastic waves of displacement – longitudinal, vertically and horizontally polarized transverse plane nonlinear elastic waves of displacement; analysis of one-dimensional nonlinear elastic waves of displacement – cylindrical and torsional nonlinear elastic waves of displacement; analysis of two-dimensional nonlinear elastic waves of displacement – Rayleigh and Love nonlinear elastic surface waves. The book is addressed first of all to people working in solid mechanics – from the students at an advanced undergraduate and graduate level to the scientists, professional...
17. The scattering potential of partial derivative wavefields in 3-D elastic orthorhombic media: an inversion prospective
KAUST Repository
Oh, Ju-Won
2016-07-04
18. Dynamic homogenization in the Nonlocal and Local regimes for a phononic superlattice: Resonant elastic metamaterial
Directory of Open Access Journals (Sweden)
J. Flores Méndez
Full Text Available In this paper, we shall propose an elastic metamaterial based on a specific rubber/aluminum superlattice. We will calculate the frequency-dependent effective mass density and transverse elastic constant in the Local and Nonlocal homogenization regimes. Using the effective dynamic parameters, the phononic dispersion calculations of the homogenized elastic crystal show a second pass band for transverse modes where the superlattice behaves as a double-negative elastic metamaterial having simultaneously negative effective mass density and shear modulus. Which is very useful for designing resonant elastic metamaterials. Keywords: Metamaterial, Phononic crystal, Homogenization theory, Effective parameters, Dispersion relation
19. Constant Proportion Debt Obligations (CPDOs)
DEFF Research Database (Denmark)
Cont, Rama; Jessen, Cathrine
2012-01-01
be made arbitrarily small—and thus the credit rating arbitrarily high—by increasing leverage, but the ratings obtained strongly depend on assumptions on the credit environment (high spread or low spread). More importantly, CPDO loss distributions are found to exhibit a wide range of tail risk measures......Constant Proportion Debt Obligations (CPDOs) are structured credit derivatives that generate high coupon payments by dynamically leveraging a position in an underlying portfolio of investment-grade index default swaps. CPDO coupons and principal notes received high initial credit ratings from...... the major rating agencies, based on complex models for the joint transition of ratings and spreads for all names in the underlying portfolio. We propose a parsimonious model for analysing the performance of CPDO strategies using a top-down approach that captures the essential risk factors of the CPDO. Our...
20. The Hubble Constant
Directory of Open Access Journals (Sweden)
Neal Jackson
2015-09-01
Full Text Available I review the current state of determinations of the Hubble constant, which gives the length scale of the Universe by relating the expansion velocity of objects to their distance. There are two broad categories of measurements. The first uses individual astrophysical objects which have some property that allows their intrinsic luminosity or size to be determined, or allows the determination of their distance by geometric means. The second category comprises the use of all-sky cosmic microwave background, or correlations between large samples of galaxies, to determine information about the geometry of the Universe and hence the Hubble constant, typically in a combination with other cosmological parameters. Many, but not all, object-based measurements give H_0 values of around 72–74 km s^–1 Mpc^–1, with typical errors of 2–3 km s^–1 Mpc^–1. This is in mild discrepancy with CMB-based measurements, in particular those from the Planck satellite, which give values of 67–68 km s^–1 Mpc^–1 and typical errors of 1–2 km s^–1 Mpc^–1. The size of the remaining systematics indicate that accuracy rather than precision is the remaining problem in a good determination of the Hubble constant. Whether a discrepancy exists, and whether new physics is needed to resolve it, depends on details of the systematics of the object-based methods, and also on the assumptions about other cosmological parameters and which datasets are combined in the case of the all-sky methods.
1. Earthquake source model using strong motion displacement
Indian Academy of Sciences (India)
The strong motion displacement records available during an earthquake can be treated as the response of the earth as the a structural system to unknown forces acting at unknown locations. Thus, if the part of the earth participating in ground motion is modelled as a known finite elastic medium, one can attempt to model the ...
2. Spaces of constant curvature
CERN Document Server
Wolf, Joseph A
2010-01-01
This book is the sixth edition of the classic Spaces of Constant Curvature, first published in 1967, with the previous (fifth) edition published in 1984. It illustrates the high degree of interplay between group theory and geometry. The reader will benefit from the very concise treatments of riemannian and pseudo-riemannian manifolds and their curvatures, of the representation theory of finite groups, and of indications of recent progress in discrete subgroups of Lie groups. Part I is a brief introduction to differentiable manifolds, covering spaces, and riemannian and pseudo-riemannian geomet
3. Calculated Changes in the Elastic Properties of MgCNi3 at the Superconducting Transition
Directory of Open Access Journals (Sweden)
R. Abd-Shukor
2013-01-01
Full Text Available We calculated the elastic properties of MgCNi3 at the superconducting transition ( using various thermodynamic and acoustic data. From the calculations, a step discontinuity of 8 ppm in the bulk modulus, 7 ppm in the Young’s modulus, and 3 ppm in the longitudinal sound velocity ( is expected at . The step discontinuities at the transition temperature indicated the importance of lattice changes to the superconducting mechanism of MgCNi3. The Debye temperature was calculated to be 460 K. The electron-phonon coupling constants calculated in the weak and strong coupling limits of the BCS theory and the van Hove scenario showed that MgCNi3 is a moderately strong coupled superconductor.
4. Elastic membranes in confinement.
Science.gov (United States)
Bostwick, J B; Miksis, M J; Davis, S H
2016-07-01
An elastic membrane stretched between two walls takes a shape defined by its length and the volume of fluid it encloses. Many biological structures, such as cells, mitochondria and coiled DNA, have fine internal structure in which a membrane (or elastic member) is geometrically 'confined' by another object. Here, the two-dimensional shape of an elastic membrane in a 'confining' box is studied by introducing a repulsive confinement pressure that prevents the membrane from intersecting the wall. The stage is set by contrasting confined and unconfined solutions. Continuation methods are then used to compute response diagrams, from which we identify the particular membrane mechanics that generate mitochondria-like shapes. Large confinement pressures yield complex response diagrams with secondary bifurcations and multiple turning points where modal identities may change. Regions in parameter space where such behaviour occurs are then mapped. © 2016 The Author(s).
5. Shells on elastic foundations
International Nuclear Information System (INIS)
Das, Y.C.; Kedia, K.K.
1977-01-01
No realistic analytical work in the area of Shells on Elastic Foundations has been reported in the literature. Various foundation models have been proposed by several authors. These models involve one or more than one parameters to characterise the foundation medium. Some of these models cannot be used to derive the basic equations governing the behaviour of shells on elastic foundations. In the present work, starting from an elastic continuum hypothesis, a mathematical model for foundation has been derived in curvilinear orthogonal coordinates by the help of principle of virtual displacements, treating one of the virtual displacements as known to satisfy certain given conditions at its edge surfaces. In this model, several foundation parameters can be considered and it can also be used for layered medium of both finite and infinite thickness. (Auth.)
6. Aluminum oxide from trimethylaluminum and water by atomic layer deposition: The temperature dependence of residual stress, elastic modulus, hardness and adhesion
International Nuclear Information System (INIS)
Ylivaara, Oili M.E.; Liu, Xuwen; Kilpi, Lauri; Lyytinen, Jussi; Schneider, Dieter; Laitinen, Mikko; Julin, Jaakko; Ali, Saima; Sintonen, Sakari; Berdova, Maria; Haimi, Eero; Sajavaara, Timo; Ronkainen, Helena; Lipsanen, Harri
2014-01-01
Use of atomic layer deposition (ALD) in microelectromechanical systems (MEMS) has increased as ALD enables conformal growth on 3-dimensional structures at relatively low temperatures. For MEMS device design and fabrication, the understanding of stress and mechanical properties such as elastic modulus, hardness and adhesion of thin film is crucial. In this work a comprehensive characterization of the stress, elastic modulus, hardness and adhesion of ALD aluminum oxide (Al 2 O 3 ) films grown at 110–300 °C from trimethylaluminum and water is presented. Film stress was analyzed by wafer curvature measurements, elastic modulus by nanoindentation and surface-acoustic wave measurements, hardness by nanoindentation and adhesion by microscratch test and scanning nanowear. The films were also analyzed by ellipsometry, optical reflectometry, X-ray reflectivity and time-of-flight elastic recoil detection for refractive index, thickness, density and impurities. The ALD Al 2 O 3 films were under tensile stress in the scale of hundreds of MPa. The magnitude of the stress decreased strongly with increasing ALD temperature. The stress was stable during storage in air. Elastic modulus and hardness of ALD Al 2 O 3 saturated to a fairly constant value for growth at 150 to 300 °C, while ALD at 110 °C gave softer films with lower modulus. ALD Al 2 O 3 films adhered strongly on cleaned silicon with SiO x termination. - Highlights: • The residual stress of Al 2 O 3 was tensile and stable during the storage in air. • Elastic modulus of Al 2 O 3 saturated to at 170 GPa for films grown at 150 to 300 °C. • At 110 °C Al 2 O 3 films were softer with high residual hydrogen and lower density. • The Al 2 O 3 adhered strongly on the SiO x -terminated silicon
7. Effective stress law for anisotropic elastic deformation
International Nuclear Information System (INIS)
Carroll, M.M.
1979-01-01
An effective stress law is derived analytically to describe the effect of pore fluid pressure on the linearly elastic response of saturated porous rocks which exhibit anisotropy. For general anisotropy the difference between the effective stress and the applied stress is not hydrostatic. The effective stress law involves two constants for transversely isotropic response and three constants for orthotropic response; these constants can be expressed in terms of the moduli of the porous material and of the solid material. These expressions simplify considerably when the anisotropy is structural rather than intrinsic, i.e., in the case of an isotropic solid material with an anisotropic pore structure. In this case the effective stress law involves the solid or grain bulk modulus and two or three moduli of the porous material, for transverse isotropy and orthotropy, respectively. The law reduces, in the case of isotropic response, to that suggested by Geertsma (1957) and by Skempton (1961) and derived analytically by Nur and Byerlee
8. Effective constants for wave propagation through partially saturated porous media
International Nuclear Information System (INIS)
Berryman, J.G.; Thigpen, L.
1985-01-01
The multipole scattering coefficients for elastic wave scattering from a spherical inhomogeneity in a fluid-saturated porous medium have been calculated. These coefficients may be used to obtain estimates of the effective macroscopic constants for long-wavelength propagation of elastic waves through partially saturated media. If the volume average of the single scattering from spherical bubbles of gas and liquid is required to vanish, the resulting equations determine the effective bulk modulus, density, and viscosity of the multiphase fluid filling the pores. The formula for the effective viscosity during compressional wave excitation is apparently new
9. Elastic plastic fracture mechanics
International Nuclear Information System (INIS)
Simpson, L.A.
1978-07-01
The application of linear elastic fracture mechanics (LEFM) to crack stability in brittle structures is now well understood and widely applied. However, in many structural materials, crack propagation is accompanied by considerable crack-tip plasticity which invalidates the use of LEFM. Thus, present day research in fracture mechanics is aimed at developing parameters for predicting crack propagation under elastic-plastic conditions. These include critical crack-opening-displacement methods, the J integral and R-curve techniques. This report provides an introduction to these concepts and gives some examples of their applications. (author)
10. ElasticSearch server
CERN Document Server
Rogozinski, Marek
2014-01-01
This book is a detailed, practical, hands-on guide packed with real-life scenarios and examples which will show you how to implement an ElasticSearch search engine on your own websites.If you are a web developer or a user who wants to learn more about ElasticSearch, then this is the book for you. You do not need to know anything about ElastiSeach, Java, or Apache Lucene in order to use this book, though basic knowledge about databases and queries is required.
11. Hybrid elastic solids
KAUST Repository
Lai, Yun
2011-06-26
Metamaterials can exhibit electromagnetic and elastic characteristics beyond those found in nature. In this work, we present a design of elastic metamaterial that exhibits multiple resonances in its building blocks. Band structure calculations show two negative dispersion bands, of which one supports only compressional waves and thereby blurs the distinction between a fluid and a solid over a finite frequency regime, whereas the other displays super anisotropy-in which compressional waves and shear waves can propagate only along different directions. Such unusual characteristics, well explained by the effective medium theory, have no comparable analogue in conventional solids and may lead to novel applications. © 2011 Macmillan Publishers Limited. All rights reserved.
12. The law of elasticity
Directory of Open Access Journals (Sweden)
Sergio Cesare Masin
2010-01-01
Full Text Available Participants estimated the imagined elongation of a spring while they were imagining that a load was stretching the spring. This elongation turned out to be a multiplicative function of spring length and load weight-a cognitive law analogous to Hooke¿s law of elasticity. Participants also estimated the total imagined elongation of springs joined either in series or in parallel. This total elongation was longer for serial than for parallel springs, and increased proportionally to the number of serial springs and inversely proportionally to the number of parallel springs. The results suggest that participants integrated load weight with imagined elasticity rather than with spring length.
13. An elastic second skin
Science.gov (United States)
Yu, Betty; Kang, Soo-Young; Akthakul, Ariya; Ramadurai, Nithin; Pilkenton, Morgan; Patel, Alpesh; Nashat, Amir; Anderson, Daniel G.; Sakamoto, Fernanda H.; Gilchrest, Barbara A.; Anderson, R. Rox; Langer, Robert
2016-08-01
We report the synthesis and application of an elastic, wearable crosslinked polymer layer (XPL) that mimics the properties of normal, youthful skin. XPL is made of a tunable polysiloxane-based material that can be engineered with specific elasticity, contractility, adhesion, tensile strength and occlusivity. XPL can be topically applied, rapidly curing at the skin interface without the need for heat- or light-mediated activation. In a pilot human study, we examined the performance of a prototype XPL that has a tensile modulus matching normal skin responses at low strain (pharmaceutical delivery and wound dressings.
14. First principles investigation of the electronic, elastic and vibrational properties of tungsten disilicide
International Nuclear Information System (INIS)
Briquet, Ludovic G.V.; Philipp, Patrick
2013-01-01
Highlights: ► Full electronic structure description. ► Elastic properties. ► Phonon band structure and DOS. ► Analysis of vibration modes. -- Abstract: Interest in tungsten disilicide is growing due to its use as protective coatings and non-volatile memory devices but fundamental investigations on tungsten disilicide vibrational properties are lacking in literature. In particular, the phonon vibration modes have never been described. This paper presents a first-principles investigation of the vibrational properties of WSi 2 crystals. The elastic and electronic properties are considered as well. First, the electron band structure is computed. Extra electronic levels for the valence electrons as compared to previously published results are found, highlighting the need for state-of-the-art DFT calculations. It is shown that the ionicity plays only a little role in the W–Si bonding. Instead, a strong degree of covalency is found. The elastic constants are computed in good agreement with the available experimental data. The complete phonon density of state as well as band structure are presented and all vibration modes are described. The phonon vibrations are also correlated to IR and Raman investigations available in the literature
15. Interaction of Myoglobin colloids with BSA in solution: Insights into complex formation and elastic compliance.
Science.gov (United States)
Madhumitha, D; Dhathathreyan, Aruna
2017-12-01
This work focusses on the supramolecular complex formed between Myoglobin (Mb) and Bovine Serum Albumin (BSA) at colloids/solution interface at pH 4.0 and pH 7.5. Electrostatic interactions between Mb as colloids and BSA solution (pH=7.5 and 4.0) have been confirmed by Zeta potential that suggest that while Mb has a narrow interaction range, BSA has a wider interaction space. The organization of Mb colloids in BSA characterized using dilational rheological parameters show that the Mb colloids are elastic and the strong adsorbed water layers on the surface restrict the deformation, regulated by the viscoelastic surface layer. Stability of the complexes analyzed using UV-vis, Fluorescence and Circular dichroic spectroscopy indicate that there is a 1:1 interaction between Mb and BSA with a binding constant of about 10 5 M -1 . Quartz Crystal microbalance with dissipation has been used to evaluate the elastic compliance of the complexes of Mb colloids dispersed in very dilute BSA solution. The higher elastic compliance at pH=4.0 (than at pH=7.5) and the complex sizes correlate with changes in zeta potential suggesting that the mechanical properties of the protein in colloids are dependent on both the electrostatic interaction as well as the degree of hydration of the colloids. Copyright © 2017 Elsevier B.V. All rights reserved.
16. Elasticity of ɛ-FeOOH: Seismic Implications for Earth's Lower Mantle
Science.gov (United States)
Thompson, E. C.; Campbell, A.; Tsuchiya, J.
2017-12-01
We have calculated the structure and elasticity of low-spin ferromagnetic ɛ-FeOOH to 140 GPa using density functional theory calculations with a Coulombic self-interaction term (U). Using this data the elastic moduli and sound velocities of ɛ-FeOOH were calculated across the pressure stability of the hydrogen bond symmetrized structure (30 to 140 GPa). The obtained values were compared with previously published values for phase H (MgSiH2O4) and δ-AlOOH, which likely form a solid solution with ɛ-FeOOH. In contrast to these Mg- and Al- endmembers, ɛ-FeOOH has smaller diagonal and larger off-diagonal elastic constants, leading to an eventual negative pressure dependence of its shear wave velocity. Because of this behavior, iron enriched solid solutions from this system have smaller shear wave velocities than surrounding mantle and therefore are a plausible contributor to large low-shear velocity provinces (LLSVPs) which exhibit similar seismic properties. Additionally, ɛ-FeOOH has substantial shear wave polarization anisotropy. Consequently, if iron-rich solid solutions from the FeOOH-AlOOH-MgSiH2O4 system at the core-mantle boundary exhibit significant lattice preferred orientation due to the strong shear stresses which occur there, it may help explain the seismically observed SH>SV anisotropy in this region.
17. Temperature-dependent elastic anisotropy and mesoscale deformation in a nanostructured ferritic alloy
Science.gov (United States)
Stoica, G. M.; Stoica, A. D.; Miller, M. K.; Ma, D.
2014-10-01
Nanostructured ferritic alloys are a new class of ultrafine-grained oxide dispersion-strengthened steels that have promising properties for service in extreme environments in future nuclear reactors. This is due to the remarkable stability of their complex microstructures containing numerous Y-Ti-O nanoclusters within grains and along grain boundaries. Although nanoclusters account primarily for the exceptional resistance to irradiation damage and high-temperature creep, little is known about the mechanical roles of the polycrystalline grains that constitute the ferritic matrix. Here we report an in situ mesoscale characterization of anisotropic responses of ultrafine ferrite grains to stresses using state-of-the-art neutron diffraction. We show the experimental determination of single-crystal elastic constants for a 14YWT alloy, and reveal a strong temperature-dependent elastic anisotropy that leads to elastic softening and instability of the ferrite. We also demonstrate, from anisotropy-induced intergranular strains, that a deformation crossover exists from low-temperature lattice hardening to high-temperature lattice softening in response to extensive plastic deformation.
18. Un saludo constante
OpenAIRE
Salcedo Ortega, Manuela; Pontificia Universidad Javeriana, Cali
2013-01-01
La presencia familiar estará siempre en mi vida: Creo que esa unión va más allá de los lazos que creamos en ese primer abrir de ojos del nacimiento pues los lazos se fortalecen con el tiempo. Es que esos lazos van de la genética al riñón y puede que suene muy raro, pero esta es mi enfermedad, la primera y la constante, la que desaparece y reaparece, la heredada y la que cada vez que me saluda, deja su huella. Comenzó hace 16 años. Mis infecciones urinarias fueron el comienzo de muchas maluque...
19. The Hubble Constant
Directory of Open Access Journals (Sweden)
Jackson Neal
2007-09-01
Full Text Available I review the current state of determinations of the Hubble constant, which gives the length scale of the Universe by relating the expansion velocity of objects to their distance. In the last 20 years, much progress has been made and estimates now range between 60 and 75 km s^-1 Mpc^-1, with most now between 70 and 75 km s^-1 Mpc^-1, a huge improvement over the factor-of-2 uncertainty which used to prevail. Further improvements which gave a generally agreed margin of error of a few percent rather than the current 10% would be vital input to much other interesting cosmology. There are several programmes which are likely to lead us to this point in the next 10 years.
20. Modeling Pseudo-elastic Behavior of Springback
International Nuclear Information System (INIS)
Xia, Z. Cedric
2005-01-01
constant. In the context of this investigation we refer psuedoelastic behavior in the most general sense as any deviation from linearity in the unloading curve. The non-linearity leads to a hysteresis loop upon reloading. The approach is based on the non-conventional theory with a vanishing elastic region as advanced by Dafalias and Popov. The treatment is purely phenomenological where we don't distinguish between macroscopic plasticity and micro-plasticity. The macroscopic uniaxial stress-strain curve is used to define effective plastic response in the same manner as classical plasticity theory except that the nonlinearity during unloading and reloading are incorporated into plasticity. It is shown that such models can be easily formulated within the context of elastoplasticity without violating any physical mechanisms of deformation. Springback for a plane strain bending model is used to demonstrate the potential effect if such a model is applied
1. The Law of Elasticity
Science.gov (United States)
Cocco, Alberto; Masin, Sergio Cesare
2010-01-01
Participants estimated the imagined elongation of a spring while they were imagining that a load was stretching the spring. This elongation turned out to be a multiplicative function of spring length and load weight--a cognitive law analogous to Hooke's law of elasticity. Participants also estimated the total imagined elongation of springs joined…
2. Influence of elastic strain gradient on the upper limit of flexocoupling strength, spatially modulated phases, and soft phonon dispersion in ferroics
Science.gov (United States)
Morozovska, Anna N.; Eliseev, Eugene A.; Scherbakov, Christian M.; Vysochanskii, Yulian M.
2016-11-01
Using the Landau-Ginzburg-Devonshire theory, we established the role of the flexoelectric coupling between the gradients of elastic strain and polarization in the stability of spatially modulated phases in ferroics, such as incipient and proper ferroelectrics with commensurate and incommensurate long-range-ordered phases. We included the square of elastic strain gradient in the Landau-Ginzburg-Devonshire functional because this term provides the functional stability for all values of the strain gradient. Analytical expressions for polarization, strain, dielectric susceptibility, and stability threshold were derived for a one-dimensional case. The expressions show that the maximal possible values of the static flexoelectric effect coefficients (upper limits) established by Yudin, Ahluwalia, and Tagantsev without the square of elastic strain gradient and other higher order gradients terms lose their direct meaning. Considering the gradients, the temperature dependent condition for the flexocoupling magnitude exists instead of the upper limits. Also, we established that spatially modulated phases appear and become stable in commensurate ferroelectrics if the flexocoupling constant exceeds a critical value. The critical value depends on the electrostriction and elastic constants, temperature, and gradient coefficients in the Landau-Ginzburg-Devonshire functional. We calculated soft phonon dispersion in commensurate and incommensurate long-range-ordered phases of ferroelectrics with the square of elastic strain gradient, static, and dynamic flexocoupling. It appeared that the dispersion of the optical mode is slightly sensitive to the flexocoupling, and the dispersion of acoustic mode strongly depends on the coupling magnitude. Obtained results demonstrate that nontrivial differences in the dispersion of optical and acoustic modes occur with the change of flexocoupling constant. Therefore, experimental determination of soft phonon dispersion might be very informative to
3. Data-Driven Problems in Elasticity
Science.gov (United States)
Conti, S.; Müller, S.; Ortiz, M.
2018-01-01
We consider a new class of problems in elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within this Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to material data sets that are not graphs.
4. Elastic softness of hybrid lead halide perovskites
KAUST Repository
Ferreira, A. C.
2018-01-26
Much recent attention has been devoted towards unravelling the microscopic optoelectronic properties of hybrid organic-inorganic perovskites (HOP). Here we investigate by coherent inelastic neutron scattering spectroscopy and Brillouin light scattering, low frequency acoustic phonons in four different hybrid perovskite single crystals: MAPbBr3, FAPbBr3, MAPbI3 and α-FAPbI3 (MA: methylammonium, FA: formamidinium). We report a complete set of elastic constants caracterized by a very soft shear modulus C44. Further, a tendency towards an incipient ferroelastic transition is observed in FAPbBr3. We observe a systematic lower sound group velocity in the technologically important iodide-based compounds compared to the bromide-based ones. The findings suggest that low thermal conductivity and hot phonon bottleneck phenomena are expected to be enhanced by low elastic stiffness, particularly in the case of the ultrasoft α-FAPbI3.
5. The Charm and Beauty of Strong Interactions
Science.gov (United States)
El-Bennich, Bruno
2018-01-01
We briefly review common features and overlapping issues in hadron and flavor physics focussing on continuum QCD approaches to heavy bound states, their mass spectrum and weak decay constants in different strong interaction models.
6. Some aspects of thermal and elastic properties of thallium
OpenAIRE
Ramji Rao, R.; Rajput, A.
1980-01-01
A model based on Keating's approach is applied to the hcp metal thallium to work out its lattice heat capacity, third-order elastic (TOE) constants and thermal expansion. The calculated TOE constants are utilized to determine the low-temperature limit of volume thermal expansion and to investigate the pressure variation of the lattice parameters and volume of thallium. The results of calculations show good agreement with the corresponding available experimental data.
7. Correlations between elastic moduli and properties in bulk metallic glasses
International Nuclear Information System (INIS)
Wang Weihua
2006-01-01
A survey of the elastic, mechanical, fragility, and thermodynamic properties of bulk metallic glasses (BMGs) and glass-forming liquids is presented. It is found that the elastic moduli of BMGs have correlations with the glass transition temperature, melting temperature, mechanical properties, and even liquid fragility. On the other hand, the elastic constants of available BMGs show a rough correlation with a weighted average of the elastic constants for the constituent elements. Although the theoretical and physical reasons for the correlations are to be clarified, these correlations could assist in understanding the long-standing issues of glass formation and the nature of glass and simulate the work of theorists. Based on the correlation, we show that the elastic moduli can assist in selecting alloying components for controlling the elastic properties and glass-forming ability of the BMGs and thus can guide BMG design. As case study, we report the formation of the families of rare-earth-based BMGs with controllable properties
8. Non-linear elastic deformations
CERN Document Server
Ogden, R W
1997-01-01
Classic in the field covers application of theory of finite elasticity to solution of boundary-value problems, analysis of mechanical properties of solid materials capable of large elastic deformations. Problems. References.
9. First-principles study of structural, elastic, electronic, lattice dynamic and optical properties of XN (X=Ga, Al and B) compounds under pressure
International Nuclear Information System (INIS)
Fatmi, M; Ghebouli, B; Ghebouli, M A; Hieba, Z K
2011-01-01
We have applied the pseudo-potential plane wave method to study the structural, elastic, electronic, lattice dynamic and optical properties of GaN and AlN in the wurtzite lattice and BN with zinc-blende structure. We have found that all elastic constants depend strongly on hydrostatic pressure, except for C 44 in wurtzite AlN and GaN that shows a weaker dependence. AlN and GaN present a direct band gap Γ-Γ, whereas BN has an indirect band gap Γ-X. The indirect Γ-K band gap in AlN occurs at about 35 GPa. The top of the valence bands reflects the p electronic character for all structures. There is a gap between optical and acoustic modes only for wurtzite phases AlN and GaN. All peaks in the imaginary part of the dielectric function for the wurtzite lattice GaN and AlN move towards lower energies, while those in the zinc-blende BN structure shift towards higher energies with increasing pressure. The decrease of the static dielectric constant and static refractive index in zinc-blende BN is weaker and it can be explained by its higher elastic constants.
10. Elastic and magnetoelastic properties of intermetallic compound NdCo5 in the spin-reorientation region
International Nuclear Information System (INIS)
Deryagin, A.V.; Kvashnin, G.M.; Kapitonov, A.M.
1984-01-01
By the ultrasonic method the temperature dependences of elastic constants of the NdCO 5 monocrystal in the temperature range (4.2 ...350) K are determined. In the spontaneous spin-reorientation (SR) region an anomalous behaviour of all NdCO 5 elastic constants is revealed. The dependence of velocities of longitudinal elastic waves propagation along hexagonal axis on the value and orientation of the magnetic field is investigated. The influence of the magnetoelastic interaction on SR boundaries and K 1 anisotropy constant is estimated. Magnetoelastic Bsub(a)sup(theta) and Bsub(c)sup(theta) constants are calculated
11. Designing interactively with elastic splines
DEFF Research Database (Denmark)
Brander, David; Bærentzen, Jakob Andreas; Fisker, Ann-Sofie
2018-01-01
We present an algorithm for designing interactively with C1 elastic splines. The idea is to design the elastic spline using a C1 cubic polynomial spline where each polynomial segment is so close to satisfying the Euler-Lagrange equation for elastic curves that the visual difference becomes neglig...... negligible. Using a database of cubic Bézier curves we are able to interactively modify the cubic spline such that it remains visually close to an elastic spline....
12. Accurate measurements of the acoustical physical constants of synthetic alpha-quartz for SAW devices.
Science.gov (United States)
Kushibiki, Juin-ichi; Takanaga, Izumi; Nishiyama, Shouichi
2002-01-01
Accurate measurements of the acoustical physical constants (elastic constants, piezoelectric constants, dielectric constants, and density) of commercially available and widely used surface acoustic wave (SAW)-grade synthetic a-quartz are reported. The propagation directions and modes of bulk waves optimal for accurately determining the constants were selected through numerical calculations, and three principal X-, Y-, and Z-cut specimens and several rotated Y-cut specimens were prepared from a single crystal ingot to determine the constants and to confirm their accuracy. All of the constants were determined through highly accurate measurements of the longitudinal velocities, shear velocities, dielectric constants, and density. The velocity values measured for the specimens that were not used to determine the constants agreed well with those calculated from the determined constants, within a difference of +/- 0.20 m/s (+/- 0.004%).
13. Predicting elastic properties of β-HMX from first-principles calculations.
Science.gov (United States)
Peng, Qing; Rahul; Wang, Guangyu; Liu, Gui-Rong; Grimme, Stefan; De, Suvranu
2015-05-07
We investigate the performance of van der Waals (vdW) functions in predicting the elastic constants of β cyclotetramethylene tetranitramine (HMX) energetic molecular crystals using density functional theory (DFT) calculations. We confirm that the accuracy of the elastic constants is significantly improved using the vdW corrections with environment-dependent C6 together with PBE and revised PBE exchange-correlation functionals. The elastic constants obtained using PBE-D3(0) calculations yield the most accurate mechanical response of β-HMX when compared with experimental stress-strain data. Our results suggest that PBE-D3 calculations are reliable in predicting the elastic constants of this material.
14. Elastic Properties of Hard Films Multi-Layer Protective Coatings by Light Scattering
National Research Council Canada - National Science Library
Sooryakumar, R
2000-01-01
... (silicon oxynitride and ZnSe) and free-standing membranes (SiN). These harmonics provide a direct means to investigate the longitudinal and transverse sound velocities and thereby to determine the C11 and C44 elastic constants...
15. Micromechanical Prediction of the Effective Behavior of Fully Coupled Electro-Magneto-Thermo-Elastic Multiphase Composites
Science.gov (United States)
Aboudi, Jacob
2000-01-01
The micromechanical generalized method of cells model is employed for the prediction of the effective moduli of electro-magneto-thermo-elastic composites. These include the effective elastic, piezoelectric, piezomagnetic, dielectric, magnetic permeability, electromagnetic coupling moduli, as well as the effective thermal expansion coefficients and the associated pyroelectric and pyromagnetic constants. Results are given for fibrous and periodically bilaminated composites.
16. Analytic approximations for the elastic moduli of two-phase materials
DEFF Research Database (Denmark)
Zhang, Z. J.; Zhu, Y. K.; Zhang, P.
2017-01-01
Based on the models of series and parallel connections of the two phases in a composite, analytic approximations are derived for the elastic constants (Young's modulus, shear modulus, and Poisson's ratio) of elastically isotropic two-phase composites containing second phases of various volume...
17. Development of elastic properties of Cu-based shape memory alloys during martensitic transformation
Czech Academy of Sciences Publication Activity Database
Novák, Václav; Landa, Michal; Šittner, Petr
2004-01-01
Roč. 115, - (2004), s. 363 ISSN 1155-4339 Institutional research plan: CEZ:AV0Z1010914 Keywords : Cu-based shape memory alloy s * elastic properties * elastic constants * modelling Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 0.294, year: 2004
18. To optimal elasticity of adhesives mimicking gecko foot-hairs
International Nuclear Information System (INIS)
Filippov, A.E.; Popov, V.
2006-01-01
Artificial structure of a plate with elastic fibers interacting with rough fractal surface by Van der Waals forces is simulated numerically to find an optimal relation between the system parameters. The force balance equations are solved numerically for different values of elastic constant and variable surface roughness. An optimal elasticity is found to provide maximum cohesion force between the plate and surface. It is shown that high flexibility of the fibers is not always good to efficiency of the system, artificial adhesives must be made from stiff enough polymers. If the ellasticity is close to an optimum, the force is almost constant at a wide interval of the surface roughness. It is desirable to make system adaptive to wide spectrum of applications
19. First-principles study of structural stabilities, elastic and electronic properties of transition metal monocarbides (TMCs) and mononitrides (TMNs)
Energy Technology Data Exchange (ETDEWEB)
Rached, H.; Rached, D.; Benalia, S. [Laboratoire des Matériaux Magnétiques, Faculté des Sciences, Université Djillali Liabès de Sidi Bel-Abbès, Sidi Bel-Abbès 22000 (Algeria); Reshak, A.H., E-mail: [email protected] [Institute of Complex Systems, FFPW, CENAKVA, University of South Bohemia in CB, Nove Hrady 37333 (Czech Republic); Center of Excellence Geopolymer and Green Technology, School of Material Engineering, University Malaysia Perlis, 01007 Kangar, Perlis (Malaysia); Rabah, M. [Laboratoire des Matériaux Magnétiques, Faculté des Sciences, Université Djillali Liabès de Sidi Bel-Abbès, Sidi Bel-Abbès 22000 (Algeria); Khenata, R. [Laboratoire de Physique Quantique et de Modélisation Mathématique de la Matière (LPQ3M), université de Mascara, Mascara 29000 (Algeria); Bin Omran, S. [Department of Physics and Astronomy, Faculty of Science, King Saud University, Riyadh 11451 (Saudi Arabia)
2013-12-16
The structural stabilities, elastic and electronic properties of 5d transition metal mononitrides (TMNs) XN with (X = Ir, Os, Re, W and Ta) and 5d transition metal monocarbides (TMCs) XC with (X = Ir, Os, Re and Ta) were investigated using the full-potential linear muffin-tin orbital (FP-LMTO) method, in the framework of the density functional theory (DFT) within the local density approximation (LDA) for the exchange correlation functional. The ground state quantities such as the lattice parameter, bulks modulus and its pressure derivatives for the six considered crystal structures, Rock-salt (B1), CsCl (B2), zinc-blend (B3), Wurtzite (B4), NiAs (B8{sub 1}) and the tungsten carbides (B{sub h}) are calculated. The elastic constants of TMNs and TMCs compounds in its different stable phases are determined by using the total energy variation with strain technique. The elastic modulus for polycrystalline materials, shear modulus (G), Young's modulus (E), and Poisson's ratio (ν) are calculated. The Debye temperature (θ{sub D}) and sound velocities (v{sub m}) were also derived from the obtained elastic modulus. The analysis of the hardness of the herein studied compounds classifies OsN – (B4 et B8{sub 1}), ReN – (B8{sub 1}), WN – (B8{sub 1}) and OsC – (B8{sub 1}) as superhard materials. Our results for the band structure and densities of states (DOS), show that TMNs and TMCs compounds in theirs energetically and mechanically stable phase has metallic characteristic with strong covalent nature Metal–Nonmetal elements. - Highlights: • Structural stabilities, elastic, electronic properties of 5d TMNs XN are investigated. • 5d TMCs XC with (X = Ir, Os, Re and Ta) were investigated. • The ground state properties for the six considered crystal structure are calculated. • The elastic constants of TMNs and TMCs in its different stable phases are determined. • The elastic modulus for polycrystalline materials, G, E, and ν are calculated.
20. Elasticity in Elastics-An in-vitro study.
Science.gov (United States)
Kamisetty, Supradeep Kumar; Nimagadda, Chakrapani; Begam, Madhoom Ponnachi; Nalamotu, Raghuveer; Srivastav, Trilok; Gs, Shwetha
2014-04-01
Orthodontic tooth movement results from application of forces to teeth. Elastics in orthodontics have been used both intra-orally and extra- orally to a great effect. Their use, combined with good patient co-operation provides the clinician with the ability to correct both anteroposterior and vertical discrepancies. Force decay over a period of time is a major problem in the clinical usage of latex elastics and synthetic elastomers. This loss of force makes it difficult for the clinician to determine the actual force transmitted to the dentition. It's the intent of the clinician to maintain optimal force values over desired period of time. The majority of the orthodontic elastics on the market are latex elastics. Since the early 1990s, synthetic products have been offered in the market for latex-sensitive patients and are sold as nonlatex elastics. There is limited information on the risk that latex elastics may pose to patients. Some have estimated that 0.12-6% of the general population and 6.2% of dental professionals have hypersensitivity to latex protein. There are some reported cases of adverse reactions to latex in the orthodontic population but these are very limited to date. Although the risk is not yet clear, it would still be inadvisable to prescribe latex elastics to a patient with a known latex allergy. To compare the in-vitro performance of latex and non latex elastics. Samples of 0.25 inch, latex and non latex elastics (light, medium, heavy elastics) were obtained from three manufacturers (Forestadent, GAC, Glenroe) and a sample size of ten elastics per group was tested. The properties tested included cross sectional area, internal diameter, initial force generated by the elastics, breaking force and the force relaxation for the different types of elastics. Force relaxation testing involved stretching the elastics to three times marketed internal diameter (19.05 mm) and measuring force level at intervals over a period of 48 hours. The data were
1. Elastic and piezoelectric properties, sound velocity and Debye ...
Indian Academy of Sciences (India)
The phase transition, the independent elastic stiffness constants, the bulk modulus, the direct and converse piezoelectric coefficients, the longitudinal, transverse, and average sound velocities, and finally the Debye temperature under pressure are studied. The results obtained are generally lower than the available ...
2. Structural, elastic, electronic and optical properties of bi-alkali ...
Indian Academy of Sciences (India)
and efficient method for the calculation of the ground-state properties of materials [22 ... Murnaghan equation of state [28]. The optimization curves for the compounds are shown in figure 2. In the ground state, a (Å), B (GPa) and Bo are evaluated. The calculated ... the cubic structure only three elastic constants are required.
3. Determination of elastic modulus in nickel alloy from ultrasonic ...
Indian Academy of Sciences (India)
Elastic constants relate technological, structural and safety aspects to various materials phenomena and to their fundamental interatomic ... nological, structural economics and safety, to various mate- rials phenomena and to their .... lographic planes, forming clusters or embryos of the pre- cipitate; and (ii) precipitation, when ...
4. Torsional vibration of thin-walled elastic beams with doubly ...
African Journals Online (AJOL)
In this paper, the problem of analyzing the torsional vibration of thin-walled elastic beams, with open cross-sections that are doubly symmetric and traversed by moving concentrated masses at constant speeds is addressed. The mathematical model adopted accounts for both the gravitational and inertial effects of the ...
5. First-principle calculations of structural, electronic, optical, elastic ...
Indian Academy of Sciences (India)
2017-11-28
Nov 28, 2017 ... The predicted band gaps using the modified Becke–Johnson. (mBJ) exchange approximation are in fairly good agreement with the experimental data. The optical constants such as the dielectric function, refractive index, and the extinction coefficient are calculated and analysed. The independent elastic ...
6. Effect of reinforcement volume fraction on the density & elastic ...
African Journals Online (AJOL)
volume fraction composites are improved with the introduction of certain materials reinforcement such as Mo,. Pt, Cr, Fe, U, etc. ... sports equipment, luxury goods, armor and anti-armor systems ... Vit.1 matrix composites density & elastic constants as function of reinforcement volume fraction of , glass E, Fe,. Mo, Ni, Cr, Mn, ...
7. Thermoelastic waves without energy dissipation in an elastic plate to ...
African Journals Online (AJOL)
The linear theory of thermoelasticity without energy dissipation for isotropic and homogeneous materials is employed to study waves in an elastic plate. The waves are assumed to arise out of a ramp-type stress on the plate's boundary which is maintained at constant temperature. Laplace transforms are used to solve the ...
8. Structural, elastic and thermodynamic properties of Ti2SC
Indian Academy of Sciences (India)
Abstract. The structural parameters, elastic constants and thermodynamic properties of Ti2SC were investi- gated under pressure and temperature by using first-principles plane-wave pseudopotential density functional theory within the generalized gradient approximation. The obtained results are in agreement with the.
9. Synthetic Strategies for High Dielectric Constant Silicone Elastomers
DEFF Research Database (Denmark)
synthetic strategies were developed in this Ph.D. thesis, in order to create silicone elastomers with high dielectric constants and thereby higher energy densities. The work focused on maintaining important properties such as dielectric loss, electrical breakdown strength and elastic modulus...... extender’ that allowed for chemical modifications such as Cu- AAC. This route was promising for one-pot elastomer preparation and as a high dielectric constant additive to commercial silicone systems. The second approach used the borane-catalysed Piers-Rubinsztajn reaction to form spatially well...... of functional groups was identified. At a concentration of 5.6 wt% of a nitrobenzene functional group the dielectric permittivity increased 70% while at this loading important properties such as electrical breakdown strength, elastic modulus and dielectric loss were not significantly compromised. The developed...
10. Introduction to linear elasticity
CERN Document Server
Gould, Phillip L
2013-01-01
Introduction to Linear Elasticity, 3rd Edition, provides an applications-oriented grounding in the tensor-based theory of elasticity for students in mechanical, civil, aeronautical, and biomedical engineering, as well as materials and earth science. The book is distinct from the traditional text aimed at graduate students in solid mechanics by introducing the subject at a level appropriate for advanced undergraduate and beginning graduate students. The author's presentation allows students to apply the basic notions of stress analysis and move on to advanced work in continuum mechanics, plasticity, plate and shell theory, composite materials, viscoelasticity and finite method analysis. This book also: Emphasizes tensor-based approach while still distilling down to explicit notation Provides introduction to theory of plates, theory of shells, wave propagation, viscoelasticity and plasticity accessible to advanced undergraduate students Appropriate for courses following emerging trend of teaching solid mechan...
11. An elastic second skin.
Science.gov (United States)
Yu, Betty; Kang, Soo-Young; Akthakul, Ariya; Ramadurai, Nithin; Pilkenton, Morgan; Patel, Alpesh; Nashat, Amir; Anderson, Daniel G; Sakamoto, Fernanda H; Gilchrest, Barbara A; Anderson, R Rox; Langer, Robert
2016-08-01
We report the synthesis and application of an elastic, wearable crosslinked polymer layer (XPL) that mimics the properties of normal, youthful skin. XPL is made of a tunable polysiloxane-based material that can be engineered with specific elasticity, contractility, adhesion, tensile strength and occlusivity. XPL can be topically applied, rapidly curing at the skin interface without the need for heat- or light-mediated activation. In a pilot human study, we examined the performance of a prototype XPL that has a tensile modulus matching normal skin responses at low strain (appearance in a 5-point severity scale. The XPL platform may offer advanced solutions to compromised skin barrier function, pharmaceutical delivery and wound dressings.
12. Beyond the Hubble Constant
Science.gov (United States)
1995-08-01
about the distances to galaxies and thereby about the expansion rate of the Universe. A simple way to determine the distance to a remote galaxy is by measuring its redshift, calculate its velocity from the redshift and divide this by the Hubble constant, H0. For instance, the measured redshift of the parent galaxy of SN 1995K (0.478) yields a velocity of 116,000 km/sec, somewhat more than one-third of the speed of light (300,000 km/sec). From the universal expansion rate, described by the Hubble constant (H0 = 20 km/sec per million lightyears as found by some studies), this velocity would indicate a distance to the supernova and its parent galaxy of about 5,800 million lightyears. The explosion of the supernova would thus have taken place 5,800 million years ago, i.e. about 1,000 million years before the solar system was formed. However, such a simple calculation works only for relatively nearby'' objects, perhaps out to some hundred million lightyears. When we look much further into space, we also look far back in time and it is not excluded that the universal expansion rate, i.e. the Hubble constant, may have been different at earlier epochs. This means that unless we know the change of the Hubble constant with time, we cannot determine reliable distances of distant galaxies from their measured redshifts and velocities. At the same time, knowledge about such change or lack of the same will provide unique information about the time elapsed since the Universe began to expand (the Big Bang''), that is, the age of the Universe and also its ultimate fate. The Deceleration Parameter q0 Cosmologists are therefore eager to determine not only the current expansion rate (i.e., the Hubble constant, H0) but also its possible change with time (known as the deceleration parameter, q0). Although a highly accurate value of H0 has still not become available, increasing attention is now given to the observational determination of the second parameter, cf. also the Appendix at the
13. Bulk rock elastic moduli at high pressures, derived from the mineral textures and from extrapolated laboratory data
International Nuclear Information System (INIS)
Ullemeyer, K; Keppler, R; Lokajíček, T; Vasin, R N; Behrmann, J H
2015-01-01
The elastic anisotropy of bulk rock depends on the mineral textures, the crack fabric and external parameters like, e.g., confining pressure. The texture-related contribution to elastic anisotropy can be predicted from the mineral textures, the largely sample-dependent contribution of the other parameters must be determined experimentally. Laboratory measurements of the elastic wave velocities are mostly limited to pressures of the intermediate crust. We describe a method, how the elastic wave velocity trends and, by this means, the elastic constants can be extrapolated to the pressure conditions of the lower crust. The extrapolated elastic constants are compared to the texture-derived ones. Pronounced elastic anisotropy is evident for phyllosilicate minerals, hence, the approach is demonstrated for two phyllosilicate-rich gneisses with approximately identical volume fractions of the phyllosilicates but different texture types. (paper)
14. Elastic properties of Gum Metal
International Nuclear Information System (INIS)
Kuramoto, Shigeru; Furuta, Tadahiko; Hwang, Junghwan; Nishino, Kazuaki; Saito, Takashi
2006-01-01
In situ X-ray diffraction measurements under tensile loading and dynamic mechanical analysis were performed to investigate the mechanisms of elastic deformation in Gum Metal. Tensile stress-strain curves for Gum Metal indicate that cold working substantially decreases the elastic modulus while increasing the yield strength, thereby confirming nonlinearity in the elastic range. The gradient of each curve decreased continuously to about one-third its original value near the elastic limit. As a result of this decrease in elastic modulus and nonlinearity, elastic deformability reaches 2.5% after cold working. Superelasticity is attributed to stress-induced martensitic transformations, although the large elastic deformation in Gum Metal is not accompanied by a phase transformation
15. <strong>Authenticated hash tablesstrong>
DEFF Research Database (Denmark)
Triandopoulos, Nikolaos; Papamanthou, Charalampos; Tamassia, Roberto
2008-01-01
Hash tables are fundamental data structures that optimally answer membership queries. Suppose a client stores n elements in a hash table that is outsourced at a remote server so that the client can save space or achieve load balancing. Authenticating the hash table functionality, i.e., verifying...... the correctness of queries answered by the server and ensuring the integrity of the stored data, is crucial because the server, lying outside the administrative control of the client, can be malicious. We design efficient and secure protocols for optimally authenticating membership queries on hash tables: for any...... fixed constants 0 1/ε, the server can provide a proof of integrity of the answer to a (non-)membership query in constant time, requiring O(nε/logκε--1 n) time to treat updates, yet keeping the communication and verification costs constant. This is the first construction...
16. Impact loads on beams on elastic foundations
International Nuclear Information System (INIS)
Kameswara Rao, N.S.V.; Prasad, B.B.
1975-01-01
Quite often, complex structural components are idealised as beams in engineering analysis and design. Also, equations governing the responses of shallow shells are mathematically equivalent to the equations governing the responses of beams on elastic foundations. Hence with possible applications in several technical disciplines, the behaviour of beams on elastic foundations subjected to impact loads is studied in detail in the present investigation both analytically and experimentally. The analytical methods include analysis and energy method. The effect of foundation parameters (stiffness, and damping constants) on the dynamic responses of the beam-foundation system has been analysed. In modal analysis, the free-vibration equation has been solved by replacing the applied impulse by suitable initial conditions and the solution has been obtained as the linear combination of an infinite sequence of discrete eigen-vectors. In the energy method, the beam-foundation system is treated to be under forced vibrations and the forcing function has been obtained using the Hertz's law of impact. In the case of free-free end conditions of the beam, the rigid body modes and the elastic modes have been superposed to obtain the total response. The responses predicted using modal analysis are higher than those obtained using energy method. From the present study it is observed that model analysis is preferable to energy method. (Auth.)
17. Mechanics of magnetic fluid column in strong magnetic fields
International Nuclear Information System (INIS)
Polunin, V.M.; Ryapolov, P.A.; Platonov, V.B.
2017-01-01
Elastic-and magnetic properties of magnetic fluid confined by ponderomotive force in a tube fixed in horizontal position are considered. The system is placed in a strong magnetic field under the influence of external static and dynamic perturbations. An experimental setup has been developed. A theoretical basis of the processes of magnetic colloid elastic deformation has been proposed. The values of the static ponderomotive elasticity coefficient and the elasticity coefficient under dynamic action are experimentally determined. The calculations of the saturation magnetization for two magnetic fluid samples, carried out according to the equation containing the dynamic elasticity coefficient, are in good agreement with the experimental magnetization curve. The described method is of interest when studying magnetophoresis and aggregation of nanoparticles in magnetic colloids.
18. Elastic and transport properties of topological semimetal ZrTe
Science.gov (United States)
Guo, San-Dong; Wang, Yue-Hua; Lu, Wan-Li
2017-11-01
Topological semimetals may have substantial applications in electronics, spintronics, and quantum computation. Recently, ZrTe was predicted as a new type of topological semimetal due to the coexistence of Weyl fermions and massless triply degenerate nodal points. In this work, the elastic and transport properties of ZrTe are investigated by combining the first-principles calculations and semiclassical Boltzmann transport theory. Calculated elastic constants prove the mechanical stability of ZrTe, and the bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio also are calculated. It is found that spin-orbit coupling (SOC) has slightly enhanced effects on the Seebeck coefficient, which along the a(b) and c directions for pristine ZrTe at 300 K is 46.26 μVK-1 and 80.20 μVK-1, respectively. By comparing the experimental electrical conductivity of ZrTe (300 K) with the calculated value, the scattering time is determined as 1.59 × 10-14 s. The predicted room-temperature electronic thermal conductivity along the a(b) and c directions is 2.37 {{Wm}}-1{{{K}}}-1 and 2.90 {{Wm}}-1{{{K}}}-1, respectively. The room-temperature lattice thermal conductivity is predicted as 17.56 {{Wm}}-1{{{K}}}-1 and 43.08 {{Wm}}-1{{{K}}}-1 along the a(b) and c directions, showing very strong anisotropy. Calculated results show that isotope scattering produces an observable effect on lattice thermal conductivity. To observably reduce lattice thermal conductivity by nanostructures, the characteristic length should be smaller than 70 nm, based on cumulative lattice thermal conductivity with respect to the phonon mean free path (MFP) at 300 K. It is noted that the average room-temperature lattice thermal conductivity of ZrTe is slightly higher than that of isostructural MoP, which is due to larger phonon lifetimes and smaller Grüneisen parameters. Finally, the total thermal conductivity as a function of temperature is predicted for pristine ZrTe. Our works provide valuable
19. Density functional calculations of elastic properties of portlandite, Ca(OH)(2)
DEFF Research Database (Denmark)
Laugesen, Jakob Lund
2005-01-01
The elastic constants of portlandite, Ca(OH)(2), are calculated by use of density functional theory. A lattice optimization of an infinite (periodic boundary conditions) lattice is performed on which strains are applied. The elastic constants are extracted by minimizing Hooke's law of linear...... elasticity, applying a least-square method. Young's modulus and bulk modulus are calculated from the stiffness matrix. The results are compared with the Brillouin zone spectroscopy results of F. Holuj et al. [F. Holuj, M. Drozdowski, M. Czajkowski, Brillouin spectrum of Ca(OH)(2), Solid State Commun., 56 (12...
20. Form finding in elastic gridshells
Science.gov (United States)
Baek, Changyeob; Sageman-Furnas, Andrew O.; Jawed, Mohammad K.; Reis, Pedro M.
2018-01-01
Elastic gridshells comprise an initially planar network of elastic rods that are actuated into a shell-like structure by loading their extremities. The resulting actuated form derives from the elastic buckling of the rods subjected to inextensibility. We study elastic gridshells with a focus on the rational design of the final shapes. Our precision desktop experiments exhibit complex geometries, even from seemingly simple initial configurations and actuation processes. The numerical simulations capture this nonintuitive behavior with excellent quantitative agreement, allowing for an exploration of parameter space that reveals multistable states. We then turn to the theory of smooth Chebyshev nets to address the inverse design of hemispherical elastic gridshells. The results suggest that rod inextensibility, not elastic response, dictates the zeroth-order shape of an actuated elastic gridshell. As it turns out, this is the shape of a common household strainer. Therefore, the geometry of Chebyshev nets can be further used to understand elastic gridshells. In particular, we introduce a way to quantify the intrinsic shape of the empty, but enclosed regions, which we then use to rationalize the nonlocal deformation of elastic gridshells to point loading. This justifies the observed difficulty in form finding. Nevertheless, we close with an exploration of concatenating multiple elastic gridshell building blocks.
1. Elastic and viscoplastic properties
International Nuclear Information System (INIS)
Lebensohn, R.A.
2015-01-01
In this chapter, we review crystal elasticity and plasticity-based self-consistent theories and apply them to the determination of the effective response of polycrystalline aggregates. These mean-field formulations, which enable the prediction of the mechanical behaviour of polycrystalline aggregates based on the heterogeneous and/or directional properties of their constituent single crystal grains and phases, are ideal tools to establish relationships between microstructure and properties of these materials, ubiquitous among fuels and structural materials for nuclear systems. (author)
2. Mathematical foundations of elasticity
CERN Document Server
Marsden, Jerrold E
1994-01-01
This advanced-level study approaches mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is directed to mathematicians, engineers and physicists who wish to see this classical subject in a modern setting with examples of newer mathematical contributions. Prerequisites include a solid background in advanced calculus and the basics of geometry and functional analysis.The first two chapters cover the background geometry ― developed as needed ― and use this discussion to obtain the basic results on kinematics and dynamics of con
3. Series Elastic Actuators.
Science.gov (United States)
1995-01-01
7.2 Planetary rover 75 7.3 Biped Robot 76 8 Conclusions 77 8.1 Review of Thesis 77 8.2 Further Work 77 List of Figures 1-1 Schematic of...have only four degrees of freedom, and a simple gripper. 75 76 CHAPTER 7. APPLICATIONS Figure 7-1: Photograph of robot arm 7.3 Biped Robot ...Another group at MIT is building a biped walking robot using series elastic actuators. The design of the actuators differs in that instead of using a
4. Strongly correlating liquids and their isomorphs
OpenAIRE
Pedersen, Ulf R.; Gnan, Nicoletta; Bailey, Nicholas P.; Schröder, Thomas B.; Dyre, Jeppe C.
2010-01-01
This paper summarizes the properties of strongly correlating liquids, i.e., liquids with strong correlations between virial and potential energy equilibrium fluctuations at constant volume. We proceed to focus on the experimental predictions for strongly correlating glass-forming liquids. These predictions include i) density scaling, ii) isochronal superposition, iii) that there is a single function from which all frequency-dependent viscoelastic response functions may be calculated, iv) that...
5. Mathematical methods in electro-magneto-elasticity
CERN Document Server
Bardzokas, DI; Filshtinsky, LA
2007-01-01
The mechanics of Coupled Fields is a discipline at the edge of modern research connecting Continuum Mechanics with Solid State Physics. It integrates the Mechanics of Continuous Media, Heat Conductivity and the theory of Electromagnetism that are usually studied separately. For an accurate description of the influence of static and dynamic loadings, high temperatures and strong electromagnetic fields in elastic media and constructive installations, a new approach is required; an approach that has the potential to establish a synergism between the above mentioned fields. Throughout the book a vast number of problems are considered: two-dimensional problems of electro-magneto-elasticity as well as static and dynamical problems for piecewise homogenous compound piezoelectric plates weakened by cracks and openings. The boundary conditions, the constructive equations and the mathematical methods for their solution are thoroughly presented, so that the reader can get a clear quantitative and qualitative understandi...
6. Converging shocks in elastic-plastic solids.
Science.gov (United States)
Ortega, A López; Lombardini, M; Hill, D J
2011-11-01
We present an approximate description of the behavior of an elastic-plastic material processed by a cylindrically or spherically symmetric converging shock, following Whitham's shock dynamics theory. Originally applied with success to various gas dynamics problems, this theory is presently derived for solid media, in both elastic and plastic regimes. The exact solutions of the shock dynamics equations obtained reproduce well the results obtained by high-resolution numerical simulations. The examined constitutive laws share a compressible neo-Hookean structure for the internal energy e=e(s)(I(1))+e(h)(ρ,ς), where e(s) accounts for shear through the first invariant of the Cauchy-Green tensor, and e(h) represents the hydrostatic contribution as a function of the density ρ and entropy ς. In the strong-shock limit, reached as the shock approaches the axis or origin r=0, we show that compression effects are dominant over shear deformations. For an isothermal constitutive law, i.e., e(h)=e(h)(ρ), with a power-law dependence e(h) is proportional to ρ(α), shock dynamics predicts that for a converging shock located at r=R(t) at time t, the Mach number increases as M is proportional to [log(1/R)](α), independently of the space index s, where s=2 in cylindrical geometry and 3 in spherical geometry. An alternative isothermal constitutive law with p(ρ) of the arctanh type, which enforces a finite density in the strong-shock limit, leads to M is proportional to R(-(s-1)) for strong shocks. A nonisothermal constitutive law, whose hydrostatic part e(h) is that of an ideal gas, is also tested, recovering the strong-shock limit M is proportional to R(-(s-1)/n(γ)) originally derived by Whitham for perfect gases, where γ is inherently related to the maximum compression ratio that the material can reach, (γ+1)/(γ-1). From these strong-shock limits, we also estimate analytically the density, radial velocity, pressure, and sound speed immediately behind the shock. While the
7. Design guidance for elastic followup
Energy Technology Data Exchange (ETDEWEB)
Naugle, F.V.
1983-01-01
The basic mechanism of elastic followup is discussed in relation to piping design. It is shown how mechanistic insight gained from solutions for a two-bar problem can be used to identify dominant design parameters and to determine appropriate modifications where elastic followup is a potential problem. It is generally recognized that quantitative criteria are needed for elastic followup in the creep range where badly unbalanced lines can pose potential problems. Approaches for criteria development are discussed.
8. Elastic emission polishing
Energy Technology Data Exchange (ETDEWEB)
Loewenthal, M.; Loseke, K.; Dow, T.A.; Scattergood, R.O.
1988-12-01
Elastic emission polishing, also called elastic emission machining (EEM), is a process where a stream of abrasive slurry is used to remove material from a substrate and produce damage free surfaces with controlled surface form. It is a noncontacting method utilizing a thick elasto-hydrodynamic film formed between a soft rotating ball and the workpiece to control the flow of the abrasive. An apparatus was built in the Center, which consists of a stationary spindle, a two-axis table for the workpiece, and a pump to circulate the working fluid. The process is controlled by a programmable computer numerical controller (CNC), which presently can operate the spindle speed and movement of the workpiece in one axis only. This apparatus has been used to determine material removal rates on different material samples as a function of time, utilizing zirconium oxide (ZrO{sub 2}) particles suspended in distilled water as the working fluid. By continuing a study of removal rates the process should become predictable, and thus create a new, effective, yet simple tool for ultra-precision mechanical machining of surfaces.
9. Heavy ion elastic scatterings
International Nuclear Information System (INIS)
Mermaz, M.C.
1984-01-01
Diffraction and refraction play an important role in particle elastic scattering. The optical model treats correctly and simultaneously both phenomena but without disentangling them. Semi-classical discussions in terms of trajectories emphasize the refractive aspect due to the real part of the optical potential. The separation due to to R.C. Fuller of the quantal cross section into two components coming from opposite side of the target nucleus allows to understand better the refractive phenomenon and the origin of the observed oscillations in the elastic scattering angular distributions. We shall see that the real part of the potential is responsible of a Coulomb and a nuclear rainbow which allows to determine better the nuclear potential in the interior region near the nuclear surface since the volume absorption eliminates any effect of the real part of the potential for the internal partial scattering waves. Resonance phenomena seen in heavy ion scattering will be discussed in terms of optical model potential and Regge pole analysis. Compound nucleus resonances or quasi-molecular states can be indeed the more correct and fundamental alternative
10. Elastic anisotropy in multifilament Nb$_3$Sn superconducting wires
CERN Document Server
Scheuerlein, C; Alknes, P; Arnau, G; Bjoerstad, R; Bordini, B
2015-01-01
The elastic anisotropy caused by the texture in the Nb3Sn filaments of PIT and RRP wires has been calculated by averaging the estimates of Voigt and Reuss, using published Nb3Sn single crystal elastic constants and the Nb3Sn grain orientation distribution determined in both wire types by Electron Backscatter Diffraction. At ambient temperature the calculated Nb3Sn E-moduli in axial direction in the PIT and the RRP wire are 130 GPa and 140 GPa, respectively. The calculated E-moduli are compared with tensile test results obtained for the corresponding wires and extracted filament bundles.
11. Ultrasonic velocity and elastic moduli of heavy metal tellurite glasses
Energy Technology Data Exchange (ETDEWEB)
Afifi, Hesham; Marzouk, Samier
2003-05-26
Longitudinal and transverse ultrasonic waves velocities in lead tungsten tellurite glasses have been measured using the pulse-echo method at 5 MHz frequency and at room temperature (300 K). The elastic properties; longitudinal modulus, shear modulus, Young's modulus, bulk modulus and Poisson's ratio together with the microhardness, softening temperature, and Debye temperature are found to be rather sensitive to the glass composition. Information about the structure of the glass can be deduced after calculating the average stretching force constant and the average ring size. A comparison between the experimental elastic moduli data obtained in this study and those calculated theoretically by other models has been discussed.
12. Elastic properties of silicon nitride ceramics reinforced with graphene nanofillers
Czech Academy of Sciences Publication Activity Database
Seiner, Hanuš; Ramírez, C.; Koller, M.; Sedlák, Petr; Landa, Michal; Miranzo, P.; Belmonte, M.; Osendí, M. I.
2015-01-01
Roč. 87, December (2015), s. 675-680 ISSN 0264-1275 R&D Projects: GA ČR GB14-36566G Institutional support: RVO:61388998 Keywords : multilayer graphene * graphene oxide (GO) * silicon nitride * elastic constants * elastic modulus * shear modulus Subject RIV: JI - Composite Materials Impact factor: 3.997, year: 2015 http://www.sciencedirect.com/science/article/pii/S0264127515302938/pdfft?md5=571e00fd7f976e9b66ed789ae2a868b2&pid=1-s2.0-S0264127515302938-main.pdf
13. Resonant Column Tests and Nonlinear Elasticity in Simulated Rocks
Science.gov (United States)
Sebastian, Resmi; Sitharam, T. G.
2018-01-01
Rocks are generally regarded as linearly elastic even though the manifestations of nonlinearity are prominent. The variations of elastic constants with varying strain levels and stress conditions, disagreement between static and dynamic moduli, etc., are some of the examples of nonlinear elasticity in rocks. The grain-to-grain contact, presence of pores and joints along with other compliant features induce the nonlinear behavior in rocks. The nonlinear elastic behavior of rocks is demonstrated through resonant column tests and numerical simulations in this paper. Resonant column tests on intact and jointed gypsum samples across varying strain levels have been performed in laboratory and using numerical simulations. The paper shows the application of resonant column apparatus to obtain the wave velocities of stiff samples at various strain levels under long wavelength condition, after performing checks and incorporating corrections to the obtained resonant frequencies. The numerical simulation and validation of the resonant column tests using distinct element method are presented. The stiffness reductions of testing samples under torsional and flexural vibrations with increasing strain levels have been analyzed. The nonlinear elastic behavior of rocks is reflected in the results, which is enhanced by the presence of joints. The significance of joint orientation and influence of joint spacing during wave propagation have also been assessed and presented using the numerical simulations. It has been found that rock joints also exhibit nonlinear behavior within the elastic limit.
14. Elastic properties of polycrystalline rare earth-cobalt Laves compounds
International Nuclear Information System (INIS)
Klimker, H.; Rosen, M.
1978-01-01
The elastic moduli of the RCo 2 compounds have been measured in the temperature range between 4.2 to 300 K. A magneto-elastic lattice softening was observed in the magnetically ordered state. This effect is particularly large in NdCo 2 , SmCo 2 , and TbCo 2 . In addition NdCo 2 and HoCo 2 exhibit spin reorientations at 12 and 42 K, respectively, which appear as a narrow dip in the elastic moduli. At the Curie temperature of these compounds a prominent anomaly in the adiabatic compressibility is observed. The shape of the anomalies in the elastic moduli of these compounds is indicative of a first order transition observed in DyCo 2 , HoCo 2 , and ErCo 2 . The Curie temperature deduced from the elastic moduli is in satisfactory agreement with the temperature determined previously. The anomalous behavior of the elastic constants in the paramagntic temperature range is attributed to crystal field effects of a similar character to that observed in RAl 2 compounds. (Auth.)
15. Multisoliton solutions, completely elastic collisions and non-elastic ...
Indian Academy of Sciences (India)
We discuss the nature of solitonsolutions before and after their interactions, and present their fusion (non-elastic) and elastic collisions of the soliton solutions. ... Department of Mathematics, Pabna University of Science and Technology, Pabna 6600, Bangladesh; School of Mathematics and Physics, University of ...
16. Spectrophotometric determination of association constant
DEFF Research Database (Denmark)
2016-01-01
Least-squares 'Systematic Trial-and-Error Procedure' (STEP) for spectrophotometric evaluation of association constant (equilibrium constant) K and molar absorption coefficient E for a 1:1 molecular complex, A + B = C, with error analysis according to Conrow et al. (1964). An analysis of the Charg...
17. Nonlinear elastic behavior of rocks revealed by dynamic acousto-elastic testing
Science.gov (United States)
Shokouhi, Parisa; Riviere, Jacques; Guyer, Robert; Johnson, Paul
2017-04-01
Nonlinear elastic behavior of rocks is studied at the laboratory scale with the goal of illuminating observations at the Earth scale, for instance during strong ground motion and earthquake slip processes. A technique called Dynamic Acousto-Elastic Testing (DAET) is used to extract the nonlinear elastic response of disparate rocks (sandstone, granite and soapstone). DAET is the dynamic analogous to standard (quasi-static) acousto-elastic testing. It consists in measuring speed of sound with high-frequency low amplitude pulses (MHz range) across the sample while it is dynamically loaded with a low frequency, large amplitude resonance (kHz range). This particular configuration provides the instantaneous elastic response over a full dynamic cycle and reveals unprecedented details: instantaneous softening, tension/compression asymmetry as well as hysteretic behaviors. The strain-induced modulation of ultrasonic pulse velocities ('fast dynamics') is analyzed to extract nonlinearity parameters. A projection method is used to extract the harmonic content and a careful comparison of the fast dynamics response is made. In order to characterize the rate of elastic recovery ('slow dynamics'), we continue to monitor the ultrasonic wave velocity for about 30 minutes after the low-frequency resonance is turned off. In addition, the frequency, pressure and humidity dependences of the nonlinear parameters are reported for a subset of samples. We find that the nonlinear components can be clustered into two categories, which suggests that two main mechanisms are at play. The first one, related to the second harmonic, is likely related to the opening/closing of microstructural features such as cracks and grain/grain contacts. In contrast, the second mechanism is related to all other nonlinear parameters (transient softening, hysteresis area and higher order harmonics) and may arise from shearing mechanisms at grain interfaces.
18. Effect of van der Waals interactions on the structural and elastic properties of black phosphorus
DEFF Research Database (Denmark)
Appalakondaiah, S.; Vaitheeswaran, G.; Lebègue, S.
2012-01-01
constant is significantly larger than the C11 and C33 parameters, implying that black phosphorus is stiffer against strain along the a axis than along the b and c axes. From the calculated elastic constants, the mechanical properties, such as bulk modulus, shear modulus, Young's modulus, and Poisson...
19. Structures and Elastic Moduli of Polymer Nanocomposite Thin Films
Science.gov (United States)
Yuan, Hongyi; Karim, Alamgir; University of Akron Team
2014-03-01
Polymeric thin films generally possess unique mechanical and thermal properties due to confinement. In this study we investigated structures and elastic moduli of polymer nanocomposite thin films, which can potentially find wide applications in diverse areas such as in coating, permeation and separation. Conventional thermoplastics (PS, PMMA) and biopolymers (PLA, PCL) were chosen as polymer matrices. Various types of nanoparticles were used including nanoclay, fullerene and functionalized inorganic particles. Samples were prepared by solvent-mixing followed by spin-coating or flow-coating. Film structures were characterized using X-ray scattering and transmission electron microscopy. Elastic moduli were measured by strain-induced elastic buckling instability for mechanical measurements (SIEBIMM), and a strengthening effect was found in certain systems due to strong interaction between polymers and nanoparticles. The effects of polymer structure, nanoparticle addition and film thickness on elastic modulus will be discussed and compared with bulk materials.
20. Elastic, dynamical, and electronic properties of LiHg and Li3Hg: First-principles study
Science.gov (United States)
Wang, Yan; Hao, Chun-Mei; Huang, Hong-Mei; Li, Yan-Ling
2018-04-01
The elastic, dynamical, and electronic properties of cubic LiHg and Li3Hg were investigated based on first-principles methods. The elastic constants and phonon spectral calculations confirmed the mechanical and dynamical stability of the materials at ambient conditions. The obtained elastic moduli of LiHg are slightly larger than those of Li3Hg. Both LiHg and Li3Hg are ductile materials with strong shear anisotropy as metals with mixed ionic, covalent, and metallic interactions. The calculated Debye temperatures are 223.5 K and 230.6 K for LiHg and Li3Hg, respectively. The calculated phonon frequency of the T2 g mode in Li3Hg is 326.8 cm-1. The p states from the Hg and Li atoms dominate the electronic structure near the Fermi level. These findings may inspire further experimental and theoretical study on the potential technical and engineering applications of similar alkali metal-based intermetallic compounds.
1. Questions about elastic waves
CERN Document Server
Engelbrecht, Jüri
2015-01-01
This book addresses the modelling of mechanical waves by asking the right questions about them and trying to find suitable answers. The questions follow the analytical sequence from elementary understandings to complicated cases, following a step-by-step path towards increased knowledge. The focus is on waves in elastic solids, although some examples also concern non-conservative cases for the sake of completeness. Special attention is paid to the understanding of the influence of microstructure, nonlinearity and internal variables in continua. With the help of many mathematical models for describing waves, physical phenomena concerning wave dispersion, nonlinear effects, emergence of solitary waves, scales and hierarchies of waves as well as the governing physical parameters are analysed. Also, the energy balance in waves and non-conservative models with energy influx are discussed. Finally, all answers are interwoven into the canvas of complexity.
2. Approximation by planar elastic curves
DEFF Research Database (Denmark)
Brander, David; Gravesen, Jens; Nørbjerg, Toke Bjerge
2016-01-01
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven...
3. Nonlinear Elasticity of Doped Semiconductors
Science.gov (United States)
2017-02-01
AFRL-RY-WP-TR-2016-0206 NONLINEAR ELASTICITY OF DOPED SEMICONDUCTORS Mark Dykman and Kirill Moskovtsev Michigan State University...2016 4. TITLE AND SUBTITLE NONLINEAR ELASTICITY OF DOPED SEMICONDUCTORS 5a. CONTRACT NUMBER FA8650-16-1-7600 5b. GRANT NUMBER 5c. PROGRAM...vibration amplitude. 15. SUBJECT TERMS semiconductors , microresonators, microelectromechanical 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF
4. Strongly nonlinear oscillators analytical solutions
CERN Document Server
Cveticanin, Livija
2014-01-01
This book provides the presentation of the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. The book presents the original author’s method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter is considered. Special attention is given to the one and two mass oscillatory systems with two-degrees-of-freedom. The criteria for the deterministic chaos in ideal and non-ideal pure nonlinear oscillators are derived analytically. The method for suppressing chaos is developed. Important problems are discussed in didactic exercises. The book is self-consistent and suitable as a textbook for students and also for profess...
5. Elastic and magnetic properties of bilayer manganates. La2-2xSr1+2xMn2O7 for x=0.35 and 0.40
International Nuclear Information System (INIS)
Imaduddin, Agung; Nakanishi, Yoshiki; Shimomura, Kota; Kanazawa, Hisanori; Nakamura, Masatoshi; Yoshimoto, Noriyuki; Yoshizawa, Masahito
2002-01-01
We have grown single crystal of LSMO(327) for x=0.35 and x=0.40 and investigated the elastic properties systematically, together with the electrical resistivity, magnetoresistance and magnetization. Elastic anomaly around the magnetic transition temperature T c , at which remarkable metal-insulator transition occurs, has been found in the longitudinal as well as shear elastic constants for x=0.35 and 0.40. They all exhibit a dip around T c and a remarkable hardening below T c implying strong couplings between magnetic moments and elastic strains. Furthermore, it was found that the amount of the elastic softening of C 33 and C 66 become smaller with increasing x, which implies that the ration of dγ polarization, d(3z 2 -r 2 )/d(x 2 -y 2 ) is decreasing with increasing x. We will discuss obtained results in terms of the 3d electric state of Mn ions in connection with a degree of carrier concentration, dγ-band width and crystalline electric field (CEF) effect. (author)
6. Blocky inversion of multichannel elastic impedance for elastic parameters
Science.gov (United States)
Mozayan, Davoud Karami; Gholami, Ali; Siahkoohi, Hamid Reza
2018-04-01
Petrophysical description of reservoirs requires proper knowledge of elastic parameters like P- and S-wave velocities (Vp and Vs) and density (ρ), which can be retrieved from pre-stack seismic data using the concept of elastic impedance (EI). We propose an inversion algorithm which recovers elastic parameters from pre-stack seismic data in two sequential steps. In the first step, using the multichannel blind seismic inversion method (exploited recently for recovering acoustic impedance from post-stack seismic data), high-resolution blocky EI models are obtained directly from partial angle-stacks. Using an efficient total-variation (TV) regularization, each angle-stack is inverted independently in a multichannel form without prior knowledge of the corresponding wavelet. The second step involves inversion of the resulting EI models for elastic parameters. Mathematically, under some assumptions, the EI's are linearly described by the elastic parameters in the logarithm domain. Thus a linear weighted least squares inversion is employed to perform this step. Accuracy of the concept of elastic impedance in predicting reflection coefficients at low and high angles of incidence is compared with that of exact Zoeppritz elastic impedance and the role of low frequency content in the problem is discussed. The performance of the proposed inversion method is tested using synthetic 2D data sets obtained from the Marmousi model and also 2D field data sets. The results confirm the efficiency and accuracy of the proposed method for inversion of pre-stack seismic data.
7. Atomistic calculations of interface elastic properties in noncoherent metallic bilayers
International Nuclear Information System (INIS)
Mi Changwen; Jun, Sukky; Kouris, Demitris A.; Kim, Sung Youb
2008-01-01
The paper describes theoretical and computational studies associated with the interface elastic properties of noncoherent metallic bicrystals. Analytical forms of interface energy, interface stresses, and interface elastic constants are derived in terms of interatomic potential functions. Embedded-atom method potentials are then incorporated into the model to compute these excess thermodynamics variables, using energy minimization in a parallel computing environment. The proposed model is validated by calculating surface thermodynamic variables and comparing them with preexisting data. Next, the interface elastic properties of several fcc-fcc bicrystals are computed. The excess energies and stresses of interfaces are smaller than those on free surfaces of the same crystal orientations. In addition, no negative values of interface stresses are observed. Current results can be applied to various heterogeneous materials where interfaces assume a prominent role in the systems' mechanical behavior
8. The elasticity of soap bubbles containing wormlike micelles.
Science.gov (United States)
Sabadini, Edvaldo; Ungarato, Rafael F S; Miranda, Paulo B
2014-01-28
Slow-motion imaging of the rupture of soap bubbles generally shows the edges of liquid films retracting at a constant speed (known as the Taylor-Culick velocity). Here we investigate soap bubbles formed from simple solutions of a cationic surfactant (cetyltrimethylammonium bromide - CTAB) and sodium salicylate. The interaction of salicylate ions with CTAB leads to the formation of wormlike micelles (WLM), which yield a viscoelastic behavior to the liquid film of the bubble. We demonstrate that these elastic bubbles collapse at a velocity up to 30 times higher than the Taylor-Culick limit, which has never been surpassed. This is because during the bubble inflation, the entangled WLM chains stretch, storing elastic energy. This extra energy is then released during the rupture of the bubble, yielding an additional driving force for film retraction (besides surface tension). This new mechanism for the bursting of elastic bubbles may have important implications to the breakup of viscoelastic sprays in industrial applications.
9. Nonlinear elasticity of disordered fiber networks
Science.gov (United States)
Feng, Jingchen; Levine, Herbert; Mao, Xiaoming; Sander, Leonard M.
One of the most striking mechanical properties in disordered biopolymer gels is strong nonlinearities. In the case of athermal gels (such as collagen- I) the nonlinearity has long been associated with a crossover from a bending dominated to a stretching dominated regime of elasticity. The physics of this crossover is related to the existence of a central-force isostatic point and to the small bending modulus for most gels. This crossover induces scaling behavior for the elastic moduli. In particular, for linear elasticity such a scaling law has been demonstrated by Broedersz et al. We generalize the scaling to the nonlinear regime with a two-parameter scaling law involving three critical exponents. We do numerical testing of the scaling law for two disordered lattice models, and find a good scaling collapse for the shear modulus in both the linear and nonlinear regimes. We compute all the critical exponents for the two lattice models and discuss the applicability of our results to real systems.
10. Bardeen-Cooper-Schrieffer universal constants generalized
International Nuclear Information System (INIS)
Hazaimeh, A.H.
1992-01-01
Weak- and moderate-coupling BCS superconductivity theory is shown to admit a more general T c formula, wherein T c approaches zero somewhat faster than with the familiar BCS T c -formula. This theory leads to a departure from the universal behavior of the gap-to-T c ratio and is consistent with some recent empirical values for exotic superconductors. This ratio is smaller than the universal BCS value of 3.53 in a way which is consistent with weak electron-boson coupling. Similarly, other universal constants related to specific heat and critical magnetic field are modified. In this dissertation, The author investigates the latter constants for weak-coupling and moderate-coupling and carry out detailed comparisons with experimental data for the cuprates and with the corresponding predictions of strong-coupling theory. This effort is to elucidate the nature of these superconductors with regards to coupling strength within an electron-boson mechanism
11. An incremental theory of magneto-elastic hysteresis in pseudo-cubic ferro-magnetostrictive alloys
International Nuclear Information System (INIS)
Armstrong, W.D.
2003-01-01
This paper presents an incremental hysteretic magneto-elastic constitutive theory of pseudo-cubic ferro-magnetostrictive alloys, which may be used to predict the magneto-elastic response of these materials under quite general applied magnetic field and stress processes. These processes may include fully saturated major loop, unsaturated minor loop or more general types of magnetic field-stress processes. Comparisons between model results and a set of high quality measurements show that the model is capable of qualitative agreement with the experimental behavior. However, the experimental data shows a strongly decreasing magnetization and magnetostriction hysteresis as the minor loop cyclic applied field amplitude becomes smaller. It appears that irreversible domain wall motion continuously activates as the applied magnetic field is reduced in the range from 79 to 39 kA/m. Minor loop processes which do not have a sufficiently low minimum applied magnetic field value do not activate irreversible domain wall translation events and any measured magnetization changes must be due to one or more reversible mechanisms such as domain wall bowing and reversible domain rotation. The present model only includes an irreversible domain wall translation mechanism, therefore the model hysteresis widths remain constant with decreasing minor loop cyclic applied field amplitude
12. π-meson elastic form factor
International Nuclear Information System (INIS)
Gajewski, W.; Swiecki, M.
1975-01-01
This paper is devoted to the description of the present status of our knowledge about π meson elastic form factor. In the first section there is given a definition of the form factor on the basis of electromagnetic interactions, and its extension to weak and strong interactions. Next, there are reviewed theoretical models, which on the assumption of analyticity predict features of π meson form factor. In the last sections the results of experimental investigaton of π meson form factor are described with the stress on the results of the recent experiment made on Serpukhov accelerator in order to measure the π meson electromagnetic radius. (author)
13. Testing strong interaction theories
International Nuclear Information System (INIS)
Ellis, J.
