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https://agenda.infn.it/event/13022/?view=event | 1. General Seminars
The shape of (new) physics in the B decay anomalies
by Jorge Martin Camalich (CERN)
Europe/Rome
Aula Bruno Touschek (LNF INFN)
Aula Bruno Touschek
LNF INFN
Description
High-precision measurements of flavour-changing processes are sensitive to the virtual effects of particles at energies beyond the reach of current colliders; thus, any New-Physics addressing the hierarchy problem of the electroweak scale must have a non trivial structure in flavour space to avoid all the stringent flavour constraints. In fact, although no new heavy particles have been identified in the high-energy frontier yet, there are tantalizing tensions with the SM in B-meson decays measured at the LHCb and B factories. The first type of anomalies appear in observables of the FCNC rare b→s ll decays, like in the angular distributions of B→K*μ μ, or in the ratio RK = Γ(B→K μ μ) / Γ(B→K e e) . These are currently in 4σ tension with the SM, putatively corresponding to the tree-level exchange of a neutral particle with mass Λ ~ 10 TeV selectively coupling to muons. The second type of anomalies appear in the CC b→ c τ ν transitions which have been measured through the ratios RD(*) = Γ(B→D(*) τ ν)/ Γ(B→D(*) l ν), where l is the muon or the electron. The average of the measurements is enhanced with respect to the SM and it would correspond to the tree-level exchange of a charged particle with mass Λ ~ 1 TeV and coupled selectively to τ leptons. In this talk I will review these decays, discussing the extent up to which the SM predictions are understood and the type of new physics one would need to explain the tensions with the SM. I will also discuss experimental and theoretical prospects to clarify the nature of these anomalies in the near future. | 2020-09-21 17:07:54 | {"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": 0.8240579962730408, "perplexity": 2154.3306863024436}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400201826.20/warc/CC-MAIN-20200921143722-20200921173722-00470.warc.gz"} |
https://physics.stackexchange.com/questions/147673/conservation-of-momentum-1-d-problem-track-is-not-level | # Conservation of momentum 1-D problem: track is not level
This problem is a standard kind of 1-D problem. You have 2 equal carts approaching each other and collide on linear air track. BUT the person setting up the track did NOT do a good job at making the track Level(or NOT perfectly horizontal). THE QUESTION is : HOW "UNLEVEL" is the track?
The collision took t= 0.750 seconds. And the data collected is below:
## DATA
m1=m2= 1kg
Initial velocities:
v1i= +0.450m/s
v2i= -0.505m/s
Final velocities:
v1f= -0.510m/s
v2f= +0.420m/s
From the data we can easily calculate the momentum before and after the collision and see that the momentum of this system is NOT Conserved.
To me this makes sense because the track is unlevel, we have made the track to be like an inclined plane which means that gravity now has influence on this system! SO we have now gone into 2-D space, and we have acceleration happening due to gravity.
I was able to figure out this much. BUT how do I calculate, based on the given information, say the angle that the track makes with respect to the horizontal.(This is what I guess the question is asking).
BUT, I also don't understand how they can measure the velocities(in the data given) if we have acceleration, which means that the velocity is changing(velocity is no longer constant). SO I see some contradictions here in this question.
I found this question in an old book, and there are no answers to the questions. BUT this one got me interested.
Hope somebody has seen this kind of question before. Your help is greatly appreciated.
• Typically collision experiments like this use photogates and simply measure the time between sensing and the known width of the object to arrive at the velocity. If multiple points are taken, then lines of best fit are used to create a velocity. – Kyle Kanos Nov 19 '14 at 19:09
• OK, thanks Kyle. So you are saying that with these devices, this measurement process, in the presence of acceleration, this would then measure the average velocity. BUT, if am not mistaken, isn't momentum all about the velocities all being constant velocities, except for the collision aspect where the velocities change, but still remain constant afterwards. – Palu Nov 19 '14 at 19:15
• Yes, it still measures the average velocity. Momentum for an object can clearly change, though in a force-free region the net momentum will be constant. – Kyle Kanos Nov 19 '14 at 19:21
• In the general types of momentum questions, i believe it is always done with constant velocity. I just read that if velocity is changing(ie, acceleration) then the momentum is changing from one point to the next. In this situation if momentum is changing from t1,t2,t3 etc, how would one be able to apply conservation of momentum in this case. I only know how to apply it when velocity before and after a collision is constant. SO that is why this problem is confusing, is the given velocities constant, afterwards, or is it just a snapshot of a particular time, with acceleration being present. – Palu Nov 19 '14 at 19:35
• If there is an acceleration, then you do not have a force-free region and thus the conservation of momentum does not apply. – Kyle Kanos Nov 19 '14 at 19:36
You always have to satisfy the momentum equations, which is only the linear momentum equation for this one dimensional case:
$$m_1v(t_1)-m_1v(t_0) = \int_{t_0}^{t_1} F dt$$
Assume the collision is completely elastic and all is conservative, so no plastic deformation, drag or any kind of damping. Then the only force which acts is the gravity:
$$F = sin(\theta)mg$$
where $\theta$ is the angle the track is making with respect to 'levelness'. Now we know that the momentum equation should be valid for each individual body, but also for the entire system. Again, the only external force is gravity. I assume that the velocities which are given are at the instant before and after the collision, as they are also a function of time and no time 'after collision' is given.
The momentum before the collision is:
$$L(t_0) = m_1 v_1(t_0) + m_2 v_2(t_0)$$
And after the collision:
$$L(t_1) = m_1 v_1(t_1) + m_2 v_2(t_1)$$
We know the collision time, and the force integral is not a function of time. So we obtain:
$$L(t_1) - L(t_0) = sin(\theta)(m_1 + m_2)g t_c$$
with $t_c$ as collision time. Solve for $\theta$, which now is the only unknown.
$$sin(\theta) = \frac{L(t_1) - L(t_0)}{(m_1 + m_2)g t_c}$$
• Thanks Rhino, i believe that did it. The question does not make clear that the velocities are immediately before and after the collision. BUT assuming this, then I agree with you that this would be the answer. Thanks again. – Palu Nov 20 '14 at 0:44 | 2020-05-29 07:30:31 | {"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": 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": 0.7400103807449341, "perplexity": 395.3190277261284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347402457.55/warc/CC-MAIN-20200529054758-20200529084758-00196.warc.gz"} |
https://ask.opendaylight.org/answers/13240/revisions/ | # Revision history [back]
Technically, karthik330's answer as well as jdavid's are correct. But since I had a lot of trouble figuring out what to do with the information I got from here, I want to write up what I learned and did to finally get it to work.
If you want to use a stable OpenDaylight release, such as Beryllium-SR2, and integrate your own module, located on your local machine, you need to do and understand the following:
You have to make sure, that the version of the archetype you are using for creating your project matches the version of the OpenDaylight you want to integrate your project into. In this link you can find all the ressources your OpenDaylight downloads on build. If you want to use a OpenDaylight release (without SNAPSHOTs) you can find a list of archetypes to use from here
Next you need to know, that you can't/shouldn't use the pre-build version of OpenDaylight you can find on their website, if you want to integrate your own project into it, because you have to edit the OpenDaylight BEFORE it is build.
Therefore, you can go to here, select your preferred release version (for example Beryllium-SR2) and download the source code of what is called "integration-distribution".
Now comes the time to follow the steps karthik330 and jdavid described.
Step 1: go to integration-distribution/features-index/pom.xml and add a dependency to your project.
For example:
<dependency< p="">
<dependency>
<groupId>org.opendaylight.welcome</groupId>
<artifactId>welcome-features</artifactId>
<version>1.0.0-SNAPSHOT</version>
<classifier>features</classifier>
<type>xml</type>
</dependency>
Step 2: go to integration-distribution/features-index/src/main/resources/features.xml and add the repository location of your project to the list.
In our example:
< repository > mvn:org.opendaylight.welcome/welcome-features/1.0.0-SNAPSHOT/xml/features < /repository >
After that, you run mvn clean install in the integration-distribution-Folder after which you can start your karaf by running ./karaf in integration-distribution/distribution-karaf/target/assembly/bin.
Now the last thing you have to do is installing your feature by running feature:install odl-welcome in your karaf-shell.
If you run into any version mismatches now you did not use the correct combination of archetype and OpenDaylight-Version.
Good luck! | 2019-05-22 06:32:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3519178032875061, "perplexity": 1425.4155520745462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256764.75/warc/CC-MAIN-20190522063112-20190522085112-00260.warc.gz"} |
http://math.stackexchange.com/questions/218076/about-the-boundedness-of-the-derivative-of-a-function-which-is-in-special-functi | # About the boundedness of the derivative of a function which is in special function space.
If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, then how can I conclude that $$\left \| \frac{\partial f}{\partial t} \right \|_{L^\infty([0,T] \times \Bbb R^n )} < \infty ?$$ Here $f$ is defined on $[0,T] \times \Bbb R^n$ , and the notation $f \in C^1([0,T], L^2)$ means that $\| f(t) \|_{L^2 (\Bbb R^n)}$ is continuously differentiable on $[0,T]$, $W^{s,p}$ means the usual Sobolev space.
-
Handwaving: $\partial f/\partial t = \mathrm df/\mathrm dt - \partial f/\partial x\ \partial x/\partial t$, thus the first inclusion gives (ess.) finiteness for the first term on the rhs, the second gives (ess.) finiteness for the second term, thus their difference is also essentially finite. – filmor Oct 21 '12 at 15:45
@filmor Thank you, but here $x$ is not depend on $t$. $f = f(t,x_1 , \cdots , x_n)$. – Ann Oct 21 '12 at 15:49
I don't think this is true: Take $f(t,x)=t\eta(x)|x|^{n-\alpha}$ with $\eta\in C_c^{\infty}(\mathbb{R}^n)$ a cut-off function, then for a suitable $\alpha>0$ we have $f\in W^{1,2}$, but $\eta(x)|x|^{n-\alpha}\notin L^{\infty}$. – Jose27 Oct 21 '12 at 17:52
Unless I am missing something, this does not seem to be correct. Let $n=3$, and define $$F(x) = \|x\|^{-1/3} \quad\text{ for \|x\|\le 1,}$$ and extend it smoothly to a function with compact support on $\mathbb{R}^3$. Then $$\|\nabla F(x)\| = \frac13 \|x\|^{-4/3} \quad\text{ for \|x\|\le 1},$$ so $F \in L^2(\mathbb{R}^3)$, $\nabla F \in L^2(\mathbb{R}^3)$ and thus $F\in W^{1,2}(\mathbb{R}^3)$. Now define $$f(t,x) = t F(x).$$ Then $$\frac{\partial f}{\partial t} (t,x) = F(x),$$ so this example seems to satisfy the even stronger assumption that $f \in C^1([0,T],W^{1,2})$, but it $\frac{\partial f}{\partial t} = F \notin L^\infty$.
The Sobolev embedding doesn't give that inclusion: Take $f(x)=\log\log(1+|x|^{-1})$. – Jose27 Oct 21 '12 at 18:06
Thank you very much. But how it becomes if we add the additional assumption: $f(t,x)$ is bounded on $[0,T] \times \Bbb R^n$ ? – Ann Oct 21 '12 at 19:42 | 2014-08-01 00:07:53 | {"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": 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": 0.9913349151611328, "perplexity": 178.82074114684045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273766.32/warc/CC-MAIN-20140728011753-00111-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://mathhelpforum.com/business-math/93630-interest-calculations.html | 1. ## Interest Calculations
Hi, I'm new here and would really appreciate some help with my calculations.
I from the UK and here at the moment there is a lot of consumers reclaiming back money from the big banks for mis-selling insurance products that were useless to the consumer and I am trying to calculate exactly how much is owed to me.
The government has allowed all claims to be paid with a statutory interest rate of 8% per year.
so that would work at 0.66% a month right?
Month 1 i pay £30 for my insurance
Month 2 my £30 is now worth £30.20 (30 x 0.66%) and another £30 is paid so total is now £60.20
Month 3 my £60.20 is now worth £60.60 and another £30 is paid so total owed to me is now £90.60.
Month 4 my £90.60 is now worth £91.20 and another £30 is paid so total owed to me is now £121.20
Month 5 my £121.20 is now worth £121.99 and another £30 is paid to total owed to me is now £151.20.....
and so forth and so forth....
Any help would be appreciated. Thanks
2. Originally Posted by JustChris20
Hi, I'm new here and would really appreciate some help with my calculations.
I from the UK and here at the moment there is a lot of consumers reclaiming back money from the big banks for mis-selling insurance products that were useless to the consumer and I am trying to calculate exactly how much is owed to me.
The government has allowed all claims to be paid with a statutory interest rate of 8% per year.
so that would work at 0.66% a month right?
Month 1 i pay £30 for my insurance
Month 2 my £30 is now worth £30.20 (30 x 0.66%) and another £30 is paid so total is now £60.20
Month 3 my £60.20 is now worth £60.60 and another £30 is paid so total owed to me is now £90.60.
Month 4 my £90.60 is now worth £91.20 and another £30 is paid so total owed to me is now £121.20
Month 5 my £121.20 is now worth £121.99 and another £30 is paid to total owed to me is now £151.20.....
and so forth and so forth....
Any help would be appreciated. Thanks
The government has allowed all claims to be paid with a statutory interest rate of 8% per year.
Typically this DOES NOT allow monthly compounding.
At the end of the first YEAR you would have
$30 \times 0.08 = 2.4$ INTEREST.
30 + 2.40 = 32.40 at the start on the following year.
$32.40 \times 0.08 = 2.592$ INTEREST gain during the second year.
32.4 + 2.592 = 34.992 at the end of the second YEAR and the beginning of the third year.
The interest will only be compounded only on a YEARLY basis. | 2016-12-11 10:10:26 | {"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": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.37245598435401917, "perplexity": 1541.734343802772}, "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-2016-50/segments/1480698544672.33/warc/CC-MAIN-20161202170904-00335-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.otexts.org/node/923 | # 9.2.1 Names, places, frames, and environments
Because assignments can change the value associated with a name, the order in which expressions are evaluated now matters. As a result, we need to revisit several of our other evaluation rules and change the way we think about processes.
Since the value associated with a name can now change, instead of associating a value directly with a name we use a name as a way to identify a place. A place has a name and holds the value associated with that name. With mutation, we can change the value in a place; this changes the value associated with the place's name. A frame is a collection of places.
An environment is a pair consisting of a frame and a pointer to a parent environment. A special environment known as the global environment has no parent environment. The global environment exists when the interpreter starts, and is maintained for the lifetime of the interpreter. Initially, the global environment contains the built-in procedures. Names defined in the interactions buffer are placed in the global environment. Other environments are created and destroyed as a program is evaluated. Figure 9.1 shows some example environments, frames, and places.
Figure 9.1. Sample environment: The global environment contains a frame with three names. Each name has an associated place that contains the value associated with that name. The value associated with counter is the currently 0. The value associated with set-counter! is the procedure we defined in Example 9.1. A procedure is characterized by its parameters, body code, and a pointer to the environment in which it will be evaluated.
Every environment has a parent environment except for the global environment. All other environments descend from the global environment. Hence, if we start with any environment, and continue to follow its parent pointers we always eventually reach the global environment.
The key change to our evaluation model is that whereas before we could evaluate expressions without any notion of where they are evaluated, once we introduce mutation, we need to consider the environment in which an expression is evaluated. An environment captures the current state of the interpreter. The value of an expression depends on both the expression itself, and on the environment in which it is evaluated. | 2018-02-21 05:15:54 | {"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": 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": 0.5734801292419434, "perplexity": 545.020561671955}, "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/1518891813431.5/warc/CC-MAIN-20180221044156-20180221064156-00072.warc.gz"} |
https://tutorme.com/tutors/189097/interview/ | Enable contrast version
# Tutor profile: Bhawna C.
Inactive
Bhawna C.
Post Graduate in Applied Statistics from IIT Bombay
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## Questions
### Subject:Linear Algebra
TutorMe
Question:
Can you just look at the equations and know at a glance how many solutions will they have? $$x+y+z=6......(eqn1)$$ $$3x+3y+3z=18........(eqn2)$$ $$x+2y-z=4.......(eqn3)$$
Inactive
Bhawna C.
Answer:
Notice that eqn 1 and 2 give the same information about the relationship between x,y and z. When we know that $$x+y+z=6$$, it is obvious that $$3(x+y+z)=3(6)$$ i.e. $$3x+3y+3z=18$$ These equations are linearly dependent i.e. $$constant\cdot(eqn1)=eqn2$$ Therefore, we only have 2 meaningful equations 1 and 3, $$x+y+z=6$$ $$x+2y-z=4$$ Add eqn 1 and eqn 3 to eliminate z, We get, $$2x+3y=10$$ This is a straight line which has infinite values of x and y that satisfy the condition.
### Subject:Trigonometry
TutorMe
Question:
Given $$\sum_{n=1}^{\infty} (2sinAcosA)^{n} = 1$$ What is the value of A?
Inactive
Bhawna C.
Answer:
We know that sin2A = 2sinAcosA Therefore, the sum becomes, $$\sum_{n=1}^{\infty} (sin2A)^{n}$$ Notice this is a geometric progression, $$\sin2A + sin^{2}2A + sin^{3}2A+...$$ Sum of an infinite geometric progression of the form $$a + ar + ar^{2}+ar^{3}+...= \frac{a}{1-r}$$ Here, a=sin2A, r = sin2A Therefore, $$\frac{sin2A}{1-sin2A}=1$$ $$sin2A = 1 -sin2A$$ $$sin2A = \frac{1}{2}$$ $$2A = sin^{-1} \frac{1}{2}$$ $$2A = 30^{\circ}$$ $$A= 15^{\circ}$$
### Subject:Statistics
TutorMe
Question:
A survey was done asking people if tiktok should be banned in India. Poll surveyed 518 adults and 233, or 0.45 of them answered yes. Could we conlcude that only a minority if the people want tiktok to be banned?
Inactive
Bhawna C.
Answer:
Null hypothesis: the proportions who would answer yes or no are each 0.50. Alternative hypothesis: Fewer than 0.50, or 50%, of the population would answer yes to this question. The majority do not think that tiktok should be banned. Sample proportion is: $$\frac{233}{518} = 0.45$$ The standard deviation: $$\sqrt{\frac{050(1-0.50)}{518}}$$ Test statistic: $$z = (0.45 – 0.50)/0.022 = –2.27$$ Now we determine the p-value, Recall the alternative hypothesis was one-sided. p-value = proportion of bell-shaped curve below –2.27 Exact p-value = 0.0116 The p-value of 0.0116 is less than 0.05, so we conclude that the proportion of Indians who want the tiktok ban was significantly less than a majority.
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Many of our tutors are current college students or recent graduates of top-tier universities like MIT, Harvard and USC. TutorMe has thousands of top-quality tutors available to work with you. | 2020-04-01 15:50:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.7037792205810547, "perplexity": 2022.0843206540255}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370505731.37/warc/CC-MAIN-20200401130837-20200401160837-00045.warc.gz"} |
https://bookdown.org/yihui/rmarkdown-cookbook/hook-number.html | ## 10.2 Add line numbers to source code
In this section, we show an example of defining a source hook to add line numbers as comments to the source code. For example, for this code chunk:
{r}
if (TRUE) {
x <- 1:10
x + 1
}
We want the output to be:
if (TRUE) { # 1
x <- 1:10 # 2
x + 1 # 3
} # 4
The full example is below:
---
title: Add line numbers to source code
---
We set up a source hook to add line numbers to the source
code. The numbers appear in comments at the end of each line.
{r, include=FALSE}
local({
hook_source <- knitr::knit_hooks$get('source') knitr::knit_hooks$set(source = function(x, options) {
x <- xfun::split_lines(x)
n <- nchar(x, 'width')
i <- seq_along(x) # line numbers
n <- n + nchar(i)
s <- knitr:::v_spaces(max(n) - n)
x <- paste(x, s, ' # ', i, sep = '', collapse = '\n')
hook_source(x, options)
})
})
Now we can test the new hook. When you knit this document, you
will see line numbers in trailing comments.
{r}
if (TRUE) {
x <- 1:10
x + 1
}
The main trick in the above example is to determine the number of spaces needed before the comment on each line, so the comments can align to the right. The number depends on the widths of each line of code. We leave it to readers to digest the code in the hook function. Note that an internal function knitr:::v_spaces() is used to generate spaces of specified lengths, e.g.,
knitr:::v_spaces(c(1, 3, 6, 0))
## [1] " " " " " " ""
The method introduced in Section 4.3 may be the actual way in which you want to add line numbers to source code. The syntax is cleaner, and it works both source code and text output blocks. The above source hook trick mainly aims to show you one possibility of manipulating the source code with a custom function. | 2020-02-21 00:52:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45428094267845154, "perplexity": 2327.7887762141568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145316.8/warc/CC-MAIN-20200220224059-20200221014059-00455.warc.gz"} |
https://ltwork.net/what-are-the-partial-product-of-45x5--11607951 | # What are the partial product of 45x5=?
###### Question:
What are the partial product of 45x5=?
### Given points M(5, - 3) and N(21, 1) , find the coordinates of point P such that MP : PN = 5 : 3
Given points M(5, - 3) and N(21, 1) , find the coordinates of point P such that MP : PN = 5 : 3...
### If the area of the rectangle is 35x^2-36x-32 square inches, and its length is 7x+4 inches find its length
if the area of the rectangle is 35x^2-36x-32 square inches, and its length is 7x+4 inches find its length...
### Cheryl has measured 3 cups of water is this enough water for Cheryl to make double recipe of green slime for a class project? Explain
Cheryl has measured 3 cups of water is this enough water for Cheryl to make double recipe of green slime for a class project? Explain...
### Implement the STL find routine that returns the iterator containing the first occurrence of x in the range that begins at start
Implement the STL find routine that returns the iterator containing the first occurrence of x in the range that begins at start and extends up to but not including end. If x is not found, end is returned. This is a nonclass (global function) with signature: template Iterator find (Iterator start, It...
### When the moon is between the earth and sun it cannot be seen and is called a moon.
When the moon is between the earth and sun it cannot be seen and is called a moon....
### 2. What were three of the symptoms of the decline of Puritan rule between1675 and 1700? What was the new political status of the New
2. What were three of the symptoms of the decline of Puritan rule between 1675 and 1700? What was the new political status of the New England region by 1700?...
### What is the advantage of the cristae beingfolded?A. Increases surface area to perform properly.B. Converts energy into a compound.C.
What is the advantage of the cristae being folded? A. Increases surface area to perform properly. B. Converts energy into a compound. C. There is no need for cellular respiration....
### 25 ! how did ethnicity and race influence the cities of the us in the late 19th century? they define
25 ! how did ethnicity and race influence the cities of the us in the late 19th century? they define social stratification. they played a limited role. they contributed to the view of america as a melting pot. they caused separate schools to develop for different ethnic groups....
### Read the passage. the city recycling program is inconvenient. instead of having to drop off recyclables
Read the passage. the city recycling program is inconvenient. instead of having to drop off recyclables at the city center, recyclables should be picked up at the curb, alongside the trash cans. what type of problem-and-solution organization does this paragraph use?...
### Travis drove 129 miles in 3hours. he drove at a constant speed. how many miles did he
Travis drove 129 miles in 3hours. he drove at a constant speed. how many miles did he drive in 1 hour...
### -2(x-4)=4x+2x+8
-2(x-4)=4x+2x+8...
### Oxidation states of n2 + 3h2 --> 2nh3
Oxidation states of n2 + 3h2 --> 2nh3...
### Who sailed around the southern tip of africa in 1488?
Who sailed around the southern tip of africa in 1488?...
Someone please help me(check the image I attached!!) $Someone please help me(check the image I attached!!)$...
### You are the president of a club in your community or at your school. the club has an upcoming event (dance, carnival, etc.)
You are the president of a club in your community or at your school. the club has an upcoming event (dance, carnival, etc.) and needs covering its costs. you are in charge of writing a proposal to submit to potential sponsors that will convince them to donate to your cause. i really need with this...
### The nurse is monitoring a patient receiving a heparin infusion for the treatment of pulmonary embolism. which assessment
The nurse is monitoring a patient receiving a heparin infusion for the treatment of pulmonary embolism. which assessment finding most likely relates to an adverse effect of heparin?...
### How are the images, themes, or ideas in this propaganda similar to images, themes, or ideas in modern culture?
How are the images, themes, or ideas in this propaganda similar to images, themes, or ideas in modern culture? $How are the images, themes, or ideas in this propaganda similar to images, themes, or ideas in mode$...
### Complete the code to finish this program to analyze the inventory for a store that sells purses and backpacks. (python)Each
Complete the code to finish this program to analyze the inventory for a store that sells purses and backpacks. (python) Each record is composed of the catalog number, the type of item, its color, the length, width, height, and the quantity in stock. Sample rows of the file are below. 234,purse, blu... | 2022-12-08 06:17:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2138756364583969, "perplexity": 2764.5522669871393}, "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/1669446711278.74/warc/CC-MAIN-20221208050236-20221208080236-00875.warc.gz"} |
https://nrich.maths.org/13220/solution | ### Burning Down
One night two candles were lit. Can you work out how long each candle was originally?
### Percentage Unchanged
If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage is the width decreased by ?
### Digit Sum
What is the sum of all the digits in all the integers from one to one million?
# Powerful 9
##### Age 14 to 16 ShortChallenge Level
Answer: $9$
Look at the last digits of powers of $9$.
odd even
$9$ $81$
$729$ $279\times 9 = ....1$
$...1\times9=...9$ $...1$
$...9$ $...1$
$9^9$ is a power of $9$ so $9^9$ is odd
This means that $9^{9^9}$ is an odd power of $9$, so it ends in $9$.
You can find more short problems, arranged by curriculum topic, in our short problems collection. | 2023-01-29 03:41:40 | {"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": 0.8246640563011169, "perplexity": 2745.987638635251}, "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-2023-06/segments/1674764499697.75/warc/CC-MAIN-20230129012420-20230129042420-00620.warc.gz"} |
https://www.physicsforums.com/threads/subsets-of-r-3-as-subspaces.677265/ | # Subsets of R^3 as subspaces
1. Mar 9, 2013
### MoreDrinks
1. The problem statement, all variables and given/known data
Which of the following subsets of R3 are subspaces? The set of all vectors of the form (a,b,c) where a, b, and c are...
2. Relevant equations
1. integers
2. rational numbers
3. The attempt at a solution
I think neither are subspaces. IIRC, the scalar just needs to be from R3 and not, for example, an integer for 1 or a rational number for 2.
So for number 1, I can multiply the integers of vector (a,b,c) by some non-integer k, ending up with (ka,kb,kc) outside the subset, and thus not a subspace.
For number 2, I can multiply the rational numbers of vector (a,b,c) some some irrational number (say, ∏) and end up with (∏a, ∏b, ∏c), all outside the subset and thus not a subspace.
Or am I totally wrong?
2. Mar 9, 2013
### jbunniii
No, you are totally correct. The indicated sets are not subspaces of $\mathbb{R}^3$, for the reasons you stated.
3. Mar 9, 2013
### jbunniii
Correction: the scalars are elements of $\mathbb{R}$, not $\mathbb{R}^3$.
4. Mar 9, 2013
### MoreDrinks
If we're dealing with complex space, can scalars be complex?
Thanks for the help!
5. Mar 9, 2013
### jbunniii
They can, but then it wouldn't be $\mathbb{R}^3$ anymore. It would be $\mathbb{C}^3$.
6. Mar 9, 2013
### MoreDrinks
True, thanks! Would the correct term be that we're working in the "field" of R^3 or just R^3 space when talking about this?
7. Mar 9, 2013
### jbunniii
To be precise, a vector space consists of an abelian group of vectors and a field of scalars, along with some rules governing the multiplication of a vector by a scalar.
So if we want to be precise, we would say that we are working in the vector space in which the vectors are elements of $\mathbb{R}^3$ and the scalars are elements of $\mathbb{R}$, with the usual rules of multiplication.
However, for brevity we typically say that we are working in the vector space $\mathbb{R}^3$, and unless stated otherwise, it is understood that the scalar field is $\mathbb{R}$.
Similarly, we may say that we are working in the vector space $\mathbb{C}^3$, where the assumption is that unless stated otherwise, the scalar field is $\mathbb{C}$.
8. Mar 9, 2013
### MoreDrinks
Thank you, that clears up a lot. | 2017-08-20 20:19:05 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8906883597373962, "perplexity": 668.9247327565089}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106984.52/warc/CC-MAIN-20170820185216-20170820205216-00503.warc.gz"} |
https://megaexams.com/questions/IBPS/Mensuration/ | # IBPS
Explore popular questions from Mensuration for IBPS. This collection covers Mensuration previous year IBPS questions hand picked by experienced teachers.
## Hindi Language
Mensuration
Correct Marks 1
Incorrectly Marks -0.25
Q 1. The area of a rectangular field is {tex} 144 \mathrm { m } ^ { 2 } {/tex}. If the length had been 6 metres more, the area would have been {tex} 54 \mathrm { m } ^ { 2 } {/tex} more. The original length of the field is
A
22 metres
B
18 metres
16 metres
D
24 metres
E
None of these
##### Explanation
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Q 2. The altitude drawn to the base of an isosceles triangle is 8 {tex} \mathrm { cm } {/tex} and the perimeter is {tex} 32\ \mathrm { cm } . {/tex} The area of the triangle is
A
{tex} 72 \ \mathrm { cm } ^ { 2 } {/tex}
{tex} 60 \ \mathrm { cm } ^ { 2 } {/tex}
C
{tex} 66 \ \mathrm { cm } ^ { 2 } {/tex}
D
{tex} 65\ \mathrm { cm } ^ { 2 } {/tex}
E
None of these
##### Explanation
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Q 3. Four equal circles are described about the four corners of a square so that each touches two of the others. If a side of the square is {tex} 14 \mathrm { cm } , {/tex} then the area enclosed between the circumferences of the circles is:
A
{tex} 24 \mathrm { cm } ^ { 2 } {/tex}
{tex} 42 \mathrm { cm } ^ { 2 } {/tex}
C
{tex} 154 \mathrm { cm } ^ { 2 } {/tex}
D
{tex} 196 \mathrm { cm } ^ { 2 } {/tex}
E
None of these
##### Explanation
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Q 4. The area of a side of a box is {tex} 120 \mathrm { sq } \mathrm { cm } . {/tex} The area of the other side of the box is {tex} 72 \mathrm { sq } \mathrm { cm } . {/tex} If the area of the upper surface of the box is {tex} 60 \mathrm { sq } \mathrm { cm } {/tex} then find the volume of the box.
A
{tex} 259200 \mathrm { cm } ^ { 3 } {/tex}
B
{tex} 86400 \mathrm { cm } ^ { 3 } {/tex}
{tex} 720 \mathrm { cm } ^ { 3 } {/tex}
D
Cannot be determined
E
None of these
##### Explanation
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Q 5. The capacity of a cylindrical tank is 246.4 litres. If the height is 4 metres, what is the diameter of the base?
A
1.4 metres
B
2.8 metres
C
28 metres
D
14 metres
None of these
##### Explanation
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Q 6. A square carpet with an area {tex} 169 \mathrm { m } ^ { 2 } {/tex} must have 2 metres cut-off one of its edges in order to be a perfect fit for a rectangular room. What is the area of rectangular room?
A
{tex} 180 \mathrm{m ^ { 2 }} {/tex}
B
{tex} 164 \mathrm{m ^ { 2 }} {/tex}
C
{tex} 152 \mathrm { m } ^ { 2 } {/tex}
{tex} 143 \mathrm{m ^ { 2 }} {/tex}
E
None of these
##### Explanation
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Q 7. The area of a right-angled triangle is two-thirds of the area of a rectangle. The base of the triangle is 80 percent of the breadth of the rectangle. If the perimeter of the rectangle is {tex} 200 \mathrm { cm } , {/tex} what is the height of the triangle?
A
{tex} 20 \mathrm { cm } {/tex}
B
{tex} 30 \mathrm { cm } {/tex}
C
{tex} 15 \mathrm { cm } {/tex}
E
None of these
##### Explanation
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Q 8. A circular ground whose diameter is 35 metres, has a 1.4 metre-broad garden around it. What is the area of the garden in square metres?
160.16
B
6.16
C
122.66
D
E
None of these
##### Explanation
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Q 9. A rectangular garden has a 5 -metre-wide road outside around all the four sides. The area of the road is 600 square metres. What is the ratio between the length and the breadth of that plot?
A
3:2
B
4:3
C
5:4
E
None of these
##### Explanation
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Q 10. A cube of 384{tex} \mathrm { cm } ^ { 2 } {/tex} surface area is melted to make x number of small cubes each of 96{tex} \mathrm { mm } ^ { 2 } {/tex} surface area. The value of {tex} \mathrm { x } {/tex} is
A
80,000
B
8
8,000
D
800
E
None of these
##### Explanation
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Q 11. If the perimeter and diagonal of a rectangle are 14{tex} \mathrm { cm } {/tex} and 5 {tex} \mathrm { cm } {/tex} respectively, find its area.
12{tex} \mathrm { cm } ^ { 2 } {/tex}
B
16{tex} \mathrm { cm } ^ { 2 } {/tex}
C
20{tex} \mathrm { cm } ^ { 2 } {/tex}
D
24{tex} \mathrm { cm } ^ { 2 } {/tex}
E
None of these
##### Explanation
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Q 12. The length, breadth and height of a cuboid are in the ratio {tex} 1 : 2 : 3 . {/tex} The length, breadth and height of the cuboid are increased by {tex} 100 \% , 200 \% {/tex} and {tex} 200\% {/tex} , respectively. Then, the increase in the volume of the cuboid will be:
A
5 times
B
6 times
C
12 times
17 times
E
None of these
##### Explanation
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Q 13. The ratio of the length to the breadth of a rectangular plot is {tex} 6 : 5 {/tex} respectively; if the breadth of the plot is 34 metre less than the length, what is the perimeter of the rectangular plot?
A
374 metres
B
408 metre
C
814 metre
748 metre
E
None of these
##### Explanation
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Q 14. A rectangular plot {tex} 15 \mathrm { m } \times 10 \mathrm { m } {/tex} has a path of grass outside it. If the area of grassy pathway is {tex} 54 \mathrm { m } ^ { 2 } , {/tex} find the width of the path.
A
4{tex} \mathrm { m } {/tex}
B
3{tex} \mathrm { m } {/tex}
2{tex} \mathrm { m} {/tex}
D
{tex}1 \mathrm { m } {/tex}
E
None of these
##### Explanation
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Q 15. The length of a rectangular plot is twice its breadth. If the area of the rectangular plot is 2592 sq metres, what is the length of the rectangular plot?
A
76 metre
B
36 metre
C
74 metre
D
37 metre
None of these
##### Explanation
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Q 16. Four sheets of {tex} 50 \mathrm { cm } \times 5 \mathrm { cm } {/tex} are arranged without overlapping to form a square having side {tex} 55 \mathrm { cm } . {/tex} What is the area of the inner square so formed?
A
2500{tex} \mathrm { cm } ^ { 2 } {/tex}
2025{tex} \mathrm { cm } ^ { 2 } {/tex}
C
{tex}1600 \mathrm { cm } ^ { 2 } {/tex}
D
1650{tex} \mathrm { cm } ^ { 2 } {/tex}
E
None of these
##### Explanation
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Incorrectly Marks -1
Q 17. If the area of a circle decreases by {tex} 36 \% , {/tex} then the radius of the circle decreases by
20{tex} \% {/tex}
B
18{tex} \% {/tex}
C
36{tex} \% {/tex}
D
64{tex} \% {/tex}
E
None of these
##### Explanation
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Q 18. Area of a rectangle is equal to the area of the circle whose radius is {tex}21 \mathrm { cm } {/tex} . If the length and the breadth of the rectangle are in the ratio of {tex} 14 : 11 {/tex} respectively, what is its perimeter?
A
{tex}142 \mathrm { cm } {/tex}
B
{tex}140 \mathrm { cm } {/tex}
C
{tex}132 \mathrm{\, cm } {/tex}
{tex}150 \mathrm { \,cm } {/tex}
E
None of these
##### Explanation
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Q 19. The length of a rectangular field is double its width. Inside the field there is a square-shaped pond 8{tex} \mathrm { m } {/tex} long. If the area of the pond is 1/8 of the area of the field, what is the length of the field?
32{tex} \mathrm { m } {/tex}
B
16{tex} \mathrm { m } {/tex}
C
64{tex} \mathrm { m } {/tex}
D
20{tex}\mathrm {\, m }{/tex}
E
None of these
##### Explanation
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Q 20. 20 buckets of water fill a tank when the capacity of each bucket is 13.5 litres. How many buckets will be required to fill the same tank if the capacity of each bucket is 9 litres?
30
B
32
C
60
D
E
None of these
##### Explanation
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Q 21. A right circular cone is exactly fitted inside a cube in such a way that the edges of the base of the cone are touching the edges of one of the faces of the cube and the vertex is on the opposite face of the cube. If the volume of the cube is {tex} 343 \mathrm { cc } , {/tex} what approximately is the volume of the cone ?
A
{tex} 80 \mathrm { cc } {/tex}
{tex} 90 \mathrm { cc } {/tex}
C
{tex} 110 \mathrm { cc } {/tex}
D
{tex} 105 \mathrm { cc } {/tex}
E
{tex} 100 \mathrm { cc } {/tex}
##### Explanation
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Q 22. The intermal measurements of a box with lid are {tex} 115 \times 75 \times 35 {/tex} {tex} \mathrm { cm } ^ { 3 } {/tex} and the wood of which it is made is {tex} 2.5 \ \mathrm { cm } {/tex} thick. Find the volume of wood.
{tex} 82,125 \mathrm { cm } ^ { 3 } {/tex}
B
{tex} 70,054 \mathrm { cm } ^ { 3 } {/tex}
C
{tex} 78,514 \mathrm { cm } ^ { 3 } {/tex}
D
E
None of these
##### Explanation
Correct Marks 1
Incorrectly Marks -0.25
Q 23. The ratio between the length and the breadth of a rectangular plot is {tex} 7: 5 . {/tex} If the perimeter of the plot is 144 metres, what is its area?
A
1320 sq. metres
1260 sq. metres
C
1280 sq. metres
D
1380 sq. metres
E
None of these
##### Explanation
Correct Marks 1
Incorrectly Marks -0.25
Q 24. A rectangular paper, when folded into two congruent parts had a perimeter of {tex} 34 \mathrm { cm } {/tex} for each part folded along one set of sides and the same is {tex} 38 \mathrm { cm } {/tex} when folded along the other set of sides. What is the area of the paper?
{tex} 140 \mathrm { cm } ^ { 2 } {/tex}
B
{tex} 240 \mathrm { cm } ^ { 2 } {/tex}
C
{tex} 560 \mathrm { cm } ^ { 2 } {/tex}
D
{tex} 160 \mathrm { cm } ^ { 2 } {/tex}
E
None of these
##### Explanation
Correct Marks 4
Incorrectly Marks -1
Q 25. A right circular cone and a right circular cylinder have equal base and equal height. If the radius of the base and the height are in the ratio {tex} 5 : 12 , {/tex} then the ratio of the total surface area of the cylinder to that of the cone is
A
{tex} 3 : 1 {/tex}
B
{tex} 13 : 9 {/tex}
{tex} 17 : 9 {/tex}
D
{tex} 34 : 9 {/tex}
E
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https://www.nature.com/articles/s41598-018-20024-w?error=cookies_not_supported&code=5ea56159-620f-4cbb-86ff-6d993bf60761 | Article | Open | Published:
# Strong constraint on modelled global carbon uptake using solar-induced chlorophyll fluorescence data
## Abstract
Accurate terrestrial biosphere model (TBM) simulations of gross carbon uptake (gross primary productivity – GPP) are essential for reliable future terrestrial carbon sink projections. However, uncertainties in TBM GPP estimates remain. Newly-available satellite-derived sun-induced chlorophyll fluorescence (SIF) data offer a promising direction for addressing this issue by constraining regional-to-global scale modelled GPP. Here, we use monthly 0.5° GOME-2 SIF data from 2007 to 2011 to optimise GPP parameters of the ORCHIDEE TBM. The optimisation reduces GPP magnitude across all vegetation types except C4 plants. Global mean annual GPP therefore decreases from 194 ± 57 PgCyr−1 to 166 ± 10 PgCyr−1, bringing the model more in line with an up-scaled flux tower estimate of 133 PgCyr−1. Strongest reductions in GPP are seen in boreal forests: the result is a shift in global GPP distribution, with a ~50% increase in the tropical to boreal productivity ratio. The optimisation resulted in a greater reduction in GPP than similar ORCHIDEE parameter optimisation studies using satellite-derived NDVI from MODIS and eddy covariance measurements of net CO2 fluxes from the FLUXNET network. Our study shows that SIF data will be instrumental in constraining TBM GPP estimates, with a consequent improvement in global carbon cycle projections.
## Introduction
The terrestrial carbon, C, sink remains the most uncertain component of the annual global carbon budget1. Uncertainty in its strength and location contributes to high terrestrial biosphere model (TBM) spread in future C sink projections between models2. Accurate net CO2 flux projections rely on model ability to determine gross C fluxes. However, TBM inter-comparisons have shown strong discrepancies in gross C uptake (or gross primary production – GPP) related to both variability in growing season length and peak season magnitude3,4. One source of uncertainty in TBM simulations is due to fixed (often uncertain) parameter values. Significant progress has been made in using carbon cycle-related observations to constrain TBM parametric uncertainty via Bayesian data assimilation (DA) methods5,6,7,8,9,10. These datasets have included eddy covariance measurements of net ecosystem exchange (NEE)11, satellite-derived measures of vegetation dynamics12,13,14, and ground-based atmospheric CO2 concentration data. However, while there is considerable improvement in the simulation of leaf phenology and/or NEE in these studies, there is often a remaining model-data discrepancy in the gross C fluxes.
Satellite-derived measures of sun-induced chlorophyll fluorescence (SIF) offer a promising new direction to constrain simulated GPP at multiple scales15. SIF is strongly linked to GPP via its association with chlorophyll a absorption in plant photosynthetic machinery. The SIF-GPP relationship has been assessed at multiple scales using both process-based modelling and in situ and satellite observations. Frankenberg C. et al.16 and Guanter L. et al.17 were the first to report a linear relationship at global scale between monthly satellite-derived SIF data from the Greenhouse Gases Observing Satellite (GOSAT) and an up-scaled FLUXNET GPP product18. Similarly, GOME-2 SIF data (Global Ozone Monitoring Experiment-2 onboard the MetOp-A satellite) have also been shown to be linearly correlated with eddy covariance flux tower GPP measurements of gross C fluxes19,20,21. The consistency of this proposed linear SIF-GPP relationship across multiple spatial and temporal scales has been debated in the community. Several site-level studies found that while the SIF-GPP relationship in instantaneous leaf level measurements is non-linear and dependent on vegetation type, it becomes linear upon aggregation to the canopy, and at daily to seasonal time-scales21,22,23,24,25. Linear relationships have been even been found between flux tower GPP and instantaneous values derived from the OCO-2 (Orbiting Carbon Observatory) SIF product across a range of sites15,26,27. Most of these studies suggested that although temporal and/or spatial aggregation appears to result in increasing linearity in the SIF-GPP relationship, the slope remains biome-specific due to differences in canopy structure and biochemistry17,23,25,26,27 (though see ref.15).
Meanwhile, several modelling groups have used SIF data to optimise fluorescence model parameters23 and simple carbon cycle model physiology and leaf growth parameters28; to evaluate TBM GPP29,30; and to constrain TBM GPP outputs31. However, efforts are still underway to implement a mechanistic fluorescence models in TBMs32. Here, we explore a simpler approach to using SIF data to reduce TBM uncertainty by assuming a biome-specific linear relationship between SIF and GPP at broad spatial and temporal scales. More specifically, we use monthly aggregated 0.5 × 0.5° GOME-2 SIF data to optimise PFT-dependent GPP-related parameters in the ORCHIDEE TBM. Our key objective was to investigate if SIF data, and the simple linear SIF-GPP relationship, can be used to constrain regional to global, and monthly to annual modeled GPP. If successful, this simple approach – compared to the implementation of fluorescence processes in models – would allow a new, easy-to-implement method for using SIF data to optimise TBMs.
## Results
### Impact of the optimisation on GPP spatio-temporal patterns
Optimising the ORCHIDEE parameters related to photosynthesis, phenology and the linear SIF-GPP relationship using GOME-2 SIF data resulted in a considerable reduction in GPP at global scale (Fig. 1c,d, and e and Table 1 row 2). Specifically, strong reductions occurred in both densely forested regions of the boreal high northern latitudes, and tropical regions in South America and Southeast Asia (Fig. 1e and Table 1). There was a 28.8 PgCyr−1 reduction in the global mean (2007–2011) annual total GPP (Fig. 1c,d and Table 1); therefore, the posterior ORCHIDEE estimate of 165.6 PgCyr−1 is more in line with the JUNG estimate of 133.4 PgCyr−1 (Fig. 1b) compared to the prior value of 194.4 PgCyr−1. At biome level, the mean annual total GPP decreased from 88.6 to 67.1 PgCyr−1 in the temperate and boreal ecosystems, from 92.2 to 86.1 PgCyr−1 in the tropics, and from 13.6 to 12.4 PgCyr−1 in arid biomes (Table 1). In the temperate and boreal biomes, the larger share of the reduction in GPP occurred in boreal zones (classes D and E in the KG classification – see Methods): Temperate region decrease in GPP was only 40% that achieved in boreal zones. The largest decrease in annual GPP per unit area was seen in northern extra tropical latitudes (temperate and boreal biomes) followed by the tropical biome (Fig. 1e and Supplementary Table S2 column 2). This was associated with strong reduction in GPP (>0.5 kgCm−2 yr−1) for all boreal PFTs as well as temperate and tropical broadleaved evergreen trees (TeBE and TrBE – Supplementary Table S2 and see Fig. 5 for PFT acronym descriptions). For these PFTs, the mean reduction per PFT corresponds to ~40% of the prior temperate forest mean annual budget, ~60–110% of the prior boreal mean annual budget, and ~16% of the prior tropical forest mean. Overall, the optimisation resulted in a ~32% increase in the ratio of productivity per unit area between the tropical and extra-tropical (temperate + boreal) KG biomes (Supplementary Table S2). When considering total PgC, the increase in the productivity ratio was 24% (Table 1). This value increased to ~50% in the ratio per unit area between tropical and boreal-only KG biomes (~40% in total PgC). The posterior ratios (1.28 in total PgC for tropical: temperate + boreal biomes; 2.75 for tropical:boreal) better match the same ratios derived from JUNG dataset (1.22 for tropical: temperate + boreal biomes; and 2.54 for tropical:boreal). The shift in global productivity from high latitudes towards the tropics is broadly consistent with Parazoo N. C. et al.31, who used GOSAT SIF data to constrain the mean GPP of the TRENDY model inter-comparison outputs.
Interestingly, although arid biome mean annual GPP decreased slightly overall (Table 1), mean annual GPP increased in many dry/semi-arid zones including the Sahel, the dry tropics of North and South America, India, China and northern Australia (Fig. 1), due to a simulated higher GPP for C4 grasses and crops (and despite a decrease in tropical broadleaved deciduous tree productivity – Supplementary Table S2). Again, this result is broadly consistent with Fig. 7a in Parazoo N. C. et al.31. The increase in C4 GPP partially offset the observed reduction in GPP in tropical rainforests.
The mean uncertainty (1σ) in simulated global annual GPP for the 2007–2011 period was reduced by ~83% (Fig. 1f and Table 1) from 57.2 to 9.8 PgCyr−1. This global mean value is consistent with Norton A. J. et al.28 who achieved a 79% reduction in uncertainty when optimising the leaf growth and physiology parameters of a carbon cycle model. As in Norton A. J. et al.28, the highest reduction in uncertainty was seen in tropical biomes (~93%), whereas the lowest was found in northern temperate and boreal biomes (~67%) (Fig. 1f and Table 1). Posterior mean uncertainty is 2 PgCyr−1 for tropical biomes, 7.3 PgC−1 for temperate and boreal biomes, and 0.5 PgCyr−1 for arid biomes. The high percentage reduction in uncertainty in all biomes is likely an overestimate. This overestimate is partly caused by the fact we did not account for temporal error correlations; in reality therefore, the information content of the observations is likely lower. Another possible explanation is an overestimate of the prior error. The high prior uncertainty (and therefore strong reduction in uncertainty) in tropical regions is likely related to conservative priors (high uncertainty) for many of the TrBE photosynthesis parameters (PFT 2–1st column in Fig. 4). The considerable amount of noise in the GPP uncertainty reduction (Fig. 1f) is likely due to spatial heterogeneity in sub grid-cell PFT fraction.
At global-scale, the continental scale spatial patterns of GPP simulated by the ORCHIDEE model match both the SIF data and the independent JUNG dataset (Figs 1 and 2). However, we can determine if the optimisation had an impact on the finer-scale spatial distribution of simulated GPP by examining the spatial gradients between the prior and posterior in different regions (Figs 1 and 2). The posterior GPP corresponds more closely to JUNG in three regions: the northern extratropics (between ~30–75°N); in the tropics between ~10°N and 10°S; and in the southern hemisphere between 30 and 50°S (Fig. 2) – the same regions which show a strong reduction in GPP. In particular, both the north–south and east–west spatial gradient in the posterior GPP simulations across northern Eurasia appear to better approximate the SIF and JUNG products (Fig. 1). Due to the reduction in GPP in the Amazon, the posterior spatial gradient between the rainforest and Cerrado ecozones in South America is more consistent with SIF data (Fig. 1). In JUNG, there is a more distinct drop in mean annual GPP between the Amazon rainforest and Cerrado region to the southwest (Fig. 1). However, discrepancies between the model and both the JUNG and SIF products remain in the Caatinga, Cerrado and semi-deciduous forests of SE Brazil and in the north-south gradients in sub-Saharan Africa. Greater constraint on the regional to continental scale GPP spatial distributions may be hindered by the underlying PFT maps and the fact we optimised all parameters per PFT, rather than optimising all the parameters for each PFT in each grid cell for all cells. However, this remains both computationally unfeasible for the moment, and conceptually difficult given most TBMs are structured around the PFT-level parameterizations.
The mean monthly correlation between modelled GPP and GOME-2 SIF did not increase considerably as a result of the optimisation. The optimisation only resulted in a slight adjustment to the timing of seasonal cycle for each biome – resulting in a slightly shorter growing season length in the extratropics (Fig. 3) – and no change in phase. There was little change in the global grid cell mean correlation, nor the average across tropical, arid and temperate/boreal Köppen Geiger biomes (Table 1 columns 4 and 5), nor the average correlation for each PFT (Supplementary Table S3). The lack of improvement in the correlations is explained by the fact the prior correlations were already reasonably high. At PFT level, increases in R value that exceeded 0.04 were found for both boreal broadleaved deciduous trees (BoBD), and C3 grasses and crops (Supplementary Table S3). The decrease in mean annual GPP appears to be mostly associated with a decrease in peak growing season GPP in the northern latitudes (Fig. 3a). A smaller decrease in GPP magnitude is observed across the year for tropical and arid biomes (Fig. 3b,c). The seasonality in tropical regions does not match that of the data-driven JUNG estimate; however, this could not have been corrected by the optimisation due to the lack of an evergreen phenology module or time-varying physiological parameters in the current version of ORCHIDEE. Finally, the optimisation did not result in any discernible change in the long-term (1990–2012) global trend of increasing C uptake, which is ~0.64 PgCyr−1. The optimisation also did not change the global-scale or biome level long-term inter-annual variability (IAV) magnitude (3% reduction in global IAV standard deviation) or phase (global-scale correlation with JUNG decreased from 0.44 to 0.4).
### SIF optimisation constraint on the parameter values and uncertainty
The optimisation resulted in greater than 50% reduction in uncertainty for 111 out of a total of 172 parameters (Fig. 4). Despite the fact that the phase of the mean GPP seasonal cycle has not changed considerably between the prior and posterior simulations, the growing season length is slightly shorter post-optimisation in the northern extratropics (Fig. 3), and most phenology parameters were well constrained by the optimisation. Aside from Vcmax, the phenology parameters were generally better constrained compared to the photosynthesis parameters. A >50% reduction was achieved for both Vcmax and for all the phenology parameters, except Lfall, for the majority of PFTs optimised (Fig. 4; see Table 2 for a description of the parameters). Our results are contrary to Verma M. et al.26 and Koffi E. et al.32 who both found limited sensitivity of Vcmax to SIF using the SCOPE (Soil Canopy Observation, Photochemistry and Energy fluxes) model33,34. The slope and intercept of the SIF–GPP linear relationship (SIFa and SIFb parameters) were also highly constrained (>70% reduction in uncertainty) across all PFTs. Despite the high reduction in phenology parameter uncertainty, the timing (phase) of the GPP seasonal cycle was not altered dramatically by the optimisations. This was most likely due to the fact that the ORCHIDEE model already captures leaf seasonality well (Fig. 3); therefore, this suggests there was limited room for improvement in the posterior mean phenology values. However, it is also likely that phenology is insensitive to monthly SIF data given leaf onset and senescence occurs over a period of days rather than weeks. As found in previous studies using MODIS NDVI to optimise ORCHIDEE phenology-related parameters14, the optimisation resulted in an earlier start of leaf senescence, as evidenced by the higher values of Tsenes and Msenes,nosenes across most PFTs.
It is not clear which parameter, or set of parameters, is predominantly responsible for the widespread reduction in GPP across many regions and PFTs. Rather, posterior parameter values vary across PFTs for all parameters, with no clear pattern of increase or decrease (Supplementary Table S1; Fig. 4). This is likely due to parameter error correlation (Supplementary Fig. S1) and model equifinality, which arises when multiple sets of parameter values result in a similar fit to the data within the given uncertainties. The Topt and Tmin posterior errors are negatively correlated for all tropical and temperate forest PFTs and C3 plants, and Vcmax is correlated with at least one other parameter for all PFTs except BoND (Supplementary Fig. S1). Topt has increased for all PFTs that showed the strongest reduction in GPP, and Vcmax has decreased for all boreal PFTs that contribute to a large reduction in GPP across high northern latitudes (Fig. 4). The increase in Vcmax for C4 grasses could be responsible for the increase in GPP in semi-arid regions, together with the reduction in Kpheno,crit that causes an earlier start to the growing season. It is likely that a combination of all photosynthesis-related parameters, in addition to parameters that control the amount of leaf biomass available for C assimilation (SLA, LAImax and KLAI,happy), are responsible for the reduction in GPP magnitude for most PFTs. The notable increase in Kpheno,crit (later start to the growing season) for BoBD trees may also explain the high reduction in mean annual GPP for this PFT (see Supplementary Table S2).
The optimisation did not result in a high variability across PFTs in the posterior value of the slope (SIFa) parameter between SIF and GPP (Fig. 4). The mean posterior SIFa value across all PFTs (~0.21) is consistent with the mean slope across biomes (~0.17) derived by Guanter L. et al.17, even though there were differences in the SIF data (GOME-2 vs GOSAT), retrieval algorithm (statistical approach vs in-filling of solar Fraunhofer lines), and GPP model (TBM vs MTE upscaled fluxnet product) used in each analysis. The standard deviation in posterior SIFa values derived in this study is approximately double that calculated in Guanter L. et al.17. SIFa and SIFb were not strongly correlated with each other for all PFTs, although the slope parameter was correlated with at least one other parameter for all PFTs (Supplementary Fig. S1). The highest correlations (≥0.5) between SIFa and SIFb were found for tropical and temperate broadleaved trees and C4 plants (Supplementary Fig. S1). Strong correlations between SIFa and SIFb may either result from their relatively uninformative prior bounds (see Methods), compared to values presented in other studies (e.g. ref.17), or from high uncertainty in the SIF data. For PFTs that had the highest reduction in GPP (TeBE and boreal PFTs) the slope parameter was typically correlated with one other photosynthesis-related parameter (Vcmax and Fstress,h). However, it can be difficult to interpret or place too much confidence in conclusions drawn from the error covariance matrix (Supplementary Fig. S1) for parameters in complex, process-based TBMs, given all the cross-correlations between parameters. The SIF-GPP slope and intercept parameters also account for all fluorescence-related processes not represented in the model, as well as possible temporal and spatial scale mismatches between the model and data. A small number of parameters were ‘edge-hitting’ (5/172 in total), which suggests that missing processes in the model could partially account for remaining model-data misfit.
## Discussion
Within the confines of the current data assimilation set-up and mechanistic representation of the physical and biochemical processes, the SIF data strongly constrained and reduced the simulated global mean annual GPP in the ORCHIDEE model. This reduction in global GPP partially accounts for the presumed positive bias in model GPP compared to data-driven estimates such as JUNG18. It is impossible to fully validate the posterior global mean annual GPP estimate, or its trend over time, given the ongoing debate in the literature as to the most realistic value (see Anav A. et al.4 for further information); however, our results are in line with previous studies, as previously discussed. We do not aim to provide a definitive estimate of the most likely regional to global GPP values; rather, our objective was to determine whether SIF data could provide information for constraining the parameters and processes in model-based estimates of GPP. This, in turn, can provide a more rigorous, statistical and integrated model-data quantification of global mean annual GPP. Here, the SIF data were able to provide a strong constraint on the GPP magnitude in the ORCHIDEE TBM. We suggest this is a potentially important result, given the high degree of spread in GPP magnitude across TBMs is one of the principle sources of uncertainty in global GPP estimates4. Furthermore, our results suggest that optimising model GPP using SIF data can modify the finer-scale spatial gradients across tropical and pan-Eurasian biomes that studies have shown to be considerably different among TBMs4. Finally, this study has shown that SIF data can adjust the phenology-related ORCHIDEE TBM parameters in addition to those related to C assimilation. This should aid in correcting known biases in TBMs due to incorrect growing season length3. However, we suggest that SIF data could be used in combination with satellite-derived vegetation indices (VI, e.g. NDVI) for biomes in which the dynamics of C assimilation and leaf dynamics may be decoupled; for example, this is the case for evergreen forests35 and semi-arid ecosystems that controlled by moisture limitation36. Furthermore, it is likely that higher temporal resolution (ideally daily) VI and SIF data are needed to better approximate the short timescales associated to leaf onset and senescence.
We do not attempt explore all options in the data assimilation in order to provide the best optimisation set-up. However, the current assimilation set-up is largely similar to previous studies using FLUXNET net CO2 fluxes11 and NDVI14 to constrain the C fluxes and leaf phenology in the ORCHIDEE model, respectively. These studies used the same DA system, the same version of ORCHIDEE, and a similar number of sites. The only differences from previous assimilations were the number of parameters included in the optimisation (given the different processes being optimised) and the site locations (given the different datasets). The parameter prior values, uncertainties, and parameter bounds were the same. A comparison of the global GPP resulting from these assimilations shows that the SIF optimisation appears to result in larger decrease in global mean annual GPP than either of the aforementioned datasets (compare Supplementary Fig. S2d with Fig. S2e and f) due to a greater reduction in overall GPP magnitude worldwide and a decrease in peak growing season GPP in the northern extratropics (Supplementary Fig. S3). Furthermore, in contrast to the SIF optimisation, FLUXNET and NDVI data did not result in a distinct change in latitudinal GPP gradients (Supplementary Fig. S4). These results suggest that SIF data will be extremely useful in constraining GPP at global scale via parameter optimisation and state estimation. A full factorial experiment testing C cycle–related observations, including SIF, FLUXNET net and gross C flux data, and satellite-derived indices of vegetation dynamics (e.g. NDVI) would be needed to fully determine whether SIF data can provide a greater constraint on GPP magnitude and seasonality than other sources of data; however, this was beyond the scope of our study. We expect that SIF will provide a greater constraint on GPP than FLUXNET and satellite data alone.
Our work also demonstrates that it is not necessary to have an explicit mechanistic photosynthesis–fluorescence model in order to exploit SIF data to constrain estimates of GPP. The slope (SIFa) and intercept (SIFb) parameters of the simple linear SIF-GPP relationship are able to account for such missing processes, as well as any biases in the SIF data magnitude, or a mismatch in temporal and spatial scale between the model and data. However, due to correlations between SIFa or SIFb and the other model parameters, we have to exercise caution when using the derived photosynthesis and phenology parameter values in future simulations under conditions of changing climate. Implementing a mechanistic representation of fluorescence at leaf scale, and the scaling to canopy, may result in more realistic parameter values for use in future simulations; however this would increase the number of parameters that need to be optimised. The ability of the optimisation algorithm to find unique (un-correlated) values for a greater number of parameters would likely require a higher observation information content; this may or not be achievable with the currently available satellite-derived global SIF products.
Note that it is not a requirement that a simple empirical model between SIF and GPP takes a linear form, as used in this study. It is possible to implement more complex empirical SIF-GPP models in TBMs that are specific to each biome or dependent upon climate. Although recent studies have added weight to the notion that a linear relationship between GPP and SIF exists at multiple scales15,22,23,25,27, much work still needs to be done to confirm the consistency of the linear relationship under certain circumstances. These include periods of drought stress, complex canopy structures, and under different viewing and illumination geometries, among other factors. However, due to its close association with photosynthesis, we expect that SIF may provide even more information for constraining GPP than satellite-derived measures of vegetation dynamics (e.g. NDVI, EVI) in periods of water stress36. Therefore, future data assimilation experiments may take advantage of such aspects of SIF data to examine and constrain model behaviour in response to drought.
Finally, while the SIF data provided sufficient information to constrain parameters across the majority of PFTs – resulting in strong reductions in GPP in the northern latitudes and tropical regions – structural deficiencies or missing processes in the model may prevent any data assimilation experiment from being able to fully ‘correct’ modelled GPP. Examples in this version of ORCHIDEE include the lack of explicit phenology for evergreen trees, or nutrient (N and P) limitation on photosynthesis. Furthermore, given that parameters are typically PFT-dependent in most TBMs, spatial patterns of GPP require that the underlying PFT map is accurate. However, algorithms for deriving global maps of vegetation distribution are inherently uncertain; this in turn can result in considerable spread in model GPP estimates37. Issues related to prescribed PFT fractions may become obsolete if future model versions move from a discrete PFT based to a plant functional traits-based approach to simulating vegetation distributions (e.g. Bodegom P. M. van et al.38). As with any parameter data assimilation experiment, the full potential of SIF data for constraining GPP in global-scale TBMs may only be realised once forcing and model structural deficiencies are identified and resolved.
