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https://newsgroup.xnview.com/viewtopic.php?f=34&t=3598&p=17066
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## move or copy file information/name missing
Ideas for improvements and requests for new features in XnView Classic
Moderators: XnTriq, xnview
loth
Posts: 173
Joined: Thu Aug 11, 2005 3:21 pm
Location: Kölle
### move or copy file information/name missing
hi,
a small issue.
when copy/move a picture to another folder via ALT+M/ALT+C
the filename (complete filename means: not only filename, but filename.ext) should/could be shown.
now, the user doesnt know, what filename he/she is moving/copying
loth
helmut
Posts: 8214
Joined: Sun Oct 12, 2003 6:47 pm
Location: Frankfurt, Germany
### Re: move or copy file information/name missing
loth wrote:hi,
a small issue.
when copy/move a picture to another folder via ALT+M/ALT+C
the filename (complete filename means: not only filename, but filename.ext) should/could be shown.
now, the user doesnt know, what filename he/she is moving/copying
That's true, there's no filename displayed. But you can select and move/copy many files at a time, so if you want to display filenames, you need a larger list which will make the dialog large.
Perhaps for one file the name could be shown, and for many files the number of files could be shown.
loth
Posts: 173
Joined: Thu Aug 11, 2005 3:21 pm
Location: Kölle
### Re: move or copy file information/name missing
helmut wrote:That's true, there's no filename displayed. But you can select and move/copy many files at a time, so if you want to display filenames, you need a larger list which will make the dialog large.
the dialog is way too small, anyway. and its not resizeable, like any "good" window should be ;->
i'd like to add this point to my request ;-) please make the copy/move dialog larger or (even better) resizeable.
but you're right. if more than one filename is shown it could be a larger list.
suggestion: only display the first/last X filenames, or (even better)
make a scrollable list with scrollbars, if needed (i.e. more than 10 filenames to show) see my picture..
helmut wrote: Perhaps for one file the name could be shown, and for many files the number of files could be shown.
also a good idea.
lets hear other ideas (maybe)
Code: Select all
+----------------------------------+
|Move: X file(s) ~| < title bar
+----------------------------------+
| ~|
|list ~|
| of ~|
| all ~|
| the ~|
| paths ~|
| ~|
| ~|
| ~|
| ~|
| ~|
| ~|
| ~|
| ~|
+----------------------------------+
|X:\pics\bla\foo\bar\ |^| < pulldown menu, where to move
+----------------------------------+
|new folder | move | cancel | < buttons
+----------------------------------+
|moveing files: ~| < new, spectacular, filelist :-)
|C:\fdgfdgd\filename1.jpg ~|
|C:\fdgfdgd\filename2.jpg ~|
|D:\dfgfdgdgfdgd\filename1.jpg ~|
|X:\dfhgfdhfdgfdgd\filename1.jpg ~|
+----------------------------------+
~ is a vertical scrollbar
loth
helmut
Posts: 8214
Joined: Sun Oct 12, 2003 6:47 pm
Location: Frankfurt, Germany
### Re: move or copy file information/name missing
loth wrote:...
the dialog is way too small, anyway. and its not resizeable, like any "good" window should be ;->
i'd like to add this point to my request please make the copy/move dialog larger or (even better) resizeable.
I second this.
loth wrote:but you're right. if more than one filename is shown it could be a larger list.
...
Wow, we have more and more inventive screen designers, here. And in ASCII art.
The simple approach "1 file name or number of files" is still under discussion. Already, some comments regarding your dialog draft: I'd prefer the file list to be at the top, the buttons of the dialogs should always be at the bottom or to the right of the dialog. So the dialog might look like this:
Code: Select all
+----------------------------------+
|Move: X file(s) ~| < title bar
+----------------------------------+
|[x] Show files | < New option for showing/hiding the file list
|Files: ~| < file list
|C:\fdgfdgd\filename1.jpg ~|
|C:\fdgfdgd\filename2.jpg ~|
|D:\dfgfdgdgfdgd\filename1.jpg ~|
|X:\dfhgfdhfdgfdgd\filename1.jpg ~|
+----------------------------------+
| ~|
|list ~|
| of ~|
| all ~|
| the ~|
| paths ~|
| ~|
| ~|
| ~|
| ~|
| ~|
| ~|
| ~|
| ~|
+----------------------------------+
|X:\pics\bla\foo\bar\ |^| < pulldown menu, where to move
+----------------------------------+
|new folder | move | cancel | < buttons
+----------------------------------+
Please note the option "Show files" which will show/hide the file list. This will satisfy both users who do not need the file list and those who need it. Sure enough, the setting of this checkbox has to be remembered.
loth
Posts: 173
Joined: Thu Aug 11, 2005 3:21 pm
Location: Kölle
### Re: move or copy file information/name missing
helmut wrote:Wow, we have more and more inventive screen designers, here. And in ASCII art.
thats, because i cannot add pictures to this forum :-) so i must be creative in ascii-art :-) otherwise i would use photoshop.
helmut wrote:The simple approach "1 file name or number of files" is still under discussion.
hmm...i dont exactly know if i understand, but i cannot see a real/true advantage of a dialog which only says, how many files it will move/copy.
i'm always interested of *which* files are copied/moved.
helmut wrote:Already, some comments regarding your dialog draft: I'd prefer the file list to be at the top, the buttons of the dialogs should always be at the bottom or to the right of the dialog.
Please note the option "Show files" which will show/hide the file list. This will satisfy both users who do not need the file list and those who need it. Sure enough, the setting of this checkbox has to be remembered.
great :-)
i support that.
loth
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2020-07-05 14:24:01
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https://pypi.org/project/odoo12-addon-spec-driven-model/12.0.3.0.0/
|
Skip to main content
Tools for specifications driven mixins (from xsd for instance)
## Intro
This module is a databinding framework for Odoo and XML data: it allows to go from XML to Odoo objects back and forth. This module started with the GenerateDS pure Python databinding framework and is now being migrated to xsdata. So a good starting point is to read the xsdata documentation here
But what if instead of only generating Python structures from XML files you could actually generate full blown Odoo objects or serialize Odoo objects back to XML? This is what this module is for!
First you should generate xsdata Python binding libraries you would generate for your specific XSD grammar, the Brazilian Electronic Invoicing for instance, or UBL.
Second you should generate Odoo abstract mixins for all these pure Python bindings. This can be achieved using xsdata-odoo. An example is OCA/l10n-brazil/l10n_br_nfe_spec for the Brazilian Electronic Invoicing.
## SpecModel
Now that you have generated these Odoo abstract bindings you should tell Odoo how to use them. For instance you may want that your electronic invoice abstract model matches the Odoo res.partner object. This is fairly easy, you mostly need to define an override like:
from odoo.addons.spec_driven_model.models import spec_models
class ResPartner(spec_models.SpecModel):
_inherit = [
'res.partner',
'partner.binding.mixin',
]
Notice you should inherit from spec_models.SpecModel and not the usual models.Model.
Field mapping: You can then define two ways mapping between fields by overriding fields from Odoo or from the binding and using _compute= , _inverse= or simply related=.
Relational fields: simple fields are easily mapped this way. However what about relational fields? In your XSD schema, your electronic invoice is related to the partner.binding.mixin not to an Odoo res.partner. Don’t worry, when SpecModel classes are instanciated for all relational fields, we look if their comodel have been injected into some existing Odoo model and if so we remap them to the proper Odoo model.
Field prefixes: to avoid field collision between the Odoo fields and the XSD fields, the XSD fields are prefixed with the name of the schema and a few digits representing the schema version (typically 2 digits). So if your schema get a minor version upgrade, the same fields and classes are used. For a major upgrade however new fields and classes may be used so data of several major versions could co-exist inside your Odoo database.
## StackedModel
Sadly real life XML is a bit more complex than that. Often XML structures are deeply nested just because it makes it easier for XSD schemas to validate them! for instance an electronic invoice line can be a nested structure with lots of tax details and product details. In a relational model like Odoo however you often want flatter data structures. This is where StackedModel comes to the rescue! It inherits from SpecModel and when you inherit from StackedModel you can inherit from all the generated mixins corresponding to the nested XML tags below some tag (here invoice.line.binding.mixin). All the fields corresponding to these XML tag attributes will be collected in your model and the XML parsing and serialization will happen as expected:
from odoo.addons.spec_driven_model.models import spec_models
class InvoiceLine(spec_models.StackedModel):
_inherit = [
'account.move.line',
'invoice.line.binding.mixin',
]
_stacked = 'invoice.line.binding.mixin'
All many2one fields that are required in the XSD (xsd_required=True) will get their model stacked automatically and recursively. You can force non required many2one fields to be stacked using the _force_stack_paths attribute. On the contrary, you can avoid some required many2one fields to be stacked using the stack_skip attribute.
## Hooks
Because XSD schemas can define lot’s of different models, spec_driven_model comes with handy hooks that will automatically make all XSD mixins turn into concrete Odoo model (eg with a table) if you didn’t inject them into existing Odoo models.