1979-01-01
The author discusses possible tests of the current theories of the strong interaction, in particular, quantum chromodynamics. High energy e + e - interactions should provide an excellent means of studying the strong force. (W.D.L.)
14. Magnetic and elastic properties of the antiferromagnet uranium mononitride
International Nuclear Information System (INIS)
Van Doorn, C.F.
1976-10-01
The magnetic and elastic properties of antiferromagnetic uranium mononitride single crystals are studied in the thesis from the measurements of the temperature dependences of the magnetic susceptibility, electrical resistivity and elastic constants. The elastic constants C 11 , C 12 and C 44 were determined in the temperature interval 4 to 300 K by ultrasonic measurements of the five possible wave velocities in the [100] and [110] directions. A test for internal consistency was also made. A dip of about 9 percent occurs in C 11 at a temperature of 5 to 6 K lower than the Neel temperature T(N) (equals about 53 K). Starting at T(N), a renormalization in C 44 is proportional to the square of the sublattice magnetization also occurs. Both these results agree with model calculations which include spin-phonon interactions. The investigation of this anomaly was extended by measuring the electrical resistivity of a sample cut from the same crystal as that on which the elasticity was measured. No anomalous behavior was observed at the temperature where C 11 displays its anomaly. However, a discontinuity in the temperature derivative of the resistance was found at T(N). The possible effect of a magnetic field on the resistivity, as well as on the elasticity, was investigated without any measurable effect. The magnetic susceptibility was measured with a Foner magnetometer between 4 and 1 000 K. It was found that above the Neel temperature the paramagnetic susceptibility followed a revised Curie-Weiss law. In an attempt to ascertain the ionic state of the 5f-uranium ion in UN, use was made of the experimentally determined Weiss constant, spin disorder resistivity and Knight shift. A calculation was made that gave a good representation of the ratio of the experimental susceptibilities along the [100] and [110] directions in the ordered region [af
15. Varying Constants, Gravitation and Cosmology.
Science.gov (United States)
Uzan, Jean-Philippe
2011-01-01
Fundamental constants are a cornerstone of our physical laws. Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to matter. This will induce a violation of the universality of free fall. Thus, it is of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy. We detail the relations between the constants, the tests of the local position invariance and of the universality of free fall. We then review the main experimental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, solar system observations, meteorite dating, quasar absorption spectra, stellar physics, pulsar timing, the cosmic microwave background and big bang nucleosynthesis. At each step we describe the basics of each system, its dependence with respect to the constants, the known systematic effects and the most recent constraints that have been obtained. We then describe the main theoretical frameworks in which the low-energy constants may actually be varying and we focus on the unification mechanisms and the relations between the variation of different constants. To finish, we discuss the more speculative possibility of understanding their numerical values and the apparent fine-tuning that they confront us with.
16. Varying Constants, Gravitation and Cosmology
Directory of Open Access Journals (Sweden)
Jean-Philippe Uzan
2011-03-01
Full Text Available Fundamental constants are a cornerstone of our physical laws. Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to matter. This will induce a violation of the universality of free fall. Thus, it is of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy. We detail the relations between the constants, the tests of the local position invariance and of the universality of free fall. We then review the main experimental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, solar system observations, meteorite dating, quasar absorption spectra, stellar physics, pulsar timing, the cosmic microwave background and big bang nucleosynthesis. At each step we describe the basics of each system, its dependence with respect to the constants, the known systematic effects and the most recent constraints that have been obtained. We then describe the main theoretical frameworks in which the low-energy constants may actually be varying and we focus on the unification mechanisms and the relations between the variation of different constants. To finish, we discuss the more speculative possibility of understanding their numerical values and the apparent fine-tuning that they confront us with.
17. Elastic properties of Ti-24Nb-4Zr-8Sn single crystals with bcc crystal structure
International Nuclear Information System (INIS)
Zhang, Y.W.; Li, S.J.; Obbard, E.G.; Wang, H.; Wang, S.C.; Hao, Y.L.; Yang, R.
2011-01-01
Research highlights: → The single crystals of Ti2448 alloy with the bcc crystal structure were prepared. → The elastic moduli and constants were measured by several resonant methods. → The crystal shows significant elastic asymmetry in tension and compression. → The crystal exhibits weak nonlinear elasticity with large elastic strain ∼2.5%. → The crystal has weak atomic interactions against crystal distortion to low symmetry. - Abstract: Single crystals of Ti2448 alloy (Ti-24Nb-4Zr-8Sn in wt.%) were grown successfully using an optical floating-zone furnace. Several kinds of resonant methods gave consistent Young's moduli of 27.1, 56.3 and 88.1 GPa and shear moduli of 34.8, 11.0 and 14.6 GPa for the , and oriented single crystals, and C 11 , C 12 and C 44 of 57.2, 36.1 and 35.9 GPa respectively. Uniaxial testing revealed asymmetrical elastic behaviors of the crystals: tension caused elastic softening with a large reversible strain of ∼4% and a stress plateau of ∼250 MPa, whereas compression resulted in gradual elastic stiffening with much smaller reversible strain. The crystals exhibited weak nonlinear elasticity with a large elastic strain of ∼2.5% and a high strength, approaching ∼20% and ∼30% of its ideal shear and ideal tensile strength respectively. The crystals showed linear elasticity with a small elastic strain of ∼1%. These elastic deformation characteristics have been interpreted in terms of weakened atomic interactions against crystal distortion to low crystal symmetry under external applied stresses. These results are consistent with the properties of polycrystalline Ti2448, including high strength, low elastic modulus, large recoverable strain and weak strengthening effect due to grain refinement.
18. High pressure structural, elastic and vibrational properties of green energetic oxidizer ammonium dinitramide
Science.gov (United States)
Yedukondalu, N.; Ghule, Vikas D.; Vaitheeswaran, G.
2016-08-01
Ammonium DiNitramide (ADN) is one of the most promising green energetic oxidizers for future rocket propellant formulations. In the present work, we report a detailed theoretical study on structural, elastic, and vibrational properties of the emerging oxidizer under hydrostatic compression using various dispersion correction methods to capture weak intermolecular (van der Waals and hydrogen bonding) interactions. The calculated ground state lattice parameters, axial compressibilities, and equation of state are in good accord with the available experimental results. Strength of intermolecular interactions has been correlated using the calculated compressibility curves and elastic moduli. Apart from this, we also observe discontinuities in the structural parameters and elastic constants as a function of pressure. Pictorial representation and quantification of intermolecular interactions are described by the 3D Hirshfeld surfaces and 2D finger print maps. In addition, the computed infra-red (IR) spectra at ambient pressure reveal that ADN is found to have more hygroscopic nature over Ammonium Perchlorate (AP) due to the presence of strong hydrogen bonding. Pressure dependent IR spectra show blue- and red-shift of bending and stretching frequencies which leads to weakening and strengthening of the hydrogen bonding below and above 5 GPa, respectively. The abrupt changes in the calculated structural, mechanical, and IR spectra suggest that ADN might undergo a first order structural transformation to a high pressure phase around 5-6 GPa. From the predicted detonation properties, ADN is found to have high and low performance characteristics (DCJ = 8.09 km/s and PCJ = 25.54 GPa) when compared with ammonium based energetic oxidizers (DCJ = 6.50 km/s and PCJ = 17.64 GPa for AP, DCJ = 7.28 km/s and PCJ = 18.71 GPa for ammonium nitrate) and well-known secondary explosives for which DCJ = ˜8-10 km/s and PCJ = ˜30-50 GPa, respectively.
19. bessel functions for axisymmetric elasticity problems of the elastic
African Journals Online (AJOL)
HOD
. ) ( ) r. (. ) ( ). The governing partial differential equation for axisymmetric elasticity problems are the strain- displacement equations, the differential equations of equilibrium and the material constitutive laws, subject to the displacement and ...
20. From the Rydberg constant to the fundamental constants metrology
International Nuclear Information System (INIS)
Nez, F.
2005-06-01
This document reviews the theoretical and experimental achievements of the author since the beginning of his scientific career. This document is dedicated to the spectroscopy of hydrogen, deuterium and helium atoms. The first part is divided into 6 sub-sections: 1) the principles of hydrogen spectroscopy, 2) the measurement of the 2S-nS/nD transitions, 3) other optical frequency measurements, 4) our contribution to the determination of the Rydberg constant, 5) our current experiment on the 1S-3S transition, 6) the spectroscopy of the muonic hydrogen. Our experiments have improved the accuracy of the Rydberg Constant by a factor 25 in 15 years and we have achieved the first absolute optical frequency measurement of a transition in hydrogen. The second part is dedicated to the measurement of the fine structure constant and the last part deals with helium spectroscopy and the search for optical references in the near infrared range. (A.C.)
1. High energy elastic hadron scattering
International Nuclear Information System (INIS)
Fearnly, T.A.
1986-04-01
The paper deals with the WA7 experiment at the CERN super proton synchrotron (SPS). The elastic differential cross sections of pion-proton, kaon-proton, antiproton-proton, and proton-proton at lower SPS energies over a wide range of momentum transfer were measured. Some theoretical models in the light of the experimental results are reviewed, and a comprehensive impact parameter analysis of antiproton-proton elastic scattering over a wide energy range is presented. A nucleon valence core model for high energy proton-proton and antiproton-proton elastic scattering is described
2. The first-principles calculations for the elastic properties of Zr2Al under compression
International Nuclear Information System (INIS)
Yuan Xiaoli; Wei Dongqing; Chen Xiangrong; Zhang Qingming; Gong Zizheng
2011-01-01
Graphical abstract: The calculated elastic constants C ij as a function of pressure P. Display Omitted Research highlights: → It is found that the five independent elastic constants increase monotonically with pressure. C 11 and C 33 vary rapidly as pressure increases, C 13 and C 12 becomes moderate. However, C 44 increases comparatively slowly with pressure. Figure shows excellent satisfaction of the calculated elastic constants of Zr 2 Al to these equations and hence in our calculation, the Zr 2 Al is mechanically stable at pressure up to 100 GPa. - Abstract: The first-principles calculations were applied to investigate the structural, elastic constants of Zr 2 Al alloy with increasing pressure. These properties are based on the plane wave pseudopotential density functional theory (DFT) method within the generalized gradient approximation (GGA) for exchange and correlation. The result of the heat of formation of Zr 2 Al crystal investigated is in excellent consistent with results from other study. The anisotropy, the shear modulus, and Young's modulus for the ideal polycrystalline Zr 2 Al are also studied. It is found that (higher) pressure can significantly improve the ductility of Zr 2 Al. Moreover, the elastic constants of Zr 2 Al increase monotonically and the anisotropies decrease with the increasing pressure. Finally, it is observed that Zr d electrons are mainly contributed to the density of states at the Fermi level.
3. Structural, elastic, electronic and optical properties of bi-alkali ...
Indian Academy of Sciences (India)
The structural parameters, elastic constants, electronic and optical properties of the bi-alkali antimonides (Na 2 KSb, Na 2 RbSb, Na 2 CsSb, K 2 RbSb, K 2 CsSb and Rb 2 CsSb) were calculated using state-of-the-art density functional theory. Different exchange-correlation potentials were adopted to predict the physical ...
4. Structural, elastic, electronic and optical properties of bi-alkali ...
Indian Academy of Sciences (India)
The structural parameters, elastic constants, electronic and optical properties of the bi-alkali antimonides (Na2KSb, Na2RbSb, Na2CsSb, K2RbSb, K2CsSb and Rb2CsSb) were calculated using state-of-the-art density functional theory. Different exchange-correlation potentials were adopted to predict the physical properties.
5. Structural, elastic, electronic and optical properties of bi-alkali
Indian Academy of Sciences (India)
The structural parameters, elastic constants, electronic and optical properties of the bi-alkali antimonides (Na 2 KSb, Na 2 RbSb, Na 2 CsSb, K 2 RbSb, K 2 CsSb and Rb 2 CsSb) were calculated using state-of-the-art density functional theory. Different exchange-correlation potentials were adopted to predict the physical ...
6. Constant fields and constant gradients in open ionic channels.
Science.gov (United States)
Chen, D P; Barcilon, V; Eisenberg, R S
1992-05-01
Ions enter cells through pores in proteins that are holes in dielectrics. The energy of interaction between ion and charge induced on the dielectric is many kT, and so the dielectric properties of channel and pore are important. We describe ionic movement by (three-dimensional) Nemst-Planck equations (including flux and net charge). Potential is described by Poisson's equation in the pore and Laplace's equation in the channel wall, allowing induced but not permanent charge. Asymptotic expansions are constructed exploiting the long narrow shape of the pore and the relatively high dielectric constant of the pore's contents. The resulting one-dimensional equations can be integrated numerically; they can be analyzed when channels are short or long (compared with the Debye length). Traditional constant field equations are derived if the induced charge is small, e.g., if the channel is short or if the total concentration gradient is zero. A constant gradient of concentration is derived if the channel is long. Plots directly comparable to experiments are given of current vs voltage, reversal potential vs. concentration, and slope conductance vs. concentration. This dielectric theory can easily be tested: its parameters can be determined by traditional constant field measurements. The dielectric theory then predicts current-voltage relations quite different from constant field, usually more linear, when gradients of total concentration are imposed. Numerical analysis shows that the interaction of ion and channel can be described by a mean potential if, but only if, the induced charge is negligible, that is to say, the electric field is spatially constant.
7. Foam-Driven Fractures of an Elastic Matrix
Science.gov (United States)
Lai, Ching-Yao; Smiddy, Sam; Stone, Howard
2015-11-01
We report an experimental study of foam-driven fractures in an elastic matrix. When a pressurized foam is constantly injected into a gelatin matrix with a constant flow rate, the foam generates a disc-like fracture which is commonly observed in liquid-driven fractures. Compare to liquid-driven fractures, foam-driven fractures grow faster with time. We investigate how the rheological behaviour of foams affects the fracture characteristics by varying the air volume fraction of the foam, the types and concentration of surfactants in the foam. Foam-fracturing reduces the environmental costs of hydraulic fracturing, which inspires this laboratory study.
8. Learning Read-constant Polynomials of Constant Degree modulo Composites
DEFF Research Database (Denmark)
Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt
2011-01-01
Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class \\textACC0ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class...... is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...
9. Dynamic visco-elastic properties of dental composite resins.
Science.gov (United States)
Mesquita, Renata V; Axmann, Detlef; Geis-Gerstorfer, Jürgen
2006-03-01
This study aimed to examine the visco-elastic properties of dental composites by dynamic mechanical analysis under the influence of clinically relevant temperatures and variable frequencies, after being stored in air or distilled water for up to 3 months. Two direct (Diamond Lite and Grandio) and two indirect (Artglass and Vita Zeta LC) composites were used. Samples were immediately tested (baseline) or stored at 37 degrees C, either in air or distilled water for 1 day, 7 or 90 days before testing. During dynamic testing, elastic modulus, viscous modulus and loss tangent were determined over a frequency range from 0.1 to 10 Hz at constant temperatures between 5 and 55 degrees C. Results were analyzed by one-way ANOVA and Turkey's-test. Elastic and viscous moduli were higher for direct than for indirect composites. No such evidence was found for loss tangent. Only the elastic modulus showed statistically relevant differences in the direct and indirect materials groups: Grandio showed higher modulus than Diamond Lite, while Artglass had higher modulus than Vita Zeta LC. The elastic modulus reduced with increasing temperature and decreasing frequency, while the loss tangent showed the opposite trend. The influence of temperature and frequency on viscous modulus was not conclusive. The elastic modulus was more sensitive to moisture than viscous modulus and loss tangent but all three properties showed no overall consistent trend in the results following the storage periods. Dynamic mechanical analysis was a valuable tool to characterize the visco-elastic properties of dental composites, thus giving us a greater insight into material behavior.
10. Ultrabroadband elastic cloaking in thin plates.
Science.gov (United States)
Farhat, Mohamed; Guenneau, Sebastien; Enoch, Stefan
2009-07-10
Control of waves with metamaterials is of great topical interest, and is fueled by rapid progress in broadband acoustic and electromagnetic cloaks. We propose a design for a cloak to control bending waves propagating in isotropic heterogeneous thin plates. This is achieved through homogenization of a multilayered concentric coating filled with piecewise constant isotropic elastic material. Significantly, our cloak displays no phase shift for both backward and forward scattering. To foster experimental efforts, we provide a simplified design of the cloak which is shown to work in a more than two-octave frequency range (30 Hz to 150 Hz) when it consists of 10 layers using only 6 different materials overall. This metamaterial should be easy to manufacture, with potential applications ranging from car industry to anti-earthquake passive systems for smart buildings, depending upon the plate dimensions and wavelengths.
11. Price and income elasticities of residential energy demand in Germany
International Nuclear Information System (INIS)
Schulte, Isabella; Heindl, Peter
2017-01-01
We apply a quadratic expenditure system to estimate price and expenditure elasticities of residential energy demand (electricity and heating) in Germany. Using official expenditure data from 1993 to 2008, we estimate an expenditure elasticity for electricity of 0.3988 and of 0.4055 for space heating. The own price elasticity for electricity is −0.4310 and −0.5008 in the case of space heating. Disaggregation of households by expenditure and socio-economic composition reveals that the behavioural response to energy price changes is weaker (stronger) for low-income (top-income) households. There are considerable economies of scale in residential energy use but scale effects are not well approximated by the new OECD equivalence scale. Real increases in energy prices show a regressive pattern of incidence, implying that the welfare consequences of direct energy taxation are larger for low income households. The application of zero-elasticities in assessments of welfare consequences of energy taxation strongly underestimates potential welfare effects. The increase in inequality is 22% smaller when compared to the application of disaggregated price and income elasticities as estimated in this paper. - Highlights: • We estimate price, income, and expenditure elasticities for residential energy demand in Germany. • We differentiate elasticities by income groups and household type. • Electricity and space heating are necessary goods since the expenditure elasticities are smaller than unity. • Low-income households show a weaker reaction to changing prices when compared to high-income households. • Direct energy taxation has regressive effects, meaning that larger burdens fall upon low-income households.
12. Reliable measurement of elastic modulus of cells by nanoindentation in an atomic force microscope
KAUST Repository
Zhou, Zhoulong
2012-04-01
The elastic modulus of an oral cancer cell line UM1 is investigated by nanoindentation in an atomic force microscope with a flat-ended tip. The commonly used Hertzian method gives apparent elastic modulus which increases with the loading rate, indicating strong effects of viscoelasticity. On the contrary, a rate-jump method developed for viscoelastic materials gives elastic modulus values which are independent of the rate-jump magnitude. The results show that the rate-jump method can be used as a standard protocol for measuring elastic stiffness of living cells, since the measured values are intrinsic properties of the cells. © 2011 Elsevier Ltd.
13. Chirality-dependent anisotropic elastic properties of a monolayer graphene nanosheet.
Science.gov (United States)
Guo, Jian-Gang; Zhou, Li-Jun; Kang, Yi-Lan
2012-04-01
An analytical approach is presented to predict the elastic properties of a monolayer graphene nanosheet based on interatomic potential energy and continuum mechanics. The elastic extension and torsional springs are utilized to simulate the stretching and angle variation of carbon-carbon bond, respectively. The constitutive equation of the graphene nanosheet is derived by using the strain energy density, and the analytical formulations for nonzero elastic constants are obtained. The in-plane elastic properties of the monolayer graphene nanosheet are proved to be anisotropic. In addition, Young's moduli, Poisson's ratios and shear modulus of the monolayer graphene nanosheet are calculated according to the force constants derived from Morse potential and AMBER force field, respectively, and they were proved to be chirality-dependent. The comparison with experimental results shows a very agreement.
14. Astronomical optics and elasticity theory
CERN Document Server
Lemaitre, Gerard Rene
2008-01-01
Astronomical Optics and Elasticity Theory provides a very thorough and comprehensive account of what is known in this field. After an extensive introduction to optics and elasticity, the book discusses variable curvature and multimode deformable mirrors, as well as, in depth, active optics, its theory and applications. Further, optical design utilizing the Schmidt concept and various types of Schmidt correctors, as well as the elasticity theory of thin plates and shells are elaborated upon. Several active optics methods are developed for obtaining aberration corrected diffraction gratings. Further, a weakly conical shell theory of elasticity is elaborated for the aspherization of grazing incidence telescope mirrors. The very didactic and fairly easy-to-read presentation of the topic will enable PhD students and young researchers to actively participate in challenging astronomical optics and instrumentation projects.
15. Uniqueness theorems in linear elasticity
CERN Document Server
Knops, Robin John
1971-01-01
The classical result for uniqueness in elasticity theory is due to Kirchhoff. It states that the standard mixed boundary value problem for a homogeneous isotropic linear elastic material in equilibrium and occupying a bounded three-dimensional region of space possesses at most one solution in the classical sense, provided the Lame and shear moduli, A and J1 respectively, obey the inequalities (3 A + 2 J1) > 0 and J1>O. In linear elastodynamics the analogous result, due to Neumann, is that the initial-mixed boundary value problem possesses at most one solution provided the elastic moduli satisfy the same set of inequalities as in Kirchhoffs theorem. Most standard textbooks on the linear theory of elasticity mention only these two classical criteria for uniqueness and neglect altogether the abundant literature which has appeared since the original publications of Kirchhoff. To remedy this deficiency it seems appropriate to attempt a coherent description ofthe various contributions made to the study of uniquenes...
16. DNA Bending elasticity
Science.gov (United States)
Sivak, David Alexander
DNA bending elasticity on length scales of tens of basepairs is of critical importance in numerous biological contexts. Even the simplest models of DNA bending admit of few simple analytic results, thus there is a need for numerical methods to calculate experimental observables, such as distance distributions, forces, FRET efficiencies, and timescales of particular large-scale motions. We have implemented and helped develop a coarse-grained representation of DNA and various other covalently-linked groups that allows simple calculation of such observables for varied experimental systems. The simple freely-jointed chain (FJC) model and extremely coarse resolution proved useful in understanding DNA threading through nanopores, identifying steric occlusion by other parts of the chain as a prime culprit for slower capture as distance to the pore decreased. Enhanced sampling techniques of a finer resolution discrete wormlike chain (WLC) model permitted calculation of cyclization rates for small chains and identified the ramifications of a thermodynamically-sound treatment of thermal melts. Adding treatment of double-stranded DNA's helical nature and single-stranded DNA provided a model system that helped demonstrate the importance of statistical fluctuations in even highly-stressed DNA mini-loops, and allowed us to verify that even these constructs show no evidence of excitation-induced softening. Additional incorporation of salt-sensitivity to the model allowed us to calculate forces and FRET efficiencies for such mini-loops and their uncircularized precursors, thereby furthering the understanding of the nature of IHF binding and bending of its recognition sequence. Adding large volume-excluding spheres linked to the ends of the dsDNA permits calculation of distance distributions and thus small-angle X-ray scattering, whereby we demonstrated the validity of the WLC in describing bending fluctuations in DNA chains as short as 42 bp. We also make important connections
17. Yielding in a strongly aggregated colloidal gel: 2D simulations and theory
Science.gov (United States)
Roy, Saikat; Tirumkudulu, Mahesh
2015-11-01
We investigated the micro-structural details and the mechanical response under uniaxial compression of the strongly aggregating gel starting from low to high packing fraction.The numerical simulations account for short-range inter-particle attractions, normal and tangential deformation at particle contacts,sliding and rolling friction, and preparation history. It is observed that in the absence of rolling resistance(RR),the average coordination number varies only slightly with compaction whereas it is significant in the presence of RR. The particle contact distribution is isotropic throughout the consolidation process. In both cases, the yield strain is constant with the volume fraction. The modulus values are very similar at different attraction, and with and without RR implying that the elastic modulus does not scale with attraction.The modulus was found to be a weak function of the preparation history. The increase in yield stress with volume fraction is a consequence of the increased elastic modulus of the network. However, the yield stress scales similarly both with and without RR. The power law exponent of 5.4 is in good agreement with previous simulation results. A micromechanical theory is also proposed to describe the stress versus strain relation for the gelled network.
18. From the Rydberg constant to the fundamental constants metrology; De la constante de Rydberg a la metrologie des constantes fondamentales
Energy Technology Data Exchange (ETDEWEB)
Nez, F
2005-06-15
This document reviews the theoretical and experimental achievements of the author since the beginning of his scientific career. This document is dedicated to the spectroscopy of hydrogen, deuterium and helium atoms. The first part is divided into 6 sub-sections: 1) the principles of hydrogen spectroscopy, 2) the measurement of the 2S-nS/nD transitions, 3) other optical frequency measurements, 4) our contribution to the determination of the Rydberg constant, 5) our current experiment on the 1S-3S transition, 6) the spectroscopy of the muonic hydrogen. Our experiments have improved the accuracy of the Rydberg Constant by a factor 25 in 15 years and we have achieved the first absolute optical frequency measurement of a transition in hydrogen. The second part is dedicated to the measurement of the fine structure constant and the last part deals with helium spectroscopy and the search for optical references in the near infrared range. (A.C.)
19. Systematics of constant roll inflation
Science.gov (United States)
Anguelova, Lilia; Suranyi, Peter; Wijewardhana, L. C. R.
2018-02-01
We study constant roll inflation systematically. This is a regime, in which the slow roll approximation can be violated. It has long been thought that this approximation is necessary for agreement with observations. However, recently it was understood that there can be inflationary models with a constant, and not necessarily small, rate of roll that are both stable and compatible with the observational constraint ns ≈ 1. We investigate systematically the condition for such a constant-roll regime. In the process, we find a whole new class of inflationary models, in addition to the known solutions. We show that the new models are stable under scalar perturbations. Finally, we find a part of their parameter space, in which they produce a nearly scale-invariant scalar power spectrum, as needed for observational viability.
20. Integrodifferential relations in linear elasticity
CERN Document Server
Kostin, Georgy V
2012-01-01
This work treats the elasticity of deformed bodies, including the resulting interior stresses and displacements.It also takes into account that some of constitutive relations can be considered in a weak form. To discuss this problem properly, the method of integrodifferential relations is used, and an advanced numerical technique for stress-strain analysis is presented and evaluated using various discretization techniques. The methods presented in this book are of importance for almost all elasticity problems in materials science and mechanical engineering.
1. Elastic and viscous properties of the nematic dimer CB7CB
Science.gov (United States)
Babakhanova, Greta; Parsouzi, Zeinab; Paladugu, Sathyanarayana; Wang, Hao; Nastishin, Yu. A.; Shiyanovskii, Sergij V.; Sprunt, Samuel; Lavrentovich, Oleg D.
2017-12-01
We present a comprehensive set of measurements of optical, dielectric, diamagnetic, elastic, and viscous properties in the nematic (N) phase formed by a liquid crystalline dimer. The studied dimer, 1,7-bis-4-(4'-cyanobiphenyl) heptane (CB7CB), is composed of two rigid rodlike cyanobiphenyl segments connected by a flexible aliphatic link with seven methyl groups. CB7CB and other nematic dimers are of interest due to their tendency to adopt bent configurations and to form two states possessing a modulated nematic director structure, namely, the twist-bend nematic, NTB, and the oblique helicoidal cholesteric, C hOH , which occurs when the achiral dimer is doped with a chiral additive and exposed to an external electric or magnetic field. We characterize the material parameters as functions of temperature in the entire temperature range of the N phase, including the pretransitional regions near the N -NTB and N-to-isotropic (I) transitions. The splay constant K11 is determined by two direct and independent techniques, namely, detection of the Frederiks transition and measurement of director fluctuation amplitudes by dynamic light scattering (DLS). The bend K33 and twist K22 constants are measured by DLS. K33, being the smallest of the three constants, shows a strong nonmonotonous temperature dependence with a negative slope in both N-I and N -NTB pretransitional regions. The measured ratio K11/K22 is larger than 2 in the entire nematic temperature range. The orientational viscosities associated with splay, twist, and bend fluctuations in the N phase are comparable to those of nematics formed by rodlike molecules. All three show strong temperature dependence, increasing sharply near the N -NTB transition.
2. On the identification of the eggshell elastic properties under quasistatic compression
Directory of Open Access Journals (Sweden)
Jana Simeonovová
2004-01-01
Full Text Available The problem of the identification of the elastic properties of eggshell, i.e. the evaluation of the Young's modulus and Poisson's ratio is solved. The eggshell is considered as a rotational shell. The experiments on the egg compression under quasistatic loading have been conducted. During these experiments a strain on the eggshell surface has been recorded. By the mutual comparison between experimental and theoretical values of strains the influence of the elastic constants has been demonstrated.
3. Role of interparticle friction and particle-scale elasticity in the shear-strength mechanism of three-dimensional granular media.
Science.gov (United States)
Antony, S J; Kruyt, N P
2009-03-01
The interlink between particle-scale properties and macroscopic behavior of three-dimensional granular media subjected to mechanical loading is studied intensively by scientists and engineers, but not yet well understood. Here we study the role of key particle-scale properties, such as interparticle friction and particle elastic modulus, in the functioning of dual contact force networks, viz., strong and weak contacts, in mobilizing shear strength in dense granular media subjected to quasistatic shearing. The study is based on three-dimensional discrete element method in which particle-scale constitutive relations are based on well-established nonlinear theories of contact mechanics. The underlying distinctive contributions of these force networks to the macroscopic stress tensor of sheared granular media are examined here in detail to find out how particle-scale friction and particle-scale elasticity (or particle-scale stiffness) affect the mechanism of mobilization of macroscopic shear strength and other related properties. We reveal that interparticle friction mobilizes shear strength through bimodal contribution, i.e., through both major and minor principal stresses. However, against expectation, the contribution of particle-scale elasticity is mostly unimodal, i.e., through the minor principal stress component, but hardly by the major principal stress. The packing fraction and the geometric stability of the assemblies (expressed by the mechanical coordination number) increase for decrease in interparticle friction and elasticity of particles. Although peak shear strength increases with interparticle friction, the deviator strain level at which granular systems attain peak shear strength is mostly independent of interparticle friction. Granular assemblies attain peak shear strength (and maximum fabric anisotropy of strong contacts) when a critical value of the mechanical coordination number is attained. Irrespective of the interparticle friction and elasticity
4. Effect of price elasticity of demand in monopolies with gradient adjustment
International Nuclear Information System (INIS)
2015-01-01
Highlights: •A monopoly with isoelastic demand function is studied. •Reduced rationality monopolist uses gradient adjustment. •If marginal cost is small, increasing elasticity leads to stable dynamics. •For large marginal cost, dynamic can be unstable for both small and large elasticity. -- Abstract: We study a monopolistic market characterized by a constant elasticity demand function, in which the firm technology is described by a linear total cost function. The firm is assumed to be boundedly rational and to follow a gradient rule to adjust the production level in order to optimize its profit. We focus on what happens on varying the price elasticity of demand, studying the effect on the equilibrium stability. We prove that, depending on the relation between the market size and the marginal cost, two different scenarios are possible, in which elasticity has either a stabilizing or a mixed stabilizing/destabilizing effect
5. The πHe3H3 coupling constant estimation using the Chew-Low equation
International Nuclear Information System (INIS)
Mach, R.; Nichitiu, F.
1975-01-01
In this paper it is presented an estimation of the πHe 3 H 3 coupling constant using the Chew-Low equation and a semi-phenomenological analysis of the π -+ He 3 elastic differential cross sections at 98, 120, 135 and 156 MeV
6. Amphiphile regulation of ion channel function by changes in the bilayer spring constant
DEFF Research Database (Denmark)
Lundbæk, Jens August; Koeppe, R.E.; Andersen, Oluf Sten
2010-01-01
effect of amphiphiles, at concentrations often used in biological research, on the bilayer elastic response to a change in the hydrophobic length of an embedded protein. The effects of structurally diverse amphiphiles can be described by changes in a phenomenological bilayer spring constant (H...
7. Elastic properties of Ni2MnGa from first-principles calculations
International Nuclear Information System (INIS)
Ozdemir Kart, S.; Cagin, T.
2010-01-01
Research highlights: In this study, we have performed spin-polarized total energy calculations aiming to develop microscopic understanding of magnetic shape memory behavior of Ni 2 MnGa. This paper is devoted to determine the mechanical properties of Ni 2 MnGa in both austenitic and martensitic structures. To the best of our knowledge, this work presents the elastic constants of Ni 2 MnGa in the structure of 5M martensite, for the first time. We have also re-calculated elastic constants for cubic and nonmodulated (NM) structures by using the potential with e/a = 7.5. The elastic constants are predicted by straining the cubic L2 1 , 5M pseudo-tetragonal and NM tetragonal martensitic structures. Because of the special significance of the isotropic bulk modulus, shear modulus, Young's modulus and Poisson's ratio for technological applications, we have also calculated these quantities from the elastic constants. - Abstract: Elastic properties of Ni 2 MnGa in both austenitic and martensitic structures are determined by using ab initio methods based on density functional theory (DFT) within the spin-polarized generalized-gradient approximation. The tetragonal shear elastic constant C' takes a very small value in the austenitic phase, indicating the elastic instability results in a phase transition to martensitic structure. Isotropic mechanical properties such as bulk modulus, shear modulus, Young's modulus and Poisson's ratio are predicted. The trend of the Debye temperatures calculated for three structures of Ni 2 MnGa is comparable with that of the experiment.
8. Non-constant retardation coefficient
International Nuclear Information System (INIS)
Wang Zhiming; Gu Zhijie; Yang Yue'e; Li Shushen
2004-12-01
Retardation coefficient is one of the important parameters used in transport models describing radionuclide migration in geological media and usually regarded as a constant in the models. The objectives of the work are to understand: (1) Whether the retardation coefficient, R d , is a constant? (2) How much effect is R d on calculated consequence if R d is not constant? (3) Is the retardation coefficient derived from distribution coefficient, k d , according to conventional equation suitable for safety assessment? The objectives are achieved through test and analysis of the test results on radionuclide migration in unsaturated loess. It can be seen from the results that retardation coefficient, R d , of 85 Sr is not constant and increases with water content, θ, under unsaturated condition. R d , of 85 Sr derived from k d according to conventional equation can not be used for safety assessment. R d , used for safety assessment should be directly measured, rather than derived from k d . It is shown from calculation that the effect of R d on calculated consequence is very considerable. (authors)
9. Universal relation between spectroscopic constants
Indian Academy of Sciences (India)
(3) The author has used eq. (6) of his paper to calculate De. This relation leads to a large deviation from the correct value depending upon the extent to which experimental values are known. Guided by this fact, in our work, we used experimentally observed De values to derive the relation between spectroscopic constants.
10. Elasticity of Pargasite Amphibole: A Hydrous Phase at Mid Lithospheric Discontinuity
Science.gov (United States)
Peng, Y.; Mookherjee, M.
2017-12-01
Mid Lithospheric Discontinuity (MLD) is characterized by a low shear wave velocity ( 3 to 10 %). In cratons, the depth of MLD varies between 80 and 100 km. The reduction of the shear wave velocity at MLD is similar to what is observed in the lithosphere-asthenosphere boundary (LAB). Such low velocity at MLD could be caused by partial melting, temperature induced grain boundary sliding, changes in the elastic anisotropy, and/or metasomatism which may lead to the formation of hydrous phases including mica and amphibole. Thus, it is clear that in order to assess the role of metasomatism at MLD, we need better constraints on the elasticity of hydrous phases. However, such elasticity data are scarce. In this study, we explore elasticity of pargasite amphibole [NaCa2(Mg4Al)(Si6Al2)O22(OH)2] using density functional theory (DFT) with local density approximation (LDA) and generalized gradient approximation (GGA). We find that the pressure-volume results can be adequately described by a finite strain equation with the bulk modulus, K0 being 102 and 85 GPa for LDA and GGA respectively. We also determined the full elastic constant tensor (Cij) using the finite difference method. The bulk modulus, K0 determined from the full elastic constant tensor is 104 GPa for LDA and 87 GPa for GGA. The shear modulus, G0 determined from the full elastic constant tensor is 64 GPa for LDA and 58 GPa for GGA. The bulk and shear moduli predicted with LDA are 5 and 1 % stiffer than the recent results [1]. In contrast, the bulk and shear moduli predicted with GGA are 12 and 10 % softer compared to the recent results [1]. The full elastic constant tensor for pargasite shows significant anisotropy. For instance, LDA predicts compressional (AVP) and shear (AVS) wave anisotropy of 22 and 20 % respectively. At higher pressure, elastic moduli stiffen. However, temperature is likely to have an opposite effect on the elasticity and this remains largely unknown for pargasite. Compared to the major mantle
11. Theoretical study of structural, elastic and thermodynamic properties of CZTX (X = S and Se) alloys
Energy Technology Data Exchange (ETDEWEB)
Bensalem, S., E-mail: [email protected] [Centre de Développement des Energies Renouvelables, CDER, BP 62 Route de l’Observatoire Bouzaréah, 16340 Algiers (Algeria); Département de Physique, Faculté des Sciences, Université de Sétif 1, 19000 Sétif (Algeria); Chegaar, M. [Département de Physique, Faculté des Sciences, Université de Sétif 1, 19000 Sétif (Algeria); Laboratoire d’Optoélectronique et Composants, Université de Sétif 1, 19000 Sétif (Algeria); Maouche, D.; Bouhemadou, A. [Laboratoire de Développement de Nouveaux Matériaux et leurs Caractérisations, Université de Sétif 1, 19000 Sétif (Algeria)
2014-03-15
Highlights: • CZTX (X = S, Se) alloys are relatively new absorbers for solar cells applications. • Elastic and thermodynamic properties of these alloys are not well understood. • The considered types “kesterite and stannite” can coexist in experimental samples. • Elastic and thermodynamic properties of both types have been investigated. • Coexistence of both types does not influence the behavior of CZTX-based devices. -- Abstract: By means of first-principles calculation approach, structural parameters, elastic and thermodynamic properties of Copper–Zinc–Tin–(Sulphide, Selenide) or Cu{sub 2}ZnSnX{sub 4} (X = S and Se) alloys for the kesterite (KS) and stannite (ST) types have been investigated. The calculated lattice parameters are in good agreement with experimental reported data. The elastic constants are calculated for both types of both compounds using the static finite strain scheme; the pressure dependence of elastic constants is predicted. The bulk modulus, anisotropy factor, shear modulus, Young’s modulus, Lame’s coefficient and Poisson’s ratio have been estimated from the calculated single crystalline elastic constants. The analysis of B/G ratio shows that Cu{sub 2}ZnSnX{sub 4} or CZTX compounds behave as ductile. Through quasi-harmonic approximation, the temperature dependence of some thermodynamic functions and lattice heat capacity of both compounds for both types have been performed.
12. Photo-elastic properties of the wing imaginal disc of Drosophila.
Science.gov (United States)
Schluck, T; Aegerter, C M
2010-10-01
In the study of developmental biology, the physical properties and constraints of the developing tissues are of great importance. In spite of this, not much is known about the elastic properties of biologically relevant tissues that are studied in biology labs. Here, we characterize properties of the wing imaginal disc of Drosophila, which is a precursor organ intensely studied in the framework of growth control and cell polarity. In order to determine the possibility of measuring mechanical stresses inside the tissue during development, we quantify the photo-elastic properties of the tissue by direct mechanical manipulation. We obtain a photo-elastic constant of 2 x 10(-10) Pa(-1).
13. Elastic models for the non-Arrhenius relaxation time of glass-forming liquids
DEFF Research Database (Denmark)
Dyre, Jeppe
-time elastic properties are all determined by just one effective, temperature-dependent force constant). We finally discuss the connection between the elastic models and two well-established research fields of condensed-matter physics: point defects in crystals and solid-state diffusion.......We first review the phenomenology of viscous liquids and the standard models used for explaining the non-Arrhenius average relaxation time. Then the focus is turned to the so-called elastic models, arguing that these models are all equivalent in the Einstein approximation (where the short...
14. Elastic models for the Non-Arrhenius Relaxation Time of Glass-Forming Liquids
DEFF Research Database (Denmark)
Dyre, J. C.
2006-01-01
-time elastic properties are all determined by just one effective, temperature-dependent force constant). We finally discuss the connection between the elastic models and two well-established research fields of condensed-matter physics: point defects in crystals and solid-state diffusion.......We first review the phenomenology of viscous liquids and the standard models used for explaining the non-Arrhenius average relaxation time. Then the focus is turned to the so-called elastic models, arguing that these models are all equivalent in the Einstein approximation (where the short...
15. The influence of elastic subsystem on phase transitions in ferromagnets with competitive exchange and single-ion anisotropies
International Nuclear Information System (INIS)
Freedman, Yu.A.; Klevets, F.N.; Matunin, D.A.
2006-01-01
The influence of planar and bulk elastic interactions on the phase states of an ultrathin ferromagnetic film with anisotropic exchange interaction is investigated for different relationships among the material constants. It is shown that when the elastic interactions, with competing exchange and single-ion anisotropies, and the magnetic dipole interaction are taken into account, a cascade of phase transitions appears. Furthermore, taking the 'planar' elastic interaction into account leads to realization of an additional phase, with an easy axis in the film plane. This state is absent in the case of a bulk elastic subsystem
16. Structural phase transition and elastic properties of mercury chalcogenides
Energy Technology Data Exchange (ETDEWEB)
Varshney, Dinesh, E-mail: [email protected] [School of Physics, Vigyan Bhavan, Devi Ahilya University, Khandwa Road Campus, Indore 452001 (India); Shriya, S. [School of Physics, Vigyan Bhavan, Devi Ahilya University, Khandwa Road Campus, Indore 452001 (India); Khenata, R. [Laboratoire de Physique Quantique et de Modelisation Mathematique (LPQ3M), Departement de Technologie, Universite de Mascara, 29000 Mascara (Algeria)
2012-08-15
Pressure induced structural transition and elastic properties of ZnS-type (B3) to NaCl-type (B1) structure in mercury chalcogenides (HgX; X = S, Se and Te) are presented. An effective interionic interaction potential (EIOP) with long-range Coulomb, as well charge transfer interactions, Hafemeister and Flygare type short-range overlap repulsion extended up to the second neighbor ions and van der Waals interactions are considered. Emphasis is on the evaluation of the pressure dependent Poisson's ratio {nu}, the ratio R{sub BT/G} of B (bulk modulus) over G (shear modulus), anisotropy parameter, Shear and Young's modulus, Lame constant, Kleinman parameter, elastic wave velocity and thermodynamical property as Debye temperature. The Poisson's ratio behavior infers that Mercury chalcogenides are brittle in nature. To our knowledge this is the first quantitative theoretical prediction of the pressure dependence of elastic and thermodynamical properties explicitly the ductile (brittle) nature of HgX and still awaits experimental confirmations. Highlights: Black-Right-Pointing-Pointer Vast volume discontinuity in phase diagram infers transition from ZnS to NaCl structure. Black-Right-Pointing-Pointer The shear elastic constant C{sub 44} is nonzero confirms the mechanical stability. Black-Right-Pointing-Pointer Pressure dependence of {theta}{sub D} infers the softening of lattice with increasing pressure. Black-Right-Pointing-Pointer Estimated bulk, shear and tetragonal moduli satisfied elastic stability criteria. Black-Right-Pointing-Pointer In both B3 and B1 phases, C{sub 11} and C{sub 12} increase linearly with pressure.
17. Stabilized power constant alimentation; Alimentation regulee a puissance constante
Energy Technology Data Exchange (ETDEWEB)
Roussel, L. [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires
1968-06-01
The study and realization of a stabilized power alimentation variable from 5 to 100 watts are described. In order to realize a constant power drift of Lithium compensated diodes, we have searched a 1 per cent precision of regulation and a response time minus than 1 sec. Recent components like Hall multiplicator and integrated amplifiers give this possibility and it is easy to use permutable circuits. (author) [French] On decrit l'etude et la realisation d'une alimentation a puissance constante reglable dans une gamme de 5 a 100 watts. Prevue pour le drift a puissance constante des diodes compensees au lithium, l'etude a ete menee en vue d'obtenir une precision de regulation de 1 pour cent et un temps de reponse inferieur a la seconde. Des systemes recents tels que multiplicateurs a effet Hall et circuits integres ont permis d'atteindre ce but tout en facilitant l'emploi de modules interchangeables. (auteur)
18. Constant-bandwidth constant-temperature hot-wire anemometer.
Science.gov (United States)
Ligeza, P
2007-07-01
A constant-temperature anemometer (CTA) enables the measurement of fast-changing velocity fluctuations. In the classical solution of CTA, the transmission band is a function of flow velocity. This is a minor drawback when the mean flow velocity does not significantly change, though it might lead to dynamic errors when flow velocity varies over a considerable range. A modification is outlined, whereby an adaptive controller is incorporated in the CTA system such that the anemometer's transmission band remains constant in the function of flow velocity. For that purpose, a second feedback loop is provided, and the output signal from the anemometer will regulate the controller's parameters such that the transmission bandwidth remains constant. The mathematical model of a CTA that has been developed and model testing data allow a through evaluation of the proposed solution. A modified anemometer can be used in measurements of high-frequency variable flows in a wide range of velocities. The proposed modification allows the minimization of dynamic measurement errors.
19. High precision fundamental constants at the TeV scale
CERN Document Server
Moch, S.; Alekhin, S.; Blumlein, J.; de la Cruz, L.; Dittmaier, S.; Dowling, M.; Erler, J.; Espinosa, J.R.; Fuster, J.; Garcia i Tormo, X.; Hoang, A.H.; Huss, A.; Kluth, S.; Mulders, M.; Papanastasiou, A.S.; Piclum, J.; Rabbertz, K.; Schwinn, C.; Schulze, M.; Shintani, E.; Uwer, P.; Zerf, N.
2014-01-01
This report summarizes the proceedings of the 2014 Mainz Institute for Theoretical Physics (MITP) scientific program on "High precision fundamental constants at the TeV scale". The two outstanding parameters in the Standard Model dealt with during the MITP scientific program are the strong coupling constant $\\alpha_s$ and the top-quark mass $m_t$. Lacking knowledge on the value of those fundamental constants is often the limiting factor in the accuracy of theoretical predictions. The current status on $\\alpha_s$ and $m_t$ has been reviewed and directions for future research have been identified.
20. Ionisation constants of inorganic acids and bases in aqueous solution
CERN Document Server
Perrin, D D
2013-01-01
Ionisation Constants of Inorganic Acids and Bases in Aqueous Solution, Second Edition provides a compilation of tables that summarize relevant data recorded in the literature up to the end of 1980 for the ionization constants of inorganic acids and bases in aqueous solution. This book includes references to acidity functions for strong acids and bases, as well as details about the formation of polynuclear species. This text then explains the details of each column of the tables, wherein column 1 gives the name of the substance and the negative logarithm of the ionization constant and column 2
1. High precision fundamental constants at the TeV scale'
International Nuclear Information System (INIS)
Moch, S.; Weinzierl, S.
2014-05-01
This report summarizes the proceedings of the 2014 Mainz Institute for Theoretical Physics (MITP) scientific program on ''High precision fundamental constants at the TeV scale''. The two outstanding parameters in the Standard Model dealt with during the MITP scientific program are the strong coupling constant α s and the top-quark mass m t . Lacking knowledge on the value of those fundamental constants is often the limiting factor in the accuracy of theoretical predictions. The current status on α s and m t has been reviewed and directions for future research have been identified.
2. Molecular dynamics simulations of Gay-Berne nematic liquid crystal: Elastic properties from direct correlation functions
International Nuclear Information System (INIS)
Stelzer, J.; Trebin, H.R.; Longa, L.
1994-08-01
We report NVT and NPT molecular dynamics simulations of a Gay-Berne nematic liquid crystal using generalization of recently proposed algorithm by Toxvaerd [Phys. Rev. E47, 343, 1993]. On the basis of these simulations the Oseen-Zoher-Frank elastic constants K 11 , K 22 and K 33 as well as the surface constants K 13 and K 24 have been calculated within the framework of the direct correlation function approach of Lipkin et al. [J. Chem. Phys. 82, 472 (1985)]. The angular coefficients of the direct pair correlation function, which enter the final formulas, have been determined from the computer simulation data for the pair correlation function of the nematic by combining the Ornstein-Zernike relation and the Wienier-Hopf factorization scheme. The unoriented nematic approximation has been assumed when constructing the reference, isotropic state of Lipkin et al. By an extensive study of the model over a wide range of temperatures, densities and pressures a very detailed information has been provided about elastic behaviour of the Gay-Berne nematic. Interestingly, it is found that the results for the surface elastic constants are qualitatively different than those obtained with the help of analytical approximations for the isotropic, direct pair correlation function. For example, the values of the surface elastic constants are negative and an order of magnitude smaller than the bulk elasticity. (author). 30 refs, 9 figs
3. Earthquake source model using strong motion displacement as ...
Indian Academy of Sciences (India)
Earthquake source model using strong motion displacement as response of finite elastic media. R N IYENGAR* and SHAILESH KR AGRAWAL**. *Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India. e-mail: [email protected]. **Central Building Research Institute, Roorkee, India.
4. Earthquake source model using strong motion displacement as ...
Indian Academy of Sciences (India)
The strong motion displacement records available during an earthquake can be treated as the response of the earth as the a structural system to unknown forces acting at unknown locations. Thus, if the part of the earth participating in ground motion is modelled as a known finite elastic medium, one can attempt to model the ...
5. Appraisal of elastic follow up
International Nuclear Information System (INIS)
Roche, R.L.
1981-01-01
Elastic computations are widely used in structural analysis, and their results are used when material behaviour is non elastic. The current practice is the partition of the computed stress between primary and secondary stress. The basic characteristic of primary stress is that it is not self limiting. On the contrary the basic characteristic of a secondary stress is that it is self limiting, and failure from one application of the stress is not to be expected. It must be emphasized that self limitation is not sufficient and that it is also necessary that strains are small enough to avoid any material disorder. Unfortunately, elastic computations do not give real strain distribution and computed strain in highly stressed areas can be magnified under conditions of plastic temperature is high enough, an undesirable amount of creep occurs in areas of reduced strength and failure can happen. In creep range, to avoid elastic follow up, the most important part of elastically computed stress is considered as primary. This practice is over conservative, and the aim of this paper is to provide indications to choise what fraction of a self limiting stress can be considered as secondary. (orig./GL)
6. Magneto-elastic coupling model of deformable anisotropic superconductors
Science.gov (United States)
Li, Yingxu; Kang, Guozheng; Gao, Yuanwen
2017-04-01
We develop a magneto-elastic (ME) coupling model for the interaction between the vortex lattice and crystal elasticity. The anisotropies in superconductivity and elasticity are simultaneously included in the GL theory frame. Under this consideration, the expression of the free energy unifies the different forms of the classical results. Concerning the ME effect on the magnetization, the theory can give a satisfying description for the field dependence of magnetization near the upper critical field. The contribution of the ME interaction to the magnetization is comparable to the vortex-lattice energy, in materials with relatively strong pressure dependence of the critical temperature. While the magnetization components along different vortex frame axes are strain dependent, the magnetization ratio is independent of the ME interaction. It is stressed that the GL description of the magnetization ratio is applicable only if the applied field moderately close to the upper critical field.
7. Matrix elasticity directs stem cell differentiation in 3D too
Science.gov (United States)
Zajac, Allison; Rehfeldt, Florian; Discher, Dennis
2009-03-01
Microenvironments appear important in stem cell lineage specification but can be difficult to adequately characterize or control with soft tissues. Naive mesenchymal stem cells (MSCs) are shown here to specify lineage andcommit to phenotypes with extreme sensitivity to tissue level elasticity. Soft matrices that mimic brain are neurogenic, stiffer matrices that mimic muscle are myogenic, and comparatively rigid matrices that mimic collagenous bone prove osteogenic. During the initial week in culture, reprogramming of these lineages is possible with addition of soluble induction factors, but after several weeks in culture, the cells commit to the lineage specified by matrix elasticity, consistent with the elasticity-insensitive commitment of differentiated cell types. Inhibition of nonmuscle myosin II blocks all elasticitydirected lineage specification--without strongly perturbing many other aspects of cell function and shape. The results have significant implications for understanding physical effects of the in vivo microenvironment and also for therapeutic uses of stem cells.
8. Elastic and inelastic cross section measurements with the ATLAS detector
CERN Document Server
Stark, Simon Holm; The ATLAS collaboration
2017-01-01
The total pp cross section is a fundamental property of the strong interaction which can not be calculated in perturbative QCD but only described based on phenomenological models. The ATLAS collaboration has measured the total inelastic proton-proton cross section and the diffractive part of the inelastic cross section at 13 TeV in special data sets taken with low beam currents and using forward scintillators. More precise measurements of the total pp cross section and the elastic and inelastic contributions have been extracted from measurements of the differential elastic cross section using the optical theorem. The ATLAS Collaboration has performed this measurement in elastic data collected with high beta* optics at 8 TeV centre-of-mass energy with the ALFA Roman Pot detector.
9. Strongly Correlated Topological Insulators
Science.gov (United States)
2016-02-03
Strongly Correlated Topological Insulators In the past year, the grant was used for work in the field of topological phases, with emphasis on finding...surface of topological insulators. In the past 3 years, we have started a new direction, that of fractional topological insulators. These are materials...in which a topologically nontrivial quasi-flat band is fractionally filled and then subject to strong interactions. The views, opinions and/or
10. Cosmological Constant and Local Gravity
CERN Document Server
Bernabeu, Jose; Mavromatos, Nick E
2010-01-01
We discuss the linearization of Einstein equations in the presence of a cosmological constant, by expanding the solution for the metric around a flat Minkowski space-time. We demonstrate that one can find consistent solutions to the linearized set of equations for the metric perturbations, in the Lorentz gauge, which are not spherically symmetric, but they rather exhibit a cylindrical symmetry. We find that the components of the gravitational field satisfying the appropriate Poisson equations have the property of ensuring that a scalar potential can be constructed, in which both contributions, from ordinary matter and $\\Lambda > 0$, are attractive. In addition, there is a novel tensor potential, induced by the pressure density, in which the effect of the cosmological constant is repulsive. We also linearize the Schwarzschild-de Sitter exact solution of Einstein's equations (due to a generalization of Birkhoff's theorem) in the domain between the two horizons. We manage to transform it first to a gauge in whic...
11. Evolution of the solar constant
International Nuclear Information System (INIS)
Newman, M.J.
1978-01-01
The ultimate source of the energy utilized by life on Earth is the Sun, and the behavior of the Sun determines to a large extent the conditions under which life originated and continues to thrive. What can be said about the history of the Sun. Has the solar constant, the rate at which energy is received by the Earth from the Sun per unit area per unit time, been constant at its present level since Archean times. Three mechanisms by which it has been suggested that the solar energy output can vary with time are discussed, characterized by long (approx. 10 9 years), intermediate (approx. 10 8 years), and short (approx. years to decades) time scales
12. Potential constants of CF4
International Nuclear Information System (INIS)
Jones, L.H.; Kennedy, C.; Ekberg, S.
1978-01-01
The infrared spectra of the 12 C, 13 C, and 14 C isotopic species of CF 4 have been observed at a resolution of 0.06 cm -1 . In addition to the fundamentals ν 3 and ν 4 a number of combination bands have been observed. Using these results, combined with Raman data in the literature, we have calculated the quadratic valence force field, in terms of force constants as well as compliance constants, with considerably better precision than previously obtained. Interaction displacement coordinates have been calculated and show that stretching one CF bond leads, for minimum energy near equilibrium, to opening up of the angles between the other three bonds as well as to their contraction
13. Equation of state, nonlinear elastic response, and anharmonic properties of diamond-cubic silicon and germanium. First-principles investigation
Energy Technology Data Exchange (ETDEWEB)
Wang, Chenju [Sichuan Univ., Chengdu (China). Inst. of Atomic and Molecular Physics; Institute of Fluid Physics, Sichuan (China). National Key Laboratory of Shock Wave and Detonation Physics; Gu, Jianbing [Institute of Fluid Physics, Sichuan (China). National Key Laboratory of Shock Wave and Detonation Physics; Sichuan Univ., Chengdu (China). College of Physical Science and Technology; Kuang, Xiaoyu [Sichuan Univ., Chengdu (China). Inst. of Atomic and Molecular Physics; Xiang, Shikai [Institute of Fluid Physics, Sichuan (China). National Key Laboratory of Shock Wave and Detonation Physics
2015-10-01
Nonlinear elastic properties of diamond-cubic silicon and germanium have not been investigated sufficiently to date. Knowledge of these properties not only can help us to understand nonlinear mechanical effects but also can assist us to have an insight into the related anharmonic properties, so we investigate the nonlinear elastic behaviour of single silicon and germanium by calculating their second- and third-order elastic constants. All the results of the elastic constants show good agreement with the available experimental data and other theoretical calculations. Such a phenomenon indicates that the present values of the elastic constants are accurate and can be used to further study the related anharmonic properties. Subsequently, the anharmonic properties such as the pressure derivatives of the second-order elastic constants, Grueneisen constants of long-wavelength acoustic modes, and ultrasonic nonlinear parameters are explored. All the anharmonic properties of silicon calculated in the present work also show good agreement with the existing experimental results; this consistency not only reveals that the calculation method of the anharmonic properties is feasible but also illuminates that the anharmonic properties obtained in the present work are reliable. For the anharmonic properties of germanium, since there are no experimental result and other theoretical data till now, we hope that the anharmonic properties of germanium first offered in this work would serve as a reference for future studies.
14. First-Principles Study of the Nonlinear Elasticity of Rare-Earth Hexaborides REB6 (RE = La, Ce
Directory of Open Access Journals (Sweden)
Xianshi Zeng
2017-10-01
Full Text Available The complete set of independent second- and third-order elastic constants of rare-earth hexaborides LaB 6 and CeB 6 are determined by the combination method of first-principles calculations and homogeneous deformation theory. The ground-state lattice parameters, second-order elastic constants, and bulk modulus are in reasonable agreement with the available experimental data. The third-order elastic constant of longitudinal mode C 111 has a larger absolute value than other shear modes, showing the contribution to lattice vibrations from longitudinal modes to be greater. The pressure derivatives of the second-order elastic constants related to the third-order elastic constants are calculated to be positive for the two hexaborides, which are consistent with those of their polycrystalline bulk modulus and shear modulus. Furthermore, the effect of pressure on the structural stability, mechanical property, and elastic anisotropy of the two hexaborides are investigated, showing a reduction in mechanical stability and an increase in ductility and anisotropy with increasing pressure.
15. The elastic and magnetic properties of a single-crystal Gd-40%Y alloy
International Nuclear Information System (INIS)
Palmer, S.B.; Isci, C.; Hukin, D.
1977-01-01
The five independent single-crystal elastic constants of hexagonal Gd-40%Y have been measured in the temperature range 4.2 to 300 K and in magnetic fields of up to 7 T. This temperature and magnetic field range covers the different magnetic states of the material and has allowed the magnetic phase diagram to be constructed from the anomalies present in the elastic constants and associated ultrasonic attenuation. At low temperatures and low fields the material does not follow Dy and Tb-50%Ho in transforming from an antiferromagnetic to a ferromagnetic phase, but exhibits a variety of more complicated magnetic structures. (author)
16. Subcritical crack growth in an aging plate with variable elastic modulus
Science.gov (United States)
Gavrilov, G. V.
2010-10-01
The paper addresses subcritical growth of a crack in a thin isotropic plate made of an aging viscoelastic material with time-dependent elastic modulus. The behavior of the material is described by Arutyunyan's creep theory. To simulate fracture, a modified Leonov-Panasyuk-Dugdale model and a critical crack opening displacement criterion are used. An equation describing the subcritical growth of the crack is derived assuming that Poisson's ratio is constant. As an example, the critical loads are determined, and curves of subcritical crack growth are plotted for a specific material. The results are compared with the case of constant elastic modulus
17. Exploring the Local Elastic Properties of Bilayer Membranes Using Molecular Dynamics Simulations
DEFF Research Database (Denmark)
Pieffet, Gilles; Botero, Alonso; Peters, Günther H.J.
2014-01-01
of mean force (PMF) allowed us to dissect the elastic contribution. With this information, we calculated an effective linear spring constant of 44 +/- 4 kJ.nm-2.mol-1 for the DOPC membrane, in agreement with experimental estimates. The membrane deformation profile was determined independently during...... the stretching process in molecular detail, allowing us to fit this profile to a previously proposed continuum elastic model. Through this approach, we calculated an effective membrane spring constant of 42 kJ-2.mol-1, which is in good agreement with the PMF calculation. Furthermore, the solvation energy we...
18. A Simple Method to Measure the Twist Elastic Constant of a Nematic Liquid Crystal
Science.gov (United States)
2015-01-01
as 180° super- twisted nematic (STN) cell. Next, we assume the helical twisting power ( HTP ) of chiral dopant is also unknown, same as K22. To solve...threshold voltages of these two 180° STN cells, both K22 and HTP can be obtained simultaneously. In the whole process, there is no need to measure...Equation (1), if we sub- stitute ϕ = π and pitch length P = 1/( HTP · c) (where c is chiral concentration), then the critical voltage can be rewritten
19. Determination of transverse elastic constants of wood using a cylindrically orthotropic model
Science.gov (United States)
John C. Hermanson
2003-01-01
The arrangement of anatomical elements in the cross section of a tree can be characterized, at least to a first approximation, with a cylindrical coordinate system. It seems reasonable that the physical properties of wood in the transverse plane, therefore, would exhibit behaviour that is associated with this anatomical alignment. Most of the transverse properties of...
20. Elastic constants and Debye temperature of wz-AlN and wz-GaN ...
Indian Academy of Sciences (India)
2014-09-05
Sep 5, 2014 ... B P Pandey, V Kumar and Eduardo Menendez Proupin varying the wide band gap [4,5]. These two are also very prominent binary compounds in extreme conditions of high pressure and temperature. Therefore, experimental and theo- retical investigations are of great interest to physicists and researchers.
1. Effective medium approximation for elastic constants of porous solids with microscopic heterogeneity
International Nuclear Information System (INIS)
Berryman, J.G.
1986-01-01
Formulas for the scattering from an inhomogeneous sphere in a fluid-saturated porous medium are used to construct a self-consistent effective medium approximation for the coefficients in Biot's equations of poroelasticity [J. Acoust. Soc. Am. 28, 168 (1956)] when the material constituting the porous solid frame is not homogeneous on the microscopic scale. The discussion is restricted to porous materials exhibiting both macroscopic and microscopic isotropy. Brown and Korringa [Geophysics 40, 608 (1975)] have previously found the general form of these coefficients. The present results give explicit estimates of all the coefficients in terms of the moduli of the solid constituents. The results are also shown to be completely consistent with the well-known results of Gassmann and of Biot and Willis, as well as those of Brown and Korringa
2. Temperature variation of higher-order elastic constants of MgO
Indian Academy of Sciences (India)
Magnesium oxide is the simplest oxide, and has been a subject of intense experi- mental and theoretical study. Oxides and silicates make up the bulk of the Earth's mantle and crust, and thus it is important to understand and predict their behav- ior. An important feature of MgO is the non-rigid behavior of the O. 2− ion, which.
3. Ultrasonic Determination of Combinations of Third-Order Elastic Constants of Small Cubic Single Crystals
Science.gov (United States)
1981-05-01
are extended to Professor A. etwnhowitch of the U niversite Pierre et Marie Curie , Paris, France, for litranini the <d ard K/nK samples, and to Mr. [W...two members of the perovskite family, CsCdF 3 and KZnF 3. Professor A. Zarembowitch and others at the Universit6 Pierre et Marie Curie , Paris, France...and Irene A. Stegun, eds., Handbook of Mathematical Functions. National Bureau of Standards, 1964. Bains, J. A., Jr., "Variations of Combinations of
4. Elastic constants and Debye temperature of wz-AlN and wz-GaN ...
Indian Academy of Sciences (India)
3Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile. ∗. Corresponding author. E-mail: [email protected]. MS received 20 October 2013; revised 12 January 2014; accepted 28 January 2014. DOI: 10.1007/s12043-014-0785-7; ePublication: 5 September 2014. Abstract.
5. Effects of hydrogen on the single crystalline elastic constants of niobium
Energy Technology Data Exchange (ETDEWEB)
Schlader, Daniel Michael [Iowa State Univ., Ames, IA (United States)
1977-06-01
A special hydriding system was designed and constructed to satisfy conditions for hydriding niobium. This system controlled the temperature and hydrogen atmosphere surrounding the niobium while ultrasonic measurements were recorded. Ultrasonic wave velocities were determined by measurement of the times for ultrasonic pulses to transit and then echo through known dimensions of test specimens. The method which was employed is commonly known as the pulse-echo-overlap method. This study confirmed the general trends of earlier investigations. In this study C' continued to decrease and C44 continued to increase up to 4.69 atomic percent hydrogen which is the maximum concentration which has yet been examined. In the case of the niobium-hydrogen system the Snoek effect may well be a contributory factor to the decrease of C' with increasing hydrogen concentration. However, crystallographic considerations preclude this effect from contributing a concentration dependence to C44 or B. The observation of the present work implies that other factors must also be contributing to the overall behavior.
6. Effects of hydrogen on the single crystalline elastic constants of niobium
International Nuclear Information System (INIS)
1977-06-01
A special hydriding system was designed and constructed to satisfy conditions for hydriding niobium. This system controlled the temperature and hydrogen atmosphere surrounding the niobium while ultrasonic measurements were recorded. Ultrasonic wave velocities were determined by measurement of the times for ultrasonic pulses to transit and then echo through known dimensions of test specimens. The method which was employed is commonly known as the pulse-echo-overlap method. This study confirmed the general trends of earlier investigations. In this study C' continued to decrease and C 44 continued to increase up to 4.69 atomic percent hydrogen which is the maximum concentration which has yet been examined. In the case of the niobium-hydrogen system the Snoek effect may well be a contributory factor to the decrease of C' with increasing hydrogen concentration. However, crystallographic considerations preclude this effect from contributing a concentration dependence to C 44 or B. The observation of the present work implies that other factors must also be contributing to the overall behavior
7. Tip-splitting instabilities in the channel Saffman-Taylor flow of constant viscosity elastic fluids
International Nuclear Information System (INIS)
Vlad, D. H.; Maher, J. V.
2000-01-01
Boger fluids are used to study viscous fingering growth in viscoelastic fluids in channel Hele-Shaw flow. We have found that the viscous finger growing in the Boger fluid is unstable to tip splitting at high velocities, in a regime where a Newtonian viscous finger is stable. No fracturelike instabilities were observed. We show that the viscoelastic normal stress differences arising in shear and extensional flow reach very high values at shear and extensional rates comparable to those achieved at the tip of the finger at the onset of tip splitting, and the fluid becomes highly anisotropic. The viscoelastic stress could affect the dynamics of the finger and induce the tip-splitting instability. (c) 2000 The American Physical Society
8. Strong Cosmic Censorship
Science.gov (United States)
Isenberg, James
2017-01-01
The Hawking-Penrose theorems tell us that solutions of Einstein's equations are generally singular, in the sense of the incompleteness of causal geodesics (the paths of physical observers). These singularities might be marked by the blowup of curvature and therefore crushing tidal forces, or by the breakdown of physical determinism. Penrose has conjectured (in his Strong Cosmic Censorship Conjecture) that it is generically unbounded curvature that causes singularities, rather than causal breakdown. The verification that AVTD behavior'' (marked by the domination of time derivatives over space derivatives) is generically present in a family of solutions has proven to be a useful tool for studying model versions of Strong Cosmic Censorship in that family. I discuss some of the history of Strong Cosmic Censorship, and then discuss what is known about AVTD behavior and Strong Cosmic Censorship in families of solutions defined by varying degrees of isometry, and discuss recent results which we believe will extend this knowledge and provide new support for Strong Cosmic Censorship. I also comment on some of the recent work on Weak Null Singularities'', and how this relates to Strong Cosmic Censorship.
9. Self-Organization of Polymeric Fluids in Strong Stress Fields
Directory of Open Access Journals (Sweden)
A. V. Semakov
2015-01-01
Full Text Available Analysis of literature data and our own experimental observations have led to the conclusion that, at high deformation rates, viscoelastic liquids come to behave as rubbery materials, with strong domination by elastic deformations over flow. This can be regarded as a deformation-induced fluid-to-rubbery transition. This transition is accompanied by elastic instability, which can lead to the formation of regular structures. So, a general explanation for these effects requires the treatment of viscoelastic liquids beyond critical deformation rates as rubbery media. Behaviouristic modeling of their behaviour is based on a new concept, which considers the medium as consisting of discrete elastic elements. Such a type of modeling introduces a set of discrete rotators settled on a lattice with two modes of elastic interaction. The first of these is their transformation from spherical to ellipsoidal shapes and orientation in an external field. The second is elastic collisions between rotators. Computer calculations have demonstrated that this discrete model correctly describes the observed structural effects, eventually resulting in a “chaos-to-order” transformation. These predictions correspond to real-world experimental data obtained under different modes of deformation. We presume that the developed concept can play a central role in understanding strong nonlinear effects in the rheology of viscoelastic liquids.
10. Photodissociation constant of NO2
International Nuclear Information System (INIS)
Nootebos, M.A.; Bange, P.
1992-01-01
The velocity of the dissociation of NO 2 into ozone and NO mainly depends on the ultraviolet sunlight quantity, and with that the cloudiness. A correct value for this reaction constant is important for the accurate modelling of O 3 - and NO 2 -concentrations in plumes of electric power plants, in particular in the case of determination of the amount of photochemical summer smog. An advanced signal processing method (deconvolution, correlation) was applied on the measurements. The measurements were carried out from aeroplanes
11. Spherically symmetric elasticity in relativity
Energy Technology Data Exchange (ETDEWEB)
Carot, J [Departament de Fisica, Universitat de les Illes Balears, Cra Valldemossa pk 7.5, E-07122 Palma (Spain); Brito, I; Vaz, E G L R, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected]
2010-05-01
The relativistic theory of elasticity is reviewed within the spherically symmetric context with a view towards the modelling of star interiors possessing elastic properties such as the ones expected in neutron stars. Emphasis is placed on generality in the main sections of the paper, and the results are then applied to specific examples. Along the way, a few general results for spacetimes admitting isometries are deduced, and their consequences are fully exploited in the case of spherical symmetry relating them next to the the case in which the material content of the spacetime is some elastic material. This paper extends and generalizes the pioneering work by Magli and Kijowski [1], Magli [2] and [3], and complements, in a sense, that by Karlovini and Samuelsson in their interesting series of papers [4], [5] and [6].
12. Searching for Kaprekar's constants: algorithms and results
OpenAIRE
Walden, Byron L.
2005-01-01
We examine some new results on Kaprekar's constants, specifically establishing the unique 7-digit (in base 4) and 9-digit (in base 5) Kaprekar's constants and showing that there are no 15-, 21-, 27-, or 33-digit Kaprekar's constants.
13. Adaptive elastic networks as models of supercooled liquids
Science.gov (United States)
Yan, Le; Wyart, Matthieu
2015-08-01
The thermodynamics and dynamics of supercooled liquids correlate with their elasticity. In particular for covalent networks, the jump of specific heat is small and the liquid is strong near the threshold valence where the network acquires rigidity. By contrast, the jump of specific heat and the fragility are large away from this threshold valence. In a previous work [Proc. Natl. Acad. Sci. USA 110, 6307 (2013), 10.1073/pnas.1300534110], we could explain these behaviors by introducing a model of supercooled liquids in which local rearrangements interact via elasticity. However, in that model the disorder characterizing elasticity was frozen, whereas it is itself a dynamic variable in supercooled liquids. Here we study numerically and theoretically adaptive elastic network models where polydisperse springs can move on a lattice, thus allowing for the geometry of the elastic network to fluctuate and evolve with temperature. We show numerically that our previous results on the relationship between structure and thermodynamics hold in these models. We introduce an approximation where redundant constraints (highly coordinated regions where the frustration is large) are treated as an ideal gas, leading to analytical predictions that are accurate in the range of parameters relevant for real materials. Overall, these results lead to a description of supercooled liquids, in which the distance to the rigidity transition controls the number of directions in phase space that cost energy and the specific heat.
14. Investor response to consumer elasticity
International Nuclear Information System (INIS)
Grenaa Jensen, Stine; Meibom, Peter; Ravn, H.F.; Straarup, Sarah
2004-01-01
In the Nordic electricity system there is considerable uncertainty with respect to the long-term development in production capacity. The process towards liberalisation of the electricity sector started in a situation with a large reserve margin, but this margin is gradually vanishing. Since the potential investors in new production capacity are unaccustomed with investments under the new regime it is unknown if and when investments will take place. The electricity price is the key market signal to potential investors. The price is settled as a balance between supply and demand, and it is generally assumed that the demand side has an important role in this, and increasingly so. However, since consumers have not earlier had the incentive to respond to electricity prices, no reliable estimate of demand elasticity is known. The purpose of the present study is to analyse the role of electricity demand elasticity for investments in new electricity production capacity. Electricity price scenarios generated with a partial equilibrium model (Balmorel) are combined with a model of investment decisions. In this, various scenarios concerning the development in the demand elasticity are used. The simulated investment decisions are taken in a stochastic, dynamic setting, where a key point is the timing of the investment decision in relation to the gathering of new information relative to the stochastic elements. Based on this, the consequences of the development in consumer price elasticity for investments in a base load and a peak load plant are investigated. The main result of the analysis is that peak load investments can be made unprofitable by the development in consumer price elasticity, such that an investor will tend to wait with his peak load investment, until the development in consumer price elasticity has been revealed. (au)
15. Density functional study of elastic and vibrational properties of the Heusler-type alloys Fe2VAl and Fe2VGa
DEFF Research Database (Denmark)
Kanchana, V.; Vaitheeswaran, G.; Ma, Yanming
2009-01-01
agree well with the experimental values. The elastic constants of Fe2VAl and Fe2VGa are predicted. From the elastic constants the shear modulus, Young's modulus, Poisson's ratio, sound velocities, and Debye temperatures are obtained. By analyzing the ratio between the bulk and shear moduli, we conclude...
16. Complex variable methods in elasticity
CERN Document Server
England, A H
2003-01-01
The plane strain and generalized plane stress boundary value problems of linear elasticity are the focus of this graduate-level text, which formulates and solves these problems by employing complex variable theory. The text presents detailed descriptions of the three basic methods that rely on series representation, Cauchy integral representation, and the solution via continuation. Its five-part treatment covers functions of a complex variable, the basic equations of two-dimensional elasticity, plane and half-plane problems, regions with circular boundaries, and regions with curvilinear bounda
17. Water hammer in elastic pipes
International Nuclear Information System (INIS)
Gale, J.; Tiselj, I.
2002-01-01
One dimensional two-fluid six-equation model of two-phase flow, that can be found in computer codes like RELAP5, TRAC, and CATHARE, was upgraded with additional terms, which enable modelling of the pressure waves in elastic pipes. It is known that pipe elasticity reduces the propagation velocity of the shock and other pressure waves in the piping systems. Equations that include the pipe elasticty terms are used in WAHA code, which is being developed within the WAHALoads project of 5't'h EU research program.(author)
18. CONFERENCE: Elastic and diffractive scattering
International Nuclear Information System (INIS)
White, Alan
1989-01-01
Elastic scattering, when particles appear to 'bounce' off each other, and the related phenomena of diffractive scattering are currently less fashionable than the study of hard scattering processes. However this could change rapidly if unexpected results from the UA4 experiment at the CERN Collider are confirmed and their implications tested. These questions were highlighted at the third 'Blois Workshop' on Elastic and Diffractive Scattering, held early in May on the Evanston campus of Northwestern University, near Chicago
19. Theoretical study of elastic, mechanical and thermodynamic properties of MgRh intermetallic compound
Directory of Open Access Journals (Sweden)
S. Boucetta
2014-03-01
Full Text Available In the last years, Magnesium alloys are known to be of great technological importance and high scientific interest. In this work, density functional theory plane-wave pseudo potential method, with local density approximation (LDA and generalized gradient approximation (GGA are used to perform first-principles quantum mechanics calculations in order to investigate the structural, elastic and mechanical properties of the intermetallic compound MgRh with a CsCl-type structure. Comparison of the calculated equilibrium lattice constant and experimental data shows good agreement. The elastic constants were determined from a linear fit of the calculated stress–strain function according to Hooke's law. From the elastic constants, the bulk modulus B, shear modulus G, Young's modulus E, Poisson's ratio σ, anisotropy factor A and the ratio B/G for MgRh compound are obtained. The sound velocities and Debye temperature are also predicted from elastic constants. Finally, the linear response method has been used to calculate the thermodynamic properties. The temperature dependence of the enthalpy H, free energy F, entropy S, and heat capacity at constant volume Cv of MgRh crystal in a quasi-harmonic approximation have been obtained from phonon density of states and discussed for the first report. This is the first quantitative theoretical prediction of these properties.
20. ELASTIC AND SAFETY CLUTCH WITH RADIAL TAPERED ROLLER AND METALLIC ELASTIC ELEMENTS AXIALLY ARRANGED
OpenAIRE
STROE Ioan
2014-01-01
The paper presents a new type of clutch named Elastic and Safety Clutch that can accomplish the functions of the elastic and those of the safety clutches, but it is not a combined clutch. The proposed clutch is an elastic and safety clutch with metallic intermediate elements. The paper presents the elastic and safety clutch with radial tapered roller and metallic elastic elements axially arranged. The design and verification computing relations of the elastic and safety...
1. Effective medium theory for elastic matrix composites containing dispersed particulates
International Nuclear Information System (INIS)
Jhon, M.S.; Metz, R.J.; Freed, K.F.
1988-01-01
We describe a new, effective medium theory to study the wave propagation and mechanical properties of a composite system with dispersed particulates. One main emphasis here is in formulating the theory and in analyzing the structure of the contribution of the fillers to the elastic response. By constructing the elastic propagator (whose fluid mechanical counterpart is known as the Oseen tensor), we show that an analogy between the theoretical description of the particulate system and of suspension rheology exists when the former corresponds to a high-rigidity solid matrix (or, analogously, when the Poisson ratio is close to 1/2) in steady state. The effective Lame constants for this case are derived by combining this analogy with the theory developed by Freed and Muthukumar for the rheology of a suspension of spheres. The analogy is also useful in our new prediction of the phenomenon of elastic screening, the possible existence of a cutoff frequency below which elastic waves cannot propagate in the filler system
2. Numerical simulation of ultrasonic wave propagation in elastically anisotropic media
International Nuclear Information System (INIS)
Jacob, Victoria Cristina Cheade; Jospin, Reinaldo Jacques; Bittencourt, Marcelo de Siqueira Queiroz
2013-01-01
The ultrasonic non-destructive testing of components may encounter considerable difficulties to interpret some inspections results mainly in anisotropic crystalline structures. A numerical method for the simulation of elastic wave propagation in homogeneous elastically anisotropic media, based on the general finite element approach, is used to help this interpretation. The successful modeling of elastic field associated with NDE is based on the generation of a realistic pulsed ultrasonic wave, which is launched from a piezoelectric transducer into the material under inspection. The values of elastic constants are great interest information that provide the application of equations analytical models, until small and medium complexity problems through programs of numerical analysis as finite elements and/or boundary elements. The aim of this work is the comparison between the results of numerical solution of an ultrasonic wave, which is obtained from transient excitation pulse that can be specified by either force or displacement variation across the aperture of the transducer, and the results obtained from a experiment that was realized in an aluminum block in the IEN Ultrasonic Laboratory. The wave propagation can be simulated using all the characteristics of the material used in the experiment valuation associated to boundary conditions and from these results, the comparison can be made. (author)
3. A micromechanics model of the elastic properties of human dentine
Energy Technology Data Exchange (ETDEWEB)
Kinney, J. H. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Balooch, M. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Marshall, G. W. [Univ. of California, San Francisco, CA (United States). Dept. of Restorative Dentistry; Marshall, S. J. [Univ. of California, San Francisco, CA (United States). Dept. of Restorative Dentistry
1999-10-01
A generalized self-consistent model of cylindrical inclusions in a homogeneous and isotropic matrix phase was used to study the effects of tubule orientation on the elastic properties of dentin. Closed form expressions for the five independent elastic constants of dentin were derived in terms of tubule concentration, and the Young's moduli and Poisson ratios of peri- and intertubular dentin. An atomic force microscope (AFM) indentation technique determined the Young's moduli of the peri- and intertubular dentin as approximately 30 GPa and 15 GPa, respectively. Over the natural variation in tubule density found in dentin, there was only a slight variation in the axial and transverse shear moduli with position in the tooth, and there was no measurable effect of tubule orientation. We conclude that tubule orientation has no appreciable effect on the elastic behavior of normal dentin, and that the elastic properties of healthy dentin can be modeled as an isotropic continuum with a Young's modulus of approximately 16 GPa and a shear modulus of 6.2 GPa.