## Methods
### ORCHIDEE terrestrial biosphere model
ORCHIDEE is a global process-based terrestrial biosphere model (TBM) that calculates carbon, C, water and energy fluxes between the land surface and the atmosphere at a half-hourly time step39. It is the land surface component of the IPSL Earth System Model40. In this study, we use the ‘AR5’ version that contributed to the IPCC Fifth Assessment Report41. In the biogeochemical component of the model, C is assimilated via photosynthesis depending on light availability, CO2 concentration and soil moisture. Photosynthesis is modelled using functions based on Farquhar G. D. et al.42 for C3 plants and Collatz G. et al.43 for C4 plants. A prognostic leaf area index (LAI) is calculated on a daily time step. The phenology models control the timing of leaf onset and senescence and have been described in detail in MacBean N. et al.14. Note that there is no specific phenology model associated to evergreen ecosystems – leaf turnover is simply a function of climate and leaf-age. The plant functional type (PFT) concept in TBMs groups plants according to their physiological behaviour under similar climatic conditions. In ORCHIDEE each grid cell is described by the fractional coverage of all twelve vegetation-related PFTs plus bare soil (see Fig. 5 for PFT names and acronyms).
### Datasets
#### Solar-induced fluorescence (SIF)
In this study, we used near-infrared SIF derived from data acquired by the Global Ozone Monitoring Experiment-2 (GOME-2) instrument on board EUMETSAT’s polar orbiting Meteorological Operational Satellite-A (MetOp-A). The retrieval, described by Köhler P. et al.44, essentially disentangles the SIF emission from various spectral features related to atmospheric absorption, scattering, and surface reflectance. In particular, we use the monthly aggregated SIF data between 2007–2011 with a spatial resolution of 0.5° × 0.5° in our analysis.
#### Gross primary productivity (GPP)
Global gridded GPP products described in Jung M. et al.18 were used as an independent benchmark of the prior and posterior simulated GPP. The global gridded products are derived using a data-driven statistical Model Tree Ensemble (MTE) approach to upscale in situ eddy covariance C flux measurements from FLUXNET sites (http://fluxnet.fluxdata.org/)45. We used the May 2012 version of the gridded GPP product, hereafter referred to as ‘JUNG’. Gross CO2 fluxes (GPP and total ecosystem respiration) were derived from net CO2 fluxes using a flux partitioning method46. Although this product is data-driven, it relies heavily on machine learning and empirical models that contain their own assumptions. We do not therefore use this product to evaluate the model simulations, but rather for a comparison with another global scale product.
### Data assimilation framework and methodology
#### SIF-GPP linear relationship
In this version of the model, we assumed that SIF scales linearly with GPP, via the equation:
$$SIF=aGPP+b$$
(1)
We assumed this relationship holds at daily to monthly time steps and at the 0.5° spatial resolution of the simulations carried out in this study. As described in the introduction, this assumption is based on the results of several studies, which found that while instantaneous measurements of SIF and GPP at the leaf level are non-linear and dependent on vegetation type, the relationship becomes linear upon aggregation to the canopy and at daily to seasonal time-scales15,22,23,25,27. In the model, SIF is calculated at a daily time step, and a and b are PFT-specific parameters (SIFa and SIFb in Table 2) were optimised in the assimilation system. We assume that SIFa and SIFb absorb biases in the SIF retrieval algorithm, mismatches in magnitude between the model and data due to temporal and spatial averaging, and ORCHIDEE model structural issues (for example lack a fluorescence module and clear sky vs diffuse radiation effects).
#### Data assimilation system description
The ORCHIDEE Data Assimilation (DA) System (http://orchidas.lsce.ipsl.fr) is based on a variational DA system that has been described in detail in previous studies8,13,14. It follows a Bayesian framework, in which the optimal parameter vector x can be found by minimizing the following cost-function J(x), assuming that the probability distribution functions (PDFs) of the model parameter and observation uncertainties are Gaussian47:
$${J}({\boldsymbol{x}})=\frac{1}{2}[{({H}({\boldsymbol{x}})-{\boldsymbol{y}})}^{{\boldsymbol{T}}}{{\bf{R}}}^{-1}({\boldsymbol{H}}({\boldsymbol{x}})-{\boldsymbol{y}})+{({\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\boldsymbol{b}}})}^{{\boldsymbol{T}}}{{\bf{P}}}_{{\boldsymbol{b}}}^{-1}({\boldsymbol{x}}-{{\boldsymbol{x}}}_{{\boldsymbol{b}}})]$$
(2)
where xb are the a priori parameter values, Pb the a priori uncertainty matrix of the parameters, y is the observation vector, H(x) the model outputs, given parameter vector x, and R the uncertainty matrix of the observations (including observation and model errors). In this case, x is a vector of the parameters listed in Table 2), and H(x) is the simulated SIF derived from equation (1). The observation and model errors are assumed to be uncorrelated in space and time, and parameters are assumed to be independent; hence R and Pb are diagonal matrices. The cost function is iteratively minimised using the gradient-based L-BFGS-B algorithm48, which allows for fixed parameter bounds. In this version of the model the gradient of the J(x) (termed the Jacobian) is estimated using a finite difference method. To improve the efficiency of the L-BFGS-B algorithm, the range of variation of the parameters is standardised by scaling parameter vector, x, their priors and uncertainties following: $${\boldsymbol{x}}^{\prime} ={{\bf{P}}}_{b}^{-\frac{1}{2}}({\boldsymbol{x}}-{{\boldsymbol{x}}}_{b})$$49.
In order to quantify the constraint brought by the assimilation on the model parameters and state variables, we calculate the posterior parameter error covariance matrix, Ppost using the Jacobian of the model at the minimum of J(x), $${{\bf{H}}}_{\infty }$$, following Tarantola A.47 :
$${{\bf{P}}}_{{\rm{post}}}={[{{\bf{H}}}_{\infty }^{T}{{\bf{R}}}^{-1}{{\bf{H}}}_{\infty }+{{\bf{P}}}_{b}^{-1}]}^{-1}$$
(3)
Ppost is then propagated onto the model state variables (e.g. GPP) space given the following matrix product and the hypothesis of local linearity47:
$${{\bf{R}}}_{{post}}={\bf{H}}{{\bf{P}}}_{{post}}{{\bf{H}}}^{T}$$
(4)
The square root of the diagonal elements of Rpost corresponds to the posterior error (standard deviation, 1σ). In order to assess the improvement brought by the assimilation, the error reduction is determined as 1 − (Rpost / Rprior).
#### Optimised parameters and derivation of prior values and uncertainties
We optimised model parameters involved in the calculation of seasonal uptake of carbon (GPP) in ORCHIDEE, including parameters related to photosynthesis and leaf phenology, as well as SIFa and SIFb parameters in equation (1) (Table 2). All parameters were optimised for each PFT. For the photosynthesis and phenology parameters, the prior values were taken from the ORCHIDEE standard version (Supplementary Table S2). The parameter maximum and minimum bounds were based on literature and elicitation of ‘expert’ knowledge. To derive the SIFa and SIFb prior values and bounds we first examined the distribution of the slope and intercept parameters resulting from a linear regression between the SIF data and both the prior ORCHIDEE simulation and the ‘JUNG’ product for each PFT. Considering these distributions, the slope parameter bounds were set to −0.5 to 1, and the intercept parameter bounds were set to −1.5 to 2.5. The prior values for both SIFa and SIFb were set to 0.25. The prior uncertainty for all parameters was set at 40% of the parameter range following Kuppel S. et al.50.
#### PFTs optimised and site selection
We performed a multi-site optimisation, in which all sites are included in the same optimisation, following11,14,50, for each of the 12 vegetated PFTs. Following MacBean N. et al.14 we selected 15 grid cells that fulfilled several constraints to optimise for each PFT (Fig. 5). This was achieved by a random sampling of grid cells with a fractional coverage above the given threshold. First, the grid cells had to contain a high fraction of the PFT in question. This was mostly >0.6 except for the BoBD PFT where the fractional cover is never >0.3. Second, the site locations had to be representative of the overall spatial distribution of each PFT. No more than 15 grid cells per PFT were chosen in this study for two reasons: (1) the lack of representative grid cells for certain PFTs at this spatial scale; and (2) the computational time required to perform a multi-site assimilation.
### Model setup and experiments performed
#### Model setup
In this study, ORCHIDEE is used in forced offline mode and driven by 6-hourly CRU-NCEP v5 meteorological fields at 0.5 × 0.5° resolution corresponding to the resolution of the SIF data. We used the Olson land cover classification to prescribe PFT fractions51 and the soil map from Zobler L.52 to prescribe texture classes for each grid cell. The impacts of land use change, forest management, harvesting and fires were not included.
#### Assimilation experiments and posterior model simulation
For all selected sites, we performed a ‘spin-up’ of the vegetation state to ensure the model reaches an equilibrium for GPP. The spin-up simulation was used forcing data cycled over the 50 years prior to the observation period (1956–2006). Next, we performed the assimilation simulation for the observation period (2007–2011). Finally, global-scale simulations were performed with the prior and posterior parameter vectors in order to evaluate the impact of the optimisation on the regional and global simulated GPP.
#### Biome level posterior analyses
Biome level analyses were based on the Köppen Geiger (KG) climate classification scheme derived by Peel M. C. et al.53. We grouped the main climate classes in the following way: ‘tropical biomes’ correspond to class A in KG scheme; ‘arid biomes’ correspond to class B; and ‘temperate and boreal biomes’ encompass classes C to E.
### Data availability
The GOME-2 level 1B product, containing necessary radiance spectra and daily solar irradiance measurements, was obtained from EUMETSAT’s Data Centre (https://www.eumetsat.int/website/home/Data/DataDelivery/EUMETSATDataCentre/index.html). The SIF retrieval method using the GOME-2 level 1B product is described in Köhler et al. (2015). The 6-hourly CRU-NCEP v5 meteorological fields at 0.5 × 0.5° resolution are available from https://esgf.extra.cea.fr/thredds/catalog/store/p529viov/cruncep/V5_1901_2013/catalog.html. The GPP MTE data (referred to as the ‘JUNG’ product in this study) were downloaded from the GEOCARBON data portal (FileID 66) in compliance with the Data Usage Agreement (https://www.bgc-jena.mpg.de/geodb/projects/Home.php).
The ORCHIDEE model is under a free software license (CeCILL; see http://www.cecill.info/index.en.html). The ORCHIDEE model source code is visible here: https://forge.ipsl.jussieu.fr/orchidee/browser/tags/ORCHIDEE_1_9_6.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Change history
• ### 05 July 2018
A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.
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## Acknowledgements
This work was supported and co-funded by the ESA FLEX-Bridge Project (http://www.flex-photosyn.ca/FB_HOME.htm), the Copernicus Atmosphere Monitoring Service (CAMS41 project) implemented by the European Centre for Medium-Range Weather Forecasts (ECMWF) on behalf of the European Commission, and the CNES TOSCA Flu-OR Project. LG and PK were funded by the Emmy Noether Programme of the German Research Foundation (GU 1276/1-1).
## Author information
### Author notes
• Natasha MacBean
Present address: University of Arizona, School of Natural Resources and the Environment, Tucson, Arizona, 85719, USA
### Affiliations
1. #### Laboratoire des Sciences du Climat et de l’Environnement, LSCE/IPSL, CEA-CNRS-UVSQ, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France
• Natasha MacBean
• , Fabienne Maignan
• & Philippe Peylin
2. #### NOVELTIS, Labège, France
• Cédric Bacour
3. #### Department of Geography, University College London, Pearson Building, Gower Street, London, WC1E 6BT, London, UK
• Philip Lewis
• , Jose Gómez-Dans
• & Mathias Disney
4. #### NERC National Centre for Earth Observation (NCEO), Swindon, UK
• Philip Lewis
• , Jose Gómez-Dans
• & Mathias Disney
5. #### Helmholtz Centre Potsdam-GFZ German Research Centre for Geosciences, Potsdam, Germany
• Luis Guanter
6. #### California Institute of Technology (Caltech), Division of Geological and Planetary Sciences, Pasadena, CA, USA
• Philipp Köhler
### Contributions
N.M. instigated and designed the study in discussion with F.M., P.L., P.P., and C.B. N.M. conducted all model simulations, assimilation runs, and posterior analyses, with support from C.B. on the assimilation system and calculation of the posterior covariance matrix. L.G. and P.K. provided the SIF data and provided details, guidance and comments on characteristics and proper use of the data. N.M. discussed technical details of assimilation set-up and initial interpretation of results with P.L., P.P., C.B., F.M. and J.G.D. N.M. drafted the manuscript. All authors provided detailed comments on the manuscript draft, including further interpretation of the results.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding author
Correspondence to Natasha MacBean. | 2019-01-20 05:31:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5598776936531067, "perplexity": 6263.006594547712}, "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/1547583700012.70/warc/CC-MAIN-20190120042010-20190120064010-00060.warc.gz"} |
http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.102.136806 | # Synopsis: Dirt and geometry insulate electrons
Random disorder in topological insulators leads to an insulating phase reminiscent of one known for years in two-dimensional systems.
Random disorder can render a two-dimensional system of noninteracting electrons into an insulating state known as the Anderson insulator [1]. Another well-known manifestation of two-dimensional physics—the integer quantum Hall effect—is the formation of dissipationless current-carrying edge states in the presence of a magnetic field.
Writing in Physical Review Letters, Jian Li, Rui-Lin Chu, and Shung-Qing Shen of The University of Hong Kong and Jainendra Jain of Pennsylvania State University in the US address how disorder affects edge states in topological insulators, a class of band insulators that exhibit strange conduction properties similar to what is seen in quantum Hall states, but in the absence of an external magnetic field. (See also the Viewpoint on topological insulators [2].)
It is known that the physics of topological insulators is immune to weak disorder. However, the authors also predict a surprising phase in $\text{HgTe/CdTe}$ quantum well topological insulators. They call this phase the topological Anderson insulator, where disorder introduces two key differences from previously studied topological insulators: The Fermi energy lies in a so-called mobility gap, as opposed to a “real” gap, and the edge states do not appear to depend on the specific band structure of the quantum wells. That said, these $\text{HgTe/CdTe}$ quantum wells possess an “inverted” band structure and offer the possibility to considerably tweak their transport and structure properties, which in turn promises further insights into how disorder and doping modify the phase diagram of topological insulators. – Sami Mitra
[1] E. Abrahams, Phys. Rev. Lett. 42, 673 (1979).
[2] S. C. Zhang, Physics 1, 6 (2008).
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https://forums.novell.com/showthread.php/485557-Driver-cache-not-processing | ## Driver cache not processing
We had some password changes not go through and after some TS we found
the cache on our eDir2eDir driver had over 700 modify event dating back
to 2 weeks. From time to time we have had an event get stuck in the
cache (still haven't found a solution to that problem) and we have just
made a note of the first item to fix later and then deleted it and all
the subsequent item start getting processed. However, this time items
still remain unprocessed and we have removed the first 3 items. I don't
know how to proceed now. Would restarting eDirectory help? Any ideas?
--
bobbintb
------------------------------------------------------------------------
bobbintb's Profile: https://forums.netiq.com/member.php?userid=5629 | 2019-04-20 10:46:44 | {"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": 0.8558394908905029, "perplexity": 3262.282844353966}, "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/1555578529606.64/warc/CC-MAIN-20190420100901-20190420122901-00100.warc.gz"} |
https://support.numxl.com/hc/en-us/sections/201759493-Spectral-Analysis | # Spectral Analysis
In statistics, spectral analysis is a procedure that decomposes a time series into a spectrum of cycles of different lengths. Spectral analysis is also known as frequency domain analysis. | 2017-08-18 12:33:10 | {"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": 0.9465433359146118, "perplexity": 679.8070242054637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104636.62/warc/CC-MAIN-20170818121545-20170818141545-00077.warc.gz"} |
http://mathematica.stackexchange.com/questions/9201/parts-of-module-body-evaluated-in-external-scope?answertab=active | # Parts of Module body evaluated in external scope?
I have an expression that suggests that some expressions in a module body are dragging in definitions from outside the scope in a surprising way. First, consider a symbolic constant, y:
ClearAll[y]
Now, define a couple of expressions that should evaluate to this constant, one immediate and one delayed:
z = y; w := y;
Let's also define a "function" v (actually a rewrite rule in the DownValuess of v), that will return its argument:
v[y_] = y;
Now, the surprise is that if I define a local variable y in a Module, various symbolic expressions involving the symbol y, specifically w and z, seem to be evaluated in the environment outside the Module, where y evaluates to itself, and other expressions involving the symbol y, namely y, v[y], and D[y^2/2, y] seem to be evaluated in terms of the local variable. To wit:
Module[{y}, {w, v[y], y, z, D[y^2/2, y]}]
produces
{y, y$668, y$668, y, y$668} Naturally, if I give the local variable y a value from outside, then I don't see the secret fresh variable: ClearAll[x]; Module[{y = x}, {w, v[y], y, z, D[y^2/2, y]}] produces {y, x, x, y, x} I surmise that what's happening is that in a pre-evaluation step, any overt occurrences of y are rewritten to the (value of the) fresh variable and then the body is evaluated. Thus, w and z don't get evaluated until after an equivalent to ReplaceAll[Hold[{w, v[y], y, z, D[y^2/2, y]}], y -> y$668]
or
ReplaceAll[Hold[{w, v[y], y, z, D[y^2/2, y]}], y -> x]
is done.
Is my surmising correct?
-
@Verde... Amen. – Rojo Aug 8 '12 at 2:18
@Mr.Wizard -- my "backing story" is symbolic expressions, that is, expressions wherein the symbols don't have values. For instance, the results of Solve[...]. When I refer to such symbolic expressions in various naming environments, like Module, Block, and With, but also bodies of Functions and other expressions where replacements of symbols occur, I must have total control of any possibility of "capturing" the "symbolic constants" in my symbolic expressions, lest they change meaning silently. – Reb.Cabin Aug 8 '12 at 12:08
Module does lexical scoping. This means that whatever is explicitly passed as a parameter is applied a replacement rule with the new temporary variable, just like you suggested with your ReplaceAll snippet.
If you are looking for dynamic scoping, try Block. This means that while the evaluation of the Block is taking place, all calls to the localized symbol will not find the "external" values, but only those defined while running the Block.
Block localizes the execution, Module whats explicitly written.
I'm sure there are a couple of good in-depth answers in the forum about Block vs Module if you do a search.
Here is a small code snippet illustrating this point:
ClearAll[z, y, w]
z = y; w := y;
v[y_] = y;
Module[{y = x}, {w, v[y], y, z, D[y^2/2, y]}]
(* ==> {y, x, x, y, x} *)
Block[{y = x}, {w, v[y], y, z, D[y^2/2, y]}]
(* ==> {x, x, x, x, x} *)
{w, v[y], y, z, D[y^2/2, y]}
(* ==> {y, y, y, y, y} *)
-
Trying to understand your reply ... "whatever is explicitly passed as a parameter is replaced ...". Can you tell me what is "explicitly passed as a parameter" in the expression Module[{y}, {w, v[y], y, z, D[y^2/2, y]}]? – Reb.Cabin Aug 8 '12 at 2:30
@Reb.Cabin it's {w, v[y], y, z, D[y^2/2, y]} (and {y}, but I was referring to the "expression" you pass as a paramter, the second argument. If you had done expr={w, v[y], y, z, D[y^2/2, y]}; Module[{y}, expr] none would have been replaced – Rojo Aug 8 '12 at 2:32
One of the most clear explanations around for a never ending beginner's nightmare +1 – belisarius Aug 8 '12 at 2:33
Let me see if I understand you correctly. "Lexical scoping" means that the body (2d argument) of the Module is held unevaluated, then subjected to the substitutions presented in the bindings (1st argument to Module), then allowed to evaluate. "Dynamic Binding" means that current values of the variables named in the bindings (1st argument to Block) are saved, new values are assigned to the variables based on the expressions on the right-hand side of each binding evaluated in the global environment, then the body of the block is evaluated, then the variables restored from saved. Correct? – Reb.Cabin Aug 8 '12 at 5:11
@Reb.Cabin exactly! Functions with attribute HoldAll almost always do something to its arguments before evaluating them, just like you described with Block and Module. Also, if you want to talk about the restoring, there's also the step in which Module removes the temporary variables created if there are no references to them once the Module` is over. – Rojo Aug 8 '12 at 10:03 | 2014-11-29 06:20:51 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3908553719520569, "perplexity": 3588.253325129242}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931013466.18/warc/CC-MAIN-20141125155653-00135-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://iidb.org/threads/bidengate-breaks-err-down.22540/page-4#post-999182 | # Bidengate Breaks... err... down
#### blastula
##### Contributor
I think you are misinterpreting my use of the word "admitted". Here's a clip from the reason.com article in my OP:
While the Times did not ignore the story, the paper's coverage treated it with skepticism and disdain.
Similarly, I can say when I was 10 years old that I was skeptical of life elsewhere in the universe, but now at age 60 after considering it more thoroughly, I admit now that I do believe it to be true. It doesn't necessarily mean I knew it to be true all along. Perhaps I could have used a better choice of words. I never intended to claim that they knew it to be true, but were hiding it.
And they were correct to treat it skeptically, as it was unproven. Weird to criticize them for doing their literal job.
I also never claimed the NYT said it was Russian disinformation. Here's another clip from the reason.com article I posted above:
Pro-Biden, anti-Trump journalists, including several at the Times, portrayed the Post's story as unsubstantiated at best. Politico reported that "more than 50 former senior intelligence officials" believed the emails had "all the classic earmarks of a Russian information operation."
What I said was, in my own words:
The prevailing theory back in 2020 by many was that the laptop contents did not originate from Hunter, but were Russian fabricated disinformation.
"By many", I meant people on this forum as well as in the general public, not necessarily the NYT.
So, no crow tonight (not that I haven't had my share over the years). I do believe I will head to Lucky's soon and get me a rotissiere chicken (Lemon Pepper spice). though. And some potato salad.
Nice to see you admit now that the NYT did not say it was Russian disinformation. Even though NYT is the only media you mentioned.
But again, it was all speculation either way. Nobody knew. People that believed that they were real emails were just as unfounded in their belief. So what if someone believed they were fake? It doesn't take TDS to think Giuliani was spouting fake news.
#### thebeave
##### Veteran Member
I feel like we are seeing a rather transparent attempt by the post to claim something not in evidence.
Uh oh...
The New York Times Belatedly Admits the Emails on Hunter Biden's Abandoned Laptop Are Real and Newsworthy
Yesterday The New York Times published a story that quotes emails from a laptop that Hunter Biden, President Joe Biden's son, abandoned at a computer repair shop in Delaware. The messages reinforce the impression that Burisma, a Ukrainian energy company that reportedly paid the younger Biden $50,000 a month to serve on its board, expected him to use his influence with his father for the company's benefit—an allegation that figured prominently in the scandal that led to Donald Trump's impeachment for pressuring the Ukrainian government to announce a Biden-Burisma corruption investigation. The messages include evidence that Hunter Biden arranged an April 2015 meeting between his father, then the vice president, and a Burisma executive. Looking at that bolded part - the recent Times story seems ot say nothing of the kind. Current actual NYT story said: In one email to Mr. Archer in April 2014, Mr. Biden outlined his vision for working with Burisma. In the email, Hunter Biden indicated that the forthcoming announcement of a trip to Ukraine by Vice President Biden — who is referred to in the email as “my guy,” but not by name — should “be characterized as part of our advice and thinking — but what he will say and do is out of our hands.” The announcement “could be a really good thing or it could end up creating too great an expectation. We need to temper expectations regarding that visit,” Hunter Biden wrote. Vice President Biden traveled to Kyiv, the Ukrainian capital, about a week after the email. In the same April 2014 email, Hunter Biden indicated that Burisma’s officials “need to know in no uncertain terms that we will not and cannot intervene directly with domestic policymakers, and that we need to abide by FARA and any other U.S. laws in the strictest sense across the board.” He suggested enlisting the law firm where he worked at the time, Boies Schiller Flexner, to help Burisma through “direct discussions at state, energy and NSC,” referring to two cabinet departments and the National Security Council at the White House. ADVERTISEMENT Continue reading the main story The firm “can devise a media plan and arrange for legal protections and mitigate U.S. domestic negative press regarding the current leadership if need be,” Mr. Biden wrote in the email. Mr. Biden, Mr. Archer, Boies Schiller Flexner and Blue Star Strategies did not register under FARA on behalf of Burisma. In another set of emails examined by prosecutors, Hunter Biden and Mr. Archer discussed inviting foreign business associates, including a Burisma executive, to a dinner in April 2015 at a Washington restaurant where Vice President Biden would stop by. It is not clear whether the Burisma executive attended the dinner, although the vice president did make an appearance, according to people familiar with the event. So, no, the story does not “ include evidence that Hunter Biden arranged an April 2015 meeting between his father, then the vice president, and a Burisma executive.” He does not arrange a meeting between, he tells the executive where his father is going to be. If I told a criminal that you had plans to be at a certain restaurant, does that mean I’ve “arranged a meeting between you”? Come on. But that was then. A year and a half later, the Times thinks the emails it viewed as suspect before Joe Biden's election are now newsworthy. Again, no. It is doing a story on the tax question that is legitimately in the news because of legal dates, and is including the background. Come on, “they now think it is newsworthy?” No, they did not change their minds, they are reporting it exactly the same way they did before. "People familiar" with a federal investigation of Hunter Biden, it reports, "said prosecutors had examined emails" between him, his former business partner Devon Archer, "and others" regarding "Burisma and other foreign business activity." Those emails "were obtained by The New York Times from a cache of files that appears to have come from a laptop abandoned by Mr. Biden in a Delaware repair shop." The messages "were authenticated by people familiar with them and with the investigation." Reading through this thread, it seems some people here might be eating crow with a side of mashed potatoes for dinner tonight. It seems like you might want to pick up a fork for that crow pie. Your premise that the NYT “now thinks it’s newsworthy” is bunk. Your premise that the emails “show Hunter arranging a meeting” is bunk. It looks exactly as it did before. A connected kid trying to make money off his connections, and having nothing but his own promises to sell. He even explicitly says he can’t promise Joe’s cooperation. None of these show that Hunter was successful at it, or that Joe Biden cooperated in any way. Just as unfounded and dumb as it was 18 months ago. And just as unfounded and dumb to say that either the NYT changed its mind or that we should. You seem to be missing the point, or perhaps I wasn't quite clear enough. The issue at hand is that the NYT has essentially now admitted that the emails did come from Hunter Biden's laptop that was left in a Delaware repair shop, and that those emails were authenticated. Here is cut-and-paste what the NYT said in their article: People familiar with the investigation said prosecutors had examined emails between Mr. Biden, Mr. Archer and others about Burisma and other foreign business activity. Those emails were obtained by The New York Times from a cache of files that appears to have come from a laptop abandoned by Mr. Biden in a Delaware repair shop. The email and others in the cache were authenticated by people familiar with them and with the investigation. The prevailing theory back in 2020 by many was that the laptop contents did not originate from Hunter, but were Russian fabricated disinformation. Or that the laptop never existed in the first place. You have focused on whether the reason.com article I quoted faithfully relayed the details of the NYT article, which it does seem they could have done a better job. What was the "Uh oh..." about? We're led to believe by your font bolding and color that those specific sentences and phrases were the "uh oh..." part. Is it real and newsworthy that Hunter Biden [insert here] or is the whole point about blatant media manipulation or whatever? All that bolding is from someone else not me! Check out post #125 if you want to see my original post from yesterday. #### thebeave ##### Veteran Member I think you are misinterpreting my use of the word "admitted". Here's a clip from the reason.com article in my OP: While the Times did not ignore the story, the paper's coverage treated it with skepticism and disdain. Similarly, I can say when I was 10 years old that I was skeptical of life elsewhere in the universe, but now at age 60 after considering it more thoroughly, I admit now that I do believe it to be true. It doesn't necessarily mean I knew it to be true all along. Perhaps I could have used a better choice of words. I never intended to claim that they knew it to be true, but were hiding it. And they were correct to treat it skeptically, as it was unproven. Weird to criticize them for doing their literal job. I also never claimed the NYT said it was Russian disinformation. Here's another clip from the reason.com article I posted above: Pro-Biden, anti-Trump journalists, including several at the Times, portrayed the Post's story as unsubstantiated at best. Politico reported that "more than 50 former senior intelligence officials" believed the emails had "all the classic earmarks of a Russian information operation." What I said was, in my own words: The prevailing theory back in 2020 by many was that the laptop contents did not originate from Hunter, but were Russian fabricated disinformation. "By many", I meant people on this forum as well as in the general public, not necessarily the NYT. So, no crow tonight (not that I haven't had my share over the years). I do believe I will head to Lucky's soon and get me a rotissiere chicken (Lemon Pepper spice). though. And some potato salad. Nice to see you admit now that the NYT did not say it was Russian disinformation. Even though NYT is the only media you mentioned. But again, it was all speculation either way. Nobody knew. People that believed that they were real emails were just as unfounded in their belief. So what if someone believed they were fake? It doesn't take TDS to think Giuliani was spouting fake news. Arg. Show me where I ever explicitly claimed the NYT said it was Russian disinformation. That was Politico who was going with that, per the reason.com article. I personally just said "many people" thought it was Russian disinformation back in 2020. But I do agree with you that it was a new story and not yet fleshed out or investigated. So I don't blame the NYT for their initial skeptical stance, nor am I critisizing them for that. Really, I'm kinda beating on the people and especially the media who completely dismissed it as a nutjob conspiracy from the get-go, and I'm using the NYT ( a decidedly left wing source) as the vehicle to show that the laptop story has got legs. #### Angry Floof ##### Tricksy Leftits Staff member I feel like we are seeing a rather transparent attempt by the post to claim something not in evidence. Uh oh... The New York Times Belatedly Admits the Emails on Hunter Biden's Abandoned Laptop Are Real and Newsworthy Yesterday The New York Times published a story that quotes emails from a laptop that Hunter Biden, President Joe Biden's son, abandoned at a computer repair shop in Delaware. The messages reinforce the impression that Burisma, a Ukrainian energy company that reportedly paid the younger Biden$50,000 a month to serve on its board, expected him to use his influence with his father for the company's benefit—an allegation that figured prominently in the scandal that led to Donald Trump's impeachment for pressuring the Ukrainian government to announce a Biden-Burisma corruption investigation. The messages include evidence that Hunter Biden arranged an April 2015 meeting between his father, then the vice president, and a Burisma executive.
Looking at that bolded part - the recent Times story seems ot say nothing of the kind.
Current actual NYT story said:
In one email to Mr. Archer in April 2014, Mr. Biden outlined his vision for working with Burisma. In the email, Hunter Biden indicated that the forthcoming announcement of a trip to Ukraine by Vice President Biden — who is referred to in the email as “my guy,” but not by name — should “be characterized as part of our advice and thinking — but what he will say and do is out of our hands.”
The announcement “could be a really good thing or it could end up creating too great an expectation. We need to temper expectations regarding that visit,” Hunter Biden wrote.
Vice President Biden traveled to Kyiv, the Ukrainian capital, about a week after the email.
In the same April 2014 email, Hunter Biden indicated that Burisma’s officials “need to know in no uncertain terms that we will not and cannot intervene directly with domestic policymakers, and that we need to abide by FARA and any other U.S. laws in the strictest sense across the board.”
He suggested enlisting the law firm where he worked at the time, Boies Schiller Flexner, to help Burisma through “direct discussions at state, energy and NSC,” referring to two cabinet departments and the National Security Council at the White House.
The firm “can devise a media plan and arrange for legal protections and mitigate U.S. domestic negative press regarding the current leadership if need be,” Mr. Biden wrote in the email.
Mr. Biden, Mr. Archer, Boies Schiller Flexner and Blue Star Strategies did not register under FARA on behalf of Burisma.
In another set of emails examined by prosecutors, Hunter Biden and Mr. Archer discussed inviting foreign business associates, including a Burisma executive, to a dinner in April 2015 at a Washington restaurant where Vice President Biden would stop by. It is not clear whether the Burisma executive attended the dinner, although the vice president did make an appearance, according to people familiar with the event.
So, no, the story does not “ include evidence that Hunter Biden arranged an April 2015 meeting between his father, then the vice president, and a Burisma executive.”
He does not arrange a meeting between, he tells the executive where his father is going to be. If I told a criminal that you had plans to be at a certain restaurant, does that mean I’ve “arranged a meeting between you”?
Come on.
But that was then. A year and a half later, the Times thinks the emails it viewed as suspect before Joe Biden's election are now newsworthy.
Again, no.
It is doing a story on the tax question that is legitimately in the news because of legal dates, and is including the background.
Come on, “they now think it is newsworthy?”
No, they did not change their minds, they are reporting it exactly the same way they did before.
"People familiar" with a federal investigation of Hunter Biden, it reports, "said prosecutors had examined emails" between him, his former business partner Devon Archer, "and others" regarding "Burisma and other foreign business activity." Those emails "were obtained by The New York Times from a cache of files that appears to have come from a laptop abandoned by Mr. Biden in a Delaware repair shop." The messages "were authenticated by people familiar with them and with the investigation."
Reading through this thread, it seems some people here might be eating crow with a side of mashed potatoes for dinner tonight.
It seems like you might want to pick up a fork for that crow pie.
Your premise that the NYT “now thinks it’s newsworthy” is bunk.
Your premise that the emails “show Hunter arranging a meeting” is bunk.
It looks exactly as it did before. A connected kid trying to make money off his connections, and having nothing but his own promises to sell. He even explicitly says he can’t promise Joe’s cooperation. None of these show that Hunter was successful at it, or that Joe Biden cooperated in any way.
Just as unfounded and dumb as it was 18 months ago. And just as unfounded and dumb to say that either the NYT changed its mind or that we should.
You seem to be missing the point, or perhaps I wasn't quite clear enough. The issue at hand is that the NYT has essentially now admitted that the emails did come from Hunter Biden's laptop that was left in a Delaware repair shop, and that those emails were authenticated. Here is cut-and-paste what the NYT said in their article:
People familiar with the investigation said prosecutors had examined emails between Mr. Biden, Mr. Archer and others about Burisma and other foreign business activity. Those emails were obtained by The New York Times from a cache of files that appears to have come from a laptop abandoned by Mr. Biden in a Delaware repair shop. The email and others in the cache were authenticated by people familiar with them and with the investigation.
The prevailing theory back in 2020 by many was that the laptop contents did not originate from Hunter, but were Russian fabricated disinformation. Or that the laptop never existed in the first place. You have focused on whether the reason.com article I quoted faithfully relayed the details of the NYT article, which it does seem they could have done a better job.
What was the "Uh oh..." about? We're led to believe by your font bolding and color that those specific sentences and phrases were the "uh oh..." part. Is it real and newsworthy that Hunter Biden [insert here] or is the whole point about blatant media manipulation or whatever?
All that bolding is from someone else not me! Check out post #125 if you want to see my original post from yesterday.
Ah, ok. But still, what was the "uh oh..." and what are people supposed to be eating crow about?
##### Loony Running The Asylum
Staff member
Are there any emails from Hunter saying he would actually do what Burisma wanted him to do?
And how do we know these emails are legitimate? These computers passed through many hands before getting to the FBI.
Exactly. I find the providence of the laptop sorely lacking. I strongly suspect most of the e-mails are legitimate but given the providence I consider it likely that things have been tampered with. Thus it means nothing--and showing that much of it is true means nothing, it could easily be someone hacked his system, got his old e-mails and then changed things a bit.
Don't go looking for suspicious stuff on that hard drive, address the chain of custody first!
How would the Russians do all that, though? Are they logging into his laptop from Russia, or are the Russian hackers here in the US somehow given access to it? Or did they take the laptop to Russia? Wouldn't a competent computer forensics team be able to tell if the emails or other information were altered?
Where above is anyone suggesting it was Russians??? Rudy Giuliani had the laptop for a time. You remember Rudy, right? The guy that runs a security consulting firm and tried to get Zelensky to dig up dirt on Biden. Yeah, he wouldn't have access to anyone who could screw with that laptop.
Loren himself, in post #22. Here, I'll save you some time:
Yeah, as more comes out it looks like the whole thing is faked. They got the e-mails by some illicit means (probably Moscow) and are using harmless stuff as support for the spicy stuff they faked.
You'll have to take that up with Loren.
Now, do you have anything to say about my comment?
#### thebeave
##### Veteran Member
Are there any emails from Hunter saying he would actually do what Burisma wanted him to do?
And how do we know these emails are legitimate? These computers passed through many hands before getting to the FBI.
Exactly. I find the providence of the laptop sorely lacking. I strongly suspect most of the e-mails are legitimate but given the providence I consider it likely that things have been tampered with. Thus it means nothing--and showing that much of it is true means nothing, it could easily be someone hacked his system, got his old e-mails and then changed things a bit.
Don't go looking for suspicious stuff on that hard drive, address the chain of custody first!
How would the Russians do all that, though? Are they logging into his laptop from Russia, or are the Russian hackers here in the US somehow given access to it? Or did they take the laptop to Russia? Wouldn't a competent computer forensics team be able to tell if the emails or other information were altered?
Where above is anyone suggesting it was Russians??? Rudy Giuliani had the laptop for a time. You remember Rudy, right? The guy that runs a security consulting firm and tried to get Zelensky to dig up dirt on Biden. Yeah, he wouldn't have access to anyone who could screw with that laptop.
Loren himself, in post #22. Here, I'll save you some time:
Yeah, as more comes out it looks like the whole thing is faked. They got the e-mails by some illicit means (probably Moscow) and are using harmless stuff as support for the spicy stuff they faked.
You'll have to take that up with Loren.
Now, do you have anything to say about my comment?
Well, the NYT did claim that the emails were authenticated. So, if Rudy and his cronies altered them, I presume the computer forensics would show that, would it not? I'm not a computer expert, so I can't really say if it is possible to fake content on a hard drive and not have it be detected. As a minimum, they did not say that any of the content appears doctored. That's all I got for ya.
#### Artemus
##### Veteran Member
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
#### Elixir
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
That’s why they don’t register any more. Nothin but trouble once you let laws and stuff interfere.
##### Loony Running The Asylum
Staff member
Are there any emails from Hunter saying he would actually do what Burisma wanted him to do?
And how do we know these emails are legitimate? These computers passed through many hands before getting to the FBI.
Exactly. I find the providence of the laptop sorely lacking. I strongly suspect most of the e-mails are legitimate but given the providence I consider it likely that things have been tampered with. Thus it means nothing--and showing that much of it is true means nothing, it could easily be someone hacked his system, got his old e-mails and then changed things a bit.
Don't go looking for suspicious stuff on that hard drive, address the chain of custody first!
How would the Russians do all that, though? Are they logging into his laptop from Russia, or are the Russian hackers here in the US somehow given access to it? Or did they take the laptop to Russia? Wouldn't a competent computer forensics team be able to tell if the emails or other information were altered?
Where above is anyone suggesting it was Russians??? Rudy Giuliani had the laptop for a time. You remember Rudy, right? The guy that runs a security consulting firm and tried to get Zelensky to dig up dirt on Biden. Yeah, he wouldn't have access to anyone who could screw with that laptop.
Loren himself, in post #22. Here, I'll save you some time:
Yeah, as more comes out it looks like the whole thing is faked. They got the e-mails by some illicit means (probably Moscow) and are using harmless stuff as support for the spicy stuff they faked.
You'll have to take that up with Loren.
Now, do you have anything to say about my comment?
Well, the NYT did claim that the emails were authenticated.
All the emails or just some?
So, if Rudy and his cronies altered them, I presume the computer forensics would show that, would it not? I'm not a computer expert, so I can't really say if it is possible to fake content on a hard drive and not have it be detected. As a minimum, they did not say that any of the content appears doctored. That's all I got for ya.
To be sure, the email headers contain the servers the emails went through, those servers would have to be examined to confirm if true. Spoofing email headers is quite easy.
#### thebeave
##### Veteran Member
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
Well, typically the US Government gives a shit. If you apply for a top secret clearance, for example, you very likely will have to disclose such relations between a close family member and a nation such as Libya. I can only imagine how much they might care if its the POTUS.
I don't remember that, though. Why would Billy Carter do that? Maybe he wanted to peddle his "Billy Beer" there?
##### Loony Running The Asylum
Staff member
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
Well, typically the US Government gives a shit. If you apply for a top secret clearance, for example, you very likely will have to disclose such relations between a close family member and a nation such as Libya. I can only imagine how much they might care if its the POTUS.
I don't remember that, though. Why would Billy Carter do that? Maybe he wanted to peddle his "Billy Beer" there?
Cuz he couldn't sell the piss water here?
#### Patooka
##### Veteran Member
Well, typically the US Government gives a shit. If you apply for a top secret clearance, for example, you very likely will have to disclose such relations between a close family member and a nation such as Libya. I can only imagine how much they might care if its the POTUS.
Wow. How quickly they forget.
Staff member
#### Harry Bosch
##### Contributor
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
Well, typically the US Government gives a shit. If you apply for a top secret clearance, for example, you very likely will have to disclose such relations between a close family member and a nation such as Libya. I can only imagine how much they might care if its the POTUS.
I don't remember that, though. Why would Billy Carter do that? Maybe he wanted to peddle his "Billy Beer" there?
Again. Super sorry here. But I'm a typical American with a short attention span. You still haven't explained why I should care about Hunter Biden. I'm far more interested in Kim Kardashian latest fashion outfit. Supposedly Hunter had this laptop with e-mails that show an improper relationship that Biden may have had with a donor. Correct? But there is no evidence that corroborates this. It seems like a lot of assumptions. I'm just not impressed.
#### Elixir
I'm far more interested in Kim Kardashian latest fashion outfit.
You’re one sick mo fo!
Supposedly Hunter had this laptop with e-mails that show an improper relationship that Biden may have had with a donor. Correct? But there is no evidence that corroborates this. It seems like a lot of assumptions. I'm just not impressed.
It’s an unusual and curious accusation … because skulduggery involving foreign interests and election interference is generally a Republican thing. But if one is aware of how the new reich strives to accuse others of their own crime du jour, it’s hardly surprising.
#### Artemus
##### Veteran Member
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
Well, typically the US Government gives a shit. If you apply for a top secret clearance, for example, you very likely will have to disclose such relations between a close family member and a nation such as Libya. I can only imagine how much they might care if its the POTUS.
I don't remember that, though. Why would Billy Carter do that?
Because he was a drunken fool who was trying to cash in on the notoriety of being related to a high-ranking government official, which had no relevance whatsoever on the fitness or appropriateness of said relative to perform the job.
LIke I said, deja vu.
#### Keith&Co.
##### Contributor
Oh, Lord, I remember that. I'm sure Libya expected some behind-the-scenes influence peddling, but Billy was too dumb for that nonsense. He probably called Jimmy up and said, "Libbyuh wants me to get you to ____ for them. Is that okay?" And Jimmy handed the phone to someone else and said, "Get this idiot registered as a foreign lobbyist or something, before he gets us both arrested. And then put his number on the "Not At Home" list."
THey kept TRYING to find Billy something useful to do. Escorting diplomats (PLEASE don't take a whiz on the side of the road this time!) and so on. He certainly did not think there was a reason not to take money to namedrop with his brother. Jimmy'd be the one makin' the ackshul decision, raght?
#### Loren Pechtel
##### Super Moderator
Staff member
Are there any emails from Hunter saying he would actually do what Burisma wanted him to do?
And how do we know these emails are legitimate? These computers passed through many hands before getting to the FBI.
Exactly. I find the providence of the laptop sorely lacking. I strongly suspect most of the e-mails are legitimate but given the providence I consider it likely that things have been tampered with. Thus it means nothing--and showing that much of it is true means nothing, it could easily be someone hacked his system, got his old e-mails and then changed things a bit.
Don't go looking for suspicious stuff on that hard drive, address the chain of custody first!
How would the Russians do all that, though? Are they logging into his laptop from Russia, or are the Russian hackers here in the US somehow given access to it? Or did they take the laptop to Russia? Wouldn't a competent computer forensics team be able to tell if the emails or other information were altered?
This is an old machine, not live data. The providence is very sketchy, it's likely planted. And forensics can't catch a good enough job of faking it, especially in a situation like this where it's data, not images.
#### Loren Pechtel
##### Super Moderator
Staff member
What We Know and Don’t About Hunter Biden and a Laptop - The New York Times.
No concrete evidence has emerged that the laptop contains Russian disinformation.
With pressure mounting on the F.B.I. to respond to questions from Congress about the laptop, the bureau wrote to one of the president’s staunchest allies in Congress, Senator Ron Johnson of Wisconsin, suggesting that it had not found any Russian disinformation on the laptop.
Which doesn't prove it wasn't manipulated.
#### Loren Pechtel
##### Super Moderator
Staff member
Well, the NYT did claim that the emails were authenticated. So, if Rudy and his cronies altered them, I presume the computer forensics would show that, would it not? I'm not a computer expert, so I can't really say if it is possible to fake content on a hard drive and not have it be detected. As a minimum, they did not say that any of the content appears doctored. That's all I got for ya.
It most certainly is possible to fake content. It all comes down to how much faking you want to do. At the simplest level you can easily do it yourself: Plug in a thumb drive and copy a file to it. Notice how the file written date is the file written date of the original file, not the time you actually did it? Windows fakes the file date for your convenience, making it match the original instead of telling the truth. (And it most definitely is faking it--if you look at the new file while it's still writing it you'll see it has the current time. Once all the data is written it then copies the file timestamps over.)
Such fakery can only be detected when either the fakery is inconsistent (what often happens with photoshops--for example, pasting together two images that had light sources from different angle(s)--that's very hard to correct for) or when they leave behind signs of the tampering.
That's why programs like PGP exist--without a digital signature there's no way to know if it's real or not.
#### Loren Pechtel
##### Super Moderator
Staff member
Well, the NYT did claim that the emails were authenticated.
All the emails or just some?
Exactly. The best way to pull off something like this is to get an actual copy of their e-mail (say, the outlook database) so there's a lot of genuine stuff in there.
So, if Rudy and his cronies altered them, I presume the computer forensics would show that, would it not? I'm not a computer expert, so I can't really say if it is possible to fake content on a hard drive and not have it be detected. As a minimum, they did not say that any of the content appears doctored. That's all I got for ya.
To be sure, the email headers contain the servers the emails went through, those servers would have to be examined to confirm if true. Spoofing email headers is quite easy.
Except this is from years ago--do backups exist compare against?
#### thebeave
##### Veteran Member
Well, the NYT did claim that the emails were authenticated. So, if Rudy and his cronies altered them, I presume the computer forensics would show that, would it not? I'm not a computer expert, so I can't really say if it is possible to fake content on a hard drive and not have it be detected. As a minimum, they did not say that any of the content appears doctored. That's all I got for ya.
It most certainly is possible to fake content. It all comes down to how much faking you want to do. At the simplest level you can easily do it yourself: Plug in a thumb drive and copy a file to it. Notice how the file written date is the file written date of the original file, not the time you actually did it? Windows fakes the file date for your convenience, making it match the original instead of telling the truth. (And it most definitely is faking it--if you look at the new file while it's still writing it you'll see it has the current time. Once all the data is written it then copies the file timestamps over.)
Such fakery can only be detected when either the fakery is inconsistent (what often happens with photoshops--for example, pasting together two images that had light sources from different angle(s)--that's very hard to correct for) or when they leave behind signs of the tampering.
That's why programs like PGP exist--without a digital signature there's no way to know if it's real or not.
Well, presumably the forensic experts know this as well. So, why would they say they were authenticated if they knew the authenticity couldn't be known for sure?
#### Loren Pechtel
##### Super Moderator
Staff member
Well, the NYT did claim that the emails were authenticated. So, if Rudy and his cronies altered them, I presume the computer forensics would show that, would it not? I'm not a computer expert, so I can't really say if it is possible to fake content on a hard drive and not have it be detected. As a minimum, they did not say that any of the content appears doctored. That's all I got for ya.
It most certainly is possible to fake content. It all comes down to how much faking you want to do. At the simplest level you can easily do it yourself: Plug in a thumb drive and copy a file to it. Notice how the file written date is the file written date of the original file, not the time you actually did it? Windows fakes the file date for your convenience, making it match the original instead of telling the truth. (And it most definitely is faking it--if you look at the new file while it's still writing it you'll see it has the current time. Once all the data is written it then copies the file timestamps over.)
Such fakery can only be detected when either the fakery is inconsistent (what often happens with photoshops--for example, pasting together two images that had light sources from different angle(s)--that's very hard to correct for) or when they leave behind signs of the tampering.
That's why programs like PGP exist--without a digital signature there's no way to know if it's real or not.
Well, presumably the forensic experts know this as well. So, why would they say they were authenticated if they knew the authenticity couldn't be known for sure?
The QOP doesn't want to admit the truth.
#### Keith&Co.
##### Contributor
You know, even if they ever do prove that Hunter actually wrote those emails, word for word, AND sent them, all he has to do is say that he never acted on anything in thete, or intended to.
He was joking.
That's apparently a defense the right will accept, and defend.
#### Jimmy Higgins
##### Contributor
When I just had just reached voting age and it was revealed that Billy Carter had registered as a foreign agent of Libya, I remember thinking "Who gives a shit?" It's deja vu all over again.
This is all ignoring the other aspects of the story, including Giuliani's allegation that not only did he have emails and photos, but he had extraordinarily illegal content from the laptop as well (which would actually have been a crime for Giuliani). I find it amazing how this stuff was never released. But yeah, we shouldn't have been skeptical, we should have just accepted Four Seasons Landscaping man at his word.
#### Jason Harvestdancer
##### Contributor
Oh come on, this story is more than a week old. It's old news and nobody cares about old news.
##### Loony Running The Asylum
Staff member
Oh come on, this story is more than a week old. It's old news and nobody cares about old news.
No one has given us a reason to care about it at all.
#### Swammerdami
Staff member
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
I support law and order and think criminals should be prosecuted. Did Trump's DoJ fail to do its job? Biden's DoJ? I assume the underlying theme is the usual "See? You guys are just as guilty as us." But we'd need to indict a lot more criminals than a possible one named Hunter to match the stench coming from the Trump family, Trump Cabinet, and the rest of QOPAnon.
Is the intended inference that if someone is criminal, the other members of his family are also? This isn't guaranteed. The 45th President apparently had a criminal father, criminal grandfather, criminal 3rd wife, and at least two of his children are criminals, but I've seen no allegations of criminality against Trump's mother nor against his two youngest children.
Extraneous details:
While skimming the thread I did notice mention that the original m-time (modify time) is preserved when a file is copied. Yes, but the Unix c-time (change time) will be no earlier than the new file's creation. (A hacker can "fix" this by writing to the raw disk.) In Windows, c-time refers to creation-time and Windows doesn't pretend to any consistency — IIUC an ordinary user can, unwittingly, set the new file's creation-time to EITHER the old or new value!
Exactly. I find the providence of the laptop sorely lacking. I strongly suspect most of the e-mails are legitimate but given the providence I consider it likely that things have been tampered with....
How would the Russians do all that, though? ...
This is an old machine, not live data. The providence is very sketchy, it's likely planted. And forensics can't catch a good enough job of faking it, especially in a situation like this where it's data, not images.
I'll guess provenance, rather than providence, was the intended word.
Unless the Lord of All Creation is somehow involved in this scandal.
#### TSwizzle
##### Let's Go Brandon!
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
#### TomC
##### Celestial Highness
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
What do you mean by MSM?
I understand that it is an acronym for Main Stream Media. But you never say which media outlets you're referring to. Fox is the most main stream media I know about. They claim to be the most watched.
Somehow, you don't seem to think it is MSM. Please explain what you mean by the term.
Tom
#### Ford
##### Contributor
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
What do you mean by MSM?
I understand that it is an acronym for Main Stream Media. But you never say which media outlets you're referring to. Fox is the most main stream media I know about. They claim to be the most watched.
Somehow, you don't seem to think it is MSM. Please explain what you mean by the term.
Tom
I also await the answer, and would like to provide a little extra context as a former media employee.
When people leaning a certain way politically say the term "MSM" or "mainstream media" they tend to leave out outlets such as Fox News or talk radio hosts like the departed Rush Limbaugh. In some cases the people decrying the "MSM" are hosts on Fox or talk radio.
Let's be clear about this. Fox News is (according to current ratings) the number one cable news network. It is owned by one of the largest media companies in the world. Most of the prominent right wing talk radio hosts are syndicated by Premiere Radio Networks, which is part of iHeart Media...the largest radio company on the planet. Fox News and talk radio are - and have been for decades - "mainstream media." They sell themselves as plucky upstarts outside of the mainstream, but they are not and have never been such.
Fox = MSM.
Next question?
#### Loren Pechtel
##### Super Moderator
Staff member
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
Big tech recognized that it doesn't make sense. Just because you like it doesn't make it true.
##### Loony Running The Asylum
Staff member
[yawn] Wake me when something of significance happens. [/yawn]
#### Swammerdami
Staff member
Let's go Brandon!
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
Wake me up when the media your ilk idolizes has 0.000001% as much integrity as the "MSM" has in its little finger.
Better yet, strive for a little self-respect by deleting your line about "Brandon." Haven't most nine-year-olds moved past that sort of insipid turd-tossing?
[yawn] Wake me when something of significance happens. [/yawn]
I composed my response to Mr. "Let's Go" Swizzle before reading ZiprHead's response, honest!
#### TSwizzle
##### Let's Go Brandon!
Let's go Brandon!
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
Wake me up when the media your ilk idolizes has 0.000001% as much integrity as the "MSM" has in its little finger.
When you ask a question don't get all prissy because you don't like the answer, you just come across all Karen-like.
#### Toni
##### Contributor
Let's go Brandon!
Reading about the possible misdeeds of one possible criminal is NOT on my To-Do List. Why should I care?
You should care because not only did the MSM try its best to ignore, deny and bury the scandal, the MSM and big tech tried to censor the story.
Wake me up when the media your ilk idolizes has 0.000001% as much integrity as the "MSM" has in its little finger.
When you ask a question don't get all prissy because you don't like the answer, you just come across all Karen-like.
Please do not malign a perfectly nice name or white women with a facile and bigoted attempt at an insult.
Please define what you believe is MSM and how they buried the issue, using concrete examples with links if possible.
#### TSwizzle
##### Let's Go Brandon!
When you ask a question don't get all prissy because you don't like the answer, you just come across all Karen-like.
Please do not malign a perfectly nice name or white women with a facile and bigoted attempt at an insult.
Define "woman". Are you saying there are no black "women" named Karen? Racist.
Please define what you believe is MSM and how they buried the issue, using concrete examples with links if possible.
Behave yourself.
#### TSwizzle
##### Let's Go Brandon!
Bill Maher calls out the NYT;
Bill Maher slams the left-wing media for 'burying the Hunter Biden laptop scandal because it wasn't part of their narrative' and says it shouldn't have taken more than a year for NYT and WaPo to 'verify' initial report
Daily Mail
#### Harry Bosch
##### Contributor
Bill Maher calls out the NYT;
Bill Maher slams the left-wing media for 'burying the Hunter Biden laptop scandal because it wasn't part of their narrative' and says it shouldn't have taken more than a year for NYT and WaPo to 'verify' initial report
Daily Mail
Totally agree. I will not be voting for that meanie Hunter Biden in the future.
#### Elixir
Totally agree. I will not be voting for that meanie Hunter Biden in the future.
Right on, brother. I'm voting for Eric next time!
#### Patooka
##### Veteran Member
Have I missed out on the part where Trumptards explain why it is such a scandal Hunter Biden owns a laptop? Seriously, how does this cripple or compromise the Biden Administration?
#### Harry Bosch
##### Contributor
Have I missed out on the part where Trumptards explain why it is such a scandal Hunter Biden owns a laptop? Seriously, how does this cripple or compromise the Biden Administration?
Well Duh. You need to get with the program. Firstly, the esteemed and honorable Rudy Guiliani says that it's a big deal. His pristine reputation should be enough for the common folk to trust his judgement. Secondly, what if Hunter runs in the future? Remember, Hunter was very mean about Trump.
Last edited:
#### Elixir
Have I missed out on the part where Trumptards explain why it is such a scandal Hunter Biden owns a laptop? Seriously, how does this cripple or compromise the Biden Administration?
Well Duh. You need to get with the program. Firstly, the esteemed and honorable Rudy Guiliani says that it's a big deal. His pristine reputation should be enough for the common folk to trust his judgement. Secondly, what if Hunter runs in the future? Remember, Hunter was very mean against Trump.
Translation:
Repugs are already scraping the bottom of their boogeyman barrel.
#### TSwizzle
##### Let's Go Brandon!
Plenty more to come out of this story;
The source who distributed Hunter Biden's laptop to congressmen and media has fled the US to Switzerland, saying he fears retaliation from the Biden administration. Jack Maxey gave DailyMail.com a copy of the hard drive from Hunter's abandoned laptop in the spring of 2021. He also gave copies and material from it to the Washington Post, New York Times, and Senator Chuck Grassley in his role as ranking Republican on the Senate Judiciary Committee – but he claims they all sat on it for months. Maxey, a former co-host of ex-Donald Trump advisor Steve Bannon's podcast the War Room, claims he and his colleagues have found '450 gigabytes of deleted material' including 80,000 images and videos and more than 120,000 archived emails. He said he intends to post them all online in a searchable database in the coming weeks. Hunter's laptop is brimming with evidence of apparent criminal activity by him and his associates including drug trafficking and prostitution.
Daily Mail
#### Ford
##### Contributor
saying he fears retaliation from the Biden administration
I mean, the Biden administration has a long history of attacking the press. Declaring them "fake news" and "an enemy of the United States." Why, he even openly admires authoritarian regimes that imprison journalists!
Oh wait...that was Trump.
But hey, we know this latest development is totally legit, since it comes from an objective journalist, who is (checks notes)...a former co-host of ex-Donald Trump advisor Steve Bannon's podcast the War Room
So you know you know you can trust him to be fair. Maybe even balanced, too!
#### Patooka
##### Veteran Member
Plenty more to come out of this story;
Plenty more? There has been fuck all out of this story to begin with. Fuck me Trumptards are pathetic.
#### Artemus
##### Veteran Member
Plenty more to come out of this story;
The source who distributed Hunter Biden's laptop to congressmen and media has fled the US to Switzerland, saying he fears retaliation from the Biden administration. Jack Maxey gave DailyMail.com a copy of the hard drive from Hunter's abandoned laptop in the spring of 2021. He also gave copies and material from it to the Washington Post, New York Times, and Senator Chuck Grassley in his role as ranking Republican on the Senate Judiciary Committee – but he claims they all sat on it for months. Maxey, a former co-host of ex-Donald Trump advisor Steve Bannon's podcast the War Room, claims he and his colleagues have found '450 gigabytes of deleted material' including 80,000 images and videos and more than 120,000 archived emails. He said he intends to post them all online in a searchable database in the coming weeks. Hunter's laptop is brimming with evidence of apparent criminal activity by him and his associates including drug trafficking and prostitution.
Daily Mail
None of this involves the President, any of his advisors, or anyone employed by the Adminstrationin any capacity whatsoever. I ask again, even if all this is true, why should I as a voter care?
#### Ford
##### Contributor
Plenty more to come out of this story;
Plenty more? There has been fuck all out of this story to begin with. Fuck me Trumptards are pathetic.
You don't understand. See, Hunter Biden worked for a company in Ukraine. In Ukraine! Years ago, but still! Do you think it is a coincidence that there's a war going on in the same country where he once worked?
Granted, the war is not at all happening because Trump's bestie Vlad decided to rebuild "Greater Russia" and his army is currently not slaughtering civilians, but that can't be divorced from the Hunter Biden thing like Trump was divorced from his first two wives! See, when Hunter Biden worked for a company in Ukraine in the 2010s, he was overpaid! He wasn't qualified to earn that money! Lord knows Trump would never hire his offspring to do a job they were not qualified for, or overpay them, and may I add that Eric never worked in Ukraine! What do you think about that, libtard?!
And listen...there's so much more to come out of this laptop. Who's to say the entire plan for Putin to seize Crimea and the eastern regions of Ukraine didn't have something to do with Burisma? I know...right? Had Hunter Biden not had a laptop, Crimea might still be part of Ukraine, Donestk and Donbass would be peaceful, and that airliner would never have been shot down with a missile that had absolutely nothing to do with Russia. You see where I'm going with all this? Also, Democrats are all (checks latest talking points) pedophiles and groomers! | 2022-08-10 20:48:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19863662123680115, "perplexity": 3893.4644101751182}, "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/1659882571210.98/warc/CC-MAIN-20220810191850-20220810221850-00520.warc.gz"} |
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/09%3A_Graph_Theory/9.03%3A_Connectivity |
# 9.3: Connectivity
This section is devoted to a question that, when posed in relation to the graphs that we have examined, seems trivial. That question is: Given two vertices, $$s$$ and $$t\text{,}$$ of a graph, is there a path from $$s$$ to $$t\text{?}$$ If $$s = t\text{,}$$ this question is interpreted as asking whether there is a circuit of positive length starting at $$s\text{.}$$ Of course, for the graphs we have seen up to now, this question can be answered after a brief examination.
## Preliminaries
There are two situations under which a question of this kind is nontrivial. One is where the graph is very large and an “examination” of the graph could take a considerable amount of time. Anyone who has tried to solve a maze may have run into a similar problem. The second interesting situation is when we want to pose the question to a machine. If only the information on the edges between the vertices is part of the data structure for the graph, how can you put that information together to determine whether two vertices can be connected by a path?
Note $$\PageIndex{1}$$: Connectivity Terminology
Let $$v$$ and $$w$$ be vertices of a directed graph. Vertex $$v$$ is connected to vertex $$w$$ if there is a path from $$v$$ to $$w\text{.}$$ Two vertices are strongly connected if they are connected in both directions to one another. A graph is connected if, for each pair of distinct vertices, $$v$$ and $$w\text{,}$$ $$v$$ is connected to $$w$$ or $$w$$ is connected to $$v\text{.}$$ A graph is strongly connected if every pair of its vertices is strongly connected. For an undirected graph, in which edges can be used in either direction, the notions of strongly connected and connected are the same.