Table of contents
### Usage
See my detailed OCA Days explanations here: https://www.youtube.com/watch?v=6gFOe7Wh8uA
You are also encouraged to look at the tests directory which features a full blown example from the famous PurchaseOrder.xsd from Microsoft tutorials.
### Known issues / Roadmap
Migrate from generateDS to xsdata; see the xsdata Pull Requests in the repo.
### Bug Tracker
Bugs are tracked on GitHub Issues. In case of trouble, please check there if your issue has already been reported. If you spotted it first, help us smashing it by providing a detailed and welcomed feedback.
Do not contact contributors directly about support or help with technical issues.
• Akretion
## Maintainers
This module is maintained by the OCA.
OCA, or the Odoo Community Association, is a nonprofit organization whose mission is to support the collaborative development of Odoo features and promote its widespread use.
Current maintainer:
This module is part of the OCA/l10n-brazil project on GitHub.
You are welcome to contribute. To learn how please visit https://odoo-community.org/page/Contribute.
## Download files
Download the file for your platform. If you're not sure which to choose, learn more about installing packages.
### Source Distributions
No source distribution files available for this release. See tutorial on generating distribution archives.
### Built Distribution
odoo12_addon_spec_driven_model-12.0.3.0.0-py3-none-any.whl (55.6 kB view hashes)
Uploaded py3
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2023-02-04 10:27:15
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http://openstudy.com/updates/50dccb60e4b0d6c1d5430433
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## ksaimouli Group Title f(x) is even and g(x) is odd one year ago one year ago
1. ksaimouli
$\int\limits_{0}^{5}f(x)dx=8$
2. ksaimouli
$\int\limits_{0}^{5}g(x)=4$
3. ksaimouli
find$\int\limits_{-5}^{5}[f(x)+g(x)]dx$
4. ksaimouli
@wio
5. ksaimouli
$\int\limits_{-5}^{0}[f(x)+g(x)] +\int\limits_{0}^{5}[f(x)+g(x)] dx$
6. ksaimouli
@abb0t
7. ksaimouli
after that i have no idea
8. ksaimouli
i know the 2nd part 12
9. ksaimouli
-+12
10. ksaimouli
well even mean f(-x)=f(x) will that mean$\int\limits_{-5}^{0}f(x)=-8$
11. ksaimouli
because$-\int\limits_{5}^{0}f(x)=-8$
12. ksaimouli
even aplies if negative in f(x)? f(-x)=f(x)
13. KingGeorge
Why should $-\int\limits_{5}^{0}f(x)=-8$be true? I think this should be 8 and not -8.
14. ksaimouli
because that is the rule right to flip flop - should be their
15. KingGeorge
$\int_0^5 f(x)=8$$\int_5^0 f(x)=-8$$-\int_5^0 f(x)=8$
16. ksaimouli
$-\int\limits_{-5}^{0}f(x)=$
17. KingGeorge
But you're certainly on the right path to solving this. You know$\int\limits_{0}^{5}[f(x)+g(x)] dx=12$So you just need to find $\int\limits_{-5}^{0}[f(x)+g(x)]dx =\int\limits_{-5}^0f(x)\;dx+\int\limits_{-5}^0g(x)\;dx$First, lets start with the f(x) part.
18. zepp
I drove by to say hello to @KingGeorge!
19. KingGeorge
Since $$f(x)=f(-x)$$, we have that \begin{aligned} \int\limits_{-5}^0f(x)\;dx&=\int\limits_{-5}^0f(x)\;dx \\ &=\int\limits_{-5}^0f(-x)\;dx\\ &=\int\limits_{5}^0f(-u)\;(-du)\qquad\text{this is a u-sub for }u=-x.\\ &=\int\limits_0^5f(-u)\;du\\ &=\int\limits_0^5f(u)\;du\\ &=8 \end{aligned}
20. KingGeorge
This is basically the same thing you do for $$g(x)$$. Instead we have $$-g(x)=g(-x)$$.If we repeat the above argument, we get to the point $\int\limits_0^5g(-u)\;du$Substitute for $$-g(u)$$, and we get $-\int\limits_0^5g(u)\;du=-4$
21. KingGeorge
Did this all make sense? And hello to you too @zepp!
22. ksaimouli
i i got it i did little bit different i took u=-x and then -du=dx
23. KingGeorge
There is a typo in my work two posts above. The second line should not be there. (The line that reads $$=\int_{-5}^0f(-x)\;dx$$).
24. KingGeorge
That works as well.
25. ksaimouli
|dw:1356649511965:dw|
26. ksaimouli
ohk thx a lot
27. KingGeorge
You're welcome.
28. ksaimouli
i will work on this if i have nay questions can i post on this wall
29. KingGeorge
sure.
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2014-09-22 02:24:09
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http://meetings.aps.org/Meeting/DFD07/Event/72684
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### Session JF: Biofluids X: Flying
3:35 PM–6:11 PM, Monday, November 19, 2007
Salt Palace Convention Center Room: 151 G
Chair: Kenneth Breuer, Brown University
Abstract ID: BAPS.2007.DFD.JF.7
### Abstract: JF.00007 : Exception to Triantafyllou's Strouhal number rule of flapping
4:53 PM–5:06 PM
Preview Abstract MathJax On | Off Abstract
#### Authors:
Triantafyllou and Triantafyllou (Sci. Amer. 1995) have shown that fish caudal fins have a preferred Strouhal number of 0.25-0.35 for efficient swimming. Strouhal number is defined as \textit{fA/U}, where $f$ is flapping frequency, $A$ is the peak-to-peak flapping amplitude at the tip of the caudal fin, and $U$ is stream velocity. Although this preference was attributed to efficient swimming, they did not measure the efficiency of fish swimming. Later Biorobotic experiments by Bandyopadhyay et al. (JFE 2000) have suggested that while Strouhal number is the dominant factor, another yet unidentified factor is also involved in efficiency. Rohr and Fish (JEB 2004) have shown that in captive cetaceans the most common range of Strouhal number is 0.20-0.30---slightly lower than that given by Triantafyllou and Triantafyllou. We have carried out the measurements of efficiency and forces produced by a single penguin-like fin. We show that for a single fin, in the range of maximum efficiency of 0.55-0.60, the Strouhal number is 0.28-0.55. Here Strouhal number is defined with amplitude $A$ equal to the arc length traversed by the point on the fin which divides the swept area in two. In addition to Strouhal number, the pitch amplitude also determines the regime of high efficiency, with the peak of efficiency seen at lower Strouhal numbers for low pitch amplitudes and at higher Strouhal numbers at higher pitch amplitudes.
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2013-06-20 04:25:51
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https://physics.paperswithcode.com/paper/entanglement-and-matrix-elements-of
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## Entanglement and matrix elements of observables in interacting integrable systems
30 Dec 2019 · LeBlond Tyler, Mallayya Krishnanand, Vidmar Lev, Rigol Marcos ·
We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and we contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems... This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ Hamiltonian is very close to, or the same as, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ Hamiltonian at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate-to-eigenstate fluctuations vanish in a power-law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of $\omega=E_{\alpha}-E_{\beta}$ ($\{E_{\alpha}\}$ are the eigenenergies) proportional to $1/D$, where $D$ is the Hilbert space dimension. read more
PDF Abstract
# Code Add Remove Mark official
No code implementations yet. Submit your code now
# Categories
Statistical Mechanics Quantum Gases Quantum Physics
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2021-09-22 17:05:39
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http://openstudy.com/updates/5156132be4b07077e0c0209f
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Here's the question you clicked on:
## aussy123 Group Title I have a question, can someone answer me. Metal awarded. one year ago one year ago
• This Question is Closed
1. aussy123 Group Title
I have a problem and it ask for tan2theta, and theta is in the 1st quad, but when I work out the problem I get a negative result. Do that number automatically turns positive?
2. aussy123 Group Title
does*
3. aussy123 Group Title
do you know anything about tan, sin and cos?
4. completeidiot Group Title
$\tan^2 \theta$ $0^\circ<\theta<90^\circ$ whats the question?
5. completeidiot Group Title
or is it $\tan (2\theta)$?
6. aussy123 Group Title
$\tan2\Theta$
7. completeidiot Group Title
what is the original question?
8. aussy123 Group Title
Given: sin =4/5 and cos x =-5/13 ; evaluate the following expression. (Theta: Quadrant I, x : Quadrant II.) tan 2 Theta a.) 8/3 b.) 24/7 c.) -24/7
9. aussy123 Group Title
my outcome is -24/7 but theta is in the first quad and tan is positive in the first quad
10. Mertsj Group Title
If theta is greater than 45 and in the first quadrant, then 2 theta is in the second quadrant where the tangent is negative.
11. completeidiot Group Title
yes theta is in the first quadrant, but you're not finding tan(theta) you're finding tan(2*theta) and its possible for 2*theta to be in the second quadrant as mertsj just stated
12. aussy123 Group Title
So my result stays at -24/5
13. aussy123 Group Title
*-24/7 sorry
14. completeidiot Group Title
assuming you didnt make any mistakes in your calculation, then that would be the answer note- i didnt actually do the problem
15. aussy123 Group Title
okay Thank You both, now I can start my quiz with no problem. fingers crossed!