4. Does maltose influence on the elasticity of SOPC membrane?
Energy Technology Data Exchange (ETDEWEB)
Genova, J; Zheliaskova, A; Mitov, M D, E-mail: [email protected] [Institute of Solid State Physics, Bulgarian Academy of Sciences, 72, Tzarigradsko Chaussee Blvd., 1784 Sofia (Bulgaria)
2010-11-01
Thermally induced shape fluctuations of giant quasi-spherical lipid vesicles are used to study the influence of the disaccharide maltose, dissolved in the aqueous solution, on the curvature elasticity k{sub c} of a lipid membrane. The influence of the carbohydrate solute is investigated throughout a considerably wide interval of concentrations. The values of the bending elastic modulus for 200 mM and 400 mM of maltose in the water solution are obtained. The data for k{sub c} in presence of maltose is compared with previously obtained results for this constant for the most popular hydrocarbons: monosaccharides glucose and fructose and disaccharides sucrose and trehalose. It is shown that the presence of maltose, dissolved in the aqueous phase surrounding the membrane does not influence on the bending elasticity with the increase of its concentration in the aqueous solution. Up to our knowledge this is the first sugar that does not show decrease of the bending elastic modulus of the lipid membrane, when present in the water surrounding it in concentration up to 400 mM.
5. Determination of the pion-nucleon coupling constant and s-wave scattering lengths
CERN Document Server
Samaranayake, V K
1972-01-01
Presently available values of D/sub +or-/, the real parts of the pi /sup +or-/p elastic scattering amplitudes in the forward direction in the laboratory frame, obtained by extrapolation of experimental data to the forward direction, have been fitted up to a pion lab. kinetic energy of 2 GeV using forward dispersion relation. A substantial number of data points have to be discarded to obtain a reasonable goodness of fit. Above 300 MeV the values of D/sub +or-/ obtained from the CERN phase shift analysis are strongly favoured compared with those from the Saclay analysis. The final results for the pion-nucleon coupling constant and s-wave scattering lengths are: 10/sup 3/f/sup 2 /=76.3+or-2.0, 10/sup 3/D/sub +/( mu )=-102.4+or-5.2, 10/sup 3/D/sub - /( mu )=104.8+or-5.4, 10/sup 3/(a/sub 1/-a/sub 3/)=270.6+or-11.3, 10 /sup 3/(a/sub 1/+2a/sub 3/)=3.1+or-8.0. The errors quoted take account of experimental uncertainties and also attempt to include systematic errors arising from the unphysical continuum and from the v...
6. Elastic scattering of 40Ar and 84Kr on 209Bi and 238U at 7.2 and 8.5 MeV/N
International Nuclear Information System (INIS)
Birkelund, J.R.; Huizenga, J.R.; Freiesleben, H.; Wolf, K.L.; Unik, J.P.; Viola, V.E. Jr.
1976-01-01
Cross sections for elastic scattering of 40 Ar on targets of 209 Bi and 238 U were measured at energies of 286 and 340 MeV. Cross sections for the elastic scattering of 84 Kr on 209 Bi were measured at energies of 600 and 712 MeV. These experimental elastic scattering data were fitted with optical and Fresnel models. The total reaction cross section deduced from the Fresnel model by the one-quarter point technique agrees within a few percent with the result from the optical model. The Fresnel interaction radius and the optical model strong absorption radius are found to be approximately equal and qualitatively reproduced by the sum of the half-density electron scattering radii of the two heavy ions and a constant of 3.2+-0.3 fm. A method of estimating total reaction cross sections for heavy ions is presented. Some observations on the real and imaginary potentials of very heavy ions are presented
7. Strong Arcwise Connectedness
OpenAIRE
Espinoza, Benjamin; Gartside, Paul; Kovan-Bakan, Merve; Mamatelashvili, Ana
2012-01-01
A space is n-strong arc connected' (n-sac) if for any n points in the space there is an arc in the space visiting them in order. A space is omega-strong arc connected (omega-sac) if it is n-sac for all n. We study these properties in finite graphs, regular continua, and rational continua. There are no 4-sac graphs, but there are 3-sac graphs and graphs which are 2-sac but not 3-sac. For every n there is an n-sac regular continuum, but no regular continuum is omega-sac. There is an omega-sac ...
8. Abortion: Strong's counterexamples fail
DEFF Research Database (Denmark)
Di Nucci, Ezio
2009-01-01
This paper shows that the counterexamples proposed by Strong in 2008 in the Journal of Medical Ethics to Marquis's argument against abortion fail. Strong's basic idea is that there are cases--for example, terminally ill patients--where killing an adult human being is prima facie seriously morally......'s scenarios have some valuable future or admitted that killing them is not seriously morally wrong. Finally, if "valuable future" is interpreted as referring to objective standards, one ends up with implausible and unpalatable moral claims....
9. Elastic and Electrical Properties Evaluation of Low Resistivity Pays in Malay Basin Clastics Reservoirs
Science.gov (United States)
Almanna Lubis, Luluan; Ghosh, Deva P.; Hermana, Maman
2016-07-01
The elastic and electrical properties of low resistivity pays clastics reservoirs in Malay Basin are strongly dependent on the complex nature of the clay content, either dispersed or laminated/layered. Estimating the hydrocarbon pore volume from conventional electrical log, i.e. resistivity log, is quite a challenge. The low elastic impedance contrast also found as one of the challenge thus create a problem to map the distribution of the low resistivity reservoirs. In this paper, we evaluate the electrical properties and elastic rock properties to discriminate the pay from the adjacent cap rock or shale. Forward modeling of well log responses including electrical properties are applied to analyze the nature of the possible pays on laminated reservoir rocks. In the implementation of rock properties analysis, several conventional elastic properties are comparatively analyzed for the sensitivity and feasibility analysis on each elastic parameters. Finally, we discussed the advantages of each elastic parameters in detail. In addition, cross-plots of elastic and electrical properties attributes help us in the clear separation of anomalous zone and lithologic properties of sand and shale facies over conventional elastic parameter crossplots attributes. The possible relationship on electrical and elastic properties are discussed for further studies.
10. Coexistence of elastic fibers with hyaluronic acid in the human urethral sphincter complex: a histological study.
Science.gov (United States)
Hinata, Nobuyuki; Murakami, Gen; Abe, Shin-ichi; Shibata, Shunichi; Morizane, Shuichi; Honda, Masashi; Isoyama, Tadahiro; Sejima, Takehiro; Takenaka, Atsushi
2013-10-01
To promote the prevention and treatment of urethral sphincteric dysfunction, we examined the distribution of elastic fibers around the urethral sphincter complex and the histological localization of hyaluronic acid in relation to elastic fiber architecture. Using elastica-Masson staining as well as biotinated hyaluronic acid binding protein, we examined specimens of the urethral sphincter complex obtained from 14 elderly Japanese cadavers, including 10 men and 4 women. As a control, we also observed other striated muscles in male cadavers. Elastic fibers were densely distributed throughout the submucosal and smooth muscle layers along the entire length of the male urethra, including the prostatic urethra. The levator ani fascia and rhabdosphincter also contained abundant elastic fibers. An intramuscular elastic net was seen in the rhabdosphincter but not in other striated muscles. Strong staining for hyaluronic acid was evident in the submucosa and smooth muscle sphincter of the urethra but not in the levator ani fascia or rhabdosphincter, suggesting that elastic fibers and hyaluronic acid might interact at the former sites. Gender related differences in the distribution of elastic fibers and hyaluronic acid were noted with a much lower density of elastic fibers and hyaluronic acid staining in women than in men. Urethral sites where elastic fibers and hyaluronic acid coexist could be targeted for the prevention and treatment of urethral sphincteric insufficiency. These findings should improve our understanding of the human urethral sphincter complex. Copyright © 2013 American Urological Association Education and Research, Inc. Published by Elsevier Inc. All rights reserved.
11. Low power constant fraction discriminator
International Nuclear Information System (INIS)
Krishnan, Shanti; Raut, S.M.; Mukhopadhyay, P.K.
2001-01-01
This paper describes the design of a low power ultrafast constant fraction discriminator, which significantly reduces the power consumption. A conventional fast discriminator consumes about 1250 MW of power whereas this low power version consumes about 440 MW. In a multi detector system, where the number of discriminators is very large, reduction of power is of utmost importance. This low power discriminator is being designed for GRACE (Gamma Ray Atmospheric Cerenkov Experiments) telescope where 1000 channels of discriminators are required. A novel method of decreasing power consumption has been described. (author)
12. Can coupling constants be related
International Nuclear Information System (INIS)
Nandi, Satyanarayan; Ng, Wing-Chiu.
1978-06-01
We analyze the conditions under which several coupling constants in field theory can be related to each other. When the relation is independent of the renormalization point, the relation between any g and g' must satisfy a differential equation as follows from the renormalization group equations. Using this differential equation, we investigate the criteria for the feasibility of a power-series relation for various theories, especially the Weinberg-Salam type (including Higgs bosons) with an arbitrary number of quark and lepton flavors. (orig./WL) [de
13. Measurement of Newton's gravitational constant
International Nuclear Information System (INIS)
Schlamminger, St.; Holzschuh, E.; Kuendig, W.; Nolting, F.; Pixley, R. E.; Schurr, J.; Straumann, U.
2006-01-01
A precision measurement of the gravitational constant G has been made using a beam balance. Special attention has been given to determining the calibration, the effect of a possible nonlinearity of the balance and the zero-point variation of the balance. The equipment, the measurements, and the analysis are described in detail. The value obtained for G is 6.674 252(109)(54)x10 -11 m 3 kg -1 s -2 . The relative statistical and systematic uncertainties of this result are 16.3x10 -6 and 8.1x10 -6 , respectively
14. Exact constants in approximation theory
CERN Document Server
Korneichuk, N
1991-01-01
This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are base
15. The effect of inclusions on macroscopic composite elasticity: A systematic finite-element analysis of constituent and bulk elastic properties
International Nuclear Information System (INIS)
Yoneda, A; Sohag, F H
2010-01-01
The bulk physical properties of composite systems are difficult to predict - even when the properties of the constituent materials in the system are well known. We conducted a finite-element method simulation to examine the inclusion effect by substituting an inclusion phase (second phase) into a host phase (first phase). We have organized the simulation results as a function of the elasticity of host and inclusion phases. In this procedure, special attention was paid to the initial change of elastic constants as the inclusion volume ratio was varied. To accomplish this, we introduced a new parameter D ij defined as the derivatives of the normalized stiffness elastic constant over the inclusion volume ratio. We succeeded in obtaining useful systematic formulations for D ij . These formulations are expected to be applicable to the study of composite systems in many disciplines, such as geophysics, mechanics, material engineering, and biology. The present results provide much more effective constraints on the physical properties of composite systems, like rocks, than traditional methods, such as the Voigt-Reuss bounds.
16. Impact loads on beams on elastic foundations
International Nuclear Information System (INIS)
Kameswara Rao, N.S.V.; Prasad, B.B.
1975-01-01
The behavior of beams on elastic foundations subjected to impact loads is studied in detail. The effect of foundation parameters (stiffness, and damping constants) on the dynamic response of the beam-foundation system has been analyzed. In modal analysis, the free-vibration equation has been solved by replacing the applied impulse by suitable initial conditions and the solution has been obtained as the linear combination of an infinite sequence of discrete eigen-vectors. In the energy method, the beam-foundation system is treated to be under forced vibrations and the forcing function has been obtained using the Hertz's law of impact. In the case of free-free end conditions of the beam, the rigid body modes and the elastic modes have been superposed to obtain the total response. A model of an Euler-Bernoulli beam resting on Winkler foundation has been subjected to impact loads using an impact testing machine and the responses such as deflection, strain velocity and acceleration have been measured with electronic instrumentation suited for shock and vibration. The analytical and experimental results are found to be in good agreement, thus showing the applicability of modal analysis and energy method to impact problems. The effect of foundation modulus is to increase the natural frequencies of the system. However, this effect diminishes as the mode number increases. The responses such as deflection, strain and stresses in the beam increase with increasing velocities of impact. If the system is overdamped the motion is aperiodic, and if the system is underdamped the motion is oscillatory. The beam response is greatly reduced by the damping in the foundation medium. From the present study it is observed that modal analysis is preferable to energy method
17. A strong comeback
International Nuclear Information System (INIS)
Marier, D.
1992-01-01
This article presents the results of a financial rankings survey which show a strong economic activity in the independent energy industry. The topics of the article include advisor turnover, overseas banks, and the increase in public offerings. The article identifies the top project finance investors for new projects and restructurings and rankings for lenders
18. Nonlinear theory of elastic shells
International Nuclear Information System (INIS)
Costa Junior, J.A.
1979-08-01
Nonlinear theory of elastic shells is developed which incorporates both geometric and physical nonlinearities and which does not make use of the well known Love-Kirchhoff hypothesis. The resulting equations are formulated in tensorial notation and are reduced to the ones of common use when simplifying assumptions encountered in the especific litterature are taken. (Author) [pt
19. Duration of an Elastic Collision
Science.gov (United States)
de Izarra, Charles
2012-01-01
With a pedagogical goal, this paper deals with a study of the duration of an elastic collision of an inflatable spherical ball on a planar surface suitable for undergraduate studies. First, the force generated by the deformed spherical ball is obtained under assumptions that are discussed. The study of the motion of the spherical ball colliding…
20. Nonlinear elastic properties of superconducting antiperovskites MNNi 3 (M =Zn, Cd, Mg, Al, Ga, and In) from first principles
KAUST Repository
Liu, Lili
2014-05-22
We present theoretical studies for the third-order elastic constants (TOECs) of superconducting antiperovskites MNNi 3 (M = Zn, Cd, Mg, Al, Ga, and In) using the density functional theory (DFT) and homogeneous deformation method. From the nonlinear least-square fitting, the elastic constants are extracted from a polynomial fit to the calculated strain-energy data. Calculated second-order elastic constants (SOECs) are compared with the previous theoretical calculations, and a very good agreement was found. The nonlinear effects often play an important role when the finite strains are larger than approximately 2.5 %. Besides, we have computed the pressure derivatives of SOECs and provided rough estimations for the Grüneisen constants of long-wavelength acoustic modes by using the calculated TOECs. © 2014 Springer Science+Business Media New York.
1. Nanoadhesion of a Power-Law Graded Elastic Material
Science.gov (United States)
Chen, Shao-Hua; Chen, Pei-Jian
2010-10-01
The Dugdale—Barenblatt model is used to analyze the adhesion of graded elastic materials at the nanoscale with Young's modulus E varying with depth z according to a power law E = E0(z/c0) (0 < k < 1) while Poisson's ratio v remains a constant, where E0 is a referenced Young's modulus, k is the gradient exponent and c0 is a characteristic length describing the variation rate of Young's modulus. We show that, when the size of a rigid punch becomes smaller than a critical length, the adhesive interface between the punch and the graded material detaches due to rupture with uniform stresses, rather than by crack propagation with stress concentration. The critical length can be reduced to the one for isotropic elastic materials only if the gradient exponent k vanishes.
2. Measurement of jet production with the ATLAS detector and extraction of the strong coupling constant
CERN Document Server
Marceca, Gino; The ATLAS collaboration
2017-01-01
The inclusive-jet cross-section at 8 TeV and the inclusive-jet and dijet cross-sections at 13 TeV with the ATLAS detector are presented. NLO QCD calculations, and NNLO for the inclusive-jet measurement at 13 TeV, are compared to the measurements. The extraction of $\\alpha_{s}$ from the measurement of the transverse energy-energy correlation at 8 TeV with the ATLAS detector is also presented.
3. Multi-jet event rates in deep inelastic scattering and determination of the strong coupling constant
Czech Academy of Sciences Publication Activity Database
Adloff, C.; Anderson, M.; Andreev, V.; Cvach, Jaroslav; Grab, C.; Herynek, Ivan; Hladký, Jan; Mehta, A.; Reimer, Petr; Taševský, Marek
1999-01-01
Roč. 6, - (1999), s. 575-585 ISSN 1434-6044 R&D Projects: GA AV ČR IAA1010821 Institutional research plan: CEZ:AV0Z1010920 Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 6.872, year: 1999
4. Heart transplantation and arterial elasticity
Directory of Open Access Journals (Sweden)
2013-12-01
Full Text Available Monica Colvin-Adams,1 Nonyelum Harcourt,1 Robert LeDuc,2 Ganesh Raveendran,1 Yassir Sonbol,3 Robert Wilson,1 Daniel Duprez11Cardiovascular Division, University of Minnesota, Minneapolis, MN, USA; 2Division of Biostatistics University of Minnesota, Minneapolis, MN, USA; 3Cardiovascular Division, St Luke's Hospital System, Sugar Land, TX, USAObjective: Arterial elasticity is a functional biomarker that has predictive value for cardiovascular morbidity and mortality in nontransplant populations. There is little information regarding arterial elasticity in heart transplant recipients. This study aimed to characterize small (SAE and large (LAE artery elasticity in heart transplant recipients in comparison with an asymptomatic population free of overt cardiovascular disease. A second goal was to identify demographic and clinical factors associated with arterial elasticity in this unique population.Methods: Arterial pulse waveform was registered noninvasively at the radial artery in 71 heart transplant recipients between 2008 and 2010. SAEs and LAEs were derived from diastolic pulse contour analysis. Comparisons were made to a healthy cohort of 1,808 participants selected from our prevention clinic database. Multiple regression analyses were performed to evaluate associations between risk factors and SAE and LAE within the heart transplant recipients.Results: LAE and SAE were significantly lower in heart transplant recipients than in the normal cohort (P <0.01 and P < 0.0001, respectively. Female sex and history of ischemic cardiomyopathy were significantly associated with reduced LAE and SAE. Older age and the presence of moderate cardiac allograft vasculopathy were also significantly associated with reduced SAE. Transplant duration was associated with increased SAE.Conclusion: Heart transplants are associated with peripheral endothelial dysfunction and arterial stiffness, as demonstrated by a significant reduction in SAE and LAE when compared with a
5. The elasticity anisotropy in the basal atomic planes of Mg(OH)2 and Ca(OH)2 associated with auxetic elastic properties of the hydrogen sub-lattice
International Nuclear Information System (INIS)
Harutyunyan, Valeri S.; Abrahamyan, Aren A.; Aivazyan, Ashot P.
2013-01-01
Graphical abstract: To the out-of-plane strain ε x induced in the (0 0 0 1) atomic planes of Mg(OH) 2 , the contributions of constituent octahedral layers ε x (1) and interlayers ε x (2) are of opposite sign. Highlights: ► Elasticity anisotropy of rare earth metal hydroxides is theoretically analyzed. ► Elastic anisotropy within (0 0 0 1) atomic planes is studied from energy consideration. ► The out-of-plane Poisson’s ratios of octahedral layers and interlayers are of opposite sign. ► Auxeticity of the hydrogen sublattice (interlayers) results from weak interlayer bonding. ► The obtained expression for the in-plane Young’s modulus results in useful conclusions. - Abstract: Within the framework of the Hook’s generalized law and using the experimental data for characteristic crystallographic parameters and stiffness constants available from literature, the individual elastic properties of constituent octahedral layers and interlayers of the (0 0 0 1) atomic planes in the Mg(OH) 2 and Ca(OH) 2 crystal lattices are theoretically quantified from intermolecular interaction energy. It is shown that, under uniaxial type of deformation applied along the (0 0 0 1) basal planes, in the “load-deformation response” the octahedral layers and interlayers exhibit the positive and negative Poisson’s ratio, respectively. Manifestation of such a type strong elastic anisotropy in the basal atomic planes and auxetic elastic behavior of the hydrogen sub-lattice (interlayers) upon applied uniaxial load result from a large difference in the strength of bonding within octahedral layers and interlayers. The intermolecular binding energy is contributed both by “hydroxyl–hydroxyl” and “metal atom–hydroxyl” dispersion interactions, whereas the Young’s modulus in the direction parallel to a (0 0 0 1) plane is practically contributed only by the former interaction. For this Young’s modulus, an approximate analytical expression is derived, which is
6. Monte Carlo simulation for relationship between magnetic Barkhausen noise and elastic stress of steel
Directory of Open Access Journals (Sweden)
Liang Li
2016-06-01
Full Text Available Monte Carlo simulations were performed for three-dimensional Ising model to study the relationships between magnetic Barkhausen noise and elastic stress of steel. The magnetization process was simulated and the dimensionless magnetic Barkhausen noise was calculated by the differentiation of magnetization. Coupling constant of energy exchange in Ising model is considered to be inversely proportional to applied tensile stress. The simulation results show that as coupling constant decreases, the magnetic Barkhausen noise increases, as proved by the experimental results.
7. Investigation of the elastic properties of LiKSO4 as a function of temperature and pressure
International Nuclear Information System (INIS)
Quirion, G; Abu-Kharma, M; Sergienko, I A; Bromberek, M; Clouter, M; Mroz, B
2003-01-01
In spite of the large number of reports on the physical properties of LiKSO 4 , its low-temperature phase diagram is still not well defined. One possible reason for this lack of reliable data below 100 K might be that LiKSO 4 crystals often break into many pieces when cooled below 80 K under atmospheric pressure. We have found that it is possible to thermally cycle LiKSO 4 crystals, particularly at temperatures below 80 K, as long as a minimum pressure of about 0.5 kbar is maintained. Thus, we successfully measured the temperature dependence of the sound velocity between 4 and 300 K for pressures up to 7 kbar. Over that temperature range, we clearly identify five different phase transitions (37, 48, 65, 185, 195 K) which correspond to those observed by other groups using different techniques. However, our results also show that both phase transitions below 50 K are strongly suppressed at pressures greater than 3 kbar. A Landau model of the free energy, based on the group theory, is also presented in order to explain the elastic and dielectric properties of LiKSO 4 above 100 K. To support our analysis, we show how this model accounts for the temperature dependence of the strains, polarizations, dielectric susceptibility and elastic constants
8. Actin filaments growing against an elastic membrane: Effect of membrane tension
Science.gov (United States)
Sadhu, Raj Kumar; Chatterjee, Sakuntala
2018-03-01
We study the force generation by a set of parallel actin filaments growing against an elastic membrane. The elastic membrane tries to stay flat and any deformation from this flat state, either caused by thermal fluctuations or due to protrusive polymerization force exerted by the filaments, costs energy. We study two lattice models to describe the membrane dynamics. In one case, the energy cost is assumed to be proportional to the absolute magnitude of the height gradient (gradient model) and in the other case it is proportional to the square of the height gradient (Gaussian model). For the gradient model we find that the membrane velocity is a nonmonotonic function of the elastic constant μ and reaches a peak at μ =μ* . For μ state and the membrane energy keeps increasing with time. For the Gaussian model, the system always reaches a steady state and the membrane velocity decreases monotonically with the elastic constant ν for all nonzero values of ν . Multiple filaments give rise to protrusions at different regions of the membrane and the elasticity of the membrane induces an effective attraction between the two protrusions in the Gaussian model which causes the protrusions to merge and a single wide protrusion is present in the system. In both the models, the relative time scale between the membrane and filament dynamics plays an important role in deciding whether the shape of elasticity-velocity curve is concave or convex. Our numerical simulations agree reasonably well with our analytical calculations.
9. Impact of two relaxation times on thermal, P and SV waves at interface with magnetic field and temperature dependent elastic moduli
Science.gov (United States)
Khan, Ambreen Asfar; Zaman, Akbar; Yaseen, Sundas
2018-03-01
In this article, two models of the generalized thermo-elastic theory are used to see the influence on the refraction and reflection of the plane waves at the interface under a constant magnetic field. The elasticity modulus depends on the reference temperature. The elasticity modulus is considered as a linear function of reference temperature. The resulting problem is solved by using the boundary conditions at the interface. The matrix equations have been solved numerically.
10. Strong Electroweak Symmetry Breaking
CERN Document Server
Grinstein, Benjamin
2011-01-01
Models of spontaneous breaking of electroweak symmetry by a strong interaction do not have fine tuning/hierarchy problem. They are conceptually elegant and use the only mechanism of spontaneous breaking of a gauge symmetry that is known to occur in nature. The simplest model, minimal technicolor with extended technicolor interactions, is appealing because one can calculate by scaling up from QCD. But it is ruled out on many counts: inappropriately low quark and lepton masses (or excessive FCNC), bad electroweak data fits, light scalar and vector states, etc. However, nature may not choose the minimal model and then we are stuck: except possibly through lattice simulations, we are unable to compute and test the models. In the LHC era it therefore makes sense to abandon specific models (of strong EW breaking) and concentrate on generic features that may indicate discovery. The Technicolor Straw Man is not a model but a parametrized search strategy inspired by a remarkable generic feature of walking technicolor,...
11. Elastic least-squares reverse time migration
KAUST Repository
Feng, Zongcai
2016-09-06
Elastic least-squares reverse time migration (LSRTM) is used to invert synthetic particle-velocity data and crosswell pressure field data. The migration images consist of both the P- and Svelocity perturbation images. Numerical tests on synthetic and field data illustrate the advantages of elastic LSRTM over elastic reverse time migration (RTM). In addition, elastic LSRTM images are better focused and have better reflector continuity than do the acoustic LSRTM images.
12. Plasmons in strong superconductors
International Nuclear Information System (INIS)
Baldo, M.; Ducoin, C.
2011-01-01
We present a study of the possible plasmon excitations that can occur in systems where strong superconductivity is present. In these systems the plasmon energy is comparable to or smaller than the pairing gap. As a prototype of these systems we consider the proton component of Neutron Star matter just below the crust when electron screening is not taken into account. For the realistic case we consider in detail the different aspects of the elementary excitations when the proton, electron components are considered within the Random-Phase Approximation generalized to the superfluid case, while the influence of the neutron component is considered only at qualitative level. Electron screening plays a major role in modifying the proton spectrum and spectral function. At the same time the electron plasmon is strongly modified and damped by the indirect coupling with the superfluid proton component, even at moderately low values of the gap. The excitation spectrum shows the interplay of the different components and their relevance for each excitation modes. The results are relevant for neutrino physics and thermodynamical processes in neutron stars. If electron screening is neglected, the spectral properties of the proton component show some resemblance with the physical situation in high-T c superconductors, and we briefly discuss similarities and differences in this connection. In a general prospect, the results of the study emphasize the role of Coulomb interaction in strong superconductors.
13. Combined effect of structural softening and magneto-elastic coupling on elastic coefficients of Ni-Mn-Ga austenite
Czech Academy of Sciences Publication Activity Database
Seiner, Hanuš; Heczko, Oleg; Sedlák, Petr; Bodnárová, Lucie; Novotný, Michal; Kopeček, Jaromír; Landa, Michal
2013-01-01
Roč. 577, November 2013 (2013), S131-S135 ISSN 0925-8388 R&D Projects: GA ČR GAP107/10/0824; GA ČR(CZ) GA101/09/0702; GA ČR(CZ) GAP107/11/0391; GA MŠk(CZ) 1M06031 Institutional research plan: CEZ:AV0Z20760514; CEZ:AV0Z10100520 Keywords : Ni2MnGa * elastic constants of Ni-Mn-Ga austenite * magnetic shape memory effect * martensitic transformation * elastic softening Subject RIV: BM - Solid Matter Physics ; Magnetism; BM - Solid Matter Physics ; Magnetism (FZU-D) Impact factor: 2.726, year: 2013 http://www.sciencedirect.com/science/article/pii/S0925838812000539
14. The fundamental constants a mystery of physics
CERN Document Server
Fritzsch, Harald
2009-01-01
The speed of light, the fine structure constant, and Newton's constant of gravity — these are just three among the many physical constants that define our picture of the world. Where do they come from? Are they constant in time and across space? In this book, physicist and author Harald Fritzsch invites the reader to explore the mystery of the fundamental constants of physics in the company of Isaac Newton, Albert Einstein, and a modern-day physicist
15. Omnidirectional antenna having constant phase
Energy Technology Data Exchange (ETDEWEB)
Sena, Matthew
2017-04-04
Various technologies presented herein relate to constructing and/or operating an antenna having an omnidirectional electrical field of constant phase. The antenna comprises an upper plate made up of multiple conductive rings, a lower ground-plane plate, a plurality of grounding posts, a conical feed, and a radio frequency (RF) feed connector. The upper plate has a multi-ring configuration comprising a large outer ring and several smaller rings of equal size located within the outer ring. The large outer ring and the four smaller rings have the same cross-section. The grounding posts ground the upper plate to the lower plate while maintaining a required spacing/parallelism therebetween.
16. AELAS: Automatic ELAStic property derivations via high-throughput first-principles computation
Science.gov (United States)
Zhang, S. H.; Zhang, R. F.
2017-11-01
The elastic properties are fundamental and important for crystalline materials as they relate to other mechanical properties, various thermodynamic qualities as well as some critical physical properties. However, a complete set of experimentally determined elastic properties is only available for a small subset of known materials, and an automatic scheme for the derivations of elastic properties that is adapted to high-throughput computation is much demanding. In this paper, we present the AELAS code, an automated program for calculating second-order elastic constants of both two-dimensional and three-dimensional single crystal materials with any symmetry, which is designed mainly for high-throughput first-principles computation. Other derivations of general elastic properties such as Young's, bulk and shear moduli as well as Poisson's ratio of polycrystal materials, Pugh ratio, Cauchy pressure, elastic anisotropy and elastic stability criterion, are also implemented in this code. The implementation of the code has been critically validated by a lot of evaluations and tests on a broad class of materials including two-dimensional and three-dimensional materials, providing its efficiency and capability for high-throughput screening of specific materials with targeted mechanical properties. Program Files doi:http://dx.doi.org/10.17632/f8fwg4j9tw.1 Licensing provisions: BSD 3-Clause Programming language: Fortran Nature of problem: To automate the calculations of second-order elastic constants and the derivations of other elastic properties for two-dimensional and three-dimensional materials with any symmetry via high-throughput first-principles computation. Solution method: The space-group number is firstly determined by the SPGLIB code [1] and the structure is then redefined to unit cell with IEEE-format [2]. Secondly, based on the determined space group number, a set of distortion modes is automatically specified and the distorted structure files are generated
17. On Elasticity Measurement in Cloud Computing
Directory of Open Access Journals (Sweden)
Wei Ai
2016-01-01
Full Text Available Elasticity is the foundation of cloud performance and can be considered as a great advantage and a key benefit of cloud computing. However, there is no clear, concise, and formal definition of elasticity measurement, and thus no effective approach to elasticity quantification has been developed so far. Existing work on elasticity lack of solid and technical way of defining elasticity measurement and definitions of elasticity metrics have not been accurate enough to capture the essence of elasticity measurement. In this paper, we present a new definition of elasticity measurement and propose a quantifying and measuring method using a continuous-time Markov chain (CTMC model, which is easy to use for precise calculation of elasticity value of a cloud computing platform. Our numerical results demonstrate the basic parameters affecting elasticity as measured by the proposed measurement approach. Furthermore, our simulation and experimental results validate that the proposed measurement approach is not only correct but also robust and is effective in computing and comparing the elasticity of cloud platforms. Our research in this paper makes significant contribution to quantitative measurement of elasticity in cloud computing.
18. Thermodynamic parameters of elasticity and electrical conductivity ...
African Journals Online (AJOL)
The thermodynamic parameters (change in free energy of elasticity, DGe; change in enthalpy of elasticity, DHe; and change in entropy of elasticity, DSe) and the electrical conductivity of natural rubber composites reinforced separately with some agricultural wastes have been determined. Results show that the reinforced ...
19. The Price Elasticity of Residential Energy Use,
Science.gov (United States)
household energy- consumption behavior : The difference between the own-price elasticity of total consumption and that of saturation is a measure of the responsiveness of ’conservation’ to price....estimates of the own-price elasticities of total consumption but almost surely will produce erroneous estimates of the cross-price elasticities. As regards
20. short communication thermodynamic parameters of elasticity
African Journals Online (AJOL)
a
ABSTRACT. The thermodynamic parameters (change in free energy of elasticity, AGe; change in enthalpy of elasticity, AHe; and change in entropy of elasticity, ASe) and the electrical conductivity of natural rubber composites reinforced separately with some agricultural wastes have been determined. Results show that the ...
1. Electronic structure, elasticity, bonding features and mechanical behaviour of zinc intermetallics: A DFT study
Energy Technology Data Exchange (ETDEWEB)
Fatima, Bushra, E-mail: [email protected]; Acharya, Nikita; Sanyal, Sankar P. [Department of Physics, Barkatullah University, Bhopal, 462026 (India)
2016-05-06
The structural stability, electronic structure, elastic and mechanical properties of TiZn and ZrZn intermetallics have been studied using ab-initio full potential linearized augmented plane wave (FP-LAPW) method within generalized gradient approximation for exchange and correlation potentials. The various structural parameters, such as lattice constant (a{sub 0}), bulk modulus (B), and its pressure derivative (B’) are analysed and compared. The investigation of elastic constants affirm that both TiZn and ZrZn are elastically stable in CsCl (B{sub 2} phase) structure. The electronic structures have been analysed quantitatively from the band structure which reveals the metallic nature of these compounds. To better illustrate the nature of bonding and charge transfer, we have also studied the Fermi surfaces. The three well known criterion of ductility namely Pugh’s rule, Cauchy’s pressure and Frantsevich rule elucidate the ductile nature of these compounds.
2. Double dividend effectiveness of energy tax policies and the elasticity of substitution. A CGE appraisal
International Nuclear Information System (INIS)
Sancho, Ferran
2010-01-01
There is a considerable body of literature that has studied whether or not an adequately designed tax swap, whereby an ecotax is levied and some other tax is reduced while keeping government income constant, may achieve a so-called double dividend, that is, an increase in environmental quality and an increase in overall efficiency. Arguments in favor and against are abundant. Our position is that the issue should be empirically studied starting from an actual, non-optimal tax system structure and by way of checking the responsiveness of equilibria to revenue neutral tax regimes under alternate scenarios regarding technological substitution. With the use of a CGE model, we find that the most critical elasticity for achieving a double dividend is the substitution elasticity between labor and capital whereas the elasticity that would generate the highest reduction in carbon dioxide emissions is the substitution elasticity among energy goods. (author)
3. Double dividend effectiveness of energy tax policies and the elasticity of substitution: A CGE appraisal
Energy Technology Data Exchange (ETDEWEB)
Sancho, Ferran, E-mail: [email protected] [Departament d' Economia, Universitat Autonoma de Barcelona, 08193-Bellaterra (Spain)
2010-06-15
There is a considerable body of literature that has studied whether or not an adequately designed tax swap, whereby an ecotax is levied and some other tax is reduced while keeping government income constant, may achieve a so-called double dividend, that is, an increase in environmental quality and an increase in overall efficiency. Arguments in favor and against are abundant. Our position is that the issue should be empirically studied starting from an actual, non-optimal tax system structure and by way of checking the responsiveness of equilibria to revenue neutral tax regimes under alternate scenarios regarding technological substitution. With the use of a CGE model, we find that the most critical elasticity for achieving a double dividend is the substitution elasticity between labor and capital whereas the elasticity that would generate the highest reduction in carbon dioxide emissions is the substitution elasticity among energy goods.
4. Double dividend effectiveness of energy tax policies and the elasticity of substitution. A CGE appraisal
Energy Technology Data Exchange (ETDEWEB)
Sancho, Ferran [Departament d' Economia, Universitat Autonoma de Barcelona, 08193-Bellaterra (Spain)
2010-06-15
There is a considerable body of literature that has studied whether or not an adequately designed tax swap, whereby an ecotax is levied and some other tax is reduced while keeping government income constant, may achieve a so-called double dividend, that is, an increase in environmental quality and an increase in overall efficiency. Arguments in favor and against are abundant. Our position is that the issue should be empirically studied starting from an actual, non-optimal tax system structure and by way of checking the responsiveness of equilibria to revenue neutral tax regimes under alternate scenarios regarding technological substitution. With the use of a CGE model, we find that the most critical elasticity for achieving a double dividend is the substitution elasticity between labor and capital whereas the elasticity that would generate the highest reduction in carbon dioxide emissions is the substitution elasticity among energy goods. (author)
5. An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials
Science.gov (United States)
Nemeth, Michael P.
2011-01-01
An in-depth tutorial on the constitutive equations for elastic, anisotropic materials is presented. Basic concepts are introduced that are used to characterize materials, and notions about how anisotropic material deform are presented. Hooke s law and the Duhamel-Neuman law for isotropic materials are presented and discussed. Then, the most general form of Hooke s law for elastic anisotropic materials is presented and symmetry requirements are given. A similar presentation is also given for the generalized Duhamel-Neuman law for elastic, anisotropic materials that includes thermal effects. Transformation equations for stress and strains are presented and the most general form of the transformation equations for the constitutive matrices are given. Then, specialized transformation equations are presented for dextral rotations about the coordinate axes. Next, concepts of material symmetry are introduced and criteria for material symmetries are presented. Additionally, engineering constants of fully anisotropic, elastic materials are derived from first principles and the specialized to several cases of practical importance.
6. Structural stability, elastic and thermodynamic properties of Au-Cu alloys from first-principles calculations
Science.gov (United States)
Kong, Ge-Xing; Ma, Xiao-Juan; Liu, Qi-Jun; Li, Yong; Liu, Zheng-Tang
2018-03-01
Using first-principles calculations method based on density functional theory (DFT) with the Perdew-Burke-Ernzerhof (PBE) implementation of the generalized gradient approximation (GGA), we investigate the structural, elastic and thermodynamic properties of gold-copper intermetallic compounds (Au-Cu ICs). The calculated lattice parameters are in excellent agreement with experimental data. The elastic constants show that all the investigated Au-Cu alloys are mechanically stable. Elastic properties, including the shear modulus, Young's modulus, Poisson's ratio and Pugh's indicator, of the intermetallic compounds are evaluated and discussed, with special attention to the remarkable anisotropy displayed by Au-Cu ICs. Thermodynamic and transport properties including the Debye temperature, thermal conductivity and melting point are predicted from the averaged sound velocity and elastic moduli, using semi-empirical formulas.
7. Constitutive relations in multidimensional isotropic elasticity and their restrictions to subspaces of lower dimensions
Science.gov (United States)
Georgievskii, D. V.
2017-07-01
The mechanical meaning and the relationships among material constants in an n-dimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke's law) to subspaces of lower dimension caused by the conditions that an m-dimensional strain state or an m-dimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elasti | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8897624015808105, "perplexity": 3185.5744498590348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863489.85/warc/CC-MAIN-20180620065936-20180620085936-00023.warc.gz"} |
https://chemistry.stackexchange.com/questions/61253/overall-equilibrium-expression-for-competitive-equilibria | # Overall equilibrium expression for competitive equilibria
I have a case where I have two reactions that depend on a common reactant. The reactions can be written as:
I) $\ce{A + B <=> AB}$
II) $\ce{A + C <=> AC}$
We can thus write equilibrium expressions for both reactions:
I) $K_\mathrm{I} = \frac{[\ce{AB}]}{[\ce{A}][\ce{B}]}$
II) $K_\mathrm{II} = \frac{[\ce{AC}]}{[\ce{A}][\ce{C}]}$
How can I combine these equations to get an equation that relates the equilibrium concentration of B to that of C?
Secondarily, can this be extended to give an equation that relates the equilibrium concentrations of B and C to the initial concentrations of A, B and C?
I already know the equilibrium and rate constants for both reactions in isolation. If it is relevant, A is a solid with a fixed number of surface sites, to which B and are adsorbed. B and C are in aqueous solution.
OK, so essentially we have $[A],[B],[C],[AB],\text{ and }[AC]$, and a few equations: three conservation laws and two equilibrium constants. \left\{ \begin{align} [A]+[AB]+[AC] & = A_0\\ [B]+[AB] & = B_0\\ [C]+[AC] & = C_0\\ {[AB]\over[A][B]}& =K_1 \\ {[AC]\over[A][C]}& =K_2 \end{align} \right. With 5 unknowns and 5 equations, this should be solvable (because the nature manages to solve it somehow, if not for other reason). There is no guarantee, however, that the solution will be nice. Indeed, if we deal with the system exactly as it stands now, it is equivalent to a certain 4th degree algebraic equation which is better solved numerically.
Things get somewhat more manageable if we may disregard something. For example, if A is a relatively minor component, so that $[A]<<[B]$, then we may assume $[B]\approx B_0$ (and the same for C). All of a sudden, the problem becomes linear: $$[AB]=K_1[A]B_0 \\ [AC]=K_2[A]C_0 \\ [A]+K_1[A]B_0+K_2[A]C_0=A_0$$ which yields $$[AB]={K_1B_0\over1+K_1B_0+K_2C_0} \\ [AC]={K_2C_0\over1+K_1B_0+K_2C_0}$$
Whether or not this approximation is realistic in your particular case is up to you.
How about this? Note that since both AB and AC affect the concentrations of B and C, it's hard to get a relationship between B and C that doesn't involve these species. You can express the relationship in terms of ratios though.
$$[\mathrm{A}] = \frac{[\mathrm{AB}]}{[\mathrm{B}]K_{\mathrm{I}}}$$ $$[\mathrm{A}] = \frac{[\mathrm{AC}]}{[\mathrm{C}]K_{\mathrm{II}}}$$
$$\frac{[\mathrm{AB}]}{[\mathrm{B}]K_{\mathrm{I}}} = \frac{[\mathrm{AC}]}{[\mathrm{C}]K_{\mathrm{II}}}$$
$$\frac{[\mathrm{B}]}{[\mathrm{C}]} = \frac{K_{\mathrm{II}}}{K_{\mathrm{I}}}\frac{[\mathrm{AB}]}{[\mathrm{AC}]}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9872615337371826, "perplexity": 404.9655539092384}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526714.15/warc/CC-MAIN-20190720214645-20190721000645-00380.warc.gz"} |
https://www.jiskha.com/questions/568325/Simplify-the-following-expression-and-rewrite-it-in-an-equivalent-form-with-positive | # Math
Simplify the following expression, and rewrite it in an equivalent form with positive exponents.
24x^3y^-3/72x^-5y^-1
My answer was going to be:
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3. 👁 48
1. ashley, I think you need a tutor in person, not online help screens.
do this in parts
x part: x^3/x^-5= x^8
y part: y^-3/y^-1=1/y^2
check that.
1. 👍 0
2. 👎 0
2. I completely agree with you. I am a visual learner and this book is so hard for me to understand it isn't even funny. I am so stressed out and i thank you for your help really.
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posted by Ashley
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More Similar Questions | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9030871987342834, "perplexity": 3127.834138692537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232259015.92/warc/CC-MAIN-20190526085156-20190526111156-00473.warc.gz"} |
http://megasoft-rapid.com/Idaho/error-propagations.html | Address 505 S Florence St, Grangeville, ID 83530 (208) 983-5284 http://www.compunet.biz
error propagations Lucile, Idaho
Answer: we can calculate the time as (g = 9.81 m/s2 is assumed to be known exactly) t = - v / g = 3.8 m/s / 9.81 m/s2 = 0.387 This is the most general expression for the propagation of error from one set of variables onto another. In this case, expressions for more complicated functions can be derived by combining simpler functions. The problem might state that there is a 5% uncertainty when measuring this radius.
Schließen Ja, ich möchte sie behalten Rückgängig machen Schließen Dieses Video ist nicht verfügbar. Retrieved 2016-04-04. ^ "Strategies for Variance Estimation" (PDF). National Bureau of Standards. 70C (4): 262. In the following examples: q is the result of a mathematical operation δ is the uncertainty associated with a measurement.
How would you determine the uncertainty in your calculated values? October 9, 2009. For , and , so (9) For division of quantities with , and , so (10) Dividing through by and rearranging then gives (11) For exponentiation of quantities with (12) and JSTOR2281592. ^ Ochoa1,Benjamin; Belongie, Serge "Covariance Propagation for Guided Matching" ^ Ku, H.
It will be interesting to see how this additional uncertainty will affect the result! Each covariance term, σ i j {\displaystyle \sigma _ σ 2} can be expressed in terms of the correlation coefficient ρ i j {\displaystyle \rho _ σ 0\,} by σ i When the errors on x are uncorrelated the general expression simplifies to Σ i j f = ∑ k n A i k Σ k x A j k . {\displaystyle In matrix notation, [3] Σ f = J Σ x J ⊤ . {\displaystyle \mathrm {\Sigma } ^{\mathrm {f} }=\mathrm {J} \mathrm {\Sigma } ^{\mathrm {x} }\mathrm {J} ^{\top }.} That
Uncertainty components are estimated from direct repetitions of the measurement result. Retrieved 3 October 2012. ^ Clifford, A. Structural and Multidisciplinary Optimization. 37 (3): 239–253. Keith (2002), Data Reduction and Error Analysis for the Physical Sciences (3rd ed.), McGraw-Hill, ISBN0-07-119926-8 Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN0-471-59995-6 Taylor, J.
The propagation of error formula for $$Y = f(X, Z, \ldots \, )$$ a function of one or more variables with measurements, $$(X, Z, \ldots \, )$$ Sensitivity coefficients The partial derivatives are the sensitivity coefficients for the associated components. Example: An angle is measured to be 30°: ±0.5°. Consider a length-measuring tool that gives an uncertainty of 1 cm.
Journal of the American Statistical Association. 55 (292): 708–713. Sometimes, these terms are omitted from the formula. Knowing the uncertainty in the final value is the correct way to officially determine the correct number of decimal places and significant figures in the final calculated result. Function Variance Standard Deviation f = a A {\displaystyle f=aA\,} σ f 2 = a 2 σ A 2 {\displaystyle \sigma _{f}^{2}=a^{2}\sigma _{A}^{2}} σ f = | a | σ A
JCGM. Journal of Sound and Vibrations. 332 (11): 2750–2776. Or in matrix notation, f ≈ f 0 + J x {\displaystyle \mathrm σ 6 \approx \mathrm σ 5 ^ σ 4+\mathrm σ 3 \mathrm σ 2 \,} where J is Note that these means and variances are exact, as they do not recur to linearisation of the ratio.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. In a probabilistic approach, the function f must usually be linearized by approximation to a first-order Taylor series expansion, though in some cases, exact formulas can be derived that do not Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Propagation of Error http://webche.ent.ohiou.edu/che408/S...lculations.ppt (accessed Nov 20, 2009).
Now a repeated run of the cart would be expected to give a result between 36.1 and 39.7 cm/s. Kategorie Bildung Lizenz Standard-YouTube-Lizenz Mehr anzeigen Weniger anzeigen Wird geladen... The exact covariance of two ratios with a pair of different poles p 1 {\displaystyle p_{1}} and p 2 {\displaystyle p_{2}} is similarly available.[10] The case of the inverse of a In both cases, the variance is a simple function of the mean.[9] Therefore, the variance has to be considered in a principal value sense if p − μ {\displaystyle p-\mu }
ISBN0470160551.[pageneeded] ^ Lee, S. Berkeley Seismology Laboratory. A. (1973). p.2.
Derivation of Exact Formula Suppose a certain experiment requires multiple instruments to carry out. The value of a quantity and its error are then expressed as an interval x ± u. Since f0 is a constant it does not contribute to the error on f. Retrieved 2016-04-04. ^ "Strategies for Variance Estimation" (PDF).
doi:10.1016/j.jsv.2012.12.009. ^ Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". In this case, expressions for more complicated functions can be derived by combining simpler functions. This ratio is very important because it relates the uncertainty to the measured value itself. The system returned: (22) Invalid argument The remote host or network may be down.
This is easy: just multiply the error in X with the absolute value of the constant, and this will give you the error in R: If you compare this to the Section (4.1.1). Authority control GND: 4479158-6 Retrieved from "https://en.wikipedia.org/w/index.php?title=Propagation_of_uncertainty&oldid=742325047" Categories: Algebra of random variablesNumerical analysisStatistical approximationsUncertainty of numbersStatistical deviation and dispersionHidden categories: Wikipedia articles needing page number citations from October 2012Wikipedia articles needing Since f0 is a constant it does not contribute to the error on f. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9868572354316711, "perplexity": 2002.7997441281914}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039741979.10/warc/CC-MAIN-20181114104603-20181114130603-00395.warc.gz"} |
http://cbsephysics.in/electricity-cbse-class-x-quick-revision-question-bank/ | # ELECTRICITY CBSE CLASS X :: QUICK REVISION & QUESTION BANK
ELECTRICITY
FORE VIEW:
This chapter deals with Electric Charge, Electric Current, Electric Circuits and Electric Potential and Potential Difference, Ohms Law, Concepts of Resistance and Resistivity, Combination of Resistors, Heating Effect of current and Electric Power.
Full attempt is made to explain every concept so that it is very easy for the students and apply in appropriate situations (reasoning, solving numericals etc)
Quick Revision
• Electric charge: this is the fundamental property of protons and electrons which gives rise to electric force between them.
• Electric current: the time rate of flow of charge through any cross section of a conductor is the measure of current
Electric current = total charge flowing/time taken
• Unit of electric current = ampere
• One ampere: the current through a wire is said to be one ampere if one coulomb of charge flows through it in one second.
• Direction of electric current is taken to be the direction of flow of positive charge or opposite to the direction of flow of negative charge (electron)
• Charge of an electron is = -1.6 x 10-19 C and charge of proton is : +1.6 x 10-19C (Q)
• Net charge on any body is given by
Q = ± ne where
Where,
e = electric charge
n = number of charges
Qn: 1. Define one Ampere?
2. Calculate the number of electrons constituting one coulomb of charge?
we know, Q = ±ne,
here Q=1C
e= 1.6 x 10-19 C
n= 1/1.6 x 10-19 = 6.25 x 1018
• To set the electrons in motion in an electric circuit, we use a cell or battery. Current flows in a circuit from positive terminal to negative terminal of the cell.
• Current in the circuit is measured by an ammeter
• Define Electric Potential Difference
Electric potential difference between any two points in an electric circuit is the amount of work done (W) in moving a unit charge from one point to another.
• Potential difference V = work done (W) / charge (Q)
• Unit of potential difference = volt
• Define 1 Volt:
The potential difference between any two points in an electric circuit is said to be 1 volt if 1 joule of work is done in moving 1 coulomb of charge from one point to another.
• Potential Difference (p.d) is measured by a volt meter
• Ammeter is connected in series in the circuit and voltmeter is connected parallel across the points between which p.d is to be measured.
QN :
How much energy is given to each coulomb of charge passing through a 6V battery?
Ans: V= W/Q Therefore , W= VQ V=6V Q=1C
Therefore, W = 1J
• Ohm’s law: The current through the conductor is directly proportional to the potential difference across the ends of the conductor provided the physical conditions such as temperature, pressure etc. are kept constant.
V ∝ I V = IR Where R= resistance of the conductor
• Resistance: It is the opposition offered by the conductor to the flow of charge trough it.
• Unit of resistance – ohm (Ω).
• Define one ohm :
We know V=IR When V=1V, I=IA Then R=1Ω.
The resistance of a conductor is said to be one ohm if 1ampere of current flows through it when the p.d across it is 1volt.
• Factors on which the resistance of the conductor depends:
(i) R is directly proportional to the length of conductor (l).
(ii) R is inversely proportional to the area of cross section (A).
R ∝ l /A R =ρl / A.
When l= 1m A=1m2
Then R= ρ
Resistivity of the material of conductor is defined as the resistance of conductor of unit length and unit area of cross section
• Unit of Resistivity: Ωm.
• Both resistance and resisitivity of conductor depend on temperature (they increase with increase in temperature). It is because as the temperature is increased the electrons in the conductor undergo so much of collision with ions which increases the resistance.
• Resistivity depends on temperature, material of the conductor and not on the dimension of the conductor.
Questions:
1. Compare the resistance of a conductor at 30oC and at 70oC
Ans: R70o>R30o (Resistance of the conductor increases with the increase in temperature)
2. A copper wire is stretched to twice its length. What happens to its resistivity?
Combination of resistances: when two or more resistances are connected in series the net or equivalent resistance is equal to the sum of individual resistances.
The total resistance in the above diagram is
R = R1+R2+R3 [derivation given in N.C.E.R.T]
• If R1>R2>R3 then R series>R1
• Equivalent resistance of series combination of ‘n’ equal resistances (each of value r) is given by Rs = nr
• Resistances in parallel :
• When two or more resistors are connected in parallel then the reciprocal of equivalent or net resistance is equal to the sum of the reciprocal of the individual resistances
1/Rparallel = 1/R1+1/R2 +1/R3
• If R1>R2>R3 then, Rparallel • Equivalent resistance of parallel combination of ‘n’ equal resistances (each of value r) is given by Rp = r/n
Qn :
1) How many 176 resistances (in parallel) are required to carry 5A on a 220V line?
Ans:
Resistance required in the circuit Rp = V/I = 220V/5V = 44 .
Resistance of each resistance r = 176 .
If ‘n’ resistors each of resistance ‘r’ are connected in parallel then Rp = r/n i.e. is 44 = 176/n Therefore, 176/44 = 4
2) Which circuit has low resistance?
1. Three resistance R1, R2, R3 in series
Or
2. Three resistances R1, R2, R3 in parallel?
Ans:
Rparallel • Advantages of connecting electrical devices in parallel with the battery over connecting them in series.
 When a number of electrical devices are connected in parallel, each device gets the same potential difference as provided by the battery and it keeps on working even if other devices fail to work, whereas in the case of devices connecting in series, when one device fails to work, all the other devices stop working.
 Parallel circuit is helpful when each device has different resistances and requires different current for its operation as in this case the current divides itself through different devices. This is not so in series circuit where some current flows through all devices irrespective of their resistances.
HEATING EFFECT OF ELECTRIC CURRENT
• Whenever a current is passed through a conductor it becomes hot after sometime which means that electrical energy is converted to heat energy. This effect is called heating effect of electric current.
JOULES LAW
• Whenever a current I flows through a conductor resistance R for a time t, the amount of heat produced is given by heat H  I2Rt
ELECTRIC ENERGY
• The total work done by the source in maintaining the current in the circuit for a given time is called the electric energy consumed in the circuit.
W = VIt = I2Rt = V2t/R
ELECTRIC POWER
• The rate at which work is done by the source (battery) in maintaining the current in the circuit is called the electric power of the circuit.
P = W/t = VI = I2R = V2/R
UNIT OF ELECTRIC POWER [WATT]
P = VI When V = 1 Volt and I = 1 Ampere, Then P = 1 Watt.
Definition of 1 Watt:
The power of an electric circuit is said to be one watt if one ampere of current flows through it when the potential difference across it is 1 Volt.
COMMERCIAL UNIT OF ELECTRIC ENERGY – KWh
Definition of 1 KWh: It is the energy consumed when a an electrical device of power 1 KW works for one hour: 1 KWh = 3.6 x 106 J
Questions
1. What determines the rate at which energy is delivered by current?
Electric Power.
2. An electric motor takes 5A from a 220V line. Determine the power of the motor and the energy consumed in 2h.
I = 5A, t = 2h = 7200 s.
Energy consumed W = VIt = 220 x 5 x 7200 = 792 x 104 J
Power P = VI = 220 x 5 = 1100 W
3. Why are Cu and Al wires usually employed for electrical transmission?
Copper and aluminium possess low resistivity and also cheaper than silver which has the least resistivity of all metals
4. Why does the cord of an electric heater not glow while the heating element does?
The cord of an electric heater is made of thick copper wire and has much lower resistance than its element. For the same current (I) flowing through the cord and the element, heat produced (I2 Rt) in the element is much more than that produced in the cord. Consequently, the element becomes very hot and glows whereas the cord does not become hot and as such does not glow.
Important formulae to remember
• Charge on a body
Q = ne Where n is number of charge carries & e electronic charge 1.6 x 10-19 C
• Current I = Q/t Q – Charge T – time
• Potentials difference V = W/Q Where W – Work done Q – Charge
• Ohms law V = IR
• V – Potential different access the conductor
• I – Current through the conductor
• R – Resistance of conductor
• Resistance
R = ρl/A
Where ρ – Specific resistance of the material of wire
l – Length of wire
A – Area of cross section of wire
• Combination of resistances
• R Series = R1 + R2 + R3
• 1/Rparallel = 1/R1 + 1/R2 + 1/R3
• Electric energy supplied or electric work done is given by
• W = VIt = I2Rt = V2 t /R
• Electric power P = VI = I2R = V2/R
• Commercial unit of electrical energy 1 KWh = 3.6 x 106 J
QUESTION BANK
Very short answer questions (1 mark)
1. Give the unit of (a) Charge (b) Current
2. Define current
3. Name the unit of (a) electrical resistance (b) resistivity
4. Define One Ohm
5. Define Resistivity
6. What is the resistance of a torch bulb rated at 2.5 V and 500 mA?
7. Two resistances of each 2 ohm are connected in parallel. Find their equalent resistance.
8. On what factors does the resistivity of a material depend?
9. Plot a graph between the Potential difference V and current I through a conductor
10. What happens to the resistance of the circuit if the current through it is doubled?
1. Two wires of same material are having length L and 2L. Compare their resistance and resistivity.
2. Why are coils of electric toaster and electric iron made of an alloy rather then a pure metal?
3. Two wires are of same length and radius but one of them is copper and the other is of iron. Which will have more resistance? (Given the resistivity of copper = 1.62 x 10 -8 ohm meter and resistivity of iron = 10 x 10-8 ohm meter.
4. Define 1KWh. Give the relation between 1kwh and Joule.
5. State which has a higher resistance. A 50W or 25W lamp. Also find the ratio of their resistances.
6. A wire of resistance 5 Ohm is spent in the form of closed circle. What is the resistance between 2 points at the ends of any diameter of the circle?
7. Calculate the amount of charge that would flow in one hour through the element of an electric iron drawing a current of 0.4 amps.
8. A electric toaster of resistance 20 Ohm takes a current of 5A. Calculate the heat developed in 30 s.
9. A bulb is rated at 5V, 100mA. Calculate its (1) Power (2) Resistance
10. Name two special characteristics of a heater coil.
1. Define resistance and resistivity. Give the relation between them. Explain the dependence of resistance on temperature.
2. With the help of neat circuit, derive the expression for the equalent resistance of 3 resistances connected in series.
3. With the help of neat circuit, derive the expression for the equivalent resistance of 3 resistances connected in parallel
4. (a ) Draw the circuit consisting of a battery of five 2V cells, 5ohm resistor, 10 ohm resistor, 15 ohm resistor and a plug key. All connected in series (b) Calculate the current passing through the above circuit when key is closed.
5. Two identical resistors each of resistance 2 Ohm are connected in turn (1) in series (2) in parallel to a battery of 12 V. Calculate the ratio of power consumed in two cases.
6. A piece of wire is redrawn by pulling it until its length is tripled. Compare the new resistance with the original value.
7. An electric kettle is rated 500W, 200V. IT is used to heat 400 gm of water for 30 secs. Assuming the voltage to be 220V calculate the rise in temperature of water. Specific heat capacity of water is 4200 J/Kg ºC.
8. In an experiment the current flowing through a resistor and potential difference across it are measured. The values are given below. Show that these values confirm Ohm’s Law and also find the resistance of the resistor.
I (ampere) I(ampere) 1.0 1.0 2 1.5 2.0 2.0 2.5 2.5 3.0 3.0
V (volt) V(volt) 4.0 4.0 6.0 6.0 8.0 8.0 10.0 10.0 12.0 12.0
9. A heater draws 1100 W at 220V. (a) Find the resistance of the heater (b) Calculate the energy in KWh consumed in a week if the heater is used daily for 4 hours. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8651182055473328, "perplexity": 1197.0464323821425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738660350.13/warc/CC-MAIN-20160924173740-00161-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/128i1/C2xM4(2).C4.html | Copied to
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## G = C2×M4(2).C4order 128 = 27
### Direct product of C2 and M4(2).C4
direct product, p-group, metabelian, nilpotent (class 3), monomial
Series: Derived Chief Lower central Upper central Jennings
Derived series C1 — C4 — C2×M4(2).C4
Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×M4(2).C4
Lower central C1 — C2 — C4 — C2×M4(2).C4
Upper central C1 — C2×C4 — C23×C4 — C2×M4(2).C4
Jennings C1 — C2 — C2 — C2×C4 — C2×M4(2).C4
Generators and relations for C2×M4(2).C4
G = < a,b,c,d | a2=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >
Subgroups: 300 in 230 conjugacy classes, 172 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C23, C23, C23, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C2×C8.C4, M4(2).C4, C22×M4(2), C22×M4(2), C2×M4(2).C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, M4(2).C4, C22×C4⋊C4, C2×M4(2).C4
Smallest permutation representation of C2×M4(2).C4
On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)
(1 30 17 10 5 26 21 14)(2 29 18 9 6 25 22 13)(3 28 19 16 7 32 23 12)(4 27 20 15 8 31 24 11)
G:=sub<Sym(32)| (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,30,17,10,5,26,21,14)(2,29,18,9,6,25,22,13)(3,28,19,16,7,32,23,12)(4,27,20,15,8,31,24,11)>;
G:=Group( (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31), (1,30,17,10,5,26,21,14)(2,29,18,9,6,25,22,13)(3,28,19,16,7,32,23,12)(4,27,20,15,8,31,24,11) );
G=PermutationGroup([[(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31)], [(1,30,17,10,5,26,21,14),(2,29,18,9,6,25,22,13),(3,28,19,16,7,32,23,12),(4,27,20,15,8,31,24,11)]])
44 conjugacy classes
class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 8A ··· 8X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4
44 irreducible representations
dim 1 1 1 1 1 2 2 2 4 type + + + + + - - image C1 C2 C2 C2 C4 D4 Q8 Q8 M4(2).C4 kernel C2×M4(2).C4 C2×C8.C4 M4(2).C4 C22×M4(2) C2×M4(2) C22×C4 C22×C4 C24 C2 # reps 1 4 8 3 16 4 3 1 4
Matrix representation of C2×M4(2).C4 in GL6(𝔽17)
16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
0 16 0 0 0 0 1 0 0 0 0 0 0 0 4 9 0 0 0 0 10 13 0 0 0 0 7 4 0 1 0 0 13 4 13 0
,
1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 16 1 0 0 0 0 1 0 1 0 0 0 0 0 0 16
,
10 16 0 0 0 0 16 7 0 0 0 0 0 0 1 0 2 0 0 0 0 0 1 16 0 0 10 0 16 0 0 0 10 13 16 0
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,10,7,13,0,0,9,13,4,4,0,0,0,0,0,13,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,1,0,10,10,0,0,0,0,0,13,0,0,2,1,16,16,0,0,0,16,0,0] >;
C2×M4(2).C4 in GAP, Magma, Sage, TeX
C_2\times M_4(2).C_4
% in TeX
G:=Group("C2xM4(2).C4");
// GroupNames label
G:=SmallGroup(128,1647);
// by ID
G=gap.SmallGroup(128,1647);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations
×
𝔽 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8467667102813721, "perplexity": 1312.3558491555175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585561.4/warc/CC-MAIN-20211023033857-20211023063857-00124.warc.gz"} |
http://terrytao.wordpress.com/tag/probabilistic-method/ | You are currently browsing the tag archive for the ‘probabilistic method’ tag.
The celebrated Szemerédi-Trotter theorem gives a bound for the set of incidences ${I(P,L) := \{ (p,\ell) \in P \times L: p \in \ell \}}$ between a finite set of points ${P}$ and a finite set of lines ${L}$ in the Euclidean plane ${{\bf R}^2}$. Specifically, the bound is
$\displaystyle |I(P,L)| \ll |P|^{2/3} |L|^{2/3} + |P| + |L| \ \ \ \ \ (1)$
where we use the asymptotic notation ${X \ll Y}$ or ${X=O(Y)}$ to denote the statement that ${X \leq CY}$ for some absolute constant ${C}$. In particular, the number of incidences between ${n}$ points and ${n}$ lines is ${O(n^{4/3})}$. This bound is sharp; consider for instance the discrete box ${P := \{ (a,b) \in {\bf Z}^2: 1 \leq a \leq N; 1 \leq b \leq 2N^2 \}}$ with ${L}$ being the collection of lines ${\{ (x,mx+b): m, b \in {\bf Z}, 1 \leq m \leq N, 1 \leq b \leq N^2 \}}$. One easily verifies that ${|P|=2N^3}$, ${|L| = N^3}$, and ${|I(P,L)| = N^4}$, showing that (1) is essentially sharp in the case ${|P| \sim |L|}$; one can concoct similar examples for other regimes of ${|P|}$ and ${|L|}$.
On the other hand, if one replaces the Euclidean plane ${{\bf R}^2}$ by a finite field geometry ${F^2}$, where ${F}$ is a finite field, then the estimate (1) is false. For instance, if ${P}$ is the entire plane ${F^2}$, and ${L}$ is the set of all lines in ${F^2}$, then ${|P|, |L|}$ are both comparable to ${|F|^2}$, but ${|I(P,L)|}$ is comparable to ${|F|^3}$, thus violating (1) when ${|F|}$ is large. Thus any proof of the Szemerédi-Trotter theorem must use a special property of the Euclidean plane which is not enjoyed by finite field geometries. In particular, this strongly suggests that one cannot rely purely on algebra and combinatorics to prove (1); one must also use some Euclidean geometry or topology as well.
Nowadays, the slickest proof of the Szemerédi-Trotter theorem is via the crossing number inequality (as discussed in this previous post), which ultimately relies on Euler’s famous formula ${|V|-|E|+|F|=2}$; thus in this argument it is topology which is the feature of Euclidean space which one is exploiting, and which is not present in the finite field setting. Today, though, I would like to mention a different proof (closer in spirit to the original proof of Szemerédi-Trotter, and also a later argument of Clarkson et al.), based on the method of cell decomposition, which has proven to be a very flexible method in combinatorial incidence geometry. Here, the distinctive feature of Euclidean geometry one is exploiting is convexity, which again has no finite field analogue.
Roughly speaking, the idea is this. Using nothing more than the axiom that two points determine at most one line, one can obtain the bound
$\displaystyle |I(P,L)| \ll |P| |L|^{1/2} + |L|, \ \ \ \ \ (2)$
which is inferior to (1). (On the other hand, this estimate works in both Euclidean and finite field geometries, and is sharp in the latter case, as shown by the example given earlier.) Dually, the axiom that two lines determine at most one point gives the bound
$\displaystyle |I(P,L)| \ll |L| |P|^{1/2} + |P| \ \ \ \ \ (3)$
(or alternatively, one can use projective duality to interchange points and lines and deduce (3) from (2)).
An inspection of the proof of (2) shows that it is only expected to be sharp when the bushes ${L_p := \{ \ell \in L: \ell \ni p \}}$ associated to each point ${p \in P}$ behave like “independent” subsets of ${L}$, so that there is no significant correlation between the bush ${L_p}$ of one point and the bush of another point ${L_q}$.
However, in Euclidean space, we have the phenomenon that the bush of a point ${L_p}$ is influenced by the region of space that ${p}$ lies in. Clearly, if ${p}$ lies in a set ${\Omega}$ (e.g. a convex polygon), then the only lines ${\ell \in L}$ that can contribute to ${L_p}$ are those lines which pass through ${\Omega}$. If ${\Omega}$ is a small convex region of space, one expects only a fraction of the lines in ${L}$ to actually pass through ${\Omega}$. As such, if ${p}$ and ${q}$ both lie in ${\Omega}$, then ${L_p}$ and ${L_q}$ are compressed inside a smaller subset of ${L}$, namely the set of lines passing through ${\Omega}$, and so should be more likely to intersect than if they were independent. This should lead to an improvement to (2) (and indeed, as we shall see below, ultimately leads to (1)).
More formally, the argument proceeds by applying the following lemma:
Lemma 1 (Cell decomposition) Let ${L}$ be a finite collection of lines in ${{\bf R}^2}$, let ${P}$ be a finite set of points, and let ${r \geq 1}$. Then it is possible to find a set ${R}$ of ${O(r)}$ lines in ${L}$, plus some additional open line segments not containing any point in ${P}$, which subdivide ${{\bf R}^2}$ into ${O(r^2)}$ convex regions (or cells), such that the interior of each such cell is incident to at most ${O(|L|/r)}$ lines.
The deduction of (1) from (2), (3) and Lemma 1 is very quick. Firstly we may assume we are in the range
$\displaystyle |L|^{1/2} \ll |P| \ll |L|^2 \ \ \ \ \ (4)$
otherwise the bound (1) follows already from either (2) or (3) and some high-school algebra.
Let ${r \geq 1}$ be a parameter to be optimised later. We apply the cell decomposition to subdivide ${{\bf R}^2}$ into ${O(r^2)}$ open convex regions, plus a family ${R}$ of ${O(r)}$ lines. Each of the ${O(r^2)}$ convex regions ${\Omega}$ has only ${O(|L|/r)}$ lines through it, and so by (2) contributes ${O( |P \cap \Omega| |L|^{1/2}/r^{1/2} + |L| / r )}$ incidences. Meanwhile, on each of the lines ${\ell}$ in ${R}$ used to perform this decomposition, there are at most ${|L|}$ transverse incidences (because each line in ${L}$ distinct from ${\ell}$ can intersect ${\ell}$ at most once), plus all the incidences along ${\ell}$ itself. Putting all this together, one obtains
$\displaystyle |I(P,L)| \leq |I(P,L \cap R)| + O( |P| |L|^{1/2}/r^{1/2} + |L| r).$
We optimise this by selecting ${r \sim |P|^{2/3} / |L|^{1/3}}$; from (4) we can ensure that ${r \leq |L|/2}$, so that ${|L \cap R| \leq |L|/2}$. One then obtains
$\displaystyle |I(P,L)| \leq |I(P,L \cap R)| + O( |P|^{2/3} |L|^{2/3} ).$
We can iterate away the ${L \cap R}$ error (halving the number of lines each time) and sum the resulting geometric series to obtain (1).
It remains to prove (1). If one subdivides ${{\bf R}^2}$ using ${r}$ arbitrary lines, one creates at most ${O(r^2)}$ cells (because each new line intersects the existing lines at most once, and so can create at most ${O(r)}$ distinct cells), and for a similar reason, every line in ${L}$ visits at most ${r}$ of these regions, and so by double counting one expects ${O(|L|/r)}$ lines per cell “on the average”. The key difficulty is then to get ${O(|L|/r)}$ lines through every cell, not just on the average. It turns out that a probabilistic argument will almost work, but with a logarithmic loss (thus having ${O( |L| \log |L| / r )}$ lines per cell rather than ${O(|L|/r)}$); but with a little more work one can then iterate away this loss also. The arguments here are loosely based on those of Clarkson et al.; a related (deterministic) decomposition also appears in the original paper of Szemerédi and Trotter. But I wish to focus here on the probabilistic approach.)
It is also worth noting that the original (somewhat complicated) argument of Szemerédi-Trotter has been adapted to establish the analogue of (1) in the complex plane ${{\bf C}^2}$ by Toth, while the other known proofs of Szemerédi-Trotter, so far, have not been able to be extended to this setting (the Euler characteristic argument clearly breaks down, as does any proof based on using lines to divide planes into half-spaces). So all three proofs have their advantages and disadvantages. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 103, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9433194994926453, "perplexity": 157.0199640273079}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507444209.14/warc/CC-MAIN-20141017005724-00342-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.khanacademy.org/kmap/numbers-and-operations-j/no231-rational-exponents-radicals/exponent-properties-review/v/multiplying-and-dividing-powers-with-integer-exponents | If you're seeing this message, it means we're having trouble loading external resources on our website.
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# Multiplying & dividing powers (integer exponents)
For any base a and any integer exponents n and m, aⁿ⋅aᵐ=aⁿ⁺ᵐ. For any nonzero base, aⁿ/aᵐ=aⁿ⁻ᵐ. These are worked examples for using these properties with integer exponents.
## Want to join the conversation?
• in how did he get 1/4 3
• When you have a negative power, you are taking the reciprocal of the number, and keep the power. So 2^(-2)=1/2^2. So 4^(-3)=1/4^3
• If negative exponents such as 10^-5 is equal to 1/10^5, what would fractions with negative exponents such as 1/10^-5 be equal to?
• Apply the same rule you have cited. As you put it (10)^-5 = 1/(10)^5
The expression in question is 1/(10)^-5.
Lets see! We can write (10)^-5 as 1/(10)^5 (as you wrote).
So 1/(10)^-5 can essentially be written as 1/(1/10^5)
Which is nothing but 10^5 itself( We're basically taking the reciprocal of 1/10^5)
So 1/(10)^-5 =10^5
Cheers!
EDIT: Since perhaps that's a bit long, you can remember it for a general case as:
1/a^-m = a^m
where a and m can be any of positive or negative integers(but not zero!)
Hope that helps!
• How do you divide exponents by exponents? I kinda really don't understand that part.
• An easier way to think about this is to treat the multiplication sign as an addition sign and treat the division sign as a subtraction sign. I'll put an example down below! :)
Therefore, 4_^-3 x 4_^5 is equal to 4_^2.
You would add -3 + 5, which is equal to 2. Then keep the 4 and put the 2 as the exponent!
• how do you do it when both powers are negative and you are multiplying.
• when both powers are negative, and you are multiplying,the negatives cancel eachother out so you would get a positive power.
• For the dividing part, how did you make the exponent of 12^-5 positive and the exponent of x^5 negative?
• The rule for dividing same bases is x^a/x^b=x^(a-b), so with dividing same bases you subtract the exponents. In the case of the 12s, you subtract -7-(-5), so two negatives in a row create a positive answer which is where the +5 comes from. In the x case, the exponent is positive, so applying the rule gives x^(-20-5).
If you want to use two different laws of exponents, you can use the negative exponent rule, if you move an exponent from numerator to denominator (or from denominator to numerator), you have to change the sign. So 12^-5 in the denominator would be the same as 12^5 in the numerator and x^5 in the denominator would be x^-5 in the numerator. Then you would have to use the rule for multiplying same bases shown as x^a * x^b=x^(a+b). Thus, x^-7*x^5 (as moved above) you still get 12^(-7+5) and x^-20 * x^-5 = x^(-20-5).
• what if you don't have the same base? like if you have 5 to the power of 3 times 6 to the power of 2?
• You can't combine the exponents. The bases don't match.
• Isn't multiplying exponents the same as adding exponents then? Because for multiplying exponents you add the exponents and for adding exponents you add the exponents. What is the difference?
• Your terminology is a little off...
If you are multiplying a common base, then you add the exponents. For example: x^7 * x^2 = x^(7+2) = x^11
There is no multiplication of the exponents in this problem. The exponents are beind added. The base values "x" are what is being multiplied.
Multiplying exponents occurs when you have an expression that involves and exponent and that expression is raised to an exponent. For example: (x^7)^2 = x^(7*2) = x^14
Hope this helps.
• What is an integer?
• An integer is a whole number and cannot be a fraction/decimal.
Some examples of numbers that are integers: 3, -403, -7, 1000
Some examples of numbers that are NOT integers: 3.56, -9.41, -30789.99, 0.87 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9322049617767334, "perplexity": 1103.6643593788513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500080.82/warc/CC-MAIN-20230204012622-20230204042622-00422.warc.gz"} |
https://www.physicsforums.com/threads/scalar-product-to-find-angle-between-two-vectors.185055/ | # Scalar product to find angle between two vectors
1. Sep 16, 2007
### noeinstein
Use the definition of scalar product, a·b = ab cos , and the fact that a·b = axbx + ayby + azbz (see Problem 46) to calculate the angle between the two vectors given by a = 2.0 i + 6.0 j + 2.0 k and b = 4.0 i + 3.0 j + 6.0 k.
AdotB= 8i + 18j + 12k
A=sqrt(2^2 + 6^2 + 2^2)=6.63
B=sqrt(4^2 + 3^2 + 6^2)=7.81
Θ=acos(38/51.78)
Θ=42.79=WRONG
2. Sep 16, 2007
### genneth
3. Sep 16, 2007
### noeinstein
I went over it again and got the same answer.
4. Sep 16, 2007
### genneth
Then why do you think it's wrong?
5. Sep 16, 2007
### noeinstein
because the online assignment is giving me a big fat red X. lol
6. Sep 16, 2007
### genneth
7. Sep 16, 2007
### noeinstein
Never mind. Got it. I switched the values of ay and by. Thanks.
8. Sep 16, 2007
### noeinstein
:d.. dah
Similar Discussions: Scalar product to find angle between two vectors | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.847572386264801, "perplexity": 3881.4559875331142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806426.72/warc/CC-MAIN-20171121204652-20171121224652-00670.warc.gz"} |
http://tex.stackexchange.com/questions/8857/how-to-type-special-accented-letters-in-latex/8884 | # How to type special/accented letters in LaTeX?
How to type these special letters from European languages in latex? ä, é, and L'?
-
Save your file as UTF-8 and put
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern} % load a font with all the characters
Then you can just type the characters normally into your source file.
Or, use XeLaTeX or LuaLaTeX which accepts UTF-8 input natively. In that case you need to add
\usepackage{fontspec}
If your text editor doesn't support UTF-8 encoded files, you should probably get another editor. But if you're stuck with one, you can also use:
\usepackage[latin1]{inputenc} % for PCs
\usepackage[applemac]{inputenc} % for Macs
and save the files in the default encoding for your machine.
-
Good advice. It's a good idea to load lmodern after switching the font encoding so that you get a font with all the extra accents and letters in it. – Will Robertson Jan 13 '11 at 7:29
You can type texdoc lshort in a command line (Command Prompt on Windows, Terminal on Linux/Mac OS X). Then have a look at Table 2.2 in Section 2.4.8. I'll quote it for you here.
-
You can also type them directly and use the inputenc package, which makes your source a lot more readable. – Alan Munn Jan 13 '11 at 3:25
Do you know how to type "L'" as well? Thanks. I typed "L'" and "L\'", but both does ways do not display "L'" when it generates the pdf file. – user2918 Jan 13 '11 at 3:28
Assuming you typed "\L" and not "L\", if you're still not getting the character its because of the font encoding or the font itself (which may not have the glyph.) It should work with a standard TeX distribution. Did you try my suggestion too? – Alan Munn Jan 13 '11 at 4:24
THe problem with using character modification is that spelling no longer works properly....or maybe there's a way around that too...let me ask a new questions. – Yossi Farjoun Jan 13 '11 at 11:57
The "L'" should work with \usepackage[T1]{fontenc} and \v L – Ulrike Fischer Jan 13 '11 at 13:00
You can use Detexify. Just draw your symbol, and it will figure out what you need to type! Much easier than plowing through endless symbol tables :).
-
The package selinput was not yet mentioned. Because I found good answers regarding its use on other places in TeX.SX let me first link to them:
• Heiko Oberdiek (author of selinput):
The input encoding depends on the editor that is used to write the TeX file … If the user is troubled to find the right encoding, then package selinput can help, …
(Encoding problems with custom LaTeX class (only on Windows 7/8). Why?)
• For languages other than English you can choose a semiautomatic input selection. … The selinput package is part of the oberdiek bundle. It will select the right input encoding in dependence of the file encoding.
(German character not rendered to pdf)
You could choose another way of input encoding by the selinput package from the oberdiek bundle. It chooses the right encoding by some glyphs from your language correspondingly to the encoding of the source file.
(How to use spanish accents?)
The hardest part can be to find out, what has actually to be written in \SelectInputMappings, read the package documentation for this. You are not forced to add every letter not in ASCII range, but you have to add some distinctive characters for your language. Below I added the letters from question.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{selinput}
\SelectInputMappings{%
eacute={é},
Lcaron={Ľ},
}
\begin{document}
äéĽ
\end{document}
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8890634775161743, "perplexity": 3201.723193437896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936465599.34/warc/CC-MAIN-20150226074105-00087-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://www.ipm.ir/ViewPaperInfo.jsp?PTID=15471&school=Particles | ## “School of Particles”
Back to Papers Home
Back to Papers of School of Particles
Paper IPM / Particles / 15471
School of Particles and Accelerator
Title: Complexity and Behind the Horizon Cut Off
Author(s):
1 Amin Akhavan 2 Mohsen Alishahiha 3 Ali Naseh 4 Hamed Zolfi
Status: Published
Journal: JHEP
No.: 090
Vol.: 12
Year: 2018
Supported by: IPM
Abstract:
Motivated by TT deformation of a conformal field theory we compute holographic complexity for a black brane solution with a cut off using "complexity=action" proposal. In order to have a late time behavior consistent with Lloyd's bound one is forced to have a cut off behind the horizon whose value is fixed by the boundary cut off. Using this result we compute holographic complexity for two dimensional AdS solutions where we get expected late times linear growth. It is in contrast with the naively computation which is done without assuming the cut off where the complexity approaches a constant at the late time. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8623508214950562, "perplexity": 3364.0810846493137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572908.71/warc/CC-MAIN-20220817122626-20220817152626-00407.warc.gz"} |
http://tex.stackexchange.com/questions/26521/how-to-change-the-spacing-between-figures-tables-and-text | # How to change the spacing between figures/tables and text?