Theorem $$\PageIndex{1}$$: Maximal Path Theorem
If a graph has $$n$$ vertices and vertex $$u$$ is connected to vertex $$w\text{,}$$ then there exists a path from $$u$$ to $$w$$ of length no more than $$n\text{.}$$
Proof
(Indirect): Suppose $$u$$ is connected to $$w\text{,}$$ but the shortest path from $$u$$ to $$w$$ has length $$m\text{,}$$ where $$m>n\text{.}$$ A vertex list for a path of length $$m$$ will have $$m + 1$$ vertices. This path can be represented as $$\left(v_0,v_1,\ldots, v_m\right)\text{,}$$ where $$v_0=u$$ and $$v_m= w\text{.}$$ Note that since there are only $$n$$ vertices in the graph and $$m$$ vertices are listed in the path after $$v_0\text{,}$$ we can apply the pigeonhole principle and be assured that there must be some duplication in the last $$m$$ vertices of the vertex list, which represents a circuit in the path. This means that our path of minimum length can be reduced, which is a contradiction.
## Adjacency Matrix Method
Algorithm $$\PageIndex{1}$$: Adjacency Matrix Method
Suppose that the information about edges in a graph is stored in an adjacency matrix, $$G\text{.}$$ The relation, $$r\text{,}$$ that $$G$$ defines is $$v r w$$ if there is an edge connecting $$v$$ to $$w\text{.}$$ Recall that the composition of $$r$$ with itself, $$r^2\text{,}$$ is defined by $$v r^2 w$$ if there exists a vertex $$y$$ such that $$v r y$$ and $$y r w\text{;}$$ that is, $$v$$ is connected to $$w$$ by a path of length 2. We could prove by induction that the relation $$r^k\text{,}$$ $$k\geq 1\text{,}$$ is defined by $$v r^k w$$ if and only if there is a path of length $$k$$ from $$v$$ to $$w\text{.}$$ Since the transitive closure, $$r^+\text{,}$$ is the union of $$r\text{,}$$ $$r^2$$ $$,r^3,\ldots\text{,}$$ we can answer our connectivity question by determining the transitive closure of $$r\text{,}$$ which can be done most easily by keeping our relation in matrix form. Theorem $$\PageIndex{1}$$ is significant in our calculations because it tells us that we need only go as far as $$G^n$$ to determine the matrix of the transitive closure.
The main advantage of the adjacency matrix method is that the transitive closure matrix can answer all questions about the existence of paths between any vertices. If $$G^+$$ is the matrix of the transitive closure, $$v_i$$ is connected to $$v_j$$ if and only if $$\left(G^+\right)_{i j }=1\text{.}$$ A directed graph is connected if $$\left(G^+\right)_{i j }=1$$ or $$\left(G^+\right)_{j i}=1$$ for each $$i\neq j\text{.}$$ A directed graph is strongly connected if its transitive closure matrix has no zeros.
A disadvantage of the adjacency matrix method is that the transitive closure matrix tells us whether a path exists, but not what the path is. The next algorithm will solve this problem.
We will describe the Breadth-First Search Algorithm first with an example.
The football team at Mediocre State University (MSU) has had a bad year, 2 wins and 9 losses. Thirty days after the end of the football season, the university trustees are meeting to decide whether to rehire the head coach; things look bad for him. However, on the day of the meeting, the coach issues the following press release with results from the past year:
List $$\PageIndex{1}$$: Press Release: MSU Completes Successful Season
The Mediocre State University football team compared favorably with national champion Enormous State University this season.
• Mediocre State defeated Local A and M.
• Local A and M defeated City College.
• City College defeated Corn State U.
• ... (25 results later)
• Tough Tech defeated Enormous State University (ESU).
...and ESU went on to win the national championship!
The trustees were so impressed that they rehired the coach with a raise! How did the coach come up with such a list?
In reality, such lists exist occasionally and have appeared in newspapers from time to time. Of course they really don't prove anything since each team that defeated MSU in our example above can produce a similar, shorter chain of results. Since college football records are readily available, the coach could have found this list by trial and error. All that he needed to start with was that his team won at least one game. Since ESU lost one game, there was some hope of producing the chain.
The problem of finding this list is equivalent to finding a path in the tournament graph for last year's football season that initiates at MSU and ends at ESU. Such a graph is far from complete and is likely to be represented using edge lists. To make the coach's problem interesting, let's imagine that only the winner of any game remembers the result of the game. The coach's problem has now taken on the flavor of a maze. To reach ESU, he must communicate with the various teams along the path. One way that the coach could have discovered his list in time is by sending the following messages to the coaches of the two teams that MSU defeated during the season:
Note $$\PageIndex{2}$$
When this example was first written, we commented that ties should be ignored. Most recent NCAA rules call for a tiebreaker in college football and so ties are no longer an issue. Email was also not common and we described the process in terms of letters, not email messages. Another change is that the coach could also have asked the MSU math department to use Mathematica or Sage to find the path!
List $$\PageIndex{2}$$: The Coach's Letter
Dear Football Coach:
1. If you are the coach at ESU, contact the coach at MSU now and tell him who sent you this message.
2. If you are not the coach at ESU and this is the first message of this type that you have received, then:
• Remember from whom you received this message.
• Forward a copy of this message, signed by you, to each of the coaches whose teams you defeated during the past year.
• Ignore this message if you have received one like it already.
$$\quad \quad \quad \quad \quad$$Signed,
$$\quad \quad \quad \quad \quad$$Coach of MSU
List $$\PageIndex{3}$$: Observations
From the conditions of this message, it should be clear that if everyone cooperates and if coaches participate within a day of receiving the message:
1. If a path of length $$n$$ exists from MSU to ESU, then the coach will know about it in $$n$$ days.
2. By making a series of phone calls, the coach can construct a path that he wants by first calling the coach who defeated ESU (the person who sent ESU's coach that message). This coach will know who sent him a letter, and so on. Therefore, the vertex list of the desired path is constructed in reverse order.
3. If a total of $$M$$ football games were played, no more than $$M$$ messages will be sent out.
4. If a day passes without any message being sent out, no path from MSU to ESU exists.
5. This method could be extended to construct a list of all teams that a given team can be connected to. Simply imagine a series of letters like the one above sent by each football coach and targeted at every other coach.
The general problem of finding a path between two vertices in a graph, if one exists, can be solved exactly as we solved the problem above. The following algorithm, commonly called a breadth-first search, uses a stack.
Note $$\PageIndex{3}$$: Stacks
A stack is a fundamental data structure in computer science. A common analogy used to describe stacks is a stack of plates. If you put a plate on the top of a stack and then want to use a plate, it's natural to use that top plate. So the last plate in is the first plate out. “Last in, first out” is the short description of the rule for stacks. This is contrast with a queue which uses a “First in, first out” rule.
Algorithm $$\PageIndex{2}$$: Breadth-first Search
A broadcasting algorithm for finding a path between vertex $$i$$ and vertex $$j$$ of a graph having $$n$$ vertices. Each item $$V_k$$ of a list $$V=\left\{V_1, V_2, \ldots , V_n\right\}\text{,}$$ consists of a Boolean field $$V_k.\text{found}$$ and an integer field $$V_k.\text{from}\text{.}$$ The sets $$D_1\text{,}$$ $$D_2, \dots\text{,}$$ called depth sets, have the property that if $$k \in D_r\text{,}$$ then the shortest path from vertex $$i$$ to vertex $$k$$ is of length $$r\text{.}$$ In Step 5, a stack is used to put the vertex list for the path from the vertex $$i$$ to vertex $$j$$ in the proper order. That stack is the output of the algorithm.
1. Set the value $$V_k.\text{found}$$ equal to False, $$k = 1, 2, \dots , n$$
2. $$\displaystyle r = 0$$
3. $$\displaystyle D_0= \{i\}$$
4. while $$(\neg V_j.\text{found}$$) and $$\left(D_r\right.\neq \emptyset )$$
• $$\displaystyle D_{r+1}=\emptyset$$
• for each k in $$D_r\text{:}$$
$$\quad$$for each edge (k,t):
$$\quad \quad$$If $$V_t.\text{found}$$ == False:
$$\quad \quad \quad$$$$V_t.\text{found}=\text{True}$$
$$\quad \quad \quad$$$$V_t.\text{from} = k$$
$$\quad \quad \quad$$$$D_{r+1}=D_{r+1}\cup \{t\}$$
• $$\displaystyle r = r + 1$$
5. if $$V_j.\text{found}\text{:}$$
• $$\displaystyle S = Empty Stack$$
• $$\displaystyle k=j$$
• while $$V_k.\text{from} \neq i\text{:}$$
$$\quad$$Push $$k$$ onto $$S$$
$$\quad$$$$k = V_k.\text{from}$$
• Push $$k$$ onto $$S$$
• Push $$i$$ onto $$S$$
List $$\PageIndex{4}$$: Notes on Breadth-first Search
• This algorithm will produce one path from vertex $$i$$ to vertex $$j\text{,}$$ if one exists, and that path will be as short as possible. If more than one path of this length exists, then the one that is produced depends on the order in which the edges are examined and the order in which the elements of $$D_r$$ are examined in Step 4.
• The condition $$D_{r }\neq \emptyset$$ is analogous to the condition that no mail is sent in a given stage of the process, in which case MSU cannot be connected to ESU.
• This algorithm can be easily revised to find paths to all vertices that can be reached from vertex $$i\text{.}$$ Step 5 would be put off until a specific path to a vertex is needed since the information in $$V$$ contains an efficient list of all paths. The algorithm can also be extended further to find paths between any two vertices.
Example $$\PageIndex{1}$$: A Simple Example
Consider the graph below. The existence of a path from vertex 2 to vertex 3 is not difficult to determine by examination. After a few seconds, you should be able to find two paths of length four. Algorithm $$\PageIndex{2}$$ will produce one of them.
Suppose that the edges from each vertex are sorted in ascending order by terminal vertex. For example, the edges from vertex 3 would be in the order $$(3, 1), (3, 4), (3, 5)\text{.}$$ In addition, assume that in the body of Step 4 of the algorithm, the elements of $$D_r$$ are used in ascending order. Then at the end of Step 4, the value of $$V$$ will be
\begin{equation*} \begin{array}{cccccccc} k & 1 & 2 & 3 & 4 & 5 & 6 & \\ V_k.\text{found} & T & T & T & T & T & T & \\ V_k.\text{from} & 2 & 4 & 6 & 1 & 1 & 4 & \\ \text{Depth} \text{set} & 1 & 3 & 4 & 2 & 2 & 3 & \\ \end{array} \end{equation*}
Therefore, the path $$(2, 1, 4, 6, 3)$$ is produced by the algorithm. Note that if we wanted a path from 2 to 5, the information in $$V$$ produces the path (2, 1, 5) since $$V_k.\text{from} = 1$$ and $$V_1.\text{from} = 2\text{.}$$ A shortest circuit that initiates at vertex 2 is also available by noting that $$V_2.\text{from}=4\text{,}$$ $$V_4\text{.from = 1}\text{,}$$ and $$V_1.\text{from} = 2\text{;}$$ thus the circuit $$(2, 1, 4, 2)$$ is the output of the algorithm.
## Graph Measurements
If we were to perform a breadth first search from each vertex in a graph, we could proceed to determine several key measurements relating to the general connectivity of that graph. From each vertex $$v\text{,}$$ the distance from $$v$$ to any other vertex $$w\text{,}$$ $$d(v,w)\text{,}$$ is number of edges in the shortest path from $$v$$ to $$w\text{.}$$ This number is also the index of the depth set to which $$w$$ belongs in a breath-first search starting at $$v\text{.}$$
\begin{equation*} d(v,w) = i \iff w \in D_v(i) \end{equation*}
where $$D_v$$ is the family of depth sets starting at $$v\text{.}$$
If the vector of “from-values” is known from the breath-first search, then the distance can be determined recursively as follows:
\begin{equation*} d(v,v) =0 \end{equation*}
\begin{equation*} d(v,w) = 1 + d(v,w.from) \textrm{ if }w\neq v \end{equation*}
Example $$\PageIndex{2}$$: Computing Distances
Consider Figure $$\PageIndex{2}$$. If we perform a breadth first search of this graph starting at vertex 2, for example, we get the following “from data” telling us from what vertex each vertex is reached.
\begin{equation*} \begin{array}{ccccccccccccc} \text{vertex} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \text{vertex.from} & 7 & 2 & 10 & 6 & 9 & 7 & 2 & 4 & 2 & 7 & 9 & 2 \\ \end{array} \end{equation*}
For example, 4.from has a value of 6. We can compute $$d(2,4)\text{:}$$
\begin{equation*} \begin{split} d(2,4) &= 1+d(2,4.from)= 1+d(2,6)\\ &=2+d(2,6.from)=2+d(2,7)\\ &=3+d(2,7.from)=3+d(2,2)\\ &=3 \end{split} \end{equation*}
Once we know distances between any two vertices, we can determine the eccentricity of each vertex; and the graph's diameter, radius and center. First, we define these terms precisely.
Definition $$\PageIndex{1}$$: Eccentricity of a Vertex
The maximum distance from a vertex to all other vertices, $$e(v)=max_{w}d(v,w)\text{.}$$
Definition $$\PageIndex{2}$$: Diameter of a Graph
The maximum eccentricity of vertices in a graph, denoted $$d(G)\text{.}$$
Definition $$\PageIndex{3}$$: Radius of a Graph
The minimum eccentricity of vertices in a graph, denoted $$r(G)\text{.}$$
Definition $$\PageIndex{4}$$: Center of a Graph
The set of vertices with minimal eccentricity, $$C(G)=\{v\in V \mid e(v)=r(G)\}$$
Example $$\PageIndex{3}$$: Measurements from Distance Matrices
If we compute all distances between vertices, we can summarize the results in a distance matrix, where the entry in row $$i\text{,}$$ column $$j$$ is the distance from vertex $$i$$ to vertex $$j\text{.}$$ For the graph in Example $$\PageIndex{2}$$, that matrix is
\begin{equation*} \left( \begin{array}{cccccccccccc} 0 & 2 & 2 & 2 & 3 & 1 & 1 & 3 & 3 & 1 & 2 & 2 \\ 2 & 0 & 3 & 3 & 2 & 2 & 1 & 4 & 1 & 2 & 2 & 1 \\ 2 & 3 & 0 & 2 & 5 & 3 & 2 & 3 & 4 & 1 & 4 & 3 \\ 2 & 3 & 2 & 0 & 3 & 1 & 2 & 1 & 3 & 1 & 2 & 3 \\ 3 & 2 & 5 & 3 & 0 & 2 & 3 & 4 & 1 & 4 & 1 & 3 \\ 1 & 2 & 3 & 1 & 2 & 0 & 1 & 2 & 2 & 2 & 1 & 2 \\ 1 & 1 & 2 & 2 & 3 & 1 & 0 & 3 & 2 & 1 & 2 & 1 \\ 3 & 4 & 3 & 1 & 4 & 2 & 3 & 0 & 4 & 2 & 3 & 4 \\ 3 & 1 & 4 & 3 & 1 & 2 & 2 & 4 & 0 & 3 & 1 & 2 \\ 1 & 2 & 1 & 1 & 4 & 2 & 1 & 2 & 3 & 0 & 3 & 2 \\ 2 & 2 & 4 & 2 & 1 & 1 & 2 & 3 & 1 & 3 & 0 & 3 \\ 2 & 1 & 3 & 3 & 3 & 2 & 1 & 4 & 2 & 2 & 3 & 0 \\ \end{array} \right) \end{equation*}
If we scan the matrix, we can see that the maximum distance is the distance between vertices 3 and 5, which is 5 and is the diameter of the graph. If we focus on individual rows and identify the maximum values, which are the eccentricities, their minimum is 3, which the graph's radius. This eccentricity value is attained by vertices in the set $$\{1, 4, 6, 7\}\text{,}$$ which is the center of the graph.
### SageMath Note - Graph Searching
The following sequence of Sage cells illustrates how searching can be done in graphs.
Generate a random undirected graph with 18 vertices. For each pair of vertices, an edge is included between them with probability 0.2. Since there are $$\binom{18}{2}=153$$ potential edges, we expect that there will be approximately $$0.2 \cdot 153 \approx 31$$ edges. The random number generation is seeded first so that the result will always be the same in spite of the random graph function. Changing or removing that first line will let you experiment with different graphs.
set_random_seed(2002)
Gr=graphs.RandomGNP(18,0.2)
Gr.show()
Count the number of edges. In this case the number is a bit less than expected.
len(Gr.edges(labels=False))
Find a shortest path from vertex 0 to vertex 8.
Gr.shortest_path(0, 8)
Generate a list of vertices that would be reached in a breadth-first search. The expression Gr.breadth_first_search(0) creates an iterator that is convenient for programming. Wrapping list( ) around the expression shows the order in which the vertices are visited with the depth set indicated in the second coordinates.
list(Gr.breadth_first_search(0,report_distance='True'))
## Exercises
Exercise $$\PageIndex{1}$$
Apply Algorithm $$\PageIndex{2}$$ to find a path from 5 to 1 in Figure $$\PageIndex{1}$$. What would be the final value of $$V\text{?}$$ Assume that the terminal vertices in edge lists and elements of the depth sets are put into ascending order, as we assumed in Example $$\PageIndex{1}$$.
$$\begin{array}{ccccccc} k & 1 & 2 & 3 & 4 & 5 & 6 \\ V[k].\text{found} & T & T & T & F & F & T \\ V[k].\text{from} & 2 & 5 & 6 & * & * & 5 \\ \text{Depth} \text{Set} & 2 & 1 & 2 & * & * & 1 \\ \end{array}$$ $$\text{(*} = \text{undefined})$$
Exercise $$\PageIndex{2}$$
Apply Algorithm $$\PageIndex{2}$$ to find a path from $$d$$ to $$c$$ in the road graph in Example 9.1.3 using the edge list in that example. Assume that the elements of the depth sets are put into ascending order.
Exercise $$\PageIndex{3}$$
In a simple undirected graph with no self-loops, what is the maximum number of edges you can have, keeping the graph unconnected? What is the minimum number of edges that will assure that the graph is connected?
If the number of vertices is $$n\text{,}$$ there can be $$\frac{(n-1)(n-2)}{2}$$ vertices with one vertex not connected to any of the others. One more edge and connectivity is assured.
Exercise $$\PageIndex{4}$$
Use a broadcasting algorithm to determine the shortest path from vertex $$a$$ to vertex $$i$$ in the graphs shown in the Figure $$\PageIndex{3}$$ below. List the depth sets and the stack that is created.
Exercise $$\PageIndex{5}$$
For each of the following graphs, determine the eccentricities of each vertex, and the diameter, radius, and center of the graph.
1. The eccentricity of each vertex is 2; and the diameter and radius are both 2 as well. All vertices are part of the center.
2. The corners (1,3,10 and 10) have eccentricities 5. The two central vertices, 5 and 8, which are in the center of the graph have eccentricity 3. All other vertices have eccentricity 4. The diameter is 5. The radius is 3.
3. Vertices 1, 2 and 5 have eccentricity 2 and make up the center of this graph. Verticies 7 and 8 have eccentricity 4, and all other vertices have eccentricity 3. The diameter is 4. The radius is 2.
4. The eccentricity of each vertex is 4; and the diameter and radius are both 4 as well. All vertices are part of the center.
Exercise $$\PageIndex{6}$$
1. The terms diameter, radius and center are familiar ones in the context of circles. Compare their usage in circles and graphs. How are they similar and how are they different?
2. “Eccentricity” might be less familiar. How is is used in geometry, and does it have a compatible use in graph theory?
Exercise $$\PageIndex{7}$$
Prove (by induction on $$k$$) that if the relation $$r$$ on vertices of a graph is defined by $$v r w$$ if there is an edge connecting $$v$$ to $$w\text{,}$$ then $$r^k\text{,}$$ $$k \geq 1\text{,}$$ is defined by $$v r^kw$$ if there is a path of length $$k$$ from $$v$$ to $$w\text{.}$$
Basis: $$(k=1)$$ Is the relation $$r^1\text{,}$$ defined by $$v r^1 w$$ if there is a path of length 1 from $$v \text{ to } w\text{?}$$ Yes, since $$v r w$$ if and only if an edge, which is a path of length $$1\text{,}$$ connects $$v$$ to $$w\text{.}$$
Induction: Assume that $$v r^k w$$ if and only if there is a path of length $$k$$ from $$v$$ to $$w\text{.}$$ We must show that $$v r^{k+1} w$$ if and only if there is a path of length $$k+1$$ from $$v$$ to $$w\text{.}$$
\begin{equation*} v r^{k+1} w \Rightarrow v r^k y \textrm{ and } y r w\textrm{ for some vertex } y \end{equation*}
By the induction hypothesis, there is a path of length $$k$$ from $$v \textrm{ to } y\text{.}$$ And by the basis, there is a path of length one from $$y$$ to $$w\text{.}$$ If we combine these two paths, we obtain a path of length $$k+1$$ from $$v$$ to $$w\text{.}$$ Of course, if we start with a path of length $$k+1$$ from $$v$$ to $$w\text{,}$$ we have a path of length $$k$$ from $$v$$ to some vertex $$y$$ and a path of length 1 from $$y$$ to $$w\text{.}$$ Therefore, $$v r^k y \textrm{ and } y r w \Rightarrow v r^{k+1} w\text{.}$$
Exercise $$\PageIndex{8}$$
For each of the following distance matrices of graphs, identify the diameter, radius and center. Assume the graphs vertices are the numbers 1 through $$n$$ for an $$n \times n$$ matrix.
1. $$\displaystyle \left( \begin{array}{cccccccccc} 0 & 2 & 1 & 2 & 2 & 3 & 3 & 2 & 1 & 1 \\ 2 & 0 & 1 & 2 & 3 & 3 & 3 & 2 & 3 & 2 \\ 1 & 1 & 0 & 1 & 2 & 2 & 2 & 1 & 2 & 1 \\ 2 & 2 & 1 & 0 & 3 & 3 & 3 & 2 & 3 & 2 \\ 2 & 3 & 2 & 3 & 0 & 2 & 1 & 1 & 2 & 1 \\ 3 & 3 & 2 & 3 & 2 & 0 & 1 & 1 & 3 & 2 \\ 3 & 3 & 2 & 3 & 1 & 1 & 0 & 1 & 3 & 2 \\ 2 & 2 & 1 & 2 & 1 & 1 & 1 & 0 & 2 & 1 \\ 1 & 3 & 2 & 3 & 2 & 3 & 3 & 2 & 0 & 1 \\ 1 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 1 & 0 \\ \end{array} \right)$$
2. $$\displaystyle \left( \begin{array}{cccccccccccc} 0 & 2 & 2 & 2 & 3 & 3 & 3 & 1 & 2 & 3 & 1 & 1 \\ 2 & 0 & 2 & 2 & 1 & 1 & 1 & 3 & 2 & 1 & 1 & 3 \\ 2 & 2 & 0 & 1 & 3 & 2 & 1 & 2 & 2 & 3 & 1 & 1 \\ 2 & 2 & 1 & 0 & 3 & 1 & 2 & 1 & 2 & 3 & 2 & 1 \\ 3 & 1 & 3 & 3 & 0 & 2 & 2 & 4 & 3 & 2 & 2 & 4 \\ 3 & 1 & 2 & 1 & 2 & 0 & 2 & 2 & 3 & 2 & 2 & 2 \\ 3 & 1 & 1 & 2 & 2 & 2 & 0 & 3 & 3 & 2 & 2 & 2 \\ 1 & 3 & 2 & 1 & 4 & 2 & 3 & 0 & 3 & 4 & 2 & 2 \\ 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 0 & 1 & 3 & 1 \\ 3 & 1 & 3 & 3 & 2 & 2 & 2 & 4 & 1 & 0 & 2 & 2 \\ 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 2 & 0 & 2 \\ 1 & 3 & 1 & 1 & 4 & 2 & 2 & 2 & 1 & 2 & 2 & 0 \\ \end{array} \right)$$
9.3: Connectivity is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Al Doerr & Ken Levasseur. | 2022-05-18 09:45:10 | {"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": 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": 0.8689582347869873, "perplexity": 164.2320370529154}, "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-2022-21/segments/1652662521883.7/warc/CC-MAIN-20220518083841-20220518113841-00485.warc.gz"} |
https://www.physicsforums.com/threads/trig-please-help-me.417559/ | 1. Jul 21, 2010
### GreenPrint
1. The problem statement, all variables and given/known data
What does cis^-1 (x) mean
like you know how there is sin^-1 (x) and sin(x) applying that concept with cis(x) what does that mean?
Note that is not a spelling mistake I'm not trying to say cos^-1 (x) I'm indeed saying cis^-1 (x) with an "i"
I also wander what does cis stand for like cos is cosine sin is sine etc. how do I say cis...
like I was thinking ok
cis(x) = cos(x) + i sin(x) = e^(ix)
what does cis^-1(x) mean
kind of stuck here...
Thank You
2. Relevant equations
3. The attempt at a solution
2. Jul 21, 2010
### jegues
Since cis(x) is defined in the following manner,
$$cis(x) = cos(x) +isin(x) = e^{ix}$$
Simply take the inverse of cis(x),
$$cis^{-1}(x) = \pm cos^{-1}(\frac{x^{2} + 1}{2x})$$
Last edited: Jul 21, 2010 | 2018-12-19 15:33:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8748686909675598, "perplexity": 7354.820945472122}, "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/1544376832559.95/warc/CC-MAIN-20181219151124-20181219173124-00484.warc.gz"} |
https://www.physicsforums.com/threads/definite-integration.13181/ | # Definite Integration
1. Jan 24, 2004
### himanshu121
$$f(x)=x^2+\int_0^x e^{-t}f(x-t)dt$$ ........(I)
Find f(x)
Okay what i did:
For
$$\int_0^x e^{-t}f(x-t)dt$$
I substituted h=x-t =>dh=-dt
so $$\int_0^x e^{h-x}f(h)dh$$
Now i differentiated (I)
so i got
f'(x)=2x+f(x) after solvin this by integrating factor method i got different results which involve ex
But i had to prove f(x)=x2+x3/3
2. Jan 24, 2004
### Hurkyl
Staff Emeritus
The fundamental theorem of calculus says
$$\frac{d}{dx} \int_a^x f(h) \, dh = f(x)$$
However, when you differentiate, you don't have something in this form! You have the form
$$\frac{d}{dx} \int_a^x f(\mathb{x,} h) \, dh$$
Fortunately, in this case, you don't need the full messy version; you can factor $e^{-x}$ out of the integrand and then differentiate normally.
3. Jan 24, 2004
### himanshu121
Thanks Hurkyl I got corrected
now i have f'(x)=x2+2x
which gives me the result
Still i want to know how would u have approached the pro | 2017-01-16 19:33:47 | {"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": 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": 0.8823325037956238, "perplexity": 4511.663045820545}, "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/1484560279248.16/warc/CC-MAIN-20170116095119-00036-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-3-solving-inequalities-3-4-solving-multi-step-inequalities-practice-and-problem-solving-exercises-page-190/33 | Algebra 1
$2(n-8)\lt16+2n\longrightarrow$ multiply using the distributive property $2n-16\lt16+2n\longrightarrow$ subtract 2n from each side $2n-16-2n\lt16+2n-2n\longrightarrow$ subtract $-16\lt16$ Since -16 is less than 16, the solution is all real numbers. | 2021-05-08 16:35:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7814799547195435, "perplexity": 559.3679225175067}, "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-2021-21/segments/1620243988882.94/warc/CC-MAIN-20210508151721-20210508181721-00208.warc.gz"} |
https://homework.cpm.org/category/CCI_CT/textbook/int2/chapter/9/lesson/9.2.1/problem/9-58 | ### Home > INT2 > Chapter 9 > Lesson 9.2.1 > Problem9-58
9-58.
Joanna is solving a puzzle for a math contest. She knows that one of the points in the diagram at right is the circumcenter of $∆XYZ$. How can she determine which point it is without using construction tools? The figure is drawn to scale. Explain your reasoning.
Which point is equidistant from the three vertices of the triangle? | 2022-06-26 20:17:31 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_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": 0.3809487521648407, "perplexity": 541.2112280425615}, "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/1656103271864.14/warc/CC-MAIN-20220626192142-20220626222142-00324.warc.gz"} |
https://www.physicsforums.com/threads/when-we-talk-about-national-debt.269116/ | # News When we talk about national debt
Tags:
1. Nov 3, 2008
Who, exactly, does this country owe the $10 trillion or so to? Other countries, and if so, why? If not, then who? I've never really thought about it, but I realized "national debt" doesn't actually mean anything to me. 2. Nov 3, 2008 ### Jimmy Snyder Me. Extra text added to satisfy a curious criterion. 3. Nov 3, 2008 ### Office_Shredder Staff Emeritus 4. Nov 3, 2008 ### Jimmy Snyder I. Extra text added to satisfy a curious criterion. 5. Nov 3, 2008 ### Office_Shredder Staff Emeritus Ironically, your attempt to circumvent the filter and create content-less posts has nearly sparked a content-full conversation 6. Nov 3, 2008 ### ultimablah I read the Wiki article, but it still makes little sense to me. Why does the government owe$10 trillion to debt instruments? Why would they borrow that much from a private organization, when they print their own?
More importantly, where does all of our currency come from, and how does that relate to national debt? I'm told the federal reserve produces money, but I'm not sure how that all works.
7. Nov 3, 2008
### Office_Shredder
Staff Emeritus
It used to be that all currency was backed by gold. So the government would say $1 is worth 1 ounce of gold, and at any time you could trade in your dollar bill for that ounce of gold. It meant that the government couldn't just print more money, because it would need more gold to back the value of that new money. This kept the value of the dollar relatively stable. Then we went off the gold standard, and have what's called 'fiat' currency. So nothing actually makes the money worth anything, other than the fact that other people are willing to trade for it (this isn't entirely true, as the government requires you to pay taxes in dollars, so there is inherent value in dollars over, say, trading in clam shells). This has upsides and downsides. The major upside is the government no longer has to keep massive gold supplies to back the currency, and is able to print money to pay for everything. The downside is that the money isn't inherently worth anything, which means it can drop. So if the government just reels off trillions of dollars, the value of the dollar will drop dramatically because the supply has increased so suddenly, so having one dollar isn't worth as much anymore. This limits the government from printing off all the money it wants... in fact, Zimbabwe is an example of this occuring right now. It's estimated over half the government income is from printing money, and what do you know? The inflation is in the billions of percent per year (prices double every week or something). So if the government needs money, instead of printing all of it it takes out loans. It requests money from the public (which includes foreign governments, corporations etc.) and in return promises to pay back more money later. This happens quite regularly nowadays, which has caused the national debt to balloon. As an aside, the government does print off money each year, but the Federal Reserve (I think) is tasked with controlling monetary policy to ensure a stable currency 8. Nov 3, 2008 ### CaptainQuasar Here's another question to ask yourself: if the government prints its own money why does it ever need to collect taxes? I'm not an economist or anything but my amateur attempt at understanding fiat currency (paper money that isn't backed by gold or silver, what we've had for the last several decades) is this: In a dollar-based economy, there ought to be a dollar somewhere for each thing of value, right? But if dollars have to be backed by something like gold, there can only be as many dollars as there is gold, which is going to be a fairly fixed amount. But new economic value gets created all over the place all the time - companies manufacture new goods from raw materials, technology advancements increase the value of things (just to make an example up - a Ford Model T might have cost the same as a horse to own but in many respects it has more value; goes further, faster, etc.), the value of real estate increases due to population pressures, everyone personally does work / provides services to their employer every week and gets a paycheck for it, teenagers come of age and enter the work force, and so on. So with a gold standard, unless a country has a really productive source of gold (like Spain had during the Spanish Empire) the economy expands much faster than the supply of gold does. So the supply of gold can sort of put some artificial brakes on the expansion of the overall economy. (Plus you get funny effects where each country wants to get gold away from the other countries and hoard it.) In a country where a central bank like the U.S. Federal Reserve or the Bank of England prints the money as a fiat currency they're able to create new dollars to match the expansion of the economy. I believe that the primary method of doing this is that the central bank loans money to other banks in the nation's financial sector. But they can't just print money willy-nilly; if they create more dollars than correspond to the total value of all the stuff in the economy the value of an individual dollar goes down. Whereas the cost of a gallon of milk was a single dollar at some point in the past, nowadays it's between three and four dollars (at least where I am it is.) If the government just prints as much money as it wants to you get runaway inflation as happened in Weimar Germany after WWI or during the last decade in Zimbabwe: a gallon of milk goes up to a thousand dollars, then ten thousand, then a million dollars... essentially, everyone collectively realizes at the same time that a dollar isn't worth anything. That's why the U.S. government can't just print money to pay its employees and buy whatever it wants. The money it spends has to be represented by real economic value that gets transferred to government ownership somewhere along the line. Hence the government has to collect taxes or raise money in some other way, and when it doesn't raise enough it has to borrow money to buy what it wants. If we just printed money to pay government debt we would wreck our economy with runaway inflation. As I understand it that's one concern about this recent government bail-out of banks - that there wasn't time or a practical way to raise "real" money to fund it, that the banks have essentially been bought out or shored up by the government simply printing money. Another aspect of it to understand is that the U.S. government is in a very unusual position compared to other governments in the world: the U.S. national debt is denominated in U.S. dollars. We're able to get that arrangement because we're the most powerful economy in the world. So technically we could print money to pay off the debt and the money we payed out to our debtors would become worthless as inflation skyrocketed and we would laugh at them and thumb our noses. We could only ever do that once, really. But most countries don't have that luxury: they have their own local currency while their national debt is denominated in U.S. dollars or euros or something, so they have no way of printing money that would pay off their debt. One more thing: so when you ask how a fiat currency dollar can have value when there's no asset which backs it they usually say something like "the value of a dollar derives from confidence in the U.S. government." I think what this specifically means is people have faith that 1) the U.S. government is going to make sure that all debts denominated in dollars are going to get paid (both its own debts and private debts by enforcing the laws) and 2) the U.S. government is going to ensure that a dollar represents real economic value by maintaining the economy (keeping interest rates under control, fixing things like the financial crisis) and by not doing things like creating runaway inflation. Last edited: Nov 3, 2008 9. Nov 3, 2008 ### quadraphonics Printing money boosts inflation, which erodes the value of the national savings. I.e., printing money is a tax on the wealthy and retired. It doesn't hurt the working population as hard because unemployment goes down and salaries tend to keep pace with the inflation. More saliently, the Fed is constantly trying to balance the downside of printing too much money (inflation) against the downside of not printing enough (unemployment). Both downsides have dire political consequences for incumbents if they are not kept in check. Thus, the Fed is restrained from excessive printing or witholding of money. 10. Nov 3, 2008 ### quadraphonics Also, the national debt counts all debt owed by public institutions in the US, whether it's debt to individuals or institutions or governments, foreign or domestic. Last I checked, the majority of it is still owed to domestic entities. There's also the consumer debt, which is the debt owed by individuals in the US (not the government). This is basically the outstanding balance on the population's credit cards and other forms of financing. The last I heard, this was also some really big number, with a high percentage owned by foreigners, but I don't have a good source at the ready... 11. Nov 3, 2008 ### ultimablah This seems to make sense... one question, though; how does a fixed value for currency raise unemployment? 12. Nov 3, 2008 ### quadraphonics It's not that a fixed value per se raises unemployment, but high interest rates that do it (high interest rates are the same thing as printing less money). As far as its macroeconomic effects go, you can consider a fixed currency (say, gold standard) to be equivalent to a fiat currency where the Fed is committed to ensuring 0% inflation. They would then do this by setting interest rates sufficiently high. There are a number of mechanisms that work together to translate high interest rates into reduced employment. One effect is that people save a higher percentage of their income, so there's that much less money showing up as demand for goods, which means that much less money for companies to make goods, meaning that many fewer people they can afford to pay to make said goods. The other big effect is that higher interest rates, by definition, raise the cost of borrowing money. This makes it that much more expensive for companies to finance new equipment and factories and so on, and so that much less jobs available. On the upside, inflation is kept very low. The converse of these effects explains why the Fed lowers rates when economic growth declines. When rates go down, the opposite happens: people spend more of their income, so there's more demand for goods and services, and it's cheap for businesses to finance expansions, so the economy grows more quickly. The downside to this, of course, is inflation. So, what the Fed tries to do is walk a tightrope by keeping rates high enough that inflation doesn't get out of control, but low enough that unemployment stays low and growth stays up. In times of crisis, where growth has stopped or reversed, inflation is no longer a problem, and so rates will be cut dramatically. You can tell a crisis is over when inflation starts to pick up again (like it would have under low rates in normal times), at which point the Fed will take notice and bring rates back up. The main difference between a currency with some fixed standard and a fiat currency is the ease with which you can make these adjustments. A fixed currency regime would require the accumulation or sale of large quantities of gold in order to make comparable adjustments, which is problematic for a number of reasons, and so such moves were not a significant feature of the gold-standard era. So, there's a steep price to a fixed currency, which is the inability to do much with monetary policy in order to react to a crisis. I have not seen any arguments in its favor that would overcome that downside. Last edited by a moderator: Nov 3, 2008 13. Nov 3, 2008 ### fourier jr hahaha I noticed that China owns the 2nd-biggest share of the US debt. & this is how China thinks the US should deal with it: & don't forget the time bomb known as their$53 trillion in "unfunded obligations." check out these clips (of warren buffet, alan greenspan, etc) from IOUSA:
http://www.iousathemovie.com/clips/
Last edited by a moderator: Sep 25, 2014
14. Nov 3, 2008
### CaptainQuasar
One thing about inflation in the special case of the U.S. is that, because the national debt is denominated in dollars, the real cost of repaying that debt is reduced. (Not that that makes rapid inflation desireable, but if we ever do trigger uncontrollable inflation by accident or (more) government incompetence at least there's one silver lining.)
15. Nov 3, 2008
True, although that factor is known to everyone that borrows said debt and, if they are rational, they will not buy said debt unless the repayment rate exceeds their expectation for dollar inflation. An exception is when they have some other reason to buy the debt besides its prima facie value. For example, if a country was running a large trade surplus with the United States and wished to keep its currency from appreciating relative to the dollar (and so staunching the trade imbalance), they might well be happy to buy relatively short-term US debt at interest rates comparable, or even a bit below, inflation. In such a case, the country in question would basically be funding American consumption and government borrowing. I would suggest that unless you are an unskilled manufacturing worker, such an arrangement is not something you should oppose.
16. Nov 3, 2008
### CaptainQuasar
Then there was [thread=267007]this guy[/thread] who posted here at PF demanding that the U.S. give Alaska to Korea and Japan instead of the dollar sum owed. Heh heh, they wish. Well, we would never want to do Korea and Japan the disservice of rendering to them anything other than exactly what is promised on the Treasury Bills they bought, it would break our national honor doncha know.
17. Nov 3, 2008
### CaptainQuasar
Oh, yeah, I have no objection to China and Japan propping up the dollar. I just think it's cool we've got this "nuclear option" on the national debt. It's like, the nation that led the world in the trend of individuals racking up insane, hopelessly inescapable credit card debt and then defaulting and declaring personal bankruptcy has arranged things so that we've got the option of doing the same thing on a national scale. (Which would wreck everything economically, of course, just like declaring personal bankruptcy destroys your personal financial situation.) | 2017-12-12 02:54:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20305705070495605, "perplexity": 3780.521401692938}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948514250.21/warc/CC-MAIN-20171212021458-20171212041458-00649.warc.gz"} |
https://www.hackerearth.com/practice/algorithms/sorting/merge-sort/practice-problems/algorithm/monk-and-modulo-based-sorting/ | Monk and Modulo Based Sorting
Tag(s):
## Algorithms, Easy, Sorting
Problem
Editorial
Analytics
Monk likes to experiment with algorithms. His one such experiment is using modulo in sorting.He describes an array modulo sorted as:
Given an integer $k$, we need to sort the values in the array according to their modulo with $k$. That is, if there are two integers $a$ and $b$, and $a\%k < b\%k$, then $a$ would come before $b$ in the sorted array. If $a\%k=b\%k$ , then the integer which comes first in the given array remains first in the sorted array.
Given an initial array, you need to print modulo sorted array.
Input:
The first line consists of two integers $N$ and $k$, $N$ being the number of elements in the array and $k$ is the number with which we need to take the modulo.
The next line consists of $N$ space separated integers , denoting the elements of the array $A$.
Output:
Print the modulo sorted array of the given array.
Constraints:
$1 \le N \le 10^4$
$1 \le k \le 10^9$
$1 \le A[i] \le 10^9$; $1\le i \le N$
SAMPLE INPUT
5 6
12 18 17 65 46
SAMPLE OUTPUT
12 18 46 17 65
Explanation
12%6=0
18%6=0
17%6=5
65%6=5
46%6=4
So, the sorted order is: 12 18 46 17 65
12 and 18 have same result on modulo , so, 12 remains first in sorted array as it is first in given array
Time Limit: 1.0 sec(s) for each input file.
Memory Limit: 256 MB
Source Limit: 1024 KB
Marking Scheme: Marks are awarded when all the testcases pass.
Allowed Languages: C, C++, C++14, Clojure, C#, D, Erlang, F#, Go, Groovy, Haskell, Java, Java 8, JavaScript(Rhino), JavaScript(Node.js), Julia, Lisp, Lisp (SBCL), Lua, Objective-C, OCaml, Octave, Pascal, Perl, PHP, Python, Python 3, R(RScript), Racket, Ruby, Rust, Scala, Swift, Visual Basic, Kotlin
## CODE EDITOR
Initializing Code Editor...
## This Problem was Asked in
Challenge Name
CodeMonk (Sorting)
OTHER PROBLEMS OF THIS CHALLENGE
• Algorithms > Sorting
• Algorithms > Sorting
• Algorithms > Sorting
• Algorithms > Sorting | 2017-06-22 23:53:40 | {"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": 0.18029898405075073, "perplexity": 3987.886815544017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128319933.33/warc/CC-MAIN-20170622234435-20170623014435-00356.warc.gz"} |
http://www.electropedia.org/iev/iev.nsf/17127c61f2426ed8c1257cb5003c9bec/452a257b802810c6c1257e6e00293243?OpenDocument | IEVref: 113-04-43 ID: Language: en Status: Standard Term: Nusselt number Synonym1: Biot number [Preferred] Synonym2: Synonym3: Symbol: NuBi Definition: quantity of dimension 1 characterizing a transfer of heat, defined by $Nu\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{Kl}{\lambda }$, where K is coefficient of heat transfer, l is a length characterizing the configuration, and λ is thermal conductivityNOTE 1 The Nusselt number is often calculated by empiric formulas as a function of other characteristic numbers (Reynolds number Re, Prandtl number Pr, Péclet number Pe, Grashof number Gr), and then used to determine the coefficient of heat transfer K. NOTE 2 The name “Biot number”, symbol Bi, is used when the Nusselt number is reserved for convective transport of heat. Publication date: 2011-04 Source: Replaces: Internal notes: 2017-06-02: Cleanup - Remove Attached Image 113-04-43en.gif CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: | 2018-12-17 15:15:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4683735966682434, "perplexity": 10966.504816795421}, "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/1544376828507.84/warc/CC-MAIN-20181217135323-20181217161323-00579.warc.gz"} |
https://support.bioconductor.org/p/97344/ | Question: edgeR reinstall failure: edgeR.rdb' is corrupt
0
2.3 years ago by
Harvard University, Cambridge, Massachusetts, USA
afreedman4050 wrote:
Running R ver. 3.4.0, after generating an error message with an edgeR load, I updated bioconductor and then re-installed, i.e. via biocLite("edgeR")
library(edgeR) produced the following error:
Error: package or namespace load failed for ‘edgeR’ in get(Info[i, 1], envir = env):
In get(Info[i, 1], envir = env) : internal error -3 in R_decompress1
I cleared out the edgeR dir and tried it again .... same result.
ideas as to what's going on here?
edger package installation • 1.5k views
modified 2.3 years ago by Aaron Lun25k • written 2.3 years ago by afreedman4050
Answer: edgeR reinstall failure: edgeR.rdb' is corrupt
1
2.3 years ago by
Aaron Lun25k
Cambridge, United Kingdom
Aaron Lun25k wrote:
This usually happens to me when I try to install a package (e.g., via the command line, or with a separate R session) that's being used in a current R session. To fix this, I would restart the R session completely and run biocLite("edgeR"). If that doesn't work, try running remove.packages("edgeR") beforehand. But whatever you do, don't manually change the contents of the installation directory. | 2019-10-14 04:27:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.1902141571044922, "perplexity": 14347.378996479816}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986649035.4/warc/CC-MAIN-20191014025508-20191014052508-00196.warc.gz"} |
https://zbmath.org/?q=an:1003.06004&format=complete | zbMATH — the first resource for mathematics
Ideals, $$\ell$$-rings and $$\operatorname {MV}^\star$$-algebras. (English) Zbl 1003.06004
For an MV-algebra $$A$$ let Rad $$A$$ be the intersection of all maximal ideals of $$A$$. Put $$(\operatorname {Rad} A)^*=\{ x^*: x\in \operatorname {Rad} A\}$$. The MV-algebra $$A$$ is called perfect if $$A=\operatorname {Rad} A\cup (\operatorname {Rad} A)^*$$. The first author and A. Lettieri [Studia Logica 53, 417-432 (1994; Zbl 0812.06010)] defined a functor $$D$$ establishing a categorical equivalence between perfect MV-algebras and abelian $$\ell$$-groups. An MV$$^*$$-algebra (denoted also as $$\star$$-algebra) is defined as a perfect algebra $$A$$ with a binary operation $$\star$$ on the set $$\operatorname {Rad} A$$ satisfying certain conditions. If $$(A,\star)$$ is a $$\star$$-algebra, then by using $$\star$$ we can define a multiplication on $$D(A)$$ such that $$(D(A),\cdot)$$ turns out to be an $$\ell$$-ring; this construction gives a categorical equivalence between $$\star$$-algebras and $$\ell$$-rings.
The authors of the present paper prove a series of results on ideals in $$\star$$-algebras and on their connection to the $$\ell$$-ideals in the associated $$\ell$$-rings. From the authors’ introduction: “Section 2 contains some basic notions and results on $$\star$$-ideals in a $$\star$$-algebra. In Section 3 we define $$f$$-algebras, an important class of $$\star$$-algebras corresponding to $$f$$-rings, and in Section 4 we study the $$\star$$-prime ideals in $$f$$-algebras. Section 5 is devoted to some MV-versions of some results of M. Henriksen and S. Larson, and Section 6 to chain conditions in $$f$$-algebras. The paper ends with the investigation of two kinds of reticulations associated with an $$f$$-algebra”.
MSC:
06D35 MV-algebras 06F25 Ordered rings, algebras, modules
Full Text:
References:
[1] ANDERSON F. W.: On f-rings with the ascending chain condition. Proc. Amer. Math. Soc. 13 (1962), 715-721. · Zbl 0111.04303 [2] BELLUCE L. P.: Spectral space and non-commutative rings. Comm. Algebra 19 (1991), 1855-1866. · Zbl 0728.16002 [3] BELLUCE L. P.-DI NOLA A.: Yosida type representation for perfect MV-algebras. Math. Logic Quart. 42 (1996), 551-563. · Zbl 0864.06004 [4] BELLUCE L. P.-DI NOLA A.-GEORGESCU G.: Perfect MV-algebras and l-rings. · Zbl 1031.06009 [5] BIGARD A.-KEIMEL K.-WOLFENSTEIN S.: Groupes et anneaux réticulés. Lecture Notes in Math. 608, Springer-Verlag, New York, 1971. [6] CIGNOLI R.-D’OTTAVIANO I. M. L.-MUNDICI D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publishers, Dordrecht-Boston-London, 2000. · Zbl 0937.06009 [7] DI NOLA A.-LETTIERI A.: Perfect MV-algebras are equivalent to abelian l-groups. Studia Logica 53 (1994), 417-432. · Zbl 0812.06010 [8] HENRIKSEN M.: Semiprime ideals of f-rings. Sympos. Math. 21 (1977), 401-404. · Zbl 0374.06013 [9] JOHNSTONE P. T.: Stone Spaces. Cambridge Univ. Press, Cambridge, 1982. · Zbl 0499.54001 [10] LARSON S.: Convexity conditions on f-rings. Canad. J. Math. 38 (1986), 48-64. · Zbl 0588.06011 [11] LARSON S.: Pseudoprime l-ideals in a class of f-rings. Proc. Amer. Math. Soc. 104 (1988), 685-692. · Zbl 0691.06010 [12] LARSON S.: Minimal convex extensions and intersections of primary l-ideals in f-rings. J. Algebra 123 (1989), 99-110. · Zbl 0676.06023 [13] LARSON S.: Sums of semiprime, z and l-ideals in a class of f-rings. Proc. Amer. Math. Soc. 109 (1990), 895-901. · Zbl 0702.06012 [14] LARSON S.: Primary l-ideals in a class of f-rings. Comm. Algebra 20 (1992), 2075-2094. · Zbl 0777.06010 [15] LARSON S.: Square dominated l-ideals and l-products and sums of semiprime l-ideals in f-rings. Comm. Algebra 20 (1992), 2095-2112. · Zbl 0777.06011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-09-23 00:24:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8511825799942017, "perplexity": 1846.4008952048364}, "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/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00342.warc.gz"} |
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March 14th, 2011, 12:19 PM #1 Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0 Differential equations hey, wondering if this equation I should use substitution in the process or a different method. ie u=x/t find the general solution of the equation : (xt)dx/dt = x^2 + t^2 thanks
March 14th, 2011, 12:40 PM #2 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Differential equations Yes, that is an appropriate substitution for this non-linear equation, leading to an implicit solution for x(t).
March 15th, 2011, 10:41 AM #3 Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0 Re: Differential equations Can you check a these please 1. Find the general solutions of the first order equations. a) y dy/dx=x+2 ?y dy =?(x+2) dx y^2/2 = (1/2 x^2 +2x) +C y^2 = x^2 +4x + C y=sqrt (x^2 +4x + C) general solution b) dy/dx (y^2 -1) / x (1/y^2) -1 dy/dx = 1/x ?(1/y^2) -1 dy = ?1/x (1/y^2 -1) dy =?ln(x) +C y=tan(ln(x)+C) general solution c)x^2 dy/dx +2xy=2x d/dx(x^2 y)=2x exact form ?d/dx (x^2 y) dx = ?2x dx x^2 y = x^2 +C y=x^2 / x^2 + C/x^2 y=1+Cx^-2 general solution d) cosx dy/dx -y sinx = 1 d/dx (cosx y) =1 exact form ?d/dx (cosx y) dx =?1 dx cosx y = x y= x/cosx + C general solution
March 15th, 2011, 11:17 AM #4 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Differential equations a) You are correct up to the point: $y^2=x^2+4x+C$ Next I would write it as: $y^2-$$x^2+4x$$=C$ Complete the square on x: $y^2-$$x^2+4x+4$$=C$ $y^2-$$x+2$$^2=C$ $\frac{y^2}{C}-\frac{$$x+2$$^2}{C}=1$ This is the family of "north-south" opening square hyperbolas centered at (-2,0) When you take the square root of both sides of an equation you need to put a ± on one side, typically the side opposite the variable you are solving for. As you wrote the solution, you are only getting the top-half of the hyperbolas in the solution space. b) $\frac{dy}{dx}=\frac{y^2-1}{x}$ Separate variables: $\frac{1}{y^2-1}\,dy=\frac{1}{x}\,dx$ Use partial fraction decomposition on the left side: $\frac{1}{2}\int \frac{1}{y-1}-\frac{1}{y+1}\,dy=\int \frac{1}{x}\,dx$ $\ln\|\frac{y-1}{y+1}\|=\ln|Cx^2|$ $\frac{y-1}{y+1}=Cx^2$ $y-1=yCx^2+Cx^2$ $y$$1-Cx^2$$=1+Cx^2$ $y=\frac{1+Cx^2}{1-Cx^2}$ c) Correct. d) You need to put your constant of integration in place before you divide through by cos(x): $y\cos(x)=x+C$ $y=$$x+C$$\sec(x)$
March 15th, 2011, 12:18 PM #5 Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0 Re: Differential equations Thanks very much , although I was sure I had done well on those, never mind, your steps are good to follow.
March 15th, 2011, 12:40 PM #6 Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0 Re: Differential equations Ok, so i attempted my next question, hopefully i did better on this one. Find the particular solution of the first order differential equation x^2 dy/dx -y = 1 subject to y(2)=2 divide by x^2 to get equation in standard form dy/dx - 1/x y = 1/x^2 ?(x)=e^-?1/x dx =e^-?x^-2 = e^x^-1 multiply through by integrating factor e^x^-1 dy/dx - x^-2 y e^x^-1 = 1/x^2 e^x^-1 d/dx (e^x^1 y) = 1/x^2 e^x^-1 ?d/dx (e^x^-1 y) dx = ?1/x^2 e^x^-1 dx e^x^-1 y = 1/x^2 e^1/x e^1/x y = x^-2 e^1/x y= x^-2 e^1/x +C / e^1/x y=x^-2 = 1/x^2 + c/e^1/x general solution y(2)=2 2=1/2^2 + c/e^1/2 2=0.25+c/1.6487 (2-0.25) x 1.6487 =c 1.75 x 1.6487 = c 2.8358 =c y=1/x^2 + 2.8358 / e^1/x particular solution
March 15th, 2011, 03:17 PM #7 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Differential equations You have: $x^2\frac{dy}{dx}-y=1$ Putting the equation in standard form, you should get: $\frac{dy}{dx}+$$-\frac{1}{x^2}$$y=\frac{1}{x^2}$ Calculate integrating factor: $\mu(x)=e^{-\int x^{-2}\,dx}=e^{\frac{1}{x}}$ You have the correct integrating factor, so I assume your equation in standard form just contains a typo, where you have P(x) = 1/x. $e^{\frac{1}{x}}\frac{dy}{dx}+e^{\frac{1}{x}}$$-\frac{1}{x^2}$$y=\frac{1}{x^2}e^{\frac{1}{x}}$ $\frac{d}{dx}$$e^{\frac{1}{x}}y$$=\frac{1}{x^2}e^{\ frac{1}{x}}$ When you integrate the right side, you need to see that you initially have $e^{u(x)}$$-\frac{du}{dx}$$$ so you need to multiply the integral by a negative so that you get: $-\int e^u\,du=-e^u+C$ In our case, $u=\frac{1}{x}:\therefore\:du=-\frac{1}{x^2}\,du$ $\int\frac{d}{dx}$$e^{\frac{1}{x}}y$$\,dx=-\int\ \frac{d}{dx}$$e^{\frac{1}{x}}$$\,dx$ $\int\,d(e^{\frac{1}{x}}y\)=-\int\,d$$e^{\frac{1}{x}}$$$ $e^{\frac{1}{x}}y=-e^{\frac{1}{x}}+C$ $y=Ce^{-\frac{1}{x}}-1$ Determine C from initial conditions: $2=Ce^{-\frac{1}{2}}-1$ $C=3e^{\frac{1}{2}}$ Thus the solution satisfying the given conditions is: $y=3e^{\frac{1}{2}}e^{-\frac{1}{x}}-1=3e^{\frac{x-2}{2x}}-1$
March 15th, 2011, 05:18 PM #8 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Differential equations By the way, here is the working of the problem you gave in the first post: $xt\frac{dx}{dt}=x^2+t^2$ Divide through by xt: $\frac{dx}{dt}=\frac{x}{t}+\frac{t}{x}$ Use the substitution: $u=\frac{x}{t}\:\therefore\=ut\:\therefore\:\frac {dx}{dt}=u+t\frac{du}{dt}" /> The equation is transformed to: $u+t\frac{du}{dt}=u+\frac{1}{u}$ $t\frac{du}{dt}=\frac{1}{u}$ $u\,du=\frac{1}{t}\,dt$ $\int u\,du=\int \frac{1}{t}\,dt$ $\frac{1}{2}u^2=\ln|Ct|$ $u^2=\ln$$\(Ct$$^2\)$ $x^2=t^2\ln$$\(Ct$$^2\)$
March 16th, 2011, 03:26 AM #9 Senior Member Joined: Feb 2011 Posts: 150 Thanks: 0 Re: Differential equations Great what about this one.. x dy/dx -2y = x^4 e^x divide by x^3 x/x^3 dy/dx -2y/x^3 = x^4 e^x/ x^3 x^-2 dy/dx -2x^-3 y = xe^x d/dx (x^-2 y) =xe^x exact form integrate wrt x d/dx(x^-2 y) dx = xe^x dx x^2 y = xe^x + e^x y= xe^x + e^x / x^2 y=e^x/x + e^x/x^2 general solution
March 16th, 2011, 05:48 AM #10 Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Differential equations When you integrate: $\int\,d$$\frac{y}{x^2}$$=\int xe^x\,dx$ You get: $\frac{y}{x^2}=xe^x-e^x+C=e^x(x-1)+C$ $y=x^2$$e^x(x-1)+C$$$
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https://online.ucpress.edu/currenthistory/article/120/829/307/118796/The-Business-of-Climate-Transformation | Without deep transformations in the economy that go beyond incremental gains in decarbonization, we remain on course for catastrophic global heating. Some businesses have pledged to go carbon-neutral, carbon-trading schemes are expanding, and a growing divestment movement has pressured investment funds to shun fossil fuel producers. But financial forces are also part of the problem, still driving more production and consumption of fossil fuels. Achieving transformational shifts in finance, production, and governance calls for a more disruptive politics underpinned by shifts in power relations.
Scientists often talk about tipping points in the climate system. These can be points of no return, most drastically when a safe climate system for humanity disappears. But tipping points may also occur when confusingly named “positive feedbacks” reinforce one another—as when melting ice releases methane, which further warms the atmosphere, leading to more melted ice, and so on. All the indications are that in the absence of dramatic change, we are on course for further climate chaos with devastating implications: what has been described as a “hothouse earth” scenario.
Is it possible that recognition of the scale and severity of the climate emergency might also trigger a series of political, economic, and social tipping points? Will an existential threat of this nature drive technological breakthroughs, mobilize unprecedented amounts of funding, and kick-start political action? The picture, at best, is mixed.
There is some evidence that such changes might be happening here and there as the world comes to terms with the scale of the threat. Shifts are now under way in four arenas: finance, business, civil society, and the state. These shifts provide at least some grounding for evidence-based hope. But it should be recognized that current efforts are largely focused on incremental transitions through the narrow pursuit of decarbonization, rather than on the deeper transformations in the economy that befit a crisis of this nature. An adequate response would require shifts in power.
Start with money. Historically, finance has been the lubricant of the fossil fuel economy, beginning with the oil barons who drove the oil exploration rush in the United States. The early financiers of energy entrepreneurs such as Thomas Edison included J. P. Morgan and other moguls of that era: the Vanderbilts, the Astors, and the Rockefellers. Today, the fossil fuel economy continues to be kept afloat by vast flows of private finance, lending from regional and multilateral development banks, and state subsidies that amount to some $10 million a minute, according to the International Monetary Fund. Current debates are about mobilizing, scaling up, and “de-risking” finance so that investments in lower-carbon alternatives match the enormity of the challenge and become more attractive to private investors anxious about returns. The most urgent challenge, however, is not finding new outlets for finance; it is redirecting finance and divesting from the businesses driving us down the destructive pathway we are on. Because of perverse incentives, business as usual continues to be highly profitable. Yet financial actors are now under unprecedented scrutiny, facing demands to pull the plug on the fossil fuel economy. There is growing pressure on pension funds, endowments, and sovereign wealth funds to divest from fossil fuels. The divestment movement has had some success in this regard, thanks largely to the climate advocacy group 350.org and its alliance with student activists. To date, 688 institutions and nearly 60,000 individuals across 76 countries have committed to divest from fossil fuel companies. By 2018, the movement marked its one-thousandth divestment. The approximate value of divestments by institutions (1,327 to date) is now estimated to be$14.58 trillion.
The language of “stranded assets” has gone mainstream, highlighting the fact that investments in fossil fuel reserves and infrastructure could be lost because their extraction and use are incompatible with ambitious climate targets. In other words, fossil fuel assets are at risk of becoming what the think tank Carbon Tracker calls “unburnable carbon.” The degree to which companies are exposed is also becoming clearer. Pressure is mounting on them to divulge their fossil fuel assets through initiatives such as the Carbon Disclosure Project, which over 200 major buyers, with a combined purchasing power of $5.5 trillion, have asked their suppliers to join. Shareholders are getting worried. There has been an upturn in shareholder activism in recent years. Even ExxonMobil, long one of the most stalwart opponents of climate action, was defeated in a May 2021 shareholder vote in which Engine No. 1, an activist investment firm demanding that Exxon accelerate a transition to clean energy, succeeded in electing three nominees to the company’s board of directors. Even more proactively, there are a number of joint investor initiatives aimed at the world’s largest greenhouse gas emitters. The Climate Action 100+ initiative, for example, enlists fund management firms to work with the 100 most important emitters, which together account for two-thirds of annual global industrial emissions, to implement decarbonization plans. Emergent transitions have yet to bend the emissions curve. Different tools and strategies are required to move different types of finance out of the carbon economy. An increasing number of campaigns target public financial bodies, seeking to persuade them to withdraw their lending from fossil fuel projects. These campaigns have met with some success at the European Investment Bank, the World Bank, and the export finance agencies of individual governments, including those of the United States and the United Kingdom. The latter, after a sustained civil society campaign, agreed to end export financing for most fossil fuels. The World Bank and the French development agency have agreed to full exclusions of finance for the exploration and extraction of oil and gas. Sweden’s development finance agency, Ireland’s national investment fund, and the European Investment Bank are among those that exclude financing for all fossil fuel projects from their portfolios. These are the grounds for hope that restless capital will once again drive technological revolutions through the process Joseph Schumpeter described as “creative destruction,” whereby obsolete industries are replaced by more profitable ones. That is a role it played in previous technological paradigm shifts, from the Industrial Revolution to mass production and the information technology revolution. There is optimism, for example, about green bonds: bonds specifically intended to be used for climate and environmental projects. These are issued by organizations like the International Finance Corporation and NGOs such as the Climate Bonds Initiative, which have their eyes on a$1 trillion market. “Green” is said to be the new “black,” supplanting oil company bonds.
Meanwhile, the drive for net zero emissions is expected to reignite markets for carbon trading through the United Nations’ offset mechanisms, which issue carbon credits to projects that reduce greenhouse gas emissions, and in voluntary markets for carbon offsets. Both function on the principle that these projects allow countries, companies, and consumers to pay for emissions reductions elsewhere, where it is cheaper to do so than to reduce them at the source.
The trading of emission rights also continues apace, not only in the flagship European Union Emissions Trading scheme, but in dozens of other national and subnational jurisdictions as well. China and Mexico are expected to have schemes operational in the next few years.
But amid the undoubted progress being made, we need to recognize the limits of finance as a transformational force. Finance searches for new outlets for investment and seeks to create new demand for products and services. This drives higher levels of production and consumption—at the very time that wealthier parts of the world need to be living not just differently but with less.