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2014-07-24 15:22:50
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https://space.stackexchange.com/questions/37162/how-much-would-cost-to-take-a-kg-to-the-same-distance-of-the-planet-than-leo-in
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# How much would cost to take a kg to the same distance of the planet than LEO in Mars?
In Mars it's needed a much lower velocity to attain orbit. How would this impact the costs of taking 1 kg to the orbit of the planet, at the same distance of the planet than Low Earth Orbit but in Mars?
The cost in money terms depends rather dramatically on your assumptions. If your rocket for launching from Mars has to be transported from Earth, then it's going to be pretty pricey, since a very much larger rocket will have been needed to launch it from Earth. On the other hand if you are imagining that Mars has been colonised and has a thriving local rocket-manufacturing industry, it could be quite cheap.
What we can give though is the velocity needed to reach that orbit, and from that, and some assumptions about the launcher we can make a guess at how big it will need to be, and how much fuel it will need.
The velocity is given on wikipedia as 4.1 km/s. This tells us, via the rocket equation
$${\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}}}$$
the mass ratio (the ratio of starting mass $$m_0$$ on the surface of Mars to mass $$m_f$$ put in orbit (including empty fuel tanks, rocket engines, etc. as well as your 1kg payload). We can try a few plausible fuel combinations which will give us different values for $$I_{sp}$$:
• A solid rocket motor, such as a space shuttle solid rocket booster, with an $$I_{sp}$$ of 250, gives us a mass ratio of 5.3
• Standard storable hypergolic liquid propellants (widely used for long space missions), $$I_{sp}$$ about 340, mass ratio 3.3
• Liquid methane and liquid oxygen (SpaceX proposal) $$I_{sp}$$ about 360, mass ratio 3.1
• Liquid hydrogen and liquid oxygen (highest $$I_{sp}$$ commonly used) $$I_{sp}$$ about 450, mass ratio 2.5.
So, assuming an optimistic 1kg of rockets and tanks for your 1kg payload, you'd need somewhere between 5 and 11kg of rocket on the surface of Mars.
This ignores atmospheric resistance, among other things, which would be a big problem for a rocket this small, even, I suspect, on Mars.
Let's check the orbital velocities of low orbits around Earth and Mars.
Mars is 1/2 the diameter and 1/8 the mass. We can use the vis-viva equation to calculate the velocity of a circular orbit.
$$v = \sqrt{\frac{GM}{a}}$$
GM (m^3/s^2) R (m) a = R + 400 km v (m/s)
Earth 3.986E+14 6.378E+06 6.778E+06 7905
Mars 4.283E+13 3.396E+06 3.796E+06 3551
It's a little harder to estimate the difference in the delta-v penalty to lift out of each planet's gravity well by 400 km because it depends in part on how fast you do it which means it depends on the specific rocket and how much you limit acceleration, but it's probably safe to estimate that for the small, uncrewed payload it will be roughy half on Mars than it is on Earth as well.
So for the same payload and exhaust velocity, using the rocket equation in the same way as in @SteveLinton's answer we can estimate that the mass of the rocket will be lower by the factor:
$$\frac{m_M}{m_E} = \exp\left( \frac{v_M - v_E}{v_{ex}} \right)$$
Assuming the exhaust velocity is moderate at 3000 m/s the Mars rocket's propellant is only 13% as massive as the Earth's rocket. Of course this is only a rough approximation, but we can say that it's going to be a much smaller rocket. (for 4000 m/s the ratio is about 21%).
Now the cost of putting that rocket on Mars to begin with, or building and fueling it there is going to be astronomical!
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2020-01-26 08:18:40
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https://math.stackexchange.com/questions/1660245/evaluating-lim-x-to-infty-frac1x-int-2x-mt-cdot-ftdt-where-mx
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# Evaluating $\lim_{x\to\infty}\frac{1}{x}\int_2^x M(t)\cdot f'(t)dt$, where $M(x)$ is Mertens functions
Let $\mu(n)$ the Möbius function. I know that combining Abel summation formula, the Prime Number Theorem and l'Hôpital's rule I can deduce $$\lim_{x\to\infty}\frac{1}{x}\sum_{2\leq n\leq x}\mu(n)\cdot\frac{\log\log n}{{\log\log (n+1)}}= 0-\lim_{x\to\infty}\frac{1}{x}\int_2^x M(t)\cdot f'(t)dt,$$ where $M(x)$ is the Mertens function, and $f(x)=\frac{\log\log x}{{\log\log (x+1)}}$. I know that $M(t)$ can be bounded by the obvious $t$, thus RHS should be
$$\lim_{x\to\infty}\frac{1}{x}O\left(\int_2^x \frac{(t+1)\log(t+1)\cdot\log\log(t+1)-t\log t\cdot\log\log t)}{(t+1)\log t\cdot\log(t+1)\cdot\log^2\log(t+1)}dt\right).$$
Question. Is there a reasonable way to compute (or compute improving my bound for Mertens function without additional more tedious computations) the limit in RHS? I say, if it is possible, determine if there is convergence and, if there is a limit compute it without do the more hardest computations. Thanks in advance.
Now with the BOUNTY I am looking an approach that provide us the behaviour of the first series, thus RHS, thus if it neccesary tedious computations your answer will be welcome. Thanks.
• My goal with this exercise is to know with more or less precission how work with a situation of this kind, and if it is possible impove the bound for Mertens function that I've used, but it isn't neccesary give the more best result. The more important is that I want to know how work easily, if it is possible, with complicated functions doing easy computations. Thanks
– user243301
Feb 17 '16 at 18:24
• I ask: It is possible to get a bound or an asymptotic quickly, without the more tedious computations, and compute the asymptotic behaviour of RHS correctly? If there is a limit od RHS, **comopute it in the sense that say if at least is finite or $\infty$, since I know that we are work with the bound encoded in the big oh notation.
– user243301
Feb 18 '16 at 6:52
Note that holds $$\frac{\log\left(\log\left(n\right)\right)}{\log\left(\log\left(n+1\right)\right)}\sim1$$ as $n\rightarrow\infty$ so your sum is very close to $M\left(x\right)$. Then $$\frac{1}{x}\sum_{2\leq n\leq x}\mu\left(n\right)\frac{\log\left(\log\left(n\right)\right)}{\log\left(\log\left(n+1\right)\right)}\sim\frac{1}{x}\sum_{2\leq n\leq x}\mu\left(n\right)=\frac{1}{x}\left(M\left(x\right)-1\right)\rightarrow0$$ as $x\rightarrow\infty$ since PNT is equivalent to $$\frac{M\left(x\right)}{x}\stackrel{x\rightarrow\infty}{\longrightarrow}0.$$ Maybe it's interesting to claim a related result which can be found in the Terence Tao blog (with the references).
Proposition: Let $f:\mathbb{N}\rightarrow\mathbb{C}$ be a bounded function such that $$\sum_{n\leq x}f\left(pn\right)\overline{f\left(qn\right)}=o_{p,q}\left(x\right)$$ for any distinct $p,q$ prime numbers, then $$\sum_{n\leq x}\mu\left(n\right)f\left(n\right)=o\left(x\right).$$
• Very thanks much, you make that learn mathematics to be easy. Very thanks much for the reference. Can you add in a comment a definition for $o_{p,q}(x)$ or a toy example of use of the proposition?
• @user243301 My pleasure. With $o_{p,q}(x)$ I intend that the decay rate of the error term may depend on $p$ and $q$. An example, from the blog of Tao, is $$\sum_{n\leq x}\mu(n)e^{2\pi i\alpha n}=o(x)$$ where $\alpha$ is irrational. I forgot to explain that $p,q$ are prime numbers. Mar 18 '16 at 16:40
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2021-09-28 13:32:34
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https://slinberg.net/post/adding-edit-this-page-links/
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In config/_default/params.yaml, find the section edit_page and:
☞︎ Set repo_url to the URL of your repository (here, github)
☞︎ Set content_dir to the directory of your site’s content, content by default)
☞︎ Set repo_branch to your repo’s published branch; I use master rather than main)
Then set the editable fields as needed for the types of content you want these links to be present in.
Here’s the code from this site:
edit_page:
repo_url: 'https://github.com/slinberg-umass/slinberg.net'
content_dir: 'content'
repo_branch: master
editable:
page: true
post: true
book: true
That’s all you need to do; when you rebuild, if the edit_page parameters are not empty, Hugo will insert the link as directed by the templates. This is the bit of triggering code:
{{ partial "page_edit" . }}
It is contained in various inherited templates such as themes/github.com/wowchemy/wowchemy-hugo-modules/wowchemy/layouts/partials/page_footer.html, and the code is in themes/github.com/wowchemy/wowchemy-hugo-modules/wowchemy/layouts/partials/page_edit.html.
Look at the code to see, among other things, that it can be overridden on a specific page or post by adding editable: false to its yaml header.