I was wondering if there is any way to lessen the spaces between figures and text. In my paper I have a lot of figures (graphics) and it is inconvenient for me to have default (large) spaces between text and graphics. Is there any command I could use to change the default settings?
-
Maybe you'd benefit from grouping figures with `subfig`? – N.N. Aug 25 '11 at 15:08
Here's a related answer: Space after float with h. It also shows a graphic of where the lengths play a role - from the `layouts` package documentation. – Werner Aug 25 '11 at 15:47
## 1 Answer
Change one or more of the following lengths:
• `\textfloatsep` — distance between floats on the top or the bottom and the text;
• `\floatsep` — distance between two floats;
• `\intextsep` — distance between floats inserted inside the page text (using `h`) and the text proper.
The command used to change them is `\setlength`:
``````\setlength{\textfloatsep}{10pt plus 1.0pt minus 2.0pt}
``````
The default values in the `article` document class with the `10pt` option are:
• `\textfloatsep`: `20.0pt plus 2.0pt minus 4.0pt`;
• `\floatsep`: `12.0pt plus 2.0pt minus 2.0pt`;
• `\intextsep`: `12.0pt plus 2.0pt minus 2.0pt`.
You can get them yourself with `\showthe\textfloatsep` or `\the\textfloatsep` etc.
The `plus` and `minus` parts allow the space to stretch or shrink (the greater they are, the more it stretches or shrinks when needed). It's not recommended to leave out the `plus` and `minus` parts, as it leaves LaTeX less typesetting choices to select from and the output might look worse.
When typesetting in two column mode, two more lengths are available:
• `\dbltextfloatsep` — distance between a float spanning both columns and the text;
• `\dblfloatsep` — distance between two floats spanning both columns.
Remember that too little space will, again, make the document look worse.
-
"\intextsep — distance between floats inserted inside the page text (using h) and the text proper." so if I use H(capital H) which option should I change ? – kronos Aug 25 '11 at 15:38
@kronos: It is the same for `h` and `H`. – Andrey Vihrov Aug 25 '11 at 15:44
@AndreyVihrov One important question remains: How can I set this locally? I know I can use `\begingroup`, so my question is rather, does that change for the whole document if I place `\setlength{\textfloatsep}{10pt plus 1.0pt minus 2.0pt}` before the figure? Also maybe there is a different way to obtain local changes? – TomM Jan 3 at 22:56 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9194344878196716, "perplexity": 1925.3871496503143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400378956.26/warc/CC-MAIN-20141119123258-00136-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://www.gnu.org/software/archimedes/manual/html/node25.html | Next: Coupling Simplified MEP model Up: Physical Models employed in Previous: The Choice of the Contents
# The Simplified MEP Model
From the vast literature, it is well known that Monte Carlo method is the best way for obtaining very accurate simulations for the transport of electrons in semiconductor devices. Even if this method is very accurate, it has the price of very high simulation time. This is why we introduce the MEP model in this release of GNU Archimedes. In this way, as we will see, we will have the possibility of coupling MEP and Monte Carlo in order to make Monte Carlo method faster.
As you can see from the precedent papers reported in the introduction of this chapter, the MEP model is a very advanced hydrodynamical model for both electrons and holes in Silicon devices. For our purpose we will need only a simplified version of it. This because, we only need simple initial conditions for the Monte Carlo method. In the following we report a sketch of this model. A paper is under construction and will be refered in the next versions of this manual. The MEP model is based on the closure of the semiclassical Boltzmann equation by means of the maximum entropy principle. Using the relaxation time approximation (for only the moments and not for the energy moment) and using the so-called Liotta-Mascali distribution function which has the following form
(5.26)
we get the following hydrodynamical model for electrons, which we will call the Simplified MEP model.
(5.27) (5.28) (5.29)
where is a function of the electrons energy, as you can see from the precedent papers. This function is computed numerically and reads:
(5.30)
Furthermore, we have the following relations:
(5.31) (5.32)
For the moment relaxation time we have the following relations which are taken from the Baccarani model:
(5.33)
where
(5.34)
with the low field mobility and the lattice temperature. It is very easy to see how to adapt everything to Silicon heavy holes, so we do not report the Simplified MEP model for them.
Subsections
Next: Coupling Simplified MEP model Up: Physical Models employed in Previous: The Choice of the Contents | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8417284488677979, "perplexity": 551.0272208472072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997877644.62/warc/CC-MAIN-20140722025757-00074-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://www.computer.org/csdl/trans/tc/1975/05/01672853-abs.html | Subscribe
Issue No.05 - May (1975 vol.24)
pp: 560-562
null Siu-Chong Si , Department of Electrical Engineering, Lehigh University
ABSTRACT
A compatibility relationship on network paths is defined in such a way that the maximal compatibles are isomorphic to the products in the "structure and parity-observing output function" (SPOOF), a subscripted Boolean expression for the network output that uniquely specifies the network structure. For a given network, the path compatibility relations are easy to find, as are the maximal compatibles and hence the SPOOF for the network output. Because the collection of all the path compatibility relations, conveniently displayed in matrix form, completely characterizes the network, the compatibility matrix can be used for a variety of purposes.
INDEX TERMS
Compatibility, maximum compatibles, logic-network representation, SPOOF, sum-of-products form, diagnostics, fault testing.
CITATION
null Siu-Chong Si, A.K. Susskind, "A Method for Obtaining SPOOF's", IEEE Transactions on Computers, vol.24, no. 5, pp. 560-562, May 1975, doi:10.1109/T-C.1975.224260 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9285674095153809, "perplexity": 2907.445435110377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131297689.58/warc/CC-MAIN-20150323172137-00073-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/285782/how-many-ascending-and-descending-numbers-are-between-1000-and-9999 | # How many ascending and descending numbers are between 1000 and 9999?
I'm working on some homework right now, and I've gotten stumped. Here's the question I'm on:
How many of the 9000 four-digit integers 1000, 1001, 1002, . . . , 9998, 9999 have four distinct digits that are either increasing (as in 1347 and 6789) or decreasing (as in 6421 and 8653)?
Through some Googling I've already found what they answer may be, but I have no idea how to arrive at the answer, so I wouldn't be learning anything from this hurdle like I'm supposed to. I can use all the help I can get on this one since I don't even know where to start. I think I have to apply the formula for permutations to this somehow, but I don't know what adjustments I have to make for it to work right.
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If you have a number xyza, where x, y, and z are known, and "a" unknown, and xyza is monotonic (either increasing or decreasing), how many choices do you have for a? – Doug Spoonwood Jan 24 '13 at 11:32
Have you tried some smaller example, e.g. calculating such numbers between $10$ and $55$ in base $6$ for example and then try to find a general rule. – Julian Kuelshammer Jan 24 '13 at 11:56
@DougSpoonwood - Okay, I think I see what you mean, I would have 10 choices there, unless I wasn't counting the zeroes for the ascending group. I don't quite understand how x, y, and z are known though. – Dave Jan 24 '13 at 12:08
@JulianKuelshammer - I'll give that a try. That has worked for me on some of the other permutation puzzles we've been doing, but I didn't think it would offer any insight on this one. This is a much more difficult problem than the others due to some base concept I seem to be missing. – Dave Jan 24 '13 at 12:09
Once you've chosen the four digits, there's only one way to arrange them increasing, and one way decreasing, so it comes down to, how many ways are there to choose four digits? Oh, there's one little tricky bit: if you choose zero, you can't do increasing.
-
Hmm, that's a good nudge for me, and helps with part of what I was confused about, but without considering the 0 for increasing permutations I see it as P(9000, 4), which I KNOW isn't right. 9000! would be an absolutely ridiculous number such that dividing it by 4! wouldn't bring it anywhere near the range I'm looking for. – Dave Jan 24 '13 at 11:43
Ack! You're choosing $4$ digits (from 10), not $4$ numbers from $9000$. – Gerry Myerson Jan 24 '13 at 11:59
Okay, 4 digits from 10. Well, ascending it would be four digits from 9 since no zeroes work. Descending will work with zeroes, though, so that could be 4 digits from 10. So it seems maybe I'm looking at (9 4) + (10 4)? Also, I really don't understand how this could work this way. How does solving for choosing four digits get me the number of compositions that are only ascending? – Dave Jan 24 '13 at 12:05
I don't know what you mean by the notation $(9\ 4)$. You choose the four digits. As I wrote, there is only one way to arrange the four digits you have chosen so that they are ascending. So the number of 4-digit integers with increasing digits is the same as the number of ways of choosing 4 digits. Right? – Gerry Myerson Jan 24 '13 at 12:10
Better notation to use there might be P(9,4) for ascending, as it's how I usually see nPr written when formula editors are not available, but it could also be written as 9P4 I suppose. So lets say I choose 1234. They are already arranged as ascending. So I use P(9,4). It seems to me that this wouldn't include combinations such as, say, 5678. As I said in reply to one of the comments above, there is something very basic that my mind hasn't accepted yet. – Dave Jan 24 '13 at 12:17
If you're looking for how many 4 digit numbers are increasing or decreasing between 1000 and 9999, the answer has been provided here: How many of the 9000 four digit integers have four digits that are increasing?
If you (or since this was posted 3 years ago, if a discrete math student) are looking for how many non-decreasing or non-increasing 4-digit numbers there are between 1000 and 9999, I believe I have the answer. *the difference is that non-decreasing can have 1226 or 7888 while increasing cannot have repetition.
$\binom{10}{4}$ is the correct answer for decreasing combinations, while $\binom{10}{4} - {9 \choose 3}$ is the answer for increasing combinations because you can't begin the 4 digit segment with (or include) a zero.
The same idea applies to non-decreasing / non-increasing values, this time we allow for repetition, so for every integer after the 1st integer, we have +1 choices in digits.
For decreasing numbers, 3210 was the only option where the 1st digit was 3. This is reflected by $\binom{3}{3}$. Now, since we can have 4-digit numbers like 3332, 3200, and 3111, we have $\binom{3+3}{3}$ combinations. We then have to subtract 1 because 3333 is not a non-increasing number.
So, this same principle applies. We get $\binom{10+3}{4}$ + ($\binom{10+3} {4} - \binom{9+3}{3}$) - 10 = 1200. We subtract ten because 1111,2222,...9999 are not valid options for either non-increasing nor non-decreasing. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8832979202270508, "perplexity": 258.3054033271583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396106.25/warc/CC-MAIN-20160624154956-00163-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/physics-for-scientists-and-engineers-a-strategic-approach-with-modern-physics-4th-edition/chapter-9-work-and-kinetic-energy-exercises-and-problems-page-228/24 | ## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)
We can find the work done on the particle as it moves from x = 0 to x = 3.14 m: $W = \int_{0}^{3.14}~F_x~dx$ $W = \int_{0}^{3.14}~(0.250~N)~sin(\frac{x}{2.00~m})~dx$ $W = -(0.50~J)~cos(\frac{x}{2.00~m})~\Big\vert_{0}^{3.14~m}$ $W = 0-(-0.50~J)(1)$ $W = 0.50~J$ The work done on the particle as it move from x = 0 to x = 3.14 m is 0.50 J. Then, we use the work-energy theorem to find the particle's speed at x = 3.14 m; $KE_2 = KE_1+W$ $\frac{1}{2}mv_2^2 = \frac{1}{2}mv_1^2+W$ $v_2^2 = \frac{mv_1^2+2W}{m}$ $v_2 = \sqrt{\frac{mv_1^2+2W}{m}}$ $v_2 = \sqrt{\frac{(0.15~kg)(2.00~m/s)^2+(2)(0.50~J)}{0.15~kg}}$ $v_2 = 3.27~m/s$ The particle's speed at x = 3.14 m is 3.27 m/s. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8442549705505371, "perplexity": 109.05709707677967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592309.94/warc/CC-MAIN-20180721032019-20180721052019-00413.warc.gz"} |
http://math.stackexchange.com/questions/251953/i-am-trying-to-map-r-into-the-r-such-that-my-map-fixes-some-finite-set-of-ratio | # I am trying to map R into the R, such that my map fixes some finite set of rational numbers and sends one element from R\Q into Q.
I am not sure how to do this map. I don't know how I can place in particular that element that is in R/Q to R while still being an bijection.
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Do you mean $\Bbb{R/Q}$ or $\Bbb{R\setminus Q}$? – Asaf Karagila Dec 6 '12 at 0:53
Hi, I meant R\Q. Thanks – Jmaff Dec 6 '12 at 0:56
You can just use straight lines between the points. Say you want to fix $0,1,2$ and send $\sqrt 2$ to $\frac 32$. Then you have have
$$f(x)=\begin {cases} x & x \le 1 \\\\ 1+\frac {.5}{\sqrt 2-1}(x-1) & 1 \lt x \le \sqrt 2 \\\\ \frac 32+\frac {.5}{2-\sqrt 2}(x-\sqrt 2) & \sqrt 2 \lt x \le 2 \\\\x & x \gt 2\end {cases}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8769190311431885, "perplexity": 423.46837086019667}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802767198.25/warc/CC-MAIN-20141217075247-00068-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.aanda.org/articles/aa/full_html/2010/09/aa13461-09/aa13461-09.html | Free Access
Issue A&A Volume 517, July 2010 A5 6 Astrophysical processes https://doi.org/10.1051/0004-6361/200913461 23 July 2010
A&A 517, A5 (2010)
## MAGIC observation of the GRB 080430 afterglow
J. Aleksic1 - H. Anderhub2 - L. A. Antonelli3 - P. Antoranz4 - M. Backes5 - C. Baixeras6 - S. Balestra4 - J. A. Barrio4 - D. Bastieri7 - J. Becerra González8 - J. K. Becker5 - W. Bednarek9 - A. Berdyugin10 - K. Berger9 - E. Bernardini11 - A. Biland2 - R. K. Bock12,7 - G. Bonnoli13 - P. Bordas14 - D. Borla Tridon12 - V. Bosch-Ramon14 - D. Bose4 - I. Braun2 - T. Bretz15 - D. Britzger12 - M. Camara4 - E. Carmona12 - A. Carosi3 - P. Colin12 - S. Commichau2 - J. L. Contreras4 - J. Cortina1 - M. T. Costado8,16 - S. Covino3 - F. Dazzi17,26 - A. De Angelis17 - E. de Cea del Pozo18 - R. De los Reyes4,28 - B. De Lotto17 - M. De Maria17 - F. De Sabata17 - C. Delgado Mendez8,27 - M. Doert5 - A. Domínguez19 - D. Dominis Prester20 - D. Dorner2 - M. Doro7 - D. Elsaesser15 - M. Errando1 - D. Ferenc21 - E. Fernández1 - R. Firpo1 - M. V. Fonseca4 - L. Font6 - N. Galante12 - R. J. García López8,16 - M. Garczarczyk1 - M. Gaug8 - N. Godinovic20 - F. Goebel12,29 - D. Hadasch18 - A. Herrero8,16 - D. Hildebrand2 - D. Höhne-Mönch15 - J. Hose12 - D. Hrupec20 - C. C. Hsu12 - T. Jogler12 - S. Klepser1 - T. Krähenbühl2 - D. Kranich2 - A. La Barbera3 - A. Laille21 - E. Leonardo13 - E. Lindfors10 - S. Lombardi7 - F. Longo17 - M. López7 - E. Lorenz2,12 - P. Majumdar11 - G. Maneva22 - N. Mankuzhiyil17 - K. Mannheim15 - L. Maraschi3 - M. Mariotti7 - M. Martínez1 - D. Mazin1 - M. Meucci13 - J. M. Miranda4 - R. Mirzoyan12 - H. Miyamoto12 - J. Moldón14 - M. Moles19 - A. Moralejo1 - D. Nieto4 - K. Nilsson10 - J. Ninkovic12 - R. Orito12 - I. Oya4 - R. Paoletti13 - J. M. Paredes14 - M. Pasanen10 - D. Pascoli7 - F. Pauss2 - R. G. Pegna13 - M. A. Perez-Torres19 - M. Persic17,23 - L. Peruzzo7 - F. Prada19 - E. Prandini7 - N. Puchades1 - I. Puljak20 - I. Reichardt1 - W. Rhode5 - M. Ribó14 - J. Rico24,1 - M. Rissi2 - S. Rügamer15 - A. Saggion7 - T. Y. Saito12 - M. Salvati3 - M. Sánchez-Conde19 - K. Satalecka11 - V. Scalzotto7 - V. Scapin17 - T. Schweizer12 - M. Shayduk12 - S. N. Shore25 - A. Sierpowska-Bartosik9 - A. Sillanpää10 - J. Sitarek12,9 - D. Sobczynska9 - F. Spanier15 - S. Spiro3 - A. Stamerra13 - B. Steinke12 - N. Strah5 - J. C. Struebig15 - T. Suric20 - L. Takalo10 - F. Tavecchio3 - P. Temnikov22 - D. Tescaro1 - M. Teshima12 - D. F. Torres24,18 - N. Turini13 - H. Vankov22 - R. M. Wagner12 - V. Zabalza14 - F. Zandanel19 - R. Zanin1 - J. Zapatero6 - A. de Ugarte-Postigo3
1 - IFAE, Edifici Cn., Campus UAB, 08193, Bellaterra, Spain
2 - ETH Zurich, 8093, Switzerland
3 - INAF National Institute for Astrophysics, 00136 Rome, Italy
5 - Technische Universität Dortmund, 44221 Dortmund, Germany
6 - Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
8 - Inst. de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain
9 - University of ódz, 90236 Lodz, Poland
10 - Tuorla Observatory, University of Turku, 21500 Piikkiö, Finland
11 - Deutsches Elektronen-Synchrotron (DESY), 15738 Zeuthen, Germany
12 - Max-Planck-Institut für Physik, 80805 München, Germany
13 - Università di Siena, and INFN Pisa, 53100 Siena, Italy
14 - Universitat de Barcelona (ICC/IEEC), 08028 Barcelona, Spain
15 - Universität Würzburg, 97074 Würzburg, Germany
16 - Depto. de Astrofisica, Universidad, 38206 La Laguna, Tenerife, Spain
17 - Università di Udine, and INFN Trieste, 33100 Udine, Italy
18 - Institut de Ciències de l'Espai (IEEC-CSIC), 08193 Bellaterra, Spain
19 - Inst. de Astrofísica de Andalucía (CSIC), 18080 Granada, Spain
20 - Croatian MAGIC Consortium, Institute R. Boskovic, University of Rijeka and University of Split, 10000 Zagreb, Croatia
21 - University of California, Davis, CA-95616-8677, USA
22 - Inst. for Nucl. Research and Nucl. Energy, 1784 Sofia, Bulgaria
23 - INAF/Osservatorio Astronomico and INFN, 34143 Trieste, Italy
24 - ICREA, 08010 Barcelona, Spain
25 - Università di Pisa, and INFN Pisa, 56126 Pisa, Italy
26 - Supported by INFN Padova
27 - Now at: Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain
28 - Now at: Max-Planck-Institut für Kernphysik, 69029 Heidelberg, Germany
29 - Deceased
Received 13 October 2009 / Accepted 17 April 2010
Abstract
Context. Gamma-ray bursts are cosmological sources emitting radiation from the gamma-rays to the radio band. Substantial observational efforts have been devoted to the study of gamma-ray bursts during the prompt phase, i.e. the initial burst of high-energy radiation, and during the long-lasting afterglows. In spite of many successes in interpreting these phenomena, there are still several open key questions about the fundamental emission processes, their energetics and the environment.
Aims. Independently of specific gamma-ray burst theoretical recipes, spectra in the GeV/TeV range are predicted to be remarkably simple, being satisfactorily modeled with power-laws, and therefore offer a very valuable tool to probe the extragalactic background light distribution. Furthermore, the simple detection of a component at very-high energies, i.e. at 100 GeV, would solve the ambiguity about the importance of various possible emission processes, which provide barely distinguishable scenarios at lower energies.
Methods. We used the results of the MAGIC telescope observation of the moderate resdhift () GRB 080430 at energies above about 80 GeV, to evaluate the perspective for late-afterglow observations with ground based GeV/TeV telescopes.
Results. We obtained an upper limit of erg cm-2 s-1 for the very-high energy emission of GRB 080430, which cannot set further constraints on the theoretical scenarios proposed for this object also due to the difficulties in modeling the low-energy afterglow. Nonetheless, our observations show that Cherenkov telescopes have already reached the required sensitivity to detect the GeV/TeV emission of GRBs at moderate redshift ( ), provided the observations are carried out at early times, close to the onset of their afterglow phase.
Key words: radiation mechanisms: non-thermal - gamma-ray burst: individual: GRB 080430
## 1 Introduction
GRB 080430 was detected by the Swift satellite (Gehrels et al. 2004) on April 30, 2008 at 19:53:02 UT (Guidorzi et al. 2008a). The prompt emission lasted 16 s (Stamatikos et al. 2008) allowing to assign this event to the long duration class (Kouveliotou et al. 1993). X-ray and optical counterparts were discovered and followed-up by many groups. Optical spectroscopy was rapidly carried out allowing to derive a redshift of z = 0.758. The redshift estimate has been revised recently with a more accurate wavelength calibration (de Ugarte Postigo et al. 2008; Cucchiara & Fox 2008, and de Ugarte Postigo et al., in preparation, hereinafter DEUG10). The relatively modest redshift made it an interesting target for the major atmospheric gamma-ray imaging Cherenkov (MAGIC) telescope observations. In the past, upper limits for several gamma-ray bursts (GRBs) at energies greater than about 100 GeV were reported both for single event observations and for a sample of events (e.g. Albert et al. 2006; Aharonian et al. 2009; Albert et al. 2007; Tam et al. 2006). In this paper we try to predict the very-high energy (VHE) flux for GRB 080430 by modeling the detected X-ray and optical afterglow and adopting as a reference the cosmological fireball model (Zhang 2007; Piran 1999). In Sect. 2 we report the results of the MAGIC observation, in Sect. 3 we discuss the lower energy afterglow, in Sect. 4 we introduce the adopted modeling scenario for the VHE flux, in Sect. 5 we discuss the effect of extragalactic background light (EBL) attenuation and finally, in Sect. 6, conclusions and considerations about future perspectives are drawn. Throughout the paper we assume a cosmology with , and h0 = 0.71. At the redshift of the GRB the luminosity distance is 4.8 Gpc ( cm, corresponding to a distance modulus mag). All errors are unless stated otherwise. Throughout this paper the convention Qx = Q/10x has been adopted in CGS units. Results presented in this paper supersede those reported in Covino et al. (2009b).
## 2 MAGIC observations
GRB 080430 occurred while the Sun was still above the horizon at the MAGIC site (Roque de los Muchachos, N, W). The MAGIC observation started immediately after sunset, at 21:12:14 UTC and ended at 23:52:30 UTC. The observation was disturbed by clouds. The beginning of the observation was at T0 + 4753 s, well after the end of the prompt emission phase. The observation with MAGIC started at a zenith angle of , reaching at the end. The data set was divided into two time intervals. Results from the first time interval, giving the lower energy upper limit, are used in this context. Analysis of the dataset, in the energy bin from 80 up to 125 GeV with the spectral parameters derived in Sect. 4, gave a 95% CL upper limit of erg cm-2 s-1 (under the assumption of steady emission) or a fluence limit of erg cm-2 for a time interval of 6258 s from 21:12:14 to 22:56:32 UTC. These limits contain a 30% systematic uncertainty on the absolute detector efficiency. Limits at higher energies are less important for the present analysis due to intense EBL absorption above 100 GeV (see Sect. 5). It is important to note that at that time, the sum trigger hardware upgrade (Garczarczyk et al. 2009; Albert et al. 2008) which allows the MAGIC telescope to carry out reliable observations with lower energy threshold was not yet available for GRB observations. Therefore the lowest obtained upper limit is a factor two higher than in later cases (e.g. Gaug et al. 2009).
## 3 Afterglow light-curve and spectral energy distribution
It is not our purpose to discuss here the physics of the afterglow of this event, which will be discussed in detail in DEUG10. Nevertheless, preliminary results shows that this afterglow can not be satisfactory described within any common referred scenario. In particular the early afterglow is puzzling, likely requiring two distinct components with separated time evolution. However, at the epoch of the MAGIC observations (about 8 ks from the high-energy event), the afterglow seems to have entered a more stable phase although other components, as late prompt emission, can still be contributing (Ghisellini et al. 2007). Analysis of the spectral (from optical to X-rays) information shows that the afterglow can be described as due to the interaction of a relativistic outflow with the circumburst medium surrounding the progenitor (Zhang 2007; Piran 1999). The outflow is relativistic and shocks form with consequent particle acceleration. Details of the acceleration process are not known and it is usually assumed that electrons follow a power-law distribution in energy with a slope p. Numerical simulations suggest it should be (Achterberg et al. 2001; Vietri 2003) although other scenarios predict a wider range which is indeed supported by the analysis of several afterglows (Ellison & Double 2004; Shen et al. 2006). The late-afterglow of GRB 080430 can be characterized by a constant circumburst density environment with typical number density cm-3. The electron distribution index turns out to be . Given the afterglow spectral properties, it is possible to predict the time decay, which in the optical, is well consistent with the predictions. On the contrary, X-ray data (Guidorzi et al. 2008b) show a much milder decay than expected. It is difficult to attribute this behaviour to a specific physical ingredients. Common additions to the reference model (Zhang 2007), which may or may not modify the spectrum involve late energy injection, structured jets, flares, circumburst density variations, etc. (see e.g. Zhang et al. 2006; Panaitescu 2006, for comprehensive discussions about these factors). It is clearly well beyond the scope of this paper to discuss these issues in detail, which are indeed still not well settled. We therefore model the VHE emission assuming the afterglow could be described in the context of the standard afterglow model (Zhang 2007; Piran 1999). Finally, we comment possible modifications induced by additional phenomena which in general can even increase the expected VHE flux.
In order to characterize the afterglow spectrum we must compute the synchrotron injection, , and cooling, , frequency values. The injection frequency is where most of the synchrotron emission occurs and the cooling frequency identifies where electrons cool effectively. In case of constant circumburst medium (Fan & Piran 2006; Yost et al. 2003) we have:
(1)
and
(2)
where z is the redshift of the source, n the medium particle density, the kinetic energy of the outflow, t the time delay after the GRB and . We will assume for the micro-physical parameters , the fraction of total energy going to electrons, and , the fraction of total energy going to magnetic fields, the values of 0.1 and 0.01, respectively. These figures are typical values measured during late-time afterglows (see e.g. Yost et al. 2003; Panaitescu & Kumar 2002) and in agreement with the results of the analysis of GRB 080430. The relation for the cooling frequency is approximate since we are neglecting the possible role played by additional inverse Compton (IC) cooling. We will consider this issue again in Sect. 4.
The total energy can be derived from the burst isotropic energy with some assumptions about the spectrum and by correcting it for the fireball radiative efficiency . We estimate as the integral of the burst spectral model (Stamatikos et al. 2008) in the 1 - 104 keV band (Amati et al. 2002), the energy range covering most of the prompt emission of GRBs. In this energy band the spectrum of a burst is typically described by a Band function (Band et al. 1993):
(3)
In order to calculate the integral we need to know the two power-law photon indices and and the peak energy . Unfortunately, the Swift-BAT energy range is often too narrow for a direct measurement. We therefore run a set of integrations by varying and derive the corresponding using the 15 - 150 keV BAT fluence to normalize the spectrum. In each integration, depending on the value of , we identify the observed photon index with one of the two indices of the Band function, fixing the other one to a canonical value (1 and 2.3 for the low- and the high-energy power-laws, respectively). The chosen values of and are those satisfying the Amati relation (Amati et al. 2002). According to this method, we estimate keV, and erg. The errors are estimated to be about 30% for both quantities mainly due to the lack of observational data better constraining the prompt emission spectrum. Peak energies for Swift GRBs can also be estimated following a correlation between peak energy and spectral parameters as measured by Swift-BAT instrument (Sakamoto et al. 2009) yielding consistent results with our analysis. The derived total energy is typical for cosmological GRBs, although it is common to observe events substantially more energetic (Sakamoto et al. 2008). The relatively low estimated peak energy can allow one to classify this event as X-Ray Flash or X-Ray Rich (Zhang 2007, and references therein). If we then assume a radiative efficiency of 10%, we find the total kinetic energy going to the outflow erg. The radiative efficiency during the prompt emission phase can vary among individual bursts (Zhang et al. 2007). A satisfactory treatment of the prompt emission phase emission process is still lacking. We choose 10% as a conservative limit recalling it can be higher for events characterized by a shallow decay phase (Nousek et al. 2006) in the X-rays as it might be the case for GRB 080430.
Summing up, for modeling the high energy emission of the GRB 080430 afterglow, we have applied these parameters: energy erg, , , , the circumburst medium density profile cm-3 and the redshift . Our observation was at ksec after the burst onset. At this epoch we have Hz and Hz. The afterglow synchrotron emission is in the so called slow-cooling'' regime (i.e. the synchrotron cooling frequency is above the synchrotron injection frequency) as confirmed by the modeling of the spectral energy distribution (SED) from optical to X-rays (DEUG10) and usually expected at the epoch of the observations for typical afterglows (Zhang & Mészáros 2004).
## 4 Synchrotron-Self Compton during the afterglow
The analysis of the high-energy emission from the various phases of a GRB has been considered by many authors as a powerful diagnostic tool of GRB physics (Aharonian et al. 2008; Fan & Piran 2008; Falcone et al. 2008; Fan 2009; Murase et al. 2009; Covino et al. 2009a; Dermer & Fryer 2008; Panaitescu 2008; Xue et al. 2009; Galli & Piro 2008; Kumar & Barniol Duran 2009; Gilmore et al. 2010; Le & Dermer 2009). In the present case, the most important emission process to consider is essentially the Synchrotron-Self Compton (SSC). Due to the long delay between the MAGIC observations and the GRB onset (about two hours) any residual prompt emission can be ruled out. Superposed to the SSC component, external inverse Compton (EIC) processes could also play a role and will be briefly mentioned later. We do not consider here hadronic models (Pe'er & Waxman 2004; Böttcher & Dermer 1998) in our discussion. They could, however, be of special interest if GRBs are important sources of cosmic-rays.
Once the parameters of the lower-energy synchrotron emission are known, it is possible to predict the SSC component with good reliability. Among the many possible choices, we followed the recipe described by Fan & Piran (2008).
The SSC process essentially generates a new spectral component superposed to the underlying synchrotron spectrum, with the same global shape up to a cut-off frequency:
(4)
where is the fireball bulk motion Lorentz factor, the electron mass, c the speed of light and h the Planck constant. Above this frequency the SSC emission is no more in the Thomson regime and becomes much weaker (Klein-Nishina regime). For typical bulk motion Lorentz factors ( at the afterglow onset, Molinari et al. 2007) the SSC emission of the afterglow is in the Thomson regime.
Assuming we are in a constant density circumburst environment, the predicted SSC spectrum is characterized by two typical frequencies (Fan & Piran 2008) as the synchrotron afterglow spectrum (Sect. 3):
(5)
(6)
where is the rest frame synchrotron to magnetic field energy density ratio.
Defining , it can be shown (Sari & Esin 2001) that:
(7)
The synchrotron injection to cooling synchrotron frequency ratio for the slow-cooling case is:
(8)
The numerical factor in front of Eq. (8) is not exactly the one derived from Eqs. (1) and (2) since, as already mentioned in Sect. 3, IC cooling also affects the location of the synchrotron cooling frequency making the problem numerically difficult to solve. We now apply an approximate solution fully adequate for our goals (see Fan & Piran 2008, for a full discussion). With our parameters Eq. (8) becomes and . Equations (5) and (6) become Hz (5 keV) and Hz (310 MeV). The cooling SSC frequency is at much lower energy than the band covered by the MAGIC observations ( GeV). We are therefore in the spectral range where the SSC spectrum is softer, following a power-law behaviour, .
In order to derive the expected flux density at the MAGIC energy we have to compute the flux density at the typical SSC frequency (Fan & Piran 2008) at the epoch of the MAGIC observation:
(9)
where is the luminosity distance of the source, Gpc ( cm). With our parameters, erg cm-2 s-1 MeV-1 which is much lower than the synchrotron flux at the same frequency, , well within the Swift-XRT energy range with these parameters.
Then, finally, from the peak energy to the MAGIC band we have to extrapolate the SSC spectrum as:
(10)
and, again with our parameters, erg cm-2 s-1 MeV-1. The flux integrated in the MAGIC band, the parameter to be compared to the reported upper limits, can be well approximated by at about 90 GeV, and we have erg cm-2 s-1 at the epoch of the MAGIC observation, ks from the burst.
Any uncertainty in the underlying afterglow parameters affects of course the VHE predictions. Some of these uncertainties have, however, a rather limited (considering the present observational limits) impact because one of the relevant factors, the ratio between the injection and cooling synchrotron frequency, is constrained by the afterglow SED and uncertainties for micro-physical parameters should still keep the ratio close to the observed value. The ratio drives the importance of the IC component and the position of the cooling SSC frequency, i.e. where the VHE flux begins to decrease steeply moving toward higher energies. The total energy on the contrary is estimated assuming an efficiency for the GRB prompt emission process. This is a weakly known factor given that at present no satisfactory description of the GRB prompt emission process exists (Lyutikov 2009). It is therefore possible (Zhang 2007) that the efficiency is substantially higher, modifying the total energy and therefore the expected flux. Circumburst matter density has an important effect on the expected SSC flux. With the present afterglow data it can essentially only be estimated coupled to the micro-physical parameters. A higher density would make the SSC component more important and possibly detectable at lower energies (see e.g. Harrison et al. 2001). However, the value of the circumburst density derived for afterglows with data allowing a detailed modeling is consistent with the value we report for GRB 080430 (Panaitescu 2005).
A milder than expected temporal decay in the X-rays band together with the consistency of the observed SED with the reference afterglow model prediction, raises some concern about the reliability of the adopted theoretical scenario. A shallower afterglow decay showing a synchrotron spectrum can be explained with late-time energy injection in the outflow (Zhang et al. 2006; Panaitescu 2006). In this case the VHE SSC flux temporal decay could be slowed in a way related to the time evolution of the energy injection (Fan & Piran 2008; Galli & Piro 2007; Gou & Mészáros 2007; Wei & Fan 2007). However, the lack of a similar behaviour at optical wavelengths do not fully support this possibility since energy injection should affect the afterglow evolution in any band. It could be possible that the X-ray afterglow is affected by the occurrence of a late-time slow flares which could be barely detectable at lower energies. Such a flare can produce a detectable VHE emission although likely with a longer and smoother time evolution due to the interaction of the flare photons with the outflow accelerated electrons (see Fan & Piran 2008), i.e. probably at later time than the MAGIC observations. Finally, we mention that micro-physical parameters can evolve in time. Their evolution could affect the position and time-evolution of the SSC injection and cooling frequencies and as consequence the expected VHE flux. However, a satisfactory theoretical framework for these possible modifications of the reference afterglow model is still lacking, leaving the introduction of these ingredients purely phenomenological and likely beyond the scope of this paper.
Figure 1:Predictions at different time delays from the high-energy event for the SSC emission during the afterglow of GRB 080430. Black triangles are 95% CL upper limits derived by MAGIC at various energies. Lines of a same color show the same SSC model, but a different absorption model of the gamma-rays by the EBL. The blue lines correspond to the MAGIC observation window. Open with DEXTER
## 5 Extragalactic background light attenuation
Gamma-rays in the GeV energy regime are absorbed through pair-production processes with the EBL. The precise light content of the EBL is strongly debated. We have to rely on many different models, the predictions of which at span a wide range of optical depths, from less than 1 up to 6 (Fan & Piran 2008). Moreover, the MAGIC collaboration recently published a striking observational result (Albert et al. 2008) suggesting that the EBL attenuation could be much lower than previously assumed. Thus at the redshift of GRB 080430 ( ) and at the MAGIC energy ( GeV) an optical depth not far from unity is possible. We included four representative models from Kneiske et al. (2004), Franceschini et al. (2008) and Gilmore et al. (2009) and show the range of possible absorbed spectra in Fig. 1. The blue lines correspond to the MAGIC observation delay, the other lines show the spectrum at earlier observation times, in principle easily accessible to IACTs. On average, we can assume an attenuation of the received flux from the afterglow of GRB 080430 of the order a factor 3 or even less, allowing us to estimate erg cm-2 s-1 as the predicted flux in the MAGIC band. As a matter of fact, our choice is possibly very conservative as Gilmore et al. (2009) described models, in agreement with the observations reported in Albert et al. (2008), with an optical depth as low as at the same conditions of these MAGIC observations.
## 6 Discussions
The prediction of the expected SSC flux for an afterglow is not straightforward since it is required to know, or at least to reliably estimate, the parameters of the underlying afterglow (see Fig. 1). In the case of GRB 080430 the sampling of the X-ray and optical afterglow allowed us to estimate the various afterglow parameters to derive meaningful predictions for the expected SSC flux. However, a satisfactory modeling of the GRB 080430 can not be obtained within the standard fireball scenario. At least two different components are required for the early-time afterglow, as discussed in detail in DEUG10. Our present discussion is based on the assumption that one of these components is the regular afterglow (i.e. the forward-shock Zhang 2007; Piran 1999) which is the main responsible for the late-afterglow emission although other components are likely playing a role.
The results appear to be well below the reported upper limits. Furthermore, our assumed low opacity for the EBL is in agreement with current observations (see also Gilmore et al. 2009). At any rate, this pilot case shows fairly interesting perspectives for a late-afterglow detection at high energies.
In general, to increase the flux expected from a GRB afterglow (for SSC) it is mandatory to try to decrease the observation energy (due to the dependence above the cooling SSC frequency), which is also very important for the minimization of the EBL attenuation. If the telescope sum trigger hardware upgrade had already been implemented before the observations, a limit above an energy of 45 GeV would have been obtained (see also Gaug et al. 2009). At these energies, the strong effect of the EBL could probably be neglected and the low energy threshold together with the expected performances of MAGIC II would undoubtedly increase the chances of positive detections.
As a matter of fact, GRB 080430 was an average event in terms of energetics. More energetic GRBs are indeed relatively common, and due to the positive dependence on the isotropic energy of a GRB, much higher fluxes than in the present case can be foreseen. This is also true if we consider the uncertainty in the present total energy determination, which is based on an average value for the prompt emission efficiency.
The time delay of the observation from the GRB has a clear impact, essentially because the observed SSC component is strictly related to the underlying synchrotron component which rapidly decays in intensity with time, depending on the specific environment and micro-physical parameters. Equation (10) goes roughly with t-1.1 which means that had MAGIC been able to start observations right at the start of the late afterglow phase (e.g. at T0 + 1 ks), the flux predictions would have increased by more than an order of magnitude. The time delay of about two hours, coupled with the poor observing conditions, were more than enough to depress the observed flux and raise the reported upper limits.
Given the uncertainties in the modeling of the afterglow, many possible modifications to the standard afterglow model (Zhang 2007; Piran 1999) can be applied. In some scenarios, substantially higher VHE flux can be predicted (e.g. Panaitescu 2008; Murase et al. 2010), which makes observations at VHE energies powerful potential diagnostic tools.
The case of GRB 080430 in this pilot study demonstrates that if three conditions are met: 1) a moderate redshift ( ), 2) start of observations right at the beginning of the afterglow phase or even during the prompt emission and 3) the use of the MAGIC sum trigger enabling reaching energy thresholds below 50 GeV, detection is within reach. The recent detection of 30 GeV photons during the prompt or afterglow phases of GRB 090510 (Abdo et al. 2009) and GRB 090902B (de Palma et al. 2009b,a) by the Fermi satellite (Band et al. 2009) indeed shows that, with a threshold energy of a few tens of GeV and with the collecting area of a ground-based Cherenkov telescope, GRB VHE astrophysics is becoming a promising observational field.
Acknowledgements
We would like to thank the Instituto de Astrofisica de Canarias for the excellent working conditions at the Observatorio del Roque de los Muchachos in La Palma. The support of the German BMBF and MPG, the Italian INFN and Spanish MICINN is gratefully acknowledged. This work was also supported by ETH Research Grant TH 34/043, by the Polish MNiSzW Grant N N203 390834, and by the YIP of the Helmholtz Gemeinschaft. We also thank Yizhong Fan for continuous theoretical support. Lorenzo Amati, Cristiano Guidorzi, Alessandra Galli, Daniele Malesani and Ruben Salvaterra for useful discussions. We finally remark the very constructive report from the referee which helped to substantially improve the paper.
## Footnotes
... telescope
http://wwwmagic.mpp.mpg.de/
... that
Here we deliberately ignore the possibility to have higher order IC components which could be effective in cooling the electron population.
## All Figures
Figure 1:Predictions at different time delays from the high-energy event for the SSC emission during the afterglow of GRB 080430. Black triangles are 95% CL upper limits derived by MAGIC at various energies. Lines of a same color show the same SSC model, but a different absorption model of the gamma-rays by the EBL. The blue lines correspond to the MAGIC observation window. Open with DEXTER In the text | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9371033906936646, "perplexity": 2194.0004674921524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267155817.29/warc/CC-MAIN-20180919024323-20180919044323-00468.warc.gz"} |
http://www.ms.uky.edu/~guillou/research.html | Research
## Articles and Preprints
(all available on the ArXiv)
1. The Klein four slices of positive suspensions of HF2 (joint with C. Yarnall, submitted, available on the ArXiv)
We describe the slices of positive integral suspensions of the equivariant Eilenberg-Mac Lane spectrum HF2 for the constant Mackey functor over the Klein four-group C2×C2.
2. The cohomology of C2-equivariant A(1) and the homotopy of koC2 (Joint with M. A. Hill, D. C. Isaksen, and D. C. Ravenel, submitted, available on the ArXiv)
We compute the cohomology of the subalgebra AC2(1) of the C2-equivariant Steenrod algebra AC2. This serves as the input to the C2-equivariant Adams spectral sequence converging to the RO(C2)-graded homotopy groups of an equivariant spectrum koC2. Our approach is to use simpler C-motivic and R-motivic calculations as stepping stones.
3. Models of G-spectra as presheaves of spectra (Joint with J. P. May, submitted, available on the ArXiv)
Restricting to the case of a finite group, we give a presentation for G-spectra as spectral Mackey functors. In other words, we describe how to build G-spectra out of fixed point data, which are determined by finite G-sets and nonequivariant spectra.
4. Enriched model categories and presheaf categories (Joint with J. P. May, submitted, available on the ArXiv)
We study enriched model categories. One of the main questions is when one can replace a given V-model category by a category of presheaves with values in V.
5. A symmetric monoidal and equivariant Segal infinite loop space machine (joint with J. P. May, M. Merling, and A. Osorno, to appear in the Journal of Pure and Applied Algebra, available on the ArXiv)
We construct a new variant of the equivariant Segal machine that starts from the category of finite sets rather than from the category of finite G-sets. In contrast to the machine in [MMO], the new machine gives a lax symmetric monoidal functor from equivariant gamma spaces to orthogonal G-spectra. Even non-equivariantly, this gives an appealing new variant of the Segal machine. This new variant makes the equivariant generalization of the theory essentially formal, hence is likely to be applicable in other contexts.
6. Unstable operations in étale and motivic cohomology (Joint with C. Weibel, to appear in Transactions of the AMS, available on the ArXiv)
We classify all étale cohomology operations on Hetn(-,μ⊗i), showing that they were all constructed by Epstein. We also construct operations Pa on the mod-ℓ motivic cohomology groups Hp,q, differing from Voevodsky's operations; we use them to classify all motivic cohomology operations on Hp,1 and H1,q and suggest a general classification.
7. Enriched model categories in equivariant contexts (Joint with J. P. May and J. Rubin, to appear in Homotopy, Homology, and its Applications, available on the ArXiv)
We study enriched model categories in equivariant contexts, using the perspective developed in "Enriched model categories and presheaf categories".
8. Permutative G-categories and equivariant infinite loop space theory (Joint with J. P. May, Algebraic & Geometric Topology, 2017)
This article supplies results from equivariant infinite loop space theory that are needed in our paper on G-spectra. The equivariant Barratt-Priddy-Quillen theorem is one of the central results, and we rederive the tom Dieck splitting of the fixed points of equivariant suspension spectra from a category-level decomposition.
9. Chaotic categories and equivariant classifying spaces (Joint with J. P. May and M. Merling, Algebraic & Geometric Topology, 2017)
We give simple and precise models of equivariant classifying spaces. We need these models for the paper below on equivariant infinite loop space theory, but the models are of independent interest in equivariant bundle theory.
10. The eta-inverted motivic sphere over R (joint with D. C. Isaksen, Algebraic & Geometric Topology, 2016)
We use an Adams spectral sequence to calculate the R-motivic stable homotopy groups after inverting eta. We also explore some of the Toda bracket structure of the eta-inverted R-motivic stable homotopy groups.
11. The motivic Adams vanishing line of slope 1/2 (joint with D. C. Isaksen, New York Journal of Mathematics, 2015)
We establish a motivic version of Adams' vanishing line of slope 1/2 in the cohomology of the motivic Steenrod algebra over Spec(C).
12. The eta-local motivic sphere (joint with D. C. Isaksen, Journal of Pure and Applied Algebra, 2015)
We compute the h1-localized cohomology of the motivic Steenrod algebra over C. This serves as the input to an Adams spectral sequence that computes the motivic stable homotopy groups of the eta-local motivic sphere. We compute some of the Adams differentials, and we state a conjecture about the remaining differentials.
13. h1-localized motivic May spectral sequence charts (joint with D. C. Isaksen, available on the ArXiv)
Charts of the motivic May spectral sequence for ExtA[h1-1] through the Milnor-Witt 66-stem.
14. Strictification of categories weakly enriched in symmetric monoidal categories (Theory and Applications of Categories, 2010)
We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a "many 0-cells" version of the strictification of bimonoidal categories to strict ones.
15. The motivic fundamental group of the punctured projective line (Journal of K-Theory, 2010)
We describe a construction of an object associated to the fundamental group of the projective line minus three points in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products of Steinberg symbols in the motivic cohomology of the ground field. This work was part of my 2008 Ph.D. thesis under Peter May at the University of Chicago.
## Notes
Course notes for a Topics Course on Hopf Algebras (Spring 2017). Hopf Algebras, cohomology of Hopf algebras, Cartan-Eilenberg spectral sequence.
Course notes for Homotopy Theory (Spring 2015). Fiber bundles, Serre spectral sequence.
Course notes for Homotopy Theory (Spring 2011). Fibrations, cofibrations, homotopy excision.
University of Illinois Suminar 2010 and 2011.
Proseminar talk notes (from graduate school):
Some old notes on Category Theory from a warmup program for graduate students at the University of Chicago. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8463959693908691, "perplexity": 1330.7541219283853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156554.12/warc/CC-MAIN-20180920175529-20180920195929-00374.warc.gz"} |
http://wiki-math.univ-mlv.fr/pdmp/doku.php/events:pdmp_marne_2017 | # Piecewise Deterministic Markov Processes and Sampling
The goal of the workshop is to bring together people working on PDMPs and on various Monte Carlo methods with a momentum and/or a memory: Hamiltonian Monte Carlo, lifted Markov chains, zig-zag process, etc.
### When ?
In 2017, January 25-27. The workshop will start Wednesday at 2pm and end Friday at noon (see the detailed program below).
### Where ?
In Cermics, École des Ponts ParisTech, Marne-la-Vallée, near Paris. Access directions are available in French or in English; the workshop will take place in Bât. Coriolis.
### Registration
Registration is free but mandatory. Please register on dedicated page.
### Program
#### Wednesday, Jan. 25
13h-14h: Registration, coffee
• 14h-14h50: J. Bierkens, Zig-Zag Sampling for Doubly Intractable Distributions
• 15h-15h50: P. Monmarché, Piecewise deterministic simulated annealing
Coffee break
• 16h30-17h20: S. Vollmer, The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method
#### Thursday, Jan 26
9h30-10h: Morning coffee
• 10h-10h50: O. Mangoubi, Rapid mixing bounds for Hamiltonian Monte Carlo and geodesic walks, via positive curvature and momentum
• 11h-11h50: K. Życzkowski, Sampling quantum states with hit and run algorithm
Lunch break
• 14h-14h50: F. Panloup, Weighted Multilevel Langevin simulation of invariant distributions
• 15h-15h50: A. Durmus, High dimensional sampling with the Unadjusted Langevin Algorithm
Coffee Break
• 16h30-17h20: B. Cloez, Properties of stochastic algorithms through homogeneous-time Markov process properties
#### Friday, Jan. 27
• 9h-9h50: P. Fearnhead, Continuous-time Importance Sampling, and Bayesian Analysis of Big Data
Coffee break
• 10h30-11h20 : S. Gadat, Second order processes reinforced by their memory
• 11h30-12h20: A. Duncan, The PDMP as a Monte Carlo method and sampling from restricted domains
### Abstracts
##### J. Bierkens: Zig-Zag Sampling for Doubly Intractable Distributions
In important models in Bayesian statistics the computation of the likelihood function is intractable. The corresponding posterior distributions are referred to as doubly intractable distributions. For example this situation occurs when inferring the temperature in an Ising model, or the parameters in an exponential random graph model (ERGM).
Existing methodology to deal with such models rely on the exchange algorithm of Møller (2006) and variations thereof. In order for this class of algorithms to be asymptotically exact it is necessary to draw perfect samples from the forward model using e.g. the Propp & Wilson (1996) methodology.
It turns out that, when applying the Zig-Zag Sampler (Bierkens et al., 2016) to problems with intractable likelihood, it suffices to obtain unbiased estimates from the forward model, which greatly enlarges the scope of possible models which can be dealt with, as well provides significant gains in computational efficiency.
This is currently work in progress in collaboration with Antonietta Mira (Lugano) and Gareth Roberts (Warwick).
##### B. Cloez: Properties of stochastic algorithms through homogeneous-time Markov process properties
In this talk, we consider some stochastic algorithms represented by an inhomogeneous (discrete time) Markov chain. We are interested in its long time behavior. We provide sufficient conditions to ensure that some of its asymptotic properties can be related to the ones of a homogeneous (continuous time) Markov process. Renowned examples such as a bandit algorithms, Monte-Carlo method, decreasing step Euler schemes are included in our framework. Our results are related to functional limit theorems, but the approach differs from the standard “Tightness/Identification” argument; our method is unified and based on the notion of pseudotrajectories on the space of probability measures. This based on the ODE-method on the probability space and can be seen as a “Markov-method”. Among the approximated Markov process, there is some classical diffusion processes but also some PDMP like switching ODE systems or some processes with jumps.
##### A. Duncan: The PDMP as a Monte Carlo method and sampling from restricted domains
While piecewise deterministic Markov processes (PDMPs) have been around for over 50 years, their potential use as a continuous time Monte Carlo (CTMC) method has only recently been explored. The success of this methodology has motivated a number of theoretical questions on PDMPs, particularly related to long-time behaviour. Focusing on the one-dimensional case, in the first part of this talk, we explore some of these questions, and in particular characterise the performance of PDMP samplers and compare them to other continuous time Monte Carlo schemes. In the second part of this talk, we return to the multidimensional case and explore the effectiveness of PDMP schemes to sample from distributions supported on bounded or semibounded domains.
##### A. Durmus: High dimensional sampling with the Unadjusted Langevin Algorithm
Recently, the problem of designing MCMC sampler adapted to high-dimensional distributions and with sensible theoretical guarantees has received a lot of interest. The applications are numerous, including large-scale inference in machine learning, Bayesian nonparametrics, Bayesian inverse problem, aggregation of experts among others. When the density is L-smooth (the log-density is continuously differentiable and its derivative is Lipshitz), we will advocate the use of a “rejection- free” algorithm, based on the discretization of the Euler diffusion with either constant or decreasing stepsizes. We will present several new results allowing convergence to stationarity under different conditions for the log-density (from the weakest, bounded oscillations on a compact set and super-exponential in the tails to the log concave). When the density is strongly log-concave, the convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality for Lipschitz functions will be reported. Finally, based on optimzation techniques we will propose new methods to sample from high dimensional distributions. In particular, we will be interested in densities which are not continuously differentiable. Some Monte Carlo experiments will be presented to support our findings.
##### P. Fearnhead: Continuous-time Importance Sampling, and Bayesian Analysis of Big Data
I will discuss how versions of importance sampling, that aim to unbiasedly estimate transition densities of continuous-time stochastic processes, can be constructed. These importance sampling algorithms are based upon simulating piecewise deterministic Markov processes. One application of these ideas is their use within a sequential Monte Carlo (SMC) algorithm that samples from a target distribution of interest. Excitingly, for Bayesian applications, these SMC algorithms have the potential to scale well to big data settings: as the computational cost per effective sample size may not increase with the amount of data.
##### S. Gadat: Second order processes reinforced by their memory
Second order methods are expected to be optimal for convex minimization problems. In this work, we develop a second order stochastic dynamical system inspired from the Heavy Ball differential equation. After some remainders on the deterministic system, we study a stochastic adaptation that leads to a second order Markov process, degenerated on the kinetic component. The regularity of such a system is shown as well as its ergodicity. We then study the large deviation properties associated to the invariant measures of the system at low temperature.
##### O. Mangoubi: Rapid mixing bounds for Hamiltonian Monte Carlo and geodesic walks, via positive curvature and momentum.
Most existing methods for analyzing geometric” MCMC algorithms such as Hamiltonian Monte Carlo (HMC) and geodesic walks use conductance bounds. Unfortunately, conductance bounds cannot capture improvements obtained from momentum or positive curvature. To take advantage of positive curvature and momentum, our analysis of HMC and geodesic walks instead uses probabilistic coupling bounds, obtained via comparison geometry.
Specifically, we obtain rapid mixing bounds for HMC in an important class of strongly log-concave target distributions, showing that HMC is faster than many competitor algorithms such as the Langevin MCMC algorithm in this regime.
We also introduce the geodesic walk, an implementable Markov chain that achieves a volume-measure stationary distribution on a Riemannian manifold $\mathcal{M}$ without computationally intensive Metropolis-Hastings corrections. For positively-curved $\mathcal{M}$, we show that the geodesic walk has mixing time $\mathcal{O}^*(\frac{\mathfrak{M}_2}{\mathfrak{m}_2})$, where $\mathfrak{M}_2$ and $\mathfrak{m}_2$ are, respectively, upper and lower bounds on the sectional curvature of $\mathcal{M}$.
(Joint work with Aaron Smith)
##### P. Monmarché: Piecewise deterministic simulated annealing
In order to sample a given probability law with density exp(-V/T), many ergodic Markov processes are available, and the efficiency of the corresponding algorithms is related to the speed of convergence to equilibrium of these processes. Kinetic processes (that is, processes (X,Y) where X is the position and Y=dX/dt is the velocity) are good candidates, since the velocity acts as an “instantaneous memory”: the process have some inertia and can't go back immediatly to the place it has just been, thus exploring the space more efficiently. We will introduce a velocity jump process, which is a kinetic PDMP designed in such a way it samples a given target law, and we will study its speed of convergence, in particular in the regime T→0, and give conditions for a simulated annealing procedure to succeed.”
##### F. Panloup: Weighted Multilevel Langevin simulation of invariant distributions
In this joint work with G. Pagès, we investigate a weighted Multilevel Richardson-Romberg extrapolation for the ergodic approximation of invariant distributions of diffusions. We first obtain some rate of convergence results under weak confluence assumptions on the diffusion. In a second part, under stronger confluence assumptions and with the help of some second order expansions of the asymptotic error, we go deeper in the study by optimizing the choice of the parameters involved by the method. In particular, for a given $\varepsilon>0$, we exhibit some semi-explicit parameters for which the number of iterations of the Euler scheme required to attain a Mean-Squared Error lower than $\varepsilon^2$ is about $\varepsilon^{-2}\log(\varepsilon^{-1})$. Finally, we discuss the so-called weak-confluence assumption and provide some numerical tests on a high dimensional diffusion motivated by a statistical problem.
##### S. Vollmer: The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method
Markov chain Monte Carlo methods have become standard tools in statistics to sample from complex probability measures. Many available techniques rely on discrete-time reversible Markov chains whose transition kernels build up over the Metropolis-Hastings algorithm. We explore and propose several original extensions of an alternative approach introduced recently in Peters and de With (2012) where the target distribution of interest is explored using a continuous-time Markov process. In the Metropolis-Hastings algorithm, a trial move to a region of lower target density, equivalently “higher energy”, than the current state can be rejected with positive probability. In this alternative approach, a particle moves along straight lines continuously around the space and, when facing a high energy barrier, it is not rejected but its path is modified by bouncing against this barrier. The resulting non-reversible Markov process provides a rejection-free MCMC sampling scheme. We propose several original techniques to simulate this continuous-time process exactly in a wide range of scenarios of interest to statisticians. When the target distribution factorizes as a product of factors involving only subsets of variables, such as the posterior distribution associated to a probabilistic graphical model, it is possible to modify the original algorithm to exploit this structure and update in parallel variables within each clique. We present several extensions by proposing methods to sample mixed discrete-continuous distributions and distributions restricted to a connected smooth domain. We also show that it is possible to move the particle using a general flow instead of straight lines. We demonstrate the efficiency of this methodology through simulations on a variety of applications and show that it can outperform Hybrid Monte Carlo schemes in interesting scenarios.
##### K. Życzkowski : Sampling quantum states with hit and run algorithm
We introduce a notion of a set $X$ accessible in $k$ steps with respect to a given collection of vectors. This property leads to an algorithm which may be considered as a derandomization of the hit and run algorithm applied to generate a sequence of random points covering $X$ uniformly.
The set $\Omega_N$ of quantum states of size $N$ consist of positive hermitian density of order $N$ with a fixed trace, $Tr(\rho)=1$. For $N=2$ this set is equivalent to a $3$-disk, called Bloch ball, while in higher dimensions the structure of this convex set is more complicated. To study composed systems we take $N=M^2$. In the theory of quantum entanglement a special role is played by a subset of $\Omega_{M^2}$ containing states with positive partial transpose, $(T\otimes 1)\rho >0$. To analyze spectral properties of such density matrices we apply a variant of the 'hit and run' algorithm and formulate a conjecture of the average spectral density of states in this subset. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8651919960975647, "perplexity": 754.6678169895164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370505730.14/warc/CC-MAIN-20200401100029-20200401130029-00092.warc.gz"} |
https://www.physicsforums.com/threads/coordinate-geometry.153915/ | # Coordinate Geometry
1. Jan 31, 2007
### Harmony
The points Q moves such that the length of the tangent from Q to the circle
$$x^2+y^2+4x+8y+9=0$$ is equal to the distance of Q from the origin ). Determine the locus of Q.
I am basically clueless about this question...but I will try to provide as many work I have done on this question.
I assume that we are required to find the locus of point Q. I reckon that locus of Q is a curve, since the word tangent is only suitable for curves. Hence, I will have to find the gradient of the tangent of the curve in order to solve for the tangent equation, then finding the perpendicular distance of the tangent to the circle, and equate it to the distance from Q to the origin.
But I found that the above methods seems overcomplicated and not likely to be the solution. I check the answer, but to my surprise, the locus of Q is a linear equation $$4x+8y+9=0$$. Is the answer wrong? And how should I approach this question?
2. Jan 31, 2007
### jing
Best way is to start with a picture. Work out the centre and radius of the circle, sketch it on a set of axes. Plot an arbitrary point Q. Draw the tangent from Q to the circle. Join Q to the origin. Work out the distances required and equate them.
Similar Discussions: Coordinate Geometry | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8543099761009216, "perplexity": 208.34023313187478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891485.97/warc/CC-MAIN-20180122153557-20180122173557-00234.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=1131127&postcount=7 | View Single Post
Mentor P: 4,499 Ok, let's just look at one row, because it's equivalent, and do out all the math. First, say you picked a jack. There is a 2/3 chance of this happening. Then Monty reveals a card. 1/2 the time, you reshuffle. 1/2 the time, he reveals the other jack, and you switch to win. So you win 1/3 of the time here, and reshuffle 1/3 of the time. Then, suppose you pick a queen instead. There is a 1/3 chance of this happening. Every single time, you lose. So there is a 1/3 chance of you losing Doc Al, the problem with your 100 card example is that if he picks 98 jacks in a row, it's probably because he was picking out of a pile of 99 jacks, and you had the queen the entire time (actually, half the time that's true) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8647476434707642, "perplexity": 630.8713127090292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00281-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/111452-rearranging-exponential-eq-help.html | # Math Help - rearranging exponential eq. help
1. ## rearranging exponential eq. help
Hi Guys can anyone oint me in the right direction of making the exponetial part the subject. ie i need to solve for T
(theta)=(Theta0) [1-e^-t/T}
20 = 190x[1-e^40/T]
im ok with transfroming eq's but not sure of the rules of this one
any help appreciated
2. Is there a difference between "theta" and "Theta0"? I'm not sure what the first equation really is, it's confusing. I imagine that "Theta0" is the original theta value. I will solve for the second equation instead (by the way, is the time negative? -t = 40?)
$
20 = 190x(1 - e^{\frac{40}{T}})
$
$1 - \frac{20}{190x} = e^{\frac{40}{T}}$
$ln(1 - \frac{20}{190x}) = \frac{40}{T}$
$
T = \frac{40}{ln(1 - \frac{20}{190x})}
$
Patrick
3. Hi Patrick
thanks for taking the time to reply
That is spot on i did it another way working back over and got T = 360
but couldnt work out how to set it out like you have
Thankyou very much for your help | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8802798390388489, "perplexity": 1695.2811199267358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042981576.7/warc/CC-MAIN-20150728002301-00281-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://indico.math.cnrs.fr/event/381/?view=lecture | Série de Courts Exposés
# Combinatorial Lefschetz Section Theorems
## by Prof. Karim Alexander ADIPRASITO (Free University Berlin & IHÉS)
Europe/Paris
Amphitéâtre Léon Motchane (IHES)
### Amphitéâtre Léon Motchane
#### IHES
Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description
Intuitively, the classical variants of the Lefschetz Section Theorem relate a complex algebraic variety X to the intersection of X with a hyperplane H transversal to X (or, alternatively, to an ample divisor D of X). They are tremendously useful to compute invariants of the variety. However, Lefschetz Section Theorems also hold for spaces that are constructed combinatorially rather than algebraically. Among other things, I will introduce Lefschetz theorems for -- certain real subspace arrangements and their complements, -- toric arrangements and their complements and, -- matroids and smooth tropical varieties (joint work with Anders Björner). These theorems translate results of Lefschetz, Hamm-Le, Andreotti--Frankel, Bott--Thom, Akizuki--Kodaira--Nakano and Kodaira--Spencer to a combinatorial setting.
Contact | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8088140487670898, "perplexity": 4806.714351342413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347391277.13/warc/CC-MAIN-20200526160400-20200526190400-00078.warc.gz"} |
http://generalpac.com/current-transformers/introduction-to-current-transformers-part-2-ct-polarity | # Introduction to Current Transformers Part 2: CT Polarity
### Resources Section
This video tutorial does not items in the resources section
Do you have a question? Click on the "Questions & Answers forum" and ask away!
Introduction to current transformers Part 2.
In part 1, we introduced the current transformer and we gave an example where we would want to use a CT. In part 2, we're going to talk about polarity marks. In part 3, we're going to talk about current transformer ratio.
Let's begin by drawing a Delta connected transformer like such. This is line A, this is line B, and this is line C.Then on the secondary side, suppose we had a wye connected transformer that was grounded.
On this wye connected side – on the secondary side of the power transformer, we have CTs connected on each phase. Let's suppose there is a relay that we want this CT to connect to. And on this relay, we have inputs for I_a, I_b, and I_c. And we have two contacts for each phase.
Okay so we've created our CTs on the secondary side of the power transformer, the wye connected side. We have our relay. Let's now talk about current flow.
In a steady state system, let's assume the current flows from the source which is on the left hand side. So here is our source. To the load. Suppose the load was on the right hand side. And the current flows from the source to the load in our steady state system.
The current will flow on each line, then on the secondary side, it will flow like this. Let's zoom back in to the CT that's connected to phase A. Okay so suppose the current in our steady state flows in this particular direction. That is key.
No in terms of polarity marks, we typically see them in two places. One is on the line side and the other on the CT side. To interpret this, we have to follow what's called the dot convention. In our dot convention, we have two rules.
Current into the dot and current out of the dot. The way to understand this is – if current flows into the primary side of this transformer – the primary side. So this is our primary current. If my primary current flows into the dot on the primary side, then the secondary current must flow out of the dot on the secondary side.
So let's connect our CTs in this particular way. Again, primary current flows into the dot on the primary side – then secondary current must flow out of the dot on the secondary side *(illustrated in the image above)*. Secondary current flows like this – it goes through this polarity mark here. Then out of the polarity mark and then completes this circulating current. Because there is a dot on this of the CT – then we would expect this contact here on the relay is the polarity side of the contact. So these two sides should match up.
Again, primary current flows into the dot on the primary side. Then secondary current must flow out of the dot on the secondary side *(illustrated in the image above)*. That's the rule of the dot convention.
And suppose that we have a dot on this side of the transformer – how we would be interpret that? Okay so it's fairly simple. The way that we would do it – the current that flows into the dot on the primary side must – then secondary current must flow out of the dot on the secondary side *(illustrated in the image below)*. So the dot comes out like that. So let's say the current flows this way.
Because we have the dot on this side now, we should expect the polarity side of relay to be like this – this is the polarity side contact of our relay. And that's how we should think about it. Okay so current that's flowing into the dot on the primary side, must flow out of the dot on the secondary side *(illustrated in the image above)*.
So now the question is – what if we had current flowing in a different direction on our primary side. Very simple – we just need a very good explanation. Current that's flowing like this. When current is flowing in this direction, we know that the CT comes first, then we have the dot *(illustrated in the image below)*.
So in this particular configuration, the current is flowing out of the dot on the primary side, which means current must flow into the dot on the secondary side *(illustrated in the image above)*. So it's going to be make a circulating path like this. So current that flows out of the dot on the primary side means that current must flow into the dot on the secondary side.
Now what if our dot on our secondary side was over here. How would we interpret that with the same current flow? Well first of all on our relay, we know that on our relay, this contact will be on polarity side because it matches with our CT.
So now current flows out of the dot on the primary side, means that current flows into the dot on the secondary side *(illustrated in the image above)*. So it'll make the connection like this. So in terms of the dot connection, there is really two rules that we need to memorize. What's goes into the dot must flow out of the dot.
Now as a teaser to other parts of this series, I'll show you show three current transformers are connected to the relay – the most typically example.
So this is a very typical arrangement of the current transformer connected in Wye. We'll talk about this particular of arrangement in later modules. In part 2, we talked about current transformer and the polarity markings of the CTs. We talked about different arrangements of these polarity marks and how to interpret them. And then as a teaser, we drew the wye connected three phase CTs.
In part 3, we'll talk about current transformer ratio and how to relate primary current with secondary current.
This module was brought to you by generalpac.com. Making Power system protection, automation, and controls intuitive. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8126832842826843, "perplexity": 604.487958070633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320368.57/warc/CC-MAIN-20170624235551-20170625015551-00007.warc.gz"} |
https://www.cfd-online.com/W/index.php?title=Wall_shear_stress&diff=13554&oldid=13551 | # Wall shear stress
(Difference between revisions)
Revision as of 08:22, 22 November 2011 (view source)Zonder (Talk | contribs) (add unit)← Older edit Revision as of 18:16, 23 November 2011 (view source)Peter (Talk | contribs) m (Reverted edits by Zonder (talk) to last revision by Peter)Newer edit → Line 4: Line 4: Where $\mu$ is the [[Dynamic viscosity|dynamic viscosity]], $u$ is the flow velocity parallell to the wall and $y$ is the distance to the wall. Where $\mu$ is the [[Dynamic viscosity|dynamic viscosity]], $u$ is the flow velocity parallell to the wall and $y$ is the distance to the wall. - - The SI unit of the kinematic viscosity is $Pa$ or $\frac{kg}{m\cdot s^2}$.
## Revision as of 18:16, 23 November 2011
The wall shear stress, $\tau_w$, is given by:
$\tau_w = \mu \left(\frac{\partial u}{\partial y} \right)_{y=0}$
Where $\mu$ is the dynamic viscosity, $u$ is the flow velocity parallell to the wall and $y$ is the distance to the wall. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9815467000007629, "perplexity": 1908.3480603581982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281151.11/warc/CC-MAIN-20170116095121-00327-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/483392/m-set-interior-point-probability-on-the-real-axis | # M-set interior point probability on the real axis
For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are inside hyperbolic regions of one of the bulbs on the real axis. What is the probability that a random point on the real axis, greater than -2, and less then 0.25, is in the interior of the m-set? Is that value $\lt 1$, and perhaps arbitrarily small for other regions on the real axis?
Of course I realize, numerical calculations are not particularly relevant for a mathematical proof, but I did numerical experiments on the region from $[-2,-1.40115518909205060052]$, or from -2 to the Feigenbaum limit point of the tip of the main cardiod. It seems like in this region, there's approximately a 10% chance that a random point is inside a hyperbolic region. I experimented by iterating $z_n\mapsto z_{n-1}^2+z_0$, where z is a random real number in that region and $z_0=z+i10^{-30}$. 90% of the values quickly iterate to infinity. Using a much larger imaginary $\delta=i10^{-5}$ did not not change the results. I did a similar analysis on the region from -1.625 to -1.615, which appears to have an even smaller probability of landing in the interior of a mini-Mandelbrot bulb. There are an infinite number of mini-Mandelbrots in that region, but the largest bug in that region is around 1/1000th the size of the region, and the next two bugs are around 1/4000th the size of the region and the bugs rapidly get smaller after that.
Anyway, clearly someone much smarter than I am has proved something about this. I conjecture that most of these random points are not inside the m-set, but rather are chaotic random points on the boundary of the m-set, and that these chaotic boundary points are a super-set of the Misiurewicz points, and that most of the points in the real valued region $[-2,-1.40115518909205060052]$ are chaotic, and not in the interior of the m-set.
-
My goal was to have someone else tell my why my intuition was correct, along with a suggested book/link with a proof in it. But I think the key to answering the question is to analyze the behavior of the m-set near a Misiurewicz on the real axis. I think this can be used to show that a random point on the real axis $[-2,0.25]$ has a probability of less than one of being in the interior of the m-set, since I can show that in the neighborhood of Misiurewicz points on the real axis, that probability can get arbitrarily small as you zoom in on the Misiurewicz point.
This isn't a proof because I assume a lemma that the length of the mini-Mandelbrot is proportional to the square of the reciprocal of the slope of $z_n(x)$ at the real axis, where the location of the bugs are determined by $z_n(x)=0$. This explains why, as you zoom in to the Misiurewicz point, the mini-Mandelbrot bugs get smaller in a very predictable way. The Misiurewicz point I analyzed is C=-1.54368901269207636157, which is the real root of $z^3 + 2z^2 + 2z + 2$, generated by iterating starting with $z_1(x)=x$ and $z_{n+1}\mapsto z_n^2+x$, and solving $z_3=z_2$. Lets look at the behavior on the m-set on the real axis in the neighborhood of this Misiurewicz point; the first graph goes from -1.645 to -1.47, which contains two main Mandelbrot bugs, with period5 on the left, and period6 on the right. We can solve numerically for the hyperbolic centers of these two bugs by solving the algebraic equations, $z_5=0$ and solving $z_6=0$. In the window of interest, there is one solution for each. I marked the solutions the six largest bugs, for $z_5=0$, $z_6=0$, $z_7=0$ and $z_8=0$, $z_9=0$, and $z_{10}=0$. There are two additional solutions for $z_9=0$ and $z_{10}=0$, so I marked the closer in slightly smaller bug as well, as z9b and z10b. I also marked the Misiurewicz point, in yellow. Below this image, I show a similar image, zoomed in two steps towards the bugs z9b and z10b, showing the largest mini-Mandelbrots in that zoomed in region. These mini-Mandelbrots are observably much smaller. But how much smaller? And why?
The next image is zoomed in by two steps, between the main bugs with period9=z9b above and period10=z10b above.
To start, lets look at two more images. The first shows the region from z9b to z10b; same as the second Mandelbrot image. I show the graph of $z_9(x)$, $z_{10}(x)$, $z_{11}(x)$, $z_{12}(x)$, $z_{13}(x)$, and $z_{14}(x)$, and the real axis. The zeros of these six graphs are the locations of the biggest mini-Mandelbrots in the zoomed in region. Next, I show another graph zoomed in two more steps, zooming in on the z13 to z14 bug region. The second graph is close to converging, and similar graphs can be made, zooming in arbitrarily. I marked the mini-Mandelbrot locations as zero crossings. Also, notice that all of the curves intersect at the Misiurewicz point, which is also marked. One can imagine an infinite number of these curves, of which only the first six with zero-crossings with the smallest slopes are shown in each graph. The scale factor converges to approximately 0.35491x, as you zoom in. The scale factor can be computed by generating the Taylor series for $z_n(x)$ for x centered at the Misiurewicz point, and generating the ratio of the derivative to the derivative of $z_{n+2}(x)$. With a little algebra this can be shown to be equal to the reciprical of $4z_3^2$, by developing the Taylor series at c, and iterating $z_{n+1}=z_n^2+x+c$, with $z_0=0$. zooming in another two steps, self similarity takes over.
I think these last two graphs give more of a sense of the self-similar behavior of the Mandelbrot set as you zoom in; but the graphs don't show the size of the mini-Mandelbrots, only the location of the mini-Mandelbrots, which are the zero crossings. Now, I stated an unproven lemma for the length of a mini-Mandelbrot $l(z_n=0) \approx \frac{1}{(z_n')^2}$. My rational is that at the hyperbolic center, we know $z_n(x)$ has a zero. Next, a Taylor series for $z_n$ is centered at the hyperbolic center itself, $z_n(x) = a_1 x+ a_2 x^2 + ....$. A similar Taylor series, also centered at the same hyperbolic center is generated for 2n, which by definition also has a zero at the hyperbolic center; with a little algebra, we can show it has the same first derivative, $z_{2n}(x) = a_1 x+ b_2 x^2 + b_3 x^3....$. Numerically, I observe that $b_2 \approx k a_1^3$, where k is a number greater than 1.5 and less than 2. $z_{2n}$ has two zeros; the second zero is very close to the first zero, and gives the length of the mini-Mandelbrot since it is the distance between the main cardioid, and the 2x cardioid. Since the Taylor series for $z_{2n}$ is centered at the first zero, we need an estimate for the second zero which is $z_{2n 2} \approx -\frac{a_1}{b_2} \approx \frac{a_1}{1.8a_1^3} \approx \frac{0.6}{a_1^2}$.
More pictures might help, showing the mini-Mandelbrot cardioid, and graphs of the two functions, $z_n$ and $z_{2n}$; and there's a lot of algebra I didn't show, that explains the lemma, but doesn't prove it. The lemma then predict that each of the bugs would scale by $\approx \frac{1}{(a_1)^2}$, where $a_1$ is the derivative of $z_n$ at the zero crossing. This matches the numerical data closely. The window size is scaling by approximately 0.35491x each iteration, but the bug length scaling by the square, 0.12596x, so relatively speaking, the bugs are getting smaller by 0.35491x each iteation. If the relative length of all the bugs in the window gets smaller by approximately 0.354921 each iteration, than the sum of the lengths of all of the bugs in the window also get smaller by the same ratio, and the total length of the bugs, relative to the window size, goes to zero as you zoom in. Assuming my logic is correct, than most of the points on the real axis, between $[-2,-1.4011552]$ are neither inside hyperbolic components, nor are they Misiurewicz points. They are simply random points, with zero width, on the real axis filigree that is the boundary of the m-set. There will be a hyperbolic component arbitrarily close to such random points, but I don't think these random points are adjacent to a hyperbolic component.
Numerical data showing the size of the six biggest bugs, in the window,
relative to the window size, as you zoom in by 2.81761x each iteration.
window -1.64500000000000000000 to -1.47000000000000000000
z5 -1.62541372512330373744 0.0378913101884657946067
z6 -1.47601464272842989752 0.0463018986609053128810
z7 -1.57488913975230096982 0.00655041322134289247183
z8 -1.52181723167125099520 0.00414679373793390113281
z9 -1.59568096343974577478 0.00119381469602047654212
z10 -1.50171683940963313793 0.00210630478418442001747
window -1.57964594647609667764 to -1.51753557148707102871
z7 -1.57488913975230096982 0.0184562130551868531476
z8 -1.52181723167125099520 0.0116838596493203506804
z9 -1.55528270076858318477 0.00276214675652270816184
z10 -1.53624327128231050351 0.00126848512492673185395
z11 -1.56362963108720118812 0.000659268351688881292083
z12 -1.53005645596148015452 0.000452102609375223546519
window -1.55645071921082531485 to -1.53440672674638312697
z9 -1.55528270076858318477 0.00778252719415849789938
z10 -1.53624327128231050351 0.00357403890898100087413
z11 -1.54790376180395502683 0.00106684073199442759725
z12 -1.54109410649138282038 0.000422764278591095275479
z13 -1.55107190019127551232 0.000289916617048784854801
z14 -1.53901837190200957968 0.000134635051612048004546
window -1.54821835199138716484 to -1.54039457691583425088
z11 -1.54790376180395502683 0.00300589278573849461110
z12 -1.54109410649138282038 0.00119116570728342799192
z13 -1.54520178169265663017 0.000393399151035709301528
z14 -1.54277519227761074816 0.000146158186626937596148
z15 -1.54636280570666495530 0.000113098359842964666488
z16 -1.54205531050306852211 0.0000447031218672100244229
window -1.54529654961969193113 to -1.54251976331242116562
z13 -1.54520178169265663017 0.00110842756050681389305
z14 -1.54277519227761074816 0.000411810147084660629635
z15 -1.54422856011952779613 0.000141929100569532803183
z16 -1.54336574221033890777 0.0000513163804751993036231
z17 -1.54464648152450039760 0.0000417496147766831044556
z18 -1.54311265996486419402 0.0000154599913144917732600
window -1.54389150716018304821 to -1.54354172740913823104
z17 -1.54388090052770975288 0.000142892198283435926925
z18 -1.54357443364491605528 0.0000510913999908299379366
z19 -1.54375717344627227934 0.0000180484964150937955160
z20 -1.54364836925566699769 0.00000642435636305903336541
z21 -1.54381025766971924338 0.00000537595932368788837315
z22 -1.54361666220344091840 0.00000192022411050607955763
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I found this link on the web, which agrees with my answer, and has foototes with a proof. https://sites.google.com/site/fabstellar84/fractals/real_chaos
This page is about an investigation that I conducted on the Mandelbrot Needle... my initial hypothesis about the needle turned out to be false...
... The Mandelbrot Needle is the part of the Mandelbrot Set between the Myrberg-Feigenbaum point and c == -2. Also, for each window of non-chaotic behavior in the chaotic region of the Logistic map, there is a miniature copy of the Mandelbrot Set in the Mandelbrot Needle.
I hypothesized that the chaotic region of the Logistic map was made completely out of an infinite number of tiny little windows of non-chaotic behavior. If that were true, then the set of all chaotic parameters would have a Lebesgue measure of 0, meaning it would be somewhat similar to the Cantor set. And, the Mandelbrot Needle would be made completely out of tiny little Mandelbrot Set copies aligned back to front. After investigating the Mandelbrot Needle for a week, I learned that back in 1981, M. V. Jacobson proved that the chaotic parameters in the chaotic region have a finite Lebesgue measure. This means my hypothesis is completely false!
Although the chaotic parameters in the Logistic map have a finite Lebesgue measure, they are infinitely discontinuous, which was proved by J. Graczyk and G. Swiatek. This means that if you examine any interval of the chaotic region, no matter how small, there will always be windows of non-chaotic behavior. As a result, if you take a picture of the Mandelbrot Needle anywhere at any magnification, there will always be miniature Mandelbrot Set copies in the picture. This means that the chaotic points in the Mandelbrot Needle form a so called "fat" fractal. The chaotic point set is somewhat similar to a Smith-Volterra-Cantor set. But, instead of removing predetermined length pieces from line segments, you make the set by removing all points from the Mandelbrot Needle that are found in the interiors of the miniature Mandelbrot Set copies.