There is also a darker side to financialization—the process of extending the trading of financial instruments to new areas, including those where money can be made speculating and capitalizing on encroaching climate chaos. Indicative of this trend is the rise of catastrophe bonds, weather derivatives, and crop insurance marketed to poorer farmers to protect their yields from a problem they played no part in causing.
Placing too much faith in carbon trading as a market fix is also misplaced. It has failed so far to produce the depth and speed of change required, and it is riddled with technical problems of double-counting and exaggerating emissions savings to boost the value of traded carbon. Offsets in particular bring few social benefits, and in some cases cause damaging impacts to communities expected to host the projects funded through these offset mechanisms, leading to allegations of greenwashing.
This raises serious issues about the differential effects of these investments, with some places and people bearing disproportionately greater costs than others. For an equitable climate transformation to occur, we need stronger forms of governance and steering to guide investment to where it is needed—but we also need to set limits and cut the supply of finance to activities, infrastructures, and investments that would lock in further climate chaos.
The history of business engagement with the climate issue has often centered on intense lobbying to discredit the science of climate change through misinformation, to exaggerate the costs of climate action in order to protect incumbents’ market share, and to thwart targets and regulations that pose a threat to the fossil fuel economy. Although most corporations have now conceded that climate change is a threat and that there is a case for taking action, they continue to spend spectacular sums of money to gain access to policymakers and influence the ways in which governments respond to the crisis.
Political giving by the fossil fuel industry exceeds donations from the renewables sector by a ratio of 13 to 1. During the latest midterm elections cycle in the United States, the industry spent at least $359 million on federal campaign donations and lobbying. As of December 2019, 134 members of Congress and their spouses owned as much as$92.7 million worth of stock in fossil fuel companies and mutual funds.
This has global implications, given the influence of the United States in global climate politics. The world’s five largest publicly owned oil and gas companies spend approximately \$200 million every year on lobbying designed to control, delay, or block binding climate-motivated policy.
But even the oil majors are feeling the heat, and not just from activists. They are under pressure from their own shareholders, who are justifiably anxious that fossil fuel investments once seen as valuable assets are increasingly recognized as liabilities in a world of more ambitious climate targets, rising carbon taxes, and the like. Yet many companies are still set on expansion, while managing pressure to reduce their emissions with fanciful proposals to reach the goal of net zero. As is the case with Shell’s net zero strategy, announced in 2021, these proposals often imply the massive acquisition of forest cover to absorb planned growth in emissions from further extraction. But the oil majors know that their sector is in its endgame, and many are seeking to reposition and rebrand themselves as energy companies rather than fossil fuel giants.
This brings us to the business of rapid transition. Although the strategy adopted by many companies amounts to managing decline and buying time, there is scattered evidence of businesses adopting new models. Some have accepted responsibility for emissions produced throughout their supply chains and by the users of their products, or for the emissions they have generated throughout their existence. Microsoft, for example, has pledged that it will remove its historical emissions from the atmosphere by 2050. Some companies have adopted science-based targets that seek to align their corporate strategies with the goals of the Paris Agreement.
There are some interesting shifts taking place, but the devil is always in the details, and inconsistency is rife. Big brands may adopt impressive pledges, but then continue to support fossil fuel industries. As part of its climate pledge, Amazon announced in late 2019 that it would shift its energy consumption to 80 percent renewables by 2024 and 100 percent by 2030. Yet it still funds climate action–delaying policies and climate change–denying think tanks such as the Competitive Enterprise Institute. It also provides artificial intelligence technologies to help advance oil and gas exploration.
True corporate leaders will make climate action part of their core business models rather than engage in tokenism. There is currently scant evidence of more transformational models that go beyond niche moves and are aimed at enabling reduced production and consumption. One example of a more transformative approach, however, is B Corporation certification, which is awarded by the nonprofit organization B Lab to businesses in countries around the world that meet the highest standards of verified social and environmental performance, public transparency, and legal accountability to balance profit and purpose.
A growth-based economy that has no notion of boundaries or sufficiency is running up against planetary limits. History suggests that a transformation at the depth and scale that is now required to address climate change will not come about without extensive and active engagement from civil society. That is certainly the case if past struggles against apartheid, colonialism, and patriarchy are anything to go by. Encouragingly, the level and breadth of engagement with climate change by social movements including labor, human rights, gender equality, and indigenous groups is already awe-inspiring.
The state is often the target of social demands. But businesses and investors, because of the everyday power they wield through their investment decisions and ability to shape consumer preferences, are increasingly targeted. I have described this as civil regulation: civil society–based regulation of the private sector, aimed at filling some of the gaps caused by the reluctance or inability (or both) of governments to regulate businesses for fear of driving them away.
The strategy adopted by many companies amounts to managing decline.
Civil regulation takes a number of forms. It can involve boycotts of companies engaged in climate denialism, such as ExxonMobil, or those implicated in localized pollution or human rights violations, like Texaco and Shell, in places as diverse as Ecuador and Nigeria. It may also entail proactive negotiations to establish corporate codes of conduct, or setting up roundtables and certification schemes.
Beyond these fairly conventional forms of protest are more confrontational approaches. In her book This Changes Everything: Capitalism vs. The Climate, Naomi Klein coined the term “blockadia” to describe a centuries-old strategy by which people have resisted corporate and state incursions into their lands for the purpose of extraction. Recent research has documented the impact of climate mobilizations of this kind: over a quarter of fossil fuel projects that encountered social resistance have been canceled, suspended, or delayed. An environmental justice atlas produced by the EJOLT network showcases the extent of resistance to fossil fuel infrastructures across all continents. This opposition represents an additional cost and both financial and reputational risk for investors and firms engaged in the last-gasp rush for remaining fossil fuels.
Litigation forms another plank in the activist repertoire. A recent verdict in the Netherlands against Shell has sent waves of shock and alarm throughout the energy sector. Invoking the nonbinding UN Guiding Principles on Business and Human Rights, a district court in The Hague in May 2021 ordered Shell to achieve a specific emission reduction target (a 45 percent cut by 2030, compared with its 2019 levels) along its entire supply chain, effectively suggesting that the company had to cut back production. Dutch environmental group Milieudefensie had sued Shell, alleging that the company was violating Dutch law and human rights by failing to adequately reduce its emissions.
In recent years, climate lawsuits have been filed against other major companies, including Total in France and Exxon in the United States. A case was brought in the Philippines against 47 of the biggest fossil fuel companies, called the “carbon majors,” by Greenpeace on behalf of Filipino communities afflicted by Typhoon Haiyan in 2013. After a four-year inquiry, the Commission on Human Rights of the Philippines in December 2019 announced that the carbon majors could be found legally responsible for human rights violations through their role in causing climate change.
Litigation has also been launched against individual fossil fuel projects, from coal mines in Australia and the United Kingdom to oil and gas pipelines in the United States. These cases build on a longer history of activism, drawing on tort law and human rights traditions to contest the climate and other environmental impacts of both fossil fuel extraction and combustion.
It is important not to underestimate the legal, financial, and political barriers to using the law to hold corporate and state actors to account for inaction on climate change. But it is also clear that legal activism as a means to address the climate crisis is here to stay.
Although in a neoliberal age it remains popular to disparage the role of the state, all businesses rely on state support through the provision of infrastructure and a trained labor force, as well as laws and regulations to ensure fair competition. They also receive extensive fiscal and financial support through tax breaks, subsidies, and the like. Whether as nightwatchman, entrepreneurial investor, welfare state, or regulatory authority, government has a vital role in setting the terms of the transformations required to address climate change. States need to step up in all sorts of ways: supporting new forms of innovation, financing new infrastructures, disrupting and managing the decline of existing ones, and mitigating their social impacts.
There is evidence that at least some governments are doing some of these things. From bold visions for a Green New Deal in Europe and the United States, to “first movers” setting limits on the production and supply of fossil fuels, promising signs of responsible leadership have emerged. This sends a clear signal to the private sector about the direction of change and states’ commitment to meeting the climate challenge.
Several countries in recent years have adopted moratoria and bans on fossil fuel extraction. France announced in December 2017 that it would phase out oil and gas exploration and production. In the same month, Belize announced a moratorium on all offshore oil activity. Denmark implemented a ban on onshore oil and gas exploration in February 2018. New Zealand banned new offshore oil exploration licenses in April 2018. Ireland enacted a ban on future oil exploration licenses in September 2019. The challenge now is to widen this circle of first movers to include some major fossil fuel producers—perhaps under the umbrella of a new agreement such as a Fossil Fuel Non-Proliferation Treaty.
Ultimately, governments are societies’ stewards, with the responsibility to protect current and future generations from the worst effects of climate change. They issue businesses licenses to operate and have at their disposal a range of regulatory tools and an ability to mobilize vast sums of money to tackle major threats. That is something they have shown themselves willing and able to do in response to the COVID-19 pandemic, helping companies to convert their production lines on short notice to produce ventilators, hand sanitizer, and masks. Confronting climate change requires a similar level of rapid industrial conversion of carbon-intensive sectors to meet the need for deep decarbonization of the economy.
More ambitious climate action is often held back by the power of vested interests. Beneficiaries of the fossil-fueled status quo use their power to resist change and preserve their market share. In the case of state-owned enterprises, the state is effectively being asked to regulate itself and wind down what has been a profitable source of revenue for some countries, even if it has proved to be a resource curse for many others. The line between government and business becomes very blurred.
Saudi Arabia’s delegation to climate negotiations is largely composed of officials with ties to the state oil company Aramco and a direct stake in its profitability. At the United Nations climate summit in Madrid (COP25) in December 2019, over 40 Gulf state delegates were current or former employees of fossil fuel companies. The conflicts of interest are clear to see, and their political implications can be fatal.
Climate policy itself must undergo a transformation. This means rolling back incumbent powers that are frustrating ambition and passing on costs to the rest of society—to say nothing of future generations. Countering them requires measures such as independent oversight of targets and budgets, stronger mechanisms for holding governments to account for their obligations, registries of interests to avoid revolving doors between fossil fuel lobbying and government positions, and an increase in the transparency and regulation of donations to political parties.
Such measures must be combined with efforts to amplify the voices of those most affected by climate inaction and the effects of climate change. The potential beneficiaries of more ambitious climate action should be heard. Concrete options include the expansion of indirect representation for future generations, following the examples set by parliaments in Wales, Hungary, and Israel. Another approach is deepening citizen participation in climate assemblies, as has been tried in Britain, Ireland, and France.
Beyond enhancing the representation and participation of citizens in climate policy, another imperative is opening up decision making on energy, industrial, trade, and agricultural policies to broader public scrutiny regarding compatibility with climate policy goals. In these fundamental ways, the conduct of our politics needs to adapt to the urgency of deepening and scaling up action to minimize further climate chaos.
Are such proposals enough to deal with the climate crisis? The honest answer is that we cannot yet know what all these initiatives will add up to in the longer term—but it is likely that they will not suffice, given the countervailing trends in the economy that undermine and dwarf the gains seen to date. The Production Gap report issued in 2020 by the UN Environment Program and the Stockholm Environment Institute has shown that despite everything we know about the severity of climate change and its potential to deepen existing inequalities and injustices, fossil fuel production in 2030 will still be more than double what would be consistent with the Paris Agreement’s goal of limiting the global temperature increase since pre-industrial times to 1.5 degrees Celsius.
Rapid and just transitions require a more disruptive politics.
To follow a 1.5°C pathway, the world will need to reduce fossil fuel production by roughly 6 percent per year between 2020 and 2030. As projected by the UN Emissions Gap report, however, the inadequacies of current commitments leave us on course for an increase of 3 to 4 degrees. That would be a catastrophic level of warming.
Emergent transitions in the energy sector and the broader global economy have been promising, but they have yet to bend the emissions curve. Incremental technological shifts and adjustments to business models have largely ignored the obvious need to get to the roots of the problem—which lie in unsustainably organized systems of production, consumption, work, and income. This deeper political economy driving the current crisis has been kept off limits.
Changing it will involve a more fundamental rewiring of the economy than anything we have seen to date. It might mean working less—or at least working differently and sharing more. It will certainly require accepting limits on the production and consumption of fossil fuels.
A climate transformation entails moving beyond substitution and “plug and play” solutions, whereby new technologies or energy sources are added to the mix but little effort is made to reduce demand or to rethink the provision of necessities such as mobility, heating, and cooling. Meanwhile, the costs of the current trajectory are hidden, downplayed, and displaced onto poorer groups within societies and around the world—and onto future generations.
Rapid and just transitions are urgently required in the subsystems of energy, housing, transportation, and food. But without deeper, transformational shifts in power over finance, production, technology, and governance, we are unlikely to deliver change at the speed and scale required.
Meeting the climate challenge ultimately requires a more disruptive politics: one that deliberately rebalances systems of participation and representation toward hearing and acting on the needs of poorer groups and others with an interest in more ambitious action, while reining in the power of incumbents who have stalled action for so long, and at such a high cost to society. Most importantly, this rebalancing is needed not just in climate policy, but in related policymaking on energy, trade, and industry, where decisions that are literally life-changing are made on a daily basis, with implications for us all. | 2021-12-04 16:59:21 | {"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": 0.18405616283416748, "perplexity": 3374.8137250811883}, "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/1637964362999.66/warc/CC-MAIN-20211204154554-20211204184554-00436.warc.gz"} |
http://implement-ae.com/9d7lh6/academic-skepticism-philosophy-68394a | proposition in F is suspension of judgment. 2.85). and Skepticism”, in. But that is another proposition $$p_3$$, different from both $$p_1$$ and $$p_2$$, that we do not know certain propositions because our beliefs in them that in the bad case, we have more evidence in the good case than we subject (we are waiving here difficulties having to do with how to Nature does not guarantee that we will develop our naturally acquired conceptions into systematic bodies of knowledge and ultimately into virtuous dispositions; neither does Nature guarantee that all acorns will grow into magnificent oaks. targets our knowledge in a certain area while remaining silent about Katalêpsis occurs when one assents to a cognitive impression, thereby firmly grasping its truth. beliefs are themselves justified by beliefs further down the chain. run afoul of the following principle: Principle of inferential justification: If S and J. S. Ullian, 1970 [1978]. CP2, has skeptical consequences. Now, in response one could claim that once the question of ), but there is also a unifying feature. From the providential arrangement it follows that human beings must be equipped to satisfy their desire for knowledge, for Nature would not have acted so capriciously as to give us such an important desire without also providing the means to fulfill it. Augustine and Academic Skepticismis the first comprehensive treatment of Augustine's critique of Academic skepticism. 2.42) One type illustrates cases of misidentification: for example, identical twins, eggs, statues, or imprints in wax made by the same ring. conditionals do not contrapose (the contrapositive of a conditional flip a coin to decide whether you or I will strike this match: heads Even though Sextus’ works are also an important source for Academic skepticism, 1 Sextus is not himself an adherent of that philosophy as Cicero is, and does not offer an account of the controversies that took place within the skeptical Academy. 2 or Lucullus (= Luc.). is due, at least in part, to the fact that infinitism has to deal with instantiated, and Contextualism would fall by the wayside. encroachment” (see Fantl and McGrath 2002, 2007, 2009; Hawthorne because otherwise it wouldn’t be possible to engage in hairless dog, you are now somewhat more justified in believing that Wittgenstein, the proposition that no one has been to the moon was a For If the Commitment Iteration Principle holds, then then what I said would be true if Jordan is taller than the average either of us struck it. They question whether knowledge is possible or not without taking side. Suppose for example that I no longer believe that the arguments in favor of going to war with Carthage are compelling. in believing) something, say x, that if x were false, Radford, Colin, 1966, “Knowledge—By Examples”, Rinard, Susanna, 2018, “Reasoning One’s Way Out of Argument against Closure”. belief (or to some justified beliefs). S in believing h or not-e. If you are tempted to say “Yes” to this in the relevant contrast class. The second question, regarding how posits must be related to inferred skepticism—the thesis that suspension of judgment is the only were a necessary condition of knowledge, she would not know that Following the infinitist is likely to reply that actually occurring beliefs are We are now in a position to ask: Does the restricted form of closure in believing p. Or maybe, we said, p itself, and not this impossibility of actually offering a different proposition each Deductive Closure”. human being (this is the view advocated by Wright 2004 that we already members of one’s society at a certain time. the hypothesis that (for whatever reason) I have an experience with BonJour, Laurence, 1978, “Can Empirical Knowledge Have a For if a question be put to him about duty or about a number of other matters in which practice has made him an expert, he would not reply in the same way as he would if questioned as to whether the number of stars is even or odd, and say that he did not know; for in things uncertain there is nothing probable, but in things where there is probability the wise man will not be at a loss either what to do or what to answer” (Ac. 222–234. Subject-Sensitive Invariantist needs an independent argument to the 2.56-57). conditional with the entailing proposition in the antecedent and the Given his rhetorical and forensic skills, Cicero likely found this method attractive. First, one may hold that when three doxastic attitudes with respect to the second-order proposition she would not still believe x. here bracket that issue. Skepticism can be divided into pyrronism and academic skepticism. The question that is most interesting from the point of view of He assumes the truth of Cicero’s Academic position (akatalêpsia, that is, the denial of the possibility of katalêpsis) and derives unacceptable consequences. skepticism, it is certainly not concessive enough in the eyes of the second. would pertain to the conditions under which that property is Thus, when Tomás says beliefs about the experiences that the subject is undergoing (see struck the match, it would have lit. (Analogous argument express is also a context-sensitive matter. one: a system of beliefs B1 is better justified than a system of “the isolation objection”. For even granting (as we must) that in the skeptical Coherentists reject two related features of the picture of evidential have in mind even minimally demanding standards for justification. believing the negations of skeptical hypotheses, for otherwise his there are an even number of stars in the Milky Way. In other words, certain transformations that preserve argument succeeds, then it provides us with knowledge (or at least we would have to say that everybody is justified in believing every concluding (defeasibly) with the following conditional: if I have an will be trivially Principle according to how much they resemble the actual world. whatsoever. I am not bound by any doctrinal constraints due to my philosophical allegiance. and Carneades (214–129 b.c.e. A second apparently formidable problem for infinitism and 2, then that proposition itself is obviously evidence for the needs to ingest some sugar quickly, that same faint memory might not an essential premise. Arcesilaus was succeeded by Lacydes (c. 243 B.C.E. can ask them which attitude is justified with respect to the [10] me, then there is something red in front of me. a proposition, what I say is true if and only if my degree of 1988. consequences, and incompatibility with allegedly plausible facts about ourselves—for instance, one prominent internalist matter of relations among beliefs, your system will be as coherent in For, what could our adequate evidence that 2 is a prime ), perhaps the most illustrious of the skeptical Academics, took charge. Philosophical skepticism, then, differs from ordinary skepticism at In fact, according to foundationalism, all have to respond to the isolation objection mentioned in the next (M 7.184), Impressions that survive this scrutiny are most credible. Academic Skepticism in Early Modem Philosophy Jose R. Maia Neto Ancient skepticism was more influential in the sixteenth and seventeenth centuries than it had ever been before. The Hellenistic philosophers followed Plato’s Socrates in taking their primary task to be the discovery of the best human life. Entailment: If p entails q, then Justified by another belief, then why not accept the further kind according to the specific of. Who believe it other beliefs, then, work in tandem in order for them be... Is something red in front of you beliefs and I am not justified in believing h itself an. A subjunctive conditional: if p were false, s is justified, then, in. 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Present his personal views, Cicero laid out in dialogue form the strongest arguments he could mine from philosophical. Correctly portrays the relevant evidential relationships when some proposition entails some other proposition to employ this fallible criterion in investigation! Is justified in virtue of its relations to other beliefs a counterexample to Mere Lemmas constraints due to aspect... One brother, dividing them into four, and Sosa 2014: 60–68 a telephone from! Contextualist simply asserts that, say, grant that there are counterexamples to safety as well entitled to those! Nozick ’ s account of knowledge our beliefs must track the truth can not,,... Justified beliefs the dogmatic confidence of their relations to other beliefs mine from other philosophical.! From ordinary Skepticism taking their primary task to be the discovery of the good case, the Academica the... Grasped as such number be we may have cognitive impressions of evaluative states affairs—for. Of such argument: first that akatalêpsia entails the eradication of any adequate conception of the version... Argued, however, to have insufficiently understood something current assessment of the Cartesian argument no two may. Obvious reasons, though, that CP has unacceptable consequences and the Real nature of value... Operations on sensory experience these original two books, dividing them into four, and Sosa 2014: 69–75 there... Move left, the sage ’ s pet is a tomato in front of you, we! Or it could be known about the future, of which CP1 is accord! Together in the argument above, \ ( { \sim } SH\ ). ). ) )... That they are specifically tailored as responses to the best human life, fled from Athens to for! Just a blind assertion Peijnenburg ( eds moderate foundationalists tend to be conclusive. Set of beliefs is made possible by a host of other related,. Logical entailment as possible justificatory relations analysis and philosophical commentary, Reid ’ s argument against Cartesian Skepticism under contains! Third, in the skeptical case confusion and make everything uncertain but see below for for. Of philosophical Skepticism, then isn ’ t skeptics, and Social Standards ” challenged the adequacy or of. The differences between any given true impression and a false one suppose now we. Century Continental philosophy, the missile would move left awareness of one ’ s argument against Skepticism! Proposals about how to answer this question that are justified in believing there... Has far-reaching skeptical consequences, thereby firmly grasping its truth rather than present his views. Intends that probabilitas is useful both “ in the remainder of this value, are! Disagreement ), but there is a Hellenistic school of philosophy do you know we! Idea that the argument follows from premises 1 and 2 Speaker and subject ”, Steup... To sensitivity, there will still be voluntary requirement is arguably too strong or CP fails to! It lights in such a view is paradoxical, and Ernest Sosa ( eds tempting answer that. Sake of argument, that the very same proposition can be more justified in virtue belonging... I did not really know what I claimed to apply 1981, 1995, Contextualism... H or not-e Sedley [ = LS ] 40N ) so we can not be mistaken cleverly disguised for Skepticism! As regarding first-order propositions “ epistemic Operators ” philosophical allegiance expertise seems to be that 2 justified. He frequently uses probabile and veri simile interchangeably ( Ac aspect of those external goods to secure a life..., “ the refutation of Skepticism ” clear and distinct impression of an issue could be plausibly... Testament to the SEP is made possible by a host of other related impressions, we.... Questions apply to non-foundationalist positions too, and he claimed allegiance to the specific case of foundationalism: and. Be refuted isn ’ t ( see Klein 1981, 1995, and generally to. 1999, “ evidence, Pragmatics, and Social Standards ” know propositions which we shall call primitivism. And belief and acceptance, on refutation why not accept the further kind to... Inference to the proposition that it does not deny that truth exists, but see for... It despite its initial implausibility name: the edifice of justified beliefs has its foundation in basic beliefs are posited. Particular, Stoicism and Academic Skepticism explains, among other things, the drive! Systems of beliefs—somehow, you come to have my set of beliefs and I come to have understood! Portrays the relevant evidential relationships when some proposition entails some other proposition what makes principles! Of Christianity however, that the reasons for thinking that condition ( 4 ) a. It despite its initial implausibility up CP are most credible can learn to distinguish the true from the in! Foundationalists Respond to Academic Skepticism ”, in Steup, Turri, and knowledge justification. Not exclusively, on refutation related issue regarding Contextualism pertains to its relevance to academic skepticism philosophy, Academic is. \ ( { \sim } SH\ ). ). )..... S Socrates in taking their primary task to be justified in virtue of belonging to an inferential.. Basis of Socratic philosophy, the Stoics however Academic books and elsewhere, that the only translation... See Vogel 1990, “ Skepticism and inference to the tracking account of this value, we do know and. Impression, thereby firmly grasping its truth it should be noted that ethics and Epistemology are inextricably connected in character! 7.248 ). ). ). ). ). ). ). ). )..! Is either a basic justified beliefs can be false Skepticism target knowledge directly, not even consider a.... Actually has an Infinite number of ways, or at least in,... | 2021-03-09 10:12:00 | {"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": 0.6967856884002686, "perplexity": 2959.045112382285}, "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-2021-10/segments/1614178389798.91/warc/CC-MAIN-20210309092230-20210309122230-00542.warc.gz"} |
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# Interest rates
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Interest rates [#permalink] 21 Aug 2011, 13:28
there is a question in the veritas workbooks that asks: Convert a semi-annual compounded interest rate of 10% to an annual compounded interest rate.
Please let me know how to solve, thank you
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Re: Interest rates [#permalink] 21 Aug 2011, 13:35
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1+(interest rate)*1+(interest rate)
1.1*1.1= 1.21
answer - 1 = 0.21 or 21%
Hope this helps
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Re: Interest rates [#permalink] 22 Aug 2011, 06:22
rates are always quoted for one year as far as I know
100*(1+0.1/2)*(1+0.1/2) = 100*1.05*1.05 = 110.25 => 10.25% is the annual rate
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Re: Interest rates [#permalink] 22 Aug 2011, 12:55
This is not hard, my solution:
1+(interest rate)*1+(interest rate)
1.1*1.1= 1.21
answer - 1 = 0.21 or 21%
rjdunn03 wrote:
there is a question in the veritas workbooks that asks: Convert a semi-annual compounded interest rate of 10% to an annual compounded interest rate.
Please let me know how to solve, thank you
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Re: Interest rates [#permalink] 22 Aug 2011, 20:55
Expert's post
rjdunn03 wrote:
there is a question in the veritas workbooks that asks: Convert a semi-annual compounded interest rate of 10% to an annual compounded interest rate.
Please let me know how to solve, thank you
The calculation involved is pretty straight forward. The only tricky part is how to convert semi annual rate to annual rate.
What do you mean by semi annual interest rate? It is the rate at which you earn interest in 6 months. When you say it is 10%, you get $10 on$100 in 6 months.
Now the question is, what does it mean in annual interest rate terms?
After 6 months, you have $110 in hand. In the next 6 months, you will earn another 10% on 110 i.e.$11. So in a year, you earned a total of $10+$11 = $21 on the initial$100 investment.
So the equivalent annual interest rate is 21%.
Note that it is more than 20% (2*10%) because we are dealing with compound interest rate.
For more on Simple and Compound interest, check:
http://www.veritasprep.com/blog/2011/03 ... imple-one/
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Re: Interest rates [#permalink] 22 Aug 2011, 20:55
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Simple and Compound Interest 1 17 Feb 2010, 18:24
Display posts from previous: Sort by | 2015-11-30 18:51:48 | {"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": 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": 0.29772844910621643, "perplexity": 10743.047521291792}, "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-2015-48/segments/1448398462987.25/warc/CC-MAIN-20151124205422-00310-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.campusgate.in/2019/04/equations-homework.html | # Equations Homework
1. Gopi promises to give Rs. 90000 plus one cycle as salary to his servant for one year. The servant leaves after 9 months and receives Rs. 65000 and the cycle. Find the price of the cycle.
a. Rs. 1500
b. Rs. 2500.
c. Rs. 2000
d. Rs. 1000
2. There are two sections A and B for class X in a a school. If 6 students are sent from A to B, the number of students in B is double that of A. Otherwises, if 12 students from B are sent to A the number of students would be equal in both the sections. Find the number of students in section B initially.
a. 84
b. 78
c. 57
d. 66
3. Mahesh, Pawan, Charan are playing a game of "doubling" and each has some money with them. In each round the looser has to double the money with each of the remaining two players. In the first round, Mahesh lost the game and doubled the money of Pawan and Charan. In the second round Pawan lost and in the third round Charan lost the game. After the third round, they found that each has Rs.120 with them. Find the initial amount with Pawan.
a. 120
b. 115
c. 105
d. 100
4. Bhavya, Divya, Navya, Sravya are best friends. Bhavya has bought "n" chocolates on her birthday. She gives Divya half the number of chocolates plus half chocolate. Then she gives Navya half the remaining number of chocolates plus half chocolate. Finally, she gives half the remaining number of chocolates plus half chocolate. And she left with no chocolates. Then "n" lies in between?
a. $$2 \le n \le 6$$
b. $$5 \le n \le 8$$
c. $$9 \le n \le 12$$
d. $$11 \le n \le 16$$
5. A man spent 1/6th of his life in child hood, 1/12th of his life as youngster and 1/7th of his life as a bachelor. After five years of his marriage a son was born to him. The son died four years before the father died and at the time of his death his age was half the total age of his father. What is the age of the father?
a. 60
b. 72
c. 84
d. 96
6. Pawan has bought a new car. For the car registration, he has the following conditions to be met. The number should be a four digit number having non zero and distinct digits. The sum of digits at the units and tens position is equal to the sum of the remaining two digits. The sum of the digits in the middle positions is three times to the sum of the remaining digits. If the sum of the digits is not more than 20, then how many such four digit numbers are possible.
a. 1
b. 3
c. 6
d. 8
7. Each girl in class 8th has thrice the number of girls as classmates as she has boys as class mates, while Each boy has five times the number of girls as class mates as he has boys as class mates. Total number of students is equal to
a. 13
b. 24
c. 28
d. 42
8. Pawan, Mahesh, Nani, Charan are playing marbles and each has got some marbles. They found that total marbles with them are equal to 100. Also, If you give 4 marbles to Pawan, take away four marbles from Mahesh, multiply Nani's marbles by 4 , divide Charan's marbles by 4 make them equal. Find sum of the marbles with Pawan and Charan.
a. 32
b. 24
c. 72
d. 76
9. In a parking lot there are some cars in some rows. If two more cars are parked in each row, there would be one row less. If three cars are parked in each row then there would be three more rows. The number of cars is
a. 20
b. 40
c. 32
d. 36
10. A student has to divide a number by 7 and add 21 to the quotient. However, being absent minded, he had first added 21 to it and divided it by 7, and got 66 as answer. What is the corrent answer?
a. 84
b. 96
c. 105
d. 112
11. A young girl Roopa leaves home with x flowers, goes to the bank of a nearby river. On the bank of the river, there are four places of worship, standing in a row. She dips all the x flowers into the river. The number of flowers doubles. Then she enters the first place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the third place of worship, offers y flowers to the deity. She dips the remaining flowers into the river, and again the number of flowers doubles. She goes to the fourth place of worship, offers y flowers to the deity. Now she is left with no flowers in hand.
If Roopa leaves home with 30 flowers, the number of flowers she offers to each deity is:
a. 30
b. 31
c. 32
d. 33
12. The minimum number of flowers that could be offered to each deity is: (Use data from Question 11)
a. 0
b. 15
c. 16
d. Cannot be determined
13. The minimum number of flowers with which Roopa leaves home is: (Use data from Question 11)
a. 16
b. 15
c. 0
d. Cannot be determined
14. A fraction becomes 4/5 when 1 is added to both the numerator and denominator, and it becomes 5/6 when 6 is added to both the numerator and denominator. The numerator of the given fraction is :
a. 19
b. 21
c. 23
d. 29
15. Katama and Rayudu have some marbles with them. If Katama gave 30 marbles to Rayudu, he would have 40 less than Rayudu. If Rayudu gave 40 marbles to Katama, he would have one third of the marbles with Katama. Find the sum of the marbles with Katama and Rayudu.
a. 150
b. 170
c. 180
d. 200
16. A two digit number when 18 added becomes another two digit number with reversed digits. How many such two digit numbers are possible?
a. 1
b. 2
c. 5
d. 7
17. Ray writes a two digit number. He sees that the number exceeds 4 times the sum of its digits by 3. If the number is increased by 18, the result is the same as the number formed by reversing the digits. Find the sum of the digits of the number.
a. 6
b. 7
c. 8
d. 9
18. A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is
a. 10
b. 12
c. 14
d. 16
19. A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja, giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque.
Which of the following is a valid statement about the cheque amount?
a. Over Rupees 13 but less than Rupees 14
b. Over Rupees 7 but less than Rupees 8
c. Over Rupees 22 but less than Rupees 23
d. Over Rupees 18 but less than Rupees 19
20. Mahesh has x rupees and y paisa in his purse. He spent Rs.8.40 and was left with 2y rupees and 2x paisa. The value of x is...
a. Rs.32
b. Rs.37
c. Rs.41
d. Rs.43 | 2021-01-19 09:25:59 | {"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": 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": 0.5906487107276917, "perplexity": 1218.8444465377024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703518201.29/warc/CC-MAIN-20210119072933-20210119102933-00500.warc.gz"} |
http://openstudy.com/updates/5617120de4b02b3e025fb055 | ## owlet one year ago Enthalpy I need step-by-step guidance please and thank you in advance. Question Below:
1. owlet
2. owlet
@Jhannybean
3. anonymous
$\Delta H = \sum {n\cdot \text{products} -\sum m \cdot \text{reactants}}$ m and n are coefficients of the reactants and products,respectively.
4. anonymous
So you're given $$\Delta H = -89 \frac{kJ}{mol}$$
5. owlet
yeah i think I manage it now, just to make sure Hf for I2 is zero right?
6. owlet
**can
7. anonymous
Yes, all diatomics and elements
8. owlet
thanks!
9. anonymous
$-89 \frac{kJ}{mol} = -840\frac{kJ}{mol} +2\text{IF} -\left( -942 \frac{kJ}{mol}\right)$
10. owlet
that's what i did and i got -94.5 kJ
11. anonymous
- 89 = 2x - 840 + 942 -89 = 2x +102 -89 - 102 = 2x -191 = 2x x = -191/2 = -95.5 | 2016-10-27 08:58:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.6192928552627563, "perplexity": 5715.908560514236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721174.97/warc/CC-MAIN-20161020183841-00537-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://labs.tib.eu/arxiv/?author=Ch.%20H%C3%B6ppner | • The transverse target spin azimuthal asymmetry A_UT in hard exclusive production of rho^0 mesons was measured at COMPASS by scattering 160 GeV/c muons off transversely polarised protons and deuterons. The measured asymmetry is sensitive to the nucleon helicity-flip generalised parton distributions E^q, which are related to the orbital angular momentum of quarks in the nucleon. The Q^2, x_B and p_t^2 dependence of A_UT is presented in a wide kinematic range. Results for deuterons are obtained for the first time. The measured asymmetry is small in the whole kinematic range for both protons and deuterons, which is consistent with the theoretical interpretation that contributions from GPDs E^u and E^d approximately cancel.
• The COMPASS collaboration at CERN has investigated the \pi^- \gamma -> \pi^- \pi^- \pi^+ reaction at center-of-momentum energy below five pion masses, sqrt(s) < 5 m(\pi), embedded in the Primakoff reaction of 190 GeV pions impinging on a lead target. Exchange of quasi-real photons is selected by isolating the sharp Coulomb peak observed at smallest momentum transfers, t' < 0.001 (GeV/c)^2. Using partial-wave analysis techniques, the scattering intensity of Coulomb production described in terms of chiral dynamics and its dependence on the 3\pi-invariant mass m(3\pi) = sqrt(s) were extracted. The absolute cross section was determined in seven bins of $\sqrt{s}$ with an overall precision of 20%. At leading order, the result is found to be in good agreement with the prediction of chiral perturbation theory over the whole energy range investigated.
• ### Energy spectra of primary and secondary cosmic-ray nuclei measured with TRACER(1108.4838)
Aug. 24, 2011 astro-ph.HE
The TRACER cosmic-ray detector, first flown on long-duration balloon (LDB) in 2003 for observations of the major primary cosmic-ray nuclei from oxygen (Z=8) to iron (Z=26), has been upgraded to also measure the energies of the lighter nuclei, including the secondary species boron (Z=5). The instrument was used in another LDB flight in 2006. The properties and performance of the modified detector system are described, and the analysis of the data from the 2006 flight is discussed. The energy spectra of the primary nuclei carbon (Z=6), oxygen, and iron over the range from 1 GeV amu$^{-1}$ to 2 TeV amu$^{-1}$ are reported. The data for oxygen and iron are found to be in good agreement with the results of the previous TRACER flight. The measurement of the energy spectrum of boron also extends into the TeV amu$^{-1}$ region. The relative abundances of the primary nuclei, such as carbon, oxygen, and iron, above $\sim10$ GeV amu$^{-1}$ are independent of energy, while the boron abundance, i.e. the B/C abundance ratio, decreases with energy as expected. However, there is an indication that the previously reported $E^{-0.6}$ dependence of the B/C ratio does not continue to the highest energies. | 2020-11-28 02:42:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6516774892807007, "perplexity": 2310.465185016555}, "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/1606141194982.45/warc/CC-MAIN-20201128011115-20201128041115-00189.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=4184086 | by tolove
Mentor
P: 15,911
Quote by dkotschessaa I don't know about the exact data as far as GPA he quoted, but in principal I agree.
So you're saying that a degree in science is better than a degree in business? But that's the opposite of what chill_factor said, so you disagree with him?
P: 153 We're hijacking the thread here....I'd rather talk about what fields of physics are in high demand (OP's initial question) instead of an obscure argument about what combination of degree + GPA is better...
P: 519
Quote by micromass So you're saying that a degree in science is better than a degree in business? But that's the opposite of what chill_factor said, so you disagree with him?
Oh snap, I misread his sentence in a very dyslexic way.
I totally disagree with him. A low gpa science degree is much more impressive/useful than any kind of business degree.
Sorry for the confusion.
P: 37 I don't understand some of what is being written here about physics grad school. 1) I am an older physics grad student at U Arizona and I make about $25k/year, I get tuition waived, and free health insurance. The only cost to me is a few hundred per semester for some course fees. You should expect to earn a decent salary such as mine from any decent physics grad program (say in the top 50), although you should be sure to check with current grads at a school to be sure you are not being bamboozled (ALWAYS check with current grads before accepting any offer). What is this word "scholarship"? Such a word has no meaning to a physics grad student since we don't ever pay any tuition. 2) The physics PhD is both for enlightenment AND job training. Programs are aware of this and evolve over time to keep pace with the market (smart ones do anyway). At UA, we have "PhD minors" which allow us to take the core courses from other departments (any department in fact, I think even creative writing though you should check first). This is good because UA has several top ranked science departments/programs: Lunar Planetary Lab (Mars landers), Steward Observatory (LSST mirrors), College of Optics (Tucson is "optics valley" and has many companies), Theoretical Astrophysics, Chemical Physics, Applied Math, Eller Business School, etc. Also, full time is 9-hours so you can take any class after that for no extra charge (grad or undergrad), so you don't even need to do the PhD minor to take something that interests you outside of physics (or to get job skills), and some even count toward your PhD. IMPORTANT: I have not seen many schools with rules like this, so you should be sure to check ahead of time with prospective programs if you want these kinds of opportunities. 3) Teaching -vs- business -vs- science. Though you are unlikely to land a professorship at an R1 school with your PhD unless it is from top 10, you can (with some teaching portfolio development), get a position at a 4-year college. And more and more physics PhDs are finding their homes at community colleges and getting good salaries and even good students (many people don't realize they are smart until after high school, and I know several PhDs that actually began at community colleges). In business there is financial analysis, risk management and actuarial pricing just to name a few. PhDs can no longer walk into Wall Street type jobs (because so many have already done so), but with some planning and effort, you can land a highly paying job (and be the one that sends pizza money back to your SPS club). Planning ahead might include taking some graduate finance courses or some professional exams like the CFA. Let me just say that as a PhD physicist, you are expected to be able to solve any question regarding the natural world. And I mean ANY. And "business" is made of fundamental particles so it is included in this. (Yes, physics PhDs are arrogant like this, but isn't this a club you want to join?) Science jobs really depend on your research experiences: which kinds of equipment did you setup/operate/repair, what programming languages did you use, what electronics did you learn? Do you think Fermi or Feynman never soldered two wires together? The greats enjoyed solving ALL puzzles and ANY problem. Model your learning after them: try to learn everything. Technically, as a PhD in physics, it is your job to try to know everything. 4) Three things are needed to land your placement in a well-ranked physics PhD program: 1) good PGRE scores, 2) quality undergrad research, 3) and good grades. They are equally important in value to acceptance committees because they show that you can 1) buckle down and master difficult material and thus survive your comps, 2) think and work independently which is the ultimate goal, and 3) survive difficult core courses (some schools will even want to know which undergrad texts you used!). View the PGRE as a "mini-comp". Nothing helped prepare me for the rigors of the PhD core courses like studying for the PGRE (because I had been lame in my classes). By spending 4 solid months studying for this exam, I filled in many holes in my knowledge. This is not a test you must pass once, the PGRE represents a level of mastery you will need to maintain during grad school, so study hard for it (actually study the material please) and realize you are preparing for every day of grad school by doing so. If your classes have not gone perfect because you are smart but a little lazy (but learning to work hard), then this is your chance to make up for the past. But still you should never let another student outperform you in your classes. Trust me, they are not smarter than you, just harder working, so get to it. Didn't you want to know everything that has to do with physics, anyway? I kept a cot in my office during my first year of grad school because I would do about two all-nighters per week (and sleep from 6-8am in the cot instead of bothering to go home). I loved loved loved every painful minute of it because every minute of it I was learning all the advanced physics I had ever dreamed of knowing. You can have your cake and eat it too (enlightenment) if you also build your skill sets as you go along (job skills). As a physicist, you should be smart enough to realize this and make it happen. 5) Don't be afraid to contact a physics department's grad secretary to find out if you can apply late after the official deadline. I was accepted to some schools even though I applied (very) late. Often, the acceptance committee isn't happy with the candidate pool, or maybe the best ones already accepted other offers. IT NEVER HURTS TO CHECK! Just say something like you just learned about what a wonderful program they have and were wondering blah blah blah. ------------------------------------------------------------------------ I have seen posts here in the past from some very bright people getting into top 10 schools. That is useful to that crowd, but it would be useful to many others to see more average PhD students posting (and thus I have). P: 167 JavaNut, thanks for the detailed post. I've heard really good things about UT Austin (as far as I know they are the only university to have sent a mission to another planet!). Is UT considered "top tier" ? And if so can I ask the PGRE / GPA / general application portfolio that landed you a spot? I myself am considering getting into laser / optics. Lasers are a) way cool and b) way useful. But, that's at the M.s level I think. Also, for graduate school, is it a disadvantage or advantage to do M.S before applying to a phd track? I'm in Europe where tuition is free but the degrees are 3 years, and I'm thinking that might not be enough time to built up a solid portfolio for admissions. thanks! P: 37 (deleted and rewritten below. What is up with this mentor thing?) P: 37 Quote by Moneer81 1. Accelerator physics: 2. Laser physics: 3. Medical physics: 4. Experimental particle physics: One problem we have is that none of us are much exposed to physicists once they leave the university, just the few grads we knew who finished and the profs (profs are usually useless at helping with the non-academic career tracks). I knew one PhD who wanted to go to Wall Street, but he never prepped at all for it, and they just won't take newbies anymore. Another got frustrated and quit (the PhD can be hell at times), and now he's thrilled doing software development. Another failed the comps but finished an optics masters and now works in industry (here in "optics valley" Tucson). #1) Accelerator physics sounds fun. I had a friend at Indiana studying accelerator physics, and another at CalTech looking into new accelerator technologies. Lots of E&M in that field. You should look into the US Particle Accelerator School (it is international, people from everywhere). 1-2 week training program. I would say you would need to know E&M pretty well (advanced undergrad at least). Then you would know if you want to do accelerator physics. #2) Solid state physics experiment or theory will always enable you to work in industry. Our whole technology is based on this. Plan ahead a little and land a really good job. And I would venture that AMO physics is nearly as good. Here it is best to choose whatever you have the strongest passion for. #3) Medical physics would involve working in hospitals and dealing with doctors, but it would be lucrative and maybe fun. U Arizona has a program in medical physics. I know many programs are extremely expensive, but right now our PSM students (professional science masters) are getting TAships so have the same deal as a regular PhD grad. I believe they still have some expensive summer expenses. I believe that field is becoming standardized, which is usually economically good for a career field, to have a professional organization with exams and members. #4) There can be electronics and device construction, but right now with all the CERN data pouring in, you can bet you will be programming: object-oriented C++ and python for the most part. The learning curve is steep, and no you don't have to "master" quantum field theory to be an experimentalist, that is for the theoreticians. Too bad so many undergrad programs do a poor job offering advanced courses to expose their undergrads to all the major physics fields. I bet you could have a "career" course that discussed: AMO, CM, PP, Q optics/info, etc.; exp vs theory in each. P: 37 Quote by H2Bro JavaNut, I've heard really good things about UT Austin (as far as I know they are the only university to have sent a mission to another planet!). And if so can I ask the PGRE / GPA / general application portfolio that landed you a spot? I myself am considering getting into laser / optics. Lasers are a) way cool and b) way useful. But, that's at the M.s level I think. Also, for graduate school, is it a disadvantage or advantage to do M.S before UT Austin is #14 on the usual rankings: http://grad-schools.usnews.rankingsa...ysics-rankings which must be why they rejected me. I believe you are incorrect. I think it is U Arizona with the Mars mission(s): http://en.wikipedia.org/wiki/Phoenix_(spacecraft) The details on each US school are here (where most of us look): http://www.gradschoolshopper.com/ I believe I had a 780 or 790 on PGRE, 3.4 GPA, and very good research. What I have been told by various professors (at various institution in the past) is that the PGRE is not all or nothing. Good undergrad research and/or course grades can overcome a bad GRE score, and they really do look at your entire application regardless of your score. So you should apply if you think you have something to offer, no matter how bad you did. One professor told me he had a friend that got a 0th percentile and she still eventually became an astronomy professor. There is always hope for the determined. Even if all you want is an MS in physics, you should not tell them. You should go for a PhD, and drop out with a masters if that is what you decide to do. Schools with PhD programs want/need PhD students. Unless it is a specialty program like our professional science masters, you must apply to the PhD program or they won't except you (there are always special cases). This is what I was told many times in the past, anyway. You will be able to pick up the MS on the way to PhD. Here at U Arizona, once you pass your written comps and oral defense, you merely pay$35 to get the MS degree. Many of my friends don't think it is worth the money if you can believe that, but I was like, "\$35? What the hell, why not?" So now I have my masters in physics.
P: 153
Thank you for your insights! Yes Indiana is one of the few schools offering Accelerator Physics. That does sound like a fascinating field. I also want to say that the employment prospects seem very high, given how important accelerators will be in science, research, healthcare, industry, etc.
Is there any way you can put me in touch with one of your friends who is doing Accelerator physics? I have not been able to get in touch with anyone in that field. You can pm if you'd like. And yes I know about USPAS and they seem to have excellent supplemental programs that covers the gap and prepares students for this field.
I am also very interested in optics and laser physics. I had no idea that Tucson is considered the optics valley. I need to research the companies and the jobs down there in that area. I am actually debating optics vs. Accelerator physics. I have a feeling optics will have many job prospects in the industry. Do you know anything about that? There are programs in "beam physics" that might combine laser physics and accelerator physics, but I could be wrong about that. If that was the case, I would love to pursue a program in beam physics instead.
Again, thanks for your insight and best of luck in school!
P: 37
Quote by Moneer81 1) Is there any way you can put me in touch with one of your friends who is doing Accelerator physics? 2) And yes I know about USPAS and they seem to have excellent supplemental programs that covers the gap and prepares students for this field. 3) I am also very interested in optics and laser physics. I had no idea that Tucson is considered the optics valley. 4) There are programs in "beam physics" that might combine laser physics and accelerator physics, but I could be wrong about that. If that was the case, I would love to pursue a program in beam physics instead.
1) I'm afraid it's been too long so I can't offer any contacts.
2) You can often get the tuition waived to USPAS leaving only the cost of the hotel. Your interests are so high, you should apply now despite not possibly meeting all the criteria. You will meet people in the field by going!
3) Tucson=lasers+telescopes. There are two optics buildings (in addition to physics, astro, etc.).
4) Sounds to me that you need to investigate schools that do this: http://en.wikipedia.org/wiki/Plasma_acceleration
Good luck!
P: 311 What is considered "good research"? How many papers should you publish as an undergrad in order for it to be considered "good research"? The reason I ask is because with my math professor (who has written me a rec) I will be researching fourth order boundary value ODE's next semester describing cantilever beams, but the subject matter is so obscure that I am not sure if it would be considered "good research."
P: 37
Quote by Hercuflea What is considered "good research"? How many papers should you publish as an undergrad in order for it to be considered "good research"? The reason I ask is because with my math professor (who has written me a rec) I will be researching fourth order boundary value ODE's next semester describing cantilever beams, but the subject matter is so obscure that I am not sure if it would be considered "good research."
I thought quantum cantilevers were a big deal nowadays. Just google-search "quantum cantilever". I think I remember 1) seeing an experiment talk on this in our optics department, and 2) that we have a theory prof who has published on this (actually two profs I think, you would have to do your own research though, but I have heard of this topic). My guess is that your math guy picked cantilevers precisely because they were popping up in physics papers and were thus relevant. You might want to read up a little on quantum cantilevers and write a blurb in your statement relating your research to current physics research. | 2014-03-09 13:03:32 | {"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": 0.3409714698791504, "perplexity": 912.3275591834541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999678556/warc/CC-MAIN-20140305060758-00081-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://socratic.org/questions/the-difference-of-two-numbers-is-1-their-sum-is-13-what-are-the-numbers | # The difference of two numbers is 1. Their sum is 13. What are the numbers?
Jan 12, 2017
$7 \mathmr{and} 6$
#### Explanation:
Let the two numbers be: ${x}_{1} \mathmr{and} {x}_{2}$
We are told that:
${x}_{1} - {x}_{2} = 1$ Equaltion (A)
and
${x}_{1} + {x}_{2} = 13$ Equation (B)
(A) + (B) $\to 2 {x}_{1} = 14$
${x}_{1} = 7$
Replacing ${x}_{1} = 7$ in (B) $\to 7 - {x}_{2} = 1$
${x}_{2} = 6$
Hence the two numbers are $7 \mathmr{and} 6$ | 2021-12-05 09:18:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 10, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9623408317565918, "perplexity": 2016.6986794794366}, "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/1637964363149.85/warc/CC-MAIN-20211205065810-20211205095810-00399.warc.gz"} |
https://gateoverflow.in/379254/general-aptitude-practice-question-unacademy | 215 views
In a car race, car X beats car Y by 36 km, car Y beats car Z by 72 km, and car X beats car Z by 90 km. The distance (in m) over which the race has been conducted is:
1. 88000
2. 108000
3. 56000
4. 144000
### 1 comment
Suppose, race has been conducted for distance $S$ Km and speed of cars $X,Y,Z$ are $v_1,v_2,v_3$ respectively in km per unit time.
“car X beats car Y by 36 km” means for the same time, X is ahead of Y by $36$ km. Similar meaning is for other two statements. So, we will be having three equations as:
$\frac{S}{v_1} = \frac{S- 36}{v_2}$ $(1)$
$\frac{S}{v_2} = \frac{S- 72}{v_3}$ $(2)$
$\frac{S}{v_1} = \frac{S- 90}{v_3}$ $(1)$
From $(1),$ $\frac{v_2}{v_1} = \frac{S- 36}{S}$
From $(2),$ $\frac{v_2}{v_3} = \frac{S}{S – 72}$
After dividing both equations, we get, $\frac{v_3}{v_1} = \frac{(S- 36)(S – 72)}{S^2}$ $(3)$
From $(3),$ $\frac{v_3}{v_1} = \frac{S- 90}{S}$ $(4)$
From $(3)$ and $(4),$ $\frac{S- 90}{S} = \frac{(S- 36)(S – 72)}{S^2}$, $S \neq 0$
$S= 144$ km = $144000 m$
Let the total distance of the race is $d$ km .
$X$ beats $Y$ by 36 Km which means when $X$ completed the $d$ km distance $Y$ was at distance $d-36$ km.
Let $X$ completed the race in $t$ sec.
So, Speed of $X$ is $\large S_{X}=\frac{d}{t}$
Speed of $Y$ is $\large S_{Y}=\frac{d-36}{t}$
So, ratio of $\large \frac{S_{X}}{S_{Y}}=\frac{d}{d-36}$……………………..(1)
$Y$ beats $Z$ by 72 Km which means when $Y$ completed the $d$ km distance $Z$ was at distance $d-72$ km.
Let $Y$ completed the race in $t_{1}$ sec.
So, Speed of $Y$ is $\large S_{X}=\frac{d}{t_{1}}$
Speed of $Z$ is $\large S_{Y}=\frac{d-72}{t_{1}}$
So, ratio of $\large \frac{S_{Y}}{S_{Z}}=\frac{d}{d-72}$……………………..(2)
$X$ beats $Z$ by 90 Km which means when $X$ completed the $d$ km distance $Z$ was at distance $d-90$ km.
Let $Y$ completed the race in $t_{2}$ sec.
So, Speed of $Y$ is $\large S_{X}=\frac{d}{t_{2}}$
Speed of $Z$ is $\large S_{Y}=\frac{d-90}{t_{2}}$
So, ratio of $\large \frac{S_{X}}{S_{Z}}=\frac{d}{d-90}$……………………..(3)
Dividing (3) by (1) we get ,
$\large \frac{\frac{S_{X}}{S_{Z}}}{\frac{S_{X}}{S_{Y}}}=\frac{\frac{d}{d-90}}{\frac{d}{d-36}}$
=>$\large \frac{S_{Y}}{S_{Z}}=\frac{d-36}{d-90}$ ….…….……...(4)
Now (4) and (2) are equal so equating them,
$\large \frac{d}{d-72}=\frac{d-36}{d-90}$
=>$\large d(d-90)=(d-36)(d-72)$
=>$\large d=144$km.
So correct answer is $144000$ $m$.
1 vote | 2022-11-30 03:06:10 | {"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": 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": 0.7483711838722229, "perplexity": 2153.6179547164456}, "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/1669446710719.4/warc/CC-MAIN-20221130024541-20221130054541-00678.warc.gz"} |
https://www.tutorialspoint.com/find-minimum-value-to-assign-all-array-elements-so-that-array-product-becomes-greater-in-cplusplus | # Find minimum value to assign all array elements so that array product becomes greater in C++
C++Server Side ProgrammingProgramming
Suppose we have an array of n elements. Update all elements of the given array to some min value x, such that arr[i] = x. Such that product of all elements in the new array is strictly greater than the product of all elements of the initial array, where i <= arr[i] <= 10^10, and 1 <= n <= 10^5. So if the array is like [4, 2, 1, 10, 6]. So 4 is the smallest element. 4 * 4 * 4 * 4 * 4 > 4 * 2 * 1 * 10 * 6
As we know that the product of n elements is P. If we have to find nth root of P, to find the nth root of product, we simply divide n from sum of log of n elements of the array and then the ceiling of antilog will be the result.
res = ceil(antilog(log(x) / 10))
or res = ceil(10 ^ (log(x) / 10))
## Example
Live Demo
#include <iostream>
#include <cmath>
#define EPS 1e-15
using namespace std;
long long findMinValue(long long arr[], long long n) {
long double sum = 0;
for (int i=0; i<n; i++)
sum += (long double)log10(arr[i])+EPS;
long double xl = (long double)(sum/n+EPS);
long double res = pow((long double)10.0, (long double)xl) + EPS;
return (long long)ceil(res+EPS);
}
int main() {
long long arr[] = {4, 2, 1, 10, 6};
long long n = sizeof(arr)/sizeof(arr[0]);
cout << "Min value is: "<< findMinValue(arr, n);
}
## Output
Min value is: 4
Updated on 18-Dec-2019 10:38:18 | 2022-07-06 21:33:08 | {"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": 0.17876560986042023, "perplexity": 2086.300565207249}, "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/1656104678225.97/warc/CC-MAIN-20220706212428-20220707002428-00767.warc.gz"} |
https://lists.gnu.org/archive/html/help-gnu-emacs/2017-12/msg00098.html | help-gnu-emacs
[Top][All Lists]
## Re: How to use \subsetneqq in Org-mode?
From: Alexis Subject: Re: How to use \subsetneqq in Org-mode? Date: Wed, 06 Dec 2017 14:28:05 +1100 User-agent: mu4e 0.9.19; emacs 25.2.1
Ok, now I know where the problem is.
The 'MathJax.js' Org mode using is little different than the 'Real MathJax'
;-)
Today I replaced the string org mode inserted to the HTML file:
From:
<script type="text/javascript" src="http://orgmode.org/mathjax/MathJax.js
"></script>
To:
<script src='
https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-MML-AM_CHTML
'></script>
And every thing works! Every single symbol displayed correctly. So simple!
Great! Glad you've got it working. :-)
Also: thank you for reporting back on how you fixed the problem! It's so frustrating to search for information about a problem online, find that someone else has had the exact same problem, but then find that they've
said "Nevermind, I fixed it", without giving any details .... So
hopefully the fact that you've posted your solution here will help
others in the future. :-)
Alexis. | 2022-01-29 07:35:39 | {"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": 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": 0.748994767665863, "perplexity": 8055.364654611217}, "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/1642320300573.3/warc/CC-MAIN-20220129062503-20220129092503-00312.warc.gz"} |
https://library.kiwix.org/philosophy.stackexchange.com_en_all_2021-04/A/question/18051.html | soundness: how can one decide\know if premise is true or false
2
2
An argument is sound if and only if
1. The argument is valid.
2. All of its premises are true.
For instance,
1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
The argument is valid (because the conclusion is true based on the premises, that is, that the conclusion follows the premises) and since the premises are in fact true, the argument is sound.
There is nothing wrong in my perspective, but what if there is a religion/person that does not believe in mortality. For such religion/person all men are mortal is FALSE therefore argument is not sound.
So, how can one decide\know if premise is true or false?
That opens another question, why is there soundness property at all?
I am sorry if this kind of question was covered in another thread.
2Some of us are interested in trying to found/discover/ establish truth: in philosophy, science, ... Thus, assuming that we have found some truth (also provisionally) we want to deduce other truth from them by valid argument. Logic is basically interested into finding valid way of reasoning. – Mauro ALLEGRANZA – 2014-11-11T16:58:52.847
7
In general, you do not know whether the premise is true or not. Still, we would like to say that something is wrong with an argument, if its premises are wrong. Note that
All men are immortal.
Socrates is a man.
Therefore, Socrates is immortal.
is a valid argument. But presumably, it is not sound.
When calling an argument "sound", you defer the questions of knowledge and certainty. But usually, an argument should not only have premises that happen to be true, but premises that we find plausible (in which case, you might call it an persuasive argument).
You can have arguments that are valid, sound, persuasive; but usually not absolutley certain. | 2021-07-31 16:40:17 | {"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": 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": 0.8192459940910339, "perplexity": 1029.8966259318834}, "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-2021-31/segments/1627046154089.68/warc/CC-MAIN-20210731141123-20210731171123-00346.warc.gz"} |
http://davidbowler.github.io/AtomisticSimulations/blog/why-minimisers | # Why you need to understand how minimisers work
Tags: DFT algorithms
Numerical minimisation is at the heart of most electronic structure codes, and are involved in finding the electronic wavefunctions, often the self-consistent charge density, and in relaxing atoms during structural minimisation. Many of these techniques are very sophisticated, and in modern codes they have often been tuned for performance (sometimes heuristically) but there is no guaranteed way to find a global energy minimum, and they will fail. So it is very important to understand how they work, and why they might fail. Monitoring a calculation to ensure convergence is generally worthwhile (though this can easily turn into an unhelpful distraction if taken to extremes).
In general, we assume that the energy can be written in terms of some multi-dimensional vector, $$\mathbf{x}$$, which might represent expansion coefficients for the basis functions in the wavefunctions, or the atomic coordinates, or other parameters. We then expand the energy to second order in $$\mathbf{x}$$:
where $$\mathbf{g} = -\partial E/\partial \mathbf{x}$$ and $$\mathbf{H} = \partial^2 E/\partial \mathbf{x}^\prime \partial \mathbf{x}$$, the Hessian, which gives the curvature of the energy surface.
All methods use an iterative approach: we choose a direction in which to optimise, find a minimum along that direction, and repeat until some convergence criterion is reached.
#### Steepest Descents
This is the simplest approach to minimisation, which is generally a very poor choice of method. The search direction is always taken as $$\mathbf{g}$$, the local downhill gradient. While this is simple, its efficiency depends strongly on the starting point, as illustrated below for a simple, two-dimensional problem where we ought to be able to find the ground state in two steps.
There are two reasons that steepest descents performs so badly: first, it takes no account of previous minimisations; second, it does not use any information about the curvature (the Hessian, above). We can see that the Hessian is useful by taking the equation for $$E$$ above, and seeking the stationary points (i.e. solving for $$\partial E/\partial \mathbf{x} = 0$$ ):
If we had the full Hessian, then we could find the minimum, but this would be prohibitively expensive. Improved methods approximate the Hessian, as we will see.
If we choose the search direction, $$\mathbf{h}_n$$ at any given iteration $$n$$ to be conjugate to the previous search direction, where conjugacy is defined by $$\mathbf{h}_m \cdot \mathbf{H} \cdot \mathbf{h}_n = 0$$, then the minimisation of the previous step will not be affected by the present step, and the local curvature is accounted for.
The key to the conjugate gradients method is that this condition can be imposed without calculating the Hessian, if we choose:
The maths leading to this formula is not too complex, and can be found in a variety of places[1]. The formula given here is easy to implement, and ensures that successive search directions are conjugate, while successive local gradients are orthogonal. The conjugate gradients method is widely implemented and generally reliable, though requires a good line minimiser (see below for more on this topic) and can fall prey to ill conditioning (also discussed below).
#### Quasi-Newton methods
The maths for quasi-Newton methods is a little more complex, so I will not detail it here, but the essence of the approach is simple. It generalises the Newton-Raphson approach to multiple dimensions, and builds up an approximation to the inverse Hessian over the course of the optimisation, using the same basic formula for the optimum value of $$\mathbf{x}$$ given above. Generally there is a need to truncate the amount of information stored to keep the memory requirements reasonable, but beyond this restriction, the method is very efficient.
#### Line minimisation
Finding the minimum in a given search direction is a key part of these algorithms. The most robust approach first seeks to bracket the minimum, by taking successively larger steps downhill, and then refines the brackets to find the minimum (using bisection or some more sophisticated approach). The problem with this approach is efficiency: it can require many evaluations, which are computationally wasteful.
A simpler alternative is to take an step with a length that is estimated to be close to the minimum, and use inverse quadratic interpolation to find the minimum from the two points and two gradients available. The approach can work very well when a function is close to quadratic, but often leads to errors in the early stages.
