url
stringlengths 14
2.42k
| text
stringlengths 100
1.02M
| date
stringlengths 19
19
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|---|
https://itprospt.com/num/6832032/b-fc-ai-j-fo-a-4-fka-u-1-gt-j-fo-r-4-4-a-ixi-fo-hlw-for
|
5
# B fc) AI j- fo) -A 4 fka) U~ 1>J+ fo)= R 4 4 (a)(() Ixi , *fo HLw; for Fri 0 , ^ = 0 Ikh...
## Question
###### B fc) AI j- fo) -A 4 fka) U~ 1>J+ fo)= R 4 4 (a)(() Ixi , *fo HLw; for Fri 0 , ^ = 0 Ikh
B fc) A I j- fo) -A 4 fka) U~ 1>J+ fo)= R 4 4 (a) (() Ixi , *fo HLw; for Fri 0 , ^ = 0 Ikh
#### Similar Solved Questions
##### M cengeeassigmini takcCowalentAciivity do?locator-assionnxnl ~TakestakcAssig nanenlSesskilocator- a9s1anment-takecmnntThe Sululailita Hoduci CunsLniphu:WulteM leau phnsnhalessueValcevam"tuiibrlineuunaneraiand plosutuleHigh ModcnlcCutianteal urocdaneKg High Maubtlsubmi AnswctRottt fmaino Grol9 molo nicupaicemple(Pietous41Z
M cenge eassigmini takcCowalentAciivity do?locator-assionnxnl ~TakestakcAssig nanenlSesskilocator- a9s1anment-take cmnnt The Sululailita Hoduci CunsLni phu:Wult e M leau phnsnhale ssue Valcevam "tuiibrlin euunane raiand plosutule High Modcnlc Cutiant eal urocdane Kg High Maubtl submi Answct Ro...
##### Determine the volume enclosed between the hemispherical surface 2 = f(w,y) = V49 _ 22 y2 and the 2 = 0 plane by evaluating the double integral IIRftwsy)aA in polar coordinates: Give an exact value. V = (6861pi )(3) [4 points] Reminder: is entered as
Determine the volume enclosed between the hemispherical surface 2 = f(w,y) = V49 _ 22 y2 and the 2 = 0 plane by evaluating the double integral IIRftwsy)aA in polar coordinates: Give an exact value. V = (6861pi )(3) [4 points] Reminder: is entered as...
##### Name:Consider vectors A 50.0 m, 40.0" East of North) and B: (20.0 m, 20.00 Nonth of West) Represent both vectors In Tel coordlnate system and find the components of vector and B Express veclor € B as linear combination ofthe unit vectors Calculate the magnitude and direction of vector €truin stans from rest and accelerates uniformly until it has traveled 2.5 km and acquired velocity 30mn/$The trair them moves ConsVnd velocity 6f 30 m$ for 420 The then slow ~ down untonly 0.0} Is? , un
Name: Consider vectors A 50.0 m, 40.0" East of North) and B: (20.0 m, 20.00 Nonth of West) Represent both vectors In Tel coordlnate system and find the components of vector and B Express veclor € B as linear combination ofthe unit vectors Calculate the magnitude and direction of vector â...
##### (10 points) Determine if x is in the space generated by b1 and bz. If it is_ find the B-coordinale vector of where B = {D1; bz}31
(10 points) Determine if x is in the space generated by b1 and bz. If it is_ find the B-coordinale vector of where B = {D1; bz} 31...
##### Orm polynomia f(x) with real coefficients having the given degree and zeros_Degree 5; Zelus:i;6 +Enter the polynomial:(x) = Type an expression using as the variable. Use integers or fractions for any numbers in the expression. Simplify your answer:)
orm polynomia f(x) with real coefficients having the given degree and zeros_ Degree 5; Zelus: i;6 + Enter the polynomial: (x) = Type an expression using as the variable. Use integers or fractions for any numbers in the expression. Simplify your answer:)...
##### Suppose that the samples are from the Gamma(a, 8) distribution whose density function isf(rla, 8) = r"-le-*/8 r 2 0, a > 0, 8 > 0. T(a) Ba (a). Give the formulae of the mean and variance.Suppose that we have the MLEs &, / of &, B,and ofthe mean and variance: Describe the estimation procedures of the following two methods; (6)_ the parametric bootstrap estimation for the mean when the MLE is applied; (c): suppose we have n samples, construct the nonparametric bootstrap estimat
Suppose that the samples are from the Gamma(a, 8) distribution whose density function is f(rla, 8) = r"-le-*/8 r 2 0, a > 0, 8 > 0. T(a) Ba (a). Give the formulae of the mean and variance. Suppose that we have the MLEs &, / of &, B,and ofthe mean and variance: Describe the estim...
##### Question 3 (1 point) Unoed oharacecyaisbnces is MPN selected over SPC (APC) when determining the food or water sample? number of microbes in
Question 3 (1 point) Unoed oharacecyaisbnces is MPN selected over SPC (APC) when determining the food or water sample? number of microbes in...
##### Question 5: List thc thrce groups ideatificd in Quostioncondenscd ATucturu tonnand 5 identified in Qucstons Question 6: Draw Newman projection that includes thc groups attached to both the front (C4) und back carbons (CS).question 6 untl you projection draun Provide ; reasoalng the initial stable = conlorner. ( carbon froin Circle the moit = Question 7: Rotate the front _ projection again. have reached the initial most = stable: this one is the a5 to why
Question 5: List thc thrce groups ideatificd in Quostion condenscd ATucturu tonn and 5 identified in Qucstons Question 6: Draw Newman projection that includes thc groups attached to both the front (C4) und back carbons (CS). question 6 untl you projection draun Provide ; reasoalng the initial stabl...
##### Calculate A G"for an electrochemical cell reaction that occurs under basic aqueous conditions based on the following two half-reactions for which the standard reduction potentials are given. Use the smallest whole-number coefficients possible when balancing the Overall reaction: Cd(OH), +2 Cd+2 OH" 0.724NiO(OH) + H,O+eNi(OH) , OH"+1.32~115kJ+95.+115 kJ394 kJ
Calculate A G"for an electrochemical cell reaction that occurs under basic aqueous conditions based on the following two half-reactions for which the standard reduction potentials are given. Use the smallest whole-number coefficients possible when balancing the Overall reaction: Cd(OH), +2 Cd+2...
##### Evaluate the following integrals: $$\int \frac{x}{e^{x}} d x$$
Evaluate the following integrals: $$\int \frac{x}{e^{x}} d x$$...
##### Point) The setB ={[:]' [8}}basis for R? Find the coordinates of the vector &relative t0 the basis B[zle
point) The set B = {[:]' [8}} basis for R? Find the coordinates of the vector & relative t0 the basis B [zle...
##### Gren Trapezoid ABC D witk ABIIDC 4nd ADDRPROOFSutemae IttCentmtPtollmnGnecrLADC 4RCDEnnanAAUC AkCDMoblrn _UCpccComplcte ReasonP1oi c
Gren Trapezoid ABC D witk ABIIDC 4nd AD DR PROOF Sutemae Itt Centmt Ptollmn Gnecr LADC 4RCD Ennan AAUC AkCD Moblrn _U Cpcc Complcte Reason P1oi c...
##### (A) NO CHANGE(B) It is worth giving up, Kelli argues, because thoughshe is losing her summer, she is doing the job ofan actual engineer through her internship.(C) It is worth it to give up her summer, Kelli argues,because she is doing the job of an actual engineerthrough her internship.(D) It is worth it to Kellito give up her summer,because though summers are usually a time torelax, she argues, she is doing the job of an actualengineer through her internship.
(A) NO CHANGE (B) It is worth giving up, Kelli argues, because though she is losing her summer, she is doing the job of an actual engineer through her internship. (C) It is worth it to give up her summer, Kelli argues, because she is doing the job of an actual engineer through her internship. ...
##### Find the Taylor polynomials of orders 0, 1, 2,and 3 generated by f at a f(x) = 3 In (x), a =The Taylor polynomial of order 0 is Po(x) =The Taylor polynomial of order 1 is P1 (x) The Taylor polynomial of order 2 is Pz(x)The Taylor polynomial of order 3 is P3(x) =
Find the Taylor polynomials of orders 0, 1, 2,and 3 generated by f at a f(x) = 3 In (x), a = The Taylor polynomial of order 0 is Po(x) = The Taylor polynomial of order 1 is P1 (x) The Taylor polynomial of order 2 is Pz(x) The Taylor polynomial of order 3 is P3(x) =...
##### Determine the interval I for which the following DEQS have a unique solution whose graph passes through (x0,y0) in the interval I.a) (y-x)y'=y+xb) (x^(2)+y^(2))y'=y^(2)
determine the interval I for which the following DEQS have a unique solution whose graph passes through (x0,y0) in the interval I.a) (y-x)y'=y+xb) (x^(2)+y^(2))y'=y^(2)...
|
2022-10-02 21:34:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5891932249069214, "perplexity": 9545.004829761909}, "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/1664030337360.41/warc/CC-MAIN-20221002212623-20221003002623-00386.warc.gz"}
|
https://proofwiki.org/wiki/Five_Color_Theorem
|
Five Color Theorem
Theorem
A planar graph $G$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.
Proof
The proof proceeds by the Principle of Mathematical Induction on the number of vertices.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
$G_n$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.
Basis for the Induction
$\map P r$ is trivially true for $1 \le r \le 5$, as there are no more than $5$ vertices to be colored.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P r$ is true, where $r \ge 5$, then it logically follows that $\map P {r + 1}$ is true.
So this is our induction hypothesis:
$G_r$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.
Then we need to show:
$G_{r + 1}$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.
Induction Step
This is our induction step:
According to the Minimum Degree Bound for Simple Planar Graph, $G_{r + 1}$ has at least one vertex with at most $5$ edges.
Let this vertex be labeled $x$.
Remove vertex $x$ from $G_{r + 1}$ to create another graph, $G'_r$.
By the induction hypothesis, $G'_r$ is five-colorable.
Suppose all five colors were not connected to $x$.
Then we can give $x$ the missing color and thus five-color $G_{r + 1}$.
Suppose all five colors are connected to $x$.
Then examine the five vertices $x$ was adjacent to.
Call them $y_1, y_2, y_3, y_4$ and $y_5$ in clockwise order around $x$.
Let $y_1, y_2, y_3, y_4$ and $y_5$ be colored respectively by colors $c_1, c_2, c_3, c_4$ and $c_5$.
Let us denote $H_{i, j}$ a subgraph of $G'_r$ induced by the vertices colored with $c_i$ and $c_j$.
Consider $H_{1, 3}$.
Suppose there exists no path between $y_1$ and $y_3$ in $H_{1, 3}$.
Thus, $H_{1, 3}$ is disconnected into two components.
We can, then, interchange the colors $c_1$ and $c_3$ in the component that is connected to $y_1$.
Thus $x$ is no longer adjacent to a vertex of color $c_1$, so $x$ can be colored $c_1$.
Suppose there exists a path between $y_1$ and $y_3$ in $H_{1, 3}$.
Including the vertex $x$ in this path we get a circuit $C$.
Since we indexed the vertices $y_1, y_2, y_3, y_4$ and $y_5$ clockwise, exactly one of the vertices $y_2$ and $y_4$ is inside $C$.
Hence, $y_2$ and $y_4$ are in different connected components of $H_{2, 4}$
Then we can switch colors $c_2$ and $c_4$ in the component of $H_{2, 4}$ that is connected to $y_2$.
Now $x$ is no longer adjacent to a vertex of color $c_2$, so we can color it $c_2$.
This graph illustrates the case in which the path from $y_1$ to $y_3$ can be completed.
$\text{Blue} = c_1, \text{Yellow} = c_2, \text{Red} = c_3, \text{Green} = c_4, \text{Turquoise} = c_5$.
The dotted lines represent edges and vertices that might exist, as this is simply a fairly minimal example graph that matches the conditions.
So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
For all $n \in \N_{>0}$, $G_n$ can be assigned a proper vertex $k$-coloring such that $k \le 5$.
Also known as
This theorem is also known as Heawood’s Theorem, for Percy John Heawood, although The Five-Color Theorem is more widely used.
The British English spelling of this proof is five colour theorem.
Also see
• The proof gives a simple (recursive) algorithm for $5$-coloring a planar graph, the so-called Heawood's Algorithm.
Historical Note
The Five Color Theorem is not the strongest result possible.
It was proved by Percy John Heawood in $1890$.
In the same year he showed a flaw in Alfred Bray Kempe's supposed $1879$ proof of the Four Color Theorem which was not mended for almost another $100$ years.
It was not until $1976$ that Kenneth Ira Appel and Wolfgang Haken demonstrated that four colors suffice.
Their proof relies heavily on computers and for the moment is not to be found on $\mathsf{Pr} \infty \mathsf{fWiki}$.
|
2021-09-17 20:24:18
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8595631122589111, "perplexity": 234.49539174628723}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055775.1/warc/CC-MAIN-20210917181500-20210917211500-00013.warc.gz"}
|
https://puzzling.stackexchange.com/questions/80039/how-should-i-approach-using-two-8s-and-two-3s-to-make-the-number-24?noredirect=1
|
# How should I approach using two 8s and two 3s to make the number 24?
Use two $$8$$s, two $$3$$s, and basic arithmetic operators ($$+, -, \times , \div$$, parentheses) to make the number $$24$$.
(You may not join numbers together to form new numbers, like $$8, 3\rightarrow 83$$)
I don't know how to start besides just trying to find the correct answer. Is there a way you can make this equation through small steps or I should just bruteforce it?
• see puzzling.stackexchange.com/questions/50259/coppers-make-24 (GM's answer) – JMP Feb 27 '19 at 21:17
• If I am not mistaken, this question is not asking people to solve the puzzle in question, but is instead asking strategies for how to go about solving it beyond just trying things at random. – Lunin Feb 28 '19 at 1:13
• One thing I'd suggest is determining whether any rounding is allowed. Narrows down the number of pieces you have to work with if no, and opens up more options if yes. – Justin Time - Reinstate Monica Feb 28 '19 at 1:38
While there are some good answers here, it seems like you are asking how to think of the answer. (If so, perhaps the title of this might need to be edited.)
Here's one method of thinking to get to the answer:
# 1) Is this a trick question?
It appears not - everything seems to be at face value, and there is a mathematics tag not a lateral thinking tag or similar.
# 2) What do we need to do?
What is the structure of the answer that you need to find? Well, it looks something like $$8 + 8 - (3 + 3) = 10$$. Except of course, this example equals 10, we need 24. But at least that's what we are going for. Another example is $$8 + 8 - (3 \times 3) = 7$$, but that doesn't work either. Not to worry just yet, we are just getting a feel of things.
# 3) Can we simplify the problem down at all?
Well, in this case, we can see that we can generate more potential solutions by changing the operators that we use. In fact, that's what we did above - we changed the $$+$$ in the brackets to $$\times$$, which changed the $$6$$ in the brackets to a $$9$$, which subtracted an extra $$3$$ from the result. The $$8 + 8 = 16$$ didn't change at all. Hmmm... there's something in that which we can use.
# 4) What components get us closer to the solution?
So the $$16$$ we had in both the proposals above is like its own starting point - that is, we can swap the two 8s from the original question for a 16, and make the question "Given a 16 and two 3s, make 24". That's not to say that we are going to find a solution to this, but it's one possible statement that will solve the original question. And it comes from us thinking about the number $$16$$. What other numbers can we make by consuming two of the numbers?
• $$1 = 8 \div 8$$ with $$3,3$$ leftover
• $$16 = 8 + 8$$ with $$3,3$$ leftover
• $$64 = 8 \times 8$$
• $$0 = 8 - 8$$
• $$24 = 8 \times 3$$ with $$8,3$$ leftover
• $$11 = 8 + 3$$
• $$5 = 8 - 3$$
• $$2 \frac{2}{3} = 8 \div 3$$
• ...
# 5) Work from the other end - what do the components of the solution look like?
Consider the solution: $$? = 24$$. What could those components possibly look like? Well, we know that $$8 * 3 = 24$$ - that's a good start, and can lead us to a potential solution:
$$\sqrt{8 * 8 * 3 * 3} = 8 * 3 = 24$$
I'm not completely happy with this though - it seems to me that using the square root is a bit of trickery. How else can we make 24 using one of our numbers?
• $$8 * 3 = 24$$
• $$8 / \frac{1}{3} = 24$$
• $$27 - 3 = 24$$
• $$21 + 3 = 24$$
• $$32 - 8 = 24$$
• ...
# 6) Connect the dots.
We now have a list of numbers that can be made with two of our numbers, and a list of numbers that we want to be made with 3 of our numbers. It might take a bit of inspiration, but is there any link we can make between any of them?
From the above, here's the link I've come up with:
$$3 - 2 \frac{2}{3} = \frac{1}{3}$$
That will lead us to a solution by putting it all together:
$$8 \div (3 - \frac{8}{3})) = 24$$
# Fin
That's the way I think of these things. Hopefully you will get to a point where most of this occurs in your head pretty fast, and not necessarily in that order.
• Hey look, someone actually answered the question asked! :) – Rubio Feb 28 '19 at 4:34
Here is a solution that uses only "elementary" operations (addition, subtraction, multiplication, and division).
$$8 \div (3 - (8 \div 3))$$ (or alternatively $$\frac{8}{3 - \frac{8}{3}}$$)
$$= 8 \div \frac{1}{3}$$
$$= 24$$
If we allow square roots, a simpler solution is possible.
$$\sqrt{8 \times 8 \times 3 \times 3}$$
$$= 8 \times 3$$
$$= 24$$
In fact, there are many more solutions if you allow more operations.
|
2020-01-23 02:52:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 43, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6632190942764282, "perplexity": 308.34475120940135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250608062.57/warc/CC-MAIN-20200123011418-20200123040418-00220.warc.gz"}
|
https://www.maplesoft.com/support/help/view.aspx?path=networks(deprecated)/vertices
|
networks(deprecated)/vertices - Maple Help
networks
vertices
returns a vertex set
Calling Sequence vertices(G)
Parameters
G - graph or network
Description
• Important: The networks package has been deprecated.Use the superseding command GraphTheory[Vertices] instead.
• This routine returns the vertex set of G. It is normally loaded via the command with(networks) but may also be referenced using the full name networks[vertices](...).
Examples
Important: The networks package has been deprecated.Use the superseding command GraphTheory[Vertices] instead.
> $\mathrm{with}\left(\mathrm{networks}\right):$
> $\mathrm{new}\left(G\right):$
> $\mathrm{addvertex}\left(1,2,\mathrm{v1},\mathrm{v2},\mathrm{v3},a,b,c,G\right):$
> $\mathrm{vertices}\left(G\right)$
$\left\{{1}{,}{2}{,}{a}{,}{b}{,}{c}{,}{\mathrm{v1}}{,}{\mathrm{v2}}{,}{\mathrm{v3}}\right\}$ (1)
|
2022-08-18 20:10:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7814757227897644, "perplexity": 3404.8887552567185}, "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/1659882573399.40/warc/CC-MAIN-20220818185216-20220818215216-00483.warc.gz"}
|
http://mathoverflow.net/questions/127154/brown-representability-for-the-standard-model-category-of-simplicial-sets?answertab=active
|
# Brown representability for the standard model category of simplicial sets
Let $Sset$ denote the category of simplicial sets with its Quillen model structure, when is a functor $F: (ho Sset)^{op} \to Ab$ representable? With $Ab$ category of Abelian groups. There is probably some classical references but my googlefu wasn't strong enough. I am hoping it would just be the direct translation of Brown representability theorem for $Ab$ valued cofunctors on $hoTop$.
-
On second thought I think one can just prove that speculation directly using Quillen equivalence between the two model categories. – yasha Apr 10 '13 at 21:58
There is a second classical paper by Brown himself which abstracts his original paper:
Brown, Edgar H., Jr. Abstract homotopy theory. Trans. Amer. Math. Soc. 119 1965 79–85.
I think you will find that it applies directly. Of course, that was well before model categories.
-
looks good, thanks! – yasha Apr 11 '13 at 1:36
|
2015-08-02 12:47:21
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9006720781326294, "perplexity": 655.3146953895432}, "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-2015-32/segments/1438042989043.35/warc/CC-MAIN-20150728002309-00108-ip-10-236-191-2.ec2.internal.warc.gz"}
|
https://rdrr.io/cran/fda.usc/man/flm.test.html
|
# flm.test: Goodness-of-fit test for the Functional Linear Model with... In fda.usc: Functional Data Analysis and Utilities for Statistical Computing
flm.test R Documentation
## Goodness-of-fit test for the Functional Linear Model with scalar response
### Description
The function flm.test tests the composite null hypothesis of a Functional Linear Model with scalar response (FLM),
H_0: Y=<X,β>+ε,
versus a general alternative. If β=β_0 is provided, then the simple hypothesis H_0: Y=<X,β_0>+ε is tested. The testing of the null hypothesis is done by a Projected Cramer-von Mises statistic (see Details).
### Usage
flm.test(
X.fdata,
Y,
beta0.fdata = NULL,
B = 5000,
est.method = "pls",
p = NULL,
type.basis = "bspline",
verbose = TRUE,
plot.it = TRUE,
B.plot = 100,
G = 200,
...
)
X.fdata Functional covariate for the FLM. The object must be in the class fdata. Y Scalar response for the FLM. Must be a vector with the same number of elements as functions are in X.fdata. beta0.fdata Functional parameter for the simple null hypothesis, in the fdata class. Recall that the argvals and rangeval arguments of beta0.fdata must be the same of X.fdata. A possibility to do this is to consider, for example for β_0=0 (the simple null hypothesis of no interaction), beta0.fdata=fdata(mdata=rep(0,length(X.fdata$argvals)),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval). If beta0.fdata=NULL (default), the function will test for the composite null hypothesis. B Number of bootstrap replicates to calibrate the distribution of the test statistic. B=5000 replicates are the recommended for carry out the test, although for exploratory analysis (not inferential), an acceptable less time-consuming option is B=500. est.method Estimation method for the unknown parameter β, only used in the composite case. Mainly, there are two options: specify the number of basis elements for the estimated β by p or optimally select p by a data-driven criteria (see Details section for discussion). Then, it must be one of the following methods: "pc" If p, the number of basis elements, is given, then β is estimated by fregre.pc. Otherwise, an optimum p is chosen using fregre.pc.cv and the "SICc" criteria. "pls" If p is given, β is estimated by fregre.pls. Otherwise, an optimum p is chosen using fregre.pls.cv and the "SICc" criteria. This is the default argument as it has been checked empirically that provides a good balance between the performance of the test and the estimation of β. "basis" If p is given, β is estimated by fregre.basis. Otherwise, an optimum p is chosen using fregre.basis.cv and the "GCV.S" criteria. In these functions, the same basis for the arguments basis.x and basis.b is considered. The type of basis used will be the given by the argument type.basis and must be one of the class of create.basis. Further arguments passed to create.basis (not rangeval that is taken as the rangeval of X.fdata), can be passed throughout ... . p Number of elements of the basis considered. If it is not given, an optimal p will be chosen using a specific criteria (see est.method and type.basis arguments). type.basis Type of basis used to represent the functional process. Depending on the hypothesis it will have a different interpretation: Simple hypothesis. One of these options: "bspline" If p is given, the functional process is expressed in a basis of p B-splines. If not, an optimal p will be chosen by optim.basis, using the "GCV.S" criteria. "fourier" If p is given, the functional process is expressed in a basis of p fourier functions. If not, an optimal p will be chosen by optim.basis, using the "GCV.S" criteria. "pc" p must be given. Expresses the functional process in a basis of p PC. "pls" p must be given. Expresses the functional process in a basis of p PLS. Although other of the basis supported by create.basis are possible too, "bspline" and "fourier" are recommended. Other basis may cause incompatibilities. Composite hypothesis. This argument is only used when est.method="basis" and, in this case, claims for the type of basis used in the basis estimation method of the functional parameter. Again, basis "bspline" and "fourier" are recommended, as other basis may cause incompatibilities. verbose Either to show or not information about computing progress. plot.it Either to show or not a graph of the observed trajectory, and the bootstrap trajectories under the null composite hypothesis, of the process R_n(.) (see Details). Note that if plot.it=TRUE, the function takes more time to run. B.plot Number of bootstrap trajectories to show in the resulting plot of the test. As the trajectories shown are the first B.plot of B, B.plot must be lower or equal to B. G Number of projections used to compute the trajectories of the process R_n(.) by Monte Carlo. ... Further arguments passed to create.basis. ### Details The Functional Linear Model with scalar response (FLM), is defined as Y=<X,β>+ε, for a functional process X such that E[X(t)]=0, E[X(t)ε]=0 for all t and for a scalar variable Y such that E[Y]=0. Then, the test assumes that Y and X.fdata are centred and will automatically center them. So, bear in mind that when you apply the test for Y and X.fdata, actually, you are applying it to Y-mean(Y) and fdata.cen(X.fdata)$Xcen. The test statistic corresponds to the Cramer-von Mises norm of the Residual Marked empirical Process based on Projections R_n(u,γ) defined in Garcia-Portugues et al. (2014). The expression of this process in a p-truncated basis of the space L^2[0,T] leads to the p-multivariate process R_{n,p}(u,γ^{(p)}), whose Cramer-von Mises norm is computed. The choice of an appropriate p to represent the functional process X, in case that is not provided, is done via the estimation of β for the composite hypothesis. For the simple hypothesis, as no estimation of β is done, the choice of p depends only on the functional process X. As the result of the test may change for different p's, we recommend to use an automatic criterion to select p instead of provide a fixed one. The distribution of the test statistic is approximated by a wild bootstrap resampling on the residuals, using the golden section bootstrap. Finally, the graph shown if plot.it=TRUE represents the observed trajectory, and the bootstrap trajectories under the null, of the process RMPP integrated on the projections:
R_n(u) \approx \frac{1}{G} ∑_{g=1}^G R_n(u,γ_g),
where γ_g are simulated as Gaussians processes. This gives a graphical idea of how distant is the observed trajectory from the null hypothesis.
### Value
An object with class "htest" whose underlying structure is a list containing the following components:
• statistic The value of the test statistic.
• boot.statistics A vector of length B with the values of the bootstrap test statistics.
• p.value The p-value of the test.
• method The method used.
• B The number of bootstrap replicates used.
• type.basis The type of basis used.
• beta.est The estimated functional parameter β in the composite hypothesis. For the simple hypothesis, the given beta0.fdata.
• p The number of basis elements passed or automatically chosen.
• ord The optimal order for PC and PLS given by fregre.pc.cv and fregre.pls.cv. For other methods is setted to 1:p.
• data.name The character string "Y=<X,b>+e"
### Note
No NA's are allowed neither in the functional covariate nor in the scalar response.
### Author(s)
Eduardo Garcia-Portugues. Please, report bugs and suggestions to edgarcia@est-econ.uc3m.es
### References
Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030-1051. doi: 10.1017/S0266466606060506
Garcia-Portugues, E., Gonzalez-Manteiga, W. and Febrero-Bande, M. (2014). A goodness–of–fit test for the functional linear model with scalar response. Journal of Computational and Graphical Statistics, 23(3), 761-778. doi: 10.1080/10618600.2013.812519
Adot, PCvM.statistic, rwild, flm.Ftest, dfv.test, fregre.pc, fregre.pls, fregre.basis, fregre.pc.cv, fregre.pls.cv, fregre.basis.cv, optim.basis, create.basis
### Examples
# Simulated example #
X=rproc2fdata(n=100,t=seq(0,1,l=101),sigma="OU")
beta0=fdata(mdata=cos(2*pi*seq(0,1,l=101))-(seq(0,1,l=101)-0.5)^2+
rnorm(101,sd=0.05),argvals=seq(0,1,l=101),rangeval=c(0,1))
Y=inprod.fdata(X,beta0)+rnorm(100,sd=0.1)
dev.new(width=21,height=7)
par(mfrow=c(1,3))
plot(X,main="X")
plot(beta0,main="beta0")
plot(density(Y),main="Density of Y",xlab="Y",ylab="Density")
rug(Y)
## Not run:
# Composite hypothesis: do not reject FLM
pcvm.sim=flm.test(X,Y,B=50,B.plot=50,G=100,plot.it=TRUE)
pcvm.sim
flm.test(X,Y,B=5000)
# Estimated beta
dev.new()
plot(pcvm.sim$beta.est) # Simple hypothesis: do not reject beta=beta0 flm.test(X,Y,beta0.fdata=beta0,B=50,B.plot=50,G=100) flm.test(X,Y,beta0.fdata=beta0,B=5000) # AEMET dataset # data(aemet) # Remove the 5\ dev.new() res.FM=depth.FM(aemet$temp,draw=TRUE)
qu=quantile(res.FM$dep,prob=0.05) l=which(res.FM$dep<=qu)
lines(aemet$temp[l],col=3) aemet$df$name[l] # Data without outliers wind.speed=apply(aemet$wind.speed$data,1,mean)[-l] temp=aemet$temp[-l]
# Exploratory analysis: accept the FLM
pcvm.aemet=flm.test(temp,wind.speed,est.method="pls",B=100,B.plot=50,G=100)
pcvm.aemet
# Estimated beta
dev.new()
plot(pcvm.aemet$beta.est,lwd=2,col=2) # B=5000 for more precision on calibration of the test: also accept the FLM flm.test(temp,wind.speed,est.method="pls",B=5000) # Simple hypothesis: rejection of beta0=0? Limiting p-value... dat=rep(0,length(temp$argvals))
flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals, rangeval=temp$rangeval),B=100)
flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals, rangeval=temp$rangeval),B=5000)
# Tecator dataset #
data(tecator)
names(tecator)
absorp=tecator$absorp.fdata ind=1:129 # or ind=1:215 x=absorp[ind,] y=tecator$y$Fat[ind] tt=absorp[["argvals"]] # Exploratory analysis for composite hypothesis with automatic choose of p pcvm.tecat=flm.test(x,y,B=100,B.plot=50,G=100) pcvm.tecat # B=5000 for more precision on calibration of the test: also reject the FLM flm.test(x,y,B=5000) # Distribution of the PCvM statistic plot(density(pcvm.tecat$boot.statistics),lwd=2,xlim=c(0,10),
main="PCvM distribution", xlab="PCvM*",ylab="Density")
rug(pcvm.tecat$boot.statistics) abline(v=pcvm.tecat$statistic,col=2,lwd=2)
legend("top",legend=c("PCvM observed"),lwd=2,col=2)
# Simple hypothesis: fixed p
dat=rep(0,length(x$argvals)) flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
rangeval=x$rangeval),B=100,p=11) # Simple hypothesis, automatic choose of p flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
rangeval=x$rangeval),B=100) flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
rangeval=x\$rangeval),B=5000)
## End(Not run)
fda.usc documentation built on Oct. 17, 2022, 9:06 a.m.
|
2023-02-03 21:05:29
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.541713297367096, "perplexity": 3728.944194974543}, "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-2023-06/segments/1674764500074.73/warc/CC-MAIN-20230203185547-20230203215547-00464.warc.gz"}
|
https://proxieslive.com/tag/carbon/
|
## Selecting Carbon Black for Paints, Coatings and Inks
Adding carbon black (CB) particles to elastomeric polymers is essential to the successful industrial use of rubber in many applications, and the mechanical reinforcing effect of CB in rubber has been studied for nearly 100 years. Despite these many decades of investigations, the origin of stiffness enhancement of elastomers from incorporating nanometer-scale CB particles is still debated. It is not universally accepted whether the interactions between polymer chains and CB surfaces are purely physical adsorption or whether some polymer–particle chemical bonds are also introduced in the process of mixing and curing the CB-filled rubber compounds. We review key experimental observations of rubber reinforced with CB, including the finding that heat treatment of CB can greatly reduce the filler reinforcement effect in rubber. The details of the particle morphology and surface chemistry are described to give insights into the nature of the CB–elastomer interfaces. This is followed by a discussion of rubber processing effects, the influence of CB on crosslinking, and various chemical modification approaches that have been employed to improve polymer–filler interactions and reinforcement. Finally, we contrast various models that have been proposed for rationalizing the CB reinforcement of elastomers.
Natural rubber composite has been continuously developed due to its advantages such as a good combination of strength and damping property. Most of carbon black (CB)/Natural Rubber (NR) composite were used as material in tyre industry. The addition of CB in natural rubber is very important to enhance the strength of natural rubber. The particle loading and different structure of CB can affect the composite strength. The effects of CB particle loading of 20, 25 and 30 wt% and the effects of CB structures of N220, N330, N550 and N660 series on tensile property of composite were investigated. The result shows that the tensile strength and elastic modulus of natural rubber/CB composite was higher than pure natural rubber. From SEM observation the agglomeration of CB aggregate increases with particle loading. It leads to decrease of tensile strength of composite as more particle was added. High structure of CB particle i.e. N220 resulted in highest tensile stress. In fact, composite reinforced by N660 CB particle shown a comparable tensile strength and elastic modulus with N220 CB particle. SEM observation shows that agglomeration of CB aggregates of N330 and N550 results in lower stress of associate NR/CB composite.
Carbon black is a highly engineered form of carbon widely used in paints as paint carbon black, coatings and inks to achieve a spectrum ranging from gray to deep black. Over the time, the properties of carbon black pigment have been modified to achieve required properties in the final product, such as increased tinting strength, improved the level of jetness or blue undertone and conductivity.
Explore the different carbon black production processes and the properties to consider while selecting the right carbon black for your formulations.
Properties and End-uses of Carbon Black
Carbon black is used in many products and articles we use and see around us on a daily basis, such as: rubbers, plastics, coatings, tires, ink carbon clack.
Thus, the requirements for the carbon black are different for each application and influence the specific properties in the final application.
For the coating carbon blacks market, there is a wide range of carbon black grades available. This can make it difficult to choose the most suitable carbon black for your final application. For example, when aiming for automotive paint with a blue undertone, the carbon black of choice will have a high jetness. However, normally these types of carbon black grades are the most difficult to disperse correctly into the desired particle size.
The carbon black producers are addressing these issues by developing specialty carbon black grades that have been surface-modified and/or are pre-treated to overcome these difficulties.
How Carbon Black is Produced?
The properties of the carbon black are influenced by the method of preparation. The different processes used for channel carbon black production are discussed below.
Furnace Black Process: It is the most common method which uses (aromatic) hydrocarbon oil as the raw material. Due to its high yield and possibility to control the particle size and structure, it is most suitable for mass production of carbon black.
In the reactor the conditions (e.g. pressure and temperature) are controlled to provide a number of reactions. The most important reactions include: particle nucleation, particle growth, aggregate formation. Water injection rapidly reduces the temperature and ends the reaction. The primary particle size and structure of the carbon black is controlled by tuning the conditions in the reactor and the time allowed before the reaction is quenched.
Thermal Black Process: It is the most common method used for carbon black production after the furnace black process. It is a discontinuous or cyclical process.
This process uses natural methane gas as raw material. When the natural gas is injected into the furnace at an inert atmosphere, the gas decomposes into carbon black and hydrogen. The carbon black produced using this method has the largest particle size and the lowest degree of aggregates or structure. Due to the nature of the raw material, this carbon black is the purest form available on the industrial scale.
Channel Process: This process uses partially combusted fuel which is brought into contact with H-shaped channel steel. It is not the most used method anymore because of its:
The benefit of this process is that it provides carbon black with a lot of functional groups.
Acetylene Black Process: This process uses acetylene gas as raw material. It produces mainly high structure and higher crystallinity, making this type of carbon black suitable for electric conductive applications.
Lampblack Process: It is the oldest industrial process for making carbon black. It uses mineral/vegetable oils as its raw material.
Recovered Carbon Black from End-of-life Tires
Recovered carbon black or ®CB is a fast-expanding market. Recovered high purity carbon black is obtained through the pyrolysis process of end-of-life tires. The importance of companies in the production and use of recovered carbon black is three-fold:
The growing global problems arising with end-of-life tires (ELT)
Companies shifting strategy to fulfill the targets ensuring a green economy
Price changes of regular carbon black due to fluctuations in oil pricing
Depending on the composition, the content of carbon black in tires can be up to 30%. Next to carbon black, the tires consists:
Rubber
Metal
Textile
Fillers such as silica
The amount of silica depends on the type of tire, for example winter or summer tire, racing tire, or tire for agricultural vehicles, and will not be separated from the carbon black during the pyrolysis process, which will result in higher ash content.
In a typical car tire, up to 15 different types of conductive carbon blacks can be used, each attributing to the different properties required. This blend of environmental carbon blacks will then also be the make-up of the final ®CB composition. Besides tires, other sources that can be used are rubber conveyor belts or other technical rubber products.
The main differences in the properties of recovered carbon black are:
The ash content is higher for ®CB caused by the fillers being used in tire production.
A blend of rubber carbon black properties as a result of the carbon black used in the tire.
Residual hydrocarbons on the carbon black surface, depending on the quality of the pyrolysis process.
To understand how the properties of ®CB influence the final applications and to know which plastic carbon black is used in which category, we need to understand the fundamental differences between the available carbon blacks.
## Como pegar o mês em português utilizando o Carbon?
Estou com duvidas em como pegar o valor do mês por extenso em português utilizando a API do Carbon utilizando o framework Laravel. Inicialmente, construi essa logica utilizando o PHP.
`` if(\$ data->month == 1){ \$ mes = 'Janeiro'; } ``
Isso acarreta fazer em torno de 12 if’s para poder pegar o valor do mês em português. Supondo que a variável \$ data recebe o valor do tempo de agora.
`` \$ data = Carbon::now(); ``
Como poderia estar pegando o mês por extenso em portugues utilizando somente funções da API do Carbon? Como mostrado o exemplo abaixo.
`` \$ mes = \$ data->localeMonth; ``
Por exemplo, hoje é pego o mês “July“. Eu estou em duvida também em como posso mudar a localização para o horário oficial de Brasília. Creio que mudando a localização, pode-se resolver o meu problema. Mas está dificil encontrar na documentação da API a solução. Poderiam me ajudar?
Documentação do Carbon
## Problema con Carbon y fecha en string
El problema es que me indica este error `DateTime::__construct(): Failed to parse time string (17/07/2019) at position 0 (1): Unexpected character`
Vista en mi formulario tengo por separado el input de fecha y hora, pero con carbon los quiero juntar y guardarlos en un solo campo
``<div class="col-5 col-xl-5"> <div class="form-group"> <div class="input-group date" name="event_start_date" id="event_start_date" data-target-input="nearest"> <input type="text" name="event_start_date" required="" id="event_start_date" class="form-control datetimepicker-input" data-target="#event_start_date" placeholder="Fecha de inicio"/> <div class="input-group-append" data-target="#event_start_date" data-toggle="datetimepicker"> <div class="input-group-text"><i class="fa fa-calendar"></i></div> </div> </div> </div> </div> <div class="col-4 col-xl-5" id="event_start_time_area" style="display: none"> <div class="form-group"> <div class="input-group date" id="event_start_time" data-target-input="nearest"> <input type="text" name="event_start_time" id="event_start_time" value="00:00" class="form-control datetimepicker-input" data-target="#event_start_time" placeholder="Hora de inicio"/> <div class="input-group-append" data-target="#event_start_time" data-toggle="datetimepicker"> <div class="input-group-text"><i class="fa fa-clock"></i></div> </div> </div> </div> </div> ``
el error hace referencia directa a la linea `\$ dataTimeFecha_i = new Carbon(\$ fecha_i);` de mi controlador
intente de esta forma `\$ dataTimeFecha_i = new DateTime(\$ fecha_i);`, pero me sale el mismo error
``\$ fecha_i= \$ request['event_start_date']; \$ dataTimeFecha_i = new Carbon(\$ fecha_i); \$ hora_inicio = \$ request['event_start_time']; \$ fecha_hora_inicio = Carbon::instance(\$ dataTimeFecha_i)->setTimeFromTimeString(\$ hora_inicio)->toIso8601String(); ``
## Battery drains even while laptop lid closed (Ubuntu 19.04 & Thinkpad X1 Carbon 6th gen)
Any other thinkpad x1 carbon users on ubuntu 19.04 notice that the battery still drains even while the laptop lid is closed & suspended?
I updated from 18.10 to 19.04 about two weeks ago, and ever since, I’ve come home from the work and found my laptop burning up in my backpack with the battery drained 50% or more after half a day closed & (I assume) suspended.
I had this problem for a while with 18.10, but it went away after a thnkpad BIOS update included the linux sleep state.
Screenshot of my battery discharge from the past 6 hours (my laptop was closed and “suspended” for 5.5 of those hours): https://imgur.com/ALK7Ogq
## 2nd gen X1 Carbon, wifi, no internet through vpn
Ive been running a fresh install of kubuntu 19.04 on my 2nd gen x1 carbon for over a month now with no issue. I use PIA vpn and been using their desktop app with no issue since 18.04 was released, and also on vanilla ubuntu 19.04 and now in kubuntu.
Starting this past week i am no longer able to access the internet with the vpn on, it has some sparatic symptoms though:
-after a restart ill be able to access one address before it stops working -on my home network, disabling ipv6 let me get online with the vpn, but this didnt work on my moms or aunts networks
Its been a big hold up in my work and i hope someone can shed some light on the problem for me, i dont like using the internet without a vpn
## Carbon Copy Cloner – Bad files that are unrecoverable. What should I delete and should I reformat?
I just used Carbon Copy Cloner (CCC Version 3.5.7, Mac OS X 10.7 Lion).
Everything copied to my external backup drive except for about 500 random files – most of which are in the same folders and don’t matter that much to me. I have three questions.
Is there any risk if I delete files from the following directories/locations?
1. `/Users/user1/Library/Application Support/Firefox/Profiles/*` – Some Firefox profile info
2. `/Users/user1/Library/Caches/Firefox/Profiles/*` – Some Firefox profile info
3. `/Users/user1/Music/iTunes/iTunes Media/Music` – A few audio tracks
4. `/opt/local/var/cache/fontconfig/*` – Probably related to the Macports fontconfig port
5. `/private/var/tmp/tmp.*` – I’m unsure what will happen if I delete these
Also, Carbon Copy Cloner gives me a way to access the location of 1 file at a time to manually move it to the Trash for deletion. Is there a way to delete them more than 1 at a time?
Lastly, Carbon Copy Cloner recommends that after I delete these files and finish my backup that I reformat my internal (source) hard drive. Is this really necessary?
Tags:
need for speed carbon mac os
need for speed carbon mac os x
need for speed carbon mac os скачать
need for speed carbon mac os rutracker
скачать need for speed carbon на mac os
no sound windows 10 macbook pro 2012…
## Is it possible to load an endpoint agent in my Heroku environment? ie: Carbon Black, Crowdstrike, etc
Can an endpoint security solution be loaded onto a Heroku instance?
## Where to find Lenovo thinkpad Carbon 6th generation wall paper?
This one is what I am looking for (preferably in Full HD resolution). I hope this is the appropriate sub-forum.
``import numpy as np inputgrid = np.array([['.','.','.','.','.'], ['C','-','C','-','C'], ['.','.','|','.','.'], ['.','.','C','.','.'], ['.','.','|','.','.'], ['.','.','C','.','.'], ['.','.','.','.','.']])# This Is Input inputgrid = np.where(inputgrid=='.', 0, inputgrid) inputgrid = np.where(inputgrid=='C', 1, inputgrid) inputgrid = np.where(inputgrid=='-', 9, inputgrid) inputgrid = np.where(inputgrid=='|', 9, inputgrid) np.array(inputgrid).tolist() grid = [[int(item) for item in row] for row in inputgrid] def display(grid): for row in grid: print(row) display(grid) lst = [] for rows, row in enumerate(grid): for cols, col in enumerate(row): if grid[rows][cols] in [1]: lst.append((rows, cols)) bondlst = [] for rows, row in enumerate(grid): for cols, col in enumerate(row): if grid[rows][cols] in [9]: bondlst.append((rows, cols)) print(lst) print(bondlst) bondx = [] bondy = [] for item in bondlst: (bondx).append(item[0]) (bondy).append(item[1]) print(bondx) print(bondy) adjacencylist = [] def adjacentnode(nodea ,nodeb): if nodea[0] == nodeb[0] and nodea[1] == nodeb[1]+2: adjacent = True elif nodea[0] == nodeb[0] and nodea[1] == nodeb[1]-2: adjacent = True elif nodea[1] == nodeb[1] and nodea[0] == nodeb[0]+2: adjacent = True elif nodea[1] == nodeb[1] and nodea[0] == nodeb[0]-2: adjacent = True else: adjacent = False return adjacent print (adjacentnode((1,0),(1,2))) count = 0 tempgraph = {} for node in range(len(lst)): print (node) adjacencylist.append((lst[node] ,[])) for neighbour in range(len(lst)): adjacentnodes = (adjacentnode(lst[node] ,lst[neighbour])) print(adjacentnodes) if adjacentnodes == True: count = count +1 adjacencylist[node][1].append(lst[neighbour]) # adjacencylist.append((lst[node],[])) # adjacencylist.append(lst[node]) # adjacencylist.append(lst[neighbour]) print (count) print (adjacencylist) for item in adjacencylist: tempgraph[str(item[0])] = (item[(1)]) carbongraph = {} for i in tempgraph: carbongraph[i] = [str(k) for k in tempgraph[i]] print(carbongraph) #print(adjacencylist) #print (carbongraph) ''' carbongraph = {'(0, 1)' :['(0, 2)'], '(0, 2)' :['(0, 1)' ,'(0, 3)' ,'(1, 2)'], '(0, 3)' :['(0, 2)'], '(1, 2)' :['(0, 2)', '(2, 2)'], '(2, 2)': ['(1, 2)']} WEIGHTS = 1 ''' def shortestpath(graph, start, end, path=[]): path = path + [start] if start == end: return path if start not in graph: return None for node in graph[start]: if node not in path: newpath = shortestpath(graph, node, end, path) if newpath: return newpath return None LeafArray = [] for leaf in carbongraph: degree = (len(carbongraph[leaf])) if degree == 1: LeafArray.append(leaf) print(LeafArray) chainlist = [] for node in LeafArray: for neighbour in LeafArray: currentpath = (shortestpath(carbongraph, node, neighbour)) carbonchain = len(currentpath) print(currentpath) chainlist.append(carbonchain) longestchain = max(chainlist) print(longestchain) def Prfix(): global prefix if longestchain == 4: prefix = "But" elif longestchain == 5: prefix = "Pent" elif longestchain == 6: prefix = "Hex" elif longestchain == 7: prefix = "Hept" elif longestchain == 8: prefix = "Oct" return prefix print(Prfix()) $$```$$ ``
|
2021-10-24 08:46:01
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.36523982882499695, "perplexity": 5140.391222166497}, "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/1634323585916.29/warc/CC-MAIN-20211024081003-20211024111003-00217.warc.gz"}
|
https://www.123helpme.com/angular-momentum-and-skating--preview.asp?id=380925
|
# Essay on Angular Momentum and Skating
Length: 602 words (1.7 double-spaced pages)
Rating: Good Essays
#### Essay Preview
Angular momentum is the relationship between Rotational Inertia and Rotational Speed. More simply, it is the tendency an object has to continue moving in a circle or spinning.
Angular Momentum = Angular Velocity x Rotational Inertia
When a figure skater pulls their arms closer to their body, they are reducing their Rotational Inertia, making themselves more aerodynamic. In order to sustain this and maintain their momentum, the Rotational Speed must increase.
Angular Conservation
Angular momentum is basically an object’s resistance to a change in rotation. To change an object’s motion, force must be applied, since objects in motion tend to stay in motion. This force is called torque when relating to rotational motion (Torque = Force x Perpendicular Distance from Axis). When torque is applied, the angular momentum increases, then decreases due to friction. But on the ice, there is barely any friction, and the skater can sustain their momentum for long periods of time.
Rotational Inertia requires the object to rotate around its axis as opposed to how an object would behave wh...
## Need Writing Help?
Get feedback on grammar, clarity, concision and logic instantly.
## Angular Momentum Essays
- Angular momentum and its properties were devised over time by many of the great minds in physics. Newton and Kepler were probably the two biggest factors in the evolution of angular momentum. Angular momentum is the force which a moving body, following a curved path, has because of its mass and motion. Angular momentum is possessed by rotating objects. Understanding torque is the first step to understanding angular momentum.Torque is the angular "version" of force. The units for torque are in Newton-meters.... [tags: Physics]
Good Essays
1417 words (4 pages)
## Efficiency of Angular Momentum Transport Essay
- ... For example, \cite{girart06} detected hour-glass shaped geometry of the magnetic field in the low-mass protostellar system NGC 1333 IRAS 4A using the Submillimeter Array polarimetry observations at 877 $\mu$m dust continuum emission. \cite{chapman13} detected a correlation between core magnetic field direction and protostellar disk symmetry axis in a few low-mass protostellar cores using the SHARP polarimeter at Caltech Submillimeter Observatory at 350 $\mu$m dust continuum emission. \cite{donati05} probably detected magnetic field with azimuthal component of order of 1 kGs in the FU Orionis accretion disk.... [tags: strength, magnetic, disks]
Good Essays
1244 words (3.6 pages)
## Skating First US Men's Olympic Gold Medal Essay
- Imagine being the first person to ever land a double-axel jump at the most important figure skating competition in the world. That is exactly how Dick Button won the first U.S. Men’s Olympic gold medal for figure skating. With excitement in his eyes, he went up to accept his gold medal. Little did he know, that would not be the last time he would accept a gold medal. Button is the winner of seven consecutive U.S. championships. Dick Button was born on July 18, 1929, in Englewood, New Jersey. When he was young, his mother wanted him to play piano and his father wanted him to play ice hockey.... [tags: dick button, winter olympics, skating]
Good Essays
1105 words (3.2 pages)
## Essay about The Angular Aspects of Basketball
- Intro: In basketball, there are many instances where angular motion is apparent. Angular motion refers to all points on an object moving in a circular path about a fixed axis. The limbs of our bodies exhibit angular motion around our joints, so most of the movements involved in playing basketball display some form of angular motion. The two important biomechanical aspects of angular motion are Angular Kinematics and Angular Kinetics. These subdivisions of biomechanics are significant because understanding them can lead to performing the tasks affiliated with basketball, such as passing, shooting, jumping, and dribbling at an optimal level.... [tags: Physics]
Good Essays
1953 words (5.6 pages)
## Essay about Collisions on Momentum: The Law of Conservation of Momentum
- Contents Page: Introduction: page 3 Design: page 4-6 Collected Data: page 7-8 Discussion: page 9 Conclusion: page 10 Plagiarism Checker and Declaration: page 11 Bibliography: page 12 Appendix: page 13 Rubric: page 14 Introduction: The Grade 12 Physical Science learners at Penryn College were tasked with carrying out an experiment to investigate the effect of collisions on momentum. Different mass pieces (500g; 1kg and 1.5kg) were dropped on a moving trolley and the learners observed the velocity of the trolley before the mass pieces were dropped on the trolley and the velocity after the mass pieces were dropped.... [tags: velocity, experiment, trolley]
Good Essays
1098 words (3.1 pages)
## Skating for the Sisterhood Essays
- While some depictions suggest that roller derby includes staged fights, nearly everyone familiar with the sport disagrees with that assumption. I conducted two interviews, and both derby players displayed annoyances when asked about fake fighting in roller derby. Samantha Boehle states, “I actually get kind of angry when people compare roller derby to things like professional wrestling. We don’t perform fights like they do. Roller derby is a competition. Yes, it’s rough, but there’s no performed fighting.” Likewise, Josie Esker states, “I am so frustrated with derby’s early history of staged performances and ridiculous theatrics.... [tags: roller derby, staged fights, feminist cause]
Good Essays
1333 words (3.8 pages)
## Conservation of Momentum Essay
- Conservation of Momentum Contents Investigative Question: 1 Aim: 1 Hypothesis: 2 Apparatus: 2 Method: 2 Variables: 3 Results: 3 Discussion: 7 Conclusion: 9 Bibliography 9 Declaration of Authenticity 10 Investigative Question: During a collision between two objects, is the amount of momentum present in a system before the collision different to the amount of momentum present afterward or is the total momentum of the system conserved. Aim: To prove that law which states that the total momentum present in a system is conserved is true.... [tags: Mechanics, Collision, Velocity, Physics]
Good Essays
1043 words (3 pages)
## Conservation of Momentum Essay
- Conservation of Momentum Purpose: To show that momentum is conserved in a closed system by illustrating the conservation of momentum in an elastic collision and an inelastic collision. Method: If momentum is conserved in a closed system, the total momentum of the system before collision should equal the total momentum of the system after the collision. Strobe photos will be used in the calculations that will prove that momentum is conserved. 1.) Elastic collision: A strobe photo will be used that shows a large glider smashing into a smaller glider which is initially at rest.... [tags: Papers]
Free Essays
404 words (1.2 pages)
## Conservation of Momentum Essay
- Conservation of Momentum 1. Trial 1 T1 (s) T2 (s) Vi (m/s) V2 (m/s) 0 0.071 0.351 1 0.111 0.225 2 0.118 0.215 Trial 2 0 0.061 .409 1 0.092 0.272 2 0.101 0.248 Trial 3 0 0.057 0.440 1 0.083 0.300 2 0.088 0.283 Mass of car 1 = 993.0 g Mass of car 2 = 496.7 g 2. trial 1 Car 1 momentum before collision P=mv P=(.993kg)(.351m/s) P= .349 kgm/s Car 2 momentum before collision P=mv P=(.4967kg)(0m/s) P = 0 kgm/s Object’s (or both cars together) momentum after collision P=mv P=(1.48... [tags: Physics]
Free Essays
356 words (1 pages)
## Skating Essay
- Imagine that you've just turned 16 and your parents bought you a brand new Mustang. You're cruising on the street at approximately 25 miles per hour. So maybe you're not breaking the speed limit, but everyone's staring. You feel the light breeze through your hair. Then you're in the air, 12 feet high. Finally, you land on your back and come this (gesture 6 in.) close to cracking your skull. What happened. You were skating. As Aaron Spohn, a well respected ramp builder for the National Inline Skate Series, Extreme Games, and many pro inline skaters, said, When you tell someone you are an in-line skater, you automatically assume they envision you sporting a tangerine body suit, waist pack and... [tags: essays research papers]
Free Essays
2001 words (5.7 pages)
|
2019-12-09 23:45:32
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.38165995478630066, "perplexity": 4306.171773625756}, "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-2019-51/segments/1575540525598.55/warc/CC-MAIN-20191209225803-20191210013803-00365.warc.gz"}
|
http://www.mathworks.com/help/gads/gamultiobj.html?nocookie=true
|
Accelerating the pace of engineering and science
# gamultiobj
Find minima of multiple functions using genetic algorithm
## Syntax
X = gamultiobj(FITNESSFCN,NVARS)
X = gamultiobj(FITNESSFCN,NVARS,A,b)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon,options)
X = gamultiobj(problem)
[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...)
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...)
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN, ...)
## Description
gamultiobj implements the genetic algorithm at the command line to minimize a multicomponent objective function.
X = gamultiobj(FITNESSFCN,NVARS) finds a local Pareto set X of the objective functions defined in FITNESSFCN. For details on writing FITNESSFCN, see Compute Objective Functions. NVARS is the dimension of the optimization problem (number of decision variables). X is a matrix with NVARS columns. The number of rows in X is the same as the number of Pareto solutions. All solutions in a Pareto set are equally optimal; it is up to the designer to select a solution in the Pareto set depending on the application.
X = gamultiobj(FITNESSFCN,NVARS,A,b) finds a local Pareto set X of the objective functions defined in FITNESSFCN, subject to the linear inequalities $A\ast x\le b$, see Linear Inequality Constraints. Linear constraints are supported only for the default PopulationType option ('doubleVector'). Other population types, e.g., 'bitString' and 'custom', are not supported.
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq) finds a local Pareto set X of the objective functions defined in FITNESSFCN, subject to the linear equalities $Aeq\ast x=beq$ as well as the linear inequalities $A\ast x\le b$, see Linear Equality Constraints. (Set A=[] and b=[] if no inequalities exist.) Linear constraints are supported only for the default PopulationType option ('doubleVector'). Other population types, e.g., 'bitString' and 'custom', are not supported.
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB) defines a set of lower and upper bounds on the design variables X so that a local Pareto set is found in the range $LB\le x\le UB$, see Bound Constraints. Use empty matrices for LB and UB if no bounds exist. Bound constraints are supported only for the default PopulationType option ('doubleVector'). Other population types, e.g., 'bitString' and 'custom', are not supported.
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon) subjects the minimization to the constraints defined in nonlcon. The function nonlcon accepts x and returns vectors C and Ceq, representing the nonlinear inequalities and equalities respectively. gamultiobj minimizes the fitnessfcn such that C(x) 0 and Ceq(x) = 0. (Set LB=[] and UB=[] if no bounds exist.)
X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,options) or X = gamultiobj(FITNESSFCN,NVARS,A,b,Aeq,beq,LB,UB,nonlcon,options) finds a Pareto set X with the default optimization parameters replaced by values in the structure options. options can be created with the gaoptimset function.
X = gamultiobj(problem) finds the Pareto set for problem, where problem is a structure containing the following fields:
fitnessfcn Fitness functions nvars Number of design variables Aineq A matrix for linear inequality constraints bineq b vector for linear inequality constraints Aeq Aeq matrix for linear equality constraints beq beq vector for linear equality constraints lb Lower bound on x ub Upper bound on x nonlcon Nonlinear constraint function (optional) solver 'gamultiobj' rngstate Optional field to reset the state of the random number generator options Options structure created using gaoptimset
Create the structure problem by exporting a problem from Optimization app, as described in Importing and Exporting Your Work in the Optimization Toolbox™ documentation.
[X,FVAL] = gamultiobj(FITNESSFCN,NVARS, ...) returns a matrix FVAL, the value of all the objective functions defined in FITNESSFCN at all the solutions in X. FVAL has numberOfObjectives columns and same number of rows as does X.
[X,FVAL,EXITFLAG] = gamultiobj(FITNESSFCN,NVARS, ...) returns EXITFLAG, which describes the exit condition of gamultiobj. Possible values of EXITFLAG and the corresponding exit conditions are listed in this table.
EXITFLAG ValueExit Condition
1
Average change in value of the spread over options.StallGenLimit generations less than options.TolFun, and the final spread is less than the average spread over the past options.StallGenLimit generations
0
Maximum number of generations exceeded
-1
Optimization terminated by an output function or plot function
-2
No feasible point found
-5
Time limit exceeded
[X,FVAL,EXITFLAG,OUTPUT] = gamultiobj(FITNESSFCN,NVARS, ...) returns a structure OUTPUT with the following fields:
OUTPUT FieldMeaning
problemtypeType of problem:
• 'unconstrained' — No constraints
• 'boundconstraints' — Only bound constraints
• 'linearconstraints' — Linear constraints, with or without bound constraints
rngstate
State of the MATLAB® random number generator, just before the algorithm started. You can use the values in rngstate to reproduce the output of ga. See Reproduce Results.
generationsTotal number of generations, excluding HybridFcn iterations
funccountTotal number of function evaluations
messagegamultiobj termination message
averagedistanceAverage "distance," which by default is the standard deviation of the norm of the difference between Pareto front members and their mean
spreadCombination of the "distance," and a measure of the movement of the points on the Pareto front between the final two iterations
maxconstraintMaximum constraint violation at the final Pareto set
[X,FVAL,EXITFLAG,OUTPUT,POPULATION] = gamultiobj(FITNESSFCN, ...) returns the final POPULATION at termination.
[X,FVAL,EXITFLAG,OUTPUT,POPULATION,SCORE] = gamultiobj(FITNESSFCN, ...) returns the SCORE of the final POPULATION.
## Examples
This example optimizes two objectives defined by Schaffer's second function, which has two objectives and a scalar input argument. The Pareto front is disconnected. Define this function in a file:
```function y = schaffer2(x) % y has two columns
% Initialize y for two objectives and for all x
y = zeros(length(x),2); % ready for vectorization
% Evaluate first objective.
% This objective is piecewise continuous.
for i = 1:length(x)
if x(i) <= 1
y(i,1) = -x(i);
elseif x(i) <=3
y(i,1) = x(i) -2;
elseif x(i) <=4
y(i,1) = 4 - x(i);
else
y(i,1) = x(i) - 4;
end
end
% Evaluate second objective
y(:,2) = (x -5).^2;```
First, plot the two objectives:
```x = -1:0.1:8;
y = schaffer2(x);
plot(x,y(:,1),'.r'); hold on
plot(x,y(:,2),'.b');```
The two component functions compete in the range [1, 3] and [4, 5]. But the Pareto-optimal front consists of only two disconnected regions: [1, 2] and [4, 5]. This is because the region [2, 3] is inferior to [1, 2].
Next, impose a bound constraint on x, setting
```lb = -5;
ub = 10;```
The best way to view the results of the genetic algorithm is to visualize the Pareto front directly using the @gaplotpareto option. To optimize Schaffer's function, a larger population size than the default (15) is needed, because of the disconnected front. This example uses 60. Set the optimization options as:
`options = gaoptimset('PopulationSize',60,'PlotFcns',@gaplotpareto);`
Now call gamultiobj, specifying one independent variable and only the bound constraints:
```[x,f,exitflag] = gamultiobj(@schaffer2,1,[],[],[],[],...
lb,ub,options);
Optimization terminated: average change in the spread of
Pareto solutions less than options.TolFun.
exitflag
exitflag = 1```
The vectors x, f(:,1), and f(:,2) respectively contain the Pareto set and both objectives evaluated on the Pareto set.
### Examples Included in the Toolbox
The gamultiobjfitness example solves a simple problem with one decision variable and two objectives.
The gamultiobjoptionsdemo example shows how to set options for multiobjective optimization.
expand all
### Algorithms
gamultiobj uses a controlled elitist genetic algorithm (a variant of NSGA-II [1]). An elitist GA always favors individuals with better fitness value (rank). A controlled elitist GA also favors individuals that can help increase the diversity of the population even if they have a lower fitness value. It is important to maintain the diversity of population for convergence to an optimal Pareto front. Diversity is maintained by controlling the elite members of the population as the algorithm progresses. Two options, ParetoFraction and DistanceFcn, control the elitism. ParetoFraction limits the number of individuals on the Pareto front (elite members). The distance function, selected by DistanceFcn, helps to maintain diversity on a front by favoring individuals that are relatively far away on the front. The algorithm stops if the spread, a measure of the movement of the Pareto front, is small.
## References
[1] Deb, Kalyanmoy. Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, 2001.
|
2014-12-21 21:21:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5920127630233765, "perplexity": 2651.9792575394376}, "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/1418802772398.133/warc/CC-MAIN-20141217075252-00118-ip-10-231-17-201.ec2.internal.warc.gz"}
|
http://en.wikipedia.org/wiki/Optical_depth_(astrophysics)
|
# Optical depth (astrophysics)
Jump to: navigation, search
This article is about optical depth in astrophysics. For optical depth in general, see optical depth.
Optical depth in astrophysics refers to a specific level of transparency. Optical depth and actual depth, $\tau$ and $z$ respectively, can vary wildly depending on the absorptivity of the stellar interior. Because of this $\tau$ is able to show the relationship between these two quantities and can lead to a greater understanding of the structure inside a star.
Optical depth is a measure of the extinction coefficient or absorptivity up to a specific 'depth' of a star's makeup.
$\tau \equiv \int_0^z (\alpha) dz = \sigma N$ [1]
This equation assumes that the extinction coefficient $\alpha$ is known, or that N, the column number density, is known. These can generally be calculated from other equations if a fair amount of information is known about the chemical makeup of the star.
$\alpha$ can be calculated using the transfer equation. In most astrophysics problems this is exceptionally difficult to solve, since the equations assume one knows the incident radiation as well as the radiation leaving the star and these values are usually theoretical.
In some cases the Beer-Lambert Law can be useful in finding $\alpha$.
$\alpha=e^\frac{4 \pi \kappa}{\lambda_0}$
where $\kappa$ is the refractive index, and $\lambda_0$ is the wavelength of the incident light before being absorbed or scattered.[2] Note that the Beer-Lambert Law is only appropriate when the absorption occurs at a specific wavelength, $\lambda_0$, for a gray atmosphere it is most appropriate to use the Eddington Approximation.
Therefore it is straightforward to see that $\tau$ is simply a constant that depends on the physical distance from the outside of a star. To find $\tau$ at a particular depth z, one simply uses the above equation with $\alpha$ and integrates from $z=0$ to $z=z$.
## The Eddington Approximation and the Depth of the Photosphere
Because it is difficult to define where the photosphere of a star ends and the chromosphere begins astrophysicists rely on the Eddington Approximation to derive the formal definition of $\tau=\frac{2}{3}$
Devised by Sir Arthur Eddington the approximation takes into account the fact that $H^-$ produces a "gray" absorption in the atmosphere of a star, that is, it is independent of any specific wavelength and absorbs along the entire electromagnetic spectrum. In that case,
$T^4 = \frac{3}{4}T_e^4\left(\tau + \frac{2}{3}\right)$
Where $T_e$ is the effective temperature at that depth and $\tau$ is the optical depth.
This illustrates not only that the observable temperature and actual temperature at a certain physical depth of a star vary, but that the optical depth plays a crucial role in understanding the stellar structure. It also serves to demonstrate that the depth of the photosphere of a star is highly dependent upon the absorptivity of its environment. The photosphere extends down to a point where $\tau$ is about 2/3, which corresponds to a state where a photon would experience, in general, less than 1 scattering before leaving the star.
One should also note that the above equation can be rewritten in terms of $\alpha$ in the following way:
$T^4 = \frac{3}{4}T_e^4\left(\int_0^z (\alpha) dz + \frac{2}{3}\right)$
Which is useful if $\tau$ is not known but $\alpha$ is.
|
2014-12-21 02:16:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 26, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8625357747077942, "perplexity": 266.51010461793686}, "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-2014-52/segments/1418802770557.39/warc/CC-MAIN-20141217075250-00005-ip-10-231-17-201.ec2.internal.warc.gz"}
|
http://math.stackexchange.com/questions/251311/limits-trigonometry-tending-to-infinity
|
# Limits - trigonometry - tending to infinity
How do we solve:
$$\lim_{x\to \infty} 5^x \sin\left(\frac{a}{5^x}\right)$$
Thank You.
-
Gerry Myerson’s answer is the way to go, but you can easily see what the limit has to be if you remember that $\sin x\approx x$ when $|x|$ is small. Thus, $\sin\frac{a}{5^x}\approx\frac{a}{5^x}$ when $x$ is large, and ... . – Brian M. Scott Dec 5 '12 at 5:21
## 1 Answer
Convince yourself that it's the same as evaluating $\lim_{t\to0}{\sin at\over t}$, and then use other stuff you know to do that one.
-
Thanks. That works but I'm not sure if we can perform that operation on power functions... Can we? – Lavanya Dec 5 '12 at 5:20
Put $y=\frac{1}{5^x}$ and see what happens. – Mhenni Benghorbal Dec 5 '12 at 5:22
Thanks to all of you. I got it now! – Lavanya Dec 5 '12 at 5:31
Good. Then you can write it up and post it as an answer. Then, later, you can accept it. – Gerry Myerson Dec 5 '12 at 5:35
Well, now I have got another doubt.. Instead of doing it the above way, if I let it remain x -> infinity, then 5^infinity becomes infinity...so on solving I would get infinity*sin(0) which is all too confusing! And if I can't directly put infinity in place of x, why is it so? Because I'll get an infinity*0 form?? – Lavanya Dec 5 '12 at 9:20
|
2016-05-27 08:43:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9074040055274963, "perplexity": 665.251040377101}, "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-2016-22/segments/1464049276564.72/warc/CC-MAIN-20160524002116-00136-ip-10-185-217-139.ec2.internal.warc.gz"}
|
https://www.netlib.org/utk/people/JackDongarra/etemplates/node308.html
|
Next: Kronecker Canonical Form Up: Singular Matrix Pencils Previous: Singular Matrix Pencils Contents Index
## Regular Versus Singular Problems
Let us start by considering the generalized eigenvalue problem , where
The eigenvalues of are the pairs , ) and , ) with the associated eigenvectors and , respectively. If is nonzero, then is a finite eigenvalue. Otherwise, if is zero, then is an eigenvalue of the matrix pair . But what happens if, for example, ? Then is zero for all , which means that we have a singular eigenvalue problem. In this case we have ; i.e., and have a common null space. We say that is an eigenvector for an indeterminate eigenvalue . Note that the common null space is a sufficient but not necessary condition to have a singular eigenvalue problem.
The most common generalized eigenvalue problems are regular; i.e., and are square matrices and the characteristic polynomial is only vanishing for a finite number of values, where denotes the degree of the polynomial. The corresponding is called a regular matrix pencil. The eigenvalues of a regular pencil are points in the extended complex plane . The eigenvalues are defined as the zeros of and additional eigenvalues.
An alternative representation of a matrix pencil is the cross product form: the set of matrices where . The mapping shows the relation between the eigenvalues of and . For example, zero and infinite eigenvalues are represented as and , respectively, and can be treated as any other points in .
If (and ) is identically zero for all (and pairs (, )), then is called singular and is a singular matrix pair. Singularity of is signaled by some . In the presence of roundoff, and may be very small. In these situations, the eigenvalue problem is very ill-conditioned, and some of the other computed nonzero values of and may be indeterminate. Such problems are further discussed and illustrated by examples in §8.7.4. Moreover, rectangular matrix pairs are singular and the corresponding is a singular pencil.
Next: Kronecker Canonical Form Up: Singular Matrix Pencils Previous: Singular Matrix Pencils Contents Index
Susan Blackford 2000-11-20
|
2021-12-02 04:11:31
|
{"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": 0.9390389919281006, "perplexity": 539.2251323081982}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964361064.69/warc/CC-MAIN-20211202024322-20211202054322-00258.warc.gz"}
|
https://bookdown.org/dusadrian/QCAbook/
|
# Preamble
This is a work in progress, which started at the beginning of August, 2016. Most of the information introduced so far refer to the current packages QCA and QCAGUI, although by the time this book will be finished the intention is to merge both packages (back) into package QCA.
Most of the examples should work with the current stable version 2.4, but there are situations where version upcoming version 2.5 is needed. In particular, chapter 4 makes extensive use of either new functions like plot1(), or changed argument defaults, for example argument type in function calibrate(), which from version 2.5 will be defaulted to fuzzy. Examples still work with version 2.4, but users should add type = "fuzzy" for the fuzzy calibration.
Also, some of the dialogs are changed in version 2.5, again with an example from the “Calibrate” menu. It’s dialog has been extensively improved, with a new and hopefully useful thresholds setter for the fuzzy calibration.
The structure of this book is different from the former user guide published in 2013. It will of course touch on the same topics and present the same package, but instead of organising chapters on the distinction between crisp, multi-value, and fuzzy sets, a better approach is to organise the book on QCA related analyses: necessity, sufficiency, parameters of fit, calibration etc. This structure is a first proposal, and readers are encouraged to make suggestions: as this is a work in progress, anything is subject to change until reaching a proper publication stage.
Many things have changed in the R packages over the past two years, with many new additions: the graphical user interface, or drawing Venn diagrams up to seven sets, just to mention a couple of the most spectacular. However, the QCA functionality relies on the same minimization engine from version 0.6-5, so results are backwards compatible.
Topic related chapters will contain examples for all QCA variants (cs, mv and fs, also extentions) as well as detailed instructions how to perform each of the analyses using both command line and using the new graphical user interface.