M. V. Jacobson, Commun. Math. Phys. 81, 39 (1981).
J.Graczyk, G.Swiatek, Hyperbolicity in the Real Quadratic Family, Report no. PM 192 PennState (1995).
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https://www.physicsforums.com/threads/is-mond-necessary.391694/ | # Is MOND necessary?
1. Apr 2, 2010
### zachzach
Consider a star of mass m at a distance r form the center of a circular disk galaxy. Newton's law: F = GM(r)m/(r^2) where M(r) is the amount of mass inside the radius r. If we consider a uniform galaxy then density (p) is p = M/L where L is the length = 2*pi*r. So M(r) = p*2*pi*r. Setting the force of gravity equal to centripetal force (mv^2/r) you get G*2*pi*p = v^2 or
v = [2G*pi*p]^(1/2) which is a constant. Why do you need MOND theory. To me it seems Newtonian mechanics predicts a flat velocity curve.
2. Apr 2, 2010
### Jonathan Scott
Firstly, your simplified form of Newton's law only applies in certain cases such as when the mass is spherically symmetrical, or like a segment of a sphere along a diameter towards the relevant direction.
Secondly, I don't get your maths for the mass. If the galaxy is of uniform density per area of the disk, the mass inside a given radius would be proportional to the square of the radius. For the mass to be proportional to the radius, the area density would have to vary as 1/r.
3. Apr 3, 2010
### FrankPlanck
As Jonathan said, it's wrong.
Your galaxy isn't spherical, you can't use your (wrong) formulas. Fix them and find the rotation curve of the bulge.
If you want to find the disk rotation curve you should write your potential considering a cylindrical distribution (hint: Green's functions), then
$$\frac{v_{c}^{2}}{R} = \frac{\partial \phi (R,z=0)}{\partial R}$$
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http://math.stackexchange.com/questions/86164/is-this-partition-problem-strongly-np-complete | # Is this partition problem strongly NP-complete?
The Partition problem is weakly NP-complete: Given a set A of positive integers, can A be partitioned into two disjoint subsets with the same sum?
I'm interested in the hardness of this variant:
Partition problem: Given a set A of positive integers, can A be partitioned into three disjoint subsets with the same sum?
Is this variant strongly NP-complete?
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No. The pseudo-polynomial dynamic programming algorithm that shows that the "partition into 2 equal-weight subsets" is only weakly NP-hard can be generalized (straightforwardly) to "partition into $k$ equal-weight subsets" for any fixed $k$. The degree of the polynomial will depend on $k$, though.
More precisely, there are at most possible $N^k$ different statements of the form "the first $i$ input numbers can be partitioned into $k-1$ sets with weights $(a_1,a_2,...,a_{k-1})$ plus a (possibly empty) rest set". Put the truth values of these statements into a $k$-dimensional array, and fill it in dynamically by indreasing $i$. This takes $O(N^k)$ time.
Thanks Henning, So I guess if $k= O(\log n)$ then it would not be strongly NP-complete? right? – Mohammad Al-Turkistany Nov 27 '11 at 22:22
@turkistany: That would give the simple algorithm I skech a runtime of $O(n^{\log n})$, and unless I'm missing something that's not actually polynomial. There might still be some other pseudopolynomial algorithm that works for the $k=O(\log n)$ case, of course. – Henning Makholm Nov 27 '11 at 22:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8500022888183594, "perplexity": 495.1265237166542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00646-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www-old.newton.ac.uk/programmes/HRT/seminars/2008120916001.html | # HRT
## Seminar
### Jet formation in decaying two-dimensional turbulence on a rotating sphere
Yoden, S (Kyoto)
Tuesday 09 December 2008, 16:00-16:30
Seminar Room 1, Newton Institute
#### Abstract
Jet formation in decaying two-dimensional turbulence on a rotating sphere is reviewed from the view point of wave mean-flow interaction for both shallow-water case and non-divergent case as the limit of Fr (Froude number) going to zero. A series of computations are performed to confirm the behavior of zonal mean zonal flow generation on the parameter space of the rotation rate Omega and Fr. When the flow is non-divergent and Omega is large, intense retrograde circumpolar jets tend to emerge in addition to a banded structure of mean zonal flows with alternating flow directions. As Fr increases, circumpolar jets disappear and a retrograde jet emerges in the equatorial region. The appearance of the intense retrograde jets can be understood by the angular momentum transport associated with the generation, propagation, and absorption of Rossby waves. When the flow is non-divergent, long Rossby waves tend to be absorbed near the poles. In contrast, when Fr is large, Rossby waves can hardly propagate poleward and tend to be absorbed near the equator. The direction of the equatorial jet, however, is not always retrograde. Our ensemble experiments showed the emergence of a prograde jet, though less likely. This result is contrasted with the previous studies that reported retrograde equatorial jets in all the cases for shallow-water turbulence. Furthermore, a mean zonal flow induced by wave-wave interactions was examined using a weakly nonlinear model to investigate the acceleration mechanisms of the equatorial jet. The second-order acceleration is induced by the Rossby and mixed Rossby-gravity waves and its mechanisms can be categorized into two types.
#### Video
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If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8947047591209412, "perplexity": 2091.91541696501}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464054915149.6/warc/CC-MAIN-20160524015515-00006-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://hotmath.com/hotmath_help/topics/simplest-form.html | Fractions in Simplest Form
A fraction is said to be in simplest form if its numerator and denominator are relatively prime, that is, they have no common factors other than $1$. (Some books use "written in lowest terms" to mean the same thing.)
So, $\frac{5}{9}$ is in simplest form, since $5$ and $9$ have no common factors other than $1$. But $\frac{6}{9}$ is not; $6$ and $9$ have a common factor $3$.
To write $\frac{6}{9}$ in simplest form, divide both the numerator and denominator by the greatest common factor, in this case $3$:
$\frac{6\text{\hspace{0.17em}}÷\text{\hspace{0.17em}}3}{9\text{\hspace{0.17em}}÷\text{\hspace{0.17em}}3}=\frac{2}{3}$
So $\frac{6}{9}$ in simplest form is $\frac{2}{3}$.
This is known as reducing fractions. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 14, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9935401082038879, "perplexity": 110.40932731728736}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982292607.17/warc/CC-MAIN-20160823195812-00034-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://paperity.org/p/81368455/synaptic-convergence-regulates-synchronization-dependent-spike-transfer-in-feedforward | # Synaptic convergence regulates synchronization-dependent spike transfer in feedforward neural networks
Journal of Computational Neuroscience, Sep 2017
Correlated neural activities such as synchronizations can significantly alter the characteristics of spike transfer between neural layers. However, it is not clear how this synchronization-dependent spike transfer can be affected by the structure of convergent feedforward wiring. To address this question, we implemented computer simulations of model neural networks: a source and a target layer connected with different types of convergent wiring rules. In the Gaussian-Gaussian (GG) model, both the connection probability and the strength are given as Gaussian distribution as a function of spatial distance. In the Uniform-Constant (UC) and Uniform-Exponential (UE) models, the connection probability density is a uniform constant within a certain range, but the connection strength is set as a constant value or an exponentially decaying function, respectively. Then we examined how the spike transfer function is modulated under these conditions, while static or synchronized input patterns were introduced to simulate different levels of feedforward spike synchronization. We observed that the synchronization-dependent modulation of the transfer function appeared noticeably different for each convergence condition. The modulation of the spike transfer function was largest in the UC model, and smallest in the UE model. Our analysis showed that this difference was induced by the different spike weight distributions that was generated from convergent synapses in each model. Our results suggest that, the structure of the feedforward convergence is a crucial factor for correlation-dependent spike control, thus must be considered important to understand the mechanism of information transfer in the brain.
This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs10827-017-0657-5.pdf
Pachaya Sailamul, Jaeson Jang, Se-Bum Paik. Synaptic convergence regulates synchronization-dependent spike transfer in feedforward neural networks, Journal of Computational Neuroscience, 2017, 1-14, DOI: 10.1007/s10827-017-0657-5 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8916285037994385, "perplexity": 1514.7977899759558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806066.5/warc/CC-MAIN-20171120130647-20171120150647-00030.warc.gz"} |
http://mymathforum.com/advanced-statistics/341233-coin-dice-problem.html | My Math Forum Coin and Dice problem
July 21st, 2017, 06:07 AM #1 Newbie Joined: Jul 2017 From: Hyderabad Posts: 1 Thanks: 0 Hello All, Could you please help me in getting the solution for the below problem on probability? If a man alternately tosses a coin and throws a die continuously, then find the probability of getting Head on the coin before he gets 4 on the die. Thanks for your support, Lakshmi Last edited by skipjack; July 21st, 2017 at 07:59 AM.
July 21st, 2017, 07:50 AM #2 Senior Member Joined: Oct 2009 Posts: 406 Thanks: 141 So let $X(n)$ be the event that the first head is on the n'th throw. Let $Y(n)$ be the event that the first $4$ is on the $n$'th throw. Am I correct that you must find the probability $\mathbb{P}\{X July 21st, 2017, 08:15 AM #3 Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics We can re-word the problem a little bit: Find the probability that he will get only the numbers 1, 2, 3, 5 or 6 before the first time he gets a head. So the required probability is$\displaystyle \frac{1}{2} + \left( \frac{1}{2} \right) \left( \frac{5}{6} \right) \left( \frac{1}{2} \right) + \left( \frac{1}{2} \right) \left( \frac{5}{6} \right) \left( \frac{1}{2} \right) \left( \frac{5}{6} \right) \left( \frac{1}{2} \right) + ... = \frac{1/2}{1-5/12} = \frac{6}{7}$Or we could do it in another way. Find the probability that he will NOT get the number 4 before he gets a head for the first time.$\displaystyle 1 - \left[ \left( \frac{1}{2} \right) \left( \frac{1}{6} \right) + \left( \frac{1}{2} \right) \left( \frac{5}{6} \right) \left( \frac{1}{2} \right) \left( \frac{1}{6} \right) + ... \right] = 1 - \frac{1/12}{1-5/12} = 1 - \frac{1}{7} = \frac{6}{7} \$ Thanks from Lakshmi77
July 23rd, 2017, 04:52 AM #4 Math Team Joined: Jan 2015 From: Alabama Posts: 3,198 Thanks: 872 It is impossible to flip a coin "continuously". You mean "continually".
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Contact - Home - Forums - Cryptocurrency Forum - Top | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9997588992118835, "perplexity": 1348.0814712846732}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867095.70/warc/CC-MAIN-20180624215228-20180624235228-00270.warc.gz"} |
http://mathoverflow.net/questions/119157/runs-in-coin-flips/119165 | # Runs in coin flips
Let $P(j,k,n)$ be the probability of getting $j$ uniform runs of length $k$ from $n$ fair coin flips. What's the best way to compute $P$? I have no idea how difficult it might be; if it's a very complicated combinatorial argument, I'm looking more to understand the mathematical tools than to actually be able to do this explicitly in nontrivial cases. If anyone wants to show off, is $P(j,k,n,\alpha)$ for $\alpha \in [0,1]$ the weight of the coin much harder?
-
Just to clarify the question, does TTTT have three runs of length two, or none? – James Martin Jan 17 '13 at 11:46
And is $P(j,k,n)$ for exactly $j$ runs or at least $j$ runs? – Brendan McKay Jan 17 '13 at 13:42
For large $n$, this sort of question can be tackled via concentration of measure. The idea is that the typical case should be very close to the average case, so that the probability is essentially either zero or one, depending on whether the property holds on average.
One useful tool is Talagrand's inequality. Let $\Omega = \Omega_1 \times \cdots \times \Omega_n$ be a product space and let $X$ be a random variable on $\Omega$ satisfying
1. $|X(\omega) - X(\omega')| \leq c$ whenever $\omega$ differs from $\omega'$ on a single coordinate. ($X$ doesn't vary too quickly.)
2. Whenever $X(\omega) \geq t$ there is a set $I$ of $f(t)$ coordinates such that $X(\omega') \geq t$ for every $\omega'$ agreeing with $\omega$ on $I$. (If $X$ is large then there is a small certificate showing why this is the case.)
Let $m$ be the median of $X$. Then Talagrand's inequality states that $\mathbb{P}(|X-m| \geq \epsilon m) \leq 4 \exp(-\frac{\epsilon^2m^2}{4f(m)c^2})$.
If the concentration is good, then the median will in fact be close to the mean $\mu$, so there is a similar (slightly more complicated) inequality in terms of $\mu$. (This is worked out carefully in Molloy and Reed's Graph colouring and the probabilistic method.)
Let $X$ be the number of runs of length $k$ in $n$ trials. There are two possible interpretations of your question, depending on whether you want the runs to be disjoint. In either case we have $f(t) = kt$, as the $t$ runs of length $k$ are themselves a certificate.
In the case where the runs must be disjoint we can take $c=1$, so the factor on the right hand side becomes $4 \exp(-\frac{\epsilon^2m}{4k})$. So provided the median grows at some reasonable rate (and it looks like it should be linear in $n$, or almost that) there is very tight concentration of $X$ around a single value, and your $P$ is either approximately 0 or approximately 1 for large $n$ and almost all values of $j$.
If the runs do not need to be disjoint then we can take $c=k$ so that the right hand side becomes $4 \exp(-\frac{\epsilon^2m}{4k^3})$. Again, provided $m$ grows, concentration is good and there is a sharp threshold for $j$ at which the probability rapidly goes from $\approx 1$ to $\approx 0$ as $j$ increases.
The only place $\alpha$, the probability of success, enters this picture is in the calculation of the median.
This doesn't say anything about how to calculate the median (or mean), but that usually turns out to be easier than calculating the probabilities of interest directly.
-
Nice explanation. Regarding this: "provided the median grows at some reasonable rate (and it looks like it should be linear in n, or almost that)" -- in this case the mean is about $\log_2 n$, is that "reasonable" would you say? – Bjørn Kjos-Hanssen Apr 30 at 14:18
@BjørnKjos-Hanssen, there are two steps, and we're interested in the mean/median of a different quantity at each stage. At the first stage we ask how many runs of length $k$ there are. Provided the median of this random variable is large, it will be tightly concentrated. ("Large" here just means large enough to make the exponential bounds decay to zero, so larger than $k$ or $k^3$.) In this case the median is linear in $n$, so if $k$ is growing slower than this we will get concentration. – Ben Barber Apr 30 at 14:59
The second step is to decide whether the value we are concentrated near is 0 or something larger than 1. This is always problem dependent: in this case the expected number of runs of length $k$ is around $n2^{-k}$, so the threshold will be around $\log_2 n$. But to be completely explicit, this isn't the value that we wanted to be reasonably large. – Ben Barber Apr 30 at 15:00
Oh right it was the number of runs not the length of the longest run – Bjørn Kjos-Hanssen Apr 30 at 15:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9102804064750671, "perplexity": 192.32956722678247}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997892648.47/warc/CC-MAIN-20140722025812-00063-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/what-is-the-power-of-the-weightlift.129404/ | # What is the power of the weightlift?"
1. Aug 22, 2006
### danago
Hey. I had a test today. In every test, my teacher puts some questions in where people can get extra marks. The questions involve things we have yet to cover, so usually people wont answer them.
Heres one of the questions:
"A weightlifter lifts a weight of 202kg to a height of 2.7m above the ground. He does this in 1.7 seconds. What is the power of the weightlift?"
I wasnt 100% sure, but i remembered some formulas i had learnt in previous years, and chucked some things together. What i did was:
$$\displaylines{ E_p = mgh \cr = (202)(9.8)(2.7) \cr = 5344.92J \cr P = \frac{E}{t} \cr = \frac{{5344.92}}{{1.7}} \cr = 3144.07W \cr}$$
Not sure if its even close to being right, but it was worth a try atleast. Could someone please tell me what i should have done.
Thanks,
Dan.
2. Aug 22, 2006
### Delzac
From what is know of, it is correct
3. Aug 22, 2006
### danago
Ooh. That would be really good :)
Similar Discussions: What is the power of the weightlift?" | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8383722305297852, "perplexity": 1220.2571937983628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218193716.70/warc/CC-MAIN-20170322212953-00151-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://irsa.stat.umn.edu/people/inge-s-helland | Inge S. Helland
Department of Mathematics, University of Oslo
Title: A Group-Theoretical Approach Towards Envelopes
Abstract:
Dennis Cook’s envelope model represents a major contribution to the statistical theory of prediction. In its x-reduction variant it corresponds to the partial least squares model, and it is a reduction of the multivariate regression model. In my talk I will address the following question: Can such a reduction be motivated intuitively? My suggestion is to use symmetry considerations, more concretely, use group theory. Group actions on the parameter space are defined in general, and orbits of such groups are introduced. As a general principle put forward, it is claimed that every model reduction should be to an orbit or to a set of orbits of the group. This principle is motivated, and shown to give natural model reductions in several simple cases. In the multivariate regression case with a reasonable group defined on the parameter space, it leads to an envelope model. Next a generalized Pitman theorem is formulated, and used as a motivation to introduce a reasonable prior in the envelope model. The corresponding Bayes estimator is briefly described. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.907295286655426, "perplexity": 439.2956417346266}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655881763.20/warc/CC-MAIN-20200706160424-20200706190424-00457.warc.gz"} |
https://nips.cc/Conferences/2021/ScheduleMultitrack?event=28756 | Timezone: »
Poster
Few-Shot Data-Driven Algorithms for Low Rank Approximation
Piotr Indyk · Tal Wagner · David Woodruff
Tue Dec 07 04:30 PM -- 06:00 PM (PST) @
Recently, data-driven and learning-based algorithms for low rank matrix approximation were shown to outperform classical data-oblivious algorithms by wide margins in terms of accuracy. Those algorithms are based on the optimization of sparse sketching matrices, which lead to large savings in time and memory during testing. However, they require long training times on a large amount of existing data, and rely on access to specialized hardware and software. In this work, we develop new data-driven low rank approximation algorithms with better computational efficiency in the training phase, alleviating these drawbacks. Furthermore, our methods are interpretable: while previous algorithms choose the sketching matrix either at random or by black-box learning, we show that it can be set (or initialized) to clearly interpretable values extracted from the dataset. Our experiments show that our algorithms, either by themselves or in combination with previous methods, achieve significant empirical advantage over previous work, improving training times by up to an order of magnitude toward achieving the same target accuracy. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8032514452934265, "perplexity": 1062.1520536938742}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710237.57/warc/CC-MAIN-20221127105736-20221127135736-00356.warc.gz"} |
https://questaal.org/docs/code/fpcapabilities/ | # Full Potential LMTO Capabilities
### Introduction
This page should serve as a demonstration of a number of the capabilities of the lmf code and can be used as a reference for the corresponding procedures.
##### Energy bands
You can run lmf in band mode to generate energy bands along lines or planes, for generating, e.g. Fermi surfaces. Particularly useful are the color weights.
##### Partial DOS and Mulliken analysis
lmf can generate partial DOS within augmentation spheres, and construct a Mulliken analysis. DOS can be resolved by site, by site and l, or by site and lm. These options are invoked through command-line switches. For illustrations, invoke
fp/test/test.fp co 2
fp/test/test.fp fe 2
##### Charge density
lmf can generate the charge density (smooth or not, with or without cores), and the contribution to the density from a selected window of states. See –wden and –window in command-line switches
##### Core-level spectroscopy
lmf can generate EELS spectra, which involve matrix elements between core and valence electrons. The EELS option is invoked with a command-line switches. For illustrations, invoke
fp/test/test.fp fe 2
fp/test/test.fp crn 2
##### LDA+U
The LDA + U functional was built into lmf in v6.15 and later, by Walter Lambrecht. LDA + U needs in addition to the LDA, parameters U and J for selected orbitals, which are empirical. The LDA + U constructs an additional potential for a particular l subblock (m = -l..l) from the U supplied by the user, and the density-matrix, which is generated by lmf.
Two modifications must be added to the input, which are described here:
In a strictly LDA calculation, complete information is carried by the density, contained in the restart file, rst.ext. In the LDA + U case, complete information is carried by density and on-site density matrices, which are contained in file dmats.ext.
An example that illustrates LDA + U method is ErAs, which you can run by
fp/test/test.fp eras
ErAs is an interesting case because LDA puts all 4 minority f electrons in a single extremely narrow band at the Fermi level. In LDA + U the minority f is split into a 4 - and 3-manifold; see PRB 67, 035104 (2003).
##### GW
lmf is designed to work in coordination with a GW package by T. Kotani (the GW package comes separately). lmf acts both as a driver for the GW package and can also be used in a self-consistent GW cycle. An extra driver lmfgw is compiled as part of this extension. Use of this driver is described in the GW driver documentation. You need the extension package GW.version.tar.gz. Also, you will need the GW package itself. For illustrations of the driver invoke
gw/test/test.gw si
gw/test/test.gw gas
##### Spin-Orbit coupling
lmf solves the scalar Dirac hamiltonian. The dominant difference between the full Dirac hamiltonian and the scalar one is the spin orbit coupling, which can be added as a term $\lambda L \cdot S$ to the scalar Dirac Hamiltonian.
Starting with v6.15, lmf can add λL·S to the scalar Dirac hamiltonian (courtesy of A. Chantis). It is possible to include the full $L \cdot S$ or just the $L_z\cdot S_z$ part. Beginning with v7.9, $L_z\cdot S_z + (L \cdot S - L_z \cdot S_z)$ can be added where the last term is treated in an approximate manner. The approximation turns out to be rather good. See here for some description and analysis of all three approximations. $L \cdot S$ in all three forms can be combined with the self-energy read by a QSGW calculation. In the $L_z\cdot S_z$ and approximate $L \cdot S$ forms the effect of $S_O$ coupling can be passed through to the GW code, to include its effect on the self-energy.
Use HAM_SO to turn on SO coupling.
For illustrations of all three kinds of approaches, try
fp/test/test.fp felz
fp/test/test.fp gasls
##### Local orbitals
In v6.12 and later, local orbitals may be added to the basis set. These orbitals are important when energy bands over a very wide energy window are required, when high accuracy is needed for shallow (semi-)core states, or for energies far above the Fermi energy. Examples of the former occur in oxides: bond lengths are small and cations with shallow p orbitals extend somewhat beyond the augmentation radius.
Local orbitals are presented in one of two flavors. The first, conventional type of local orbital is constructed by solving the radial Schrödinger at a different linearization energy than the usual valence states, and then subtracting off a particular linear of the valence wave function $\phi$ and energy derivative $\dot{\phi}$ such that the local orbital’s value and slope vanish at the augmentation radius. Thus
• A local orbital is strictly confined to the augmentation sphere, and has no envelope function at all.
• When taken in linear combination with the valence augmentation functions, it can solve the Schrödinger equation exactly for linearization the energy chosen (that is, for the spherical potential that defines the wave functions).
• It turns the linear method into a quadratic one. Of course, it would be possible, and more accurate, to construct a LMTO orbital of a different principal quantum number complete with envelope function; however, there is a corresponding loss in efficiency because the additional matrix elements of the envelope function must be evaluated.
• for states with energies in the vicinity of the local orbital energy, the envelope functions of the regular valence states combine with the local orbital to make the interstitial part of the wave function.
Examples that illustrate local orbitals of this type are
fp/test/test.fp gas
fp/test/test.fp cu
The Ga 3d semi-core the high-lying As 5s state are included as local orbitals. In the Cu case, the high-lying Cu 4d is included, which is important in GW calculations.
The second, extended, kind of local orbital can only be used for semi-core states. Instead of artificially subtracting off some linear of the phi and phi-dot to make the orbital vanish at the augmentation radius, a smooth Hankel tail is attached to the orbital. The smoothing of tail is constructed to match as well as possible the kinetic energy of the semi-core state. This type of orbital has the advantage that the valence envelope function need not “carry” the tail of the semi-core state. Its drawback is that more things can “go wrong,” namely it may fail to do a good job of fitting the kinetic energy.
An example that illustrates the second kind of local orbital is
fp/test/test.fp srtio3
The Sr 4p and Ti 3p semi-core states are included as local orbitals. In the first part of the test, they are included as local orbitals of the first type; then the last step is recalculated using local orbitals of the second type.
##### Floating Orbitals
In v6.15 and later, floating orbitals may be added to the basis set. These orbitals can be important when very accurate calculations are needed in open systems, e.g. when reliable energy bands are needed for a wide energy window. These orbitals differ from the usual smooth Hankels in that they are not centered at an atom. They are augmented just as the other orbitals, but there is no augmentation radius and thus no “head” sphere.
The following illustrate the inclusion of floating orbitals in the basis:
fp/test/test.fp te
fp/test/test.fp gaslc
##### Augmented Plane Waves
Starting with v6.17, Augmented Plane Waves can also be included in the basis. They play the same role as floating orbitals do, but APWs are superior because they are simpler to use (there are no parameters and they do not need to be located at any particular site), and the control over convergence is more systematic.
The following illustrate the inclusion of APWs in the basis:
fp/test/test.fp te
fp/test/test.fp srtio3
fp/test/test.fp felz 4
##### Parallel Implementation
Two separate parallel versions of lmf have been made (courtesy of A. T. Paxton). One parallelizes over k-points, and is the most efficient for scaling; the other parallelizes many points at a lower level. | {"extraction_info": {"found_math": true, "script_math_tex": 10, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8307732939720154, "perplexity": 1620.2010302578728}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948515311.25/warc/CC-MAIN-20171212075935-20171212095935-00386.warc.gz"} |
http://viavca.in2p3.fr/2010c_o_s_m_o_v_i_a__forum_sd24fsdf4zerfzef4ze5f4dsq34sdteerui45788789745rt7yr68t4y54865h45g4hfg56h45df4h86d48h48t7uertujirjtiorjhuiofgrdsqgxcvfghfg5h40yhuyir/viewtopic.php?f=9&t=3717&sid=f60ae6cc6d17389cc7a2d51c119cc449&p=6229 | ## F-L. Julie: Gravitational radiation from BBH in ESGB gravity
Gravitational radiation can provide important information, when other means of observations are impossible. Cosmological and astrophysical sources for gravitational waves, related with astroparticle physics, news on the development of gravitational wave experiments are the topic of discussions here.
Moderator: Maxim Khlopov
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Only topics, specified in the description of this forum can be posted here. Other topics will be either removed or moved to an appropriate forum.
### F-L. Julie: Gravitational radiation from BBH in ESGB gravity
VIA APC Theory Seminar of Felix-Louis Julie "Gravitational radiation from a binary black hole coalescence in Einstein-scalar-Gauss-Bonnet gravity" took place on 07.04.2020. His presentation is attached and the record is in VIA library
http://viavca.in2p3.fr/felix_louis_julie.html
The following questions were adressed in discussion during this talk
Yan LIU: may I ask what's this \epsilon<<1 mean?
Dani: Is it obvious why A is of order epsilon^2 and phi is linear in epsilon?
Geoffrey: Why the Gauss-Bonnet term does not contribute to the mass?
O.M. Lecian: why can You say that the pole could be the sign of a naked singularity?'
O.M. Lecian: in which case (for the calculation of what PN order) is there the necessity for a large range of redshifts?
Geoffrey: How do you know that the relation between H and H_EOB is quadratic for arbitrary theories such as Gauss-Bonnet here?
Houri Ziaeepour: In observations of GW we ignore mass of binaryies. Can degeneracy of parameters smear Gauss-Bonnet or other Mod. GR models?
You are welcome to continue discussion in replies to this post
Attachments
VIA_presentation_FLJfin.pdf
Presentation by Felix-Luis Julie
[i]Хлопов Максим Юрьевич
Maxim Khlopov[/i]
Maxim Khlopov | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8401007056236267, "perplexity": 3912.3547617165837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949701.0/warc/CC-MAIN-20230401032604-20230401062604-00508.warc.gz"} |
http://philpapers.org/s/%20finite | Search results for 'finite' (try it on Scholar)
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Bibliography: The Infinite in Philosophy of MathematicsBibliography: Infinitesimals and Probability in Philosophy of ProbabilityBibliography: Infinite Decision Theory in Philosophy of ActionBibliography: Indefinite Descriptions in Philosophy of Language
1. David J. Chalmers (1996). Does a Rock Implement Every Finite-State Automaton? Synthese 108 (3):309-33.score: 18.0
Hilary Putnam has argued that computational functionalism cannot serve as a foundation for the study of the mind, as every ordinary open physical system implements every finite-state automaton. I argue that Putnam's argument fails, but that it points out the need for a better understanding of the bridge between the theory of computation and the theory of physical systems: the relation of implementation. It also raises questions about the class of automata that can serve as a basis for understanding (...)
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2. Panu Raatikainen (2000). The Concept of Truth in a Finite Universe. Journal of Philosophical Logic 29 (6):617-633.score: 18.0
The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.
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3. Cédric Dégremont & Nina Gierasimczuk (2011). Finite Identification From the Viewpoint of Epistemic Update. Information And Computation 209 (3):383-396.score: 18.0
Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of (...)
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4. Raymond J. Nelson (1975). Behaviorism, Finite Automata, and Stimulus-Response Theory. Theory and Decision 6 (August):249-67.score: 18.0
In this paper it is argued that certain stimulus-response learning models which are adequate to represent finite automata (acceptors) are not adequate to represent noninitial state input-output automata (transducers). This circumstance suggests the question whether or not the behavior of animals if satisfactorily modelled by automata is predictive. It is argued in partial answer that there are automata which can be explained in the sense that their transition and output functions can be described (roughly, Hempel-type covering law explanation) while (...)
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5. Jonas R. Becker Arenhart (2012). Finite Cardinals in Quasi-Set Theory. Studia Logica 100 (3):437-452.score: 18.0
Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to them. According to (...)
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6. Michał Kozak (2009). Distributive Full Lambek Calculus has the Finite Model Property. Studia Logica 91 (2):201 - 216.score: 18.0
We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus ( DFL ) whose algebraic semantics is the class of distributive residuated lattices ( DRL ). The problem was left open in [8, 5]. We use the method of nuclei and quasi–embedding in the style of [10, 1].
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7. Eric Rosen (1997). Modal Logic Over Finite Structures. Journal of Logic, Language and Information 6 (4):427-439.score: 18.0
We investigate properties of propositional modal logic over the classof finite structures. In particular, we show that certain knownpreservation theorems remain true over this class. We prove that aclass of finite models is defined by a first-order sentence and closedunder bisimulations if and only if it is definable by a modal formula.We also prove that a class of finite models defined by a modal formulais closed under extensions if and only if it is defined by a -modal (...)
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8. Lauri Hella, Phokion G. Kolaitis & Kerkko Luosto (1996). Almost Everywhere Equivalence of Logics in Finite Model Theory. Bulletin of Symbolic Logic 2 (4):422-443.score: 18.0
We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures (...)
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9. Ehud Hrushovski (2013). On Pseudo-Finite Dimensions. Notre Dame Journal of Formal Logic 54 (3-4):463-495.score: 18.0
We attempt to formulate issues around modularity and Zilber’s trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9].
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10. Jean-Luc Nancy (2003). A Finite Thinking. Stanford University Press.score: 18.0
This book is a rich collection of philosophical essays radically interrogating key notions and preoccupations of the phenomenological tradition. While using Heidegger’s Being and Time as its permanent point of reference and dispute, this collection also confronts other important philosophers, such as Kant, Nietzsche, and Derrida. The projects of these pivotal thinkers of finitude are relentlessly pushed to their extreme, with respect both to their unexpected horizons and to their as yet unexplored analytical potential. A Finite Thinking shows that, (...)
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11. Jay Newhard (2004). Disquotationalism, Minimalism, and the Finite Minimal Theory. Canadian Journal of Philosophy 34 (1):61 - 86.score: 18.0
Recently, Paul Horwich has developed the minimalist theory of truth, according to which the truth predicate does not express a substantive property, though it may be used as a grammatical expedient. Minimalism shares these claims with Quine’s disquotationalism; it differs from disquotationalism primarily in holding that truth-bearers are propositions, rather than sentences. Despite potential ontological worries, allowing that propositions bear truth gives Horwich a prima facie response to several important objections to disquotationalism. In section I of this paper, disquotationalism is (...)
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12. score: 18.0
A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in (...)
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13. Stan Gudder (2006). Quantum Mechanics on Finite Groups. Foundations of Physics 36 (8):1160-1192.score: 18.0
Although a few new results are presented, this is mainly a review article on the relationship between finite-dimensional quantum mechanics and finite groups. The main motivation for this discussion is the hidden subgroup problem of quantum computation theory. A unifying role is played by a mathematical structure that we call a Hilbert *-algebra. After reviewing material on unitary representations of finite groups we discuss a generalized quantum Fourier transform. We close with a presentation concerning position-momentum measurements in (...)
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14. M. Krynicki & K. Zdanowski (2005). Theories of Arithmetics in Finite Models. Journal of Symbolic Logic 70 (1):1-28.score: 18.0
We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in finite (...)
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15. Ross Willard (2000). A Finite Basis Theorem for Residually Finite, Congruence Meet-Semidistributive Varieties. Journal of Symbolic Logic 65 (1):187-200.score: 18.0
We derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given $m and a finite algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less than m. Direct download (6 more) My bibliography Export citation 16. Pierre Cartier (2012). How to Take Advantage of the Blur Between the Finite and the Infinite. Logica Universalis 6 (1-2):217-226.score: 18.0 In this paper is presented and discussed the notion of true finite by opposition to the notion of theoretical finite. Examples from mathematics and physics are given. Fermat’s infinite descent principle is challenged. Direct download (4 more) My bibliography Export citation 17. Georg Gottlob (1997). Relativized Logspace and Generalized Quantifiers Over Finite Ordered Structures. Journal of Symbolic Logic 62 (2):545-574.score: 18.0 We here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized to an oracle in C. We show that this is not (...) Direct download (6 more) My bibliography Export citation 18. Sven Ove Hansson (2012). Finite Contractions on Infinite Belief Sets. Studia Logica 100 (5):907-920.score: 18.0 Contractions on belief sets that have no finite representation cannot be finite in the sense that only a finite number of sentences is removed. However, such contractions can be delimited so that the actual change takes place in a logically isolated, finite-based part of the belief set. A construction that answers to this principle is introduced, and is axiomatically characterized. It turns out to coincide with specified meet contraction. Direct download (5 more) My bibliography Export citation 19. Ian Hodkinson & Martin Otto (2003). Finite Conformal Hypergraph Covers and Gaifman Cliques in Finite Structures. Bulletin of Symbolic Logic 9 (3):387-405.score: 18.0 We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques-thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a (...) conformal hypergraph. In terms of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF. (shrink) Direct download (8 more) My bibliography Export citation 20. Mauro Gattari (2005). Finite and Physical Modalities. Notre Dame Journal of Formal Logic 46 (4):425-437.score: 18.0 The logic Kf of the modalities of finite, devised to capture the notion of 'there exists a finite number of accessible worlds such that . . . is true', was introduced and axiomatized by Fattorosi. In this paper we enrich the logical framework of Kf: we give consistency properties and a tableau system (which yields the decidability) explicitly designed for Kf, and we introduce a shorter and more natural axiomatization. Moreover, we show the strong and suggestive relationship between (...) Direct download (4 more) My bibliography Export citation 21. Merrie Bergmann (2005). Finite Tree Property for First-Order Logic with Identity and Functions. Notre Dame Journal of Formal Logic 46 (2):173-180.score: 18.0 The typical rules for truth-trees for first-order logic without functions can fail to generate finite branches for formulas that have finite models–the rule set fails to have the finite tree property. In 1984 Boolos showed that a new rule set proposed by Burgess does have this property. In this paper we address a similar problem with the typical rule set for first-order logic with identity and functions, proposing a new rule set that does have the finite (...) Direct download (4 more) My bibliography Export citation 22. Gerd Sebald (2011). Crossing the Finite Provinces of Meaning. Experience and Metaphor. Human Studies 34 (4):341-352.score: 18.0 Schutz’s references to literature and arts in his theoretical works are manifold. But literature and theory are both a certain kind of a finite province of meaning, that means they are not easily accessible from the paramount reality of everyday life. Now there is another kind of referring to literature: metaphorizing it. Using it, as may be said with Lakoff and Johnson, to understand and to experience one kind of thing in terms of another. Literally metapherein means “to carry (...) No categories Direct download (7 more) My bibliography Export citation 23. Ross Willard (1994). Hereditary Undecidability of Some Theories of Finite Structures. Journal of Symbolic Logic 59 (4):1254-1262.score: 18.0 Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms. Direct download (6 more) My bibliography Export citation 24. M. Carmen Sánchez (1998). Rational Choice on Non-Finite Sets by Means of Expansion-Contraction Axioms. Theory and Decision 45 (1):1-17.score: 18.0 The rationalization of a choice function, in terms of assumptions that involve expansion or contraction properties of the feasible set, over non-finite sets is analyzed. Schwartz's results (1976), stated in the finite case, are extended to this more general framework. Moreover, a characterization result when continuity conditions are imposed on the choice function, as well as on the binary relation that rationalizes it, is presented. No categories Direct download (4 more) My bibliography Export citation 25. C. J. van Alten (2005). The Finite Model Property for Knotted Extensions of Propositional Linear Logic. Journal of Symbolic Logic 70 (1):84-98.score: 18.0 The logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: γ, xn → y / γ, xm → y. It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the (...) Direct download (3 more) My bibliography Export citation 26. score: 18.0 . The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the (...) Direct download (5 more) My bibliography Export citation 27. Radosav Dordević, Miodrag Rašković & Zoran Ognjanović (2004). Completeness Theorem for Propositional Probabilistic Models Whose Measures Have Only Finite Ranges. Archive for Mathematical Logic 43 (4):557-563.score: 18.0 A propositional logic is defined which in addition to propositional language contains a list of probabilistic operators of the form P ≥s (with the intended meaning ‘‘the probability is at least s’’). The axioms and rules syntactically determine that ranges of probabilities in the corresponding models are always finite. The completeness theorem is proved. It is shown that completeness cannot be generalized to arbitrary theories. No categories Direct download (3 more) My bibliography Export citation 28. Olivier Finkel (2008). Topological Complexity of Locally Finite Ω-Languages. Archive for Mathematical Logic 47 (6):625-651.score: 18.0 Locally finite omega languages were introduced by Ressayre [Formal languages defined by the underlying structure of their words. J Symb Log 53(4):1009–1026, 1988]. These languages are defined by local sentences and extend ω-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ω-languages are analytic sets, the class LOC ω of locally finite ω-languages meets all finite levels of the Borel hierarchy and there exist some locally (...) No categories Direct download (4 more) My bibliography Export citation 29. Murdoch J. Gabbay (2012). Finite and Infinite Support in Nominal Algebra and Logic: Nominal Completeness Theorems for Free. Journal of Symbolic Logic 77 (3):828-852.score: 18.0 By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atoms-abstraction. The results are of interest in their (...) Direct download (5 more) My bibliography Export citation 30. score: 18.0 This book gives a comprehensive overview of central themes of finite model theory â expressive power, descriptive complexity, and zero-one laws â together with selected applications relating to database theory and artificial intelligence, especially constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, emphasizing the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of and hierarchies within first-order, second-order, fixed-point, and (...) Translate to English | Direct download (2 more) My bibliography Export citation 31. Xingxing He, Jun Liu, Yang Xu, Luis Martínez & Da Ruan (2012). On Α-Satisfiability and its Α-Lock Resolution in a Finite Lattice-Valued Propositional Logic. Logic Journal of the Igpl 20 (3):579-588.score: 18.0 Automated reasoning issues are addressed for a finite lattice-valued propositional logic LnP(X) with truth-values in a finite lattice-valued logical algebraic structure—lattice implication algebra. We investigate extended strategies and rules from classical logic to LnP(X) to simplify the procedure in the semantic level for testing the satisfiability of formulas in LnP(X) at a certain truth-value level α (α-satisfiability) while keeping the role of truth constant formula played in LnP(X). We propose a lock resolution method at a certain truth-value level (...) No categories Direct download My bibliography Export citation 32. Laurence Kirby (2008). A Hierarchy of Hereditarily Finite Sets. Archive for Mathematical Logic 47 (2):143-157.score: 18.0 This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy. No categories Direct download (3 more) My bibliography Export citation 33. John Krueger (2014). Strongly Adequate Sets and Adding a Club with Finite Conditions. Archive for Mathematical Logic 53 (1-2):119-136.score: 18.0 We continue the study of adequate sets which we began in (Krueger in Forcing with adequate sets of models as side conditions) by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω 2 with finite conditions, thereby showing that a version of the forcing posets of Friedman (Set theory: Centre (...) No categories Direct download (3 more) My bibliography Export citation 34. Arnold W. Miller (2011). A Dedekind Finite Borel Set. Archive for Mathematical Logic 50 (1-2):1-17.score: 18.0 In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if${B\subseteq 2^\omega}$is a G δσ -set then either B is countable or B contains a perfect subset. Second, we prove that if 2 ω is the countable union of countable sets, then there exists an F σδ set${C\subseteq 2^\omega}$such that C is uncountable but contains no perfect subset. Finally, (...) No categories Direct download (4 more) My bibliography Export citation 35. Randolph Sloof (2004). Finite Horizon Bargaining With Outside Options And Threat Points. Theory and Decision 57 (2):109-142.score: 18.0 We characterize equilibrium behavior in a finite horizon multiple-pie alternating offer bargaining game in which both agents have outside options and threat points. In contrast to the infinite horizon case the strength of the threat to delay agreement is non-stationary and decreases over time. Typically the delay threat determines equilibrium proposals in early periods, while the threat to opt out characterizes those in later ones. Owing to this non-stationarity both threats may appear in the equilibrium shares immediately agreed upon (...) No categories Direct download (4 more) My bibliography Export citation 36. Matthew Smedberg (2013). A Dense Family of Well-Behaved Finite Monogenerated Left-Distributive Groupoids. Archive for Mathematical Logic 52 (3-4):377-402.score: 18.0 We construct a family$\fancyscript{F}$, indexed by five integer parameters, of finite monogenerated left-distributive (LD) groupoids with the property that every finite monogenerated LD groupoid is a quotient of a member of$\fancyscript{F}$. The combinatorial abundance of finite monogenerated LD groupoids is encoded in the congruence lattices of the groupoids$\fancyscript{F}$, which we show to be extremely large. No categories Direct download (3 more) My bibliography Export citation 37. Frank Wagner (2001). Fields of Finite Morley Rank. Journal of Symbolic Logic 66 (2):703-706.score: 18.0 If K is a field of finite Morley rank, then for any parameter set$A \subseteq K^{eq}$the prime model over A is equal to the model-theoretic algebraic closure of A. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl($\emptyset\$ ).
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38. score: 16.0
In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no (...)
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39. Takahito Aoto & Hiroyuki Shirasu (1999). On the Finite Model Property of Intuitionistic Modal Logics Over MIPC. Mathematical Logic Quarterly 45 (4):435-448.score: 15.0
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40. J.‐M. Brochet (1993). The Finite Cutset Property. Mathematical Logic Quarterly 39 (1):158-164.score: 15.0
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41. Wojciech Buszkowski (2002). Finite Models of Some Substructural Logics. Mathematical Logic Quarterly 48 (1):63-72.score: 15.0
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42. Anuj Dawar, Kees Doets, Steven Lindell & Scott Weinstein (1998). Elementary Properties of the Finite Ranks. Mathematical Logic Quarterly 44 (3):349-353.score: 15.0
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43. Peter C. Fishburn (1990). Unique Nontransitive Measurement on Finite Sets. Theory and Decision 28 (1):21-46.score: 15.0
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44. Martin Grohe (1996). Some Remarks on Finite Löwenheim‐Skolem Theorems. Mathematical Logic Quarterly 42 (1):569-571.score: 15.0
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45. Frieder Haug (1994). On Preservation of Stability for Finite Extensions of Abelian Groups. Mathematical Logic Quarterly 40 (1):14-26.score: 15.0
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46. Shawn Hedman & Wai Yan Pong (2011). Quantifier-Eliminable Locally Finite Graphs. Mathematical Logic Quarterly 57 (2):180-185.score: 15.0
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47. B. Herrmann & W. Rautenberg (1992). Finite Replacement and Finite Hilbert‐Style Axiomatizability. Mathematical Logic Quarterly 38 (1):327-344.score: 15.0
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48. Ai‐ni Hsieh & James G. Raftery (2006). A Finite Model Property for RMImin. Mathematical Logic Quarterly 52 (6):602-612.score: 15.0
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49. Laurence Kirby (2010). Substandard Models of Finite Set Theory. Mathematical Logic Quarterly 56 (6):631-642.score: 15.0
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https://astrobites.org/2015/05/29/circumplanetary-disks-soon/ | # Circumplanetary disks… soon!
### Circumplanetary disks – signposts of planet formation
Disks seem to be ubiquitous in astrophysical literature. Everywhere you look, disks, disks, disks. From galactic disks, protostellar disks, protoplanetary disks, debris disks, etc. And now, circumplanetary disks (CPD)! These are thought to form as a side effect of planet formation, when massive planets of ~ Jupiter size are still embedded in a massive protoplanetary disk. When these forming planets accumulate enough mass (~ 1 Jupiter mass), they evacuate the space around them and open up a gap, free of gaseous material. The protoplanetary disk is then separated into two distinct parts, which are by then only connected by the planetary wake (the gas stream around the planet, connecting the inner and outer disk, centered on the planet, see Figure 1), which still enables the planet to draw some material from the outer part. This material then falls down to the young planet and forms a CPD around it. So far, so good. Unfortunately, as the author’s of today’s paper nicely state, the CPDs have “eluded unambiguous detection so far”. This is mainly because of the fact that the emission of CPDs is very weak in comparison with the emission from the surrounding protoplanetary disk. However, this might have changed in the meantime…
### How we think they work
With the rise of ALMA, a new and very powerful interferometer in the Chilean desert, things change. In their work, Perez et al. try to assess whether it is now finally possible to probe and image these very early stages of planet formation and get a glimpse of a CPD in action. To do so, they run fluid dynamics computer models of the stage of planet formation at which CPDs are expected to form (see Figure 1).
Figure 1: Projected density in the disk at the end of of each simulation of its evolution. The planets in all simulations have formed a CPD, shown by a zoom-in in the upper right corners, which are much more pronounced in the two left runs (isothermal) in comparison with the adiabatic one on the right. The local time of the simulations is given in units of the orbital time of the planet, which means that SPH1 was stopped after 10 orbits and SPH2/3 after 50 orbits of the planet. Source: Perez et al. (2015)
The models differ a bit from each other and deliver different answers to the question. The first (SPH1) is a model with a 1 Jupiter mass planet and an extremely high resolution. The second and third (SPH2/3) feature a 5 Jupiter mass planet. Furthermore, SPH1 and SPH2 are using a locally isothermal equation of state. In principle, this means that the gas does not change its temperature at all, which is equivalent to perfect cooling. Physically speaking, this assumes that when the temperature of the gas would rise because of friction or pressure effects, the energy difference is completely and instantaneously radiated away. The SPH3 simulation features an adiabatic assumption, which means that there is no cooling at all and the temperature can rise very much. They use these two extrema (a very cool CPD, isothermal runs, and a very hot CPD, adiabatic run) and from them calculate how this would look like on the sky through a powerful interferometer like ALMA. This is shown in Figure 2.
Figure 2: Predictions for measuring the emissions from the $^{13}CO$ isotope, based on the performed simulations. The colouring goes from zero (white) to high (black) emission. Each panel is centered on the star (cross) , shows different components of the disk and has the position of the CPD indicated with a circle. As you can see the CPD can be spotted best for the SPH2 run. Source: Perez et al. (2015)
### Can you spot it in REAL data?
As you can see the SPH2 simulation reveals distinct spots at the position of the CPD. A more thorough analysis of the features of the coolest CPD (SPH2 simulation) reveals that for such a model the signal is clear enough that features like this will pop up in data for 1 and 5 Jupiter mass planets for the current capability of ALMA! Of course, there is still work to do to understand the systematics and uncertainties coming from real data. Anyway, the analysis clearly shows that for good circumstances we are well within the detection limits. So, let’s go out and find it…! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8321928977966309, "perplexity": 1143.2646349077481}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578596541.52/warc/CC-MAIN-20190423074936-20190423100936-00131.warc.gz"} |
http://www.affairsguru.com/tag/power-indices-and-surds/ | ## POWER INDICES AND SURDS RULES OR FORMULAS | GOVT EXAMS
POWER INDICES AND SURDS RULES OR FORMULAS Importance : 1 or 2 questions from ‘Surds and Indices’ have essentially been asked in every exam. In order to accuracy in your calculations, you will require complete practice of this chapter. Scope of questions : Asked questions are based on basic concepts, completely arithmetic and without language like to […] | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8579050302505493, "perplexity": 3645.962794259505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573065.17/warc/CC-MAIN-20190917081137-20190917103137-00530.warc.gz"} |
http://tej.canspeak.net/2010/03/he-talks-as-if-he-were-expert-in.html | ## 2010/03/09
### He talks as if he were an expert in economics. - 3/9( 火）
Good morning all,
お送りいたします。現在Italiaの時間は0:20分です。
******************************
We must be ready for success!
******************************
You behave as if you were a star.
あなたは あなたがまるで スターであるかのようにふるまうね.
ように...する' と仮定して話す方法です. 詳しいは 第2部を参考してください.
He talks as if he were an expert in economics.
Have a good day today!
Seq 317 ( unit 15 仮定法 ) : lesson 15 < 'まるで ...であるかのように.. 'の as if + 仮定法 >
1. as if = as though + 仮定法
' as if (= as though) + 仮定法 ' を使います. (いま現実はそうではない) を表すためには
(1) He talks as if he were an expert in economics.
(2) You look as if you had seen a ghost!
(1) 彼はまるで自分が経済専門家であるかのように話す！
(2) あなたはまるで幽霊でもみたかのような顔色だよ！
(2)は以前のことなので仮定法過去完了をつかっています !
You behave as if you were a star.
She looked as though she had never met me before.
as if 代わりに as though を使っても同じ意味です.
as if + 仮定法で, be 動詞は were だが, 会話では wasも使います..
Have a good day today!
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http://billset.blogspot.com/ | ## Thursday, February 25, 2016
### UCI Summer School, part 7: Sacks forcing (Brent Cody)
This lecture will introduce some basic properties of Sacks forcing for uncountable inaccessible cardinals, and examine an Easton support iteration of such forcing.
The Sacks forcing on $\omega$ adds a real of minimal constructibility degree, and crucially satisfies a fusion property. Although this was reviewed in the summer school, I'm going to omit the discussion for this post.
Instead we will start with Sacks forcing on uncountable cardinals, which traces back to Kanamori (1980), where using $\diamond_\kappa$ it was shown that long products and iterations of $\mathrm{Sacks}(\kappa)$ preserve $\kappa^+$.
Definition: We say $p\subseteq 2^{<\kappa}$ is a perfect $\kappa$-tree if:
1. If $s\in p$ and $t\subseteq s$ then $t\in p$.
2. If $\langle s_\alpha:\alpha<\eta\rangle$ is a sequence of nodes in $p$, then $s=\bigcup_{\alpha<\eta} s_\alpha\in p$.
3. For every $s\in p$ there is $t\supset s$ with $t\frown 0, t\frown 1\in p$.
4. Let $\mathrm{Split}(p)=\{s\in p: s\frown 0, s\frown 1\in p\}$. Then for some unique club $C(p)\subseteq \kappa$, we have $$\mathrm{Split}(p)=\{s\in p: \mathrm{length}(s)\in C(p)\}.$$
$\mathrm{Sacks}(\kappa)$ is the poset of perfect $\kappa$-trees ordered by inclusion. We think of the generic subset of $\kappa$ added by $\mathrm{Sacks}(\kappa)$ as the intersection of the trees in the generic filter.
The only surprising thing in the generalization is (4): splitting happens for every node on certain levels, which form a club in $\kappa$.
Exercise: $\mathrm{Sacks}(\kappa)$ is $<\kappa$-closed.
Assume $\kappa>\omega$ is inaccessible. Then $\mathrm{Sacks}(\kappa)$ is $\kappa^{++}$-c.c. We will really only consider this case.
Definition: $\mathrm{Split}_\alpha(p)$ is the set of all nodes $s\in p$ with $\mathrm{length}(s)=\beta_\alpha$, where $\langle \beta_\alpha:\alpha<\kappa\rangle$ is an enumeration of $C(p)$, i.e., the level of $p$ at the $\alpha$th member of $C(p)$.
For $p,q\in \mathrm{Sacks}(\kappa)$, write $p\le_\beta q$ iff $p\le q$ and $\mathrm{Split}_\alpha(p)=\mathrm{Split}_\alpha(q)$ for all $\alpha<\beta$.
A descending sequence $\langle p_\alpha:\alpha<\kappa\rangle$ in $\mathrm{Sacks}(\kappa)$ is a fusion sequence if for all $\alpha<\kappa$, $p_\alpha\le_\alpha p_\alpha$.
Lemma (fusion lemma): If $\langle p_\alpha:\alpha<\kappa\rangle$ is a fusion sequence, then $p=\bigcap_{\alpha<\kappa} p_\alpha$ is a lower bound in $\mathrm{Sacks}(\kappa)$.
Proof: exercise. Hint: show that any node $p$ in the intersection is in a cofinal branch of the intersection.
This important lemma affords us a kind of $\kappa^+$ closure, with the catch that we require more of our decreasing sequence. We can see this in action in the next lemma.
Lemma: $\mathrm{Sacks}(\kappa)$ preserves $\kappa^+$.
Proof: If $\dot{f}$ is the name of a function $\kappa\rightarrow \kappa^+$, then we will find $q\le p$ with $q\Vdash \mathrm{ran}(\dot{f})$ bounded.
Let $p_0=p$. Given $p_\alpha$, for each $s\in \mathrm{Split}_\alpha(p_\alpha)$, let $\bar{r}^s_\alpha\le (p_\alpha)_s$ be such that $\bar{r}^s \Vdash \dot{f}(\alpha)=\eta^s_\alpha$. Here the $(p)_s$ means the subtree of $p$ of nodes compatible with $s$.
Note $\bigcup\{\bar{r}^s_\alpha:s\in \mathrm{Split}_\alpha(p_\alpha)\}$ might not be a condition by the requirement on splitting levels. Let $C=\bigcap \{C(\bar{r}^s_\alpha:s\in \mathrm{Split}_\alpha(p_\alpha)\}$ and thin each $\bar{r}^s_\alpha$ to some $r^s_\alpha\le \bar{r}^s_\alpha$ with $C(r^s_\alpha)=C$.
At limits $\gamma<\kappa$, let $p_\gamma=\bigcap_{\alpha<\gamma} p_\alpha$ by the fusion lemma. This defines a fusion sequence where the limit forces that the range of $f$ is bounded.
Exercise: Suppose ${}^\kappa M\subseteq M$, for an inner model $M$. Suppose $\mathrm{Sacks}(\kappa)\in M\subseteq V$. If $G$ is $V$-generic for $\mathrm{Sacks}(\kappa)$ then ${}^\kappa M[G]\subseteq M[G]$ in $V[G]$.
Note: This holds for $\kappa^+$-c.c. forcing, but $\mathrm{Sacks}(\kappa)$ is not $\kappa^+$-c.c.
Now we will see what happens when we iterate these Sacks forcings with Easton support below, and at, a measurable cardinal $\kappa$. Think of this like a Sacks forcing version of the Kunen-Paris iteration, where we use the nice fusion property to replace the $\gamma^+$ closure of the factors there.
Theorem (Friedman-Thompson 2008): Assume GCH holds. Suppose $\kappa$ is measurable and let $\mathbb{P}$ be the length $\kappa+1$ Easton support iteration with $\mathbb{Q}_\gamma=\mathrm{Sacks}(\gamma)$ (computed in $V^{\mathbb{P}_\gamma}$) for $\gamma\le \kappa$ inaccessible, and $\mathbb{Q}_\gamma$ is trivial forcing otherwise. Then if $G\ast H$ is $V$-generic for $\mathbb{P}= \mathbb{P}_\kappa\ast \dot{\mathbb{Q}}_\kappa$, then every normal ultrapower lifts to $V[G\ast H]$ (and in a particularly interesting way!)
Proof: Let $j:V\rightarrow M$ be a normal ultrapower by $U\in V$. Then $j(\mathbb{P}_\kappa=\mathbb{P}_\kappa\ast \dot{\mathbb{Q}}_\kappa\ast \dot{\mathbb{P}}_{\kappa+1,j(\kappa)}$. We get the actual $\dot{\mathbb{Q}}_\kappa$ factor at the $\kappa$ step by using the $\kappa$ closure of the ultrapower.
Using this closure further, and the last exercise, ${}^\kappa M[G\ast H]\subseteq M[G\ast H]$ in $V[G\ast H]$, so $M[G\ast H] \vDash \dot{\mathbb{P}}_{\kappa+1,j(\kappa)}\textrm{ is }\le \kappa-\textrm{closed.}$ So there are $\kappa^+$ maximal antichains of $\mathbb{P}_{\kappa,j(\kappa)}$ in $M[G][H]$. We can now build as usual a generic $G_{\kappa+1,j(\kappa)}\in V[G\ast H]$ for $\mathbb{P}_{\kappa+1,j(\kappa)}$ over $M[G\ast H]$. Lift to $j:V[G]\rightarrow M[j(G)]$.
Now we have to lift $j$ through $\mathbb{Q}_\kappa=\mathrm{Sacks}(\kappa)$. Using the Silver method, $jH$ has size $\kappa^+$, but the target model $M[j(G)]$ does not have this much closure.
The crucial point is to just take $t:=\bigcap jH$. We claim that $t$ is a "tuning fork": by this we mean that $t$ consists of a single branch up to the level $\kappa$, at which point it splits into two branch which are cofinal (and that's everything in $t$).
1. The function $f:\kappa\rightarrow 2$ determined by $H$ is in $t$, and this is everything in $t$ below $\kappa$.
2. Every condition in $jH$ splits at $\kappa$ since for each $p\in H$, $p$ splits at club many levels below $\kappa$, and therefore $j(p)$ splits at level $\kappa$. Therefore, $f\frown 0,f\frown 1\in t$.
3. Since $H$ is a filter, $t$ is cofinal in $j(\kappa)$.
4. We will argue that $t$ does not split anywhere else. Given a club $C\subseteq \kappa$, $D_C:=\{p\in \mathrm{Sacks}(\kappa): C(p)\subseteq C\}$ is dense. So there must be $p_C\in H$ so that $C(p_C)\subseteq C$. Now we have:
Claim: $X=\bigcap \{j(C):C\subseteq \kappa \textrm{ club in } V[G]\}=\{\kappa\}$.
Proof of Claim: Clearly $\kappa\in X$. For the other inclusion, suppose $\alpha\in X$, $\alpha>\kappa$. Then choose $f:\kappa\rightarrow \kappa$, $f\in V[G]$ so that $j(f)(\kappa)=\alpha$. Then let $C_f=\{\nu<\kappa: f\nu\subseteq \nu\}$ is club, but $\alpha\not\in j(C_f)$ since $\alpha$ is not a closure point of $j(f)$ ($\kappa<\alpha$ maps to $\alpha$). This proves the claim.
Let $t_0, t_1$ be the leftmost and rightmost branches through $t$, respectively. Let $K_0=\{p\in j(\mathbb{Q}_\kappa):t_0\subseteq p\}$. Clearly $jH\subseteq K_0$.
It remains to show that $K_0$ is $M[j(G)]$-generic for $j(\mathbb{Q}_\kappa)$. Let $D$ be a dense open subset of $j(\mathbb{Q}_\kappa$ in $M[j(G)]$. Then there is a sequence $\vec{D}=\langle D_\alpha:\alpha<\kappa\rangle \in V[G]$ such that $j(\vec{D})_\kappa=D$, where each $D_\alpha$ is a dense open subset of $\mathbb{Q}_\kappa$.
Claim: Every condition $p\in \mathrm{Sacks}(\kappa)$ can be extended to $q_\infty \le p$ so that for every $\alpha<\kappa$ there is $\beta<\kappa$ so that for any node $s\in \mathrm{Split}_\beta(q_\infty)$, the condition $(q_\infty)_s$ meets $D_\alpha$.
Proof of Claim: exercise, a fusion argument.
Let $q_\infty\in H$ be as in the claim, using genericity of $H$. By elementarity, $j(q_\infty)$ has the property that at some splitting level of $j(q_\infty)$, say $\beta<j(\kappa)$, any node $s\in \mathrm{Split}_\beta(j(q_\infty))$ is such that $(j(q_\infty))_s$ meets $D$. Now we can just take $s$ to be $t_0\upharpoonright \delta_\beta$, where $\delta_\beta$ is the $\beta$th splitting level of $j(q_\infty)$.
Therefore $K_0$ is generic as claimed, and it is in $V[G\ast H]$, so $j$ lifts to
$$j:V[G\ast H]\rightarrow M[j(G)\ast j(H)].$$
## Friday, February 19, 2016
### The rearrangement inequality is everywhere
Recently, I've been talking with John Susice about some elementary Olympiad-style problems. I told him that in high school I was taught that virtually every inequality problem that appeared in this setting follows from the Rearrangement Inequality. This states that if $x_1\le \cdots\le x_n$ and $y_1\le \cdots\le y_n$ are real numbers, then the expression
$$x_1 y_{\sigma(1)}+\cdots+x_n y_{\sigma(n)}$$
for $\sigma$ a permutation on $[n]$ is maximized when $\sigma$ is the identity permutation and minimized when $\sigma$ is the reversing permutation. To me, this neatly isolates a useful and general principle that seems kind of obvious in hindsight.
The proof of the $n=2$ case of the inequality, which is all I'll need below, is just to expand $(x_2-x_1)(y_2-y_1)\ge 0$. This case also implies the AM-GM inequality (taking $x_1=y_1=\sqrt{a}$ and $x_2=y_2=\sqrt{b}$).
Now sometime later, John told me an interesting problem about factoring numbers. He had a solution using a trick similar to Euler's product, but for me this was a chance to test my thesis about the rearrangement inequality:
Problem: Prove that every natural number $n$ has more factors congruent to 1 (mod 4) than factors congruent to 3 (mod 4).
Solution: By induction. Clearly this holds for 1 and for primes. Now suppose $n$ is composite, so write $n=pq$ where $p,q<n$. For any integer $k$, let $r_k$ denote the number of factors it has which are congruent to 1 (mod 4) and $s_k$ the number of factors congruent to 3 (mod 4).
The product of two 1 (mod 4) numbers is still 1 (mod 4), and the product of two 3 (mod 4) numbers is 3 (mod 4). On the other hand, the product of a 1 (mod 4) and a 3 (mod 4) is a 3 (mod 4), and even factors can never be 1 or 3 (mod 4). This proves that $r_n=r_pr_q+s_ps_q$, which must be greater than $s_n=r_ps_q+r_qs_p$ by the rearrangement inequality!
### UCI Summer School, part 6: the number of normal measures (Brent Cody)
Sorry for the delay, loyal readers. Here is the beginning of Brent's part of the summer school.
Assume that is consistent that there is a measurable cardinal.
Question: How many normal measures can a measurable cardinal carry?
Let $\mathrm{NM}(\kappa)$ denote the set of normal measures on $\kappa$. Trivially, $\#\mathrm{NM}(\kappa)\le 2^{2^\kappa}$.
One interesting case happens in a canonical inner model.
Theorem (Kunen '71): In $L[U]$, there is exactly one normal measure (on $\kappa$, the unique measurable cardinal).
We can also realize the other extreme:
Theorem (Kunen-Paris '71): There is a forcing extension in which $\#\mathrm{NM}(\kappa)=2^{2^\kappa}$.
We will prove this one later today. In the middle, we have
Theorem (Mitchell '74): It is consistent relative to a measurable $\kappa$ of order $\delta$ that $\#\mathrm{NM}(\kappa)=2^{2^\kappa}$.
And for one case, we can lower the large cardinal assumption used.
Theorem (Apter-Cummings-Hamkins '07): It is consistent relative to a measurable cardinal that that $\#\mathrm{NM}(\kappa)=\kappa^+$.
Finally, we can do it in all cases.
Theorem (Friedman-Magidor '09): Assume GCH. Suppose $\kappa$ is measurable and let $\mu\le \kappa^{++}$ be a cardinal. Then in a cofinality-preserving forcing extension, $\#\mathrm{NM}(\kappa)=\mu$.
The goal eventually will be to show the proof of this result.
Lemma: Suppose $j:V\rightarrow M$ is the ultrapower by a normal measure on $\kappa$. Let $G$ be $V$-generic for $\mathbb{P}$. Assume that in $V[G]$,
$$j_0:V[G]\rightarrow M[j_0(G)]$$
and
$$j_1:V[G]\rightarrow M[j_1(G)]$$
are elementary embeddings extending $j$. Then the following are equivalent:
1) $j_0=j_1$.
2) $j_0(G)=j_1(G)$.
3) The normal measure $U_0$ derived from $j_0$ is equal to the normal measure $U_1$ derived from $j_1$.
Proof: Exercise.
Exercise (Levy-Solovay): Show that every normal measure extends uniquely to a normal measure in any forcing extension by small forcing.
As promised, we will now prove the Kunen-Paris Theorem.
Theorem (Kunen-Paris '71): There is a forcing extension in which $\#\mathrm{NM}(\kappa)=2^{2^\kappa}$.
Proof: By a preparation forcing if necessary, assume that $2^\kappa=\kappa^+$. Let $\mathbb{P}$ be the length $\kappa+1$ Easton support iteration that forces at cardinal stages $\gamma\le \kappa$ with $\mathbb{Q}_\gamma:=\mathrm{Add}(\gamma^+,1)$ (computed in the extension by $\mathbb{P}_\gamma$), trivial forcing at other stages. This is a standard way of forcing the GCH to hold below $\kappa$.
Let $G\ast H$ be $V$-generic for $\mathbb{P}=\mathbb{P}_\kappa\ast \dot{\mathbb{Q}}_\kappa$. Let $j:V\rightarrow M$ be the ultrapower by a normal measure. Factor $j(\mathbb{P}_\kappa)\simeq \mathbb{P}_\kappa \ast \dot{\mathbb{P}}_{\kappa,j(\kappa)}$.
Since ${}^\kappa M[G]\subseteq M[G]$ in $V[G]$ (a name exercise which uses closure under $\kappa$-sequences of $M$ in $V$), we have that in $M[G]$, $\mathbb{P}_{\kappa,j(\kappa)}$ is $\le \kappa$-closed (it is the composition of increasingly closed posets starting with $\mathbb{Q}_\kappa=\mathrm{Add}(\kappa^+,1)$.
Furthermore, computing in $V[G]$, $\mathbb{P}_{\kappa,j(\kappa)}$ has at most $\kappa^+$ maximal antichains inside $M[G]$. since $|j(\kappa)|=\kappa^+$ by our cardinal arithmetic assumption, and $\mathbb{P}_{\kappa,j(\kappa)}$ is $j(\kappa)$-c.c. of size $j(\kappa)$ in $M[G]$. By enumerating the maximal antichains of $M[G]$ in order-type $\kappa^+$, we can meet them one by one, with the closure of the poset in $M[G]$, noticing that closure of the model $M[G]$ gives that the proper initial segments of this enumeration are in $M[G]$.
Since $jG\subseteq G\ast G_{\kappa,j(\kappa)}$, we can lift the embedding to
$$j^+:V[G]\rightarrow M[j(G)].$$
Using the Lemma/Exercise above, different choices of $j(G)$ will give rise to different embeddings which will give different normal measures. By passing to finer antichains, we can assume in the enumeration $\langle A_\alpha:\alpha<\kappa^+\rangle$ that if $i<j$, then $A_j$ (strictly) refines $A_i$. Looking at how we built $G_{\kappa,j(\kappa)}$, we can form the tree of attempts to build the generic, noticing that at each level there are incompatible ways to extend the generic so far to meet the maximal antichain. This gives $2^{\kappa^+}=2^{2^\kappa}$ many different generics, and hence different normal measures. $\Box$
So it's not hard to force many normal measures. It is harder to force so that there are few normal measures.
We will need to use Hamkins's Gap Forcing Theorem which gives a sufficient condition for an ultrapower embedding in a generic extension to be the lift of a ground model embedding.
Definition: A forcing $\mathbb{P}$ admits a closure point at $\delta$ if it factors as $\mathbb{P}\simeq \mathbb{Q}\ast \dot{\mathbb{R}}$ where $\mathbb{Q}$ is nontrivial, $|\mathbb{Q}|\le \delta$, and $\Vdash_{\mathbb{Q}} \mathbb{R} \textrm{ is }<\delta-\textrm{closed}$.
Theorem (Hamkins '01, Gap Forcing Theorem): If $V\subseteq V[G]$ admits a closure point at $\delta$ and $j:V[G]\rightarrow M[j(G)]$ is an ultrapower in $V[G]$ with $\mathrm{crit}(j)>\delta$, then $j\upharpoonright V:V\rightarrow M$ is a definable class in $V$.
Finally, we prove one more of the theorems in the introduction.
Theorem (Apter-Cummings-Hamkins '07): It is consistent relative to a measurable cardinal that that $\#\mathrm{NM}(\kappa)=\kappa^+$.
Idea of the proof: Start with at least $\kappa^+$ normal measures on $\kappa$ (e.g., by Kunen-Paris forcing). Force with $\mathrm{Col}(\kappa^+,2^{2^\kappa})$, and show that no new normal measures are added.
Proof: Again we assume $2^\kappa=\kappa^+$. Start with $\#\mathrm{NM}(\kappa)\ge \kappa^+$. Let $\mathbb{P}=\mathrm{Add}(\omega,1)\ast \dot{\mathrm{Col}}(\kappa^+, 2^{2^\kappa})$. The point of the Cohen forcing is to give $\mathbb{P}$ a closure point below $\kappa$. Suppose $c\ast G\subseteq \mathbb{P}$ is $V$-generic. Then every normal measure in $V$ generates a normal measure in $V[c]$ by Levy-Solovay, and these remain normal measures in $V[c][G]$ since $\mathrm{Col}(\kappa^+, 2^{2^\kappa})$ is $\le \kappa$-closed in $V[c]$. So there are at least $\kappa^+$ normal measures in $V[c][G]$.
To show the other inequality, suppose $U$ is a normal measure on $\kappa$ in $V[c][G]$. Let
$$j:V[c][G]\rightarrow M[c][j(G)]$$
be the ultrapower. By the gap forcing theorem, $j\upharpoonright V:V\rightarrow M$ is a definable class in $V$. So we can lift to $j\upharpoonright V[c]:V[c]\rightarrow M[c]$. We can now define $U$ in $V[c]$, as the derived measure from this embedding. $\Box$
## Tuesday, September 1, 2015
### UCI Summer School, part 5 (Monroe Eskew)
This is just a placeholder, for now. My notes for this part are quite rough, so it will be a while before I will try to record it here. The next installment of these notes will cover Brent Cody's lectures on some results about the number of normal measures.
### UCI Summer School, part 4: Measure algebras (Monroe Eskew)
Here are some more applications of the ideas we have been considering.
Definition: $\mathcal{B}$ is a measure algebra if it is a complete Boolean algebra equipped with some function $\mu:\mathcal{B}\rightarrow [0,1]$ with $\mu(0)=0, \mu(1)=1, \mu(b)>0$ for $b\neq 0$, and $\mu$ countably additive (i.e., if $\langle b_i:i<\omega\rangle$ is an antichain, then $\mu(\sum b_i)=\sum \mu(b_i)$.
Exercise: All measure algebras are c.c.c.
Example: Let $\kappa$ be a cardinal. We will describe a topology on ${}^\kappa 2$. Fix some $x\in [\kappa]^{<\omega}$ and $s:x\rightarrow 2$ (i.e., $s$ is a finite domain partial function from $\kappa$ to $2$). Then basic open sets are of the form $\mathcal{O}_s=\{r\in {}^\kappa 2: \forall \alpha\in x(r(\alpha)=s(\alpha))\}$. Let $\mathcal{B}_\kappa$ be the $\sigma$-algebra generated by these basic open sets, and define $\mu$ on $\mathcal{B}_\kappa$ by setting $\mu(\mathcal{O}_s=\frac{1}{2^{|s|}}$ (standard theorems from real analysis give that $\mu$ extends uniquely to a countably additive probability measure on $\mathcal{B}_\kappa$.
Define $\mathrm{Null}=\{A\in\mathcal{B}_\kappa: \mu(A)=0\}$. Then $\mathcal{R}_\kappa:=\mathcal{B}_\kappa/\mathrm{Null}$ is a measure algebra.
Exercise: Prove that $\mathcal{R}_\kappa$ forces $2^\omega\ge \kappa$.
Exercise: If $\mathcal{A}$ is a measure algebra and $\Vdash_A \dot{\mathcal{B}}$ is a measure algebra, then $\mathrm{r.o.}(\mathcal{A}\ast \dot{\mathcal{B}})$ is a measure algebra. (Note: this is not as easy as it may seem at first since for example $\mathcal{A}$ might even add new reals which can be measures of elements of $\mathcal{B}$! We use r.o. for the Boolean completion here since the letter $\mathcal{B}$ is overloaded).
Exercise: If $\mathcal{A}$ is a complete sublagebra of a measure algebra $\mathcal{B}$, then $\mathcal{A}$ is a measure algebra.
Continuing along this line,
Theorem: If $\mathcal{B}$ is a measure algebra, $\mathcal{A}$ a complete subalgebra of $\mathcal{B}$, and $G\subseteq \mathcal{A}$ is generic over $V$, then in $V[G]$ we have that $\mathcal{B}/G$ is a measure algebra.
Note that in $V[G]$, $G$ is a filter on $\mathcal{B}$, so $\mathcal{B}/G=\{[b]_G:b\in \mathcal{B}\}$. We use $G^*$ for the dual ideal. It's important to distinguish between the orderings of the two Boolean algebras here, and will be good to see how to translate between them using the forcing relation.
Lemma: If $\mathcal{B}$ is complete and $\mathcal{A}$ is a complete subalgebra and $G\subseteq \mathcal{A}$ is generic, then $\mathcal{B}/G$ is complete in $V[G]$.
Proof of Lemma: Suppose $\langle [b_\alpha]_G:\alpha<\kappa\rangle \in P(\mathcal{B}/G)\cap V[G]$. For each $\alpha<\kappa$, let $X_\alpha:=\{b:1\Vdash_{\mathcal{A}} [b]_{\dot{G}}\le [\dot{b}_\alpha]_{\dot{G}}\}$. Let $c_\alpha=\sum X_\alpha\in V$ (meet taken in $\mathcal{B}$).
We claim that $1\Vdash_{\mathcal{A}} [c_\alpha]_{\dot{G}}=[\dot{b}_\alpha]_{\dot{G}}$ for each $\alpha$--this suffices to prove the lemma. The proof of the claim is straightforward but a little tedious. First we show $1\Vdash_{\mathcal{A}} [c_\alpha]_{\dot{G}}\le[\dot{b}_\alpha]_{\dot{G}}$. If this doesn't hold, then there are $d,p$ so that $p\in \mathcal{A}$, $d\wedge p\neq 0$, and
$$p\Vdash [\check{d}]\le [\check{c}_\alpha] \textrm{ and }[\check{d}]\wedge [\dot{b}_\alpha]=0.$$
The first conjunct implies that $p\wedge d\le c_\alpha$ in $\mathcal{B}$, and since $c_\alpha$ is a lub for $X_\alpha$, there is some $b\in X_\alpha$ so that $p\wedge d\wedge b\neq 0$. So $1\Vdash [p\wedge d\wedge b]\le [b_\alpha]$ by the definition of $b\in X_\alpha$. But the second conjunct gives $p\wedge d\wedge b_\alpha=0$, contradiction.
Now to show $1\Vdash_{\mathcal{A}} [c_\alpha]_{\dot{G}}\ge[\dot{b}_\alpha]_{\dot{G}}$, assume for a contradiction that there are $p,a\in \mathcal{A}$ so that $p\Vdash [\check{a}]\le [\dot{b}_\alpha]$ and $p\Vdash [a\wedge \neg c_\alpha]\neq 0$. Now $p$ forces $[p\wedge a \wedge \neg c_\alpha]\le [\dot{b}_\alpha]$. Trivially, $\neg p$ forces $[p\wedge a \wedge \neg c_\alpha]=0$. So it's just outright forced that $[p\wedge a \wedge \neg c_\alpha]\le [b_\alpha]$ and thus $p\wedge a\wedge \neg c_\alpha \in X_\alpha$, which contradicts $c_\alpha$ is an upper bound for $X_\alpha$, completing the proof of the claim and the lemma. $\Box$
Proof of Theorem: Let $\mu$ be a measure on $\mathcal{B}$, $\mathcal{A}$ a complete subalgebra of $\mathcal{B}$. Define
$$\mu(b\mid a)=\frac{\mu(a \wedge b)}{\mu(a)}.$$
(We say the measure of $b$ conditioned on $a$).
Definition: For $a\in A,b\in B, \epsilon>0$, say $a$ is $\epsilon$-stable for $b$ if for all $x\le a$ in $\mathcal{A}$, $|\mu(b\mid x)-\mu(b\mid a)|<\epsilon$.
Lemma: For all $b\in B$ and for all $\epsilon>0$ the set $\{a\in A:a \textrm{ is }\epsilon-\textrm{stable for} b\}$ is dense in $A$.
Proof: Exercise. An interesting one.
In $V[G]$, let $\nu:\mathcal{B}/G\rightarrow [0,1]$ be given by $\nu([b])=r$ if for every $\epsilon>0$ there is some $a\in G$ so that $a$ is $\epsilon$-stable for $b$ and $|\mu(b\mid a)-r|<\epsilon$. The idea is that $G$ could add new reals, so we can only have approximations to the measure of $[b]$ using ground model reals attached to the members of $\mathcal{A}$.
We can check that this is well-defined: if $[b]_G=[c]_G$, then some $a\in G$ forces $b\Delta c\in G^*$. This means that $a\perp (b\Delta c)$, so $\mu(a\wedge (b\Delta c)=0$. Therefore $\mu(b\mid x)=\mu(c\mid x)$ for all $x\le a$. Suppose $r_0\neq r_1$ both satisfy $\nu(b)=r_i$. Take $\epsilon<|r_1-r_0|$. Let $a_0,a_1\in G$ be $\epsilon/4$-stable for $b$ with $|\mu(b\mid a_i)-r_i|<\epsilon/4$. Now take $a\le a_0, a_1$ in $\mathcal{A}$. By a triangle inequality argument, we have $|r_1-r_0|<\epsilon$, a contradiction.
Exercise: Check that $\nu(b)>0$ for all $b\neq_G 0$.
Exercise: Check that $\nu$ is countably additive. First prove that it is finitely additive.
$\Box$.
Now suppose $P(Z)/I$ is a measure algebra. The duality theorem (ccc case) says that for any $\theta$, $\mathcal{R}_\theta\ast P(Z)/\bar{I}\cong P(Z)/I\ast j(\mathcal{R}_\theta)$,
where $j:V\rightarrow M$ is the generic embedding in $V[G]$, $G$ generic for $P(Z)/I$.
The right hand side of this isomorphism is a measure algebra, since $P(Z)/I$ is a measure algebra by assumption, and $j(\mathcal{R}_\theta)$ is a measure algebra of $M$, a model which is closed under countable sequences (and so has all the countable sequences to witness countable additivity and completeness of the Boolean algebra).
We have a map $e:\mathcal{R}_\theta\rightarrow \mathrm{r.o.}(P(Z)/I\ast j(\mathcal{R}_\theta)$, so $\mathcal{R}_\theta$ is isomorphic to a complete subalgebra of the right hand side. Now if $H\subseteq \mathcal{R}_\theta$ is generic, then $B/e''H$ is a measure algebra. Therefore $P(Z)/\bar{I}$ is also a measure algebra.
A real-valued measurable cardinal is a cardinal $\kappa$ which carries a $\kappa$-additive probability measure on all subsets of $\kappa$ which gives measure 0 to singletons. It is atomless if every set of positive measure has a subset of strictly smaller positive measure.
Corollary: If $\langle \kappa_i:i<\theta\rangle$ is a sequence of measurable cardinals, then if $\gamma=\sup \kappa_i$, $\mathcal{R}_\gamma$ forces all $\kappa_i$ to be atomless real-valued measurable cardinals (RVMs).
We note the fact that if $\kappa$ is atomlessly RVM, then $2^\omega\ge \kappa$, so we can't get class many RVMs.