But we’re not doing that in this case, so falling off the bottom of the content here, we get the groovy link to the page source below (note, you’ll need to fork the repo to actually see it, it’s more for me than for you):
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2021-07-25 17:57:37
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http://math.stackexchange.com/questions/107221/relationship-between-nullspace-and-row-column-space
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# Relationship between nullspace and row/column space
Suppose I have a $3\times 3$ matrix $A$, whose null space is a line through the origin in $3$-space. Can the row or column space of $A$ also be a line through the origin ?
-
Hint: The rank is the dimension of the rowspace is the dimension of the columnspace. – Arturo Magidin Feb 8 '12 at 21:29
Do you know rank nullity theorem? – Davide Giraudo Feb 8 '12 at 21:30
I know that $rank (A) + \text{nullity} (A) = n$, however I do not see how this will help me. I feel so stupid now =( – N3buchadnezzar Feb 8 '12 at 21:32
What is the dimension of a line through the origin? – Davide Giraudo Feb 8 '12 at 21:35
I think I might have gotten this now. The dimension is three right? – N3buchadnezzar Feb 8 '12 at 21:42
Since the null-space of $A$ is a line, which is a 1-dimensional subspace, the rank-nullity theorem tells us, that the rank of the matrix, which is the dimension of its row/column-space, is 2 and therefore the column-space cannot be a line, but a plane, a 2-dimensional subspace.
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2015-01-26 10:56:35
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https://www.ademcetinkaya.com/2022/11/budapest-se-index-stock-price-prediction.html
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Budapest SE Index Research Report
## Summary
In this paper a Bayesian regularized artificial neural network is proposed as a novel method to forecast financial market behavior. Daily market prices and financial technical indicators are utilized as inputs to predict the one day future closing price of individual stocks. The prediction of stock price movement is generally considered to be a challenging and important task for financial time series analysis. We evaluate Budapest SE Index prediction models with Ensemble Learning (ML) and Paired T-Test1,2,3,4 and conclude that the Budapest SE Index stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period: The dominant strategy among neural network is to Hold Budapest SE Index stock.
## Key Points
1. Market Risk
2. How can neural networks improve predictions?
3. Can neural networks predict stock market?
## Budapest SE Index Target Price Prediction Modeling Methodology
We consider Budapest SE Index Stock Decision Process with Ensemble Learning (ML) where A is the set of discrete actions of Budapest SE Index stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4
F(Paired T-Test)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Ensemble Learning (ML)) X S(n):→ (n+8 weeks) $∑ i = 1 n a i$
n:Time series to forecast
p:Price signals of Budapest SE Index stock
j:Nash equilibria (Neural Network)
k:Dominated move
a:Best response for target price
For further technical information as per how our model work we invite you to visit the article below:
How do AC Investment Research machine learning (predictive) algorithms actually work?
## Budapest SE Index Stock Forecast (Buy or Sell) for (n+8 weeks)
Sample Set: Neural Network
Stock/Index: Budapest SE Index Budapest SE Index
Time series to forecast n: 23 Nov 2022 for (n+8 weeks)
According to price forecasts for (n+8 weeks) period: The dominant strategy among neural network is to Hold Budapest SE Index stock.
X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)
Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)
Z axis (Yellow to Green): *Technical Analysis%
## Adjusted IFRS* Prediction Methods for Budapest SE Index
1. An entity's estimate of expected credit losses on loan commitments shall be consistent with its expectations of drawdowns on that loan commitment, ie it shall consider the expected portion of the loan commitment that will be drawn down within 12 months of the reporting date when estimating 12-month expected credit losses, and the expected portion of the loan commitment that will be drawn down over the expected life of the loan commitment when estimating lifetime expected credit losses.
2. If an entity previously accounted at cost (in accordance with IAS 39), for an investment in an equity instrument that does not have a quoted price in an active market for an identical instrument (ie a Level 1 input) (or for a derivative asset that is linked to and must be settled by delivery of such an equity instrument) it shall measure that instrument at fair value at the date of initial application. Any difference between the previous carrying amount and the fair value shall be recognised in the opening retained earnings (or other component of equity, as appropriate) of the reporting period that includes the date of initial application.
3. When an entity separates the foreign currency basis spread from a financial instrument and excludes it from the designation of that financial instrument as the hedging instrument (see paragraph 6.2.4(b)), the application guidance in paragraphs B6.5.34–B6.5.38 applies to the foreign currency basis spread in the same manner as it is applied to the forward element of a forward contract.
4. Hedge effectiveness is the extent to which changes in the fair value or the cash flows of the hedging instrument offset changes in the fair value or the cash flows of the hedged item (for example, when the hedged item is a risk component, the relevant change in fair value or cash flows of an item is the one that is attributable to the hedged risk). Hedge ineffectiveness is the extent to which the changes in the fair value or the cash flows of the hedging instrument are greater or less than those on the hedged item.
*International Financial Reporting Standards (IFRS) are a set of accounting rules for the financial statements of public companies that are intended to make them consistent, transparent, and easily comparable around the world.
## Conclusions
Budapest SE Index assigned short-term Caa2 & long-term B1 forecasted stock rating. We evaluate the prediction models Ensemble Learning (ML) with Paired T-Test1,2,3,4 and conclude that the Budapest SE Index stock is predictable in the short/long term. According to price forecasts for (n+8 weeks) period: The dominant strategy among neural network is to Hold Budapest SE Index stock.
### Financial State Forecast for Budapest SE Index Budapest SE Index Stock Options & Futures
Rating Short-Term Long-Term Senior
Outlook*Caa2B1
Operational Risk 4569
Market Risk4070
Technical Analysis3547
Fundamental Analysis4035
Risk Unsystematic3767
### Prediction Confidence Score
Trust metric by Neural Network: 88 out of 100 with 800 signals.
## References
1. Bai J, Ng S. 2017. Principal components and regularized estimation of factor models. arXiv:1708.08137 [stat.ME]
2. M. J. Hausknecht and P. Stone. Deep recurrent Q-learning for partially observable MDPs. CoRR, abs/1507.06527, 2015
3. S. Proper and K. Tumer. Modeling difference rewards for multiagent learning (extended abstract). In Proceedings of the Eleventh International Joint Conference on Autonomous Agents and Multiagent Systems, Valencia, Spain, June 2012
4. Mazumder R, Hastie T, Tibshirani R. 2010. Spectral regularization algorithms for learning large incomplete matrices. J. Mach. Learn. Res. 11:2287–322
5. Dudik M, Langford J, Li L. 2011. Doubly robust policy evaluation and learning. In Proceedings of the 28th International Conference on Machine Learning, pp. 1097–104. La Jolla, CA: Int. Mach. Learn. Soc.
6. A. Tamar and S. Mannor. Variance adjusted actor critic algorithms. arXiv preprint arXiv:1310.3697, 2013.
7. Batchelor, R. P. Dua (1993), "Survey vs ARCH measures of inflation uncertainty," Oxford Bulletin of Economics Statistics, 55, 341–353.
Frequently Asked QuestionsQ: What is the prediction methodology for Budapest SE Index stock?
A: Budapest SE Index stock prediction methodology: We evaluate the prediction models Ensemble Learning (ML) and Paired T-Test
Q: Is Budapest SE Index stock a buy or sell?
A: The dominant strategy among neural network is to Hold Budapest SE Index Stock.
Q: Is Budapest SE Index stock a good investment?
A: The consensus rating for Budapest SE Index is Hold and assigned short-term Caa2 & long-term B1 forecasted stock rating.
Q: What is the consensus rating of Budapest SE Index stock?
A: The consensus rating for Budapest SE Index is Hold.
Q: What is the prediction period for Budapest SE Index stock?
A: The prediction period for Budapest SE Index is (n+8 weeks)
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2022-12-03 08:12:04
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https://docs.google.com/document/d/e/2PACX-1vTykt8GRFZx0_E9y3IpJv6rTvZpiEKblCff0CoZ5VUZ31rKQIdLlfkroirUtfPkJoMsLnSC-emF5W1_/pub
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Epa Registration Number Lookup, You might good luck out as well as encounter several websites using a supposedly free solution. Those websites generally existing poor, improperly managed details. When it pertains to finding a contact number’s point of beginning, these service providers are not exactly trustworthy ones.,__________________________________________________________________________ ,# 3 - It has to be dependable and also provide precise results, , , , Click Here Now To See If Your Partner Has Been Cheating On You
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2018-02-24 01:15:05
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https://feature.engineering/dialog-state-tracking-models/
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Thoughts on some of the latest and greatest as of mid-2018.
1. Towards E2E RL of Dialogue Agents for Information aka. KB-Infobot (Dhingra et al.)
a. Process: Follows the format of many traditional DST systems. In particular, this includes a belief tracker, policy manager, database operator and text generator. This paper focuses on the database querying portion by introducing a soft KB-lookup mechanism that selects items across a probability distribution rather than a hard KB-lookup which is non-differentiable.
It assumes access to a domain-specific entity-centric knowledge base (EC-KB) where all items are split intro triples of (h,r,t) which stands for head, relation, tail. This is very similar to querying based on Key-Value Pair as in Key-Value Retrieval Networks for Task-Oriented Dialogue.