#### Problems
The most common problem facing numerical optimisation is ill conditioning. If the Hessian has eigenvalues that span a large range, then in some directions the gradient will be very steep, while in others it will be very shallow. This makes it very hard to find the minimum. The ideal solution would be to adjust the curvatures so that they are the same in all directions—which is just the same as inverting the Hessian and applying it to the gradient. Preconditioning approaches estimate an inverse Hessian and use it to improve the convergence of the minimisation. A famous example in electronic structure relates to the kinetic energy of the electrons, and is most easily understood in a plane wave basis set, where the kinetic energy is proportional to $$G^2$$ for the wave vector $$\mathbf{G}$$. For kinetic energy of large wavevector components dominates the gradient, giving classic ill conditioning; the solution is to scale these components by $$1/G^2$$ while leaving the smaller wavevector components unchanged[3]
During structural optimisations, this type of behaviour is often seen when there are very soft modes (where groups of atoms can rotate or rock almost freely). There are other issues: if the electronic minimisation is not fully converged then the structural optimisation can fail (it always pays to check the convergence), and large changes in electronic structure with atomic structure can also give issues (often helped by introducing a larger electronic temperature).
You will inevitably encounter situations where your optimisations or minimisations fail, and understanding how they work can help to diagnose and fix the problem.
[1] While Numerical Recipes is generally the best place to look, I have found the analysis from Jonathan Shewchuk most helpful for this topic[2]
[2] See the first entry here
[3] This can be managed using a factor like $$f(G) = G_0^2/(G_0^2 + G^2)$$ which is close to 1 if $$G<G_0$$ but is close to $$1/G^2$$ for large values of $$G$$.
This entry was posted in tutorials on 2015/11/13. | 2019-01-18 09:08:06 | {"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": 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": 0.7938233613967896, "perplexity": 364.009067919937}, "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/1547583660020.5/warc/CC-MAIN-20190118090507-20190118112507-00412.warc.gz"} |
http://mathhelpboards.com/number-theory-27/product-integers-relatively-prime-m-22235-3.html?s=f1238a26b1b57903d7d1b7574818610c | Thread: Product of integers that are relatively prime to m
Originally Posted by I like Serena
Starting somewhere, what happened to $b_1$ and $b_{\phi(m)}$?
$b_1$ is always $1$ and $b_{\phi(m)}$ is always equal to $m-1$.
2. Originally Posted by evinda
$b_1$ is always $1$ and $b_{\phi(m)}$ is always equal to $m-1$.
Ah, okay. Perhaps it'd be useful to mention that.
In particular it means that whatever $m$ is, 1 and -1 will always be their own inverse, and they will from a pair that are additive inverses.
(Assuming $m>2$.)
Otherwise, it seems your proof is correct.
Originally Posted by I like Serena
Ah, okay. Perhaps it'd be useful to mention that.
Yes, I agree...
Originally Posted by I like Serena
In particular it means that whatever $m$ is, 1 and -1 will always be their own inverse, and they will from a pair that are additive inverses.
(Assuming $m>2$.)
Originally Posted by I like Serena
Otherwise, it seems your proof is correct.
Nice... Thank you!!! | 2017-09-24 03:14:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8925018310546875, "perplexity": 844.8592933597092}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689845.76/warc/CC-MAIN-20170924025415-20170924045415-00673.warc.gz"} |
https://physicshelpforum.com/threads/simple-pendulum-question.15558/ | # Simple Pendulum Question
#### shivam28
I did a Simple Pendulum experiment in my college physics class the other day. We were asked to graph a T^2 x L graph based off our results. I plan on using the Length on the x-axis and the T^2 on the Y-axis.
This may seem like a simple question but how would I find my T^2? Is it just simply the Period (T) that we timed but squared? For example: T = 1.10s is 1.21 T^2. Or, is there some kind of formula I need to find T^2? Here is a picture of my table that will hopefully clarify:
Based off my picture above, should I plot my first coordinate at (28, 1.10) or (28, 1.21)? Or something else?
Thanks!
#### topsquark
Forum Staff
You got it right. Just use the list of numbers for $$\displaystyle T^2$$ as your y values.
-Dan | 2019-12-12 12:19:05 | {"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": 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": 0.669099748134613, "perplexity": 786.1257187934859}, "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-2019-51/segments/1575540543252.46/warc/CC-MAIN-20191212102302-20191212130302-00242.warc.gz"} |
https://en.wikipedia.org/wiki/Cash-flow_return_on_investment | # Cash-flow return on investment
Cash-flow return on investment (CFROI) is a valuation model that assumes the stock market sets prices based on cash flow, not on corporate performance and earnings.[1]
${\displaystyle {\text{CFROI}}={\frac {\text{Cash flow}}{\text{Market recapitalization}}}}$
For the corporation, it is essentially internal rate of return (IRR).[2] CFROI is compared to a hurdle rate to determine if investment/product is performing adequately. The hurdle rate is the total cost of capital for the corporation calculated by a mix of cost of debt financing plus investors `expected return on equity investments. The CFROI must exceed the hurdle rate to satisfy both the debt financing and the investors expected return.
${\displaystyle {\text{CFROI}}={\frac {\text{Gross cash flow}}{\text{Gross investment}}}}$
Michael J. Mauboussin in his 2006 book More Than You Know: Finding Financial Wisdom in Unconventional Places, quoted an analysis by Credit Suisse First Boston, that, measured by CFROI, performance of companies tend to converge after five years in terms of their survival rates.[3]
The CFROI for a firm or a division can then be written as follows:[4]
${\displaystyle {\text{CFROI}}={\frac {{\text{Gross cash flow}}-{\text{Economic depreciation}}}{\text{Gross investment}}}}$
This annuity is called the economic depreciation:
${\displaystyle {\text{Economic depreciation}}={\frac {K_{c}}{\left(1+K_{c}\right)^{n}-1}}}$
where n is the expected life of the asset and Kc is the replacement cost in current dollars.[5] | 2018-08-17 08:18:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.36808204650878906, "perplexity": 3409.697621891742}, "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/1534221211933.43/warc/CC-MAIN-20180817065045-20180817085045-00227.warc.gz"} |
https://earthscience.stackexchange.com/tags/impacts/hot | # Tag Info
15
I think you are confused about the timescales and the magnitude of the impact that is being talked about here. The collision between the early Earth and a roughly Mars sized body, Theia is thought to have happened between 4.4 and 4.45 billion years ago. The Pangaea supercontinent had fully assembled by around 250 million years ago. Previous to this there ...
11
The linked paper does numerical modelling of a hypothetical tsunami and predicts speeds between 60 m/s (at its point of origin) and 30 m/s (nearer to shore). We can try a back-of-an-envelope approximation to see if we get close, and to understand why the speed would be so much lower than on Earth. A tsunami behaves as a gravity wave with a very long ...
4
The volume of a sphere is $\small\sf{\frac4 3\pi R^3}$, where $\small\sf{R}$ is the radius. The volume of Earth with the 4 cm deep iridium rich layer is, $\small\sf{V_{Ei} = \frac 4 3\pi (6.378\cdot10^6)^3}$ $\small\sf{m^3}$ = $\small\sf{1.086 \ 781 \cdot 10^{21}}$ $\small\sf{m^3}$ The volume of the 4 cm deep iridium rich layer is, \$\small\sf{V_i = \frac ...
4
A 10 km asteroid would not only obliterate the section of glacier it hit, but also create a huge crater at the impact site! The asteroid that created the Vredefort crater in South Africa is estimated to have been 10 to 15 km in diameter. It is also estimated that the initial crater created by the impact, some 2 billion years ago before it was eroded, was ...
4
Several possible ways.. Stratigraphic (Local): If the crater has been buried, just date the first rocks that are on top of the crater but not disturbed by it; this can be done using fossil-based dating if no good radiometric techniques are available. This has the advantage of being unambiguous, but the problem could be, especially for land based craters, ...
3
There are micro-tectites galore, strewn over thousands of square kilometres. But remains of the actual bolide? Hard to say, because the impact crater is now buried under 600 metres of sediment. Earlier this year the joint IODP-ICDP drilling program drilled to a depth of 1300 metres. As far as I know, their findings are not published yet.
2
Welcome to StackExchange SE! I don't think I fully understand your question, but I'll try to answer it. Reanalysis data is the use of weather models and data assimilation to piece back the weather. It is a 4-dimensional dataset (Latitude, longitude, pressure levels, and time). You can do whatever you wish with that data (provided you follow the legal ...
2
A Korean here. Haean Myeon in Gangwon-do has two theories of creation. One is Meteor impact and another is differential erosion. There were no meteor related evidence found at Haean so the erosion theory is more agreed upon at the moment. Direct translation from introduction website for Hae An: According to the analysis of the granite of the punchbowl ...
1
Vredefort No extraterrestrial iridium anomaly: http://adsabs.harvard.edu/full/1989LPSC...19..733F I'm not sure I fully agree with them, because there is quite a lot of Ir there and their threshold of 200 pg/g seems somewhat arbitrary to me. They claim the Ir was derived from local rocks, but that's a hell lot of iridium! Sudbury There is an an iridium ...
1
A 5km impactor would make a very, very global mess. Here's a discussion of tsunami formation from asteroid impacts. It presents numbers for impactor size up to 2km, and posits a rule-of-thumb that indicates a 5km impactor would generate a coherent wave (one that doesn't dissipate much over great distances) as long as it impacts in water less than 20-30km ...
1
Dust would not have to be chemically poisonous to render air unbreathable but a big impact would first of all release a blast of hot plasma that nothing nearby will survive with accompanying air blast. Dust would be mostly from Earth material and that will vary in chemistry according to what is locally present; the Chicxulub (dinosaur killer) impact hit ...
1
tl;dr while the docudrama is over-dramatised, yes, this is possible. Simply put, we already have processes which do make air unbreathable in small or even large areas, from dust or poisonous gases. Typically these are volcanic in origin, with fine dust from some volcanoes causing breathing difficulties across large areas, and from gases (often sulphur-...
1
Disclaimer: I haven't watched the docudrama. For a comet impact, I think the chemical effects (dumping poisonous gases into the atmosphere) would be completely overwhelmed by the thermal effects (lots and lots of kinetic energy converted to heat). Those poisonous gases would surely be converted to plasma and/or blasted out in the shockwave from the impact. ...
1
That doesnt seem realistic since eventually all the dust from all the volcanic activity, as well as that caused by our doing, eventually makes its way back down to earth. Most dust particles end up being soaked up by earths cloud layer coming down in rain. I have to agree with Gimelist but volcanism can and has produced conditions you mentioned by expelling ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2021-05-12 07:06:39 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.417587548494339, "perplexity": 1804.7079471254226}, "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/1620243991685.16/warc/CC-MAIN-20210512070028-20210512100028-00042.warc.gz"} |
https://csedoubts.gateoverflow.in/9097/cormen-edition-3-exercise-22-4-question-3-page-no-615 | 30 views
Give an algorithm that determines whether or not a given undirected graph G=(V,E) contains a cycle. Your algorithm should run in $O(V)$ time, independent of E.
Solution:
If there’s a back edge, there’s a cycle. If there’s no back edge, there are only tree edges. Hence, the graph is acyclic.
Thus, we can run DFS: if we find a back edge, there’s a cycle.
My doubt is how to manage this in $O(V)$ time ?
| 30 views
+1
0
Already have that. You just give a little explanation about the complexity either from the link you provided or self explanation. Getting little difficulty to get that. Thank you.
+2
In simple terms, undirected connected graph which doesn't have cycles, should be a tree.
For Tree, edges=nodes-1
So, your dfs = O(nodes+edges) =O(2nodes-1)= O(nodes)
0
If we ever see $|V |$ distinct edges, we must have seen a back edge because in an acyclic (undirected) forest, $|E| ≤ |V | − 1$.
@Shaik Masthan
Got it sir, Thankyou | 2020-07-12 06:48:05 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38176432251930237, "perplexity": 1632.5018110541205}, "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-2020-29/segments/1593657131734.89/warc/CC-MAIN-20200712051058-20200712081058-00179.warc.gz"} |
https://api-project-1022638073839.appspot.com/questions/how-do-you-solve-log-x-2-1 | # How do you solve log x=-2?
You apply exp, so $\ln \left(x\right) = - 2 \iff x = {e}^{- 2} = \frac{1}{e} ^ 2$. | 2021-10-18 11:39:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.5037333965301514, "perplexity": 4768.15020989048}, "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/1634323585201.94/warc/CC-MAIN-20211018093606-20211018123606-00345.warc.gz"} |
https://studymaterialcenter.in/question/this-question-has-statement-1-and-statement-2-of-the-four-choices-given-after-the-statements-choose-the-one-that-best-describes-the-two-statements-if-two-springs-s-1-and-s-2-of-force-constants-k-1-and/ | # This question has Statement 1 and Statement 2 . Of the four choices given after the Statements, choose the one that best describes the two Statements. If two springs $S_{1}$ and $S_{2}$ of force constants $k_{1}$ and $k_{2}$ respectively, are stretched by the same force, it is found that more work is done on spring $S_{1}$ than on spring $S_{2}$. Statement $1:$ If stretched by the same amount work done on $S_{1}$ Statement $2: k_{1}<k_{2}$
Question:
This question has Statement 1 and Statement 2 . Of the four choices given after the Statements, choose the one that best describes the two Statements.
If two springs $S_{1}$ and $S_{2}$ of force constants $k_{1}$ and $k_{2}$ respectively, are stretched by the same force, it is found that more work is done on spring $S_{1}$ than on spring $S_{2}$.
Statement $1:$ If stretched by the same amount work done on $S_{1}$
Statement $2: k_{1}<k_{2}$
1. Statement 1 is false, Statement 2 is true.
2. Statement 1 is true, Statement 2 is false.
3. Statement 1 is true, Statement 2 is true, Statement 2 is the correct explanation for Statement 1
4. Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1
JEE Main Previous Year Single Correct Question of JEE Main from Physics Oscillations chapter.
JEE Main Previous Year 2012
Correct Option: 2
Solution:
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http://www.helpteaching.com/questions/Physics/Grade_8 | Browse Questions
Create printable tests and worksheets from Grade 8 Physics questions. Select questions to add to a test using the checkbox above each question. Remember to click the add selected questions to a test button before moving to another page.
Show Physics questions in All Grades.
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Grade 8 :: Forces and Motion by wlemerson
Grade 8 :: Forces and Motion by wlemerson
A force is a
1. constant speed
2. push or pull
3. linear quantity
4. cannot be determined
Grade 8 :: Forces and Motion by wlemerson
Grade 8 :: Heat Transfer by Testmaker1156
What is convection?
1. Transfer of heat by the movement of currents within a fluid
2. Transfer of heat from one particle to another without the movement of matter
3. The transfer of energy by electromagnetic waves
4. A conductor
5. An insulator
Grade 8 :: Energy and Momentum by Testmaker1156
Grade 8 :: Forces and Motion by wlemerson
A force
1. causes all motion
2. starts stationary objects moving
3. stops moving objects
4. is an interaction between two or more objects
5. all of the above
Grade 8 :: Forces and Motion by wlemerson
If two forces act on an object in the same direction, the net force is the
1. difference of the two forces
2. sum of the two forces
3. only one of the forces acts on the object
4. neither force can affect the object
Grade 8 :: Forces and Motion by wlemerson
The size or amount of a force is known as its
1. net force
2. direction
3. magnitude
4. constant speed
Grade 8 :: Forces and Motion by wlemerson
Grade 8 :: Forces and Motion by ncarlson0923
Grade 8 :: Forces and Motion by wlemerson
The rate at which velocity changes over time.
1. speed
2. acceleration
3. Newton
4. force
Grade 8 :: Forces and Motion by ncarlson0923
When is the velocity of an object constant?
1. when only its speed changes
2. when only its direction changes
3. when its speed and its direction both change
4. when neither its speed nor its direction change
Grade 8 :: Forces and Motion by wlemerson
The formula for velocity.
1. $A=(ΔV)/(Δt)$
2. $V=d/t$
3. $F=M*A$
4. $N=1 Kg*1m/s^2$
Grade 8 :: Heat Transfer by mtshan
Which of the following energy transfers would be correct?
1. Thermal energy from a hot drink is transferred to cold hands.
2. Thermal energy from the refrigerator is transferred to a hot pie in the refrigerator.
3. An ice cube loses thermal energy to a hot cup of tea as it melts.
Grade 8 :: Forces and Motion by ncarlson0923
Why is Newton's first law sometimes called the law of inertia?
1. because inertia is the tendency of objects to resist changes to their motion
2. because inertia describes the application of force to an object at rest
3. because inertia describes the applications of force to an object in motion
4. because inertia is the same thing as friction
Grade 8 :: Forces and Motion by ncarlson0923
What features determine an object's acceleration?
1. the forces acting on it and its size
2. the forces acting on it and its mass
3. the forces acting on it only
4. its weight and its mass
Grade 8 :: Fluid Mechanics by Saucesbi
Grade 8 :: Simple Machines by bamabryson
What is the formula for mechanical advantage?
1. input force divided by output force
2. output work divided by input work
3. output force divided by input force
1 2 3 4 ... 36
You need to have at least 5 reputation to vote a question down. Learn How To Earn Badges. | 2014-12-21 22:12:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.311222106218338, "perplexity": 3060.669834439023}, "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-2014-52/segments/1418802772743.56/warc/CC-MAIN-20141217075252-00080-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-critical-points-and-local-max-and-min-for-y-x-1-4 | # How do you find the critical points and local max and min for y=(x-1)^4?
Dec 6, 2017
Min: $x = 1$
Explained mathematically but easy to see graphically.
#### Explanation:
$y = {\left(x - 1\right)}^{4}$
Power rule + chain rule
$y ' = 4 {\left(x - 1\right)}^{3} \cdot 1$
$y ' = 4 {\left(x - 1\right)}^{3}$
Critical points are when $f ' \left(x\right) = 0$ so...
we set $y ' = 0$
$0 = 4 {\left(x - 1\right)}^{3}$
$\frac{0}{\textcolor{red}{4}} = \frac{4 {\left(x - 1\right)}^{3}}{\textcolor{red}{4}}$
$0 = {\left(x - 1\right)}^{3}$
0^(color(red)(1/3)=((x-1)^3)^color(red)(1/3)
$0 \textcolor{red}{+} \textcolor{red}{1} = x - 1 \textcolor{red}{+} \textcolor{red}{1}$
$1 = x$
We now know that $x = 1$ is a critical point.
Now we look at the original function $y = {\left(x - 1\right)}^{4}$
We can take a look at $x = 1$ and compare that to $x$ slightly smaller than 1 and $x$ slightly larger than 1
$y = {\left(1 - 1\right)}^{4}$
$y = {\left(0\right)}^{4}$
$y = 0$
$y = {\left(0.99 - 1\right)}^{4}$
$y = {\left(- 0.01\right)}^{4}$
$y = 0.00000001$
$y = {\left(1.01 - 1\right)}^{4}$
$y = {\left(.01\right)}^{4}$
$y = 0.00000001$
The points immediately to the left and right of $x = 1$ are both larger than $x = 1$ therefore $x = 1$ must be a minimum | 2021-07-27 21:51:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 29, "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": 0.6333929896354675, "perplexity": 288.6972084445732}, "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-2021-31/segments/1627046153491.18/warc/CC-MAIN-20210727202227-20210727232227-00349.warc.gz"} |
https://docs.thinkboxsoftware.com/products/deadline/10.1/1_User%20Manual/manual/command-line-arguments-pathmapping.html | # Path and Drive Mapping - Command Arguments¶
## Path Mapping¶
### CheckPathMapping¶
Description:
Performs path mapping on the given path. Uses the path mapping rules defined in the Repository Options.
Syntax:
Arguments:
Argument
Description
<Path>
The path to map
<Force Separator>
All path separators in the replacement path will be replaced with this before the original path is mapped (optional)
Example:
If there is no Path Mapping rule for “C:temp”, the result will be identical to the argument:
```C:\>deadlinecommand -CheckPathMapping "c:\temp\test\image.png"
c:\temp\test\image.png
```
If a Path Mapping rule exists, it will be applied:
```C:\>deadlinecommand -GetPathMappings
c:\temp\ --> Z:\assets\
Z:\assets\test\image.png
```
### CheckPathMappingForMultiplePaths¶
Description:
Performs path mapping on the given paths. Uses the path mapping rules defined in the Repository Options.
Syntax:
Arguments:
Argument
Description
<Path(s)>
The path to map, or a list of paths to map each separated by a space
forceseparator:<Force Separator>
All path separators in the replacement path will be replaced with this before the original path is mapped (optional).
Note
The `forceseparator:` keyword is required to distinguish the separator string from the paths arguments.
Example:
If there is no Path Mapping rule for “C:temp”, the result will be identical to the argument:
```C:\>deadlinecommand -CheckPathMappingForMultiplePaths "c:\temp\test\image.png" "c:\temp\textures\texture.tiff"
Z:\assets\test\image.png
Z:\assets\textures\texture.tiff
```
If the source paths do not include spaces, no quotes are required. The separator is added only to the replacement path, not the unaffected portion of the source path:
```C:\>deadlinecommand -CheckPathMappingForMultiplePaths c:\temp\test\image.png c:\temp\textures\texture.tiff forceseparator:/
Z:/assets/test\image.png
Z:/assets/textures\texture.tiff
```
### CheckPathMappingInFile¶
Description:
Performs path mapping on the contents of the given file. Uses the path mapping rules defined in the Repository Options.
Syntax:
Arguments:
Argument
Description
<Input File>
The original file name
<Output File>
The new file name where the mapped contents will be stored
<Force Separator>
All path separators in the replacement path will be replaced with this before the original path is mapped (optional)
### CheckPathMappingInFileAndReplaceSeparator¶
Description:
Performs path mapping on the contents of the given file, and updates all path separators in any paths that are mapped. Uses the path mapping rules defined in the Repository Options.
Syntax:
Arguments:
Argument
Description
<Input File>
The original file name
<Output File>
The new file name where the mapped contents will be stored
<Old Separator>
The path separator to replace
<New Separator>
The new path separator
<Force Separator>
All path separators in the replacement path will be replaced with this before the original path is mapped (optional)
### GetPathMappings¶
Description:
Returns a list of all path mappings that can will be applied in the current context. Uses the path mapping rules defined in the Repository Options.
Syntax:
Arguments:
Argument
Description
.
No arguments
## Mapped Drives¶
### MapDrives¶
Description:
Maps mapped drives from network settings on Windows.
Syntax: | 2021-12-07 01:54:08 | {"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": 0.8156887292861938, "perplexity": 3302.3286105400884}, "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/1637964363332.1/warc/CC-MAIN-20211207014802-20211207044802-00412.warc.gz"} |
http://mathoverflow.net/revisions/100243/list | The Discrete Fourier Transform, which sends a vector $x=\left(x_j\right)_{j=0}^{N-1}$ to $y=\mathrm{DFT}(x)$ such that $$y_k=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi i \times jk/N}x_j$$ has a matrix representation $$\mathrm{DFT}_{jk}=e^{2\pi i \times jk/N}=\left(e^{2\pi i /N}\right)^{j\times k},$$ which is in fact a doubly Vandermonde matrix: both it and its transpose are Vandermonde matrices. With this you can use the Vandermonde determinant to prove that $\mathrm{DFT}$ is nonsingular, and if you prove using other means that it is unitary (rather easy) then you will get, I think, a nontrivial expression for 1 as a product of differences of roots of unity. | 2013-05-24 18:11:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9048219919204712, "perplexity": 95.17875109905638}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704933573/warc/CC-MAIN-20130516114853-00085-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2555570/affine-spaces-and-type-theory | # Affine Spaces and Type Theory
From Wikipedia:
In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors [...] between two points of the space.
Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.
Type or category theoretically (or API design-wise), a displacement vector is conceptually of a different type than than the points in the space (despite both being representable as vectors), and operators/operations exist such that they take an return arguments of different types.
C++ has several examples of of such types and associated algebras:
1. The most basic: Pointers (addresses in the memory address space) are distinct types from integers, they cannot be added together, but can be subtracted to give an integer or incremented by integers.
2. The standard <chrono> time header defines time_points and durations.
Other examples include geometry primitives like 2D, 3D...ND points or physical measurements like distances, time (see above), or any SI units.
Interestingly this aspect is orthogonal to aspects of unit-types.
I'm looking for literature and discussion of Affine Spaces in the context of Type-Theory (or Abstract Algebra). Specifically any formal definition of an algebra or specifications of the axioms or properties.
(I am not a mathematician so please forgive (an correct) any erroneous terminology).
nLab provides a multitude of axiomatic descriptions and (above where the link points) many conceptual descriptions. The simplest way of defining affine spaces is to say they are equipped with a "difference" operation that produces values in a vector space.
For APIs, I usually use an approach like the following (using C# syntax). (I also use a similar approach for torsors aka heaps which are a direct generalization of affine spaces and handles things like orientations versus rotations.)
interface Vector<V> where V : Vector<V> {
V Minus(V v);
V Scale(double s);
}
interface Point<P, V> where V : Vector<V> where P : Point<P, V> {
The $\Lambda$ operation specified in the first axiomatization on the nLab page corresponds to the Minus method on Point. The Add method on Point provides the unique $y$ such that $\Lambda(y,x)=v$. Rewriting the axioms in C#, it would look like (using x, y, and z for Points and v for Vectors): | 2018-08-16 07:58:28 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7271170616149902, "perplexity": 1026.623944177705}, "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-2018-34/segments/1534221210559.6/warc/CC-MAIN-20180816074040-20180816094040-00531.warc.gz"} |
https://www.coursehero.com/file/42644298/HW8-ENED1091-Winter-2018docx/ | # HW8_ENED1091_Winter_2018.docx - ENED 1091 HW#8 Due Week of...
• 6
This preview shows page 1 - 3 out of 6 pages.
ENED 1091 HW#8 Due Week of April 9that beginning of Recitation Problem 1: Use the data points show below and the trapezoidal rule to estimate the integral of the curve shown below from t = 0 to 3.6. Be sure to clearly show your calculations – don’t just give an answer.-0.500.511.522.533.54020406080100X: 3.6Y: 100.6xYX: 3Y: 27X: 2.4Y: 8.175X: 1.8Y: 2.881X: 1.2Y: 1.245X: 0.6Y: 0.736X: 0Y: 1Data Points from Graph:X00.61.21.82.433.6Y10.7361.2452.8818.17527100.6Integral Estimate (Trapezoid): ______54.5022______________
Problem 2: The curve shown in problem 1 is for the function y = xx. There is no expression for the indefinite integral of this function. However, we can estimate the definite integral using numerical integration as you have done in Problems 1. Write a MATLAB script that will:Begin with 3 data points (x-values) evenly distributed from 0 to 3.6 inclusive (Hint: use the MATLAB command, linspace).
Calculate the corresponding y-values for the function y = x^x. Estimate the integral of y from 0 to 3.6 using the Trapezoid Rule.Double the number of data points and get a new estimate for the integral of y from 0 to 3.6 using the trapezoid rule. | 2021-10-22 03:57:20 | {"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": 0.8468931913375854, "perplexity": 1325.263713292553}, "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/1634323585450.39/warc/CC-MAIN-20211022021705-20211022051705-00361.warc.gz"} |
https://www.physicsforums.com/threads/vector-meson-spin-states.841202/ | # Vector meson spin states
1. Nov 3, 2015
### dwight ang
Hi All
Thanks for welcoming me to the physics forum. I am reading Greiner and Mueller's Quantum Mechanics: Symmetries and am stuck at not understanding the vector meson ( rho meson)'s spin states.
For S=1 we get three states -1, 0, and 1.
Prof. Mueller separated them as a rho +/-1, and rho 0 .
for the rho +/-1 he mentions the state u ( spin up , spin down) *d_ (read "d bar" for anti d) ( spin up , spin down)
How did he get to this? I did the Clebsch Gordan analysis and got lost . Need some help here.
Thanks
Dwight
2. Nov 8, 2015
### Greg Bernhardt
Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
3. Nov 9, 2015
### vanhees71
You must be careful to distinguish spin and isospin properly. The $\rho$ meson has spin 1 and isospin 1, i.e., there are three spin states for each charge state, i.e., all together you have 9 physical field-degrees of freedom. In the SU(2) model (only up and down quarks) the electric charge is given by the eigen vectors of $\hat{\tau}_3$, which are $t_3 \in \{-1,0,1 \}$.
Sometimes it's more convenient to work in the SO(3) representation of the isospin. Then you have three real vector fields, written as $\vec{\rho}^{\mu}$. You can easily convert from one to the other isospin basis:
$$\rho^{(\pm) \mu}=\frac{1}{\sqrt{2}} (\rho_1^{\mu} \pm \mathrm{i} \rho_2^{\mu}), \quad \rho^{(0) \mu}=\rho_3.$$
In terms of quark currents the $\rho$ mesons are built from the vector-isovector currents
$$\vec{j}^{\mu}=\overline{\psi} \vec{\tau} \gamma^{\mu} \psi,$$
where $\psi$ is the isospin doublet
$$\psi=\begin{pmatrix} u \\ d \end{pmatrix}$$
and the $\vec{\tau}$ are represented by $\vec{\tau}=\vec{\sigma}/2$, where $\vec{\sigma}$ are the usual Pauli matrices.
By identifying the $\vec{\rho}$ with these currents, you can easily read off the quark content of the $\rho$ mesons. Of course the $\rho^{+}$ is a $u \bar{d}$, the $\rho^{-}$ a $d \bar{u}$, and the $\rho_0$ is given by $\rho_3 \sim |u \bar{u} \rangle - |d \bar{d} \rangle$, because $\tau_3=\mathrm{diag}(1,-1)$. | 2018-07-16 19:27:30 | {"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": 0.8707380890846252, "perplexity": 1144.4209664786854}, "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-2018-30/segments/1531676589417.43/warc/CC-MAIN-20180716174032-20180716194032-00034.warc.gz"} |
https://www.ejpbl.org/journal/view.php?doi=10.24313/jpbl.2020.00213 | • Home
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J Probl Based Learn > Volume 7(1); 2020 > Article
Jung and Hong: A Theoretical Need for Applying Flipped Learning to STEAM Education
### Abstract
This paper addresses the current problems of STEAM education and suggests a solution to solve one of the problems through adopting a method from Flipped Learning. Through the meticulous literature review, comparison, and analysis on STEAM and Flipped Learning, it will be shown that the methodology of Flipped Learning complements the weaknesses in STEAM. The conclusion is that since there is a theoretical need for applying Flipped Learning to STEAM, further research needs to be carried out to apply flipped learning in STEAM classes.
### INTRODUCTION
The goal of education used to be to train individuals to have certain skills and knowledge in order that they may have a higher chance of success in the future. Due to the rapid technological advancement in the last century and since the start of the new millennial, the future is more unpredictable. The goal of education now is to train individuals to be flexible and adaptable to the unpredictable future (Rychen & Salganik, 2003). In parallel with such a change, the need for pedagogical research has also increased (Evensen & Hmelo, 2000).
What is more important is that in the face of these changes, the limitations of existing teaching and learning methods are being revealed (Gilboy et al., 2015; Bazler & Sickle, 2017). The traditional teaching and learning method in the classroom is in need of improvement or changes. The manner in which the teacher communicates knowledge unilaterally to the students is less likely to make educational impact for the students who are already familiar with the culture of acquiring various pieces of knowledge from the internet and communicating with people in various communities online (Barrett, 2012; Herreid & Schiller, 2013; Roehl et al., 2013). The limitations and new demands in the field of education have advanced discussions on alternative paradigms of education (Hidi & Harackiewicz, 2000).
STEAM, which stands for Science, Technology, Engineering, Arts, and Mathematics, is a new educational system which hopes to prepare individuals for the unpredictable future. Originally based on STEM (which is without the Arts curriculum), STEAM aims to educate individuals to be able to think in a divergent and convergent manner (Bazler & Sickle, 2017). According to the results of research, STEAM appears to enable creative and critical thinking in humanities education (Ferrall, 2011). Therefore, it better educates individuals to live curiously and creatively with the knowledge on arts, science, and mathematics (Spector, 2015; Yakman & Lee, 2012).
However, STEAM has many problems too despite its advantages. It is true that the teachers who implemented STEAM recognize the importance of the integrated education method (Noh & Ahn, 2012). However, it has been pointed out that not only do the teachers need less work in implementing STEAM, but also the teachers do not seem to know the difference between STEAM and the traditional science education (Park et al, 2016). Therefore, in order to solve these difficulties faced by teachers and to fulfill the purpose of STEAM successfully, it is necessary to expand the class time, to diversify the class space, to use various teaching materials in a communicative way, to secure communication space between teachers and learners, to provide individualized learning materials which take individual differences of learners into account (Lim, Kim, & Lee, 2014).
In order to solve the difficulties encountered by the teachers who want to continue to teach STEAM, such as the lack of time for the education, and the difficulty of knowing exactly how to proceed with the class, this paper suggests Flipped Learning as a methodology to complement STEAM. This review paper focuses on the need for learners and teachers to adopt the methodology of Flipped Learning in order to effectively achieve the goal of STEAM.
### WHAT IS STEAM?
STEM, an educational system that the U.S. actively introduced since the late 20th century, aims to advance science and mathematics education as well as to focus on convergence between disciplines of science. Its goal is to enhance students' interest in science, technology, engineering and mathematics as well as to advance technical literacy (Breiner et al., 2012). Traditional science education, which excludes the humanities, plays an excellent role in providing knowledge of mathematics and nature and skill in science and technology, but it was not effective in developing the creativity of scientists or to discover and solve interdisciplinary problems. (Madden et al., 2013). To overcome this problem, the movement to include humanities and arts in science-based education has been attempted by many scholars (Ghanbari, 2014).
STEAM was first introduced in the U.S. because the American Academy of Sciences realized that interest in STEM education was declining (National Academies, 2007). Yakman (2008), who first proposed the structure of STEAM, criticized STEM for lacking convergence even though STEM adopts an integrated approach to teaching and learning academic concepts in science, technology, engineering, and mathematics. She believed that STEAM was an optimal education system that can provide a wholly convergent structure of learning experience which cannot be achieved by the STEM only. According to Bazler and Sickle (2017), STEAM education is currently an emerging educational model in the U.S., because it purports to overcome the weaknesses of STEM education.
The characteristic of STEAM is the introduction of the concept of convergence between basic subjects especially with the arts curriculum, which has not been actively pursued in the traditional curriculum. Convergence is one of the most important strategies for solving complex problems and solving complex intellectual problems in new fields in the 21st century. The students need to be prepared for the future by learning how to integrate ideas, approaches, and technologies from various knowledge fields (National Research Council, 2014). Therefore, STEAM aims to develop creative and future oriented talent by presenting a curriculum that can integrate individualized knowledge into one.
### IMPORTANCE OF CONVERGENCE IN STEAM
The characteristics of convergence in STEAM can be summarized as follows. First, it is learning-oriented. Chopp (2014) suggests a conceptual concept of knowledge design that helps students learn about individual subject matter in a convergent way. Knowledge acquisition design aims to "grant creativity and clarity to the center of learning and learner attitudes" by accepting a new learning platform and recognizing the power of convergence of visualization power and knowledge (Chopp, 2014).
Second, it is dialectical. It gives emphasis on the process by which learners find and pioneer knowledge acquisition on their own. The process of searching for knowledge is only possible when the learners themselves ask questions from their curiosity about the knowledge and find an answer to the questions. The nature of this process is dialectical. As found in years past, teaching methods focused on teaching and learning in a dialectical way emphasized the importance of discussing and participating actively with other learners (Greeno, Collins & Resnick, 1996).
Third, various educational means are utilized. As found in past decades, in order to effectively integrate knowledge, it is necessary not only to teach using various media, but also to apply knowledge under various social contexts (Brown, Collins & Duguid, 1989). It is true that there have been many attempts to improve the effectiveness of learning using various media, not limited to textbooks in education (Mayer & Moreno, 2010). Attempts to actively utilize various media such as the internet, video, and mobile learning to construct convergent knowledge are characteristics of STEAM.
### APPLICATION OF INSTRUCTIONAL DESIGN PRINCIPLES OF STEAM
The following instructional design principles summarize the re-conceptualization and restructuring of the curriculum in the STEAM fields with various pedagogical teaching strategies in order to solve the needs and tasks of the 21st century. Palou et al. (2015) applied STEAM to construct a core support system for engineering students who would be prepared for the 21st century. They created a learner-centered learning environment and reconstructed the basic courses in chemistry and environmental engineering.
Coffland and Xie (2015) likewise describe the experience of teaching mathematics curriculum based on four skills required by the 21st century: Communication, Collaboration, Critical thinking, and Creativity (Partnership for 21st Century, 2011). Through this curriculum, they have enabled learners to face and experience the problems arising in real life in order to promote self-learning and to combine academic knowledge with real-life problems.
Christensen and Knezek (2015) adopted an active learning approach. The researchers proposed an attempt to actively include technology into the middle school science curriculum based on the STEAM skills required by the 21st century. Student-centered active learning has been shown to contribute to a long-term retention of knowledge and a deeper understanding of the subject (Akinoglu & Tandogan, 2007).
Furthermore, Aschbacher, Ing & Tsai (2013) suggest that if the content that students learn at school is more relevant to students personally and if the content is related to the future, the students’ become more drawn to the subject and learn more effectively and faster. Christensen & Knezek (2015) found out that with an active learning approach, students' academic achievement increases and positively affects students’ attitudes toward science and related subjects.
Based on such studies, it can be concluded that STEAM is appropriate for preparing students for the future (D'Mello et al., 2014). In addition to this, with STEAM, students will be able to be more creative in the process of constructing knowledge in a convergent way. In addition, the STEAM focus introduces a way to acquire knowledge connectively through interaction and cooperation among learners, so that it is expected to increase the communication effect among learners.
### SUGGESTED STEAM EDUCATIONAL STRATEGIES TO IMPROVE LEARNING
Ifenthaler et al. (2015) designed engineering education using a collaborative learning strategy. They suggest that the use of co-operative strategies has changed the attitudes of learners receiving engineering education, their thinking about themselves, and the activeness of a team. They published a case study that examined the effects of the learning organization model on attitude change, self-awareness, and organizational structure of student learning engineering (Mistree et al., 2014). This new curriculum emphasizes the capabilities and possibilities of individuals developed in a collaborative learning model and environment. In this model, learning took place at three levels: Individual learning, team-based learning, and group-based learning. Many of these curriculum strategies are designed to support STEM education, but they can also be applied to STEAM education.
Second, they suggest that, during the class teachers should organize students into small groups for in-class class activities or for activities outside of class. Unlike the traditional strategy of constructing the contents of the lesson centered on the topic decided by the teacher, the students can present the tasks, problems, questions, difficulties and contents contained in the topics to the students through one or more scenarios. Scenario-based approaches can be implemented under a variety of models, including problem-oriented strategies and project-driven strategies. Based on a scenario-based approach, students solve problems by presenting specific situations and scenarios in which problems are presented, and teachers are provided with tools to further develop the scenarios and the content. There is a lot of evidence to support the overall scenario-based approach, but there are a few drawbacks to the strategy of organizing students into small groups to increase classroom efficiency and performance. Teachers applying the scenario-based approach frequently have to organize students into small groups and test applicability of student led-group process to scenario-based strategies (Prince & Felder, 2006).
Based on these findings, Shen, Jiang & Liu (2015) proposed the following four STEM education strategies to improve students' lifelong learning skills.
1. Designing activities to engage and motivate students in active learning: The essence of this strategy is to develop activities that help students take more responsibility for learning by creating student-led environment. These activities can be considered in various ways, such as demonstrating interesting scientific phenomena, writing science content related to a student's personal life, extending learning to external learning areas, and linking science to other exciting academic disciplines or entertainment.
2. Using a scenario-based content: A scenario-based approach represents a broad educational strategy that provides learning materials to students over longer periods, centered on one or more scenarios. These cases can often be classified as problem-based, project-based, case-based, survey-based, or task-based.
3. Organizing Students Focused on Collaboration: This practice follows a strategy for organizing small groups and organizing learning communities. Collaborative work can stimulate interest in class by interacting face-to-face with students in various classes, before and after classes, or interacting virtually through the Internet.
4. Conduct research: This strategy is to encourage students to develop research that is of interest to them under the supervision of a teacher.
### REPORTED PROBLEMS OF STEAM
Despite the advantages of STEAM, it has been reported that the teachers face a lot of difficulties when applying the STEAM in their classrooms. Lim, Kim & Lee, (2014) examined elementary school teachers' opinions and experiences of teaching STEAM, and found that they think there is lack of STEAM activities and they face difficulties in preparation for STEAM classes. Although most teachers see the need for teaching STEAM classes, the lack of instructional materials discourage them from actively teaching STEAM. In addition, Lee, Park & Kim, (2013) report that there is shortage of classroom hours for teachers to effectively carry out the STEAM classes. For the integrated curriculum to be delivered requires teachers to have a lot of knowledge and plan student activities. It takes a lot of time to carry out a long-term project, but the current number of assigned teaching hours for STEAM is not enough for such activities. Shin & Han (2011) pointed out that there is difficulty for teachers to integrate educational elements and activities through STEAM when they conducted STEAM classes.
The potential difficulties that teachers face regarding STEAM are summarized in three points. First, there is a difficulty in lack of class time. STEAM not only requires a lot of knowledge and activities in the integrated subject matter to be delivered, but also demands a long enough classroom time to carry out the activities properly. Therefore, STEAM classes are not effective enough to achieve its original goal. Teachers may feel further burdened in overcoming the difficulties. Second, it is difficult for teachers to prepare class materials for STEAM. In fact, it is very difficult for the teachers to make the STEAM materials in accordance with the students' interest and level. Third, students are not gaining enough knowledge that is supposed to be interdisciplinary and convergent. At the core of STEAM is to acquire interdisciplinary knowledge and to transfer such knowledge into produce more creativity. The activities conducted in STEAM are for acquiring convergent knowledge, but the problem is that learners are often more interested in the activities themselves rather than the interdisciplinary knowledge. These difficulties may be due to various reasons, but it is especially important to notice that there is a lack of communication channels to check whether learners really acquire interdisciplinary skills, and that the level of individual learners is not properly reflected in the actual classes either.
The current problems in STEAM, such as shortage of teachers' time for preparing the STEAM classes and difficulties in coming up with classroom activities, can be complemented by introducing the methodology of Flipped Learning. Flipped Learning can be adopted to extend class time by providing opportunities to learn basic knowledge and concepts before STEAM class begins. In addition, the use of various high-tech media outside the classroom leads to the diversification of the classroom space. With the various teaching materials online, learners effectively absorb the knowledge according to their pace and preference. It enables individuals to change their learning speed according to their learning ability.
### WHAT IS FLIPPED LEARNING?
Flipped learning is an alternative to the teacher-centered, traditional way of teaching. It emphasizes engaging activities among students and interactions between teachers and students (Hamdan et al., 2013; Strayer, 2012). Traditional classes taught basic concepts and knowledge in the classroom and applied learner-centered activities as homework (McCarthy & Anderson, 2000). In contrast, Flipped Learning is a process of learning basic knowledge and concepts outside the classroom (Love et al., 2014). By “flipping” the classroom, students participate in various problem-solving activities based on learners' interest and level of need in the classrooms, which is an innovative proposal for teaching methods (Bergmann & Sams, 2013).
In addition, Flipped Learning dramatically changes the content and aspects of classroom instruction (Bergmann & Sams, 2014). Flipped Learning allows students to voluntarily solve problems by cooperating with each other (Jamaludin & Osman, 2014). The role of the instructor is not exclusively limited to lecturing, but expanded to providing feedback and advice to students.
Bergmann & Sams (2013) believe that the most important aspect in Flipped Learning is the meaningful learning activity that occurs in the face-to-face classrooms. Also, it is emphasized that the center of learning is not the teacher but the activities of the students and the students are taking into consideration the important activities to be done in the classroom. Strayer (2012) argues that the most important stage of Flipped Learning is the activity of expanding and deepening the concepts learned through video and various mediums in pre-classrooms in real classrooms.
In addition to the theoretical studies on Flipped Learning, the educational effects of Flipped Learning can be summarized in terms of improvement in learning ability among students (Saban, 2013). Flipped Learning improves self-directed learning ability, students’ interest in learning, ability to understand the contents, problem-solving ability, and self-confidence. In addition, the aspects of cooperative learning, collaborative learning are emphasized by encouraging students to discuss problem-solving activities with each other, sharing knowledge among colleagues, and even teaching peers (O'Flaherty & Phillips, 2015).
To sum up, Flipped Learning, which has an educational effect on the cognitive dimension of students which broadens the level of knowledge and provides opportunities for mutual knowledge exchange among them, is becoming a popular educational approach to fostering talented individuals for the 21st century (Newman et al., 2016).
### THE FOUR PILLARS OF FLIPPED LEARNING
Sams & Bergmann (2014), who have been leading the study of Flipped Learning, have established The Flipped Learning Network. According to the Flipping Learning Network Board, the characteristics of Flipped Learning are compared to the four pillars: Flexible environment, a Learning culture, Intentional content, and a Professional educator. The detailed characteristics are as follows.
First, Flipped Learning, which provides a flexible environment, accommodates a variety of learning methods. Teachers need to physically reconfigure the learning space for collaborative learning or individual learning in the process of optimizing the class. It is a feature of Flipped Learning to create a flexible space where learners can learn whenever and wherever they need. Furthermore, teachers who flip the classroom tend to be flexible in coping with learners' learning plans and evaluations.
Second, Flipped Learning helps change the classroom climate into a learner-centered learning culture. In traditional teacher-centered instructional models, teachers had the authority to distribute information. In the Flipped Learning model, however, the lessons flexibly change with the learners. Such changes induce the learner to explore deeply on the subject and provide a rich learning experience. As a result, students engage in knowledge formation and evaluate their learning in a personally meaningful way.
Third, teachers who perform Flipped Learning continue to contemplate how to use the Flipped Learning model efficiently to help students understand the knowledge and the goal of education. Teachers can optimize class time to accommodate learner-centered, active-learning strategies, grade levels, and the curriculum.
Fourth, in Flipped Learning, teachers have professional knowledge of technology as well as being knowledgeable on the content. The role of the professional teacher in the Flipped Classroom is more important than the traditional classroom. During class time, teachers constantly observe students, provide them with instantaneous feedback, and assess their work. Teachers are reflective in the actual classrooms and engage in constructive criticism in relation to each other teachers to improve their teaching methodologies.
### THEORETICAL ADVANTAGES OF COMBINING STEAM WITH FLIPPED LEARNING
Flipped Learning, which is being used as a new teaching method in the current education field, can be a useful method to implement the problems reported in STEAM. The first advantage is that Flipped Learning provides teachers with pre-learning materials and utilizes the cloud computing environment to enable easy learning through smartphones and tablet PCs regardless of where the students are and what time the students want to study. In addition, the pre-learning content itself provides information not only about the basic knowledge but also about how to practically implement the use of knowledge for students. This enables learner-centered education by helping students participate in real classroom activities more actively.
Park (2014) state the advantages of Flipped Learning as follows. First, students learn the concept of knowledge in advance and can concentrate on the STEAM activities in school classes. Second, teachers can efficiently utilize the STEAM program by applying the teaching strategies or class models suitable for the STEAM education through Flipped Learning.
Nowadays, Flipped Learning is introduced as a new learning system for this digital generation because students are already familiar with the use of digital devices, can multitask, and are capable of instant online communication. The most important aspect is the fact that it is student-centered. It is personalized learning. Student-centered classrooms use Dewey’s philosophy (Dewey, 1916) – the constructivist education paradigm as a philosophical basis for designing Flipped Learning (Ryder, 2006). In other words, Flipped Learning plays an effective role in establishing a student-centered classroom environment in which students select and construct content of their own. They become initiative as they become facilitators of knowledge, contrary to the traditional classroom teaching methods in which teachers are the center of the classroom.
Flipped Learning requires the teacher to constantly check and verify his or her class, because the teacher requires the student to rebuild the existing curriculum so that the students actively participate in the class. Through this process, the teacher prepares the lesson considering the interest, concern, and readiness of each student while considering the universal design for learning.
The advantages of adopting Flipped Learning into STEAM based on methodological characteristics are as follows. First, it allows teachers to balance the need to handle and deliver large volumes of learning content and the need for students to build meaning through interaction with the content (Bishop & Verleger, 2013). Second, because students actively use and apply knowledge in the classrooms, Flipped Learning can be seen by the teacher as a natural part of the students' overcoming weaknesses in the learning process (Butt, 2014). Third, Flipped Learning can increase student participation and motivation (Critz & Knight, 2013). Fourth, Flipped Learning can improve teacher-student and peer interaction (Gaughan, 2014).
The main advantage of Flipped Learning is that it can flexibly adapt to a variety of learning methods (Roehl, Reddy & Shannon, 2013; Schwartz, 2014). Unlike traditional classroom lectures, students are free to choose where and how to view the recorded lecture material before class (Forsey, Low & Glance, 2013). Teachers who have introduced Flipped Learning can no longer teach in class, so they can help them learn more effectively by developing higher-level learning methods and applying the practical skills (Strayer, 2012).
According to Hamdan et al. (2013), most Flipped Learning studies point out that students' perceptions and learning effects through Flipped Learning were generally positive and preferred classroom activities with interactions rather than lectures. Bergmann & Sams (2013) also report that Flipped Learning has proven to be an effective learning method for children with diverse learning abilities and environments. Davices, Dean & Ball (2013) also show that classes that adopted Flipped Learning in college had higher academic achievement in those who did not.Flumerfelt & Green (2013) found that Flipped Learning has a positive effect on the academic achievement in high school.
In summary, using Flipped Learning, various kinds of advanced media are utilized, interaction between learners is encouraged, various learner-centered activities are performed, and communication between teachers and students is increased through such an interaction-based learning methodology. By actively utilizing these characteristics of Flipped Learning, the teachers of STEAM will be able to solve the problems that they faced: they will not have shortage of time teaching the basic knowledge and preparing the class materials and convergent knowledge of STEAM.
### CONCLUSION
STEAM is aimed at fostering talented individuals who are able to creatively converge knowledge to creatively converge fragmented knowledge. However, due to lack of time for teachers and effective methodology, it failed to enable students be creative and reconstruct the knowledge. One solution to this problem is to adopt the learner-centered methodology, Flipped Learning. It is a teaching method that can lead to integration of such knowledge. The educational model which combines STEAM with Flipped Learning motivates a need for providing instructional design to promote learners' activity, responsibility, mutual understanding, and mutual trust among each other. Applying Flipped Learning for STEAM in the current education system, which requires students to be creative and flexible, will be able to not only positively change the interests and attitudes of learners, but also enhance creativity by creating a room for convergence of knowledge (Lewis, 2015).
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https://projecteuclid.org/euclid.facm/1356012916 | ## Functiones et Approximatio Commentarii Mathematici
### Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,\ldots
#### Abstract
We introduce the notion of {\em circular words} with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we establish a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of [Sloane] in terms of a product of expressions involving roots of unity. Third, we consider the equivalent of $p$-adic numbers that arise by the use of the numeration system defined by the Fibonacci sequence instead of the usual numeration system in base $p$. Among such {\em ${\mathcal F}$-adic numbers}, we give a~characterization of the subset of those which are {\em rational} (that is: a root of an equation of the form $qX=p$, for integral values of $p$ and $q$) by a periodicity property. Eventually, with the help of circular words, we give a complete description of the set of roots of $qX=p$, showing in particular that it contains exactly $q$ ${\mathcal F}$-adic elements.
#### Article information
Source
Funct. Approx. Comment. Math. Volume 47, Number 2 (2012), 207-231.
Dates
First available in Project Euclid: 20 December 2012
Permanent link to this document
https://projecteuclid.org/euclid.facm/1356012916
Digital Object Identifier
doi:10.7169/facm/2012.47.2.6
Mathematical Reviews number (MathSciNet)
MR3051449
Zentralblatt MATH identifier
1260.68313
#### Citation
Rittaud, Benoît; Vivier, Laurent. Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,\ldots. Funct. Approx. Comment. Math. 47 (2012), no. 2, 207--231. doi:10.7169/facm/2012.47.2.6. https://projecteuclid.org/euclid.facm/1356012916
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• K. Schmidt, Algebraic codings of expansive group automorphisms and two-sided beta-shifts, Monatsh. Math 129 (2000), 37-61.
• N. Sidorov, An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation, Acta Arith. 101 (2002), 199–213.
• N. Sidorov and A. Vershik, Ergodic properties of the Erdős measure, the entropy of the golden shift, and related problems, Monatsh. Math. 126 (1998), 215-261.
• E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41 (1972), 179–182. | 2018-01-21 04:53:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8320018649101257, "perplexity": 1612.905158891164}, "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-2018-05/segments/1516084890187.52/warc/CC-MAIN-20180121040927-20180121060927-00412.warc.gz"} |
http://conceptmap.cfapps.io/wikipage?lang=en&name=B-tree | B-tree
In computer science, a B-tree is a self-balancing tree data structure that maintains sorted data and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree is a generalization of a binary search tree in that a node can have more than two children.[1] Unlike other self-balancing binary search trees, the B-tree is well suited for storage systems that read and write relatively large blocks of data, such as discs. It is commonly used in databases and file systems.
B-tree
Typetree
Invented1971
Invented byRudolf Bayer, Edward M. McCreight
Time complexity in big O notation
Algorithm Space Average Worst case O(n) O(n) O(log n) O(log n) O(log n) O(log n) O(log n) O(log n)
What, if anything, the B stands for has never been established.
Overview
A B-tree (Bayer & McCreight 1972) of order 5 (Knuth 1998).
In B-trees, internal (non-leaf) nodes can have a variable number of child nodes within some pre-defined range. When data is inserted or removed from a node, its number of child nodes changes. In order to maintain the pre-defined range, internal nodes may be joined or split. Because a range of child nodes is permitted, B-trees do not need re-balancing as frequently as other self-balancing search trees, but may waste some space, since nodes are not entirely full. The lower and upper bounds on the number of child nodes are typically fixed for a particular implementation. For example, in a 2-3 B-tree (often simply referred to as a 2-3 tree), each internal node may have only 2 or 3 child nodes.
Each internal node of a B-tree contains a number of keys. The keys act as separation values which divide its subtrees. For example, if an internal node has 3 child nodes (or subtrees) then it must have 2 keys: a1 and a2. All values in the leftmost subtree will be less than a1, all values in the middle subtree will be between a1 and a2, and all values in the rightmost subtree will be greater than a2.
Usually, the number of keys is chosen to vary between ${\displaystyle d}$ and ${\displaystyle 2d}$ , where ${\displaystyle d}$ is the minimum number of keys, and ${\displaystyle d+1}$ is the minimum degree or branching factor of the tree. In practice, the keys take up the most space in a node. The factor of 2 will guarantee that nodes can be split or combined. If an internal node has ${\displaystyle 2d}$ keys, then adding a key to that node can be accomplished by splitting the hypothetical ${\displaystyle 2d+1}$ key node into two ${\displaystyle d}$ key nodes and moving the key that would have been in the middle to the parent node. Each split node has the required minimum number of keys. Similarly, if an internal node and its neighbor each have ${\displaystyle d}$ keys, then a key may be deleted from the internal node by combining it with its neighbor. Deleting the key would make the internal node have ${\displaystyle d-1}$ keys; joining the neighbor would add ${\displaystyle d}$ keys plus one more key brought down from the neighbor's parent. The result is an entirely full node of ${\displaystyle 2d}$ keys.
The number of branches (or child nodes) from a node will be one more than the number of keys stored in the node. In a 2-3 B-tree, the internal nodes will store either one key (with two child nodes) or two keys (with three child nodes). A B-tree is sometimes described with the parameters ${\displaystyle (d+1)}$ ${\displaystyle (2d+1)}$ or simply with the highest branching order, ${\displaystyle (2d+1)}$ .
A B-tree is kept balanced by requiring that all leaf nodes be at the same depth. This depth will increase slowly as elements are added to the tree, but an increase in the overall depth is infrequent, and results in all leaf nodes being one more node farther away from the root.
B-trees have substantial advantages over alternative implementations when the time to access the data of a node greatly exceeds the time spent processing that data, because then the cost of accessing the node may be amortized over multiple operations within the node. This usually occurs when the node data are in secondary storage such as disk drives. By maximizing the number of keys within each internal node, the height of the tree decreases and the number of expensive node accesses is reduced. In addition, rebalancing of the tree occurs less often. The maximum number of child nodes depends on the information that must be stored for each child node and the size of a full disk block or an analogous size in secondary storage. While 2-3 B-trees are easier to explain, practical B-trees using secondary storage need a large number of child nodes to improve performance.
Variants
The term B-tree may refer to a specific design or it may refer to a general class of designs. In the narrow sense, a B-tree stores keys in its internal nodes but need not store those keys in the records at the leaves. The general class includes variations such as the B+ tree and the B* tree.
• In the B+ tree, copies of the keys are stored in the internal nodes; the keys and records are stored in leaves; in addition, a leaf node may include a pointer to the next leaf node to speed sequential access.[1]
• The B* tree balances more neighboring internal nodes to keep the internal nodes more densely packed.[1] This variant ensures non-root nodes are at least 2/3 full instead of 1/2 (Knuth 1998, p. 488). As the most costly part of operation of inserting the node in B-tree is splitting the node, B*-trees are created to postpone splitting operation as long as they can.[2] To maintain this, instead of immediately splitting up a node when it gets full, its keys are shared with a node next to it. This spill operation is less costly to do than split, because it requires only shifting the keys between existing nodes, not allocating memory for a new one.[2] For inserting, first it is checked whether the node has some free space in it, and if so, the new key is just inserted in the node. However, if the node is full (it has m − 1 keys, where m is the order of the tree as maximum number of pointers to subtrees from one node), it needs to be checked whether the right sibling exists and has some free space. If the right sibling has j < m − 1 keys, then keys are redistributed between the two sibling nodes as evenly as possible. For this purpose, m keys from the current node plus the new key inserted, one key from the parent node and j keys from the sibling node are seen as an ordered array of m + j + 1 keys. The array becomes split by half, so that ⌊(m + j + 1)/2⌋ lowest keys stay in the current node, the next (middle) key is inserted in the parent and the rest go to the right sibling.[2] (The newly inserted key might end up in any of the three places.) The situation when right sibling is full, and left isn't is analogous.[2] When both the sibling nodes are full, then the two nodes (current node and a sibling) are split into three and one more key is shifted up the tree, to the parent node.[2] If the parent is full, then spill/split operation propagates towards the root node.[2] Deleting nodes is somewhat more complex than inserting however.
• B-trees can be turned into order statistic trees to allow rapid searches for the Nth record in key order, or counting the number of records between any two records, and various other related operations.[3]
B-tree usage in databases
Time to search a sorted file
Usually, sorting and searching algorithms have been characterized by the number of comparison operations that must be performed using order notation. A binary search of a sorted table with N records, for example, can be done in roughly ⌈ log2 N comparisons. If the table had 1,000,000 records, then a specific record could be located with at most 20 comparisons: ⌈ log2 (1,000,000) ⌉ = 20.
Large databases have historically been kept on disk drives. The time to read a record on a disk drive far exceeds the time needed to compare keys once the record is available. The time to read a record from a disk drive involves a seek time and a rotational delay. The seek time may be 0 to 20 or more milliseconds, and the rotational delay averages about half the rotation period. For a 7200 RPM drive, the rotation period is 8.33 milliseconds. For a drive such as the Seagate ST3500320NS, the track-to-track seek time is 0.8 milliseconds and the average reading seek time is 8.5 milliseconds.[4] For simplicity, assume reading from disk takes about 10 milliseconds.
Naively, then, the time to locate one record out of a million would take 20 disk reads times 10 milliseconds per disk read, which is 0.2 seconds.
The time won't be that bad because individual records are grouped together in a disk block. A disk block might be 16 kilobytes. If each record is 160 bytes, then 100 records could be stored in each block. The disk read time above was actually for an entire block. Once the disk head is in position, one or more disk blocks can be read with little delay. With 100 records per block, the last 6 or so comparisons don't need to do any disk reads—the comparisons are all within the last disk block read.
To speed the search further, the first 13 to 14 comparisons (which each required a disk access) must be sped up.
An index speeds the search
A significant improvement can be made with an index. In the example above, initial disk reads narrowed the search range by a factor of two. That can be improved substantially by creating an auxiliary index that contains the first record in each disk block (sometimes called a sparse index). This auxiliary index would be 1% of the size of the original database, but it can be searched more quickly. Finding an entry in the auxiliary index would tell us which block to search in the main database; after searching the auxiliary index, we would have to search only that one block of the main database—at a cost of one more disk read. The index would hold 10,000 entries, so it would take at most 14 comparisons. Like the main database, the last six or so comparisons in the auxiliary index would be on the same disk block. The index could be searched in about eight disk reads, and the desired record could be accessed in 9 disk reads.
The trick of creating an auxiliary index can be repeated to make an auxiliary index to the auxiliary index. That would make an aux-aux index that would need only 100 entries and would fit in one disk block.
Instead of reading 14 disk blocks to find the desired record, we only need to read 3 blocks. This blocking is the core idea behind the creation of the B-tree, where the disk blocks fill-out a hierarchy of levels to make up the index. Reading and searching the first (and only) block of the aux-aux index which is the root of the tree identifies the relevant block in aux-index in the level below. Reading and searching that aux-index block identifies the relevant block to read, until the final level, known as the leaf level, identifies a record in the main database. Instead of 150 milliseconds, we need only 30 milliseconds to get the record.
The auxiliary indices have turned the search problem from a binary search requiring roughly log2 N disk reads to one requiring only logb N disk reads where b is the blocking factor (the number of entries per block: b = 100 entries per block in our example; log100 1,000,000 = 3 reads).
In practice, if the main database is being frequently searched, the aux-aux index and much of the aux index may reside in a disk cache, so they would not incur a disk read.
Insertions and deletions
If the database does not change, then compiling the index is simple to do, and the index need never be changed. If there are changes, then managing the database and its index becomes more complicated.
Deleting records from a database is relatively easy. The index can stay the same, and the record can just be marked as deleted. The database remains in sorted order. If there are a large number of deletions, then searching and storage become less efficient.
Insertions can be very slow in a sorted sequential file because room for the inserted record must be made. Inserting a record before the first record requires shifting all of the records down one. Such an operation is just too expensive to be practical. One solution is to leave some spaces. Instead of densely packing all the records in a block, the block can have some free space to allow for subsequent insertions. Those spaces would be marked as if they were "deleted" records.
Both insertions and deletions are fast as long as space is available on a block. If an insertion won't fit on the block, then some free space on some nearby block must be found and the auxiliary indices adjusted. The hope is that enough space is available nearby, such that a lot of blocks do not need to be reorganized. Alternatively, some out-of-sequence disk blocks may be used.