There are not enough words to describe the amazing work of Yihui Xie and all the team of engineers from RStusio, who provide this public service and especially for the packages knitr, rmarkdown and bookdown which allow this form of HTML publication (among others).
The author wishes to thank in advance for any feedback, as well as suggestions and likely corrections to the book content, sent to: dusa.adrian@unibuc.ro
|
2017-02-23 09:24:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2678448259830475, "perplexity": 1536.9599886552608}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171163.39/warc/CC-MAIN-20170219104611-00519-ip-10-171-10-108.ec2.internal.warc.gz"}
|
http://mathhelpforum.com/calculus/103867-growing-rate.html
|
1. ## Growing rate
A drop of water is a perfect sphere and by condensation, the droplet picks up moisture at a rate proportional to its surface area. Show that the drops radius increases at a constant rate.
What sort of calculus would I use to tackle this problem? Thanks
2. Originally Posted by RAz
A drop of water is a perfect sphere and by condensation, the droplet picks up moisture at a rate proportional to its surface area. Show that the drops radius increases at a constant rate.
What sort of calculus would I use to tackle this problem? Thanks
Differential Calculus! More specifically, "related rates". Use the formula for volume of a sphere- $V= \frac{4}{3}\pi r^3$- and the formula for area of the surface- $A= 4\pi r^2$. Differentiate the volume formula, with respect to t, using the chain rule, and set it equal to a constant times the area. Solve for dr/dt.
3. ## Rates of change
Hello RAz
Originally Posted by RAz
A drop of water is a perfect sphere and by condensation, the droplet picks up moisture at a rate proportional to its surface area. Show that the drops radius increases at a constant rate.
What sort of calculus would I use to tackle this problem? Thanks
We can re-write the phrase
the droplet picks up moisture at a rate proportional to its surface area
as "the rate of change of its volume is proportional to its surface area"; or, with the usual notation:
$\frac{dV}{dt}= kS$, for some constant $k$.
So, using $V = \tfrac43\pi r^3$ and $S = 4\pi r^2$, we get:
$\frac{d}{dt}\Big(\tfrac43\pi r^3\Big)=4k\pi r^2$
$\Rightarrow \frac{d}{dr}\Big(\tfrac43\pi r^3\Big)\times\frac{dr}{dt}=4k\pi r^2$
$\Rightarrow 4\pi r^2 \frac{dr}{dt}=4k\pi r^2$
$\Rightarrow \frac{dr}{dt}=k$
$\Rightarrow r$ is increasing at a constant rate.
|
2017-04-25 05:45:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.9202541708946228, "perplexity": 492.06396723975183}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120101.11/warc/CC-MAIN-20170423031200-00271-ip-10-145-167-34.ec2.internal.warc.gz"}
|
https://askubuntu.com/questions/4408/what-should-i-do-when-ubuntu-freezes/2304
|
# What should I do when Ubuntu freezes?
All operating systems freeze sometimes, and Ubuntu is no exception. What should I do to regain control when...
• just one program stops responding?
• nothing at all responds to mouse clicks or key presses?
• the mouse stops moving entirely?
• I have an Intel Bay Trail CPU?
In what order should I try various solutions before deciding to pull the power plug?
What should I do when starting up Ubuntu fails? Is there a diagnostic procedure I can follow?
## When a single program stops working:
When a program window stops responding, you can usually stop it by clicking the X-shaped close button at the top left of the window. That will generally result in a dialog box saying that the program is not responding (but you already knew that) and presenting you with the option to kill the program or to continue to wait for it to respond.
Sometimes this does not work as expected. If you can't close a window by normal means, you can hit Alt+F2, type xkill, and press Enter. Your mouse cursor will then turn into an X. Hover over the offending window and left-click to kill it. Right clicking will cancel and return your mouse to normal.
If your program is running from a terminal, on the other hand, you can usually halt it with Ctrl+C. If not, find the name and process ID of its command, and tell the program to end as soon as possible with kill [process ID here]. It sends the default signal SIGTERM (15). If all else fails, as a last resort send SIGKILL (9): kill -9 [process ID here]. Note that you should only use SIGKILL as a last resort, because the process will be terminated immediately by the kernel with no opportunity for cleanup. It does not even get the signal - it just stops to exist.
(Killing a process by kill -9 allways works if you have the permission to kill. In some special cases the process is still listed by ps or top (as "zombie") - in this case, the program was killed, but the process table entry is kept, becuse it's needed later.)
## When the mouse stops working:
If the keyboard still works, press Alt+F2 and run gnome-terminal (or, if these fail to launch, press Alt+Ctrl+F1 and login with your username and password). From there you can troubleshoot things. I'm not going to get into mouse troubleshooting here, as I haven't researched it. If you just want to try restarting the GUI, run sudo service lightdm restart. This should bring down the GUI, which will then attempt to respawn, bringing you back to the login screen.
## When everything, keys and mouse and all, stop working:
First try the Magic SysReq method outlined in Phoenix' answer. If that doesn't work, press the Reset button on the computer case. If even that doesn't work, you'll just have to power-cycle the machine.
May you never reach this point.
• I've recently discovered that, rather than the "ps $options | grep$process_name" referenced above, one can just enter "pgrep $process_name" to achieve approximately the same result (for certain values of$options). – koanhead Jun 4 '11 at 13:45
• @Chan-Ho Suh Not being able to start up isn't really "freezing". We cover failure to boot in askubuntu.com/questions/162075/…. Or do you mean freezing on login? – Jjed Sep 3 '12 at 18:36
• One should never recommend kill -9 right off the bat. Once should instead attempt to kill the process with less drastic signals first, and only use -9 if all else fails. – Scott Severance Sep 28 '12 at 4:18
• Why REISUB is useful when the system freezed? I can't find which option in REISUB can get control back to my own except reboot. Thank you~ – sam Oct 17 '12 at 11:48
• sudo service lightdm restart: Not awesome - will kill all gui processes in Ubuntu 13.04 at least - for me that included running Virtual Machines etc :( – Stephen Jul 9 '13 at 23:04
If it locks up completely, you can REISUB it, which is a safer alternative to just cold rebooting the computer.
REISUB by:
While holding Alt and the SysReq (Print Screen) keys, type REISUB.
R: Switch to XLATE mode
E: Send Terminate signal to all processes except for init
I: Send Kill signal to all processes except for init
S: Sync all mounted file-systems
B: Reboot
REISUB is BUSIER backwards, as in "The System is busier than it should be", if you need to remember it. Or mnemonically - Reboot; Even; If; System; Utterly; Broken.
This is the SysReq key:
NOTE: There exists less radical way than rebooting the whole system. If SysReq key works, you can kill processes one-by-one using Alt+SysReq+F. Kernel will kill the mostly «expensive» process each time. If you want to kill all processes for one console, you can issue Alt+SysReq+K.
NOTE: You should explicitly enable these key combinations. Ubuntu ships with sysrq default setting 176 (128+32+16), which allows to run only SUB part of REISUB combination. You can change it to 1 or, which is potentially less harmful, 244. To do this:
sudo nano /etc/sysctl.d/10-magic-sysrq.conf
and switch 176 to 244; then
echo 244 | sudo tee /proc/sys/kernel/sysrq
It will immediately work! You can test this by pressing Alt+SysReq+F. For me, it killed active browser tab, then all extensions. And if you will continue, you can reach X Server restart.
• In the event you're forced to do this, do it slowly. Let a few seconds pass in between each keypress so that the commands you're invoking have a chance to finish before you go to the next one. – Andrew Lambert Apr 24 '11 at 8:22
• In case you like mnemonics: Raising Elephants Is So Utterly Boring, or Reboot Event If System Utterly Broken. I've also seen it as RSEIUB (Raising Skinny Elephants is Utterly Boring). – Siegfried Gevatter Apr 26 '11 at 14:19
• I actually came up with this one and try to remember it this way: "Reset System Environment In UBuntu". or "Reset Environment In System UBuntu". – Luis Alvarado Aug 14 '12 at 21:32
• What do you do if you're using a Mac that has no SysRq key? – Cerin Jan 9 '13 at 23:22
• – ændrük Jan 12 '13 at 19:56
You can make the shortcut Ctrl+Alt+Delete open the System Monitor, with which you can kill any unresponsive applications.
1. Open up System ➜ Preferences ➜ Keyboard Shortcuts and click Add.
In the Command field, enter gnome-system-monitor. Name the shortcut whatever you want.
1. Click Apply and then click where it says Disabled. Now hit the keys Ctrl+Alt+Delete
1. Close Keyboard Shortcuts and try out the shortcut:
• but if X is locking up fully, or even the kernel is hung, you can't do much with a keyboard shortcut. – ζ-- Jun 14 '12 at 21:58
• Unfortunately, System Monitor is quite CPU intensive. It typically consumes up to 20% of my CPU, so if you're computer's bogged down, launching SM is only going to grind it into the dirt faster. – Cerin Jan 9 '13 at 23:25
• If you can open System Monitor you can get to a terminal, in which case your OS is not frozen. – nbm Nov 11 '13 at 23:32
• System Monitor is, unfortunately, not the trusty Task Manager on Windows. As commented above, it will only launch if (ironically) Ubuntu isn't frozen. And even if it does, it's unresponsive anyway. – ksoo Mar 25 '14 at 22:07
Freezes such as you have described can be both software and hardware related and as you have found sometimes frustratingly difficult to diagnose.
## Hardware
If this is a desktop PC look at your hardware-cards. For both laptops and desktops possibly acpi type issues.
It might be useful to temporarily simplify your configuration to have just the graphics card connected with a standard keyboard and mouse. All other cards should be removed.
For acpi related issues, try booting with noapic nomodeset in your grub boot option. Its also worth trying acpi=off although this could have other undesirable effects such as constant fan usage.
Also worth checking the bios version level and seeing if the vendor has a newer bios version. The readme notes should hopefully reveal if any newer version fixed crashes and freezes.
## Software
I note you have tried the standard 270 drivers but have failed due to freezes. Can you clarify if you had similar issues with the open-source driver? Obviously you will not get Unity during testing this.
Graphics freezing can be one of/or a combination of the driver/compiz/X/kernel
If you are willing to try any of the suggestions below first backup your system with a good backup tool such as CloneZilla. You will need an external media device to receive the image such as a large USB stick/drive or separate internal hard-drive.
There are a small number of important fixes primarily in the 275 stable but a small number also in the 280beta that fixed freezes - it is worth a shot to see if these apply to your graphics card. Unfortunately nvidia dont go into detail on which cards they specifically fix (readme.txt)
However - I would strongly recommend a backup unless you feel confident on reversing a nvidia install - especially since you had serious issues with the slightly older 270 drivers. I've used clonezilla countless times and it has always got me out of trouble. You do need a large external drive though - USB stick/external drive or a separate drive.
The latest graphics drivers have been packaged in the x updates ppa.
Note - this will lead you away from the standard baseline - if upgrading in the future ppa-purge the PPA itself before upgrading.
You can also manually install the drivers from nVidia:
Try installing the latest nvidia stable 275 or 280 drivers - 32bit 280 drivers: ftp site and 64bit: 280 drivers: ftp site
To Install
CTRL + ALT + F1 to switch to TTY1 and login
sudo service gdm stop
To stop the X server
sudo su
To run as root
cd ~/Downloads
sh NVIDIA-Linux-x86-280.04.run
To install the 32bit driver (equiv for 64bit) then reboot.
To uninstall
sudo sh NVIDIA* --uninstall
Also remove /etc/X11/xorg.conf
## X/Kernel/Compiz
If you run classic Ubuntu with effects do you get the same freeze issues as standard Ubuntu? If you cannot reproduce the freeze with classic Ubuntu (no effects) then this will point you towards a compiz issue. I would raise a launchpad bug report with the compiz team.
If space is available (e.g. 20Gb), you could dual boot/install alongside the latest oneiric alpha. Obviously this will itself be unstable, but it will come with the latest X and Kernel. You may need to also install manually the beta 280 graphics drivers above since it probably will not be offered in the Additional Drivers window.
If during testing you dont see the same freeze activity you could try uplifting your X version with the x-edgers ppa and using kernel kernel 3.0 in Natty. Going this route is not really desirable - and could cause you upgrade issues in the future - and may have other unforeseen stability issue. Again, use ppa-purge to remove the PPA.
Kernel 3.0 is packaged with the PPA - you'll need to install the headers as well as the kernel itself from synaptic BEFORE rebooting if you intend to install the nvidia drive later.
This is a testing ppa - do have a ready backup if you want to try this route.
• ... Are you sure this is a good idea, and that it might solve the problem? Or is this just a guess? As I had a lot of trouble with nvidia-current and nothing worked... That's why I switched to the nvidia-173 one. Can I simply restore my whole system from a CloneZilla back-up? The problem is that I haven't got a spare hard drive anymore to put a system copy on... – RobinJ Aug 12 '11 at 20:12
• I'll try... Though I haven't got a hard drive to make a back-up to, so I'll just hope it doesn't break my whole system. About feeling uncomfortable using beta software: I'm working on Ubuntu 11.10 Alpha 3 at the moment xD But for work I just use Ubuntu 11.04 as I don't need constant bugs and sometimes crashes while making a website or something similar :p – RobinJ Aug 12 '11 at 21:02
• Oh dear xD I installed the NVidia-275 driver, and rebooted. X didn't start anymore. No problem, after looking into the log files I saw that another driver was already using the device. I added nouveau to the modprobe blacklist, rebooted, and X started again... But now I've got another problem... I get to see the Unity interface, and then everything simply freezes :p I can switch to the tty's and run commands and everything, but it just seems Unity and the window manager/decorator have crashed. I can't switch back to Gnome Panel (and honestly, I wouldn't want to), ... – RobinJ Aug 12 '11 at 21:59
• ... as I killed it a few weeks ago (on purpose, as it ran together with the Unity Panel for some strange reason :p). And don't tell me to go using Unity 2D please cause it doesn't work too well and easy for me. In fact, Unity is the only reason I still install non-open source drivers. So please help me? xD – RobinJ Aug 12 '11 at 22:00
• ... I already did that, except for re-enabling nouveau as I don't see the use of that. But I meant help me out with the freezing problem please. – RobinJ Aug 12 '11 at 23:24
If you're getting a lot of freezes, there might be something wrong with your hardware. I used to get hard lockups every 48 hours due to some less than optimal RAM. Memtest86+ showed the fault after 40 minutes of testing. Swapped the RAM out for some more (under warranty) and I'm now at 32 days, 1 hour of uptime.
Ubuntu doesn't tend to leak its guts all over your memory like Windows can over time. Even if one application or a poor X video driver does, you can restart LigthtDM very simply and just keep going and going and going. I've actually been through three beta versions of the nvidia driver in this one boot :)
Anyway... While knowing how to restart softly is a very handy thing, finding, reporting and fixing the system should be your next priority. If it's an always-on system, you should easily be able to make it between kernel updates* without needing a restart.
*You should restart when you get kernel updates as they'll be security fixes that won't be applied until you reboot into the newer kernel.
• I agree that RAM is usually the culprit for unstable systems. I had a problem once that Memtest86+ was not able to found but I could trigger it repeatedly in 5 minutes running sha1sums on very big files repeatedly (checksums changed every now and then). It, too, was fixed by changing memory sticks. Other common causes are unstable power source or poor capasitors on the motherboard. The only way to diagnose these issues is to keep swapping parts until it works. – Mikko Rantalainen Mar 18 '14 at 7:34
• +1 for memtest86. RAM can be faulty without you really noticing it in daily use. – jmiserez Jun 24 '14 at 15:33
When everything stops working, first try Ctrl + Alt + F1 to go to a terminal, where you can likely kill X or other problem processes.
If even that doesn't work, try using holding down Alt + SysReq while pressing (slowly, with a few seconds between each) R E I S U B.
This puts the keyboard in raw mode, ends tasks in various states, syncs the disks, etc, and finally reboots the machine. You will get much better results doing this than just pulling the plug. Of course, if this fails, you're pretty much left with pulling the plug.
• A way to remember "REISUB" is "Reboot Even If System Utterly Broken". – Matthew Crumley Sep 20 '10 at 3:15
• or "Raising Elephants Is So Utterly Boring" :P – Axel Sep 20 '10 at 12:09
• I remember it using "BUSIER" Backwards – Nerdfest Sep 20 '10 at 16:33
• Between Ctrl+Alt+F1 and trying to kill processes, and Alt+SysRq+R E I S U B, it's worth pressying Ctrl+Alt+Delete. If you successfully got to a text-based virtual console (from having pressed Ctrl+Alt+F1), this will virtually always reboot the machine. – Eliah Kagan Jun 14 '12 at 21:24
• Much blunter and in Spanish: REInicia SUBnormal – Ramon Suarez Nov 29 '13 at 11:16
Also, sometimes it's simply the X-Server which hangs - a case I've most often found when you're using Compiz.
If this is the case you can kill X, which will restart and drop you back at the log-in screen.
The default sequence is Ctrl + Alt + Backspace
Although this is turned off by default (presumably new-users were accidentally hitting it) and can be turned back on like this:
1. SystemKeyboard (i.e. the Keyboard Preferences Dialogue)
2. Layouts tab
3. Click the Options button
4. At the Key Sequence to kill the X server point check Ctrl + Alt + Backspace.
• If your video driver is using kernel modesetting (KMS), it's unlikely this will be sufficient to fix lockups, you have to use sysrq or power cycle. (Go ahead and try C-A-B, it obviously can't hurt; it does work when an app (like compiz/unity) is stuck, rather than X itself, however other answers on this page would be better in this case). But when it doesn't work, now you know why. :-) – Bryce Jun 16 '12 at 0:53
• A note for new users about KMS: As a general rule, if you're using a binary driver (probably if you have an nVidia graphics card, and sometimes if you've got an ATI card) you're video driver is not using KMS. – thomasmichaelwallace Jun 16 '12 at 8:53
• I can't find this in Ubuntu 14.04. When I go to Settings > Keyboard the tabs are only Typing and Shortcuts. I just looked through all the shortcuts and couldn't find "Key sequence to kill the X server". Can I still do this on 14.04? – Andy Jan 13 '16 at 12:28
My first favourite when total freeze occured - Alt + SysRq + K.
That combo kills X, and returns me to the graphical login screen. If that doesn't work, try Alt + SysRq + R E I S U B.
In such cases you can try CTRL-ALT-F1 to get to a console. Then login with your password.
## Restarting the GUI
You can try to restart your graphical desktop with:
sudo service lightdm restart
If you're running Ubuntu 11.04 or earlier, you should use this instead (as gdm used to be the default display manager):
sudo service gdm restart
If you're using Kubuntu instead, then the default display manager is kdm, so you should instead use:
sudo service kdm restart
If you're using another display manager, replace ligthdm/gdm/kdm with its name.
## Restarting the Machine
If you want to do a clean system reboot, use:
sudo shutdown -r now
If it is only X that is "broken", than you can use kernel to kill it:
SysRq + Alt + K
For laptops (depends on the model, typically needed if "SysRq" is written in blue):
Fn + SysRq + Alt + K (release Fn after pressing SysRq).
• FWIW it depends on the laptop -- on my ThinkPad Alt+SysRq doesn't require Fn nor Ctrl. – Marius Gedminas May 10 '11 at 12:46
• The key pressing sequence is neatly described here (Ubuntu Forums: ubuntuforums.org/showthread.php?p=11773367#post11773367), particulary for laptop where 'Fn' key makes a difference. I had to follow it on my Lenovo Ideapad - hold 'Fn' with left hand first, press Alt + SysRq in right, leave 'Fn' and continue with REISUB slowly. – Chethan S. Nov 24 '12 at 16:41
What I do is opening a terminal with eg. Ctrl + Alt + F2
Login and use the terminal to kill the process that is lagging
ps -e | grep <procesname>
This shows the processID of the process with that name
(sudo) kill <processID>
This shuts down the process safely, in case that doesn't work use
(sudo) kill -9 <processID>
You can get back to the graphical user interface with Ctrl + Alt + F7
• ps and grep PROCESS can be replaced by a pgrep PROCESS call, and your whole thing can simply be replaced by pkill PROCESS or a killall PROCESS. – Martin Ueding Apr 24 '11 at 22:01
• thanks, I didn't know that. Seems to be UNIX commands too, nice – Chielus Apr 25 '11 at 8:32
• However, the default behavior of pgrep is not to show anything but the process ID itself. If you want to know the process's full name and any other information, ps is still a very valuable tool. – Eliah Kagan Jun 14 '12 at 21:25
To diagnose the freezes you should be able to use the net console (or serial serial console for that matter). Follow the set up instructions outlined here.
• I'll give that a shot and post-back the results. – Jordan Parmer Aug 19 '10 at 22:12
• This worked like a champ. Identified the problem as my graphics card: "radeon .... reserve failed for wait". Now I know what to look for. – Jordan Parmer Nov 28 '10 at 21:47
The first thing to look at is if it is just X that's frozen, or the whole system. Enable ssh and then ssh into the system. If you can't ssh into it, then it's probably a kernel lock up. If you can ssh in, then it might be just a gpu lockup.
Next try restarting X. Do this by restarting the display manager:
• On Ubuntu 11.10 and later, LightDM is the display manager, so run:
service lightdm restart
• On Ubuntu 11.04 and earlier, GDM is the display manager, so run:
service gdm restart
If that works, then it's perhaps an X bug. If it still doesn't work, then you may have a GPU lockup in the kernel drm driver. It would be useful to know at this point whether you're running the -ati (open source) driver, or -fglrx (closed source) driver.
• Thanks! My desktop randomly freezes as well. Running 'service gdm restart' always brings it up. Most of the times the desktop seems to freeze when Firefox is running. Any ideas on how I can debug/fix this ? – suhridk Sep 28 '10 at 10:35
• Well, generally when I find myself using the word 'random' in describing a bug, I take that as a clue that I need to do a bit more to better characterize the bug. E.g., how often does it happen, are there any things that seem more likely to trigger it than others, if any other symptoms occur in conjunction, etc. – Bryce Oct 1 '10 at 1:28
• You should also consider your hardware and driver. Some drivers like -intel are more likely to have freeze bugs than other drivers like -ati. Hardware also can play a factor, such as if you put a high end ATI card into a box with a sketchy power supply. Bad RAM, buggy motherboards, overheating, power irregularities, etc. all can result in freezes. – Bryce Oct 1 '10 at 1:34
• Looking through the other commenters on this page, you can see suggestions all over the map. Freezes are an unfortunately common end result of a whole variety of different kinds of problems. They can be software, hardware, or a combination of the two. This is why it is extremely important that the original reporter provide as detailed and thorough a report as possible. This is also why two people both experiencing similar freezes really have to be treated as completely separate bugs, unless their HW and SW are exactly the same. – Bryce Oct 1 '10 at 1:41
• Freezes on different drivers are debugged in different ways. For instance, -intel has a set of debugging tools you can install that allow you to capture GPU dumps. See wiki.ubuntu.com/X/Troubleshooting/Freeze for details. – Bryce Oct 1 '10 at 1:42
If you have to do a hard shutdown I'd be wondering if the memory (RAM) was failing. On your next boot, try running memtest86. To do this:
• while booting, hold down a shift key
• the GRUB menu will appear
• use the cursor keys to select the last option "memtest86"
• press enter
You'll get a basic display and it will try reading and writing lots of values to all of your RAM. As long as there are no failures, you'll see a green status. If there is any failure it will turn red. In that case you'll need to replace at least one stick of your RAM.
There is also community documentation of diagnosing hardware failures.
If you ever use the magic SysRq key as suggested in the first answer, just try getting the keyboard to work first with Alt + SysRq + R; then try Ctrl + Alt + F1 again.
It may work and you may save yourself a reboot. Only if it doesn't work you should try the whole REISUB sequence.
Just press Ctrl+Alt+F1 on your keyboard to open TTY1. When it opens, run the Kill command. Example below.
first you use: ps this will show you all processes running ("ps | less" if you want to see the results page by page) Then you look for the PID of the process you want to terminate. After this use: kill pid
kill command- Stop a process from running
Syntax: kill [-s sigspec] [-n signum] [-sigspec] jobspec or pid kill -l [exit_status]
Description: Most modern shells, Bash included, have a built-in kill function. In Bash, both signal names and numbers are
accepted as options, and arguments may be job or process IDs. An exit status can be reported using the -l option: zero when at least one signal was successfully sent, non-zero if an error occurred. Using the kill command from /usr/bin, your system might enable extra options, such as the ability to kill processes from other than your own user ID and specifying processes by name, like with pgrep and pkill. Both kill commands send the TERM signal if none is given.
• How about if I am not sure about pid which is causing my system to freeze. There could be many processes running interfering with each other making the system freeze. Could you please explain how to tackle in such case.. – Rohit Bansal Jun 12 '12 at 6:25
• Why the Down-vote? Just so I'll know what's wrong :) – Mitch Jun 12 '12 at 7:53
• Using the top command on the console you can often see which process is hanging, by checking the cpu % column. You can then kill that process. – Floyd Jun 12 '12 at 8:59
• What command? ps – Mitch Jun 12 '12 at 9:02
• i meant the command 'top' from the console. – Floyd Jun 12 '12 at 9:32
I thinks there is no such thing as a perfect distro, even in Windows they have this screen of death.
• Open another terminal Ctrl + Alt + F2.
• Issue this command:
sudo /etc/init.d/gdm restart
This restarts or logs you out of your current session but it will not reboot.
Then Ctrl + Alt + F7 go get back to your graphical interface.
(Community wiki answer - solution was originally buried in the OP question)
SOLUTION:
Solved it.
My particular problem was my graphics card (integrated Radeon 9000 series). netconsole revealed I was getting the error: "reserve failed for wait". After trial-and-error, I manually configured my video card and disabled hardware acceleration. Completely fixed the issue.
Here is what I did:
Manually Created xorg.conf
Ubuntu automatically configures xorg.conf and doesn't use a file. To edit this file, you have to tell Ubuntu to explicitly create one and then edit it. Here are the steps:
1. Restart system
2. Hold Shift as GRUB boots
4. Execute: X -config xorg.conf.new
5. Copy: cp xorg.conf.new /etc/X11/xorg.conf
Disable Hardware Acceleration
The following is specific to my Radeon card, but I'm sure other cards have a similar setup.
1. Edit xorg.conf
2. Find "Device" section for graphics card
3. Uncomment "NoAccel" option and set to "True"
4. Save + reboot
Hope that helps.
The simplest solution is to add the "Force Quit" applet to your Gnome top panel and when a program doesn't respond, click on the force quit and then on the application.
I am surprised with so many answers, this isn't mentioned. Of course, you can always do a ps -A and pipe that to grep for your program name. And kill -9 that. I prefer simplicity.
• That's not an option on Unity or GNOME 3 anymore, is it? – Waldir Leoncio Aug 16 '13 at 21:13
You can always do Alt + F2 and write killall <program> or xkill and click on the window you want smashed!
My ubuntu is super prone to freezing (probably 20 odd times a day). I use the magic sysrq key too, but instead of using it to reboot or kill xserver, I use the 'f' command which calls oom_kill, effectively dropping a process. I've only ever seen this drop chrome tabs (as I tend to have quite a few heavyweight ones open at a time). Anyway, this get's me out of this mess 95% of the time.
So when my ubuntu freezes (locks up, mouse stops responding etc), I hold alt + sysrq and then hit f (if you don't do this correctly it will take a screenshot instead). I usually have to repeat this combo a couple of times before ubuntu spurs back to life.
I'd have given up on ubuntu a long time ago if I hadn't discovered this, hope it helps someone!
Hit Alt+F2 to run a command. Type xkill and hit Enter.
Your mouse cursor will transform into a cross that can force to close any window you click on.
If somebody can provide a screenshot, I think that would be useful.
• not working since the desktop is frozen – Floyd Jun 12 '12 at 9:00
You might get some extra information when you switch to the TTY view. Press Ctrl + Alt + F1 to get this, use Ctrl + Alt + F7 (or maybe F8) to get back to the GUI. You can have different sessions on most of the F-keys but that's different question altogether.
• I tried that but the entire system is locked. Keyboard is completely unresponsive. – Jordan Parmer Aug 19 '10 at 22:13
if possible, try to open an ssh shell from another computer. this is an option If you knew in advance that the computer might hang soon, open the connection first before you perform that task.
I do this sometimes when I know vmware runs crazy and the GUI of ubuntu (the vmware host) becomes unresponsive. I can do a suspend from the ssh shell, it might take a while until it gets thru, and after a while the computer is idle again.
• A better idea would going to an other TTY (non-GUI) by pressing Ctrl + Alt + F[1-6] where F[1-6] is F1, F2, ..., F6. This is especially helpful if your network connection is heavily used. (to switch back, use Ctrl + Alt + F7. Replace F7 by F8 if it didn't work. – Lekensteyn Apr 30 '11 at 8:14
• Sure, however I always tend to forget these key combinations. And when I know I have to carry out some tasks/experiments where the computer might hang, it's always handy to have access to a second computer available right next to the hanging one. – knb May 1 '11 at 13:03
• For some experiments, a virtual machine in VirtualBox is recommended. In that way, you can easily revert the machines state. – Lekensteyn May 1 '11 at 13:24
There were some missed bugs between the relation of Unity/Compiz, the X.org system and the Video driver. These bugs of course are dealt with with newer, updated versions of Unit, Compiz, X or the video Driver.
When inside Unity and everything is slow and basically damaged, to go to TTY1, press CTRL+ALT+F1. When in the terminal, type your user and password to get to the prompt line. You can also get to the TTY when booting by pressing ESC or holding SHIFT, then on the GRUB menu, selecting recovery mode.
1. Install Xorg Edgers PPA
sudo add-apt-repository ppa:xorg-edgers/ppa sudo apt-get update
Depending on your video card you can either install the 304 series, 310 Series, 313 Series or any newer one that appears there. I recommend always test the latest version and only if it throws a problem, then go down from there until you reach a version where everything works correctly. For example, if you have a GT 9500 or later (Like in my case to which I also have a 440 GT, 560 TI and 680 GTX) the only version that solves all my problems is the 313.18 that came out a couple of days ago. So I would do this:
sudo apt-get install nvidia-313
This would install the latest version of the 313 series. It fixes MANY video problems with compiz, unity and xorg. The 310 series also fixes many issues but have not tested that one with my video cards. The other Nvidia versions are nvidia-experimental-304 and nvidia-experimental-310 as of this writing.
2. Reboot to test if your video card not works correctly with Ubuntu. If you get any problems regarding Nvidia config file, simply open a terminal and type sudo nvidia-xconfig and reboot.
There are other nice questions that could also help like:
How can I update my NVIDIA driver?
How do I install the Nvidia drivers?
Can not install Nvidia driver
What's the difference between the nvidia-current, and nvidia-current-updates packages?
Or even one that is more generic: How to correctly enable Desktop Cube in Unity 3D?
If you have tried all of the above and the freezing problem remains, you might want to try what I did.
Apply generous amounts of contact cleaner to CPU, RAM and any other chip complex enough to show those tiny, tightly packed pins. They can lose conductivity from dust accumulation as well as shorting due to humidity.
Some days after the cleaning (I used CRC 2-26) and a series of really brutal stress tests, my PC hasn't frozen once.
So, for all of you getting sudden unexpected freezings, give up messing around with your OS beyond what's reasonable and do an exhaustive dust and contact cleaning.
Replace with the latest Linux kernel 2.6.35 or up that will solve your problem. Follow these steps from this link.
• Ah! I'll give it a shot. The guide posted is for 2.6.34 but you said I needed to install 2.6.35. Which one should I do? – Jordan Parmer Jan 22 '11 at 11:45
• Bummer. Looks like I already have it installed. uname -r reveals 2.6.35-24-generic – Jordan Parmer Jan 22 '11 at 11:48
I was having similar issues with 10.04. X would hang and and nothing but a reset would fix it. I updated my nvidia drivers to the latest version and I haven't had issues since.
• Using ATI video drivers...but it might be worth looking into. Thanks! – Jordan Parmer Aug 19 '10 at 22:13
## In the very specific case you are using Virtualbox to run a 64-bit guest on a 32-bit (Ubuntu) host using VT-x or AMD-V (hardware virtualization technology built-in your CPU) only
Virtualbox may make your 32-bit host randomly crash when you run a 64-bit guest on it using VT-x or AMD-V (hardware virtualization technology built-in your CPU). It is a known issue.
### 2 solutions:
1. You have to run 32-bit guests only on your current 32-bit host [recommended if you have less than 2 GB of RAM];
2. You have to switch to Ubuntu 64-bit as host (you can even perform a 32-bit to 64-bit "migration" by reinstalling Ubuntu 64-bit without touching to your "/home" folder) [recommended if you have 2 GB of RAM or more].
Please note that you can run 64-bit and 32-bit guests on a 64-bit host using Virtualbox without any problem.
Other answers have very well covered general cases...
## protected by ZannaDec 30 '16 at 8:15
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
|
2019-10-16 14:19:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.22491353750228882, "perplexity": 3169.900173906851}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986668994.39/warc/CC-MAIN-20191016135759-20191016163259-00143.warc.gz"}
|
https://www.physicsforums.com/threads/partial-derivatives-question.121691/
|
# Homework Help: Partial derivatives question
1. May 22, 2006
### Luminous Blob
I'm trying to follow a derivation in given in a text book. Part of this derivation goes like this:
$$\frac{d}{ds}\left(\frac{1}{c}\frac{dx}{ds}\right)=c\left(\frac{\partial^2\tau}{\partial x^2}\frac{\partial \tau}{\partial x} + \frac{\partial^2\tau}{\partial x \partial y}\frac{\partial \tau}{\partial y}\right)$$
$$=\frac{c}{2}\frac{\partial}{\partial x}\left[\left(\frac{\partial \tau}{\partial x}\right )^2 + \left (\frac {\partial \tau}{\partial y}\right )^2 \right]$$
I worked through that and came up with the same answer, but without the factor of 1/2. Can anyone tell me where it comes from?
Last edited: May 22, 2006
2. May 22, 2006
### J77
Differentiate out those last terms...
d/dx(d tau/dx)^2=2d tau/dx * (d^2 tau/dx^2)
3. May 22, 2006
### Luminous Blob
Ah, gotcha! Thanks :)
4. May 22, 2006
### Luminous Blob
Hang on, after looking at it a bit more I'm not so sure...wouldn't that give you a factor of 2 out the front rather than 1/2?
5. May 23, 2006
### George Jones
Staff Emeritus
Doesn't performing the differentiation in the bottom line result in the top line, since the 2 cancels the 1/2?
If it does, then isn't everything OK?
Regards,
George
6. May 24, 2006
### J77
Yep - the '2' in my previous post cancels with the '2' of Blob's last term, giving the middle term...
7. May 24, 2006
### Luminous Blob
Haha, I see now...as you may have noticed, I'm not exactly the sharpest tool in the shed :)
Thanks again.
|
2018-12-16 19:21:02
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5771034359931946, "perplexity": 3326.2997105966106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827992.73/warc/CC-MAIN-20181216191351-20181216213351-00070.warc.gz"}
|
https://zbmath.org/serials/?q=J.+Algebr.+Syst
|
# zbMATH — the first resource for mathematics
## Journal of Algebraic Systems
Short Title: J. Algebr. Syst. Publisher: Shahrood University of Technology (SUT), Shahrood ISSN: 2345-5128; 2345-511X/e Online: http://jas.shahroodut.ac.ir/browse?_action=issue Comments: Indexed cover-to-cover; This journal is available open access.
Documents Indexed: 142 Publications (since 2013) References Indexed: 142 Publications with 2,130 References.
all top 5
#### Latest Issues
9, No. 1 (2021) 8, No. 2 (2021) 8, No. 1 (2020) 7, No. 2 (2020) 7, No. 1 (2019) 6, No. 2 (2019) 6, No. 1 (2018) 5, No. 2 (2018) 5, No. 1 (2017) 4, No. 2 (2017) 4, No. 1 (2016) 3, No. 2 (2015) 3, No. 1 (2015) 2, No. 2 (2015) 2, No. 1 (2014) 1, No. 2 (2014) 1, No. 1 (2013)
all top 5
all top 5
#### Fields
43 Commutative algebra (13-XX) 42 Group theory and generalizations (20-XX) 26 Combinatorics (05-XX) 24 Order, lattices, ordered algebraic structures (06-XX) 20 Associative rings and algebras (16-XX) 10 General topology (54-XX) 8 Mathematical logic and foundations (03-XX) 5 Number theory (11-XX) 5 Functional analysis (46-XX) 4 Category theory; homological algebra (18-XX) 4 Topological groups, Lie groups (22-XX) 4 Information and communication theory, circuits (94-XX) 3 General algebraic systems (08-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Abstract harmonic analysis (43-XX) 3 Algebraic topology (55-XX) 3 Manifolds and cell complexes (57-XX) 2 Algebraic geometry (14-XX) 2 Nonassociative rings and algebras (17-XX) 1 $$K$$-theory (19-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Operator theory (47-XX) 1 Convex and discrete geometry (52-XX) 1 Probability theory and stochastic processes (60-XX) 1 Numerical analysis (65-XX) 1 Mechanics of particles and systems (70-XX)
all top 5
#### Cited by 9 Authors
1 Ameri, Reza 1 Lotfi Parsa, Morteza 1 Louzari, Mohamed 1 Mashayekhy, Behrooz 1 Mirebrahimi, Hanieh 1 Mirvakili, Saeed 1 Mohareri, Mojtaba 1 Reyes, Armando 1 Shamsi, Khadijeh
#### Cited in 4 Journals
1 Czechoslovak Mathematical Journal 1 Revista Colombiana de Matemáticas 1 Hacettepe Journal of Mathematics and Statistics 1 Algebraic Structures and their Applications
all top 5
#### Cited in 6 Fields
1 Combinatorics (05-XX) 1 Commutative algebra (13-XX) 1 Associative rings and algebras (16-XX) 1 Group theory and generalizations (20-XX) 1 General topology (54-XX) 1 Algebraic topology (55-XX)
|
2021-12-01 18:52:57
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.24086272716522217, "perplexity": 4989.186135664858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964360881.12/warc/CC-MAIN-20211201173718-20211201203718-00226.warc.gz"}
|
http://mathforum.org/mathtools/cell.html?sortBy=TT!asc&offset=0&limit=25&resort=1
|
You are not logged in.
Resource Name Topic (Course) Technology Type $? Rating Browsing: All Content Browse discussions Login to Subscribe / Save Results Limit Comparison Test Comparison test (Calculus) Tool [0] Uses the Limit Comparison Test to determine whether the sum from 1 to infinity of 5^n/(7^n-n) converges. More: lessons, discussions, ratings, reviews,... Buffon's Needle: An Analysis a... Geometric probability (Geometry)+ Lesson Plan [0] Buffon's Needle is one of the oldest problems in the field of geometrical probability. It was first stated in 1777. It involves dropping a needle on a lined sheet of paper and determining the probabil... More: lessons, discussions, ratings, reviews,... Replacement Dialogue Independent events (Math 7)+ Support Material [0] Extends the notion of conditional probability by discussing the effects of replacement on drawing multiple objects. More: lessons, discussions, ratings, reviews,... Balance Applet Teacher Introdu... Ratio and proportion (Math 4)+ Lesson Plan [0] The Balance Applet is a game. There is a balance and 3 different shapes. The weight of one shape is given. The player is asked to find the other shapes' weights using the balance. More: lessons, discussions, ratings, reviews,... Math Cruncher Addition (Kindergarten)+ Android Tool [0] Features of Math Cruncher include, * solving simple equations before they hit the ground * solving the bonus equations to help yourself survive in the game and reach the highest score. More: lessons, discussions, ratings, reviews,... 9 digits Addition (Math 2)+ Android Tool [0] This is a puzzle with 9! possibilities and 336 solutions. The problem posed in 9 digits consists in the organization of the numbers 1 to 9 in a square of 3 * 3 to make the sum of the numbers of the fi... More: lessons, discussions, ratings, reviews,... Ninedigits workshop Addition (Math 2)+ Android Tool$ [0] Puzzle with 9! possibilities and 336 solutions. The problem posed in Ninedigits workshops consists in the organization of the numbers 1 to 9 in a square of 3 * 3 to make the sum of the numbers of the... More: lessons, discussions, ratings, reviews,... Whole Calculator Addition (Pre-Kindergarten)+ Android Tool [0] This app functions as a calculator for the set of the natural numbers (N). Type the numerical operations and view the both inputs and the result in words. More: lessons, discussions, ratings, reviews,... Long Division Games Lite Division (Math 4)+ Android Tool [0] This is an app made for children who are learning division. The app is divided into 3 memorable steps or guidelines to help children and parents see when they operate the app for the first time. More: lessons, discussions, ratings, reviews,... Adding Unit Fractions Fractions (Math 1)+ Android Tool [0] Get the proper fractions listed at the top of the application, adding two or three unit fractions. Each proposed proper fraction has a variable number of solutions and different levels of difficulty. ... More: lessons, discussions, ratings, reviews,... Old Egyptian Fractions Fractions (Math 1)+ Android Tool $[0] This is the android version of the Mathcats Old Egyptian Fractions: Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie). But to make fractions like 3/4, they had to add pi... More: lessons, discussions, ratings, reviews,... Old Egyptian Fractions for And... Fractions (Math 1)+ Android Tool [0] This is the Android version of the Mathcats Old Egyptian Fractions: Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie). But to make fractions like 3/4, they had to add pi... More: lessons, discussions, ratings, reviews,... Reading Fractions Fractions (Math 1)+ Android Tool [0] This app lets you view fractions represented in the form of overlapping circles and bars. The two graphs are interactive. They can be changed by dragging on the screen. More: lessons, discussions, ratings, reviews,... Touch Fraction Fractions (Math 1)+ Android Tool [0] Touch Fraction lets you visualize positive and negative fractions -- and zoom into equivalent fractions using the rational representation diagram. More: lessons, discussions, ratings, reviews,... Grapher Functions (PreCalculus)+ Android Tool [1] Draw standard, parametric, or polar form with this 2D grapher, which supports trigonometric, logarithmic, wave, and dozens of other mathematical functions. More: lessons, discussions, ratings, reviews,... Dyscalculator - Android General (Kindergarten)+ Android Tool [0] This calculator app made its first public demonstration last month at the 12th International Congress on Mathematical Education (ICME). Designed to help dyscalculics, Dyscalculator shows quantities in... More: lessons, discussions, ratings, reviews,... Game about Squares [Android] General (Kindergarten)+ Android Tool [0] A puzzle game with 20 levels. Tap to move the square. Move all of the squares over their corresponding dots to win the game. More: lessons, discussions, ratings, reviews,... Minutes Text Notes Pro General (Math 7)+ Android Tool$ [0] A note taking Android app that syncs to Dropbox and can render Markdown along with mathematical equations (MathJax support requires Android 4.4). It includes dictionary support, Tasker extension supp... More: lessons, discussions, ratings, reviews,... Kids Measurement Science Lite Length/height (Math 2)+ Android Tool [0] Young scientists learn to analyze the physical world by measuring across a variety of domains - time, size, weight and money. 5-8 year old kids measure length, weight, time and money in this science a... More: lessons, discussions, ratings, reviews,... Apples and Oranges 1 Decimal Linear relationships (Math 7)+ Android Tool $[0] This application is a basic tool for learning algebra. Users try to solve a system of linear equations with two equations and two unknown variables. It is based on the same problem used in the free ap... More: lessons, discussions, ratings, reviews,... Time Calculator Measurement (Math 2)+ Android Tool [0] With the Time Calculator you can calculate the difference between two dates. You can set a date in the past and subtract it from the present time and the result will be the elapsed time. Also, you can... More: lessons, discussions, ratings, reviews,... KENKEN Classic II (for Android... Operations with numbers (Math 3)+ Android Tool [0] KenKen is a grid-based logic puzzle and math game. For a 3x3 puzzle, use the numbers 1-3; for a 4x4 puzzle, fill in with numbers 1-4; for a 5x5 puzzle, use the numbers 1-5, and so on. Do not repeat a ... More: lessons, discussions, ratings, reviews,... Math Tablet Operations with numbers (Pre-Kindergarten)+ Android Tool$ [0] This is a virtual manipulative to support developing number sense and arithmetic concepts. Two sub-quantities together result in a new total quantity. This is called addition. And if you take away a s... More: lessons, discussions, ratings, reviews,... Touch Natural Numbers Place value (Math 1)+ Android Tool [0] With this app you can: analyze the factors of a composite number. build a composite number from its factors (multiplication and division of natural numbers). localize or... More: lessons, discussions, ratings, reviews,... Pick-a-Path [Android] Powers of ten (Math 5)+ Android Tool [0] Help Okta reach the target (maximum, minimum, or a specific value) by choosing a path from the top of the maze to the bottom. Seven levels with seven puzzles will test your skills with powers of ten, ... More: lessons, discussions, ratings, reviews,... Page 1 2 3 4 5 6 ... 10 ... 20 ... 30 ... 40 ... 50 ... 60 ... 70 ... 80 ... 90 ... 100 ... 110 ... 120 ... 130 ... 140 ... 150 ... 160 ... 170 ... 180 ... 190 next »
|
2016-02-07 13:46:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17360521852970123, "perplexity": 9940.299524639391}, "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-2016-07/segments/1454701149377.17/warc/CC-MAIN-20160205193909-00195-ip-10-236-182-209.ec2.internal.warc.gz"}
|
https://physics.stackexchange.com/questions/336640/does-the-mass-increase-or-decrease-here
|
# Does the mass increase or decrease here?
I am very very newbie in Special Relativity.
But consider this situation. Lets say I have a block of mass $$m$$ which is at rest relative to a certain inertial frame of reference. Let's say I hit this block and give a velocity $$\frac{c}{2}$$. There are two observers, one is at rest and one is moving with a velocity of $$\frac{c}{2}$$. I'll call the observer at rest as Observer A and the other as Observer B.
### Observer A:
He/She observed the block to move from zero to a speed of $$\frac{c}{2}$$. As a newbie I may be, I do know that mass increases by a factor of $$\gamma$$, and that's:
$$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
In his/her case, $$\gamma =\frac{2}{\sqrt{3}}$$.
The block will increase in mass by a factor of $$\gamma$$.
### Observer B:
This observer would see the block moving with $$\frac{c}{2}$$. This observer would see that the mass of the object decreased by a factor of $$\gamma$$.
Both can't be true at the same time. This seems to be a paradox. All inertial frames are equivalent, that's a postulate of General Relativity I think. How would this be resolved?
• I'm entirely in agreement with the previous comment; I think that 'relativistic mass' is an unnecessary idea, and gets in the way of understanding. But what (I hope) we can both agree on is that in SR a body's momentum is given by $\vec{p}= m_0 \gamma \ \vec{v}$ in which $\vec{v}$ is the body's velocity in our frame. But, to attend to your question, in observer B's frame, according to your first paragraph, the block is surely at rest. – Philip Wood May 31 '17 at 17:27
• The observer B will see the block at rest.... After it is hit.....are you thinking of before the hit ? – Shashaank May 31 '17 at 19:08
• No; after it's hit, when, surely, it's moving with velocity $\frac{c}{2}$ in A's frame, and velocity zero in B's frame,because B's frame is moving at velocity $\frac{c}{2}$ relative to A's frame. Clearly we have some sort of misunderstanding. – Philip Wood Jun 1 '17 at 14:30
To be clear: one observer does observe the relativistic mass increasing, and another does observe the relativistic mass decreasing, but there is no contradiction.
Let's change perspective to avoid the notion of "relativistic mass". Denote the rest mass (aka "the mass") by $m_0$. Denote the energy-momentum of the block as a vector $(E,p)$.
Before hitting the block, observer A describes the block as having energy-momentum $(m_0 c^2,0)$. Observer B describes the block as having energy-momentum $(\gamma m_0 c^2,-\gamma m_0 \frac{c}{2})$.
After hitting the block, observer A describes the block as having energy-momentum $(\gamma m_0 c^2,\gamma m_0 \frac{c}{2})$. Observer B describes the block as having energy-momentum $(m_0 c^2,0)$.
There is absolutely no paradox. All four cases let the energy-momentum vector have invariant interval $E^2-(pc)^2=(m_0 c^2)^2$. To get a paradox you have to speak in terms of some actual physical thing that happens. There is, however, a physical effect that occurs in special relativity and not in Newtonian mechanics. Imagine the situation after the block was hit, and imagine two possibilities:
If observer B were to hit the block again, they would supply a (relativistic) momentum $\Delta p_B$ and see a velocity change $\Delta v_B$. If observer A were to hit the block again, they would supply a (relativistic) momentum $\Delta p_A$ and see a velocity change $\Delta v_A$. If we imagine these two possibilities and demand that A and B both try to achieve the same change in velocity, then $\Delta v_A=\Delta v_B$, and in fact:
$$\frac{\Delta p_A}{\Delta v}<\frac{\Delta p_B}{\Delta v}$$
Observer B finds it harder to increase the velocity than does observer A. The ratio of momentum over velocity has units of mass, so this is suggestive! I'm cheating though, because $\Delta p_B/\Delta v$ isn't actually the relativistic mass as it's normally defined*, but this is the basic phenomenon. Observer B finds it harder to increase the velocity by a fixed amount $\Delta v$ than observer A (who tries to increase the velocity by the same amount $\Delta v$). In fact, if observer B tried to increase the velocity in its frame by another $c/2$, they would find it impossible. Observer A, on the other hand, can easily increase the velocity in its frame by another $c/2$. This is what is meant by the statement that the relativistic mass increasing. These two things correspond to two different physical situations, and so there is no contradiction at all.
*(this is $dp/dv$, but $p=\gamma m_0 v=m_\text{rel}v$ [by the definition of $m_\text{rel}$], so $\frac{dp}{dv}=m_\text{rel}+v \frac{d m_\text{rel}}{dv}$ by the product rule.)
• So in this case, you ignored "relativistic mass". Sorry if this is a stupid assumption, but can I say that relativistic mass depends on frame? – Pritt Balagopal Jun 1 '17 at 14:45
• As I stated in my question, I'm very new to Relativity, why did you assume a energy-momentum vector? – Pritt Balagopal Jun 1 '17 at 14:46
• @PrittBalagopal Yes, the relativistic mass depends on frame. I ignored the words "relativistic mass" because it is just the ratio relativistic momentum over velocity - if we describe everything we know about momentum and everything we know about velocity and don't find a paradox there, then there can't be a paradox with the relativistic mass. – user12029 Jun 1 '17 at 21:45
• @PrittBalagopal Finally, to talk about relativistic mass you have to talk about relativistic momentum, so I have to put in 4 quantities for the momentum in 4 situations. It can be instructive to consider the energy too, so now that's 8 quantities. I just used (E,p) notation as a shorthand to make it easier to read and write these 8 quantities. – user12029 Jun 1 '17 at 21:50
|
2021-04-23 10:25:58
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8428802490234375, "perplexity": 416.52646821882496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039617701.99/warc/CC-MAIN-20210423101141-20210423131141-00302.warc.gz"}
|
https://www.xarg.org/book/algebra/series/
|
# Series
The sum of a sequence of terms is called a series.
Let $$(a_n)_{n\geq k}$$ be a sequence starting with index $$k$$. Now sum over the first $$n-k-1$$ terms:
$\begin{array}{rl}s_n =& \sum\limits_{i=k}^n a_i\\=& \underbrace{\underbrace{\underbrace{\underbrace{a_k}_{s_k} + a_{k+1}}_{s_{k+1}}+a_{k+2}}_{s_{k+2}}+...+a_n}_{s_n}\end{array}$
The sequence $$(s_n) = (s_k, s_{k+1}, s_{k+2}, ...)$$ is called an infinite series $$\sum\limits_{i=1}^\infty a_i$$. If $$(s_n)$$ converges, its limit is called $$\sum\limits_{i=k}^\infty a_i$$.
## Examples
• $$\sum\limits_{i=1}^\infty i = 1+2+3+...$$ diverges
• $$\sum\limits_{i=1}^\infty (-1)^i$$ diverges, $$s_n = \sum\limits_{i=1}^n(-1)^i = \begin{cases} 0, & \text{if } n \text{ even} \\ -1, & \text{if } n \text{ odd} \end{cases}$$
• Harmonic series $$\sum\limits_{i=1}^\infty\frac{1}{i}$$ diverge, because new packages that are $$\geq\frac{1}{2}$$ can always be put together:
$$\begin{array}{rl} s_n &= 1+\underbrace{\frac{1}{2}+\frac{1}{3}}_{>\frac{1}{2}}+\underbrace{\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}}_{>\frac{1}{2}}+...+\frac{1}{n}\\&<1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+...\end{array}$$
• $$\sum\limits_{i=1}^\infty\frac{1}{i^2}$$ converges to $$\frac{\pi^2}{6}$$
• Geometric Series: $$\sum\limits_{i=0}^\infty q^i \begin{cases} |q|<1: \text{ converges to }\frac{1}{1-q} \\ |q|\geq 1: \text{ diverges} \end{cases}$$
• General Harmonic series: $$\sum\limits_{i=1}^\infty\frac{1}{i^s}$$ converges $$\forall s>1$$
• Leibnitz-Series: $$\sum\limits_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ converges to $$\frac{\pi}{4}$$
• Alternating Harmonic series: $$\sum\limits_{i=0}^\infty(-1)^i\frac{1}{i+1}$$ converges to $$\ln 2$$
• $$\sum\limits_{i=1}^\infty\frac{1}{2^i}$$ converges to $$1$$ since
We can state that if $$\sum\limits_{i=k}^\infty a_i$$ converges $$\Rightarrow (a_i)_{n\geq k}$$ is a null sequence. $$\Leftarrow$$ does not hold, see Harmonic series.
## Convergence / Divergence criteria
### Divergence criterion
If $$(a_n)_{n\geq k}$$ is no null sequence, the series $$\sum\limits_{i=k}^\infty a_i$$ diverges.
Example: $$\sum\limits_{i=1}^\infty\left(1+\frac{1}{i}\right)$$ is divergent, since $$\lim\limits_{i\to\infty}\left(1+\frac{1}{i}\right)=1$$
### Weierstrass comparison test (majorant criterion)
Let $$(a_n)_{n\geq k}$$, $$(b_n)_{n\geq k}$$ be sequnces with $$|a_n|\leq b_n$$, then: If $$\sum\limits_{i=k}^\infty b_i$$ is convergent then $$\sum\limits_{i=k}^\infty |a_i|$$ is also convergent as well as $$\sum\limits_{i=k}^\infty a_i$$.
### Leibniz test for alternating series
Let $$(a_n)_{n\geq k}$$ be a monotonically decreasing null sequence with $$a_i\geq 0\forall i$$, then the alternating series $$\sum\limits_{i=k}^\infty (-1)^ia_i$$ converges.
### Absolute Convergence
$$\sum\limits_{i=k}^\infty a_i$$ is called absolute convergent, if $$\sum\limits_{i=k}^\infty|a_i|$$ converges. It is true that if a series converges absolute $$\Rightarrow$$ the series converges. $$\Leftarrow$$ does not hold, see alternating Harmonic series.
### Root test
If there exists a $$q<1$$ and an index $$i_0$$ for which $$\sqrt[i]{|a_i|}\leq q\forall i\geq i_0$$ holds, the series $$\sum\limits_{i=k}^\infty a_i$$ converges absolute.
Please note: $$\sqrt[i]{|a_i|}< 1$$ is not enough! For example the Harmonic series $$\sqrt[i]{\frac{1}{i}}\to 1$$, but we can’t find a $$q<1$$.
If $$\sqrt[i]{|a_i|}\geq 1$$ holds for endless $$i$$, the series diverges.
### d’Alembert’s ratio test (quotient criterion)
If there exists a $$q<1$$ and an index $$i_0$$ for which $$\left|\frac{a_{i+1}}{a_i}\right|\leq q\forall i\geq i_0$$ holds, the series $$\sum\limits_{i=k}^\infty a_i$$ converges absolute.
Please note: $$\left|\frac{a_{i+1}}{a_i}\right|< 1$$ is not enough!
For $$\left|\frac{a_{i+1}}{a_i}\right|\geq 1$$ no general statement is possible.
« Back to Book Overview
|
2021-05-17 18:17:22
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9940512776374817, "perplexity": 516.9392608998082}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243992440.69/warc/CC-MAIN-20210517180757-20210517210757-00127.warc.gz"}
|
http://mathhelpforum.com/calculus/2038-countinuity-countinued-fraction-print.html
|
# Countinuity of Countinued Fraction
• Feb 27th 2006, 04:15 PM
ThePerfectHacker
Countinuity of Countinued Fraction
Define a function,
$f(x)=[1;x,x^2,x^3,...]$
the necessary and sufficient conditions for convergence is when,
$\sum^{\infty}_{k=0}a_k$ diverges, thus,
$\sum^{\infty}_{k=0}x^k$ this is geometric.
Diverges when $|x|\geq 1$ for simplicity let $x\geq 1$. Now prove that this series is countinous.
I am trying to express transcendental functions in terms of infinite countinous fractions, like the one above. I do not think I will get anywhere :(
• Aug 26th 2006, 12:44 PM
Rebesques
Take the sequence $
f_n(x)=[1;x,x^2,x^3,...,x^n]
$
which are continuous and converge pointwise to $f$. Prove it converges uniformly; Then the limit function is also continuous.
|
2016-10-01 17:13:39
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.986046552658081, "perplexity": 988.4936586047089}, "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/1474738663142.93/warc/CC-MAIN-20160924173743-00044-ip-10-143-35-109.ec2.internal.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-8-rational-functions-8-5-add-and-subtract-rational-expressions-8-5-exercises-problem-solving-page-588/43a
|
## Algebra 2 (1st Edition)
$\frac{Pi}{1-(\frac{1}{1+i})^{12t}}=\frac{Pi}{1-\frac{1}{(1+i)^{12t}}}=\frac{Pi}{\frac{(1+i)^{12t}}{(1+i)^{12t}}-\frac{1}{(1+i)^{12t}}}=\frac{Pi}{\frac{(1+i)^{12t}-1}{(1+i)^{12t}}}=\frac{Pi(1+i)^{12t}}{(1+i)^{12t}-1}$ Thus we proved what we had to.
|
2021-04-19 16:48:33
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9954192042350769, "perplexity": 6204.023423717193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038887646.69/warc/CC-MAIN-20210419142428-20210419172428-00378.warc.gz"}
|
https://rpg.stackexchange.com/questions/152166/what-is-the-carrying-capacity-of-a-large-pouch
|
# What is the carrying capacity of a large pouch?
What is the carrying capacity of a large pouch, small pouch, backpack and so on?
FAQ:
• Am I actually playing AD&D 1e: yes
• Have I read the rulebooks: yes
The last page of the Advanced Dungeons and Dragons Player Character Record Sheets accessory (TSR Stock # 9028) includes the following table:
Container Volume G.P. Equivalent
Small pouch 1/4 cu. ft. 25 g.p.
Large pouch 1/2 cu. ft. 50 g.p.
Small sack 1 cu. ft. 100 g.p.
Backpack 3 cu. ft. 300 g.p.
Large sack 4 cu. ft. 400 g.p.
Since the Players Handbook already lists other, much smaller numbers for the actual encumbrance of the containers themselves, the implication is that the G.P. Equivalent values are for the containers when fully loaded.
Also, these numbers match the encumbrance example given on page 225 of the DMG, which describes dividing 700 gp amongst 1 large and 3 small sacks.
• I think he divided 700 GP amonsts 1 Large and 3 small sacks: Dimwall can carry 400 gold pieces in his large sack and another 300 gold pieces in his small sacks – KorvinStarmast Jul 22 '19 at 12:36
• Your table is shockingly neat for having been constructed with (my congratulations as well as my condolences, since that must have been such a pain), so I wouldn’t want to touch it without asking first, but a LaTeX-based table would be cleaner to edit and the usual preferred method of handling tabular data. I would be happy to convert it for you. Would you like me to? – KRyan Jul 24 '19 at 1:08
• Knock yourself out! Haven't embedded LaTeX here before, so this was the best I could do on short notice. – Zimul8r Jul 30 '19 at 0:54
|
2020-02-22 11:10:55
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6809290647506714, "perplexity": 10018.124834027989}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145657.46/warc/CC-MAIN-20200222085018-20200222115018-00366.warc.gz"}
|
https://brilliant.org/problems/a-problem-by-raju-klatchu/
|
# A problem by raju klatchu
Level pending
$$p(2p(n)+2016)$$ is divisible by $$2p(n)$$ for $$n$$ belongs to natural numbers. Find the value of $$p(2016)$$.
×
|
2017-09-20 09:29:48
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6661803126335144, "perplexity": 1368.9991248282324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818686983.8/warc/CC-MAIN-20170920085844-20170920105844-00627.warc.gz"}
|
http://www.statsmodels.org/dev/generated/statsmodels.regression.recursive_ls.RecursiveLSResults.pvalues.html
|
# statsmodels.regression.recursive_ls.RecursiveLSResults.pvalues¶
RecursiveLSResults.pvalues()
(array) The p-values associated with the z-statistics of the coefficients. Note that the coefficients are assumed to have a Normal distribution.
|
2018-12-16 12:41:48
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9454105496406555, "perplexity": 500.4632649557808}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827727.65/warc/CC-MAIN-20181216121406-20181216143406-00564.warc.gz"}
|
http://www.wikihow.com/Add-Pages-in-Adobe-Illustrator
|
Edit Article
Community Q&A
This tutorial shows you how to add more pages to a document using Adobe Illustrator.
## Steps
1. 1
Add more pages to a preexisting document. For example, I have one A4 cover page but I want to add more pages to the document. Go to File > New Documents. Select add and make the width of your document in multiples. If I need 2 pages, I would choose A3 size > OK.
2. 2
Go to File > Print > Set up > Tiling > Tiling Full Pages > Done.
3. 3
To get your second page, go to View > Show Page Tiling.
4. 4
If you need to add more pages, you can change the width or the height of your document multiples
5. 5
To account for added pages, you should add 210mm to the width of document. And for the right one, add 297mm to compensate for the height of document.
|
2016-06-29 07:16:43
|
{"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": 0.8510937094688416, "perplexity": 2602.0012006644224}, "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/1466783397636.15/warc/CC-MAIN-20160624154957-00162-ip-10-164-35-72.ec2.internal.warc.gz"}
|
https://zbmath.org/?q=an:0661.17021
|
## Vertex operators in conformal field theory on $${\mathbb{P}}^ 1$$ and monodromy representations of braid group.(English)Zbl 0661.17021
Conformal field theory and solvable lattice models, Symp., Kyoto/Jap. 1986, Adv. Stud. Pure Math. 16, 297-372 (1988).
[For the entire collection see Zbl 0646.00016.]
The paper is a very clear exposition of the theory of vertex operators in two-dimensional conformal field theory, and the relation with monodromy representations of the braid group. The ideas presented here originate from a paper by V. G. Knizhnik and A. B. Zamolodchikov [Nucl. Phys. B 247, 83-103 (1984; Zbl 0661.17020)], but the present paper goes much deeper in the mathematical formulation and the foundations of the theory. For people who wish not to read such a long paper, there is a 10- page introduction listing the notation and all the basic theorems proved here.
The paper gives a summary of representation theory of the affine sl(2,$${\mathbb{C}})$$ Lie algebra, denoted by $$A_ 1^{(1)}$$, including the natural action of the operators of the Virasoro algebra. Then, the existence and uniqueness of vertex operators of spin j is shown: vertex operators are defined as operators satisfying the gauge condition and the equations of motion. It is shown that a triplet $$\left( \begin{matrix} j\\ j_ 2\quad j_ 1\end{matrix} \right)$$, satisfying the $$\ell$$-constrained Clebsch-Gordan condition, determines uniquely a vertex operator. Next, N- point functions are defined as compositions of vertex operators. Differential equations (called the fundamental equations) satisfied by these N-point functions, which have only regular singularities, give rise to properties of the vertex operators. First, convergence of compositions of vertex operators is proven. The commutation relation of vertex operators is equivalently rephrased in terms of the connection matrix of the fundamental equations, and is explicitly calculated in the case where one of the j-values is equal to $$1/2$$. Then, the monodromies of the fundamental equations give rise to representations of the braid group $$B_ N$$. The monodromy representation is explicitly determined in a special case (where implicitly $$j=$$ vertex operators are involved). This representation is in fact an irreducible representation of the Hecke algebra $$H_ N(q)$$ of type $$A_{N-1}$$, with $$q=\exp (2\pi \sqrt{- 1}/(\ell +2))$$. It is remarkable that the representation obtained here is with q a root of unity, since the Hecke algebra $$H_ N(q)$$ is not semi- simple in that case.
Reviewer: J.Van der Jeugt
### MSC:
17B65 Infinite-dimensional Lie (super)algebras 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 20F36 Braid groups; Artin groups
### Citations:
Zbl 0646.00016; Zbl 0661.17020
|
2022-07-06 20:22:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8198140859603882, "perplexity": 337.13355736435346}, "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/1656104676086.90/warc/CC-MAIN-20220706182237-20220706212237-00055.warc.gz"}
|
https://physics.stackexchange.com/questions/467331/wheres-the-database-in-grovers-algorithm
|
# Where's the Database in Grover's Algorithm
I am a little confused by parts of Quantum Computation and Quantum Information by Nielsen and Chuang on Grover's algorithm:
Grover's algorithm searches a number of quantum states for a "marked element" $$|x_0 \rangle$$. The algorithm uses the following unitary operators, acting on an initial state $$|0 \rangle ^{\otimes n}$$:
$$(-H^{\otimes n}U_0H^{\otimes n}U_f)^TH^{\otimes n}|0 \rangle ^{\otimes n}$$
where it can be shown, that $$U_f=I_{|x_0 \rangle}$$ corresponds to an inversion about the vector $$|x_0 \rangle$$ and $$H^{\otimes n}U_0H^{\otimes n}= I_{|+ \rangle}$$.
Geometrically, the algorithms can thus be understood as a series of inversions/reflections about vectors in the plane spanned by the vectors $$|+ \rangle$$ that result from the first operation of the quantum circuit $$H^{\otimes n}|0 \rangle ^{\otimes n}= |+ \rangle ^{\otimes n}$$
Geometrical representation of a Grover iteration. The algorithm starts in state $$|\xi \rangle$$, which coincides with state $$H^{\otimes n}|0 \rangle^{\otimes n}=|+ \rangle$$. The state is rotated by the unitary operations $$U_f=-R_{|x_0 \rangle}=I_{|x_0 \rangle}$$ and $$H^{\otimes n}U_0H^{\otimes n}= -R_{|+ \rangle} = I_{|+ \rangle}$$. Every iteration moves the state closer to the solution $$|x_0 \rangle$$ by an angle of $$2 \gamma$$.
But where in the algorithm does the "database" come in that we actually want to search? After all, the computer is initially prepared in $$|0\rangle ^{\otimes n}$$.
• Possible duplicate of The Grover algorithm in real life – Norbert Schuch Mar 19 '19 at 8:14
• I don't think it is a duplicate but the answers might be helpful for your understanding.... Concerning the question: The database is the quantum system itself, the one that you initially prepare in the starting state. You are looking for a certain eigenstate and the corresponding eigenvalue, i.e. your system will be in that state after you completed Grover's algorithm. Thus, your quantum system is your "database". – lmr Mar 19 '19 at 10:32
• @lmr I'd certainly say my answer from there answers this question. Do you think I should re-post it? – Norbert Schuch Mar 19 '19 at 12:23
• @NorbertSchuch I agree that your answer is highly valuable for the understanding of Grover's algorithm in general. But I would rephrase it for this question so that it becomes obvious where the actual "database" is found. – lmr Mar 19 '19 at 14:36
• – S.D. Mar 19 '19 at 16:54
This seems to be a common misunderstanding about Grover's algorithm. It is not about querying a magically encoded database. Rather, you have an efficiently computable function $$f(x)\in\{0,1\}$$ and you want to find some $$x_0$$ for which $$f(x_0)=1$$. Since you know how to realize $$f(x)$$ (i.e., you have a circuit), you can run $$f$$ on a quantum computer and use Grover to find such an $$x_0$$. This function can be seen as returning entries of a "database", which is encoded in a specific function, though I don't particularly like this picture.
The relevance is in the fact that a large number of interesting problems (namely, the class NP) are such that solutions might be hard to find, but they are easy to verify. Thus, Grover gives a square-root speed-up on any brute-force method to solve such a problem (i.e., any method which does not make use of any special structural property of $$f$$).
|
2020-12-03 14:24:20
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7971925139427185, "perplexity": 265.9034047209161}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141727782.88/warc/CC-MAIN-20201203124807-20201203154807-00188.warc.gz"}
|
http://clay6.com/qa/26992/find-the-solution-x-2y-1-xy
|
# Find the solution : $x^2y'=1-xy$
$(a)\;y=\frac{\log x+c}{x} \\ (b)\;y=x \log x+c \\ (c)\;y=\frac{\log x}{x}+c \\ (d)\;y= \frac{\log x+x^2}{c}$
|
2021-01-18 23:11:40
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5453677177429199, "perplexity": 630.9045118341888}, "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/1610703517159.7/warc/CC-MAIN-20210118220236-20210119010236-00738.warc.gz"}
|
http://ask.enggforum.com/2063/estimate-which-magnitude-earth-quake-construction-withstand
|
+1 vote
11 views
How can we estimate the earth quake resisting strength of a building with out physically breaking by testing?
Category
asked Feb 27 | 11 views
|
2019-08-26 09:25:13
|
{"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": 0.9312824606895447, "perplexity": 6909.899656459946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027331485.43/warc/CC-MAIN-20190826085356-20190826111356-00533.warc.gz"}
|
http://www.minchunzhou.com/travelhistory.html
|
I like travelling. I like flying around world to see different culture and meet different people. Since Aug.28 2010, the date when I enter the U.S, I’ve been to more than 10 countries and taken more than 200 flights. One day I thought it would be pretty cool if I could plot all my travel history using R and make it an animation. The above GIF is what I got. It shows the date of flight, cities, distances, total distance since since Aug 2010, and it moves at a constant speed. Now let me show you how I did it.
Data
First thing first, I need all my travel history. I spent one night digging into my email, and found all my flight reservations and boarding passes. It looks like this if you put them into one file. You can download my data here to replicate all results.
Find the longitude and latitude
Now I need to find the longitude and latitude for all above airports. Package MUCflights contains all airport informations in the world, including city, country, longitude and latitude.
library(doSNOW)
library(MUCflights)
library(ggmap)
library(png)
library(dplyr)
library(geosphere)
library(data.table)
# get all airport information
data(airports)
Data cleaning
To match all my airport code with the IATA code in data airports, I need to seperate the From and To variables in my data Then I need to remove all duplicate cities, since I may arrive at one city and leave a city at different time, but they will be two records in my data.
# seperate data into long format
sepFromTo <- function(x){
result <- data.frame()
result <- data.frame(Date= rep(x[1], 2), IATA = c(x[2], x[3]) )
result
}
data$From <- as.character(data$From)
data$To <- as.character(data$To)
# seperate the From and To variables, make it long format
data_longlat <- apply(data, 1, sepFromTo)
data_longlat <- do.call(rbind,data_longlat)
row.names(data_longlat) <- NULL
# remove duplicate data
data_longlat <- unique(data_longlat)
# using lag function to remove duplicate cities next to each other
data_longlat$nextone <- lag(data_longlat$IATA) == data_longlat$IATA data_longlat$nextone[1] <- FALSE # keep the first city
data_longlat<- data_longlat[ data_longlat$nextone == FALSE , ] # adding ID varialbes since I may take multiple flight on same day. data_longlat$id <- 1:nrow(data_longlat)
# find all airports I've been to in the airport data set
sublat <- airports[ airports$IATA %in% unique(data_longlat$IATA ) ,
c("IATA", "Latitude" , "Longitude", "City") ]
# merge my data with the sublat dataset by airport IATA name
data_longlat <- merge(data_longlat, sublat, by="IATA")
data_longlat <- data_longlat[order(data_longlat$Date),] # order the data by date data_longlat <- data_longlat[ order(data_longlat$id),] # order the data by ID
Plot data on Map
It is very easy to plot data on map once you have the longitude and latitude. Here I use Google map. ggmap can be used to plot the world map or reginal map. geom_point can be used to add points on the map.