The belief state is the set of all p and q outputs. P is a multinomial distribution over the slot values v (and thus are vectors of size V), and each Q is a scalar probability of the user knowing the value of slot j. Each slot is summarized into an entropy statistic over a distribution, with final size 2M + 1, where the first M is the summary of all previous p, M is all the q values (which are unchanged since they are already scalar), and 1 is the summary of the current p distribution.
b. Strengths: Good job in building out the system from end-to-end, with different experiments comapred to handcrafted systems. They also test across different KB sizes.
c. Weaknesses: Final performance is not yet exciting with only 66% success rate for small KBs. They also achieve 83%, 83% and 68% for medium, large and extra large KBs respectively, which looks better but it is odd we don't see consistent trends, but rather a bump in the middle.
d. Notes: Easy to criticize (ie. multiple networks used for belief tracking), but to be fair, that is not the goal of their paper. As far as database operation is concerned, this is a great contribution!
2. E2E Task-Completion Neural Dialogue Systems with a novel User Simulator (Li et al.)
a. Process: Lays down a full framework for end-to-end dialogue. The first half is a user simulator that starts by randomly generating an agenda, and then uses a NLG module to translate that into natural language. The agenda explicitly outlines a user's goal, like which movie they want to watch and what requirements they have (i.e. at 3PM, about comedy), so the user model behaves in a consistent, goal-oriented manner. Separately, this concept is extended further in Building a Conversational Agent Overnight with Dialogue Self-Play where the authors (a) add the ability of varying user personality through a temperature mechanism and (b) greatly expand the diversity of the user dialogue by introducing Turkers paraphrasing the original text to inject human-likeness.
The second half of the framework is actual dialogue agent which includes an language understanding module to parse utterances into a semantic frame. This is then passed into the state tracker which accumulates the semantics from each utterance, and policy learner for generating the next system action. It is assumed that the output of the dialogue agent is simply the next action, rather than natural text, which is acceptable because the "user" is actually the simulator which operates under the assumption it is able to perfectly understand the agent.
b. Strengths: The other major contribution is a method of gathering data with a User Simulator so that there are enough examples for training an RL agent. By tweaking the parameters of the simulator, the auuthors are able to show that slot-level errors have a greater impact on the system performance than intent-level ones, and that slot value replacement degrades the performance most.
To understand what this means, let's start by defining the possible intent-level errors:
• Within Group Error: noise is from the same group of the real intent, where groups are either (a) inform intents (b) request intents or (c) other intents. For example, the real intent is request_theater, which calls into the request group, but the predicted intent from LU module might be request_moviename.
• Between Group Error: noise is from the different group. For example, a real intent request_moviename might be predicted as the intent inform_moviename.
• Random Error: any mixture of the two above
Similarly, let's also define the possible slot-level errors:
• Slot Deletion: slot is not recognized by the LU module
• Incorrect Slot Value: slot name is correctly recognized, but the slot value is wrong
• Incorrect Slot: both the slot and its value are incorrectly recognized
• Random Error: any mixture of the three above
Then the conclusion is that "Incorrect Slot Value" causes the biggest performance drop of all options.
c. Weaknesses: While the idea sounds promising, the RL agents were asked to explore a limited number of dialogue acts. Looking at actual dialogues, it feels like the entire system can be solved using a competent rule-based system.
d. Notes: The framework itself and the results are not particularly interesting, but the many experiments analyzing model succeess are quite insightful. They find predicting the wrong intent or slot is recoverable, but choosing the wrong value is not. This actually makes a lot of intuitive sense because if as a listener, you recieve a mismatched signal (intent implies one thing, but slot-value pair imply another), you ask for clarification. However, if you only get a wrong value, then you actually think you're correct, so you never bother to inquire further!
3. Network-based E2E Trainable Task-oriented Dialogue System (Wen et al.)
a. Process: Generally follows traditional DST framework and includes five main components:
• Intent Detection: maps parsed utterance into user-defined intents using LSTM, output is a hidden state vector
• Belief Tracker: maps parsed utterance (and previously predicted beliefs) into a distribution of beliefs for current timestep. This consists of a different tracker for every possible slot. There is also a distinction between inform slots (food type, area, price range) vs. request slots (address, phone number). A CNN is used to extract features, along with n-gram embeddings surrounding the delexicalized keywords. The key slots and values are "special", so they get their part of the embedding.
• Database Operator: takes in probability distribution of slots to calculate a binary truth vector that is 1 if the value is predicted to be important and 0 otherwise. Rules are written to make sure that a value is 1, only if that value is compatible with the query. For example, if the query is "food type", then values like "Japanese" and "Indian" are allowed, but values like "3-star rating" and "expensive" are not allowed.
• Policy Management: maps slot probability distribution, database entities, and intent hidden vector into agent actions. This part makes very little sense because it seems to be trained as a linear transform with SGD, rather than a reinforcement learning module. The output is a single vector o, but seems to be in a vector format with no interpretable meaning, as an actual action. Most critically, there is little justification for why the policy manager has the structure that it does.
• Text Generation: uses an LSTM to generate words one-by-one until EOS. (Can be replaced by a retrieval model for more human-like, but less generalizable responses). Then fills in the slots based on return values of the database entity pointer.
b. Strengths: This paper is among the first to develop an end-to-end neural system within the traditional DST framework. In doing so, they strike a good balance between imposing too much structure (ie. rule-based systems of the past) and not enough structure (ie. pure neural models). Also proposes a new way of collecting dialogue data based on a Wizard-of-Oz framework.
c. Weaknesses: Despite new dataset, the topic is still restaurant booking, so its really not that novel. Also, the method is just having person A (acting as the agent) label person B's utterance (acting as the customer) while also providing the agent output. This just sounds like a lot of work (and room for error) on the part of Person A.
Different belief trackers for each slot mean lots of data is needed. Does not allow room for agent to confirm any ambiguity. Slots are hard to identify when creating training data (and thus hard to delexicalize) which means placing a lot of confidence in the Turker to get it right. Not sure how this would perform with ambiguities found in real world data. Would have really liked to see how well the belief tracker module alone compares to hand-crafted lexicon.
d. Notes: Not sure why belief tracking (ie. semantic parsing into slot-value pairs) needs to be distinguished from intent detection. Feels like two separate networks are trained when they should be combined for weight sharing purposes.
Overall, data collection method is improved, but the lack of data issue remains largely unresolved. Also, the architecture seems overly complicated, and finally, attention helps, as always.
1. Neural Belief Tracker (Mrkšić et al.)
a. Process: Encodes three inputs using either a deep NN or a convolutional NN. It is a bit odd that an LSTM is not considered:
• Agent Utterance (ie. System Ouput) - the previous sentence, spoken by the agent.
• User Utterance: the current sentence, spoken by the user.
• Candidate Pairs: a list of all possible slot-value pairs, either inform (food type, area, price range) or request (address, phone number)
These 3 items are then passed into another layer:
• Semantic Decoding: calculates a similarity score between the user utterance and the candidate pair. Checks is the user explicitly offered/requested a specific piece of information that turn.
• Context Modeling: uses the interaction between agent utterance and candidate pair to decide if (a) system request or (b) system confirm occured. If the candidate pair passes this gating portion, then another similarity score is calculated to determine the impact of the user's answer.
• System request: "What price range would you like?", then area and food are not relevant.
• System confirm: "How about Turkish food?", then any user response is referring to food type, and not price range or area.
Finally, these two outputs are mashed together in a final network to calculate the binary (yes/no) decision about whether this candidate-pair occurred in the current timestep.
b. Strengths: Delexicalization is a process where "slots and values mentioned in the text are replaced with generic labels." However, this introduces a hidden dependency of identifying slot/value mentions in text. The authors then hit the nail on the head when they state "For toy domains, one can manually construct semantic dictionaries which list the potential rephrasings for all slot value pairs ... although, this will not scale to the rich variety of user language or to general domains." User utterances are unlabeled in real world scenarios, so slot-filling becomes impossible since we don't know which slots exist, much less which values belong to them. The solution proposed is to use dense vectors embeddings which encode meaning, and thus have a sense of semantic similarity baked in.
c. Weaknesses: Only uses n-grams, rather than full sentence encoding. Does not use attention or other means of long-term tracking. Makes Markovian assumption about influence of previous system dialogue acts. Does not report final impact on task completion rate, only performs intrinsic evaluation on belief tracking.
d. Note: This paper only deals with the belief tracking component, so evaluation is performed on predicting inform-slots and request-slots, ignoring policy manangement and dialog generation.
While GloVE performed admirably, a different set of vectors Paragram-SL999 actually did best, since these embeddings are trained on paraphrases and are specifically optimized for semantic similarity, as opposed to GloVE or Word2vec which are optimized for window-context similarity. The latter maps antonyms to similar vector spaces since the words are highly related, despite the opposing semantics.