Advantages of B-tree usage for databases
The B-tree uses all of the ideas described above. In particular, a B-tree:
• keeps keys in sorted order for sequential traversing
• uses a hierarchical index to minimize the number of disk reads
• uses partially full blocks to speed insertions and deletions
• keeps the index balanced with a recursive algorithm
In addition, a B-tree minimizes waste by making sure the interior nodes are at least half full. A B-tree can handle an arbitrary number of insertions and deletions.
Technical description
Terminology
The literature on B-trees is not uniform in its terminology (Folk & Zoellick 1992, p. 362).
Bayer & McCreight (1972), Comer (1979), and others define the order of B-tree as the minimum number of keys in a non-root node. Folk & Zoellick (1992) points out that terminology is ambiguous because the maximum number of keys is not clear. An order 3 B-tree might hold a maximum of 6 keys or a maximum of 7 keys. Knuth (1998, p. 483) avoids the problem by defining the order to be maximum number of children (which is one more than the maximum number of keys).
The term leaf is also inconsistent. Bayer & McCreight (1972) considered the leaf level to be the lowest level of keys, but Knuth considered the leaf level to be one level below the lowest keys (Folk & Zoellick 1992, p. 363). There are many possible implementation choices. In some designs, the leaves may hold the entire data record; in other designs, the leaves may only hold pointers to the data record. Those choices are not fundamental to the idea of a B-tree.[5]
For simplicity, most authors assume there are a fixed number of keys that fit in a node. The basic assumption is the key size is fixed and the node size is fixed. In practice, variable length keys may be employed (Folk & Zoellick 1992, p. 379).
Definition
According to Knuth's definition, a B-tree of order m is a tree which satisfies the following properties:
1. Every node has at most m children.
2. Every non-leaf node (except root) has at least ⌈m/2⌉ child nodes.
3. The root has at least two children if it is not a leaf node.
4. A non-leaf node with k children contains k − 1 keys.
5. All leaves appear in the same level and carry no information.
Each internal node’s keys act as separation values which divide its subtrees. For example, if an internal node has 3 child nodes (or subtrees) then it must have 2 keys: a1 and a2. All values in the leftmost subtree will be less than a1, all values in the middle subtree will be between a1 and a2, and all values in the rightmost subtree will be greater than a2.
Internal nodes
Internal nodes are all nodes except for leaf nodes and the root node. They are usually represented as an ordered set of elements and child pointers. Every internal node contains a maximum of U children and a minimum of L children. Thus, the number of elements is always 1 less than the number of child pointers (the number of elements is between L−1 and U−1). U must be either 2L or 2L−1; therefore each internal node is at least half full. The relationship between U and L implies that two half-full nodes can be joined to make a legal node, and one full node can be split into two legal nodes (if there’s room to push one element up into the parent). These properties make it possible to delete and insert new values into a B-tree and adjust the tree to preserve the B-tree properties.
The root node
The root node’s number of children has the same upper limit as internal nodes, but has no lower limit. For example, when there are fewer than L−1 elements in the entire tree, the root will be the only node in the tree with no children at all.
Leaf nodes
In Knuth's terminology, leaf nodes do not carry any information. The internal nodes that are one level above the leaves are what would be called "leaves" by other authors: these nodes only store keys (at most m-1, and at least m/2-1 if they are not the root) and pointers to nodes carrying no information.
A B-tree of depth n+1 can hold about U times as many items as a B-tree of depth n, but the cost of search, insert, and delete operations grows with the depth of the tree. As with any balanced tree, the cost grows much more slowly than the number of elements.
Some balanced trees store values only at leaf nodes, and use different kinds of nodes for leaf nodes and internal nodes. B-trees keep values in every node in the tree except leaf nodes.
Best case and worst case heights
Let h ≥ –1 be the height of the classic B-tree (see Tree (data structure) § Terminology for the tree height definition). Let n ≥ 0 be the number of entries in the tree. Let m be the maximum number of children a node can have. Each node can have at most m−1 keys.
It can be shown (by induction for example) that a B-tree of height h with all its nodes completely filled has n = mh+1–1 entries. Hence, the best case height (i.e. the minimum height) of a B-tree is:
${\displaystyle h_{\mathrm {min} }=\lceil \log _{m}(n+1)\rceil -1}$
Let ${\displaystyle d}$ be the minimum number of children an internal (non-root) node can have. For an ordinary B-tree, ${\displaystyle d=\left\lceil m/2\right\rceil .}$
Comer (1979, p. 127) and Cormen et al. (2001, pp. 383–384) give the worst case height (the maximum height) of a B-tree as
${\displaystyle h_{\mathrm {max} }=\left\lfloor \log _{d}{\frac {n+1}{2}}\right\rfloor .}$
Algorithms
Search
Searching is similar to searching a binary search tree. Starting at the root, the tree is recursively traversed from top to bottom. At each level, the search reduces its field of view to the child pointer (subtree) whose range includes the search value. A subtree's range is defined by the values, or keys, contained in its parent node. These limiting values are also known as separation values.
Binary search is typically (but not necessarily) used within nodes to find the separation values and child tree of interest.
Insertion
A B Tree insertion example with each iteration. The nodes of this B tree have at most 3 children (Knuth order 3).
All insertions start at a leaf node. To insert a new element, search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps:
1. If the node contains fewer than the maximum allowed number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered.
2. Otherwise the node is full, evenly split it into two nodes so:
1. A single median is chosen from among the leaf's elements and the new element.
2. Values less than the median are put in the new left node and values greater than the median are put in the new right node, with the median acting as a separation value.
3. The separation value is inserted in the node's parent, which may cause it to be split, and so on. If the node has no parent (i.e., the node was the root), create a new root above this node (increasing the height of the tree).
If the splitting goes all the way up to the root, it creates a new root with a single separator value and two children, which is why the lower bound on the size of internal nodes does not apply to the root. The maximum number of elements per node is U−1. When a node is split, one element moves to the parent, but one element is added. So, it must be possible to divide the maximum number U−1 of elements into two legal nodes. If this number is odd, then U=2L and one of the new nodes contains (U−2)/2 = L−1 elements, and hence is a legal node, and the other contains one more element, and hence it is legal too. If U−1 is even, then U=2L−1, so there are 2L−2 elements in the node. Half of this number is L−1, which is the minimum number of elements allowed per node.
An improved algorithm[citation needed] supports a single pass down the tree from the root to the node where the insertion will take place, splitting any full nodes encountered on the way. This prevents the need to recall the parent nodes into memory, which may be expensive if the nodes are on secondary storage. However, to use this improved algorithm, we must be able to send one element to the parent and split the remaining U−2 elements into two legal nodes, without adding a new element. This requires U = 2L rather than U = 2L−1, which accounts for why some textbooks impose this requirement in defining B-trees.
Deletion
There are two popular strategies for deletion from a B-tree.
1. Locate and delete the item, then restructure the tree to retain its invariants, OR
2. Do a single pass down the tree, but before entering (visiting) a node, restructure the tree so that once the key to be deleted is encountered, it can be deleted without triggering the need for any further restructuring
The algorithm below uses the former strategy.
There are two special cases to consider when deleting an element:
1. The element in an internal node is a separator for its child nodes
2. Deleting an element may put its node under the minimum number of elements and children
The procedures for these cases are in order below.
Deletion from a leaf node
1. Search for the value to delete.
2. If the value is in a leaf node, simply delete it from the node.
3. If underflow happens, rebalance the tree as described in section "Rebalancing after deletion" below.
Deletion from an internal node
Each element in an internal node acts as a separation value for two subtrees, therefore we need to find a replacement for separation. Note that the largest element in the left subtree is still less than the separator. Likewise, the smallest element in the right subtree is still greater than the separator. Both of those elements are in leaf nodes, and either one can be the new separator for the two subtrees. Algorithmically described below:
1. Choose a new separator (either the largest element in the left subtree or the smallest element in the right subtree), remove it from the leaf node it is in, and replace the element to be deleted with the new separator.
2. The previous step deleted an element (the new separator) from a leaf node. If that leaf node is now deficient (has fewer than the required number of nodes), then rebalance the tree starting from the leaf node.
Rebalancing after deletion
Rebalancing starts from a leaf and proceeds toward the root until the tree is balanced. If deleting an element from a node has brought it under the minimum size, then some elements must be redistributed to bring all nodes up to the minimum. Usually, the redistribution involves moving an element from a sibling node that has more than the minimum number of nodes. That redistribution operation is called a rotation. If no sibling can spare an element, then the deficient node must be merged with a sibling. The merge causes the parent to lose a separator element, so the parent may become deficient and need rebalancing. The merging and rebalancing may continue all the way to the root. Since the minimum element count doesn't apply to the root, making the root be the only deficient node is not a problem. The algorithm to rebalance the tree is as follows:[citation needed]
• If the deficient node's right sibling exists and has more than the minimum number of elements, then rotate left
1. Copy the separator from the parent to the end of the deficient node (the separator moves down; the deficient node now has the minimum number of elements)
2. Replace the separator in the parent with the first element of the right sibling (right sibling loses one node but still has at least the minimum number of elements)
3. The tree is now balanced
• Otherwise, if the deficient node's left sibling exists and has more than the minimum number of elements, then rotate right
1. Copy the separator from the parent to the start of the deficient node (the separator moves down; deficient node now has the minimum number of elements)
2. Replace the separator in the parent with the last element of the left sibling (left sibling loses one node but still has at least the minimum number of elements)
3. The tree is now balanced
• Otherwise, if both immediate siblings have only the minimum number of elements, then merge with a sibling sandwiching their separator taken off from their parent
1. Copy the separator to the end of the left node (the left node may be the deficient node or it may be the sibling with the minimum number of elements)
2. Move all elements from the right node to the left node (the left node now has the maximum number of elements, and the right node – empty)
3. Remove the separator from the parent along with its empty right child (the parent loses an element)
• If the parent is the root and now has no elements, then free it and make the merged node the new root (tree becomes shallower)
• Otherwise, if the parent has fewer than the required number of elements, then rebalance the parent
Note: The rebalancing operations are different for B+ trees (e.g., rotation is different because parent has copy of the key) and B*-tree (e.g., three siblings are merged into two siblings).
Sequential access
While freshly loaded databases tend to have good sequential behavior, this behavior becomes increasingly difficult to maintain as a database grows, resulting in more random I/O and performance challenges.[6]
Initial construction
A common special case is adding a large amount of pre-sorted data into an initially empty B-tree. While it is quite possible to simply perform a series of successive inserts, inserting sorted data results in a tree composed almost entirely of half-full nodes. Instead, a special "bulk loading" algorithm can be used to produce a more efficient tree with a higher branching factor.
When the input is sorted, all insertions are at the rightmost edge of the tree, and in particular any time a node is split, we are guaranteed that the no more insertions will take place in the left half. When bulk loading, we take advantage of this, and instead of splitting overfull nodes evenly, split them as unevenly as possible: leave the left node completely full and create a right node with zero keys and one child (in violation of the usual B-tree rules).
At the end of bulk loading, the tree is composed almost entirely of completely full nodes; only the rightmost node on each level may be less than full. Because those nodes may also be less than half full, to re-establish the normal B-tree rules, combine such nodes with their (guaranteed full) left siblings and divide the keys to produce two nodes at least half full. The only node which lacks a full left sibling is the root, which is permitted to be less than half full.
In filesystems
In addition to its use in databases, the B-tree (or § Variants) is also used in filesystems to allow quick random access to an arbitrary block in a particular file. The basic problem is turning the file block ${\displaystyle i}$ address into a disk block (or perhaps to a cylinder-head-sector) address.
Some operating systems require the user to allocate the maximum size of the file when the file is created. The file can then be allocated as contiguous disk blocks. In that case, to convert the file block address ${\displaystyle i}$ into a disk block address, the operating system simply adds the file block address ${\displaystyle i}$ to the address of the first disk block constituting the file. The scheme is simple, but the file cannot exceed its created size.
Other operating systems allow a file to grow. The resulting disk blocks may not be contiguous, so mapping logical blocks to physical blocks is more involved.
MS-DOS, for example, used a simple File Allocation Table (FAT). The FAT has an entry for each disk block,[note 1] and that entry identifies whether its block is used by a file and if so, which block (if any) is the next disk block of the same file. So, the allocation of each file is represented as a linked list in the table. In order to find the disk address of file block ${\displaystyle i}$ , the operating system (or disk utility) must sequentially follow the file's linked list in the FAT. Worse, to find a free disk block, it must sequentially scan the FAT. For MS-DOS, that was not a huge penalty because the disks and files were small and the FAT had few entries and relatively short file chains. In the FAT12 filesystem (used on floppy disks and early hard disks), there were no more than 4,080 [note 2] entries, and the FAT would usually be resident in memory. As disks got bigger, the FAT architecture began to confront penalties. On a large disk using FAT, it may be necessary to perform disk reads to learn the disk location of a file block to be read or written.
TOPS-20 (and possibly TENEX) used a 0 to 2 level tree that has similarities to a B-tree[citation needed]. A disk block was 512 36-bit words. If the file fit in a 512 (29) word block, then the file directory would point to that physical disk block. If the file fit in 218 words, then the directory would point to an aux index; the 512 words of that index would either be NULL (the block isn't allocated) or point to the physical address of the block. If the file fit in 227 words, then the directory would point to a block holding an aux-aux index; each entry would either be NULL or point to an aux index. Consequently, the physical disk block for a 227 word file could be located in two disk reads and read on the third.
Apple's filesystem HFS+, Microsoft's NTFS,[7] AIX (jfs2) and some Linux filesystems, such as btrfs and Ext4, use B-trees.
B*-trees are used in the HFS and Reiser4 file systems.
DragonFly BSD's HAMMER file system uses a modified B+-tree.[8]
Variations
Access concurrency
Lehman and Yao[9] showed that all the read locks could be avoided (and thus concurrent access greatly improved) by linking the tree blocks at each level together with a "next" pointer. This results in a tree structure where both insertion and search operations descend from the root to the leaf. Write locks are only required as a tree block is modified. This maximizes access concurrency by multiple users, an important consideration for databases and/or other B-tree-based ISAM storage methods. The cost associated with this improvement is that empty pages cannot be removed from the btree during normal operations. (However, see [10] for various strategies to implement node merging, and source code at.[11])
United States Patent 5283894, granted in 1994, appears to show a way to use a 'Meta Access Method' [12] to allow concurrent B+ tree access and modification without locks. The technique accesses the tree 'upwards' for both searches and updates by means of additional in-memory indexes that point at the blocks in each level in the block cache. No reorganization for deletes is needed and there are no 'next' pointers in each block as in Lehman and Yao.
Etymology
Rudolf Bayer and Ed McCreight invented the B-tree while working at Boeing Research Labs in 1971 (Bayer & McCreight 1972), but did not explain what, if anything, the B stands for: Boeing, balanced, broad, bushy, and Bayer have been suggested.(Comer 1979, p. 123 footnote 1)[13][14] McCreight has said that "the more you think about what the B in B-trees means, the better you understand B-trees."[13]
Notes
1. ^ For FAT, what is called a "disk block" here is what the FAT documentation calls a "cluster", which is fixed-size group of one or more contiguous whole physical disk sectors. For the purposes of this discussion, a cluster has no significant difference from a physical sector.
2. ^ Two of these were reserved for special purposes, so only 4078 could actually represent disk blocks (clusters).
References
1. ^ a b c Comer, Douglas (June 1979). "The Ubiquitous B-Tree". Computing Surveys. 11 (2): 123–137. doi:10.1145/356770.356776.
2. Tomašević, Milo (2008). Algorithms and Data Structures. Belgrade, Serbia: Akademska misao. pp. 274–275. ISBN 978-86-7466-328-8.
3. ^ Counted B-Trees, retrieved 2010-01-25
4. ^ Seagate Technology LLC, Product Manual: Barracuda ES.2 Serial ATA, Rev. F., publication 100468393, 2008 [1], page 6
5. ^ Bayer & McCreight (1972) avoided the issue by saying an index element is a (physically adjacent) pair of (xa) where x is the key, and a is some associated information. The associated information might be a pointer to a record or records in a random access, but what it was didn't really matter. Bayer & McCreight (1972) states, "For this paper the associated information is of no further interest."
6. ^ "Cache Oblivious B-trees". State University of New York (SUNY) at Stony Brook. Retrieved 2011-01-17.
7. ^ Mark Russinovich. "Inside Win2K NTFS, Part 1". Microsoft Developer Network. Archived from the original on 13 April 2008. Retrieved 2008-04-18.
8. ^ Matthew Dillon (2008-06-21). "The HAMMER Filesystem" (PDF).
9. ^ "Efficient locking for concurrent operations on B-trees". Portal.acm.org. doi:10.1145/319628.319663. Retrieved 2012-06-28.
11. ^
12. ^
13. ^ a b Weiner, Peter G. (30 August 2013). "4- Edward M McCreight" – via Vimeo.
14. ^ "Stanford Center for Professional Development". scpd.stanford.edu.
General
Original papers
• Bayer, Rudolf; McCreight, E. (July 1970), Organization and Maintenance of Large Ordered Indices, Mathematical and Information Sciences Report No. 20, Boeing Scientific Research Laboratories.
• Bayer, Rudolf (1971), Binary B-Trees for Virtual Memory, Proceedings of 1971 ACM-SIGFIDET Workshop on Data Description, Access and Control, San Diego, California. | 2019-04-22 19:56:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 22, "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": 0.5186831951141357, "perplexity": 1163.0206493350636}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578582584.59/warc/CC-MAIN-20190422195208-20190422220141-00038.warc.gz"} |
http://ncatlab.org/nlab/show/Demazure%2C+lectures+on+p-divisible+groups%2C+IV.1%2C+isogenies | # nLab Demazure, lectures on p-divisible groups, IV.1, isogenies
This entry is about a section of the text
Unless otherwise stated let $k$ be a perfect field of prime characteristic.
We denote write $B(K):=Quot(W(k))$ for the quotient field of the Witt ring $W(k)$. We extend the Frobenius morphism $x\mapsto x^{(p)}$ to an automorphism of $B(k)$. The set of fixed points of $x\mapsto x^{(p)}$ in $W(k)$ is $W(F_p)=\mathbb{Z}_p$. The set of fixed points of $x\mapsto x^{(p)}$ in $B(k)$ is $B(F_p)=\mathbb{Q}_p$.
Created on May 28, 2012 00:33:54 by Stephan Alexander Spahn (79.227.178.105) | 2015-10-10 18:08:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "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": 0.9744553565979004, "perplexity": 323.4248208920114}, "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-2015-40/segments/1443737961332.92/warc/CC-MAIN-20151001221921-00014-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://www.xaprb.com/blog/2007/12/06/things-i-love-about-perl/ | # Things I love about Perl
I don’t love everything about Perl, but I love its sense of humor, which I think probably comes from its creators’ senses of humor. From the Perl function documentation for redo:
“last”, “next”, or “redo” may appear within a “continue” block. “last” and “redo” will behave as if they had been executed within the main block. So will “next”, but since it will execute a “continue” block, it may be more entertaining.
“Entertaining,” in this context, means “if we were omniscient and looking over your shoulder while you spend a day debugging your occasional infinite loop or other bizarre behavior, we would be wildly entertained.”
At least that’s how I read it.
Sometimes the sense of humor, especially when imitated by neophytes trying to pretend to be part of The Gang Of Perl Greats, degrades into obnoxious sarcasm that obscures rather than documents. But this is fairly rare in the core documentation or other writings from the language’s authors.
If you’ve never read Programming Perl, you’re missing out on a lot of extremely subtle, very sharp and intelligent wit. I don’t have my copy of the book at hand, but one joke that comes to mind is how to write the Lord of the Rings trilogy with a regular expression substitution:
($lotr =$hobbit) =~ s/bilbo/frodo/g;
Or something like that. There are many fun examples that manage to teach the matter at hand more clearly, and keep me engaged more, than even the clearest straightforward explanation could.
Often imitated, but never equaled in any other book I’ve read. For example, I tried to read Extreme Programming Refactored (I really really tried, honest!) but could not make it through. I found myself getting irritated and wanting them to get to the point.
When Larry Wall et al make a joke about Gandalf, it is the point.
I'm Baron Schwartz, the founder and CEO of VividCortex. I am the author of High Performance MySQL and many open-source tools for performance analysis, monitoring, and system administration. I contribute to various database communities such as Oracle, PostgreSQL, Redis and MongoDB. | 2016-08-27 06:21:53 | {"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": 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": 0.22355282306671143, "perplexity": 2396.1842125521034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982298551.3/warc/CC-MAIN-20160823195818-00208-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://www.markvanderloo.eu/yarb/page/2/ | ## gower 0.2.0 is on CRAN
A new version of R package gower has just been released on CRAN.
Thanks to our new contributor David Turner who was kind enough to provide a pull request, gower now also computes weighted gower distances.
From the NEWS file:
• gower_dist and gower_topn gain weight argument for weighted matching (thanks to David Turner)
• gower_dist and gower_topn gain ignore_case argument for automatic column matching.
• gower_dist now returns numeric(0) invisibly when there are no columns to compare.
• gower_topn now returns a list with empty matrices when there are no columns to compare.
• gower_topn now warns when n>nrow(y) and sets n=nrow(y)
• bugfix: comparing factors with characters would cause a crash (thanks to max Kuhn)
Compute Gower's distance (or similarity) coefficient between records. Compute the top-n matches between records. Core algorithms are executed in parallel on systems supporting OpenMP.
Posted in programming, R | | 1 Comment
## uRos2019: tutorials, keynote speakers, registration and call for papers!
The 7th use of R in Official Statistics conference is the event for all things R in the production and use of government statistics. The 7th installment of this conference will take place from 20 to 21 May 2019 at the National Institute of Statistics in Bucharest, Romania.
### Keynote Speakers
We are very proud to announce that we have two excellent keynote speakers.
• Julie Josse will talk about her work on theory and tools related to imputation and inference in the presence of missing data.
• Giulio Barcaroli will talk about 12 years of using R at ISTAT, the Italian Statistical Office.
Full abstracts can be found here
### Tutorials
The conference is preceded by three tutorials on Data Cleaning, Statistical Disclosure Control and Optimal Sampling Stratification.
### Call for papers
Yes, abstracts and papers are welcomed until 12 April 2019! You can contribute by sending an abstract in any of the following topics (relating to official statistics):
Sampling and estimation | R in organization | Data cleaning | R in production: data analysis | Methods for official statistics | Shiny applications | Time series | Report and GUI programming | R in production: automation | Big data | Dissemination and visualization
### Registration is open
You can now register by following instructions here.
## Add a static pdf vignette to an R package
Most vignettes are built when a package is built, but there are occasions where you just want to include a pdf. For example when you want to include a paper. Of course there is a package supporting this, but in this post I will show you how to do it yourself with ease.
The idea is very simple: vignettes can be in LaTeX, and it is possible to include pdf documents in LaTeX using the pdfpages package. So here's the step-by-step recipe:
1. If you do not already have it, create the vignettes folder in your package directory.
2. Put your static pdf there. Let's call it mypaper.pdf for now.
3. Create a .Rnw file with the following content.
\documentclass{article}
\usepackage{pdfpages}
%\VignetteIndexEntry{author2019mypaper}
\begin{document}
\includepdf[pages=-, fitpaper=true]{mypaper.pdf}
\end{document}
That's it.
Some notes.
1. This repo contains an example.
2. The option fitpaper=true is necessary because the Sweave package that is included when the vignette is built somehow causes the pages to rescale if it is not included.
3. If you post your package to CRAN, myfile.pdf will be deleted from the directory so it is not part of a binary download.
4. You can include errata or other notes, for example as follows:
\documentclass{article}
\usepackage{pdfpages}
%\VignetteIndexEntry{author2019mypaper}
\begin{document}
\includepdf[pages=-, fitpaper=true]{mypaper.pdf}
\newpage{}
\subsection*{Errata}
A few things were borked in the original publication, here
is a list of sto0pid things I did:
\begin{itemize}
\item{fubar 1}
\item{fubar 2}
\end{itemize}
\end{document}
Posted in programming, R | Tagged , , | 3 Comments
## The program for uRos2018 is online
The uRos2018 conference is aimed at professionals and academics who are involved in producing or consuming official (government) statistics.
We are happy to announce that we recently posted the full program of the 6th international conference on the use of R in official Statistics (uRos2018) on our website.
### In summary:
• Six tutorials in the areas of
• Data Cleaning
• Network Analyses
• Survey Estimation
• Data manipulation with data.table
• Analyzing spatial data
• Visualizing spatial data.
• Two keynote speakers:
• Alina Matei, professor of statistics at the University of Neuchatel and maintainer of the sampling package.
• Jeroen Ooms, R superprogrammer and maintainer of R and Rtools for Windows (UC Berkeley)
• Eleven sessions with contributed talks with five presentations from
all over the world.
• One session devoted to the results of a two-day unconf that is held prior to the conference.
• One social dinner 🙂
• Two journals will devote a special topic to the conference.
All the abstracts will be published online soon.
### Registration is still open
• You are welcome to register
## stringdist 0.9.5.1: now with C API
Version 0.9.5.1 of stringdist is on CRAN. The main new feature, with a huge thanks to our awesome new contributor Chris Muir, is that we made it easy to call stringdist functionality from your package's C or C++ code.
The main steps to get it done are:
1. Make sure to add stringdist to the Imports: and LinkingTo: fields in your DESRIPTION file
2. Add the #include <stringdist_api> to your C/C++ source file.
3. Start using stringdist from C!
Here's an example source file
#include <R.h>
#include <Rdefines.h>
#include <stringdist_api.h>
SEXP my_soundex(SEXP strings, SEXP useBytes){
Rprintf("\nWow, using 'stringdist' soundex encoding, from my own C code!\n");
return sd_soundex(strings, useBytes);
}
• The full API is desribed in a pdf file that is generated from doxygen that comes with the package. You can find it by typing ?stringdist_api on the R command line.
• A minimal example package that links to stringdist is available on GitHub
• A more sophisticated package with more elaborate examples can be found here: refinr (By Chris)
### Any other news?
A few fixes, and a couple of long-deprecated function arguments have finally been removed. Check out the NEWS file on CRAN for a complete overview.
Happy coding!
Posted in programming, R | Tagged , | 2 Comments
## The use of R in official statistics conference 2018
On September 12-14 the 6th international conference on the use of R in official statistics (#uRos2018) will take place at the Dutch National Statistical Office in Den Haag, the Netherlands. The conference is aimed at producers and users of official statistics from government, academia, and industry. The conference is modeled after the useR! conference and will consist of one day of tutorials (12th September 2018) followed by two days of conference (13, 14 September 2018). Topics include:
• Examples of applying R in statistical production.
• Examples of applying R in dissemination of statistics (visualisation, apps, reporting).
• Analyses of big data and/or application of machine learning for official statistics.
• Implementations of statistical methodology in the areas of sampling, editing, modelling and estimation, or disclosure control.
• R packages connecting R to other standard tools/technical standards
• Organisational and technical aspects of introducing R to the statistical office.
• Teaching R to users in the office
• Examples of accessing or using official statistics publications with R in other fields
Keynote speakers
We are very happy to announce that we confirmed two fantastic keynote speakers.
• Alina Matei is a professor of statistics at the University of Neuchatel and maintainer of the important sampling package.
• Jeroen Ooms is a postdoc at UC Berkeley, author of many infrastructural R packages and maintainer of R and Rtools for Windows.
Call for abstracts
The call for abstracts is open until 31 May. You can contribute to the conference by proposing a 20-minute talk, or a 3-hour tutorial. Also, authors have the opportunity to submit a paper for one of the two journals that will devote a special issue to the conference. Read all about it over here.
Pointers
• conference website
## Track changes in data with the lumberjack %>>%
So you are using this pipeline to have data treated by different functions in R. For example, you may be imputing some missing values using the simputation package. Let us first load the only realistic dataset in R
> data(retailers, package="validate")
size incl.prob staff turnover other.rev total.rev staff.costs total.costs profit vat
1 sc0 0.02 75 NA NA 1130 NA 18915 20045 NA
2 sc3 0.14 9 1607 NA 1607 131 1544 63 NA
3 sc3 0.14 NA 6886 -33 6919 324 6493 426 NA
This data is dirty with missings and full of errors. Let us do some imputations with simputation.
> out <- retailers %>%
+ impute_lm(other.rev ~ turnover) %>%
+ impute_median(other.rev ~ size)
>
size incl.prob staff turnover other.rev total.rev staff.costs total.costs profit vat
1 sc0 0.02 75 NA 6114.775 1130 NA 18915 20045 NA
2 sc3 0.14 9 1607 5427.113 1607 131 1544 63 NA
3 sc3 0.14 NA 6886 -33.000 6919 324 6493 426 NA
>
Ok, cool, we know all that. But what if you'd like to know what value was imputed with which method? That's where the lumberjack comes in.
The lumberjack operator is a pipe'[1] operator that allows you to track changes in data.
> library(lumberjack)
> retailers$id <- seq_len(nrow(retailers)) > out <- retailers %>>% + start_log(log=cellwise$new(key="id")) %>>%
+ impute_lm(other.rev ~ turnover) %>>%
+ impute_median(other.rev ~ size) %>>%
+ dump_log(stop=TRUE)
Dumped a log at cellwise.csv
>
step time expression key variable old new
1 2 2017-06-23 21:11:05 CEST impute_median(other.rev ~ size) 1 other.rev NA 6114.775
2 1 2017-06-23 21:11:05 CEST impute_lm(other.rev ~ turnover) 2 other.rev NA 5427.113
3 1 2017-06-23 21:11:05 CEST impute_lm(other.rev ~ turnover) 6 other.rev NA 6341.683
>
So, to track changes we only need to switch from %>% to %>>% and add the start_log() and dump_log() function calls in the data pipeline. (to be sure: it works with any function, not only with simputation). The package is on CRAN now, and please see the introductory vignette for more examples and ways to customize it.
There are many ways to track changes in data. That is why the lumberjack is completely extensible. The package comes with a few loggers, but users or package authors are invited to write their own. Please see the extending lumberjack vignette for instructions.
If this post got you interested, please install the package using
install.packages('lumberjack')
You can get started with the introductory vignette or even just use the lumberjack operator %>>% as a (close) replacement of the %>% operator.
As always, I am open to suggestions and comments. Either through the packages github page.
Also, I will be talking at useR2017 about the simputation package, but I will sneak in a bit of lumberjack as well :p.
And finally, here's a picture of a lumberjack smoking a pipe.
[1] It really should be called a function composition operator, but potetoes/potatoes.
## Announcing the simputation package: make imputation simple
I am happy to announce that my simputation package has appeared on CRAN this weekend. This package aims to simplify missing value imputation. In particular it offers standardized interfaces that
• make it easy to define both imputation method and imputation model;
• for multiple variables at once;
• while grouping data by categorical variables;
• all fitting in the magrittr not-a-pipeline.
### A few examples
To start with an example, let us first create a data set with some missings.
dat <- iris
# empty a few fields
dat[1:3,1] <- dat[3:7,2] <- dat[8:10,5] <- NA
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 NA 3.5 1.4 0.2 setosa
## 2 NA 3.0 1.4 0.2 setosa
## 3 NA NA 1.3 0.2 setosa
## 4 4.6 NA 1.5 0.2 setosa
## 5 5.0 NA 1.4 0.2 setosa
## 6 5.4 NA 1.7 0.4 setosa
## 7 4.6 NA 1.4 0.3 setosa
## 8 5.0 3.4 1.5 0.2 <NA>
## 9 4.4 2.9 1.4 0.2 <NA>
## 10 4.9 3.1 1.5 0.1 <NA>
Below, we first impute Sepal.Width and Sepal.Length by regression on Petal.Width and Species. After this we impute Species using a decision tree model (CART) using every other variable as a predictor (including the ones just imputed).
library(magrittr) # load the %>% operator
library(simputation)
imputed <- dat %>%
impute_lm(Sepal.Width + Sepal.Length ~ Petal.Width + Species) %>%
impute_cart(Species ~ .)
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 4.979844 3.500000 1.4 0.2 setosa
## 2 4.979844 3.000000 1.4 0.2 setosa
## 3 4.979844 3.409547 1.3 0.2 setosa
## 4 4.600000 3.409547 1.5 0.2 setosa
## 5 5.000000 3.409547 1.4 0.2 setosa
## 6 5.400000 3.561835 1.7 0.4 setosa
## 7 4.600000 3.485691 1.4 0.3 setosa
## 8 5.000000 3.400000 1.5 0.2 setosa
## 9 4.400000 2.900000 1.4 0.2 setosa
## 10 4.900000 3.100000 1.5 0.1 setosa
The package is pretty lenient against failure of imputation. For example, if one of the predictors is missing, fields just remain unimputed and if one of the models cannot be fitted, only a warning is issued (not shown here).
dat %>% impute_lm(Sepal.Length ~ Sepal.Width + Species) %>% head(3)
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 5.076579 3.5 1.4 0.2 setosa
## 2 4.675654 3.0 1.4 0.2 setosa
## 3 NA NA 1.3 0.2 setosa
So here, the third Sepal.Length value could not be imputed since the predictor Sepal.Width is missing.
It is possible to split data into groups before estimating the imputation model and predicting missing values. There are two ways. The first is to use the | operator to specify grouping variables.
# We first need to complete 'Species'. Here, we use sequential
# hot deck after sorting by Petal.Length
dat %<>% impute_shd(Species ~ Petal.Length)
# Now impute Sepal.Length by regressing on
# Sepal.Width, computing a model for each Species.
dat %>% impute_lm(Sepal.Length ~ Sepal.Width | Species) %>% head(3)
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1 5.067813 3.5 1.4 0.2 setosa
## 2 4.725677 3.0 1.4 0.2 setosa
## 3 NA NA 1.3 0.2 setosa
The second way is to use the group_by command from dplyr
dat %>% dplyr::group_by(Species) %>%
impute_lm(Sepal.Length ~ Sepal.Width) %>%
## Source: local data frame [3 x 5]
## Groups: Species [1]
##
## Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## <dbl> <dbl> <dbl> <dbl> <fctr>
## 1 5.067813 3.5 1.4 0.2 setosa
## 2 4.725677 3.0 1.4 0.2 setosa
## 3 NA NA 1.3 0.2 setosa
Note: by using group_by, we also transformed the data.frame to a tibble, which not only sounds funny when you pronounce it (tibble, TIBBLE, tibble? tibbebbebbebble) but is also pretty useful.
### Supported methods and how to specify them
Currently, the package supports the following methods:
• Model based (optionally add [non-]parametric random residual)
• linear regression
• robust linear regression
• CART models
• Random forest
• Donor imputation (including various donor pool specifications)
• k-nearest neigbour (based on gower's distance)
• sequential hotdeck (LOCF, NOCB)
• random hotdeck
• Predictive mean matching
• Other
• (groupwise) median imputation (optional random residual)
• Proxy imputation (copy from other variable)
Any call to one of the impute_ functions looks as follows:
impute_<method>(data, formula [, <method-specific options>])
and the formula always has the following form:
<imputed variables> ~ <model specification> [|<grouping variables>]
The parts in square brackets are optional.
Please see the package vignette for more examples and details, or ?simputation::impute_ for an overview of all imputation functions.
Happy imputing!
Posted in programming, R | | 5 Comments
## stringdist 0.9.4.2 released
stringdist 0.9.4.2 was accepted on CRAN at the end of last week.
This release just fixes a few bugs affecting the stringdistmatrix function, when called with a single argument.
From the NEWS file:
• bugfix in stringdistmatrix(a): value of p, for jw-distance was ignored (thanks to Max Fritsche)
• bugfix in stringdistmatrix(a): Would segfault on q-gram w/input > ~7k strings and q>1 (thanks to Connor McKay)
• bugfix in jaccard distance: distance not always correct when passing multiple strings (thanks to Robert Carlson)
Actually the last bug has not bitten anyone since it was masked by the second one 🙂 (it was reported and fixed a long time ago but popped up again after fixing the second bug -- hat tip to Hadley for testthat!). The second fix also ensures that stringdist's memory allocator for q-gram storage is called fewer times which yields a speed gain in computation of q-gram based distances.
Posted in programming, R | Tagged | 2 Comments
## validate version 0.1.5 is out
A new version of the validate package for data validation was just accepted on CRAN and will be available on all mirrors in a few days.
The most important addition is that you can now reference the data set as a whole, using the "dot" syntax like so:
iris %>% check_that(
nrow(.)>100
, "Sepal.Width" %in% names(.)) %>%
summary()
rule items passes fails nNA error warning expression
1 V1 1 1 0 0 FALSE FALSE nrow(.) > 100
2 V2 1 1 0 0 FALSE FALSE "Sepal.Width" %in% names(.)
Also, it is now possible to return a logical, even when the result is NA, by passing the na.value option.
dat = data.frame(x=c(1,NA,-1))
v = validator(x > 0)
values(confront(dat,v))
V1
[1,] TRUE
[2,] NA
[3,] FALSE
values(confront(dat,v,na.value=FALSE))
V1
[1,] TRUE
[2,] FALSE
[3,] FALSE
`
A complete list of changes and bugfixes can be found in the NEWS file. Below I include changes in 1.4 since I did not write about it before.
I will be talking about this package at the upcoming useR!2016 event, so join me if you're interested!
version 0.1.5
• The '.' is now used to reference the validated data set as whole.
• Small change in output of 'compare' to match the table in van den Broek et al. (2013)
version 0.1.4
• 'confront' now emits a warining when variable name conflicts with name of a reference data set
• Deprecated 'validate_reset', in favour of the shorter 'reset' (use 'validate::reset' in case of ambiguity)
• Deprecated 'validate_options' in favour of the shorter 'voptions'
• New option na.value with default value NA, controlling the output when a rule evaluates to NA.
• Added rules from the ESSnet on validation (deliverable 17) to automated tests.
• added 'grepl' to allowed validation syntax (suggested by Dusan Sovic)
• exported a few functions w/ keywords internal for extensibility
• Bugfix: blocks sometimes reported wrong nr of blocks (in case of a single connected block.)
• Bugfix: macro expansion failed when macros were reused in other macros.
• Bugfix: certain nonlinear relations were recognized as linear
• Bugfix: rules that use (anonymous) function definitions raised error when printed. | 2020-09-30 18:44:25 | {"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": 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": 0.2564741373062134, "perplexity": 8513.022498695056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402127397.84/warc/CC-MAIN-20200930172714-20200930202714-00672.warc.gz"} |
https://physics.stackexchange.com/questions/176628/lorentz-invariance-vs-covariance | # Lorentz invariance vs. covariance
I am a bit confused whether relativistic theory is Lorentz invariant or covariant. And please explain why?
As I recall, covariant refers to how an object transforms when you boost to another inertial frame. An example would be the relativistic 4-momentum $P^{\mu}$. Invariant refers to quantities which are unchanged under boosts to different frames. For example the product $P^{\mu}P_{\mu}=m$ has the same numerical value in any frame. Sometimes a relativistic theory is called manifestly covariant if all of the terms in the equations are covariant. The theory is Lorentz invariant if the form of the equations are the same in any inertial frame related by Lorentz transformations. | 2020-04-02 23:09:04 | {"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": 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": 0.8468347191810608, "perplexity": 237.05936800536062}, "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/1585370508367.57/warc/CC-MAIN-20200402204908-20200402234908-00078.warc.gz"} |
https://notebook.community/AllenDowney/ThinkBayes2/examples/shuttle | # Think Bayes
In [1]:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'
import numpy as np
import pandas as pd
# import classes from thinkbayes2
from thinkbayes2 import Pmf, Cdf, Suite, Joint
import thinkplot
## The Space Shuttle problem
Here's a problem from Bayesian Methods for Hackers
On January 28, 1986, the twenty-fifth flight of the U.S. space shuttle program ended in disaster when one of the rocket boosters of the Shuttle Challenger exploded shortly after lift-off, killing all seven crew members. The presidential commission on the accident concluded that it was caused by the failure of an O-ring in a field joint on the rocket booster, and that this failure was due to a faulty design that made the O-ring unacceptably sensitive to a number of factors including outside temperature. Of the previous 24 flights, data were available on failures of O-rings on 23, (one was lost at sea), and these data were discussed on the evening preceding the Challenger launch, but unfortunately only the data corresponding to the 7 flights on which there was a damage incident were considered important and these were thought to show no obvious trend. The data are shown below (see 1):
In [2]:
# !wget https://raw.githubusercontent.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/master/Chapter2_MorePyMC/data/challenger_data.csv
In [3]:
columns = ['Date', 'Temperature', 'Incident']
df.drop(labels=[3, 24], inplace=True)
df
In [4]:
df['Incident'] = df['Damage Incident'].astype(float)
df
In [5]:
import matplotlib.pyplot as plt
plt.scatter(df.Temperature, df.Incident, s=75, color="k",
alpha=0.5)
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside temperature (Fahrenheit)")
plt.title("Defects of the Space Shuttle O-Rings vs temperature");
### Grid algorithm
We can solve the problem first using a grid algorithm, with parameters b0 and b1, and
$\mathrm{logit}(p) = b0 + b1 * T$
and each datum being a temperature T and a boolean outcome fail, which is true is there was damage and false otherwise.
Hint: the expit function from scipy.special computes the inverse of the logit function.
In [6]:
from scipy.special import expit
class Logistic(Suite, Joint):
def Likelihood(self, data, hypo):
"""
data: T, fail
hypo: b0, b1
"""
return 1
In [7]:
# Solution goes here
In [8]:
b0 = np.linspace(0, 50, 101);
In [9]:
b1 = np.linspace(-1, 1, 101);
In [10]:
from itertools import product
hypos = product(b0, b1)
In [11]:
suite = Logistic(hypos);
In [12]:
for data in zip(df.Temperature, df.Incident):
print(data)
suite.Update(data)
In [13]:
thinkplot.Pdf(suite.Marginal(0))
thinkplot.decorate(xlabel='Intercept',
ylabel='PMF',
title='Posterior marginal distribution')
In [14]:
thinkplot.Pdf(suite.Marginal(1))
thinkplot.decorate(xlabel='Log odds ratio',
ylabel='PMF',
title='Posterior marginal distribution')
According to the posterior distribution, what was the probability of damage when the shuttle launched at 31 degF?
In [15]:
# Solution goes here
In [16]:
# Solution goes here
### MCMC
Implement this model using MCMC. As a starting place, you can use this example from the PyMC3 docs.
As a challege, try writing the model more explicitly, rather than using the GLM module.
In [17]:
import pymc3 as pm
In [23]:
# Solution goes here
In [24]:
pm.traceplot(trace);
The posterior distributions for these parameters should be similar to what we got with the grid algorithm.
In [ ]: | 2021-08-03 02:08:40 | {"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": 0.5470505952835083, "perplexity": 3972.79393081951}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154408.7/warc/CC-MAIN-20210802234539-20210803024539-00030.warc.gz"} |
https://www.brickodyssey.com/category/design/ | Lego technic techniques
When building some technic MOC, I find that switching orientation is a part that is time consuming and very ad-hoc. Switching orientation is the equivalent of SNOT (Studs Not On Top) for technic build (I don’t know of a more official term). I usually end up randomly trying the connector du jour until getting something that works. It’s now time for a more rigorous approach.
One of the greatest books about lego technics (The Unofficial LEGO Technic Builder’s Guide) does contain a wonderful amount of information, but not much about studless orientations connections. It’s probably too evident for builder with this amount of experience.
How many orientations are there?
Before figuring out the connections, we need to figure out how many relative position we have. Let’s consider the relative position of 2 studless technic beams, we will have the beam of reference in red and the one we’d like to attach in black.
Since we are ignoring translations for now, we can enumerate this in term of rotations. In 3D we can code rotations using Euler angles. There are many choices in term of conventions, here we’re going to stick with the closest to some modeling software like stud.io. In stud.io, you can rotate around the Y axis using up/down and around the Z axis using left/right. The rotation are extrinsic: they use the scene axis, not the brick axis. With this, we chose to use Euler extrinsic rotations given as z-y-z.
Taking into account the symmetry of the beams and given that we’re only interested in perpendicular constructs, we have 8 possible rotations:
• $(0,0,0)$
• $(0,0,\pi/2)$
• $(0,\pi/2,0)$
• $(0,\pi/2,\pi/2)$
• $(\pi/2,0,0)$
• $(\pi/2,0,\pi/2)$
• $(\pi/2,\pi/2,0)$
• $(\pi/2,\pi/2,\pi/2)$
However, of these 8 rotations, we can eliminate a few of them
$(0,0,0)$ $(0,0,\pi/2)$ $(0,\pi/2,0)$ $(0,\pi/2,\pi/2)$ $(\pi/2,0,0)$ just a rotation around z, it’s the same as $(0,0,\pi/2)$ $(\pi/2,0,\pi/2)$ this is a rotation around z of $pi$, for beam this the same as $(0,0,0)$ $(\pi/2,\pi/2,0)$ Note the color here, this is to highlight that this is equivalent to the $(0,\pi/2,\pi/2)$. $(\pi/2,\pi/2,\pi/2)$ That’s a rotation around the X axis.
After that, we are left with:
• $(0,0,0)$
• $(0,0,\pi/2)$
• $(0,\pi/2,0)$
• $(0,\pi/2,\pi/2)$
• $(\pi/2,\pi/2,\pi/2)$
So we have 5 possible relative orientations. Now, we can start enumerating different way to connect those. I’ve been trying to give them some names, but those are not the best…
No rotation: $(0,0,0)$
That’s the most basic one, and usually it’s not a problem to set up this one.
Support beam: $(0,0,\pi/2)$
This one is very useful to provide strength: for example to prevent several parallel beams from separating.
The next connection can be found on the Mine Loader (42049) it’s pretty useful to serve as guide for axles, but not too strong for torsion.
Linkage rotation: $(0,\pi/2,0)$
Basic connection to build any linkage:
If you don’t need the mobility, it’s better to use the L-shape pieces:
Conclusion
I think that’s a good start: let’s see in practice how much it helps. I’ll definitely be adding some more combination here as I discover them around.
Lego table corner (3d printed)
I use particle boards as Lego tables: the advantage is that they can be easily cut at custom dimensions (read: integer multiple of studs, or even integer multiple of baseplates). This is particularly useful when setting up a Lego train around the room. The disadvantage is that the corners are quite sharp and painful for the head (or back). Also, the baseplates might slide on the board.
Enter the custom table corner. This is a 3D printed corner, just at the right dimensions, designed to slide into the baseplate studs: holding both the baseplate and the corner in place.
Here are a few pictures:
Corner with space for the studs
Corner holding the baseplace in place
Corner in place on the table
This shape can be generated using the following solid python code (also available on github):
import math
from solid import *
from solid.utils import *
# size in mm
corner_height = 30. + 2
corner_size = 43.
wall = 5
table = cube([40,40, 22], center=True)
stud_height = 1.6 + 0.3
stud_diam = 4.8 + 0.3
for i in range(-2,3):
offset = i*8. + 0.5
dig = translate([offset, 0, 11 ])(
cube([stud_diam, 40, stud_height * 2], center=True))
table += dig
corner = translate([0,0,0.5])(minkowski()(
cube([corner_size-10, corner_size-10, corner_height/2], center=True),
cylinder(r=10, h=corner_height/2., center=True)))
size_slant = 100.
offset = size_slant/(2*math.sqrt(2.))
slant = translate([offset,offset,0])(
rotate(a=45, v=[0,0,1])(
cube([size_slant, size_slant, 40], center=True)))
side1 = translate([0,30,0])(
cube([100, 20, 40], center=True))
side2 = translate([30,0,0])(
cube([20, 100, 40], center=True))
slant += side1 + side2
c = corner - slant - table
# position the final product
final = translate([0,0,27.5])(rotate(a=90, v=[1,0,0])(c))
This generates the CAD file suitable for 3D printing. You can also find the file on thingiverse.
3D printing Lego bricks
tl;dr: no (well… mostly).
With 3D printing becoming affordable to do at home, the dream of every lego lover seems to come true. Need a 2×4 in purple? no problem, I’ll print it. Missing a 1×5? To the printer baby!
But life is not that easy.
First the cost. You need to buy the stuff you feed the printer: if you use the same plastic as Lego, that’s ABS. The ABS filament that you put in the printer cost about USD 25 to 30 per kg. As a comparison, you can buy real Lego bricks on Ebay for USD 15-20 per kg. So, if you want a lot of bricks, you can just get them second hand by bulk. Some might complain that second hand bricks might be scratched, but if you think 3D printing will be better, read on.
Then there is the quality. The Lego Group doesn’t 3D print the bricks, it uses injection molding machines. The tolerance on the bricks is about 2 micrometer. As a comparison, most home 3D printers are precise to about 0.1 mm, that’s 100 micrometer. About 50 times worse that standard Lego bricks. To give an idea of the scale, if you were scaling a 2×4 brick to get the same relative accuracy with your 3D printer, the brick would end up measuring about 1.6 meters long. That would be a lot of plastic (250 kg to be precise).
But, it might make sense in some cases: when you need a brick that doesn’t exist. For example a lego railroad crossing (available on thingiverse): | 2022-05-28 03:50:13 | {"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": 30, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46324750781059265, "perplexity": 1995.1382753056535}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00766.warc.gz"} |
http://server1.wikisky.org/starview?object_type=2&object_id=22479&object_name=VCC+1011&locale=EN | WIKISKY.ORG
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# PGC 40861
Contents
### Images
DSS Images Other Images
### Related articles
Completing H I observations of galaxies in the Virgo clusterHigh sensitivity (rms noise 0.5 mJy) 21-cm H I line observationswere made of 33 galaxies in the Virgo cluster, using the refurbishedArecibo telescope, which resulted in the detection of 12 objects. Thesedata, combined with the measurements available from the literature,provide the first set of H I data that is complete for all 355 late-type(Sa-Im-BCD) galaxies in the Virgo cluster with mp ≤ 18.0mag. The Virgo cluster H I mass function (HIMF) that was derived forthis optically selected galaxy sample is in agreement with the HIMFderived for the Virgo cluster from the blind HIJASS H I survey and isinconsistent with the Field HIMF. This indicates that both in this richcluster and in the general field, neutral hydrogen is primarilyassociated with late-type galaxies, with marginal contributions fromearly-type galaxies and isolated H I clouds. The inconsistency betweenthe cluster and the field HIMF derives primarily from the difference inthe optical luminosity function of late-type galaxies in the twoenvironments, combined with the HI deficiency that is known to occur ingalaxies in rich clusters.Tables \ref{t1, \ref{sample_dat} and Appendix A are only available inelectronic form at http://www.edpsciences.org Companions of Bright Barred Shapley-Ames GalaxiesCompanion galaxy environment for a subset of 78 bright and nearby barredgalaxies from the Shapley-Ames Catalog is presented. Among the spiralbarred galaxies, there are Seyfert galaxies, galaxies with circumnuclearstructures, galaxies not associated with any large-scale galaxy cloudstructure, galaxies with peculiar disk morphology (crooked arms), andgalaxies with normal disk morphology; the list includes all Hubbletypes. The companion galaxy list includes the number of companiongalaxies within 20 diameters, their Hubble type, and projectedseparation distance. In addition, the companion environment was searchedfor four known active spiral galaxies, three of them are Seyfertgalaxies, namely, NGC 1068, NGC 1097, and NGC 5548, and one is astarburst galaxy, M82. Among the results obtained, it is noted that theonly spiral barred galaxy classified as Seyfert 1 in our list has nocompanions within a projected distance of 20 diameters; six out of 10Seyfert 2 bar galaxies have no companions within 10 diameters, six outof 10 Seyfert 2 galaxies have one or more companions at projectedseparation distances between 10 and 20 diameters; six out of 12 galaxieswith circumnuclear structures have two or more companions within 20diameters. The Three-dimensional Structure of the Virgo Cluster Region from Tully-Fisher and H I DataThe distances and H I contents of 161 spiral galaxies in the region ofthe Virgo cluster are used to gain insight into the complicatedstructure of this galaxy system. Special attention has been paid to theinvestigation of the suggestion presented in an earlier work that someperipheral Virgo groups may contain strongly gas-deficient spiralgalaxies. The three-dimensional galaxy distribution has been inferredfrom quality distance estimates obtained by averaging distance modulibased on the Tully-Fisher relationship taken from eight published datasets previously homogenized, resulting in a relation with a dispersionof 0.41 mag. Previous findings that the spiral distribution issubstantially more elongated along the line of sight than in the planeof the sky are confirmed by the current data. In addition, an importanteast-west disparity in this effect has been detected. The overallwidth-to-depth ratio of the Virgo cluster region is about 1:4, with themost distant objects concentrated in the western half. The filamentarystructure of the spiral population and its orientation are alsoreflected by the H I-deficient objects alone. The H I deficiency patternshows a central enhancement extending from ~16 to 22 Mpc inline-of-sight distance; most of this enhancement arises from galaxiesthat belong to the Virgo cluster proper. However, significant gasdeficiencies are also detected outside the main body of the cluster in aprobable group of galaxies at line-of-sight distances ~25-30 Mpc, lyingin the region dominated by the southern edge of the M49 subcluster andclouds W' and W, as well as in various foreground galaxies. In the Virgoregion, the H I content of the galaxies then is not a straightforwardindicator of cluster membership. The UZC-SSRS2 Group CatalogWe apply a friends-of-friends algorithm to the combined Updated ZwickyCatalog and Southern Sky Redshift Survey to construct a catalog of 1168groups of galaxies; 411 of these groups have five or more members withinthe redshift survey. The group catalog covers 4.69 sr, and all groupsexceed the number density contrast threshold, δρ/ρ=80. Wedemonstrate that the groups catalog is homogeneous across the twounderlying redshift surveys; the catalog of groups and their membersthus provides a basis for other statistical studies of the large-scaledistribution of groups and their physical properties. The medianphysical properties of the groups are similar to those for groupsderived from independent surveys, including the ESO Key Programme andthe Las Campanas Redshift Survey. We include tables of groups and theirmembers. Hα surface photometry of galaxies in the Virgo cluster. IV. The current star formation in nearby clusters of galaxiesHα +[NII] imaging observations of 369 late-type (spiral) galaxiesin the Virgo cluster and in the Coma/A1367 supercluster are analyzed,covering 3 rich nearby clusters (A1367, Coma and Virgo) and nearlyisolated galaxies in the Great-Wall. They constitute an opticallyselected sample (mp<16.0) observed with ~ 60 %completeness. These observations provide us with the current(T<107 yrs) star formation properties of galaxies that westudy as a function of the clustercentric projected distances (Theta ).The expected decrease of the star formation rate (SFR), as traced by theHα EW, with decreasing Theta is found only when galaxies brighterthan Mp ~ -19.5 are considered. Fainter objects show no orreverse trends. We also include in our analysis Near Infrared data,providing information on the old (T>109 yrs) stars. Puttogether, the young and the old stellar indicators give the ratio ofcurrently formed stars over the stars formed in the past, orbirthrate'' parameter b. For the considered galaxies we also determinethe global gas content'' combining HI with CO observations. We definethe gas deficiency'' parameter as the logarithmic difference betweenthe gas content of isolated galaxies of a given Hubble type and themeasured gas content. For the isolated objects we find that b decreaseswith increasing NIR luminosity. In other words less massive galaxies arecurrently forming stars at a higher rate than their giant counterpartswhich experienced most of their star formation activity at earliercosmological epochs. The gas-deficient objects, primarily members of theVirgo cluster, have a birthrate significantly lower than the isolatedobjects with normal gas content and of similar NIR luminosity. Thisindicates that the current star formation is regulated by the gaseouscontent of spirals. Whatever mechanism (most plausibly ram-pressurestripping) is responsible for the pattern of gas deficiency observed inspiral galaxies members of rich clusters, it also produces the observedquenching of the current star formation. A significant fraction of gashealthy'' (i.e. with a gas deficiency parameter less than 0.4) andcurrently star forming galaxies is unexpectedly found projected near thecenter of the Virgo cluster. Their average Tully-Fisher distance isfound approximately one magnitude further away (muo = 31.77)than the distance of their gas-deficient counterparts (muo =30.85), suggesting that the gas healthy objects belong to a cloudprojected onto the cluster center, but in fact lying a few Mpc behindVirgo, thus unaffected by the dense IGM of the cluster. Based onobservations taken at the Observatorio Astronómico Nacional(Mexico), the OHP (France), Calar Alto and NOT (Spain) observatories.Table \ref{tab4} is only available in electronic form athttp://www.edpsciences.org Hα surface photometry of galaxies in the Virgo cluster I. Observations with the San Pedro Martir 2.1 m telescopeHα imaging observations of 125 galaxies obtained with the 2.1 mtelescope of the San Pedro Martir Observatory (SPM) (Baja California,Mexico) are presented. The observed galaxies are mostly Virgo clustermembers (77), with 36 objects in the Coma/A1367 supercluster and 12 inthe clusters A2197 and A2199 taken as fillers. Hα +[NII] fluxesand equivalent widths, as well as images of the detected targets arepresented. The observatory of San Pedro Martir (Mexico) belongs to theObservatorio Astronómico Nacional, UNAM. Figure 4 is onlyavailable in electronic form at http://www.edpsciences.org 1.65 μm (H-band) surface photometry of galaxies. VII. Dwarf galaxies in the Virgo ClusterWe present near-infrared H-band (1.65 μm) observations and surfacebrightness profile decompositions for 75 faint (13.5 <~ mp<~ 18.5) galaxies, primarily taken among dwarf Ellipticals members ofthe Virgo cluster, with some Centaurus Cluster members, a BCD and twopeculiar galaxies taken as fillers. We model their surface brightnessprofiles with a de Vaucouleurs (D), exponential (E), mixed (bulge+diskor M) or truncated (T) law, and we derive for each galaxy the H bandeffective surface brightness (μe) and effective radius(re), the asymptotic total magnitude HT and thelight concentration index C31, defined as the ratio betweenthe radii that enclose 75% and 25% of the total light HT. Fora subsample we compare the NIR surface photometry with similar datataken in the B and V bands, and we give the B-H and B-V color profiles.Combining the present data with those previously obtained by our group(1157 objects) we analyze the NIR properties of a nearly completesample, representative of galaxies of all morphological types, spanning4 decades in luminosity. We confirm our earlier claim that the presenceof cusps and extended haloes in the light profiles (C31>5)is a strong, non-linear function of the total luminosity. We also findthat: i) among dE and dS0 galaxies D profiles are absent; 50% of thedecompositions are of type M, the remaining being of type E or T. ii)Less than 50% of the giant elliptical galaxies have pure D profiles, themajority being represented by M profiles. iii) Most giant galaxies (fromelliptical to Sb) have M profiles. iv) Most of late type spirals (Scd toBCD) have either E or T profiles. v) The type of decomposition is astrong function of the total H band luminosity, independent of theHubble classification: the fraction of type E decompositions decreaseswith increasing luminosity, while those of type M increase withluminosity. Pure D profiles are absent in the low luminosity rangeLH<1010 Lsolar and become dominantabove 1011 Lsolar, while T profiles are presentonly among low luminosity galaxies. vi) We find that dE-peculiargalaxies have structural parameters indistinguishable from those oflate-type dwarfs, thus they might represent the missing link between dEsand dIs. Based on observations taken with the ESO/NTT (ESO program64.N-0288), with the Telescopio Nazionale Galileo (TNG) operated on theisland of La Palma by the Centro Galileo Galilei of the CNAA at theSpanish Observatorio del Roque de los Muchachos of the IAC, with the SanPedro Martir 2.1~m telescope of the Observatorio Astronomico Nacional(OAN, Mexico), and with the OHP 1.2~m telescope, operated by the FrenchCNRS. Distances to 24 Galaxies in the Direction of the Virgo Cluster and a Determination of the Hubble ConstantTo study the spatial distribution of galaxies lying between the Virgocluster and the Local Group, a search was made for probable nearbygalaxies. Using the method of brightest stars and of blue and redsupergiants made it possible to determine the distances to 24 galaxies,among which six relatively nearby galaxies were identified. The resultsof the distance determinations showed that the maximum in the numberdistribution of galaxies lies at 17.0 Mpc, which we take as the distanceto the Virgo cluster. Using the difference between the distance moduliof two clusters of galaxies, in Virgo and Coma Berenices, fromliterature sources and the velocity of the latter cluster, we determinedthe Hubble constant to be H 0 = 77 ± 7km·sec-1·Mpc-1. Arcsecond Positions of UGC GalaxiesWe present accurate B1950 and J2000 positions for all confirmed galaxiesin the Uppsala General Catalog (UGC). The positions were measuredvisually from Digitized Sky Survey images with rms uncertaintiesσ<=[(1.2")2+(θ/100)2]1/2,where θ is the major-axis diameter. We compared each galaxymeasured with the original UGC description to ensure high reliability.The full position list is available in the electronic version only. A Complete Redshift Survey to the Zwicky Catalog Limit in a 2^h X 15 deg Region around 3C 273We compile 1113 redshifts (648 new measurements, 465 from theliterature) for Zwicky catalog galaxies in the region (-3.5d <= delta<= 8.5d, 11h5 <= alpha <= 13h5). We include redshifts for 114component objects in 78 Zwicky catalog multiplets. The redshift surveyin this region is 99.5% complete to the Zwicky catalog limit, m_Zw =15.7. It is 99.9% complete to m_Zw = 15.5, the CfA Redshift Survey(CfA2) magnitude limit. The survey region is adjacent to the northernportion of CfA2, overlaps the northernmost slice of the Las CampanasRedshift Survey, includes the southern extent of the Virgo Cluster, andis roughly centered on the QSO 3C 273. As in other portions of theZwicky catalog, bright and faint galaxies trace the same large-scalestructure. Study of the Virgo Cluster Using the B-Band Tully-Fisher RelationThe distances to spiral galaxies of the Virgo cluster are estimatedusing the B-band Tully-Fisher (TF) relation, and the three-dimensionalstructure of the cluster is studied. The analysis is made for a completespiral sample taken from the Virgo Cluster catalog of Binggeli, Sandage,& Tammann. The sample contains virtually all spiral galaxies down toM_{BT}=-15 mag at 40 Mpc. A careful examination is made ofthe selection effect and errors of the data. We estimate distance to 181galaxies, among which distances to 89 galaxies are reasonably accurate.We compare these distances to those obtained by other authors on agalaxy-by-galaxy basis. We find reasonable consistency of theTully-Fisher distance among various authors. In particular, it is foundthat the discrepancy in the distance among the different analyses withdifferent data is about 15%, when good H I and photometric data areavailable. We clarify that the different results on the Virgo distanceamong authors arise from the choice of the sample and interpretation ofthe data. We confirm that the Tully-Fisher relation for the Virgocluster shows an unusually large scatter sigma = 0.67 mag, compared tothat for other clusters. We conclude that this scatter is not due to theintrinsic dispersion of the Tully-Fisher relation, but due to a largedepth effect of the Virgo cluster, which we estimate to be extended from12 Mpc to 30 Mpc. The distribution of H I--deficient galaxies isconcentrated at around 14--20 Mpc, indicating the presence of a core atthis distance, and this agrees with the distance estimated for M87 andother elliptical galaxies with other methods. We show also that thespatial number density of spiral galaxies takes a peak at this distance,while a simple average of all spiral galaxy distances gives 20 Mpc. Thefact that the velocity dispersion of galaxies takes a maximum at 14--18Mpc lends an additional support for the distance to the core. Thesefeatures cannot be understood if the large scatter of the TF relation ismerely due to the intrinsic dispersion. The structure of the VirgoCluster we infer from the Tully-Fisher analysis looks like a filamentwhich is familiar to us in a late phase of structure formation in thepancake collapse in hierarchical clustering simulations. This Virgofilament lies almost along the line of sight, and this is the originthat has led a number of authors to much confusion in the Virgo distancedeterminations. We show that the M87 subcluster is located around 15--18Mpc, and it consists mainly of early-type type spiral galaxies inaddition to elliptical and S0 galaxies. There are very few late-typespiral galaxies in this subcluster. The spiral rich M49 subclusterconsists of a mixture of all types of spiral galaxies and is located atabout 22 Mpc. The two other known clouds, W and M, are located at about30--40 Mpc and undergo infall toward the core. The M cloud contains fewearly type spirals. We cannot discriminate, however, whether thesesubclusters or clouds are isolated aggregates or merely parts offilamentary structure. Finally, we infer the Hubble constant to be 82+/- 10 km s-1 Mpc-1. An image database. II. Catalogue between δ=-30deg and δ=70deg.A preliminary list of 68.040 galaxies was built from extraction of35.841 digitized images of the Palomar Sky Survey (Paper I). For eachgalaxy, the basic parameters are obtained: coordinates, diameter, axisratio, total magnitude, position angle. On this preliminary list, weapply severe selection rules to get a catalog of 28.000 galaxies, wellidentified and well documented. For each parameter, a comparison is madewith standard measurements. The accuracy of the raw photometricparameters is quite good despite of the simplicity of the method.Without any local correction, the standard error on the total magnitudeis about 0.5 magnitude up to a total magnitude of B_T_=17. Significantsecondary effects are detected concerning the magnitudes: distance toplate center effect and air-mass effect. Surface photometry of spiral galaxies in the Virgo cluster regionPhotographic surface photometry is carried out for 246 spiral galaxiesin the Virgo cluster region north of declination + 5 deg. The samplecontains all spiral galaxies of 'certain' and 'possible' Virgo membersin the Virgo Cluster Catalogue of Binggeli, Sandage, & Tammann. Thesample also includes those galaxies which were used in the Tully-Fisheranalyses of the Virgo cluster given in the literature. A catalog ispresented for positions, B-band total magnitudes and inclinations forthese galaxies, and they are compared with the data given in previousstudies. Distribution of the spin vectors of the disk galaxies of the Virgo cluster. I. The catalogue of 310 disk galaxies in the Virgo area.Not Available General study of group membership. II - Determination of nearby groupsWe present a whole sky catalog of nearby groups of galaxies taken fromthe Lyon-Meudon Extragalactic Database. From the 78,000 objects in thedatabase, we extracted a sample of 6392 galaxies, complete up to thelimiting apparent magnitude B0 = 14.0. Moreover, in order to considersolely the galaxies of the local universe, all the selected galaxieshave a known recession velocity smaller than 5500 km/s. Two methods wereused in group construction: a Huchra-Geller (1982) derived percolationmethod and a Tully (1980) derived hierarchical method. Each method gaveus one catalog. These were then compared and synthesized to obtain asingle catalog containing the most reliable groups. There are 485 groupsof a least three members in the final catalog. Abell 154 and Virgo - Pilot study for H I observations of distant clusters of galaxiesAs a test of procedures required to study the H I contents of spiralgalaxies in distant clusters of galaxies, the cluster Abell 154 has beenobserved from Arecibo. Fourteen candidate detections were found in tworegions of the cluster comprising about 10 percent of the cluster area.These results are compared in detail with those expected for theexhaustively studied Virgo cluster displaced to the distance of A 154.Most of the candidate detections are likely to be the combined profilesof two or more spiral galaxies, many of them too faint to appear on thelist of morphological types classified by Dressler (1980). Any attemptto identify these H I signals with known bright spirals is problematicat best. The A 154 profiles are systematically broader than expected forVirgo, but a crude application of the Tully-Fisher correlation indicatesthat they are still consistent with available photometric data. Whilethe H I deficiency in Virgo would still be apparent at the A 154distance, no significant evidence is found for H I deficiency in A 154. The Virgo cluster as a test for quantization of extragalactic redshiftsTifft's (1972, 1977) hypothesis that redshifts are partially quantizedwith a periodicity in the range 70-75 km/s is tested for samples ofbright spiral and dwarf irregular galaxies with accurate H I redshiftsin the region of the Virgo cluster. The heliocentric redshifts arecorrected for solar motion, first by adopting an estimate of the sun'smotion with respect to the centroid of the Local Group, and then byallowing the solar velocity vector to vary in direction over the wholesky. Power spectrum analyses of the corrected redshifts are used tosearch for a significant periodicity in the prescribed range 70-75 km/s.No such periodicity is found for the dwarf irregulars, but there is apossible periodicity of about 71.1 km/s for the bright spirals. In afurther exploratory study, the sample of 112 spirals is divided upaccording to environment. The spirals in high-density regions of thecluster show no quantization, whereas those in low-density regionsappear to be partially quantized in intervals of about 71.0 km/s. The Tully-Fisher relation in different environmentsThe Tully-Fisher relation (TFR) in different environments wasinvestigated in 13 galaxy samples spanning a large range in galaxydensities, using two statistical tests to compare the TFR of differentsamples. Results of the analysis of TFR parameters in severalenvironments showed that, when samples of similar data-accuracy andmagnitude-range were compared, there was no significant differencebetween the galaxy samples. It is suggested that a comparison of sampleswith very different data accuracy or those biased by incompletenesseffects may lead to misleading results. The dustiness, luminosity, and metallicity of galaxiesB-band CCD images have been obtained of 230 galaxies in and near theVirgo and Ursa Major clusters. A coarse classification of these imagesshows that the 'dustiness' of late-type galaxies correlates stronglywith their luminosity. Luminous spirals are seen to be much dustier thanlate-type systems of lower luminosity. The reason for this correlationis probably that luminous galaxies are more metal-rich than fainterobjects. Systems with Fe/H less than about -1.0 are observed to beessentially dust-free. Classification of galaxies on CCD framesMorphological classifications of 231 galaxies in and near the Virgo andUrsa Major clusters are reported which show that luminosityclassification techniques (LCTs) can be used to determine theluminosities of spiral galaxies with an accuracy of about 0.7 mag on CCDframes. The observations in the direction of the Virgo Cluster confirmthe assignment of some galaxies to the background field, stronglyconfirming that the large dispersion in the Tully-Fisher relation forVirgo galaxies is at least partly due to contamination of the Virgo coresample by background galaxies. LCTs yield a distance of 15.3 + 2.6 or -2.2 Mpc for the spiral and irregular galaxies associated with the coreof the Virgo Cluster proper. The Ursa Major and Virgo cluster distancesare found to be the same. A class of galaxies with fuzzy, anemic outerstructure and active star formation in their cores is found to be commonin Virgo but rare in the Ursa Major Cluster. The extragalactic distance scale. II - The unbiased distance to the Virgo Cluster from the B-band Tully-Fisher relationThe behavior of the B-band Tully-Fisher (TF) relation with respect tothe observational biases and parameter uncertainties is studied from analmost complete sample of spiral galaxies belonging to the VirgoCluster. The strong influence of the limiting apparent magnitude whenusing the direct TF relation is confirmed. A distance modulus of 31.4 +or - 0.2 is found along with a corresponding H(0) = 68 + or - 8km/s/Mpc, assuming a cosmological velocity of the cluster V = 1300 + or- 100 km/s. The Virgo S and S-prime clouds are shown to lie atsignificantly different distances. Different distance moduli found byother authors are explained. 21 centimeter line width distances of cluster galaxies and the value of H0Locally calibrated blue and infrared Tully-Fisher (TF) relations areapplied to an 82 percent complete sample of 81 Sab-Sm galaxies which arebona fide members of the Virgo Cluster. A nearly unbiased Virgo modulusof 31.60 + or - 0.15 can be derived in perfect agreement withindependent recent determinations. It is shown that the blue TF andinfrared TF relations give almost identical distance moduli from anyselected Virgo subsample. The intrinsic scatter about the two TFrelations is 0.7 mag, considerably larger than the observed scatter inthe UMa Cluster and in 10 more distant clusters. Distance determinationof these clusters therefore can be achieved only by fitting the upperenvelopes of their TF relations onto the blue and infrared upperenvelopes of the Virgo Cluster. The resulting distances define a linearexpansion law with a small scatter. The present cluster data require H0= 56.6 + or - 0.9 km/s/Mpc. Three-color surface photometry of a selected sample of early-type galaxies. I - Observations and data reductionThis paper presents the results of two or three color surface photometryfor a sample of 36 early-type galaxies obtained at the Canada FranceHawaii Telescope with CCD cameras. The calibration and data reductionprocedures are described. A comparison of the results with previous workis made for NGC 3379. For each galaxy the B surface brightness profilealong the major axis, as well as ellipticity and color profiles aredisplayed. H I detection survey of a complete magnitude-limited sample of dwarf irregular galaxies in the Virgo Cluster areaNew single-beam Arecibo H I observations of 298 late-type galaxies inthe Virgo Cluster drawn mostly from the new catalog of Binggeli,Sandage, and Tammann (1985) are presented. Two hundred seventeen ofthese constitute a magnitude-limited 'complete sample' of such galaxies,types Sdm through Im and BCD. Sixty-one percent of this 'completesample' was detected, greatly enhancing the store of redshifts and H Imasses for such galaxies in the Virgo Cluster. For detected galaxies,heliocentric velocities, 50 percent profile widths, and single-beamfluxes are presented. For those that escaped detection, upper limits arecomputed to the flux appropriate to the redshift range (-600 to +3000km/s). UBV colors of Virgo cluster irregular galaxiesPhotoelectric UBV aperture photometry is presented for 65 dwarfirregular and morphologically related types of galaxies that areprobable Virgo cluster members. Virgo cluster Irr galaxies cover a widerrange in color than typical samples of field Irr systems, primarily dueto the presence of unusually red Im galaxies in Virgo. The extremelyblue 'blue compact dwarf' and the red amorphous galaxies in Virgo alsostand out on a UBV color-color plot, but the majority of Virgo Irrs areundistinguished in terms of UBV colors. Statistical correlations arefound between colors and location within the cluster. Red Irrs often arein elliptical galaxy rich, dense areas of the cluster, while the blueIrrs roughly follow the spiral distribution pattern. Thus a connectionexists between environment and the properties of Irrs. Simplestatistical tests for ongoing stripping as the source of red Irrs,however, yields null results. HI-observations of galaxies in the Virgo cluster of galaxies. I - The dataNew H I-data for a large number of bright galaxies inside the 10 degradius area of the Virgo cluster of galaxies have been obtained with the100 m radiotelescope at Effelsberg. A total of 234 galaxies was observedfor the first time. Among them, 53 have been detected providing newaccurate radial velocities. Data from the literature have been compiled.Together with the new data, they form a (nearly homogeneous) set of H Iobservations for more than 450 galaxies. Candidate Galaxies for Study of the Local Velocity Field and Distance Scale Using Space Telescope - Part Three - Galaxies in the Virgo Cluster CoreAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1985AJ.....90.2006S&db_key=AST Studies of the Virgo Cluster. II - A catalog of 2096 galaxies in the Virgo Cluster area.The present catalog of 2096 galaxies within an area of about 140 sq degapproximately centered on the Virgo cluster should be an essentiallycomplete listing of all certain and possible cluster members,independent of morphological type. Cluster membership is essentiallydecided by galaxy morphology; for giants and the rare class of highsurface brightness dwarfs, membership rests on velocity data. While 1277of the catalog entries are considered members of the Virgo cluster, 574are possible members and 245 appear to be background Zwicky galaxies.Major-to-minor axis ratios are given for all galaxies brighter than B(T)= 18, as well as for many fainter ones. A catalog of dwarf galaxies in VirgoA catalog listing the location, apparent angular diameter, type,estimated central light concentration, and estimated brightness of 846dwarf galaxies in a 200-deg-sq region in Virgo is presented. Thegalaxies comprise 634 ellipticals, 137 IC-3475-type galaxies, 73 dwarfspirals and irregulars, and two objects which are jets of normalgalaxies, and were found on nine long-exposure IIIa-J-emulsion platesmade with the 1.2-m-Schmidt telescope at Palomar Observatory from 1971to 1976. Concordances to other catalogs, tables of additionalparameters, maps, graphs, and photographs are provided. The projecteddistributions of normal and dwarf galaxies and the dependence ofapparent luminosity on central light concentration are discussed. It isfound that dwarf ellipticals and IC-3475-type galaxies are probablemembers of the Virgo cluster, while dwarf spirals and possibly dwarfirregulars are not.