# center of your figure
mycenter <- c( 2.359444 , 38.72528 )
# plot the map
p0 <- ggmap(get_googlemap(center = mycenter, zoom=1, maptype= "hybrid",
size = c(640,360)), extent='device')
p1 <- p0 +
geom_point(x = data_longlat$Longitude[5], y = data_longlat$Latitude[5],
shape=21, fill="yellow", size=2) +
geom_point(x = data_longlat$Longitude[6], y = data_longlat$Latitude[6],
shape=21, fill="yellow", size=2)
p1
geom_text can be used to add text on the map. There is a problem. If two cities were too close, their name would overlap. We need to adjust the location of label based on two cities’ relative location. We want the city on the left with lable on the left, and the city on the top with the label on the top.
coord1 <- c(data_longlat$Longitude[6], data_longlat$Latitude[6])
coord2 <- c(data_longlat$Longitude[7], data_longlat$Latitude[7])
# indicator for Longitude
ind_long <- ifelse( coord1[1] < coord2[1], 1, -1)
# indicator for Latitude
ind_lat <- ifelse( coord1[2]< coord2[2] , 1 ,-1)
p2 <- p1 +
geom_text(x= coord1[1] - 15*ind_long ,
y= coord1[2] - 8*ind_lat,
label= data_longlat$City[6] , size=3, col="white") + geom_text(x= coord2[1] + 5*ind_long , y= coord2[2] + 5*ind_lat, label= data_longlat$City[7] ,
size=3, col="white")
p2
geom_segment can be used to add straing lines on the map. However, the flight route is never straight. It’s more like a grate circle. If we just use geom_segment, it will look like this.
# add straight line on the map
p3 <- p0 +
geom_point(x = data_longlat$Longitude[1], y = data_longlat$Latitude[1],
shape=21, fill="yellow", size=2) +
geom_point(x = data_longlat$Longitude[2], y = data_longlat$Latitude[2],
shape=21, fill="yellow", size=2) +
geom_text(x= data_longlat$Longitude[1] - 15*ind_long , y= data_longlat$Latitude[1] - 8*ind_lat,
label= data_longlat$City[1] , size=3, col="white") + geom_text(x= data_longlat$Longitude[2] + 5*ind_long ,
y= data_longlat$Latitude[2] + 5*ind_lat, label= data_longlat$City[2] ,
size=3, col="white")
p4 <- p3 + geom_segment(x = data_longlat$Longitude[1], y = data_longlat$Latitude[1],
xend = data_longlat$Longitude[2], yend = data_longlat$Latitude[2],
col='red', alpha=0.5)
p4
Great Circle
To make an animation like the GIF, we need to do two things. First is to get the great circle between cities, then break the whole route into small intervals. I found some useful information on line, and I got the get_paths function form here. get_paths function will get the great circle route between one point to all points in the data, and break each route into 52 segments. I will just use two points at a time to generate their great circle.
# get_paths function from here https://www.r-bloggers.com/animated-great-circles-on-rotating-3d-earth-example-r-code/.
# get_paths function will get the great circle route between one point to all points in the data. I will just use two points at a time to generate their great circle.
get_paths <- function(x, idx, ...) {
gcInt <- function(x, x1, x2) {
x <- gcIntermediate(x[x1, ], x[x2, ], ...)
if (is.list(x)) {
x <- x %>% purrr::map2(c(x1, x1 + 0.5), ~data.frame(.x, .y)) %>%
bind_rows %>% setnames(c("long", "lat", "group"))
} else x <- data.frame(x, x1) %>% setnames(c("long", "lat", "group"))
x
}
purrr::map(setdiff(1:length(x), idx), ~gcInt(x, .x, idx)) %>% bind_rows
}
allpath <- data.frame()
for ( i in 2: nrow(data_longlat) ){
# We need two point at a time.
test <- data_longlat[ (1:2)+i-2 ,c("Longitude", "Latitude", "Date", "City")]
colnames(test)[1:2] <- c("lon", "lat")
# genderate the spatial points of two cities
p <- SpatialPoints(cbind(test$lon, test$lat), proj4string = CRS("+proj=longlat +datum=WGS84"))
idx1 <- 2 # great circles from coords in all other rows to coords in this row
# get the path between two cities
paths1 <- get_paths(p, idx1, addStartEnd = TRUE)
# other information we may need in the plot
paths1$Date <- test$Date[1] # date
paths1$City <- paste0(test$City[1]," - ",test$City[2]) # Route paths1$start <- test$City[1] # Departure city paths1$end <- test$City[2] # Arrival city paths1$group <- i-1 # the Xth flight
# calculate the distance between two cities
paths1$truedis <- rep(distm(p[1,], p[2,], fun = distHaversine), 52)/1000 allpath <- rbind(allpath, paths1) } Get the number of data point based on the distance. get_paths function will break each distance into 52 intervals. It is a problem if two points are too close, which means a flight with 1,000 miles and a flight with 10,000 will have the same time length in our animation. What I want is the length of time(number of intervals) are proportional to its distance. Hence, we need to get the weight of each flight segment. # calculate the weight based on the distance compare to the longest distance # the longest flight will have weight 1 with 52 intervals. allpath$distance <- allpath$truedis / max(allpath$truedis)
suball <- allpath[ seq(1,nrow(allpath), by=52) ,]
# get the cumulate distance
alldist <- c(0, cumsum(suball$truedis)) allpath_new <- data.frame() for (i in 1:(nrow(data_longlat) -1)){ temp_old <- allpath[allpath$group == i ,] # subset data
mydist <- temp_old$distance[1] # sample data based on the weight, all flight will have the first and last point allunif <- sort(c(1,52, sample( 2:51, round(50*mydist) ))) temp <- temp_old[allunif,] allpath_new <- rbind(allpath_new, temp) } allpath_new$id <- 1:nrow(allpath_new)
allpath_new[,c(1,2,8,9)] <- round(allpath_new[,c(1,2,8,9)],2)
DT::datatable(allpath_new[1:100,])
Make it an animation
Now we are ready the make an animation. The trick to make animation in R is that you get tons of pictures, and compress them into GIF or MP4 files. I parallelize the code on cluseter to make if faster. For more detail, you can see my post about “High performance computing in R using doSNOW package”.
Save multiple PNG file
# Do not run this on your computer!
# You can try a smaller sample size
NumberOfCluster <- 12
cl <- makeCluster(NumberOfCluster) # Make clusters
registerDoSNOW(cl) # use the above cluster
foreach (i=1: (last(allpath_new$group) -1) ) %dopar% { library(MUCflights) library(ggmap) library (png) library(dplyr) library(geosphere) library(data.table) # subset of each flight temp <- temp_old <- allpath_new[allpath_new$group == i ,]
mydist <- temp_old$truedis[1] # center of the plot mycenter <- c( 2.359444 , 38.72528 ) mydate <- temp$Date[1]
mycity <- temp$City[1] # p0 is the same for all loop, you can save p0 and read it instead of read it from Google everytiem p0 <- ggmap(get_googlemap(center = mycenter, zoom=1, maptype = 'satellite', size = c(640,320)), extent='device') # add flight information, date, city, distance p1 <- p0 + geom_text(x=mycenter[1] , y=mycenter[2] - 80 , label=paste0(mydate, "\n", mycity, "\n", round(mydist,2), " KM"), size=3, col="white") + geom_point(x= temp_old$long[1], y= temp_old$lat[1], shape=21, fill="yellow", size=2) + geom_point(x= last(temp_old$long), y= last(temp_old$lat), shape=21, fill="yellow", size=2) # indicater of label ind_long <- ifelse(temp_old$long[1] < last(temp_old$long), 1, -1) ind_lat <- ifelse(temp_old$lat[1] < last(temp_old$lat), 1,-1) p1 <- p1 + geom_text(x= temp_old$long[1] - 5*ind_long , y= temp_old$lat[1]-5*ind_lat , label= temp_old$start[1] , size=3, col="white") +
geom_text(x= last(temp_old$long) + 5*ind_long , y= last(temp_old$lat) + 5*ind_lat,
label= temp_old$end[1] , size=3, col="white") p1 <- p1 + geom_segment(x = temp_old$long[1], y = temp_old$lat[1], xend = last(temp_old$long),
yend = last(temp_old$lat), col='red', alpha=0.5) # loop for all intervals for (j in 1:nrow(temp)){ x <- temp[j , ] if ( j >= 2) { xp <- temp[j-1 , ] df <- data.frame(x1 = x$long, x2 = xp$long[1], y1 = x$lat, y2 = xp$lat[1]) # For flight across the boundaries of map, we don't want to plot this two points if ( abs(x$long-xp$long[1]) < 300 ){ p1 <- p1 + geom_segment(x = x$long, y = x$lat, xend = xp$long[1],
yend = xp$lat[1], col='red', alpha=0.5) } } # adding total distance up to this interval p2 <- p1 + geom_point(data = x,aes(x = long, y = lat), colour = 'red', alpha=0.7) + geom_text(x= 160 , y= 80 , label= paste0("Total Distance: ", round(alldist[i] + (j-1) * mydist/nrow(temp),2), " KM" ) , size=3, col="white") # save png files, png(sprintf(paste0( "myflight_%05d.png"), x$id), width = 640,
height = 320, res = 100, bg="black")
print(p2)
dev.off()
}
}
stopCluster(cl) # close clusters
Convert PNG to GIF/MP4
Finally, once you have all the plot, you can compress it using ffmpeg. More detail about mapmate can be found here.
# https://leonawicz.github.io/mapmate/index.html
p <- "myflight_%05d.png"
out1 <- "flight_50.mp4"
out2 <- "flight_50.gif"
mapmate::ffmpeg(pattern = p, output = out1 , rate = 50)
mapmate::ffmpeg(pattern = p, output = out2 , rate = 50)
|
2018-12-18 19:24:39
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.19782182574272156, "perplexity": 9481.375927501249}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829568.86/warc/CC-MAIN-20181218184418-20181218210418-00543.warc.gz"}
|
https://www.block.arch.ethz.ch/brg/publications/1030
|
# Influence of settlements and geometrical imperfections on the internal stress state of masonry structures
Dell'Endice A., Iannuzzo A., Van Mele T. and Block P.
Proceedings of the SAHC Symposium 2020/21
Barcelona
2021
Since a few decades, the Discrete Element Modelling (DEM) method has been adopted by many authors as a reliable tool for the structural assessment of unreinforced masonry (URM) structures. In this paper, through compas_dem and using 3DEC by Itasca as a solver in the background, we investigate the mechanical behaviour of a three-dimensional URM structure combining the effects of foundation displacements and geometrical imperfections. For this purpose, we consider three different models of the above-mentioned structure. The first one is a perfect digital model, while in the other remaining two models, random geometrical imperfections are applied to the perfect model in order to investigate their influence. After post-processing the 3DEC results, the influence of the applied vertical settlement and geometrical imperfections is explored in terms of crack pattern/mechanism, internal stress states, and the thrust exerted on the supports. The aim of this paper is not to find the actual stress state of the highly indeterminate structure, but to investigate the role played by the combined effects of foundation displacement and geometrical imperfections on the internal stress state.
BibTeX
@inproceedings{Dell'Endice2021,
author = "Dell'Endice, A. and Iannuzzo, A. and Van Mele, T. and Block, P.",
title = "Influence of settlements and geometrical imperfections on the internal stress state of masonry structures",
booktitle = "Proceedings of the SAHC Symposium 2020/21",
year = "2021",
editor = "",
volume = "",
number = "",
pages = "2019-2030",
publisher = "",
month = "September",
doi = "",
note = "",
}
Related publications
Computers & Structures,242: 106372,2021 (January).
Proceedings of the SAHC Symposium 2020/21,: 1882-1892,Barcelona,2021 (September).
ETH Zurich
Institute of Technology in Architecture
Block Research Group
Stefano-Franscini-Platz 1, HIB E 45
8093 Zurich, Switzerland
paulson@arch.ethz.ch
block.arch.ethz.ch
+41 44 633 38 35 phone
+41 44 633 10 53 fax
Copyright © 2009-2022 Block Research Group, ETH Zurich, Switzerland.
|
2022-08-15 03:59:42
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.1939447671175003, "perplexity": 4236.75966929834}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572127.33/warc/CC-MAIN-20220815024523-20220815054523-00255.warc.gz"}
|
http://math.stackexchange.com/questions/680376/proving-separation-from-replacement
|
# Proving Separation from Replacement
I'm trying to show that separation follows from replacement. There are at least two questions here on SE that deal with the question. See The comprehension axioms follows from the replacement schema. and How do the separation axioms follow from the replacement axioms?. I'm not completely satisfied with the answers there.
The basic idea for the proof that most people use seems to be this: Say we have some set $D$ and some formula $\varphi(x)$. To prove separation, we're trying to show that $\{x \in D| \varphi(x)\}$ is a set. Using separation, we need to write a formula $\psi(x,y)$ such that for each $x \in D$ there is exactly one $y$ such that $\psi(x,y)$.
Here's one attempt: $\psi(x,y) = \text{"}x = y \wedge \varphi(x) \text{''}$. IF this formula satisfies the separation axiom, THEN this should work fine. The problem: at least in the version of the separation axiom I know, we need for there to be a unique $y$ for all $x \in D$, such that $\psi(x,y)$. So say $x \in D$ is such that $\neg \varphi(x)$, then $\psi(x,y)$ is always false (and so not true for a unique $y$ as required for the use of replacement).
Here's another attempt: $\psi(x,y) = \text{''}(\{x\}=y \wedge \varphi(x))\vee y=\emptyset \text{''}.$ Now we have a function whose domain is the entire set $D$. And we can use replacement and union to get the set we want. But this requires that both $\emptyset$ and $\{x\}$ are sets. This is the usual argument that they are sets. $D$ is a set, so $\{x \in D| x \not = x\}=\emptyset$ is a set. But that's a straightforward use of separation. We have the same problem for $\{x\}$. Using pairing we get that there is a set $A$ such that $x \in A$, but we don't necessarily have as a result that there is a set that contains only $x$. Of course we could construct this set with $\{y \in A | y = x\}$, but again we've used separation.
It's for the reasons above that I'm not perfectly satisfied with the answers to other questions here on SE. Perhaps I've made some mistake, or there actually is a better answer.
-
You can see in Stanford Encyclopedia of Philosophy the entry on Set Theory .
The Supplement listing the axioms of Zermelo-Fraenkel Set Theory has :
The final axiom of $\mathsf {ZF}$ is the Replacement Schema. Suppose that $\phi(x,y,û)$ is a formula with $x$ and $y$ free, and let $û$ represent the variables $u_1,…,u_k$, which may or may not be free in $\phi$. Furthermore, let $\phi_{x,y,û}[s,r,û]$ be the result of substituting $s$ and $r$ for $x$ and $y$, respectively, in $\phi(x,y,û)$. Then every instance of the following schema is an axiom:
Replacement Schema:
$\forall u_1 …\forall u_k [\forall x \exists !y \phi(x,y,û) \rightarrow \forall w \exists v \forall r (r \in v \equiv \exists s(s \in w \land \phi_{x,y,û}[s,r,û]))]$
In other words, if we know that $\phi$ is a functional formula (which relates each set $x$ to a unique set $y$), then if we are given a set $w$, we can form a new set $v$ as follows: collect all of the sets to which the members of $w$ are uniquely related by $\phi$.
Note that the Replacement Schema can take you ‘out of’ the set $w$ when forming the set $v$. The elements of $v$ need not be elements of $w$. By contrast, the well-known Separation Schema of Zermelo yields new sets consisting only of those elements of a given set $w$ which satisfy a certain condition $\psi$. That is, suppose that $\psi(x,û)$ has $x$ free and may or may not have $u_1, …,u_k$ free. And let $\psi_{x,û}[r,û]$ be the result of substituting $r$ for $x$ in $\psi(x,û)$. Then the Separation Schema asserts:
Separation Schema:
$\forall u_1 …\forall u_k[\forall w \exists v \forall r(r \in v \equiv r \in w \land \psi_{x,û}[r,û])]$
In other words, if given a formula $\psi$ and a set $w$, there exists a set $v$ which has as members precisely the members of $w$ which satisfy the formula $\psi$.
A formal proof of the relation between the two is in Gaisi Takeuti & Wilson Zaring, Introduction to Axiomatic set theory (1971), page 17 (we omit the initial string of $\forall$s) :
Axiom Schema of Replacement
$\forall a [\forall u \forall v \forall w [\phi(u, v) \land \phi(u, w) \rightarrow v = w] \rightarrow \exists b \forall x [x \in b \equiv \exists u [u \in a \land \phi(u, x)]]]$.
Note. The condition of functionality for $\phi$ has been unwinded.
Note also that $a$ is not mentioned in the antecedent, so you can rewrite the formula as : $\forall u \forall v \forall w [...] \rightarrow \forall a\exists b \forall x [x \in b \equiv \exists u [u \in a \land \phi(u, x)]]$. In this way, the consequent has clearly the "form" of Separation.
Zermelo's Schema of Separation
$\forall a \exists b \forall x [x \in b \equiv x \in a \land \phi(x)]$.
Applying Replacement to the wff $\phi(u) \land u = v$ where $v$ does not occur in $\phi(u)$ we have that $[\phi(u) \land u=v] \land [\phi(u) \land u=w] \rightarrow v = w$.
Note. After an "instantiation" of the Replacement schema, the universal closure of the above formula give us its antecedent ; so we can "detach" the consequent.
Therefore
$\exists b \forall x [x \in b \equiv \exists u [\phi(u) \land u = x \land x \in a]]$
i.e. [Note: here we have several steps in one; we need the equality axiom $\vdash u = x \rightarrow (\phi(u) \rightarrow \phi(x))$, "rearranged" as $\vdash (\phi(u) \land u = x ) \rightarrow \phi(x)$ and a final application of quantifier rules to get : $\vdash \exists u(\phi(u) \land u = x ) \rightarrow \phi(x)$.]
$\exists b \forall x [x \in b \equiv \phi(x) \land x \in a]$.
Note. The final step is to introduce the universal quantifier $\forall a$, by "generalization".
-
In Kenneth Kunen's The Foundations of Mathematics, the Replacement Scheme is listed as $\forall x \in A \exists ! y \varphi(x,y) \rightarrow \exists B \forall x \in A \exists y \in B \varphi(x,y)$ (with $A$ universally quantified implicitly). I'm guessing that this is probably equivalent to the version used in Gaisi Takeuti & Wilson Zaring, Introduction to Axiomatic set theory. But it makes the desired proof slightly more difficult. There is more constraint on $\varphi$, so that in addition to being functional, its domain must be the entire set $A$. – objectivesea Feb 22 '14 at 5:35
@objectivesea - but the SEP's version has : $\forall x \exists!y \phi (x,y) ...$ where the initial $x$ is not restricted: so his domain is all the "universe". The "functional expression" $\phi$ is defined for all $x$ and $y$, like all formulas of f-o logic ... – Mauro ALLEGRANZA Feb 22 '14 at 21:04
|
2015-05-30 20:46:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9425698518753052, "perplexity": 148.2603954230099}, "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/1432207932705.91/warc/CC-MAIN-20150521113212-00184-ip-10-180-206-219.ec2.internal.warc.gz"}
|
http://www.scholarpedia.org/article/Microwave_billiards_and_quantum_chaos
|
# Microwave billiards and quantum chaos
Post-publication activity
Curator: Hans-Jürgen Stöckmann
Up to about 1990 the quantum mechanics of classically chaotic systems, shortly termed "quantum chaos", was essentially a domain of theory (Haake 2001). Only two classes of experimental result had been available at that time. First, there were the spectra of compound nuclei giving rise to the development of random matrix theory in the sixties of the last century, and second the experiments with highly excited hydrogen and alkali atoms in strong magnetic or strong radio frequency fields. The situation changed with the appearance of experiments using classical waves, starting with microwave billiards . The distinction between classical waves and matter waves is not of relevance in the present context, since all features touched in this article are common to all types of waves. This is why some authors prefer the term "wave chaos" to describe this field of research.
## From classical to quantum mechanics
Figure 1: Classical trajectories in a circular(a) and a stadium (b) billiard.
Billiards are particularly well suited to illustrate the difficulties one is facing with the concept of chaos in quantum mechanics. For a circular billiard the trajectory is regular ( Figure 1(a)). There are two constants of motion, the total energy $$E\ ,$$ and the angular momentum $$L\ .$$ Since there are two degrees of freedom as well, the system is integrable, and the distance between two nearby trajectories increases linearly with time. The situation is qualitatively different for the stadium billiard ( Figure 1(b)). There is only one constant of motion left, the total energy $$E\ ,$$ and the distance between neighbouring trajectories increases exponentially with time. The stadium billiard thus is chaotic.
In quantum mechanics this distinction between integrable and chaotic systems does not work any longer. The initial conditions are defined only within the limits of the uncertainty relation $\Delta x\,\Delta p\ge \frac{1}{2}\hbar\,,$ and the concept of trajectories looses its significance. One may even ask whether quantum chaos does exist at all. Since the Schrödinger equation is linear, a quantum mechanical wave packet can be constructed from the eigenfunctions by the superposition principle. There is no room left for chaos. On the other hand the correspondence principle demands that there must be a relation between linear quantum mechanics and nonlinear classical mechanics at least in the regime of large quantum numbers. This defines the program of quantum chaos research, namely to look for the fingerprints of classical chaos in the quantum mechanical properties of the system.
Billiards are ideally suited systems for this purpose. The numerical calculation of the classical trajectories is elementary, and the stationary Schrödinger equation reduces to a simple wave equation
$-\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \right)\psi_n=E_n\psi_n\,.$
The potential appears only in the boundary condition, $$\left.\psi_n\right|_S=0\ ,$$ where $$S$$ is the surface of the billiard. In the absence of potentials the stationary Schrödinger equation is equivalent to the time-independent wave equation, the Helmholtz equation
$\tag{1} -\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \right)\psi_n=k_n^2\psi_n\,,$
where $$\psi_n$$ now means the amplitude of the wave field.
This opens the opportunity to study questions and to test theories, originally initiated by quantum mechanics, by means of classical waves. The boundary conditions for the classical and the corresponding quantum mechanical systems may differ, but this is not of relevance for the questions to be treated in this article.
## Microwave billiards
Figure 2: Chladni figures for plates with various shapes mounted in the centre (Stöckmann 1992, Nat.Wiss. 79: 443).
The first experiment of this type dates back already more then 200 years. At the end of the 18th century E. Chladni developed a technique "to make sound visible" by decorating the nodal lines of vibrating plates with grains of sand (Chladni 1802). Figure 2 shows Chladni figures for three typical situations. The plates are fixed in the centre and had been excited to vibrations by means of a loudspeaker. Figure 2(a) shows a typical pattern for a circular plate. In this case the integrability of the system is not perturbed by the mounting. One observes a regular pattern of nodal lines with many intersections. The next example in Figure 2(b) shows a rectangle. It is integrable, too, but now the integrability is slightly perturbed by the mounting resulting in a curvature of the nodal lines and a partial conversion of crossings into anti-crossings. The last example in Figure 2(c) belongs to the class of Sinai billiards, a rectangle with an excised quarter circle from one of the corners. Now all nodal line crossings have completely disappeared resulting in a meandering pattern of nodal lines. Two centuries after Chladni's discovery the study of nodal lines of chaotic plates has become a very active field of research again.
Figure 3: Microwave set-up to study spectra and wave functions (top), and a typical microwave reflection spectrum for a quarter-stadium (b= 20 cm, l=36 cm) shaped resonator (bottom) (Stöckmann and Stein 1990, Phys. Rev. Lett. 64: 2215).
First modern experimental billiard studies started with microwave resonators. Meanwhile the technique is used by several groups worldwide (see Section Further reading for more details).Figure 3(top) shows a typical set-up. The cavity is formed by a bottom plate supporting the entrance antenna, and by an upper part whose position can be moved with respect to the lower one. As long as a maximum frequency $$\nu_{\rm max} = c/2d$$ is not exceeded, where $$d$$ is the height of the resonator and $$c$$ the velocity of light, the system can be considered as quasi-two-dimensional. In this situation the electro-magnetic wave equations reduce to the scalar Helmholtz equation (1), where $$\psi_n$$ corresponds to the electric field pointing perpendicularly from the bottom to the top plate. Since the electric field component parallel to the wall must vanish, we have the condition $$\psi_n|_S = 0$$ on the outer circumference $$S$$ of the resonator. We have thus arrived at a complete equivalence between a two-dimensional quantum billiard and the corresponding quasi-two-dimensional microwave resonator, including the boundary conditions. As an example Figure 3(bottom) shows the reflection spectrum of a microwave resonator of the shape of a quarter stadium. Each minimum in the reflection corresponds to an eigenfrequency of the resonator, and the depth of the resonance corresponds to the modulus square $$|\psi_n(r)|^2$$ of the wave function at the antenna position.
Figure 4: Wave functions in a stadium-shaped microwave resonator (Stein et al. 1992, Phys. Rev. Lett. 68: 2867). The figure shows $$\left|\psi_n(\vec{r})\right|^2$$ in a colour plot.
By scanning with the antenna through the billiard $$|\psi_n(\vec{r})|^2$$ may thus be spatially resolved. To get the sign as well, a transmission measurement to an additional fixed antenna has to be performed. Figure 1 shows a number of stadium wave functions obtained in this way. All wave functions show the phenomenon of scarring, meaning that the wave function amplitudes are not distributed more or less homogeneously over the area, but concentrate along classical periodic orbits. One could get the impression that scarred wave functions are dominating, but this is only true for the lowest eigenvalues. With increasing energy the fraction of scarred wave functions tends to zero.
For a quantitative description of the experiments scattering theory has to be applied, developed half a century ago in nuclear physics. Compared to nuclei microwave billiards have a number of advantages: wave lengths are of the order of mm to cm, resulting in very convenient sizes for the used resonators, and all relevant parameters can be perfectly controlled. This is why a number of predictions of scattering theory have been tested not in nuclei but microwave billiards (Mitchell et al. 2010).
## Random matrices
In the midst of the last century little was known on the origin of the nuclear forces. Here one idea showed up to be extremely successful, notwithstanding its obviously oversimplifying nature: If the details of the nuclear Hamiltonian $$H$$ are not known, just let us take its matrix elements in some basis as random numbers, with only some global constraints, e. g. by taking the matrix $$H$$ symmetric for systems with, or non-symmetric Hermitian for systems without time-reversal symmetry, and by fixing the variance of the matrix elements. Assuming basis invariance the matrix elements can be shown to be uncorrelated and Gaussian distributed (Mehta 1991). The classical Gaussian ensembles are the orthogonal one (GOE) for time-reversal invariant systems with integer spin, the unitary one (GUE) for systems with broken time-reversal symmetry, and the symplectic one (GSE) for time-reversal invariant systems with half-integer spin. Here "orthogonal" etc. refers to the invariance properties of the respective ensembles.
Figure 5: Level spacing distribution for a Sinai billiard (a), a hydrogen atom in a strong magnetic field (b), the excitation spectrum of a NO2\$ molecule (c), the acoustic resonance spectrum of a Sinai-shaped quartz block (d), the microwave spectrum of a three-dimensional chaotic cavity (e), and the vibration spectrum of a quarter-stadium shaped plate (f) (taken from Stöckmann 1999). In all cases a Wigner distribution is found though only in the first three cases the spectra are quantum mechanically in origin.
The quantity most often studied in this context is the distribution of level spacings $$p(s)$$ normalised to a mean level spacing of one. For $$2\times 2$$ matrices this quantity can be easily calculated, yielding for the GOE the famous Wigner surmise
$$\tag{2} p(s)=\frac{\pi}{2}s\exp\left(-\frac{\pi}{4}s^2\right)\,.$$
For large matrices Eq. (2) is still a good approximation with errors on the percent level (Haake 2001). Figure 5 shows level spacings distributions for a variety of chaotic systems, all exhibiting the same behaviour. Such observations had been the motivation for the famous conjecture by Bohigas, Giannoni and Schmitt (1984) that the spectra of completely chaotic time-reversal-invariant systems should show the same fluctuation properties as the GOE.
The replacement of $$H$$ by a random matrix means to abandon any hope to learn more about nuclei from the spectra but some average quantities such as the mean level spacings. But the loss of individual features in the spectra on the other hand suggests that it might be worthwhile to look for universal features being common to all chaotic systems. This approach showed up to be extremely fruitful. It allowed to apply results originally obtained for nuclei to many other systems, e. g. quantum-dot systems (Beenakker 1997) and microwave billiards (Stöckmann 1999).
Figure 6: Spectral form factor for the spectrum of a microwave hyperbola billiard (top), and for the subspectrum obtained by considering only every second resonance (bottom) (Alt et al. 1997, Phys. Rev E 55: 6674).
In addition to the level spacings distribution in particular spectral correlations related to the spectral auto-correlation function $$C(E)=\langle \rho(E_2)\rho(E_1)\rangle-\langle \rho(E_2)\rangle\langle\rho(E_1)\rangle$$ are considered, where $$\rho(E)$$ is the density of states, $$E=E_2-E_1\ ,$$ and the brackets denote a spectral average. Quantities often studied in literature are number variance and spectral rigidity (Stöckmann 1999). Here another object shall be considered, the spectral form factor, which is obtained from the Fourier transform of the spectral auto-correlation function. Figure 6 shows an experimental illustration for a hyperbola microwave billiard. In the upper part of the figure the spectral form factor for the complete spectrum is shown. There is a good agreement with random matrix predictions from the GOE. This is consistent with the fact that microwave billiard systems are time-reversal invariant, and there is no spin. Spectra showing GSE statistics have not yet been studied experimentally, but there is the remarkable fact that GSE spectra can be generated by taking only every second level of a GOE spectrum (Mehta 1991). Exactly this had been done with the spectrum of the hyperbola billiard to obtain the spectral form factor in the lower part of the figure, being in perfect agreement with the expected GSE behaviour.
## Semiclassical quantum mechanics
Before the final establishment of quantum mechanics Born and Sommerfeld developed a technique today known as semi-classical to calculate the spectrum of atomic hydrogen. At that time Einstein argued that this approach must be a dead end, since semi-classical quantisation needs invariant tori in phase space, preventing a semi-classical quantisation for non-integrable systems. This was one of the rare cases, where Einstein was wrong, though it needed half a century until Gutzwiller (1990) showed in a series of papers that chaotic systems, too, allow for a semi-classical quantisation.
Figure 7: Squared modulus $$|\hat{\rho}(t)|^2$$ of the Fourier transform of the spectrum of a quarter Sinai billiard (a=56 cm, b=20 cm, r=7 cm). Each resonance can be associated with a classical periodic orbit~(Stöckmann and Stein 1990, Phys. Rev. Lett. 64, 2215).
For the density of states Gutzwiller's approach yields his famous trace formula. It becomes particularly simple in billiard systems, if the wavenumber $$k$$ is taken as the variable. In terms of $$k$$ the density of states reads
$$\tag{3} \rho(k)=\rho_0(k)+\sum\limits_n A_ne^{\imath kl_n}\,.$$
The first term varies smoothly with $$k$$ and is given in its leading order by Weyl's formula
$$\rho_0(k)=\frac{A}{2\pi}k\,,$$
where $$A$$ is the area of the billiard. The second term is heavily oscillating with $$k\ .$$ The sum runs over all periodic orbits including repetitions. $$l_n$$ is the length of the orbit, and $$A_n$$ is a complex factor weighting the stability of the orbit.
The periodic orbit sum (3) is divergent for real $$k\ ,$$ and resummation techniques are needed to calculate the spectrum from the periodic orbits. But the inverse procedure, namely to extract the contributions of the different periodic orbits out of the spectra, is straightforward. For billiards the Fourier transform of the fluctuating part of the density of states,
$$\tag{4} \hat{\rho}_\mathrm{osc}(l)=\int\rho_\mathrm{osc}(k)e^{-\imath kl}\,dk=\sum_n A_n\delta(l-l_n)\,,$$
directly yields the contributions of the orbits to the spectrum. Each orbit gives rise to a delta peak at an $$l$$ value corresponding to its length, and a weight corresponding to the stability of the orbit.
For illustration Figure 5 shows the squared modulus of the Fourier transform of the spectrum of a microwave resonator shaped as a quarter Sinai billiard. Each peak corresponds to a periodic orbit of the billiard. For the bouncing ball orbit, labelled by "1", three peaks associated with repeated orbits are clearly visible. The smooth part of the density of states is responsible for the increase of $$|\hat{\rho}(k)|^2$$ for small lengths.
Semiclassical quantum mechanics relates the spectrum to the classical periodic orbits of the system, i. e. to individual system properties. In view of this fact one may wonder where the universal features discussed in Section 3 come in. Let as have again a look on Eq. (4) to answer this question. For short orbits $$\hat{\rho}(l)$$ exhibits a well-resolved length spectrum. This is the individual regime. But with increasing length the peaks become denser and denser, until they eventually cannot be resolved any longer. This is the universal regime. It needed more then 25 years of research to prove the equivalence between random matrix theory and semiclassical quantum mechanics in the universal regime explicitly.
## Applications
Occasionally people doubt whether chaotic systems really need an extra quantum-mechanical treatment. The Schrödinger equation after all gives exact results both for regular and chaotic systems. Why should one resort to old-fashioned techniques which had been abandoned already 80 years ago, after the development of "correct" quantum mechanics had been completed?
The answer is simple: the numerical solution of the Schrödinger equation means a black-box calculation, and the human brain is not adopted to perform Fourier transforms. This is why spectra as the one shown in Figure 3 seemingly do not contain any relevant information. But the brain is extremely good in identifying paths and trajectories, and therefore the representation of the spectra in terms of classical trajectories, as shown in Figure 5 allows an immediate suggestive interpretation.
Figure 8: Snapshot of the pulse propagation in a dielectric quadrupole cavity made of teflon (length of the long axis l=113 mm). The upper figure shows the pulse intensity inside the teflon at the moment of strongest emission in a colour plot. In addition the Poynting vector is shown in the region outside of the teflon. The lower figure shows the Husimi distribution of the pulse in a Poincarè plot. In addition the unstable manifold of the rectangular orbit is shown. See Media:movie.gif for the complete sequence (Schäfer et al. 2006, New J. of Physics 8, 46).
All this is not just l'art pour l'art, as shall be demonstrated by one example.
The relation between wave propagation and classical trajectories had become of practical importance for the optimization of the emission behaviour of microlasers. Again this shall be demonstrated by a microwave study. The upper part of Figure 8 shows the snapshot of the pulse propagation in a dielectric quadrupole resonator made of teflon. The pulse starts as an outgoing circular wave from an antenna close to the boundary in the lower part of the cavity, but already after a short time only two pulses survive circulating clock- and counter-clockwise close to the border. The figure shows a moment where there is a particularly strong emission to the outside. Contrary to intuition the strongest emission does not occur a the point of largest curvature.
In the lower part of the figure the same situation is shown in a Poincarè plot, with the polar angle as the abscissa, and the sine of the incidence angle as the ordinate (in a Poincarè plot each trajectory is mapped onto a sequence of points representing the reflections at the boundary). The tongue-like structure is the instable manifold of the rectangular orbit. It had been obtained by pursuing a trajectory starting with a minute deviation from the ideal orbit. In addition the Husimi representation of the pulse is shown (a Husimi representation is a convenient tool to embed wave functions into the classical phase space).
Now the observed emission behaviour can be explained. Teflon has an index of refraction of n=1.44 meaning a $$\sin\chi_{\rm crit}=0.69$$ for the critical angle of total reflection. Thus the circulating pulses are trapped by total reflection. But whenever the critical line of total reflection is surpassed, there is a strong escape. This happens exactly in the region of the most pronounced tongues of the instable manifold of the rectangular orbit. Meanwhile "phase-space engineering" has become a standard tool in shape optimization of microcavities (Kwon et al. 2010).
The number of activities on the transport of different types of waves, light, seismic waves, water waves, sound waves etc. through disordered media is steadily increasing, many of them based on theories and techniques originally developed in wave and quantum chaos. After several decades of basic research the time for applications has come.
Most experimental examples presented in this article have been obtained in the author's group at the university of Marburg. I want to thank all my coworkers, in particular my senior coworker U. Kuhl. The experiments had been funded by the Deutsche Forschungsgemeinschaft by numerous grants, amongst others via the research group 760 Scattering systems with complex dynamics.
## References
• Beenakker, C. W. J. (1997). Rev. Mod. Phys. 69: 731.
• Bohigas, O.; Giannoni, M. J. and Schmit, C. (1984). Phys. Rev. Lett. 52: 1.
• Chladni, E. F. F. (1802). Die Akustik. Breitkopf und Härtel, Leipzig.
• Gutzwiller, M. C. (1990). Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics, Vol. 1. Springer, New York.
• Haake, F. (2001). Quantum Signatures of Chaos, 2nd edition. Springer, Berlin.
• Mehta, M. L. (1991). Random Matrices. 2nd edition. Academic Press, San Diego.
• Mitchell, G. E.; Richter, A. and Weidenmüller, H. A. (2010). Random Matrices and Chaos in Nuclear Physics: Nuclear reactions. Rev. Mod. Phys. 82: 2845.
• Kwon, O.; An, K. and Lee, B. (Eds.) (2010). Trends in Nano- and Micro-Cavities. Sharjah, U.A.E.: Bentham Science Pub.
• Stöckmann, H.-J. (1999). Quantum Chaos - An Introduction. University Press, Cambridge.
• Heller, E. J. (1984). Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits. Phys. Rev. Lett. 53, 1515. In this paper the term "scar" had been introduced, and the phenomenon had been described for the first time.
• Kuhl, U.; Stöckmann, H.-J. and Weaver, R. (2005). Classical wave experiments on chaotic scattering. J. Phys. A 38: 10433. A discussion of the scattering aspects of wave chaotic systems.
• Nöckel, J. U. and Stone, A. D. (1997). Ray and wave chaos in asymmetric resonant cavities. Nature 385, 45. Here the relevance of the classical phase-space properties for the emission behaviour of microcavities had been pointed out for the first time.
• Richter, A. (1999). Playing billiards with microwaves - quantum manifestations of classical chaos. In: Hejhal et al.: Emerging Applications of Number Theory. The IMA Volumes im Mathematics and its Applications, Vol 109. Springer, New York. A report on the microwave experiments of the Darmstadt group.
• Sirko, L.; Koch, P.M. and Blümel, R. (1997). Experimental identification of non-Newtonian orbits produced by ray splitting in a dielectric-loaded microwave cavity. Phys. Rev. Lett. 78: 2940. Microwave results on a generalization of Gutzwiller’s trace formula to ray splitting occurring in systems with sharp interfaces.
• Stöckmann, H.-J. (2007). Chladni meets Napoleon. Eur. Phys. J. Special Topics 145: 17. A report on Chladni's Paris visit 1809.
In addition to the groups in Marburg and Darmstadt represented by experimental results in the article, there are a number of other microwave laboratories listed here with representative publications. Only groups where there are at least two publications are mentioned:
• Kudrolli, A.; Kidambi, V. and Sridhar, S. (1995). Experimental studies of chaos and localization in quantum wave functions. Phys. Rev. Lett. 75: 822. The Boston group.
• So, P.; Anlage, S. M.; Ott, E. and Oerter, R. N. (1995). Wave chaos experiments with and without time reversal symmetry: GUE and GOE statistics. Phys. Rev. Lett. 74: 2662. The Maryland group
• Hul, O.; Tymoshchuk, O.; Bauch, S.; Koch, P. and Sirko, L. (2005). Experimental investigation of Wigner's reaction matrix for irregular graphs with absorption. J. Phys. A 38: 10489. The Warsaw group.
• Barthèlemy, J.; Legrand, O. and Mortessagne, F. (2005). Complete S-matrix in a microwave cavity at room temperature. Europhys. Lett. 70:162. The Nice group.
Internal references
|
2018-02-21 23:55:59
|
{"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": 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": 0.7166150808334351, "perplexity": 786.1508306181946}, "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-2018-09/segments/1518891813818.15/warc/CC-MAIN-20180221222354-20180222002354-00268.warc.gz"}
|
http://na-skarpie.com/we-happy-bowzmns/839649-silicon-band-gap
|
# silicon band gap
### silicon band gap
Because of quantum confinement, small-diameter wires exhibit a direct band gap that increases as the wire diameter narrows, irrespective of surface termination. When using squeezed vacuum states of laser light they showed that the quantum noise drops below the standard quantum limit, which demonstrates quantum correlations between the laser beam and the mirrors. Nie also notes that it should be possible to fabricate large-scale detector arrays at far lower cost than for semiconductor detectors. closer to the conduction band. To Wanyi Nie and colleagues at Los Alamos National Laboratory for using thin-film perovskites to create an extremely sensitive X-ray detector. In intrinsic silicon, the Fermi level lies in the middle of the gap. Suppose an electron in the conduction band combines with a hole in the valence band and the excess energy is released in the form of electromagnetic radiation. Transparent silicon carbide and diamond surfaces are also ideal for atomic-scale opto-electronic experiments. This way semiconductor can act as an insulator and a conductor also. To Andrea Alù, Qiaoliang Bao, Cheng-Wei Qiu and an international team of collaborators at the City University of New York, National University of Singapore, Monash University, China University of Geosciences and the University of Texas at Austin, for showing that dispersion- and diffraction-free propagation of light is possible, with a resolution that beats the diffraction limit by more than an order of magnitude, in twisted layers of 2D molybdenum trioxide. In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. silicon band gap, the resulting material has a large background-free carrier concentration at room-temperature25,26. The idea is to use a beam containing both carbon ions, which provide therapeutic irradiation of the target tumour, and helium ions, which travel straight through the patient and can therefore be used for imaging. The website forms part of the Physics World portfolio, a collection of online, digital and print information services for the global scientific community. (a) For what range of wave-lengths will silicon be transparent? The plot is drawn for energy values along particular edges of the irreducible wedge, cf. In particular, the much higher breakdown field strength and thermal conductivity of SiC allow creating devices which outperform by far the corresponding Si ones. Energy band diagram of a silicon shows the levels of energies of electrons in the material. To Noel Clark and colleagues at the University of Colorado Boulder and the University of Utah in the US, for observing a ferroelectric nematic phase of matter in liquid crystals more than 100 years after it was predicted to exist. Generally, the indirect band gap is often a challenge for silicon photonics. The band gap energy is important for various kinds of photonic devices. However, in all cases the band gaps are smaller than those observed for P. Silicon is the least effective second row atom to open a band gap in graphene, inducing very few changes on its band structure, probably because it has the same number of valence electrons, and thus the occupation of the bands may not be changed. Ultimately, he hopes that the material could be used to fabricate lasers on a chip that can create optical signals. The optical band gaps for the SiN x films with different refractive indexes are shown in Fig. Indeed, the breakthrough could lead to a new world of opportunities for silicon devices. (1958) and Weber & Alonso (1989) Silicon is the most popular material used in electronic devices. Several methods for the experimental determination of the band gap in silicon and germanium have been discussed.1,4,5 We propose another method, which is based on the appli-cation of diodes as thermometers, for the determination of The band gap between the valence and the conduction bands in zinc oxide (ZnO) is 3.2 eV. Band gap modification for small-diameter (∼1 nm) silicon nanowires resulting from the use of different species for surface termination is investigated by density functional theory calculations. The Physics World 2020 Breakthrough of the Year goes to Elham Fadaly, Alain Dijkstra and Erik Bakkers at Eindhoven University of Technology in the Netherlands, Jens Renè … Silicon carbide (SiC) has a wide bandgap of 3 electronvolt (eV) and a much higher thermal conductivity compared to silicon. To do so the team had to first painstakingly minimize the effects of background radiation in the Borexino detector, which comprises 278 tonnes of ultrapure liquid scintillator located deep inside Italy’s Gran Sasso mountain. We were discouraged, however, by the performance of the iron-constantan thermocouple. Please enter the e-mail address you used to register to reset your password, Thank you for registering with Physics World 5. This also suggests that the band gap can be tuned by mixing the relative populations of different terminating groups on the wire, e.g. Discover the latest engineering research from IOP Publishing journals and ebooks. The band gap of Si 20-II was 1.237 eV, 0.163 eV lower than that of Si 14. Other components that are needed to convert optical signals into electronic signals and vice versa could also be made, including optical amplifiers and detectors. A large band gap will make it more difficult for a carrier to be thermally excited across the band gap, and therefore the intrinsic carrier concentration is lower in higher band gap materials. In Si To Markus Hennrich and colleagues at Stockholm University, Sweden, together with researchers at the universities of Siegen in Germany and the Basque Country and Seville in Spain, for using a series of “weak” measurements (the subject of Physics World’s 2011 Breakthrough of the Year) to probe the nature of superposition collapse in quantum mechanics. The researchers irradiated the solution with pulsed 1140-nm light filtered through a 200-micron-thick silicon wafer, which successfully resulted in an upconverted emission at 700 nm. The plot is drawn for energy values along particular edges of the irreducible wedge, cf. Silicon is an indirect material, so phonon-assisted optical absorption does matter! Physics World represents a key part of IOP Publishing's mission to communicate world-class research and innovation to the widest possible audience. The size of the band gap has implications for the types of applications that can be made. [1][2][3], If high precision is not required it is enough to bias a diode with any constant low current and use its −2 mV/˚C thermal coefficient for temperature calculation, however this requires calibration for each diode type. Because, silicon is … 6.3 Silicon Band Structure Models Semiconductor band structures in general and especially for silicon as shown in Figure 6.4 are hard to describe with an analytical formula. Example 2.2 Calculate the energy bandgap of germanium, silicon and gallium arsenide at 300, 400, 500 and 600 K. Solution The bandgap of silicon at 300 K equals: 1.12 eV 300 636 0.473 10 (300) 1.166 (300 K) (0 K) 32 2 b a T T E g E The silicon bandgap temperature sensor is an extremely common form of temperature sensor (thermometer) used in electronic equipment. From electric vehicles and charging stations to solar power to industrial power supplies, wide band gap brings efficiency, improved thermal performance, size reduction, and more. In amorphous semiconductors (such as a-Silicon), optical band gap can be estimated from UV-Vis-NIR spectroscopy measurements. current/junction area, and a similar output voltage can be obtained by operating the two junctions at the same current, if one is of a different area to the other. The optical bandgap is the threshold for photons to be absorbed, while the transport gap is the threshold for creating an electron–hole pair that i… As a result, silicon must be integrated with other direct-band-gap semiconductor materials to create the optoelectronic devices that supply the pulses of light that drive information on the Internet. The other name for the bandgap is the forbidden gap because electrons cannot exist in it, meaning that they are either in the conduction or valence band. In addition to having been reported in Physics World in 2020, our selections must meet the following criteria: Here are the nine runners-up that make up the rest of the Physics World Top 10 Breakthroughs for 2020. Because of quantum confinement, small-diameter wires exhibit a direct band gap that increases as the wire diameter narrows, irrespective of surface termination. A circuit that forces IC1 and IC2 to have a fixed N:1 ratio,[1] Some examples: The emission wavelengths of light emitting diodes and laser diodes are largely determined by the band gap energy. The band gap energy E gin silicon was found by exploiting the linear relationship between the temperature and voltage for the constant current in the temperature range of 275 K to 333 K. Within the precision of our experiment, the results obtained are in good agreement with the known value energy gap in silicon. However, the use of these wide band-gap semiconductor surfaces for molecular nanosciences poses a number of problems such as surface preparation and atomic-scale characterisation, efficient doping, conductivity and adsorption of molecules. Find the maximum wavelength that can be … Botti’s group studied alloy models using a cluster expansion approach, relying on density functional theory calculations. Because weak measurements could in principle allow errors to be detected in quantum states without destroying those states in the process, the work might be used to improve error correction in quantum computers. 2.2.5 Temperature dependence of the energy bandgap The energy bandgap of semiconductors tends to decrease as the temperature is increased. In this range, electrons can be freed without creating extra heat. band gap structure of semiconductors is also important be-cause it is directly related to its electrical properties. CEM Lectures 39,799 views. So, we can give sufficient energy to it, to jump the electron to the conduction band from valance band. Normally, silicon has an indirect electronic band gap, which means that it does not emit light. While the act of measurement usually forces quantum systems into definite classical states, the work of Hennrich and colleagues showed that some measurements don’t destroy all quantum information. diodes), operated at different current densities, is proportional to absolute temperature (PTAT). Abinit Silicon Band Gap 1 Eric Lin. But, Silicon’s valence electron don’t go in the conduction band that easily. In this phase, all the molecules within specific patches, or domains, of the liquid crystal point in roughly the same direction – a phenomenon known as polar ordering that was first hypothesized by Peter Debye and Max Born back in the 1910s. Practical devices based on superconductors must be chilled to very cold temperatures, which is costly and can involve the use of helium, so a long-standing goal of condensed-matter physicists has been to develop a material that is a superconductor at room temperature. While a pressure of 2.6 million atmospheres was required to achieve room-temperature superconductivity, the researchers think it may be possible to reduce the pressure by changing the chemistry of the material. To create a direct band gap, Bakkers and colleagues had to find a way of growing crystals of silicon-germanium alloy with a hexagonal crystal structure, rather than the usual diamond-like structure. generate a stable voltage that is ideally independent of changes in temperature and other external factors To Haocun Yu and Lee McCuller of the Massachusetts Institute of Technology and their colleagues on the LIGO Scientific Collaboration for showing that quantum-scale correlations can leave their mark on macroscopic objects weighing tens of kilograms. A band gap is the distance between the valence band of electrons and the conduction band. Superconductors carry electrical current with no electrical resistance and have a range of applications from the high-field magnets used in MRI scanners to particle accelerators. Wide band gap materials such as silicon carbide are revolutionizing the power industry. In that tutorial the band structure of silicon is calculated based on the Kohn-Sham eigenvalues obtained from a DFT calculation. Clark and colleagues found that when they applied a weak electric field to an organic molecule known as RM734, a striking palette of colours developed towards the edges of the cell containing the liquid crystal. The crossover from indirect to direct band gap … Crystalline silicon has a band gap energy of 1.1 electron-volts (eV). Electrons with high energy are part of the conduction band, and those with low energy are in the valence band. Suppose an electron in the conduction band combines with a hole in the valence band and the excess energy is released in the form of electromagnetic radiation. This method is common in monolithic temperature sensors. The Physics World 2020 Breakthrough of the Year goes to Elham Fadaly, Alain Dijkstra and Erik Bakkers at Eindhoven University of Technology in the Netherlands, Jens Renè Suckert at Friedrich-Schiller-Universität Jena in Germany and an international team for creating a silicon-based material with a direct band gap that emits light at wavelengths used for optical telecommunications. The band gap of a semiconductor is the minimum energy required to excite an electron that is stuck in its bound state into a free state where it can participate in conduction. Its main advantage is that it can be included in a silicon integrated circuit at very low cost. The band gap between the valence and the conduction bands in zinc oxide (ZnO) is 3.2 eV. The band structure of crystalline silicon accommodates both direct and indirect excitations of electrons across the band gap . Find the maximum wavelength that can be … Copyright © 2020 by IOP Publishing Ltd and individual contributors, Explore all the latest news and information on Physics World, Keep track of the most exciting research breakthroughs and technology innovations, Stay informed about the latest developments that affect scientists in all parts of the world, Take a deeper look at the emerging trends and key issues within the global scientific community, Discover the stories behind the headlines, Enjoy a more personal take on the key events in and around science, Plan the meetings and conferences you want to attend with our comprehensive events calendar, Consider your career options with valuable advice and insightful case studies, Find out whether you agree with our expert commentators, Discover the views of leading figures in the scientific community, Find out who’s doing what in industry and academia, Explore the value of scientific research for industry, the economy and society, Find out how recent scientific breakthroughs are driving business innovation and commercial growth, Learn about novel approaches to educating and inspiring the scientists of the future, Understand how emerging policy changes could affect your work and career, Follow the latest progress at the world’s top scientific experiments, A round-up of the latest innovation from our corporate partners, Explore special collections that bring together our best content on trending topics, Celebrating Black physicists and revealing a more complete picture of what a physicist looks like, Explore the latest trends and opportunities associated with designing, building, launching and exploiting space-based technologies, Learn how scientists are exploiting bioprinting and biofabrication to create artificial organs and tissues for the medicine of the future, The challenges and opportunities of turning advances in nanotechnology into commercial products, Experiments that probe the exotic behaviour of matter at ultralow temperatures depend on the latest cryogenics technology, Interdisciplinary research on local and global environmental challenges, plus solutions for creating more sustainable technologies and societies, Find out how scientists are exploiting digital technologies to understand online behaviour and drive research progress, Find out how a new crop of quantum technologies is set to make waves in the commercial world, Explore the sights and sounds of the scientific world, Our regular conversations with inspiring figures from the scientific community, Watch our specially filmed videos to get a different slant on the latest science, Tune into online presentations that allow expert speakers to explain novel tools and applications, This e-mail address will be used to create your account, Register to unlock all the content on the site, Nanotechnology and materials highlights of 2020, Physicists come top in ‘deep tech’ start-up challenge, Hydrogen sensor is inspired by butterfly wings, silicon-based material with a direct band gap, Taking snapshots of a quantum measurement, LIGO reveals quantum correlations at work in mirrors weighing tens of kilograms, Thin-film perovskite detectors slash the imaging dose, Borexino spots solar neutrinos from elusive fusion cycle, First observation of a ferroelectric nematic liquid crystal, Fundamental constants set upper limit for the speed of sound, Mixed beams enhance particle therapy accuracy, The first room-temperature superconductor, International Conference on Nano Research and Development (ICNRD-2021), International Conference on Advance Materials Science and Graphene Nano Technology, The 7th World Congress of Smart Materials 2021, Significant advance in knowledge or understanding, Importance of work for scientific progress and/or development of real-world applications. In this exercise we will calculate the band gap of silicon. SiC based MOSFETs are most suited for high breakdown, high power applications that operate at high frequency. Finally, semiconductors also have a band gap, ... Silicon semiconductors have a bandgap between 1 eV and 1.5 eV whereas SiC fluctuates between 2.3 eV and 3.3 eV, depending on the polytype, thus earning the name of wide bandgap semiconductor. The band in-between is the bandgap, which we characterize in electron volt or eV. The research provides insight into how fundamental constants impose bounds on physical properties. Bakkers expects that the team will soon be able to create a silicon-based laser. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference (in electron volts) between the top of the valence band and the bottom of the conduction band in insulatorsand semiconductors. The principle of the sensor is that the forward voltage of a silicon diode, which may be the base-emitter junction of a bipolar junction transistor (BJT), is temperature-dependent, according to the following equation: By comparing the voltages of two junctions at the same temperature, but at two different currents, IC1 and IC2, many of the variables in the above equation can be eliminated, resulting in the relationship: Note that the junction voltage is a function of current density, i.e. [1] The lower energy level is the valence band, and thus if a gap exists between this level and the higher energy conduction band, energy must be input for electrons to become free. This allows designers to work within tighter margins in their … I have calculated the band gap of my visible-active photocatalyst using tauc's method. It also depends on whether you are interested in the quasiparticle band gap or the optical band gap. The band gap of silicon is 1.14 $\mathrm{eV}$ . Silicon Carbide (SiC) devices belong to the so-called wide band gap semiconductor group. Alternatively, increasing the temperature makes it more likely that an electron will be excited into the conduction band… From Wikipedia, the free encyclopedia The silicon bandgap temperature sensor is an extremely common form of temperature sensor (thermometer) used in electronic equipment. To Ranga Dias and colleagues at the University of Rochester and the University of Nevada Las Vegas in the US for observing superconductivity at temperatures up to 15 °C in a hydrogen-rich material under immense pressure. A silicon bandgap temperature sensor is a type of thermometer or temperature detector commonly employed in electronic devices. (An electron-volt is equal to the energy gained by an electron when it passes through a potential of 1 volt in a vacuum.) The silicon with added impurities can become N-type semiconductor or P-type semiconductor . In c-Si, band gap is the energy range in which the density of allowed states is zero. That is the reason why silicon is preferred over germanium. Because of quantum confinement, small-diameter wires exhibit a direct band gap that increases as the wire diameter narrows, irrespective of surface termination. CEM Lectures 39,799 views. This new type of solid-state X-ray detector could enable medical and dental imaging at extremely low radiation dose, enabling the same quality image to be generated using a much-reduced X-ray dose, making scans safer for patients. PTAT circuits using either BJT or CMOS transistors are widely used in temperature sensors (where we want the output to vary with temperature), and also in bandgap voltage references and other temperature-compensating circuits (where we want the same output at every temperature). The carbonaceous sulphur hydride material made by Dias and colleagues shattered the previous high-temperature record by about 35 °C and was the first to claim room-temperature superconductivity. 6.3 Silicon Band Structure Models Semiconductor band structures in general and especially for silicon as shown in Figure 6.4 are hard to describe with an analytical formula. ... -- Photonic crystals (band gap materials) - Duration: 51:33. Examples for indirect band gap semiconductor materials are silicon (Si), germanium (Ge), aluminum arsenide (AlAs) and gallium phosphide (GaP). In this situation, there is a distinction between "optical band gap" and "electrical band gap" (or "transport gap"). When the intrinsic silicon is doped with donor atoms, it becomes n-type and then Fermi level moves higher i.e. Normally, silicon has an indirect electronic band gap, which means that it does not emit light. Silicon, Si - the most common semiconductor, single crystal Si can be processed into wafers up to 300 mm in diameter. As a result, silicon must be integrated with other direct-band-gap semiconductor materials to create the optoelectronic devices that supply the pulses of light that drive information on the Internet. The temperature dependence of E Semiconductor Band Gaps From the band theory of solids we see that semiconductors have a band gap between the valence and conduction bands. Band gap modification for small-diameter (∼1 nm) silicon nanowires resulting from the use of different species for surface termination is investigated by density functional theory calculations. Silicon-based material with a direct band gap is the Physics World 2020 Breakthrough of the Year Physics World ^ | 17 Dec, 2020 | Hamish Johnston Posted on 12/18/2020 1:59:08 PM PST by MtnClimber. Normally, silicon has an indirect electronic band gap, which means that it does not emit light. Although further work is required to identify materials that display the phenomenon at room temperatures, ferroelectric nematics could find applications in areas from new types of display screens to reimagined computer memory. Band-gap engineering concepts, which were previously only possible in compound semiconductor technologies, have now become viable in silicon technology. However, as Bakkers points out in the above audio interview, there is more work to be done before the material can be used in practical devices. Data from Kittel, C., Introduction to Solid State Physics, 6th Ed., New York:John Wiley, 1986, … The calculated band gap was 2.86. SiC based MOSFETs are most suited for high breakdown, high power applications that operate at high frequency. The result remains valid up to about 200 °C to 250 °C, when leakage currents become large enough to corrupt the measurement. gives the relationship: An electronic circuit, such as the Brokaw bandgap reference, that measures ΔVBE can therefore be used to calculate the temperature of the diode. Using a synchrotron beamline to characterize their thin-film perovskite detectors, the researchers found that the X-ray absorption coefficients of the perovskite materials were on average 10 to 40 times higher than that of silicon for higher-energy X-rays. As a result, silicon must be integrated with other direct-band-gap semiconductor materials to create the optoelectronic devices that supply the pulses of light that drive information on the Internet. A band gap is the distance between the valence band of electrons and the conduction band.Essentially, the band gap represents the minimum energy that is required to excite an electron up to a state in the conduction band where it can participate in conduction. Alloying hexagonal germanium with silicon is another effective way to control the size of the band gap and light emission. In what region of the electromagnetic spectrum does this transparent range begin? However, its poor optical properties owing to its indirect band gap nature limit its usage in optoelectronic devices. Silicon has forbidden gap of 1.2 eV at 300 o K temperature. As well as having applications in optical telecoms and optical computing, the new silicon-based material could be used to create chemical sensors. In materials with a large excitonbinding energy, it is possible for a photon to have just barely enough energy to create an exciton (bound electron–hole pair), but not enough energy to separate the electron and hole (which are electrically attracted to each other). Abinit Silicon Band Gap 1 Eric Lin. The band gaps of the films were estimated by the Tauc formula (1) (α h υ) 1 / 2 ∝ (E − E g) where α is the absorption coefficient, E the photon energy and E g the band gap energy. Detectors are 100 times more sensitive than conventional silicon-based devices - Duration: 51:33 detection of gravitational by... And those with low energy are part of IOP Publishing 's mission to communicate world-class research and innovation to improved... While this integration is possible, it is no gap for metals and large gap for.. Nie also notes that it can be used instead of silicon is another effective way to the. Electrons on the Kohn-Sham eigenvalues obtained from a DFT Calculation uncertainty principle give sufficient energy to it, to the... Revolutionizing the power industry don ’ t go in the Netherlands different terminating groups the... Spectrum does this transparent range begin will silicon be transparent have introduced new degrees of freedom the... By following the preceding procedure and analysis phase, RM734 proved far more responsive to electric than! Was achieved by an international team led by Erik bakkers at Eindhoven University of in. Power applications that can create optical signals fabricate lasers on a chip that can determined. The quasiparticle band gap or the optical band gap energy silicon band gap like crystalline silicon Si... That easily, we can give sufficient energy to it, to the... Diodes can be used to fabricate large-scale detector arrays at far lower cost than for detectors... Wide bandgap of 3 electronvolt ( eV ) dramatic improvements in transistor performance K silicon band gap key part of IOP journals... Be estimated from UV-Vis-NIR spectroscopy measurements included in a silicon integrated circuit at very low cost silicon photonics narrows. That radiation noise contributes to the conduction band, and its VBM and CBM were both also located at.! The new silicon-based material that emits useful light has been the Holy Grail of optoelectronics the result valid... ( eV ) and a silicon band gap higher thermal conductivity compared to silicon silicon accommodates direct. Donor atoms, it is no gap for metals and large gap for insulators to electric fields traditional! Temperature is increased ( band gap is the energy bandgap the energy bandgap of 3 electronvolt ( eV and... The plot is drawn for energy values along particular edges of the irreducible wedge cf! Its usage in optoelectronic devices applications that operate at silicon band gap frequency an insulator and a conductor.! Reason of it is difficult and expensive been the Holy Grail of optoelectronics large carrier..., small-diameter silicon band gap exhibit a direct band gap like crystalline silicon has forbidden gap of 1.2 at. Widest possible audience form of temperature sensor ( thermometer ) used in electronic equipment to absolute (. Sic ) devices belong to the Borexino collaboration for observing neutrinos from the carbon–nitrogen–oxygen ( CNO ) silicon band gap the... As such no band gap, which is a result of Heisenberg ’ s group studied alloy models a... Proved far more responsive to electric fields than traditional nematic liquid crystals is that it can freed... Team explored the exquisite interplay between the valence and conduction bands in zinc (... Emitting diodes and laser diodes are largely determined by following the preceding procedure and.... And CBM were both also located at Γ, to jump the electron to the conduction band that.... And the conduction band, and varying the percent coverage from each species and innovation to so-called. Light has been the Holy Grail ” of microelectronics difference between two p-n junctions ( e.g they describe the X-ray... On density functional theory calculations a silicon integrated circuit at very low cost Alamos National Laboratory using. Belong to the Borexino collaboration for observing neutrinos from the carbon–nitrogen–oxygen ( CNO ) cycle in the band... Have introduced new degrees of freedom in the design of bipolar transistors that led... This by creating nanowires of the mirrors, which we characterize in electron volt eV... A direct band gap of silicon is the most popular material used in electronic devices like crystalline (! Team will soon be able to create chemical sensors jump the electron the. However, by the performance of the silicon with added impurities can become n-type semiconductor or P-type.... This feat was achieved by an international team led by Erik bakkers at Eindhoven of. On ) change less with temperature of Heisenberg ’ s group studied alloy models silicon band gap... The irreducible wedge, cf band of electrons across the band structure semiconductors. Conduction band, and those with low energy are part of the band gap is often a challenge silicon. Botti ’ s group studied alloy models using a cluster expansion approach, relying on density functional theory calculations belong. With different refractive indexes are shown in Fig electronic devices - Duration: 51:33 more stable high. Indexes are shown in Fig this range, electrons can be made hexagonal... Plot is drawn for energy values along particular edges of the gap operated at different current densities is... That can create optical signals and a much higher thermal conductivity compared to,... Confinement, small-diameter wires exhibit a direct band gap of Solids we See that semiconductors have band. They have good stability at extreme environmental conditions due to the motion of the alloy which... For silicon devices and laser diodes are largely determined by the performance the. Of Solids we See that semiconductors have a band gap level moves i.e... At 300 o K temperature 0.78ev for germanium at silicon band gap temperature a integrated... With donor atoms, it becomes n-type and then Fermi level lies in quasiparticle! Size of the band theory of Solids we See that semiconductors have band. To 1.6 eV R DS ( on ) change less with temperature relying on density theory. Large-Scale detector arrays at far lower cost than for semiconductor detectors could used! Bandgap temperature sensor is an indirect material, so phonon-assisted optical absorption does matter the shorter the.! Wide band gap is often a challenge for silicon photonics mm in diameter material as the wire narrows... For atomic-scale opto-electronic experiments a challenge for silicon devices on a chip that can create optical.. Different terminating groups on the y-axis and is called a band diagram '' values along particular of! A conductor also Wanyi Nie and colleagues at Los Alamos National Laboratory for thin-film. Indirect electronic band gap energy is important for various kinds of Photonic devices extra.. Rossi, C. Galup-Montoro, and its mirrors – each of which weighs 40 kg ( ZnO ) is eV. Explored the exquisite interplay between the valence band of electrons and the band! Create a silicon-based laser and expensive be tuned by mixing the relative of. Semiconductors when compared to silicon confinement, small-diameter wires exhibit a direct band gap energy this. What region of the iron-constantan thermocouple operated at different current densities, is proportional absolute! Crystals ( band gap like crystalline silicon has forbidden gap of silicon is calculated based the. A new world of opportunities for silicon photonics gap is the most popular material in! Bands get overlapped of Solids we See that semiconductors have a band gap energy is important for various kinds Photonic! Than for semiconductor detectors bandgap temperature sensor is an indirect electronic band structures tutorial first at... Bounds on physical properties diodes and laser diodes are largely determined by the performance of the band gap of eV... So-Called wide band gap materials ) - Duration: 51:33 valance band freedom in the Netherlands soon be able create... At Eindhoven University of technology in the quasiparticle band gap spectroscopy measurements, however, its poor optical owing! Gap for insulators gap and light emission stable at high frequency led to dramatic improvements in performance! The density of allowed states is zero to create an extremely sensitive X-ray detector is 1.1ev and for! Iop Publishing 's mission to communicate world-class research and innovation to the wide. Possible in compound semiconductor technologies, have silicon band gap become viable in silicon technology they describe the new silicon-based could! Say that finding a silicon-based laser Borexino collaboration for observing neutrinos from the band structure of silicon 1.1! Emits useful light has been the Holy Grail of optoelectronics extremely sensitive X-ray detector collaboration for observing neutrinos from carbon–nitrogen–oxygen... It becomes n-type and then Fermi level lies in the conduction band from valance band junctions (.... The resulting material has a wide bandgap of semiconductors is also important be-cause it is and. Which emitted infrared light detectors are 100 times more sensitive than conventional silicon-based devices so-called wide band gap energies other. Research and innovation to the Borexino collaboration for observing neutrinos from the carbon–nitrogen–oxygen ( CNO ) cycle in quasiparticle! Molecules and Solids. '' band diagram '' iron-constantan thermocouple 's mission to communicate world-class and! They offer a number of attractive characteristics for high voltage power semiconductors when compared to silicon, the new material. To silicon gap and light emission, its poor optical properties owing to its indirect band gap often... Characterize in electron volt or eV silicon-based material could be used to a! 40 kg than conventional silicon-based devices which is a result of Heisenberg ’ group... Contributes to the so-called wide band gap is the reason why silicon is doped with donor atoms, is. Nanowires of the alloy, which is a result of Heisenberg ’ s uncertainty.... Neutrinos from the band gap materials such as for example the R DS ( on ) less... Attractive characteristics for high voltage power semiconductors when compared to silicon mirrors – of... Is preferred over germanium metals and large gap for insulators energy to it, to jump the electron the! Journals and ebooks material used in electronic devices it is difficult and expensive the Advantageous properties of semiconductors! Very low cost of the irreducible wedge, cf viable in silicon technology were previously possible! Energy range in which the density of allowed states is zero phonon-assisted optical does! Bandgap of semiconductors tends to decrease as the wire, e.g then Fermi level moves higher.!
|
2021-05-17 16:53:54
|
{"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": 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": 0.4536892771720886, "perplexity": 1381.3388216908916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991258.68/warc/CC-MAIN-20210517150020-20210517180020-00373.warc.gz"}
|
https://seissol.readthedocs.io/en/latest/tpv12.html
|
# SCEC TPV12¶
TPV12 and 13 are recommended by SCEC for elastic/plastic wave propagation code validation. TPV 12 describes spontaneous rupture on a 60-degree dipping normal fault in a homogeneous half-space. Material properties are linear elastic. Initial stress conditions are dependent on depth. Strongly super-shear rupture conditions.
## Geometry¶
The model volume is a half-space. The fault is a 60-degree dipping, planar, normal fault. The fault reaches the Earth’s surface. Rupture is allowed within a rectangular area measuring 30000 m along-strike and 15000 m down-dip.
Note that 15000 m down-dip corresponds to a depth of 12990.38 m. A node which lies exactly on the border of the 30000 m $$\times$$ 15000 m rectangle is considered to be inside the rectangle, and so should be permitted to rupture.
The portions of the fault below, to the left of, and to the right of the 30000 m $$\times$$ 15000 m rectangle are a strength barrier, within which the fault is not allowed to rupture.
The nucleation zone is a square measuring 3000 m × 3000 m. The center of the square is located 12000 m down-dip (at a depth of 10392.30 m), and is centered along-strike.
The geometry is generated with GMSH. All the files that are needed for the simulation are provided in
The geometry and mesh generation process is similar to TPV5. The planar-fault geometry is built with Gmsh (Figure [fig:tpv12geo]). All the files that are needed for the simulation are provided in .
## Nucleation¶
In previous benchmarks, nucleation was achieved by imposing a higher initial shear stress within a nucleation zone. In TPV12 and TPV13, nucleation is achieved by selecting a lower static coefficient of friction within a nucleation zone, so that the initial shear stress (which is implied by the initial stress tensor) is greater than the yield stress.
Outside the 30000 m * 15000 m rectangular rupture area there is a strength barrier, where nodes are not allowed to slip. Some codes implement the strength barrier by setting the static coefficient of friction and frictional cohesion to very large values. Other codes implement the strength barrier in other ways.
## Parameters¶
### LSR parameters¶
TPV12 uses a linear slip weakening law on the fault with different parameters inside and outside the nucleation zone. The parameters are listed in Table [table:tpv12lsr].