1. Global-Locally Self-Attentive Dialogue State Tracker (Zhong, Xiong, Socher)
a. Process: Past DST models perform poorly when predicting on rare slot-value pairs since each slot requires its own tracker. Thus, this paper uses global modules to share parameters between estimators for each slot and local modules to learn slot-specific feature representations. The overall process is very similar to the NBT model above, where both include encoders for agent utterance, user utterance, and candidate slot-value pairs. Both also produce a similarity score based on user text (semantic decoder) and based on agent's utterance (context decoder), which are then fed into a final binary-decision mixture model. The key differences are
• Global-Local Encoding: rather than using a deep neural network or CNN, the three encoders operate with a two step LSTM/attention process. In more detail, given the input of a tokenized utterance X, the encoder performs a mapping of f(X) -> H, c where H which is the result of a BiLSTM encoder and c is the result of a self-attention on the encoding.
• Mixture model: What makes this unit special is that rather than just one encoder with attention, as typically found in Learning End-to-End Goal-Oriented Dialog, the encoder operates on two levels. Namely, there is one encoder for the global level where the weights are shared across slots, and an encoder for the local level, where the weights are retrained for each slot. The results of these two encoders are then combined in an interpolation mechanism where the mixture strength, beta, is a hyperparameter to be tuned on the dev set.
b. Strengths: The model takes into account global context so that weights can be shared across different slots. Uses an RNN as the encoder which makes sense.
c. Weaknesses: No clear explanation of where proposed slot-value pairs come from. I think the assumption is that the model cycles through all possible pairs.
d. Notes: Would be interesting to encode with hierarchical encoders and to also see how the model performs on extrinsic measures. Given the resources of the group, it would have been nice for them to release a new dataset.
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2019-12-15 21:14:19
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http://www.ck12.org/algebra/Equations-with-Radicals-on-Both-Sides/asmtpractice/Equations-with-Radicals-on-Both-Sides-Practice/r1/
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# Equations with Radicals on Both Sides
## Simplify and square both sides to solve
%
Progress
MEMORY METER
This indicates how strong in your memory this concept is
Progress
%
Equations with Radicals on Both Sides Practice
By CK-12
MAT.ALG.838.1
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2017-04-28 06:47:28
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https://assertnotmagic.com/2017/01/18/assert-not-magic/
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# Assert Not Magic
First post on the new layout with the new domain name! Woo! I’d like to take a minute to explain the significance behind the name. See, my criteria for selecting a good name came down to two things. I wanted it to be significant in my life – something that reflected my experience and point of view, and something that is somewhat of a theme in my life. The second thing is that it had to be a domain name that was available unique name.
So, “assert_not magic?” you say. “What does it mean?” you say. Well, at the most literal level, it comes from testing in Ruby. Much like the snippet on my blog’s banner, you might see something like this:
As is hopefully clear, assert_not, as the opposite of assert, just checks to make sure something is false.
The second part is a play on a Ruby idiom: methods that return boolean values generally end in a question mark. Things like list.empty? or number.even? are common.
Altogether, the phrase generally means, “make sure there’s no magic afoot.” And this leads to the heart of the meaning. There is no magic! (As far as you know.) And that is a very comforting fact. “Why?!” you ask, disappointedly. Because, if there is no magic, then everything that happens has a distinct and find-out-able cause. This is a mantra that has popped up in every aspect of my life. Math classes? No, there’s no magic rules or mystery. This equation was derived from somewhere. There is a logic behind it and I don’t have to memorize anything. Engineering? This mold won’t assemble right. That’s not the end of the investigation. There is a reason, somewhere in the design, for it not assembling right. We just have to figure out what that is. The code won’t run! I’ve checked everything and it must just be a glitch. Not possible. There’s still an explainable error somewhere.
Anyways, I just wanted to share that and explain the name of the blog. Keep assert_not magic? in mind with me as I keep learning!
Author: Ryan Palo | Tags: not-magic soft-skills
Like my stuff? Have questions or feedback for me? Want to mentor me or get my help with something? Get in touch! To stay updated, subscribe via RSS
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2018-05-20 23:41:27
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https://docs.mantidproject.org/nightly/fitting/fitfunctions/DiffSphere.html
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$$\renewcommand\AA{\unicode{x212B}}$$
# DiffSphere¶
## Description¶
This fitting function models the dynamics structure factor of a particle undergoing continuous diffusion but confined to a spherical volume. According to Volino and Dianoux 1,
$S(Q,E\equiv \hbar \omega) = A_{0,0}(Q\cdot R) \delta (\omega) + \frac{1}{\pi} \sum_{l=1}^{N-1} (2l+1) A_{n,l} (Q\cdot R) \frac{x_{n,l}^2 D/R^2}{[x_{n,l}^2 D/R^2]^21+\omega^2}$
$A_{n,l} = \frac{6x_{n,l}^2}{x_{n,l}^2-l(l+1)} [\frac{QRj_{l+1}(QR) - lj_l(QR)}{(QR)^2 - x_{n,l}^2}]^2$
Because of the spherical symmetry of the problem, the structure factor is expressed in terms of the $$j_l(z)$$ spherical Bessel functions. Furthermore, the requirement that no particle flux can escape the sphere leads to the following boundary condition2:
$\frac{d}{dr}j_l(rx_{n,l}/R)|_{r=R}=0 \,\,\,\, \forall l$
The roots of this set of equations are the numerical coefficients $$x_{n,l}$$.
The fit function DiffSphere has an elastic part modeled by fitting function ElasticDiffSphere, and an inelastic part modeled by InelasticDiffSphere.
## Attributes (non-fitting parameters)¶
Name
Type
Default
Description
NumDeriv
Q
f0.Q
f0.WorkspaceIndex
f1.Q
f1.WorkspaceIndex
$$NumDeriv$$ (boolean, default=true) carry out numerical derivative - $$Q$$ (double, default=1.0) Momentum transfer
## Properties (fitting parameters)¶
Name
Default
Description
f0.Height
1.0
Scaling factor to be applied to the resolution.
f0.Centre
0.0
Shift along the x-axis to be applied to the resolution.
2.0
f1.Intensity
1.0
scaling factor
2.0
f1.Diffusion
0.05
Diffusion coefficient, in units of A^2*THz, if energy in meV, or A^2*PHz if energy in ueV
f1.Shift
0.0
Shift in domain
Categories: FitFunctions | QuasiElastic
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2023-04-02 05:45:52
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https://archive.lib.msu.edu/crcmath/math/math/e/e101.htm
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## Elliptic Integral of the Second Kind
Let the Modulus satisfy . (This may also be written in terms of the Parameter or Modular Angle .) The incomplete elliptic integral of the second kind is then defined as
(1)
A generalization replacing with gives
(2)
To place the elliptic integral of the second kind in a slightly different form, let
(3) (4)
so the elliptic integral can also be written as
(5)
The complete elliptic integral of the second kind, illustrated above as a function of the Parameter , is defined by
(6) (7) (8) (9)
where is the Hypergeometric Function and is a Jacobi Elliptic Function. The complete elliptic integral of the second kind satisfies the Legendre Relation
(10)
where and are complete Elliptic Integrals of the First and second kinds, and and are the complementary integrals. The Derivative is
(11)
(Whittaker and Watson 1990, p. 521). If is a singular value (i.e.,
(12)
where is the Elliptic Lambda Function), and and the Elliptic Alpha Function are also known, then
(13)
See also Elliptic Integral of the First Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value
References
Abramowitz, M. and Stegun, C. A. (Eds.). Elliptic Integrals.'' Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.
Spanier, J. and Oldham, K. B. The Complete Elliptic Integrals and '' and The Incomplete Elliptic Integrals and .'' Chs. 61 and 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
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2021-12-02 04:38:07
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https://www.physicsforums.com/threads/crash-reconstruction-question.153137/
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# Crash reconstruction question
1. Jan 26, 2007
### Colm2243
What formula(s) would I use to find the initial velocity of a vehicle that has impacted a second vehicle in a same direction crash. There are no tire marks on the roadway and no braking involved. Thanks.
2. Jan 26, 2007
### Moridin
In physics, there is something called translation symmetry, meaning that the fundamental properties of the Universe does not depend on where you are. Out of this symmetry, a phenomena known as momentum arises. In non-relativistic velocities, this is a good approximation.
From your post, I assume that the first car (called car A) hits the second car (car B) from behind. No matter if some kinetic energy is converted to heat energy via friction, the sum of the momentum for car A and the momentum of car B before the impact is the same as the sum of the momentum for car A and the momentum for car B after the impact.
$$m_{A(before)} v_{A(before)} ~+~ m_{B(before)}v_{B(before)}~ = m_{A(after)} v_{A(after)} ~+~ m_{B(after)} v_{B(after)}$$
Solve for the velocity for A before the crash if you have access to the other values.
Last edited: Jan 26, 2007
3. Jan 26, 2007
### Colm2243
I don't have the values for pre collision velocity on either vehicle. How can I determine this? A wall was impacted after the two vehicles became stuck together and after impact with the wall vehicle two separated and rolled over and skid to a stop 40 feet after separation on it' s roof.
4. Jan 27, 2007
### cesiumfrog
Translation symmetry only exists in the Newtonian limit?!?