However, if $\kappa$ is strongly compact, then $\mathcal{R}_\kappa$ forces that for all regular $\lambda\ge \kappa$, there is a countably additive real-valued probably measure $\mu_\lambda$ on $\lambda$ giving measure 0 to all subsets of size $<\lambda$.
## Friday, August 7, 2015
### UCI Summer School, part 3: Applications of Duality Theorem (Monroe Eskew)
We now turn towards applications of the duality theorem. It is recommended that the reader recalls the notation ($I,j,K,J,\hat{H},e,\iota$, etc.) from the previous lecture before proceeding.
The basic idea is that one uses the isomorphism there:
$$\mathcal{B}(\dot{\mathbb{P}\ast P(Z)/J})\equiv \mathcal{B}(P(Z)/I\ast j(\mathbb{P})/\dot{K})$$
to calculate the quotient algebra $P(Z)/J$ as $\mathcal{B}(P(Z)/I\ast j(\mathbb{P})/\dot{K})/e''H$.
As discussed near the end of the last lecture, under certain assumptions, the statement of the duality theorem becomes somewhat simpler. The first examples will fall into this case.
Special Case: If $I$ is $\kappa$-complete and $\mathbb{P}$ is $\kappa$-c.c., then the hypothesis of the duality theorem holds, $K=\{0\}$ and $J=\bar{I}$, the ideal generated by $I$ in $V^\mathbb{P}$.
Exercise: Show that in the above case, if $p\in \mathbb{P}$ and $A\in (P(Z)/I)\cap V$, then
$$\iota(p,\check{A})=(A,j(\dot{p})).$$
A further simplification will be that we will usually start with a measurable cardinal $\kappa$ and take $I$ to be the dual to the measure on $\kappa$, so $P(\kappa)/I$ is the trivial Boolean algebra.
A measurable cardinal $\kappa$ has a 2-saturated, $\kappa$-complete ideal, namely the dual to the measure on $\kappa$, and under GCH every cardinal $\kappa$ carries a $\kappa^{++}$-saturated, $\kappa$-complete ideal, namely the ideal of bounded subsets. This motivates the following natural questions, which are the main focus of this lecture:
Question: Suppose $\mu\le \kappa^+$ is a regular cardinal. Is it consistent that there is a cardinal $\kappa$ which is not measurable, but still $\kappa$ carries a $\mu$-saturated, $\kappa$-complete ideal? (Here we want the amount of saturation to be exactly $\mu$).
Digression: does the answer change if we require $\kappa$ to be a successor cardinal?
For the case where $\kappa$ is a successor cardinal, $\kappa^+$-saturation is the strongest we can hope to achieve.
Exercise: Prove using the method of generic ultrapowers that if $\kappa$ is a successor cardinal then there is no $\kappa$-complete, $\kappa$-saturated ideal on $\kappa$.
Kunen showed that if $\kappa$ is a successor cardinal, then getting a $\kappa^+$-saturated ideal on $\kappa$ requires large cardinals much stronger than a measurable, although we won't do this argument here (you can find it in this previous Specinar post. We now return to the original question.
Answer to question 1, if $\mu<\kappa$ ($\mu$ regular): We will use the basic technique of computing the quotient algebra $P(\kappa)/J$ in $V[H]$ using the duality theorem. Start with $\kappa$ measurable in the ground model. Let $\theta\ge \kappa$, and consider $\mathrm{Add}(\mu,\theta)$ which adds $\theta$ Cohen subsets of $\mu$. $\mathrm{Add}(\mu,\theta)$ is $\kappa$-c.c., and under the GCH it is even $\mu^+$-c.c. Let $I=U^*$, where $U$ is a $\kappa$-complete normal ultrafilter on $\kappa$ (here the star means taking the dual ideal). Let $j:V\rightarrow M$ be the ultrapower embedding.
The duality theorem gives the isomorphism:
$$\mathrm{Add}(\mu,\theta)\ast P(\kappa)/\bar{I}\cong P(\kappa)/I\ast \mathrm{Add}(\mu,j(\theta))\cong\mathrm{Add}(\mu,j(\theta)),$$
since $P(\kappa)/I$ is trivial.
If $H$ is generic for $\mathrm{Add}(\mu,\theta)$ over $V$, then
$$e''H=\{(1,j(p)):p\in H\}.$$
So
$$P(\kappa)/\bar{I}\cong \mathrm{Add}(\mu,j(\theta))/e''H\cong \mathrm{Add}(\mu,j(\theta)).$$
In $V[H]$, $P(\kappa)/\bar{I}\cong \mathcal{B}(\mathrm{Add}(\mu,j(\theta))$, so $\bar{I}$ is $(\mu^{<\mu})^+$ saturated. Furthermore, it is easy to check that $\bar{I}$ is $\kappa$-complete, and $2^\mu\ge \kappa$ in $V[H]$, so $\kappa$ is not measurable. This answers question 1 for the case where $\mu<\kappa$. $\Box$
We might ask what large cardinal properties of $\kappa$ are implied by this ideal hypothesis.
Proposition: If $\kappa$ carries a $\kappa$-complete $\mu$-saturated ideal for some $\mu<\kappa$, then:
1. $\kappa$ is weakly Mahlo
2. $\kappa$ has the tree property.
Exercise: Prove (1) of the proposition using generic ultrapowers.
Proof of Proposition (2): Suppose $T$ is a $\kappa$-tree. If $G\subseteq P(\kappa)/I$ is generic, then in $V[G]$, $T$ has a branch $b$ given by taking any member of level $\kappa$ of $j(T)$, where $j$ is the generic ultrapower embedding. Now for each $\alpha<\kappa$, $S_\alpha=\{x\in T_\alpha: \exists p(p\Vdash \check{x}\in \dot{b})\}<\mu$ by the saturation. Now $\bigcap_{\alpha<\kappa} S_\alpha$ is a $\kappa$-tree all of whose levels have size $<\mu<\kappa$. It is well-known (or a good exercise) that such trees have cofinal branches. $\Box$
Definition: An ideal $I$ is nowhere prime if there is no $I$-positive set $A$ so that $I\upharpoonright A$ is prime (i.e., dual to an ultrafilter).
Exercise: Show that if there is a nowhere prime, $\kappa$-complete, $\mu^+$-saturated ideal, where $\mu<\kappa$, then $2^\mu\ge \kappa$.
We continue with Question 1 with other arrangements of $\mu$ and $\kappa$.
Answer to question 1, if $\mu=\kappa^+$: Start with $\kappa$ measurable with $2^\kappa=\kappa^+$, $U$ a normal ultrafilter and $j:V\rightarrow M$ the ultrapower embedding. Let $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha<\kappa\rangle$ be the Easton support iteration where for regular $\alpha$, $\Vdash_{\mathbb{P}_\alpha} \dot{Q}=\mathrm{Add}(\alpha,1)$. It is straightforward to verify that $\mathbb{P}_\kappa$ has the $\kappa$-c.c., so we are in the special case again.
Note that $j(\mathbb{P}_\kappa)=\mathbb{P}_\kappa\ast \mathbb{Add}(\alpha,1)\ast (\mathbb{P}_{\kappa+1,j(\kappa)})^M$ (We use the notation $\mathbb{P}_{\xi,j(\kappa)}$ for $j(\mathbb{P}_\kappa)_{\xi,j(\kappa)}$). The tail part is computed differently in $M$ than in $V$, e.g., the support is on $M$-regular cardinals.
If $G_\kappa\subseteq \mathbb{P}_\kappa$ is generic over $V$, then the special case of the duality theorem says that in $V[G_\kappa]$, $P(\kappa)/\bar{I}\cong (\mathbb{P}_{\kappa,j(\kappa)})^M$. However, $P(\kappa)/\bar{I}\cong(\mathbb{P}_{\kappa,j(\kappa)})^M$ does not have the $\kappa^+$-c.c. since there are $M$-regular cardinals between $\kappa^+$ and $j(\kappa)$. So $\bar{I}$ is not $\kappa^+$-saturated.
Now we could also satisfy the hypothesis of the duality theorem of adding a $j(\mathbb{P}_\kappa)=P(\kappa)/J\cong \mathrm{Add}(\alpha,1)\ast (\mathbb{P}_{\kappa+1,j(\kappa)})^M$ generic filter $\hat{H}$ over $M$ in a different way. By a standard technique, a $j(\mathbb{P})$-generic over $M$ exists in $V[G_{\kappa+1}]$ (where $G_{\kappa+1}$ is $\mathbb{P}_\kappa\ast \mathrm{Add}(\alpha,1)$-generic). This is because we clearly get a generic for the initial part $\mathbb{P}_\kappa\ast \mathrm{Add}(\alpha,1)$, just $G_{\kappa+1}$ itself. For the tail, $(\mathbb{P}_{\kappa+1,j(\kappa)})^{M[G_{\kappa+1}]}$ is is $j(\kappa)$-c.c. of size $j(\kappa)$ in $M[G_{\kappa+1}]$, so $M[G_{\kappa+1}]$ thinks the poset has at most $j(\kappa)$ maximal antichains. In $V[G_{\kappa+1}]$, $|j(\kappa)|=\kappa^+$, and the poset is $\kappa^+$-closed (in $M[G_{\kappa+1}]$, but also in $V[G_{\kappa+1}]$ by the agreement between these models). So we can construct a generic by hand in $V[G_{\kappa+1}]$ by enumerating all of the maximal antichains in $M[G_{\kappa+1}]$. This completes the construction of $\hat{H}$ in the extension by $\mathrm{Add}(\kappa,1)$.
In this construction, we have that $j(\mathbb{P})/K\cong \mathbb{P}_\kappa\ast \mathrm{Add}(\kappa,1)$, since the Boolean algebra homomorphism $j(\mathbb{P}_\kappa)\rightarrow \mathbb{P}_\kappa\ast \mathrm{Add}(\kappa,1)$ given by $p\mapsto \|p\in \hat{H}\|$. has kernel exactly $K$ as defined in the last lecture, and the map is surjective since the codomain completely embeds into the domain. So in the duality theorem calculation, we obtain an ideal $J$ so that $P(\kappa)/J\cong \mathrm{Add}(\kappa,1)$. So $J$ is a $\kappa^+$-saturated ideal on $\kappa$. $\Box$
Note that $\kappa$ is inaccessible.
Exercise: Prove that $\kappa$ is weakly compact in $V[G_\kappa]$. (Hint: use the tree property characterization.)
Exercise: Prove that $\kappa$ is not measurable in $V[G_\kappa]$, but it is measurable in $V[G_{\kappa+1}]$.
Remark: By forcing with $(\mathbb{P}_{\kappa,j(\kappa)})^{M[G_{\kappa+1}]}$ instead of just $\mathrm{Add}(\kappa,1)$ to add the $j(\mathbb{P})$-generic, we can get a nowhere prime $\kappa$ complete $\kappa^+$-saturated ideal on $\kappa$ in $V[G_{\kappa+1}]$.
Answer to question 1, if $\mu=\kappa$: We will find an example so that $\kappa$ is not weakly compact (compare to earlier results for saturation below $\kappa$), and in fact the quotient algebra is isomorphic to a $\kappa$-Suslin tree.
In the exercises, we will describe how to construct, for $\alpha$ regular with $\alpha^{<\alpha}=\alpha$, a forcing $\mathbb{Q}_\alpha$ which adds an $\alpha$-Suslin tree $\dot{T}_\alpha$ so that $\mathcal{B}(\mathbb{Q}_\alpha\ast \dot{T}_\alpha)\cong\mathrm{Add}(\alpha,1)$. This is due to Kunen.
In the construction for $\mu=\kappa^+$ we got a model (there called $V[G_\kappa]$) where there was an inaccessible $\kappa$ and an ideal $J$ on $\kappa$ so that
$$P(\kappa)/J\cong\mathrm{Add}(\kappa,1)\cong \mathbb{Q}_\kappa\ast \dot{T},$$
where $\dot{T}$ is the $\kappa$-Suslin tree added by $\mathbb{Q}_\kappa$.
Now start with this to be our ground model $V$. Let $H\subseteq \mathbb{Q}_\kappa$ be generic. We want to show that in $V[H]$, there is an ideal $J_1$ on $\kappa$ so that $P(\kappa)/J_1\cong T$.
If $G\subseteq \mathrm{Add}(\kappa,1)$ is generic over $V$, then take in $V[G]$ an embedding
$$j:V\rightarrow M$$
which was constructed before. We want to extend the embedding to $V[H]$.
Now $G\in M$ since $M$ is closed under $\kappa$-sequences in $V[G]$. We can extend $j$ to $V[G]$ by constructing a generic $\hat{G}$ for $\mathrm{Add}(j(\kappa))^M$ over $M$ with $\hat{G}\upharpoonright \kappa=G$ (using the standard method; cf the second exercise following previous construction).
In $V$, by duality theorem there are $J_1$ and $K$ so that
$$\mathbb{Q}_\kappa\ast P(\kappa)/J_1\cong P(\kappa)/J\ast j(\mathbb{Q}_\kappa)/K.$$
In this case, $K$ is a maximal ideal since the $j(\mathbb{Q}_\kappa)$-generic over $M$ is already just added by $P(\kappa)/J$. So in $V[H]$, $P(\kappa)/J_1\cong T$.
Now we turn to Kunen's forcing construction. Conditions in Kunen's forcing $\mathbb{Q}$ are normal trees of successor ordinal height $<\kappa$ which are homogeneous: for all $t\in T$ not on the top level, $T_t\cong T$, where $T_t$ is the tree $\{s\in T: t\le_T s\}$ with the order inherited from $T$.
Exercise:
1. Show that Kunen's forcing is $\kappa$-strategically closed. Hint: the strategy will go by choosing a particular branch through each of the small trees chosen in a play of the game so far.
2. Show that $\mathbb{Q}\ast \dot{T}$ has a $\kappa$-closed dense subset, and deduce that $\mathbb{Q}\ast \dot{T}\cong \mathrm{Add}(\kappa)$.
3. Show that $\dot{T}$ is a Suslin tree.
$\Box$
We will do one last application to construct a precipitous ideal on a cardinal $\kappa$ which is not measurable so that its quotient algebra is $\kappa^+$ closed.
Start with $\kappa$ measurable and $2^\kappa>\kappa^+$. We will use the Easton support iteration $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha<\kappa\rangle$, where $\dot{\mathbb{Q}}_\alpha=\dot{\mathrm{Add}(\alpha^+,1)}$ for inaccessible $\alpha<\kappa$ (and is trivial forcing otherwise).
Then $\mathbb{P}_\kappa$ is $\kappa$-c.c., and forces that for all inaccessible $\alpha<\kappa$, $2^\alpha=\alpha^+$ (this is a standard coding trick that was assigned as an exercise in one of Spencer Unger's lectures here). By duality,
$$\mathbb{P}_\kappa\ast P(\kappa)/\bar{I}\equiv j(\mathbb{P}_\kappa).$$
If $G_\kappa\subseteq \mathbb{P}_\kappa$ is generic, then $j(\mathbb{P}_\kappa)/e''G_\kappa\equiv \mathbb{P}_{\kappa,j(\kappa)}$. Since $M[G_\kappa]$ is closed under $\le \kappa$ sequences in $V[G_\kappa]$, this tail is $\kappa^+$-closed forcing over $V[G_\kappa]$.
However, GCH holds at every inaccessible $\alpha<\kappa$ and fails at $\kappa$ in $V[G_\kappa]$. By a reflection argument, $\kappa$ cannot be measurable in $V[G_\kappa]$. $\Box$
Exercise: Show that if $H\subseteq P(\kappa)/\bar{I}$ is generic, then $\kappa$ is measurable in $V[G_\kappa\ast H]$.
## Monday, July 27, 2015
### UCI Summer School part 2: Duality Theorem (Monroe Eskew)
The Duality Theorem gives a general technique for forcing to make an ideal whose quotient algebra has various properties. It appears in Matthew Foreman's "Calculating quotient algebras of generic embeddings." My version of these notes omits a lot of the dots which indicate that certain objects are just names in a forcing extension. This is for aesthetic reasons, and hopefully does not lead to confusion.
Duality Theorem: Suppose $I$ is a precipitous ideal on $Z$ and $\mathbb{P}$ is any partial order. If: there is a further generic extension of the extension by $P(Z)/I$ so that if $j:V\rightarrow M\subseteq M\subseteq V[G]$ is the ultrapower embedding from $G\subseteq P(Z)/I$, there is $H\subseteq \mathbb{P}$ generic over $V$ and $\hat{H}\subseteq j(\mathbb{P})$ generic over $M$ and some extension of $j$ to $\hat{j}:V[H]\rightarrow M[\hat{H}]$.
Then: there is a $\mathbb{P}$-name for an ideal $J$ on $Z$ and a $P(Z)/I$-name for an ideal $K$ on $j(\mathbb{P})$ and a canonical isomorphism $$\iota:\mathcal{B}(\dot{\mathbb{P}\ast P(Z)/J})\equiv \mathcal{B}(P(Z)/I\ast j(\mathbb{P})/\dot{K}).$$
So a very general statement of lifting a generic ultrapower map to a forcing extension gives a useful isomorphism for computing $P(Z)/J$ in the generic extension by $\mathbb{P}$. We remark that in some cases, this will be an equivalence.
Note: In what follows, we tacitly identify all of the posets involved with their Boolean completions. Occasionally for emphasis, this identification will be explicit.
Proof: Assume (1). There is some $A\in I^+$ and some $P(Z)/I$-name for a forcing $\dot{\mathbb{Q}}$ so that
$$A\Vdash_{P(Z)/I}(\Vdash_\dot{\mathbb{Q}} \dot{H_0}\subseteq j(\mathbb{P}) \textrm{ is generic over }M \textrm{ and }H:=j^{-1}[H_0]\subseteq \mathbb{P}\textrm{ is generic over }V).$$
Note that the set $\{p\in\mathbb{P}:\quad \Vdash_{P(A)/I\ast \mathbb{Q}} j(p)\not\in H_0\}$ cannot be dense, since it is the complement of the generic $j^{-1}[H_0]$. So there is $p_0\in \mathbb{P}$ so that for all $p\le p_0$, $\| j(p)\in H_0 \|_{P(A)/I\ast \mathbb{Q}}\neq 0$. We will constrain ourselves to work below this $p_0$ in $\mathbb{P}$ and below $A$ in $P(Z)/I$. For simplicity, assume that $A=Z$ and $p_0=1_\mathbb{P}$.
In $V^{P(Z)/I}$, define
$$K=\{p\in j(\mathbb{P}):\quad \Vdash_{\mathbb{\dot{Q}^G}} p\not\in H_0\}.$$
Let $G\ast h$ be generic for $P(Z)/I\ast j(\mathbb{P})/K$. From the $j(\mathbb{P})/K$-generic $h$, we can define a $j(\mathbb{P})$-generic $\hat{H}=\{p:[p]_K\in h\}$.
Claim: The following properties of $H_0$ are also true of $\hat{H}$:
1. $\Vdash_{P(Z)/I\ast \mathbb{Q}} \hat{H}$ is $j(\mathbb{P})$-generic over $M$.
2. $\Vdash_{P(Z)/I\ast \mathbb{Q}} j^{-1}[\hat{H}]$ is $\mathbb{P}$-generic over $V$.
3. For all $p\in \mathbb{P}$, $\not\Vdash_{P(Z)/I\ast \mathbb{Q}} j(p)\not\in \hat{H}$.
Proof of Claim: For (1), suppose $D\in M$ is open dense in $j(\mathbb{P})$. Then $\{[d]_K:d\in D\textrm{ and }d\not\in K\}$ is dense in $j(\mathbb{P})/K$. Otherwise, there would exist $p\in j(\mathbb{P})/K$ so that $p\wedge d\in K$ for all $d\in D$. But this is impossible because we could then force with $\mathbb{Q}$ over $V[G]$ to get a generic $H_0$ containing $p$ (as $p\not\in K$), and then $H_0\cap D=\emptyset$, contradicting genericity of $H_0$ over $M$.
The remaining parts of the claim can be checked similarly, and are left as an exercise. $\Box$.
Now let $e:\mathbb{P}\rightarrow \mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})$ be defined by $e(p)=\|j(p)\in \hat{H}\|$.
By (3) of the claim above, $\mathrm{ker}(e)=0$. Also $e$ preserves Boolean operations simply by the elementarity of $j$. By (2) of the claim, $e$ is a regular embedding (maps maximal antichains pointwise to maximal antichains).
Exercise: $e:\mathbb{P}\rightarrow \mathbb{Q}$ is a regular embedding iff for every $q\in\mathbb{Q}$ there is $p\in \mathbb{P}$ so that for every $p'\le p$, $e(p')$ is compatible with $q$.
Thus, if $H\subseteq \mathbb{P}$ is generic over $V$, we can force with the quotient $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})/e"H$ over $V[H]$ to obtain a generic $G\ast h$ for $P(Z)/I\ast j(\mathbb{P})/K$. By the definition of $e$, we have $j_G"H\subseteq \hat{H}$, where $\hat{H}$ is defined from $h$ as before. So we can extend the embedding $j_G$ to $\hat{j}:V[H]\rightarrow M[\hat{H}]$.
In $V[H]$ we can finally define $J=\{A\subseteq Z:1\Vdash [\mathrm{id}]\not\in \hat{j}(A)\}$, where the forcing is with the quotient $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K})/e"H$. In $V$, let
$$\iota(p,\dot{A})=e(p)\wedge \|[id]\in\hat{j}(\dot{A})\|.$$
Exercise: $\iota$ is order and incompatibility preserving.
It remains to show that the range of $\iota$ is dense. So take an arbitrary $(B,\dot{q})\in P(Z)/I\ast j(P)/K$. By strengthening this condition, we may assume without loss of generality that there is $f:Z\rightarrow \mathbb{P}$ in $V$ so that $B\Vdash [[f]_M]_K=\dot{q}$.
By regularity of $e$ (using the characterization in the exercise), there is a $p$ so that for all $p'\le p$, $e(p')\wedge (B,\dot{q})\neq 0$. Let $\dot{A}$ be a $\mathbb{P}$-name for a subset of $Z$ such that $p \Vdash \dot{A}=\{z\in B: f(z)\in H\}$ and $\neg p \Vdash \dot{A}\in J^+$.
We check that $(p,\dot{A})$ is actually a condition, which involves checking that $p\Vdash \dot{A}\in J^+$. So take a generic $G\ast \dot{h}$ containing $e(p)\wedge (B,\dot{q})$ (which is nonzero by choice of $p$). Then clearly $B\in G$ and since $[[f]_M]_K=q\in h$, we have $[f]_M=j(f)([\mathrm{id}])\in\hat{H}$. Therefore $[\mathrm{id}]\in \hat{j}(A)$, so this generic $G\ast \dot{h}$ shows that it is not forced by $\mathcal{B}(P(Z)/I\ast \dot{j(\mathbb{P})/K}/e"H$ that $[\mathrm{id}]\not\in \hat{j}(A)\}$.
By definition $\iota:=\iota(p,\dot{A})$ forces $j(p)\in \hat{H}$ and $[\mathrm{id}]\in\hat{j}(\dot{A})$. Since $\hat{j}$ extends $j$ and it's forced that $\dot{A}\subseteq B$, $\iota$ must force $B\in G$. And since $j(p)\Vdash_{j(\mathbb{P})} j(\dot{A})=j(\{z:j(f)(z)\in \hat{H}\})$, $\iota$ must force $\dot{q}=[j(f)(\mathrm{id})]_K\in h$. $\Box$
Remark: Suppose $K$ as in the Duality Theorem is forced to be principal, i.e., there is $m$ so that $$\Vdash K=\{p\in j(\mathbb{P}:p\le \neg m\}.$$
Then the Duality Theorem is easily seen to be an equivalence.
We can compute some nice properties of the ideal $J$ arising from the previous theorem.
Proposition: Using the notation of the previous theorem, $J$ is forced to be precipitous, with the same completeness as $I$. If $I$ is normal, then $J$ is also normal. Also, if $\bar{G}\subseteq P(Z)/J$ is generic over $V[H]$ and $G\ast h=\iota[H\ast \bar{G}]$ and $\hat{j}:V[H]\rightarrow M[\hat{H}]$ are as before, then $V[H]^Z/{\bar{G}}=M[\hat{H}]$ and $\hat{j}$ is the ultrapower embedding.
Finally, we relate $J$ to the ideal in $V[H]$ generated by $I$.
Proposition: Suppose $K$ as in the Duality Theorem is forced to be principal, with $m$ so that $\Vdash K=\{p\in j(\mathbb{P}):p\le \neg m\}.$ Suppose further that there exist $f$ and $A$ so that $A\Vdash \dot{m}=[f]_G$ and $\dot{B}$ is a $\mathbb{P}$-name for $\{z\in A: f(z)\in \dot{H}\}$. Then $\bar{I}\upharpoonright B=J\upharpoonright B$, and $A\setminus B\in J$, where $\bar{I}$ is the ideal in $V[H]$ generated by $I$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.987612783908844, "perplexity": 171.40950829937617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886107487.10/warc/CC-MAIN-20170821022354-20170821042354-00084.warc.gz"} |
https://www.physicsforums.com/threads/please-help-me.79508/ | 1. Jun 18, 2005
### mahtab_lovelygirl
i have lots of question about physic....
i hope u guys can help me. i really appreciate :)
1)
a) now that i am 50 and semi-mature. you'll never catch me bungee jumping. but let's say someone my mass(80kg) and height (1.83m) did and they used a 20m bungee cord with a k value of 500 n/m. what is the minimum height that the lunatic would need in order that he/she wont get killed?
b) if the same person was gently lowered down and the bungee cord came to equilibruim, far far would the cord stretch?
c)what is the person's speed when they have stretched the cord 8m? consider energy transfor motion.
d) what is the acceleration when the cord has been stretched?
2) an electron moving threough an electric field of 475 v/m and a magnetic field of oil experiences no force. if the electron's direction and the directions of the electric and magnetic fields are all mutually perpendicular, what is the speed of the electron? start with a sketch :P
i really need them as soon as possible for my final exam:(
2. Jun 18, 2005
### pete worthington
Welcome to Physics Forum..........We can help you, but you must do the work...
1a) Conservation of energy. Initial potential-gravitational energy will transform into final potential-spring energy. You have all the variables except height. As far as the persons height, that depends on how/where the person jumps from. The length of the cord must also be taken into account. Draw a picture and label positions.
1b) Hookes Law. That equation should be easy to find.
1c) Go back to your drawing. At this position you have potential-gravitational, kinetic and potential-spring. The unknown is velocity.
1d) Hookes Law and Newton's Second Law combined. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8243610858917236, "perplexity": 1396.6979776487335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188553.49/warc/CC-MAIN-20170322212948-00598-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://stats.stackexchange.com/questions/50404/bayesian-inference-notation-confusion | # Bayesian Inference Notation Confusion
In Bayesian Inference the following notation is quite common:
$P(H|D) = \frac{P(D|H)P(H)}{P(D)}$
where $D$ is data and $H$ is hypothesis. Moreover $P(D)$ is represented as total probability.
$P(D) = \sum P(D|H)P(H)dH$
However in my opinion notation seems to be quite sloppy. It is confusing on two fronts:
1) In order for this to work H should be a partitioning of sample space of $D$ by definition of total probability. But if you look at this point of view, let's say $D\sim N(m,v)$ then its sample space is $-\infty$ to $\infty$. I don't see why a hypothesis should partition this sample space, and what is that partition exactly? Let's say that my hypothesis is $D \sim N(m',v)$. In reality there should be a selector function behind the scenes, but I cannot really put my finger on it.
2) I think that distribution of $D$ is independent of what I select as a hypothesis, because my selection doesn't have the power to change reality. Therefore $P(D|H) = P(D)$. And inference cannot be done.
My interpretation is that in reality the notation is trying to partition a sample space of hypotheses. But how that should be properly written in a rigorous notation I don't know. Or my interpretation is wrong.
Any help appreciated.
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add comment
## 2 Answers
The easiest way to imagine this is to think of the sample space as the Cartesian product $\Omega = \mathcal{D} \times \mathcal{H}$, with measure $\mathbb{P}(D, H) := \mathbb{P}(H) \mathbb{P}(D|H)$. In this way, each $D$ or $H$ correspond to subsets of pairs in $\Omega$ - both $\mathcal{D}$ and $\mathcal{H}$ partition the sample space differently. In this simple scheme,
1. $\mathbb{P}(H)$ is the prior measure over the members of the partition $\mathcal{H}$,
2. $\mathbb{P}(D|H)$ is measure of $D$ relative to the subset $H$.
Note that a particular observation $D \in \mathcal{D}$ can be regarded as a subset $\{(D, H): H \in \mathcal{H} \}$ containing all the pairs consistent with $D$. This way of reasoning can easily be generalized to multiple observations. With this in mind, your answers are:
1) No, your hypotheses do not partition the set of observations.
2) Yes, the distribution of $D$ is independent of your model about the world represented by the sample space (, the $\sigma$-algebra) and the probability measure. Bayesian probability theory is just a model of how to systematically update your beliefs given your prior assumptions plus data.
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Thank you Pedro. Could we make a concrete toy example? Let's say in reality D is distributed with N(m,v) and I have two hypothesis N(m,v) and N(m',v) with equal probability. What will be omega? – Cagdas Ozgenc Feb 20 '13 at 16:14
In this case you have two hypotheses: $H_1 = m_1$ and $H_2 = m_2$ (note that $v$ is irrelevant). Your data space is $\mathcal{D} = \mathbb{R}$. Hence, the sample space is $\Omega = \mathcal{D} \times \mathcal{H} = \{ (x,m) : x \in \mathbb{R}, m \in \{m_1, m_2\} \}$. – Pedro A. Ortega Feb 20 '13 at 16:19
Thanks. That made great sense now. In this case I think it is ok to say that choice of an H is partitioning omega, isn't it? The odd situation is that the total probability has nothing to do with real D. – Cagdas Ozgenc Feb 20 '13 at 16:34
Yes, it is OK to say that a choice of an $H$ is partitioning the sample space. But no, it doesn't have anything to do with the real $D$, but that's ok, since you don't know the real $D$. What matters is that there should be at least one of the hypotheses that matches the real $D$. – Pedro A. Ortega Feb 20 '13 at 16:37
One follow up question. If I have sequential observations, can I make a sequential update in the following manner: calculate P(H|D) and plug it back to the place of P(H) for next observation. Or do I have to make a bigger cartesian space as I observe more data and calculate P(DxD...|H)? – Cagdas Ozgenc Feb 20 '13 at 16:42
show 3 more comments
Starting with your comment in part 2: $P(D|H)$ is not to be read as "the probability of getting this data given that I select hypothesis H". It is saying "probability of getting this given that hypothesis H is true". Clearly therefore $P(D|H) + P(D|¬H) = P(D)$
In other words, the partition can always in principle be between H is true and H is not true. By the law of the excluded middle, this must cover everything.
However, you are right to observe that if you are summing up different models that could explain the data, then there is no guarantee (or even reason to believe) that you have got "everything" covered (i.e. in practice it is usually impossible to know $P(D|¬H)$). That is why typically Bayesian analysis doesn't try to talk about the absolute probability of the model being correct given the data but only the relative probabilities of competing models.
So although I can in theory I can state:
$P(H_1 | D) = \frac{P(D|H_1) P(H_1)}{P(D)}$
$P(H_2 | D) = \frac{P(D|H_2) P(H_2)}{P(D)}$
When I do my analsis there is an implicit conditioning variable that is $M =$ "One of my models is correct". Perhaps being strict one should write:
$P(H_1 | D, M) = \frac{P(D|H_1) P(H_1|M)}{P(D|M)}$
$P(H_2 | D, M) = \frac{P(D|H_2) P(H_2|M)}{P(D|M)}$
Noting that $P(H) = P(H,M)$ since if H is true then M is true.
Now, however, notice that defined like this my priors must logically add up to 1. Often people don't have priors that add up to one. We could incorporate this by saying $P(H_1) = P(H_1|M)P(M)$. Writing our priors like this would then give us:
$P(H_1 | D, M) = \frac{P(D|H_1) P(H_1)}{P(D|M)P(M)}$
$P(H_2 | D, M) = \frac{P(D|H_2) P(H_2)}{P(D|M)P(M)}$
Of course none of this makes the slightest difference, because we are only going to compare the models, so we calculate:
$\frac{P(H_1 | D, M)}{P(H_2 | D, M)} = \frac{P(D|H_1,M) P(H_1)}{P(D|M)P(M)} \cdot \frac{P(D|M)P(M)}{P(D|H_2,M) P(H_2)} = \frac{P(D|H_1) P(H_1)} {P(D|H_2) P(H_2)}$
And all that intricacy cancels out and we are back at the original expressions.
Is it just laziness not writing the extra conditional variables that exist? Not really no. To me this is just another facet of the conditionality principle.
Edit
To take your example, suppose that $H_1$ is that $D \sim N(m,v)$ and $H_2$ is that $D \sim N(m', v)$ then clearly these two do not partition the space fully. Since we cannot evaluate $P(D|¬H_1 \cup H_2)$ we cannot work out the true $P(D)$ and so cannot ever get the absolute probability that either model is correct.
What we can do though is establish the ratio of those probabilities, since the unpartitioned part of the space cancels out (see above). This could then tell us that say, that $P(H_1|D)$ is 10 times more likely than $P(H_2|D)$.
In order to derive this, we had to write down expressions involving $P(D)$, even though we knew we'd never actually be able to calculate them.
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"Clearly therefore P(D|H)+P(D|¬H)=P(D)". I don't agree with this statement. If my hypothesis is D~N(m',v), then probability of data given my hypothesis is true is P(D|H) = N(m',v). In that case what is P(D|not H)? 1-N(m',v)? How is that going to add up to N(m,v) which is real P(D). – Cagdas Ozgenc Feb 20 '13 at 15:51
That is my point, it is impossible to know "not H". But you can still write down an expression for it. $P(D)$ cannot usually be calculated, unless you happen to have a situation where "H" and "not H" are known and perfectly partition the universe. If your hypothesis is $D\sim N(m,v)$ then you clearly can't define the probability of the data NOT coming from the model - it could be anything! – Corone Feb 20 '13 at 16:04
Bayesian inference is all about chosing the most likely model, given the data. Not about evaluating the absolute probability that the model is correct. Remember "all models are wrong; some are useful". In reality $P(H)=0$, but what we really mean is not "H is true" but "H is close enough". – Corone Feb 20 '13 at 16:07
@CagdasOzgenc I've added a bit based on your example that might make things a little clearer? – Corone Feb 20 '13 at 16:21
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https://socratic.org/questions/59d589b411ef6b49f1232c89 | Chemistry
Topics
# Question 32c89
Oct 7, 2017
$\text{15.7 u}$
#### Explanation:
The idea here is that each isotope will contribute to the average atomic mass of the element in proportion to its abundance.
In other words, the more abundant an isotope is, the more its atomic mass will contribute to the average atomic mass of the element, i.e. the closer the average atomic mass of the element will be to the atomic mass of the isotope.
In this case, you know that element $\text{Z}$ has two stable isotopes
• $\text{Isotope 1: " "15.0 u", 30% = 3/10 color(white)(.)"abundance}$
• $\text{isotope 2: " "16.0 u", 70% = 7/10color(white)(.)"abundance}$
Even without doing any calculation, you should be able to say that average atomic mass of $\text{Z}$ will be closer in value to $\text{16.0 u}$ than to $\text{15.0 u}$ because the second isotope is more abundant than the first one.
To actually calculate the average atomic mass, use the atomic masses of the isotopes and their decimal abundances.
"avg. atomic mass" = overbrace("15.0 u" * 3/10)^(color(blue)("the contribution of isotope 1")) + overbrace("16.0 u" * 7/10)^(color(blue)("the contribution of isotope 2"))#
This will get you
$\text{avg. atomic mass" = "4.5 u" + "11.2 u}$
$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{avg. atomic mass = 15.7 u}}}}$
As predicted, the average atomic mass of element $\text{Z}$ is closer in value to the atomic mass of the second isotope because this isotope is more abundant.
##### Impact of this question
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https://wiki.seg.org/index.php?title=Dictionary:Eikonal_equation&diff=prev&oldid=26224 | # Difference between revisions of "Dictionary:Eikonal equation"
(ī kōn’ ∂l) A form of the wave equation for harmonic waves in which the local velocity ${\displaystyle V}$ is compared to a reference velocity ${\displaystyle V_{R}}$(analogous to comparing a velocity to the speed of light in vacuum):
${\displaystyle \left(\nabla \phi \right)^{2}=\left({\frac {V}{V_{R}}}\right)^{2}=n^{2}}$,
where n is an index of refraction and is the wave function. Valid only where the variation of properties is small within a wavelength, sometimes called the ‘‘high-frequency condition.’’
More commonly in geophysical literature, the eikonal equation (for scalar waves) is written in terms of medium velocity only ${\displaystyle V(\mathbf {x} )}$ where ${\displaystyle \mathbf {x} =(x_{1},x_{2},x_{3})}$, as
${\displaystyle \left(\nabla \phi (\mathbf {x} )\right)^{2}={\frac {1}{V^{2}(\mathbf {x} )}}.}$
Solutions to the eikonal equation yield a high-frequency or large-wavenumber asymptotic representation of the wave field as a family of rays, represented by ray position and ray direction---the so-called kinematic aspect of wave propagation. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8997637033462524, "perplexity": 1100.1889663569084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347391277.13/warc/CC-MAIN-20200526160400-20200526190400-00256.warc.gz"} |
http://maths.david.olivier.name/algebraic-infinite-sums/ | # Algebraic infinite sum notation
On an infinite-dimensional vector space one usually defines a linear combination as a finite sum; for instance, if is a basis of the -vector space , one may write, for some finite part of :
I find this notation cumbersome, because it seems to make the sum dependent on the arbitrary choice of the finite set , while in fact it doesn’t, provided contains all the nonzero values to be added.
For instance, if you wish to add two such linear combinations written with different finite sets and , you cannot do so directly, without first arbitrarily choosing some other finite subset of containing :
while making excuses for the necessity to define the new ‘s and ‘s as zero and for the arbitrary choice of which doesn’t change the result.
I find it more practical to accept writing sums of an arbitrary collection of objects, such as:
provided we know in advance that only for a finite number of are the values nonzero. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.952873706817627, "perplexity": 375.7457425419815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710712.51/warc/CC-MAIN-20221129232448-20221130022448-00525.warc.gz"} |
http://math.stackexchange.com/questions/207153/a-question-about-independent-random-variable-and-probability-distribution | # A question about independent random variable and probability distribution
Suppose that $X_1,...,X_n$ are Random Variables and given that there exist an $k$ where k is an integer and $1\le k\le n-1$ s.t. the joint distribution $F_{X_1,...,X_k}$ are independent to $F_{X_k+1,...,X_n}$, prove that for all $1\le r \le k\le m\le n-1$ the joint distribution of $X_1,...,X_r$ is independent to joint distibutions $X_{m+1},...,X_n$
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What did you try? – Did Oct 4 '12 at 13:54 i tried to use contradiction, but not sure how to get the contrary – Mathematics Oct 4 '12 at 14:04 I fail to see how a proof by contradiction would help. More to the point: what is the conclusion you try to reach, that is, what are you trying to prove? – Did Oct 4 '12 at 16:59
You could try to use the fact that if $X$ and $Y$ are independent random variables, and if $f$ and $g$ are two (measurable ...) functions, then also $f(X)$ is independent from $g(Y)$.
but here is about the the joint pmf not pmf of a random variable – Mathematics Oct 8 '12 at 8:36 kjetil: +1. – Did Oct 8 '12 at 10:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8570063710212708, "perplexity": 117.68146705191246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706964363/warc/CC-MAIN-20130516122244-00070-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/143860-solved-derivatives-real-valued-functions.html | # Math Help - [SOLVED] Derivatives of real valued functions
1. ## [SOLVED] Derivatives of real valued functions
Let f and g be twice differentiable real-valued functions defined on $\mathbb{R}$. If f'(x)>g'(x) $\forall x>0$, which of the following must be true for all x>0?
(a) f(x)>g(x)
(b) f''(x)>g''(x)
(c) f(x)-f(0)>g(x)-g(0)
(d) f'(x)-f'(0)>g'(x)-g'(0)
(e) f''(x)-f''(0)>g''(x)-g''(0)
The answer is c but I thought it was d. Can someone show me how to prove it is c?
2. Integrate $f'>g'$ from 0 to x.
3. Originally Posted by maddas
Integrate $f'>g'$ from 0 to x.
That is just f(x)-f(0) and same for g
4. $f(x)-f(0) = \int_0^x f' > \int_0^x g' = g(x)-g(0)$...
edit: to see its not (d), take $f(x)=2x$ and $g(x)=x$.
5. Then why wouldn't the derivative of f'(x)>g'(x) also be an answer?
6. (a) is not true, take $f(x) = 2x$ and $g(x) = x+1$.
(b), (d), and (e) are not true, take $f(x)=2x$, $g(x) = x$.
7. Originally Posted by dwsmith
Let f and g be twice differentiable real-valued functions defined on $\mathbb{R}$. If f'(x)>g'(x) $\forall x>0$, which of the following must be true for all x>0?
(c) f(x)-f(0)>g(x)-g(0)
Can someone show me how to prove it is c?
We know the derivative of $f-g$ is $f '-g '>0$.
Therefore $f-g$ is increasing or $f(x)-g(x)>f(0)-g(0)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 17, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.965813398361206, "perplexity": 715.6894976901145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657104119.19/warc/CC-MAIN-20140914011144-00055-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://mathematica.meta.stackexchange.com/questions/235/latex-latex-or-mathrm-latex | # LaTeX, $\LaTeX$ or $\mathrm{\LaTeX}$?
What is the preferred way to write the name "LaTeX" on this site? A few questions use $\LaTeX$ in order to get the LaTeX name; however that looks wrong to me because it makes the letters italics, which isn't the case with the "proper" typeset name. You can get that name with $\mathrm{\LaTeX}$ which looks quite perfect (at least where LaTeX is enabled) but is a lot to type. And of course there exists the standard textual form LaTeX matching the name of the macro, and quite usual where you are restricted to pure text.
Use whichever way you like best. Even when writing $\LaTeX$, post titles are searchable, so this has no disadvantage. I wouldn't bother too much trying to get it to look excellent: it doesn't look perfect when MathML rendering is used anyway (and I need to use MathML rendering, otherwise long posts are painfully slow to edit when they have formulae) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9523410797119141, "perplexity": 969.1333945798201}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540564599.32/warc/CC-MAIN-20191213150805-20191213174805-00063.warc.gz"} |
http://mathhelpforum.com/advanced-applied-math/40592-gravitational-potential-energy.html | # Math Help - Gravitational Potential Energy
1. ## Gravitational Potential Energy
I have a question that reads as the following:an object is dropped from the top of a high building and falls under the force of gravity alone. Which option gives the graph of the potential energy U of the object as a function of the distance x through which it has fallen?
My first instinct was for it to be a graph curving towards the x axis(like a half a rainbow)
But another one that seems feasible is a graph worth a straight line from the U(x) axis down onto the x axis
2. Originally Posted by thermalwarrior
I have a question that reads as the following:an object is dropped from the top of a high building and falls under the force of gravity alone. Which option gives the graph of the potential energy U of the object as a function of the distance x through which it has fallen?
My first instinct was for it to be a graph curving towards the x axis(like a half a rainbow)
But another one that seems feasible is a graph worth a straight line from the U(x) axis down onto the x axis | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.827253520488739, "perplexity": 373.4386813942433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999642168/warc/CC-MAIN-20140305060722-00026-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3463101/finding-discriminant-of-a-monic-polynomial | # Finding discriminant of a monic polynomial.
I have now engaged in studying Galois Theory from NPTEL online lecture series which encompasses Finite Fields and Galois Theory. While watching the $$48$$-th lecture on Discriminant of a Polynomial a proposition has been discussed which I failed to understand properly.
Before going to the main proposition let us first define formally the discriminant of a polynomial.
Let $$K$$ be a field. Let $$f_n$$ denote general monic polynomial of degree $$n$$ i.e. it is of the form $$f_n = (X-X_1)(X-X_2) \cdots (X-X_n).$$
Let $$V(X_1,X_2, \cdots, X_n)$$ denote the Vandermonde deteminant in $$X_1,X_2, \cdots X_n.$$ So $$V(X_1,X_2, \cdots , X_n) = \prod\limits_{1 \leq i < j \leq n} (X_j - X_i).$$ Now the discriminant of $$f_n$$ is denoted by $$D(f_n)$$ and it is defined as $$D(f_n):= {V(X_1,X_2, \cdots , X_n)}^2 = \prod\limits_{1 \leq i < j \leq n} {(X_j - X_i)}^2.$$
Now let us take any monic polynomial $$f \in K[X]$$ of degree $$n.$$ Let $$f=X^n + a_1 X^{n-1} + \cdots + a_n.$$ Then by Kronecker's theorem $$\exists$$ a finite field extension $$L|K$$ such that $$f$$ splits completely into linear factors in $$L[X].$$ Let $$x_1,x_2, \cdots , x_n$$ be the zeros of $$f$$ lying in $$L.$$ Then it is clear that $$(-1)^r a_r = S_r (x_1,x_2,\cdots , x_n)$$ for $$r=1,2, \cdots , n$$ where $$S_r$$ is the $$r$$-th elementary symmetric polynomial in $$n$$-variables $$X_1,X_2, \cdots , X_n$$ i.e. $$S_r = \sum\limits_{1 \leq i_1 < i_2 < \cdots < i_r \leq n} X_{i_1} X_{i_2} \cdots X_{i_n}$$ for $$r=1,2, \cdots , n.$$ Now the discriminant of $$f$$ is denoted by $$D(f)$$ and is defined as \begin{align*} D(f) & = D(f_n) (-a_1, \cdots , (-1)^r a_r , \cdots , (-1)^n a_n ) \\ & = D(f_n) (S_1(x_1,x_2, \cdots , x_n), S_2(x_1,x_2, \cdots , x_n), \cdots , S_n (x_1,x_2, \cdots , x_n)). \end{align*}
By Fundamental Theorem of Symmetric Polynomials it is easy to show that $$D(f) \in K.$$ Now let us come back to the main proposition.
$$\textbf {Proposition} :$$ Let $$f(X) \in K[X]$$ be a monic polynomial of degree $$n$$ and $$x_1,x_2, \cdots , x_n \in L$$ be all zeros of $$f$$ in a finite field extension $$L|K.$$ Then $$D(f)= {V(x_1,x_2, \cdots , x_n)}^2 = \prod\limits_{1 \leq i < j \leq n} (x_j - x_i)^2.$$
In the proof of the above proposition the instructor wrote down an equality without giving any proper reasoning behind it. He said that $$D(f_n) (-a_1, \cdots , (-1)^r a_r , \cdots ,(-1)^n a_n ) = D(f_n) (x_1,x_2, \cdots , x_n).$$
But why is it always the case? The thing what he wrote implies $$D(f_n)(x_1,x_2, \cdots , x_n) = D(f_n) (S_1(x_1,x_2, \cdots ,x_n), S_2(x_1,x_2. \cdots , x_n), \cdots , S_n (x_1,x_2, \cdots ,x_n)).$$
But I don't understand why it necessarily holds. For instance let $$K= \Bbb Q$$ and $$L=\Bbb Q (\sqrt 2).$$ Let $$f=X^2-2 \in \Bbb Q[x].$$ Then $$f$$ splits completely into linear factors in $$L[X].$$ The zeros of $$f$$ are $$\pm \sqrt 2 \in L.$$ Let $$x_1 = \sqrt 2$$ and $$x_2 = -\sqrt 2.$$ Then $$S_1(x_1,x_2) = x_1 + x_2 = \sqrt 2 - \sqrt 2 = 0$$ and $$S_2(x_1,x_2) = x_1x_2 = \sqrt 2 (- \sqrt 2) = -2.$$ If the equality holds then we must have $$D(f_2)(\sqrt 2 , - \sqrt 2) = D(f_2) (0,-2).$$ But $$D(f_2) (\sqrt 2, - \sqrt 2) = 8 \neq 4 = D(f_2) (0,-2).$$ So the equality is in general false. So ultimately we get a false proof of the above proposition.
How do I manage to overcome the mistake in the lecture to prove the above proposition? Any suggestion regarding this will be highly appreciated.
Source $$:$$ https://youtu.be/PPI_3yVTHzQ?list=PLOzRYVm0a65dsCb_gMYe3R-ZGs53jjw02&t=1219
What I observed is that the actual problem lies in the definition of discriminant of a monic polynomial. Below is a way to prove the desired proposition by redefining the discriminant of a monic polynomial properly in the following way $$:$$
Let us first state the following theorem due to Jacobi without proof (the proof is very simple thoough!)
Theorem $$:$$ Let $$V = V(X_1,X_2, \cdots , X_n) = \prod\limits_{1 \leq i < j \leq n} (X_j - X_i) \in K[X_1,X_2, \cdots , X_n),$$ the Vandermonde's determinant in $$n$$ unknowns $$X_1,X_2, \cdots , X_n.$$ Then for any $$\sigma \in S_n$$ $$\sigma (V) = \text{sgn} (\sigma)\ V$$ where $$\text {sgn} (\sigma)$$ is defined as follows $$:$$
$$\text {sgn} (\sigma) = \left\{ \begin{array}{ll} 1 & \quad \text {if}\ \sigma\ \text {is even} \\ -1 & \quad \text {if}\ \sigma\ \text{is odd} \end{array} \right.$$
With the help of the above theorem it is easy to see that $$D(f_n),$$ the discriminant of the general monic polynomial of degree $$n,$$ is fixed by every permutation $$\sigma \in S_n.$$ Because $$D(f_n) = V^2 = \prod\limits_{1 \leq i < j \leq n} (X_j - X_i)^2 \in K[X_1,X_2, \cdots , X_n].$$ So for any $$\sigma \in S_n$$ when it extends to an automorphism of $$K(X_1,X_2, \cdots ,X_n)$$ defined by $$X_i \mapsto X_{\sigma(i)}$$ for all $$i=1,2,\cdots , n$$ and leaving all elements of $$K$$ fixed then we have $$\sigma (D(f_n)) = \sigma (V^2) = {\sigma (V)}^2 = V^2,$$ because for any permuatation $$\sigma \in S_n$$ we have $${\text {sgn}(\sigma)}^2 = 1.$$ This shows that $$D(f_n)$$ is a symmetric polynomial in $$X_1,X_2, \cdots , X_n.$$ So by Fundamental theorem of Symmetric Polynomials (also known as Newton's theorem) it follows that $$\exists$$ $$D \in K[X_1,X_2, \cdots , X_n]$$ such that $$D(f_n) = D(S_1,S_2, \cdots , S_n)$$ where $$S_i$$ is the $$i$$-th elementary symmetric polynomial in $$X_1,X_2, \cdots , X_n.$$ Now let $$f = X^n + a_1 X^{n-1} + \cdots + a_n \in K[X]$$ be a monic polynomial. Let us denote discriminant of $$f$$ by $$\text {Disc} (f)$$ (for avoiding confusion with $$D$$ I already defined). Then $$\text {Disc} (f)$$ is defined as follows $$:$$ $$\text {Disc} (f) : = D(-a_1, \cdots , (-1)^i a_i, \cdots , (-1)^na_n).$$
With the help of the revised definition of Discriminant of a Monic Polynomial it is now very easy to prove the desired proposition.
Let $$x_1,x_2, \cdots , x_n$$ be the zeros of $$f$$ lying in some finite field extension $$L|K.$$ Then we first note that $$S_r (x_1,x_2, \cdots , x_n) = (-1)^r a_r$$ for $$r=1,2, \cdots , n.$$ Then we have \begin{align*} \prod\limits_{1 \leq i < j \leq n} (x_j - x_i)^2 & = D(f_n) (x_1,x_2, \cdots , x_n)\\ & = D(S_1(x_1,x_2, \cdots , x_n), S_2(x_1,x_2, \cdots , x_n), \cdots , S_n(x_1,x_2, \cdots , x_n))\\ & = D(-a_1, \cdots , (-1)^i a_i , \cdots , (-1)^na_n)\\ & = \text {Disc} (f). \end{align*}
So we have $$\text {Disc} (f) = \prod\limits_{1 \leq i < j \leq n} (x_j - x_i)^2 = {V(x_1,x_2, \cdots , x_n)}^2,$$ as required.
This completes the proof of the proposition.
QED
• Yeah sorry for the typo. Fixed it. Can you please check now @ancientmathematician? – math maniac. Dec 5 '19 at 8:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 100, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991675615310669, "perplexity": 113.71514251724236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738864.9/warc/CC-MAIN-20200812024530-20200812054530-00361.warc.gz"} |
http://math.stackexchange.com/questions/147514/a-set-together-with-its-subset?answertab=active | # A set together with its subset
Are there any particular term for a pair $(U;A)$ where $U$ is a set and $A\in\mathscr{P}U$? That is, saying informally, $(U;A)$ is a set $A$ together with a set $U$ on which $A$ is defined ($A$ is defined as a subset of $U$).
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One could consider this pair as the inclusion $A\subset U$... – Simon Markett May 20 '12 at 20:05
How about extension? But I don't think there is a standard term for that. The closer I know of is pointed set. – lhf May 20 '12 at 20:45
You could call it a structure with one unary relation, or simply a unary relation $A$ on $U$. – Trevor Wilson Oct 12 '12 at 20:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9770900011062622, "perplexity": 346.88514501910197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989301.17/warc/CC-MAIN-20150728002309-00104-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/carbon-dioxide-and-global-warming.893991/ | # B Carbon dioxide and global warming
Tags:
1. Nov 19, 2016
### resurgance2001
Hi
I have a simple question about global warming. The percentage of carbon dioxide in the earth atmosphere is about 0.04% . My question is how is such a small percentage of this gas able to have such a powerful affect on global warming? Thanks
2. Nov 19, 2016
### CWatters
I think it's wrong to focus on the percentage...
If you could somehow travel to another planet and bring back a load of nitrogen and release it into the atmosphere you could reduce the percentage of the atmosphere that is C02, however it wouldn't change the absolute amount of C02 in the atmosphere.
Just for info C02 isn't the biggest cause of the greenhouse effect..
https://en.wikipedia.org/wiki/Greenhouse_effect
3. Nov 19, 2016
### mathman
The answer is that it is simply a quantitative result. The earth receives a lot of energy from the sun. Most of it gets radiated back. A small average difference is enougth to increase the temperature of the earth.
4. Nov 19, 2016
### resurgance2001
So I am guessing that it is 'relativity' easy to calculate the amount of heat that the Earth receives from the Sun and then also the relative amount of infrared radiation that effectively becomes trapped due to individual greenhouse gases. I will check out the wiki article. I guess I am looking for something that goes someway to introducing the mathematics that is used in modelling climate change, not something at post graduate level but something that is most likely at an intermediate graduate level. Thanks
5. Nov 19, 2016
### resurgance2001
I understand that methane can have a much greater effect than CO2 but I have also been reading somewhere in some paper recently that at according to some climate modelling, CO2 still has the greatest effect. I think the author of the paper I was reading did experiments (computer modelling) where they removed the CO2 from their model and they found that other gases such as H20 were not sufficient to sustain a greenhouse gas effect and the Earth jus got seriously colder. Thats parphrasing it roughly.
6. Nov 19, 2016
### Staff: Mentor
The density of carbon dioxide in the atmosphere is 1020/m3. Do you see how such a huge number can have a powerful effect?
It doesn't help to compare the amount of CO2 to the amount of N2 if you want to measure the effect of CO2 on the climate. The amount of CO2 we have is sufficient to absorb a significant fraction of infrared radiation going through the atmosphere, only the absolute concentration matters here.
Earth would be significantly colder without CO2, but even colder without water vapor.
7. Nov 19, 2016
### bigfooted
I tried to find a decent website and I found that this one was explaining it rather well:
https://scienceofdoom.com/2009/11/28/co2-an-insignificant-trace-gas-part-one/
8. Nov 20, 2016
### resurgance2001
Thanks - yes it is beginning to make more sense. I would really like to see the numbers a bit, as in the basic models used to estimate the amount of Infrared absorbed and then re-emitted by those molecules. I understand the idea that the re-emitted IR is emitted in all different directions so that on average less of what was absorbed is transmitted back out into space, but I would like to see how it is calculated. Thanks
9. Nov 20, 2016
### CWatters
The absorption spectra of greenhouse gasses can be measured rather than just calculated.
10. Nov 20, 2016
### rootone
It's not plausible to do a precise calculation of atmospheric warming since many parts of Earth are subject to local changes of cloud cover and ocean currents which are not entirely predictable.
It's more practical to measure what is actually happening over a period of time, then build models based on understood thermodynamics that explain the data.
From there it's possible to extrapolate trends within a reasonable margin of error.
You might find this NASA article interesting.
http://climate.nasa.gov/news/2264/twelve-years-of-satellite-data-help-decode-climate-change/
11. Nov 20, 2016
### resurgance2001
Thanks - I will check out that link.
Draft saved Draft deleted
Similar Discussions: Carbon dioxide and global warming | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8023905754089355, "perplexity": 703.2514117759998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102891.30/warc/CC-MAIN-20170817032523-20170817052523-00716.warc.gz"} |
https://im.kendallhunt.com/HS/students/1/1/12/index.html | # Lesson 12
Standard Deviation
• Let’s learn about standard deviation, another measure of variability.
### 12.1: Notice and Wonder: Measuring Variability
What do you notice? What do you wonder?
### 12.2: Investigating Standard Deviation
Use technology to find the mean and the standard deviation for the data in the dot plots.
1. What do you notice about the mean and standard deviation you and your partner found for the three dot plots?
2. Invent some data that fits the conditions. Be prepared to share your data set and reasoning for choice of values.
Partner 1
Partner 2
Dot plots:
Dot plots:
Conditions:
• 10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 6.
• 10 numbers with a standard deviation three times greater than the data in the first row.
• 10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Conditions:
• 10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 12.
• 10 numbers with a standard deviation four times greater than the data in the first row.
• 10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
### 12.3: Investigating Variability
Begin with the data:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
1. Use technology to find the mean, standard deviation, median, and interquartile range.
2. How do the standard deviation and mean change when you remove the greatest value from the data set? How do they change if you add a value to the data set that is twice the greatest value?
3. What do you predict will happen to the standard deviation and mean when you remove the least value from the data set? Check to see if your prediction was correct.
4. What happens to the standard deviation and mean when you add a value to the data set equal to the mean? Add a second value equal to the mean. What happens?
5. Add, change, and remove values from the data set to answer the question: What appears to change more easily, the standard deviation or the interquartile range? Explain your reasoning.
How is the standard deviation calculated? We have seen that the standard deviation behaves a lot like the mean absolute deviation and that is because the key idea behind both is the same.
1. Using the original data set, calculate the deviation of each point from the mean by subtracting the mean from each data point.
2. If we just tried to take a mean of those deviations what would we get?
3. There are two common ways to turn negative values into more useful positive values: take the absolute value or square the value. To find the MAD we find the absolute value of each deviation, then find the mean of those numbers. To find the standard deviation we square each of the deviations, then find the mean of those numbers. Then finally take the square root of that mean. Compute the MAD and the standard deviation of the original data set.
### Summary
We can describe the variability of a distribution using the standard deviation. The standard deviation is a measure of variability that is calculated using a method that is similar to the one used to calculate the MAD, or mean absolute deviation.
A deeper understanding of the importance of standard deviation as a measure of variability will come with a deeper study of statistics. For now, know that standard deviation is mathematically important and will be used as the appropriate measure of variability when mean is an appropriate measure of center.
Like the MAD, the standard deviation is large when the data set is more spread out, and the standard deviation is small when the variability is small. The intuition you gained about MAD will also work for the standard deviation.
### Glossary Entries
• standard deviation
A measure of the variability, or spread, of a distribution, calculated by a method similar to the method for calculating the MAD (mean absolute deviation). The exact method is studied in more advanced courses.
• statistic
A quantity that is calculated from sample data, such as mean, median, or MAD (mean absolute deviation). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8929623961448669, "perplexity": 353.4566486344445}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587794.19/warc/CC-MAIN-20211026011138-20211026041138-00516.warc.gz"} |
https://susmost.com/mclangmuir.html | # Tutorial - Monte Carlo simulation of the Langmuir adsorption model with nearest neighbor interactions¶
In this tutorial we provide an example of applying the Monte Carlo method as implemented in SuSMoST to simulate the Langmuir adsorption model with interaction between nearest neighbors.
Please have a look at the following script mc_langmuir.py and read below what the script does.
## Libraries¶
In this block we include libraries. Here numpy and mpi4py are standard libraries. load_lattice_task and mc are the modules of the susmost general library required for setting up the structure of the lattice model and applying the Monte Carlo methods to simulate that.
from susmost import load_lattice_task
from susmost import mc
import numpy as np
from mpi4py import MPI
## Constants, Parameters and Model¶
Please download the archive Langmuir_model.zip, where all files needed for the model definition are provided. Here we exploit the following assumption of the Langmuir adsorption model: - All adsorption sites are equivalent (assume the surface is homogeneous);
• Each site can hold at most one molecule of A (mono-layer coverage only).
As a model of the surface we use the square lattice with a distance between nearest neighbor sites. See the sample_empty_unit_cell.xyz file. Only two xyz-files for adsorption complexes are available. They are adsorbate_state_0.xyz and adsorbate_state_1.xyz for empty and occupied states of the site, respectively. In this example we take into account the interaction between molecules adsorbed at nearest neighbor sites. Therefore, there is a single pair configuration sample_001.xyz with non-zero interaction energy. The value of such interaction is provided in the energy file.
lt = load_lattice_task('./')
lt.set_property('coverage', {'occupied':1.0} )
L = 60 # lattice size
temperatures = [400, 500, 700, 1000] # K
kB = 0.008314 # kJ/(mol*K)
Function load_lattice_task load the lattice model into the SuSMoST. lt.set_property states the property coverage for the adsorption complex named “occupied” equal to 1. Note that name of the adsorption complex is included in the head of the corresponding xyz-file. In this case, you may see ac_name=occupied in the top of the adsorbate_state_1.xyz file.
Also we are setting up the key constants and model parameters:
• L is the linear size of the lattice in a units;
• temperatures is a list of temperatures for parallel tempering Monte Carlo simulation;
• kB is Gas constant in kJ/(mol*K). If k_B is measured in kJ/K, then k_B is the Boltzmann constant .
## Setting Up the Monte Calo Simulation¶
Here we perform the Monte Carlo simulation of the Langmuir model with interaction between nearest neighbors in Grand Canonical Ensemble for a given amount of Monte Carlo steps at fixed values of chemical potential and different temperatures, see temperatures. Firstly, we set the state energies of the available adsorption complexes as lt.set_ads_energy. The state energy of an empty site is assumed to be zero, so we ignore it. And the state energy for an occupied site is set equal to the chemical potential mu. Note that in this tutorial, the chemical potential mu means the differences between molar Gibbs free energy of the gas phase and adsorption layer $$\mu_g - \mu_a \approx-RTlnp$$.
for mu in np.arange(20., -80.-0.0001, -5.0 ):
# for mu in np.arange(30., 0.-0.0001, -2.0 ):
m = mc.make_metropolis(lt, L, temperatures, kB)
mc.run(m, log_periods_cnt=10, log_period_steps = 1000*m.cells_count, relaxation_steps = 100000*m.cells_count, \
traj_fns = ["T={}.xyz".format(T) for T in temperatures])
mc.stat_digest(m)
make_metropolis function creates a task m for Monte Carlo simulation. Here we run the m simulation for each value of the chemical potential with relaxation time - 100 000 Monte Carlo steps (MCS) and 10 000 MCS for thermal averaging over the 10 points. Recall that MCS is the amount of attempts to switch the state of the lattice equaled to the amount of the lattice sites. Every 1000 MCS in the production run we save the xyz-snapshots having 10 snapshots at the end of the simulation at each considered temperature. stat_digest prints summary on sample statistics.
## Calculation the thermodynamic averages¶
We have obtained as simple averages the potential energy of the adlayer and the surface coverage at fixed temperatures (400, 500, 700, 1000 K) changing the chemical potential:
• from 20 to -80 with step = -5 kJ/mol for the repulsive interaction between nearest neighbors ($$\epsilon = 10$$ kJ/mol);
• from 30 to 0 with step = -2 kJ/mol for the attractive interaction between nearest neighbors ($$\epsilon = -5$$ kJ/mol).
Also we have calculated the differential heat of adsorption and heat capacity. All averages are calculated using 10 points. Results are saved in file.
E_2 = 0
covE = 0
cov_2 = 0
stat = np.mean(param_log,axis=0)[[0, -2, -3]] # coverage, H, U
for log_row in param_log:
E_2 += log_row[-3]**2.0
covE += log_row[0]*log_row[-3]
cov_2 += log_row[0]**2.0
E_2 /= len(param_log)
covE /= len(param_log)
cov_2 /= len(param_log)
Qd = -(covE - stat[0]*stat[2])/(cov_2-stat[0]**2.0)
Cp = (L^2)*(E_2 - stat[2]*stat[2])/(T*T*kB)
where
• E_2 is the squared potential energy per lattice site;
• covE is the value of surface coverage multiplied by energy;
• cov_2 is the squared coverage;
• Qd is the differential heat of adsorption calculated with equation 1
$Q_{d} = \frac{\langle\theta \cdot E\rangle-\langle \theta \rangle \cdot \langle E \rangle}{\langle \theta^2 \rangle-{\langle \theta \rangle}^2}$
## Results¶
Fig.1 shows typical results of the simulation of the Langmuir model with the repulsive interactions between nearest neighboring particles. On the A and C panel we can see the adsorption isotherms and heat capacities vs. the chemical potential calculated at different temperatures. There are three plateaus on the isotherms those correspond to lattice gas, chess-like and close-packed phases of the adlayer. Two peaks of the heat capacity are related to the phase transitions from the lattice gas to chess-like structure and further transition to the close-packed monolayer. The differential heat of adsorption and potential energy of the molecular layer as functions of surface coverage are shown on the panels B and D. Stable phases of the adlayer are revealed as steps on the differential heat curves and inflection points on the potential energy curves.
Fig. 1. Results of the simulation at $$\epsilon = 10$$ kJ/mol.
The animations of the adsorption layer with decrease of the chemical potential (increase of the gas phase pressure) at 400 K and 1000 K are shown in the Fig. 2 and Fig.3.
Fig. 2. Filling the monolayer at $$RT/\epsilon = 0.33$$.
Fig. 3. Filling the monolayer at $$RT/\epsilon = 0.83$$.
Fig.4 and Fig. 5 demonstrates a behavior of the adlayer with attractive interactions between nearest neighbors $$\epsilon = -5$$ kJ/mol at different temperatures $$RT/|\epsilon| = 0.66$$ and $$RT/|\epsilon| = 1.66$$ with increase of the gas phase pressure. A surface condensation as the first order phase transition occurs in the adlayer only at the low temperature. At the high temperature, for example $$RT/\epsilon = 1.66$$, the close packed phase appears as a result of continuous filling of the surface when the gas phase pressure increases.
Fig. 4. Filling the monolayer at $$RT/|\epsilon| = 0.66$$.
Fig. 5. Filling the monolayer at $$RT/|\epsilon| = 1.66$$.
The adsorption isotherms and heat capacity as a function of the chemical potential of the adlayer calculated at various temperatures are shown in the Fig.5A and Fig.6C, correspondingly. It is seen that the isotherms at high temperatures are smooth, but have the step at lower ones. The heat capacity tends to infinity in the point of the first order phase transition, which we observed at low temperatures. At high temperatures there is only a finite step on the heat capacity. And it disappears with further increasing of temperature. The Fig.6B,D represent the behavior of the potential energy and differential heat of adsorption vs. surface coverage.
Fig. 6. Results of the simulation at $$\epsilon = -5$$ kJ/mol.
## References¶
1
J.E. González, A.J. Ramirez-Pastor, V.D. Pereyra, Adsorption of Dimer Molecules on Triangular and Honeycomb Lattices, Langmuir. 17 (2001) 6974–6980. doi:10.1021/la010465i. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8603714108467102, "perplexity": 1600.0137669261874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703533863.67/warc/CC-MAIN-20210123032629-20210123062629-00204.warc.gz"} |
https://rms-hg.com/victoria/density-function-of-arcsine-example.php | Density function of arcsine example
Arcsine Distribution by Donald Schaefer AppAdvice
Fourier transform of distributions Cross Validated. 27/07/2013В В· A lecture with examples for joint probability density functions., The shorthand X в€јarcsin is used to indicate that the random variable X has the arcsin distribution. An arcsin random variable X has probability density function f(x)= 1.
M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1
Excel ASIN Function excelfunctions.net. A New Generalisation of Sam-Solai’s Multivariate symmetric Arcsine Distribution of Kind-1* ,then the density function of Sam-Solai’s multivariate conditional, On the first homework, we considered a good example of This is referred to as the arcsine distribution, since the corresponding CDF is C(α) = 1 2 +sin.
The following characterization of the arcsine density is established: For example, for a general random orthogonal with respect to any weight function w 26/12/2017В В· Arcsin is the abbreviation of arcsine which is a trigonometric function that takes the opposite leg of a right triangle as well as the hypotenuse of the same triangle
and X is a continuous random variable with density function f. Here is another example. Suppose that X has the density f(x)= x 2 arcsin(y) is defined to take arcsine: Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin-1
Use arcsin when you know the sine of an angle and want to know the actual angle. See also Inverse functions - trigonometry. Example - using arcsin to find an angle The shorthand X в€јarcsin is used to indicate that the random variable X has the arcsin distribution. An arcsin random variable X has probability density function f(x)= 1
It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a Probability/Transformation of Probability Densities. with grey arrows for two example is equal to the area under the probability density function,
ArcSinDistribution {x It is named for the functional form of its cumulative distribution function For example, the standard arc sine distribution How to transform an arcsine distribution to a normal distribution? Is there a function that can transform this distribution to a normal distribution?
For example, in various Graph of the density function (left) and the cumulative distribution function (right) for an arcsine distributed random variable. Function Reference. This page contains a list of all the functions you can use in Sisense ’s formula editor. Statistical Functions Average. Avg()
On the п¬Ѓrst homework, we considered a good example of This is referred to as the arcsine distribution, since the corresponding CDF is C(О±) = 1 2 +sin Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability
ArcSinDistribution {x It is named for the functional form of its cumulative distribution function For example, the standard arc sine distribution arcsine: Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin-1
Distribution 22.3 introduced the functions: the GAMS built-in functions. For example, PRIVACY POLICY gams/missing_trig_functions_arccos_arcsin_tan.txt The arcsine is the inverse operation of the sine function. This lesson will give a definition of the arcsine and provide examples of how it can be...
The following characterization of the arcsine density is established: For example, for a general random orthogonal with respect to any weight function w distributions related to arcsine density MAKOTO MAEJIMA1, of L evy measures where the modi ed Bessel function K 0 plays an important role. 2. Arcsine transformation A
M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 function of the following probability density functions. (a) and x= Л‡ arcsin(y);2Л‡ It is a one-to-one function. Here is an example: How to use the arcsine function in the Algebra Coach. Type arcsin(x) into the textbox, where x is the argument.
The following characterization of the arcsine density is established: For example, for a general random orthogonal with respect to any weight function w M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 function of the following probability density functions. (a) and x= Л‡ arcsin(y);2Л‡
ArcSinDistribution {x It is named for the functional form of its cumulative distribution function For example, the standard arc sine distribution scipy.stats.arcsine Examples >>> from scipy.stats Probability density function. logpdf(x, loc=0, scale=1) Log of the probability density function.
For example, in various Graph of the density function (left) and the cumulative distribution function (right) for an arcsine distributed random variable. Evaluates the Arcsin distribution PDF. This function evaluates the PDF of the Arcsin distribution with given argument, Example 1 #include
The class type arcsine_distribution represents an arcsine probability distribution function. The arcsine distribution is For example: arcsine_distribution The shorthand X в€јarcsin is used to indicate that the random variable X has the arcsin distribution. An arcsin random variable X has probability density function f(x)= 1
arcsine: Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin-1 Examples # NOT RUN { A <- Arcsine() # A is a Arcsine distribution with shape1 = 1 and shape2 = 1. r(A)(3) # three random number generated from this distribution, e.g
Examples # NOT RUN { A <- Arcsine() # A is a Arcsine distribution with shape1 = 1 and shape2 = 1. r(A)(3) # three random number generated from this distribution, e.g M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 function of the following probability density functions. (a) and x= Л‡ arcsin(y);2Л‡
6. Distribution and Quantile Functions Suppose that X has a continuous distribution on в„ќ with density function f that is symmetric p. For example, Excel Asin Function Examples. The following spreadsheet shows the Excel Asin Function, used to calculate the arcsine of four different values. Formulas: A; 1
The next section presents an interesting property of the arcsine distribution. density function of arcsine distribution of arcsine distribution and It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a
How can I apply an arcsine transformation to my data using Statistica ? (example: arcsin(-0.38) Within this group you will find the Arcsin function; Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability
MA 381 Section 8.1 Joint Probability Density Functions. It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a, 27/07/2013В В· A lecture with examples for joint probability density functions..
An interesting property of the arcsine distribution and
gamsmissing_trig_functions_arccos_arcsin_tan [GAMS. arcsine: Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin-1, The leads in coin tossing distribution is also known as the discrete arcsine distribution. Syntax: LET = Compute the arcsine probability density function..
The Arcsine Transformation Has the time come for retirement?. distributions related to arcsine density MAKOTO MAEJIMA1, of L evy measures where the modi ed Bessel function K 0 plays an important role. 2. Arcsine transformation A, Examples # NOT RUN { A <- Arcsine() # A is a Arcsine distribution with shape1 = 1 and shape2 = 1. r(A)(3) # three random number generated from this distribution, e.g.
Arcsine Article about Arcsine by The Free Dictionary
The Arcsine Distribution MIT OpenCourseWare. ArcSinDistribution {x It is named for the functional form of its cumulative distribution function For example, the standard arc sine distribution The class type arcsine_distribution represents an arcsine probability distribution function. The arcsine distribution is For example: arcsine_distribution.
• Arcsine Article about Arcsine by The Free Dictionary
• Leads in Coin Tossing PDF itl.nist.gov
• Arcsine Distribution 1.66.0 - boost.org
• The proof that \( g \) is a valid probability density function explains the name arcsine distribution, as does the form of the distribution function given next. \ scipy.stats.arcsine Examples >>> from scipy.stats Probability density function. logpdf(x, loc=0, scale=1) Log of the probability density function.
M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 function of the following probability density functions. (a) and x= Л‡ arcsin(y);2Л‡ Distribution 22.3 introduced the functions: the GAMS built-in functions. For example, PRIVACY POLICY gams/missing_trig_functions_arccos_arcsin_tan.txt
It is a one-to-one function. Here is an example: How to use the arcsine function in the Algebra Coach. Type arcsin(x) into the textbox, where x is the argument. Distribution 22.3 introduced the functions: the GAMS built-in functions. For example, PRIVACY POLICY gams/missing_trig_functions_arccos_arcsin_tan.txt
Learn how to create probability plots in R for both didactic purposes and for data analyses. cumulative density function : qname( ) For example, rnorm(100, ArcSinDistribution {x It is named for the functional form of its cumulative distribution function For example, the standard arc sine distribution
Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability Probability/Transformation of Probability Densities. with grey arrows for two example is equal to the area under the probability density function,
Use arcsin when you know the sine of an angle and want to know the actual angle. See also Inverse functions - trigonometry. Example - using arcsin to find an angle Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability
Examples # NOT RUN { A <- Arcsine() # A is a Arcsine distribution with shape1 = 1 and shape2 = 1. r(A)(3) # three random number generated from this distribution, e.g Arcsine Distribution. Arcsin is the abbreviation of arcsine which is a trigonometric function that takes the opposite leg of a Typical example: arcsin(y
The next section presents an interesting property of the arcsine distribution. density function of arcsine distribution of arcsine distribution and and X is a continuous random variable with density function f. Here is another example. Suppose that X has the density f(x)= x 2 arcsin(y) is defined to take
How to transform an arcsine distribution to a normal distribution? Is there a function that can transform this distribution to a normal distribution? The Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk for example, [15]. The corresponding density function f arcsin.x/ D F0 arcsin
For a binomial distribution, variance is a function and two of the most common variance-stabilizing transformations are the logit and arcsine For example Examples # NOT RUN { A <- Arcsine() # A is a Arcsine distribution with shape1 = 1 and shape2 = 1. r(A)(3) # three random number generated from this distribution, e.g
Function Reference. This page contains a list of all the functions you can use in Sisense ’s formula editor. Statistical Functions Average. Avg() For a binomial distribution, variance is a function and two of the most common variance-stabilizing transformations are the logit and arcsine For example
Some Properties of the Arcsine Distribution Taylor & Francis
A characterization of the arcsine distribution. cfS_Arcsine(t) evaluates the characteristic function cf(t) of the symmetric zero-mean Arcsine distribution on the interval (-1,1) (U-shaped distribution with mean = 0, 10/11/2012В В· #13 Probability density function of a monotone transformed variable using change of variable method Example 2 - Duration:.
#13 Probability density function of a monotone transformed
Arcsine distribution Wikipedia. In probability theory the Arcsine Distribution is the probability distribution whose cumulative distribution function is 0 = x = 1 and whose probability density, It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a.
ArcSinDistribution {x (CDF), which is a normalized arc sine function. Its probability density function (PDF) For example, the standard arc The Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk for example, [15]. The corresponding density function f arcsin.x/ D F0 arcsin
Excel Asin Function Examples. The following spreadsheet shows the Excel Asin Function, used to calculate the arcsine of four different values. Formulas: A; 1 Distribution 22.3 introduced the functions: the GAMS built-in functions. For example, PRIVACY POLICY gams/missing_trig_functions_arccos_arcsin_tan.txt
The next section presents an interesting property of the arcsine distribution. density function of arcsine distribution of arcsine distribution and Levy’s arcsine law in that a trader spends in "black" or "red" then the distribution of his P & L is given by the arcsine probability density function
and X is a continuous random variable with density function f. Here is another example. Suppose that X has the density f(x)= x 2 arcsin(y) is defined to take arcsine: Any of several single-valued or multivalued functions that are inverses of the sine function. Symbol: arcsin, sin-1
M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 function of the following probability density functions. (a) and x= Л‡ arcsin(y);2Л‡ How to transform an arcsine distribution to a normal distribution? Is there a function that can transform this distribution to a normal distribution?
On the п¬Ѓrst homework, we considered a good example of This is referred to as the arcsine distribution, since the corresponding CDF is C(О±) = 1 2 +sin Evaluates the Arcsin distribution PDF. This function evaluates the PDF of the Arcsin distribution with given argument, Example 1 #include
for 0 ≤ x ≤ 1, and whose probability density function is = (−) on Arcsine distribution is closed under translation and scaling by a positive factor It is a one-to-one function. Here is an example: How to use the arcsine function in the Algebra Coach. Type arcsin(x) into the textbox, where x is the argument.
The next section presents an interesting property of the arcsine distribution. density function of arcsine distribution of arcsine distribution and You do the statistics on the transformed numbers. For example, the mean of the untransformed data is \(18.9\); the mean of the square-root transformed data is \(3.89
How can I apply an arcsine transformation to my data using Statistica ? (example: arcsin(-0.38) Within this group you will find the Arcsin function; Excel Asin Function Examples. The following spreadsheet shows the Excel Asin Function, used to calculate the arcsine of four different values. Formulas: A; 1
cfS_Arcsine(t) evaluates the characteristic function cf(t) of the symmetric zero-mean Arcsine distribution on the interval (-1,1) (U-shaped distribution with mean = 0 Answer to Python Homework 4: Arcsine Laws The purpose of this python homework is to explore the so-called Arcsine laws numerically...