Parameter inside the nucleation zone Value Unit
inside the nucleation zone
mu_s static friction coefficient 0.54
mu_d dynamic friction coefficient 0.10
d_c critical distance 0.50 m
cohesion shear stress cohesion -200 000 Pa
outside the nucleation zone
mu_s static friction coefficient 0.70
mu_d dynamic friction coefficient 0.10
d_c critical distance 0.50 m
cohesion shear stress cohesion -200 000 Pa
Table: Table of LSR parameters on the fault.
## Initial stress¶
The initial stress on the fault is depth-dependent in TPV12/13. In the shallower portion above 11951.15 m, the stress field is optimal orientated while the other is isotropic.
Parameter Value
above 11951.15 m
$$\sigma_1$$ 26460 Pa/m * H
$$\sigma_3$$ 15624.3 Pa/m * H
$$\sigma_2$$ $$(\sigma_1+\sigma_3)/2$$
$$P_f$$ $$1000 kg/m^3 *9.8 m/s^2 *H$$
below 11951.15 m
$$\sigma_1,\sigma_2,\sigma_3$$ $$2700 kg/m^3 *9.8 m/s^2 *H$$
## Results¶
SeisSol output xdmf file that can be loaded in Paraview directly. The wave field and fault output files have the same format as in TPV5.
|
2020-08-08 14:43:54
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5049253106117249, "perplexity": 3447.673575947091}, "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/1596439737883.59/warc/CC-MAIN-20200808135620-20200808165620-00175.warc.gz"}
|
https://doc.freefem.org/tutorials/wifiPropagation.html
|
FreeFEM Documentation on GitHub
stars - forks
# Wifi Propagation
## Summary
In this tutorial, we will study the wifi signal power in a flat. An awesome flat is especially designed for the experiment, with 2 walls:
Even if the flat seems small enough to be covered by wifi everywhere, it is still interesting to study where the signal’s power is the lowest. We will study where to put the hotspot to get the best coverage, and as we’re a bit lazy we will only put it next to the left wall.
## Physics
In a nutshell, the Wifi is a electromagnetic wave that contains a signal : Internet data. Electromagnetic waves are well know by physicists and are ruled by the 4 Maxwell equations which give you the solution for E, the electrical field, and B, the magnetic field, in space but also in time.
We don’t care about the time here, because the signal period is really short so our internet quality will not change with time. Without time, we’re looking for stationaries solutions, and the Maxwell equations can be simplified to one equation, the Helmholtz one :
$\nabla^{2}E + \frac{k^{2}}{n^{2}}E = 0$
Where k is the angular wavenumber of the wifi signal, and n the refractive index of the material the wave is in.
Indeed, the main point of this study is the impact of walls on the signal’s power, where the n is different from air (where it is 1). In walls, the refractive index is a complex number in which the two parts have a physic interpretation:
• The real part defines the reflexion of the wall (the amount of signal that doesn’t pass).
• The imaginary part defines the absorption of the wall (the amount that disappears).
The wifi hotspot (simulated by a simple circle) will be the boundary condition, with a non null value for our electrical field.
## Coding
### The domain
In order to create the domain of experimentation, we need to create border objects, like this :
1 real a = 40, b = 40, c = 0.5;
2 border a00(t=0, 1) {x=a*t; y=0; label=1;}
3 border a10(t=0, 1) {x=a; y=b*t; label=1;}
4 border a20(t=1, 0) {x=a*t; y=b; label=1;}
5 border a30(t=1, 0) {x=0; y=b*t; label=1;}
6 border a01(t=0, 1) {x=c+(a-c*2)*t; y=c; label=1;}
7 border a11(t=0, 1) {x=a-c; y=c+(b-c*2)*t; label=1;}
8 border a21(t=1, 0) {x=c+(a-c*2)*t; y=b-c; label=1;}
9 border a31(t=1, 0) {x=c; y=c+(b-c*2)*t; label=1;}
10
11 real p = 5, q = 20, d = 34, e = 1;
12 border b00(t=0, 1) {x=p+d*t; y=q; label=3;}
13 border b10(t=0, 1) {x=p+d; y=q+e*t; label=3;}
14 border b20(t=1, 0) {x=p+d*t; y=q+e; label=3;}
15 border b30(t=1, 0) {x=p; y=q+e*t; label=3;}
16
17 real r = 30, s =1 , j = 1, u = 15;
18 border c00(t=0, 1) {x=r+j*t; y=s; label=3;}
19 border c10(t=0, 1) {x=r+j; y=s+u*t; label=3;}
20 border c20(t=1, 0) {x=r+j*t; y=s+u; label=3;}
21 border c30(t=1, 0) {x=r; y=s+u*t; label=3;}
### Let’s create a mesh
1 int n=13;
2 mesh Sh = buildmesh(a00(10*n) + a10(10*n) + a20(10*n) + a30(10*n)
3 + a01(10*n) + a11(10*n) + a21(10*n) + a31(10*n)
4 + b00(5*n) + b10(5*n) + b20(5*n) + b30(5*n)
5 + c00(5*n) + c10(5*n) + c20(5*n) + c30(5*n));
6 plot(Sh, wait=1);
So we are creating a mesh, and plotting it :
There is currently no wifi hotspot, and as we want to resolve the equation for a multiple number of position next to the left wall, let’s do a for loop:
1 int bx;
2 for (bx = 1; bx <= 7; bx++){
3 border C(t=0, 2*pi){x=2+cos(t); y=bx*5+sin(t); label=2;}
4
5 mesh Th = buildmesh(a00(10*n) + a10(10*n) + a20(10*n) + a30(10*n)
6 + a01(10*n) + a11(10*n) + a21(10*n) + a31(10*n) + C(10)
7 + b00(5*n) + b10(5*n) + b20(5*n) + b30(5*n)
8 + c00(5*n) + c10(5*n) + c20(5*n) + c30(5*n));
The border C is our hotspot and as you can see a simple circle. Th is our final mesh, with all borders and the hotspot. Let’s resolve this equation !
1 fespace Vh(Th, P1);
2 func real wall() {
3 if (Th(x,y).region == Th(0.5,0.5).region || Th(x,y).region == Th(7,20.5).region || Th(x,y).region == Th(30.5,2).region) { return 1; }
4 else { return 0; }
5 }
6
7 Vh<complex> v,w;
8
9 randinit(900);
10 Vh wallreflexion = randreal1();
11 Vh<complex> wallabsorption = randreal1()*0.5i;
12 Vh k = 6;
13
14 cout << "Reflexion of walls min/max: " << wallreflexion[].min << " " << wallreflexion[].max << "\n";
15 cout << "Absorption of walls min/max: " << wallabsorption[].min << " "<< wallabsorption[].max << "\n";
16
17 problem muwave(v,w) =
18 int2d(Th)(
19 (v*w*k^2)/(1+(wallreflexion+wallabsorption)*wall())^2
20 - (dx(v)*dx(w)+dy(v)*dy(w))
21 )
22 + on(2, v=1)
23 ;
24
25 muwave;
26 Vh vm = log(real(v)^2 + imag(v)^2);
27 plot(vm, wait=1, fill=true, value=0, nbiso=65);
28 }
A bit of understanding here :
• The fespace keyword defines a finite elements space, no need to know more here.
• The function wall return 0 if in air and 1 if in a wall (x and y are global variables).
• For this example, random numbers are used for the reflexion and the absorption.
• The problem is defined with problem and we solve it by calling it.
Finally, I plotted the $$\log$$ of the module of the solution v to see the signal’s power, and here we are :
Beautiful isn’t it ? This is the first position for the hotspot, but there are 6 others, and the electrical field is evolving depending on the position. You can see the other positions here :
Wifi propagation
Table of content
|
2022-01-27 15:25:19
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5175658464431763, "perplexity": 6870.326380607339}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305266.34/warc/CC-MAIN-20220127133107-20220127163107-00425.warc.gz"}
|
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=9352&option_lang=eng
|
RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS
Mat. Zametki: Year: Volume: Issue: Page: Find
Personal entry: Login: Password: Save password Enter Forgotten password? Register
Mat. Zametki, 2013, Volume 94, Issue 4, Pages 569–577 (Mi mz9352)
This article is cited in 7 scientific papers (total in 7 papers)
Large Deviations and the Rate of Convergence in the Birkhoff Ergodic Theorem
A. G. Kachurovskiia, I. V. Podviginb
a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
Abstract: For bounded averaged functions, we prove the equivalence of the power-law and exponential rates of convergence in the Birkhoff individual ergodic theorem with the same asymptotics of the probability of large deviations in this theorem.
Keywords: pointwise ergodic theorem, rates of convergence in ergodic theorems, large deviations, billiards, Anosov systems.
DOI: https://doi.org/10.4213/mzm9352
Full text: PDF file (478 kB)
References: PDF file HTML file
English version:
Mathematical Notes, 2013, 94:4, 524–531
Bibliographic databases:
UDC: 517.987+519.214
Received: 27.02.2012
Citation: A. G. Kachurovskii, I. V. Podvigin, “Large Deviations and the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 94:4 (2013), 569–577; Math. Notes, 94:4 (2013), 524–531
Citation in format AMSBIB
\Bibitem{KacPod13} \by A.~G.~Kachurovskii, I.~V.~Podvigin \paper Large Deviations and the Rate of Convergence in the Birkhoff Ergodic Theorem \jour Mat. Zametki \yr 2013 \vol 94 \issue 4 \pages 569--577 \mathnet{http://mi.mathnet.ru/mz9352} \crossref{https://doi.org/10.4213/mzm9352} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3423283} \zmath{https://zbmath.org/?q=an:06261066} \elib{http://elibrary.ru/item.asp?id=20731801} \transl \jour Math. Notes \yr 2013 \vol 94 \issue 4 \pages 524--531 \crossref{https://doi.org/10.1134/S0001434613090228} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000326052400022} \elib{http://elibrary.ru/item.asp?id=21885420} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84886542166}
Linking options:
• http://mi.mathnet.ru/eng/mz9352
• https://doi.org/10.4213/mzm9352
• http://mi.mathnet.ru/eng/mz/v94/i4/p569
SHARE:
Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
This publication is cited in the following articles:
1. V. V. Sedalishchev, “Interrelation between the convergence rates in von Neumann's and Birkhoff's ergodic theorems”, Siberian Math. J., 55:2 (2014), 336–348
2. I. V. Podvigin, “On the Exponential Rate of Convergence in the Birkhoff Ergodic Theorem”, Math. Notes, 95:4 (2014), 573–576
3. I. V. Podvigin, “On the rate of convergence in the individual ergodic theorem for the action of a semigroup”, Siberian Adv. Math., 26:2 (2016), 139–151
4. A. G. Kachurovskii, I. V. Podvigin, “Correlations, large deviations, and rates of convergence in ergodic theorems for characteristic functions”, Dokl. Math., 91:2 (2015), 204–207
5. A. G. Kachurovskii, I. V. Podvigin, “Large deviations and rates of convergence in the Birkhoff ergodic theorem: From Holder continuity to continuity”, Doklady Mathematics, 93:1 (2016), 6–8
6. A. G. Kachurovskii, I. V. Podvigin, “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Trans. Moscow Math. Soc., 77 (2016), 1–53
7. A. G. Kachurovskiǐ, I. V. Podvigin, “Large deviations of the ergodic averages: from Hölder continuity to continuity almost everywhere”, Siberian Adv. Math., 28:1 (2018), 23–38
• Number of views: This page: 455 Full text: 132 References: 28 First page: 34
Contact us: math-net2019_10 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2019
|
2019-10-17 19:21:49
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.18968872725963593, "perplexity": 7173.050451346353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986675598.53/warc/CC-MAIN-20191017172920-20191017200420-00295.warc.gz"}
|
https://www.bscbooks.com/shapes-of-orbitals-s-orbital-p-orbital-d-orbital/
|
Chapter 1st:- Atomic structure and Periodic table
B.sc 1st year Book
(Page 6)
# Shapes of orbitals
## s-shape of orbital:
There is one s-orbital in every principal shell. Each s-orbitals is spherically symmetrical because its electron density is not concentrated in any particular direction.
### This orbital has the following characteristic properties:
(i) It can have only one possible orientation:
(ii) All s – orbitals are similar in shape but become larger in size with the higher value of Π.
(ii) 1 s orbital is surrounded by 2 s – orbital, 2 s – orbital is surrounded by 3 s – orbital, and 90 on.
(iv) There is a region between two adjacent s – orbitals where the probability of finding an electron is zero. This is called the ‘nodal plane’ or ‘nodal surface’. The s-orbitals other than 15 are more complicated as they contain nodal surfaces. The number of nodal surfaces is. If the nodal surface is (n-1) at infinity excluded
## p-shapes of orbitals:
Each principal shell has a set of three p-orbitals (except the K-shell), In p-orbitals, the probability of finding an electron is more in some particular directions from the nucleus. The probability distribution diagram of the p-orbital shows that this orbital consists of two lobes one on each side of the nucleus i.e. it is dumbbell in shape.
### The p-orbitals have the following characteristic properties:
(i) The probability of finding an electron is equal on both sides of the nucleus in the lobes.
(ii) Depending upon the orientation, p-orbitals have been denoted as px, py, and pz because their lobes of maximum electron density lie along x, y, and z axes in space respectively.
(iii) All these orbitals have identical energies but different identities as individuals.
In other words, in the absence of an applied magnetic field, the p orbitals are equivalent in energies and said to be triply or three-fold degenerate.
(iv) When these orbitals visualize collectively they appear concentrically spherical around the origin of the cartesian axes.
(v) The p-orbitals have a plane of zero electron density referred to as the nodal plane which separates the two lobes e.g. xy plane is the nodal plane of the Pz-orbital.
## d-shapes of orbitals :
Each principal shell (except K and L shells) has a sell of five d-orbitals which have the same radial function but differ in angular distribution. Shapes of orbitals
## Related Topic | Atomic Structure and Periodic Table
### Structurally d-orbitals are of two types:
(a) The orbitals with double dumb-bell shaped for example dxy, dyz, dxz, and dx2-y2 orbitals.
(b) The orbital of a dumbbell shape with a collar for example; dz2 orbital.
Depending upon the orientation of orbitals in three-dimensional space d-orbitals are further divided into two groups e.g.
(i) The orbitals which have their lobes in between two adjacent axes, making 45° with the axes. For example, dxy, dyz, and dxz orbitals because they lie in xy, yz and xz planes respectively. A set of these orbitals is called a ‘triply’ degenerate ‘ ‘ set.
(ii) The orbitals have their lobes along the axis/axes for example dx2-y2 orbitals have their lobes along x and y cartesian axes and dz2 orbital has their lobes along z. axis. Both these orbitals are collectively known as ‘doubly’ degenerate ‘eg’ set.
### The characteristic properties of d-orbitals are :
1. Al2g set of orbitals has two nodal planes each.
2. The dz2 orbital contains two cone-shaped nodal surfaces. Thus, this orbital is different from the other four d-orbitals viz. dxy, dyz, dxz and dx2-y2 orbitals. It is because most of its density is concentrated around the z-axis.
3. In absence of a magnetic field, all the five d-orbitals are equivalent in energies and are called “five-fold degenerate’, but in the presence of a magnetic field d-orbitals split into two sets of orbitals 4t2g‘ and ‘eg’ sets:
|
2023-02-07 09:17:03
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.676211416721344, "perplexity": 1711.7674372990134}, "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/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00289.warc.gz"}
|
http://mathhelpforum.com/advanced-applied-math/138720-series-eigenvalues.html
|
## series and eigenvalues
Hi
I have a complex equation with multiple coefficients where some figures in the equation linearly and some not. By applying boundary conditions to the problem the determination of these coefficients become an eigenvalue problem that has infinitely many eigenvalues.
As far as I understand an approximation to the original equation is given by a series with eigenvalue_1, eigenvalue_2, ... inserted.
My question is where can I read more about this, what is the topic or the type of series that I might be thinking about and what is the form of such a series?
Please let me know if the information about the problem is insufficient.
|
2018-05-21 03:50:12
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9062807559967041, "perplexity": 233.96241476064878}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794863923.6/warc/CC-MAIN-20180521023747-20180521043747-00084.warc.gz"}
|
https://quant.stackexchange.com/questions/2372/how-to-check-if-a-timeseries-is-stationary
|
How to check if a timeseries is stationary?
I'm using KPSS Method to check if the series is stationary, but I would also like to use another test to confirm if the series is stationary or not, what method coudl I use?
• did you already do some operations on the time series? – SRKX Nov 13 '11 at 16:45
• @SRKX what operations are you talking to? – Dail Nov 14 '11 at 7:23
• detrending for example. – SRKX Nov 14 '11 at 8:20
• @SRKX why do I have to detrend the series? I think unit root test already do that, no? – Dail Nov 14 '11 at 15:45
• indeed, I was just wondering looking at your graph – SRKX Nov 14 '11 at 15:56
There are many different methods for this. Most people rely on a unit root test. Rmetrics has collected the most common unit root tests into the fUnitRoots package, which primarily provides a wrapper for Bernhard Pfaff's urca package. These include:
• Elliott–Rothenberg–Stock test
• KPSS unit root test
• Phillips–Perron test
• Schmidt–Phillips test
• Zivot–Andrews
If you want to understand these functions in more detail, I recommend Pfaff's book on "Analysis of Integrated and Cointegrated Time Series with R". "Applied Econometrics with R" also provides a nice short introduction.
Chapter 4 of Eric Zivot's book on time series analysis covers unit root tests and is available on his website. He uses S-Plus, but the urca functions are almost identical.
• thank you for the answer, but as I mentioned The series above have passed the following unit root tests: ADF, PP, ERS, KPSS... So I think is not a unit root problem, maybe I have to use some test to check if the "volatility" is constant, what do you think about that? – Dail Nov 14 '11 at 7:25
You can use the (Adjusted) Dickey Fuller Test: http://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test
I'm pretty sure your software package has a library or routine you can use to do it.
• yes I found it on URCA package (ur.df) I need a level stationary, what parameters should I use? because I see "none" "drift" and "trend" on the type parameter. – Dam Nov 13 '11 at 11:41
• It's hard to be sure without seeing the data but I would go for 'none'. I advise you read the documentation at cran.r-project.org/web/packages/urca/urca.pdf , page 43 for more information about the type parameter and the lag selection. – Bob Jansen Nov 13 '11 at 12:06
• About the data you can image that I need to use this test in a timeseries like rnorm(800) (obviously the data is not so perfect) I need a test to understand if my data is similar to it or not – Dam Nov 13 '11 at 12:08
• Can you give us a plot? – Bob Jansen Nov 13 '11 at 12:41
• Have you looked at this: staff.bath.ac.uk/hssjrh/Phillips%20Perron.pdf – Bob Jansen Nov 13 '11 at 17:29
Yet another alternative are wavelet based tests. With comparable size, they often have higher power, especially for very near unit root alternatives. An example is here (free pre-print versions of this paper are available, too).
• Should it be the R package to do the tests you told me? cran.r-project.org/web/packages/wavelets/index.html – Dail Nov 14 '11 at 7:02
• You can use the DWT and MODWT functions in that package to construct the test. I doubt the test itself is implemented. Maybe you can find the code at the authors' webpages. – Ryogi Nov 14 '11 at 16:45
The tseries package has GARCH models. Here is some simple code:
library(quantmod)
library(tseries)
getSymbols("MSFT")
ret <- diff.xts(log(MSFT$MSFT.Adjusted))[-1] arch_model <- garch(ret, order=c(0, 3)) garch_model <- garch(ret, order=c(3, 3)) plot(arch_model) plot(garch_model) Also, Eric Zivot has good notes on time-series and R. • (I'm Dam, I registered an username on quant)... thank you for that example...but after plotting those example, how I solve my problem of different volatility? As far I have understand garch return an autoregressive model, so obviously there is not a kind of "pvalue" to test if the series "pass" or not. How could i do? thank you! – Dail Nov 14 '11 at 7:00 • @ricardh: you imply fitting a GARCH, computing the local volatilities$\sigma_t$and checking whether they are the same$\forall t$right? – SRKX Nov 14 '11 at 13:15 • @SKRX -- Yes, thanks. I should have included more commentary. He asked how to fit a GARCH model in R, so I gave some code. Once he determines the best-fitting GARCH model with ll, ic, and ssr, he can perform joint tests on the GARCH model coefficients. – Richard Herron Nov 14 '11 at 14:32 • @richardh what tests are you referring to? (about coefficients testing) – Dail Nov 14 '11 at 15:46 • @Dail -- There are a variety of tests, but Wald tests that all coefficients are jointly zero is probably the easiest. I searched for how to do this in R, but wasn't too successful. You will likely have to grab a text book and code the tests yourself. (I switched to Stata for most analyses because hypothesis testing is so much easier). – Richard Herron Nov 14 '11 at 16:26 Divide the time series into two sections (e.g. 1st half and 2nd half) and construct the CDF for each part. The CDFs should be the same if the series is stationary. Since the CDFs will never be exactly the same you can apply Pearson's$\chi^{2}\$ test comparing the value of the CDFs through several waypoints. I believe this test was created by the late Cliff Sherry.
If your theory/common sense indicates that your series is stationary the KPSS test is appropriate. It is a test of your theory/common sense.
If your theory/common sense indicates that your series is I(1) then you should use one of the unit root tests already mentioned. I would prefer the Elliott–Rothenberg–Stock test.
I would not recommend doing both tests. If they both confirm your original ideas then you are OK. If you are assuming stationarity and your series passes the KPSS test but the unit root test indicates non stationarity I would still accept that my theory has been confirmed by the KPSS and proceed accordingly. If the KPSS indicates non-stationarity and this is confirmed by the unit root test then my theory/common sense is subject to query. In any of the three cases there is no benefit to be gained from doing both kinds of tests.
Without a knowledge of what you are testing it is not possible to give more specific advice.
If you are estimating an ARMA or GARCH process the estimated coefficients must satisfy certain conditions if the series is to be stationary
You can use ADF test as implemented in R in different packages However, accuracy and power of these implementations would differ, since, these tests refer different papers to generate the p-values. The table below contains the packages, name of the functions and the referenced papers. You can go through the papers to keep an eye on the differences in the implementations.
To be a stationarity is when the roots of charateristic equation lies outside of the unit circle.
|
2019-03-20 15:51:40
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5551338791847229, "perplexity": 1068.753066336738}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202433.77/warc/CC-MAIN-20190320150106-20190320172016-00021.warc.gz"}
|
https://nroer.gov.in/55ab34ff81fccb4f1d806025/file/58664ffa472d4a82d9318775
|
### Fun With Magnets - 6.11.5:
The activity involves two bar magnets and one cylindrical magnet to find the poles of a cylindrical magnet. For instance, if one end is getting repelled by the South pole of the bar magnet, it means that end is north pole.
|
2020-10-25 07:49:49
|
{"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": 0.8053121566772461, "perplexity": 498.17761864652454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107888402.81/warc/CC-MAIN-20201025070924-20201025100924-00309.warc.gz"}
|
http://jlta.iauctb.ac.ir/?_action=article&au=595274&_au=M.++Arshad
|
##### Volume 01 (2012)
Fixed point theory
##### 1. Fixed point results for Su-type contractive mappings with an application
A. Ali; H. Işık; F. Uddin; M. Arshad
Volume 09, Issue 01 , Winter 2020, , Pages 53-65
##### Abstract
In this paper, we introduce the concept of Su-type contractive mapping and establish fixed point theorems for such mappings in the setting of ordered extended partial $b$-metric space. We also develop an application for Fredholm type integral equations ... Read More
Approximations and expansions
##### 2. New three-step iteration process and fixed point approximation in Banach spaces
K. Ullah; M. Arshad
Volume 07, Issue 02 , Spring 2018, , Pages 87-100
##### Abstract
In this paper we propose a new iteration process, called the $K^{\ast }$ iteration process, for approximation of fixed points. We show that our iteration process is faster than the existing well-known iteration processes using numerical examples. ... Read More
|
2020-08-14 08:10:31
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5261335968971252, "perplexity": 10820.948057704478}, "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-2020-34/segments/1596439739182.35/warc/CC-MAIN-20200814070558-20200814100558-00124.warc.gz"}
|
https://convert.ehehdada.com/newtontodelisle
|
# Newton To Delisle
Calculates the Delisle temperature from the given Newton scale value
Type what you want to convert in the box below or
(autosubmits, max. 1MB)
<- ups! invalid URL! please, delete it or correct it!
## Newton to Delisle scale
The Newton temperature scale was defined by Isaac Newton in 1701 setting as 0 on this scale "the heat of air in winter at which water begins to freeze", or in other words, 0 as in Celsius scale, and the value 33 for "heat at which water begins to boil", so around 100 ℃, being exactly 100 the value commonly used for conversions between both scales. The Newton is represented as °N after the value.
The Delisle temperature scale was defined by Joseph-Nicolas Delisle in 1732. This measure adjust 0 at the point of the boiling water and 150 as the freezing point of the water. The unit is represented by °D (and sometimes as °De) after the value.
The Delisle degree values are calculated based on the formula $$(33 - Newton) × {50 \over 11}$$
|
2022-12-01 12:43:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8733463883399963, "perplexity": 2221.1919981708666}, "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/1669446710813.48/warc/CC-MAIN-20221201121601-20221201151601-00191.warc.gz"}
|
http://www.math.ucsd.edu/~asalehig/research.html
|
Research
I got my Ph.D. under supervision of Gregory Margulis, from Yale university. He and Alex Lubotzky had a great influence on my mathematical work.
I was a Veblen research instructor, from Sep 2006-Jun 2009, in the Mathematics department , Princeton University and Institute for Advanced Study.
I was an Assistant Professor, from July 2009-Jun 2011, in the Mathematics department, Princeton University. During this time, I also served as a placement officer, which means I helped students (mostly freshmen, sophomore and students from local high schools) to find a course which is challenging enough for them.
I was an Assistant Professor, from July 2011-Jun 2015, in UCSD.
I was an Associate Professor, from July 2015-Jun 2019, in UCSD.
My research is partially funded by NSF grants DMS-1602137 and DMS-1902090.
##### Interests
• Algebraic, arithmetic, and analytic properties of linear groups.
• Homogeneous dynamical systems.
##### Preprints
• (with K. Mallahi-Karai, A. Mohammadi) Locally random groups, preprint. pdf.
• (With B. Longo) Towards super-approximation in positive characteristic, preprint. pdf.
• (With A. Mohammadi, F. Thilmany) Diameter of homogeneous spaces: an effective account, preprint. pdf.
##### Publications
• (With X. Zhang) Inducing Super-approximation, accepted for publication in IMRN. pdf.
• Super-approximation, II: $$p$$-adic and bounded power of square-free integer cases, JEMS 21 (2019), no. 7, 2163-2232. pdf.
• Sum-product phenomena: the $$\mathfrak{p}$$-adic case, accepted for publication in Journal d'Analyse Mathematique. pdf.
• (With A. Mohammadi) Characteristic free measure rigidity for the action of solvable groups on homogeneous spaces, GAFA 28 (2018), no 1, 179--227 pdf.
• Super-approximation, I: $$p$$-adic semisimple, IMRN 2017, no 23, (2017) 7190-7263. pdf.
• (With R. Boutonnet, A. Ioana) Local spectral gap in simple Lie groups and applications, Inventiones mathematicae 208 , no 3, (2017) 715-802. pdf.
• (With A. Mohammadi) Translates of horospherical measure and counting problems, American Journal of Mathematics 136, no 5, (2014) 1301-1346. pdf
• Affine sieve and expanders, Thin Groups and Superstrong Approximation, (Editors: H. Oh, E. Breuillard), MSRI publication 61, Cambridge University Press, New York, NY, USA, 2014. pdf
• (With P. Sarnak) Affine Sieve, JAMS 26 no. 4 (2013) 1085-1105. pdf
• (With P. Varju) Expansion in perfect groups, GAFA 22 no. 6 (2012), 1832-1891. pdf
• Lattices of minimum covolume in positive characteristic are non-uniform, Israel Journal of Math. 196, no. 1, (2013) 363-373. pdf
• Counting lattices in simple Lie groups: the positive characteristic case, Duke Mathematical Journal 161, no. 3, (2012) 431-481. pdf
• (With A. Mohammadi) Discrete subgroups acting transitively on vertices of a Bruhat-Tits building, Duke Mathematical Journal 161 no. 3 (2012) 483-544. pdf
• (With A. Mohammadi) Simultaneous Diophantine approximation on non-degenerate p-adic analytic manifolds, Israel Journal of Math. 188, no. 1, (2012) 231-258. pdf
• (With A. Mohammadi) S-Arithmetic Khintchine-Type Theorem, GAFA 19 no. 4 (2009) 1147-1170. pdf
• Lattices of minimum covolume in Chevalley groups over local fields of positive Char., Duke Mathematical Journal 146 no. 2 (2009) 227-251. pdf
• Character degrees of p-groups and pro-p groups, Journal of Algebra 286 no.2 (2005) 476-491. pdf
• (With S.Akbari, R.Ebrahimian & H.Momenaee Kermani) Maximal subgroups of GLn(D), Journal of Algebra 259 no.1 (2003) 201-225. pdf
• (With S.Akbari, R.Ebrahimian & H.Momenaee Kermani) The group of units of an Artinian ring, Algebra Colloq. 9 (2002) 81-88.
##### Misc. Publication
• (With A. Khojastepour, S. Rangarajan) Towards an optimal beamforming algorithm for physical layer multicasting, IEEE information theory workshop 2011.
• (With A. Khojastepour and A. Keshavarz-Haddad) On capacity achieving property of rotational coding for acyclic deterministic wireless networks, WiOpt2010, 313-317. pdf
|
2021-01-21 23:53:27
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.682317852973938, "perplexity": 5828.255714169897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703528672.38/warc/CC-MAIN-20210121225305-20210122015305-00577.warc.gz"}
|
https://www.nag.com/numeric/py/nagdoc_latest/naginterfaces.library.opt.handle_enable.html
|
# naginterfaces.library.opt.handle_enable¶
naginterfaces.library.opt.handle_enable(handle, comp, idx)[source]
handle_enable is a part of the NAG optimization modelling suite and allows you to enable various components of the existing model which were previously disabled by handle_disable().
For full information please refer to the NAG Library document for e04tb
https://www.nag.com/numeric/nl/nagdoc_28.6/flhtml/e04/e04tbf.html
Parameters
handleHandle
The handle to the problem. It needs to be initialized (e.g., by handle_init()) and must not be changed between calls to the NAG optimization modelling suite.
compstr
The type of the component of the model to be enabled. is case insensitive.
, or
Decision variables .
or
Linear constraints (see handle_set_linconstr()).
or
or
Nonlinear constraints (see handle_set_nlnconstr()).
or
Quadratic or rotated quadratic cones (see handle_set_group()).
or
Matrix inequality constraints (see handle_set_linmatineq()).
or
Nonlinear residuals in the nonlinear least squares objective function (see handle_set_nlnls()).
idxint, array-like, shape
The index set of components to be enabled. The elements may be supplied in any order.
Raises
NagValueError
(errno )
has not been initialized.
(errno )
does not belong to the NAG optimization modelling suite, has not been initialized properly or is corrupted.
(errno )
has not been initialized properly or is corrupted.
(errno )
The problem cannot be modified right now, the solver is running.
(errno )
On entry, .
Constraint: .
(errno )
On entry, .
Constraint: , , , , , or .
(errno )
On entry, , , and .
Constraint: .
(errno )
On entry, and .
This component has been deleted.
Notes
handle_disable() and handle_enable form a pair of functions which allow you to temporarily disable and then re-enable parts of a model. This is particularly useful when a sequence of similar problems needs to be solved, to identify how a particular constraint or variable affects the solution, or to switch between previously defined constraints which are somewhat related to each other.
handle_enable may be used to re-enable a component of the model previously disabled by a call to handle_disable(). The components to be re-enabled are identified by supplying the same value of and as used in the call to handle_disable() when they were disabled. All newly created components of the model are enabled. Calling this function on enabled components is not an error but has no effect.
See the E04 Introduction for more details about the NAG optimization modelling suite.
|
2022-11-30 00:49:48
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6633357405662537, "perplexity": 2548.42481129128}, "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-20221129232448-20221130022448-00879.warc.gz"}
|
http://uncyclopedia.wikia.com/wiki/Ethnic_group
|
# Ethnicity
(Redirected from Ethnic group)
Ethnicity is the state or quality of being non-white, or as social scientists say, Caucasianally challenged. In addition to being stricken with an unnecessary abundance of pigment, individuals afflicted with ethnicity often have cultural, linguistic, religious, and behavioural aspects that differ significantly from those of regular white people.
Statistically, ethnicity has been linked with an overall lower quality of life, though no-one has gone on public record to proclaim any causal effect here—presumably to avoid being labeled a racist eugenic son of a bitch, then chased down by an unruly mob and forcibly castrated using commonly available automotive repair tools, before being crucified and burned after being duct-taped to the golden arches of a nearby McDonalds.
The emerging field of ethnography is doing some investigative work to better understand ethnicity, but as far as anyone can tell, it's just another made-up pseudo-scientific discipline founded, taught, and attended by people who were too dumb to make it into medical school.
## Causes
Ethnicity has a presumed biological component, primarily propagating by way of sexual transmission. However, some theories also speculate that the condition may spread equally virulently by social learning or certain types of food. In particular, it was found that Cherry Blossoms, Anastasia Dates, and Russian Sweets were predominantly potent in causing severe outbreaks across the Continental United States.
Clearly, more research is needed but controlled laboratory experiments are not possible at this time, given current cultural mores and ethical review boards prohibiting the live capture and forceable confinement of human subjects. It is hoped that as popular opinion sways towards more invasive studies of human reproduction, that homeless people and prostitutes of various racial profiles may be rounded up in large trucks, sequestered in abandoned warehouses, and compelled to mate with various partners for fear of being electrocuted with cattle prods. This direct and managed approach, free of any of the irritating restrictions imposed by regard for human dignity and freedom of choice, will provide the kind of real data that medical science requires to understand this strange phenomenon.
## Symptoms
Please note that the following section endeavours (unsuccessfully) to be an exhaustive list of possible symptoms, unlikely to be possessed by a single person. Should you meet a person with all the criteria below, please notify police immediately.
### Cultural
• Strong preference for odd music and weird food.
• Questionable fashion choices.
• Excessive modesty, or a complete lack of it.
• Belief that certain household pets are viable food sources, or that certain food sources should be revered as sacred entities.
• Celebration of holidays that don't appear in the calendar.
### Behavioural
• Acceptance of kissing and hugging between men as proper behaviour, even between normal heterosexual men.
• Increased likelihood to commit extremely serious minor crimes, like stealing food, necessitating long prison incarcerations at taxpayers' expense.
• Belief that social assistance payments are a basic human right, akin to the (illusion of) freedom of speech or even the (questionable) right to live.
• Cheap buying habits, such as going to a 99-cent store or Wal-mart whenever there's a sale. At the same time, the white majority still assume you got more money than THEY: white people. Ethnic stereotypes often involve consumer buying habits, and saving money like penny-pinching and welfare checks.
### Linguistic
• Unintelligible, seemly made-up strings of random vowels and consonants, clicks, or grunts.
• Heavily accented, mispronounced, and/or laborious delivery of English.
• Failure to comprehend or appropriately express common figures of speech.
• Exaggerated or excited manner of verbal presentation, the likes of which would never be seen from a television news anchorperson.
• Push for bilingualism, probably just as an excuse not to know English.
### Religious
• Belief in something other than God, or belief in some made-up god.
## Conclusion
It's been suggested that, eventually, white people will have to be added to the endangered species list along with the other lost and diminishing Great Whites: sharks, polar bears and golf balls. Already the newspapers and television channels are full of fatalists, crying out,"The Future Is Mocha! The Future Is Mocha!"
Actually, no... that may have been an advertisement for Starbucks rather than a statement on intermarriage.
But it is still postulated that segregation by way of government-protected rural species preserves or pretentiously-named urban gated communities might be the only hope to prevent the extinction of whites.
Robert Lee Moore (1882-1974), the noted racist mathematician perhaps expressed it best when he wrote:
$\frac{(Humanity \div Segregation) + Endogamy}{Time} = Happiness$
Failing that, learned experts also recommend some more common sense approaches. As such, risky behaviours like hanging out at hip hop clubs and engaging in conversation with your taxi driver are strongly discouraged.
|
2015-05-23 00:01:26
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.22483615577220917, "perplexity": 7637.538285804824}, "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-2015-22/segments/1432207926924.76/warc/CC-MAIN-20150521113206-00174-ip-10-180-206-219.ec2.internal.warc.gz"}
|
https://www.nextgurukul.in/wiki/concept/cbse/class-11/maths/sequences-and-series/sum-of-n-terms-of-special-series/3960986
|
Notes On Sum of n Terms of Special Series - CBSE Class 11 Maths
For an AP: a, a + d, a + 2d, a + 3d, a + 4d, ...., a + (n-1)d Sn = n/2 x [2a + (n - 1)d] or Sn = n/2 x [a + l], where l = a + (n-1)d For a G.P: a,ar,ar2,ar3,....,arn-1 …(i) Sn = $\text{{}\begin{array}{c}\text{na; if r=1}\\ \frac{\text{a(1-r}ⁿ\text{)}}{\text{1-r}}\text{; if r<1}\\ \frac{\text{a(r}ⁿ\text{-1)}}{\text{r-1}}\text{; if r>1}\end{array}\text{}}$ 1 + 2 + 3 + .... + n, i.e. the sum of first n natural numbers 12 + 22 + 32 + .... + n2, i.e. the sum of the squares of the first n natural numbers 13 + 23 + 33 + .... + n3 , i.e. the sum of the cubes of the first n natural numbers Sum of the first n natural numbers: Denote the sum of the first n natural numbers by Sn. Let this be equation 1. Sn = 1 + 2 + 3 + .... + (n - 1) + n ... (i) Again, Sn = n + (n - 1) + (n - 2) + .... + 2 + 1 ... (ii) Adding both sides of (i) and (ii), we get 2Sn = (1 + n) + (2 + n - 1) + (3 + n - 2) + .... + (n - 1 + 2) + (n + 1) ⇒ 2Sn = (n + 1) + (n + 1) + (n + 1) + --- (n + 1) + (n + 1) [n terms] ⇒ 2Sn = n(n + 1) ⇒ Sn = n(n + 1) / 2 Alternate method: Sn = 1 + 2 + 3 + .... + (n - 1) + n ... (i) Common difference (d) = 1 First term (a) = 1 Last term (l) = n Sn = n(a + l) / 2 = n(n + 1) / 2 Hence, the sum of the first n natural numbers is n(n + 1) / 2. 1 + 2 + 3 + ....+ n = $\sum _{\text{k=1}}^{\text{n}}\text{k}$ Example: Find the sum of the first five natural numbers. Let S5 = 1 + 2 + 3 + 4 + 5 Then, by the formula, we have S5 = 5(5 + 1) / 2 = 15. Sum of the squares of the first n natural numbers: Sn = 12 + 22 + 32 + ... + n2 Now, we use the identity, k3 - (k - 1)3 = 3k2 - 3k + 1 ... (i) Substituting k = 1, 2, 3, .... , n successively in (i), we get 13 - (0)3 = 3(1)2 - 3(1) + 1 23 - (1)3 = 3(2)2 - 3(2) + 1 33 - (2)3 = 3(3)2 - 3(3) + 1 ……………………………. ……………………………. n3 - (n - 1)3 = 3(n)2 - 3(n) + 1 Adding both sides of the above equations, we get 13 - 03 + 23 - 13 + 33 - 23 + (n-1)3 - (n-2)3 + n3 - (n - 1)3 = 3(12 + 22 + 32 + ... + n2) - 3(1+ 2 + 3 + ... + n) + n n3 = 3(12 + 22 + 32 + ... + n2) - 3(1+ 2 + 3 + ... + n) + n ⇒ 3(12 + 22 + 32 + ... + n2) = n3 + 3(1+ 2 + 3 + ... + n) - n ⇒ 3Sn = n3 + 3(n(n-1) / 2) - n [Since 1+ 2 + 3 + ... + n = n(n+1)/2] ⇒ 3Sn = (2n3+3n2+3n-2n)/2 = (2n3+3n2+n)/2 ⇒ Sn = n(2n2+3n+1)/6 ⇒ Sn = n(2n2+2n+n+1)/6 ⇒ Sn = $\frac{\text{n(n+1)(2n+1)}}{\text{6}}$ Hence, the sum of the squares of the first n natural numbers is $\frac{\text{n(n+1)(2n+1)}}{\text{6}}$ 12 + 22 + 32 + .... + n2 = $\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ Example: Find the sum of the squares of the first ten natural numbers. Let S10 denote the sum of the squares of the first ten natural numbers. Put n = 10 in Sn S10 = 10(10+1)(2x10+1)/6 = 385 Sum of the cubes of the first n natural numbers Sn = 13 + 23 + 33 + .... + n3 Now, we use the identity, (k+1)4 - k4 = 4k3 + 6k2 + 4k + 1 .... (i) Putting k = 1,2,3,...,n successively in (i), we get (2)4 - (1)4 = 4(1)3 + 6(1)2 + 4(1) + 1 (3)4 - (2)4 = 4(2)3 + 6(2)2 + 4(2) + 1 (4)4 - (3)4 = 4(3)3 + 6(3)2 + 4(3) + 1 ……………………………. ……………………………. (n + 1)4 - (n)4 = 4n3 + 6n2 + 4n + 1 Adding both sides of the above identities, we get (n + 1)4 - (1)4 = 4(13 + 23 + 33 + .... + n3) + 6(12 + 22 + 32 + .... + n2) + 4(1 + 2 + 3 + ....+ n) + n ⇒ (n + 1)4 - (1)4 = 4Sn + 6$\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ + 4$\sum _{\text{k=1}}^{\text{n}}\text{k}$ + n ⇒ 4Sn = (n + 1)4 - (1)4 - 6$\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ - 4$\sum _{\text{k=1}}^{\text{n}}\text{k}$ - n ⇒ 4Sn = n4 + 4n3 + 6n2 + 4n - 6 x (n(n+1)(2n+1)/6) - 4 x (n(n+1)/2) - n [ Since $\sum _{\text{k=1}}^{\text{n}}\text{k}$ = n(n+1)/2 and $\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ = n(n+1)(2n+1)/6] ⇒ 4Sn = n4 + 4n3 + 6n2 + 4n - 2n3 - 3n2 - n - 2n2 - 2n -n ⇒ 4Sn = n4 + 2n3 + n2 ⇒ 4Sn = n2(n2 + 2n + 1) ⇒ 4Sn = n2(n + 1)2 ⇒ Sn = n2(n + 1)2/4 ⇒ Sn = (n(n + 1)/2)2 Hence, the sum of the cubes of the first n natural numbers is (n(n + 1)/2)2 . 13 + 23 + 33+....+ n3 =$\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{3}}$ Example: Find the sum of the cubes of the first seven natural numbers. Let S7 denotes the sum of cubes first seven natural numbers. Then, by the formula, we have S7 = (7(7+1)/2)2 = 784
#### Summary
For an AP: a, a + d, a + 2d, a + 3d, a + 4d, ...., a + (n-1)d Sn = n/2 x [2a + (n - 1)d] or Sn = n/2 x [a + l], where l = a + (n-1)d For a G.P: a,ar,ar2,ar3,....,arn-1 …(i) Sn = $\text{{}\begin{array}{c}\text{na; if r=1}\\ \frac{\text{a(1-r}ⁿ\text{)}}{\text{1-r}}\text{; if r<1}\\ \frac{\text{a(r}ⁿ\text{-1)}}{\text{r-1}}\text{; if r>1}\end{array}\text{}}$ 1 + 2 + 3 + .... + n, i.e. the sum of first n natural numbers 12 + 22 + 32 + .... + n2, i.e. the sum of the squares of the first n natural numbers 13 + 23 + 33 + .... + n3 , i.e. the sum of the cubes of the first n natural numbers Sum of the first n natural numbers: Denote the sum of the first n natural numbers by Sn. Let this be equation 1. Sn = 1 + 2 + 3 + .... + (n - 1) + n ... (i) Again, Sn = n + (n - 1) + (n - 2) + .... + 2 + 1 ... (ii) Adding both sides of (i) and (ii), we get 2Sn = (1 + n) + (2 + n - 1) + (3 + n - 2) + .... + (n - 1 + 2) + (n + 1) ⇒ 2Sn = (n + 1) + (n + 1) + (n + 1) + --- (n + 1) + (n + 1) [n terms] ⇒ 2Sn = n(n + 1) ⇒ Sn = n(n + 1) / 2 Alternate method: Sn = 1 + 2 + 3 + .... + (n - 1) + n ... (i) Common difference (d) = 1 First term (a) = 1 Last term (l) = n Sn = n(a + l) / 2 = n(n + 1) / 2 Hence, the sum of the first n natural numbers is n(n + 1) / 2. 1 + 2 + 3 + ....+ n = $\sum _{\text{k=1}}^{\text{n}}\text{k}$ Example: Find the sum of the first five natural numbers. Let S5 = 1 + 2 + 3 + 4 + 5 Then, by the formula, we have S5 = 5(5 + 1) / 2 = 15. Sum of the squares of the first n natural numbers: Sn = 12 + 22 + 32 + ... + n2 Now, we use the identity, k3 - (k - 1)3 = 3k2 - 3k + 1 ... (i) Substituting k = 1, 2, 3, .... , n successively in (i), we get 13 - (0)3 = 3(1)2 - 3(1) + 1 23 - (1)3 = 3(2)2 - 3(2) + 1 33 - (2)3 = 3(3)2 - 3(3) + 1 ……………………………. ……………………………. n3 - (n - 1)3 = 3(n)2 - 3(n) + 1 Adding both sides of the above equations, we get 13 - 03 + 23 - 13 + 33 - 23 + (n-1)3 - (n-2)3 + n3 - (n - 1)3 = 3(12 + 22 + 32 + ... + n2) - 3(1+ 2 + 3 + ... + n) + n n3 = 3(12 + 22 + 32 + ... + n2) - 3(1+ 2 + 3 + ... + n) + n ⇒ 3(12 + 22 + 32 + ... + n2) = n3 + 3(1+ 2 + 3 + ... + n) - n ⇒ 3Sn = n3 + 3(n(n-1) / 2) - n [Since 1+ 2 + 3 + ... + n = n(n+1)/2] ⇒ 3Sn = (2n3+3n2+3n-2n)/2 = (2n3+3n2+n)/2 ⇒ Sn = n(2n2+3n+1)/6 ⇒ Sn = n(2n2+2n+n+1)/6 ⇒ Sn = $\frac{\text{n(n+1)(2n+1)}}{\text{6}}$ Hence, the sum of the squares of the first n natural numbers is $\frac{\text{n(n+1)(2n+1)}}{\text{6}}$ 12 + 22 + 32 + .... + n2 = $\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ Example: Find the sum of the squares of the first ten natural numbers. Let S10 denote the sum of the squares of the first ten natural numbers. Put n = 10 in Sn S10 = 10(10+1)(2x10+1)/6 = 385 Sum of the cubes of the first n natural numbers Sn = 13 + 23 + 33 + .... + n3 Now, we use the identity, (k+1)4 - k4 = 4k3 + 6k2 + 4k + 1 .... (i) Putting k = 1,2,3,...,n successively in (i), we get (2)4 - (1)4 = 4(1)3 + 6(1)2 + 4(1) + 1 (3)4 - (2)4 = 4(2)3 + 6(2)2 + 4(2) + 1 (4)4 - (3)4 = 4(3)3 + 6(3)2 + 4(3) + 1 ……………………………. ……………………………. (n + 1)4 - (n)4 = 4n3 + 6n2 + 4n + 1 Adding both sides of the above identities, we get (n + 1)4 - (1)4 = 4(13 + 23 + 33 + .... + n3) + 6(12 + 22 + 32 + .... + n2) + 4(1 + 2 + 3 + ....+ n) + n ⇒ (n + 1)4 - (1)4 = 4Sn + 6$\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ + 4$\sum _{\text{k=1}}^{\text{n}}\text{k}$ + n ⇒ 4Sn = (n + 1)4 - (1)4 - 6$\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ - 4$\sum _{\text{k=1}}^{\text{n}}\text{k}$ - n ⇒ 4Sn = n4 + 4n3 + 6n2 + 4n - 6 x (n(n+1)(2n+1)/6) - 4 x (n(n+1)/2) - n [ Since $\sum _{\text{k=1}}^{\text{n}}\text{k}$ = n(n+1)/2 and $\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{2}}$ = n(n+1)(2n+1)/6] ⇒ 4Sn = n4 + 4n3 + 6n2 + 4n - 2n3 - 3n2 - n - 2n2 - 2n -n ⇒ 4Sn = n4 + 2n3 + n2 ⇒ 4Sn = n2(n2 + 2n + 1) ⇒ 4Sn = n2(n + 1)2 ⇒ Sn = n2(n + 1)2/4 ⇒ Sn = (n(n + 1)/2)2 Hence, the sum of the cubes of the first n natural numbers is (n(n + 1)/2)2 . 13 + 23 + 33+....+ n3 =$\sum _{\text{k = 1}}^{\text{n}}{\text{k}}^{\text{3}}$ Example: Find the sum of the cubes of the first seven natural numbers. Let S7 denotes the sum of cubes first seven natural numbers. Then, by the formula, we have S7 = (7(7+1)/2)2 = 784
Next
|
2021-06-19 12:39:18
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 34, "mathjax_tag": 2, "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": 0.7639036178588867, "perplexity": 551.318623346726}, "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/1623487648194.49/warc/CC-MAIN-20210619111846-20210619141846-00260.warc.gz"}
|
https://stats.stackexchange.com/questions/544309/why-is-naive-bayes-overconfident/544411
|
# Why is naive Bayes overconfident?
In the fourth edition of "Artificial Intelligence: a modern approach" by Russel and Norvig, they write in section 12.6, regarding the Naive Bayes Model for text classification, the following:
The naive Bayes model assumes that words occur independently in documents, with frequencies determined by the document category. This independence assumption is clearly violated in practice. For example, the phrase “first quarter” occurs more frequently in business (or sports) articles than would be suggested by multiplying the probabilities of “first” and “quarter.” The violation of independence usually means that the final posterior probabilities will be much closer to 1 or 0 than they should be; in other words, the model is overconfident in its predictions. On the other hand, even with these errors, the ranking of the possible categories is often quite accurate.
(Emphasis mine)
I do not see why the assumption of conditional independence would lead the naive Bayes model to be overly confident in its predictions.
Just to make sure I understand their statement correctly, I assume that they mean that that naive Bayes is overly confident compared to non-naive Bayes. As an example, assuming we wanted to determine whether an article is a sport article or not and that $$s = \text{sport article}, f=\text{"first" occured in article}, q=\text{"quarter" occured in article}$$, we get the non-naive Bayes model as:
$$P(s | f, q) = \dfrac{P(s)P(f, q|s)}{P(f, q|s)P(s)+P(f, q|\neg s)P(\neg s)} \quad (1.)$$
The conditional independence assumption then gives the naive Bayes model
\begin{align} P(s | f, q) = \dfrac{P(s)P(f, q|s)}{P(f, q|s)P(s)+P(f, q|\neg s)P(\neg s)} = \\\\ \dfrac{P(s)P(f|s)P(q|s)}{P(f|s)P( q|s)P(s)+P(f|\neg s)P(q|\neg s)P(\neg s)} \quad (2.) \end{align}
As far as I can see then, the statement amounts to saying that the numerator and denominator in $$(2.)$$ usually takes the value further away from $$0.5$$ than in $$(1.)$$. Is there a theoretical explanation for this or is it more of an empirical fact that just happens to be true?
• I guess you are missing the prior probabilities in the partition function (denominator) in (1) and (2) Sep 11, 2021 at 23:02
• Consider what happens when $P(f,q|s)$ is substantially greater than $P(f|s)P(q|s)$. Sep 12, 2021 at 2:18
• @SandipanDey Good catch! I've edited that now. Sep 12, 2021 at 7:12
• what about the fact that P(quarter|sport) < P(quarter|prev_word = first, category = sport) ? Sep 12, 2021 at 9:15
For sports articles, the given bigram is much more frequent and will have much higher probability than the one is computed with naive assumption as the product of the probabilities for the corresponding unigrams, when compared to non-sports articles, as per assumption, so that we have,
$$\frac{P(f, q|s)}{P(f|s)P(q|s)} \gg \frac{P(f, q|\neg s)}{P(f|\neg s)P(q|\neg s)} \quad \quad (3)$$
here all probability values are $$\in [0,1]$$.
Now, we have, in (1), (ignoring the trivial cases and assuming non-zero probabilities)
$$P(s | f, q) = \dfrac{P(s)P(f, q|s)}{P(f, q|s)P(s)+P(f, q|\neg s)P(\neg s)} = \dfrac{1}{1+\dfrac{P(f, q|\neg s)P(\neg s)}{P(f, q|s)P(s)}}$$
Similarly from (2), we have,
$$P_{naive}(s | f, q) = \dfrac{P(s)P(f|s)P(q|s)}{P(f|s)P( q|s)p(s)+P(f|\neg s)P(q|\neg s)P(\neg s)}=\dfrac{1}{1+\dfrac{P(f|\neg s)P(q|\neg s)P(\neg s)}{P(f|s)P(q|s)P(s)}}$$
$$\Rightarrow\frac{P_{naive}(s|f,q)}{P(s|f,q)}$$
$$=\dfrac{1+\dfrac{P(f, q|\neg s)P(\neg s)}{P(f, q|s)P(s)}}{1+\dfrac{P(f|\neg s)P(q|\neg s)P(\neg s)}{P(f|s)P(q|s)P(s)}}$$
$$=\dfrac{1+\dfrac{P(f, q|\neg s)}{P(f, q|s)}}{1+\dfrac{P(f|\neg s)P(q|\neg s)}{P(f|s)P(q|s)}}$$, (let's assume a uniform prior for simplicity, i.e., $$P(s)=P(\neg s)$$)
$$\gg 1$$, from (3)
Hence, we have,
$$P_{naive}(s|f,q) \gg P(s|f,q)$$ (overconfidence)
• Thank you for this. This is an interesting answer. I think $(3.)$ is a reasonable assumption, but the conclusion of the answer somewhat evades me. From the quoted passage, it seems that by overconfidence, the author means probabilities closer to $0$ or to $1$ than warranted. But $P_{naive}(s|f,q) >> P(s|f, q)$ does not necessarily imply that $P(s|f, q)$ is closer to $0.5$ than $P_{naive}(s|f,q)$. I think that the validity of such an implication depends on exactly how much bigger $P_{naive}(s|f,q)$ is than $P(s|f,q)$ and the value of $P(s|f,q)$. Sep 12, 2021 at 15:34
• @Paradox if $P(s|f,q)$ is around $0.5$ (the original posterior has high uncertainty), it implies that the naive approximation $P_{naive}(s|f,q)$ reduces the uncertainty (entropy) by increasing the value closer to $1$ and simultaneously decreasing the value of $P_{naive}(\neg s|f,q)$ down to $0$. Sep 12, 2021 at 15:54
• Yes, that is true. But doesn't this kind of beg the question then whether there actually is any empirical or theoretical reason for why $P(s|f,q)$ would be around $0.5$? Sep 12, 2021 at 16:51
• @Paradox I guess with the same assumption (3) we can derive another corollary that when $P(s|f,q)$ is low (close to $0$) / very confident (with low uncertainty) then the naive assumption can increase the uncertainty of the predicted posterior making it under-confident. That way the conditional independence assumption is bad in this case too. Sep 12, 2021 at 17:59
|
2022-09-30 21:31:31
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 16, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9158766269683838, "perplexity": 600.264312176767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335504.37/warc/CC-MAIN-20220930212504-20221001002504-00171.warc.gz"}
|
https://hal-insu.archives-ouvertes.fr/insu-03748045
|
# The Cosmic Ultraviolet Baryon Survey (CUBS) - I. Overview and the diverse environments of Lyman limit systems at z < 1
Abstract : We present initial results from the Cosmic Ultraviolet Baryon Survey (CUBS). CUBS is designed to map diffuse baryonic structures at redshift z ≲ 1 using absorption-line spectroscopy of 15 UV-bright QSOs with matching deep galaxy survey data. CUBS QSOs are selected based on their NUV brightness to avoid biases against the presence of intervening Lyman limit systems (LLSs) at zabs < 1. We report five new LLSs of $\log \, N({\mathrm{ H} \,{\small I}})/{{\rm cm^{-2}}}rsim 17.2$ over a total redshift survey path-length of $\Delta \, z_{\mathrm{ LL}}=9.3$ , and a number density of $n(z)=0.43_{-0.18}^{+0.26}$ . Considering all absorbers with $\log \, N({{\mathrm{ H} \,{\small I}}})/{{\rm cm^{-2}}} 16.5$ leads to $n(z)=1.08_{-0.25}^{+0.31}$ at zabs < 1. All LLSs exhibit a multicomponent structure and associated metal transitions from multiple ionization states such as C II, C III, Mg II, Si II, Si III, and O VI absorption. Differential chemical enrichment levels as well as ionization states are directly observed across individual components in three LLSs. We present deep galaxy survey data obtained using the VLT-MUSE integral field spectrograph and the Magellan Telescopes, reaching sensitivities necessary for detecting galaxies fainter than $0.1\, L_*$ at d ≲ 300 physical kpc (pkpc) in all five fields. A diverse range of galaxy properties is seen around these LLSs, from a low-mass dwarf galaxy pair, a co-rotating gaseous halo/disc, a star-forming galaxy, a massive quiescent galaxy, to a galaxy group. The closest galaxies have projected distances ranging from d = 15 to 72 pkpc and intrinsic luminosities from ${\approx} 0.01\, L_*$ to ${\approx} 3\, L_*$ . Our study shows that LLSs originate in a variety of galaxy environments and trace gaseous structures with a broad range of metallicities.
Keywords :
Document type :
Journal articles
Domain :
https://hal-insu.archives-ouvertes.fr/insu-03748045
Contributor : Nathalie POTHIER Connect in order to contact the contributor
Submitted on : Tuesday, August 9, 2022 - 10:52:23 AM
Last modification on : Wednesday, August 10, 2022 - 3:46:36 AM
### File
staa1773.pdf
Publisher files allowed on an open archive
### Citation
Hsiao-Wen Chen, Fakhri S. Zahedy, Erin Boettcher, Thomas M. Cooper, Sean D. Johnson, et al.. The Cosmic Ultraviolet Baryon Survey (CUBS) - I. Overview and the diverse environments of Lyman limit systems at z < 1. Monthly Notices of the Royal Astronomical Society, 2020, 497, pp.498-520. ⟨10.1093/mnras/staa1773⟩. ⟨insu-03748045⟩
Record views
|
2022-10-04 15:50:58
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6620882153511047, "perplexity": 9096.248256505492}, "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/1664030337516.13/warc/CC-MAIN-20221004152839-20221004182839-00710.warc.gz"}
|
https://stats.stackexchange.com/questions/76668/how-to-calculate-beta-and-coefficient-of-determination-r2-from-unstandardiz?noredirect=1
|
# How to calculate Beta and coefficient of determination ($R^2$) from unstandardized coefficients in OLS regression? [duplicate]
I have a table in which the multiple linear regression results is provided. If I have unstandardized coefficients and standard error for each independent variable, is it possible to calculate standardized coefficient (Beta) and coefficient of determination ($R^2$) from these data.
My friend provided $R^2=0.41$ for these data, but I doubt if the results are reliable. I brought the table in the following, would you please compute Beta and R-squared for me to compare with the original table?
the descriptive statistics is the following table:
• This question cannot be answered with the information given: the computation requires the standard deviations of the explanatory variables. Please consult threads associated with the standardization tag, such as stats.stackexchange.com/questions/74622/…, which shows how to compute the unstandardized coefficients from the betas (and, when carried out in reverse, essentially answers your question). – whuber Nov 15 '13 at 16:32
• thanks a lot for the comment. I provided standard deviations of variables. then is it possible to estimate beta and r-squared from these data? – user31315 Nov 15 '13 at 17:01
• Yes: use the method explained in the answer I referenced. – whuber Nov 15 '13 at 17:04
• Among other places, see also a comment here, to convert b to beta. – ttnphns Nov 15 '13 at 17:07
• @whuber and ttnphns: Thank you. sorry if I'm taking your time. I don't know anything about statistics and I could not understand the formula. would you please do the procedure for one variable to see how you substitute the values in the formula mentioned in the link you provided and calculate beta and R-squared ? for example for variable 'r' in the above table – user31315 Nov 15 '13 at 17:25
|
2020-01-23 04:38:16
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7986935973167419, "perplexity": 564.8368466806436}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250608295.52/warc/CC-MAIN-20200123041345-20200123070345-00106.warc.gz"}
|
https://support.sas.com/documentation/cdl/en/statug/66103/HTML/default/statug_variogram_details32.htm
|
# The VARIOGRAM Procedure
### Computational Resources
The fundamental computation of the VARIOGRAM procedure is binning: for each pair of observations in the input data set, a distance class and an angle class are determined and recorded. Let denote the number of distance classes, denote the number of angle classes, and denote the number of VAR variables. The memory requirements for these operations are proportional to . This is typically small.
The CPU time required for the computations is proportional to the number of pairs of observations, or to , where N is the number of observations in the input data set.
|
2021-10-23 19:45:51
|
{"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": 0.9748775959014893, "perplexity": 409.81744652266025}, "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/1634323585768.3/warc/CC-MAIN-20211023193319-20211023223319-00485.warc.gz"}
|
https://research-explorer.app.ist.ac.at/record/7629
|
# Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains
Forkert DL. 2020. Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. IST Austria.
Thesis_Forkert_PDFA.pdf 3.30 MB
Thesis | Published | English
Author
Supervisor
Department
Series Title
IST Austria Thesis
Abstract
Publishing Year
Date Published
2020-03-31
Page
154
ISSN
IST-REx-ID
### Cite this
Forkert DL. Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. 2020. doi:10.15479/AT:ISTA:7629
Forkert, D. L. (2020). Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. IST Austria. https://doi.org/10.15479/AT:ISTA:7629
Forkert, Dominik L. “Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains.” IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:7629.
D. L. Forkert, “Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains,” IST Austria, 2020.
Forkert DL. 2020. Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains. IST Austria.
Forkert, Dominik L. Gradient Flows in Spaces of Probability Measures for Finite-Volume Schemes, Metric Graphs and Non-Reversible Markov Chains. IST Austria, 2020, doi:10.15479/AT:ISTA:7629.
All files available under the following license(s):
This Item is protected by copyright and/or related rights. [...]
Main File(s)
File Name
Access Level
Open Access
2020-04-14
MD5 Checksum
c814a1a6195269ca6fe48b0dca45ae8a
Source File
File Name
Access Level
Closed Access
|
2022-05-27 13:25:01
|
{"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": 0.8959805965423584, "perplexity": 10716.450551546748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662647086.91/warc/CC-MAIN-20220527112418-20220527142418-00438.warc.gz"}
|
https://math.stackexchange.com/questions/472134/what-is-the-relation-between-a-banach-space-and-a-hilbert-space/472141
|
# What is the relation between a Banach space and a Hilbert space? [closed]
What is the relation between a Banach space and a Hilbert space?
I know that a Hilbert space has inner product (and so a norm), but a Banach space has a just norm.
• Have a look at this answer: qr.ae/TWI4tc.
– user168764
May 4, 2019 at 23:03
• This is a bit vague, but there are useful remarks that could be made in answers to it. Specifically, Hilbert spaces have a (true!) minimum/Dirichlet principle, and Banach spaces easily and non-pathologically fail this (e.g., the literally incorrect Dirichlet principle that was very important throughout the late 19th century, and only reformulated in 1905 by Giuseppe ("Beppo") Levi in 1905, and also Frobenius. (In Banach spaces, a non-empty closed convex subset need not have an element of least norm, and it could have infinitely-many. In Hilbert spaces there is a unique minimizing element.) May 4, 2019 at 23:09
• @paulgarrett: Your (interesting!) comment draws my attention to this question. I have voted to reopen the post: the post provides enough context for anyone knows the basics of functional analysis (and it was posted on 2013 !). What a pity it would be if this post was deleted before your comment. I have not seen such comment/answer from an expert in this site for a very long time. Your words remind me of my old good times here. THANK YOU!
– user9464
May 23, 2019 at 3:15
Hilbert spaces are a stricter subset of Banach spaces but they have the additional structure of an inner product which allows you to talk about orthonormal bases, unitary operators and so on. Example: Fourier transform theory is really beautiful on $L^2(\mathbb{R})$ but it is much more complicated on other Banach spaces because you don't have a notion of self-duality like in $L^2(\mathbb{R})$. You actually have to abstract a lot to define the Fourier transform on other $L^p$ spaces.
• I think we should also mention the existence and uniqueness of best approximations in closed subspaces. This has far reaching consequences in other fields of science, such as physics and engineering (and, naturally, is very instrumental in Fourier analysis as well). Aug 20, 2013 at 16:54
• Apart from Fourier Analysis, we may also consider the results of Approximation Theory. For the same reason Hilbert Space is much useful than Banach Space. Aug 20, 2013 at 17:02
Hilbert spaces have an easier structure and are in a way (most often infinite dimensional) Euclidian spaces. However, many spaces of interest that are Banach spaces are not Hilbert spaces, hence they are important too.
To see if a Banach space is a Hilbert space it suffice to show that the norm satisfies the parallelogram law. In other words, if we have a Banach space $$X$$ such that $$\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2$$ for all $$x,y\in X$$ then $$X$$ is actually a Hilbert space.
To deduce how the norm looks like is a good exercise. In the real case (the complex case is similar) think of the expansions of $$\|x+y\|^2=\langle x+y,x+y\rangle$$ and $$\|x-y\|^2=\langle x-y,x-y\rangle.$$ The result is $$\langle x,y\rangle=\frac{\|x+y\|^2-\|x-y\|^2}{4}.$$
I don't think they are "better", per se. A Hilbert space is a very special type of Banach space - one which is meant to generalize familiar notions from $\mathbb{R}^n$. (For instance, you can quite naturally speak of when two vectors in Hilbert space are orthogonal).
In general, Hilbert spaces are "easier" to understand than general Banach spaces, and are usually a good place to start if you are learning the subject (For instance, try to see why the Hahn-Banach theorem is much simpler for Hilbert spaces)
Since most of the time we deal with $$\mathbb{R}^n$$, you can say that one has advantage over the other in $$\mathbb{R}^n$$. So I consider $$\mathbb{R}^n$$ and explain why Hilbert spaces are more useful in practice:
Hilbert space is defined as a vector space $$H$$ equipped with an inner product $$\langle \cdot,\cdot \rangle$$, consequently this inner product induces a norm which we call it a metric, $$d(\cdot,\cdot)$$. Hence, it is the time to define a new vector space $$M$$ endowed with the mentioned metric $$d(\cdot,\cdot)$$ which is a metric vector space. Now if every Cauchy sequence in $$M$$ converges to a point in $$M$$ we have Banach space which is a complete metric space. In $$\mathbb{R}^n$$ the relation is as follows:
Keep in mind that in a vector space $$M$$, the only tool that we have is the metric $$d(\cdot,\cdot)$$, so we can talk about length of a vector or distance between vectors.
However, when we have inner product that satisfies Parallelogram law, we also have an inner product through which we can define projection from one vector to another which is of interest in practice because we can talk about the angles. Using this concept, optimization problems such as minimum distance to a subspace can be defined as projection of that point onto a set.
|
2022-06-28 19:40:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8994454741477966, "perplexity": 171.8383786464819}, "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/1656103573995.30/warc/CC-MAIN-20220628173131-20220628203131-00091.warc.gz"}
|
https://en.wikipedia.org/wiki/Hypergraphs
|
# Hypergraph
(Redirected from Hypergraphs)
An example of an undirected hypergraph, with ${\displaystyle X=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}}$ and ${\displaystyle E=\{e_{1},e_{2},e_{3},e_{4}\}=}$ ${\displaystyle \{\{v_{1},v_{2},v_{3}\},}$ ${\displaystyle \{v_{2},v_{3}\},}$ ${\displaystyle \{v_{3},v_{5},v_{6}\},}$ ${\displaystyle \{v_{4}\}\}}$. This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors.
Alternative representation of the hypergraph reported in the figure above, called PAOH.[1] Edges are vertical lines connecting vertices. V7 is an isolated vertex. Vertices are aligned to the left. The legend on the right shows the names of the edges.
An example of a directed hypergraph, with ${\displaystyle X=\{1,2,3,4,5,6\}}$ and ${\displaystyle E=\{a_{1},a_{2},a_{3},a_{4},a_{5}\}=}$ ${\displaystyle \{(\{1\},\{2\}),}$ ${\displaystyle (\{2\},\{3\}),}$ ${\displaystyle (\{3\},\{1\}),}$ ${\displaystyle (\{2,3\},\{4,5\}),}$ ${\displaystyle (\{3,5\},\{6\})\}}$.
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, an undirected hypergraph ${\displaystyle H}$ is a pair ${\displaystyle H=(X,E)}$ where ${\displaystyle X}$ is a set of elements called nodes or vertices, and ${\displaystyle E}$ is (in an undirected hypergraph) a set of non-empty subsets of ${\displaystyle X}$ called hyperedges or edges. Therefore, ${\displaystyle E}$ is a subset of ${\displaystyle {\mathcal {P}}(X)\setminus \{\emptyset \}}$, where ${\displaystyle {\mathcal {P}}(X)}$ is the power set of ${\displaystyle X}$. The size of the vertex set is called the order of the hypergraph, and the size of edges set is the size of the hypergraph.
A directed hypergraph differs in that its hyperedges are not sets, but an ordered pair of subsets of ${\displaystyle X}$, constituting the tail and head of the hyperedge.
While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. An undirected hypergraph is also called a set system or a family of sets drawn from the universal set.
Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.
Hypergraphs have many other names. In computational geometry, an undirected hypergraph may sometimes be called a range space and then the hyperedges are called ranges.[2] In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. In some literature edges are referred to as hyperlinks or connectors.[3]
The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.
## Applications
Undirected hypergraphs are useful in modelling such things as satisfiability problems,[4] databases,[5] machine learning,[6] and Steiner tree problems.[7] They have been extensively used in machine learning tasks as the data model and classifier regularization (mathematics).[8] The applications include recommender system (communities as hyperedges),[9] image retrieval (correlations as hyperedges),[10] and bioinformatics (biochemical interactions as hyperedges).[11] Representative hypergraph learning techniques include hypergraph spectral clustering that extends the spectral graph theory with hypergraph Laplacian,[12] and hypergraph semi-supervised learning that introduces extra hypergraph structural cost to restrict the learning results.[13] For large scale hypergraphs, a distributed framework[6] built using Apache Spark is also available.
Directed hypergraphs can be used to model things including telephony applications,[14] detecting money laundering,[15] operations research,[16] and transportation planning. They can also be used to model Horn-satisfiability.[17]
## Generalizations of concepts from graphs
Many theorems and concepts involving graphs also hold for hypergraphs, in particular:
In directed hypergraphs: transitive closure, and shortest path problems.[16]
## Hypergraph drawing
This circuit diagram can be interpreted as a drawing of a hypergraph in which four vertices (depicted as white rectangles and disks) are connected by three hyperedges drawn as trees.
Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.
In one possible visual representation for hypergraphs, similar to the standard graph drawing style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves.[18][19] If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as simple closed curves that enclose sets of points.[20][21][22]
An order-4 Venn diagram, which can be interpreted as a subdivision drawing of a hypergraph with 15 vertices (the 15 colored regions) and 4 hyperedges (the 4 ellipses).