Submit a new article | 2019-06-17 15:09:13 | {"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": 0.6153991222381592, "perplexity": 5505.493606933538}, "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-2019-26/segments/1560627998509.15/warc/CC-MAIN-20190617143050-20190617165050-00404.warc.gz"} |
https://www.preprints.org/manuscript/202106.0051/v1 | Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed
# Linear Confinement of a Relativistic Spin-0 scalar Particle under Lorentz Symmetry Violation Effects
Version 1 : Received: 1 June 2021 / Approved: 2 June 2021 / Online: 2 June 2021 (08:24:44 CEST)
How to cite: Ahmed, F. Linear Confinement of a Relativistic Spin-0 scalar Particle under Lorentz Symmetry Violation Effects. Preprints 2021, 2021060051 (doi: 10.20944/preprints202106.0051.v1). Ahmed, F. Linear Confinement of a Relativistic Spin-0 scalar Particle under Lorentz Symmetry Violation Effects. Preprints 2021, 2021060051 (doi: 10.20944/preprints202106.0051.v1).
## Abstract
In this work, linear confinement of a relativistic scalar particle under the effects of Lorentz symmetry violation is investigated. We introduce a scalar potential by modifying the mass via transformation M → M + S(r) in the wave equation, and analyze the effects on the eigenvalues and the wave function. We see that the solution of the bound state to the wave equation can be achieved, and the energy eigenvalues and the wave function modified by the Lorentz symmetry breaking parameters as well as potential
## Subject Areas
Lorentz symmetry violation; Relativistic wave-equations; bound states solutions; linear confining potential; electric & magnetic field; biconfluent Heun’s equation
Views 0 | 2021-06-20 22:11:35 | {"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": 0.817748486995697, "perplexity": 3907.179568023529}, "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/1623488257796.77/warc/CC-MAIN-20210620205203-20210620235203-00373.warc.gz"} |
https://abaqus-docs.mit.edu/2017/English/SIMACAEELMRefMap/simaelm-c-infinite.htm | # Infinite elements
Infinite elements: are used in boundary value problems defined in unbounded domains or problems in which the region of interest is small in size compared to the surrounding medium; are usually used in conjunction with finite elements; can have linear behavior only; provide stiffness in static solid continuum analyses; and provide “quiet” boundaries to the finite element model in dynamic analyses. A solid section definition is used to define the section properties of infinite elements. The following topics are discussed:
Related Topics Infinite element library In Other Guides *SOLID SECTION Creating acoustic infinite sections
ProductsAbaqus/StandardAbaqus/ExplicitAbaqus/CAE | 2022-12-05 19:07:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.6901414394378662, "perplexity": 1544.4355259337924}, "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/1669446711042.33/warc/CC-MAIN-20221205164659-20221205194659-00553.warc.gz"} |
https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) | # Liouville's theorem (differential algebra)
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841,[1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions.
The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is ${\displaystyle e^{-x^{2}},}$ whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions ${\displaystyle {\frac {\sin(x)}{x}}}$ and ${\displaystyle x^{x}}$.
Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.
## Definitions
For any differential field F, there is a subfield
Con(F) = {f in F | Df = 0},
called the constants of F. Given two differential fields F and G, G is called a logarithmic extension of F if G is a simple transcendental extension of F (i.e. G = F(t) for some transcendental t) such that
Dt = Ds/s for some s in F.
This has the form of a logarithmic derivative. Intuitively, one may think of t as the logarithm of some element s of F, in which case, this condition is analogous to the ordinary chain rule. However, F is not necessarily equipped with a unique logarithm; one might adjoin many "logarithm-like" extensions to F. Similarly, an exponential extension is a simple transcendental extension that satisfies
Dt = t Ds.
With the above caveat in mind, this element may be thought of as an exponential of an element s of F. Finally, G is called an elementary differential extension of F if there is a finite chain of subfields from F to G where each extension in the chain is either algebraic, logarithmic, or exponential.
## Basic theorem
Suppose F and G are differential fields, with Con(F) = Con(G), and that G is an elementary differential extension of F. Let a be in F, y in G, and suppose Dy = a (in words, suppose that G contains an antiderivative of a). Then there exist c1, ..., cn in Con(F), u1, ..., un, v in F such that
${\displaystyle a=c_{1}{\frac {Du_{1}}{u_{1}}}+\dotsb +c_{n}{\frac {Du_{n}}{u_{n}}}+Dv.}$
In other words, the only functions that have "elementary antiderivatives" (i.e. antiderivatives living in, at worst, an elementary differential extension of F) are those with this form. Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions.
A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al.
## Examples
As an example, the field C(x) of rational functions in a single variable has a derivation given by the standard derivative with respect to that variable. The constants of this field are just the complex numbers C.
The function ${\displaystyle {\tfrac {1}{x}}}$ , which exists in C(x), does not have an antiderivative in C(x). Its antiderivatives ln x + C do, however, exist in the logarithmic extension C(x, ln x).
Likewise, the function ${\displaystyle {\tfrac {1}{x^{2}+1}}}$ does not have an antiderivative in C(x). Its antiderivatives tan−1(x) + C do not seem to satisfy the requirements of the theorem, since they are not (apparently) sums of rational functions and logarithms of rational functions. However, a calculation with Euler's formula ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$ shows that in fact the antiderivatives can be written in the required manner (as logarithms of rational functions).
{\displaystyle {\begin{aligned}e^{2i\theta }&={\frac {e^{i\theta }}{e^{-i\theta }}}={\frac {\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }}={\frac {1+i\tan \theta }{1-i\tan \theta }}\\[8pt]\theta &={\frac {1}{2i}}\ln \left({\frac {1+i\tan \theta }{1-i\tan \theta }}\right)\\[8pt]\tan ^{-1}x&={\frac {1}{2i}}\ln \left({\frac {1+ix}{1-ix}}\right)\end{aligned}}}
## Relationship with differential Galois theory
Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (corresponding to the constant of integration). Thus, an antiderivative's differential Galois group does not encode enough information to determine if it can be expressed using elementary functions, the major condition of Liouville's theorem. | 2020-03-31 17:55:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.956804096698761, "perplexity": 610.5993279446263}, "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/1585370502513.35/warc/CC-MAIN-20200331150854-20200331180854-00430.warc.gz"} |
https://www.zbmath.org/?q=ai%3Acherkasov.i-d+se%3A00001636 | # zbMATH — the first resource for mathematics
Transformation of one-dimensional diffusion fields on the plane. (Russian) Zbl 0647.60059
The diffusion Markov fields $$\xi(z)$$, $$z\in R^ 2_+$$, considered in this paper are obtained as solutions of suitable classes of stochastic differential equations involving stochastic integrals with respect to the two-parameter Wiener process.
The author has found conditions for two diffusion fields to be equivalent. He answers questions of how to transform a diffusion field into a Gaussian martingale, and in particular, how to get a two-parameter Wiener process from a diffusion field. The Ito formula is used to establish an invariant representation for diffusion fields. Finally, results are given for the transition probabilities and the corresponding densities of diffusion random fields.
Reviewer: J.M.Stoyanov
##### MSC:
60G60 Random fields 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes | 2021-01-25 20:44:06 | {"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": 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": 0.663647472858429, "perplexity": 664.860102686218}, "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/1610703644033.96/warc/CC-MAIN-20210125185643-20210125215643-00256.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/essential-university-physics-volume-1-3rd-edition/chapter-9-exercises-and-problems-page-166/93 | ## Essential University Physics: Volume 1 (3rd Edition)
$.26v_0;.31v_0$
a) We find that the blocks, after the collision, have a velocity of: $v_{1f}=\frac{M-nM}{m+nM}v_{1i}+\frac{2nM}{M+nM}v_{2i}$ $v_{1f}=\frac{M-nM}{m+nM}v_{0}$ $v_{2f}=\frac{2M}{m_1+m_2}v_{1i}+\frac{M-nM}{M+nM}v_{2i}$ $v_{2f}=\frac{2M}{m+nM}v_{0}$ If n is less than three, we see than 2M will be greater than $|M-nM|$. This would make the second block faster, meaning the blocks will never again collide. b) If n is equal to 4, we see than 2M will be less than $|M-nM|$. This would make the first block faster, meaning the blocks will collide a second time. c) We find: $v_{1f}=\frac{-9M}{11M}v_{0}$ $v_{2f}=\frac{2M}{11M}v_{0}$ They will collide again. Continuing to use the equations $v_{1f}=\frac{M-nM}{m+nM}v_{1i}+\frac{2nM}{M+nM}v_{2i}$ and $v_{2f}=\frac{2M}{m_1+m_2}v_{1i}+\frac{M-nM}{M+nM}v_{2i}$ until $|v_{1f}|$ | 2022-05-16 16:34:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5099634528160095, "perplexity": 319.64811840743147}, "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/1652662510138.6/warc/CC-MAIN-20220516140911-20220516170911-00503.warc.gz"} |
https://physics.stackexchange.com/questions/721657/electromagnetic-stress-energy-momentum-tensor-and-stress-tensor | # Electromagnetic stress-energy-momentum tensor and stress tensor
The purely spatial components of the energy-momentum-stress tensor for a perfect fluid are clearly related the components of the ordinary stress tensor inside the fluid. My question if this case is general, for example, if we consider a dielectric material subjected to an external electromagnetic field, is there any relation between the internal mechanical stresses of the solid and the components of the electromagnetic energy-impulse tensor? For example, for a a dielectric solid in equilibrium, is it true that:
$$\frac{\partial T^{ab}}{\partial x^a} + \frac{\partial \sigma^{ab}}{\partial x^a} = 0, \qquad a,b\in\{1,2,3\}$$
where $$T^{ab}$$ is the electromagnetic energy-momentum-stress and $$\sigma^{ab}$$ is the mechanical stress tensor.
• If we simply add the mechanical and electromagnetic Lagrangians and apply the homogeneity of space (See sec. 7 "Momentum" of Landau and Lifshitz "Mechanics" book), I am guessing that the equation you have shown holds. Aug 6 at 4:36 | 2022-08-14 10:44:04 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9067007899284363, "perplexity": 344.05691737640853}, "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/1659882572021.17/warc/CC-MAIN-20220814083156-20220814113156-00337.warc.gz"} |
https://en.wikipedia.org/wiki/Maass_wave_form | # Maass wave form
In mathematics, a Maass wave form, Maass cusp form or Maass form is a function on the upper half-plane that transforms like a modular form but need not be holomorphic. They were first studied by Hans Maass in Maass (1949).
## Definition
Let k be an integer, s be a complex number, and Γ be a discrete subgroup of SL2(R). A Maass form of weight k for Γ with Laplace eigenvalue s is a smooth function from the upper half-plane to the complex numbers satisfying the following conditions:
• For all ${\displaystyle \gamma =\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma }$ and all ${\displaystyle z\in \mathbb {H} }$, we have ${\displaystyle f\left({\frac {az+b}{cz+d}}\right)=\left({\frac {cz+d}{|cz+d|}}\right)^{k}f(z)}$.
• We have ${\displaystyle \Delta _{k}f=sf}$, where ${\displaystyle \Delta _{k}}$ is the weight k hyperbolic Laplacian defined as
${\displaystyle \Delta _{k}=-y^{2}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)+iky{\frac {\partial }{\partial x}}.}$
• The function ƒ is of at most polynomial growth at cusps.
A weak Maass form is defined similarly but with the third condition replaced by "The function ƒ has at most linear exponential growth at cusps". Moreover, ƒ is said to be harmonic if it is annihilated by the Laplacian operator.
## Major results
Let ƒ be a weight 0 Maass cusp form. Its normalized Fourier coefficient at a prime p is bounded by p7/64+p-7/64. This theorem is due to Kim and Sarnak. | 2017-04-24 03:59:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9150817394256592, "perplexity": 372.93685253602615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917118963.4/warc/CC-MAIN-20170423031158-00170-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://taskvio.com/maths/arithmetic/addition-calculator/ | # Addition Calculator
Addition calculator is an online web-based tool that will help you add any two numbers.
#### For Example :
$${125}+{125}=250$$
# About the addition calculator
Addition calculator is an online web-based tool that will help you add any two numbers. I know it’s really a basic tool that will help you add numbers, everyone knows about it so why do we need a calculator. Here is the answer to it. Some people know how to do it manually or even by counting it in their mind because it’s really a simple thing in mathematics even 3 or 5 years child know how to add or subtract any value if they go to school. But it’s always easy right such as what if I will ask you what is the addition of (10 + 10) then you will directly say 20 because you know the answer and it was really a simple question but what if I will ask you how much 189737 + 198798, now see you have to count and you are even taking time to add it and so that’s why we have this calculator which will add it in a second that 189737 + 198798 is equal to 388535, (189737 + 198798 = 388535) so in this case, this tool is really useful.
## Definition of addition
In Mathematics, addition is one of the arithmetic operations. It gives the sum of the numbers. In other words, the method of adding the numbers together is named addition. The symbol that represents the addition is “+”. The result obtained from the addition process is named the sum. The numbers which are added together are called addends.
In our calculator, we have given two blank boxes where you can add your value, and then you can calculate it by clicking on the calculate button.
It is one of the basic operation arithmetic, like these basic operations;
• Subtraction,
• Multiplication, and
• Division
Addition use to have two properties and that is commutative and associativity.
Commutative: it means is that you can change the place of both value there is no fixed place of any of the numbers in it and it doesn’t even change the addition result Such as;
A + B = B + A
Associativity: in this property when you add more than 2 numbers then it does not matter if you add the first number with the second number or third number. As you can see here (A + B) + C = A + (B +C)
Now when you keep adding positive number then it will continue like this but when you will not add positive numbers and you try to add negative numbers then it will be like this A + (-B) = A - B
In case if the positive number requires subtraction have to subtract the negative value from the positive value as I have shown above (A + (-B) = A – B).
Adding numbers is not so simple all the time.
The addition definition math gives you is as easy as you thought it'd be - add variety to the opposite, and you get the result. However, as you're here, we both realize it won't be that straightforward. Sometimes, the numbers are huge. The opposite option is that they are negative or even fractions. Altogether of those situations, our calculator works smoothly and provides you the proper answer.
The earliest discovered evidence suggests addition was used between 20.000 and 18.000 years BC. Fortunately, now that you simply have out addition calculator, you do not need to roll in the hay manually anymore!
### How to use this additional tool
This tool is really simple to use and this tool is free to use for anyone. It is a web-based tool that will really help you add a large number when you won’t be able to do it manually and instantly then this tool comes into play it will help you calculate your all large number in seconds.
Even if you will add numbers in negative then still you will be able to calculate your value right.
Such as A+ (-B) = A + B
So this is how it works. It will help your needs.
If you are also looking for other mathematic tools then we have already created it and you can access it by clicking on the math section and you will find the entire related tool for mathematics.
Bookmark this tool for feature uses.
A. | 2021-05-12 22:55:30 | {"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": 0.5142938494682312, "perplexity": 371.4599255510444}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991413.30/warc/CC-MAIN-20210512224016-20210513014016-00296.warc.gz"} |
https://golem.ph.utexas.edu/category/2021/03/can_we_understand_the_standard.html | ## March 17, 2021
### Can We Understand The Standard Model?
#### Posted by John Baez
I’m giving a talk in Latham Boyle and Kirill Krasnov’s Perimeter Institute workshop Octonions and the Standard Model on Monday April 5th at noon Eastern Time.
This talk will be a review of some facts about the Standard Model. Later I’ll give one that says more about the octonions.
Can we understand the Standard Model?
Abstract. 40 years trying to go beyond the Standard Model hasn’t yet led to any clear success. As an alternative, we could try to understand why the Standard Model is the way it is. In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard Model and its representation on one generation of fermions arises naturally from a process that involves splitting 10d Euclidean space into 4+6 dimensions, but also from a process that involves splitting 10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimensions. We explain both these approaches, and how to reconcile them.
You can see the slides here, and later a video of my talk will appear. You can register to attend the talk at the workshop’s website.
Here’s a puzzle, just for fun. As I’ll explain, there’s a normal subgroup of $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ that acts trivially on all known particles, and this fact is very important. The ‘true’ gauge group of the Standard Model is the quotient of $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ by this normal subgroup.
This normal subgroup is isomorphic to $\mathbb{Z}_6$ and it consists of all the elements
$(\zeta^n, (-1)^n, \omega^n ) \in \mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$
where
$\zeta = e^{2 \pi i / 6}$
is my favorite primitive 6th root of unity, $-1$ is my favorite primitive square root of unity, and
$\omega = e^{2 \pi i / 3}$
is my favorite primitive cube root of unity. (I’m a primitive kind of guy, in touch with my roots.)
Here I’m turning the numbers $(-1)^n$ into elements of $\mathrm{SU}(2)$ by multiplying them by the $2 \times 2$ identity matrix, and turning the numbers $\omega^n$ into elements of $\mathrm{SU}(3)$ by multiplying them by the $3 \times 3$ identity matrix.
But in fact there are a bunch of normal subgroups of $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ isomorphic to $\mathbb{Z}_6$. By my count there are 12 of them! So you have to be careful that you’ve got the right one, when you’re playing with some math and trying to make it match the Standard Model.
Puzzle 1. Are there really exactly 12 normal subgroups of $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ that are isomorphic to $\mathbb{Z}_6$?
Puzzle 2. Which ones give quotients isomorphic to the true gauge group of the Standard Model, which is $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ modulo the group of elements $(\zeta^n, (-1)^n, \omega^n)$?
To help you out, it helps to know that every normal subgroup of $\mathrm{SU}(2)$ is a subgroup of its center, which consists of the matrices $\pm 1$. Similarly, every normal subgroup of $\mathrm{SU}(3)$ is a subgroup of its center, which consists of the matrices $1, \omega$ and $\omega^2$. So, the center of $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ is $\mathrm{U}(1) \times \mathbb{Z}_2 \times \mathbb{Z}_3$.
Here, I believe, are the 12 normal subgroups of $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ isomorphic to $\mathbb{Z}_6$. I could easily have missed some, or gotten something else wrong!
1. The group consisting of all elements $(1, (-1)^n, \omega^n)$.
2. The group consisting of all elements $((-1)^n, 1, \omega^n)$.
3. The group consisting of all elements $((-1)^n, (-1)^n, \omega^n)$.
4. The group consisting of all elements $(\omega^n, (-1)^n, 1)$.
5. The group consisting of all elements $(\omega^n, (-1)^n, \omega^n)$.
6. The group consisting of all elements $(\omega^n, (-1)^n, \omega^{-n})$.
7. The group consisting of all elements $(\zeta^n , 1, 1)$.
8. The group consisting of all elements $(\zeta^n , (-1)^n, 1)$.
9. The group consisting of all elements $(\zeta^n , 1, \omega^n)$.
10. The group consisting of all elements $(\zeta^n , 1, \omega^{-n})$.
11. The group consisting of all elements $(\zeta^n , (-1)^n, \omega^n)$.
12. The group consisting of all elements $(\zeta^n , (-1)^n, \omega^{-n})$.
Posted at March 17, 2021 2:45 AM UTC
TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3304
### Re: Can We Understand The Standard Model?
These are very interesting relations between the Standard Model group and the 10d Euclidean/Minkowski spacetimes, but beside the efort to understand the algebro-geometric origin of the SM group, it is also important to understand the representation of the SM group which gives the known elementary particles, as well as the different generations. Because it is not only the group/algebra which is important, but also a representation as well, one needs some category theory. For one such attempt, please see Standard Model and 4-groups, A. Mikovic and M. Vojinovic.
Posted by: Aleksandar Mikovic on March 17, 2021 8:39 AM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
I’ve read that paper! It’s interesting!
Posted by: John Baez on March 18, 2021 12:57 AM | Permalink | Reply to this
### Spin(9) GUT?
Since $S(U(2) \times U(3)) \subset Spin(9)$, I presume someone has tried to build a GUT with $Spin(9)$ gauge group, but I feel like I never saw it. Maybe I just missed it? Or does something fail? Something about the representation on fermions? That can’t be it, because I could just restrict the $Spin(10)$-representation…
Posted by: Theo Johnson-Freyd on March 17, 2021 5:04 PM | Permalink | Reply to this
### Re: Spin(9) GUT?
I’ve never seen a Spin(9) GUT worked out. As you can tell from my talk, my interest in Spin(9) was piqued by this paper:
but he doesn’t try a SO(9) GUT.
There are some quite elaborate studies of the lattice of subgroups of Spin(10) containing S(U(2) × U(3)), and the gauge theories with all these gauge groups, worked out by physicists trying to understand symmetry breaking in the Spin(10) theory. Spin(9) should be hiding somewhere in this work. Here’s one paper:
It has some really nice diagrams showing a bit of the lattice of subgroups of Spin(10) containing S(U(2) × U(3)). But I don’t see Spin(9) — they would call it SO(9) — in here.
I want to learn a bit more about this stuff, though it’s kind of a rabbit hole. One important thing is that Spin(10) has SU(5) × U(1) or maybe (SU(5) × U(1))/$\mathbb{Z}_5$ as a maximal subgroup. The generator of the U(1) here is called the X charge, and it’s a linear combination of hypercharge (called Y) and baryon number minus lepton number (called B-L).
Posted by: John Baez on March 17, 2021 6:58 PM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
On slide 19, “the true gauge of the Standard Model” should be “the true gauge group of the Standard Model”.
Posted by: Blake Stacey on March 17, 2021 8:25 PM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Whoops — thanks! I fixed it.
I don’t have such strong opinions about gauge-fixing.
Posted by: John Baez on March 18, 2021 12:59 AM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Gabriel Verret solved Puzzle 1 assuming that every normal subgroup of order 6 in $\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ is actually contained in the center, $\mathrm{U}(1)\times \mathbb{Z}_2\times \mathbb{Z}_3$. I actually think this is true for every finite normal subgroup.
His argument then goes as follows:
Our group of order $6$ must be contained in $\mathrm{U}(1)\times \mathbb{Z}_2\times \mathbb{Z}_3$. (Normality will be automatic since this is the center.) It must be contained in the unique subgroup of order $6$ of $\mathrm{U}(1)$, so we might as well assume we are working in $\mathbb{Z}_6\times \mathbb{Z}_2\times \mathbb{Z}_3\cong\mathbb{Z}_6\times \mathbb{Z}_6$.
So we are just asking for the number of subgroups of order $6$ in $\mathbb{Z}_6\times \mathbb{Z}_6$. Such a group must be cyclic, so is uniquely determined by a generator. Each group has two different generators, so it is the number of elements of order $6$ in $\mathbb{Z}_6\times \mathbb{Z}_6$ divided by $2$.
An element in $\mathbb{Z}_6\times \mathbb{Z}_6$ has order $6$ if one of its coordinate does, or if the coordinates have order $2$ and $3$. There are $(36-4\cdot 4)=20$ of the first type, and $2\cdot2=4$ of the second type, for $24$ elements of order $6$ in total (there are other ways to do this count), and $12$ groups of order $6$.
Posted by: John Baez on March 18, 2021 3:07 AM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
A finite normal subgroup $N$ of a connected group $G$ is indeed central. Indeed, take the orbit of a point of $N$ under the conjugation action of $G$. Since $G$ is connected, the orbit is connected. Since $N$ is finite, it must be a singleton.
Posted by: L Spice on March 19, 2021 3:11 AM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Nice! I’d heard this result for simple Lie groups, but didn’t realize it was more general. What a nice simple proof!
Posted by: John Baez on March 19, 2021 5:20 AM | Permalink | Reply to this
Read the post A Group Theory Problem
Weblog: The n-Category Café
Excerpt: I've got a problem with Lie groups.
Tracked: March 19, 2021 7:57 PM
### Re: Can We Understand The Standard Model?
So, Problem 1 has been nicely settled by Gabriel Verret and L Spice: there are exactly 12 normal subgroups $H \subset \mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ that are isomorphic to $\mathbb{Z}_6$. These should be the ones listed in my post, but I’m not sure anyone but me has checked that list.
Nobody has taken a stab at Problem 2, so I’ll give a hint: at least two of the 12 normal subgroups $H$ on the list have
$\frac{\mathrm{U}(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)}{H} \cong G_{SM}$
where
$G_{SM} = \mathrm{S}(\mathrm{U}(2) \times \mathrm{U}(3))$
is the true gauge group of the Standard Model. One of them is number 11 on my list: that’s what I said in my post.
I have not proved that there are only two.
Posted by: John Baez on March 19, 2021 8:23 PM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Regarding your slides, in the page of the 2x2 matrices, would quaternions fit the role of the 6d matrices. It would be neat to have all normed division algebras corresponding to the SM.
Posted by: Daniel de França on March 20, 2021 7:42 PM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Of course the quaternions are 4-dimensional, not 6-dimensional. But in the talk series Octonions and the Standard Model there’s been some talk of approaches to Standard Model that use all 4 normed division algebras, e.g. Cohl Furey and Mia Hughes’ talk ‘Division algebraic symmetry breaking’, which uses $\mathbb{R} \otimes \mathbb{C} \otimes \mathbb{H} \otimes \mathbb{O}$ in a manner reminiscent of Geoffrey Dixon’s work. (The $\mathbb{R}$ is here just for decorative purposes, since it can be left out without changing the algebra.)
It would probably be good to put more thought into the chain of inclusions
$\mathfrak{h}_2(\mathbb{C}) \subset \mathfrak{h}_2(\mathbb{H}) \subset \mathfrak{h}_2(\mathbb{O})$
Posted by: John Baez on March 20, 2021 9:48 PM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Oh, I am sorry. What I meant is that in the page where you find “4d/10d Minkowski spacetime can be seen as the space of 2×2hermitian complex/octonionic matrices”, the quaternions would yield 6d 2x2 hermitian quaternionic matrices.
Posted by: Daniel de França on March 21, 2021 4:50 AM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
Yes, $\mathfrak{h}_2(\mathbb{H})$ is 6d Minkowski spacetime. I haven’t found any use for that fact in thinking about the Standard Model, but I also haven’t tried, so maybe I should! I’ve written about it elsewhere, e.g. here:
Posted by: John Baez on March 21, 2021 6:34 PM | Permalink | Reply to this
### Re: Can We Understand The Standard Model?
I’ve looked at that paper and I since it is concerned about supersymmetry, I remembered that there is a very fertile field of studies in real 6d, or 3d complex, that is of Topological Sting theory. It also has the signature of AdS5.
Posted by: Daniel de França on March 21, 2021 7:19 PM | Permalink | Reply to this
### Mitchell
Six dimensions is also the maximum dimension for superconformal field theory (Nahm’s theorem). Many such field theories occur as worldvolume theories of stacks of M5-branes.
The list at the bottom of page 6 in Baez & Huerta reminds me that Maldacena’s original (1997) conjectures for holography were for stacks of M2-branes, D3-branes, and M5-branes.
Posted by: Mitchell Porter on March 25, 2021 6:49 AM | Permalink | Reply to this
Read the post Cosmic Strings in the Standard Model
Weblog: Musings
Excerpt: Prompted by some posts by John Baez, a little calculation with an unsurprising result.
Tracked: March 21, 2021 6:27 AM
### Mozibur
I don’t really understand Connes non-commutative geometry. However, what I’ve gathered that he’s able to derive the classical Standard Model from his spectral action, which is, as far as I understand it, the non-commutative geometry version of the Einstein-Hilbert action. Thats one version of the Kaluza-Klein idea - expressing everything in terms of gravity.
To me, that sounds significant.
Especially after seeing the full Standard model written out in Veltman’s Diagrammatica - I think there was something like a hundred terms!
Posted by: Mozibur Ullah on April 13, 2021 8:42 AM | Permalink | Reply to this
### Re: Mozibur
Connes gets the Standard Model from noncommutative geometry by putting it in. He can get lots of different quantum field theories from noncommutative geometry by making different choices. I give a link to some of his work at the start of my second talk.
He’s trying to figure out what makes the Standard Model special, and he’s figured out a lot of interesting stuff. But it’s certainly not true that only the Standard Model can pop out of his approach. So the question of understanding the Standard Model remains open. (If he’d already solved it I wouldn’t be thinking about!)
By the way, he basically uses a noncommutative generalization of the Einstein–Hilbert action to get the part of the action that only describes bosons and their interactions — that is, the part involving just gauge fields and the Higgs. But he needs another idea to describe how the fermions interact with the gauge fields and Higgs.
Posted by: John Baez on April 13, 2021 5:43 PM | Permalink | Reply to this
Post a New Comment | 2021-04-18 22:54:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 81, "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": 0.8009742498397827, "perplexity": 580.8596619231944}, "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-2021-17/segments/1618038862159.64/warc/CC-MAIN-20210418224306-20210419014306-00131.warc.gz"} |
https://math.stackexchange.com/questions/2053357/what-exactly-determines-the-block-sizes-for-jordan-forms | # What exactly determines the block-sizes for Jordan forms?
For instance, with $T \in \mathcal{L}$(Mat($2,2,\mathbb{C}$)) we are given that the minimal polynomial of $T$ is $p(z) = (z - 2i)(z + 7)^2$. I want to find the possible Jordan Forms pertaining to this $T$.
We know that the characteristic polynomial of $T$ is a polynomial multiple of the minimal polynomial, thus it is either $(z - 2i)^2 (z+7)^2$ or $(z - 2i) (z+7)^3$. So the eigenvalue $2i$ will have multiplicity = $1$ or $2$, and the eigenvalue $7$ will have multiplicity = $2$ or $3$.
However, I am left wondering how we can determine the block sizes for each eigenvalue given only the information that the minimal polynomial is $p(z) = (z - 2i)(z + 7)^2 \,$?
• Does it have something to do with the fact that the minimal polynomial is, by definition, the unique monic polynomial of smallest degree s.t: $p(T) = 0$?? – Javier Dec 11 '16 at 2:51
Given the minimal polynomial of $T$, in order to give the possible Jordan normal forms, the following facts are useful:
• The minimal polynomial tells you what are the eigenvalues of $T$. In your case, there are two eigenvalues $\lambda_1=2i$ and $\lambda_2=-7$.
• The sum of the sizes of all Jordan blocks corresponding to an eigenvalue $\lambda_i$ is its algebraic multiplicity, which is given by the characteristic polynomial of $T$.
• The minimal polynomial divides the characteristic polynomial.
• Given an eigenvalue $\lambda_i$, its multiplicity in the minimal polynomial is the size of its largest Jordan block.
So what can you tell about the Jordan form of $T$ according to what we have above?
You have exploited the third bullet point to give the possible characteristic polynomials:
$$p_1(z)=(z-2i)^2(z+7)^2,\quad p_2(z)=(z-2i)(z+7)^3.$$
Now suppose it is $p_1$. ($p_2$ for your exercise.)
• The diagonal must consist of $2i$ and $-7$.
• Each of $2i$ and $-7$ appears twice in the diagonal.
• The biggest block for $-2i$ is $(-2i)$ and the one for $-7$ is $\begin{pmatrix}-7&1\\0&-7\end{pmatrix}$.
So you can conclude that the Jordan form for $T$ should be $$\begin{pmatrix} -2i&&&\\ &-2i&&\\ &&-7&1\\ &&0&-7 \end{pmatrix}.$$
Hint:
For a given eigenvalue $\lambda$, let $d_i=\dim \ker (T-\lambda I)^i$. A general result is the following:
The subspaces $\ker (T-\lambda I)^i$ constitute an eventually constant, non-decreasing sequence, and $\;d_i-d_{i-1}\;(i\ge 1)$ is equal to the number of Jordan blocks of size $\ge i$. | 2019-05-25 22:03:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9548500776290894, "perplexity": 96.7387675165792}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232258451.88/warc/CC-MAIN-20190525204936-20190525230936-00051.warc.gz"} |
https://goldensyrupgames.com/blog/2023-01-14-gobgp-windows/ | # BGP on Windows Desktop
I love seeing BGP pushed to the edges of its intended use-case, and I’d been wanting to learn Go for a while. I’d also been frustrated that I couldn’t ingest BGP routes on my Windows desktop - it’s only included in the server OS versions. So I decided that my holiday project would be extending GoBGP to get my network’s BGP routes into my Windows 10 desktop’s routing table.
I’m also a huge fan of “Why not?” projects. This one falls firmly in that camp!
I think this is also the first instance of BGP running on a Windows desktop SKU? I hadn’t been able to find any other way before this (otherwise I’d have used it) but it could just be The Algorithm thinking I’m looking specifically for Windows Server BGP.
This was a fun project. I mistakenly assumed GoBGP worked more or less like Free Range Routing with Zebra, that it already modified the route table of the host it’s running on. This was incorrect - it was a pure route server only - so I got to add a bit more code than I otherwise would have.
Go has great cross-platform support, except when you use some unabstracted details.
GoBGP uses some unabstracted details.
The Windows build was failing with some incompatible syscall concepts. I searched the GoBGP issue tracker and found that joshuaclausen had already found the same issues and proposed a fix (thanks Josh), but it was never merged because there was some undesired MD5 code in the commits.
From here I started looking at the failing files and worked out the project’s cross-platform support pattern: it’s basically <filename>_<platform>.go with different build constraints to only use each file when building for the selected platform. I used Josh’s PR as a base, made the minimum required changes, made sure existing builds were unaffected and added it to upstream.
At this point GoBGP would run on Windows and act like a full route server, but not do anything with them yet. Noice.
I had routes in a structured Go format, now I needed to modify the Windows routing table. Ideally I wanted to avoid adding extra dependencies but this isn’t in the standard library at time of writing. I searched around and out of the options I found I selected the Windows routing package of the wireguard-go project, because if you can’t trust Jason Donenfeld who can you trust. I whipped up a quick simple test to confirm it worked and moved onto planning the code structure.
I reviewed the existing GoBGP code and found that the existing Zebra integration (GoBGP can’t modify the Linux routing table directly but can interface with Zebra/FRR to do it) was a good reference for it. This is where the project got bigger and I got to learn a little more than I was expecting.
### YANG
It turns out that GoBGP generates its configuration from a YANG model - a data modelling language for the enterprise networking domain. I hadn’t seen this done before but it makes good sense - the underlying BGP configuration across network devices is approximately standards-based and that’s what most of the configuration of a BGP route server will involve. Way less reinventing the wheel. I read through the overview in the spec, extended the GoBGP model and rendered it to Go code.
Sidenote: YANG is neat. It has re-use, imports with smooth modifications, comments (hi json). Section 4 of the RFC is a great, readable overview. Also - this statement:
YANG resists the tendency to solve all possible problems, limiting the problem space to allow expression of data models for network management protocols
Mmmmm.
Tracing out the Zebra integration (called zapi from here on) showed the following:
• /tools/pyang_plugins/gobgp.yang defines the user-facing configuration options for Zebra in YANG format
• pyyang converts this into Go structures in internal/config/bgp_configs.go
• If Zebra is enabled, InitialConfig() in pkg/config/config.go calls EnableZebra()
• EnableZebra() in pkg/server/server.go calls NewZebraClient() to create a new Zebra client and add it to the BgpServer singleton
• NewZebraClient() in pkg/server/zclient.go sets up the connection to Zebra and kicks off loop() as a goroutine
• loop() in pkg/server/zclient.go watches for events from BgpServer and the Zebra daemon and triggers a matching effect on the opposite side
### Implementing
I mirrored the zapi integration to get the loop up, then used the wireguard-go wrapper around the native Windows route table API to inject learned routes. Seeing BGP-learned routes in a Windows desktop OS’s routing table for the first time felt good. It felt cursed. It felt cursedgood.
After that I added an extra feature that the zapi code doesn’t have - when GoBGP shuts down cleanly the routes are removed for cleanliness. The stop code was fun to add - I got to use some of the newerish higher-level Go concurrency additions. The loop is stopped via a cancelFunc passed via context, and the loop stopping is awaited via a WaitGroup. From what I understand this can be done with basic channels, but the newer methods scale better?
You can see the full changes here.
## Using it yourself
This change hasn’t been merged to upstream yet but you can grab it from a branch on my fork (words are fun). This assumes you’re on Windows 10:
Build:
# install go from https://go.dev/doc/install
# then restart your powershell instance to ensure PATH is updated
cd ~\Documents
git clone https://github.com/GSGBen/gobgp.git
cd gobgp
go build .\cmd\gobgp\
go build .\cmd\gobgpd\
Create a config file in the same directory called gobgp.toml, changing the bits to match your external BGP peer:
# the BGP details of the desktop you're running GoBGP on
[global.config]
as = 65001
router-id = "10.1.1.10"
[[neighbors]]
[neighbors.config]
peer-as = 65002
[experimental]
[experimental.modify-host-fib]
[experimental.modify-host-fib.config]
enabled = true
Run a new powershell instance as administrator (required to modify the routing table) then start gobgp:
cd ~\Documents\gobgp\
.\gobgpd.exe -f gobgp.toml
If your BGP settings were correct you should see some routes come through in a bit. And route print should now show them!
## Limitations and extending
This currently only adds routes to the route table - I didn’t have the scope to redistribute routes because I’d want filtering in place first.
It also only injects IPv4 routes. I don’t have a full IPv6 environment any more to test with so I scoped it down. Theoretically this should just be another address family case and a few functions you could add in pkg/server/modify_host_fib_windows.go.
The full dev documentation can be found here.
## Videogames
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See full gameplay on Steam! | 2023-02-02 15:31:59 | {"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": 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": 0.2141495645046234, "perplexity": 4077.7573622044065}, "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/1674764500028.12/warc/CC-MAIN-20230202133541-20230202163541-00558.warc.gz"} |
https://stats.stackexchange.com/questions/560751/if-every-event-is-trivial-0-or-1-probability-then-every-random-variable-is-a | # If every event is trivial (0 or 1 probability), then every random variable is (a.s.) degenerate/constant. Maybe Lebesgue decomposition?
There are these: 1, 2, 3, but I wanna try different ways.
Let $$(\Omega, \mathfrak{F}, \mathbb P)$$ be such probability space with each (event) $$E \in \mathfrak F$$ having trivial probability. Consider a random variable $$X$$.
An attempt:
Show that $$X$$ is almost surely equal to $$c = \inf\{x:F_{X}(x)=1\}$$, with $$F_X$$ as cdf (cumulative distribution function or, well, just distribution function) of $$X$$.
Actually $$c$$ is finite. If $$c=\pm \infty$$, then $$(x:F_{X}(x)=1) = \emptyset$$ or $$\mathbb R$$. Then $$F_X(x) < 1 \ \forall x$$ or $$F_X(x) = 1 \ \forall x$$.
By definition of such finite c, $$P(X \le c) = F_X(c) = 1 = F_X(d) \forall \ d \ge c$$
Now either $$P(X \ge c) = 1$$, in which case we're done or $$P(X \ge c) = 0$$.
Now $$P(X \ge c) = 0 \to P(X < c) = 1 \to \exists \ d < c$$ s.t. $$P(X \le d) = 1$$
But $$\forall \ d < c$$,
$$P(X \le d) = F_X(d) < 1 \to P(X \le d) = 0$$
Question 1: Is this correct?
Question 2: Is there perhaps an even simpler way to go about this namely without guessing what the constant $$c$$ is?
Actually, what I might try to do is show the $$c$$ exists and then later (not sure if I have to 1st show $$c$$ is finite for this next part) say it's the infimum (minimum actually? idk) of when the cdf reaches 1.
I think to consider any extended real $$a$$ and then the event $$\{X=a\}$$. Well, we can't have $$P(X=a)=1$$ for more than 1 extended real $$a$$. But from this, all we can say is that $$P(X=a)=1$$ for at most one possible $$a$$.
Of course it's possible we don't have $$P(X=a)=1$$ for any $$a$$, i.e. we have $$P(X=a)=0$$ for all $$a$$ such as when $$X$$ is a continuous(/an absolutely continuous or whatever) random variable. (But I guess this is easy to show for every discrete random variable [even if we don't assume finite] ?)
But if $$X$$ were continuous then $$P(X \in (g,h)) = 1$$ means...yeah you could probably argue a contradiction here to show continuous random variables don't exist in such a probability space (and apparently neither do binomial distributions) but in higher probability of course not all random variables are either discrete or continuous.
Well, maybe this simpler approach works for only half the battle namely the at most 1 part, but the at least 1 part really requires more.
• Note: Simpler need not mean easier/less machinery. Could be more difficult/more machinery. Like perhaps Lebesgue's decomposition theorem can be used? What I understand of this theorem is that every random variable basically has a 'continuous part' and a 'discrete part' (Eg this and this). Maybe I can somehow show that in such a probability space every random variable is discrete and then (immediately?) conclude every random variable is a.s. constant?
• What is the problem with considering the events $\{X \le a\}$? They should have probability $1$ for some $a$ and $0$ for other $a$ Jan 17 at 10:23
• ah thanks! so basically that's what i'm doing in the attempt? like either they're all 1 so $X=-\infty$ a.s., they're all 0 so $X = \infty$ a.s. or they're all 1 before some $a$ and all 1 after $a$ and then this 3rd case $a$ is such $\inf$ and then we somehow rule out 1st and 2nd cases?
– BCLC
Jan 17 at 14:57
• Argue the contrapositive. That means assuming $X$ is a random variable that is not a.s. constant. Show this means there is a number $x$ for which $\Pr(X\le x) \gt 0$ and $\Pr(X\gt x) \gt 0.$ Derive a contradiction from this. Although of course you can use "machinery," that's not an effective question because it's personal and subjective: what constitutes "machinery" depends on what you know, how you know it, and how you think of it.
– whuber
Jan 17 at 15:34
• @whuber i just knew there had to be something simpler. i mean hell $c=\inf\{\cdot\}$? come on! thanks! you can post as answer Edit: wait at least for question 2. what's the answer to question 1 please?
– BCLC
Jan 17 at 21:48
• You did a good job with #1, because you thought of proving that $c$ has to exist (that is, is finite).
– whuber
Jan 17 at 22:17 | 2022-07-06 06:16:23 | {"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": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8779928684234619, "perplexity": 360.40468651857475}, "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/1656104668059.88/warc/CC-MAIN-20220706060502-20220706090502-00433.warc.gz"} |
https://stacks.math.columbia.edu/tag/06UE | Lemma 99.23.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent
1. $f$ is quasi-DM,
2. for any morphism $V \to \mathcal{Y}$ with $V$ an algebraic space there exists a surjective, flat, locally finitely presented, locally quasi-finite morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ where $U$ is an algebraic space, and
3. there exist algebraic spaces $U$, $V$ and a morphism $V \to \mathcal{Y}$ which is surjective, flat, and locally of finite presentation, and a morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ which is surjective, flat, locally of finite presentation, and locally quasi-finite.
Proof. The implication (2) $\Rightarrow$ (3) is immediate.
Assume (1) and let $V \to \mathcal{Y}$ be as in (2). Then $\mathcal{X} \times _\mathcal {Y} V \to V$ is quasi-DM, see Lemma 99.4.4. By Lemma 99.4.3 the algebraic space $V$ is DM, hence quasi-DM. Thus $\mathcal{X} \times _\mathcal {Y} V$ is quasi-DM by Lemma 99.4.11. Hence we may apply Theorem 99.21.3 to get the morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ as in (2).
Assume (3). Let $V \to \mathcal{Y}$ and $U \to \mathcal{X} \times _\mathcal {Y} V$ be as in (3). To prove that $f$ is quasi-DM it suffices to show that $\mathcal{X} \times _\mathcal {Y} V \to V$ is quasi-DM, see Lemma 99.4.5. By Lemma 99.4.14 we see that $\mathcal{X} \times _\mathcal {Y} V$ is quasi-DM. Hence $\mathcal{X} \times _\mathcal {Y} V \to V$ is quasi-DM by Lemma 99.4.13 and (1) holds. This finishes the proof of the lemma. $\square$
## Comments (0)
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https://socratic.org/questions/element-having-variable-valency-1-and-3 | # Element having variable valency 1 and 3?
Jul 10, 2017
Thallium and gold have valencies of 1 and 3.
#### Explanation:
Thallium
Thallium is about 170 times as abundant as gold in the Earth's crust.
The electron configuration of thallium is $\text{[Xe] 6s"^2 4"f"^14 "5d"^10 "6p}$.
It is easy to remove the $\text{6p}$ electron and form $\text{Tl"^"+}$.
The $\text{6s}$ subshell contains the next-most easily removed electrons.
Removal of these electrons gives $\text{Tl"^"3+}$ with electron configuration ${\text{[Xe] 4"f^14 "5d}}^{10}$.
Thus, some common compounds of thallium are:
• thallium(I) sulfide $\text{Tl"_2"S}$,
• thallium(I) sulfate ${\text{Tl"_2"SO}}_{4}$
• thallium(III) fluoride ${\text{TlF}}_{3}$
• thallium(III) oxide ${\text{Tl"_2"O}}_{3}$
Gold
The electron configuration of gold is ${\text{[Xe] 6s 4"f^14 "5d}}^{10}$.
It is easy to remove the $\text{6s}$ electron and form $\text{Au"^"+}$.
The $\text{5d}$ subshell contains the next-most easily removed electrons.
Another common state of gold is $\text{Au"^"3+}$ with electron configuration ${\text{[Xe] 4"f^14 "5d}}^{8}$.
Thus, the most common compounds of gold are
• gold(I) chloride $\text{AuCl}$
• gold(III) chloride ${\text{AuCl}}_{3}$
• chlorauric acid ${\text{HAuCl}}_{4}$
Here's a table giving the most common oxidation numbers of the elements in their compounds. | 2019-10-17 05:56:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 19, "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": 0.8425125479698181, "perplexity": 7103.59956792214}, "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/1570986672723.50/warc/CC-MAIN-20191017045957-20191017073457-00087.warc.gz"} |
http://eprints.adm.unipi.it/1341/ | # Minimization of $λ_2(Ω)$ with a perimeter constraint
Bucur, Dorin and Buttazzo, Giuseppe and Henrot, Antoine (2009) Minimization of $λ_2(Ω)$ with a perimeter constraint. Indiana University mathematics journal, 58 (6). pp. 2709-2728. ISSN 1943-5258
Full text not available from this repository.
## Abstract
We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $\gamma$ lower semicontinuous.
Item Type: Article Imported from arXiv Area01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica Dipartimenti (from 2013) > DIPARTIMENTO DI MATEMATICA dott.ssa Sandra Faita 05 Aug 2013 12:48 05 Aug 2013 12:48 http://eprints.adm.unipi.it/id/eprint/1341
### Repository staff only actions
View Item | 2018-06-23 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": 0.9224830269813538, "perplexity": 2017.2237149657444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267865181.83/warc/CC-MAIN-20180623190945-20180623210945-00041.warc.gz"} |
https://brilliant.org/problems/factorial-problem/ | # Factorial problem
Number Theory Level pending
Find the number of solutions in natural number of the equation a!+b!+c!=3^d where a≥b≥c.
× | 2017-10-22 19:19:22 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8217606544494629, "perplexity": 8245.458161314482}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825436.78/warc/CC-MAIN-20171022184824-20171022204824-00301.warc.gz"} |
https://ericmjl.github.io/blog/2018/8/1/nxviz-05-released/ | # nxviz 0.5 released!
written by Eric J. Ma on 2018-08-01
A new version of nxviz is released!
In this update, I have added a declarative interface for visualizing geographically-constrained graphs. Here, nodes in a graph have their placement constrained by longitude and latitude.
An example of how to use it is below:
In the GeoPlot constructor API, the keyword arguments node_lat and node_lon specify which node metadata are to be used to place nodes on the x- and y- axes.
By no means do I intend for GeoPlot to replace more sophisticated analysis methods; like seaborn, the interface is declarative; for me, the intent is to provide a very quick-and-dirty way for an end user to visualize graphs with spatially constrained nodes. | 2019-06-19 13:30:32 | {"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": 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": 0.17597924172878265, "perplexity": 3315.2271471189947}, "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-2019-26/segments/1560627998986.11/warc/CC-MAIN-20190619123854-20190619145854-00288.warc.gz"} |
https://tex.stackexchange.com/questions/178861/hidden-footnotes-in-presentation | # Hidden Footnotes in presentation
I have a presentation where bullet points appear after each other. In one of those I am citing a work using \footnote. Unfortunately, the command does not care about the \pause command, meaning that the citation appears already on the first slide of the frame. Down below is a minimal working example. Ideally, I would like that the citation appears simultaneously with the second bullet point.
It would be great if someone could help me out. In particular, any command which does the job is more than welcome.
Cheers, Max
\documentclass[10pt,xcolor={usenames,dvipsnames}]{beamer}
\begin{document}
\begin{frame}
\begin{itemize}
\item Unfortunately, the footnote appears already before the second point
\item \pause ... where the actual citation takes place\footnote{Citation}
\end{itemize}
\end{frame}
\end{document}
• Welcome to TeX.SX!
– user31729
May 18, 2014 at 10:14
• \only<2->{\footnote{Citation}} seems to work. May 18, 2014 at 10:24
• Thank you very much, egreg! That is what I was looking for! Max May 18, 2014 at 11:01
• @egreg: Make this an answer. OP: Please accept egreg's answer if he makes one, otherwise accept Hunsu's. Jun 8, 2016 at 8:52
\only<2->{\footnote{Citation}}
That's mean that \footnote{Citation} will appear just after you next your presentation on that slide. | 2022-05-19 05:08:19 | {"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": 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": 0.4289668798446655, "perplexity": 1427.69448215641}, "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/1652662525507.54/warc/CC-MAIN-20220519042059-20220519072059-00265.warc.gz"} |
https://stats.stackexchange.com/questions/530158/why-accuracy-is-divided-by-the-number-of-classes/530160 | # Why accuracy is divided by the number of classes?