26/12/2017В В· Arcsin is the abbreviation of arcsine which is a trigonometric function that takes the opposite leg of a right triangle as well as the hypotenuse of the same triangle Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability
It is a one-to-one function. Here is an example: How to use the arcsine function in the Algebra Coach. Type arcsin(x) into the textbox, where x is the argument. Use arcsin when you know the sine of an angle and want to know the actual angle. See also Inverse functions - trigonometry. Example - using arcsin to find an angle
Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability The shorthand X в€јarcsin is used to indicate that the random variable X has the arcsin distribution. An arcsin random variable X has probability density function f(x)= 1
Some Properties of the Arcsine Distribution constant c where X has the probability density function (pdf) given in For example if X = 1 with probability The Arcsine Transformation: Has the time come for tends to be skewed when the distribution is example provided by Hardy (2002), the arcsine transformation
Description. arcsin(x) represents the inverse of the sine function. The angle returned by this function is measured in radians, not in degrees. For example, the The class type arcsine_distribution represents an arcsine probability distribution function. The arcsine distribution is For example: arcsine_distribution
ArcSinDistribution {x (CDF), which is a normalized arc sine function. Its probability density function (PDF) For example, the standard arc In probability theory the Arcsine Distribution is the probability distribution whose cumulative distribution function is 0 = x = 1 and whose probability density
It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a On the п¬Ѓrst homework, we considered a good example of This is referred to as the arcsine distribution, since the corresponding CDF is C(О±) = 1 2 +sin
26/12/2017В В· Arcsin is the abbreviation of arcsine which is a trigonometric function that takes the opposite leg of a right triangle as well as the hypotenuse of the same triangle It is a one-to-one function. Here is an example: How to use the arcsine function in the Algebra Coach. Type arcsin(x) into the textbox, where x is the argument.
In probability theory the Arcsine Distribution is the probability distribution whose cumulative distribution function is 0 = x = 1 and whose probability density Levy’s arcsine law in that a trader spends in "black" or "red" then the distribution of his P & L is given by the arcsine probability density function
6. Distribution and Quantile Functions Suppose that X has a continuous distribution on в„ќ with density function f that is symmetric p. For example, Math and trigonometry: Returns the arcsine of a number ASINH function Returns the inverse of the cumulative distribution function for a specified beta distribution
Description. arcsin(x) represents the inverse of the sine function. The angle returned by this function is measured in radians, not in degrees. For example, the scipy.stats.arcsine Examples >>> from scipy.stats Probability density function. logpdf(x, loc=0, scale=1) Log of the probability density function.
scipy.stats.arcsine — SciPy v1.1.0 Reference Guide
The Arcsine Distribution randomservices.org. In probability theory the Arcsine Distribution is the probability distribution whose cumulative distribution function is 0 = x = 1 and whose probability density, 10/11/2012В В· #13 Probability density function of a monotone transformed variable using change of variable method Example 2 - Duration:.
Inverse sine function MuPAD - MathWorks
An interesting property of the arcsine distribution and. How can I apply an arcsine transformation to my data using Statistica ? (example: arcsin(-0.38) Within this group you will find the Arcsin function; 6. Distribution and Quantile Functions Suppose that X has a continuous distribution on в„ќ with density function f that is symmetric p. For example,.
On the п¬Ѓrst homework, we considered a good example of This is referred to as the arcsine distribution, since the corresponding CDF is C(О±) = 1 2 +sin Use arcsin when you know the sine of an angle and want to know the actual angle. See also Inverse functions - trigonometry. Example - using arcsin to find an angle
Description. arcsin(x) represents the inverse of the sine function. The angle returned by this function is measured in radians, not in degrees. For example, the For example, in various Graph of the density function (left) and the cumulative distribution function (right) for an arcsine distributed random variable.
ArcSinDistribution {x It is named for the functional form of its cumulative distribution function For example, the standard arc sine distribution It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a
For example, in various Graph of the density function (left) and the cumulative distribution function (right) for an arcsine distributed random variable. 10/11/2012В В· #13 Probability density function of a monotone transformed variable using change of variable method Example 2 - Duration:
and X is a continuous random variable with density function f. Here is another example. Suppose that X has the density f(x)= x 2 arcsin(y) is defined to take It will calculate the inverse of the lognormal cumulative lognormal distribution function at a For example, we can use the function to know the probability of a
Evaluates the Arcsin distribution PDF. This function evaluates the PDF of the Arcsin distribution with given argument, Example 1 #include
The functions Arc sin x and Arc cos x are defined in the real domain for for example, Arc sin x = characterization and influence of extraguild prey density. The shorthand X в€јarcsin is used to indicate that the random variable X has the arcsin distribution. An arcsin random variable X has probability density function f(x)= 1
It is a one-to-one function. Here is an example: How to use the arcsine function in the Algebra Coach. Type arcsin(x) into the textbox, where x is the argument. The following characterization of the arcsine density is established: For example, for a general random orthogonal with respect to any weight function w
For a binomial distribution, variance is a function and two of the most common variance-stabilizing transformations are the logit and arcsine For example The proof that \( g \) is a valid probability density function explains the name arcsine distribution, as does the form of the distribution function given next. \
Examples # NOT RUN { A <- Arcsine() # A is a Arcsine distribution with shape1 = 1 and shape2 = 1. r(A)(3) # three random number generated from this distribution, e.g ArcSinDistribution {x (CDF), which is a normalized arc sine function. Its probability density function (PDF) For example, the standard arc
How to transform an arcsine distribution to a normal distribution? Is there a function that can transform this distribution to a normal distribution? Levy’s arcsine law in that a trader spends in "black" or "red" then the distribution of his P & L is given by the arcsine probability density function | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9483637809753418, "perplexity": 695.6444155522995}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500365.52/warc/CC-MAIN-20230206212647-20230207002647-00712.warc.gz"} |
https://www.physicsforums.com/threads/physic-problem.284215/ | # Physic Problem!
1. Jan 11, 2009
### 09mcibrian
A 250 pound basketball player pushes against the floor with an additional force of 30 pounds in order to accelerate upwards after the rebound. The floor exerts a force back upon the player of ______
A) 30 pounds
B) 220 pounds
C) 250 pounds
D) 280 pounds
E) not enough info to tell
I think it is D because wouldn't you need the same force?
Last edited by a moderator: Jan 11, 2009
2. Jan 11, 2009
### Hootenanny
Staff Emeritus
Welcome to Physics Forums.
You are indeed correct. The Baseball player exerts a total force of 280 pounds (250 pounds from his weight and an additional 30 pounds to accelerate upwards), hence by Newton's 3rd law the ground must exert a force of equal magnitude on the player.
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https://dantopology.wordpress.com/category/omega-1/ | # Counterexample 106 from Steen and Seebach
As the title suggests, this post discusses counterexample 106 in Steen and Seebach [2]. We extend the discussion by adding two facts not found in [2].
The counterexample 106 is the space $X=\omega_1 \times I^I$, which is the product of $\omega_1$ with the interval topology and the product space $I^I=\prod_{t \in I} I$ where $I$ is of course the unit interval $[0,1]$. The notation of $\omega_1$, the first uncountable ordinal, in Steen and Seebach is $[0,\Omega)$.
Another way to notate the example $X$ is the product space $\prod_{t \in I} X_t$ where $X_0$ is $\omega_1$ and $X_t$ is the unit interval $I$ for all $t>0$. Thus in this product space, all factors except for one factor is the unit interval and the lone non-compact factor is the first uncountable ordinal. The factor of $\omega_1$ makes this product space an interesting example.
The following lists out the basic topological properties of the space that $X=\omega_1 \times I^I$ are covered in [2].
• The space $X$ is Hausdorff and completely regular.
• The space $X$ is countably compact.
• The space $X$ is neither compact nor sequentially compact.
• The space $X$ is neither separable, Lindelof nor $\sigma$-compact.
• The space $X$ is not first countable.
• The space $X$ is locally compact.
All the above bullet points are discussed in Steen and Seebach. In this post we add the following two facts.
• The space $X$ is not normal.
• The space $X$ has a dense subspace that is normal.
It follows from these bullet points that the space $X$ is an example of a completely regular space that is not normal. Not being a normal space, $X$ is then not metrizable. Of course there are other ways to show that $X$ is not metrizable. One is that neither of the two factors $\omega_1$ or $I^I$ is metrizable. Another is that $X$ is not first countable.
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The space $X$ is not normal
Now we are ready to discuss the non-normality of the example. It is a natural question to ask whether the example $X=\omega_1 \times I^I$ is normal. The fact that it was not discussed in [2] could be that the tool for answering the normality question was not yet available at the time [2] was originally published, though we do not know for sure. It turns out that the tool became available in the paper [1] published a few years after the publication of [2]. The key to showing the normality (or the lack of) in the example $X=\omega_1 \times I^I$ is to show whether the second factor $I^I$ is a countably tight space.
The main result in [1] is discussed in this previous post. Theorem 1 in the previous post states that for any compact space $Y$, the product $\omega_1 \times Y$ is normal if and only if $Y$ is countably tight. Thus the normality of the space $X$ (or the lack of) hinges on whether the compact factor $I^I=\prod_{t \in I} I$ is countably tight.
A space $Y$ is countably tight (or has countable tightness) if for each $S \subset Y$ and for each $x \in \overline{S}$, there exists some countable $B \subset S$ such that $x \in \overline{B}$. The definitions of tightness in general and countable tightness in particular are discussed here.
To show that the product space $I^I=\prod_{t \in I} I$ is not countably tight, we let $S$ be the subspace of $I^I$ consisting of points, each of which is non-zero on at most countably many coordinates. Specifically $S$ is defined as follows:
$S=\Sigma_{t \in I} I=\left\{y \in I^I: y(t) \ne 0 \text{ for at most countably many } t \in I \right\}$
The set $S$ just defined is also called the $\Sigma$-product of copies of unit interval $I$. Let $g \in I^I$ be defined by $g(t)=1$ for all $t \in I$. It follows that $g \in \overline{S}$. It can also be verified that $g \notin \overline{B}$ for any countable $B \subset S$. This shows that the product space $I^I=\prod_{t \in I} I$ is not countably tight.
By Theorem 1 found in this link, the space $X=\omega_1 \times I^I$ is not normal.
____________________________________________________________________
The space $X$ has a dense subspace that is normal
Now that we know $X=\omega_1 \times I^I$ is not normal, a natural question is whether it has a dense subspace that is normal. Consider the subspace $\omega_1 \times S$ where $S$ is the $\Sigma$-product $S=\Sigma_{t \in I} I$ defined in the preceding section. The subspace $S$ is dense in the product space $I^I$. Thus $\omega_1 \times S$ is dense in $X=\omega_1 \times I^I$. The space $S$ is normal since the $\Sigma$-product of separable metric spaces is normal. Furthermore, $\omega_1$ can be embedded as a closed subspace of $S=\Sigma_{t \in I} I$. Then $\omega_1 \times S$ is homeomorphic to a closed subspace of $S \times S$. Since $S \times S \cong S$, the space $\omega_1 \times S$ is normal.
____________________________________________________________________
Reference
1. Nogura, T., Tightness of compact Hausdorff space and normality of product spaces, J. Math. Soc. Japan, 28, 360-362, 1976
2. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.
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$\copyright \ 2015 \text{ by Dan Ma}$
# Normality in the powers of countably compact spaces
Let $\omega_1$ be the first uncountable ordinal. The topology on $\omega_1$ we are interested in is the ordered topology, the topology induced by the well ordering. The space $\omega_1$ is also called the space of all countable ordinals since it consists of all ordinals that are countable in cardinality. It is a handy example of a countably compact space that is not compact. In this post, we consider normality in the powers of $\omega_1$. We also make comments on normality in the powers of a countably compact non-compact space.
Let $\omega$ be the first infinite ordinal. It is well known that $\omega^{\omega_1}$, the product space of $\omega_1$ many copies of $\omega$, is not normal (a proof can be found in this earlier post). This means that any product space $\prod_{\alpha<\kappa} X_\alpha$, with uncountably many factors, is not normal as long as each factor $X_\alpha$ contains a countable discrete space as a closed subspace. Thus in order to discuss normality in the product space $\prod_{\alpha<\kappa} X_\alpha$, the interesting case is when each factor is infinite but contains no countable closed discrete subspace (i.e. no closed copies of $\omega$). In other words, the interesting case is that each factor $X_\alpha$ is a countably compact space that is not compact (see this earlier post for a discussion of countably compactness). In particular, we would like to discuss normality in $X^{\kappa}$ where $X$ is a countably non-compact space. In this post we start with the space $X=\omega_1$ of the countable ordinals. We examine $\omega_1$ power $\omega_1^{\omega_1}$ as well as the countable power $\omega_1^{\omega}$. The former is not normal while the latter is normal. The proof that $\omega_1^{\omega}$ is normal is an application of the normality of $\Sigma$-product of the real line.
____________________________________________________________________
The uncountable product
Theorem 1
The product space $\prod_{\alpha<\omega_1} \omega_1=\omega_1^{\omega_1}$ is not normal.
Theorem 1 follows from Theorem 2 below. For any space $X$, a collection $\mathcal{C}$ of subsets of $X$ is said to have the finite intersection property if for any finite $\mathcal{F} \subset \mathcal{C}$, the intersection $\cap \mathcal{F} \ne \varnothing$. Such a collection $\mathcal{C}$ is called an f.i.p collection for short. It is well known that a space $X$ is compact if and only collection $\mathcal{C}$ of closed subsets of $X$ satisfying the finite intersection property has non-empty intersection (see Theorem 1 in this earlier post). Thus any non-compact space has an f.i.p. collection of closed sets that have empty intersection.
In the space $X=\omega_1$, there is an f.i.p. collection of cardinality $\omega_1$ using its linear order. For each $\alpha<\omega_1$, let $C_\alpha=\left\{\beta<\omega_1: \alpha \le \beta \right\}$. Let $\mathcal{C}=\left\{C_\alpha: \alpha < \omega_1 \right\}$. It is a collection of closed subsets of $X=\omega_1$. It is an f.i.p. collection and has empty intersection. It turns out that for any countably compact space $X$ with an f.i.p. collection of cardinality $\omega_1$ that has empty intersection, the product space $X^{\omega_1}$ is not normal.
Theorem 2
Let $X$ be a countably compact space. Suppose that there exists a collection $\mathcal{C}=\left\{C_\alpha: \alpha < \omega_1 \right\}$ of closed subsets of $X$ such that $\mathcal{C}$ has the finite intersection property and that $\mathcal{C}$ has empty intersection. Then the product space $X^{\omega_1}$ is not normal.
Proof of Theorem 2
Let’s set up some notations on product space that will make the argument easier to follow. By a standard basic open set in the product space $X^{\omega_1}=\prod_{\alpha<\omega_1} X$, we mean a set of the form $O=\prod_{\alpha<\omega_1} O_\alpha$ such that each $O_\alpha$ is an open subset of $X$ and that $O_\alpha=X$ for all but finitely many $\alpha<\omega_1$. Given a standard basic open set $O=\prod_{\alpha<\omega_1} O_\alpha$, the notation $\text{Supp}(O)$ refers to the finite set of $\alpha$ for which $O_\alpha \ne X$. For any set $M \subset \omega_1$, the notation $\pi_M$ refers to the projection map from $\prod_{\alpha<\omega_1} X$ to the subproduct $\prod_{\alpha \in M} X$. Each element $d \in X^{\omega_1}$ can be considered a function $d: \omega_1 \rightarrow X$. By $(d)_\alpha$, we mean $(d)_\alpha=d(\alpha)$.
For each $t \in X$, let $f_t: \omega_1 \rightarrow X$ be the constant function whose constant value is $t$. Consider the following subspaces of $X^{\omega_1}$.
$H=\prod_{\alpha<\omega_1} C_\alpha$
$\displaystyle K=\left\{f_t: t \in X \right\}$
Both $H$ and $K$ are closed subsets of the product space $X^{\omega_1}$. Because the collection $\mathcal{C}$ has empty intersection, $H \cap K=\varnothing$. We show that $H$ and $K$ cannot be separated by disjoint open sets. To this end, let $U$ and $V$ be open subsets of $X^{\omega_1}$ such that $H \subset U$ and $K \subset V$.
Let $d_1 \in H$. Choose a standard basic open set $O_1$ such that $d_1 \in O_1 \subset U$. Let $S_1=\text{Supp}(O_1)$. Since $S_1$ is the support of $O_1$, it follows that $\pi_{S_1}^{-1}(\pi_{S_1}(d_1)) \subset O_1 \subset U$. Since $\mathcal{C}$ has the finite intersection property, there exists $a_1 \in \bigcap_{\alpha \in S_1} C_\alpha$.
Define $d_2 \in H$ such that $(d_2)_\alpha=a_1$ for all $\alpha \in S_1$ and $(d_2)_\alpha=(d_1)_\alpha$ for all $\alpha \in \omega_1-S_1$. Choose a standard basic open set $O_2$ such that $d_2 \in O_2 \subset U$. Let $S_2=\text{Supp}(O_2)$. It is possible to ensure that $S_1 \subset S_2$ by making more factors of $O_2$ different from $X$. We have $\pi_{S_2}^{-1}(\pi_{S_2}(d_2)) \subset O_2 \subset U$. Since $\mathcal{C}$ has the finite intersection property, there exists $a_2 \in \bigcap_{\alpha \in S_2} C_\alpha$.
Now choose a point $d_3 \in H$ such that $(d_3)_\alpha=a_2$ for all $\alpha \in S_2$ and $(d_3)_\alpha=(d_2)_\alpha$ for all $\alpha \in \omega_1-S_2$. Continue on with this inductive process. When the inductive process is completed, we have the following sequences:
• a sequence $d_1,d_2,d_3,\cdots$ of point of $H=\prod_{\alpha<\omega_1} C_\alpha$,
• a sequence $S_1 \subset S_2 \subset S_3 \subset \cdots$ of finite subsets of $\omega_1$,
• a sequence $a_1,a_2,a_3,\cdots$ of points of $X$
such that for all $n \ge 2$, $(d_n)_\alpha=a_{n-1}$ for all $\alpha \in S_{n-1}$ and $\pi_{S_n}^{-1}(\pi_{S_n}(d_n)) \subset U$. Let $A=\left\{a_1,a_2,a_3,\cdots \right\}$. Either $A$ is finite or $A$ is infinite. Let’s examine the two cases.
Case 1
Suppose that $A$ is infinite. Since $X$ is countably compact, $A$ has a limit point $a$. That means that every open set containing $a$ contains some $a_n \ne a$. For each $n \ge 2$, define $y_n \in \prod_{\alpha< \omega_1} X$ such that
• $(y_n)_\alpha=(d_n)_\alpha=a_{n-1}$ for all $\alpha \in S_n$,
• $(y_n)_\alpha=a$ for all $\alpha \in \omega_1-S_n$
From the induction step, we have $y_n \in \pi_{S_n}^{-1}(\pi_{S_n}(d_n)) \subset U$ for all $n$. Let $t=f_a \in K$, the constant function whose constant value is $a$. It follows that $t$ is a limit of $\left\{y_1,y_2,y_3,\cdots \right\}$. This means that $t \in \overline{U}$. Since $t \in K \subset V$, $U \cap V \ne \varnothing$.
Case 2
Suppose that $A$ is finite. Then there is some $m$ such that $a_m=a_j$ for all $j \ge m$. For each $n \ge 2$, define $y_n \in \prod_{\alpha< \omega_1} X$ such that
• $(y_n)_\alpha=(d_n)_\alpha=a_{n-1}$ for all $\alpha \in S_n$,
• $(y_n)_\alpha=a_m$ for all $\alpha \in \omega_1-S_n$
As in Case 1, we have $y_n \in \pi_{S_n}^{-1}(\pi_{S_n}(d_n)) \subset U$ for all $n$. Let $t=f_{a_m} \in K$, the constant function whose constant value is $a_m$. It follows that $t=y_n$ for all $n \ge m+1$. Thus $U \cap V \ne \varnothing$.
Both cases show that $U \cap V \ne \varnothing$. This completes the proof the product space $X^{\omega_1}$ is not normal. $\blacksquare$
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The countable product
Theorem 3
The product space $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ is normal.
Proof of Theorem 3
The proof here actually proves more than normality. It shows that $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ is collectionwise normal, which is stronger than normality. The proof makes use of the $\Sigma$-product of $\kappa$ many copies of $\mathbb{R}$, which is the following subspace of the product space $\mathbb{R}^{\kappa}$.
$\Sigma(\kappa)=\left\{x \in \mathbb{R}^{\kappa}: x(\alpha) \ne 0 \text{ for at most countably many } \alpha<\kappa \right\}$
It is well known that $\Sigma(\kappa)$ is collectionwise normal (see this earlier post). We show that $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$ is a closed subspace of $\Sigma(\kappa)$ where $\kappa=\omega_1$. Thus $\omega_1^{\omega}$ is collectionwise normal. This is established in the following claims.
Claim 1
We show that the space $\omega_1$ is embedded as a closed subspace of $\Sigma(\omega_1)$.
For each $\beta<\omega_1$, define $f_\beta:\omega_1 \rightarrow \mathbb{R}$ such that $f_\beta(\gamma)=1$ for all $\gamma<\beta$ and $f_\beta(\gamma)=0$ for all $\beta \le \gamma <\omega_1$. Let $W=\left\{f_\beta: \beta<\omega_1 \right\}$. We show that $W$ is a closed subset of $\Sigma(\omega_1)$ and $W$ is homeomorphic to $\omega_1$ according to the mapping $f_\beta \rightarrow W$.
First, we show $W$ is closed by showing that $\Sigma(\omega_1)-W$ is open. Let $y \in \Sigma(\omega_1)-W$. We show that there is an open set containing $y$ that contains no points of $W$.
Suppose that for some $\gamma<\omega_1$, $y_\gamma \in O=\mathbb{R}-\left\{0,1 \right\}$. Consider the open set $Q=(\prod_{\alpha<\omega_1} Q_\alpha) \cap \Sigma(\omega_1)$ where $Q_\alpha=\mathbb{R}$ except that $Q_\gamma=O$. Then $y \in Q$ and $Q \cap W=\varnothing$.
So we can assume that for all $\gamma<\omega_1$, $y_\gamma \in \left\{0, 1 \right\}$. There must be some $\theta$ such that $y_\theta=1$. Otherwise, $y=f_0 \in W$. Since $y \ne f_\theta$, there must be some $\delta<\gamma$ such that $y_\delta=0$. Now choose the open interval $T_\theta=(0.9,1.1)$ and the open interval $T_\delta=(-0.1,0.1)$. Consider the open set $M=(\prod_{\alpha<\omega_1} M_\alpha) \cap \Sigma(\omega_1)$ such that $M_\alpha=\mathbb{R}$ except for $M_\theta=T_\theta$ and $M_\delta=T_\delta$. Then $y \in M$ and $M \cap W=\varnothing$. We have just established that $W$ is closed in $\Sigma(\omega_1)$.
Consider the mapping $f_\beta \rightarrow W$. Based on how it is defined, it is straightforward to show that it is a homeomorphism between $\omega_1$ and $W$.
Claim 2
The $\Sigma$-product $\Sigma(\omega_1)$ has the interesting property it is homeomorphic to its countable power, i.e.
$\Sigma(\omega_1) \cong \Sigma(\omega_1) \times \Sigma(\omega_1) \times \Sigma(\omega_1) \cdots \ \ \ \ \ \ \ \ \ \ \ \text{(countably many times)}$.
Because each element of $\Sigma(\omega_1)$ is nonzero only at countably many coordinates, concatenating countably many elements of $\Sigma(\omega_1)$ produces an element of $\Sigma(\omega_1)$. Thus Claim 2 can be easily verified. With above claims, we can see that
$\displaystyle \omega_1^{\omega}=\omega_1 \times \omega_1 \times \omega_1 \times \cdots \subset \Sigma(\omega_1) \times \Sigma(\omega_1) \times \Sigma(\omega_1) \cdots \cong \Sigma(\omega_1)$
Thus $\omega_1^{\omega}$ is a closed subspace of $\Sigma(\omega_1)$. Any closed subspace of a collectionwise normal space is collectionwise normal. We have established that $\omega_1^{\omega}$ is normal. $\blacksquare$
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The normality in the powers of $X$
We have established that $\prod_{\alpha<\omega_1} \omega_1=\omega_1^{\omega_1}$ is not normal. Hence any higher uncountable power of $\omega_1$ is not normal. We have also established that $\prod_{\alpha<\omega} \omega_1=\omega_1^{\omega}$, the countable power of $\omega_1$ is normal (in fact collectionwise normal). Hence any finite power of $\omega_1$ is normal. However $\omega_1^{\omega}$ is not hereditarily normal. One of the exercises below is to show that $\omega_1 \times \omega_1$ is not hereditarily normal.
Theorem 2 can be generalized as follows:
Theorem 4
Let $X$ be a countably compact space has an f.i.p. collection $\mathcal{C}$ of closed sets such that $\bigcap \mathcal{C}=\varnothing$. Then $X^{\kappa}$ is not normal where $\kappa=\lvert \mathcal{C} \lvert$.
The proof of Theorem 2 would go exactly like that of Theorem 2. Consider the following two theorems.
Theorem 5
Let $X$ be a countably compact space that is not compact. Then there exists a cardinal number $\kappa$ such that $X^{\kappa}$ is not normal and $X^{\tau}$ is normal for all cardinal number $\tau<\kappa$.
By the non-compactness of $X$, there exists an f.i.p. collection $\mathcal{C}$ of closed subsets of $X$ such that $\bigcap \mathcal{C}=\varnothing$. Let $\kappa$ be the least cardinality of such an f.i.p. collection. By Theorem 4, that $X^{\kappa}$ is not normal. Because $\kappa$ is least, any smaller power of $X$ must be normal.
Theorem 6
Let $X$ be a space that is not countably compact. Then $X^{\kappa}$ is not normal for any cardinal number $\kappa \ge \omega_1$.
Since the space $X$ in Theorem 6 is not countably compact, it would contain a closed and discrete subspace that is countable. By a theorem of A. H. Stone, $\omega^{\omega_1}$ is not normal. Then $\omega^{\omega_1}$ is a closed subspace of $X^{\omega_1}$.
Thus between Theorem 5 and Theorem 6, we can say that for any non-compact space $X$, $X^{\kappa}$ is not normal for some cardinal number $\kappa$. The $\kappa$ from either Theorem 5 or Theorem 6 is at least $\omega_1$. Interestingly for some spaces, the $\kappa$ can be much smaller. For example, for the Sorgenfrey line, $\kappa=2$. For some spaces (e.g. the Michael line), $\kappa=\omega$.
Theorems 4, 5 and 6 are related to a theorem that is due to Noble.
Theorem 7 (Noble)
If each power of a space $X$ is normal, then $X$ is compact.
A proof of Noble’s theorem is given in this earlier post, the proof of which is very similar to the proof of Theorem 2 given above. So the above discussion the normality of powers of $X$ is just another way of discussing Theorem 7. According to Theorem 7, if $X$ is not compact, some power of $X$ is not normal.
The material discussed in this post is excellent training ground for topology. Regarding powers of countably compact space and product of countably compact spaces, there are many topics for further discussion/investigation. One possibility is to examine normality in $X^{\kappa}$ for more examples of countably compact non-compact $X$. One particular interesting example would be a countably compact non-compact $X$ such that the least power $\kappa$ for non-normality in $X^{\kappa}$ is more than $\omega_1$. A possible candidate could be the second uncountable ordinal $\omega_2$. By Theorem 2, $\omega_2^{\omega_2}$ is not normal. The issue is whether the $\omega_1$ power $\omega_2^{\omega_1}$ and countable power $\omega_2^{\omega}$ are normal.
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Exercises
Exercise 1
Show that $\omega_1 \times \omega_1$ is not hereditarily normal.
Exercise 2
Show that the mapping $f_\beta \rightarrow W$ in Claim 3 in the proof of Theorem 3 is a homeomorphism.
Exercise 3
The proof of Theorem 3 shows that the space $\omega_1$ is a closed subspace of the $\Sigma$-product of the real line. Show that $\omega_1$ can be embedded in the $\Sigma$-product of arbitrary spaces.
For each $\alpha<\omega_1$, let $X_\alpha$ be a space with at least two points. Let $p \in \prod_{\alpha<\omega_1} X_\alpha$. The $\Sigma$-product of the spaces $X_\alpha$ is the following subspace of the product space $\prod_{\alpha<\omega_1} X_\alpha$.
$\Sigma(X_\alpha)=\left\{x \in \prod_{\alpha<\omega_1} X_\alpha: x(\alpha) \ne p(\alpha) \ \text{for at most countably many } \alpha<\omega_1 \right\}$
The point $p$ is the center of the $\Sigma$-product. Show that the space $\Sigma(X_\alpha)$ contains $\omega_1$ as a closed subspace.
Exercise 4
Find a direct proof of Theorem 3, that $\omega_1^{\omega}$ is normal.
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$\copyright \ 2015 \text{ by Dan Ma}$
# Cp(omega 1 + 1) is monolithic and Frechet-Urysohn
This is another post that discusses what $C_p(X)$ is like when $X$ is a compact space. In this post, we discuss the example $C_p(\omega_1+1)$ where $\omega_1+1$ is the first compact uncountable ordinal. Note that $\omega_1+1$ is the successor to $\omega_1$, which is the first (or least) uncountable ordinal. The function space $C_p(\omega_1+1)$ is monolithic and is a Frechet-Urysohn space. Interestingly, the first property is possessed by $C_p(X)$ for all compact spaces $X$. The second property is possessed by all compact scattered spaces. After we discuss $C_p(\omega_1+1)$, we discuss briefly the general results for $C_p(X)$.
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Initial discussion
The function space $C_p(\omega_1+1)$ is a dense subspace of the product space $\mathbb{R}^{\omega_1}$. In fact, $C_p(\omega_1+1)$ is homeomorphic to a subspace of the following subspace of $\mathbb{R}^{\omega_1}$:
$\Sigma(\omega_1)=\left\{x \in \mathbb{R}^{\omega_1}: x_\alpha \ne 0 \text{ for at most countably many } \alpha < \omega_1 \right\}$
The subspace $\Sigma(\omega_1)$ is the $\Sigma$-product of $\omega_1$ many copies of the real line $\mathbb{R}$. The $\Sigma$-product of separable metric spaces is monolithic (see here). The $\Sigma$-product of first countable spaces is Frechet-Urysohn (see here). Thus $\Sigma(\omega_1)$ has both of these properties. Since the properties of monolithicity and being Frechet-Urysohn are carried over to subspaces, the function space $C_p(\omega_1+1)$ has both of these properties. The key to the discussion is then to show that $C_p(\omega_1+1)$ is homeopmophic to a subspace of the $\Sigma$-product $\Sigma(\omega_1)$.
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Connection to $\Sigma$-product
We show that the function space $C_p(\omega_1+1)$ is homeomorphic to a subspace of the $\Sigma$-product of $\omega_1$ many copies of the real lines. Let $Y_0$ be the following subspace of $C_p(\omega_1+1)$:
$Y_0=\left\{f \in C_p(\omega_1+1): f(\omega_1)=0 \right\}$
Every function in $Y_0$ has non-zero values at only countably points of $\omega_1+1$. Thus $Y_0$ can be regarded as a subspace of the $\Sigma$-product $\Sigma(\omega_1)$.
By Theorem 1 in this previous post, $C_p(\omega_1+1) \cong Y_0 \times \mathbb{R}$, i.e, the function space $C_p(\omega_1+1)$ is homeomorphic to the product space $Y_0 \times \mathbb{R}$. On the other hand, the product $Y_0 \times \mathbb{R}$ can also be regarded as a subspace of the $\Sigma$-product $\Sigma(\omega_1)$. Basically adding one additional factor of the real line to $Y_0$ still results in a subspace of the $\Sigma$-product. Thus we have:
$C_p(\omega_1+1) \cong Y_0 \times \mathbb{R} \subset \Sigma(\omega_1) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
Thus $C_p(\omega_1+1)$ possesses all the hereditary properties of $\Sigma(\omega_1)$. Another observation we can make is that $\Sigma(\omega_1)$ is not hereditarily normal. The function space $C_p(\omega_1+1)$ is not normal (see here). The $\Sigma$-product $\Sigma(\omega_1)$ is normal (see here). Thus $\Sigma(\omega_1)$ is not hereditarily normal.
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A closer look at $C_p(\omega_1+1)$
In fact $C_p(\omega_1+1)$ has a stronger property that being monolithic. It is strongly monolithic. We use homeomorphic relation in (1) above to get some insight. Let $h$ be a homeomorphism from $C_p(\omega_1+1)$ onto $Y_0 \times \mathbb{R}$. For each $\alpha<\omega_1$, let $H_\alpha$ be defined as follows:
$H_\alpha=\left\{f \in C_p(\omega_1+1): f(\gamma)=0 \ \forall \ \alpha<\gamma<\omega_1 \right\}$
Clearly $H_\alpha \subset Y_0$. Furthermore $H_\alpha$ can be considered as a subspace of $\mathbb{R}^\omega$ and is thus metrizable. Let $A$ be a countable subset of $C_p(\omega_1+1)$. Then $h(A) \subset H_\alpha \times \mathbb{R}$ for some $\alpha<\omega_1$. The set $H_\alpha \times \mathbb{R}$ is metrizable. The set $H_\alpha \times \mathbb{R}$ is also a closed subset of $Y_0 \times \mathbb{R}$. Then $\overline{A}$ is contained in $H_\alpha \times \mathbb{R}$ and is therefore metrizable. We have shown that the closure of every countable subspace of $C_p(\omega_1+1)$ is metrizable. In other words, every separable subspace of $C_p(\omega_1+1)$ is metrizable. This property follows from the fact that $C_p(\omega_1+1)$ is strongly monolithic.
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Monolithicity and Frechet-Urysohn property
As indicated at the beginning, the $\Sigma$-product $\Sigma(\omega_1)$ is monolithic (in fact strongly monolithic; see here) and is a Frechet-Urysohn space (see here). Thus the function space $C_p(\omega_1+1)$ is both strongly monolithic and Frechet-Urysohn.
Let $\tau$ be an infinite cardinal. A space $X$ is $\tau$-monolithic if for any $A \subset X$ with $\lvert A \lvert \le \tau$, we have $nw(\overline{A}) \le \tau$. A space $X$ is monolithic if it is $\tau$-monolithic for all infinite cardinal $\tau$. It is straightforward to show that $X$ is monolithic if and only of for every subspace $Y$ of $X$, the density of $Y$ equals to the network weight of $Y$, i.e., $d(Y)=nw(Y)$. A longer discussion of the definition of monolithicity is found here.
A space $X$ is strongly $\tau$-monolithic if for any $A \subset X$ with $\lvert A \lvert \le \tau$, we have $w(\overline{A}) \le \tau$. A space $X$ is strongly monolithic if it is strongly $\tau$-monolithic for all infinite cardinal $\tau$. It is straightforward to show that $X$ is strongly monolithic if and only if for every subspace $Y$ of $X$, the density of $Y$ equals to the weight of $Y$, i.e., $d(Y)=w(Y)$.
In any monolithic space, the density and the network weight coincide for any subspace, and in particular, any subspace that is separable has a countable network. As a result, any separable monolithic space has a countable network. Thus any separable space with no countable network is not monolithic, e.g., the Sorgenfrey line. On the other hand, any space that has a countable network is monolithic.
In any strongly monolithic space, the density and the weight coincide for any subspace, and in particular any separable subspace is metrizable. Thus being separable is an indicator of metrizability among the subspaces of a strongly monolithic space. As a result, any separable strongly monolithic space is metrizable. Any separable space that is not metrizable is not strongly monolithic. Thus any non-metrizable space that has a countable network is an example of a monolithic space that is not strongly monolithic, e.g., the function space $C_p([0,1])$. It is clear that all metrizable spaces are strongly monolithic.
The function space $C_p(\omega_1+1)$ is not separable. Since it is strongly monolithic, every separable subspace of $C_p(\omega_1+1)$ is metrizable. We can see this by knowing that $C_p(\omega_1+1)$ is a subspace of the $\Sigma$-product $\Sigma(\omega_1)$, or by using the homeomorphism $h$ as in the previous section.
For any compact space $X$, $C_p(X)$ is countably tight (see this previous post). In the case of the compact uncountable ordinal $\omega_1+1$, $C_p(\omega_1+1)$ has the stronger property of being Frechet-Urysohn. A space $Y$ is said to be a Frechet-Urysohn space (also called a Frechet space) if for each $y \in Y$ and for each $M \subset Y$, if $y \in \overline{M}$, then there exists a sequence $\left\{y_n \in M: n=1,2,3,\cdots \right\}$ such that the sequence converges to $y$. As we shall see below, $C_p(X)$ is rarely Frechet-Urysohn.
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General discussion
For any compact space $X$, $C_p(X)$ is monolithic but does not have to be strongly monolithic. The monolithicity of $C_p(X)$ follows from the following theorem, which is Theorem II.6.8 in [1].
Theorem 1
Then the function space $C_p(X)$ is monolithic if and only if $X$ is a stable space.
See chapter 3 section 6 of [1] for a discussion of stable spaces. We give the definition here. A space $X$ is stable if for any continuous image $Y$ of $X$, the weak weight of $Y$, denoted by $ww(Y)$, coincides with the network weight of $Y$, denoted by $nw(Y)$. In [1], $ww(Y)$ is notated by $iw(Y)$. The cardinal function $ww(Y)$ is the minimum cardinality of all $w(T)$, the weight of $T$, for which there exists a continuous bijection from $Y$ onto $T$.
All compact spaces are stable. Let $X$ be compact. For any continuous image $Y$ of $X$, $Y$ is also compact and $ww(Y)=w(Y)$, since any continuous bijection from $Y$ onto any space $T$ is a homeomorphism. Note that $ww(Y) \le nw(Y) \le w(Y)$ always holds. Thus $ww(Y)=w(Y)$ implies that $ww(Y)=nw(Y)$. Thus we have:
Corollary 2
Let $X$ be a compact space. Then the function space $C_p(X)$ is monolithic.
However, the strong monolithicity of $C_p(\omega_1+1)$ does not hold in general for $C_p(X)$ for compact $X$. As indicated above, $C_p([0,1])$ is monolithic but not strongly monolithic. The following theorem is Theorem II.7.9 in [1] and characterizes the strong monolithicity of $C_p(X)$.
Theorem 3
Let $X$ be a space. Then $C_p(X)$ is strongly monolithic if and only if $X$ is simple.
A space $X$ is $\tau$-simple if whenever $Y$ is a continuous image of $X$, if the weight of $Y$ $\le \tau$, then the cardinality of $Y$ $\le \tau$. A space $X$ is simple if it is $\tau$-simple for all infinite cardinal numbers $\tau$. Interestingly, any separable metric space that is uncountable is not $\omega$-simple. Thus $[0,1]$ is not $\omega$-simple and $C_p([0,1])$ is not strongly monolithic, according to Theorem 3.
For compact spaces $X$, $C_p(X)$ is rarely a Frechet-Urysohn space as evidenced by the following theorem, which is Theorem III.1.2 in [1].
Theorem 4
Let $X$ be a compact space. Then the following conditions are equivalent.
1. $C_p(X)$ is a Frechet-Urysohn space.
2. $C_p(X)$ is a k-space.
3. The compact space $X$ is a scattered space.
A space $X$ is a scattered space if for every non-empty subspace $Y$ of $X$, there exists an isolated point of $Y$ (relative to the topology of $Y$). Any space of ordinals is scattered since every non-empty subset has a least element. Thus $\omega_1+1$ is a scattered space. On the other hand, the unit interval $[0,1]$ with the Euclidean topology is not scattered. According to this theorem, $C_p([0,1])$ cannot be a Frechet-Urysohn space.
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Reference
1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
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$\copyright \ 2014 \text{ by Dan Ma}$
# Normal x compact needs not be subnormal
In this post, we revisit a counterexample that was discussed previously in this blog. A previous post called “Normal x compact needs not be normal” shows that the Tychonoff product of two normal spaces needs not be normal even when one of the factors is compact. The example is $\omega_1 \times (\omega_1+1)$. In this post, we show that $\omega_1 \times (\omega_1+1)$ fails even to be subnormal. Both $\omega_1$ and $\omega_1+1$ are spaces of ordinals. Thus they are completely normal (equivalent to hereditarily normal). The second factor is also a compact space. Yet their product is not only not normal; it is not even subnormal.
A subset $M$ of a space $Y$ is a $G_\delta$ subset of $Y$ (or a $G_\delta$-set in $Y$) if $M$ is the intersection of countably many open subsets of $Y$. A subset $M$ of a space $Y$ is a $F_\sigma$ subset of $Y$ (or a $F_\sigma$-set in $Y$) if $Y-M$ is a $G_\delta$-set in $Y$ (equivalently if $M$ is the union of countably many closed subsets of $Y$).
A space $Y$ is normal if for any disjoint closed subsets $H$ and $K$ of $Y$, there exist disjoint open subsets $U_H$ and $U_K$ of $Y$ such that $H \subset U_H$ and $K \subset U_K$. A space $Y$ is subnormal if for any disjoint closed subsets $H$ and $K$ of $Y$, there exist disjoint $G_\delta$ subsets $V_H$ and $V_K$ of $Y$ such that $H \subset V_H$ and $K \subset V_K$. Clearly any normal space is subnormal.
A space $Y$ is pseudonormal if for any disjoint closed subsets $H$ and $K$ of $Y$ (one of which is countable), there exist disjoint open subsets $U_H$ and $U_K$ of $Y$ such that $H \subset U_H$ and $K \subset U_K$. The space $\omega_1 \times (\omega_1+1)$ is pseudonormal (see this previous post). The Sorgenfrey plane is an example of a subnormal space that is not pseudonormal (see here). Thus the two weak forms of normality (pseudonormal and subnormal) are not equivalent.
The same two disjoint closed sets that prove the non-normality of $\omega_1 \times (\omega_1+1)$ are also used for proving non-subnormality. The two closed sets are:
$H=\left\{(\alpha,\alpha): \alpha<\omega_1 \right\}$
$K=\left\{(\alpha,\omega_1): \alpha<\omega_1 \right\}$
The key tool, as in the proof for non-normality, is the Pressing Down Lemma ([1]). The lemma has been used in a few places in this blog, especially for proving facts about $\omega_1$ (e.g. this previous post on the first uncountable ordinal). Lemma 1 below is a lemma that is derived from the Pressing Down Lemma.
Pressing Down Lemma
Let $S$ be a stationary subset of $\omega_1$. Let $f:S \rightarrow \omega_1$ be a pressing down function, i.e., $f$ satisfies: $\forall \ \alpha \in S, f(\alpha)<\alpha$. Then there exists $\alpha<\omega_1$ such that $f^{-1}(\alpha)$ is a stationary set.
Lemma 1
Let $L=\left\{(\alpha,\alpha) \in \omega_1 \times \omega_1: \alpha \text{ is a limit ordinal} \right\}$. Suppose that $L \subset \bigcap_{n=1}^\infty O_n$ where each $O_n$ is an open subset of $\omega_1 \times \omega_1$. Then $[\gamma,\omega_1) \times [\gamma,\omega_1) \subset \bigcap_{n=1}^\infty O_n$ for some $\gamma<\omega_1$.
Proof of Lemma 1
For each $n$ and for each $\alpha<\omega_1$ where $\alpha$ is a limit, choose $g_n(\alpha)<\alpha$ such that $[g_n(\alpha),\alpha] \times [g_n(\alpha),\alpha] \subset O_n$. The function $g_n$ can be chosen since $O_n$ is open in the product $\omega_1 \times \omega_1$. By the Pressing Down Lemma, for each $n$, there exists $\gamma_n < \omega_1$ and there exists a stationary set $S_n \subset \omega_1$ such that $g_n(\alpha)=\gamma_n$ for all $\alpha \in S_n$. It follows that $[\gamma_n,\omega_1) \times [\gamma_n,\omega_1) \subset O_n$ for each $n$. Choose $\gamma<\omega_1$ such that $\gamma_n<\gamma$ for all $n$. Then $[\gamma,\omega_1) \times [\gamma,\omega_1) \subset O_n$ for each $n$. $\blacksquare$
Theorem 2
The product space $\omega_1 \times (\omega_1+1)$ is not subnormal.
Proof of Theorem 2
Let $H$ and $K$ be defined as above. Suppose $H \subset \bigcap_{n=1}^\infty U_n$ and $K \subset \bigcap_{n=1}^\infty V_n$ where each $U_n$ and each $V_n$ are open in $\omega_1 \times (\omega_1+1)$. Without loss of generality, we can assume that $U_n \cap (\omega_1 \times \left\{\omega_1 \right\})=\varnothing$, i.e., $U_n$ is open in $\omega_1 \times \omega_1$ for each $n$. By Lemma 1, $[\gamma,\omega_1) \times [\gamma,\omega_1) \subset \bigcap_{n=1}^\infty U_n$ for some $\gamma<\omega_1$.
Choose $\beta>\gamma$ such that $\beta$ is a successor ordinal. Note that $(\beta,\omega_1) \in \bigcap_{n=1}^\infty V_n$. For each $n$, there exists some $\delta_n<\omega_1$ such that $\left\{\beta \right\} \times [\delta_n,\omega_1] \subset V_n$. Choose $\delta<\omega_1$ such that $\delta >\delta_n$ for all $n$ and that $\delta >\gamma$. Note that $\left\{\beta \right\} \times [\delta,\omega_1) \subset \bigcap_{n=1}^\infty V_n$. It follows that $\left\{\beta \right\} \times [\delta,\omega_1) \subset [\gamma,\omega_1) \times [\gamma,\omega_1) \subset \bigcap_{n=1}^\infty U_n$. Thus there are no disjoint $G_\delta$ sets separating $H$ and $K$. $\blacksquare$
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Reference
1. Kunen, K., Set Theory, An Introduction to Independence Proofs, First Edition, North-Holland, New York, 1980.
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$\copyright \ 2014 \text{ by Dan Ma}$
# Looking for another closed and discrete subspace of a product space
Let $\omega_1$ be the first uncountable ordinal. In a previous post called Looking for a closed and discrete subspace of a product space, it was shown that the product space $\mathbb{R}^c$, the product of continuum many copies of the real line $\mathbb{R}$, contains a closed and discrete subset of cardinality continuum. This example shows that a product space of uncountably many copies of a “nice” space is “big and wide” enough to hide uncountable closed and discrete sets even when the product space is separable. This post reinforces this same fact by showing that $\mathbb{R}^{\omega_1}$ contains a closed and discrete subset of cardinality $\omega_1$. It follows that for any uncountable cardinal $\tau$, the product space $\mathbb{R}^\tau$ contains an uncountable closed and discrete subset, i.e., the product of uncountably many copies of the real line $\mathbb{R}$ has uncountable extent.
Let $\omega$ be the first infinite ordinal, i.e., the set of all nonnegative integers. Consider $\omega^{\omega_1}$, the product of $\omega_1$ many copies of $\omega$ with the discrete topology. Since $\omega^{\omega_1}$ is a closed subspace of $\mathbb{R}^{\omega_1}$, it suffices to show that $\omega^{\omega_1}$ has an uncountable closed and discrete subset.
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The Construction
We now construct an uncountable closed and discrete subset of $\omega^{\omega_1}$. Let $\delta$ be an infinite ordinal such that $\omega<\delta<\omega_1$. Let $W=\left\{\alpha: \delta \le \alpha<\omega_1 \right\}$. For each $\alpha \in W$, let $Y_\alpha=\left\{\beta<\omega_1: \beta<\alpha \right\}$. We can also use interval notations: $W=[\delta,\omega_1)$ and $Y_\alpha=[0,\alpha)$. Consider $Y_\alpha$ as a space with the discrete topology. Then it is clear that $\omega^{\omega_1}$ is homeomorphic to the product space $\prod_{\alpha \in W} Y_\alpha$. Thus the focus is now on finding an uncountable closed and discrete subset of $\prod_{\alpha \in W} Y_\alpha$.
One interesting fact about the space $\prod_{\alpha \in W} Y_\alpha$ is that every function $f \in \prod_{\alpha \in W} Y_\alpha$ is a pressing down function. That is, for every $f \in \prod_{\alpha \in W} Y_\alpha$, $f(\alpha)<\alpha$ for all $\alpha \in W$. Note that $f$ is defined on $W$, a closed and unbounded subset of $\omega_1$ (hence a stationary set). It follows that for each $f \in \prod_{\alpha \in W} Y_\alpha$, there is a stationary set $S \subset W$ and there exists $\rho<\omega_1$ such that $f(\alpha)=\rho$ for all $\alpha \in S$. This fact is called the pressing down lemma and will be used below. See this post for more information about the pressing down lemma.
For each $\gamma \in W$, let $h_\gamma: Y_{\gamma+1} \rightarrow \delta$ be a one-to-one function. For each $\gamma \in W$, define $t_\gamma \in \prod_{\alpha \in W} Y_\alpha$ as follows:
$t_\gamma(\alpha) = \begin{cases} h_\gamma(\alpha) & \mbox{if } \delta \le \alpha \le \gamma \\ \gamma & \mbox{if } \gamma < \alpha <\omega_1 \end{cases}$
Note that each $t_\gamma$ is a pressing down function. Thus each $t_\gamma \in \prod_{\alpha \in W} Y_\alpha$. Let $T=\left\{t_\gamma: \gamma \in W \right\}$. Clearly $t_\gamma \ne t_\mu$ if $\gamma \ne \mu$. Thus $T$ has cardinality $\omega_1$. We claim that $T$ is a closed and discrete subset of $\prod_{\alpha \in W} Y_\alpha$. It suffices to show that for each $f \in \prod_{\alpha \in W} Y_\alpha$, there exists an open set $V$ with $f \in V$ such that $V$ contains at most one $t_\gamma$.
Let $f \in \prod_{\alpha \in W} Y_\alpha$. As discussed above, there is a stationary set $S \subset W$ and there exists $\rho<\omega_1$ such that $f(\alpha)=\rho$ for all $\alpha \in S$. In particular, choose $\mu, \lambda \in S$ such that $\mu \ne \lambda$. Thus $f(\mu)=f(\lambda)=\rho$. Let $V$ be the open set defined by:
$V=\left\{g \in \prod_{\alpha \in W} Y_\alpha: g(\mu)=g(\lambda)=\rho \right\}$
Clearly, $f \in V$. We show that if $t_\gamma \in V$, then $\gamma=\rho$. Suppose $t_\gamma \in V$. Then $t_\gamma(\mu)=t_\gamma(\lambda)=\rho$. Consider two cases: Case 1: $\delta \le \mu, \lambda \le \gamma$; Case 2: one of $\mu$ and $\lambda>\gamma$. The definition of $t_\gamma$ indicates that $t_\gamma=h_\gamma$ on the interval $[\delta, \gamma]$. Note that $h_\gamma$ is a one-to-one function. Since $t_\gamma(\mu)=t_\gamma(\lambda)$, it cannot be that $\mu, \lambda \in [\delta, \gamma]$, i.e., Case 1 is not possible. Thus Case 2 holds, say $\mu>\gamma$. Then by definition, $t_\gamma(\mu)=\gamma$. Putting everything together, $\gamma=t_\gamma(\mu)=t_\gamma(\lambda)=\rho$. Thus $V \cap T \subset \left\{t_\rho \right\}$. This concludes the proof that the set $T$ is a closed and discrete subset of $\prod_{\alpha \in W} Y_\alpha$. $\blacksquare$
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$\copyright \ 2014 \text{ by Dan Ma}$
# The normality of the product of the first uncountable ordinal with a compact factor
The product of a normal space with a compact space needs not be normal. For example, the product space $\omega_1 \times (\omega_1+1)$ is not normal where $\omega_1$ is the first uncountable ordinal with the order topology and $\omega_1+1$ is the immediate successor of $\omega_1$ (see this post). However, $\omega_1 \times I$ is normal where $I=[0,1]$ is the unit interval with the usual topology. The topological story here is that $I$ has countable tightness while the compact space $\omega_1+1$ does not. In this post, we prove the following theorem:
Theorem 1
Let $Y$ be an infinite compact space. Then the following conditions are equivalent:
1. The product space $\omega_1 \times Y$ is normal.
2. $Y$ has countable tightness, i.e., $t(Y)=\omega$.
Theorem 1 is a special case of the theorem found in [4]. The proof for the direction of countable tightness of $Y$ implies $\omega_1 \times Y$ is normal given in [4] relies on a theorem in another source. In this post we attempt to fill in some of the gaps. For the direction $2 \Longrightarrow 1$, we give a complete proof. For the direction $1 \Longrightarrow 2$, we essentially give the same proof as in [4], proving it by using a series of lemmas (stated below).
The authors in [2] studied the normality of $X \times \omega_1$ where $X$ is not necessarily compact. The necessary definitions are given below. All spaces are at least Hausdorff.
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Definitions and Lemmas
Let $X$ be a topological space. The tightness of $X$, denoted by $t(X)$, is the least infinite cardinal number $\kappa$ such that for any $A \subset X$ and for any $x \in \overline{A}$, there exists a $B \subset A$ such that $x \in \overline{B}$ and $\lvert B \lvert \le \kappa$. When $t(X)=\omega$, we say $X$ has countable tightness or is countably tight. When $t(X)>\omega$, we say $X$ has uncountably tightness or is uncountably tight. An handy example of a space with uncountably tightness is $\omega_1+1=\omega_1 \cup \left\{\omega_1 \right\}$. This space has uncountable tightness at the point $\omega_1$. All first countable spaces and all Frechet spaces have countable tightness. The concept of countable tightness and tightness in general are discussed in more details here.
A sequence $\left\{x_\alpha: \alpha<\tau \right\}$ of points of a space $X$ is said to be a free sequence if for each $\alpha<\tau$, $\overline{\left\{x_\beta: \beta<\alpha \right\}} \cap \overline{\left\{x_\beta: \beta \ge \alpha \right\}}=\varnothing$. When a free sequence is indexed by the cardinal number $\tau$, the free sequence is said to have length $\tau$. The cardinal function $F(X)$ is the least infinite cardinal $\kappa$ such that if $\left\{x_\alpha \in X: \alpha<\tau \right\}$ is a free sequence of length $\tau$, then $\tau \le \kappa$. The concept of tightness was introduced by Arkhangelskii and he proved that $t(X)=F(X)$ (see p. 15 of [3]). This fact implies the following lemma.
Lemma 2
Let $X$ be compact. If $t(X) \ge \tau$, then there exists a free sequence $\left\{x_\alpha \in X: \alpha<\tau \right\}$ of length $\tau$.
A proof of Lemma 2 can be found here.
The proof of the direction $1 \Longrightarrow 2$ also uses the following lemmas.
Lemma 3
For any compact space $Y$, $\beta (\omega_1 \times Y)=(\omega_1+1) \times Y$.
Lemma 4
Let $X$ be a normal space. For every pair $H$ and $K$ of disjoint closed subsets of $X$, $H$ and $K$ have disjoint closures in $\beta X$.
For Lemma 3, see 3.12.20(c) on p. 237 of [1]. For Lemma 4, see 3.6.4 on p. 173 of [1].
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Proof of Theorem 1
$1 \Longrightarrow 2$
Let $X=\omega_1 \times Y$. Suppose that $X$ is normal. Suppose that $Y$ has uncountable tightness, i.e., $t(Y) \ge \omega_1$. By Lemma 2, there exists a free sequence $\left\{y_\alpha \in Y: \alpha<\omega_1 \right\}$. For each $\beta<\omega_1$, let $C_\beta=\left\{y_\alpha: \alpha>\beta \right\}$. Then the collection $\left\{\overline{C_\beta}: \beta<\omega_1 \right\}$ has the finite intersection property. Since $Y$ is compact, $\bigcap_{\beta<\omega_1} \overline{C_\beta} \ne \varnothing$. Let $p \in \bigcap_{\beta<\omega_1} \overline{C_\beta}$. Consider the following closed subsets of $X=\omega_1 \times Y$.
$H=\overline{\left\{(\alpha,y_\alpha): \alpha<\omega_1 \right\}}$
$K=\left\{(\alpha,p): \alpha<\omega_1 \right\}$
We claim that $H \cap K=\varnothing$. Suppose that $(\alpha,p) \in H \cap K$. Either $p \in \overline{\left\{y_\delta: \delta< \alpha+1 \right\}}$ or $p \in \overline{\left\{y_\delta: \delta \ge \alpha+1 \right\}}$. The latter case is not possible. Note that $[0,\alpha] \times Y$ is an open set containing $(\alpha,p)$. This open set cannot contain points of the form $(\delta,p)$ where $\delta \ge \alpha+1$. So the first case $p \in \overline{\left\{y_\delta: \delta< \alpha+1 \right\}}$ must hold. Since $p \in \bigcap_{\beta<\omega_1} \overline{C_\beta}$, $p \in \overline{C_\alpha}=\overline{\left\{y_\delta: \delta \ge \alpha+1 \right\}}$, a contradiction. So $H$ and $K$ are disjoint closed subsets of $X=\omega_1 \times Y$.
Now consider $\beta X$, the Stone-Cech compactification of $X=\omega_1 \times Y$. By Lemma 3, $\beta X=\beta (\omega_1 \times Y)=(\omega_1+1) \times Y$. Let $H^*=\overline{H}$ and $K^*=\overline{K}$ (closures in $\beta X$). We claim that $(\omega_1,p) \in H^* \cap K^*$. Let $O=(\theta,\omega_1] \times V$ be an open set in $\beta X$ with $(\omega_1,p) \in O$. Note that $p \in \overline{C_\theta}=\left\{y_\delta: \delta>\theta \right\}$. Thus $V \cap \overline{C_\theta} \ne \varnothing$. Choose $\delta>\theta$ such that $y_\delta \in V$. We have $(\delta,y_\delta) \in (\theta,\omega_1] \times V$ and $(\delta,y_\delta) \in H^*$. On the other hand, $(\delta,p) \in K^*$. Thus $(\omega_1,p) \in H^* \cap K^*$, a contradiction. Since $X=\omega_1 \times Y$ is normal, Lemma 4 indicates that $H$ and $K$ should have disjoint closures in $\beta X=(\omega_1+1) \times Y$. Thus $Y$ has countable tightness.
$2 \Longrightarrow 1$
Suppose $t(Y)=\omega$. Let $H$ and $K$ be disjoint closed subsets of $\omega_1 \times Y$. The following series of claims will complete the proof:
Claim 1
For each $y \in Y$, there exists an $\alpha<\omega_1$ such that either $W_{H,y} \subset \alpha+1$ or $W_{K,y} \subset \alpha+1$ where
$W_{H,y}=\left\{\delta<\omega_1: (\delta,y) \in H \right\}$
$W_{K,y}=\left\{\delta<\omega_1: (\delta,y) \in K \right\}$
Proof of Claim 1
Let $y \in Y$. The set $V=\omega_1 \times \left\{y \right\}$ is a copy of $\omega_1$. It is a known fact that in $\omega_1$, there cannot be two disjoint closed and unbounded sets. Let $V_H=V \cap H$ and $V_K=V \cap K$. If $V_H \ne \varnothing$ and $V_K \ne \varnothing$, they cannot be both unbounded in $V$. Thus the claim follows if both $V_H \ne \varnothing$ and $V_K \ne \varnothing$. Now suppose only one of $V_H$ and $V_K$ is non-empty. If the one that is non-empty is bounded, then the claim follows. Suppose the one that is non-empty is unbounded, say $V_K$. Then $W_{H,y}=\varnothing$ and the claim follows.
Claim 2
For each $y \in Y$, there exists an $\alpha<\omega_1$ and there exists an open set $O_y \subset Y$ with $y \in O_y$ such that one and only one of the following holds:
$H \cap (\omega_1 \times \overline{O_y}) \subset (\alpha+1) \times \overline{O_y} \ \ \ \ \ \ \ \ (1)$
$K \cap (\omega_1 \times \overline{O_y}) \subset (\alpha+1) \times \overline{O_y} \ \ \ \ \ \ \ \ (2)$
Proof of Claim 2
Let $y \in Y$. Let $\alpha<\omega_1$ be as in Claim 1. Assume that $W_{H,y} \subset \alpha+1$. We want to show that there exists an open set $O_y \subset Y$ with $y \in O_y$ such that (1) holds. Suppose that for each open $O \subset Y$ with $y \in O$, there is a $q \in \overline{O}$ and there exists $\delta_q>\alpha$ such that $(\delta_q,q) \in H$. Let $S$ be the set of all such points $q$. Then $y \in \overline{S}$. Since $Y$ has countable tightness, there exists countable $T \subset S$ such that $y \in \overline{T}$. Since $T$ is countable, choose $\gamma >\omega_1$ such that $\alpha<\delta_q<\gamma$ for all $q \in T$. Note that $[\alpha,\gamma] \times \left\{y \right\}$ does not contain points of $H$ since $W_{H,y} \subset \alpha+1$. For each $\theta \in [\alpha,\gamma]$, the point $(\theta,y)$ has an open neighborhood that contains no point of $H$. Since $[\alpha,\gamma] \times \left\{y \right\}$ is compact, finitely many of these neighborhoods cover $[\alpha,\gamma] \times \left\{y \right\}$. Let these finitely many open neighborhoods be $M_i \times N_i$ where $i=1,\cdots,m$. Let $N=\bigcap_{i=1}^m N_i$. Then $y \in N$ and $N$ would contain a point of $T$, say $q$. Then $(\delta_q,q) \in M_i \times N_i$ for some $i$, a contradiction. Note that $(\delta_q,q)$ is a point of $H$. Thus there exists an open $O_y \subset Y$ with $y \in O_y$ such that (1) holds. This completes the proof of Claim 2.
Claim 3
For each $y \in Y$, there exists an $\alpha<\omega_1$ and there exists an open set $O_y \subset Y$ with $y \in O_y$ such that there are disjoint open subsets $Q_H$ and $Q_K$ of $\omega_1 \times \overline{O_y}$ with $H \cap (\omega_1 \times \overline{O_y}) \subset Q_H$ and $K \cap (\omega_1 \times \overline{O_y}) \subset Q_K$.
Proof of Claim 3
Let $y \in Y$. Let $\alpha$ and $O_y$ be as in Claim 2. Assume (1) in the statement of Claim 2 holds. Note that $(\alpha+1) \times \overline{O_y}$ is a product of two compact spaces and is thus compact (and normal). Let $R_{H,y}$ and $R_{K,y}$ be disjoint open sets in $(\alpha+1) \times \overline{O_y}$ such that $H \cap (\alpha+1) \times \overline{O_y} \subset R_{H,y}$ and $K \cap (\alpha+1) \times \overline{O_y} \subset R_{K,y}$. Note that $[\alpha+1,\omega_1) \times \overline{O_y}$ contains no points of $H$. Then $Q_{H,y}=R_{H,y}$ and $Q_{K,y}=R_{K,y} \cup [\alpha+1,\omega_1) \times \overline{O_y}$ are the desired open sets. This completes the proof of Claim 3.
To make the rest of the proof easier to see, we prove the following claim , which is a general fact that is cleaner to work with. Claim 4 describes precisely (in a topological way) what is happening at this point in the proof.
Claim 4
Let $Z$ be a space. Let $C$ and $D$ be disjoint closed subsets of $Z$. Suppose that $\left\{U_1,U_2,\cdots,U_m \right\}$ is a collection of open subsets of $Z$ covering $C \cup D$ such that for each $i=1,2,\cdots,m$, only one of the following holds:
$C \cap \overline{U_i} \ne \varnothing \text{ and } D \cap \overline{U_i}=\varnothing$
$C \cap \overline{U_i} = \varnothing \text{ and } D \cap \overline{U_i} \ne \varnothing$
Then there exist disjoint open subsets of $Z$ separating $C$ and $D$.
Proof of Claim 4
Let $U_C=\cup \left\{U_i: \overline{U_i} \cap C \ne \varnothing \right\}$ and $U_D=\cup \left\{U_i: \overline{U_i} \cap D \ne \varnothing \right\}$. Note that $\overline{U_C}=\cup \left\{\overline{U_i}: \overline{U_i} \cap C \ne \varnothing \right\}$. Likewise, $\overline{U_D}=\cup \left\{\overline{U_i}: \overline{U_i} \cap D \ne \varnothing \right\}$. Let $V_C=U_C-\overline{U_D}$ and $V_D=U_D-\overline{U_C}$. Then $V_C$ and $V_D$ are disjoint open sets. Furthermore, $C \subset V_C$ and $D \subset V_D$. This completes the proof of Claim 4.
Now back to the proof of Theorem 1. For each $y \in Y$, let $O_y$, $Q_{H,y}$ and $Q_{K,y}$ be as in Claim 3. Since $Y$ is compact, there exists $\left\{y_1,y_2,\cdots,y_n \right\} \subset Y$ such that $\left\{O_{y_1},O_{y_2},\cdots,O_{y_n} \right\}$ is a cover of $Y$. For each $i=1,\cdots,n$, let $L_i=Q_{H,y_i} \cap (\omega_1 \times O_y)$ and $M_i=Q_{K,y_i} \cap (\omega_1 \times O_y)$. Note that both $L_i$ and $M_i$ are open in $\omega_1 \times Y$. To apply Claim 4, rearrange the open sets $L_i$ and $M_i$ and re-label them as $U_1,U_2,\cdots,U_m$. By letting $Z=\omega_1 \times Y$, $C=H$ and $D=K$, the open sets $U_i$ satisfy Claim 4. Tracing the $U_i$ to $L_j$ or $M_j$ and then to $Q_{H,y_j}$ and $Q_{K,y_j}$, it is clear that the two conditions in Claim 4 are satisfied:
$H \cap \overline{U_i} \ne \varnothing \text{ and } K \cap \overline{U_i}=\varnothing$
$H \cap \overline{U_i} = \varnothing \text{ and } K \cap \overline{U_i} \ne \varnothing$
Then by Claim 4, the disjoint closed sets $H$ and $K$ can be separated by two disjoint open subsets of $\omega_1 \times Y$. $\blacksquare$
The theorem proved in [4] is essentially the statement that for any compact space $Y$, the product $\kappa^+ \times Y$ is normal if and only $t(Y) \le \kappa$. Here $\kappa^+$ is the first ordinal of the next cardinal that is greater than $\kappa$.
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Reference
1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
2. Gruenhage, G., Nogura, T., Purisch, S., Normality of $X \times \omega_1$, Topology and its Appl., 39, 263-275, 1991.
3. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
4. Nogura, T., Tightness of compact Hausdorff space and normality of product spaces, J. Math. Soc. Japan, 28, 360-362, 1976.
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$\copyright \ 2014 - 2015 \text{ by Dan Ma}$
# One way to find collectionwise normal spaces
Collectionwise normality is a property that is weaker than paracompactness and stronger than normality (see the implications below). Normal spaces need not be collectionwise normal. Bing’s Example G is an example of a normal and not collectionwise normal space (see the blog post “Bing’s Example G”). We discuss one instance when normal spaces are collectionwise normal, giving a way to obtain collectionwise normal spaces that are not paracompact.
$\text{ }$
$\text{paracompact} \Longrightarrow \text{collectionwise normal} \Longrightarrow \text{normal}$
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Collectionwise Normal Spaces
A normal space is one in which any two disjoint closed sets can be separated by disjoint open sets. By induction, in a normal space any finite number of disjoint closed sets can be separated by disjoint open sets. Of course, the inductive reasoning cannot be carried over to the case of infinitely many disjoint closed sets. In the real line with the usual topology, the singleton sets $\left\{x \right\}$, where $x$ is rational, are disjoint closed sets that cannot be simultaneously separated by disjoint open sets. In order to separate an infinite collection of disjoint closed sets, it makes sense to restrict on the type of collections of closed sets. A space $X$ is collectionwise normal if every discrete collection of closed subsets of $X$ can be separated by pairwise disjoint open subsets of $X$. The following is a more specific definition.
Definition
A space $X$ is collectionwise normal if for every discrete collection $\mathcal{A}$ of closed subsets of $X$, there exists a pairwise disjoint collection $\mathcal{U}=\left\{U_A: A \in \mathcal{A} \right\}$ of open subsets of $X$ such that $A \subset U_A$ for each $A \in \mathcal{A}$.
For more details about the definitions of collectionwise normality, see “Definitions of Collectionwise Normal Spaces”. The implications displayed above are repeated below. None of the arrows is reversible.
$\text{ }$
$\text{paracompact} \Longrightarrow \text{collectionwise normal} \Longrightarrow \text{normal}$
As indicated above, Bing’s Example G is an example of a normal and not collectionwise normal space (see the blog post “Bing’s Example G”). The propositions in the next section can be used to obtain collectionwise normal spaces that are not paracompact.
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When Normal implies Collectionwise Normal
Being able to simultaneously separate any discrete collection of closed sets is stronger than the property of merely being able to separate finite collection of disjoint closed sets. It turns out that the stronger property of collectionwise normality is required only for separating uncountable discrete collections of closed sets. As the following lemma shows, normality is sufficient to separate any countable discrete collection of closed sets.
Lemma 1
Let $X$ be a normal space. Then for every discrete collection $\left\{C_1,C_2,C_3,\cdots \right\}$ of closed subsets of $X$, there exists a pairwise disjoint collection $\left\{O_1,O_2,O_3,\cdots \right\}$ of open subsets of $X$ such that $C_i \subset O_i$ for each $i$.
Proof of Lemma 1
Let $\left\{C_1,C_2,C_3,\cdots \right\}$ be a discrete collection of closed subsets of $X$. For each $i$, choose disjoint open sets $U_i$ and $V_i$ such that $C_i \subset U_i$ and $\cup \left\{C_j: j \ne i \right\} \subset V_i$. Let $O_1=U_1$. For each $i>1$, let $O_i=U_i \cap V_1 \cap \cdots \cap V_{i-1}$. It follows that $O_i \cap O_j = \varnothing$ for all $i \ne j$. It is also clear that for each $i$, $C_i \subset O_i$. $\blacksquare$
We have the following propositions.
Proposition 2
Let $X$ be a normal space. If all discrete collections of closed subsets of $X$ are at most countable, then $X$ is collectionwise normal.
Proposition 3
Let $X$ be a normal space. If all closed and discrete subsets of $X$ are at most countable (such a space is said to have countable extent), then $X$ is collectionwise normal.
Proposition 4
Any normal and countably compact space is collectionwise normal.
Proposition 2 follows from Lemma 1. As noted in Proposition 3, any space in which all closed and discrete subsets are countable is said to have countable extent. It is easy to verify that $X$ has countable extent if and only if all discrete collections of closed subsets of $X$ are at most countable. If $X$ is a countably compact space, then every infinite subset of $X$ has a limit point. Thus Proposition 4 follows from the fact that any countably compact space has countable extent.
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Paracompact Spaces
One way to find a collectionwise normal space that is not paracompact is to find a non-paracompact space that satisfies Propositions 3, 4 or 5. For example, $\omega_1$, the space of all countable ordinals with the order topology, is not paracompact. Since $\omega_1$ is normal and countably compact, it is collectionwise normal by Proposition 4. For a basic discussion of $\omega_1$ as a topological space, see “The First Uncountable Ordinal”.
As the following theorem shows, paracompact spaces are collectionwise normal. Thus the class of collectionwise normal spaces includes all metric spaces and paracompact spaces.
Theorem 5
If a space $X$ is paracompact, then $X$ is collectionwise normal.
Proof of Theorem 5
Let $X$ be a paracompact space. Let $\mathcal{A}$ be a discrete collection of closed subsets of $X$. For each $x \in A$, let $O_x$ be open such that $x \in O_x$ and $O_x$ meets at most one element of $\mathcal{A}$. Let $\mathcal{O}=\left\{O_x: x \in X \right\}$. By the paracompactness of $X$, $\mathcal{O}$ has a locally finite open refinement $\mathcal{V}=\left\{V_x: x \in X \right\}$ such that $V_x \subset O_x$ for each $x \in X$.
For each $A \in \mathcal{A}$, let $W_A=\cup \left\{\overline{V}: V \in \mathcal{V} \text{ and } \overline{V} \cap A=\varnothing \right\}$ and let $U_A=X-W_A$. Each $W_A$ is a closed set since $\mathcal{V}$ is locally finite. Thus each $U_A$ is open. Furthermore, for each $A \in \mathcal{A}$, $A \subset U_A$. It is easily checked that $\left\{U_A: A \in \mathcal{A} \right\}$ is pairwise disjoint. $\blacksquare$
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Reference
1. Bing, R. H., Metrization of Topological Spaces, Canad. J. Math., 3, 175-186, 1951.
2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
3. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.
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https://www.physicsforums.com/threads/help-with-interpreting-a-derivative-of-a-given-function-geometrically.759682/ | # Help with interpreting a derivative of a given function geometrically.
1. Jun 27, 2014
### TitoSmooth
This is one of the the things I did not quite master in my calculus 1 course last semester. I understand for a function to be different on a point a. It must be defined at point a n not have any cusp or appear vertically tangent.
My question is for a general function. How to I sketch it's derivative?
What I do know: horizontal tangents are very important because these points will lie on the x-axis. Pay attention to see where the line is decreasing and increasing. Do not understand what to do with the increasing and decreasing portion. My book Stewart does not really explain this portion of derivatives in terms I can understand. Sorry for this elementary question.
2. Jun 27, 2014
### TitoSmooth
Also it must be continuous at point a
3. Jun 27, 2014
### HallsofIvy
Staff Emeritus
If, at a given point, f is increasing, then f' is positive there. If f is decreasing, f' is negative. In order to transition from "increasing" to "decreasing" the function must "level off" so the derivative there is 0 as you say- to go from positive or negative the derivative must pass through 0. If a function is "convex upward" then the second derivative is positive which means that the first derivative is increasing. Conversely, if a function is "concave downward" the derivative is decreasing.
(You added "Also it must be continuous at point a". It is not clear to me whether you intended "it" to mean the function or the derivative. If a function is not continuous at a, it cannot be differentiable so the question is moot. While the derivative of a differentiable function is not necessarily continuous, it can be shown that it must satisfy the "intermediate value property": If f'(a)= X and f'(b)= Y then f' must take on every value between X and Y at some point between a and b. That why, even if f' is not continuous, if f changes from increasing to decreasing, so that f' changes from positive to negative, f' must be 0 there.)
Similar Discussions: Help with interpreting a derivative of a given function geometrically. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9180851578712463, "perplexity": 394.4412582841229}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824820.28/warc/CC-MAIN-20171021152723-20171021172723-00401.warc.gz"} |
https://mathhelpboards.com/threads/lagrange-theorem.2770/ | # Lagrange Theorem
#### Yankel
##### Active member
Jan 27, 2012
398
Speaking of theorems, I have another question. I need to show, using Lagrange's theorem, that:
$$1.71<\sqrt{3}<1.75$$
By Lagrange's theorem I mean the one of:
f ' (c)=(f(b)-f(a)) / (b-a)
thanks !
#### MarkFL
Staff member
Feb 24, 2012
13,775
I have never seen the mean-value theorem used in such a way, but I assume we may state:
$\displaystyle f(x)=x^2\,\therefore\,f'(x)=2x$
$\displaystyle a=1.71,\,b=1.75$
Hence:
$\displaystyle c=\frac{1.75^2-1.71^2}{2(1.75-1.71)}=1.73$
$\displaystyle f(c)=1.73^2=2.9929$
Or maybe it's as simple as stating (given the monotonically increasing behavior of the function on the given interval):
$\displaystyle f(1.71)<f(\sqrt{3})<f(1.75)$
$\displaystyle 2.9241<3<3.0625$
#### Opalg
##### MHB Oldtimer
Staff member
Feb 7, 2012
2,799
Speaking of theorems, I have another question. I need to show, using Lagrange's theorem, that:
$$1.71<\sqrt{3}<1.75$$
By Lagrange's theorem I mean the one of:
f ' (c)=(f(b)-f(a)) / (b-a)
thanks !
Use the Lagrange relation (which I prefer to call the mean value theorem) $f'(c) = \dfrac{f(b)-f(a)}{b-a}$, using the function $f(x)=\sqrt x$, and taking $b=4$, $a=3$, so that $c$ has to be some point between 3 and 4. You will then need to estimate the value of $f'(c)$, using the fact that it lies between $f'(3)$ and $f'(4)$ (because $f'(x)$ is a decreasing function in this case). That will give you two inequalities for $\sqrt3$, one of which should lead quite easily to the result $\sqrt{3}<1.75$. The other one is a bit trickier, and you may find it helpful to use the fact that $\dfrac1{\sqrt3} = \dfrac{\sqrt3}3.$
#### Deveno
##### Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
it's a darn good thing 12/7 > 1.71 is all i have to say. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9817966818809509, "perplexity": 562.7534894705933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057580.39/warc/CC-MAIN-20210924201616-20210924231616-00649.warc.gz"} |
https://cms.math.ca/cjm/msc/22?page=5 | location: Publications → journals
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Search: MSC category 22 ( Topological groups, Lie groups )
Expand all Collapse all Results 101 - 125 of 133
101. CJM 2001 (vol 53 pp. 675)
Ban, Dubravka
Jacquet Modules of Parabolically Induced Representations and Weyl Groups The representation parabolically induced from an irreducible supercuspidal representation is considered. Irreducible components of Jacquet modules with respect to induction in stages are given. The results are used for consideration of generalized Steinberg representations. Category:22E50
102. CJM 2001 (vol 53 pp. 278)
Helminck, G. F.; van de Leur, J. W.
Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\KP$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau$-functions is given. Keywords:KP hierarchy, Darboux transformation, Grassmann manifoldCategories:22E65, 22E70, 35Q53, 35Q58, 58B25
103. CJM 2001 (vol 53 pp. 244)
Goldberg, David; Shahidi, Freydoon
On the Tempered Spectrum of Quasi-Split Classical Groups II We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension $E/F$ of $p$-adic fields of characteristic zero. We study the case where the Levi component $M \simeq \GL_n (E) \times U_m (F)$, with $n \equiv m$ $(\mod 2)$. This, along with earlier work, determines the poles of the local Rankin-Selberg product $L$-function $L(s, \tau' \times \tau)$, with $\tau'$ an irreducible unitary supercuspidal representation of $\GL_n (E)$ and $\tau$ a generic irreducible unitary supercuspidal representation of $U_m (F)$. The results are interpreted using the theory of twisted endoscopy. Categories:22E50, 11S70
104. CJM 2001 (vol 53 pp. 195)
Mokler, Claus
On the Steinberg Map and Steinberg Cross-Section for a Symmetrizable Indefinite Kac-Moody Group Let $G$ be a symmetrizable indefinite Kac-Moody group over $\C$. Let $\Tr_{\La_1},\dots,\Tr_{\La_{2n-l}}$ be the characters of the fundamental irreducible representations of $G$, defined as convergent series on a certain part $G^{\tralg} \subseteq G$. Following Steinberg in the classical case and Br\"uchert in the affine case, we define the Steinberg map $\chi := (\Tr_{\La_1},\dots, \Tr_{\La_{2n-l}})$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega \colon \C^{n} \times (\C^\times)^{\,n-l} \to C$. We investigate the local behaviour of $\chi$ on $C$ near $\omega \bigl( (0,\dots,0) \times (1,\dots,1) \bigr)$, and we show that there exists a neighborhood of $(0,\dots,0) \times (1,\dots,1)$, on which $\chi \circ \omega$ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on~$C$. Categories:22E65, 17B65
105. CJM 2000 (vol 52 pp. 1192)
Herb, Rebecca A.
Orbital Integrals on $p$-Adic Lie Algebras Let $G$ be a connected reductive $p$-adic group and let $\frakg$ be its Lie algebra. Let $\calO$ be any $G$-orbit in $\frakg$. Then the orbital integral $\mu_{\calO}$ corresponding to $\calO$ is an invariant distribution on $\frakg$, and Harish-Chandra proved that its Fourier transform $\hat \mu_{\calO}$ is a locally constant function on the set $\frakg'$ of regular semisimple elements of $\frakg$. If $\frakh$ is a Cartan subalgebra of $\frakg$, and $\omega$ is a compact subset of $\frakh\cap\frakg'$, we give a formula for $\hat \mu_{\calO}(tH)$ for $H\in\omega$ and $t\in F^{\times}$ sufficiently large. In the case that $\calO$ is a regular semisimple orbit, the formula is already known by work of Waldspurger. In the case that $\calO$ is a nilpotent orbit, the behavior of $\hat\mu_{\calO}$ at infinity is already known because of its homogeneity properties. The general case combines aspects of these two extreme cases. The formula for $\hat\mu _{\calO}$ at infinity can be used to formulate a theory of the constant term'' for the space of distributions spanned by the Fourier transforms of orbital integrals. It can also be used to show that the Fourier transforms of orbital integrals are linearly independent at infinity.'' Categories:22E30, 22E45
106. CJM 2000 (vol 52 pp. 1101)
Zhang, Yuanli
Discrete Series of Classical Groups Let $G_n$ be the split classical groups $\Sp(2n)$, $\SO(2n+1)$ and $\SO(2n)$ defined over a $p$-adic field F or the quasi-split classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a quadratic extension $E/F$. We prove the self-duality of unitary supercuspidal data of standard Levi subgroups of $G_n(F)$ which give discrete series representations of $G_n(F)$. Category:22E35
107. CJM 2000 (vol 52 pp. 804)
Kottwitz, Robert E.; Rogawski, Jonathan D.