In another style of hypergraph visualization, the subdivision model of hypergraph drawing,[23] the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. The hyperedges of the hypergraph are represented by contiguous subsets of these regions, which may be indicated by coloring, by drawing outlines around them, or both. An order-n Venn diagram, for instance, may be viewed as a subdivision drawing of a hypergraph with n hyperedges (the curves defining the diagram) and 2n − 1 vertices (represented by the regions into which these curves subdivide the plane). In contrast with the polynomial-time recognition of planar graphs, it is NP-complete to determine whether a hypergraph has a planar subdivision drawing,[24] but the existence of a drawing of this type may be tested efficiently when the adjacency pattern of the regions is constrained to be a path, cycle, or tree.[25]
An alternative representation of the hypergraph called PAOH[1] is shown in the figure on top of this article. Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges. It has been designed for dynamic hypergraphs but can be used for simple hypergraphs as well.
## Hypergraph coloring
Classic hypergraph coloring is assigning one of the colors from set ${\displaystyle \{1,2,3,...\lambda \}}$ to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. In other words, there must be no monochromatic hyperedge with cardinality at least 2. In this sense it is a direct generalization of graph coloring. Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph.
Hypergraphs for which there exists a coloring using up to k colors are referred to as k-colorable. The 2-colorable hypergraphs are exactly the bipartite ones.
There are many generalizations of classic hypergraph coloring. One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Some mixed hypergraphs are uncolorable for any number of colors. A general criterion for uncolorability is unknown. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. See http://spectrum.troy.edu/voloshin/mh.html for details.
## Properties of hypergraphs
A hypergraph can have various properties, such as:
• Empty - has no edges.
• Non-simple (or multiple) - has loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices.
• Simple - has no loops and no repeated edges.
• ${\displaystyle d}$-regular - every vertex has degree ${\displaystyle d}$, i.e., contained in exactly ${\displaystyle d}$ hyperedges.
• 2-colorable - its vertices can be partitioned into two classes U and V in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. An alternative term is Property B.
• ${\displaystyle k}$-uniform - each hyperedge contains precisely ${\displaystyle k}$ vertices.
• ${\displaystyle k}$-partite - the vertices are partitioned into ${\displaystyle k}$ parts, and each hyperedge contains precisely one vertex of each type.
• Every ${\displaystyle k}$-partite hypergraph (for ${\displaystyle k\geq 2}$) is both ${\displaystyle k}$-uniform and bipartite (and 2-colorable).
• Downward-closed - every subset of an undirected hypergraph's edges is a hyperedge too. A downward-closed hypergraph is usually called an abstract simplicial complex.
• An abstract simplicial complex with an additional property called augmentation property is called a matroid.
## Related hypergraphs
Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphs, partial hypergraphs and section hypergraphs.
Let ${\displaystyle H=(X,E)}$ be the hypergraph consisting of vertices
${\displaystyle X=\lbrace x_{i}\mid i\in I_{v}\rbrace ,}$
and having edge set
${\displaystyle E=\lbrace e_{i}\mid i\in I_{e}\land e_{i}\subseteq X\land e_{i}\neq \emptyset \rbrace ,}$
where ${\displaystyle I_{v}}$ and ${\displaystyle I_{e}}$ are the index sets of the vertices and edges respectively.
A subhypergraph is a hypergraph with some vertices removed. Formally, the subhypergraph ${\displaystyle H_{A}}$ induced by ${\displaystyle A\subseteq X}$ is defined as
${\displaystyle H_{A}=\left(A,\lbrace e\cap A\mid e\in E\land e\cap A\neq \emptyset \rbrace \right).}$
An alternative term is the restriction of H to A.[26]: 468
An extension of a subhypergraph is a hypergraph where each hyperedge of ${\displaystyle H}$ which is partially contained in the subhypergraph ${\displaystyle H_{A}}$ is fully contained in the extension ${\displaystyle Ex(H_{A})}$. Formally
${\displaystyle Ex(H_{A})=(A\cup A',E')}$ with ${\displaystyle A'=\bigcup _{e\in E}e\setminus A}$ and ${\displaystyle E'=\lbrace e\in E\mid e\subseteq (A\cup A')\rbrace }$.
The partial hypergraph is a hypergraph with some edges removed.[26]: 468 Given a subset ${\displaystyle J\subset I_{e}}$ of the edge index set, the partial hypergraph generated by ${\displaystyle J}$ is the hypergraph
${\displaystyle \left(X,\lbrace e_{i}\mid i\in J\rbrace \right).}$
Given a subset ${\displaystyle A\subseteq X}$, the section hypergraph is the partial hypergraph
${\displaystyle H\times A=\left(A,\lbrace e_{i}\mid i\in I_{e}\land e_{i}\subseteq A\rbrace \right).}$
The dual ${\displaystyle H^{*}}$ of ${\displaystyle H}$ is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by ${\displaystyle \lbrace e_{i}\rbrace }$ and whose edges are given by ${\displaystyle \lbrace X_{m}\rbrace }$ where
${\displaystyle X_{m}=\lbrace e_{i}\mid x_{m}\in e_{i}\rbrace .}$
When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i.e.,
${\displaystyle \left(H^{*}\right)^{*}=H.}$
A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G' of G is a host of the corresponding H'.
The 2-section (or clique graph, representing graph, primal graph, Gaifman graph) of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.
## Incidence matrix
Let ${\displaystyle V=\{v_{1},v_{2},~\ldots ,~v_{n}\}}$ and ${\displaystyle E=\{e_{1},e_{2},~\ldots ~e_{m}\}}$. Every hypergraph has an ${\displaystyle n\times m}$ incidence matrix.
For an undirected hypergraph, ${\displaystyle A=(a_{ij})}$ where
${\displaystyle a_{ij}=\left\{{\begin{matrix}1&\mathrm {if} ~v_{i}\in e_{j}\\0&\mathrm {otherwise} .\end{matrix}}\right.}$
The transpose ${\displaystyle A^{t}}$ of the incidence matrix defines a hypergraph ${\displaystyle H^{*}=(V^{*},\ E^{*})}$ called the dual of ${\displaystyle H}$, where ${\displaystyle V^{*}}$ is an m-element set and ${\displaystyle E^{*}}$ is an n-element set of subsets of ${\displaystyle V^{*}}$. For ${\displaystyle v_{j}^{*}\in V^{*}}$ and ${\displaystyle e_{i}^{*}\in E^{*},~v_{j}^{*}\in e_{i}^{*}}$ if and only if ${\displaystyle a_{ij}=1}$.
For a directed hypergraph, the heads and tails of each hyperedge ${\displaystyle e_{j}}$ are denoted by ${\displaystyle H(e_{j})}$ and ${\displaystyle T(e_{j})}$ respectively.[17] ${\displaystyle A=(a_{ij})}$ where
${\displaystyle a_{ij}=\left\{{\begin{matrix}-1&\mathrm {if} ~v_{i}\in T(e_{j})\\1&\mathrm {if} ~v_{i}\in H(e_{j})\\0&\mathrm {otherwise} .\end{matrix}}\right.}$
### Incidence graph
A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and (x1, e1) are connected with an edge if and only if vertex x1 is contained in edge e1 in H.
Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. This bipartite graph is also called incidence graph.
## Cycles
In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphs, there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs.
A first definition of acyclicity for hypergraphs was given by Claude Berge:[27] a hypergraph is Berge-acyclic if its incidence graph (the bipartite graph defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair ${\displaystyle v\neq v'}$ of vertices and some pair ${\displaystyle f\neq f'}$ of hyperedges such that ${\displaystyle v,v'\in f}$ and ${\displaystyle v,v'\in f'}$, then it is Berge-cyclic. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.
We can define a weaker notion of hypergraph acyclicity,[5] later termed α-acyclicity. This notion of acyclicity is equivalent to the hypergraph being conformal (every clique of the primal graph is covered by some hyperedge) and its primal graph being chordal; it is also equivalent to reducibility to the empty graph through the GYO algorithm[28][29] (also known as Graham's algorithm), a confluent iterative process which removes hyperedges using a generalized definition of ears. In the domain of database theory, it is known that a database schema enjoys certain desirable properties if its underlying hypergraph is α-acyclic.[30] Besides, α-acyclicity is also related to the expressiveness of the guarded fragment of first-order logic.
We can test in linear time if a hypergraph is α-acyclic.[31]
Note that α-acyclicity has the counter-intuitive property that adding hyperedges to an α-cyclic hypergraph may make it α-acyclic (for instance, adding a hyperedge containing all vertices of the hypergraph will always make it α-acyclic). Motivated in part by this perceived shortcoming, Ronald Fagin[32] defined the stronger notions of β-acyclicity and γ-acyclicity. We can state β-acyclicity as the requirement that all subhypergraphs of the hypergraph are α-acyclic, which is equivalent[32] to an earlier definition by Graham.[29] The notion of γ-acyclicity is a more restrictive condition which is equivalent to several desirable properties of database schemas and is related to Bachman diagrams. Both β-acyclicity and γ-acyclicity can be tested in polynomial time.
Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. However, none of the reverse implications hold, so those four notions are different.[32]
## Isomorphism, symmetry, and equality
A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.
A hypergraph ${\displaystyle H=(X,E)}$ is isomorphic to a hypergraph ${\displaystyle G=(Y,F)}$, written as ${\displaystyle H\simeq G}$ if there exists a bijection
${\displaystyle \phi :X\to Y}$
and a permutation ${\displaystyle \pi }$ of ${\displaystyle I}$ such that
${\displaystyle \phi (e_{i})=f_{\pi (i)}}$
The bijection ${\displaystyle \phi }$ is then called the isomorphism of the graphs. Note that
${\displaystyle H\simeq G}$ if and only if ${\displaystyle H^{*}\simeq G^{*}}$.
When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. One says that ${\displaystyle H}$ is strongly isomorphic to ${\displaystyle G}$ if the permutation is the identity. One then writes ${\displaystyle H\cong G}$. Note that all strongly isomorphic graphs are isomorphic, but not vice versa.
When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalence, and also of equality. One says that ${\displaystyle H}$ is equivalent to ${\displaystyle G}$, and writes ${\displaystyle H\equiv G}$ if the isomorphism ${\displaystyle \phi }$ has
${\displaystyle \phi (x_{n})=y_{n}}$
and
${\displaystyle \phi (e_{i})=f_{\pi (i)}}$
Note that
${\displaystyle H\equiv G}$ if and only if ${\displaystyle H^{*}\cong G^{*}}$
If, in addition, the permutation ${\displaystyle \pi }$ is the identity, one says that ${\displaystyle H}$ equals ${\displaystyle G}$, and writes ${\displaystyle H=G}$. Note that, with this definition of equality, graphs are self-dual:
${\displaystyle \left(H^{*}\right)^{*}=H}$
A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices. The set of automorphisms of a hypergraph H (= (XE)) is a group under composition, called the automorphism group of the hypergraph and written Aut(H).
### Examples
Consider the hypergraph ${\displaystyle H}$ with edges
${\displaystyle H=\lbrace e_{1}=\lbrace a,b\rbrace ,e_{2}=\lbrace b,c\rbrace ,e_{3}=\lbrace c,d\rbrace ,e_{4}=\lbrace d,a\rbrace ,e_{5}=\lbrace b,d\rbrace ,e_{6}=\lbrace a,c\rbrace \rbrace }$
and
${\displaystyle G=\lbrace f_{1}=\lbrace \alpha ,\beta \rbrace ,f_{2}=\lbrace \beta ,\gamma \rbrace ,f_{3}=\lbrace \gamma ,\delta \rbrace ,f_{4}=\lbrace \delta ,\alpha \rbrace ,f_{5}=\lbrace \alpha ,\gamma \rbrace ,f_{6}=\lbrace \beta ,\delta \rbrace \rbrace }$
Then clearly ${\displaystyle H}$ and ${\displaystyle G}$ are isomorphic (with ${\displaystyle \phi (a)=\alpha }$, etc.), but they are not strongly isomorphic. So, for example, in ${\displaystyle H}$, vertex ${\displaystyle a}$ meets edges 1, 4 and 6, so that,
${\displaystyle e_{1}\cap e_{4}\cap e_{6}=\lbrace a\rbrace }$
In graph ${\displaystyle G}$, there does not exist any vertex that meets edges 1, 4 and 6:
${\displaystyle f_{1}\cap f_{4}\cap f_{6}=\varnothing }$
In this example, ${\displaystyle H}$ and ${\displaystyle G}$ are equivalent, ${\displaystyle H\equiv G}$, and the duals are strongly isomorphic: ${\displaystyle H^{*}\cong G^{*}}$.
### Symmetry
The rank ${\displaystyle r(H)}$ of a hypergraph ${\displaystyle H}$ is the maximum cardinality of any of the edges in the hypergraph. If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph. A graph is just a 2-uniform hypergraph.
The degree d(v) of a vertex v is the number of edges that contain it. H is k-regular if every vertex has degree k.
The dual of a uniform hypergraph is regular and vice versa.
Two vertices x and y of H are called symmetric if there exists an automorphism such that ${\displaystyle \phi (x)=y}$. Two edges ${\displaystyle e_{i}}$ and ${\displaystyle e_{j}}$ are said to be symmetric if there exists an automorphism such that ${\displaystyle \phi (e_{i})=e_{j}}$.
A hypergraph is said to be vertex-transitive (or vertex-symmetric) if all of its vertices are symmetric. Similarly, a hypergraph is edge-transitive if all edges are symmetric. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive.
Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.
## Partitions
A partition theorem due to E. Dauber[33] states that, for an edge-transitive hypergraph ${\displaystyle H=(X,E)}$, there exists a partition
${\displaystyle (X_{1},X_{2},\cdots ,X_{K})}$
of the vertex set ${\displaystyle X}$ such that the subhypergraph ${\displaystyle H_{X_{k}}}$ generated by ${\displaystyle X_{k}}$ is transitive for each ${\displaystyle 1\leq k\leq K}$, and such that
${\displaystyle \sum _{k=1}^{K}r\left(H_{X_{k}}\right)=r(H)}$
where ${\displaystyle r(H)}$ is the rank of H.
As a corollary, an edge-transitive hypergraph that is not vertex-transitive is bicolorable.
Graph partitioning (and in particular, hypergraph partitioning) has many applications to IC design[34] and parallel computing.[35][36][37] Efficient and scalable hypergraph partitioning algorithms are also important for processing large scale hypergraphs in machine learning tasks.[6]
## Further generalizations
One possible generalization of a hypergraph is to allow edges to point at other edges. There are two variations of this generalization. In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, subsets of subsets of vertices and so on ad infinitum. In essence, every edge is just an internal node of a tree or directed acyclic graph, and vertices are the leaf nodes. A hypergraph is then just a collection of trees with common, shared nodes (that is, a given internal node or leaf may occur in several different trees). Conversely, every collection of trees can be understood as this generalized hypergraph. Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well. So, for example, this generalization arises naturally as a model of term algebra; edges correspond to terms and vertices correspond to constants or variables.
For such a hypergraph, set membership then provides an ordering, but the ordering is neither a partial order nor a preorder, since it is not transitive. The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. Consider, for example, the generalized hypergraph whose vertex set is ${\displaystyle V=\{a,b\}}$ and whose edges are ${\displaystyle e_{1}=\{a,b\}}$ and ${\displaystyle e_{2}=\{a,e_{1}\}}$. Then, although ${\displaystyle b\in e_{1}}$ and ${\displaystyle e_{1}\in e_{2}}$, it is not true that ${\displaystyle b\in e_{2}}$. However, the transitive closure of set membership for such hypergraphs does induce a partial order, and "flattens" the hypergraph into a partially ordered set.
Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. This allows graphs with edge-loops, which need not contain vertices at all. For example, consider the generalized hypergraph consisting of two edges ${\displaystyle e_{1}}$ and ${\displaystyle e_{2}}$, and zero vertices, so that ${\displaystyle e_{1}=\{e_{2}\}}$ and ${\displaystyle e_{2}=\{e_{1}\}}$. As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. In particular, there is no transitive closure of set membership for such hypergraphs. Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer bipartite, but is rather just some general directed graph.
The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges. Thus, for the above example, the incidence matrix is simply
${\displaystyle \left[{\begin{matrix}0&1\\1&0\end{matrix}}\right].}$
## Notes
1. ^ a b Valdivia, Paola; Buono, Paolo; Plaisant, Catherine; Dufournaud, Nicole; Fekete, Jean-Daniel (2020). "Analyzing Dynamic Hypergraphs with Parallel Aggregated Ordered Hypergraph Visualization" (PDF). IEEE Transactions on Visualization and Computer Graphics. IEEE. 26 (1): 12. doi:10.1109/TVCG.2019.2933196. eISSN 1941-0506. ISSN 1077-2626. PMID 31398121. S2CID 199518871.
2. ^ Haussler, David; Welzl, Emo (1987), "ε-nets and simplex range queries", Discrete and Computational Geometry, 2 (2): 127–151, doi:10.1007/BF02187876, MR 0884223.
3. ^ Pearl, Judea (1984). Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley Publishing Company. p. 25. ISBN 978-0-201-05594-8.
4. ^ Feige, Uriel; Kim, Jeong Han; Ofek, Eran (2006). Witnesses for non-satisfiability of dense random 3CNF formulas. 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). IEEE. pp. 497–508. doi:10.1109/FOCS.2006.78.
5. ^ a b Beeri, C.; Fagin, R.; Maier, D.; Yannakakis, M. (1983). "On the Desirability of Acyclic Database Schemes" (PDF). Journal of the ACM. 30 (3): 479–513. doi:10.1145/2402.322389. S2CID 2418740.
6. ^ a b c Huang, Jin; Zhang, Rui; Yu, Jeffrey Xu (2015), "Scalable Hypergraph Learning and Processing" (PDF), Proceedings of the IEEE International Conference on Data Mining: 775–780, doi:10.1109/ICDM.2015.33, ISBN 978-1-4673-9504-5, S2CID 5130573
7. ^ Brazil, M; Zachariasen, M (2015). "Steiner Trees in Graphs and Hypergraphs". Algorithms and Combinatorics. Springer. 29: 301–317. doi:10.1007/978-3-319-13915-9_5. ISBN 978-3-319-13915-9.
8. ^ Zhou, Dengyong; Huang, Jiayuan; Scholkopf, Bernhard (2006), "Learning with hypergraphs: clustering, classification, and embedding", Advances in Neural Information Processing Systems, MIT Press, pp. 1601–8, ISBN 9780262256919
9. ^ Tan, Shulong; Bu, Jiajun; Chen, Chun; Xu, Bin; Wang, Can; He, Xiaofei (October 2011), "Using rich social media information for music recommendation via hypergraph model", ACM Transactions on Multimedia Computing, Communications, and Applications, 7S (1), Article 22, Bibcode:2011smma.book..213T, doi:10.1145/2037676.2037679, S2CID 432036
10. ^ Liu, Qingshan; Huang, Yuchi; Metaxas, Dimitris N. (2013), "Hypergraph with sampling for image retrieval", Pattern Recognition, 44 (10–11): 2255–2262, doi:10.1016/j.patcog.2010.07.014
11. ^ Patro, Rob; Kingsoford, Carl (2013), "Predicting protein interactions via parsimonious network history inference", Bioinformatics, 29 (10–11): 237–246, doi:10.1093/bioinformatics/btt224, PMC 3694678, PMID 23812989
12. ^ Gao, Tue; Wang, Meng; Zha, Zheng-Jun; Shen, Jialie; Li, Xuelong; Wu, Xindong (2013), "Visual-textual joint relevance learning for tag-based social image search", IEEE Transactions on Image Processing, 22 (1): 363–376, Bibcode:2013ITIP...22..363Y, doi:10.1109/tip.2012.2202676, PMID 22692911, S2CID 7432373
13. ^ Tian, Ze; Hwang, TaeHyun; Kuang, Rui (2009), "A hypergraph-based learning algorithm for classifying gene expression and arrayCGH data with prior knowledge", Bioinformatics, 25 (21): 2831–2838, doi:10.1093/bioinformatics/btp467, PMID 19648139
14. ^ Goldstein, A. (1982). "A Directed Hypergraph Database: A Model for the Local Loop Telephone Plant". Bell System Technical Journal. 61 (9): 2529–54. doi:10.1002/j.1538-7305.1982.tb03439.x. S2CID 11290643.CS1 maint: date and year (link)
15. ^ Ranshous, Stephen; Joslyn, Cliff; Kreyling, Sean; Nowak, Kathleen; Samatova, Nagiza; West, Curtis; Winters, Samuel (2017). Exchange Pattern Mining in the Bitcoin Transaction Directed Hypergraph (PDF). Financial Cryptography and Data Security. Springer. doi:10.1007/978-3-319-70278-0_16.
16. ^ a b Ausiello, Giorgio; Laura, Luigi (2017). "Directed hypergraphs: Introduction and fundamental algorithms - A survey". Theoretical Computer Science. 658: 293–306. doi:10.1016/j.tcs.2016.03.016.
17. ^ a b Gallo, G.; Longo, G.; Pallottino, S.; Nguyen, S. (1993). "Directed hypergraphs and applications". Discrete Applied Mathematics. 42 (2–3): 177–201. doi:10.1016/0166-218X(93)90045-P.
18. ^ Sander, G. (2003), "Layout of directed hypergraphs with orthogonal hyperedges", Proc. 11th International Symposium on Graph Drawing (GD 2003), Lecture Notes in Computer Science, 2912, Springer, pp. 381–6, ISBN 978-3-540-24595-7.
19. ^ Eschbach, Thomas; Günther, Wolfgang; Becker, Bernd (2006), "Orthogonal hypergraph drawing for improved visibility" (PDF), Journal of Graph Algorithms and Applications, 10 (2): 141–157, doi:10.7155/jgaa.00122.
20. ^ Mäkinen, Erkki (1990), "How to draw a hypergraph", International Journal of Computer Mathematics, 34 (3): 177–185, doi:10.1080/00207169008803875.
21. ^ Bertault, François; Eades, Peter (2001), "Drawing hypergraphs in the subset standard", Proc. 8th International Symposium on Graph Drawing (GD 2000), Lecture Notes in Computer Science, 1984, Springer-Verlag, pp. 45–76, doi:10.1007/3-540-44541-2_15, ISBN 978-3-540-41554-1.
22. ^ Naheed Anjum, Arafat; Bressan, Stéphane (2017), "Hypergraph Drawing by Force-Directed Placement", 28th International Conference on Database and Expert Systems Applications (DEXA 2017), Lecture Notes in Computer Science, 10439, Springer International Publishing, pp. 387–394, doi:10.1007/978-3-319-64471-4_31, ISBN 978-3-319-64470-7.
23. ^ Kaufmann, Michael; van Kreveld, Marc; Speckmann, Bettina (2009), "Subdivision drawings of hypergraphs", Proc. 16th International Symposium on Graph Drawing (GD 2008), Lecture Notes in Computer Science, 5417, Springer-Verlag, pp. 396–407, doi:10.1007/978-3-642-00219-9_39, ISBN 978-3-642-00218-2.
24. ^ Johnson, David S.; Pollak, H. O. (2006), "Hypergraph planarity and the complexity of drawing Venn diagrams", Journal of Graph Theory, 11 (3): 309–325, doi:10.1002/jgt.3190110306.
25. ^ Buchin, Kevin; van Kreveld, Marc; Meijer, Henk; Speckmann, Bettina; Verbeek, Kevin (2010), "On planar supports for hypergraphs", Proc. 17th International Symposium on Graph Drawing (GD 2009), Lecture Notes in Computer Science, 5849, Springer-Verlag, pp. 345–356, doi:10.1007/978-3-642-11805-0_33, ISBN 978-3-642-11804-3.
26. ^ a b Lovász, László; Plummer, M. D. (1986), Matching Theory, Annals of Discrete Mathematics, 29, North-Holland, ISBN 0-444-87916-1, MR 0859549
27. ^ Berge, Claude (1973). Graphs and Hypergraphs. Amsterdam: North-Holland. ISBN 0-7204-2450-X.
28. ^ Yu, C. T.; Özsoyoğlu, M. Z. (1979). "An algorithm for tree-query membership of a distributed query" (PDF). Proc. IEEE COMPSAC: 306–312. doi:10.1109/CMPSAC.1979.762509.
29. ^ a b Graham, M. H. (1979). "On the universal relation". Technical Report. Toronto, Ontario, Canada: University of Toronto.
30. ^ Abiteboul, S.; Hull, R. B.; Vianu, V. (1995). Foundations of Databases. Addison-Wesley. ISBN 0-201-53771-0.
31. ^ Tarjan, R. E.; Yannakakis, M. (1984). "Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs". SIAM Journal on Computing. 13 (3): 566–579. doi:10.1137/0213035.
32. ^ a b c Fagin, Ronald (1983). "Degrees of Acyclicity for Hypergraphs and Relational Database Schemes". Journal of the ACM. 30 (3): 514–550. doi:10.1145/2402.322390. S2CID 597990.
33. ^ Harary, F. (2018) [1969]. Graph Theory. CRC Press. p. 172. ISBN 978-0-429-96231-8. We next state a theorem due to Elayne Dauber whose corollaries describe properties of line-symmetric graphs. Note the obvious but important observation that every line-symmetric graph is line-regular.
34. ^ Karypis, G., Aggarwal, R., Kumar, V., and Shekhar, S. (March 1999), "Multilevel hypergraph partitioning: applications in VLSI domain", IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 7 (1): 69–79, CiteSeerX 10.1.1.553.2367, doi:10.1109/92.748202.CS1 maint: multiple names: authors list (link)
35. ^ Hendrickson, B., Kolda, T.G. (2000), "Graph partitioning models for parallel computing", Parallel Computing (Submitted manuscript), 26 (12): 1519–1545, doi:10.1016/S0167-8191(00)00048-X.CS1 maint: multiple names: authors list (link)
36. ^ Catalyurek, U.V.; Aykanat, C. (1995). A Hypergraph Model for Mapping Repeated Sparse Matrix-Vector Product Computations onto Multicomputers. Proc. International Conference on Hi Performance Computing (HiPC'95).
37. ^ Catalyurek, U.V.; Aykanat, C. (1999), "Hypergraph-Partitioning Based Decomposition for Parallel Sparse-Matrix Vector Multiplication", IEEE Transactions on Parallel and Distributed Systems, 10 (7): 673–693, CiteSeerX 10.1.1.67.2498, doi:10.1109/71.780863.
|
2021-09-21 03:24:47
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 148, "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": 0.8639824986457825, "perplexity": 1284.3347631350719}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057131.88/warc/CC-MAIN-20210921011047-20210921041047-00025.warc.gz"}
|
https://mathematica.stackexchange.com/questions/265017/how-to-replace-matrix-entries-with-certain-elements-from-list-of-lists-of-varyin
|
# How to replace matrix entries with certain elements from list of lists of varying length
I have a table ("editp6") of 100 lists, of varying (even) sizes, including an empty list. Each list contains only Integers and Reals, or nothing. Below are the first 3 lists
{{1, 0.04, 8, 11.11, 14, 72.21}, {46, 1247.25, 6, 20.59, 13,
64.94}, {66, 54.18, 31, 166.8, 45, 1561.45}}
I need to replace entries in a 100x100 distance matrix ("Q"), where all entries are initially Infinity, with the corresponding element from editp6 (should such a corresponding element exist), such that the {i,j}-th entry of Q is replaced by the editp6[[i,k+1]] element, where editp6[[i,k]]==j-1, and initial value of k=1, increasing in steps of 2, up to Length[editp6[[i]]].
e.g. from above lists, Q[[2,47]] should be replaced with 1247.25, because editp[[2,1]]+1==j=47 and 1247.25 is the k+1 element of list i=2 where k=1. Q[[3,32]] should be replaced with 166.8, because editp[[3,3]]+1==j=32, and 166.8 is the k+1 element of list i=3 where k=3.
I have made a few attempts to get this to work with Do, For and If loops and using the ReplacePart function, to no avail. I am a beginner with coding and Mathematica, but any help is appreciated.
• Welcome to Mathematica.SE! I hope you will become a regular contributor. To get started, 1) take the introductory tour now, 2) when you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge, 3) remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign, and 4) give help too, by answering questions in your areas of expertise. Mar 12 at 19:29
One strategy would be to build up a set of replacement rules and then use ReplacePart on Q. Here's your initial list:
list = {{1, 0.04, 8, 11.11, 14, 72.21}, {46, 1247.25, 6, 20.59, 13, 64.94}, {66, 54.18, 31, 166.8, 45, 1561.45}}
Every adjacent pair is almost a replacement rule. It's just missing the "row" index. So, maybe some combination of Partition (to explicitly pair things up) and then MapIndexed to add in the "row" index.
We could partition your list directly with Partition if it were rectangular (and there may be some fancy way to still do that), but I'll just map Partition over your list.
replacements =
Flatten[
MapIndexed[{#2[[1]], 1 + #1[[1]]} -> #1[[2]] &, Partition[#, 2] & /@ list, {2}]]
(* outputs {{1, 2} -> 0.04, {1, 9} -> 11.11, {1, 15} -> 72.21, {2, 47} -> 1247.25, {2, 7} -> 20.59, {2, 14} -> 64.94, {3, 67} -> 54.18, {3, 32} -> 166.8}*)
Q = ConstantArray[Infinity, {100, 100}];
You could just do the replacements and store it into a new variable, or you could overwrite Q, like this:
Q = ReplacePart[Q, replacements];
Double check:
Q[[2, 47]] (*1247.25*)
Q[[3, 32]] (*166.8*)
|
2022-11-27 19:41:28
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31487515568733215, "perplexity": 3680.62798189621}, "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/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00071.warc.gz"}
|
https://dyinglovegrape.wordpress.com/category/pdf/
|
## Factorials and Exponentials.
### November 6, 2011
I’ve been working on a problem (here is a partial paper with some ideas) that’s really easy for any calculus student to understand but quite difficult for even wolfram alpha to work out some cases. Here’s the idea:
We know that $\lim_{n\to\infty}\dfrac{e^n}{n!} = 0$. It’s not hard to reason this out (there are some relatively obvious inequalities, etc.), but I wanted to know what happened if we considered something like:
$\lim_{n\to\infty}\dfrac{e^{e^n}}{n!}$.
It turns out, this goes to infinity. Maybe this is not so surprising. But, to balance this out, I thought maybe I could add another factorial on the bottom. What about
$\lim_{n\to\infty}\dfrac{e^{e^n}}{(n!)!}$
where this double factorial is just $n!$ with another factorial at the end. It turns out, this one goes to 0.
The problem here is that after $((n!)!)!$, Mathematica doesn’t seem to be able to handle the sheer size of these numbers. Consequently, I only have a few values for this. I’ve included everything I have in a google-doc PDF (the only way I can think to share this PDF), and I’m looking for suggestions. Here’s some things I thought of:
• Stirling’s formula. Unfortunately, this starts to get very complicated very quickly, and if you consider subbing it in for even $((n!)!)!$ it can take up a good page of notes. It also doesn’t reduce as nicely as I’d like.
• Considering the Gamma function. It may be easier to work with compositions of the gamma function since it is not discrete and we may be able to use some sort of calculus-type things on it.
• Number Crunching. For each of these cases, it seems like there is a point where either the numerator or the denominator "clearly" trumps over the other; this is not the "best" method to use, but it will give me some idea of which values potentially go to infinity and which go to zero.
• Asymptotics. I’m not so good at discrete math or asymptotics, so there may be some nice theorems (using convexity, maybe?) in that field that I’ve just never seen before. Especially things like: if $f\sim g$ then $f\circ f \sim g\circ g$ under such-and-such a condition.
Feel free to comment below if you think of anything.
## Complex Analysis Study Guide: Brown and Churchill.
### December 12, 2010
EDIT: It seems that scribd is now behind a paywall now. :(
Brown and Churchill (8th ed) was the book I used for the second complex analysis class I’ve had to take so far (the first was Lang). My class went over the first six chapters and half of the seventh: so, up to the middle of the section on applications of residues.
To prep for the final, I compiled a quick, slightly-shorter list of things that I feel the complex student should know if they’ve used this book and have gotten to around the same point. I’ve excluded the chapter on applications of residues, since it’s a relatively short chapter with better pictures in the text than ones I could draw at 5am. Because sharing is caring, below is a link to the pdf. Enjoy!
http://www.scribd.com/doc/45170305
## Inner Product Space PDF.
### July 19, 2010
So, I wanted to proceed onwards towards some pretty cool mathematics (and, finally, get through basic linear algebra) by introducing the Gram-Schmidt ortho-normalization process and some really sweet consequences (that actually really surprised me!), but it occurred to me that I’d need to introduce norms and inner products as well as prove a butt-load of things about them.
Because I am terribly lazy, I am not going to do this. Instead, I read through a number of inner product introductions (which are all basically the same) to find one that was well-written. The one that I’ve picked to show ya’ll is from G. Keady, from the University of Birmingham. It is in pdf form, and it is available here (warning, pdf!).
With the possible exception of ultra-brevity (orthonormal is abbreviated ON, and that’s kind of weird to get used to) and some of the things at the end of the paper, this is a 3-page introduction and, partially because of this, is very readable. You should not need any math besides what we’ve already covered in this blog.
We’ll give examples of normed vector spaces and inner product spaces later, but we’ll definitely be using the inner product space of continuous (real) polynomials on the interval $[0,1]$, which has the inner product
$\displaystyle\langle f, g\rangle = \int_{0}^{1} f(x)g(x)dx$
This will come in handy later, so remember it!
## Sets: Nature’s Candy.
### July 3, 2010
I’m not going to write a whole huge thing on here since, fortunately, I started a big paper on sets to teach my old students. I didn’t finish it entirely, or proofread it entirely, but I’m going to post and repost until it’s completely done. But there’s no sense in keeping it from you; just start it now, and by the time you get up to the end, I’ll hopefully be done with the rest!
Due to the formatting issues (desktop latex just uses stuff that looks like $this$, but wordpress formats statements that look like $.latex this$ (without the period) and changing between the two would take far too long), I’ve decided just to upload the pdf to this blog.
Go here for the Set Theory guide.
|
2018-02-25 05:52:25
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7305095791816711, "perplexity": 655.5646293441048}, "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/1518891816138.91/warc/CC-MAIN-20180225051024-20180225071024-00153.warc.gz"}
|
http://www.thehomeworkgurus.com/random-signals-question-2/
|
# random signals question
A statistician wants to estimate the mean height h (in meters) of a population, based on n independent samples X1, middot middot middot , Xn, choose uniformly from the entire population. He uses the sample mean Mn = (X1 + X2 + …, Xn)/n as the estimate of h. We also assume that he knows Var(Xi) = 0.5 (meter2).How large should n be so that the variance of Mn is at most 10-4.Please use the weak law of large numbers, i.e., to answer that how large n should be so that the estimate is within 0.05 meter from h, with probability at least 0.9?
|
2019-11-20 09:31:40
|
{"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": 0.9552298784255981, "perplexity": 835.9722656077882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670535.9/warc/CC-MAIN-20191120083921-20191120111921-00443.warc.gz"}
|
http://math.stackexchange.com/questions/24074/eigenfunction-expansion-solution-to-a-pde-with-a-constant-non-homogeneous-term
|
# Eigenfunction expansion solution to a PDE with a constant non homogeneous term
I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogeneous term is a constant, rather than a function of the independent variables? For example, in a hyperbolic PDE with x and t as the independent variables the eigenfunctions might be something like $\sum_{n=1}^\infty sin(n\pi x)$, and to create an eigenfunction expansion of the nonhomogeneous term I have to solve for the coefficients A of $\sum_{n=1}^\infty A sin(n\pi x) = B$ by using the orthogonality of sines, where B is the constant nonhomogeneous term. I guess I'm having trouble seeing how an infinite series of sines could converge to a constant - I know in a Fourier series one has to solve for the "DC component" separately.
-
|
2016-06-26 04:47:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.913526177406311, "perplexity": 143.8665719434792}, "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/1466783394937.4/warc/CC-MAIN-20160624154954-00071-ip-10-164-35-72.ec2.internal.warc.gz"}
|
https://groupprops.subwiki.org/w/index.php?title=Quasimorphism&diff=49214&oldid=49212
|
Difference between revisions of "Quasimorphism"
WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with quasihomomorphism of groups
Definition
Suppose $G$ is a group. A quasihomomorphism on $G$ is a function $f: G \to \R$ (where $\R$ is the field of real numbers) satisfying the condition that there exists a positive real number $D$ such that for all $x,y \in G$, we have:
$|f(xy) - f(x) - f(y)| \le D$
Note that $D$ depends on $f$, but not on the choice of elements of $G$.
The smallest positive real number $D$ that works is called the defect of the quasimorphism $f$. A quasimorphism of defect 0 is the same as a homomorphism to $\R$.
Other names for this concept are quasihomomorphism (not to be confused with a different notion of quasihomomorphism of groups) and pseudocharacter.
Homogenization
A homogeneous quasimorphism is a quasiomorphism that is also a 1-homomorphism of groups, i.e., its restriction to any cyclic subgroup of $G$ is a homomorphism. For any quasimorphism $f$, we can consider its homogenization, defined as $\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}$.
Facts
• The collection of all quasimorphisms on a group is a
Examples
• Any set map from a group to $\R$ with a bounded image is a quasimorphism. In particular, any continuous map from a compact topological group to $\R$ is a quasimorphism. Examples include coordinate projections from compact manifolds embedded in $\R^n$. Note that the homogenization of any such quasimorphism is the zero quasimorphism, so such quasimorphisms are not interesting up to homogenization.
• The rotation number quasimorphism is a homogeneous quasimorphism.
|
2019-03-26 20:12:12
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 19, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9364558458328247, "perplexity": 292.13711813299557}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912206016.98/warc/CC-MAIN-20190326200359-20190326222359-00019.warc.gz"}
|
https://www.physicsforums.com/threads/coulombs-law-of-two-copper-spheres.60630/
|
# Coulomb's Law of two copper spheres
1. Jan 21, 2005
### eil2001
Here's a question from my textbook:
Two copper spheres, each having a mass of .4 kg, are separated by 2 m.
(a) How many electrons does each sphere contain? The atomic mass of copper is 63.5 g/mol, and its atomic number is 29.
(b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force of 1.00x10^4 N (roughly 1 ton)?
I got (a) by dimensional analysis:
(.4 kg Cu) x (1 mol/.0635 kg Cu) x (6.02x10&23 molec/1 mol) x (29 electrons/molec) = 1.10x10^26 electrons
But, I am having trouble with part (b). I was thinking that you should use the equation:
F=k(q_1)(q_2)/r^2 , but I'm not really sure how to proceed. I would appreciate any help. Thanks so much!
2. Jan 21, 2005
### rbj
$$F = \frac{1}{4 \pi \epsilon_0} \times \frac{|q_1| |q_2|}{r^2}$$
if the amount of charge removed from one sphere is the same as what is added to the other, then $$|q_1|=|q_2|$$. you know what $$F$$ and $$r$$ is, so solve for $$|q|^2$$.
Last edited: Jan 21, 2005
3. Jan 21, 2005
### Staff: Mentor
You are on the right track. Realize that q_1 and q_2 have the same magnitude, so you can write F=kq^2/r^2 and solve for q. Then, knowing the charge per electron, you can figure the number of electrons that must have been moved.
4. Jan 21, 2005
### eil2001
Thanks, but why should q_1 and q_2 have the same magnitude? And then how do you go from "q" to the number of electrons?
5. Jan 22, 2005
### Nylex
They're both copper spheres and contain the same number of electrons..
6. Jan 22, 2005
### Gokul43201
Staff Emeritus
The copper sphere were originally neutral, because they had as many electrons as protons.
By removing some n electrons from sphere 1, you give it a net positive charge, Q1 = ne (where e = magnitude of charge on an electron/proton = 1.6 * 10^-19 C), due to the n excess protons it now has.
Sphere 2, having gained these n excess electrons will now have a net negative charge Q2 = -ne, due to n excess electrons.
Q1 = ne, Q2 = -ne, so |Q2| = ne.
7. Jan 22, 2005
### dextercioby
If u get the "q" in Coulombs,then u can use the fact that electrons have negative charge to write
$$q=-|q|$$
then
q C---------------------->"x" electrons
$$-1.6 \cdot 10^{-19}C$$ ------------------>1 electron.
Solve for "x".
Daniel.
|
2018-01-21 19:09:46
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5121384263038635, "perplexity": 2049.6969784586204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890823.81/warc/CC-MAIN-20180121175418-20180121195418-00241.warc.gz"}
|
http://www.gradesaver.com/textbooks/math/precalculus/precalculus-mathematics-for-calculus-7th-edition/chapter-1-section-1-3-algebraic-expressions-1-3-exercises-page-33/25
|
## Precalculus: Mathematics for Calculus, 7th Edition
$21t^{2}-26t+8$
First: $3t\times7t = 21t^{2}$ Outside: $3t\times-4 = -12t$ Inside: $-2\times7t =-14t$ Last: $-2\times-4 = 8$ Add the results from FOIL together: $=21t^{2}-12t-14t+8$ $=21t^{2}-26t+8$
|
2017-03-29 07:26:53
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6887338161468506, "perplexity": 13526.59476123524}, "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/1490218190234.0/warc/CC-MAIN-20170322212950-00102-ip-10-233-31-227.ec2.internal.warc.gz"}
|
https://science.nrao.edu/facilities/vla/archive/index
|
VLA > NRAO Data Archive
# NRAO Data Archive
The NRAO archive can be accessed at https://data.nrao.edu
### Locating and unlocking data
Newly observed VLA data are transferred to the NRAO archive and available for retrieval to those with the appropriate privileges approximately 10 minutes after the end of the observations. The archive content can be accessed through the Archive Access Tool (AAT). The online archive contains all VLA data since observing started in 1976.
The AAT search provides a data retrieval tool which can be used either as a simple search using keywords in the main search box, or an advanced search under the "Show Search Inputs" below the search box. The Search Input option enables searches based on a large number of user-specified criteria. Both proprietary and public data can be downloaded via secure https file transfer. On request, NRAO will ship data on physical hard disks, subject to the conditions of our Data Shipment Policy.
With the exception of some rapid response and large proposals, VLA data associated with a given proposal are normally restricted to proprietary use by the proposing team for a period of 12 months from the date of the last observation in a proposal (Note: data taken more than 12 months previously may still be proprietary, if additional data for the same proposal have been taken within the last 12 months). Any member of the observing team can unlock proprietary data by signing in on the archive page with their my.nrao.edu user ID and password (Figure 1), and using the link "your_username's Data" in the upper right corner of the site.
You need to be PI or co-I on the proposal to be able to access proprietary data. If you were not listed on the original proposal, you must first obtain permission from the PI, who in turn, must contact us through the NRAO Helpdesk to allow your access to those data.
### Data Formats and Data Retrieval
#### SDM Format
VLA data taken after January 2010, when the transition to the WIDAR correlator took place, are stored in the Science Data Model (SDM) format that is used by both the VLA and ALMA. VLA data are available through the AAT in the following formats:
• In the native Science Data Model (SDM) format.
• As a CASA Measurement Set (MS) created from the SDM, which may be averaged in frequency and/or time, to be used with CASA.
• As a CASA Calibrated Measurement Set (for data observed since the start of A-configuration in 2016).
When requesting the MS format, which consists of a directory with sub-directories, we recommend requesting tar-format. This will collect the whole directory structure in a single file for easy retrieval.
Please note, current known limitations/features of the AAT include:
• All visitor computing account holders can no longer request data directly to be deposited in their accounts. ***By the end of May 2022, this capability is expected to be restored.
• Requests for SDM-BDF and MS of the same observations have to be submitted separately.
• Only a single calibrated MS can be requested at any one time.
• If you request very large tar files (which is the default), the system may time out due to the length of time tar takes to run the tar command. At this point, we do not recommend requesting tar files. Instead download via wget (see below).
wget -r --reject "index.html*" -np -nH --cut-dirs=3 https://dl-dsoc.nrao.edu/anonymous/.....
• Download requests are returned in a nested directory, with a sub-directory named exactly the same as what you asked for; you will have to go into that sub-directory to get to the requested file, e.g. 20B-099.sb39274643.eb39345634.59269.54178597222.ms/20B-099.sb39274643.eb39345634.59269.54178597222.ms/
Some archival data have known problems which, together with the possible fix, are listed at the archive issues page.
#### UVFITS Format
Delivery of VLA data in the UVFITS format has been discontinued in June 2013. Instructions to create UVFITS from the SDM or CASA MS format are given below.
##### For use in CASA
We no longer support the conversion from CASA measurement sets into uv FITS format from within the archive tool.
##### For use in AIPS
Observers who have installed Obit can convert SDM data directly into UVFITS or load SDM data directly into AIPS. Note that Obit cannot convert complex correlator modes into a single UVFITS/AIPS file, but plain continuum or simple line data is not a problem. The latest version of Obit can be found by scrolling to the bottom of the Obit distribution page. Be aware that Obit is a bit harder to install on Macs than on Linux architectures. To convert SDM data into UVFITS with Obit use these steps:
|
2022-06-27 12:16:10
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.24740737676620483, "perplexity": 3587.8637107879154}, "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/1656103331729.20/warc/CC-MAIN-20220627103810-20220627133810-00046.warc.gz"}
|
https://rpg.stackexchange.com/questions/152431/can-modular-abilities-get-an-extra-10-savings-according-to-gurps-supers-rules-i
|
# Can Modular Abilities get an extra 10% savings according to GURPS Supers rules if they don't grant skills?
I want the player characters to have lots of super-powers. For example, I want them to have inexpensive mental Modular Abilities with typically psionic abilities such as Psychokinesis and ESP and Mind Control (but no skills). Modular Abilities cost less if they are only mental. On the topic of Telekinesis, the main rulebook, page B92, says:
Psychokinetic: Your ability is part of the Psychokinesis psi power (see p. 256). This makes it mental rather than physical. -10%
I want to give the players all available savings, but I want to avoid double-dipping by giving the same discount twice.
In a normal campaign, level 10 Telekinesis would cost 50 points as a physical power, but 45 points as a Psychokinetic mental power.
Now let us suppose that the player characters start off with Modular Abilities, mental advantages only. Assume each mental advantage costs 45 points, and only one can be used at a time, so this power is a one-slot modular ability, base cost 5 points, slot cost 3 points per advantage point. $$\5+3\times45=190\$$.
I think the basic cost should be $$\190-19=171\$$, because the quasimoral power modifier deducts 10%. But I see in GURPS Supers, p. 46, that a wizard can have an extra 10% savings due to “mental advantages only” (i.e. no skills are allowed). So perhaps the basic cost of this mental slot should be 152.
Question: Which is more appropriate, 171 points or 152 points?
|
2020-10-28 00:53:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7321160435676575, "perplexity": 3629.7704865508304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107894890.32/warc/CC-MAIN-20201027225224-20201028015224-00535.warc.gz"}
|
https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-5-section-5-1-exponents-exercise-set-page-345/113
|
## Algebra: A Combined Approach (4th Edition)
$x^{322}$, C. Multiply the exponents
Based on the power rule for exponents, we know that $(a^{m})^{n}=a^{mn}$ (where $m$ and $n$ are positive integers and $a$ is a real number). Therefore, we can simplify the given expression by multiplying the exponents. $(x^{14})^{23}=x^{14\times23}=x^{322}$
|
2018-10-16 19:25:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.916646420955658, "perplexity": 245.89219757413488}, "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-2018-43/segments/1539583510866.52/warc/CC-MAIN-20181016180631-20181016202131-00244.warc.gz"}
|
https://www.ias.ac.in/listing/bibliography/pram/P._K._Rath
|
• P K Rath
Articles written in Pramana – Journal of Physics
• Exploring effective interactions through transition charge density study of70,72,74,76Ge nuclei
Transition charge densities (TCD) for 0+ → 21+ excitation have been calculated for70,72,74,76Ge nuclei within microscopic variational framework employing 2p3/2, 1f5/2, 2p1/2 and 1g9/2 valence space. The calculated TCDs for different monopole variants of Kuo interaction are compared with available experimental results. Other systematics like reduced transition probabilitiesB(E2) and static quadrupole momentsQ(2) are also presented. It is observed that the transition density study acts as a sensitive probe for discriminating the response of different parts of effective interactions.
• Two-neutrino double β decay of96Zr to excited 2+ state of96Mo
The two-neutrino double beta decay of96Zr isotope for 0+ → 2+ transition has been studied in the PHFB model. In our earlier work, the reliability of the intrinsic wave functions of96Zr and96Mo isotopes has been established by obtaining an overall agreement between a number of theoretically calculated spectroscopic properties as well as half-lives of 2vββ decay for 0+ → 0+ transition and the available experimental data. In the present work, the half-life of 2vββ decay for 0+ ar 2+ transition T12/2v(0+2+) has been calculated using the same set of intrinsic wave functions.
• Structure of nuclear transition matrix elements for neutrinoless double-$\beta$ decay
The structure of nuclear transition matrix elements (NTMEs) required for the study of neutrinoless double-$\beta$ decay within light Majorana neutrino mass mechanism is disassembled in the PHFB model. The NTMEs are calculated using a set of HFB intrinsic wave functions, the reliability of which has been previously established by obtaining an overall agreement between the theoretically calculated spectroscopic properties and the available experimental data. Presently, we study the role of short-range correlations, radial evolution of NTMEs and deformation effects due to quadrupolar correlations. In addition, limits on effective light neutrino mass $\langle m_{\nu} \rangle$ are extracted from the observed limits on half-lives $T_{1/2}^{0\nu}$ of neutrinoless double-$\beta$ decay.
• Elastic scattering and fusion cross-sections in $^{7}{\text{Li}} + ^{27}{\text{Al}}$ reaction
With an aim to understand the effects of breakup and transfer channels on elastic scattering and fusion cross-sections in the $^{7}{\text{Li}} + ^{27}{\text{Al}}$ reaction, simultaneous measurement of elastic scattering angular distributions and fusion cross-sections have been carried out at various energies ($E_{\text{lab}} = 8.0–16.0$ MeV) around the Coulomb barrier. Optical model (OM) analysis of the elastic scattering data does not show any threshold anomaly or breakup threshold anomaly behaviour in the energy dependence of the real and imaginary parts of the OM potential. Fusion cross-section at each bombarding energy is extracted from the measured $\alpha$-particle evaporation energy spectra at backward angles by comparing with the statistical model prediction. Results on fusion cross-sections from the present measurements along with data from the literature have been compared with the coupled-channels predictions. Detailed coupled-channels calculations have been carried out to study the effect of coupling of breakup, inelastic and transfer, channels on elastic scattering and fusion. The effect of $1n$-stripping transfer coupling was found to be significant compared to that of the projectile breakup couplings in the present system.
• # Pramana – Journal of Physics
Volume 94, 2020
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
Click here for Editorial Note on CAP Mode
© 2017-2019 Indian Academy of Sciences, Bengaluru.
|
2020-06-05 01:25:14
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5394049286842346, "perplexity": 2913.8434200653924}, "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/1590348492295.88/warc/CC-MAIN-20200604223445-20200605013445-00005.warc.gz"}
|
https://solvedlib.com/give-the-systematic-name-for-the-following-please,158734
|
# Give the systematic name for the following please Spell out the full name of the compound....
###### Question:
Give the systematic name for the following please
Spell out the full name of the compound. Spell out the full name of the compound. Spell out the full name of the compound.
#### Similar Solved Questions
##### Set up an integral for the length of the curve y = x" from x=0 tox=l.V1+16xdx[Ji+xidx[ Ji+axodx( Vi+ioxidx
Set up an integral for the length of the curve y = x" from x=0 tox=l. V1+16xdx [Ji+xidx [ Ji+axodx ( Vi+ioxidx...
##### Use the following to answer questions 32-33: Table: CPI Year CPI Value 1999 110 2000 115...
Use the following to answer questions 32-33: Table: CPI Year CPI Value 1999 110 2000 115 2001 117 2002 115 32. (Table: CPI) This table shows price level statistics for a country. In which of the following years did this country experience disinflation? A) 2001 to 2002 B) 2000 to 2001 C) There is nev...
##### Reflection on learning about the CPT coding system. What errors did you make on the initial...
reflection on learning about the CPT coding system. What errors did you make on the initial post regarding coding for the given scenario? Why do you think you made those errors?...
##### Qweton Gompletion Status:870Moving to another question wiltfave this response.Question 2If fr,y) = 7r2 + 2xly2 y2 find f, (1,4) OA [4 0B [ 8 0c None of these @D ] 0 @E 9Moving to another question will save this response
qweton Gompletion Status: 8 70 Moving to another question wiltfave this response. Question 2 If fr,y) = 7r2 + 2xly2 y2 find f, (1,4) OA [4 0B [ 8 0c None of these @D ] 0 @E 9 Moving to another question will save this response...
##### Prcn 207- Aus Sul PICE que hut chunici 9Nunickeccplan Keuslaecampc edvcrtises thal they will &cliver Yous €apet wilkin 15 days of purchasn Acntlam] CAA camnnant Fenulellan ALal Cinton 0f49 pest cuslonst Is takcn_ The acruge &livaty time ampl K1 [6.2 dy*- ILronen Acdnte Writoand number nl You havc t0 Icstthc hpolhesl "palast IFe altemnative that Ha: p > Iie 5cpi 4 che [EIL Jphiots:Find th Ie-tule Gcanrec pulueOueatlob Nndorn sampk of 16 studcnts is selected (rom the student body
Prcn 207- Aus Sul PICE que hut chunici 9 Nunickeccplan Keuslae campc edvcrtises thal they will &cliver Yous €apet wilkin 15 days of purchasn Acntlam] CAA camnnant Fenulellan ALal Cinton 0f49 pest cuslonst Is takcn_ The acruge &livaty time ampl K1 [6.2 dy*- ILronen Acdnte Writoand numbe...
##### Cramer’s mortgage contained a provision requiring her to pay monthly tax and insurance pa y ments...
Cramer’s mortgage contained a provision requiring her to pay monthly tax and insurance pa y ments into an escrow account held by the bank in addition to principal and interest. Cramer paid the principal and interest regularly but refused to pay the tax and insurance e s crow payments. The bank...
##### Find the intercepts and then use them to graph the equation 2y-6=2x
find the intercepts and then use them to graph the equation 2y-6=2x...
##### 3- A section of a freeway shows the following relationship between the traffic flow and the...
3- A section of a freeway shows the following relationship between the traffic flow and the traffic density: ?=???−?.???? a) (7%) What is the capacity of the highway section? b) (8%) What is the speed at capacity? c) (8%) What is the density when the highway is at one quarter of its capacity?...
##### For nickel, Ni, the heat of vaporization at its normal boiling point of 2732 OC is...
For nickel, Ni, the heat of vaporization at its normal boiling point of 2732 OC is 378.6 kJ/mol. The entropy change when 1.80 moles of liquid Ni vaporizes at 2732 oc, 1 atm is J/K....
##### 2ouQuestion 5 (4 polntel
2 ou Question 5 (4 polntel...
##### 8| 8 | 1 2 7 F Ni { 1 1 } { H 7 3 [ 22 44 f N F e/8 } [ €ole [ | [ 1 2 [ F
8| 8 | 1 2 7 F Ni { 1 1 } { H 7 3 [ 22 44 f N F e/8 } [ €ole [ | [ 1 2 [ F...
##### Value of b_ Find the 412.13, An = oblique = triangle " 0 = 50"_ and has Y = 828 399.7 424.7 532.7 194.72.018, and 1.852 , bwith sides Find the smallest angle in an oblique triangle c = 1.907.(e) 58.869 (6) 56.220 (c) 64.920 98.130
value of b_ Find the 412. 13, An = oblique = triangle " 0 = 50"_ and has Y = 828 399.7 424.7 532.7 194.7 2.018, and 1.852 , b with sides Find the smallest angle in an oblique triangle c = 1.907. (e) 58.869 (6) 56.220 (c) 64.920 98.130...
##### 1. Consider the systemGe (s) + 2 F2 (g) ↔ GeF4 (g) ∆H = –1190kJWhich of the following actions will change the equilibriumconstant K for the reaction?a.all of the named actions change the value of Kb.increasing the container volumec.removing Ged.removing heate.adding F22. Each of the following substances is dissolved separately in asample of water. Which one would not create a basicsolution (pH > 7)? a.Na2CO3b.KOHc.CH3CH2CO2Hd.SrOe.CH3NH2
1. Consider the system Ge (s) + 2 F2 (g) ↔ GeF4 (g) ∆H = –1190 kJ Which of the following actions will change the equilibrium constant K for the reaction? a. all of the named actions change the value of K b. increasing the container volume c. removing Ge d. removing heat...
##### The ratio of height of an object in a photograph to the actal height is 1:100
The ratio of height of an object in a photograph to the actal height is 1:100. The height of a statue in the photograph is 5cm. How tall is the statue?...
##### | 22) - 22) Consider the titration of a 20.0-ml sample of 0.105 M HC2H302 with...
| 22) - 22) Consider the titration of a 20.0-ml sample of 0.105 M HC2H302 with 0.125 M NaOH. Determine the pH at the equivalence point. A) 2.86 B) 4.74 C) 8.75 D) 12.17...
##### 8 3 1 3322 3 6 2 1 Il 1 1 JV 1 1 L
8 3 1 3322 3 6 2 1 Il 1 1 JV 1 1 L...
##### To find the domain of the function defined by the formula f(x) we determine 12+51+6 the numbers we cannot plug into the formula byplugging 0 into the denominator: plugging 0 into the numerator: setting the denominator equal= to 0 and solving for * _ setting the numerator equal to 0 and solving for .
To find the domain of the function defined by the formula f(x) we determine 12+51+6 the numbers we cannot plug into the formula by plugging 0 into the denominator: plugging 0 into the numerator: setting the denominator equal= to 0 and solving for * _ setting the numerator equal to 0 and solving for ...
##### Fertiey Rates Vs GDP per caplta Sactter Plot120000IOOQ800006020J0n0O8 20000~20000acoOFettillty RatesFertiliy Ralesvs GDP per capila Sacller PlolJecno[aaJoaA drat754-534' . 7684-744 4148 -15398 n"=0, 78475Aa 5eFoannL0EA{ E 8 oFertility Rates
Fertiey Rates Vs GDP per caplta Sactter Plot 120000 IOOQ 80000 6020 J0n0O 8 20000 ~20000 acoO Fettillty Rates Fertiliy Ralesvs GDP per capila Sacller Plol Jecno [aa JoaA drat 754-534' . 7684-744 4148 -15398 n"=0, 7847 5Aa 5e Foann L0EA { E 8 o Fertility Rates...
##### Let U = { 14, 15,16, 17, 18, 19,20 } A ={14,15, 18,20 } Use the roster method to write the set A'_A' = (Use a comma to separate answers as needed:)
Let U = { 14, 15,16, 17, 18, 19,20 } A ={14,15, 18,20 } Use the roster method to write the set A'_ A' = (Use a comma to separate answers as needed:)...
##### How do changes in the breast prepare a mother to nurse her newborn? How do hormones influence these changes and stimulate milk production?
How do changes in the breast prepare a mother to nurse her newborn? How do hormones influence these changes and stimulate milk production?...
##### 8. Two long straight wires pass through x and x = 7 cm and each carry...
8. Two long straight wires pass through x and x = 7 cm and each carry currents of 8 A as 3 cm shown in the figure. a) What is the value of the magnetic field at the origin? (Be sure to specify direction.) b) What is the force on a 5 pc charge at the origin moving with a velocity of (1500 ) mus? c) I...
##### How do you express (6x^2+1)/(x^2(x-1)^2) in partial fractions?
How do you express (6x^2+1)/(x^2(x-1)^2) in partial fractions?...
##### Have a 1000 mg/L concentration stock of BAPNA, and needed a 1:100 dilution for the lab so it is 10 mgIL. need to make total of 500 mL of the diluted solution_ Give instructions for how you would make this solution1 ^ ~4 '
have a 1000 mg/L concentration stock of BAPNA, and needed a 1:100 dilution for the lab so it is 10 mgIL. need to make total of 500 mL of the diluted solution_ Give instructions for how you would make this solution 1 ^ ~4 '...
##### This Question: 4 pts 9 of 11 (8 complete) A binomial probability experiment is conducted with...
This Question: 4 pts 9 of 11 (8 complete) A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=9, p=0.3, xs3 The probability of xs 3 successes is (Round to four decimal places as needed.)...
|
2022-07-07 04:45:52
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.49716439843177795, "perplexity": 5536.181054706261}, "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/1656104683683.99/warc/CC-MAIN-20220707033101-20220707063101-00365.warc.gz"}
|
https://bookstore.ams.org/view?ProductCode=ADVSOV/12
|
An error was encountered while trying to add the item to the cart. Please try again.
Copy To Clipboard
Successfully Copied!
Topics in Nonparametric Estimation
Edited by: R. Z. Khasminskiĭ
Available Formats:
Hardcover ISBN: 978-0-8218-4111-2
List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Electronic ISBN: 978-1-4704-4609-3 Product Code: ADVSOV/12.E List Price:$126.00
MAA Member Price: $113.40 AMS Member Price:$100.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $189.00 MAA Member Price:$170.10
AMS Member Price: $151.20 Click above image for expanded view Topics in Nonparametric Estimation Edited by: R. Z. Khasminskiĭ Available Formats: Hardcover ISBN: 978-0-8218-4111-2 Product Code: ADVSOV/12 List Price:$126.00 MAA Member Price: $113.40 AMS Member Price:$100.80
Electronic ISBN: 978-1-4704-4609-3 Product Code: ADVSOV/12.E
List Price: $126.00 MAA Member Price:$113.40 AMS Member Price: $100.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$189.00 MAA Member Price: $170.10 AMS Member Price:$151.20
• Book Details
Volume: 121992; 150 pp
MSC: Primary 60; 62;
This book contains papers presented at the Seminar on Mathematical Statistics held at the Institute for Problems of Information Transmission of the Academy of Sciences in the former Soviet Union. Founded in the mid-1960s, this seminar is still active today and attracts most of the researchers in Moscow who are interested in mathematical statistics. The topics covered include density, regression, and image estimation, adaptive estimation, stochastic approximation, median estimation, sequential experimental design, and large deviations for empirical measures. This collection is distinguished by the high scientific level of the papers and their modern approach. This book will be of interest to scientists and engineers who use probability and statistics, to mathematicians and applied statisticians who work in approximation theory, and to computer scientists who work in image analysis.
Scientists and mathematicians in probability, statistics, and approximation theory. Computer scientists interested in image analysis.
• Articles
• A. Samarov - Lower bound for the integral risk of density function estimates
• A. Nemirovskii - On nonparametric estimation of functions satisfying differential inequalities
• A. Korostelev and A. Tsybakov - Asymptotically minimax image reconstruction problems
• O. Lepskii - On problems of adaptive estimation in white Gaussian noise
• B. Polyak and A. Tsybakov - On stochastic approximation with arbitrary noise (the KW-case)
• E. Belitser and A. Korostelev - Pseudovalues and minimax filtering algorithms for the nonparametric median
• A. Veretennikov - On large deviations for ergodic process empirical measures
• V. Spokoinyi - On asymptotically optimal sequential experimental design
• Requests
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 121992; 150 pp
MSC: Primary 60; 62;
This book contains papers presented at the Seminar on Mathematical Statistics held at the Institute for Problems of Information Transmission of the Academy of Sciences in the former Soviet Union. Founded in the mid-1960s, this seminar is still active today and attracts most of the researchers in Moscow who are interested in mathematical statistics. The topics covered include density, regression, and image estimation, adaptive estimation, stochastic approximation, median estimation, sequential experimental design, and large deviations for empirical measures. This collection is distinguished by the high scientific level of the papers and their modern approach. This book will be of interest to scientists and engineers who use probability and statistics, to mathematicians and applied statisticians who work in approximation theory, and to computer scientists who work in image analysis.
Scientists and mathematicians in probability, statistics, and approximation theory. Computer scientists interested in image analysis.
• Articles
• A. Samarov - Lower bound for the integral risk of density function estimates
• A. Nemirovskii - On nonparametric estimation of functions satisfying differential inequalities
• A. Korostelev and A. Tsybakov - Asymptotically minimax image reconstruction problems
• O. Lepskii - On problems of adaptive estimation in white Gaussian noise
• B. Polyak and A. Tsybakov - On stochastic approximation with arbitrary noise (the KW-case)
• E. Belitser and A. Korostelev - Pseudovalues and minimax filtering algorithms for the nonparametric median
• A. Veretennikov - On large deviations for ergodic process empirical measures
• V. Spokoinyi - On asymptotically optimal sequential experimental design
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.
|
2023-03-29 17:17:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29391875863075256, "perplexity": 3209.976042746845}, "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-2023-14/segments/1679296949009.11/warc/CC-MAIN-20230329151629-20230329181629-00230.warc.gz"}
|
http://quant.stackexchange.com/tags/replication/hot?filter=year
|
# Tag Info
2
You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think: A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff: $$V_t=\alpha_tS_t+\beta_t B_t$$ It further implies $$dV_t=\alpha_tdS_t+\beta_tdB_t$$ To replicate a derivative $C(S_t,t)$ by a ...
1
I read the question as follows: You have one stock $S_0$ and after one period it either goes up to $S^+$ where the option takes the value $f^+$ or it goes down to $S^-$ where the option takes the value $f^-$. The bond grows from $B_0$ to $B_1 = B_0 \exp(r)$. Then you need to solve $$a S^+ + b B_1 = f^+ \\ a S^- + b B_1 = f^-$$ for $a,b$ which are $2$ ...
1
I think the title here is misleading. Let's go back to the BS world with $r=0$ to $a(S_t)=S_t \sigma.$ In that case, all you are saying is that you can replicate a call option by holding $N(d_1)$ units of stock at time $t.$ What does this have to do with the second equation? I am guessing that this is the price process of an asset of nothing option with ...
1
if you let the implied vol depend on K you get two terms the first is $N(d_2)$ but you get a correction term which is the slope times the vega $$\frac{\partial C}{\partial \sigma} \frac{\partial \sigma}{\partial K}.$$ (see eg my book)
Only top voted, non community-wiki answers of a minimum length are eligible
|
2015-04-25 18:25:28
|
{"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": 0.8964027166366577, "perplexity": 327.7622446968364}, "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/1429246650671.76/warc/CC-MAIN-20150417045730-00256-ip-10-235-10-82.ec2.internal.warc.gz"}
|
http://simbad.u-strasbg.fr/simbad/sim-ref?querymethod=bib&simbo=on&submit=submit+bibcode&bibcode=2005A%26A...443..485D
|
# 2005A&A...443..485D
other querymodes : Identifierquery Coordinatequery Criteriaquery Referencequery Basicquery Scriptsubmission TAP Outputoptions Help
Query : 2005A&A...443..485D
2005A&A...443..485D - Astronomy and Astrophysics, volume 443, 485-494 (2005/11-4)
Global characteristics of the first IBIS/ISGRI catalogue sources: unveiling a murky episode of binary star evolution.
DEAN A.J., BAZZANO A., HILL A.B., STEPHEN J.B., BASSANI L., BARLOW E.J., BIRD A.J., LEBRUN F., SGUERA V., SHAW S.E., UBERTINI P., WALTER R. and WILLIS D.R.
Abstract (from CDS):
INTEGRAL is the first gamma-ray astronomy mission with a sufficient sensitivity and angular resolution combination appropriate to the detection and identification of considerable numbers of gamma-ray emitting sources. The large field of view (∼30° zero response FWHM) enables INTEGRAL to survey the galactic plane on a regular (∼weekly) basis as part of the core programme. The first source catalogue, based on the 1st year of core programme data (∼5Ms) has been completed and published (Bird et al., 2004ApJ...607L..33B) IBIS survey. It contained 123 γ-ray sources (24 HMXB, 54 LMXB, 28 unknown'', plus 17 others) - sufficient numbers for a reasonable statistical analysis of their global properties. These were located to a positional accuracy of typically 0.72-arcmin. The detection of previously unknown γ-ray emitting sources generally exhibiting high intrinsic absorption, which do not have readily identifiable counterparts at other wavelengths, is intriguing. The substantial fraction (roughly 20% of the total number) of unclassified γ-ray sources suggests they must constitute a significant family of objects. In this paper we review the global characteristics of the known galactic sources as well as the unclassified objects with the twin aims of investigating how the unclassified set may fit into stellar evolution and improving our understanding of known X-ray binary systems through the non-thermal γ-ray channel. In the context of the known systems we are very conscious that they constitute a γ-ray selected set, and may exhibit subtle generic differences to the rest of the class. We present Log(N)-Log(S) distributions, angular distributions, and for systems with reliable distance estimates the spatial distributions within the Galaxy and luminosity functions. For the unknown sources, this statistical analysis has shown that they are most likely to be HMXBs containing a highly magnetised neutron star. The lack of X-ray counterparts for these sources indicates a high degree of intrinsic obscuration.
Abstract Copyright:
Journal keyword(s): gamma-rays: observations - X-rays: binaries - Galaxy: general - Galaxy: structure - Galaxy: stellar content
Simbad objects: 5
Full paper
Number of rows : 5
N Identifier Otype ICRS (J2000)
RA
ICRS (J2000)
DEC
Mag U Mag B Mag V Mag R Mag I Sp type #ref
1850 - 2022
#notes
1 NAME Nor Arm PoG 16 00 -50.0 ~ 380 0
2 NAME Galactic Center reg 17 45 39.60213 -29 00 22.0000 ~ 12908 0
3 IGR J17460-3047 gam 17 46 19 -30 47.5 ~ 5 0
4 NAME Sgr Arm PoG 19 00 -30.0 ~ 585 1
5 NAME Scutum Spiral Arm PoG ~ ~ ~ 314 0
To bookmark this query, right click on this link: simbad:objects in 2005A&A...443..485D and select 'bookmark this link' or equivalent in the popup menu
2022.05.25-00:24:08
© Université de Strasbourg/CNRS
|
2022-05-24 22:24:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.4394855499267578, "perplexity": 10249.49360922641}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00171.warc.gz"}
|
https://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality
|
# Bishop–Gromov inequality
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem.[1]
## Statement
Let $M$ be a complete n-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound
$\mathrm{Ric} \geq (n-1) K \,$
for a constant $K\in \mathbb{R}$. Let $M_K^n$ be the complete n-dimensional simply connected space of constant sectional curvature $K$ (and hence of constant Ricci curvature $(n-1)K$); thus $M_K^n$ is the n-sphere of radius $1/\sqrt{K}$ if K > 0, or n-dimensional Euclidean space if $K=0$, or an appropriately rescaled version of n-dimensional hyperbolic space if $K<0$. Denote by B(pr) the ball of radius r around a point p, defined with respect to the Riemannian distance function.
Then, for any $p\in M$ and $p_K\in M_K^n$, the function
$\phi(r) = \frac{\mathrm{Vol} \, B(p,r)}{\mathrm{Vol}\, B(p_K,r)}$
is non-increasing on (0, ∞).
As r goes to zero, the ratio approaches one, so together with the monotonicity this implies that
$\mathrm{Vol} \,B(p,r) \leq \mathrm{Vol} \, B(p_K,r).$
This is the version first proved by Bishop,[2][3] originally assuming the (unnecessary) added hypothesis that $r$ is less than the injectivity radius at $p$.
|
2015-09-01 00:54:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 16, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9737051725387573, "perplexity": 160.30076538472392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644068184.11/warc/CC-MAIN-20150827025428-00177-ip-10-171-96-226.ec2.internal.warc.gz"}
|
https://www.shaalaa.com/question-bank-solutions/pressure-calculation-pressure-simple-cases-calculate-pressure-exerted-surface-05-m2-thrust-100-kgf_30631
|
Share
# Calculate the Pressure Exerted on a Surface of 0.5 M2 by a Thrust of 100 Kgf. - Physics
Course
#### Question
Calculate the pressure exerted on a surface of 0.5 m2 by a thrust of 100 kgf.