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2017-10-18 09:29:43
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https://uniteasy.com/post/1580/
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#### Question
Determine the angle (in radians) subtended at the centre of a circle of radius 3 cm by each of the following arcs:
arc of length 6 cm
arc of length cm
arc of length 1.5 cm
arc of length cm
Collected in the board: Circles problems
Steven Zheng posted 4 months ago
The arc length can be expressed in terms of centered angle in degrees or radians
l =\dfrac{n\pi r}{180\degree } = r\theta
In this question, the angle is required in radians. So the arc length formula to be used is
l = r\theta =3\theta
(r =3 cm)
arc of length 6 cm
l = 6 cm
then
6 = 3\theta
\theta =2
arc of length cm
l = cm
3π = 3\theta
\theta =\pi
arc of length 1.5 cm
l = 1.5 cm
1.5 = 3\theta
\theta =0.5
arc of length cm
l = 6π cm
6π =3\theta
\theta =2\pi
Steven Zheng posted 4 months ago
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2023-03-25 13:16:53
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https://infoscience.epfl.ch/record/154524
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Infoscience
Journal article
Measurement of K+K- production in two-photon collisions in the resonant-mass region
K+K- production in two-photon collisions has been studied using a large data sample of 67 fb-1 accumulated with the Belle detector at the KEKB asymmetric e+e- collider. We have measured the cross section for the process γγ → K +K- for center-of-mass energies between 1.4 and 2.4 GeV, and found three new resonant structures in the energy region between 1.6 and 2. 4 GeV. The angular differential cross sections have also been measured.
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2017-01-17 11:14:27
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http://openstudy.com/updates/515a52cde4b0507ceba2b443
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enay800 2 years ago dy/dx ln(xy)=xy^2-x
• This Question is Open
1. Azteck
What are you meant to find or solve?
2. Azteck
$\large \frac{dy}{dx} ln(xy)=xy^2-x$ $\large \frac{dy}{dx}[lnx+lny]=xy^2-x$ $\large \frac{1}{x}=xy^2-x$ $\large 1=x^2y^2-x^2$ $\large y^2=\frac{x^2+1}{x^2}$ $\large y=\pm \frac{\sqrt{x^2+1}}{x}$
3. abdullatif92211
this question can be solved using bernaoulli method
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2015-07-01 16:03:00
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https://feddebenedictus.com/2016/02/26/probability-0-is-not-impossibility/
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## Probability “0” Is Not Impossibility
LAST EDITED: July 20, 2019
## “The dart that said zero”
The probability that a dart will hit any specific point on a dartboard is zero because there are infinitely many points on the board. And yet if you throw a dart at a dartboard you’ll always hit some point (assuming you hit the dartboard). Hitting a specific point at a dartboard is highly improbable, but not impossible.
The responses to a similar statement in a previous post were mixed. Some people were delighted by its counterintuitiveness, whereas others were skeptical – what if we assume that the tip of the dart has a size that is not a mathematical point? What if it were, say, 0.01 square mm? Doesn’t that solve the problem? I’m afraid not. The assumption that the dart has a certain size wouldn’t change the probability of hitting any one point, because there would still be an infinite number of points on the board. Of course, the probability of hitting any one of those points would increase by a certain factor, but no matter how large this factor is, a probability of $\frac{1}{\infty}$ is still $\frac{1}{\infty}$: the probability remains zero.
What we can do is divide the area of the dartboard into a (finite) number of smaller areas and assume that the probability of hitting a certain area scales with the size of the area. This assumption boils down to assuming that if two areas are equal in size, then the probability of hitting them is also equal. Let’s call this the assumption of equal probabilities. At first sight it might seem silly to even pay attention to this assumption – let alone give it a name. Isn’t making such an assumption what we always do? A fair die has sides of equal size, divisions on the roulette table have equal sizes, etc. etc.
Indeed that is what we do in daily life. And yet it is good to realise that there are two fundamental problems with such an approach:
1. How do we know that equal areas have equal probabilities?
2. There are infinitely many ways to subdivide an area. How do we know which one is the correct one?
To grasp the second of these problems it might be helpful to realise that this problem occurs whenever we try to subdivide an infinite set. Just think of the set of positive integers (1, 2, 3, 4, 5…etc.) and try to subdivide it into equal subsets. These equal subsets may have any size we choose. For example, we can choose subsets each of which contains two elements, like this: (1, 2), (3, 4), (5, 6), (7, 8)…(etc.). But we might also opt for the smaller subsets (1), (2), (3), (4)…(etc.) or the larger subsets (1, 2, 3), (4, 5, 6), (7, 8, 9), (10, 11, 12)…(etc.). There are infinitely many choices, because we’ll never run out of integers! The same argument goes for areas (just assign an integer to every point) or any other infinite set.
## Conventions
Both of the questions facing the probability theorist can only be answered by adopting a suitable convention. We usually assume that equal areas have equal probability of being hit and that there is a natural way to divide up a continuous area (such as a roulette table) that is shared by everyone who looks at the situation, but these assumptions are actually conventions. The probabilistic problem of uniquely dividing up infinite sets has become known as Bertrand’s paradox.
It might seem an obvious choice that equal areas have equal probabilities of being hit. Isn’t that what we always assume? Well.. yes, but games in which this assumption seems the most obvious (such as games of dice, darts, or roulette) are chosen as games of chance precisely because the assumption of equal probability holds here. Think of this example: suppose we know that a tree in the forest is going to fall. If it falls, it does so either (slightly) to the right or (slightly) to the left. There are two possibilities, so if we know nothing about the situation, it seems natural to assume that both possibilities have a probability of $\frac{1}{2}$. But that’s odd! We just said we know nothing about the situation, and now we can make predictions about what will happen.
## Science Is A Different Game
For the scientist – perhaps working in the field of particle physics or in cosmology – it is not obvious which conventions are useful: perhaps there is a particle for which equal size does not imply equal probability. Or perhaps two cosmologists from opposite points on the earth’s surface study some galaxy without knowing the angle from which they observe the galaxy. It might be the case that the size of the galaxy as it appears to the two cosmologists differs by a factor two. Now suppose both cosmologists try to estimate the probability of light being emitted from the galaxy. If they were to naively adopt the assumption that probability scales with area, they will disagree!
Science is, in the end, a capricious affair.
Philosopher of physics at Amsterdam University College and Utrecht University, managing editor for Foundations of Physics and international paraclimbing athlete
This entry was posted in Philosophy of Mathematics, Philosophy of Physics and tagged , , , , . Bookmark the permalink.
### 3 Responses to Probability “0” Is Not Impossibility
1. AshGreen says:
Awesome! No words. You always go one step beyond.
There is so much great, useful information here. Thank you! Thank you! Thank you!
Read our guide if you wish.
Thebestpickers
Thanks again 🙂
2. Excellent post, thank you for sharing! Some thoughts:
1) actually, as Evi explained to me, dart boards consist of fibre bound together. The darts stick in between the fibres, so actually with this metaphor, there exist a finite number of potential points where the darts can end up 🙂
2) The paragraph starting with “what we can do” isn’t entirely clear – why would we “do” that? You just countered a critical comment, but it’s not entirely clear to me where the narrative is going from here.
3) I personally find the generalisation from this specific example, where you show how resolving this issue requires subjective consensus, to ‘all probability’, dubious. I think it’s quite possible that probability is objective, yet the example you provided happens to have some properties rendering probability /in that example/ subjective. So I’m not entirely convinced yet that all of probability is subjective 🙂
• fbenedictus says:
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2021-05-14 09:16:09
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https://brilliant.org/problems/too-much-of-one-variable-not-enough-of-the-other-2/
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# Too much of one variable, not enough of the other (fixed)
Number Theory Level pending
Let $$a$$ and $$b$$ be positive integers that satisfy
$\dfrac {(4a^2-1)^2}{4ab-1}$
is also an integer. How many ordered solutions for $$(a, b)$$ exist provided $$a+b \leq 1000$$?
×
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2017-10-21 12:29:18
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https://stat.unm.edu/~fletcher/bida.html
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# Bayesian Ideas and Data Analysis
Chapter 13 of this contains newer computing discussion.
## Preface
Bayesian statistics is about probability. What is the probability that a new chemotherapy treatment will be effective? What is the probability that a particular type of bank will go bankrupt? What is the probability that a positive mammogram is truly indicative of breast cancer? The Bayesian method combines expert scientific information with data using Bayes Theorem to obtain such probabilities. Dennis Lindley, a famous Bayesian, has asserted that there are two rules in Bayesian inference: (i) all uncertainty is modeled using probability and (ii) always obey the laws of probability. Consequently, a primary prerequisite for this course in a course in probability, preferably a calculus based course.
For most of what we do, probability is simply the area (or volume) under a curve (or surface) over a set. Such curves (surfaces) are defined by probability density functions. Bayesian inference requires integrating particular functions against density functions, for example, to obtain a prediction or the mean or variance of a quantity of interest. Sometimes, it is possible to evaluate integrals analytically, and we start the book with problems where the calculus is tractable. However, since most integrals that are necessary for Bayesian computation are not tractable, we turn to numerical approximation by simulation, in particular Markov chain Monte Carlo (MCMC) simulation. Appendices A and B present the basics of matrix algebra and probability for those whose skills in these areas are rusty.