The Distributions in the Invariant Trace Formula Are Supported on Characters J.~Arthur put the trace formula in invariant form for all connected reductive groups and certain disconnected ones. However his work was written so as to apply to the general disconnected case, modulo two missing ingredients. This paper supplies one of those missing ingredients, namely an argument in Galois cohomology of a kind first used by D.~Kazhdan in the connected case. Categories:22E50, 11S37, 10D40
108. CJM 2000 (vol 52 pp. 539)
Jantzen, Chris
On Square-Integrable Representations of Classical $p$-adic Groups In this paper, we use Jacquet module methods to study the problem of classifying discrete series for the classical $p$-adic groups $\Sp(2n,F)$ and $\SO(2n+1,F)$. Category:22E50
109. CJM 2000 (vol 52 pp. 449)
An Intertwining Result for $p$-adic Groups For a reductive $p$-adic group $G$, we compute the supports of the Hecke algebras for the $K$-types for $G$ lying in a certain frequently-occurring class. When $G$ is classical, we compute the intertwining between any two such $K$-types. Categories:22E50, 20G05
110. CJM 2000 (vol 52 pp. 412)
Varopoulos, N. Th.
Geometric and Potential Theoretic Results on Lie Groups The main new results in this paper are contained in the geometric Theorems 1 and~2 of Section~0.1 below and they are related to previous results of M.~Gromov and of myself (\cf\ \cite{1},~\cite{2}). These results are used to prove some general potential theoretic estimates on Lie groups (\cf\ Section~0.3) that are related to my previous work in the area (\cf\ \cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos (\cf\ \cite{5},~\cite{21}). Categories:22E30, 43A80, 60J60, 60J65
111. CJM 2000 (vol 52 pp. 438)
Wallach, N. R.; Willenbring, J.
On Some $q$-Analogs of a Theorem of Kostant-Rallis In the first part of this paper generalizations of Hesselink's $q$-analog of Kostant's multiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a $q$-analog of the Kostant-Rallis theorem is given for the real group $\SL(4,\mathbb{R})$ (that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation. Categories:22E47, 20G05
112. CJM 2000 (vol 52 pp. 306)
Cunningham, Clifton
Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals---an expression which is ideally suited for the study of the stability of those characters. Building on work of F.~Murnaghan, our proof involves Lusztig's Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of $p$-adic Lie algebras. Two applications of the main result are considered toward the end of the paper. Categories:22E50, 22E35
113. CJM 1999 (vol 51 pp. 1135)
Arthur, James
Endoscopic $L$-Functions and a Combinatorial Identity The trace formula contains terms on the spectral side that are constructed from unramified automorphic $L$-functions. We shall establish an identify that relates these terms with corresponding terms attached to endoscopic groups of $G$. In the process, we shall show that the $L$-functions of $G$ that come from automorphic representations of endoscopic groups have meromorphic continuation. Categories:22E45, 22E46
114. CJM 1999 (vol 51 pp. 1307)
Johnson, Norman W.; Weiss, Asia Ivić
Quadratic Integers and Coxeter Groups Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$. Categories:11F06, 20F55, 20G20, 20H10, 22E40
115. CJM 1999 (vol 51 pp. 952)
Deitmar, Anton; Hoffmann, Werner
On Limit Multiplicities for Spaces of Automorphic Forms Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme. Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculusCategories:11F72, 22E30, 22E40, 43A85, 58G25
116. CJM 1999 (vol 51 pp. 835)
Kim, Henry H.
Langlands-Shahidi Method and Poles of Automorphic $L$-Functions: Application to Exterior Square $L$-Functions In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate maximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several \emph{completed} automorphic $L$-functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square $L$-functions of $\GL_n$, $n$ odd, the Rankin-Selberg $L$-functions of $\GL_n\times \GL_m$, $n\ne m$, and $L$-functions $L(s,\sigma,r)$, where $\sigma$ is a generic cuspidal representation of $\SO_{10}$ and $r$ is the half-spin representation of $\GSpin(10, \mathbb{C})$. The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture. Categories:11F, 22E
117. CJM 1999 (vol 51 pp. 816)
Hall, Brian C.
A New Form of the Segal-Bargmann Transform for Lie Groups of Compact Type I consider a two-parameter family $B_{s,t}$ of unitary transforms mapping an $L^{2}$-space over a Lie group of compact type onto a holomorphic $L^{2}$-space over the complexified group. These were studied using infinite-dimensional analysis in joint work with B.~Driver, but are treated here by finite-dimensional means. These transforms interpolate between two previously known transforms, and all should be thought of as generalizations of the classical Segal-Bargmann transform. I consider also the limiting cases $s \rightarrow \infty$ and $s \rightarrow t/2$. Categories:22E30, 81S30, 58G11
118. CJM 1999 (vol 51 pp. 636)
Paul, Annegret
First Occurrence for the Dual Pairs $\bigl(U(p,q),U(r,s)\bigr)$ We prove a conjecture of Kudla and Rallis about the first occurrence in the theta correspondence, for dual pairs of the form $\bigl(U(p,q),U(r,s)\bigr)$ and most representations. Category:22E46
119. CJM 1999 (vol 51 pp. 266)
Deitmar, Anton; Hoffman, Werner
Spectral Estimates for Towers of Noncompact Quotients We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$. Categories:11F72, 58G25, 22E40
120. CJM 1999 (vol 51 pp. 164)
Tan, Victor
Poles of Siegel Eisenstein Series on $U(n,n)$ Let $U(n,n)$ be the rank $n$ quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of $U(n,n)$ has at most simple poles at the integers or half integers in certain strip of the complex plane. Categories:11F70, 11F27, 22E50
121. CJM 1999 (vol 51 pp. 130)
Savin, Gordan; Gan, Wee Teck
The Dual Pair $G_2 \times \PU_3 (D)$ ($p$-Adic Case) We study the correspondence of representations arising by restricting the minimal representation of the linear group of type $E_7$ and relative rank $4$. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of $G_2$ and $\PU_3(D)$. Categories:22E35, 22E50, 11F70
122. CJM 1998 (vol 50 pp. 1090)
Lohoué, Noël; Mustapha, Sami
Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes Let $\Delta$ be a left invariant sub-Laplacian on a Lie group $G$ and let $\nabla$ be the associated gradient. In this paper we investigate the boundness of the Riesz transform $\nabla\Delta^{-1/2}$ on Lie groups $G$ which are amenable and with exponential volume growth and on certain homogenous spaces. Categories:22E30, 35H05, 43A80, 43A85
123. CJM 1998 (vol 50 pp. 1105)
Roberts, Brooks
Tempered representations and the theta correspondence Let $V$ be an even dimensional nondegenerate symmetric bilinear space over a nonarchimedean local field $F$ of characteristic zero, and let $n$ be a nonnegative integer. Suppose that $\sigma \in \Irr \bigl(\OO (V)\bigr)$ and $\pi \in \Irr \bigl(\Sp (n,F)\bigr)$ correspond under the theta correspondence. Assuming that $\sigma$ is tempered, we investigate the problem of determining the Langlands quotient data for $\pi$. Categories:11F27, 22E50
124. CJM 1998 (vol 50 pp. 972)
Brüchert, Gerd
Trace class elements and cross-sections in Kac-Moody groups Let $G$ be an affine Kac-Moody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$ its fundamental irreducible representations and $\chi_0, \dots, \chi_r, \chi_{\delta}$ their characters. We determine the set of all group elements $x$ such that all $\pi_i(x)$ act as trace class operators, \ie, such that $\chi_i(x)$ exists, then prove that the $\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r, \chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$. In a second part, following Steinberg, we define a cross-section $C$ for the potential regular classes in $G$. We prove that the restriction $\chi|_C$ behaves well algebraically. Moreover, we obtain an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a functional identity for $\chi|_C$ which shows that $\chi|_C$ is quasi-homogeneous. Categories:22E65, 17B67
125. CJM 1998 (vol 50 pp. 356)
Gross, Leonard
Some norms on universal enveloping algebras The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$ supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism $U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak g_2)$. The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It is also shown that the algebraic dual space $U'$ is spanned by its finite rank elements if and only if $\frak g$ is nilpotent. Categories:17B35, 16S30, 22E30
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https://www.tutorialspoint.com/what-are-the-schemes-of-branch-prediction | # What are the schemes of Branch prediction?
Computer ArchitectureComputer ScienceNetwork
The branch prediction scheme used in a processor has a central impact on its execution. Therefore, some effort has been placed into promoting an effective scheme.
A prediction can be a fixed or a true prediction. In a fixed prediction the same guess is continually made, either ‘taken’ or ‘not-taken’. This is a one outcome guess. The ‘always not taken’ approach prefers the ‘not taken’ path, so the taken penalty (TP) is typically higher than the not-taken penalty (NTP).
The ‘not-taken’ scheme is easier to implement than the ‘taken’ scheme. A large number of pipelined microprocessors employ this scheme, including certain earlier processors like the i486 but also many processors which appeared at the beginning of the 1990s. Examples are the SuperSparc, the Power1 and Power2, and the α 21064 and α 21064A.
The taken penalty is expected to be less than the penalty for not-taken branches (NTP).
A true prediction has two possible outcomes, either ‘taken’ or ‘not-taken’ (fall through, sequential path). With true predictions, the difference between static and dynamic predictions, as shown in the figure. The prediction is known as static if it is based only on the code in question. If the prediction is subject to the preceding execution of the similar branch instruction, especially, the prediction relies upon the implementation history, they are dealing with dynamic prediction.
Static prediction is easier than dynamic prediction. In a static prediction technique, the branch is ‘always taken’ or the branch is ‘always not-taken’ approaches. It can make static predictions by investigating particular attributes of the object code.
Static prediction consists of the following components such as opcode-based predictions, displacement-based predictions, and compiler-directed predictions.
Opcode-based predictions are made by assuming that the branch will be ‘taken’ for certain opcodes and ‘not taken’ for others. This prediction technique is used, for instance, in the MC 88110 and PowerPC 601/603 processors.
Displacement-based predictions depend on the sign of the displacement. If D < 0, the prediction is ‘taken’, in the opposite case, D≥0, it is ‘not taken’. Here the underlying assumption is that a conditional branch with negative displacement is used as a loop-closing branch.
Finally, a static prediction can also be derived from a hint from the compiler. This kind of prediction is called compiler-directed prediction.
In contrast, dynamic prediction is based on branch history. The basic philosophy of dynamic prediction is that branches which were taken at their last occurrence (or last n occurrences) are also likely to be taken at their next occurrence.
Dynamic techniques have a higher performance potential than static schemes. But the price is a more complex implementation since the processor has to store and update the last outcomes of a large number of branches. By contrast, static schemes neglect branch history and instead make a code-based prediction on the fly.
Published on 23-Jul-2021 08:17:45 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8000065088272095, "perplexity": 2075.150202243974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304570.90/warc/CC-MAIN-20220124124654-20220124154654-00199.warc.gz"} |
http://math.stackexchange.com/questions/408090/expression-for-sum-of-k-products-of-n-variables | # Expression for sum of $k$-products of $n$ variables
Given $n$ variables there are $n \choose k$ different terms that are the product of $k$ different variables. For example, in the case that $n = 3$, the $k$-products of the variables $x_1, x_2, x_3$, are
• $k = 1$: $x_1$, $x_2$, $x_3$;
• $k = 2$: $x_1x_2$, $x_1x_3$, $x_2x_3$; and
• $k = 3$: $x_1x_2x_3$.
Can we write the sum of all the $k$-products concisely using sum and product notation? If so, how? For $n = 3$, for instance, we want an expression using $\Sigma$-$\Pi$-notation, in terms of $k$ (and generally in terms of $n$ as well), for $s_{3,k}$, the sum of the $k$-products:
• $s_{3,1} = x_1 + x_2 + x_3$;
• $s_{3,2} = x_1x_2 + x_1x_3 + x_2x_3$; and
• $s_{3,3} = x_1x_2x_3$;
This is basically just the multinomial theorem, with the coefficients and exponents neglected, and broken into several separate sums. But it's not clear to me whether it's possible in the conventional notation to express the "choice" needed to form the sums separately.
-
One possibility to express the $k$-th symmetric polynomial is the following expression: $$s_{n,k}=\sum_{A\subseteq\{1,\ldots,n\} \wedge |A|=k}\ \ \prod_{i\in A} x_i$$
As a sidenote, an "implicit" definition of these sums could look like this: $$\prod_{i=1}^n (x+x_i) = \sum_{k=0}^n s_{n,k}x^{n-k}$$ (with the silent assumption that $s_{n,0}=1$) Borrowing the generating functions notation: $$s_{n,k}=[x^{n-k}] \prod_{i=1}^n (x+x_i)$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.956863284111023, "perplexity": 380.0320208923457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986615.83/warc/CC-MAIN-20150728002306-00124-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.ck12.org/tebook/Texas-Instruments-Calculus-Teacher%27s-Edition/section/9.2/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 9.2: Infinite Geometric Series
Difficulty Level: At Grade Created by: CK-12
This activity is intended to supplement Calculus, Chapter 8, Lesson 2.
ID: 11065
Time required: 45 minutes
## Activity Overview
In this activity, students will explore infinite geometric series. They will consider the effect of the value for the common ratio and determine whether an infinite geometric series converges or diverges. They will also consider the derivation of the sum of a convergent infinite geometric series and use it to solve several problems.
Topic: Sequences & Series
• Explore geometric sequences
• Sum a geometric series
• Convergence of an infinite geometric series
Teacher Preparation and Notes
• This activity serves as an introduction to infinite geometric series. Students will need to have previously learned about finite geometric series.
• Before beginning the activity, students need to clear all lists and turn off all plots and equations. To clear all lists, press \begin{align*}2^{nd}\end{align*} LIST, scroll down to ClrAllLists and press ENTER
Associated Materials
## Problem 1 – Investigating Infinite Geometric Series
Students will explore what happens when the common ratio changes for an infinite geometric series. For each value of \begin{align*}r\end{align*}, students will create lists \begin{align*}L2\end{align*} and \begin{align*}L3\end{align*} and then scroll through \begin{align*}L3\end{align*} to determine if the series converges or diverges.
As an extension, students could change the initial value of the sequence by changing the number 200 in the formula for \begin{align*}L2\end{align*} and see if it affects the convergence or divergence of the series.
If students determine that a series converges, then they are to create and view the scatter plot. The necessary settings for Plot1 are shown on the student worksheet.
To change the window, students can press # and select ZoomStat or manually adjust it by pressing ZOOM.
1.
\begin{align*}r\end{align*} -2 -0.5 -0.25 0.25 0.5 2
Converges or Diverges Diverges Converges 133.33 Converges 160 Converges 266.667 Converges 400 Diverges
2. \begin{align*}| r | < 1\end{align*}
3. There is a horizontal asymptote at the point of convergence.
## Problem 2 – Deriving a Formula for the Sum of a Convergent Infinite Geometric Series
Students are to use the Home screen to determine the values of \begin{align*}r^n\end{align*} when \begin{align*}r = 0.7\end{align*}. They should see that as \begin{align*}n\end{align*} gets very large \begin{align*}r^n\end{align*} becomes zero when \begin{align*}| r | < 1\end{align*}.
Students are given the formula for the sum of a finite geometric series. With the information found on the worksheet, they can determine the formula for the sum of an infinite geometric series using substitution.
4. Note that even though the calculator says \begin{align*}r^{1000}\end{align*} and \begin{align*}r^{10000}\end{align*} both equal zero, they are actually very small numbers and are approximately zero.
\begin{align*}n\end{align*} 10 100 1000 10000
\begin{align*}r^n = 0.7^n\end{align*} 0.028248 3.23 E-16 0 0
5. \begin{align*}s_n = \frac{a_1(1 - 0)}{1 - r} = \frac{a_1}{1 - r}\end{align*}
## Problem 3 – Apply what was learned
In this problem, students are given a scenario relating to drug prescriptions and dosages. Students need to use the formulas shown in the previous problem to answer the questions.
They may get caught up on the first question. Explain to students that if 15% of the drug leaves the body every hour, then that means that 85% percent is still in the body.
6. a. \begin{align*}0.40 = 1 - 0.15(4)\end{align*}
b. \begin{align*}240 + 240(0.4) = 336 \ mg\end{align*}
c.
Hours 0 (1st dosage) 4 (\begin{align*}2^{nd}\end{align*} dosage) 8 (\begin{align*}3^{rd}\end{align*} dosage) 12 16
Amount in the Body 240 336 374.4 389.76 395.904
d. This is the \begin{align*}7^{th}\end{align*} dosage, so \begin{align*}S_7 = \frac{240(1 - .4^7)}{1 - 0.4} = 399.34464\end{align*}.
e. This is the \begin{align*}19^{th}\end{align*} dosage, so \begin{align*}S_{19} = \frac{240(1 - .4^{19})}{1 - 0.4} = 399.999989\end{align*}.
f. \begin{align*}S_t = \frac{240(1 - .4^t)}{1 - 0.4}\end{align*}
g. No, since \begin{align*}S = \frac{240}{1 - 0.4} = 400\end{align*}, you will not reach the minimum lethal dosage.
f. Yes, since he/she waits 2 hours, only 30% of the drug is out of his/her system, so 70% remains. This is the common ratio \begin{align*}r = 0.7\end{align*} and \begin{align*}S = \frac{240}{1 -0.7} = 800\end{align*}
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http://mathhelpforum.com/trigonometry/153190-trig-identities.html | 1. ## trig identities
I’m stuck on the following problem (see Attachment) any help would be greatly appreciated
trig identities 2.rtf
I’m not sure how it goes from here to the next line
Thanks
Kind regards
2. you need to review your post ...
3. ## Trigonometric Identities
Hi All
Im having a problem with the problem in the attached file
trig identities 2.rtf
any help would be greatly appreciated
Thanks Kind Regards
John
4. $2 \left( \sin \frac{\theta}{2} \cos \frac{\theta}{2} \cos^2 \frac{\phi}{2} + \sin \frac{\phi}{2} \cos \frac{\phi}{2} \cos^2 \frac{\theta}{2} + \sin \frac{\phi}{2} \cos \frac{\phi}{2} \sin^2 \frac{\theta}{2} + \sin \frac{\theta}{2} \cos \frac{\theta}{2} \sin^2 \frac{\phi}{2}\right)$
(I rearranged the factors in the last term to make it consistent.)
Distribute the 2:
$2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \cos^2 \frac{\phi}{2} + 2 \sin \frac{\phi}{2} \cos \frac{\phi}{2} \cos^2 \frac{\theta}{2} + 2 \sin \frac{\phi}{2} \cos \frac{\phi}{2} \sin^2 \frac{\theta}{2} + 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \sin^2 \frac{\phi}{2}$
Note that each term begins with $2\sin \frac{\theta}{2} \cos \frac{\theta}{2}$ or $2\sin \frac{\phi}{2} \cos \frac{\phi}{2}$. Recall the sine of a double angle identity: $\sin 2u = 2\sin u \cos u$ So
$2\sin \frac{\theta}{2} \cos \frac{\theta}{2} = \sin \left( 2\cdot \frac{\theta}{2} \right) = \sin \theta$, and the same for angle phi.
Replace:
$\sin \theta \cos^2 \frac{\phi}{2} + \sin \phi \cos^2 \frac{\theta}{2} + \sin \phi \sin^2 \frac{\theta}{2} + \sin \theta \sin^2 \frac{\phi}{2}$
Rearrange:
$\sin \theta \cos^2 \frac{\phi}{2} + \sin \theta \sin^2 \frac{\phi}{2} + \sin \phi \cos^2 \frac{\theta}{2} + \sin \phi \sin^2 \frac{\theta}{2}$
Factor:
$\sin \theta \left( \cos^2 \frac{\phi}{2} + \sin^2 \frac{\phi}{2} \right) + \sin \phi \left( \cos^2 \frac{\theta}{2} + \sin^2 \frac{\theta}{2} \right)$
Use the Pythagorean identity:
$\sin \theta (1) + \sin \phi (1)$
$\sin \theta + \sin \phi$
5. Thanks EUMYANG that has helped a bundle
kind regards
John | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9306464791297913, "perplexity": 2177.115499747151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320040.36/warc/CC-MAIN-20170623082050-20170623102050-00357.warc.gz"} |
https://www.physicsforums.com/threads/diophantine-equation.100748/ | # Diophantine equation
1. Nov 20, 2005
### Galileo
This problem is working on my nerves. Im trying to find all integer solutions to the equation $x^2+4=y^3$ using the PID of Gaussian integers Z.
My thoughts.
By inspection (2,2) is a solution.
Suppose (x,y) is a solution. I write the equation as $(x+2i)(x-2i)=y^3$.
I now look at the ideal (x+2i,x-2i)=(d) with d a generator. d divides x+2i and x-2i, so it also divides the difference 4i.
What I want is to find conditions under which x+2i and x-2i are coprime in Z. Then I can show that (under the conditions) x+2i has to be a third power in Z and that no solutions exist (under this condition).
Any help is appreciated.
2. Nov 22, 2005
### Galileo
Last cry for help...
3. Nov 23, 2005
### shmoe
This may be too late, but anyways:
If d is a divisor of of x+2i and x-2i, then d divides 4i like you said. What can you then say about the norm of d? If x is odd, what does this say about the norm of x+2i?
If x is even, consider the original equation mod 8. You should be able to reduce it to something easier to handle.
4. Nov 24, 2005
### Galileo
Thanks shmoe.
Here's what I came up with.
Let (x,y) be a solution. Because d|4i we have N(d)|N(4i)=16=2^4. So N(d) is 1,2,4,8 or 16. We also have d|x+2i so N(d)|x^2+4.
Suppose x is odd, then x^2+4 is odd and N(d) must be 1 so d is a unit. Then (d)=Z and (x+2i) and (x-2i) are coprime. Then x+2i and x-2i will not have any common irreducible factors, so they must both be equal to a third power in Z, because every unit is too and their product is y^3.
We then get the equation:
$$x+2i=(a+bi)^3=a(a^2-3b^2)+(3a^2-b^2)bi$$
If $b=1$, then $3a^2-1=2$ so $a=\pm 1$, yielding $x=\pm 2$, giving the solutions (2,2) and (-2,2).
(Although I should not count these, because I assumed x is odd )
For b=-1 and b=2 there are no solutions, but for b=-2 I get $x=\pm 11$ with the solutions (11,5) and (-11,5)
If x is even then y must be even and thus y^3 congruent 0 mod 8. So $x^2 \equiv 4 \pmod 8$. So x=2 or x=6 (mod 8), but Im not sure what to do with that.
Last edited: Nov 24, 2005
5. Nov 24, 2005
### shmoe
Correct. The even solutions will appear again, don't worry.
You can write x=2a, where a is odd, and y=2b. Stuff into your equation and what can you say?
6. Nov 24, 2005
### Galileo
Stuffing... I get $a^2+1=2b^3$.
So I'd say I get another diophantine equation :grumpy:
Well, at least I know a is odd, from which follows that b must also be odd. If b was even the right side would be 0 mod 4 while the left is 2 mod 4.
After puzzling I feel like I`m reducing possible solutions, but I still have an infinite number of options. I may be going the wrong way, but I let a=2m+1 and b=2k+1 and got: m(m+1)=k(4k^2+6k+3). Two solutions are ofcourse k=m=0 and k=0, m=-1, corresponding to (x,y)=(2,2) and (x,y)=(-2,2).
7. Nov 24, 2005
### shmoe
From this point: $a^2+1=2b^3$ you can do some factoring over Z. Try to get something like u^2+v^2=b^3. It might help to notice this is expressing b^3 as the norm of an element in Z.
Similar Discussions: Diophantine equation | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9420793056488037, "perplexity": 1095.2006084124773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105961.34/warc/CC-MAIN-20170820015021-20170820035021-00453.warc.gz"} |
http://mathhelpforum.com/math-topics/17092-chemistry-stuff-solutions.html | 1. ## Chemistry Stuff (Solutions)
1. Calculate the number of moles of nitrate ions in 2.20L of 2.02x10-3 mol L-1 lead(II) nitrate solution.
2. Calculate the mass of solute which must be used in order to prepare the following solution 630 mL of 1.26 molL-1 KCl (aq) from KCl (s)
3. What volume of water must be added to 150.0mL of 1.10 molL-1 sulfuric acid solution to prepare a 0.210 mol L-1 solution?
4. Commercial sulphuric acid contains 98.0% H2SO4 by mass and has a density of 1.84g mL-1. Calculate the concentration in mol L-1 of the solution.
5. A 14.2 mol L-1 solution of NaOH has a density of 1.42g mL-1. What is the percentage by mass of NaOH in the solution?
6. 150mL of a 1.01 Mol L-1 KCl solution had 11.7g of NaCl dissolved in it. Calculate the new concentration in mol L-1 of the chloride ion. Assume no volume change.
This I have no idea how to do.
Would appreciate it if someone could show me.
1. Calculate the number of moles of nitrate ions in 2.20L of 2.02x10-3 mol L-1 lead(II) nitrate solution.
Lead nitate has the formula $\mbox{Pb(NO}_3)_2$ thus every mol of lead nitrate contains two mols of nitrate ions (that is you get two nitrate ions for every molecule of lead nitrate).
$2.2$ litres of $2.02\times 10^{-3}$ mol per litre solution contains $2.2 \times 2.02\times 10^{-3}=4.04 \times 10^{-3}$ mols of lead nitrate, and so $8.08\times 10^{-3}$ mols of nitrate ions.
RonL
2. Calculate the mass of solute which must be used in order to prepare the following solution 630 mL of 1.26 molL-1 KCl (aq) from KCl (s)
The mass of 1 mol of KCl is
$(39.098~g) + (35.453~g) = 74.551~g$
So we want 630 mL of a 1.26 ml KCl(aq).
$\frac{0.630~\text{L soln}}{1} \cdot \frac{1.26~\text{mol KCl}}{1~\text{L soln}} \cdot \frac{74.551~\text{g KCl}}{1~\text{mol}} = 33.8163~\text{g KCl}$
Between my answer and CaptainBlack's, do you see what you need to do?
-Dan
4. I think so, thanks.
EDIT: Nope wait, i cant get number 4 or 6...
5. ## umm...
Originally Posted by topsquark
The mass of 1 mol of KCl is
$(39.098~g) + (35.453~g) = 74.551~g$
So we want 630 mL of a 1.26 ml KCl(aq).
$\frac{0.630~\text{L soln}}{1} \cdot \frac{1.26~\text{mol KCl}}{1~\text{L soln}} \cdot \frac{74.551~\text{g KCl}}{1~\text{mol}} = 33.8163~\text{g KCl}$
Between my answer and CaptainBlack's, do you see what you need to do?
-Dan
What have I done wrong. I did this and got a different answer:
to work out the number of moles of KCl I thought it would be 1.26 x 0.63 = 0.7938
then to work out how much KCl that is it would be 0.7938 x 74.551 = 59.1785838
final answer I got was 59.1785838g of KCl is needed
6. Originally Posted by DarkReviver
What have I done wrong. I did this and got a different answer:
to work out the number of moles of KCl I thought it would be 1.26 x 0.63 = 0.7938
then to work out how much KCl that is it would be 0.7938 x 74.551 = 59.1785838
final answer I got was 59.1785838g of KCl is needed
That looks OK to me, may be topsquark's arithmetic has gone wrong.
RonL
7. Originally Posted by DarkReviver
What have I done wrong. I did this and got a different answer:
to work out the number of moles of KCl I thought it would be 1.26 x 0.63 = 0.7938
then to work out how much KCl that is it would be 0.7938 x 74.551 = 59.1785838
final answer I got was 59.1785838g of KCl is needed
Apparently the gremlins got to my calculator again! Sorry 'bout that.
-Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9344063997268677, "perplexity": 2188.0950519871126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806771.56/warc/CC-MAIN-20171123104442-20171123124442-00710.warc.gz"} |
https://www.h2knowledgecentre.com/content/journal62 | 1900
# Steady State Analysis of Gas Networks with Distributed Injection of Alternative Gas
### Abstract
A steady state analysis method was developed for gas networks with distributed injection of alternative gas. A low pressure gas network was used to validate the method. Case studies were carried out with centralized and decentralized injection of hydrogen and upgraded biogas. Results show the impact of utilizing a diversity of gas supply sources on pressure distribution and gas quality in the network. It is shown that appropriate management of using a diversity of gas supply sources can support network management while reducing carbon emissions.
Keywords:
Related subjects:
Countries: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9646024703979492, "perplexity": 1300.4763248622714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337287.87/warc/CC-MAIN-20221002052710-20221002082710-00617.warc.gz"} |
http://outdoorssite.com/Interests/comp/os/xp/recover%20from%20a%20corrupted%20registry.htm | ### How to Recover from a Corrupted Registry that Prevents Windows XP from Starting
• Microsoft Windows XP Home Edition
• Microsoft Windows XP Professional
Article ID: Q307545
Last Reviewed:
March 22, 2002
### SUMMARY
This article describes how to recover a Windows XP system that does not start because of corruption in the registry. This procedure does not guarantee full recovery of the system to a previous state; however, you should be able to recover data when you use this procedure.
You can repair a corrupted registry in Windows XP. Corrupted registry files can cause a variety of different error messages. Please refer to the Knowledge Base for articles regarding error messages related to registry issues.
This article assumes that normal recovery methods have failed and access to the system is not available except by using Recovery Console. If an Automatic System Recovery (ASR) backup exists, it is the preferred method for recovery; it is recommended that you use the ASR backup before you try the procedure described in this article.
When you try to start or restart your Windows XP-based computer, you may receive one of the following error messages:
Windows XP could not start because the following file is missing or corrupt: \WINDOWS\SYSTEM32\CONFIG\SYSTEM
Windows XP could not start because the following file is missing or corrupt: \WINDOWS\SYSTEM32\CONFIG\SOFTWARE
Stop: c0000218 {Registry File Failure} The registry cannot load the hive (file): \SystemRoot\System32\Config\SOFTWARE or its log or alternate
The procedure described in this article uses Recovery Console, System Restore, and lists all the required steps in specific order to ensure that the process completes fully. After you complete this procedure, the system should return to a state very close to the system before the problem occurred. If you have ever run NTBackup and completed a system state backup, you do not have to follow the procedures in parts two and three; you can skip to part four.
### Part One
In part one, you boot to the Recovery Console, create a temporary folder, back up the existing registry files to a new location, delete the registry files at their existing location, and then copy the registry files from the repair folder to the System32\Config folder. When you are finished this procedure, a registry is created that you can use to boot back into Windows XP. This registry was created and saved during the initial setup of Windows XP, so any changes and settings that took place after Setup completes are lost.
To complete part one, follow these steps:
1. Boot to the Recovery Console.
2. At the Recovery Console command prompt, type the following lines, pressing ENTER after you type each line:
md tmp
copy c:\windows\system32\config\system c:\windows\tmp\system.bak
copy c:\windows\system32\config\software c:\windows\tmp\software.bak
copy c:\windows\system32\config\sam c:\windows\tmp\sam.bak
copy c:\windows\system32\config\security c:\windows\tmp\security.bak
copy c:\windows\system32\config\default c:\windows\tmp\default.bak
delete c:\windows\system32\config\system
delete c:\windows\system32\config\software
delete c:\windows\system32\config\sam
delete c:\windows\system32\config\security
delete c:\windows\system32\config\default
copy c:\windows\repair\system c:\windows\system32\config\system
copy c:\windows\repair\software c:\windows\system32\config\software
copy c:\windows\repair\sam c:\windows\system32\config\sam
copy c:\windows\repair\security c:\windows\system32\config\security
copy c:\windows\repair\default c:\windows\system32\config\default
NOTE : This procedure assumes that Windows XP is installed to the C:\Windows folder. Make sure to change C:\Windows to the appropriate windows_folder if it is a different location.
If you have access to another computer, to save time, you can copy the text in step two, and then create a text file called "Regcopy1.txt" (for example). To create this file, run the following command when you boot into Recovery Console:
batch regcopy1.txt
The Batch command in Recovery Console allows for all the commands in a text file to be sequentially processed. When you use the batch command, you do not have to manually type as many commands.
### Part Two
In part two, you copy the registry files from their backed up location by using System Restore. This folder is not available in Recovery Console and is normally not visible during normal usage. Before you start this procedure, you must change several settings to make the folder visible:
1. Start Windows Explorer.
2. On the Tools menu, click Folder options .
3. Click the View tab.
4. Under Hidden files and folders , click to select Show hidden files and folders , and then click to clear the Hide protected operating system files (Recommended) check box.
5. Click Yes when the dialog box is displayed that confirms that you want to display these files.
6. Double-click the drive where you installed Windows XP to get a list of the folders. If is important to click the correct drive.
7. Open the System Volume Information folder. This folder appears dimmed folder because it is set as a super-hidden folder.
NOTE : This folder contains one or more _restore {GUID} folders such as "_restore{87BD3667-3246-476B-923F-F86E30B3E7F8}".
NOTE: You may receive the following error message:
C:\System Volume Information is not accessible. Access is denied.
If you get this message, see the following Microsoft Knowledge Base article to gain access to this folder and continue with the procedure:
8. Open a folder that was not created at the current time. You may have to click Details on the View menu to see when these folders were created. There may be one or more folders starting with "RP x under this folder. These are restore points.
9. Open one of these folders to locate a Snapshot subfolder folder; the following path is an example of a folder path to the Snapshot folder:
C:\System Volume Information\_restore{D86480E3-73EF-47BC-A0EB-A81BE6EE3ED8}\RP1\Snapshot
10. From the Snapshot folder, copy the following files to the C:\Windows\Tmp folder:
• _REGISTRY_USER_.DEFAULT
• _REGISTRY_MACHINE_SECURITY
• _REGISTRY_MACHINE_SOFTWARE
• _REGISTRY_MACHINE_SYSTEM
• _REGISTRY_MACHINE_SAM
These files are the backed up registry files from System Restore. Because you used the registry file created by Setup, this registry does not know that these restore points exist and are available. A new folder is created with a new GUID under System Volume Information and a restore point is created that includes a copy of the registry files that were copied during part one. This is why it is important not to use the most current folder, especially if the time stamp on the folder is the same as the current time.
The current system configuration is not aware of the previous restore points. You need a previous copy of the registry from a previous restore point to make the previous restore points available again.
The registry files that were copied to the Tmp folder in the C:\Windows folder are moved to ensure the files are available under Recovery Console. You need to use these files to replace the registry files currently in the C:\Windows\System32\Config folder. Recovery Console has limited folder access and cannot copy files from the System Volume folder by default.
NOTE : The procedure described in this section assumes that you are running your computer with the FAT32 file system.
### Part Three
In part three, you delete the existing registry files, and then copy the System Restore Registry files to the C:\Windows\System32\Config folder:
1. Boot to Recovery Console.
2. At the Recovery Console command prompt, type the following lines, pressing ENTER after you type each line:
del c:\windows\system32\config\sam
del c:\windows\system32\config\security
del c:\windows\system32\config\software
del c:\windows\system32\config\default
del c:\windows\system32\config\system
copy c:\windows\tmp\_registry_machine_software c:\windows\system32\config\software
copy c:\windows\tmp\_registry_machine_system c:\windows\system32\config\system
copy c:\windows\tmp\_registry_machine_sam c:\windows\system32\config\sam
copy c:\windows\tmp\_registry_machine_security c:\windows\system32\config\security
copy c:\windows\tmp\_registry_user_.default c:\windows\system32\config\default
NOTE : Some of the preceding command lines may be wrapped for readability.
NOTE : This procedure assumes that Windows XP is installed to the C:\Windows folder. Make sure to change C:\Windows to the appropriate windows_folder if it is a different location.
If you have access to another computer, to save time, you can copy the text in step two, and then create a text file called "Regcopy1.txt" (for example).
### Part Four
1. Click Start , and then click All Programs .
2. Click Accessories , and then click System Tools .
3. Click System Restore , and then click Restore to a previous Restore Point .
### REFERENCES
For additional information about using Recovery Console, click the article numbers below to view the articles in the Microsoft Knowledge Base:
Q307654 HOW TO: Access the Recovery Console During Startup
Q216417 How to Install the Windows XP Recovery Console
Q240831 How to Copy Files from Recovery Console to Removable Media
Q314058 Description of the Windows XP Recovery Console
For additional information about System Restore, click the article numbers below to view the articles in the Microsoft Knowledge Base:
Q306084 HOW TO: Restore Windows XP to a Previous State
Q261716 System Restore Removes Files During a Restore Procedure | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8230337500572205, "perplexity": 3780.1980537459003}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607960.64/warc/CC-MAIN-20170525010046-20170525030046-00068.warc.gz"} |
http://slideplayer.com/slide/2614186/ | # We develop the formula by considering how to differentiate products.
## Presentation on theme: "We develop the formula by considering how to differentiate products."— Presentation transcript:
We develop the formula by considering how to differentiate products.
where u and v are both functions of x. Substituting for y, e.g. If ,
So, Integrating this equation, we get The l.h.s. is just the integral of a derivative, so, since integration is the reverse of differentiation, we get Can you see what this has to do with integrating a product?
The function in the integral on the l.h.s. . . .
Here’s the product . . . if we rearrange, we get The function in the integral on the l.h.s . . . is a product, but the one on the r.h.s is a simple function that we can integrate easily.
Here’s the product . . . if we rearrange, we get So, we’ve integrated ! We need to turn this method ( called integration by parts ) into a formula.
Example Generalisation Integrating: Simplifying the l.h.s.: Rearranging:
SUMMARY Integration by Parts To integrate some products we can use the formula
So, Using this formula means that we differentiate one factor, u to get
So, Using this formula means that we differentiate one factor, u to get and integrate the other , to get v
So, Using this formula means that we differentiate one factor, u to get and integrate the other , to get v e.g. 1 Find Having substituted in the formula, notice that the 1st term, uv, is completed but the 2nd term still needs to be integrated. and differentiate integrate ( +C comes later )
So, differentiate integrate and We can now substitute into the formula
So, and differentiate integrate We can now substitute into the formula The 2nd term needs integrating
e.g. 2 Find Solution: and differentiate This is a compound function, so we must be careful. integrate So,
Exercises Find 1. 2. 1. Solutions: = ò dx xe x 2.
Definite Integration by Parts
With a definite integral it’s often easier to do the indefinite integral and insert the limits at the end. We’ll use the question in the exercise you have just done to illustrate.
Using Integration by Parts
Integration by parts cannot be used for every product. It works if we can integrate one factor of the product, the integral on the r.h.s. is easier* than the one we started with. * There is an exception but you need to learn the general rule.
e.g. 3 Find Solution: What’s a possible problem? ANS: We can’t integrate Can you see what to do? If we let and , we will need to differentiate and integrate x. Tip: Whenever appears in an integration by parts we choose to let it equal u.
x cancels. e.g. 3 Find So, integrate differentiate
The r.h.s. integral still seems to be a product! BUT . . . x cancels. So,
e.g. 4 Solution: Let and The integral on the r.h.s. is still a product but using the method again will give us a simple function. We write
e.g. 4 Solution: ( 1 ) Let and So, Substitute in ( 1 )
Example e.g. 5 Find Solution: It doesn’t look as though integration by parts will help since neither function in the product gets easier when we differentiate it. However, there’s something special about the 2 functions that means the method does work.
e.g. 5 Find Solution: We write this as:
e.g. 5 Find So, where and We next use integration by parts for I2
e.g. 5 Find So, where and We next use integration by parts for I2
2 equations, 2 unknowns ( I1 and I2 ) !
e.g. 5 Find So, ( 1 ) ( 2 ) 2 equations, 2 unknowns ( I1 and I2 ) ! Substituting for I2 in ( 1 )
2 equations, 2 unknowns ( I1 and I2 ) !
e.g. 5 Find So, ( 1 ) ( 2 ) 2 equations, 2 unknowns ( I1 and I2 ) ! Substituting for I2 in ( 1 )
2 equations, 2 unknowns ( I1 and I2 ) !
e.g. 5 Find So, ( 1 ) ( 2 ) 2 equations, 2 unknowns ( I1 and I2 ) ! Substituting for I2 in ( 1 )
Exercises 1. 2. ( Hint: Although 2. is not a product it can be turned into one by writing the function as )
Solutions: and Let 1. ( 1 ) and Let For I2: Subs. in ( 1 )
2. This is an important application of integration by parts and Let So,
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http://u2commerce.com/truncation-error/truncation-error-methods.html | Home > Truncation Error > Truncation Error Methods
# Truncation Error Methods
## Contents
Noting that , we find that the global truncation error for the Euler method in going from to is bounded by This argument is not complete since it does not Their derivation of local trunctation error is based on the formula where is the local truncation error. Linear multistep methods that satisfy the condition of zero-stability have the same relation between local and global errors as one-step methods. It follows from Eq. (10) that the error becomes progressively worse with increasing t; Similar computations for bounds for the local truncation error give in going from 0.4 to 0.5 and navigate here
Then we immediately obtain from Eq. (5) that the local truncation error is Thus the local truncation error for the Euler method is proportional to the square of the step However, the central fact expressed by these equations is that the local truncation error is proportional to . This requires our increment function be sufficiently well-behaved. K.; Sacks-Davis, R.; Tischer, P. navigate here
## Local Truncation Error Euler Method
All modern codes for solving differential equations have the capability of adjusting the step size as needed. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Next: Improvements on the Up: Errors in Numerical Previous: Sources of Error Local Truncation Error for the Euler Method The analysis for estimating is more difficult than that for .
This includes the two routines ode23 and ode45 in Matlab. Generated Sun, 30 Oct 2016 18:35:41 GMT by s_hp90 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.7/ Connection In other words, if a linear multistep method is zero-stable and consistent, then it converges. Local Truncation Error Runge Kutta Thus, to reduce the local truncation error to an acceptable level throughout , one must choose a step size h based on an analysis near t = 1.
A method that provides for variations in the step size is called adaptive. Global Truncation Error Please try the request again. Since the equation given above is based on a consideration of the worst possible case, that is, the largest possible value of , it may well be a considerable overestimate of Bonuses CiteSeerX: 10.1.1.85.783. ^ Süli & Mayers 2003, p.317, calls τ n / h {\displaystyle \tau _{n}/h} the truncation error. ^ Süli & Mayers 2003, pp.321 & 322 ^ Iserles 1996, p.8;
If f has these properties and if is a solution of the initial value problem, then and by the chain rule Since the right side of this equation is continuous, is Local Truncation Error Backward Euler For simplicity, assume the time steps are equally spaced: h = t n − t n − 1 , n = 1 , 2 , … , N . {\displaystyle h=t_{n}-t_{n-1},\qquad In each step the error is at most ; thus the error in n steps is at most . Because it is more accessible, we will hereafter use the local truncation error as our principal measure of the accuracy of a numerical method, and for comparing different methods.
• The system returned: (22) Invalid argument The remote host or network may be down.
• Of course, this step size will be smaller than necessary near t = 0 .
• Then, as noted previously, and therefore Equation (6) then states that The appearance of the factor 19 and the rapid growth of explain why the results in the preceding section
## Global Truncation Error
Thus, in the definition for the local truncation error, it is now assumed that the previous s iterates all correspond to the exact solution: τ n = y ( t n However, knowing the local truncation error we can make an intuitive estimate of the global truncation error at a fixed as follows. Local Truncation Error Euler Method The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. How To Find Truncation Error Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply.
Generated Sun, 30 Oct 2016 18:35:41 GMT by s_hp90 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection http://u2commerce.com/truncation-error/truncation-error-analysis-lattice-boltzmann-methods.html Contents 1 Definitions 1.1 Local truncation error 1.2 Global truncation error 2 Relationship between local and global truncation errors 3 Extension to linear multistep methods 4 See also 5 Notes 6 A uniform bound, valid on an interval [a, b], is given by where M is the maximum of on the interval . Generated Sun, 30 Oct 2016 18:35:41 GMT by s_hp90 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.6/ Connection Truncation Error In Numerical Methods
E. (March 1985). "A review of recent developments in solving ODEs". More important than the local truncation error is the global truncation error . Another approach is to keep the local truncation error approximately constant throughout the interval by gradually reducing the step size as t increases. his comment is here Your cache administrator is webmaster.
Then, making use of a Taylor polynomial with a remainder to expand about , we obtain where is some point in the interval . Truncation Error Example Computing Surveys. 17 (1): 5–47. Your cache administrator is webmaster.
## Generated Sun, 30 Oct 2016 18:35:41 GMT by s_hp90 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection
The actual error is 0.1090418. Please try the request again. Generated Sun, 30 Oct 2016 18:35:41 GMT by s_hp90 (squid/3.5.20) Local Truncation Error Trapezoidal Method Your cache administrator is webmaster.
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. As an example of how we can use the result (6) if we have a priori information about the solution of the given initial value problem, consider the illustrative example. Nevertheless, it can be shown that the global truncation error in using the Euler method on a finite interval is no greater than a constant times h. weblink It is because they implicitly divide it by h.
These results indicate that for this problem the local truncation error is about 40 or 50 times larger near t = 1 than near t = 0 . | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9076762795448303, "perplexity": 659.7845196522993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423269.5/warc/CC-MAIN-20170720161644-20170720181644-00072.warc.gz"} |
http://mathoverflow.net/revisions/92700/list | MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
6 deleted 2 characters in body
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with $C_{p,q}$ to be a free $\mathbb{Z}$-module generated by the maps $\Delta[p]\times\Delta[q]\rightarrow E$ ($E$ is a total space of Serre fibration over $B$) which fit into the diagram
\begin{matrix} \Delta[p]\times\Delta[q] & \to & E \\ \downarrow & & \downarrow \\
\Delta[p] & \to & B \end{matrix}
with obvious differentials (coordinate by coordinate differentatials differentials as in normal singular complex).
There are two filtrations, he uses the first one (by rows) to determine the homology of the total complex, ane the second one to get $E^2_{p,q}=H_p(B,\underline{H_q}(F)$. I have a problem with the second part. He fixes $p$ and a map $c$ in the bottom row and interprets the diagram as a map $\Delta[q]\rightarrow F_c$ where $F_c$ is the image of $\Delta[p]\times\Delta[q]$ in $E$ such that the diagram commutes, which basically means that $C_{p,q}=\bigoplus_c \mathbb{Z}[F_c]$ (as a module, that's true). Now he calculates the homology of the column and says that it's $\bigoplus_c H_*(F_c)$. Why we can we apply the vertical differential here? Do we need to check some compatibility condition?
Next, we want to use that $E^1_{p,q}$ is a module of singular $p$-chains with coefficients in a local system $\underline{H_q}(F)$ and say that the horizontal differential is just the regular differential to get the desired output. But how do we know that this differential works in a nice way?
5 deleted 9 characters in body
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with $C_{p,q}$ to be a free $\mathbb{Z}$-module generated by the maps $\Delta[p]\times\Delta[q]\rightarrow E$ ($E$ is a total space of Serre fibration over $B$) which fit into the diagram
\begin{matrix} \Delta[p]\times\Delta[q] & \to & E \\ \downarrow & & \downarrow \\
\Delta[p] & \to & B \end{matrix}
with obvious differentials (coordinate by coordinate differentatials as in normal singular complex).
There are two filtrations, he uses the first one (by rows) to determine the homology of the total complex, ane the second one to get $E^2_{p,q}=H_p(B,\underline{H_q}(F)$. I have a problem with the second part. He fixes $p$ and an application a map $c$ in the bottom row and interprets the diagram as a map $\Delta[q]\rightarrow F_c$ where $F_c$ is the image of $\Delta[p]\times\Delta[q]$ in $E$ such that the diagram commutes, which basically means that $C_{p,q}=\bigoplus_c \mathbb{Z}[F_c]$ (as a module, that's true). Now he calculates the homology of the column and says that it's $\bigoplus_c H_*(F_c)$. Why we can apply the vertical differential here? Do we need to check some compatibility condition?
Next, we want to use that $E^1_{p,q}$ is a module of singular $p$-chains with coefficients in a local system $\underline{H_q}(F)$ and say that the horizontal differential is just the regular differential to get the desired output. But how do we know that this differential works in a nice way?
4 added 2 characters in body
I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C_{\bullet,\bullet}$ with $C_{p,q}$ to be a free $\mathbb{Z}$-module generated by the maps $\Delta[p]\times\Delta[q]\rightarrow E$ ($E$ is a total space of Serre fibration over $B$) which fit into the diagram
\begin{matrix} \Delta[p]\times\Delta[q] & \to & E \\ \downarrow & & \downarrow \\
\Delta[p] & \to & B \end{matrix}
with obvious differentials (coordinate by coordinate differentatials as in normal singular complex).
There are two filtrations, he uses the first one (by rows) to determine the homology of the total complex, ane the second one to get $E^2_{p,q}=H_p(B,\underline{H_q}(F)$. I have a problem with the second part. He fixes $p$ and an application $c$ in the bottom row and interprets the diagram as a map $\Delta[q]\rightarrow F_c$ where $F_c$ is the image of $\Delta[p]\times\Delta[q]$ in $E$ such that the diagram commutes, which basically means that $C_{p,q}=\bigoplus_c \mathbb{Z}[F_c]$ (as a module, that's true). Now he calculates the homology of the column and says that it's $\bigoplus_c H_*(F_c)$. Why we can apply the vertical differential here? Do we need to check some compatibility condition?
Next, we want to use that $E^1_{p,q}$ is a module of singular $p$-chains with coefficients in a local system $\underline{H_q}(F)$ and say that the horizontal differential is just the regular differential to get the desired output. But how do we know that this differential works in a nice way?
3 edited tags
2 deleted 25 characters in body
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http://mathhelpforum.com/algebra/153735-finding-power.html | # Math Help - finding power of
1. ## finding power of
Hi,
one simple question (to someone who knows), i am stuck,
how to get n from equation, if i know what x and y are:
$x = y^n$
2. You need to use logarithms...
$x = y^n$
$\ln{x} = \ln{(y^n)}$
$\ln{x} = n\ln{y}$
$n = \frac{\ln{x}}{\ln{y}}$.
3. TNX , i thought it has something to do with logarithms. I really appreciate it
4. Of course, it doesn't have to be the natural logarithm. It can be the logarithm of any base... I just always use natural logarithms because they're the most often used in practice... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8933267593383789, "perplexity": 676.2923778856573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00535-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/15356/showing-the-bibliographic-entry-in-a-popup-when-you-hover-over-the-citation-key/54831 | # Showing the bibliographic entry in a popup when you hover over the citation key
I thought it would be useful to have bibliographic entries displayed as a tooltip so readers are not interrupted by following hyperlinks or trying to find the Bibliography, so I created the following command:
\newcommand\annocite[1]{\pdfmarkupcomment[markup=Underline,subject=Citation]
{\parencite{#1}}{\fullcite{#1}}}
Unfortunately this does not work. Instead of showing the bibliographic entry in the popup, it merely displays the citation key; i.e., the \fullcite command seems to have no effect. I believe the problem is related to argument expansion. Here is a MWE:
\documentclass{article}
\usepackage{biblatex,pdfcomment,filecontents}
\bibliography{\jobname}
\newcommand\annocite[1]{\pdfmarkupcomment[markup=Underline,subject=Citation]
{\parencite{#1}}{\fullcite{#1}}}
\begin{filecontents}{\jobname.bib}
@article{Bli74,
author = {Blinder, Alan S.},
year = {1974},
title = {The economics of brushing teeth},
journaltitle = {Journal of Political Economy},
volume = {82},
number = {4},
pages = {887--891},
}
\end{filecontents}
\begin{document}
Hover over this citation: \annocite{Bli74}
The tooltip should contain the following text: \fullcite{Bli74}
\end{document}
Can anyone think of a solution, or better, a way to redefine biblatex' citation commands (e.g., \parencite, \cite) to do this automatically?
-
\fullcite is a \protected command. It cannot be expanded, because it contains something that isn't expandable, namely macro definitions. – TH. Apr 10 '11 at 1:28
You can probably duplicate \blx@citei@fullcite except save the text produced rather than typeset it and then use \pdfmarkupcomment. I don't have time to try it right now, myself. Maybe someone else can give it a go. – TH. Apr 10 '11 at 1:34
If you want to see it done right, try one of these open access articles. – Emre Aug 16 '11 at 21:57
@Emre The journal you linked i snot open-access, it seems. – matth Mar 19 '12 at 8:43
@matth Thanks for the notice. Try this PDF or search Google for recent articles. – Emre Mar 20 '12 at 4:20
The idea is to use the fancytooltips package. I will show schematically the process: to make my example compilable for everyone, I'll use the following bibliographical database (called biblio.bib):
@book{goossens93,
author = "Michel Goossens and Frank Mittlebach and Alexander Samarin",
title = "The Latex Companion A",
year = "1993",
}
@book{knuth79,
author = "Donald E. Knuth",
title = "Tex and Metafont, New Directions in Typesetting",
year = {1979{(}1950{)}},
publisher = "American Mathematical Society and Digital Press",
}
First you need to create a .pdf file containing the \keytips commands and the text for the tooltips that will be used by the fancytooltips package. To create this .pdf document you can use LaTeX: in this example I used a file (called bibtips.tex) having the following aspect:
\documentclass{article}
\usepackage{xcolor}
\usepackage{biblatex}
\usepackage[createtips]{fancytooltips}
\newcommand\MyTip[1]{%
\keytip{#1}
\fcolorbox{green!50!black}{yellow!20}{\parbox{\textwidth}{\fullcite{#1}}}\newpage%
}
\bibliography{biblio}
\begin{document}
\MyTip{goossens93}
\MyTip{knuth79}
\end{document}
You need to use the \MyTip command for every bibliographical entry that will have a tooltip in your final document.
Compile this file in the standard way: pdflatex+bibtex+pdflatex+pdflatex to generate the references, one on each page, nicely framed, and with the corresponding \keytip command. The resulting .pdf file has to be in the same directory containing your main .tex file.
Now, your main document has to be something like the following (note that the value for the filename key is exactly the name of the .pdf file obtained in the previous step):
\documentclass{article}
\usepackage{xcolor}
\usepackage{biblatex}
\usepackage[filename=bibtips,mouseover]{fancytooltips}
\bibliography{biblio}
\begin{document}
\tooltip{\cite{goossens93}}{goossens93}
\tooltip{\cite{knuth79}}{knuth79}
\end{document}
Compile this file in the standard way: pdflatex+bibtex+pdflatex+pdflatex and you'll see your citations with a blue balloon; if you move the mouse pointer to the active area, a tooltip will open displaying the complete bibliographical information corresponding to the citation.
Some remarks:
1. Of course, you can obtain fancyer tooltips by changing the aspect of the references in the file bibtips.pdf.
2. This approach doesn't work in all PDF viewers, since it requires cooperation with JavaScripts; you have to use Adobe Reader or Adobe Acrobat to see the tooltips.
-
Good solution. But could you say how to use it in XeLaTex? – filokalos Apr 10 '11 at 9:05
@filokalos: there's some incompatibility with fancytooltips and XeLaTeX (some driver option). If I have the time I will look into it. – Gonzalo Medina Apr 11 '11 at 20:28
Ok.:) Thank you. – filokalos Apr 11 '11 at 20:36
@filokalos: fancytooltips might depend on some support for directly outputting PDF code. Maybe the difference is that the pdftex engine goes directly to PDF while xetex routes through DVI? Perhaps lualatex would work? – Sharpie Apr 11 '11 at 20:46
@Gonzalo Medina: Would it be possible to deposit somewhere a compiled version of your example, please? – Stephen Aug 20 '11 at 18:14
After giving Gonzalo's nice answer a whirl, I put together some tweaks. Here's a summary.
• Tooltip creation. A tooltip for every entry in the given bib file(s) can easily be created using \nocite{*}, \AtDataInput and list processing commands from etoolbox.
• Tooltip location. By default tooltips appear at the top of the page in the presentation document, no matter where the active areas are located. We can move each tooltip closer to its active area by resizing the pages in the tooltips document and using the movetips option setting.
• Citation commands. Instead of issuing the \tooltip command directly, tooltips can be incorporated into existing citation commands via the bibhyperref format. The starred variant of \tooltip allows us to separate the active areas for citation links and tooltips.
Here's an example of how all these ideas can be implemented.
% --- tooltips document
\begin{filecontents*}{bibtooltips.tex}
\documentclass{article}
\usepackage{xcolor}
\usepackage[createtips]{fancytooltips}
\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[maxnames=2]{biblatex}
\usepackage{geometry}
% Size page a little larger than the longest tooltip
\pagestyle{empty}
\parindent=0pt
\DeclareCiteCommand{\keytipcite}
{}
{\null\vfill% Move tooltip to bottom of page
\begin{center}
\keytip{\thefield{entrykey}}%
\fcolorbox{brown!50}{yellow!10}
{\footnotesize\parbox{0.95\textwidth}
{\usedriver
{\clearfield{extrayear}% Omit extraneous fields here
\clearfield{subtitle}%
\clearfield{booksubtitle}%
\clearfield{mainsubtitle}%
\clearfield{issuesubtitle}%
\clearfield{journalsubtitle}}
{\thefield{entrytype}}}}%
\end{center}
\medskip% Leave space below tooltip to avoid obscuring text
\newpage}
{}
{}
\def\allkeys{}
\begin{document}
%\nocite{*}
\nocite{companion,cicero,baez/article,bertram,kant:kpv,kant:ku}
\nocite{aristotle:poetics,aristotle:rhetoric,aristotle:anima}
\forlistloop{\keytipcite}{\allkeys}
\end{document}
\end{filecontents*}
% --- end of tooltips document
% --- presentation document
\documentclass{article}
\usepackage[svgnames]{xcolor}
\usepackage[filename=bibtooltips,mouseover,movetips]{fancytooltips}
\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[style=numeric-comp]{biblatex}
\usepackage{hyperref}
\definecolor{tooltipcolor}{named}{Green}
% Display nothing in "extratext" area following the tooltip (by default this
% area displays a speech bubble/balloon)
\let\TooltipExtratext\relax
% Apply tooltip to "extratext" area just after inline citation links
\DeclareFieldFormat{bibhyperref}{%
\tooltip*{\bibhyperref{#1}}{\thefield{entrykey}}}
% Define new citation commands that replace citation links with tooltips
\DeclareFieldFormat{bibtooltip}{\tooltip{#1}{\thefield{entrykey}}}
\newrobustcmd*{\tooltiphook}{%
\AtNextCite{\DeclareFieldAlias{bibhyperref}{bibtooltip}}}
\newrobustcmd*{\tooltipcite}{\tooltiphook\cite}
\newrobustcmd*{\tooltipcites}{\tooltiphook\cites}
% Apply tooltip to instance where numeric-comp uses \bibhyperref instead
% of bibhyperref format
\makeatletter
\newbibmacro*{cite:dump:tooltip}{%
\ifnumgreater{\value{cbx@tempcnta}}{0}
{\ifnumgreater{\value{cbx@tempcnta}}{1}
{\bibrangedash}
{\multicitedelim}%
\tooltip*
{\bibhyperref[\cbx@lastkey]{%
\ifdef\cbx@lastprefix
{\printtext[prefixnumber]{\cbx@lastprefix}}
{}%
\printtext[labelnumber]{\cbx@lastnumber}}}
{\cbx@lastkey}}
{}%
\setcounter{cbx@tempcnta}{0}%
\global\undef\cbx@lastprefix}
\ifcsundef{abx@macro@\detokenize{cite:dump}}
{}{\renewbibmacro*{cite:dump}{\usebibmacro{cite:dump:tooltip}}}
\makeatother
\begin{document}
\null\vfill
\subsection*{Inline citations}
\textcite[10--15]{companion} showed that...
\textcites[10]{companion}[1-10]{cicero}{}{kant:kpv,kant:ku}.
Filler text \parencites(e.g.)()[10]{companion}{cicero}{baez/article}.
Filler text.\supercite{bertram}
\subsection*{Footnote citations}
Filler text.\footcites{cicero}{companion}
Filler text.\footnote{See \smartcite{cicero}.}
\subsection*{More inline citations}
\textcites[10--15]{companion}[10]{cicero} show that...
Filler text \parencites(e.g.)()[10]{companion}{cicero}{baez/article}.
Filler text \tooltipcites(e.g.)()[10]{companion}{cicero}{baez/article}.
Filler text \parencite[e.g.][]{baez/article,bertram,cicero,kant:ku,companion}.
Filler text
\cite{cicero,companion,aristotle:poetics,aristotle:rhetoric,aristotle:anima,bertram}.
\printbibliography
\end{document}
Note the following:
• I've used filecontents to just to keep all the code together.
• The above formats can apply tooltips to any standard citation label, but \tooltip*/\tooltip needs to be applied directly whenever \bibhyperref is used instead of the format. The numeric-comp style's cite:dump macro is an example of this.
• The load order of the fancytooltips and biblatex packages is intentional. If biblatex is loaded first in the presentation document, it will generate persistent "rerun LaTeX" messages.
• Documents are best viewed at their full width. Otherwise you should consider making the tooltips half the \textwidth of the presentation document.
-
When you published your answer I thought about writing a note like this, but then I forgot. Better late than never: nice answer! – Gonzalo Medina May 19 '12 at 0:55
@Audrey: \supercite{bertram} in Inline citations actually adds extra vertical space between the current and the upper line. Might it be possible to change that? – maetra Oct 24 '12 at 16:07
@maetra Odd. I didn't notice that back then. This can't get fixed without adjustments to how fancytooltips sets the "extratext" area - something that I won't have time to look into for awhile. If you're willing to part with the bibliography links, you could use \DeclareFieldAlias{bibhyperref}{bibtooltip}. Otherwise you could try contacting Robert with a stripped-down MWE (i.e. without biblatex). – Audrey Oct 25 '12 at 3:42
@Audrey I tried to analyze this a bit more and found that the line spacing is correct in the article class if you use \tooltip instead of \tooltip* in \DeclareFieldFormat{bibhyperref}, but this doesn't work in beamer, so I posted a question here: tex.stackexchange.com/questions/79168/… – maetra Oct 26 '12 at 13:11
@maetra Yes, the new bibtooltip format I defined does the same thing. I don't think the culprit here is beamer, but \tooltip*. Unlike its unstarred variant, it sets an "extratext" area following the tooltip text. – Audrey Oct 26 '12 at 14:44
Both Gonzalo Medina and Audrey posted an excellent solution. But the things are simpler now. See the fancy-preview webpage - based on the new version of fancytooltips (May 2012), preview.sty and some bash scripts you get tooltips for bibliographic entries, theorems, definitions, displayed equations etc. automatically.
Edit: Small example and screenshots are attached.
\documentclass{article}
\usepackage{amsmath,amsthm}
\newtheorem{lemma}{Lemma}
\usepackage{hyperref} % Important!
\begin{document}
\begin{lemma}[Lemma from \cite{M}]\label{lemma}
If $k<0$, then
$$\label{eq:1} x^2+k<x^2$$
for every real number $x$.
\end{lemma}
Inequality \eqref{eq:1} in Lema \ref{lemma} can be proved easily. Is
more general than \cite[Theorem 3.4]{K}.
\begin{thebibliography}{9}
\bibitem[M]{M} Me: My book related to the problem (2006), 145 p.
% The new line is important!
\bibitem{K} Karl: Karl's paper published in some minor proceedings,
a local conferrence organized by his university (2005), 23--25.
% The new line is important!
\end{thebibliography}
\end{document}
Ref M
Ref 1
Equation (1)
Lemma 1.
-
+1. Thank you for developing this. Is it possible to have it work with Xetex/Luatex and using a package, rather than a bash script? – Emre May 7 '12 at 20:03
Your answer would benefit if you made it more self contained by including a description of how to use that script and screenshots of its results. – N.N. May 7 '12 at 20:11
+1 biblatex doesn't print bibliography items via \bibitem, but this looks like a great solution for biblatex alternatives. Thanks for this (and the fancytooltips package). – Audrey May 7 '12 at 20:37
Emre: I have no experience with Xetex, but the answer is probably no, since it does not generate PDF directly. I have minor experience with luatex, but guess, it will work. – robert.marik.cz May 8 '12 at 15:10
I copied and pasted your example document into TeXMaker on my Windows 7 PC. When I compile I get the normal output, but the 'previews' don't work for me (in Adobe Reader) :( Is there something that I am missing? – User 17670 Oct 14 '12 at 11:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8226829171180725, "perplexity": 4232.182315184371}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00452-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.ck12.org/book/CK-12-Chemistry-Basic/r11/section/17.0/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Chapter 17: Thermochemistry
Difficulty Level: Basic Created by: CK-12
When a chunk of potassium metal is added to water, a vigorous reaction occurs. Hydrogen gas is produced, along with a large amount of heat. In fact, the released heat is enough to ignite the hydrogen, creating a brilliant flame. Energy is an integral component of chemical reactions. Some reactions require an input of energy, whereas others release energy as they proceed. When we burn natural gas in our furnace at home, we are oxidizing small hydrocarbons by reacting them with oxygen. This reaction produces heat, which can be used to warm the house. In this chapter, we will focus primarily on the transfer of heat and energetic changes that occur during chemical reactions.
Chapter Outline
### Chapter Summary
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Description
Difficulty Level:
Basic
Authors:
Tags:
Subjects: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8535738587379456, "perplexity": 2883.8220158432405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982982027.87/warc/CC-MAIN-20160823200942-00151-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/150805-power-poles.html | # Math Help - power of poles
1. ## power of poles
$f=\frac{z}{1-\cos z}$
the singular points are z=2pik
and poles because there limit is infinity
now i want to determine te power of the pole
g=1/f= $\frac{1-\cos z}{z}$
$g'=\frac{(-\sin z)z-(1-\cos z)}{z^2}$
$g'(2\pi k)=0$
$g''=\frac{-\sin z z^2 -(cos z -1)2z}{z^4}$
$g''(2\pi k)=0$
the book says that its a second order pole
which is not true
because $g''(2\pi k)=0$
where it should differ zero in order to be pole
2. i have solved the 2pi k part
for zero i have a problem
$f=\frac{z}{1-\cos z}$
the singular points are z=2pik and zero
i solved for z=2pik
and poles because there limit is infinity
now i want to determine te power of the pole
g=1/f= $\frac{1-\cos z}{z}$
$g'=\frac{(-\sin z)z-(1-\cos z)}{z^2}$
$g'(0)=0/0$
$g''=\frac{-\sin z z^2 -(cos z -1)2z}{z^4}$
$g''(0)=0/0$
the book says that its a first order pole
it should differ zero in order to be pole
3. Originally Posted by transgalactic
$f=\frac{z}{1-\cos z}$
the singular points are z=2pik
and poles because there limit is infinity
now i want to determine te power of the pole
g=1/f= $\frac{1-\cos z}{z}$
$g'=\frac{(-\sin z)z-(1-\cos z)}{z^2}$
$g'(2\pi k)=0$
$g''=\frac{-\sin z z^2 -(cos z -1)2z}{z^4}$
$g''(2\pi k)=0$
the book says that its a second order pole
which is not true
because $g''(2\pi k)=0$
where it should differ zero in order to be pole
It's always good to have different methods for doing the same thing, that way you can check if your answers coincide. I'm going to give you how I would do it, and then you figure out where your argument went wrong.
Take $g(z)=1-\cos(z)$ expand in Taylor series and you get that the first term is $a_1z^2$ form which it follows that at $0$ $f$ has a first order pole. Since $\cos$ is periodic and the numerator doesn't vanish at any point other than $0$, we get that the poles for $2\pi k$ with $k\neq 0$ are of order $2$
4. Reply to Jose27: I think you meant, "... and the numerator doesn't vanish...", right? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 29, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8511838912963867, "perplexity": 988.1612076336069}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430454809062.87/warc/CC-MAIN-20150501043329-00002-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/9684/how-does-qiskit-finally-implement-a-noise-model?answertab=active | How does qiskit finally implement a noise model?
I have been reading qiskit documentation for hours and I still don't get how does it implements noise in the circuit. I have understood that it works with a objects of the class QuantumError which finally gives a zip of the quantum instructions (quantum gates that can be applied to the circuit) and the probability of each instruction. Once we have this, how is it really applied to the circuit when we use the add_all_qubit_quantum_error to add the error to the noise model? What I want to do is to check if it is theoretically correct as for a quantum channel the most efficient form to apply this will be to use the Kraus representation:
$$\mathcal E(\rho)=\sum_{k=1}^M E_k \rho E_k^\dagger.$$
For example for the bit-flip case the corresponding Kraus operators E_k will be:
Therefore to add the quantum noise for the bit-flip case we will need to multiply the density matrix of our state by the Kraus operator and its hermitian conjugate as we can see in the Kraus representation of a quantum channel. However I don't understand how can this corresponds with what I have read in the documentation as finally the functions such as pauli_error and so on end up returning a QuantumError object which is finally written as a zip of instructions and probabilities which I guess are append to the circuit in someway. I have carefully read the source code of every noise function that is used to implement the different types of noise and I am not able to figure out whether it is the same as implementing the corresponding Kraus errors, I would be extremely grateful if someone could answer me.
A probability distribution over unitaries $${(p_1, U_1),..., (p_k, U_K)}$$, where you select $$U_i$$ with probability $$p_i$$, is equivalent to Kraus operators $$\sqrt{p_1}U_1,..., \sqrt{p_k}U_k$$. For example, the bit flip error can be implemented by randomly deciding whether to leave the state unchanged (with probability $$p$$) or apply an $$X$$ gate (with probability $$1-p$$). With an infinite number of shots, where each shot randomizes independently from the other shots, all measurement statistics will converge to those obtained from the density matrix. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8221974968910217, "perplexity": 269.7247027508948}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302355.97/warc/CC-MAIN-20220120160411-20220120190411-00238.warc.gz"} |
http://lists.gnu.org/archive/html/bug-lilypond/2010-05/msg00290.html | bug-lilypond
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## Re: Issue 1089 in lilypond: DynamicTextSpanner not printed
From: Xavier Scheuer Subject: Re: Issue 1089 in lilypond: DynamicTextSpanner not printed Date: Sat, 29 May 2010 16:07:44 +0200
```2010/5/28 Carl Sorensen <address@hidden>:
> I think there are two questions here, and perhaps we agree on one and
> disagree on the other.
>
> The first question is: Does the 2.10.33 ouptut match what is asked for
> from the music, i.e. a text crescendo starting on the first note, and
> ending on the second note. I think we both agree that it does not.
> If you think that the 2.10.33 output matches the input, please explain
Of course you are right, if you think LilyPond should do it in a strict
"stubborn" way. :)
>
> [...]
>
> For the record, I don't think either situation is right!
Agree.
> In my mind, it's probably best to *not* print the cresc. and to omit
> the warning, because the user will know (from the output) that
> something is missing, and then can go to the log file to figure out
> why. On the other hand, if the cresc. is printed, the user may not
> notice from the output that the extent of the text crescendo is too
> long for the music. Hence, I think that not printing is a better
> failure mode than printing, because it's more obvious.
There I don't agree anymore.
You are reasoning in a "developer-mind", where I'm reasoning as a
"lambda" user. My idea is closer to Reinhold's thoughts (but not
exactly the same, see below).
> But I don't believe that it is a regression. A regression means that
> music that used to print correctly now doesn't print correctly.
> Since it never printed correctly, it's not a regression.
> On the other hand, if the "cresc" is printed, it is not such a
> problem if it goes beyond the second note, even though it might be not
> 100% correct interpretation.
That's exacly what I think!
> In my eyes, the strict solution would be to stretch the two notes so
> that the whole "cresc" can be printed and ends on the second note.
In a strict way, yes.
In a "practical" way, no.
Imagine somebody changes the text to "cresc. poco a poco" but still
ends it on the second note (which is an abhorrence, we all agree).
Then the second note would be incredibly shifted!
No, I don't think DynamicTextSpanner should behave like \textLengthOn .
> In reality, however, text crescendi should only be used for longer
> (de-)crescendi, so using a text crescendo and ending it on the next
> note is usually a shortcut of the engraver so he does not have to keep
> track of open crescendi (if no spanner line is printed, the output
> will be the same).
Yeah, that's what I meant by saying this is a pure "theorical case"
(but however could be used in practise).
BTW when http://lists.gnu.org/archive/html/lilypond-user/2010-05/msg00171.html
will be fixed, I'll certainly come again to request
"*without dashed line*" cresc by default for \cresc ... ;D
> So, I would print the "cresc" (even though it goes beyond the second
> note), but print out a warning, too.
Yes (but since it prints the "cresc." I'm not sure the warning is
really necessary). OK, in strict way, yes it is. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8077778816223145, "perplexity": 4321.483826803077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280242.65/warc/CC-MAIN-20170116095120-00472-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/100159-finding-value-k-system-equations-has-infinite-solutions.html | # Math Help - Finding the value of k, for which the system of equations has infinite solutions
1. ## Finding the value of k, for which the system of equations has infinite solutions
The value of k, k N, for which the system of equations 10x + ky = -8 and -15x - 6y = 12, has an infinite number of solutions is ___.
2. Originally Posted by slategray
The value of k, k N, for which the system of equations 10x + ky = -8 and -15x - 6y = 12, has an infinite number of solutions is ___.
for an infinite number of solutions, both lines must be the same line (superimposed)
... the coefficients of one equation will be identical multiples of the coefficients in the other equation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9549403786659241, "perplexity": 303.8101759621239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929012.53/warc/CC-MAIN-20150521113209-00329-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/160712/construction-of-cotangent-bundle | # construction of cotangent bundle
Cotangent Bundle $T^{*}(M)=\bigcup_{m\in M} M_m^{*}$ (disjoint union of contangent space) Could any one explain me how there is a natural projection from $T^{*}(M)\rightarrow M$ given by $\pi(f)=m$ if $f\in M^{*}_m$? I am not able to understand and feel the map naturally Please explain.
-
For each point in $M$ you have a cotangent space $M_m^*$. So we now look at the disjoint union of all of them as a collection of fibers -- "above" each point in $M$ lies a cotangent space. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8395828008651733, "perplexity": 205.61358055843888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931008215.58/warc/CC-MAIN-20141125155648-00198-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/388896/consistency-and-consistency-strength-of-certain-special-cuts-in-i-delta-0 | # Consistency and consistency strength of certain special cuts in $I\Delta_0$
Recall that $$I\Delta_0$$ is the theory in the language of arithmetic that consists of the axioms of $$\mathsf{PA}$$ with induction restricted to $$\Delta_0$$ formulas (i.e., formulas where all quantifiers are bounded).
It is not too difficult to build a model $$M$$ of $$I\Delta_0$$ with a proper cut $$N \subset M$$ such that $$M \equiv N$$. Moreover, we can ensure that $$2^n$$ exists for every $$n \in N$$ and that $$2^x$$ is not a total function on $$N$$ (and therefore not a total function on $$M$$ either). It is also a standard fact that the set $$\{ x \in M : (\exists y \in N) x < 2^y\}$$ is a cut that models $$I\Delta_0$$ (this follows from the fact that it is downwards closed and closed under addition and multiplication). My question is about this last cut actually being all of $$M$$.
Let $$C$$ be a unary predicate symbol, and let $$T$$ be the theory in the language of arithmetic augmented with $$C$$ that contains $$I\Delta_0$$ and says that
• $$C$$ is downwards closed,
• $$C$$ is not all of the structure,
• $$2^x$$ is not a total function,
• $$2^x$$ is total on $$C$$,
• the induced structure on $$C$$ has the same theory as the full structure (in particular, it is a model of $$I\Delta_0$$), and
• for every $$x$$, there exists a $$y \in C$$ such that $$x < 2^y$$.
Obviously, the fifth bullet point must be stated as an axiom scheme.
Question 1. Is $$T$$ consistent?
Question 2. Does $$I\Delta_0$$ interpret $$T$$?
Question 3. Does $$T$$ interpret $$I\Delta_0 + \mathrm{Exp}$$ (where $$\mathrm{Exp}$$ is the statement that $$2^x$$ is a total function)?
Note that since $$I\Delta_0$$ does not interpret $$I\Delta_0 + \mathrm{Exp}$$, these last two questions cannot both have a positive answer.
• In the introduction, you required that $2^x$ exists in $M$ for every $x\in N$. This condition is missing in $T$ (and this will make a lot of difference). Is this omission intentional? – Emil Jeřábek Mar 31 at 7:22
• No that was a mistake. – James Hanson Mar 31 at 8:01
$$T$$ is inconsistent. The argument below is due to Robert Solovay (it is attributed to a letter from Solovay to Nelson in Visser’s Peano Basso and Peano Corto, see Lemma 3.7).
Let $$2^x_n$$ denote the iterated exponential function $$2^x_0=x$$, $$2^x_{n+1}=2^{2^x_n}$$. It is well known that the graph of $$2^x_n$$ has a well-behaved $$\Delta_0$$ definition in $$I\Delta_0$$.
Lemma (Solovay): $$I\Delta_0+\neg\mathrm{Exp}$$ proves that there exists a unique number $$n$$ such that $$2^0_n$$ exists, but $$2^0_{n+1}$$ does not.
Proof: Take $$x$$ such that $$2^x$$ does not exist, and let $$n$$ be maximal such that $$2^0_n\le x$$. Then $$2^0_{n+1}$$ either does not exist, or satisfies $$2^0_{n+1}>x$$, hence $$2^0_{n+2}$$ does not exist. Thus, $$n$$ or $$n+1$$ satisfies the conclusion of the Lemma. QED
Let us call the $$n$$ from the Lemma as Solovay’s number.
Now, observe that the axioms of $$T$$ imply that $$C=\{x:2^x\text{ exists}\}$$. Thus, working in $$T$$ (which includes $$I\Delta_0+\neg\mathrm{Exp}$$), if $$n$$ is Solovay’s number, then $$n-1$$ is Solovay’s number in $$C$$. In particular, the universe and $$C$$ disagree on the truth of the sentence “Solovay’s number is even”, contradicting the elementary equivalence axiom. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 67, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9825707077980042, "perplexity": 120.06649727363022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488551052.94/warc/CC-MAIN-20210624045834-20210624075834-00477.warc.gz"} |
http://mathoverflow.net/questions/47888/expectation-multinomial-distribution-and-monotonicity-a-conjecture | # Expectation, multinomial distribution, and monotonicity (A conjecture)
Let $n$ and $k$ be two positive integers. Let $S = \{ \mathbf{p} \in \mathbb{R}^k : \mathbf{p} \geq 0, \sum_{i=1}^k p_i = 1 \}$ (i.e., a simplex). Consider a function $\mathbf{f}:\mathbb{Z}^k \rightarrow \mathbb{R}^k$. And from $\mathbf{f}$ we may define function $\mathbf{g}: S\subset \mathbb{R}^k \rightarrow \mathbb{R}^k$ as follows $$\mathbf{g}(\mathbf{p}) = \mathbb{E}[\mathbf{f}(\mathbf{X})], \quad \mathbf{X}\sim \text{Multinomial}(n,\mathbf{p}).$$ Here $\mathbf{X}$ is a random vector that follows the multinomial distribution determined by the number of trials $n$ and probabilities $p_1,p_2,\dots,p_k$.
Conjecture: If $\mathbf{f}$ is monotone, that is, $$(\mathbf{x}-\tilde{\mathbf{x}})^T (\mathbf{f}(\mathbf{x})-\mathbf{f}(\tilde{\mathbf{x}})) \geq 0\quad \forall \mathbf{x}, \tilde{\mathbf{x}} \in \mathbb{Z}^k,$$ then $\mathbf{g}$ is also monotone, i.e., $$(\mathbf{p}-\tilde{\mathbf{p}})^T (\mathbf{g}(\mathbf{p})-\mathbf{g}(\tilde{\mathbf{p}})) \geq 0\quad \forall \mathbf{p}, \tilde{\mathbf{p}} \in S.$$
Remark: The above result should hold in the following two special cases (I omit the proofs) -
1. The function $\mathbf{f}$ is affine, namely, $\mathbf{f}(\mathbf{x})=G\mathbf{x}+\mathbf{b}$ for some matrix $G$ and vector $\mathbf{b}$.
2. The function $\mathbf{f}$ is "separable", meaning that $$\mathbf{f}(\mathbf{x}) = (f_1(x_1), \dots, f_k(x_k))^T$$ for some non-decreasing scalar functions $f_1(\cdot), \dots, f_k(\cdot)$.
But does it hold in the general case?
-
@daizhuo - can you give an example of a monotone function, possibly of interest, that is not affine or separable? – ronaf Jan 22 '11 at 4:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.994065523147583, "perplexity": 166.0948707826639}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464052946109.59/warc/CC-MAIN-20160524012226-00241-ip-10-185-217-139.ec2.internal.warc.gz"} |
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