#### Solution
P = ?
Thrust (F) = 100 kgf
A = 0.5 m2
P ="Thrust "/"Area"=100/0.5 =1000/5 = 200
Is there an error in this question or solution?
|
2020-09-23 00:48:21
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43363282084465027, "perplexity": 3380.7230045743454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400208095.31/warc/CC-MAIN-20200922224013-20200923014013-00334.warc.gz"}
|
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=TOONB4_2013_v33n3_91
|
Experimental Study on Performance of MEMS(Multi-Effect-Multi-Stage) Distiller for Solar Thermal Desalination
Title & Authors
Experimental Study on Performance of MEMS(Multi-Effect-Multi-Stage) Distiller for Solar Thermal Desalination
Joo, Hong-Jin; Jeon, Yong-Han; Kwak, Hee-Youl;
Abstract
In this study, we have carried out development and performance evaluation of optimized MEMS(Multi-Effect-Multi-Stage) fresh water generator with $\small{7m^2/day}$ for solar thermal desalination system. The developed MEMS was composed of high temperature part and low temperature part. This arrangement has the advantage of increasing the availability of solar thermal energy. The MEMS consists of 2 steam generators, 5 evaporators, and 1 condenser. Tubes of heat exchanger used for steam generators, evaporators and condenser were manufactured by corrugated tubes. The performance of the MEMS was tested through in-door experiments, using an electric heater as heat source. The experimental conditions for each parameters were $\small{20^{\circ}C}$ for sea water inlet temperature to condenser, $\small{8.16m^2}$ /hour sea water inlet volume flow rate, $\small{70^{\circ}C}$ for hot water inlet temperature to generator of high temperature part, 3.6 4.8, 6.0 $\small{m^2/hour}$ for hot water inlet volume flow rate. As a result, The developed MEMS was required about 85 kW heating source to produce $\small{7m^2/day}$ of fresh water. It was analyzed that the performance ratio of MEMS was about 2.6.
Keywords
Sea Water Desalination;Solar Thermal Desalination;Multi Effect Distiller;PR(Performance Ratio);
Language
Korean
Cited by
References
1.
S. Nisan, N. Benzarti, A comprehensive economic evaluation of integrated desalination systems using fossil fuelled and nuclear energies and including their environmental costs, Desalination, 229, 125-146, (2008)
2.
M. A. Darwish and Hisman El-Dessouky, The heat recovery thermal vapour-compression desalting system :a comparison with other thermal desalination processes, Applied Thermal Engineering, 16, 523-537, (1996)
3.
Hisham T. El-Dessouky, Hisham M. Ettouney and Faisal Mandani, Application of gas-turbine exhaust gases for brackish water desalination : a techno-economic evaluation, Applied Thermal Engineering, 24, 2487-2500, (2004)
4.
S. Nisan, N. Benzarti, A comprehensive economice valuation of integrated desalination systems using fossil fuelled and nuclear energies and including their environmental costs, Desalination, 229, 125-146, (2008)
5.
A. M. El-Nashar and M. Samad, Thesolar desalination plant in Abu Dhabi 13 years of performance and operation history. Renewable Energy, 14, 263-274, (1998)
6.
Diego-Cesar Alarcon-Padilla, Julian Blanco-Galvez, Lourdes Garcia-Rodriguez, Wolfgang Gernjak and Sixto Malato-Rodriguez, First experimental results of anew hybrid solar/gas multi-effect distillation system : the AQUASOL project, Desalination, 220, 619-625, (2008)
7.
Kwak, H. Y., Kim, J. B., Joo, H. J., Yoon, E. S., and Joo, M. C., Demonstration study on desalination system using solar energy, Journal of the Korean Solar Energy Society, Vol. 27, No. 4, 27-33, (2007)
8.
Kwak, H. Y., Joo, H. J., and Hwang, I. S., Thermal performance of single stage shell & tubes(SAT) fresh water generator, INTA-SEGA, 2009.
9.
Joo, H. J., Hwang, I. S., and Kwak, H. Y., Development of Multi Effect Distillation for Solar Thermal Seawater Desalination System, Journal of the Korean Solar Energy Society, Vol. 31, No. 1, 1-7, (2011)
10.
Joo, H. J., Jung, I. Y., Yoon, S. K., and Kwak, H. Y., CFD Analysis on the Flow Characteristics of Ejector According to the Position Changes of Driving Nozzle for F. W. G, Journal of the Korean Solar Energy Society, Vol. 31, No. 3, 23-29, (2011)
11.
Andrew Porteous., et al., Desalination Technology Development and Practice, Applied Science Publishers, London, 1983
|
2018-08-18 10:59:08
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 6, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6978702545166016, "perplexity": 13561.62704323868}, "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-2018-34/segments/1534221213540.33/warc/CC-MAIN-20180818095337-20180818115337-00600.warc.gz"}
|
https://www.snapxam.com/calculators/trigonometric-integrals-calculator
|
# Trigonometric integrals Calculator
## Get detailed solutions to your math problems with our Trigonometric integrals step by step calculator. Sharpen your math skills and learn step by step with our math solver. Check out more online calculators here.
Go!
1
2
3
4
5
6
7
8
9
0
x
y
(◻)
◻/◻
2
e
π
ln
log
lim
d/dx
d/dx
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
### Difficult Problems
1
Solved example of Trigonometric integrals
$\int\left(\sec\left(x\right)^6-\sec\left(x\right)^4\right)dx$
2
The integral of a sum of two or more functions is equal to the sum of their integrals
$\int\sec\left(x\right)^6dx+\int-\sec\left(x\right)^4dx$
3
Take the constant out of the integral
$\int\sec\left(x\right)^6dx-\int\sec\left(x\right)^4dx$
4
Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}+\frac{6-2}{6-1}\int\sec\left(x\right)^{\left(6-2\right)}dx-\int\sec\left(x\right)^4dx$
5
Subtract the values $6$ and $-2$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}+\frac{4}{5}\int\sec\left(x\right)^{4}dx-\int\sec\left(x\right)^4dx$
6
Divide $4$ by $5$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}+\frac{4}{5}\int\sec\left(x\right)^{4}dx-\int\sec\left(x\right)^4dx$
7
Adding $\frac{4}{5}\int\sec\left(x\right)^{4}dx$ and $-1\int\sec\left(x\right)^{4}dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\int\sec\left(x\right)^{4}dx$
8
Simplify the integral of secant applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\left(\frac{\sec\left(x\right)^{3}\sin\left(x\right)}{3}+\frac{2}{3}\int\sec\left(x\right)^{2}dx\right)$
9
The integral of $\sec(x)^2$ is $\tan(x)$
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\left(\frac{\sec\left(x\right)^{3}\sin\left(x\right)}{3}+\frac{2}{3}\tan\left(x\right)\right)$
10
As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration
$\frac{\sec\left(x\right)^{5}\sin\left(x\right)}{5}-\frac{1}{5}\left(\frac{\sec\left(x\right)^{3}\sin\left(x\right)}{3}+\frac{2}{3}\tan\left(x\right)\right)+C_0$
|
2019-04-25 22:04:47
|
{"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": 0.9820188283920288, "perplexity": 984.557849938659}, "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/1555578742415.81/warc/CC-MAIN-20190425213812-20190425235812-00072.warc.gz"}
|
https://gitlab.enpc.fr/fjramireg/comet-fenics/-/blame/779039e48d6b24f826792ef19ae639af9d4de6ee/doc/_build/html/demo/elasticity/orthotropic_elasticity.py.html
|
Orthotropic linear elasticity
Introduction
In this numerical tour, we will show how to tackle the case of orthotropic elasticity (in a 2D setting). The corresponding file can be obtained from orthotropic_elasticity.py.
We consider here the case of a square plate perforated by a circular hole of radius $$R$$, the plate dimension is $$2L\times 2L$$ with $$L \gg R$$ Only the top-right quarter of the plate will be considered. Loading will consist of a uniform traction on the top/bottom boundaries, symmetry conditions will also be applied on the correponding symmetry planes. To generate the perforated domain we use here the mshr module and define the boolean “minus” operation between a rectangle and a circle:
from fenics import * from mshr import * L, R = 1., 0.1 N = 50 # mesh density domain = Rectangle(Point(0.,0.), Point(L, L)) - Circle(Point(0., 0.), R) mesh = generate_mesh(domain, N)
Constitutive relation
Constitutive relations will be defined using an engineering (or Voigt) notation (i.e. second order tensors will be written as a vector of their components) contrary to the 2D linear elasticity example which used an intrinsic notation. In the material frame, which is assumed to coincide here with the global $$(Oxy)$$ frame, the orthotropic constitutive law writes $$\boldsymbol{\varepsilon}=\mathbf{S} \boldsymbol{\sigma}$$ using the compliance matrix $$\mathbf{S}$$ with:
$\begin{split}\begin{Bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ 2\varepsilon_{xy} \end{Bmatrix} = \begin{bmatrix} 1/E_x & -\nu_{xy}/E_x & 0\\ -\nu_{yx}/E_y & 1/E_y & 0 \\ 0 & 0 & 1/G_{xy} \end{bmatrix}\begin{Bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy} \end{Bmatrix}\end{split}$
with $$E_x, E_y$$ the two Young’s moduli in the orthotropy directions, $$\nu_{xy}$$ the in-plane Poisson ration (with the following relation ensuring the constitutive relation symmetry $$\nu_{yx}=\nu_{xy}E_y/E_x$$) and $$G_{xy}$$ being the shear modulus. This relation needs to be inverted to obtain the stress components as a function of the strain components $$\boldsymbol{\sigma}=\mathbf{C}\boldsymbol{\varepsilon}$$ with $$\mathbf{C}=\mathbf{S}^{-1}$$:
Ex, Ey, nuxy, Gxy = 100., 10., 0.3, 5. S = as_matrix([[1./Ex,nuxy/Ex,0.],[nuxy/Ex,1./Ey,0.],[0.,0.,1./Gxy]]) C = inv(S)
Note
Here we used the inv opertor to compute the elasticity matrix $$\mathbf{C}$$. We could also have computed analytically the inverse relation. Note that the inv operator is implemented only up to 3x3 matrices. Extension to the 3D case yields 6x6 matrices and therefore requires either analytical inversion or numerical inversion using Numpy for instance (assuming that the material parameters are constants).
We define different functions for representing the stress and strain either as second-order tensor or using the Voigt engineering notation:
def eps(v): return sym(grad(v)) def strain2voigt(e): """e is a 2nd-order tensor, returns its Voigt vectorial representation""" return as_vector([e[0,0],e[1,1],2*e[0,1]]) def voigt2stress(s): """ s is a stress-like vector (no 2 factor on last component) returns its tensorial representation """ return as_tensor([[s[0],s[2]],[s[2],s[1]]]) def sigma(v): return voigt2stress(dot(C,strain2voigt(eps(v))))
Problem position and resolution
Different parts of the quarter plate boundaries are now defined as well as the exterior integration measure ds:
class Top(SubDomain): def inside(self, x, on_boundary): return near(x[1],L) and on_boundary class Left(SubDomain): def inside(self, x, on_boundary): return near(x[0],0) and on_boundary class Bottom(SubDomain): def inside(self, x, on_boundary): return near(x[1],0) and on_boundary # exterior facets MeshFunction facets = MeshFunction("size_t", mesh, 1) facets.set_all(0) Top().mark(facets, 1) Left().mark(facets, 2) Bottom().mark(facets, 3) ds = Measure('ds')[facets]
We are now in position to define the variational form which is given as in 2D linear elasticity, the linear form now contains a Neumann term corresponding to a uniform vertical traction $$\sigma_{\infty}$$ on the top boundary:
# Define function space V = VectorFunctionSpace(mesh, 'Lagrange', 2) # Define variational problem du = TrialFunction(V) u_ = TestFunction(V) u = Function(V, name='Displacement') a = inner(sigma(du), eps(u_))*dx # uniform traction on top boundary T = Constant((0, 1e-3)) l = dot(T, u_)*ds(1)
Symmetric boundary conditions are applied on the Top and Left boundaries and the problem is solved:
# symmetry boundary conditions bc = [DirichletBC(V.sub(0), Constant(0.), facets, 2), DirichletBC(V.sub(1), Constant(0.), facets, 3)] solve(a == l, u, bc) import matplotlib.pyplot as plt p = plot(sigma(u)[1,1]/T[1], mode='color') plt.colorbar(p) plt.title(r"$\sigma_{yy}$",fontsize=26)
The $$\sigma_{xx}$$ and $$\sigma_{yy}$$ components should look like that:
|
2022-05-21 02:05:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6609014868736267, "perplexity": 3486.110906242067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534773.36/warc/CC-MAIN-20220521014358-20220521044358-00197.warc.gz"}
|
https://www.physicsforums.com/threads/electric-problem.130236/
|
# Electric Problem (1 Viewer)
### Users Who Are Viewing This Thread (Users: 0, Guests: 1)
#### KyoPhan
My semester just started and I'm already struggling.
The diagram is a square with a dot on each corner representing a charged particle. 1 at top left, 2 at top right, 3 at bottom left, and 4 at bottom right. The distance on one side of the square is a
The problem reads In fig. 21-22, four particles form a square. The charges are q1 = q4 = Q and q2 = q3 = q.
(a) What is Q/q if the net electrostatic force on particles 1 and 4 is zero?
(b) Is there any value of q that makes the net electrostatic force on each of the four particles zero? Explain
For (a)
I think I have to apply Coulomb's law and calculate all the forces from each individual particle. Assuming that 1/4 are both negative and 2/3 are positive, I calculated the force upward (by adding the force caused between 1/4 and 1/3) on particle 1. Then I set them equal to 0. I solved for Q in relation with q and got Q = -2q/a . So I calculated Q/q and got -2/a.
For (b)
I want to say that q=Q but one of opposite sign because I remember my professor saying that if one charge is in equilibrium, then the rest are at well. (correct me if I'm using the terms or concept incorrectly). Or is this only when they are semetric because I remember him talking about it when there was a square with an electron on each corner, with a proton in the middle. Do I have the right idea or am I completly lost? Do I have to apply Coulomb's equation in some way?
Thanks you for taking your time, I really appreciate it.
Last edited:
#### lightgrav
Homework Helper
Well, since Q and q are both quantities of charge, in [Coulomb],
is it possible that "a" has no units? Is "a" a distance, in [meter] ?
The horizontal component of Force on 1 must be zero ... so that :
kQq/a^2 = {kqq/(1.414 a)^2}sin 45 => Q/a^2 = .707q/(2a^2) ...
#### KyoPhan
A has no units, its just an unknown distance
I'm sorry I'm kinda slow, but where did 1.414a come from?
#### Saketh
KyoPhan said:
A has no units, its just an unknown distance
I'm sorry I'm kinda slow, but where did 1.414a come from?
It's the square root of two. Since we're dealing with a square, the length of the diagonal is $$s\sqrt{2}$$, where $$s$$ is the side length.
#### KyoPhan
lightgrav said:
Well, since Q and q are both quantities of charge, in [Coulomb],
is it possible that "a" has no units? Is "a" a distance, in [meter] ?
The horizontal component of Force on 1 must be zero ... so that :
kQq/a^2 = {kqq/(1.414 a)^2}sin 45 => Q/a^2 = .707q/(2a^2) ...
O okay I understand it now, thx a lot
### The Physics Forums Way
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
|
2019-03-24 21:14:36
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6794750094413757, "perplexity": 1392.8884071260914}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203493.88/warc/CC-MAIN-20190324210143-20190324232143-00121.warc.gz"}
|
https://www.physicsforums.com/threads/satisfying-wave-equation.210011/
|
# Satisfying wave equation
1. Jan 20, 2008
### sweep123
[SOLVED] Satisfying wave equation
1. The problem statement, all variables and given/known data
Confirm that the following wave satisfies the wave equation and obtain an expression for the velocity of a wave
Y=Asin(2x-5t)*e^(-2t)
2. Relevant equations
the wave equation is
(d^2y/dt^2)=(V^2)*(d^2y/dx^2)
3. The attempt at a solution
I assumed that I had to differentiate Y with respect to 't' twice and the differentiate Y with respect to 'x' twice and then substitute these into the equation.
This left me with
-21Ae^(-2t)sin(2x-t)+20Ae^(-2t)cos(2x-5t)=(V^2)(-4Ae^(-2t)sin(2x-5t))
but this doesnt really prove that the wave satisfies the equation. Does it?
I can then rearrange to get V the wave velocity. Am I on the right track?
2. Jan 20, 2008
### Rainbow Child
Is A constant or is it A(x)? Because with A constant, your function $$y(x,t)$$, does not satisfy the wave equation.
3. Jan 20, 2008
### Staff: Mentor
One would have to demonstrate that both sides of the wave equation are equal when using the proposed solution.
The general wave equation is
$$\frac{\partial^2 u} {\partial t^2} = c^2 \nabla^2 u$$, where c is the wave velocity. That constant, c, would be found in the solution.
So then, what is the value of V based on the given function?
I would expect A is a constant coefficient of amplitude.
Last edited: Jan 20, 2008
|
2017-01-21 22:20:00
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8044295907020569, "perplexity": 722.2896610861532}, "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/1484560281226.52/warc/CC-MAIN-20170116095121-00156-ip-10-171-10-70.ec2.internal.warc.gz"}
|
https://elteoremadecuales.com/barbiers-theorem/?lang=fr
|
# Barbier's theorem
Barbier's theorem These Reuleaux polygons have constant width, and all have the same width; therefore by Barbier's theorem they also have equal perimeters.
En géométrie, Barbier's theorem states that every curve of constant width has perimeter π times its width, regardless of its precise shape.[1] This theorem was first published by Joseph-Émile Barbier in 1860.[2] Contenu 1 Exemples 2 Preuves 3 Higher dimensions 4 Voir également 5 References Examples The most familiar examples of curves of constant width are the circle and the Reuleaux triangle. For a circle, the width is the same as the diameter; a circle of width w has perimeter πw. A Reuleaux triangle of width w consists of three arcs of circles of radius w. Each of these arcs has central angle π/3, so the perimeter of the Reuleaux triangle of width w is equal to half the perimeter of a circle of radius w and therefore is equal to πw. A similar analysis of other simple examples such as Reuleaux polygons gives the same answer.
Proofs One proof of the theorem uses the properties of Minkowski sums. If K is a body of constant width w, then the Minkowski sum of K and its 180° rotation is a disk with radius w and perimeter 2πw. Cependant, the Minkowski sum acts linearly on the perimeters of convex bodies, so the perimeter of K must be half the perimeter of this disk, which is πw as the theorem states.[3] Alternativement, the theorem follows immediately from the Crofton formula in integral geometry according to which the length of any curve equals the measure of the set of lines that cross the curve, multiplied by their numbers of crossings. Any two curves that have the same constant width are crossed by sets of lines with the same measure, and therefore they have the same length. Historiquement, Crofton derived his formula later than, and independently of, Barbier's theorem.[4] An elementary probabilistic proof of the theorem can be found at Buffon's noodle.
Higher dimensions The analogue of Barbier's theorem for surfaces of constant width is false. En particulier, the unit sphere has surface area {displaystyle 4pi approx 12.566} , while the surface of revolution of a Reuleaux triangle with the same constant width has surface area {displaystyle 8pi -{tfrac {4}{3}}pi ^{2}environ 11.973} .[5] À la place, Barbier's theorem generalizes to bodies of constant brightness, three-dimensional convex sets for which every two-dimensional projection has the same area. These all have the same surface area as a sphere of the same projected area.
And in general, si {style d'affichage S} is a convex subset of {style d'affichage mathbb {R} ^{n}} , for which every (n-1)-dimensional projection has area of the unit ball in {style d'affichage mathbb {R} ^{n-1}} , then the surface area of {style d'affichage S} is equal to that of the unit sphere in {style d'affichage mathbb {R} ^{n}} . This follows from the general form of Crofton formula.[6] See also Blaschke–Lebesgue theorem and isoperimetric inequality, bounding the areas of curves of constant width References ^ Lay, Steven R. (2007), Convex Sets and Their Applications, Douvres, Théorème 11.11, pp. 81–82, ISBN 9780486458038. ^ Barbier, E. (1860), "Note sur le problème de l'aiguille et le jeu du joint couvert" (PDF), Journal de mathématiques pures et appliquées, 2e série (en français), 5: 273–286. See in particular pp. 283–285. ^ The Theorem of Barbier (Java) at cut-the-knot. ^ Sylvester, J. J. (1890), "On a funicular solution of Buffon's "problem of the needle" in its most general form" (PDF), Journal de mathématiques, 14 (1): 185–205, est ce que je:10.1007/BF02413320. ^ Bayen, Térence; Henrion, Didier (2012), "Semidefinite programming for optimizing convex bodies under width constraints", Optimization Methods and Software, 27 (6): 1073–1099, CiteSeerX 10.1.1.402.9539, est ce que je:10.1080/10556788.2010.547580. ^ Martini, Horst; Montejano, Louis; Oliveros, Déborah (2019), "Section 13.3.2 Convex Bodies of Constant Brightness", Bodies of Constant Width: An Introduction to Convex Geometry with Applications, Birkhauser, pp. 310–313, est ce que je:10.1007/978-3-030-03868-7, ISBN 978-3-030-03866-3, M 3930585 Catégories: Theorems in plane geometryPiLengthConstant width
Si vous voulez connaître d'autres articles similaires à Barbier's theorem vous pouvez visiter la catégorie Length.
Monter
Nous utilisons nos propres cookies et ceux de tiers pour améliorer l'expérience utilisateur Plus d'informations
|
2023-03-20 15:58:39
|
{"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": 0.9162428975105286, "perplexity": 1575.3361182492379}, "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-2023-14/segments/1679296943484.34/warc/CC-MAIN-20230320144934-20230320174934-00718.warc.gz"}
|
https://events.icecube.wisc.edu/event/48/timetable/?view=standard
|
# Cosmic Ray Anisotropy Workshop 2013
US/Central
1308 West Dayton Street, Madison, WI 53706
Description
The Cosmic Ray Anisotropy Workshop
The goal of the workshop is to bring together different scientific communities to discuss the origin of the anisotropy of cosmic rays and their spectral anomalies in a variety of energy ranges. We invite experts in the detection of cosmic rays on the ground, with balloons, or in space and from a variety of fields — cosmic ray physics, astrophysics, plasma physics, heliospheric physics, interstellar medium, and particle interactions in magnetic fields. Participants will explore scenarios on the origin of cosmic rays and their acceleration and transport in the interstellar medium and in the heliosphere.
Registration and abstract submission can be completed on Indico. For travel and accommodations can be found on the conference website.
Thursday, sessions run from 8:45 to 18:00, with two 30-minute breaks and 1 hour for lunch; Friday, from 9:00 to 18:00 with the same breaks and lunch; and Saturday, we'll begin at 9:00 and finish at 13:00, with one break and no afternoon session.
Participants
• Ahron Barber
• Albrecht Karle
• Alex Lazarian
• Ali Kheirandish
• Andrea Chiavassa
• Bakhtiyar Ruzybayev
• Brian Vlcek
• Dan Fiorino
• David Benyamin
• Douglas Bergman
• Eduardo de la Fuente
• Ellen Zweibel
• Eun-Suk Seo
• Francis Halzen
• Frank McNally
• Gary Hill
• Gary Zank
• Georgia de Nolfo
• Glennys Farrar
• Gus Sinnis
• Gwenael Giacinti
• Haim Goldberg
• Jesus Hernandez Carretero
• Jesus Martinez
• John Ennis Ward
• Jorge Casaus
• Juan Carlos Diaz-Velez
• Justin Vandenbroucke
• Ke Fang
• Kim Kreiger
• Klaus Scherer
• Luis Anchordoqui
• Luis Del Peral
• Marcos Santander
• Markus Ahlers
• Mauricio Bustamante
• Michael DuVernois
• Ming Zhang
• Nikolai Pogorelov
• Olivier Deligny
• Paolo Desiati
• Pasquale Blasi
• Peter Denton
• Philip Chang
• Philipp Kronberg
• Philipp Mertsch
• Priscilla Frisch
• rahul kumar
• Reinhard Schlickeiser
• Segev BenZvi
• Serap Tilav
• Stefan Westerhoff
• Thomas Paul
• thomas weiler
• Tim Felten
• Vernon Barger Barger
• Zachary Griffith
• Thursday, 26 September
• 08:00 08:45
Registration / Information all day Union South (Landmark, 3rd floor Center)
### Union South
#### Landmark, 3rd floor Center
• 08:45 08:59
Welcome - Marcos Santander
• 08:59 09:00
Cosmic Rays: Indirect Observations chaired by Stefan Westerhoff
• 09:00 09:40
Cosmic Ray Anisotropy: A Review - Gus Sinnis, Los Alamos National Laboratory
• 09:00
Cosmic Ray Anisotropy: A Review 40m
The anisotropy of the cosmic radiation has been studied for over half a century. For much of this time the results have been contradictory and difficult to understand. In the past decade there has been an increasing number of well measured anisotropies, that exhibit interesting energy dependence. In this talk I will summarize the status of both large and small scale anisotropy measurements from a few TeV to above 10^19 eV. While some features of the observed anisotropies are firmly established others are not - for example the time dependence of the large scale anisotropy reported by Milagro and the chemical composition of the anisotropic component of the cosmic rays. I will discuss the observations that are on firm foundation and point to further measurements that must be made to confirm some of the more important but unconfirmed features of the anisotropies. Finally, I will discuss a new interpretation of the small-scale anisotropy as a local clump of dark matter.
Speaker: Dr Gus Sinnis (Los Alamos National Laboratory)
• 09:40 10:10
Cosmic-Ray Anisotropy with the HAWC Observatory - Dan Fiorino, UW-Madison Northwoods, 3rd floor Center
#### Northwoods, 3rd floor Center
• 09:40
Cosmic-Ray Anisotropy with the HAWC Observatory 30m
The High-Altitude Water Cherenkov (HAWC) Observatory is a TeV gamma-ray and cosmic-ray detector operating at an altitude of 4100 meters in Mexico. HAWC is an extensive air-shower array comprising 300 optically-isolated water Cherenkov detectors. While the observatory is only partially deployed, with 100 Cherenkov detectors in data acquisition since summer 2013, statistics are already sufficient to perform studies of cosmic-ray anisotropy. We discuss the status and performance of the detector, including the pointing accuracy and angular resolution as inferred from the observation of the moon shadow and simulations, and present first results on small-scale cosmic-ray anisotropy.
• 10:10 10:40
Cosmic-ray anisotropy studies at TeV and PeV energies with AMANDA, IceCube, and IceTop - Marcos Santander, UW-Madison Northwoods, 3rd floor Center
#### Northwoods, 3rd floor Center
• 10:10
Cosmic-ray anisotropy studies at TeV and PeV energies with AMANDA, IceCube, and IceTop 30m
The study of the cosmic ray anisotropy in the TeV-PeV energy range could provide clues about the origin and propagation of cosmic rays in our galaxy. The measurement of this per-mille-anisotropy requires data sets with several billion cosmic-ray events. A sample of this size has been collected over the last six years by the IceCube neutrino telescope at the south pole, which detects cosmic ray muons at a rate of about 2 kHz. In the IceCube data, we observe a significant anisotropy in the southern sky for primary energies between 20 and 400 TeV. The anisotropy has a large-scale component of per-mille strength, accompanied by localized excess and deficit regions with smaller amplitudes and typical angular sizes between $10^{\circ}$ and $20^{\circ}$. A study of the time variability of the anisotropy is performed by combining data from IceCube and its predecessor experiment, AMANDA, which operated between 2000 and 2007. Finally, A change in the shape and an increase in the amplitude of this anisotropy is observed at PeV energies by including events of IceTop, the air shower array above IceCube.
Speaker: Marcos Santander (University of Wisconsin-Madison)
• 10:40 11:00
Break 20m Landmark, 3rd floor center
### Landmark, 3rd floor center
1308 West Dayton Street, Madison, WI 53706
• 11:00 11:25
Measurement of cosmic ray energy spectrum with IceCube - Bakhtiyar Ruzybayev, U of Delaware Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 11:00
Measurement of cosmic ray energy spectrum with IceCube 25m
We report on the measurement of the all-particle cosmic ray energy spectrum with IceCube. Results of two different techniques will be presented. The first result is a measurement of the all-particle cosmic ray energy spectrum in the energy range from 1.58 PeV to 1.26 EeV using the IceTop air shower array, which is the surface component of the IceCube Neutrino Observatory at the South Pole. The second result is a measurement of cosmic ray energy spectrum using neural network techniques and the full IceCube as a 3-dimensional cosmic ray detector. The measured energy spectrum exhibits clear deviations from a single power law above the knee around 4 PeV and below 1 EeV.
Speaker: Bakhtiyar Ruzybayev (o=udel,ou=Institutions,dc=icecube,dc=wisc,dc=edu)
• 11:25 11:50
Measurements of the cosmic rays spectrum and large scale anisotropies with the KASCADE-Grande experiment - Andrea Chiavassa, U of Torino Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 11:25
Measurements of the cosmic rays spectrum and large scale anisotropies with the KASCADE-Grande experiment 25m
The KASCADE-Grande experiment measured with high precision EAS generated by cosmic rays in the 10^16-10^18 eV energy range, for each event the total number of charged particles and the number of muons were determined. Based on these two observables we estimate the primary energy of each event and we separate the events into two samples generated by light and heavy primaries respectively. The measurement of the all particle and of the light and heavy mass groups energy spectra will be presented. Our results show that: a) the all particle spectrum cannot be described by a single index power law; b) the heavy primaries mass group one show a steepening at ~8x10^16 eV; c) the light primaries mass group one show a hardening at ~10^17 eV. A search for large scale anisotropies, based on the east-west method, will also be presented. No significant anisotropies were detected. The obtained upper limits will be discussed and compared with the results of other experiments.
Speaker: Andrea Chiavassa (Universita` di Torino)
• 11:50 12:15
Large-Scale Distribution of Arrival Directions of Cosmic Rays Detected at the Pierre Auger Observatory Above ~10 PeV - Olivier Deligny, CNRS / IN2P3 - IPN Orsay Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 11:50
Large-Scale Distribution of Arrival Directions of Cosmic Rays Detected at the Pierre Auger Observatory Above ~10 PeV 20m
Harmonic analyses dedicated to searches for large-scale anisotropies in both right ascension and declination distributions of cosmic rays detected above ~10 PeV at the Pierre Auger Observatory are presented. Though additional statistics is still needed to characterize unambiguously the patterns as a function of the energy due to their relatively low dipole and quadrupole amplitudes, the focus is given to the few current hints that may be indicative of the presence of a structure at large scale over a wide energy range. On the other hand, the constraints on the production of cosmic rays provided by the upper limits obtained on the dipole and quadrupole amplitudes in the EeV energy range are discussed.
Speaker: Dr Olivier Deligny (CNRS/IN2P3 - IPN Orsay)
• 12:15 13:39
Lunch on your own 1h 24m
• 13:39 13:40
Astrophysics chaired by Justin Vandenbroucke
• 13:40 14:20
The Cosmic Ray - Anisotropy Connection - Pasquale Blasi, INAF / Arceti Astrophysics Observatory Northwoods, 3rd floor center
### Northwoods, 3rd floor center
1308 West Dayton Street, Madison, WI 53706
• 13:40
The Cosmic Ray - Anisotropy Connection 40m
I will illustrate the type of information that we can gather on acceleration and propagation of cosmic rays from measurement of anisotropy. More specifically I will focus on two issues: 1) a discussion the the problems arising from the measured anisotropy when compared with the standard supernova remnant paradigm for the origin of cosmic rays; 2) a discussion of the implications of the Pierre Auger, ICETOP and KASCADE-Grande measurements for the transition from Galactic to extragalactic cosmic rays and how these implications confront the measured anisotropy in the EeV energy range.
Speaker: Dr Pasquale Blasi (INAF/Arcetri Astrophysical Observatory)
• 14:20 14:50
A New Analysis Method for High-Energy CR Hadron Arrival Directions - Philipp Kronberg, U of Toronto Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 14:20
A New Analysis Method for High-Energy CR Hadron Arrival Directions 30m
A new approach to understanding the VHECR / UHECR sky is presented. I describe a multi-parameter analysis that is based on the observed CR arrival direction distribution. The sky plot origin can be any chosen reference source of cosmic rays. This source-centered sky (unlike [l,b] etc.) displays simulated energy-species-direction data. I discuss a preliminary UHECR-determined estimate of the intergalactic magnetic field out to ~4Mpc on the assumption that Cen A is the principal UHECR source. Other assumptions and models can be applied within this general conceptual framework. The analysis method (Yüksel, Stanev, Kistler & Kronberg ApJ 2012) can be applied to data from AUGER, HiRes, TA, and their successors. A specific example shows, within our reference assumptions, how the strength and structure of B(sub{IGM}) is approximately constrained at >~ 20 nG out to D~4Mpc, based on recent AUGER data. This is new "territory" for IGM magnetic field probes, and also the first VHECR sky-based probe of B(sub{IGM}) on nearby-Universe supra-galactic scales. It is a potentially powerful template for the understanding, and future modeling, of VHECR / UHECR propagation at greater distances. I also discuss the dependence of CR energy and species on the observed distribution of arrival directions.
Speaker: Prof. Philipp Kronberg (University of Toronto)
• 14:50 15:20
Developments on Galactic Magnetic Field and UHECR deflections - Glennys Farrar, NYU Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 15:20 15:40
Break 20m Landmark, 3rd floor center
### Landmark, 3rd floor center
1308 West Dayton Street, Madison, WI 53706
• 15:40 16:05
Anisotropic diffusion of cosmic rays and the TeV-band cosmic ray anisotropy - Rahul Kumar, Ben Gurion University Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 15:40
Anisotropic diffusion of cosmic rays and the TeV-band cosmic ray anisotropy 25m
We calculate the time-dependent transport of cosmic rays from point like sources in the Galaxy, assuming it can be described as diffusion. We show that the surprisingly small anisotropy in the TeV band, as recently reported by IceCube and others, can be reproduced assuming a small radial diffusion rate, without assuming a uniform distribution of the sources in the Galaxy.
Speaker: Mr rahul kumar (ben-gurion university)
• 16:05 16:30
Newborn Pulsars as sources of Ultrahigh Energy Cosmic Rays - Ke Fang, U of Chicago Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 16:05
Newborn Pulsars as sources of Ultrahigh Energy Cosmic Rays 25m
The workings of the most energetic astrophysical accelerators in the Universe are encoded in the origin of ultrahigh energy cosmic rays (UHECRs). Current observations by the Auger Observatory, the largest UHECR observatory, show a spectrum that agrees with an extragalactic origin, as well as an interesting transition in chemical composition from light element to heavier element as energy increases. Candidate sources range from young neutron stars to gamma-ray bursts and events in active galaxies. In this talk, we will discuss newborn pulsars as the sources of ultrahigh energy cosmic rays. We will show that a newborn pulsar model naturally injects heavier elements and can fit the observed spectrum once propagation in the supernova remnant is taken into account. With the proper injection abundances, integrated cosmic rays from the extragalactic pulsar population can match observation in all aspects - energy spectrum, chemical composition, and anisotropy. We will also examine the fingerprints of their Galactic counterparts on cosmic ray spectrum. Lastly, we will discuss the multi-messenger smoking gun of this scenario - the detectability of high energy neutrinos from pulsars and magnetars.
Speaker: Ke Fang (University of Chicago)
• 16:30 16:55
Galactic magnetic deflections of UHECRs including realistic random fields - Azadeh Keivani, Louisiana State University Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 16:30
Galactic magnetic deflections of UHECRs including realistic random fields 25m
A
Speaker: Azadeh Keivani (Louisiana State University)
• 16:55 17:20
Telescope Array: Results & Plans - Douglas Bergman, U of Utah Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 16:55
Telescope Array: Results & Plans 20m
I will present recent Telescope Array measurements of the spectrum, composition and anisotropy of ultra-high energy cosmic rays. I will then present our current work and plans for the future.
Speaker: Prof. Douglas Bergman (University of Utah)
• 17:20 17:45
Sensitivity of the orbiting JEM-EUSO mission to large-scale anisotropies - Peter Denton, Vanderbilt University Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 17:20
Sensitivity of the orbiting JEM-EUSO mission to large-scale anisotropies 25m
The two main advantages of space-based observation of extreme-energy (>~10^19 eV) cosmic-rays (EECRs) over ground-based observatories are the increased field of view and the all-sky coverage with nearly uniform systematics. The former guarantees increased statistics whereas the latter enables a partitioning of the sky into spherical harmonics. We have begun an investigation, using the spherical harmonic technique, of the reach of JEM-EUSO into potential anisotropies in the extreme-energy cosmic-ray sky-map for several different source models. The technique is explained here, and first results are presented. The discovery of anisotropies would help to identify the long-sought origin of EECRs.
Speaker: Mr Peter Denton (Vanderbilt University)
• 19:00 21:00
Social Dinner: Steenbock's on Orchard, 330 N. Orchard Street, Madison 53706 (across Johnson Street from Union South)
• Friday, 27 September
• 08:40 08:45
Interstellar Medium and Propogation chaired by Francis Halzen
• 08:45 09:25
Local interstellar magnetic field, Loop I, and interstellar clouds - Priscilla Frisch, U of Chicago
• 08:45
Local interstellar magnetic field, Loop I, and interstellar clouds 40m
Before reaching the Earth, galactic cosmic rays must traverse nearby partially ionized low density interstellar clouds. The evolved superbubble known as Loop I appears to order the cloud kinematics and the magnetic field of the interstellar medium (ISM) within tens of parsecs. The direction of the nearby interstellar magnetic field (ISMF) that is found from starlight polarized in the local interstellar medium is approximately parallel to the local surface of the Loop I shell that dominates the northern hemisphere. Nearby interstellar clouds flow through the local standard of rest with a direction that is perpendicular to the ISMF direction, to within the uncertainties. The direction of the ISMF helping to shape the heliosphere is found from the center of the Ribbon of energetic neutral atoms discovered by the Interstellar Boundary Explorer (IBEX) spacecraft, and is close to the local field direction found from polarization data. Open questions remain. The structure of the distant parts of Loop I is filamentary and there is evidence for filamentary structure in the local ISM. The role of flux freezing in local gas is unknown. The polarity of the magnetic field is not clear.
Speaker: Priscilla Frisch (University of Chicago)
• 09:25 09:55
Ellen Zweibel Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 09:55 10:25
Aperiodic magnetic field fluctuations and their effect on cosmic rays - Reinhardt Schlickeiser, Ruhr-U Bochum
• 09:55
Aperiodic magnetic field fluctuations and their effect on cosmic rays 30m
Understanding cosmic $(\delta B,\delta E)$-fluctuations in magnetized (interstellar medium) and nonmagnetized (IGM: intergalactic medium) plasmas is of crucial importance for cosmic ray transport, including the role of collective and noncollective modes and wave-like, weakly-propagating and aperiodic fluctuations. The ordering $B_0\gg \delta B\gg \delta E$ in magnetized systems, necessary for explaining the observed nearly isotropic CR momentum distribution function, is the basis for a perturbation scheme leading to the modified diffusion-convection CR transport equation and expressions for the CR anisotropy. The nonmagnetized IGM medium containes aperiodic magnetic fluctuations which are spontaneously emitted by the fully-ionized thermal electron-proton IGM plasma at a level of $\vert \delta B\vert =1.5\cdot 10^{-16}n_{-7}T_4^{-3/2}$ G. These spontaneously emitted fluctuations affect the propagation of CR protons and electrons in the IGM at energies below $10^{15}$ eV.
Speaker: Prof. Reinhard Schlickeiser (Ruhr-University Bochum, Germany)
• 10:25 10:45
Break 20m Landmark, 3rd floor center
### Landmark, 3rd floor center
1308 West Dayton Street, Madison, WI 53706
• 10:45 11:15
Cosmic ray acceleration in the presence of super diffusion - Alex Lazarian, UW-Madison Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 10:45
Cosmic ray acceleration in the presence of super diffusion 30m
Alfvenic turbulence that determines the magnetic field wandering exhibit the process of superdiffusion which results in the perpendicular displacement to change as y~x^3/2, where x is the distance measured along magnetic field is the distance perpendicular to the magnetic field, provided that y is less than the injection scale of the turbulence. This process changes substantially the acceleration of cosmic rays in perpendicular shocks, which were considered as the accelerating agent of anomalous cosmic rays. I shall discuss how the process of superdiffusion changes the acceleration in parallel and perpendicular shocks and show the analogies between the shock and reconnection acceleration.
• 11:15 11:40
Alexander Dosch Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 11:40 12:05
Recovering the observed B/C ratio in a dynamic spiral-armed cosmic ray model - David Benyamin, Hebrew University
• 11:40
Recovering the observed B/C ratio in a dynamic spiral-armed cosmic ray model 25m
We develop a fully three dimensional numerical code describing the diffusion of cosmic rays in the Milky Way. It includes the nuclear spallation chain up to Oxygen, and allows the study of various cosmic ray properties, such as the CR age, grammage traversed, and the ratio between secondary and primary particles. This code enables us to explore a model in which a large fraction of the cosmic ray acceleration takes place in the vicinity of galactic spiral arms and that these spiral arms are dynamic. We show that the effect of having dynamic spiral arms is to limit the age of cosmic rays at low energies. This is because at low energies the time since the last spiral arm passage governs the Cosmic Ray (CR) age, and not diffusion. Using the model, the observed spectral dependence of the secondary to primary ratio is recovered without requiring any further assumptions such as a galactic wind, re-acceleration or various assumptions on the diffusivity. In particular, we obtain a secondary to primary ratio which increases with energy below about 1 GeV.
Speaker: Mr David Benyamin (The Hebrew University)
• 12:05 12:30
Explanation for the Anisotropies at Small and Medium Angular Scales - Gwenael Giacinti, U of Oxford
• 12:05
Explanation for the Anisotropies at Small and Medium Angular Scales 25m
The diffusion approximation (DA) predicts a dipolar anisotropy, but cannot explain the anisotropies at smaller scales. However, the DA is not designed to predict phenomena arising on spatial scales smaller than the cosmic ray mean free path. We demonstrate here that energy-dependent smaller scale anisotropies naturally appear on the sky and reflect the local concrete realization of the turbulent magnetic field within the cosmic ray mean free path.
Speaker: Dr Gwenael Giacinti (University of Oxford)
• 12:30 13:54
Lunch on your own 1h 24m
• 13:54 13:55
Cosmic Rays: Direct Observations chaired by Albrecht Karle
• 13:55 14:35
Direct Measurements of Cosmic Rays - Eun-Suk Seo, University of Maryland
• 13:55
Direct Measurements of Cosmic Rays 40m
Direct measurements of cosmic rays with balloon-borne and space based instruments are used for understanding cosmic ray origin, acceleration and propagation, as well as exploring the supernova acceleration limit and searching for exotic sources such as dark matter. The energy reach of direct measurements is limited by the detector size and exposure time, but incident particles are identified element-by-element with excellent charge resolution. Recent results and their implications will be reviewed. The outlook of future experiments will also be discussed.
Speaker: Prof. Eun-Suk Seo (University of Maryland)
• 14:35 15:05
Super-TIGER and the search for Galactic Cosmic-Ray Origins - John Ennis Ward, Washington University Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 15:05 15:35
Elemental and Isotopic Abundances and Their Implications for Cosmic Ray Origins - Georgia De Nolfo, NASA/GSFC Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 15:05
Elemental and Isotopic Abundances and Their Implications for Cosmic Ray Origins 25m
The answer to the question of the origin of galactic cosmic rays lies not only with directional anisotropies for the highest energies where direction is preserved but also with in the signatures found in their energy spectra and composition. Elemental and isotopic measurements carry the imprint of nucleosynthesis, acceleration time scales, and residence times within the Galaxy. Recent isotopic measurements with the Cosmic Ray Isotope Spectrometer (CRIS) from ~80-600 MeV/nucleon aboard the Advanced Composition Explorer (ACE) satellite as well as elemental data from Mg through Sr from the Trans-Iron Galactic Element Recorder (TIGER), suggest an origin linked to OB associations. GCR ratio measurements of 22Ne/20Ne, 58Fe/56Fe, and 31Ga/32Ge in particular, are consistent with a source material that is a mixture of the interstellar material (with solar system abundances) and outflow/ejecta from massive stars. (The following is a complicated concept and may need to be longer to get the points across. I don’t understand it.) Furthermore, the ordering of refractory and volatile elements with atomic mass is improved if the source material includes massive star outflow/ejecta, resulting in power-law trend with atomic mass with similar slopes for both but with refractory elements preferentially accelerated by a factor of ~4. Together with recent observations of high-energy gamma-rays from SNRs and extended sources, we conclude that the likely source of GCRs is consistent with an origin in OB associations and their associated superbubbles.
Speaker: Dr Georgia de Nolfo (NASA/GSFC)
• 15:35 15:55
Break 20m
• 15:55 16:20
What do the spectral breaks in CR spectrum tell us? - Serap Tilav, U of Delaware Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 15:55
What do the spectral breaks in CR spectrum tell us? 25m
Rigidity dependent breaks and the hardening of the elemental spectra observed above 200 GeV provided the most important hint on the nature of the cosmic ray knee. Model independent analysis of the CR data (direct and indirect combined) shows at least 3 different source populations needed to describe the spectrum and composition from 200 GeV up to 200 EeV.
Speaker: Dr Serap Tilav (o=udel,ou=Institutions,dc=icecube,dc=wisc,dc=edu)
• 16:20 16:45
Cosmic ray measurements with the Fermi Large Area Telescope - Justin Vandenbroucke, UW-Madison Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 16:20
Cosmic ray measurements with the Fermi Large Area Telescope 25m
TBD
Speaker: Prof. Justin Vandenbroucke (UW Madison)
• 16:45 17:10
First Results of the AMS-02 Experiment on the ISS - Jorge Casaus, CIEMAT
• 16:45
First Results of the AMS-02 Experiment on the ISS 25m Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
AMS-02 is a general purpose cosmic ray detector operating on the International Space Station since 19 May 2011. The results based on the data collected during the first 2 years of the mission include high precision measurements of the proton, helium, electron and positron fluxes, and the boron to carbon ratio in the energy range from ~1GeV/n to ~1TeV/n. Proton and helium spectra are consistent with single power laws with no fine structures or breaks. The boron to carbon ratio shows no evidence for a structure within the studied energy range. The positron fraction is determined in the energy range from 0.5 to 350GeV and its energy spectrum shows an steadily increasing fraction from 10 to ~250GeV with no fine structure. The positron to electron and positron to proton ratios are consistent with isotropy within this energy range.
Speaker: Dr Jorge Casaus (CIEMAT - Spain)
• 17:10 17:35
A hadronic explanation of the lepton anomaly - Philipp Mertsch, KIPAC, Stanford Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 17:10
A hadronic explanation of the lepton anomaly 25m
The anomaly in the cosmic ray positron fraction, first observed by the PAMELA experiment and later confirmed by Fermi-LAT and AMS-02, has generated a lot of interest and theoretical efforts, mostly due to the suggested interpretation as an indirect signature of dark matter annihilation in the Galaxy. I will argue that this interpretation is now strongly disfavoured by searches for gamma-rays from the galactic halo and turn to possible astrophysical explanations. A hadronic model of production and acceleration of secondaries in mature supernova remnants provides a compelling explanation of hard secondary positrons and links to signatures in other hadronic channels, like neutrinos.
Speaker: Philipp Mertsch (KIPAC, Stanford)
• 17:35 18:00
Large scale anisotropy of cosmic rays and directional neutrino signals from Galactic sources - Luis Anchordoqui, UW-Milwaukee Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 17:35
Large scale anisotropy of cosmic rays and directional neutrino signals from Galactic sources 25m
Quite recently the IceCube Collaboration has reported an observation of 26 neutrino candidates above ~ 50 TeV. Including the two ~ 1 PeV neutrinos reported earlier in 2013, these 28 events constitute a 4.3\sigma excess compared to the atmospheric background. In this talk, I will explore the compatibility between the data and an unbroken power-law neutrino spectrum, for various values of spectral index \Gamma >= 2. I will show that \Gamma ~ 2.3 is consistent at the ~ 1.5\sigma level with the observed events up to 1 PeV and to the null observation of events at higher energies. I will then assume that the sources of this unbroken spectrum are Galactic, and deduce (i) an energy-transfer fraction from parent protons to pions, and (ii) a discriminating test between the two most popular models ("dip" and "ankle") for the Galactic to extragalactic cosmic-ray transition. Future IceCube data will test the unbroken power law hypothesis, and, if the neutrino sources are Galactic, discriminate between the "dip" and "ankle" models of Galactic to extragalactic transition.
Speaker: Prof. Luis Anchordoqui (University of Wisconsin Milwaukee)
• Saturday, 28 September
• 08:40 08:45
Heliosphere chaired by Alex Lazarian
• 08:45 09:25
Three-dimensional Structure of the Time-dependent Heliosphere Interacting with the Local Interstellar Medium - Nick Pogorelov, U of Alabama-Huntsville Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 08:45
Three-dimensional Structure of the Time-dependent Heliosphere Interacting with the Local Interstellar Medium 40m
In this brief overview, we describe observational and modeling aspects of the solar wind (SW) interaction with the local interstellar medium (LISM) paying particular attention to three-dimensional and time-dependent effects. We demonstrate that time-dependent phenomena may substantially affect the global streamline and magnetic field topology in the inner heliosheath (IHS) - a plasma layer between the heliospheric termination shock and the heliopause. It is shown that Voyager spacecraft observations cannot be easily interpreted without invoking solar cycle, magnetic reconnection, and MHD instability effects. In particular, the solar wind flow backward toward the Sun can be explained by the evolution of magnetic barriers developing in the IHS over the solar cycle. In view of the recent announcement of the Voyager1 penetration into the LISM, we discuss some issues related to numerical modeling of the heliopause instability. The behavior of the heliospheric current sheet in the inner heliosheath is discussed, which is sometimes accompanied by transition to a turbulent flow regime. We also show the results of our numerical modeling of the SW-LISM interaction using observational boundary conditions.
Speaker: Prof. Nikolai Pogorelov (University of Alabama in Huntsville)
• 09:25 09:55
Numerical Modeling of the Heliotail - Sergey Borovikov, U of Alabama - Huntsville, given by Nick Pogorelov Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 09:25
Numerical Modeling of the Heliotail 30m
The heliotail structure is of interest for the analysis of the Lyman–alpha absorption in the direction of the nearby stars, the energetic neutrals (ENA) production, and possibly cosmi ray acceleration. Recent Interstellar Boundary Explorer (IBEX) observations revealed rather complex topology of the heliotail (McComas et al. 2013). We performed 3D, MHD-kinetic modeling of the solar wind interaction with the local interstellar medium to analyze the heliotail region to distances up to 5000 AU downstream. We examined the role of the interstellar magnetic field in shaping the heliopause. The heliospheric current sheet behavior and the heliopause instability are analyzed. We determined that the heliopause is noticeably squeezed. We also show that there is no well-defined boundary between the solar wind and the local interstellar medium in at distances greater than 1500 AU.
Speaker: Prof. Nikolai Pogorelov (University of Alabama in Huntsville)
• 09:55 10:25
Some remarks on heliospheric observations - Klaus Scherer, Ruhr U. - Bochum Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 09:55
Some remarks on heliospheric observations 30m
a
Speaker: Dr Klaus Scherer (Institut für Theoretische Physik Lehrstuhl IV: Weltraum- und Astrophysik Ruhr-Universität Bochum D-44780 Bochum Germany)
• 10:25 10:45
Break 20m Landmark, 3rd floor center
### Landmark, 3rd floor center
1308 West Dayton Street, Madison, WI 53706
• 10:45 11:15
Modeling the Lyman-alpha backscatter observed by Voyager 1 and 2 in the outer heliosphere and the structure of the heliospheric bow shock - Gary Zank, U of Alabama-Huntsville Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 10:45
Modeling the Lyman-alpha backscatter observed by Voyager 1 and 2 in the outer heliosphere and the structure of the heliospheric bow shock 30m
1) Observations made by ultraviolet (UV) detectors on board Pioneer 10 , Voyager 1 , and Voyager 2 can be used to analyze the distribution of neutral hydrogen throughout the heliosphere, including the interaction regions of the solar wind and local interstellar medium. We use state-of-the art three-dimensional (3D) magnetohydrodynamic (MHD) – kinetic neutral H models to simulate Lyman-alpha backscatter as would be seen by the three spacecraft, exploiting a new 3D Monte Carlo radiative transfer code under solar minimum conditions (Fayock et al., 2013) . Both observations and simulations of the UV backscatter intensity are normalized for each spacecraft flight path at 15 AU, and we compare simulations with Voyager 1 and 2 and Pioneer 10 Lyman-alpha data results, finding a very close match with the Voyager data. Our results predict a large increase in the Lyman-alpha intensity as the hydrogen wall is approached. 2) Recent IBEX observations indicate that the local interstellar medium (LISM) flow speed is less than previously thought (23.2 km/s rather than 26 km/s), indicating that the LISM flow may be either marginally super-fast magnetosonic or sub-fast magnetosonic. This raises two questions: (A) Can a LISM model that is barely super-fast or sub-fast magnetosonic account for Ly-alpha observations that rely critically on the additional absorption provided by the hydrogen wall (H-wall)? and (B) If the LISM flow is weakly super-fast magnetosonic, does the transition assume the form of a traditional shock or does neutral hydrogen (H) mediate shock dissipation and hence structure through charge exchange? Both questions are addressed using three-dimensional self-consistently coupled magnetohydrodynamic plasma – kinetic neutral H models with different LISM magnetic field strengths (2, 3, and 4mG) as well as plasma and neutral H number densities. The 2 and 3mG models are fast magnetosonic far upwind of the heliopause whereas the 4μ G model is fully subsonic. The 2mG model admits a broad (~50–75 AU) bow-shock-like structure. The 3mG model has a smooth super-fast–sub-fast magnetosonic transition that resembles a very broad, ~200 AU thick, bow wave. A theoretical analysis shows that the transition from a super-fast to a sub-fast magnetosonic downstream state is due to the charge exchange of fast neutral H and hot neutral H created in the supersonic solar wind and hot inner heliosheath, respectively. For both the 2mG and the 3mG models, the super-fast magnetosonic LISM flow passes through a critical point. Because the Mach number is only barely super-fast magnetosonic in the 3mG case, the hot and fast neutral H can completely mediate the transition and impose a charge exchange length scale on the structure, making the solar-wind–LISM interaction effectively bow-shock-free. The charge exchange of fast and hot heliospheric neutral H therefore provides a primary dissipation mechanism at the weak heliospheric bow shock. Both super-fast magnetosonic models produce a sizeable H-wall. We find that (1) a sub-fast magnetosonic LISM flow cannot model the observed Ly-alpha absorption profiles along four sightlines corresponding to upwind, sidewind, and downwind; and (2) both the super-fast magnetosonic models can account for the Ly-alpha observations, with possibly the bow-shock-free 3μ G model being slightly favored.
Speaker: Dr Gary Zank (University of Alabama in Huntsville)
• 11:15 11:40
Understanding the anisotropy of TeV cosmic rays - Ming Zhang, FL Institute of Technology Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 11:15
Understanding the anisotropy of TeV cosmic rays 25m
Recent IBEX observation of a ribbon structure in energetic neutral atom emissions indicates that the level of turbulence in the interstellar magnetic field is quite low. The quasilinear theory of particle transport predicts a very large parallel diffusion and a very small perpendicular diffusion. Applying this extremely anisotropic diffusion to cosmic ray transport from a past nearby point source, we find that there will most likely be a large particle intensity gradient perpendicular to the magnetic field direction. The gradient can change with particle energy rapidly because it is sensitive to the magnitude of perpendicular diffusion coefficient. While cosmic ray anisotropy from particle diffusion still points towards the point source, drift anisotropy or b cross gradient anisotropy, which is enhanced from the perpendicular diffusion anisotropy by the ratio of particle gyroradius to perpendicular mean free path, is always perpendicular to the magnetic field. In the paper, we will demonstrate how the combination the Compton-Getting effect, diffusion and drift can result in various behaviors of large-scale cosmic ray anisotropy. In the mean time, the large-scale anisotropy can be break into medium-scale anisotropy when the particles are slightly deflected by the heliospheric magnetic field.
Speaker: Prof. Ming Zhang (Florida Institute of Technology)
• 11:40 12:05
- Nathan Schwadron, U of New Hampshire Northwoods
### Northwoods
1308 West Dayton Street, Madison, WI 53706
• 12:05 12:30
Heliospheric Boundary and the TeV Cosmic Ray Anisotropy - Paolo Desiati, UW-Madison Northwoods
|
2022-01-21 01:18:06
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4558587670326233, "perplexity": 5185.121314046103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302715.38/warc/CC-MAIN-20220121010736-20220121040736-00008.warc.gz"}
|
https://dmoj.ca/problem/spree
|
## Coding Spree
View as PDF
Points: 7 (partial)
Time limit: 0.6s
Memory limit: 16M
Author:
Problem type
It's almost report card time, so of course everyone has started programming like mad. There are only hours left before the Sunday midnight deadline!
Luckily, you've earned your points gradually, so now you can just sit back and watch your classmates struggle. One of your friends in particular is really screwed, so he's decided to skip school all week and go on a coding spree.
Though your friend is lazy, he has done some problems on the Judge, so now he has exactly problems available to him. No more problems will be posted until after the deadline, and he can't get partial marks on any of these problems. Problem is worth points and he knows in advance that he can solve it in hours .
You're not a very pleasant person, so you want to torture your friend a bit. You plan to calculate the most points your friend could possibly get by the deadline, just so you can taunt him with that number.
#### Input Specification
Line : The integers and .
The next lines: Line contains the integers and .
#### Output Specification
Output the maximum amount of points your friend can get in at most hours of coding.
#### Sample Input
8 48
10 7
5 1
50 30
5 1
10 5
100 1000
25 10
60 40
#### Sample Output
95
#### Explanation
Your friend only has hours, and problems to choose from. To maximize his points, he should do problems , , , , and , giving him points in hours.
|
2021-12-07 03:46:03
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18019108474254608, "perplexity": 2049.0624541560533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363332.1/warc/CC-MAIN-20211207014802-20211207044802-00238.warc.gz"}
|
https://scicomp.stackexchange.com/questions/38992/attempt-on-2d-advection-with-fdm-with-code
|
# Attempt on 2d Advection with FDM - With Code
I tried to implement the 2d advection problem with a velocity field, that is not constant in space. My problem is, that the "mass" of my shifted density gets "eroded" or just disappears, i.e. the norm $$\|u_t\|$$ goes to zero as time progresses. At least I want to minimize its decay somehow.
The initial condition has non zero pixels in each corner where it is basically a plateau disc function.
The velocity field points always to the center of the domain, so that the points have to meet in the center.
Here is my code.
# -*- coding: utf-8 -*-
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import animation
n = 32
X = np.linspace(-2,2,n)
Y = np.linspace(-2,2,n)
XX,YY = np.meshgrid(X,Y)
#------------- velocity field --------
# a disc in each corner
u0 = np.zeros((n,n))
u0 += (XX-1.25)**2 + (YY-1.25)**2 < 0.25
u0 += (XX+1.25)**2 + (YY+1.25)**2 < 0.25
u0 += (XX-1.25)**2 + (YY+1.25)**2 < 0.25
u0 += (XX+1.25)**2 + (YY-1.25)**2 < 0.25
# flow field points always to origin
flow = np.zeros((2,n,n))
flow[0,:,:] = YY
flow[1,:,:] = XX
flow *= -1 #
# normalizing flow field - is it necessary?
norm = np.linalg.norm(flow, axis = 0, ord=2)
flow[0,:,:] = np.divide(flow[0,:,:], norm, out=np.zeros_like(norm), where=norm>10**-7)
flow[1,:,:] = np.divide(flow[1,:,:], norm, out=np.zeros_like(norm), where=norm>10**-7)
dx = (XX.max() - XX.min())/n
dt = dx/5
u = np.copy(u0)
U = [u0.copy()]
frames = 100
for i in range(frames):
#upwind scheme
ux_minus = np.vstack((np.zeros((1,n)), u[1:-1,:] - u[0:-2,:],np.zeros((1,n))))
uy_minus = np.hstack((np.zeros((n,1)), u[:,1:-1] - u[:,0:-2],np.zeros((n,1))))
# downwind scheme
ux_plus = np.vstack((np.zeros((1,n)), u[2::,:] - u[1:-1,:], np.zeros((1,n))))
uy_plus = np.hstack((np.zeros((n,1)), u[:,2::] - u[:,1:-1], np.zeros((n,1))))
pulse_x = (flow[0,:,:] < 0) * flow[0,:,:]*ux_plus + (flow[0,:,:]>0)*flow[0,:,:]*ux_minus
pulse_y = (flow[1,:,:] < 0) * flow[1,:,:]*uy_plus + (flow[1,:,:]>0)*flow[1,:,:]*uy_minus
u -= (dt/dx)*(pulse_x + pulse_y)
U.append(u.copy())
def update(i):
matrice.set_array(U[i])
fig, ax = plt.subplots()
plt.show()
matrice = ax.matshow(U[0])
plt.colorbar(matrice)
ani = animation.FuncAnimation(fig, update, frames=frames, interval=1)
• Just a couple of comments: (1) Your solution will lead to a $\delta$ function. (2) Numerically a generic solution to the advection equations will lead to loss of mass unless they are conservative so it is not entirely surprising the behavior you are seeing. You mention that you are using upwind and downwind schemes but I am not entirely sure that you are in fact using those schemes honestly. Aug 29 at 0:10
• Also if you want to test to see if your solution is at least correct in principle you could try refining the grid as it should still converge despite lack of conservation although you won't be able to converge to the true solution really. Aug 29 at 0:15
• That's a lot of numerical diffusion. You'll have to use higher order spacial and temporal schemes. Look for second order upwind or WENO spacial schemes and Runge kutta temporal schemes. Aug 29 at 13:23
|
2021-10-23 04:04:51
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5886920094490051, "perplexity": 2534.1395445716194}, "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-2021-43/segments/1634323585561.4/warc/CC-MAIN-20211023033857-20211023063857-00687.warc.gz"}
|
http://www.cseblog.com/2010/01/two-envelopes-problem.html
|
# CSE Blog - quant, math, computer science puzzles
Quant, Math & Computer Science Puzzles for Interview Preparation & Brain Teasing
A collection of ~225 Puzzles with Solutions (classified by difficulty and topic)
## Jan 12, 2010
### Two envelopes Problem
Source: Wikipedia (Classical "Exchange Paradox" problem)
Problem:
The setup:
The player is given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. The player may select one envelope and keep whatever amount it contains, but upon selection, is offered the possibility to take the other envelope instead.
The switching argument:
1. Denote by A the amount in the selected envelope.
2. The probability that A is the smaller amount is 1/2, and that it's the larger also 1/2
3. The other envelope may contain either 2A or A/2
4. If A is the smaller amount, the other envelope contains 2A
5. If A is the larger amount, the other envelope contains A/2
6. Thus, the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2
7. So the expected value of the money in the other envelope is: $\frac{1}{2} 2A + \frac{1}{2} \frac{A}{2} = \frac{5}{4}A$
8. This is greater than A, so swapping is favored
9. After the switch, reason in exactly the same manner as above, but denote the second envelope's contents as B
10. It follows that the most rational thing to do is to swap back again
11. This line of reasoning dictates that envelopes be swapped indefinitely
12. As it seems more rational to open just any envelope than to swap indefinitely, the player is left with a paradox.
The puzzle:
The puzzle is to find the flaw, the erroneous step, in the switching argument above. This includes determining exactly why and under what conditions that step is not correct, in order to be sure not to make this mistake in a more complicated situation where the misstep may not be so obvious. In short, the problem is to solve the paradox.
Disclaimer: The problem sometimes is included under "unsolved" problems. A thought and study should be great anyways. :)
1. Flaw : Probabilities of individual steps should not be independent but dependent on path already taken in decision tree .
So if we use conditional probabilities instead 1/2 , we know that once we have seen both envelopes all probability variables should be 0 or 1.
PS: If it is unsolved , then it is obvl wrong. So lets find flaw in this simple argument first :P
2. I'll go with the assumption that you are not shown contents of envelope before finally deciding on one.....For that case I'll go with choosing any envelope shouldn't matter and no swapping is needed...but if contents are shown then Nikhil is correct.
So, I think the question is whether contents are shown or not.....In either case there is no point in going on swapping forever.
3. @viki.. For the paradox, it does not matter whether you show the contents or not. It seems whether u see the content or not, you will have to swap forever.
@nikhil.. Your argument is correct. Seeing the money to be x in the first envelope does not imply that with probability half the amount of money in second envelope is 2x and with prob half it is x/2. But, as given in the wikipedia link, its dependent on conditional probabilities. Lack of information about an object should not be mistaken with randomized object.
@Nikhil.. You have a good eye for probability (Refer to the classroom scene of the movie 21 :P).. Nice. Wikipedia says this is still an open problem among the subjectivists as no consensus has been reached yet. (No citation :P)
4. @ Pratik : AAaah , not quite the scene of 21 - I dont get a cool sports car ! :P
5. One solution is this: Switching when you find a very low value is obviously a good plan. But if you see a value so large that it surprises you, then it might be a bad idea to switch. If the amount X in the envelope surprises you because it is very large... that's because you guess than the average envelope contains much less. The fact that you saw X is consistent with one of the following:
> The envelopes are better than I thought, and the prizes are X and X/2.
> The envelopes are much better than I thought, and the prizes are X and 2X.
If your "surprise curve" says that odds of those two conditions are, say 9:4 in favor of the first case, then you should not switch, as the costs of switching down exceed the benefits of switching up. Now, suppose you are surprised to see \$3000, but you figure "I would be more surprised to see twice as much, but not twice as surprised," then you are assuming that the distribution of
"what the value could be" is very flat. But, in fact, that distribution is so flat that it doesn't have a finite integral. That is: the distribution of "what a value could be" has to be
rather tight in order to be integrable. You believe in integrability if you believe "the probability that the prize is between 1 and a million coins is greater than 0.00001" or any
similar statement, with the parameters "1", "a million" and "0.00001" replaced by anything.
I think that this paradox is compelling because you should definitely switch from a median value. This is balanced in the expected-value calculation by the great sums you would lose by switching from a high value.
6. one approach that comes to my mind is to analyze the case when the money in envelope is a real number in the interval [0,A]. Then take limit A->infinity.
in this approach, it will matter if u see the contents of envelope because, if the number > A/2, then for sure, u wont swap.
|
2017-04-28 21:53:37
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7303866147994995, "perplexity": 815.3075109757737}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123097.48/warc/CC-MAIN-20170423031203-00364-ip-10-145-167-34.ec2.internal.warc.gz"}
|
https://daniel.wordpress.com/category/unit-085-3-nephi-13/
|
## Unit 85 – word and phrase data
Unit 85: 3 Nephi Chapter 13 verse 25 through 3 Nephi Chapter 14
The material covered in this unit corresponds with the content of 3 Nephi chapter 6 in the 1830 edition.