I am doing simple image classification using CNN. My accuracy always equals one divided by the number of classes. For example, for one class it is 100%, for two classes it is 50%, for three it is 33%, for four 25% and so on. Could you please help me with this issue? In what conditions something like this happens? The inputs to the network are 240*64 tensors. I have tried normalizing them but it didn't help. Here is the network I am using. This network is working well for MNIST data but not for my data:
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__()
self.conv1 = nn.Conv2d(1, 10, kernel_size=1)
self.conv2 = nn.Conv2d(10, 20, kernel_size=5)
self.conv2_drop = nn.Dropout2d()
self.fc1 = nn.Linear(20*14*58, 50)
self.fc2 = nn.Linear(50, 4)
def forward(self, x):
x = F.relu(F.max_pool2d(self.conv1(x), 2))
x = F.relu(F.max_pool2d(self.conv2_drop(self.conv2(x)), 2))
x = x.view(-1, 20*14*58)
x = F.relu(self.fc1(x))
x = F.dropout(x, training=self.training)
x = self.fc2(x)
return F.log_softmax(x)
And here is the training procedure:
network.train()
pred=0
correct=0
for batch_idx, (data, target) in enumerate(train_loader):
output = network(data.double())
loss = F.nll_loss(output, target)
top_p, top_class = output.topk(1, dim=1)
pred1 = top_class.flatten().long()
loss.backward()
optimizer.step()
pred = output.data.max(1, keepdim=True)[1]
correct += pred.eq(target.data.view_as(pred)).sum()
target = np.round(target.detach())
y_pred.extend(pred.tolist())
y_true.extend(target.tolist())
CF = confusion_matrix(y_true, y_pred)
#print( skm.classification_report(y_true,y_pred))
if batch_idx % log_interval == 0:
print('Train Epoch: {} [{}/{} ({:.0f}%)]\tLoss: {:.6f}'.format(
100. * batch_idx / len(train_loader), loss.item()))
train_losses.append(loss.item())
train_counter.append(
$$$$
• It looks like your classifier is not working better than chance. – mdewey Jun 10 at 15:21
• Look at your confusion matrix. – Arya McCarthy Jun 10 at 18:52
If you find accuracy exactly equal to one of the class proportions, it may indicate that your classifier is just outputting a constant result - it labels every sample as one of the classes, and only gets the ones that are actually that class correct. This is especially true if the accuracy is equal to the majority class proportion, as it's the simplest way to "maximize" accuracy. If you find an accuracy close to one of the class proportions, your method may just be producing random results, but if you find accuracy exactly equal to a class proportion, it's a big red flag that you're just classifying everything as belonging to that class.
Take a look at your confusion matrix - is your classifier actually making different predictions, or is it outputting the same result regardless of input?
• Thanks for your answer. I've checked the confusion matrix as well as the classification report. The precision, recall, and F1 score values are updating at every iteration. However, from the four labels that I have, it seems like the true predictions for labels '1' and '3' are significantly bigger than labels '2' and '4'. Do you have any opinion on what the reason might be? It seems like the network is predicting labels '2' and '4' only a few times. – gazelle Jun 11 at 10:40
• Just an update, the problem solved with another training part and the suggested network by @Avelina in the next answer. – gazelle Jun 13 at 18:32
Whenever you see an accuracy of $$\frac{1}{NumClasses}$$ after a few epochs it is an immediate red flag that your network isn't learning anything at all. It may be outputting a constant - as Nuclear Hoagie suggested - or it could be outputting noise.
This can be due to vanishing gradients on the backwards pass, exploding gradients on the forwards pass, too much regularization, lack of normalization, or a number of other factors.
Without knowing the exact structure of your network it's difficult to point out what in particular is happening. If you update your question with a network structure we can further diagnose what the issue might be.
• Thanks, Avelina. I have updated my post with the code. I have tried normalization but it didn't help. Could the problem come from the data itself? I am using tensors as input. Could the reason be the similarity between the tensors under different labels? I have also checked the confusion matrix. It seems like the network is more likely to predict two of the labels out of four most of the time. – gazelle Jun 11 at 10:57
• @gazelle what framework are you using? I'm not familiar with the syntax used there. It doesn't look like tensorflow keras to me, nor does it look like pytorch code although I'm not very familiar with pytorch. Maybe your network architecture isn't particularly suited for the task. I tend to use exclusively 3x3 and 1x1 convolutions, with filter counts doubling after every couple convolutions, starting with 64 filters, but for mnist a lower number like 16 sounds good. – Avelina Jun 11 at 12:29
• @gazelle can you try a network architecture i specify here? its a network i'm sure will work, so if it still doesn't work we know something else in your training code is broken. conv2d( 3x3, 16 filters ) -> relu -> batch norm -> max pool 2d conv2d( 3x3, 32 filters ) -> relu -> batch norm -> max pool 2d conv2d( 3x3, 64 filters ) -> relu -> batch norm -> flatten fully connected( 128 units ) -> relu -> batch norm fully connected( <num out classes> units ) -> softmax` Also, can I ask the number of classes and the size of your input images? – Avelina Jun 11 at 12:35
• thanks for the comment @Avelina. I have applied this network but I'm still getting 25% accuracy. My inputs are tensors of size 240*64 (with one channel) and I have 4 labels. The tensors are matrices from MATLAB. I have also tried converting them to images and feed them to the network but still I am getting 25% accuracy. – gazelle Jun 11 at 19:05
• Dear @Avelina, the problem solved! Regarding your suggestion that if the accuracy is not changing then the problem is coming from the training part, I've found one of my old codes that the training part was working fine but the accuracies were so small. I've just replaced my network with your suggested network and it worked very well! Could I ask what is the reason behind this layer arrangement working so well? Or if there is a reference explaining this, I will be so happy to go through that. Thanks for all the help. – gazelle Jun 13 at 18:31 | 2021-08-02 06:18:16 | {"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": 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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6415804624557495, "perplexity": 1429.588200257869}, "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-2021-31/segments/1627046154304.34/warc/CC-MAIN-20210802043814-20210802073814-00165.warc.gz"} |
https://www.allmath.com/double-angle-identity.php | # Double angle formula calculator
To find the value of sin2x, cos2x, or tan2x, put the angle in the double angle formula calculator.
The double angle identity calculator finds the value of a double angle for any trigonometric function if the value of an angel is provided. This calculator can compute values of:
• Sin2x
• Cos2x
• Tan2x
## What is a double angle?
Double angle, as the words imply, means to increase the size of the angle to twice its size. It simply means two times of a trigonometric angle i.e 2x in terms of x.
Double angle identities are used to simplify trigonometric calculations.
## Double angle formula
The main formulas used to find a double angle are:
• Sin2x = 2 (sinx).(cosx)
• Cos2x= 2.cos2x -1
• tan2x = (2.tanx)/(1 - tan2x)
You can also calculate half angle of trigonometric identities by using our half angle identity calculator.
## How to find a double angle?
You can learn to find the double angle through an example.
Example
What is the double angle for sin 40°?
Solution
Step 1: Write the formula of sin2x.
Sin2x = 2 (sinx).(cosx)
Step 2: Put the value of the angle into the formula.
Sin2x = 2 (sin 40).(cos 40)
= 2 (0.642)(0.766)
= 0.9836 | 2021-06-19 12:22:56 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.82950758934021, "perplexity": 1976.3433683272488}, "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-20210619111846-20210619141846-00004.warc.gz"} |
https://mathstodon.xyz/@roninkoi/108206715323488111 | In 2019, I wanted have spherical harmonics $$Y_\ell^m$$ in shader, solved in real time instead of from precomputed functions. After looking at some papers, one approach was to solve them using the hypergeometric function, which gives $$P_\ell^m$$ as
$P_\ell^m(x) = \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}} {_2F_1}\left( -n, n + 1, 1-m, \frac{1-x}{2} \right).$
This requires computing a series consisting of many factorials, which is slow and results in precision problems
As it turns out, there is a better approach using Clenshaw's algorithm! A review of different methods is given in arXiv:1410.1748 [physics.chem-ph]. Now I could get up to high values of $$\ell$$ and $$m$$ without everything blowing up! From the ALPs we can then simply obtain the SHs as
$Y_\ell^m(\theta, \phi) = \sqrt{\frac{(2 \ell + 1) (\ell - m)!}{4\pi (\ell + m)!}} P_\ell^m (\cos \theta) e^{i m \phi}.$
The final code can be viewed at shadertoy.com/view/3dKGWG, which borrows from iq's shader
· · Web · · ·
@roninkoi Very nicely done! I had planned to write an improvement of Quilez's original demo after writing my own implementation of Clenshaw for other orthogonal polynomials (e.g. shadertoy.com/view/tlX3D7), but real life got in the way. Thanks for making it so I no longer have to. ;)
@tpfto Thanks! Your posts have helped me many times over the years
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon! | 2022-05-21 15:17:32 | {"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": 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": 0.5859867930412292, "perplexity": 1399.9387673148294}, "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/1652662539131.21/warc/CC-MAIN-20220521143241-20220521173241-00300.warc.gz"} |
https://wikieducator.org/The1stLawofThermodynamicsLesson4 | # The1stLawofThermodynamicsLesson4
## THE JOULE - THOMSON EXPERIMENT
The Joule experiment (see Fig. 1.5) shows that ΔT = 0 when an ideal gas expands under the condition w = O. By careful measurements Joule and Thomson found a small temperature change, usually negative, when a real gas expanded adiabatically through a porous plug. The experimental arrangement is shown in Fig. 4.1.
fig 4.1 Adiabatic expansion of a real gas through a porous plug.
Fig. 4.1. The Joule - Thomson experiment .Adiabatic expansion of a real gas through a porous plug. (a) Initial state. (b) Final state. The pressures PI and P2 are kept constant during the expansion. The work supplied to the gas on the left hand side of the porous plug is P1VI,and the work carried out by the gas on the right hand side of the porous plug is P2V2. The net work received by the gas is:
w = P1V1-P2V2 (4.2)
The process is adiabatic, q = O. From the first law we have:
ΔU =U2-U1= w = P1V1-P2V2
which gives:
U2+P2V2=U1+P1V1
From the definition of enthalpy (eq. (4.11)) we obtain:
H2 = H1 (4.3)
This means that the enthalpy is constant during the expansion. Joule and Thomson observed that the pressure change, ΔP = P2 - PI, gave a change in temperature, ΔT = T2 - T1. For most gases at room temperature one observes a positive ratio ΔT/ΔP. The differential:
(4.4)
is called the Joule - Thomson coefficient. The total differential, dH, can be expressed by the Joule ¬Thomson coeffisient. For the function H = f(P, 1), the total differential is expressed by eq. (2.12):
The last term is equal to CpdT (compare eq. (3.2)). For the Joule - Thomson experiment dH = 0, see eq. (4.3). Dividing eq. (2.12) with dT for dH = 0 we obtain:
or:
(4.5)
Thus the change in enthalpy with changes in P and T is equal to:
dH = - CpμdP + CpdT (4.6)
This equation is used in calculations of the enthalpy of real gases at high pressures when experimental values are known for the Joule - Thomson coefficient. Joule - Thomson expansion is important in refrigeration and in the liquefaction machine for condensing gases to liquids at very low temperatures.
## FUNDAMENTAL EQUATIONS
The first law of thermodynamics:
dU=dq+dw(4.7) (4.7)
For a cyclic process:
(4.8)
The reversible pressure - volume work:
(4.9)
Isothermal reversible pressure - volume work supplied to an ideal gas:
w= -nRT ln (V2/V1) (4.10)
Definition of enthalpy:
H= u+pv (4.11)
Heat capacities:
Cv= (∂U/∂T)y (4.12)
Cp = (∂H/∂T)p (4.13) | 2021-01-16 21:17:19 | {"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": 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": 0.6767938733100891, "perplexity": 2432.821568097809}, "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/1610703507045.10/warc/CC-MAIN-20210116195918-20210116225918-00515.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/112252-dual-bases.html | # Math Help - Dual Bases
1. ## Dual Bases
I am having a really hard time understanding what a dual basis is.
I understand that $V^*=\mathcal{L}(V, F)$ is the vector space of linear transformations from a vector space V onto its field of scalars F, also known as linear functionals.
I get this. What I am struggling on is how to find the dual basis and why it is defined the way it is.
We call the ordered basis $\beta ^*= \{ f_1, f_2, ..., f_n \}$ of $V^*$ that satisfies $f_i (x_j)=\delta_{ij}$, $1 \le i,j \le n$, the dual basis of $V^*$
The book only gave on example using $\mathbb{R}^2$. I followed what they were doing but it doesn't help me understand the dual basis any better.
Based on the way we set up bases for any other vector space, I quess I figured that the dual basis would somehow be a set of linear functionals such that every linear functional could be written as a linear combination of these linear functionals.
I don't know, I am lost. Can someone explain this to me better or possibly refer me to a good website to read in more depth about dual spaces and dual bases?
2. Perhaps I should be more specific in my confusion.
In a problem that I am working on I am asked a question involving the vector space $V=P( \mathbb{R} )$, the vector space of polynomials with coefficients from $\mathbb{R}$.
I know that $\beta = \{ 1, x, x^2, ... \}$ is a basis for $V$.
If I blindly apply the definition of a Dual Basis (which I mentioned previously) then I know,
$f_1(1)=1$
$f_1(x)=0$
...
and $f_2(1)=0$
$f_2(x)=1$
$f_2 \left( x^2 \right) = 0$
and so on...
What does this mean?
I don't even know if knowing what $V^*$ is will help me with this problem but I want to know.
Does this mean that $f_1$ is some linear functional that takes the first vector in my basis and maps it to 1 and every other vector in the basis to zero?
What does $V^*$ look like??
3. Originally Posted by Jen
I am having a really hard time understanding what a dual basis is.
I understand that $V^*=\mathcal{L}(V, F)$ is the vector space of linear transformations from a vector space V onto its field of scalars F, also known as linear functionals.
I get this. What I am struggling on is how to find the dual basis and why it is defined the way it is.
We call the ordered basis $\beta ^*= \{ f_1, f_2, ..., f_n \}$ of $V^*$ that satisfies $f_i (x_j)=\delta_{ij}$, $1 \le i,j \le n$, the dual basis of $V^*$
No: if $X:=\{x_i\}_{i=1}^n$ is a basis of V, then $\{f_i\}_{i=1}^n$ as defined above is THE dual basis of X in $V^*$ , or wrt the basis X.
Its importance resides,among other possible things, on the fact that it is one simple way to show that when we're dealing with finite dimensional vector spaces, then $V\cong V^*$
The book only gave on example using $\mathbb{R}^2$. I followed what they were doing but it doesn't help me understand the dual basis any better.
Based on the way we set up bases for any other vector space, I quess I figured that the dual basis would somehow be a set of linear functionals such that every linear functional could be written as a linear combination of these linear functionals.
Exactly, but NOT only: every element in V* indeed can be expressed as a lin. comb. of a dual basis, WHERE the elements of this basis get very simple and explicit values on some given basis of V.
I don't know, I am lost. Can someone explain this to me better or possibly refer me to a good website to read in more depth about dual spaces and dual bases?
Any decent linear algebra book covers all this subject, with dual and double dual spaces (V** is canonically isomorphic to V, for example), etc.
Tonio
4. Originally Posted by Jen
Perhaps I should be more specific in my confusion.
In a problem that I am working on I am asked a question involving the vector space $V=P( \mathbb{R} )$, the vector space of polynomials with coefficients from $\mathbb{R}$.
I know that $\beta = \{ 1, x, x^2, ... \}$ is a basis for $V$.
If I blindly apply the definition of a Dual Basis (which I mentioned previously) then I know,
$f_1(1)=1$
$f_1(x)=0$
...
and $f_2(1)=0$
$f_2(x)=1$
$f_2 \left( x^2 \right) = 0$
and so on...
What does this mean?
I don't even know if knowing what $V^*$ is will help me with this problem but I want to know.
Does this mean that $f_1$ is some linear functional that takes the first vector in my basis and maps it to 1 and every other vector in the basis to zero?
What does $V^*$ look like??
It's hard to know if dual basis will help you if you are not explicit about the problem you're dealing with.
V* is just a set of functions with a binary operation on it and a multiplication with scalars from a field which makes it into a linear space.
Tonio
5. Yeah, I know that you have no way of knowing if understanding the dual basis will help in a specific problem if I don't offer it up, but I don't want to post the full questions because somtimes people just answer the whole thing rather than giving hints and that doesn't help me, so I just post questions about the question.
Sorry, I asked previously what $V^*$ looked like. I meant to ask what the basis for $V^*$ looked like.
Is there a way to come up with an explicit basis for $\left( P( \mathbb{R} )\right)^*$?
I mean I know that it is an infinite dimensional basis so obviously we couldn't list every element... | 2015-09-05 00:08:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 45, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8685107231140137, "perplexity": 156.311022851416}, "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/1440645367702.92/warc/CC-MAIN-20150827031607-00178-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://2020.ieee-isit.org/Papers/ViewPaper.asp?PaperNum=1404 | # Technical Program
## Paper Detail
Paper ID O.4.4 Paper Title Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks Authors Charalambos D. Charalambous, University of Cyprus, Cyprus; Jan H. van Schuppen, Van Schuppen Control Research, Netherlands Session O.4: Multi-terminal Source Coding II Presentation Lecture Track Source Coding Manuscript Click here to download the manuscript Virtual Presentation Click here to watch in the Virtual Symposium Abstract The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) $X_1 : \Omega \rightarrow {\mathbb R}^{p_1}$ and $X_2 : \Omega \rightarrow {\mathbb R}^{p_2}$, with respect to square-error distortion at the two decoders is re-examined using (1) Hotelling's geometric approach of Gaussian RVs-the canonical variable form, and (2) van Putten's and van Schuppen's parametrization of joint distributions ${\bf P}_{X_1, X_2, W}$ by Gaussian RVs $W : \Omega \rightarrow {\mathbb R}^n$ which make $(X_1,X_2)$ conditionally independent, and the weak stochastic realization of $(X_1, X_2)$. Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions ${\bf E}\big\{||X_i-\hat{X}_i||_{{\mathbb R}^{p_i}}^2 \big\}\leq \Delta_i \in [0,\infty], i=1,2$, by the covariance matrix of RV $W$. From this then follows Wyner's common information $C_W(X_1,X_2)$ (information definition) is achieved by $W$ with identity covariance matrix, while a formula for Wyner's lossy common information (operational definition) is derived. | 2020-08-05 22:35:53 | {"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": 0.38615456223487854, "perplexity": 1862.511131143676}, "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/1596439735989.10/warc/CC-MAIN-20200805212258-20200806002258-00439.warc.gz"} |
http://meria.sumy.ua/the-pas-ocksev/4e2a72-degree-sum-formula-proof | Now let's use the formulas backwards: look at the expression below: \begin{equation*} \dfrac{\tan 285\degree - \tan 75\degree}{1 + \tan 285\degree \tan 75\degree} \end{equation*} Does it remind you of … To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas. (At this point you might ask what happens if the graph contains loops, You know the tan of sum of two angles formula but it is very important for you to know how the angle sum identity is derived in mathematics. Can we have a graph with 9 vertices and 7 edges? The degree sum formula says that if you add up the degree of all the vertices in a Let the straight line AB revolve to the point C and sweep out the . Lemma 2.2.2 The number of odd degree vertices in a graph is an even number. A vertex is incident to an edge if the vertex is one of the two vertices the edge … As given, the diagrams put certain restrictions on the angles involved: neither angle, nor their sum, can be larger than 90 degrees; and neither angle, nor their difference, can be negative. And half of a half note is a quarter note; and so on. cos. . In the world of angles, we have half-angle formulas. Topic is fram Advanced Graph theory. Modelling shows that your choice of how many households you bubble with this Christmas can make a real difference to the spread COVID-19. This sum is twice the number of edges. It Bipartite graphs, Degree Sum Formula Eulerian circuits Lecture 4. First, recall that degree means the number of edges that are incident to a vertex. It’s natural to ask what is the genus of . The degree sum formula states that, given a graph G = (V,E), the sum of the degrees is twice the number of edges. Proof. It's a formulation based on the whole note. In conclusion, By Lemma 2.2.1 x + y = 2 m. Since x is the sum of even integers, x is even, and … Vieta's formula can find the sum of the roots (3 + (− 5) = − 2) \big( 3+(-5) = -2\big) (3 + (− 5) = − 2) and the product of the roots (3 ⋅ (− 5) = − 15) \big(3 \cdot (-5)=-15\big) (3 ⋅ (− 5) = − 1 5) without finding each root directly. the sum of the degrees equals the total number of incident pairs Summing the degrees of each vertex will inevitably re-count edges. In music there is the whole note. (v, e) is twice the number of edges. Theorem: is a nonsingular curve defined by a homogeneous polynomial . DEV Community © 2016 - 2021. The Cartesian product of a set and the empty set. Step 4. Actually, for all K graphs (complete graphs), each vertex has n-1 degrees, n being the number of vertices. The diagrams can be adjusted, however, to push beyond these limits. \sum_{k=1}^n (2k-1) = 2\sum_{k=1}^n k - \sum_{k=1}^n 1 = 2\frac{n(n+1)}2 - n = n^2.\ _\square k = 1 ∑ n (2 k − 1) = 2 k = 1 ∑ n k − k = 1 ∑ n 1 = 2 2 n (n + 1) − n = n 2. Want facts and want them fast? The quantity we count is the number of incident pairs (v, e) In maths a graph is what we might normally call a network. The trigonometric formula of the tangent of a sum of two angles is derived using the Formulas of the sine and cosine. This change is done in the nominator) (Multiplied 180° with 1 … We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. We're a place where coders share, stay up-to-date and grow their careers. Built on Forem — the open source software that powers DEV and other inclusive communities. same thing, you conclude that they must be equal. When we sum the degrees of all 9 vertices we get 63, since 9 * 7 = 63. A graph G is connected if for each u;v 2V(G), G has a u;v-path (or equivalently a u;v-walk). The degree sum formula states that, given a graph = (,), ∑ ∈ = | |. Bm()x = -2m!Σ s=1 ()2 s m 1 cos 2 sx-2 m 0 x 1 The proof works Proof Our Maths in a minute series explores key mathematical concepts in just a few words. In a similar vein to the previous exercise, here is another way of deriving the formula for the sum of the first n n n positive integers. Copyright © 1997 - 2021. Prove the genus-degree formula. Does the above proof make sense? Summing 8 degrees 9 times results in 72, meaning there are 36 edges. Derivation of Sum and Difference of Two Angles | Derivation of Formulas Review at … A degree is a property involving edges. This statement (as well as the degree sum formula) is known as the handshaking lemma.The latter name comes from a popular mathematical problem, to prove that in any group of people the … You can find out more about graph theory in these Plus articles. Substituting the values, we get-n x k = 2 x 24. k = 48 / n . We will show that it is only related to the degree of athe polynomial defining . Max Max. The number of elements in a power set of size <= 1 is the size of the original set + 1 more element: the empty set . Therefore, the number of incident pairs is the sum of the degrees. By definition of the tangent: The formula implies that in any undirected graph, the number of vertices with odd degree is even. attached to two vertices. The degree sum formula is about undirected graphs, so let's talk Facebook. Hence, (Formation of the equation as per the formula) (We have Subtracted 3 from 2 that yields 1. The whole note defines the duration of all the other notes. The following corollary is immediate from the degree-sum formula. (See, for instance, this answer.) In mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum of the p-th powers of the first n positive integers ∑ = = + + + ⋯ + as a (p + 1)th-degree polynomial function of n, the coefficients involving Bernoulli numbers B j, in the form submitted by Jacob Bernoulli and published in 1713: ∑ = = + + + + ∑ =! Therefore the total number of pairs consists of a collection of nodes, called vertices, connected This requirement is irrelevant, as to any of these angles an angle with a factor of 2π can be added, and this will not affect the validity of the formula of the cosine of the difference of … This is usually the first Theorem that you will learn in Graph Theory. (finite) graph, the result is twice the number of the edges in the graph. So in the above equation, only those values of ‘n’ are permissible which gives the whole value of ‘k’. Step 5. it. The proof of the basic sum-to-product identity for sine proceeds as follows: sin (+ β) = sin cos β + cos sin β : and cos (+ β) = cos cos β − sin sin β. For the second way of counting the incident pairs, notice that each edge is Vertex v belongs to deg(v) pairs, where deg(v) (the degree of v) is the number of edges incident to it. This gives us n triangles and so the sum of … The simplest application of this is with quadratics. With the above knowledge, we can know if the description of a graph is possible. All rights reserved. The sum and difference of two angles can be derived from the figure shown below. equals twice the number of edges. In every finite undirected graph, an even number of vertices will always have odd degree The handshaking lemma is a consequence of the degree sum formula (also sometimes called the handshaking lemma) How is Handshaking Lemma useful in Tree Data structure? Take a quick trip to the foundations of probability theory. Formula 4.1.5 When m is a natural number, x is a floor function and Bm are Bernoulli numbers , Bm x- x = -2m!Σ s=1 ()2 s m 1 cos 2 sx-2 m x 0 Proof According to Formula 5.1.2 (" 05 Generalized Bernoulli Polynomials ") , the following expression holds. Vieta's Formulas can be used to relate the sum and product of the roots of a polynomial to its coefficients. Let's look at K 3, a complete graph (with all possible edges) with 3 vertices. Proof. If you have memorized the Sum formulas, how can you also memorize the Difference formulas? But each edge has two vertices incident to it. There's a neat way of proving this result, which involves That is, the half note lasts half as long as the whole note. Want to shuffle like a professional magician? In the case of K3, each vertex has two edges incident to it. Also known as the explained sum, the model sum of squares or sum of squares dues to regression. Dope. − _ − +, where − _ = − =! Can we have 9 mathematicians shake hands with 8 other mathematicians instead? University of Cambridge. We strive for transparency and don't collect excess data. First, recall that degree means the number of edges that are incident to a vertex. Now, let us check all the options one by one- For n = 20, k = 2.4 which is not allowed. Edges are connections between two vertices. These classes are calledconnected componentsof … by links, called edges. Since the sum of degrees is two times the number of edges the result must be even and the number of edges must be even too. The degree sum formula says that if you add up the degree of all the vertices in a (finite) graph, the result is twice the number of the edges in the graph. Euler's proof of the degree sum formula uses the technique of double counting: he counts the number of incident pairs (v,e) where e is an edge and vertex v is one of its endpoints, in two different ways. the graph equals the total number of incident pairs (v, e) There is an elementary proof of this. Is it possible that each mathematician shook hands with exactly 7 people at the seminar? Let x be the sum of the degrees of even degree vertices and y be the sum of the degrees of odd degree vertices. A simple proof of this angle sum formula can be provided in two ways. Using the distributive property to expand the right side we now have Vieta's Formulas are often used … DEV Community – A constructive and inclusive social network for software developers. Since the 65 degrees angle, the angle x, and the 30 degrees angle make a straight line together, the sum must be 180 degrees Since, 65 + angle x + 30 = 180, angle x must be 85 This is not a proof yet. The first constraint was nonnegativity of the angles. ( x + y) = D J D H. The side H J ¯ divides the side D F ¯ as two parts. Maths in a minute: The axioms of probability theory. … Proof:-(LONG EXPLAINATION:-) We know, Degree of one angle of a polygon equals to (formula): (Where n is the side of the polygon) Hence, In case of a triangle, n will be equal to 3 as their are 3 sides in the triangle. Second approach is to take a point in the interior of the polygon and join this point with every vertex of the polygon. where v is a vertex and e an edge attached to Since half a handshake is merely an awkward moment, we know this graph is impossible. In the degree sum formula, we are summing the degree, the number of edges incident to each vertex. degree of v. Thus, the sum of all the degrees of vertices in I had a look at some other questions, but couldn't find a fully written proof by induction for the sum of all degrees in a graph. The degree of a vertex is The ∠ J D H is x + y in the Δ J D H and write the cos of compound angle x + y in its ratio from. It there is some variation in the modelled values to the total sum of squares, then that explained sum of squares formula is used. Since the sum of degrees is twice the number of edges, we know that there will be 63 ÷ 2 edges or 31.5 edges. Let us consider the Formulas of the cosine of the sum and difference of two angles: By adding them termwise, we find: Based on this, we obtain the proof of the formula of the product of the cosine of α and cosine of β: In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive … in this case as well, we leave that for you to figure out.). tan ( x) + tan ( y) = tan ( x + y) ( 1 − tan ( x) tan ( y)) tan ( x) − tan ( y) = tan ( x − y) ( 1 + tan ( x) tan ( y)). Proof of the Sum and Difference Formulas for the Cosine. = tan(x+ y)(1−tan(x)tan(y)) = tan(x− y)(1+tan(x)tan(y)). If we have a quadratic with solutions and , then we know that we can factor it as: (Note that the first term is , not .) Now, It is obvious that the degree of any vertex must be a whole number. Observe that the relation F(u;v) that G has a u;v-path is reflexive, symmetric and transitive. leave a comment » Take a nonsingular curve in . This just shows that it works for one specific example Proof of the angle sum theorem: Or, in another way, construct a degree sequence for a graph and sum it: sum([2, 2, 2]) # 6. But now I’d like to … The angle sum tan identity is a trigonometric identity, used as a formula to expanded tangent of sum of two angles. Where coders share, stay up-to-date and grow their careers dev Community – a constructive and social. Line AB revolve to the point C and sweep out the ; V ) that G has a ;. Cartesian product of the degrees of even degree vertices in a minute series explores key mathematical in! Line AB revolve to the foundations of probability theory, you conclude that they n't... Proof of this angle sum formula Eulerian circuits Lecture 4 check all the options one one-! C and sweep out the n ’ are permissible which gives the whole of... Sum the degrees of odd degree is even to … sum of the degrees of even degree in... Graphs ( complete graphs ), each vertex has n-1 degrees, n being number. Sum the degrees of each vertex in the interior of the degrees stay up-to-date grow! Quick trip to the spread COVID-19 be solved nicely with one the and! That powers dev and other inclusive communities lemma 2.2.2 the number of edges the genus of ) G! Called vertices, connected by links, called vertices, connected by links, edges! To that vertex are attached to it _ − +, where − _ −... Of vertices vertices to prove that G has a vertex well, we increment our sum by number... The trigonometric formula of the polygon and join this point with every mathematician minus yourself and one person. As per the formula of the tangent of the degrees of even degree and... Bipartite graphs, degree sum formula is about undirected graphs, so 's! Mathematician as a vertex is the number of edges '' bit may seem arbitrary more. Grow their careers by links, called edges J + J F. this is usually the first that... Bit may seem arbitrary solved nicely with one complete graphs ), ∑ ∈ = |.! ; V ) that G has a u ; V ) that G has a vertex,. Helps to represent how well a data that has been modelled called a reference triangle to help find component! Transparency and do n't collect excess data axioms of probability theory triangle to help find each component the! Let G be a whole number with the above knowledge, we half-angle. V ( G ) intoequivalence classes F = D J D H. the side D ¯! X be the sum of squares or sum of the polygon to do so, each. Note ; and so on since 9 * 7 = 63 have 9 mathematicians shake hands choice of many. You will learn in graph theory the trigonometric formula of the equation as per formula... 8 other mathematicians instead are attached to two vertices Transcribed Image Text this... Any tree with at least two vertices find each component of the polygon a! The open source software that powers dev and other inclusive communities, to push beyond these limits series explores mathematical. And y be the sum of squares or sum of squares dues to regression,! Point with every mathematician minus yourself and one other person of all other. The foundations of probability theory question Next question Transcribed Image Text from this question twice. Vertices and n-1 edges − = so on whole number at K3, each has! Mathematicians instead only those values of ‘ k ’ that in any graph... Recall that degree means the number of incident pairs equals twice the number of edges we increment our sum the. Must be a graph is an equivalence relation, and so partitions (. Call a network edge degree sum formula proof and y be the sum of the degrees of each as... Point degree sum formula proof every vertex of the degrees of all 9 vertices and be! Complete graph ( with all possible edges ) with 3 vertices of all =. To relate the sum of degree of athe polynomial defining, each has... Out. ) the empty set to regression n't collect excess data results in 72, there. Maths a graph is an equivalence relation, and so on one of polygon! Suppose the G = (, ), each vertex will inevitably re-count edges formula states that, a... Whole number one other person F. this is usually the first Theorem that you will in... Open source software that powers dev and other inclusive communities times results 72. ¯ divides the side H J ¯ divides the side H J ¯ divides the degree sum formula proof H ¯. D H. the side H J ¯ divides the side H J ¯ divides side!, where − _ − +, where − _ = − = F an. Helps to represent how well a data that has been model has been.! A polynomial to its coefficients of probability theory complete graphs ), vertex. Means the number of vertices with odd degree vertices and n-1 edges graph is possible of,. Any undirected graph, the development of these formulas involves more than si… Bipartite graphs, so let 's at... Others which amounts to shaking hands with every mathematician minus yourself and one person! Equivalence relation, and so on 's talk Facebook the straight line AB revolve to the spread COVID-19,. Constraints on angles α and β use the degree-sum formula for vertices to prove G! Is incident to that vertex the proof works in this case as well, know... Vertices, connected by links, called vertices, connected by links, called vertices, connected by,! Templates let you quickly answer FAQs or store snippets for re-use may not have jumped out at,. A u ; v-path is reflexive, symmetric and transitive equivalence relation and! With exactly 7 people at the seminar it consists of a sum of the tangent of half! Mathematician as a vertex this puzzle can be adjusted, however, to push beyond these limits that any! Mathematician would shake the hand of 7 others which amounts to shaking hands with 7. Minute: the axioms of probability theory straight line AB revolve to the point C and out... Other notes of K3, each vertex has two vertices incident to an edge if vertex... This Christmas can make a real difference to the spread COVID-19 2 x number of odd degree vertices a! It helps to represent how well a data that has been modelled, since 9 * 7 =.! Handshake is merely an awkward moment, we are summing the degrees each edge two., this answer. ) know if the vertex is one of the.! Choice of how many households you bubble with this Christmas can make a real difference to the of! We might normally call a network relation, and the empty set 2.4 which is allowed. N'T shake hands with every mathematician minus yourself and one other person symmetric and transitive symmetric and.. Transparency and do n't collect excess data can know if the description of a graph with edges. Has n-1 degrees, n being the number of edges that are attached to two vertices of degree.. Note ; and so partitions V ( G ) intoequivalence classes social network for software developers explained. − _ = − = push beyond these limits H. the side H J ¯ divides the side D =. To push beyond these limits look at k 3, a complete graph ( with all possible edges with. Collection of nodes, called edges the set V, we leave that for you to figure.... The sum of squares dues to regression degrees 9 times results in 72, meaning there are edges... Imperfectly, and the empty set the two vertices the edge connects F = D D! And grow their careers where − _ − +, where − _ − +, where _... For re-use help find each component of the roots of a half lasts! Intoequivalence classes ’ s natural to ask what is the number of incident pairs, notice that each shook! _ = − =, imperfectly, and so on is immediate from the degree-sum formula for to. Solved nicely with one relation, and so partitions V ( G intoequivalence! Is one of the tangent: in maths a graph is what we might normally a... Way of counting the incident pairs is the genus of as the whole note increment our sum the!, and the magic behind it J + J F. this is usually the first Theorem that will! The twice the number of edges that are attached to two vertices of one. Triangle to help find each component of the degrees of even degree vertices in graph. To the point C and sweep out the of any vertex must be.... Vertex of the proof works in this case as well, we are summing the.! 36 edges axioms of probability theory to figure out. ) mathematician minus yourself and one other.... Any vertex must be a whole number shaking hands with 8 other mathematicians instead nodes, vertices. C and sweep out the component of the sum of the two vertices with 3 vertices has two must! The options one by one- for n = 20, k = 48 / n sum... Collect excess data they ca n't shake hands with 8 other mathematicians instead,! These limits the G = ( V, E ) is a connected graph with 9 we. That degree means the number of edges that are incident to an edge sum...
Corsica Ferries Cabins, 250 Euro To Naira Black Market, Moscow Snow 2021, Danganronpa Characters V2, Life Size Resin Statues, Byron Bay Beach Bure, | 2021-04-14 11:10:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8184458613395691, "perplexity": 590.601039969105}, "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/1618038077810.20/warc/CC-MAIN-20210414095300-20210414125300-00353.warc.gz"} |
https://cstheory.stackexchange.com/questions/25996/a-or-an-with-big-o-notation/26009#26009 | # "a" or "an" with big O notation?
A simple (though somewhat pedantic) question: when using big O notation in a sentence, should it be preceded with "a" or "an"? Example:
The extra loop results in a/an $\mathcal{O}(N)$ increase in cost.
The ubiquitous Cormen textbook notes on page 47:
For a given function $g(n)$, we denote by $\mathcal{O}(g(n))$ (pronounced “big-oh of g of n” or sometimes just “oh of g of n”)
which would seem to indicate "a" should precede something pronounced "big-oh". Later on the same page, however, the textbook uses the phrase "an $\mathcal{O}(n^2)$ upper bound." Which is used more in the literature? It seems like a triviality, but as someone working on the edge of theoretical computer science from a primary focus on another field, I'd like not to sound like an outsider.
• I think the common way to pronounce the offending sentence would have just “oh en increase” with no “big”, so an “an” is appropriate. Oct 8 '14 at 16:54
• Speaking of pedantry, the proper way to typeset the big-oh notation, as it has been used for over a century, is $O(\dots)$. (I observe that the Cormen textbook agrees, I don’t know why you changed it in the quote.) The caligraphic $\mathcal O$ just displays a lack of taste. Oct 8 '14 at 17:11
• (Tell me to shut up if I’m annoying.) While we are on the subject: on the other hand, a calligraphic $\mathcal P$ is an appropriate power-set notation. A depressingly large number of people abuse for that purpose the symbol for the Weierstraß elliptic function $\wp(z;\tau)$. Oct 8 '14 at 19:34
• I may be the only one, but when I am wondering about similar issues I mentally expand "$O(N)$ increase" to "order of $N$ increase". Oct 8 '14 at 20:33
• "...increases the running time by a factor of O(n)." (And pace Cormen, I've always pronounced it "order en".) Oct 12 '14 at 16:29 | 2022-01-21 20:20:46 | {"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": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7445695400238037, "perplexity": 1139.2301445579649}, "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/1642320303709.2/warc/CC-MAIN-20220121192415-20220121222415-00642.warc.gz"} |
https://python-programs.com/python-program-for-maximum-number-of-2x2-squares-that-can-be-fit-inside-a-right-isosceles-triangle/ | # Python Program for Maximum Number of 2×2 Squares That Can be Fit Inside a Right Isosceles Triangle
In the previous article, we have discussed Python Program to Find Slope of a Line
Given the base of the isosceles triangle, the task is to find the count of the maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle.
The side of the square must be parallel to the base of the given isosceles triangle.
Examples:
Example1:
Input:
Given base of triangle = 8
Output:
The maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle = 6
Explanation:
Example2:
Input:
Given base of triangle = 6
Output:
The maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle = 3
## Program for Maximum Number of 2×2 Squares That Can be Fit Inside a Right Isosceles Triangle in python:
Below are the ways to find the count of the maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle:
### Method #1: Using Mathematical Formula (Static Input)
Approach:
• Give the base of the triangle as static input and store it in a variable.
• Create a function to say count_Squares() which takes the given base of the isosceles triangle as an argument and returns the count of the maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle.
• Inside the function, subtract 2 from the given base value as it is the extra part.
• Store it in the same variable.
• Divide the given base of the triangle by 2 since each square has a base length of 2.
• Store it in the same variable.
• Calculate the value of gvn_trianglebase * (gvn_trianglebase + 1) / 2 (Mathematical Formula) and store it in another variable.
• Return the above result which is the count of the maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle.
• Pass the given base of the isosceles triangle to the count_Squares() function and print it.
• The Exit of the Program.
Below is the implementation:
# Create a function to say count_Squares() which takes the given base of the isosceles
# triangle as an argument and returns the count of the maximum number of 2*2
# squares required that can be fixed inside the given isosceles triangle.
def count_Squares(gvn_trianglebase):
# Inside the function, subtract 2 from the given base value as it is the extra part.
# Store it in the same variable.
gvn_trianglebase = (gvn_trianglebase - 2)
# Divide the given base of the triangle by 2 since each square has a base length of 2.
# Store it in the same variable.
gvn_trianglebase = gvn_trianglebase // 2
# Calculate the value of gvn_trianglebase * (gvn_trianglebase + 1) / 2
# (Mathematical Formula) and store it in another variable.
rslt = gvn_trianglebase * (gvn_trianglebase + 1) // 2
# Return the above result which is the count of the maximum number of 2*2 squares
# required that can be fixed inside the given isosceles triangle.
return rslt
# Give the base of the triangle as static input and store it in a variable.
gvn_trianglebase = 6
# Pass the given base of the isosceles triangle to the count_Squares() function
# and print it.
print("The maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle = ",
count_Squares(gvn_trianglebase))
Output:
The maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle = 3
### Method #2: Using Mathematical Formula (User Input)
Approach:
• Give the base of the triangle as user input using the int(input()) function and store it in a variable.
• Create a function to say count_Squares() which takes the given base of the isosceles triangle as an argument and returns the count of the maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle.
• Inside the function, subtract 2 from the given base value as it is the extra part.
• Store it in the same variable.
• Divide the given base of the triangle by 2 since each square has a base length of 2.
• Store it in the same variable.
• Calculate the value of gvn_trianglebase * (gvn_trianglebase + 1) / 2 (Mathematical Formula) and store it in another variable.
• Return the above result which is the count of the maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle.
• Pass the given base of the isosceles triangle to the count_Squares() function and print it.
• The Exit of the Program.
Below is the implementation:
# Create a function to say count_Squares() which takes the given base of the isosceles
# triangle as an argument and returns the count of the maximum number of 2*2
# squares required that can be fixed inside the given isosceles triangle.
def count_Squares(gvn_trianglebase):
# Inside the function, subtract 2 from the given base value as it is the extra part.
# Store it in the same variable.
gvn_trianglebase = (gvn_trianglebase - 2)
# Divide the given base of the triangle by 2 since each square has a base length of 2.
# Store it in the same variable.
gvn_trianglebase = gvn_trianglebase // 2
# Calculate the value of gvn_trianglebase * (gvn_trianglebase + 1) / 2
# (Mathematical Formula) and store it in another variable.
rslt = gvn_trianglebase * (gvn_trianglebase + 1) // 2
# Return the above result which is the count of the maximum number of 2*2 squares
# required that can be fixed inside the given isosceles triangle.
return rslt
# Give the base of the triangle as user input using the int(input()) function
# and store it in a variable.
gvn_trianglebase = int(input("Enter some random number = "))
# Pass the given base of the isosceles triangle to the count_Squares() function
# and print it.
print("The maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle = ",
count_Squares(gvn_trianglebase))
Output:
Enter some random number = 8
The maximum number of 2*2 squares required that can be fixed inside the given isosceles triangle = 6
Find a comprehensive collection of Examples of Python Programs ranging from simple ones to complex ones to guide you throughout your coding journey. | 2022-05-26 04:12:44 | {"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": 0.4262225925922394, "perplexity": 892.4813131594416}, "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/1652662601401.72/warc/CC-MAIN-20220526035036-20220526065036-00546.warc.gz"} |
https://www.yaclass.in/p/science-state-board/class-10/health-and-diseases-13761/abuses-and-lifestyle-modifications-12731/re-5c771b66-3eb5-450f-a33c-55d168256c0a | Theory:
Observing our surroundings, we can see a large number of drug users. Because of their way of life, they are not only being harmed by drugs. Addiction has a wide-ranging impact on the entire family.
Before we can learn about drug addiction, we must first understand the distinction between drug abuse and drug dependence.
The abuse was considered the early stage of inappropriate drug use, resulting in dependence. But drug dependence is a more severe issue than abuse.
When drug abuse progresses to dependence, treatment becomes more difficult. Dependence on illegal drugs can be dangerous if not treated. As the body adapts to the drugs, the dose of the drug may be increased, resulting in overdoes or death. This section will study drug dependence and drug-deaddiction.
Drug dependence:
The condition in which people who consume drugs become entirely dependent on them and cannot live without those drugs is known as drug dependence.
A picture depicting drug dependence word cloud concept
Drug dependence is of two types:
1. Physical and mental dependence
2. Psychological dependence
1. Physical and mental dependence:
It is a condition in which the individual depends on the drug for normal well-being and maintaining their physiological state.
2. Physiological dependence:
It is a feel of an individual that drugs help reduce stress.
Behavioural changes of drug users:
The various adverse effects of drug use among adolescents are as follows:
• Absence from school or college, and a drop in academic achievement.
• The victims lose interest in personal hygiene, leading to isolation, depression, fatigue and aggressive behaviour.
• Relationships with family and friends are weak.
• Diet and sleeping habits change.
• Bodyweight and appetite fluctuations.
• Always on the lookout for a quick way to earn cash to purchase drugs.
• Infections such as AIDS and Hepatitis B are common in these victims.
Drug de-addiction is a complicated and challenging task as it involves the recovery of drug addicts. It is a long and slow process.
Family members, friends, and society play a prominent role in this process.
The process of Drug-deaddiction involves $$5$$ phases as follows:
1. Detoxification:
• It is the first phase of the process.
• During this phase, the drug is gradually discontinued, and the addict is assisted in overcoming withdrawal symptoms.
• Because the addict is experiencing great physical and emotional distress during this procedure, they are given medication to help them.
2. Psychotherapy:
• During this, psychologists and counsellors provide individual and group counselling.
• The treatment includes reducing the addict's stress, teaching new methods to tackle everyday difficulties, eating a healthy diet, resting, and relaxing.
3. Counselling to family members:
During this phase, social workers advise family members on adjusting their rejecting attitudes so that the family and society accept the addict.
4. Rehabilitation:
They are given proper vocational training to live a healthy life and contribute to society.
A picture depicts stopping drug abuse. | 2022-05-25 01:08:00 | {"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": 0.24393391609191895, "perplexity": 6518.246544457991}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00273.warc.gz"} |
https://paperity.org/p/155858192/spectroscopy-and-decays-of-the-fully-heavy-tetraquarks | # Spectroscopy and decays of the fully-heavy tetraquarks
The European Physical Journal C, Aug 2018
We discuss the possible existence of the fully-heavy tetraquarks. We calculate the ground-state energy of the $$bb {\bar{b}} {\bar{b}}$$ bound state, where b stands for the bottom quark, in a nonrelativistic effective field theory framework with one-gluon-exchange (OGE) color Coulomb interaction, and in a relativized diquark model characterized by OGE plus a confining potential. Our analysis advocates the existence of uni-flavor heavy four-quark bound states. The ground state $$bb{\bar{b}}{\bar{b}}$$ tetraquark mass is predicted to be $$(18.72\pm 0.02)$$ GeV. Mass inequality relations among the lowest $$QQ\bar{Q}\bar{Q}$$ states, where $$Q\in \{c, b\}$$, and the corresponding heavy quarkonia are presented, which give the upper limit on the mass of ground state $$QQ\bar{Q}\bar{Q}$$. The possible decays of the lowest $$bb\bar{b}\bar{b}$$ are highlighted, which might provide useful references in the search for them in ongoing LHC experiments, and its width is estimated to be a few tens of MeV.
This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-018-6073-9.pdf
Muhammad Naeem Anwar, Jacopo Ferretti, Feng-Kun Guo, Elena Santopinto, Bing-Song Zou. Spectroscopy and decays of the fully-heavy tetraquarks, The European Physical Journal C, 2018, 647, DOI: 10.1140/epjc/s10052-018-6073-9 | 2019-07-19 13:13:43 | {"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": 0.7608667612075806, "perplexity": 2180.5873926592285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526237.47/warc/CC-MAIN-20190719115720-20190719141720-00316.warc.gz"} |
http://math.stackexchange.com/questions/446488/on-the-pronunciation-of-the-second-derivative | # On the pronunciation of the second derivative
I have been looking at Lancelot Hogben's Mathematics for the Million (first published in 1936).
In the chapter on calculus he says that the second derivative $\displaystyle \frac{d^2y}{dx^2}$ is
pronounced dee-two-wy-by-dee-eks-two.
The final "two" surprised me as I was taught to say "squared" and I have never heard it pronounced differently.
Assuming that it's not a typo and that Prof Hogben wasn't mistaken, did people really say "two" in those days and does anyone say "two" these days?
I suppose that for higher derivatives just saying the number rather than the power it represents may be easier.
-
Well, maybe he meant that $\dfrac{d^2y}{dx^2}$ is $\dfrac{d^2y}{dx}$ too. ;-) – Asaf Karagila Jul 18 '13 at 10:27
i've heard a lot of people saying it this way – Aang Jul 18 '13 at 10:31
I usually say "the second derivative of $y$ with respect to $x$" ... – Neal Jul 18 '13 at 11:03
I say "squared" in the place of the first "two" too: "dee-squared-wy-by-dee-eks-squared". If we can find someone who says "dee-squared-wy-by-dee-eks-two", all $2^2$ possibilities will have been covered. – ShreevatsaR Jul 18 '13 at 11:53
According to this post there are several ways to pronounce $\frac {d^2y}{dx^2}$, including: | 2015-11-28 15:14:26 | {"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": 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": 0.6474649906158447, "perplexity": 447.315231414826}, "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-2015-48/segments/1448398453553.36/warc/CC-MAIN-20151124205413-00068-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.zora.uzh.ch/id/eprint/160223/ | Observation of Correlated Azimuthal Anisotropy Fourier Harmonics in pp and p+Pb Collisions at the LHC
Abstract
The azimuthal anisotropy Fourier coefficients $(vn)$ in $8.16 TeV p+Pb$ data are extracted via long-range two-particle correlations as a function of the event multiplicity and compared to corresponding results in$pp$ and $PbPb$ collisions. Using a four-particle cumulant technique, $v_n$ correlations are measured for the first time in $pp$ and $p+Pb$ collisions. The$v2$ and $v4$ coefficients are found to be positively correlated in all collision systems. For high-multiplicity $p+Pb$ collisions, an anticorrelation of $v2$ and $v3$ is observed, with a similar correlation strength as in PbPb data at the same multiplicity. The new correlation results strengthen the case for a common origin of the collectivity seen in $p+Pb$ and $PbPb$ collisions in the measured multiplicity range.
Abstract
The azimuthal anisotropy Fourier coefficients $(vn)$ in $8.16 TeV p+Pb$ data are extracted via long-range two-particle correlations as a function of the event multiplicity and compared to corresponding results in$pp$ and $PbPb$ collisions. Using a four-particle cumulant technique, $v_n$ correlations are measured for the first time in $pp$ and $p+Pb$ collisions. The$v2$ and $v4$ coefficients are found to be positively correlated in all collision systems. For high-multiplicity $p+Pb$ collisions, an anticorrelation of $v2$ and $v3$ is observed, with a similar correlation strength as in PbPb data at the same multiplicity. The new correlation results strengthen the case for a common origin of the collectivity seen in $p+Pb$ and $PbPb$ collisions in the measured multiplicity range.
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Detailed statistics | 2020-10-20 06:46:44 | {"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": 0.7603747844696045, "perplexity": 1329.4902917806148}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107869933.16/warc/CC-MAIN-20201020050920-20201020080920-00600.warc.gz"} |
http://math.stackexchange.com/questions/334031/proof-by-induction-that-n1n2-cdots2n-frac12n3n1 | # Proof by induction that $(n+1)+(n+2)\cdots+2n=\frac{1}{2}n(3n+1)$
Prove by induction that $(n+1)+(n+2)\cdots+2n=\frac{1}{2}n(3n+1)$
I was not really sure how to do this, but I assumed that the case holds for $n=k$, therefore $\displaystyle\sum_{r=1}^kk+r=\frac{1}{2}k(3k+1)$.
Want it on this form:$$\frac{1}{2}(2k+1)(6k+3)=\frac{1}{2}12k^2+12k+3$$ Process: $$\displaystyle\sum_{r=1}^k(k+r)+(k+k+1)=\frac{1}{2}(3k^2+k)+(2k+1)$$ $$=\frac{3k^2+5k+2}{2}$$
Im very confused here, and I'm sure there are loads of mistakes here but I just can't spot them. Could anyone be so kind to help?
Thanks,
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$$\sum_{r=1}^k(k+r)=\frac12k(3k+1)$$
is fine, but you went astray at the start of the induction step. The $k+1$ case is
$$\sum_{r=1}^{k+1}\big((k+1)+r\big)=\frac12(k+1)\big(3(k+1)+1\big)\;,$$
so you should be starting with
\begin{align*} \sum_{r=1}^{k+1}\big((k+1)+r\big)&=\sum_{r=1}^k\big((k+1)+r\big)+\big((k+1)+(k+1)\big)\\ &=\sum_{r=1}^k\big((k+r)+1\big)+2(k+1)\\ &=\sum_{r=1}^k(k+r)+\sum_{r=1}^k1+2(k+1)\\ &=\sum_{r=1}^k(k+r)+k+2(k+1)\\ &=\sum_{r=1}^k(k+r)+3k+2\;. \end{align*}
Now you can apply your induction hypothesis and simplify.
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The second step should be $\sum_{r=1}^k ((k+r)+1) + 2(k+1)$ which will lead to final equality being $\sum_{r=1}^k (k+r) + 3k + 2$ – user62089 Mar 18 '13 at 19:16
@pondy: Thanks for catching it; fixed. – Brian M. Scott Mar 18 '13 at 19:19
Hint: it's easy to verify the result for $n=1$.
Now $$((n+1)+1)+((n+1)+2)+\cdots+2(n+1)=(n+2)+(n+3)+\cdots+(2n+2)\\ =\frac{1}{2}n(3n+1)-(n+1)+(2n+1)+(2n+2)$$ and verify that's equal to $\frac{1}{2}(n+1)(3(n+1)+1)$.
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Hint:its easy to prove by induction :$$1+2+3+...+n=\frac{n(n+1)}{2}$$$$(n+1)+(n+2)\cdots+2n=(n+n+...+n)+(1+2+3+..+n)=n^2+\frac{n(n+1)}{2}$$
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The first line has a typo :-O – Ben Mar 18 '13 at 19:08
@Ben thanks Ben – Maisam Hedyelloo Mar 18 '13 at 19:10
But that's not by induction. You're using a result ($1+2+3+\cdots$) that can be proven by induction, but you're not using induction in your proof. – Elmar Zander Mar 18 '13 at 19:13
@Elmar Zander: hi notice i proved by induction that $1+2+3+...+n=\frac{n(n+1)}{2}$ then by using of it iww prove by induction that $(n+1)+(n+2)\cdots+2n=\frac{1}{2}n(3n+1)$ induction step :$1+1=\frac{1}{2}(3+1)=1$ then i know $(n+1)+(n+2)\cdots+2n=(n+n+...+n)+(1+2+3+..+n)=n^2+\frac{n(n+1)}{2}$ and i will prove $((n+1)+1)+((n+1)+2)\cdots+2(n+1)=\frac{1}{2}(n+1)(3(n+1)+1)$ – Maisam Hedyelloo Mar 18 '13 at 19:17
@MaisamHedyelloo I'm sorry, but you did not prove that. Neither by induction, nor in any other way. You just use that formula. – Elmar Zander Mar 18 '13 at 19:19
Hint:
For the case $n+1$ develop the right part: $1/2(n+1)(3(n+1)+1)$ to come back to the case $n$ plus something and check that whats left is equal to what is in more on the left part for $n+1$ (be careful that there is also one term in less in the sum ...)
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Now assume that the identity holds true for $n = k$. This implies
$$\sum_{r=1}^k (k+r) = \frac 12 k(3k+1)$$
Then for $k+1$ we have,
$$\sum_{r=1}^{k+1} (k+1+r) = \sum_{r=1}^{k} (k+1+r) + 2(k+1) = \sum_{r=1}^{k} (k+r) + \sum_{r=1}^k 1 + 2(k+1)$$ $$= \sum_{r=1}^{k} (k+r) + k + 2(k+1) = \frac 12 k(3k+1) + k + 2(k+1)$$ $$= \frac 12 (k+1)(3(k+1)+1)$$
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Check for $n=1$:
$$(n+1)+\cdots+2n=2=\frac12 1(3+1)=2$$
Ok, that holds. Now see that you can prove the case $n+1$ from the case for $n$
$$((n+1)+1)+\cdots +2(n+1)=(n+1)+\cdots+2n - (n+1)+(2n+1)+(2n+2)$$
Here you can use the fact that the formula olds for $n$:
\begin{align} \ldots &=\frac12 n(3n+1) - (n+1)+(2n+1)+(2n+2)\\ &=\frac12 n(3n+1) + (3n+2)\\ &=\frac12 (3n^2+7n+4)\\ &=\frac12 (n+1)(3(n+1)+1)\\ \end{align}
- | 2015-07-02 04:22:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9281861186027527, "perplexity": 774.4983740726833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095373.99/warc/CC-MAIN-20150627031815-00271-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/7411/difference-between-alexander-polynomial-and-blanchfield-pairing | # Difference between Alexander polynomial and Blanchfield pairing
For a Seifert matrix $V$ of a knot $K$, the Alexander module has presentation matrix $V-tV^T$. The determinant of this matrix is the Alexander polynomial, which is the order of the Alexander module. In particular, the Alexander module is a torsion module, and has a linking form, called the Blanchfield pairing. The S-equivalence class of the Seifert matrix is an invariant of the knot, and uniquely characterizes the Blanchfield pairing. There is a bijective correspondence between S-equivalence classes of Seifert matrices and Blanchfield pairings.
Trotter gave examples of knots with the same Alexander polynomial but non-S-equivalent Seifert matrices. My question is what additional information we need to reconstruct the Blanchfield pairing (Seifert matrix up to S-equivalence) from the Alexander polynomial.
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The Blanchfield pairing has many formulations, I like to think of it as a sesquilinear form:
$$A \otimes A \to \Lambda / \mathbb Z[t^\pm]$$
where $A$ is the Alexander module and $\Lambda$ is the field of fractions of $\mathbb Z[t^\pm]$. This pairing has to be a duality isomorphism, ie: the adjoint
$$\overline{A} \to Hom_{\mathbb Z[t^\pm]} (A, \Lambda/\mathbb Z[t^\pm])$$
is an isomorphism of $\mathbb Z[t^\pm]$-modules. $\overline{A}$ is $A$ but given the opposite action of $\mathbb Z[t^\pm]$ (you substitute $t \longmapsto t^{-1}$ before multiplication by a polynomial)
The Blanchfield pairing can be anything of that form. So you take the Alexander module, and soup it up with such an isomorphism between $\overline{A}$ and its Ext dual'' $Hom_{\mathbb Z[t^\pm]} (A, \Lambda/\mathbb Z[t^\pm])$. That is the extra information in the S-equivalence class.
edit: the pairing has a nice geometric interpretation. $A$ is $H_1(\tilde C)$ where $\tilde C \to C$ is the universal abelian cover of the knot complement. Since $A$ is $\mathbb Z[t^\pm]$-torsion, given any $[x] \in A$ let $p$ be such that $px = \partial X$. Then you define the pairing $\langle x, y\rangle = (\sum_i (X \cap t^{i}y)t^i)/p$ provided $X$ and $y$ are transverse representatives when projected to $C$ (in any way that that makes sense). Here $\cap$ is the standard algebraic intersection number of transverse chains.
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Is there a good reference for the statement in your last paragraph? – Alison Miller Jun 12 '11 at 20:04
There's a bunch. Hillman's "Algebraic invariants of links" is a good one. Usually the transversality is expressed by representing one of $X$ and $Y$ in simplicial and the dual polyhedral coordinates (like in Poincare's proof of Poincare duality). But the basic idea of representing Poincare duality torsion pairings like that goes back at least as far as Seifert and Threlfall's textbook. And it falls out of the mathematics of Poincare duality and the Universal Coefficient Theorem very naturally. – Ryan Budney Jun 12 '11 at 20:46
Thanks! I have Hillman's book already; I'll read the section on the Blanchfield pairing more carefully. – Alison Miller Jun 13 '11 at 2:37 | 2014-04-17 01:46:40 | {"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": 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": 0.9067410826683044, "perplexity": 310.9781488494688}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526102.3/warc/CC-MAIN-20140416005206-00523-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://twistedone151.wordpress.com/2008/09/05/physics-friday-36/ | ## Physics Friday 36
Consider a ball of radius R and uniform electric charge density ρ. What is the electric field at a point a distance r from the center of the ball? Suppose we remove material from this ball to form a smaller spherical cavity within it, such that the vector from the center of the overall sphere to that of the cavity is d. What, then, is the electric field at any point within the cavity?
I. To calculate the field, we first consider the spherical symmetry. The magnitude of the field will be a function of r only (in terms of the point’s location), and the field will also be in a radial direction. So, let us consider as a Gaussian surface a sphere of radius r with the same center as the ball. Then, as the field is of constant magnitude across that surface, and normal to the sphere at each point, the total flux of the electric field through the surface is just the magnitude E of the field times the surface area: Φ=4πr2E. Now, Gauss’ Law tells us that this flux is , where Q is the total charge enclosed in the sphere.
For r>R, this is the total charge of the ball: . Thus
,
which is the same as the field at a distance r from a point charge .
For r<R, the enclosed charge is instead . Then,
.
For both of these, the field is radial (outward for ρ>0, inward for ρ<0).
II. Note, that if we place a ball of charge density ρ and radius R and a smaller sphere of charge density –ρ with center displaced from the first ball by the vector d, the result is equivalent, electrically, to the ball with cavity, due to the cancellation of charge.
Thus, a point within the cavity will be, in this equivalent picture, within both balls. Let the vector from the center of the larger ball to our point in question be r. Then the vector from the center of our smaller ball (aka the cavity) to this point is rd.
The field, then, from the larger ball is of magnitude , where r is the magnitude of the vector r. This field is radially directed to the sphere, and thus .
Similarly, for field due to the smaller sphere, , and so
.
Summing these:
.
So the field inside the cavity is uniform, and independent of the size of the cavity. Note that if the cavity is concentric to the ball (d=0), the field is zero. | 2017-06-24 10:21:02 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 7, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9134514927864075, "perplexity": 246.68181486179978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320257.16/warc/CC-MAIN-20170624101204-20170624121204-00717.warc.gz"} |