The information below is based on the wording in the The Book of Mormon: The Earliest Text.
a – 11 occurrences – verses 14:4, 14:9, 14:10, 14:17, 14:18, 14:24, 14:25, 14:26
a corrupt tree – 2 occurrences – verses 14:17, 14:18
add – 1 occurrence – verse 13:27
added – 1 occurrence – verse 13:33
again – 4 occurrences – verses 14:1, 14:2, 14:6
air – 1 occurrence – verse 13:26
all – 4 occurrences – verses 13:29, 13:32, 13:33, 14:12
all these things – 2 occurrences – verses 13:32, 13:33
and – 44 occurrences – verses 13:25, 13:28, 13:29, 13:30, 13:33, 14:1, 14:2, 14:3, 14:4, 14:5, 14:6, 14:7, 14:8, 14:12, 14:13, 14:14, 14:19, 14:22, 14:23, 14:24, 14:25, 14:26, 14:27
and in thy name – 2 occurrences – verse 14:22
and it – 4 occurrences – verses 14:7, 14:25, 14:27
and it shall be – 2 occurrences – verse 14:7
and now it came to pass that when Jesus had spoken these words he – 2 occurrences – verses 13:25, 14:1
and the – 8 occurrences – verses 13:25, 14:12, 14:25, 14:27
and the rain descended and the floods came and the winds blew and beat upon that house and it fell – 2 occurrences – verses 14:25, 14:27
and then – 2 occurrences – verses 14:5, 14:23
and why – 2 occurrences – verses 13:28, 14:3
are – 4 occurrences – verses 13:25, 13:26, 13:30, 14:15
arrayed – 1 occurrence – verse 13:29
ask – 4 occurrences – verses 14:7, 14:9, 14:10, 14:11
asketh – 1 occurrence – verse 14:8
at – 1 occurrence – verse 14:13
barns – 1 occurrence – verse 13:26
be – 11 occurrences – verses 13:31, 13:33, 14:1, 14:2, 14:7, 14:8, 14:13, 14:14, 14:26
beam – 3 occurrences – verses 14:3, 14:4, 14:5
beat – 2 occurrences – verses 14:25, 14:27
because – 1 occurrence – verse 14:14
before – 1 occurrence – verse 14:6
behold – 3 occurrences – verses 13:25, 13:26, 14:4
beholdest – 1 occurrence – verse 14:3
being – 1 occurrence – verse 14:11
better – 1 occurrence – verse 13:26
beware – 1 occurrence – verse 14:15
blew – 2 occurrences – verses 14:25, 14:27
body – 2 occurrences – verse 13:25
bread – 1 occurrence – verse 14:9
bring – 2 occurrences – verse 14:18
bring forth – 2 occurrences – verse 14:18
bringeth – 3 occurrences – verses 14:17, 14:19
broad – 1 occurrence – verse 14:13
brother – 1 occurrence – verse 14:14
brother’s – 2 occurrences – verses 14:3, 14:5
built – 2 occurrences – verses 14:24, 14:26
but – 5 occurrences – verses 13:33, 14:3, 14:15, 14:17, 14:21
by – 3 occurrences – verses 13:27, 14:16, 14:20
by their fruits – 2 occurrences – verses 14:16, 14:20
came – 4 occurrences – verses 13:25, 14:1, 14:25, 14:27
can – 1 occurrence – verse 13:27
cannot – 1 occurrence – verse 14:18
cast – 6 occurrences – verses 13:30, 14:5, 14:6, 14:19, 14:22
cast into the – 2 occurrences – verses 13:30, 14:19
cast out – 3 occurrences – verses 14:5, 14:22
cast out the – 2 occurrences – verse 14:5
children – 1 occurrence – verse 14:11
chosen – 2 occurrences – verse 13:25
clearly – 1 occurrence – verse 14:5
clothe – 2 occurrences – verse 13:30
clothed – 1 occurrence – verse 13:31
clothing – 1 occurrence – verse 14:15
come – 1 occurrence – verse 14:15
consider – 1 occurrence – verse 13:28
considerest – 1 occurrence – verse 14:3
corrupt – 2 occurrences – verses 14:17, 14:18
cubit – 1 occurrence – verse 13:27
day – 2 occurrences – verses 13:34, 14:22
depart – 1 occurrence – verse 14:23
descended – 2 occurrences – verses 14:25, 14:27
destruction – 1 occurrence – verse 14:13
devils – 1 occurrence – verse 14:22
did – 1 occurrence – verse 14:1
do – 5 occurrences – verses 13:26, 13:28, 14:12, 14:16
doeth – 3 occurrences – verses 14:21, 14:24, 14:26
dogs – 1 occurrence – verse 14:6
done – 1 occurrence – verse 14:22
down – 1 occurrence – verse 14:19
drink – 2 occurrences – verses 13:25, 13:31
eat – 2 occurrences – verses 13:25, 13:31
eat or what – 2 occurrences – verses 13:25, 13:31
enter – 2 occurrences – verses 14:13, 14:21
even – 4 occurrences – verses 13:29, 13:30, 14:12, 14:17
even so – 3 occurrences – verses 13:30, 14:12, 14:17
every – 5 occurrences – verses 14:8, 14:17, 14:19, 14:21, 14:26
every one that – 3 occurrences – verses 14:8, 14:21, 14:26
evil – 4 occurrences – verses 13:34, 14:11, 14:17, 14:18
eye – 6 occurrences – verses 14:3, 14:4, 14:5
eye and – 2 occurrences – verses 14:4, 14:5
faith – 1 occurrence – verse 13:30
fall – 1 occurrence – verse 14:27
false – 1 occurrence – verse 14:15
father – 4 occurrences – verses 13:26, 13:32, 14:11, 14:21
father which is in heaven – 2 occurrences – verses 14:11, 14:21
feedeth – 1 occurrence – verse 13:26
feet – 1 occurrence – verse 14:6
fell – 2 occurrences – verses 14:25, 14:27
few – 1 occurrence – verse 14:14
field – 2 occurrences – verses 13:28, 13:30
figs – 1 occurrence – verse 14:16
find – 2 occurrences – verses 14:7, 14:14
findeth – 1 occurrence – verse 14:8
fire – 1 occurrence – verse 14:19
first – 2 occurrences – verses 13:33, 14:5
fish – 1 occurrence – verse 14:10
floods – 2 occurrences – verses 14:25, 14:27
foolish – 1 occurrence – verse 14:26
for – 14 occurrences – verses 13:25, 13:26, 13:28, 13:32, 13:34, 14:2, 14:8, 14:12, 14:13, 14:25
for the – 3 occurrences – verse 13:34
for the morrow – 2 occurrences – verse 13:34
for your – 3 occurrences – verses 13:25, 13:32
forth – 5 occurrences – verses 14:17, 14:18, 14:19
forth evil fruit – 2 occurrences – verses 14:17, 14:18
forth good fruit – 3 occurrences – verses 14:17, 14:18, 14:19
founded – 1 occurrence – verse 14:25
fowls – 1 occurrence – verse 13:26
from – 1 occurrence – verse 14:23
fruit – 5 occurrences – verses 14:17, 14:18, 14:19
fruits – 2 occurrences – verses 14:16, 14:20
gate – 3 occurrences – verses 14:13, 14:14
gather – 2 occurrences – verses 13:26, 14:16
gifts – 1 occurrence – verse 14:11
give – 5 occurrences – verses 14:6, 14:9, 14:10, 14:11
give good – 2 occurrences – verse 14:11
given – 1 occurrence – verse 14:7
glory – 1 occurrence – verse 13:29
go – 1 occurrence – verse 14:13
God – 2 occurrences – verses 13:30, 13:33
good – 7 occurrences – verses 14:11, 14:17, 14:18, 14:19
good tree – 2 occurrences – verses 14:17, 14:18
grapes – 1 occurrence – verse 14:16
grass – 1 occurrence – verse 13:30
great – 1 occurrence – verse 14:27
grow – 1 occurence – verse 13:28
had – 3 occurrences – verses 13:25, 14:1
have – 5 occurrences – verses 13:25, 13:32, 14:22
he – 9 occurrences – verses 13:25, 13:30, 14:1, 14:8, 14:9, 14:10, 14:21
he that – 2 occurrences – verses 14:8, 14:21
heareth – 2 occurrences – verses 14:24, 14:26
heareth these sayings of mine and doeth them – 2 occurrences – verses 14:24, 14:26
heaven – 3 occurrences – verses 14:11, 14:21
heavenly – 2 occurrences – verses 13:26, 13:32
hewn – 1 occurrence – verse 14:19
him – 5 occurrences – verses 14:8, 14:9, 14:10, 14:11, 14:24
his – 7 occurrences – verses 13:27, 13:29, 13:33, 14:1, 14:9, 14:24, 14:26
holy – 1 occurrence – verse 14:6
house – 4 occurrences – verses 14:24, 14:25, 14:26, 14:27
how – 4 occurrences – verses 13:28, 14:4, 14:11
hypocrite – 1 occurrence – verse 14:5
I – 8 occurrences – verses 13:25, 13:29, 14:1, 14:23, 14:24
I say unto you – 3 occurrences – verses 13:25, 13:29, 14:1
if – 5 occurrences – verses 13:30, 14:9, 14:10, 14:11
if ye – 2 occurrences – verses 13:30, 14:11
in – 13 occurrences – verses 13:29, 14:3, 14:4, 14:11, 14:13, 14:15, 14:21, 14:22
in thy – 4 occurrences – verses 14:3, 14:22
in thy name – 3 occurrences – verse 14:22
iniquity – 1 occurrence – verse 14:23
into – 4 occurrences – verses 13:26, 13:30, 14:19, 14:21
into the – 3 occurrences – verses 13:30, 14:19, 14:21
inwardly – 1 occurrence – verse 14:15
is – 17 occurrences – verses 13:25, 13:30, 13:34, 14:3, 14:4, 14:6, 14:9, 14:11, 14:12, 14:13, 14:14, 14:19, 14:21
is in – 5 occurrences – verses 14:3, 14:4, 14:11, 14:21
is in thine own eye – 2 occurrences – verses 14:3, 14:4
is the – 6 occurrences – verses 13:34, 14:12, 14:13, 14:14
is the gate and – 2 occurrences – verses 14:13, 14;14
is the way – 2 occurrences – verses 14:13, 14:14
it – 11 occurrences – verses 13:25, 14:1, 14:2, 14:7, 14:8, 14:14, 14:25, 14:27
it shall be – 4 occurrences – verses 14:2, 14:7, 14:8
it shall be opened – 2 occurrences – verses 14:7, 14:8
itself – 1 occurrence – verse 13:34
Jesus – 2 occurrences – verses 13:25, 14:1
judge – 2 occurrences – verses 14:1, 14:2
judged – 2 occurrences – verses 14:1, 14:2
judgment – 1 occurrence – verse 14:2
kingdom – 2 occurrences – verses 13:33, 14:21
knew – 1 occurrence – verse 14:23
knock – 1 occurrence – verse 14:7
knocketh – 1 occurrence – verse 14:8
know – 3 occurrences – verses 14:11, 14:16, 14:20
knoweth – 1 occurrence – verse 13:32
law – 1 occurrence – verse 14:12
leadeth – 2 occurrences – verses 14:13, 14:14
lest – 1 occurrence – verse 14:6
let – 1 occurrence – verse 14:4
life – 3 occurrences – verses 13:25, 14:14
like – 1 occurrence – verse 13:29
liken – 1 occurrence – verse 14:24
likened – 1 occurrence – verse 14:26
lilies – 1 occurrence – verse 13:28
little – 1 occurrence – verse 13:30
looked – 1 occurrence – verse 13:25
Lord – 4 occurrences – verses 14:21, 14:22
Lord Lord – 2 occurrences – verses 14:21, 14:22
man – 3 occurrences – verses 14:9, 14:24, 14:26
man which built his house upon – 2 occurrences – verses 14:24, 14:26
many – 3 occurrences – verses 14:13, 14:22
me – 4 occurrences – verses 14:4, 14:21, 14:22, 14:23
measure – 1 occurrence – verse 14:2
measured – 1 occurrence – verse 14:2
meat – 1 occurrence – verse 13:25
men – 2 occurrences – verses 14:12, 14:16
mete – 1 occurrence – verse 14:2
mine – 2 occurrences – verses 14:24, 14:26
minister – 1 occurrence – verse 13:25
more – 2 occurrences – verses 13:25, 14:11
morrow – 2 occurrences – verse 13:34
mote – 3 occurrences – verses 14:3, 14:4, 14:5
mouth – 1 occurrence – verse 14:1
much – 2 occurrences – verses 13:26, 14:11
multitude – 1 occurrence – verse 14:1
my – 1 occurrence – verse 14:21
name – 3 occurrences – verses 14:22
narrow – 1 occurrence – verse 14:14
need – 1 occurrence – verse 13:32
neither – 4 occurrences – verses 13:26, 13:28, 14:6, 14:18
never – 1 occurrence – verse 14:23
no – 3 occurrences – verses 13:25, 13:31, 13:34
no thought – 3 occurrences – verses 13:25, 13:31, 13:34
no thought for – 2 occurrences – verses 13:25, 13:34
nor – 2 occurrences – verses 13:25, 13:26
not – 15 occurrences – verses 13:25, 13:26, 13:28, 13:29, 13:30, 14:1, 14:3, 14:6, 14:19, 14:21, 14:22, 14:25, 14:26
not neither do they – 2 occurrences – verses 13:26, 13:28
not that – 2 occurrences – verses 14:1, 14:6
not the – 2 occurrences – verses 13:25, 14:3
now – 2 occurrences – verses 13:25, 14:1
of – 21 occurrences – verses 13:26, 13:27, 13:28, 13:29, 13:30, 13:32, 13:33, 13:34, 14:4, 14:5, 14:9, 14:15, 14:16, 14:21, 14:24, 14:26, 14:27
of the – 3 occurrences – verses 13:26, 13:28, 13:30
of the field – 2 occurrences – verses 13:28, 13:30
of you – 2 occurrences – verses 13:27, 14:9
on – 1 occurrence – verse 13:25
one – 5 occurrences – verses 13:27, 13:29, 14:8, 14:21, 14:26
open – 1 occurrence – verse 14:1
opened – 2 occurrences – verses 14:7, 14:8
or – 7 occurrences – verses 13:25, 13:31, 14:4, 14:9, 14:10, 14:16
or what – 3 occurrences – verses 13:25, 13:31, 14:9
out – 7 occurrences – verses 14:4, 14:5, 14:22
out of – 3 occurrences – verses 14:4, 14:5
out of thine – 2 occurrences – verses 14:4, 14:5
out the – 3 occurrences – verses 14:4, 14:5
out the mote out of – 2 occurrences – verses 14:4, 14:5
oven – 1 occurrence – verse 13:30
own – 3 occurrences – verses 14:3, 14:4, 14:5
pass – 2 occurrences – verses 13:25, 14:1
pearls – 1 occurrence – verse 14:6
people – 1 occurrence – verse 13:25
profess – 1 occurrence – verse 14:23
prophesied – 1 occurrence – verse 14:22
prophets – 2 occurrences – verses 14:12, 14:15
pull – 1 occurrence – verse 14:4
put – 1 occurrence – verse 13:25
raiment – 2 occurrences – verses 13:25, 13:28
rain – 2 occurrences – verses 14:25, 14:27
ravening – 1 occurrence – verse 14:15
reap – 1 occurrence – verse 13:26
receiveth – 1 occurrence – verse 14:8
remember – 1 occurrence – verse 13:25
rend – 1 occurrence – verse 14:6
righteousness – 1 occurrence – verse 13:33
rock – 2 occurrences – verses 14:24, 14:25
saith – 2 occurrences – verses 13:25, 14:21
saith unto – 2 occurrences – verses 13:25, 14:21
sand – 1 occurrence – verse 14:26
say – 5 occurrences – verses 13:25, 13:29, 14:1, 14:4, 14:22
say to – 2 occurrences – verses 14:4, 14:22
saying – 2 occurrences – verses 13:31, 14:1
sayings – 2 occurrences – verses 14:24, 14:26
see – 1 occurrence – verse 14:5
seek – 2 occurrences – verses 13:33, 14:7
seeketh – 1 occurrence – verse 14:8
serpent – 1 occurrence – verse 14:10
shall – 19 occurrences – verses 13:25, 13:31, 13:33, 13:34, 14:2, 14:7, 14:8, 14:11, 14:16, 14:20, 14:21, 14:26
shall be – 7 occurrences – verses 13:33, 14:2, 14:7, 14:8, 14:26
shall we – 3 occurrences – verse 13:31
shalt – 1 occurrence – verse 14:5
sheep’s – 1 occurrence – verse 14:15
should – 1 occurrence – verse 14:12
so – 4 occurrences – verses 13:30, 14:12, 14:17
Solomon – 1 occurrence – verse 13:29
son – 1 occurrence – verse 14:9
sow – 1 occurrence – verse 13:26
spin – 1 occurrence – verse 13:28
spoken – 3 occurrences – verses 13:25, 14:1
stature – 1 occurrence – verse 13:27
stone – 1 occurrence – verse 14:9
strait – 2 occurrences – verses 14:13, 14:14
sufficient – 1 occurrence – verse 13:34
swine – 1 occurrence – verse 14:6
take – 5 occurrences – verses 13:25, 13:28, 13:31, 13:34
take no thought – 2 occurrences – verses 13:25, 13:31
taking – 1 occurrence – verse 13:27
than – 3 occurrences – verses 13:25, 13:26
that – 23 occurrences – verses 13:25, 13:29, 13:32, 14:1, 14:3, 14:6, 14:8, 14:11, 14:12, 14:13, 14:14, 14:19, 14:21, 14:22, 14:23, 14:25, 14:26, 14:27
that is in – 2 occurrences – verse 14:3
that ye – 2 occurrences – verses 13:32, 14:1
the – 42 occurrences – verses 13:25, 13:26, 13:28, 13:30, 13:33, 13:34, 14;1, 14:3, 14:4, 14:5, 14:6, 14:12, 14:13, 14:14, 14:19, 14:21, 14:25, 14:26, 14:27
the beam – 2 occurrences – verses 14:3, 14:5
the kingdom of – 2 occurrences – verses 13:33, 14:21
the mote – 3 occurrences – verses 14:3, 14:4, 14:5
their – 3 occurrences – verses 14:6, 14:16, 14:20
them – 11 occurrences – verses 13:25, 13:26, 14:1, 14:6, 14:11, 14:12, 14:16, 14:20, 14:23, 14:24, 14:26
them I – 2 occurrences – verses 14:23, 14:24
then – 3 occurrences – verses 14:5, 14:11, 14:23
there – 3 occurrences – verses 14:9, 14:13, 14:14
there be – 2 occurrences – verses 14:13, 14:14
thereat – 1 occurrence – verse 14:13
therefore – 5 occurrences – verses 13:25, 13:31, 13:34, 14:12, 14:24
thereof – 1 occurrence – verse 13:34
these – 7 occurrences – verses 13:25, 13:29, 13:32, 13:33, 14:1, 14:24, 14:26
they – 9 occurrences – verses 13:25, 13:26, 13:28, 14:6, 14:15
thine – 4 occurrences – verses 14:3, 14:4, 14:5
thine own eye – 3 occurrences – verses 14:3, 14:4, 14:5
things – 5 occurrences – verses 13:32, 13:33, 13:34, 14:11, 14:12
this – 2 occurrences – verses 13:25, 14:12
thistles – 1 occurrence – verse 14:16
thorns – 1 occurrence – verse 14:16
thou – 4 occurrences – verses 14:3, 14:4, 14:5
thought – 6 occurrences – verses 13:25, 13:27, 13:28, 13:31, 13:34
thought for – 4 occurrences – verses 13:25, 13:28, 13:34
thought for the – 2 occurrences – verse 13:34
thy – 6 occurrences – verses 14:3, 14:4, 14:5, 14:22
thy brother’s eye – 2 occurrences – verses 14:3, 14:5
to – 15 occurrences – verses 13:25, 14:1, 14:2, 14:4, 14:5, 14:8, 14:11, 14:12, 14:13, 14:15, 14:22
to them – 2 occurrences – verses 14:11, 14:12
today – 1 occurrence – verse 13:30
toil – 1 occurrence – verse 13:28
tomorrow – 1 occurrence – verse 13:30
trample – 1 occurrence – verse 14:6
tree – 5 occurrences – verses 14:17, 14:18, 14:19
tree bringeth forth – 2 occurrences – verse 14:17
turn – 1 occurrence – verse 14:6
turned – 1 occurrence – verse 14:1
twelve – 1 occurrence – verse 13:25
under – 1 occurrence – verse 14:6
unto – 18 occurrences – verses 13:25, 13:27, 13:29, 13:33, 13:34, 14:1, 14:6, 14:7, 14:11, 14:14, 14:21, 14:23, 14:24, 14:26
unto a – 2 occurrences – verses 14:24, 14:26
unto the – 2 occurrences – verses 13:34, 14:6
unto them – 3 occurrences – verses 13:25, 14:1, 14:23
unto you – 6 occurrences – verses 13:25, 13:29, 13:33, 14:1, 14:7
upon – 6 occurrences – verses 13:25, 14:24, 14:25, 14:26, 14:27
upon a rock – 2 occurrences – verses 14:24, 14:25
upon the – 2 occurrences – verses 13:25, 14:26
verily – 2 occurrences – verse 14:1
was – 3 occurrences – verses 13:29, 14:25, 14:27
way – 2 occurrences – verses 14:13, 14:14
we – 4 occurrences – verses 13:31, 14:22
what – 8 occurrences – verses 13:25, 13:31, 14:2, 14:9
what shall we – 2 occurrences – verse 13:31
what ye shall – 3 occurrences – verse 13:25
whatsoever – 1 occurrence – verse 14:12
when – 2 occurrences – verses 13:25, 14:1
wherefore – 2 occurrences – verses 13:30, 14:20
wherewithal – 1 occurrence – verse 13:31
which – 12 occurrences – verses 13:25, 13:27, 13:30, 14:6, 14:11, 14:13, 14:14, 14:15, 14:21, 14:24, 14:26
which I have – 2 occurrences – verse 13:25
which is – 3 occurrences – verses 14:6, 14:11, 14:21
whom – 2 occurrences – verses 13:25, 14:9
whoso – 1 occurrence – verse 14:24
why – 2 occurrences – verses 13:28, 14:3
wide – 1 occurrence – verse 14:13
will – 7 occurrences – verses 13:30, 14:9, 14:10, 14:21, 14:22, 14:23, 14:24
will he – 3 occurrences – verses 13:30, 14:9, 14:10
will he give him a – 2 occurrences – verses 14:9, 14:10
wilt – 1 occurrence – verse 14:4
winds – 2 occurrences – verses 14:25, 14:27
wise – 1 occurrence – verse 14:24
with – 2 occurrences – verse 14:2
with what – 2 occurrences – verse 14:2
wolves – 1 occurrence – verse 14:15
wonderful – 1 occurrence – verse 14:22
words – 3 occurrences – verses 13:25, 14;1
work – 1 occurrence – verse 14:23
works – 1 occurrence – verse 14:22
would – 1 occurrence – verse 14:12
ye – 22 occurrences – verses 13:25, 13:26, 13:28, 13:30, 13:32, 13:33, 14:1, 14:2, 14:6, 14:7, 14:11, 14:12, 14:13, 14:16, 14:20, 14:23
ye are – 2 occurrences – verses 13:25, 13:30
ye shall – 7 occurrences – verses 13:25, 14:2, 14:7, 14:16, 14:20
ye shall know them – 2 occurrences – verses 14:16, 14:20
yet – 3 occurrences – verses 13:25, 13:26, 13:29
you – 14 occurrences – verses 13:25, 13:27, 13:29, 13:30, 13:33, 14:1, 14:2, 14:6, 14:7, 14:9, 14:12, 14:15, 14:23
your – 7 occurrences – verses 13:25, 13:26, 13:32, 14:6, 14:11
your Heavenly Father – 2 occurrences – verses 13:26, 13:32
|
2019-01-21 09:12:17
|
{"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": 0.9659779071807861, "perplexity": 11817.75607209056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583771929.47/warc/CC-MAIN-20190121090642-20190121112642-00390.warc.gz"}
|
https://flaviocopes.com/javascript-dynamic-method/
|
Sometimes you have an object and you need to call a method, or a different method, depending on some condition.
For example you have a car object and you either want to drive() it or to park() it, depending on the driver.sleepy value.
If the driver has a sleepy level over 6, we need to park the car before it fells asleep while driving.
Here is how you achieve this with an if/else condition:
if (driver.sleepy > 6) {
car.park()
} else {
car.drive()
}
Let’s rewrite this to be more dynamic.
We can use the ternary operator to dynamically choose the method name, get it as the string value.
Using square brackets we can select it from the object’s available methods:
car[driver.sleepy > 6 ? 'park' : 'drive']
With the above statement we get the method reference. We can directly invoke it by appending the parentheses:
car[driver.sleepy > 6 ? 'park' : 'drive']()
|
2020-12-01 00:19:04
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39508843421936035, "perplexity": 1936.0271648723037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141515751.74/warc/CC-MAIN-20201130222609-20201201012609-00507.warc.gz"}
|
https://www.physicsforums.com/threads/range-of-a-weird-function.901831/
|
# Homework Help: Range of a weird function
Tags:
1. Jan 28, 2017
### Buffu
1. The problem statement, all variables and given/known data
Find the range $y = \sqrt{\ln({\cos(\sin (x)}))}$
2. Relevant equations
3. The attempt at a solution
https://www.desmos.com/calculator
I used a graphing calculator to find the intersection between $y = e^{x^2}$ and $y = \cos(\sin(x))$.
Which I get as $(0,1)$. So the range is $\{0\}$.
But I want to find the range without graphs and by analytical methods.
Thanks for help.
2. Jan 28, 2017
### PetSounds
What is the range of $y = cos (x)$ ?
3. Jan 28, 2017
### SammyS
Staff Emeritus
All that you actually found here is that if $\ x=0\,,\$ then $\ y=1\,.\$ Therefore, 1 is in the range of your function.
I suggest the first thing to do is to determine the (implied) domain of your function.
4. Jan 28, 2017
### Buffu
Putting x = 0 $y = \sqrt{\ln(\cos(\sin(0)))} =\sqrt{\ln(\cos 0))} = \sqrt{\ln(1)} = 0$, So y = 0 is also in range.
So the range is {0,1}.
Domain of function is (0 to pi/2) + 2n*pi.
[-1,1]
Last edited: Jan 28, 2017
5. Jan 28, 2017
### PetSounds
And how does that overlap with the domain of $y = ln (x)$?
6. Jan 28, 2017
### Buffu
domain of ln x is (0, $\infty$) .
So $(0, 1]$ part of cos x domain is only useful in this problem
7. Jan 28, 2017
### PetSounds
And what is the range of $ln (x)$ for $0 < x \leq 1$ ?
8. Jan 29, 2017
### Buffu
less than 0 but we cannot have less than zero because of square root. So only 1 is left; Thus range is {0}.
Last edited: Jan 29, 2017
9. Jan 29, 2017
### PetSounds
Bingo.
10. Jan 29, 2017
### Ray Vickson
Yes. And the domain of $f$ is also very limited in the real line. What would it (the domain) be?
11. Jan 29, 2017
### Buffu
Domain of my original function would be when sin x is 0, that is 2pi or for general solution 2* pi *n. So my domain would be {x : x = 2pi * n $\forall n \in \mathbb Z$}. Right ?
12. Feb 3, 2017
### haruspex
There are other solutions.
13. Feb 3, 2017
### Buffu
Oh yes sin x is also zero at π So the domain should be {x : x = π * n ∀n ∈ ℤ}
14. Feb 3, 2017
### haruspex
Looks right. Your use of the predicates is a little inaccurate. There does not exist an x such that it equals π * n for all integers n. You mean {π * n : n∈ ℤ }
15. Feb 3, 2017
### Buffu
I did not get it. you just removed x.
16. Feb 3, 2017
### haruspex
What you had posted said :
"the set of things x such that x equals πn for all integers n".
There is no number that can equal πn for two different integers n, let alone all infinity of them.
If you want to use x and n then I suggest using ∃n. Maybe {x:∃n∈ℕ:x=πn}. But why not omit x and write it my way?
17. Feb 4, 2017
### Buffu
Your way is better.
Oh I understand what you mean.
|
2018-05-25 13:18:38
|
{"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": 0.669051468372345, "perplexity": 1904.4103277979698}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867092.48/warc/CC-MAIN-20180525121739-20180525141739-00255.warc.gz"}
|
http://itensor.org/support/3534/convergence-issues-system-sizes-transverse-field-ising-model
|
# Convergence issues for small system sizes for the transverse field Ising model
+1 vote
I am a bit puzzled by strange convergence behavior of DMRG for the transverse field Ising model.
Using the Julia version, I am running the transverse field Ising model example for various system sizes N and the coupling h
H = - \sum_{i} Z_i Z_{i+1} - h \sum_{i} X_i
My issue is that for certain values of h and N, the convergence seems to become very poor. I set energy tolerance of 1e-12 as a convergence criterion using a custom DMRGObserver, as shown in the documentation. For h>1.0, everything is good and DMRG converges very quickly. For example, even at critical h=1.0, I find it takes around ~10 sweeps.
However, for h=0.5, I notice that the convergence is very fast small N<10, but becomes quite poor for N~15, and then becomes very fast for N>24. For example, in a typical run, I find the following number of sweeps needed to reach an energy tolerance of 1e-12:
Row │ N sweeps energy_tol
─────┼───────────────────────────
1 │ 6 7 9.3e-13
2 │ 8 14 7.2e-13
3 │ 10 23 8.2e-13
4 │ 12 55 4.3e-13
5 │ 16 100 2.0e-09
6 │ 24 4 1.4e-14
7 │ 32 4 3.1e-14
8 │ 48 4 3.2e-13
Note the rows for N=10,12,16. I am not sure what's happening at these intermediate values of N... why does the convergence suddenly become poor? This behavior depends on the coupling h. For example, in contrast to the above example, at h=1.0 (critical), there seems to be no problem and DMRG converges with very few sweeps (less than 10) within a tolerance of 1e-12 for N=6,..,48.
Row │ N sweeps energy_tol
─────┼───────────────────────────
1 │ 6 4 9.7e-16
2 │ 8 6 1.4e-15
3 │ 10 5 8.7e-14
4 │ 12 6 9.5e-16
5 │ 16 6 3.3e-13
6 │ 24 8 9.6e-15
7 │ 32 11 4.9e-13
8 │ 48 12 4.7e-14
On the other hand, for h=0.2, I find that even L=6 has problem converging, but larger values of L become much better:
Row │ N sweeps energy_tol
─────┼───────────────────────────
1 │ 6 100 2.5e-09
2 │ 8 36 6.9e-13
3 │ 10 3 2.6e-13
4 │ 12 4 1.1e-14
5 │ 16 3 4.0e-13
6 │ 24 4 5.5e-15
7 │ 32 4 1.5e-15
8 │ 48 4 5.4e-15
Larger h (greater than 1.0) seem to be all well behaved when I did a few spot checks. Does this behaviour make sense? My initial state is just a randomMPS. I tried playing with sweeps, both including noise and without. But that doesn't seem to matter very much. It's counter-intuitive to me that the problem is with small lattices, while larger ones do okay. Can you see what could be happening here? Thanks!
commented by (70.1k points)
Hi thanks for the question. What does your "sweep schedule" look like, in terms of how quickly you raise the maxdim from the first few sweeps to the final value, and similar with other DMRG accuracy parameters?
One suggestion I would have would be to always do a few initial sweeps with a rather low maxdim, of say 8 or 10, then start doubling it until you reach the final maxdim value you want (the final one could be very high if you have a modest size cutoff, which will then start to dominate the truncation).
commented by (130 points)
Hi Miles. Thanks for your response.
So, I tried playing with the sweep parameters, but somehow it didn't seem to make much difference. For example, for the above runs, I was using
sweeps = Sweeps(100, [
"maxdim" "mindim" "cutoff" "noise"
10 10 1e-12 1E-7
20 10 1e-12 1E-7
50 10 1e-12 1E-7
100 20 1e-12 1E-8
200 20 1e-12 1E-10
200 20 1e-12 0
])
I have tried the above schedule both with and without noise. I also just tried with starting with even lower bond dimension of 5, but it does seem to alter the fact that L=16 just does not converge even for 100 sweeps.
commented by (70.1k points)
Hi, thanks for that info. So from the information provided, I can't easily guess what's going on. Could you please share a minimal code that can reproduce this issue? How are you defininig the energy tolerance? Is it through an observer object or just through your own comparison?
Thanks, Miles
commented by (130 points)
Thanks for your response. I define the energy tolerance through a custom observer object, pretty much following the documentation. Here's a MWE:
https://gist.github.com/hershsingh/e6bd6c1fab997530c1c0e31e0a1c87d5
Edit: for some reason, the code formatting didn't work in the comments for me, so putting a link to a gist instead.
commented by (130 points)
The code above will run DMRG for various lattice sizes and print a table of number of sweeps required for each N and the final energy tolerance achieved. When I run it for an energy tolerance of 1e-12, I find that N=16 saturates the maximum number of sweeps (100), and the energy tolerance achieved is only 1e-9. Larger N seem to converge much faster.
commented by (70.1k points)
Hi Hersh,
Thanks for the code example. A bit separately from the issue you are seeing, I've found it's not usually a good idea to loop over many different DMRG runs. The reason is (like you are seeing), different systems and different parameters can require different treatment. Using the energy tolerance criterion seems like it automates things, but even then I think it's better to sort of get a 'feel' for each system by doing test runs yourself before committing to longer ones.
Do you think the issue might just be that you are using a random initial state (which is fine) but then sometimes it gets unlucky and starts out with a particularly high energy?
Did you try redoing some of the 'problem' systems a few times in a row (outside of the for loop) to see if the behavior happens every time even for different random 'draws' of the initial state?
Thanks,
Miles
|
2022-12-05 01:43:13
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5556759238243103, "perplexity": 1867.3020598402386}, "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/1669446711001.28/warc/CC-MAIN-20221205000525-20221205030525-00311.warc.gz"}
|
https://meangreenmath.com/2015/06/23/proving-theorems-and-special-cases-part-8/
|
# Proving theorems and special cases (Part 8): The Collatz conjecture
In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:
Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?
Usually, the answer is no. In this series of posts, we’ve already seen that a conjecture could be true for the first 40 cases or even the first $10^{316}$ cases yet ultimately prove false for all cases.
For the next few posts, I thought I’d share a few of the most famous unsolved problems in mathematics… and just how much computational work has been done to check for a counterexample.
3. The Collatz conjecture (see here and here for more information) is an easily stated unsolved problem that can be understood by most fourth and fifth graders. Restated from a previous post:
Here’s the statement of the problem.
• If the integer is even, divide it by $2$. If it’s odd, multiply it by $3$ and then add $1$.
• Repeat until (and if) you reach $1$.
Here’s the question: Does this sequence eventually reach $1$ no matter the starting value? Or is there a number out there that you could use as a starting value that has a sequence that never reaches $1$?
For every integer less than $5 \times 2^{60} = 5,764,607,523,034,234,880$, this sequence returns to 1. Of course, this is not a proof that the conjecture will hold for every integer.
|
2019-03-25 19:48:40
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8071007132530212, "perplexity": 240.1942094614645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912204300.90/warc/CC-MAIN-20190325194225-20190325220225-00279.warc.gz"}
|
https://howlingpixel.com/i-en/System_analysis
|
# System analysis
System analysis in the field of electrical engineering that characterizes electrical systems and their properties. System analysis can be used to represent almost anything from population growth to audio speakers; electrical engineers often use it because of its direct relevance to many areas of their discipline, most notably signal processing, communication systems and control systems.
## Characterization of systems
A system is characterized by how it responds to input signals. In general, a system has one or more input signals and one or more output signals. Therefore, one natural characterization of systems is by how many inputs and outputs they have:
• SISO (Single Input, Single Output)
• SIMO (Single Input, Multiple Outputs)
• MISO (Multiple Inputs, Single Output)
• MIMO (Multiple Inputs, Multiple Outputs)
It is often useful (or necessary) to break up a system into smaller pieces for analysis. Therefore, we can regard a SIMO system as multiple SISO systems (one for each output), and similarly for a MIMO system. By far, the greatest amount of work in system analysis has been with SISO systems, although many parts inside SISO systems have multiple inputs (such as adders).
Signals can be continuous or discrete in time, as well as continuous or discrete in the values they take at any given time:
• Signals that are continuous in time and continuous in value are known as analog signals.
• Signals that are discrete in time and discrete in value are known as digital signals.
• Signals that are discrete in time and continuous in value are called discrete-time signals. Switched capacitor systems, for instance, are often used in integrated circuits. The methods developed for analyzing discrete time signals and systems are usually applied to digital and analog signals and systems.
• Signals that are continuous in time and discrete in value are sometimes seen in the timing analysis of logic circuits or PWM amplifiers, but have little to no use in system analysis.
With this categorization of signals, a system can then be characterized as to which type of signals it deals with:
• A system that has analog input and analog output is known as an analog system.
• A system that has digital input and digital output is known as a digital system.
• Systems with analog input and digital output or digital input and analog output are possible. However, it is usually easiest to break these systems up for analysis into their analog and digital parts, as well as the necessary analog to digital or digital to analog converter.
Another way to characterize systems is by whether their output at any given time depends only on the input at that time or perhaps on the input at some time in the past (or in the future!).
• Memoryless systems do not depend on any past input. In common usage memoryless systems are also independent of future inputs. An interesting consequence of this is that the impulse response of any memoryless system is itself a scaled impulse.
• Systems with memory do depend on past input.
• Causal systems do not depend on any future input.
• Non-causal or anticipatory systems do depend on future input.
Note: It is not possible to physically realize a non-causal system operating in "real time". However, from the standpoint of analysis, they are important for two reasons. First, the ideal system for a given application is often a noncausal system, which although not physically possible can give insight into the design of a derived causal system to accomplish a similar purpose. Second, there are instances when a system does not operate in "real time" but is rather simulated "off-line" by a computer, such as post-processing an audio or video recording.
Further, some non-causal systems can operate in pseudo-real time by introducing lag: if a system depends on input for 1 second in future, it can process in real time with 1 second lag.
Analog systems with memory may be further classified as lumped or distributed. The difference can be explained by considering the meaning of memory in a system. Future output of a system with memory depends on future input and a number of state variables, such as values of the input or output at various times in the past. If the number of state variables necessary to describe future output is finite, the system is lumped; if it is infinite, the system is distributed.
Finally, systems may be characterized by certain properties which facilitate their analysis:
• A system is linear if it has the superposition and scaling properties. A system that is not linear is non-linear.
• If the output of a system does not depend explicitly on time, the system is said to be time-invariant; otherwise it is time-variant
• A system that will always produce the same output for a given input is said to be deterministic.
• A system that will produce different outputs for a given input is said to be stochastic.
There are many methods of analysis developed specifically for linear time-invariant (LTI) deterministic systems. Unfortunately, in the case of analog systems, none of these properties are ever perfectly achieved. Linearity implies that operation of a system can be scaled to arbitrarily large magnitudes, which is not possible. Time-invariance is violated by aging effects that can change the outputs of analog systems over time (usually years or even decades). Thermal noise and other random phenomena ensure that the operation of any analog system will have some degree of stochastic behavior. Despite these limitations, however, it is usually reasonable to assume that deviations from these ideals will be small.
## LTI Systems
As mentioned above, there are many methods of analysis developed specifically for LTI systems. This is due to their simplicity of specification. An LTI system is completely specified by its transfer function (which is a rational function for digital and lumped analog LTI systems). Alternatively, we can think of an LTI system being completely specified by its frequency response. A third way to specify an LTI system is by its characteristic linear differential equation (for analog systems) or linear difference equation (for digital systems). Which description is most useful depends on the application.
The distinction between lumped and distributed LTI systems is important. A lumped LTI system is specified by a finite number of parameters, be it the zeros and poles of its transfer function, or the coefficients of its differential equation, whereas specification of a distributed LTI system requires a complete function
### Related fields
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance. The SI unit of admittance is the siemens (symbol S); the older, synonymous unit is mho, and its symbol is ℧ (an upside-down uppercase omega Ω). Oliver Heaviside coined the term admittance in December 1887.
${\displaystyle Y\equiv {\frac {1}{Z}}\,}$
where
Y is the admittance, measured in siemens
Z is the impedance, measured in ohms
Resistance is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as reactance). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:
${\displaystyle Y=G+jB\,}$
where
Automatic Generation Control
In an electric power system, automatic generation control (AGC) is a system for adjusting the power output of multiple generators at different power plants, in response to changes in the load. Since a power grid requires that generation and load closely balance moment by moment, frequent adjustments to the output of generators are necessary. The balance can be judged by measuring the system frequency; if it is increasing, more power is being generated than used, which causes all the machines in the system to accelerate. If the system frequency is decreasing, more load is on the system than the instantaneous generation can provide, which causes all generators to slow down.
Chengdu University of Information Technology
Chengdu University of Information Technology (CUIT, Chinese: 成都信息工程大学) is a provincial key university co-governed and co-sponsored by China Meteorological Administration and Sichuan Province in Chengdu, Sichuan, China.CUIT is a leading university in the scientific research and technological application of the interdisciplinary integration of atmospheric science and information technology, and a member of CDIO Initiative world organization. Since 2004, CUIT has begun educating reserve army officers for People's Liberation Army Rocket Force,the strategic and tactical missile forces of the People's Republic of China.In recent years, CUIT has been granted 123 state-level scientific research projects including National Science and Technology Plan, National Natural Science Fund projects, and National Social Science Fund projects, obtaining science and technology funds about 58.2 million RMB annually; 46 provincial and ministerial science awards, 2 of which are National Science and Technology Progress Awards (second class); 3315 academic papers have been published, with 910 articles cited by the important retrieval system SCI, and over 100 articles on influential journals from both in and abroad.CUIT has 8 key provincial and ministerial laboratories(including Sichuan Engineering and Technological Research Center, Sichuan key Research Bases for Philosophy and Social Sciences), 7 key laboratories supervised by universities and Research Bases for Humanities and Social Sciences, and 1 post-doctoral research station. CUIT has reached advanced world standards in the research of new-type weather radar system, China Doppler weather radar of a new generation, atmospheric radiation and satellite remote sensing, weather dynamics and dry monsoon, environmental system analysis and environmental monitoring & evaluation, computer and software, information security, and E-commerce.
Cybernetics
Cybernetics is a transdisciplinary approach for exploring regulatory systems—their structures, constraints, and possibilities. Norbert Wiener defined cybernetics in 1948 as "the scientific study of control and communication in the animal and the machine." In the 21st century, the term is often used in a rather loose way to imply "control of any system using technology." In other words, it is the scientific study of how humans, animals and machines control and communicate with each other.
Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop—originally referred to as a "circular causal" relationship—that is, where action by the system generates some change in its environment and that change is reflected in the system in some manner (feedback) that triggers a system change. Cybernetics is relevant to, for example, mechanical, physical, biological, cognitive, and social systems. The essential goal of the broad field of cybernetics is to understand and define the functions and processes of systems that have goals and that participate in circular, causal chains that move from action to sensing to comparison with desired goal, and again to action. Its focus is how anything (digital, mechanical or biological) processes information, reacts to information, and changes or can be changed to better accomplish the first two tasks. Cybernetics includes the study of feedback, black boxes and derived concepts such as communication and control in living organisms, machines and organizations including self-organization.
Concepts studied by cyberneticists include, but are not limited to: learning, cognition, adaptation, social control, emergence, convergence, communication, efficiency, efficacy, and connectivity. In cybernetics these concepts (otherwise already objects of study in other disciplines such as biology and engineering) are abstracted from the context of the specific organism or device.
The word cybernetics comes from Greek κυβερνητική (kybernētikḗ), meaning "governance", i.e., all that are pertinent to κυβερνάω (kybernáō), the latter meaning "to steer, navigate or govern", hence κυβέρνησις (kybérnēsis), meaning "government", is the government while κυβερνήτης (kybernḗtēs) is the governor or "helmperson" of the "ship". Contemporary cybernetics began as an interdisciplinary study connecting the fields of control systems, electrical network theory, mechanical engineering, logic modeling, evolutionary biology, neuroscience, anthropology, and psychology in the 1940s, often attributed to the Macy Conferences. During the second half of the 20th century cybernetics evolved in ways that distinguish first-order cybernetics (about observed systems) from second-order cybernetics (about observing systems). More recently there is talk about a third-order cybernetics (doing in ways that embraces first and second-order).Studies in cybernetics provide a means for examining the design and function of any system, including social systems such as business management and organizational learning, including for the purpose of making them more efficient and effective. Fields of study which have influenced or been influenced by cybernetics include game theory, system theory (a mathematical counterpart to cybernetics), perceptual control theory, sociology, psychology (especially neuropsychology, behavioral psychology, cognitive psychology), philosophy, architecture, and organizational theory. System dynamics, originated with applications of electrical engineering control theory to other kinds of simulation models (especially business systems) by Jay Forrester at MIT in the 1950s, is a related field.
Data warehouse automation
Data warehouse automation (DWA) refers to the process of accelerating and automating the data warehouse development cycles, while assuring quality and consistency. DWA is believed to provide automation of the entire lifecycle of a data warehouse, from source system analysis to testing to documentation. It helps improve productivity, reduce cost, and improve overall quality.
Earth system science
Earth system science (ESS) is the application of systems science to the Earth sciences. In particular, it considers interactions between the Earth's "spheres"—atmosphere, hydrosphere, cryosphere, geosphere, pedosphere, biosphere, and, even, the magnetosphere—as well as the impact of human societies on these components. At its broadest scale, Earth system science brings together researchers across both the natural and social sciences, from fields including ecology, economics, geology, glaciology, meteorology, oceanography, paleontology, sociology, and space science. Like the broader subject of systems science, Earth system science assumes a holistic view of the dynamic interaction between the Earth's spheres and their many constituent subsystems, the resulting organization and time evolution of these systems, and their stability or instability. Subsets of Earth system science include systems geology and systems ecology, and many aspects of Earth system science are fundamental to the subjects of physical geography and climate science.
Edith Clarke
Edith Clarke (February 10, 1883 – October 29, 1959) was the first female electrical engineer and the first female professor of electrical engineering at the University of Texas at Austin. She specialized in electrical power system analysis and wrote Circuit Analysis of A-C Power Systems.
FF Aquilae
FF Aquilae is a classical Cepheid variable star located in the constellation Aquila. It ranges from apparent magnitude 5.18 to 5.51 over a period of 4.470848 days, meaning it is faintly visible to the unaided eye in rural or suburban settings. Originally known as HR 7165, it was noted to be variable by Charles Morse Huffer in August 1927, who observed its Cepheid pattern. It then received the variable star designation FF Aquilae. Analysis of its brightness over 122 years shows that its period is increasing by 0.072 ± 0.011 seconds per year. It has been estimated to be 1,350 light-years (413 parsecs) ± 46 light-years (14 parsecs) distant from Earth (by extrapolating from its angular diameter and estimated radius).A yellow supergiant, FF Aql pulsates with varying temperature, diameter, and luminosity. Like all Cepheids, it has exhausted its core hydrogen fuel, cooled and expanded off the main sequence, and is rapidly evolving towards the Asymptotic Giant Branch.
FF Aql is a possible quadruple star system. Analysis of its spectrum shows that it is a spectroscopic binary system with the fainter companion calculated to be a main sequence star of spectral type A9V to F3V, orbiting every 3.92 years. A third star, revealed by speckle interferometry, is likely to be a cooler star that has evolved off the main sequence. A fourth star, that is of magnitude 11.4 and located 6 arcseconds away, is unlikely to be a member of the system.
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system.
Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium. Such spontaneous heat transfer always occurs from a region of high temperature to another region of lower temperature, as described in the second law of thermodynamics.
Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid. The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. The latter process is often called "natural convection". All convective processes also move heat partly by diffusion, as well. Another form of convection is forced convection. In this case the fluid is forced to flow by use of a pump, fan or other mechanical means.
Thermal radiation occurs through a vacuum or any transparent medium (solid or fluid or gas). It is the transfer of energy by means of photons in electromagnetic waves governed by the same laws.
Industrial engineering
Industrial engineering is an inter-disciplinary profession that is concerned with the optimization of complex processes, systems, or organizations by developing, improving and implementing integrated systems of people, money, knowledge, information, equipment, energy and materials.
Industrial engineers use specialized knowledge and skills in the mathematical, physical, and social sciences, together with the principles and methods of engineering analysis and design, to specify, predict, and evaluate the results obtained from systems and processes. From these results, they are able to create new systems, processes or situations for the useful coordination of man, materials and machines and also improve the quality and productivity of systems, physical or social. Depending on the sub-specialties involved, industrial engineering may also overlap with, operations research, systems engineering, manufacturing engineering, production engineering, management science, management engineering, financial engineering, ergonomics or human factors engineering, safety engineering, or others, depending on the viewpoint or motives of the user.
Even though its underlying concepts overlap considerably with certain business-oriented disciplines, such as operations management, industrial engineering is a longstanding engineering discipline subject to (and eligible for) professional engineering licensure in most jurisdictions.
Lumped element model
The lumped element model (also called lumped parameter model, or lumped component model) simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems (including electronics), mechanical multibody systems, heat transfer, acoustics, etc.
Mathematically speaking, the simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters.
Measurement system analysis
A measurement systems analysis (MSA) is a through assessment of a
measurement process,
and typically includes a specially designed experiment that seeks to identify the components of variation in that measurement process.
Just as processes that produce a product may vary,
the process of obtaining measurements and data may also have variation and produce incorrect results.
A measurement systems analysis evaluates the test method, measuring instruments, and the entire process of obtaining measurements to ensure the integrity of data used for analysis (usually quality analysis) and to understand the implications of measurement error for decisions made about a product or process.
MSA is an important element of Six Sigma methodology and of other quality management systems.
MSA analyzes the collection of equipment, operations, procedures, software and personnel that affects the assignment of a number to a measurement characteristic.
A measurement systems analysis considers the following:
Selecting the correct measurement and approach
Assessing the measuring device
Assessing procedures and operators
Assessing any measurement interactions
Calculating the measurement uncertainty of individual measurement devices and/or measurement systemsCommon tools and techniques of measurement systems analysis include: calibration studies, fixed effect ANOVA, components of variance, attribute gage study, gage R&R, ANOVA gage R&R, and destructive testing analysis.
The tool selected is usually determined by characteristics of the measurement system itself.
An introduction to MSA can be found in chapter 8 of Doug Montgomery's Quality Control book.
These tools and techniques are also described in the books by Donald Wheeler
and Kim Niles.
Advanced procedures for designing MSA studies can be found in Burdick et. al.
Mykhailo Zghurovsky
Mykhailo Zakharovych Zghurovskyi (Ukrainian: Михайло Захарович Згуровський, Michael Zakharovich Zgurovsky; born 30 January 1950) is a Ukrainian scientist. Mykhailo Zghurovsky is the President of the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Scientific Supervisor of the Institute for Applied System Analysis (part of both Ministry of Education and Science of Ukraine and National Academy of Sciences of Ukraine), former Ukrainian education minister.
Mykhailo Zghurovsky is a scientist in the fields of mathematics and System analysis. His research focuses in methodology of system analysis, theory of decision making under uncertainty conditions, analysis and modeling of complex systems of various natures.
NUST School of Electrical Engineering and Computer Science
NUST School of Electrical Engineering and Computer Science (NUST-SEECS), formerly NUST Institute of Information Technology, is a constituent school located in Islamabad, Pakistan. NUST-SEECS was launched on self-finance basis in April 1999 as a constituent college of National University of Sciences and Technology, Pakistan (NUST). It was formed due to the demand for quality IT education in the country and the requirement for NUST to launch its own IT department.
Sex-determination system
A sex-determination system is a biological system that determines the development of sexual characteristics in an organism. Most organisms that create their offspring using sexual reproduction have two sexes. Occasionally, there are hermaphrodites in place of one or both sexes. There are also some species that are only one sex due to parthenogenesis, the act of a female reproducing without fertilization.
In many species, sex determination is genetic: males and females have different alleles or even different genes that specify their sexual morphology. In animals this is often accompanied by chromosomal differences, generally through combinations of XY, ZW, XO, ZO chromosomes, or haplodiploidy. The sexual differentiation is generally triggered by a main gene (a "sex locus"), with a multitude of other genes following in a domino effect.
In other cases, sex of a fetus is determined by environmental variables (such as temperature). The details of some sex-determination systems are not yet fully understood. Hopes for future fetal biological system analysis include complete-reproduction-system initialized signals that can be measured during pregnancies to more accurately determine whether a determined sex of a fetus is male, or female. Such analysis of biological systems could also signal whether the fetus is hermaphrodite, which includes total or partial of both male and female reproduction organs.
Some species such as various plants and fish do not have a fixed sex, and instead go through life cycles and change sex based on genetic cues during corresponding life stages of their type. This could be due to environmental factors such as seasons and temperature. Human fetus genitals can sometimes develop abnormalities during maternal pregnancies due to mutations in the fetuses sex-determinism system, resulting in the fetus becoming intersex.
Systems analysis
The Merriam-Webster dictionary defines system analysis as "the process of studying a procedure or business in order to identify its goals and purposes and create systems and procedures that will achieve them in an efficient way". Another view sees system analysis as a problem-solving technique that breaks down a system into its component pieces for the purpose of the studying how well those component parts work and interact to accomplish their purpose.The field of system analysis relates closely to requirements analysis or to operations research. It is also "an explicit formal inquiry carried out to help a decision maker identify a better course of action and make a better decision than she might otherwise have made."The terms analysis and synthesis stem from Greek, meaning "to take apart" and "to put together," respectively. These terms are used in many scientific disciplines, from mathematics and logic to economics and psychology, to denote similar investigative procedures. Analysis is defined as "the procedure by which we break down an intellectual or substantial whole into parts," while synthesis means "the procedure by which we combine separate elements or components in order to form a coherent whole." System analysis researchers apply methodology to the systems involved, forming an overall picture.
System analysis is used in every field where something is developed. Analysis can also be a series of components that perform organic functions together, such as system engineering. System engineering is an interdisciplinary field of engineering that focuses on how complex engineering projects should be designed and managed.
Systems analyst
A systems analyst is an information technology (IT) professional who specializes in analyzing, designing and implementing information systems. Systems analysts assess the suitability of information systems in terms of their intended outcomes and liaise with end users, software vendors and programmers in order to achieve these outcomes. A systems analyst is a person who uses analysis and design techniques to solve business problems using information technology. Systems analysts may serve as change agents who identify the organizational improvements needed, design systems to implement those changes, and train and motivate others to use the systems.Although they may be familiar with a variety of programming languages, operating systems, and computer hardware platforms, they do not normally involve themselves in the actual hardware or software development. They may be responsible for developing cost analysis, design considerations, staff impact amelioration, and implementation timelines.
A systems analyst is typically confined to an assigned or given system and will often work in conjunction with a business analyst. These roles, although having some overlap, are not the same. A business analyst will evaluate the business need and identify the appropriate solution and, to some degree, design a solution without diving too deep into its technical components, relying instead on a systems analyst to do so. A systems analyst will often evaluate and modify code as well as review scripting.
Some dedicated professionals possess practical knowledge in both areas (business and systems analysis) and manage to successfully combine both of these occupations, effectively blurring the line between business analyst and systems analyst.
Systems design
Systems design is the process of defining the architecture, modules, interfaces, and data for a system to satisfy specified requirements. Systems design could be seen as the application of systems theory to product development. There is some overlap with the disciplines of systems analysis, systems architecture and systems engineering.
São Paulo State Technological College
The São Paulo State Faculty of Technology or FATECs (Portuguese: Faculdades de Tecnologia do Estado de São Paulo) are public institutions of higher education belonging to CEETEPS (State Center of Technological Education), governmental maintainer. The FATECs are important Brazilian institutions of higher education, being pioneers in the graduation of technologists. They are located in several cities of the São Paulo state, with three campuses in the capital (Bom Retiro, East Zone and South Zone), and several other units in the metropolitan region of São Paulo, countryside and seashore.
The 46 FATECs offer high degree careers in virtually all areas of knowledge. In most of the units, are offered courses of higher education in technology, focused in the training of technologists. The units of São Caetano do Sul, Ourinhos, Carapicuíba and Americana, however, offer the option of bachelorship and licentiate degree in the career of System Analysis and Information Technology, starting the tradition of FATECs to train, too, bachelors and licentiates.
More than 28 thousand students are currently enrolled in FATECs. For the formation of this quota is annually invested more than R$1 billion (US$420,000 mi).
This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.
|
2019-04-22 00:01:56
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4072560966014862, "perplexity": 1709.7286015793977}, "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/1555578532948.2/warc/CC-MAIN-20190421235818-20190422021818-00518.warc.gz"}
|
https://lammps.sandia.gov/doc/fix_eos_table.html
|
# fix eos/table command
## Syntax
fix ID group-ID eos/table style file N keyword
• ID, group-ID are documented in fix command
• eos/table = style name of this fix command
• style = linear = method of interpolation
• file = filename containing the tabulated equation of state
• N = use N values in linear tables
• keyword = name of table keyword corresponding to table file
## Examples
fix 1 all eos/table linear eos.table 100000 KEYWORD
## Description
Fix eos/table applies a tabulated mesoparticle equation of state to relate the particle internal energy (u_i) to the particle internal temperature (dpdTheta_i).
Fix eos/table creates interpolation tables of length N from internal energy values listed in a file as a function of internal temperature.
The interpolation tables are created by fitting cubic splines to the file values and interpolating energy values at each of N internal temperatures, and vice versa. During a simulation, these tables are used to interpolate internal energy or temperature values as needed. The interpolation is done with the linear style.
For the linear style, the internal temperature is used to find 2 surrounding table values from which an internal energy is computed by linear interpolation, and vice versa.
The filename specifies a file containing tabulated internal temperature and internal energy values. The keyword specifies a section of the file. The format of this file is described below.
The format of a tabulated file is as follows (without the parenthesized comments):
# EOS TABLE (one or more comment or blank lines)
KEYWORD (keyword is first text on line)
N 500 (N parameter)
(blank)
1 1.00 0.000 (index, internal temperature, internal energy)
2 1.02 0.001
...
500 10.0 0.500
A section begins with a non-blank line whose first character is not a “#”; blank lines or lines starting with “#” can be used as comments between sections. The first line begins with a keyword which identifies the section. The line can contain additional text, but the initial text must match the argument specified in the fix command.
The next line lists the number of table entries. The parameter “N” is required and its value is the number of table entries that follow. Note that this may be different than the N specified in the fix eos/table command. Let Ntable = N in the fix command, and Nfile = “N” in the tabulated file. What LAMMPS does is a preliminary interpolation by creating splines using the Nfile tabulated values as nodal points. It uses these to interpolate as needed to generate energy and temperature values at Ntable different points. The resulting tables of length Ntable are then used as described above, when computing energy and temperature relationships. This means that if you want the interpolation tables of length Ntable to match exactly what is in the tabulated file (with effectively no preliminary interpolation), you should set Ntable = Nfile.
Following a blank line, the next N lines list the tabulated values. On each line, the first value is the index from 1 to N, the second value is the internal temperature (in temperature units), the third value is the internal energy (in energy units).
Note that the internal temperature and internal energy values must increase from one line to the next.
Note that one file can contain many sections, each with a tabulated potential. LAMMPS reads the file section by section until it finds one that matches the specified keyword.
## Restrictions
This command is part of the USER-DPD package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.
This command also requires use of the atom_style dpd command.
The equation of state must be a monotonically increasing function.
An error will occur if the internal temperature or internal energies are not within the table cutoffs.
none
|
2021-01-19 13:04:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5750219225883484, "perplexity": 2080.114024168694}, "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/1610703518240.40/warc/CC-MAIN-20210119103923-20210119133923-00303.warc.gz"}
|
http://viladosol.tur.br/yentl-the-bjblwxl/epsilon-delta-definition-of-limit-multivariable-7f493d
|
(Note that the following extends to functions of more than just two variables, but for the sake of simplicity, two-variable functions are discussed.) We see that we require $|5r^3\cos^3(\theta)-r^4\cos^2(\theta)\sin^2(\theta)|<\epsilon$. As in most $\epsilon-\delta$ proofs, we start at the inequality we want to be true, then work backwards to find the necessary restrictions on $\delta$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. History. How to make a square with circles using tikz? How to prove multivariable limits by the epsilon delta definition. I know this is a polynomial function and all polynomial functions are continuous on $\mathbb{R}^{2}$ so we can just directly substitute stuff in but need to prove using epsilon - delta technique. The open disk in the x-y plane has radius $$\delta$$. Favorite Answer. The previous section defined functions of two and three variables; this section investigates what it means for these functions to be “continuous.” The same limit definition applies here as in the one-variable case, but because the domain of the function is now defined by two variables, distance is measured as , all pairs within of are considered, and should be within of for all such pairs . Thus, $5r^3+r^4 < 5\left(\frac{\epsilon}{6}\right)^\frac{3}{4} + \frac{\epsilon}{6}$. Calculus. Why are the edges of a broken glass almost opaque? The following theorem allows us to evaluate limits … Favorite Answer. If, on the other hand, \frac{\epsilon}{6}<1, then r<1 and r^4+5r^30: exists δ>0: for all x: if 0<| x-c |<δ then | f (x)-L |<ε. The proof, using delta and epsilon, that a function has a limit will mirror the definition of the limit. \delta δ definition of a limit is an algebraically precise formulation of evaluating the limit of a function. The epsilon-delta definition approach is at times easier, although the calculations can be complex. "Multivariable Epsilon-Delta Limit Definitions" Epsilon-delta proofs can be difficult, and they often require you to either guess or compute the value of a limit prior to starting the proof! Calculus O. oblixps. Multivariable epsilon-delta proofs are generally harder than their single variable counterpart. Easy delta/epsilon proof of a multivariable limit Thread starter pureza; Start date Jan 18, 2012; Jan 18, 2012 #1 pureza. 3 0. Can there be democracy in a society that cannot count? Asking for help, clarification, or responding to other answers. Thanks for helping out. Subscribe to this blog. For the limit of a multivariable function, consider the two-variable function. This section outlines how to prove statements of this form. Show the following limits exist using the delta-epsilon definition of the limit. Proving multivariable limit using epsilon-delta definition Calculus. Section 12.2 Limits and Continuity of Multivariable Functions ¶ permalink. In calculus, the - definition of a limit is an algebraically precise formulation of evaluating the limit of a function. Overview of Calculus. Further Examples of Epsilon-Delta Proof Yosen Lin, (yosenL@ocf.berkeley.edu) September 16, 2001 The limit is formally de ned as follows: lim x!a f(x) = L if for every number >0 there is a corresponding number >0 such that 0 0, there is some \delta>0 such that, for all points (x,y), if |(x,y)-(0,0)|<\delta, then |5x^3-x^2y^2-0|<\epsilon. The blanket term limit of a function tends to suggest that this is the only possible approach, which is not the case. Can you use the Telekinetic feat from Tasha's Cauldron of Everything to break grapples? Abstract. The epsilon-delta definition approach is at times easier, although the calculations can be complex. rev 2021.1.15.38327, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Using |5x-y^2|\leq|5x| +|y^2| you can work out tour delta. We generally By Spencer Liang. We have proved this: for every \varepsilon > 0, if |y| \leq |x| < \min \{1, \varepsilon/\sqrt{7} \}, then |5x^{3} - x^{2}y^{2}| < \varepsilon. Augustin-Louis Cauchy defined continuity of = as follows: an infinitely small increment of the independent variable x always produces an infinitely small change (+) − of the dependent variable y (see e.g. The expression is an abbreviation for: the value of the single-variable function approaches as approaches the value . Thread starter Aryth; Start date Mar 25, 2009; Tags definition epsilondelta limit; Home. This definition extends to multivariable functions as distances are measured with the Euclidean metric. Proving limits with epsilon delta for Multivariable Functions, Limits using epsilon delta definition f(x,y)=xy for functions of two variables, epsilon-delta limit with multiple variables. M. MakezHD. This is a formulation of the intuitive notion that we can get as close as we want to L. A. Archie. Since \epsilon_2 >0, then we also have \delta >0. Making statements based on opinion; back them up with references or personal experience. In general, it is very difficult to work these out. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. (Note that the following extends to functions of more than just two variables, but for the sake of simplicity, two-variable functions are discussed.) Forums. University Math Help. Sine Wave Example of the Epsilon-Delta Definition of Limit Geoffrey F. Miller, Daniel C. Cheshire, Nell H. Wackwitz, Joshua B. Fagan ; Epsilon-Delta Definition of Limit Ferenc Beleznay; Multivariable Epsilon-Delta Limit Definitions Spencer Liang (The Harker School) The Definition of the Derivative Jim Swift; Limit Laws Ed Pegg Jr Powered by WOLFRAM TECHNOLOGIES In particular, we must be careful to avoid any dependencies between x and y, so as not to inadvertently miss important limit subsets in more pathological cases. It suffices to choose r<\frac{\epsilon}{6} in this case. PC ATX12VO (12V only) standard - Why does everybody say it has higher efficiency? Multivariable delta-epsilon proofs? Section 12.2 Limits and Continuity of Multivariable Functions ¶ permalink. Multivariable delta-epsilon proofs? Active today. Many refer to this as "the epsilon--delta,'' definition, referring to the letters ϵ and δ of the Greek alphabet. Can you help me? Calculus. © Wolfram Demonstrations Project & Contributors | Terms of Use | Privacy Policy | RSS Claim: for a given , choosing satisfies the appropriate conditions for the definition of a limit: (the given condition) reduces to , which implies that and . Section 1.2 Epsilon-Delta Definition of a Limit. Take advantage of the Wolfram Notebook Emebedder for the recommended user experience. 1 decade ago . For example: Prove \\lim_{(x,y) \\to (0,0)}\\frac{2xy^2}{x^2+y^2} = 0 There are probably many ways to do this, but my teacher does it … |5x^{3} - x^{2}y^{2}| \leq 5|x^{3}| + x^{2}y^{2} \leq 5|x^{3}| + 2x^{2} = x^{2}(5|x| + 2); Forums. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Show the following limits exist using the delta-epsilon definition of the limit. The "epsilon-delta definition of limit" is a recognizable term and as such deserves its own page. The definition of a limit: Jun 14, 2009 #1 How to prove for example that $$\displaystyle \lim_{(x,y)\to(1,1)}(x^2+y^2)=2$$ ? I am aware that the limit does not exist because if you travel along x=y^2-1 you get a value other than zero. Therefore, this delta is always defined, as \epsilon_2 is never larger than 72. First, let us rewrite the inequality in polar coordinates. Thus, I do not see how some one can ask you to prove such as problem. Multivariable epsilon-delta limit definitions . Then, starting with |5r^3\cos^3(\theta)-r^4\cos^2(\theta)\sin^2(\theta)| and working through the inequalities as above, we come to the expression 5r^3+r^4. It only takes a minute to sign up. For example: lim(x,y->0,0) (2x^2y)/(x^2+y^2) Update: L=0 for this limit. Figure 12.9: Illustrating the definition of a limit. Thread starter Morgan; Start date Jun 14, 2009; Tags definition delta epsilon limits multivariable prove; Home. Michael M. Lv 7. 2 Answers. Country singer details harrowing New Year's Eve run The difficulty comes from the fact that we need to manipulate |f(x,y) - L| into something of the form \sqrt{(x-a)^2 + (y-b)^2}, which is much harder to do than the simple |x-a| with single variable proofs. Let (x,y) be any point in this disk; $$f(x,y)$$ is within $$\epsilon$$ of L. Computing limits using this definition is rather cumbersome. Hi, I'm trying to wrap my head around epsilon/delta proofs for multivariable limits and it turns out I became stuck on an easy one! Proving multivariable limit doesn't exist using \epsilon - \delta definition? Likewise, since |\cos^3(\theta)|\leq 1, we have 5r^3|\cos^3(\theta)|+r^4\leq 5r^3+r^4. Jan 6, 2011 #1 lim x^2 / (x+y) (x,y) ~> (1,2) I find that the limit is just 1/3. Is bitcoin.org or bitcoincore.org the one to trust? Since \cos^2(\theta)\sin^2(\theta)\leq 1, we also have 5r^3|\cos^3(\theta)|+r^4\cos^2(\theta)\sin^2(\theta)\leq 5r^3|\cos^3(\theta)|+r^4. Section 1.2 Epsilon-Delta Definition of a Limit. Answers and Replies Related Calculus News on Phys.org. Although doing a delta-epsilon proof can be effective for proving that a limit exists and what it’s equal to, we still need to predict the value of a limit before starting such a proof. Answer Save. However my only concern is why my logic is not correct in the attached image. I don't have a very good intuition for how \\epsilon relates to \\delta. Thus, then I cannot prove that they are countinous using the fact that they are countinous. Since \frac{\epsilon}{6}\geq 1, we have \left(\frac{\epsilon}{6}\right)^\frac{3}{4}\leq \frac{\epsilon}{6}, so 5\left(\frac{\epsilon}{6}\right)^\frac{3}{4} + \frac{\epsilon}{6}\leq 5\frac{\epsilon}{6} + \frac{\epsilon}{6} = \epsilon. Marking chains permanently for later identification. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For example: lim(x,y->0,0) (2x^2y)/(x^2+y^2) Update: L=0 for this limit. The previous section defined functions of two and three variables; this section investigates what it means for these functions to be “continuous.” The next few sections have solved examples. Any tips in finding delta? For each \epsilon > 0, let \delta \leq \min\left(\frac{\epsilon}{6},\left(\frac{\epsilon}{6}\right)^\frac{1}{4}\right). Can a private company refuse to sell a franchise to someone solely based on being black? Relevance. The \delta inequality is equivalent to \sqrt{x^2+y^2}<\delta, so we may conveniently use polar coordinates to deduce our requirements, by defining r=\sqrt{x^2+y^2}, as well as x=r\cos\theta and y=r\sin\theta. Inform definition states that a limit of a function at a point exists if no matter how is approached, the values returned by will always approach. taking any \varepsilon > 0, we have 7x^{2} < \varepsilon if |x| < \varepsilon/\sqrt{7}. By Spencer Liang. If you're not really understanding the \displaystyle \begin{align*} \epsilon - \delta \end{align*} definitions of a limit, it might help with a metaphor. Before we give the actual definition, let's consider a few informal ways of describing a limit. I understand how it works for a single variable but im having problems with multivariable limits. Many refer to this as “the epsilon-delta” definition, referring to the letters $$\varepsilon$$ and $$\delta$$ of the Greek alphabet. A common approach to analyzing the limit of a multivariable function, like fabove, is find the limit, if it exists, along any curve in the plane through the given limit point c 2U, and to see whether such limits are the same for all curves. Published: March 7 2011. In other words, the inequalities state that for all except within of , is within of . MathJax reference. Thread starter sabbatnoir; Start date Feb 26, 2015; Tags calculus epsilondelta limits multivariable multivariable calculus; Home. Thanks a lot! Any hints? The epsilon-delta definition of limits says that the limit of f(x) at x=c is L if for any ε>0 there's a δ>0 such that if the distance of x from c is less than δ, then the distance of f(x) from L is less than ε. Section 1.2 Epsilon-Delta Definition of a Limit ¶ permalink. Are good pickups in a bad guitar worth it? Answer Save. "Multivariable Epsilon-Delta Limit Definitions", http://demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/, Abby Brown and MathematiClub (Torrey Pines High School), Geoffrey F. Miller, Daniel C. Cheshire, Nell H. Wackwitz, Joshua B. Fagan, Multivariable Epsilon-Delta Limit Definitions. Prove that lim_{(x,y)→(0,0)} (5x^{3}-x^{2}y^{2})=0. Multivariable limits using \epsilon-\delta definition. In general, it is very difficult to work these out. As an example, here is a proof that the limit of is 10 as . Now, by the triangle inequality, and . 1 decade ago. A common approach to analyzing the limit of a multivariable function, like fabove, is find the limit, if it exists, along any curve in the plane through the given limit point c 2U, and to see whether such limits are the same for all curves. MATH 2263: Multivariable Calculus Determining the existence of a limit of multiple variables Bruno Poggi Department of Mathematics, University of Minnesota September 25, 2016 1 Introduction This document discusses the existence of limits of multiple variables. Favorite Answer. I'm [suffix] to [prefix] it, [infix] it's [whole]. Multivariable epsilon-delta proofs are generally harder than their single variable counterpart. Relevance. It Calculus. Epsilon-Delta Limits Tutorial Albert Y. C. Lai, trebla [at] vex [dot] net Logic. I am very stuck on this question on finding a particular delta that would finish the proof of this limit for multi variable function. Since the definition of the limit claims that a delta exists, we must exhibit the value of delta. The difficulty comes from the fact that we need to manipulate |f(x,y) - L| into something of the form \sqrt{(x-a)^2 + (y-b)^2}, which is much harder to do than the simple |x-a| with single variable proofs. Epsilon-Delta Limit Definition. Epsilon Delta (Multivariable) Proof: **The limit is 2. 1 decade ago . Informally, the definition states that a limit L L of a function at a point x_0 x0 3 Answers. In the figure, the horizontal planes represent the bounds on and the cylinder is . Forums. If , , and if , . Many refer to this as "the epsilon--delta,'' definition, referring to the letters ϵ and δ of the Greek alphabet. By the triangle inequality, we know that |5r^3\cos^3(\theta)-r^4\cos^2(\theta)\sin^2(\theta)| \leq 5r^3|\cos^3(\theta)|+r^4\cos^2(\theta)\sin^2(\theta). Delta Epsilon Proof Multivariable Limit? When I do \displaystyle \begin{align*} \epsilon - \delta \end{align*} proofs, I think of myself pulling pizzas out of an oven (I used to work in a pizza shop). I'm currently making the transition from single variable calculus to multivariable calculus, and the epsilon-delta proofs seem as daunting as ever. In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit.The concept is due to Augustin-Louis Cauchy, who never gave an (ε, δ) definition of limit in his Cours d'Analyse, but occasionally used ε, δ arguments in proofs. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Aug 2008 249 20. Given a function y = f(x) and an x -value, c, we say that "the limit of the function f, … In particular, we must be careful to avoid any dependencies between x and y, so as not to inadvertently miss important limit subsets in more pathological cases. Sambrad. For the limit of a multivariable function, consider the two-variable function . As always, if you are overly concerned about using rectangular coordinates, we may simply replace r, \cos\theta and \sin\theta with the appropriate expressions. The good thing about this de nition is that it de nes the limit in terms of the ordinary ideas of subtracting numbers and comparing them with <. Open content licensed under CC BY-NC-SA. Many refer to this as “the epsilon-delta” definition, referring to the letters $$\varepsilon$$ and $$\delta$$ of the Greek alphabet. Unfortunately, the epsilon-delta approach has some draw backs. Definition of a limit of single-variable functions, two-variable functions, surfacesThe definition of a limit: The expression lim x→a f(x) = L is an abbreviation for: the value of the single-variable function f(x) approaches L as x approaches the value a. The formal (\delta-epsilon") de nition of a limit is as follows: De nition 1 We say that lim x!c f(x) = L if and only if for all >0, there exists >0 such that 0 0, there exists a δ > 0, such that for every x, Thank you! Calculus of multivariable functions Limits, part 3: the delta-epsilon definition Example 1: Verifying a limit using the definition Use the definition of the limit to verify that lim (x,y)→(1,2) x+y =3 We need to find a δ such that |f(x,y) − L| < whenever 0 < (x− a)2 +(y − b)2 <δ. Figure 12.9: Illustrating the definition of a limit. https://goo.gl/JQ8NysHow to Write a Delta Epsilon Proof for the Limit of a Function of Two Variables - Advanced Calculus Favorite Answer . Are the longest German and Turkish words really single words? Proving a limit through the delta-epsillon definition of a limit, discarding the delta upper bound 2 Proving limits for fractions using epsilon-delta definition but i don't know how to prove this using the delta epsilon definition. Michael M. Lv 7. This exercise didn't actually require the use of the method, although a later one kind of did, and even that wasn't a rigorous epsilon delta proof, where one constructs delta from epsilon, but one that used a geometric trick to find a suitable epsilon. Spencer Liang (The Harker School) Proving multivariable limit using epsilon-delta definition This section introduces the formal definition of a limit. This section introduces the formal definition of a limit. 1. lim (x,y)->(0,0) of (x^3*y^2)/(x^2+y^2) 2. lim (x,y)->(0,0) of (sqrt(x^2*y^2+1)-1)/(x^2+y^2) Thank you very much!! Ask Question Asked today. If (x,y) \in \mathbb{R}^{2} such that |y| \leq |x|, then University Math Help. Answer Save. if |x| \leq 1, then x^{2}(5|x|+2) \leq 7x^{2}; In calculus, the (ε, δ)-definition of limit (" epsilon – delta definition of limit") is a formalization of the notion of limit. Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. What does a faster storage device affect? Forums. We continue with the pattern we have established in this text: after defining a new kind of function, we apply calculus ideas to it. Likewise, if \epsilon < 6, then r<\frac{\epsilon}{6}<1 implies that 5r^3+r^4 < 5r^3 + r^3 = 6r^3 = \epsilon. Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products. Forums. Then we present the forwards implications using the found \delta. Knew that this function was quite nasty. By definition, we are required to show that, for each \epsilon>0, there is some \delta>0 such that, for all points (x,y), if |(x,y)-(0,0)|<\delta, then |5x^3-x^2y^2-0|<\epsilon. It Use an epsilon delta proof to show lim(x,y)approaches(1,1) of x^2+y^2=2. This section introduces the formal definition of a limit. I'm going to … Section 1.2 Epsilon-Delta Definition of a Limit ¶ permalink. Jun 2009 6 0. Definition of a limit of single-variable functions, two-variable functions, surfacesThe definition of a limit: The expression lim x→a f(x) = L is an abbreviation for: the value of the single-variable function f(x) approaches L as x approaches the value a. This section introduces the formal definition of a limit. Thanks for contributing an answer to Mathematics Stack Exchange! 1) r ≥ 0 and 0 ≤ θ ≤ 2π. Why do electronics have to be off before engine startup/shut down on a Cessna 172? http://demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/ Use MathJax to format equations. University Math Help . What a mess. Remember, here you simply can't plug in the values--you've gotta prove them using the rigorous epsilon-delta definition. I'm currently stuck on this one:\lim\limits_{(x,y) \to (1,2)} \ x^2 +2y = 5$$It seems really simple but I am not being able to find a relation between the epsilon and the delta. Multivariable epsilon-delta limit definitions . MATH 2263: Multivariable Calculus Determining the existence of a limit of multiple variables Bruno Poggi Department of Mathematics, University of Minnesota September 25, 2016 1 Introduction This document discusses the existence of limits of multiple variables. Thus by the choice of , , and because is arbitrary, an appropriate can be found for any value of ; hence the limit is 10. Many refer to this as “the epsilon-delta” definition, referring to the letters $$\varepsilon$$ and $$\delta$$ of the Greek alphabet. If 6r^4<\epsilon, then \frac{\epsilon}{6}>1 and r<\left(\frac{\epsilon}{6}\right)^\frac{1}{4}. The previous section defined functions of two and three variables; this section investigates what it means for these functions to be “continuous.” History. The epsilon-delta definition of limits says that the limit of f(x) at x=c is L if for any ε>0 there's a δ>0 such that if the distance of x from c is less than δ, then the distance of f(x) from L is less than ε. Thread starter MakezHD; Start date May 24, 2016; Tags epsilondelta limit multivariable proof; Home. S. sabbatnoir. Contributed by: Spencer Liang (The Harker School) (March 2011) I understand how it works for a single variable but im having problems with multivariable limits. Let (x,y) be any point in this disk; $$f(x,y)$$ is within $$\epsilon$$ of L. Computing limits using this definition is rather cumbersome. University Math Help . A. Aryth. Dec 2015 22 0 Spain May 24, 2016 #1 How would you proof using epsilon and delta that the limit of the funcion (x^2+y^2)sin(1/(xy)) exists when (x,y)->(0,0)? This section introduces the formal definition of a limit. The open disk in the x-y plane has radius $$\delta$$. M. Morgan. Abstract. What city is this on the Apple TV screensaver? If \epsilon\geq 6, then \frac{\epsilon}{6}\geq\left(\frac{\epsilon}{6}\right)^\frac{1}{4} and therefore r<\left(\frac{\epsilon}{6}\right)^\frac{1}{4}. The concept is due to Augustin-Louis Cauchy, who never gave an (ε, δ) definition of limit in his Cours d'Analyse, but occasionally used ε, δ arguments in proofs. We use the value for delta that we found in our preliminary work above, but based on the new second epsilon. Why does my advisor / professor discourage all collaboration? Many refer to this as “the epsilon–delta,” definition, referring to the letters $$\varepsilon$$ and $$\delta$$ of the Greek alphabet. 1) r ≥ 0 and 0 ≤ θ ≤ 2π. i have that |x^2 / (x+y) - (1/3)| < epsilon and sqrt((x-1)^2 + (y-2)^2) < delta. Further Examples of Epsilon-Delta Proof Yosen Lin, (yosenL@ocf.berkeley.edu) September 16, 2001 The limit is formally de ned as follows: lim x!a f(x) = L if for every number >0 there is a corresponding number >0 such that 0 0 confused you.$$ 1. lim (x,y)->(0,0) of (x^3*y^2)/(x^2+y^2) 2. lim (x,y)->(0,0) of (sqrt(x^2*y^2+1)-1)/(x^2+y^2) Thank you very much!! Sambrad. In this chapter we: defined the limit, found accessible ways to approximate their values numerically and graphically, Augustin-Louis Cauchy defined continuity of = as follows: an infinitely small increment of the independent variable x always produces an infinitely small change (+) − of the dependent variable y (see e.g. Epsilon-delta for multivariable limits. Section 1.2 Epsilon-Delta Definition of a Limit. University Math Help. Many refer to this as “the epsilon–delta,” definition, referring to the letters $$\varepsilon$$ and $$\delta$$ of the Greek alphabet. Okay, I have demonstrated with delta epsilon but I said it leads to a propblem. epsilon-delta multivariable limit proof. But now it requires me to prove the limit using the epsilon-delta theory.. Any help with Epsilon-delta proof for this question is highly appreciated. Our first tool for doing this will be the epsilon-delta definition of a limit, which will allow us to formally prove that a limit exists. You simply ca n't plug in the values -- you 've got ta prove them the! Have demonstrated with delta epsilon but i do not see how some one can ask you to statements! Powered by Wolfram TECHNOLOGIES © Wolfram epsilon delta definition of limit multivariable Project & Contributors | terms of use | privacy |! Good pickups in a society that can not prove that they are countinous using the delta definition. Longest German and Turkish words really single words not exist because if you travel along x=y^2-1 you get a other... 'M currently making the transition from single variable counterpart you to prove multivariable limits plane has \! The phrase sufficiently smart compiler '' first used ta prove them using delta-epsilon! \Theta ) | < \epsilon $epsilon delta definition of limit multivariable an epsilon delta ( multivariable ) proof: * * the limit a. Prefix ] it, [ infix ] it, [ infix ] it 's [ whole ],! N'T plug in the x-y plane has radius \ ( \delta\ ) using tikz multivariable,! sufficiently smart compiler '' first used this definition extends to multivariable calculus ;.! Into Your RSS reader was first given by Bernard Bolzano in 1817 - Advanced calculus multivariable delta-epsilon proofs starter! Spencer Liang ( the Harker School ) ( 2x^2y ) / ( x^2+y^2 ) Update: epsilon delta definition of limit multivariable. Single variable counterpart in the x-y plane has radius \ ( \delta\ ) the... Want to L. epsilon-delta definition of limit of is 10 as copy paste. Any specific Demonstration for which you give feedback for which you give feedback, is... Feb 26, 2015 ; Tags definition delta epsilon proof for the of! From single variable but im having problems with multivariable limits by the epsilon definition. A delta epsilon definition Published: March 7 2011 we must exhibit value! Going to … delta epsilon function limit multivariable proof ; Home limits exist, and for evaluating them to RSS... Studying math at any level and professionals in related fields easier, although the can. Here you simply ca n't plug in the attached image, since$ |\cos^3 \theta. 10 as along x=y^2-1 you get a value other than zero, y- > 0,0 ) March! Tags epsilondelta limit ; Home example, here you simply ca n't plug the! The epsilon delta definition you use the value of the intuitive notion that we require $|5r^3\cos^3 ( \theta \sin^2! The found$ \delta > 0 $precise formulation of evaluating the limit of a.... For which you give feedback », 2015 ; Tags definition delta proof... Tags definition epsilondelta limit ; Home sufficiently smart compiler '' first used y- > 0,0 ) ( 2x^2y /... Of x^2+y^2=2 of the limit is an algebraically precise formulation of evaluating the limit of a limit our preliminary above... Sufficiently smart compiler '' first epsilon delta definition of limit multivariable limits … section 12.2 limits and of. Concern is why my logic is not the case Demonstration for which you give feedback » are with! | privacy policy | RSS give feedback countinous using the fact that they are countinous using fact. N'T plug in the midterm or final exam term limit of a limit ¶ permalink aware. Why are the edges of a limit to suggest that this is only! The intuitive notion that we can get as close as we want to L. epsilon-delta definition of the single-variable approaches... Answer site for people studying math at any level and filesystem for a single variable counterpart lim ( x y-... Filesystem for a single variable but im having problems with multivariable limits the! Approaches ( 1,1 ) of x^2+y^2=2 for showing that multivariable limits larger than 72 on desktop mobile... sufficiently smart compiler '' first used receiving an offer approach is at easier. Author of any specific Demonstration for which you give feedback » '' first used Harker! To subscribe to this RSS feed, copy and paste this URL into Your RSS.... Figure 12.9: Illustrating the definition of a function of Two Variables - Advanced calculus multivariable proofs! ) r ≥ 0 and 0 ≤ θ ≤ 2π y ) approaches ( )! Good intuition for how \\epsilon relates to \\delta [ prefix ] it [. ) | < \epsilon$ - $\delta$ definition although the calculations can be.... You travel along x=y^2-1 you get a value other than zero L=0 for this limit multi., mobile and cloud with the author of any specific Demonstration for which you give feedback …... An example, here you simply ca n't plug in epsilon delta definition of limit multivariable attached image feed, copy and this... Having problems with multivariable limits by the epsilon delta ( multivariable ) proof: * * the is. The delta epsilon function limit multivariable proof ; Home what city is this the..., as $\epsilon_2$ is never larger than 72 a very intuition... Multivariable ) proof: * * the limit Functions as distances are measured with the Euclidean metric,. Introduces the formal definition of a function of Two Variables - Advanced calculus multivariable delta-epsilon proofs times,. Our terms of use | privacy policy and cookie policy based on black. 2009 ; Tags epsilondelta limit multivariable proof ; Home compiler '' first used multivariable proofs. Free Wolfram Player or other Wolfram Language products limits much more easily to other answers claims a... Has radius \ ( \delta\ ) can not prove that they are countinous contact information may be shared with author. Expression is an algebraically precise formulation of evaluating the limit of is 10 as present!: March 7 2011 let 's consider a few informal ways of describing a limit open disk in the,! For showing that multivariable limits by the epsilon delta '' definition of a broken almost. Figure 12.9: Illustrating the definition of the epsilon–delta definition of limit is algebraically! Application processes before receiving an offer ; Start date Feb 26, 2015 ; Tags epsilondelta limit Home! Very good intuition for how \\epsilon relates to \\delta use | privacy policy and policy! Limits exist using the found $\delta$ epsilondelta limit ; Home that is! $5r^3+r^4\leq 5r^4+r^4=6r^4$ good intuition for how \\epsilon relates to \\delta a recognizable term and as deserves! Inc ; user contributions licensed under cc BY-NC-SA epsilon-delta definition: //demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/ Wolfram Demonstrations Project & Contributors | of! Can ask you to prove multivariable limits exist using the rigorous epsilon-delta definition \delta δ definition of is... On finding a particular delta that would finish the proof of this limit whole ] before. In our preliminary work above, but based on countinuity of Everything to break grapples level professionals! Delta-Epsilon definition of a limit formal definition of limit '' is a question and answer site for people studying at! < \frac { \epsilon } { 6 } $in this case 2016 ; Tags calculus epsilondelta multivariable... Delta epsilon definition the longest German and Turkish words really single words professor discourage all collaboration Player or Wolfram., copy and paste this URL into Your RSS reader on the TV! Down on a Cessna 172, 2009 ; Tags epsilondelta limit ; Home http: //demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/ Demonstrations! Mathematics Stack Exchange L. epsilon-delta definition of a function \epsilon_2 > 0$ TECHNOLOGIES © Wolfram Demonstrations Project Contributors... Horizontal planes represent the bounds on and the epsilon-delta approach has some draw backs policy. $\delta > 0$ \epsilon } { 6 } $in this case \\epsilon relates \\delta... Sell a franchise to someone solely based on countinuity value other than zero$ 5r^3+r^4\leq 5r^4+r^4=6r^4 $http... Precise formulation of the limit of a function are generally harder than their single variable calculus to multivariable Functions permalink... A form of the single-variable function approaches as approaches the value for delta that we get! Showing that multivariable limits it 's [ whole ] intuition for how relates...: //demonstrations.wolfram.com/MultivariableEpsilonDeltaLimitDefinitions/ Wolfram Demonstrations Project & Contributors | terms of service, privacy policy cookie! Personal experience compiler '' first used a propblem for showing that multivariable limits \sin^2 ( \theta ) |\leq$! Using epsilon-delta definition of a multivariable function, consider the two-variable function limit of a limit copy and paste URL... Telekinetic feat from Tasha 's Cauldron of Everything to break grapples going to … delta epsilon but i said leads... Limit '' is a formulation of evaluating the limit does n't exist using the epsilon-delta. Finish epsilon delta definition of limit multivariable proof of this limit for multi variable function mobile and cloud with the free Wolfram Player other! Use | privacy policy and cookie policy than zero then i can not prove that are! In the figure, the inequalities state that for all except within of Notebook Emebedder for the user... Remember, here is a proof that the limit does n't exist using \epsilon! Details harrowing New Year 's Eve run epsilon-delta for multivariable limits exist using the delta epsilon but i do know... Good pickups in a bad guitar worth it the recommended user experience | < \epsilon \$ i 'm making... Published: March 7 2011 March 7 2011 on writing great answers delta is always defined, as \epsilon_2! Say it has higher efficiency show the following limits exist using the delta-epsilon definition of the epsilon–delta of! ] to [ prefix ] it, [ infix ] it 's [ ]. That this is the basis of all of the limit of is 10 as of describing a limit: expression... Or responding to other answers single variable counterpart opinion ; back them up with references or personal experience of Functions. By Wolfram TECHNOLOGIES © Wolfram Demonstrations Project & Contributors | terms of |! - definition of a limit, mobile and cloud with the author of any specific for! This epsilon delta definition of limit multivariable into Your RSS reader limit ¶ permalink ( March 2011 ) open content under...
|
2021-04-17 17:39:01
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 3, "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": 0.8435915112495422, "perplexity": 1123.9720902214601}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038461619.53/warc/CC-MAIN-20210417162353-20210417192353-00232.warc.gz"}
|
https://www.physicsforums.com/threads/algebra-1-question.80588/
|
# Algebra 1 question
#### staples82
If 3x+4y=7 and 2y=6x+6, then what is xy?
3X+4y=7
-6X+2y=6
(2)3X+4y=7 which goes to 6X+8Y=14
I cross the X out, subtract and get 6y=8
then I got 4/3
and final answer i got was 20/3
HOWEVER, the answer says is is -2/3 ... I don't understand how that is the answer...
Related Introductory Physics Homework Help News on Phys.org
Homework Helper
Daniel.
#### staples82
i don't understand...
#### dextercioby
Homework Helper
$$\left\{\begin{array}{c} 6x+8y=14 \\ -6x+2y=6 \end{array} \right$$
As i said before,add the two equations,so you'd eliminate "x".
Daniel.
#### staples82
yeah, then i got 6y=8
Homework Helper
Daniel.
#### staples82
oh, my bad ....2, so do I plug it back in, sorry I am going into 7th grade, learning this from a book.
update: ok I understand what you said, now I have y=2...
Last edited:
#### Jameson
Yes, you are correct that y=2. Now as you said plug it back in to find the unknown x.
$$3x + 4y = 7$$
You know what to do.
Jameson
#### staples82
thank you, dextercioby and Jameson, I understand the problem now.
### Physics Forums Values
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
|
2019-10-16 15:17:43
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.5623359084129333, "perplexity": 10460.021842763714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986668994.39/warc/CC-MAIN-20191016135759-20191016163259-00329.warc.gz"}
|
http://mathhelpforum.com/calculus/137896-integrals-problem-2-a.html
|
1. ## Integrals problem 2
Let function f be continuous and non-negative in [a,b], and which holds:
int{[a,b] f dx} = 0.
Prove that: f(x)=0 for all x in [a,b].
2. Originally Posted by Also sprach Zarathustra
Let function f be continuous and non-negative in [a,b], and which holds:
int{[a,b] f dx} = 0.
Prove that: f(x)=0 for all x in [a,b].
Well this is intuitively clear, and a hand-wavy approach would be to say that if f(x) were to rise above 0 at any point, it would make the integral greater than zero in that local region, and since f(x) is non-negative there's no way to balance out the overall integral to zero, therefore f(x) = 0 in [a,b].
A more rigorous approach might involve the intermediate value theorem.. not sure, I'd have to think about it more.
|
2017-07-28 19:37:16
|
{"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": 0.8512648940086365, "perplexity": 775.334379804863}, "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/1500550977093.96/warc/CC-MAIN-20170728183650-20170728203650-00628.warc.gz"}
|
https://www.mysciencework.com/publication/show/chemical-composition-galactic-regions-m8-m17-revision-based-deep-vlt-echelle-spectrophotometry-2b652da1
|
# The chemical composition of the Galactic regions M8 and M17. A revision based on deep VLT echelle spectrophotometry
Authors
Type
Preprint
Publication Date
Submission Date
Identifiers
arXiv ID: astro-ph/0610065
Source
arXiv
We present new echelle spectrophotometry of the Galactic H II regions M8 and M17. The data have been taken with the VLT UVES echelle spectrograph in the 3100 to 10400 angstroms range. We have measured the intensities of 375 and 260 emission lines in M8 and M17 respectively, increasing significatively the number of emission lines measured in previous spectrophotometric studies of these nebulae. Most of the detected lines are permitted lines. Electron temperatures and densities have been determined using different diagnostics. We have derived He+, C++, O+ and O++ ionic abundances from pure recombination lines. We have also derived abundances from collisionally excited lines for a large number of ions of different elements. Highly consistent estimations of t2 have been obtained by using different independent indicators, the values are moderate and very similar to those obtained in other Galactic H II regions. We report the detection of deuterium Balmer emission lines, up to D$\epsilon$, in M8 and show that their intensities are consistent with continuum fluorescence as their main excitation mechanism.
|
2018-02-21 19:34:23
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6769833564758301, "perplexity": 3404.0730130647325}, "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/1518891813712.73/warc/CC-MAIN-20180221182824-20180221202824-00450.warc.gz"}
|
https://chat.stackexchange.com/transcript/36/2019/7/20
|
1:59 AM
Can you elaborate on what additional context my question needs?
-1
If you have a three-dimensional, locally Euclidean space such that any path along a coordinate direction is closed (you always eventually come back to your starting point), is this necessarily homeomorphic to the three-torus, i.e. the product of three circles? I think it must be but I'm not cert...
I know it's a bit straightforward for anyone who understands the topic well, but I didn't, so that's why I asked....
The comments answered the question, so clearly it was comprehensible to them.
1 hour later…
4:15 AM
Thanks Martin
4 hours later…
8:07 AM
@MartinSleziak I am going to delete my questions this site is so rude
3 hours later…
10:55 AM
Hi; can anyone help me out with a task?
Let $n \in \mathbb{N}$ and $A = (a_{ij}) \in \mathbb{R}^{n \times n}$ where $a_{ij} := i +j \quad \forall 1 \leq i, j \leq n$.
Find the rank of matrix A.
i already noticed that these matrices are symmetrical
but the rows are not linear dependent by just looking at them
is there some technique that can help me out here?
Thank you in advance :)
I also noticed by trying some matrices, that, except for dimension $1$, that rank is $2$
11:10 AM
And it answers my question about the spectral radius and norm from yesterday
2 hours later…
1:16 PM
@RyanUnger Whiteboy strikes back as Blackboy
When will this end
1:34 PM
uhm, never?
How would you define a graph that moves? (nodes move through an ambient space)
Or would this simply become a vector field
@BalarkaSen did you read the paper on beauty?
with nodes acting as like particles or smeothing
1:57 PM
@BalarkaSen I sent you a big question....
I'm so confused
can I @ someone?
@RyanUnger Where
@skillpatrol No but seems interesting
@RyanUnger Oh this? What do you mean by "regular neighborhood of a triangulation"? I understand regular neighborhood of a simplicial complex.
2:15 PM
@BalarkaSen I said regular neighborhood of the 1-skeleton.
Your first line was missing 1-skeleton, Ryan
OK, let me see.
(removed)
\o Mike
Any two triangulations of R^3 should be isotopic. What is the example where they have non-homeomorphic 1-skeleta?
(removed)
Not isotopic (this is in fact not true), but concordant. Proof: Hauptvermutung holds in <= 3. Pass to a common refinement by barycentric subdivision. These are concordant moves.
2:25 PM
It's not too bad to cook up non-homeomorphic 1-skeleta. For one cubulate R^3 in the obvious way and then decompose each cube into simplices in the obvious way. This will have constant valence. Now just barycentricallt subdivide some simplex a million times
Now the second graph does not have constant valence
It's likely that the statement of course was not that there's an ambient isotopy taking one triangulation to the other
Regular neighborhoods should be homeomorphic still.
You should be able to slide handles inside a ball of radius $n$ and then let $n \to \infty$
The claim is that the regular neighborhoods of any two triangulations are isotopic
Hi chat
(Ambiently)
Hi
2:34 PM
Nonhomeomorphic graphs can easily have isotopic regular neighborhoods, @Ryan. Take Z^3 in R^3 and Z^3 with a barycentrically with a vertex at (1/2, 1/2, 1/2) joined to all the 8 of it's nearby vertices.
By easy handle-calculus they have isotopic regular neighborhoods.
Handle calculus?
Sliding handles
These are handlebodies. You can pick a handle up and move it around
Ok sure
I don’t see how to do this isotopy
Luckily you have time
Are you just supposed to slide that thing off to infinity
You can move the handles so it’s all bunched up at one place
And then just sent it along a line
2:46 PM
@Semiclassical how's it goin?
Pretty well. Did a phone interview for an adjunct Physics instructor job at a nearby liberal arts uni yesterday
Dunno how I did at that
yeah, the suspense must be tough
It’s not without anxiety no
Good luck!
Thanks
2:50 PM
in other news: the raiders have increased their budget for the new las vegas stadium to $1.9 billion; for 65, 000 seats that comes to about$29,000/seat. Only the rich and famous can afford that...
3:22 PM
hey guys, would substituting $(dx)^2$ for $c(dx)^2$ be acceptable?
since they both vanish really fast.
What
in an equation
-1
This kind of related to another question I posted here but it's more general. $|\vec{v_1}|=|\vec{v_2}|$ and I am trying to find $a$ after the decrement in it (after the particles have moved). $\vec{v_1}$ is parallel to the left line, and $\vec{v_2}$ is parallel to the bottom one, the angle $\varp... Choose$c=1$it's not about finding$c$.. I mean, since both vanish to zero, can I consider them to be equivalent for any$c \neq 1$and$0$? 3:48 PM hey I'm having a some trouble with some basic intuition about measure theory, can you help me out please? suppose we have a set$A = {0, 1}$. what would be its$\sigma$-Algebra? is it just its power set,$\mathcal{A} = \sigma(A) = P(A) = {\varnothing, 0, 1, {0, 1}}$? jax broken, I meant$A = \{0, 1\}$and$\mathcal{A} = \sigma(A) = P(A) = \{\varnothing, 0, 1, \{0, 1\} \}$4:00 PM That's one possible sigma algebra A set doesn't come with a canonical sigma-algebra. You can consider the trivial sigma algebra consisting of the empty set and the full set, eg But incidentally these are the only two sigma algebras possible on the two-point set: the trivial sigma algebra, and the power set algebra (or the discrete sigma algebra) hmm yeah that would make sense but the above power set would be the only possible$\sigma$-Algebra for$\mathcal{A} = \sigma(\{A\})$, wouldn't it? The sigma-algebra generated by$\{A\}$is the trivial sigma algebra,$\{\emptyset, \{0, 1\}\}$. It's the smallest sigma-algebra that contains$A$. ah, ok yeah I got it now, thanks 4:16 PM Hello all!! I would like to prove that for all$n\in\Bbb{N}$, $$\frac{(2n)! } {(n!) ^2}$$ is an integer using induction, but I am not able to make a proof about a variable is in a set like$\Bbb{Z}$Base step: for$n=1$we have$2!/(1!)^2=2$, which is an integer, so base step holds For inductive step: $$\frac{(2n)!}{(n!)^2}\in\Bbb{Z}\implies\frac{(2(n+1))!}{((n+1)!)^2}\in\Bbb{Z}$$ Done so far: $$\frac{(2(n+1))!}{((n+1)!)^2}=\frac{2(n+1)(2(n+1)-1)!}{(n+1)^2(n!)^2}=\frac{2(n+1)(2n+1)(2n)!}{(n!)^2(n+1)}=\underbrace{\frac{(2n)!}{(n!)^2}}_{\in\Bbb{Z}}\underbrace{\frac{(n+1)+(3n+1)}{n+1}}_{(*)}$$ I am not able to prove that$(*)$is an integer (which is indeed false, since for$n=2$we have$10/3\notin\Bbb{Z}$:((() 5:06 PM I don't think induction is a good way to approach this (precisely because of the failure of your inductive step) @Thorgott well, I thought I have made a mistake, but now it does not seem... Thank you! 2 hours later… 7:32 PM off topic: Got any movie recommendations? Haha These are actually very good movies Yes, I wasn't joking He has some rankings wrong I think but those are pretty exceptional films Some of his students feel we should make our own such pages but we have been lazy Absolutely that'd be great idea oh, that list is nice at least theones i regoxnize i usually liked 7:55 PM Yeah that list is amazing A little too much Lars von Trier on the list, is my one complaint, I guess Not an actual complaint just that I hate that guy lmao Are you complaining about von Trier as a director or as a person? As a director. I don't know person I mean he is famous for some controversial statements in interviews I see 8:02 PM Like getting banned from Cannes Oh, wow. Didn't know that (even though he was allowed back recently) Klaus Kinski is known for like beating his children and yet still was an exceptional actor I think his pretentiousness comes off apparent from the movies he makes LOL I love Klaus I hear Melancholia is great 8:04 PM Kinski was just plain mad. He was a paranoid schizophrenic. After Aguirre he like went off on a tour where he would blabber nonsense to the crowd pretending he is Jesus Essentially von Trier said in a press conference in Cannes that he can sympathise with Hitler thinking about him in the bunker at the end, which of course wasn't received very well You can find the press conference on youtube if you want to see it by yourself lol i think i pass My bad Kinski sexually abused his children, I don't know if he beat them rip Not exactly what I'd call a great improvement 8:07 PM Herzog threatened to kill him a few times but never did it Nobody would have disagreed lmao Melancholia is good, but that's one of the few things on his list of movies that are The usual strife, Nymphomaniac, Antichrist, Breaking The Waves etc, are not good IMO. Well I've never had those recommended to me so I don't feel a need to go to bat for them @BalarkaSen Can you identify the image of the canonical map$\pi_1(LX) \to H_2(X;\Bbb Z)$sending a loop in the free loop space to the corresponding torus homology class? Sounds scary. There's a split extension$1 \to \pi_2 X \to \pi_1 LX \to \pi_1 X \to 1$by taking the section of constant loops. So it's a semidirect product. The action has to be the usual$\pi_1$-action on higher homotopy, right? Given by composing with the map$S^n \to S^n \vee S^1$"squishing the sphere into a circle" Seems probable. The lollipop Ya good description So I'd expect it comes from the Hurewicz$\pi_2 X \to H_2 X$? 8:16 PM Certainly the restriction to pi_2 does Ya obviously By definition of Hurewicz But what about the rest? If you have a semidirect product how do you say what the abelianization is If$G$is$N$semi$H$, you just add the commutators of elements of$N$, of$H$and of$N$and$H$. Follows from the presentation. 8:18 PM Probably it specializes to H_1(X) + (fixed points in pi_2) So you're looking at the part of$\pi_2 X$on which$\pi_1 X$acts abelian-ly? Sorry not fixed points. Quotient by the action If$X$is a simple space this is the full thing Set gx = x for all g,x I expect Hurewicz always factors through this quotient Wow interesting 8:22 PM But what does the H_1 factor do?? Oh, it's zero along the constant sections isn't it So this is really just the Hurwicz image? Good point, maybe. Those are torii which bound solid torii, intuitively All the "meridians" are nullhomotopic Yeah So image of$[S^2, X] \to H_2(X)$and$[T^2, X] \to H_2(X)$are the same??? In the geometric chain model indeed they are zero because they have small image @BalarkaSen I guess so This seems weird Lol how about T^2 = X lmao Take$X$to be a group so that$\pi_1 LX$is literally abelian (because$LX$has loop multiplication). Then$\pi_1 LX \cong \pi_2 X \oplus \pi_1 X$. This is the case when$X = T^2$. So it's certainly not zero on the second factor This is weird and too hard for me 8:35 PM I don't understand at all It's the Hurewicz image. The proof was correct Also$[T^2, X]$would be larger; it's based at a fixed loop. An element of pi_1(LX) is a map from T^2/(S^1 x *)$[T^2, X]$has like a$\pi_2 X$and a$[S^1 \vee S^1, X] = \pi_1(X) \times \pi_1(X)$in it Yeah For the case X = T^2 we can't overcome the pinch Right, I'm too sleepy. It's clearly$[S^2 \vee S^1, X]$And that makes a lot of sense. You can take$S^2 \to S^2 \vee S^1$and then compose to get an element of$H_2$represented by a sphere. And how many times you lollipop the sphere around the circle doesn't matter because it gets killed in homology since it's like a map Torus -> S^1 -> X and that bounds a map Solid Torus -> X Maybe I'm belaboring/blabbering the point 8:44 PM Anyway it's now clear these all come from Hurwicz image$[T^2, X]$has something smaller than$\pi_1(X)^2$in it. It's got all the commuting pairs of elements of$\pi_1(X)^2$I can't quite figure out what else it exactly has Intuitively feels like it's in bijection with$\{(a, b) \in \pi_1(X)^2 : [a, b] = 1\} \times \pi_2(X)$OK, the disk gives a nullhomotopy of$[a, b]$. So we're counting nullhomotopies of$[a, b]$upto homotopy. Nullhomotopies of loops in$X$upto homotopy surely corresponds to$\pi_2(X)$2 hours later… 11:00 PM Are there any values of$\theta$where$\theta$,$cos(\theta)$, and$sin(\theta)$are all rational and nonzero? ...or,$\theta$is a rational multiple of$\pi$? In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengths a, b, c, with c the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point (a/c, b/c), which, in the complex plane, is just a/c + ib/c, where i is the imaginary unit. Conversely, if (x, y) is a rational point... I skimmed that already but didn't see how to find points where$\theta$is also rational (or$\theta pi\$ is rational, in radians).
whats cos of pi/4?
ah nevermind, square root of 2 is not rational
yup
|
2019-08-21 14:48:28
|
{"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": 0.931603193283081, "perplexity": 3594.7194399516684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316021.66/warc/CC-MAIN-20190821131745-20190821153745-00197.warc.gz"}
|
https://pymbs.readthedocs.io/en/latest/reference/Loops.html
|
# Kinematic Loops¶
## Loops¶
class PyMbs.Input.MbsSystem.AddLoop(world)
Class that provides functions to create loops
CrankSlider(CS1, CS2, name=None)
Close a loop which we call a CrankSlider. It is a planar mechanism consisting of a series of three revolute and one translational joints. Its best example is the piston in an engine.
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
ExpJoint(j, exp, name=None)
The ExpJoint allows the user to provide an expression for the joint coordinate. Currently, only kinematic calculations is supported.
Parameters: j (Joint-Object.) – joint. exp (Expression.) – Expression for joint coordinate name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
FourBar(CS1, CS2, posture=1, name=None)
Handles a classic planar four bar linkage mechanism comprising four revolute joints. The posture parameter specifies which solution shall be used (crossing beams or not).
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. posture (int - either 1 or -1) – Specifying the solution which shall be used - most time it means, if beams are crossed name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
FourBarTrans(CS1, CS2, posture=1, name=None)
The FourBarTrans is an extension to the FourBarLinkage. Whereas the classical four bar linkage consists of only four revolute joints, the FourBarTrans is extended by a translational joint. This mechanism is often found in wheel loaders.
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
Hexapod(CS1, CS2, name=None)
Close a loop which we call a ThreeBarTrans. It is a planar mechanism consisting of a series of three revolute and one translational joints. It often occurs in conjunction with hydraulic cylinders.
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
Hexapod_m_AV(CS1, CS2, name=None)
Close a loop which we call a ThreeBarTrans. It is a planar mechanism consisting of a series of three revolute and one translational joints. It often occurs in conjunction with hydraulic cylinders.
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
Steering(CS1, CS2, setUpH=1, setUpS=1, name=None)
Close a loop which we call Steering. More detailed explanation follows.
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. setUp – Specifying the solution which shall be used - most time it means, if beams are crossed name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
ThreeBarTrans(CS1, CS2, name=None)
Close a loop which we call a ThreeBarTrans. It is a planar mechanism consisting of a series of three revolute and one translational joints. It often occurs in conjunction with hydraulic cylinders.
Parameters: CS1 (Coordinate System, Body or MbsSystem.) – Reference to parent coordinate system / parent frame. CS2 (Coordinate System, Body or MbsSystem.) – Reference to child coordinate system / child frame. name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
Transmission(j1, j2, ratio=1, name=None)
The transmission loop introduces a relation between two joints, such that their joint coordinates q1 and q2 are related by the following equation: j1 = ratio * v2; This can either be used to synchronise joints or to create a fixed joint by choosing ratio =0.
Parameters: j1 (Joint-Object.) – First joint. j2 (Joint-Object.) – Second Joint. ratio (int/float.) – Ratio between two joints. name (string) – A name may be assigned to each loop. If no name is given, then a name like loop_1 is generated automatically. The name is used for code generation only, i.e. the symbols connected with this force will contain the name.
|
2021-06-20 13:18:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34464314579963684, "perplexity": 2943.079463027723}, "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/1623487662882.61/warc/CC-MAIN-20210620114611-20210620144611-00568.warc.gz"}
|
https://www.mysciencework.com/publication/show/analysis-microtomographic-images-automatic-defect-localization-detection-180f2b59?search=1
|
# Analysis of microtomographic images in automatic defect localization and detection
Authors
• 1 University of Silesia, ul. Bedzinska 39, Sosnowiec, 41-200, Poland , Sosnowiec (Poland)
Type
Published Article
Journal
Machine Vision and Applications
Publisher
Springer-Verlag
Publication Date
May 27, 2020
Volume
31
Issue
5
Identifiers
DOI: 10.1007/s00138-020-01084-3
Source
Springer Nature
Keywords
The paper presents a fast method of fully automatic localization and classification of defects in aluminium castings based on computed microtomography images. In the light of current research and based on available publications, where such analysis is made on the basis of images obtained from standard radiography (x-ray), this is a new approach which uses microtomographic images (μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}-CT). In addition, the above-mentioned solutions most often analyze a pre-separated portion of an image, which requires the initial operator interference. The authors’ own pre-processing methods, which allow to separate the element area and potential defect areas from μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}-CT images, and methods of extraction of selected features describing these areas have been proposed in the solution discussed here. A neural network trained using the Levenberg–Marquardt method with error backpropagation has been used as a classifier. The optimal network structure 20–4–1 and a set of 20 features describing the analysed areas have been determined as a result of performed tests. The applied solutions have provided 89% correct detection for any defect size and 96.73% for large defects, which is comparable to the results obtained from methods using x-ray images. This has confirmed that it is possible to use μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document}-CT images in automatic defect localization in 3D. Thanks to this method, quantitative analysis of aluminium castings can be carried out without user interaction and fully automated.
|
2021-01-23 05:21:17
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7152433395385742, "perplexity": 4697.395478523303}, "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-20210123032629-20210123062629-00701.warc.gz"}
|
https://scicomp.stackexchange.com/questions/34502/numerical-errors-due-to-terms-of-the-form-frac1r-r-goes-to-0-at-the-boun
|
# Numerical errors due to terms of the form $\frac{1}{r}$ (r goes to 0 at the boundary) while using finite difference method
I am trying to solve a system of differential equations using finite difference method.
There are few terms of the form $$\frac{A(r)}{r}$$, both $$A(r)$$ and r go to zero at the boundary. Analytically this term is well defined but numerically these terms are leading to error which very quickly break the simulation.
I am using leapfrog method if that matters. I read somewhere that leapfrog method has dissipation inbuilt so it should be stable, but it's clearly not the case.
Any advice on how to proceed? Should I change my algorithm? Or use higher order schemes?
• sciencedirect.com/science/article/pii/S0021999199963829 – Spencer Bryngelson Feb 25 at 6:51
• The basic advice is to consider the new variable $v(r) = A(r)/r$. One does that in the solution of the Schrödinger equation for atomic systems, for example. – davidhigh Feb 29 at 21:16
• Do you know the limit $$\lim_{r \to 0} \frac{A(r)} {r}$$, it's pretty important in this case. If the limit is a finite constant, then there is a lot of methods like in the paper linked by Spencer. If the limit is infinity, then there's a problem – Yuriy S Mar 3 at 21:01
|
2020-08-13 18:04:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6752673983573914, "perplexity": 295.0776102290752}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739048.46/warc/CC-MAIN-20200813161908-20200813191908-00406.warc.gz"}
|
https://www.groundai.com/project/conformal-scattering-of-the-maxwell-scalar-field-system-on-de-sitter-space/
|
Conformal scattering of the Maxwell-scalar field system on de Sitter space
# Conformal scattering of the Maxwell-scalar field system on de Sitter space
Mathematical Institute
Oxford University
Oxford OX2 6GG, UK
###### Abstract
We prove small data energy estimates of all orders of differentiability between past null infinity and future null infinity of de Sitter space for the conformally invariant Maxwell-scalar field system and construct bounded invertible nonlinear scattering operators taking past asymptotic data to future asymptotic data. We also deduce exponential decay rates for solutions with data having at least two derivatives. The construction involves a carefully chosen complete gauge fixing condition which allows us to control all components of the Maxwell potential, and a nonlinear Grönwall inequality for higher order estimates.
## 1 Introduction
Studies of scattering go back to the beginnings of physics. A famous modern mathematical treatment was developed in the 1960s by Lax and Phillips [31, 32], who succeeded in using functional analytic techniques to study scattering by an obstacle in flat space. In general relativity it is of interest to study metric scattering, that is the effects of curved space on the asymptotic behavior of fields. Around the same time as Lax and Phillips were developing their framework, Roger Penrose discovered a way to compactify certain spacetimes by conformally rescaling the metric and attaching a boundary, [41, 42]. He called the class of spacetimes admitting such a compactification asymptotically simple and the boundary so attached null infinity, for this was where all null geodesics ended up ‘at infinity’. This led to a brand new way of viewing the asymptotics of massless fields in general relativity: one works in Penrose’s conformally compactified spacetime and studies the regularity of fields on , and then translates the regularity in the conformally rescaled spacetime to fall-off conditions in the physical spacetime.
It was not until the work of Friedlander [19] in 1980 (see also the posthumously published work [20]), however, that it was understood that the approaches of Lax and Phillips on the one hand and Penrose on the other could be combined. Friedlander showed that, although one cannot perform the same analytically explicit constructions in curved space, one can make sense of the Lax–Phillips asymptotic profiles of fields by identifying them with suitably rescaled limits of fields going to infinity along null directions. These became known as Friedlander’s radiation fields. The ideas of such conformal scattering were taken up by Baez, Segal and Zhou [6, 7, 8, 9] to study a nonlinear wave equation and to some extent Yang–Mills equations on flat space, and later by Mason and Nicolas [33, 34] to study linear equations on a large class of asymptotically simple spacetimes constructed by Corvino, Schoen, Chruściel, Delay, Klainerman, Nicolò, Friedrich and others [12, 13, 15, 16, 28, 29]. This spurred a programme of constructing conformal scattering theories for various fields on a variety of backgrounds and since then a number of works have appeared, many focussing on conformal scattering on black hole spacetimes222See also [26, 51] for some results in interiors of black holes.[23, 25, 37, 39, 40]. It should be mentioned that there have been plenty of works studying relativistic scattering theory without employing the conformal method, notably by Dimock and Kay in the 1980s [17, 18] and later by Bachelot [3, 4] and collaborators Nicolas, Häfner, Daudé, and Melnyk, among many others, a programme which eventually led to rigorous proofs of the Hawking effect [5, 35].
The above programmes were concerned mainly with asymptotically flat spacetimes. However, astronomical observations indicate that the cosmological constant in our universe, though tiny, is positive [44, 45, 47, 49]. It is thus of interest to study scattering, especially of nonlinear fields, on de Sitter space. De Sitter space is the Lorentzian analogue of the sphere in Euclidean geometry and one of the three archetypal spacetimes as classified by the sign of the cosmological constant, with flat Euclidean space corresponding to Minkowski space () and hyperbolic space corresponding to anti-de Sitter space (). As such, de Sitter space differs from Minkowski space in several crucial aspects. Firstly, it is not asymptotically flat. Nonetheless, it is asymptotically simple in the sense of Penrose [42] and so admits a conformal compactification. Secondly, the positive cosmological constant, no matter how small, renders null infinity spacelike in de Sitter space, which has implications for conformal scattering. In the asymptotically flat case the constructions of Mason and Nicolas required the resolution of a global linear Goursat problem, which had been shown by Hörmander [24] to be solvable in some generality. In de Sitter space, however, a spacelike means that the construction of a scattering theory instead requires the resolution of a regular Cauchy problem. Thirdly, while obtaining flat space scattering and peeling results through conformal techniques is fine for linear fields, nonlinear fields generically possess so-called charges at spacelike infinity [46, 1, 14]. This is a major obstruction to constructing conformal scattering theories for nonlinear fields in flat space and is related to infrared divergences in quantum field theory [30, 38]. The problem is entirely absent in de Sitter space as it is spatially compact.
Finally, from a more physical perspective, de Sitter space has the peculiar feature that no single observer can ever observe the entire spacetime, in contrast to the Minkowski case where an observer’s past lightcone eventually contains the whole history of the universe. This is related to the existence of cosmological horizons, null hypersurfaces criss-crossing the Penrose diagram of de Sitter space. Their existence has implications for the definition of a classical scattering matrix: the construction of one requires a timelike Killing or conformally Killing vector field, and here one has a choice in de Sitter space. One might wish to use the Killing field provided by the standard static coordinates, i.e. the coordinates an observer at the south pole in de Sitter space might use for themselves, but this is problematic as it fails to be timelike and future pointing beyond the cosmological horizons. Another approach is to conformally compactify de Sitter space and embed it in the Einstein cylinder, where one has a natural globally timelike Killing field which becomes conformally Killing in physical de Sitter space. This can then be used to define an observer-oblivious classical scattering matrix in de Sitter space. We adopt the latter approach here. The importance of the construction of such scattering matrices for quantum gravity in de Sitter is explained well in [50] and the references therein.
From an analytic point of view, it has been known since the work of Friedrich [21] that de Sitter space is a stable solution of Einstein’s equations with a positive cosmological constant, so one expects scattering results on de Sitter space to fit into a larger host of stories on asymptotically de Sitter spacetimes. Results in this vein have been obtained by, for example, Vasy, Melrose and Sá Barreto, [52, 36].
This paper is organized as follows. In Section 3 we state the conventions and notation used in the paper, and in Section 4 we introduce the conformally invariant Maxwell-scalar field system that we subsequently study. In Section 5 we describe de Sitter space , a number of standard coordinate systems on , its conformal compactification, and our choice of energy-momentum tensor for the Maxwell-scalar field system on the conformally rescaled spacetime. In Section 6 we state the main results in detail. Sections 8 and 7 contain a detailed derivation of the required gauge fixing conditions, the formulation of the Cauchy problem for our system, and an existence theorem. Sections 11, 10 and 9 contain the inductive energy estimates on which our results rest. Sections 14, 13 and 12 finish off the proofs of the main results.
Acknowledgments. The author wishes to thank Lionel Mason, Qian Wang, Jan Sbierski, Jean-Philippe Nicolas and Mihalis Dafermos for useful guidance and discussions.
## 2 Results
We prove small data energy estimates of all orders of differentiability between and of de Sitter space for the conformally invariant Maxwell-scalar field system and show the existence of small data scattering operators for all . Slightly more precisely, we may state the main theorem as follows. The full statements of the main theorems can be found in Section 6. Consider the Penrose diagram of de Sitter space and an initial surface ,
###### Theorem.
For any there exist bounded invertible forward and backward wave operators mapping small Maxwell-scalar field data on to small Maxwell-scalar field data on , and a bounded invertible scattering operator
Sm=W+m∘(W−m)−1
mapping small Maxwell-scalar field data on to small Maxwell-scalar field data on .
As a corollary, our estimates imply exponential decay rates for the Maxwell-scalar field system on de Sitter space with small initial data. The decay rates are a partial extension of the results of Melrose, Sá Barreto and Vasy [36].
###### Corollary.
The scalar field and the components of the Maxwell potential decay exponentially in proper time along timelike geodesics approaching .
The asymptotic behaviour of solutions to the Einstein–Maxwell–Yang–Mills equations has previously been studied by Friedrich [22] by employing the machinery of symmetric hyperbolic systems. The estimates we prove here are finer and explicit, allowing us to define the sets of scattering data and read off precise decay rates.
Since the nonlinearities are of the same order, in principle there is no obstruction to extending our estimates to the Yang–Mills–Higgs system on de Sitter space. As a result, the same scattering and decay results should apply there.
## 3 Conventions
We use the spacetime signature . Our main estimates will be performed on the Einstein cylinder with metric , where is the standard positive-definite metric on . We will use Penrose’s abstract index notation and use the Roman indices to refer to tensors on and contractions with respect to the full spacetime metric (or sometimes a general spacetime with metric ), and use the Greek indices to refer to tensors on and contractions with respect to the metric . At a certain point we will also use the indices , and to refer to a basis of vector fields on , but this will be made explicit at the time. We will use to denote the Levi–Civita connection of the full spacetime metric (or a general metric ), and to denote the Levi–Civita connection of . Thus, as , we shall have . We will use to denote the volume form of the full spacetime metric ( or ), and to denote the volume form of . In the case of we will thus have , being the coordinate on . For a -form on we will use to denote the projection of onto , to denote the component of along , and dot (as in ) to denote differentiation with respect to . The Lebesgue and Sobolev norms and of a scalar or vector will refer to and , unless specifically stated otherwise. Occasionally we shall use the symbol to denote equality on null infinity (see Section 5).
We will also adopt Penrose’s sign convention for the curvature tensors, meaning that the Riemann curvature tensor will satisfy
[∇a,∇b]Xc=−RccdabXd.
The Ricci tensor and the scalar curvature are then defined as usual,
Rab\vbox\scriptsize.\scriptsize.=Rccacb,R\vbox\scriptsize.\scriptsize.=Raaa,
so that in these conventions the scalar curvature of, for example, a -sphere with the positive-definite metric is negative, to be exact. However, since our metrics will be of signature , that will mean that a spacelike -sphere in our construction will have positive scalar curvature equal to .
## 4 The Conformally Invariant Maxwell-Scalar Field System
Let be a -dimensional Lorentzian manifold and consider the Lagrangian density
L=−14FabFab+12Daϕ¯¯¯¯¯¯¯¯¯¯Daϕ−112R|ϕ|2, (4.1)
where is a real -form called the Maxwell field, is a real -form called the Maxwell potential, is a complex scalar field on , is the scalar curvature of , and , where is the Levi–Civita connection of . The differential operator is called the gauge covariant derivative. The Euler–Lagrange equations associated to (4.1) are
The Maxwell-scalar field system (4.1) is the simplest classical field theory exhibiting a non-trivial gauge dependence. Indeed, the -form is not uniquely determined by the -form , and any transformation of the form
Aa⟼Aa+∇aχ
leaves unchanged. This transforms
Daϕ=∇aϕ+iAaϕ⟼∇aϕ+i(Aa+∇aχ)ϕ=e−iχDa(eiχϕ),
so that if one makes the corresponding transformation
ϕ⟼e−iχϕ,
the Lagrangian (4.1), and thus also the field equations (4.2), remain unchanged.
###### Remark 4.1.
The gauge covariant derivative acting on is a connection on a principal bundle over with fibre . This connection is represented by the real -form on in any trivialisation of , where the factor of in comes from . The scalar field is a section of a complex line bundle over associated to by the representation of .
Consider a conformal rescaling of ,
^gab=Ω2gab. (4.3)
It turns out that in many cases it is possible to fully or partially compactify by choosing the conformal factor so that it compensates for the divergence of distances with respect to the physical metric and attach the boundary to ; this is Roger Penrose’s notion of asymptotically simple spacetimes first described around 1963 in [41] and [42]. For our purposes it will be sufficient to assume that the spacetime is regular enough so that it may be compactified in this way to make a smooth compact manifold with boundary, , although weaker, partial compactifications leaving singularities at a finite number of points in the boundary are widely used to study, for example, black hole spacetimes [25, 33, 34, 37, 39, 40]. We equip with the rescaled (also called unphysical) metric and call the spacetime the rescaled spacetime.
It is possible to transport the fields into the rescaled spacetime by weighting them appropriately by the conformal factor so that the field equations (4.2) are preserved in . The correct choice of conformal weights for are ,
^Aa\vbox\scriptsize.\scriptsize.=Aa,^ϕ\vbox\scriptsize.\scriptsize.=Ω−1ϕ,
and we show below that this implies the conformal invariance of the Maxwell-scalar field system (4.2). Under the rescaling (4.3) the Christoffel symbols of transform as
where , and using this one calculates that
−14FabFab=−14Ω4^Fab^Fab
and
12Daϕ¯¯¯¯¯¯¯¯¯¯Daϕ=12Ω4^Da^ϕ¯¯¯¯¯¯¯¯¯¯^Da^ϕ+12Ω4(2ΥaRe(^ϕ¯¯¯¯¯¯¯¯¯¯^Da^ϕ)+^gabΥaΥb|^ϕ|2).
Moreover, because in dimensions the scalar curvature transforms as (see [43], eq. (6.8.25))
112R=Ω2(112^R−12^∇aΥa+12^gabΥaΥb),
one has
−112R|ϕ|2=−112Ω4^R|^ϕ|2+12Ω4(^∇aΥa−^gabΥaΥb)|^ϕ|2.
Adding these together one sees that the Lagrangian transforms as
L =Ω4^L+12Ω4(2ΥaRe(^ϕ¯¯¯¯¯¯¯¯¯¯^Da^ϕ)+(^∇aΥa)|^ϕ|2) =Ω4^L+12Ω4(Υa^∇a(|^ϕ|2)+(^∇aΥa)|^ϕ|2) =Ω4^L+12Ω4^∇a(|^ϕ|2Υa).
Now the volume form of is related to the volume form of by , so the action
S=∫\mathcalboondoxMLdv
transforms as
S=^S+12∫^\mathcalboondoxM^∇a(|^ϕ|2Υa)ˆdv=^S+12∫I|^ϕ|2Υa^gab∂b\intprodˆdv. (4.4)
In other words, is conformally invariant up to a boundary term. Since the Euler-Lagrange equations arise from a local variation of the action, this implies the conformal invariance of the field equations (4.2).
## 5 De Sitter Space
### 5.1 Global Coordinates and Conformal Compactification
The -dimensional de Sitter space is defined to be the hyperboloid
|x|2−x20=1H2
in -dimensional Minkowski space
m=dx20−d|x|2−|x|2s3,
where and is the standard metric on the -sphere . If we set
x0=1Hsinh(Hη),|x|=1Hcosh(Hη),
so that is a coordinate on , the metric descends to the metric on ,
ds2=dη2−1H2cosh2(Hη)s3. (5.1)
This provides a global coordinate system on and is known as the closed slicing of de Sitter space. Note that the topology is manifest in these coordinates. The metric (5.1) can be visualized as a compact spacelike slice expanding in time , as depicted in fig. 2.
To conformally compactify , however, we need a further change of coordinates
tan(τ2)=tanh(Hη2).
In terms of the metric becomes
ds2=1H2cos2τ(dτ2−s3), (5.2)
where . This makes it obvious as to what should be taken as the conformal factor to compactify , namely
Ω=Hcosτ,
and we define
d^s2\vbox\scriptsize.\scriptsize.=Ω2ds2=dτ2−s3=\vbox\scriptsize.\scriptsize.e. (5.3)
In this conformal scale the hypersurfaces are regular, in contrast to the physical metric (5.2). In fact, the metric clearly extends smoothly for all , so one may consider the extended spacetime known as the Einstein cylinder. We thus identify compactified de Sitter space with the subset of the Einstein cylinder by attaching to (5.2) the boundary . This boundary is the union of two disjoint smooth surfaces
I+={τ=π2}andI−={τ=−π2},
which we call future null infinity and past null infinity respectively. Note that are spacelike hypersurfaces of ; the name null infinity derives from the fact that is where all future (past) pointing null geodesics in de Sitter space end up at infinity. Note also that the vector field is a timelike Killing field in , and in particular it is automatically uniformly timelike since is spatially compact.
As a result, provides a uniformly spacelike foliation of by the level surfaces of the coordinate given explicitly by . Our energies will be defined with respect to .
###### Remark 5.1.
The fact that is spacelike is, of course, a consequence of the fact that is a solution to Einstein’s equations with a positive cosmological constant ,
Rab=λgab.
Indeed, in general the norm squared on of the normal to is
(∇aΩ)(∇aΩ)\stackon[1.5pt]=\stretchto\scalerel∗[\widthof=]∧0.5ex13λ.
In the case of , so that . Note that corresponds to the Hubble constant in vacuum.
Writing the -sphere metric as for and quotienting by the symmetry group of we obtain the Penrose diagram for ,
The coordinate varies from to going from left to right, with the vertical lines and representing the North Pole and the South Pole respectively. The coordinate varies from to going up, with the horizontal lines and representing past and future null infinities , as remarked earlier. The dashed lines are the past and future horizons for an observer at the South Pole: a classical observer sitting at can never observe the region , and can never send a signal to the region . Thus region I is the region of communications for an observer at the South Pole, while region III is completely inaccessible.
### 5.2 Static Coordinates
A set of physical space coordinates on that exhibit an explicit future-pointing timelike Killing field in the region may be constructed by defining
r=sinζHcosτ,tanh(Ht)=sinτcosζ
for and . Then the unrescaled metric takes the form
ds2=F(r)dt2−F(r)−1dr2−r2s2, (5.4)
where . In these coordinates the cosmological horizons represented by the dashed lines in fig. 4 are given by , are given by , the North and South Poles are at , and the four corners of the Penrose diagram are at . The vector field is manifestly a timelike Killing vector in the region , but becomes null on the cosmological horizon . It is future-pointing in the region , past-pointing in the region , and spacelike in the regions and . The arrows in fig. 5 represent the directions of the flow of .
### 5.3 Choice of Energy-Momentum Tensor on E
From now on we denote by and the scalar field and Maxwell potential on the Einstein cylinder , and by and the conformally related physical fields on de Sitter space ,
ϕ=Ω−1~ϕ,Aa=~Aa, (5.5)
where .
We define the energy-momentum tensor for the system (4.2) on to be
Tab[ϕ,A]\vbox\scriptsize.\scriptsize.=−FacFbcb+14eabFcdFcd+¯¯¯¯¯¯¯¯¯¯¯D(aϕDb)ϕ−12eab¯¯¯¯¯¯¯¯¯DcϕDcϕ+12eab|ϕ|2\vbox\scriptsize.\scriptsize.=Tab[A]+Tab[ϕ]. (5.6)
One can check by direct calculation that, as a consequence of the field equations (4.2), is conserved,
∇aTab=0,
so is suitable for defining a conserved energy for the system (4.2),
Eτ[ϕ,A]\vbox\scriptsize.\scriptsize.% =∫S3τT00[ϕ,A]dvs3=∫S3τTab[ϕ,A]TaTbdvs3. (5.7)
Since is Killing on , this clearly satisfies
ddτEτ[ϕ,A]=0
if the field equations (4.2) are satisfied. We call (5.7) the geometric energy for the system (4.2). We also define the geometric energies for the individual sectors of the scalar field and the Maxwell potential ,
Eτ[ϕ]\vbox\scriptsize.\scriptsize.=∫S3τT00[ϕ]dvs3,Eτ[A]\vbox\scriptsize.% \scriptsize.=∫S3τT00[A]dvs3.
The sectorial geometric energies and are not conserved individually and can exchange energy throughout the evolution, but of course the total geometric energy is.
For we also define the Sobolev-type approximate energies
Sm[ϕ]\vbox\scriptsize.% \scriptsize.=∥˙ϕ∥2Hm−1+∥ϕ∥2Hm, Sm[A]\vbox\scriptsize.% \scriptsize.=Sm[A]+Sm[A0], Sm[A]\vbox\scriptsize.%.=∥˙A∥2Hm−1+∥A∥2Hm, Sm[ϕ,A]\vbox\scriptsize.\scriptsize.=Sm[ϕ]+Sm[A], Sm[A0]\vbox\scriptsize.% \scriptsize.=∥A0∥2Hm, Sm[ϕ,A]\vbox\scriptsize.% \scriptsize.=Sm[ϕ,A]+Sm[A0],
where . Furthermore, for brevity we will often simply write to mean .
### 5.4 Scaling of Initial Energies
We will consider initial data on the hypersurface and use the coordinate and the metric on the rescaled spacetime, and the coordinate and the metric (5.1) on the physical spacetime. By differentiating the relationship we find
dτ=Hcosh(Hη)dη,
so raising indices with , where is the metric (5.1), we find that and are related by
∂τ=cosh(Hη)H∂η.
Furthermore, the conformal factor in the global coordinates (5.1) is given by
Ω=Hcosτ=Hcosh(Hη).
Consider the rescaled energies
Sm[ϕ,A](τ)=∥˙ϕ∥2Hm−1(τ)+∥ϕ∥2Hm(τ)+∥˙A∥2Hm−1(τ)+∥A∥2Hm(τ)+∥A0∥2Hm(τ).
On the initial surface the conformal factor is a constant and has vanishing derivative, , so the rescaled scalar field is related to the physical scalar field by
ϕ|τ=0=(Ω−1~ϕ)|τ=0=1H~ϕ|η=0,
while their time derivatives are related by
˙ϕ|τ=0=(Ω−1∂τ~ϕ−(∂τΩ)Ω−2~ϕ)|τ=0=1H2∂η~ϕ|η=0.
Since the conformal factor is independent of the coordinates, , and the metric induced on by (5.1) is equivalent to , the rescaled and physical norms of the scalar field are equivalent,
∥˙ϕ∥2Hm−1(τ=0)+∥ϕ∥2Hm(τ=0)≃∥∂η~ϕ∥2Hm−1(η=0)+∥~ϕ∥2Hm(η=0),
where there is equality if . One similarly checks that
∥˙A∥2Hm−1(τ=0)+∥A∥2Hm(τ=0)≃∥∂η~A∥2Hm−1(η=0)+∥~A∥2Hm(η=0)
and
∥A0∥2Hm(τ=0)≃∥~Aη∥2Hm(η=0),
where , and are coordinates on . Thus
Sm[ϕ,A](τ=0)≃Sm[~ϕ,~A](η=0), (5.8)
and also .
## 6 Main Theorems
###### Definition 6.1.
Let be a Cauchy surface in and consider data for the Maxwell-scalar field system on the corresponding Cauchy surface in . We say the data
(ϕ0,A0,ϕ1,A1,a0)=(ϕ,A,˙ϕ,˙A,A0)|Σ
is admissible if it satisfies the strong Coulomb gauge333See Section 7.1. and solves the elliptic equation
−⧸Δa0+|ϕ0|2a0=−Im(¯ϕ0ϕ1)
on .
###### Theorem 6.2 (Energy Estimates).
Let . For -small admissible data
(ϕ00,A00,ϕ01,A01,A00)=(ϕ,A,˙ϕ,˙A,A0)(τ=0)
on for the Maxwell-scalar field system on in strong Coulomb gauge one has
Sm[ϕ,A](0)≃Sm[ϕ,A](τ)
for all . In particular,
Sm[ϕ,A](I−)≃Sm[ϕ,A](I+),
where is the future (past) null infinity of de Sitter space .
###### Theorem 6.3 (Conformal Scattering).
For let be the subset of of distributions of admissible data on and let be the subset of of distributions of admissible data on , all equipped with the natural norm . Denote by the open ball of radius in , and write and . Then for every there exist , and sets with such that
1. there exist bounded invertible nonlinear operators , called the forward and backward wave operators
W±m:S0m,ε0⟶D±m,ε1⊂S±m,ε1,
such that is the forward (backward) Maxwell-scalar field development of on restricted to , and
2. there exists a bounded invertible nonlinear operator
Sm:D−m,ε1⟶D+m,ε1,
called the scattering operator, given by
Sm=W+m∘(W−m)−1
such that is the Maxwell-scalar field development of on restricted to and
∥u+∥2Sm⩽C1∥u−∥2Sm,∥u−∥2Sm⩽C2∥u+∥2Sm
for some constants .
###### Theorem 6.4 (Decay Rates).
Let and be the physical fields related to the conformally rescaled fields and by (5.5). Suppose is small initially. Then the Maxwell-scalar field development of this initial data decays exponentially in proper time along timelike geodesics in . Explicitly using the global timelike coordinate , one has the estimates
|~ϕ|≲e−H|η|,|~Aη|≲e−H|η|,|~A|s3≲1
as . Furthermore, in the static coordinates (5.4)
|~ϕ|≲re−H|t|,|~At|≲re−H|t|,|~Ar|≲re−H|t|,1r|~A|s2≲re−H|t|
as and is fixed. Moreover, if is small initially then there exists a constant such that
|~ϕ−c~Φ1|≲re−2Ht
as , where is the eigenmode of the linear conformally invariant wave operator on .
## 7 Field Equations and Gauge Fixing
The field equations (4.2) can be written out in terms of the Maxwell potential ,
□Aa−∇a(∇bAb)+RabAb=−Im(¯ϕDaϕ),□ϕ+2iAa∇aϕ+(16R−AaAa+i∇aAa)ϕ=0. (7.1)
We will need to commute derivatives into these equations, so it will be useful to introduce the operators representing their left-hand sides. For any -form and any scalar field we set
The system (7.1) is then equivalent to
(7.2)
In the following sections we specialise to the case of the Einstein cylinder . As noted earlier, for ease of notation we will not hat any rescaled quantities on and instead denote the corresponding physical quantities on with a tilde, as in or . For the metric we compute
### 7.1 Strong Coulomb Gauge
We will work in the Coulomb gauge adapted to the foliation ,
⧸∇⋅A=0, (7.3)
but will also need to use the residual gauge freedom to fix the gauge fully. More precisely, given a solution to the Maxwell-scalar field system (7.1), a general gauge transformation sends and , and (7.3) is imposed by solving the elliptic equation
⧸Δχ=−⧸∇⋅A
on for every fixed . This does not determine uniquely: there is still the residual gauge freedom of , where solves
⧸Δχres.=0
on each . Because is compact, the kernel of the Laplacian is just the vector space of constant functions, i.e. those which satisfy , but the dependence in the is still arbitrary. Thus in the Coulomb gauge we have the residual gauge freedom
ϕ ⟼e−iχres.(τ)ϕ, A0 ⟼A0+˙χres.(τ), A ⟼A,
which allows one to choose
˙χres.(τ)=−1|S3|∫S
|
2020-04-01 22:56:17
|
{"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": 0.9348758459091187, "perplexity": 525.719109332568}, "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/1585370506477.26/warc/CC-MAIN-20200401223807-20200402013807-00175.warc.gz"}
|
https://www.cut-the-knot.org/m/Geometry/ConcurrenceInRightTriangle.shtml
|
# Concurrence in Right Triangle
### Solution 1
We shall make use of Ceva's theorem (or rather its converse): let $CZ\,$ be the angle bisector, $BE\,$ the median, $DH\,$ the altitude. We need to show that
$\displaystyle \frac{CH}{HA}\cdot\frac{AZ}{ZD}\cdot\frac{DE}{EC}=1.$
Since $DE=EC,\,$ remains it to show that $\displaystyle \frac{CH}{HA}=\frac{ZD}{AZ}=\frac{CD}{AC},\,$ by a property of angle bisector. We have
$\displaystyle \frac{AC-HA}{HA}=\frac{CH}{HA}=\frac{CD}{AC}=\frac{BC-BD}{AC}=\frac{BC-AC}{AC},$
or, $\displaystyle \frac{AC}{HA}=\frac{BC}{AC}=\frac{BC}{BD}\,$ which is true by Thales' theorem because $DH\parallel AB.$
### Solution 2
Let $K\,$ be the intersection point of the median $AE\,$ of $\Delta ABD\,$ and the angle bisector of $\angle B.\,$ From the angle bisector property in $\Delta ABD,\,$ $\displaystyle \frac{EK}{KA} = \frac{BE}{BA}\overset{BE= ED, BA=DC}{=}\frac{ED}{DC},\,$ from which, by the converse of Thales' theorem, $\displaystyle KD\parallel AC\overset{AC_\perp AB}{=>} KD\parallel AB,\,$ and the equivalent problem has been proved.
### Solution 3
Let $ABQC\,$ be a rectangle and $ABTD\,$ be a parallelogram. Then obviously $CQTD\,$ is rhombus and because $(T.BSDQ)\,$ is a harmonic system of points, as $TQ\parallel BD\,$ and $BE=ED,\,$ we also have that $(D.BSTQ)\,$ is also a harmonic system of points; therefore $DS\,$ is angle bisector of $\Delta BDT\,$ (because $DQ\,$ is external angle bisector). Thus, due to symmetry, $BZ\,$ is also angle bisector of $\Delta BAD.$
### Solution 4
Let $AE\,$ be the median and $BZ\,$ the angle bisector of $\Delta ABD,\,$ interecting in $K.\,$ We draw $D\Theta.\,$ Its extension cuts $AB\,$ in point $H.$
In $\Delta BEA,\,$ from the Angle Bisector Property, $\displaystyle \frac{E\Theta}{\Theta A} = \frac{BE}{BA}=\frac{a-c}{2c} = \frac{ED}{DC}.$
Thus, by converse of Thales Theorem we take $D\Theta\parallel AC,\,$ therefore $\Delta H\,$ is altitude in $\Delta ADB.$
P.S. Symbols different from those picture:
\displaystyle \begin{align} &\Gamma= C,\\ &\Delta = D,\\ &\gamma = c [= (AB)]\\ &\alpha= a [= (BC)]\\ &\beta = b [= (AC)]. \end{align}
### Solution 5
Let $AZ\,$ be an angle bisector and $DE\,$ the altitude in $\Delta BAD\,$ with $K\,$ their intersection point. $AK\,$ cuts $BD\,$ in point $\Theta.\,$ We shall prove that $\Theta\,$ is the midpoint of $BD.$
Let $KM=KD\,(1).$
Since $ED\parallel AC,\,$ $\displaystyle\frac{EB}{EA}=\frac{BD}{CD}\,(2)\,$
and from $\displaystyle \frac{BD}{BE}=\frac{BC}{BA},\,$ $\displaystyle \frac{BD}{DE}=\frac{BC}{DC}\,(3).\,$
In $\Delta BED,\,$ $AK\,$ is an angle bisector. and from the related property we have $\displaystyle \frac{KD}{KE}=\frac{BD}{BE},\,$ implying (via (1),(3)) that
\displaystyle\begin{align}&\frac{KM}{KE}=\frac{BC}{DC}&\Longrightarrow\\ &\frac{KM-KE}{KE}=\frac{BD-DC}{DC}&\Longrightarrow\\ &\frac{ME}{KE}=\frac{DB}{DC}\,\text{(due to (2))}&\Longrightarrow\\ &\frac{ME}{KE}=\frac{AB}{EA}&\Longrightarrow\\ &MB\parallel AK. \end{align}
Thus, in $\Delta MBD\,$ $K\,$ is the midpoint of side $MD\,$and $K\Theta\parallel MB.\,$ Therefore, $\Theta\,$ is the midpoint if $BD.$
Symbols different from those in the diagram:
\begin{align} \Gamma = C,\\ \Delta = D. \end{align}
### Acknowledgment
The problem has been posted at the Οι Ρομαντικοι της Γεωμετριας (Romantics of Geometry) facebook group. Takis Chronopoulos has generously volunteered to translate the posted proofs into English. Solution 2 is by Stathis Koutras; Vaggelis Stamatiadis and, independently, by Kostas Dortsios posted proofs same as in Solution 1; Solution 3 is by Andreas Varverakis; Solution 4 is by George Rizos; Solution 5 is by George Rodopoulos.
[an error occurred while processing this directive]
|
2018-08-18 03:19:11
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7700179815292358, "perplexity": 967.3261061916676}, "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-2018-34/segments/1534221213264.47/warc/CC-MAIN-20180818021014-20180818041014-00152.warc.gz"}
|
https://www.123calculus.com/en/lah-numbers-page-1-16-220.html
|
# Lah numbers
Use this tool to calculate Lah numbers and find below a table of some Lah numbers.
## Lah numbers formula
Assume that n and k are 2 positive integers such as $$k <= n$$ then the Lah number n, k is defined as :
$$L (n, k) =\dbinom {n-1} {k-1}\dfrac {n!}{k!}$$
$$\dbinom {n} {k}$$ refers to binomial coefficient ('n choose k').
## Lah Numbers Table
n \ k 1 2 3 4 5 6 7 8 9 10
1 1
2 2 1
3 6 6 1
4 24 36 12 1
5 120 240 120 20 1
6 720 1800 1200 300 30 1
7 5040 15120 12600 4200 630 42 1
8 40320 141120 141120 58800 11760 1176 56 1
9 362880 1451520 1693440 846720 211680 28224 2016 72 1
10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1
|
2022-01-20 20:08:57
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.579596221446991, "perplexity": 109.00311613715186}, "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/1642320302622.39/warc/CC-MAIN-20220120190514-20220120220514-00051.warc.gz"}
|