Of course the more statistics courses you have had the better. We introduce a very large number of statistical models in the book, and the more of them that you have seen in other courses, the better. However, some of us have taught a version of this course to individuals who have only had a single probability course, and another version to scientifically sophisticated students who have not had a calculus based probability course, but who did have a background involving one or more applied statistics courses.
The first five chapters of the book constitute our version of the core of a traditional Bayesian Statistics course, covering the basics of Bayesian ideas, calculations and inference, including modeling one and two sample data from traditional normal, binomial, Poisson, exponential and other sampling models. Chapter 1 is motivational, presenting a number of portraits of the power of the Bayesian approach based on scientific scenarios that we have previously encountered and subsequently address in the book. Chapter 2 presents the fundamentals of Bayesian philosophy, and its' implementation including {\it real} prior specification, data modeling, and posterior inferences via Bayes Theorem. Chapter 3 presents the interplay between probability calculus and its approximation by simulation, and the implementation of simulation via WinBUGS. Chapter 4 develops deeper foundational issues, including aspects of hypothesis testing, exchangeability, prediction, model checking and selection, diffuse" prior specification, large sample approximation to posteriors, consistency, identifiability, and hierarchical modeling. Chapter 5 handles one and two sample analysis of binomial data, including relative risk estimation, as well as inferences based on normal and Poisson sampling. Also included in Chapter 5 is a treatment of case-control sampling and traditional odds ratio estimation, and methods for sample size determination.
Chapter 6 discusses the theoretical basis for Markov chain Monte Carlo simulation, and issues related to its practical application.
Chapter 7 introduces the concept of regression modeling at an elementary level, and Chapter 8 covers binomial regression including logistic regression for correlated data using generalized linear mixed models. Chapter 9 presents methods for the general linear model (analysis of variance and regression) and Chapter 10 extends those methods to handle correlated clustered and longitudinal measurement data using linear mixed models and multivariate normal analysis. Chapter 11 carries on the discussion of generalized linear models with coverage of Poisson regression including mixed models.
Chapters 12 covers the topic of time to event data (survival analysis) based on one and two sample data that are subject to censoring. Chapter 13 continues by developing regression models for survival data, including the accelerated failure time model and the Cox proportional hazards model. The chapter concludes with a discussion of frailty models for correlated survival data.
Chapter 14 discusses the topic of binary diagnostic testing. Material on continuous-response diagnostic test data appears on the book website.
Finally, Chapter 15 covers semiparametric and nonparametric inference. This chapter covers density estimation and flexible regression modeling of mean functions. %, thus allowing for curves as a function of time in longitudinal modeling for example, and taking standard parametric regression models and %allowing for great flexibility in modeling.
We started this project with the intention of writing a compact book. We ended it with enough material for two books (Chapters 1-9 and 11 would make up volume 1 with the remaining chapters being a second volume). There are a number of different versions of courses that we envision from our book. The most elementary chapters are 1, 2, 3, 5, 7, 11 and 14. If students have already had a course in regression modeling, Chapter 7 would not be necessary, but could be treated as required reading nonetheless. The most sophisticated chapters are 4, 6, 13 and 15. For more elementary courses we envision instructor discussion of parts of Chapter 4 on an {\it as needed} basis when covering subsequent chapters. Chapters 4 and 6 would be a must for more advanced courses. Different versions of this course that we envision are:
• M.S./Ph.D. Statistics$^{**}$: Chapters 1-6, 8-9, plus selection from remaining chapters.
• Biostatistics$^{**}$: Chapters 1-4, plus selection from 5, 6 and 8-14.
• Non-Biostatistics: Chapters 1-3, 4$^*$, 5, 7, plus selection from 8, 9, 11.
• Epidemiology: Chapters 1-3, 4$^*$, $5^*$, 7, plus selection from 8, 11, 12, 13, 14.
• Second Bayesian Course: Chapters 4, 10-15 plus Topics
$^*$ indicates selection of topics. $^{**}$ Students have taken basic regression course.
Exercises are an integral part of the book. Even if students choose not to do them, they should read them. They are often strategically placed in order to reinforce the surrounding discussion.
While our book is written with an emphasis on the use of WinBUGS and R, it is certainly feasible to teach a Bayesian course without requiring either of these software packages. For instance, we illustrate the use of SAS in some examples. Historically, books have been written either without thought of analyzing real data, which was one of our motivations for writing a different kind of book, or authors envisioned data analysis but left open the question of what software might be used. %In fact, we have previously used Gauss and R software in previous versions of this course. Instructors should have no difficulty using our book if that is the case. It is of course important for statistics Ph.D. students to learn to write their own programs for analyzing data. Some universities teach Bayesian computing in a separate course. When this isn't the case, instructors can easily augment the course with assignments to write code in the language of their choice. Much of our WinBUGS code serves as a kind of precise description of modeling efforts and the types of inferences we intend to make, which can be translated into other environments like R or Matlab.
The seeds for this book were planted when the first two authors were graduate students at the University of Minnesota starting in 1974. The environment there, which was created in large part by Seymour Geisser, fostered and promoted the foundational aspects of Statistics in virtually all aspects of the program. Surprisingly, there were no Bayesian courses in their curriculum at that time, but Bayesian ideas and methods were an integral part of many key courses. So we grew up" thinking that Bayesian methods were just another way to approach the whole of inferential statistics.
Another phase of development devolved from efforts with Ed Bedrick on how to specify what we call real" priors, that allow for incorporation of input from subject matter experts in generalized linear models. Indeed, a unique aspect of this book is its' emphasis on incorporating real" prior information.
From the early 1980s, Statistics 145 at UC Davis was taught initially by the second author as a successful and purely theoretical course using the seminal book by Jim Berger (***). It evolved through the early 1990s, during a failed attempt to make the course accessible to non-statistics graduate students, and then finally succeeded with a broad audience, including Statistics students, in our opinion due to the advent of the WinBUGS software (Spiegelhalter et al., ****). A missing component in previous versions was the capability of analyzing a wide variety of simple to complex data sets on top of emphasizing the foundations of the subject.
The final version of the book was transformed by joint efforts that were lead by the first author, and which were ultimately guided by the influence of Seymour Geisser and Spiegelhalter et al. Indeed, we have emphasized the importance of prediction and ease of computation throughout.
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2022-12-04 17:46:28
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https://kb.osu.edu/dspace/handle/1811/10947
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# ON ATOM/DIATOM COLLISIONAL REORIENTATION BY POLARISED FLUORESCENCE
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/10947
Files Size Format View
1979-RH-07.jpg 103.5Kb JPEG image
Title: ON ATOM/DIATOM COLLISIONAL REORIENTATION BY POLARISED FLUORESCENCE Creators: Ibbs, K. G.; McCaffery, A. J.; Rowe, M. D. Issue Date: 1979 Publisher: Ohio State University Abstract: A single mode circularly polarised dye laser is used to selectively excite individual vibronic states of several diatoms. The result is that the Mj state distribution of the excited molecules is asymmetric. The degree of polarisation of fluorescence is a measure of this orientation and easily predicted for collision free fluorescence. Results show the polarisation of resonance fluorescence to be virtually independent of pressure. The polarisation of rotational transferred fluorescence is a function of $\Delta$J, but for $\Delta J$ small compared with J the polarisation is again high.(1) This shows the Mj distribution is largely retained during elastic and inelastic collisions. Results are shown on high J lines in the $B^{3}\Sigma - X^{3}\Sigma$ spectrum of $Se_{2}$, and small J values on the $B^{1}\pi_{u} - X^{1}\Sigma_{g}$ spectrum of $Li_{2}$. The evidence lends support for Alexander’s l-dominant model of atom/diatom collisions. (2) Description: 1. S. Jeyes, A. McCaffery, M. Rowe, and H. Kato, Chem. Phys., Lett., 48, 91 (1977). 2. M. Alexander, J. Chem. Phys., 67, 2703 (1977). Author Institution: URI: http://hdl.handle.net/1811/10947 Other Identifiers: 1979-RH-07
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2016-06-26 19:10:19
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http://math.stackexchange.com/questions/205807/finding-vector-subspaces
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# Finding vector subspaces
I've got this problem:
Let $H = \left \{ x \in \mathbb{R}^{4} \, \left| \, x_2 - x_3 + 2x_4 = 0 \right. \right \}$
Find, if possible, $a \in \mathbb{R}$ and $S, T$ vector subspaces so that $\dim(S) = \dim(T)$, $S + T^\perp = H$, $S \cap T = \left \langle (1, a, 0, -1) \right \rangle$
What I have is:
• Using the dimension theorem for vector spaces: $\dim(S+T^\perp) = \dim(S) + \dim(T^\perp) - \dim(S \cap T^\perp) = \dim(H)$. Since $H$ is a $\mathbb{R}^{4}$ vector subspace with one equation, $\dim(H) = 3$. So $\dim(S) = 2$, $\dim(T^\perp) = 2$ and $\dim(S \cap T^\perp)=1$.
• If $\dim(T^\perp) = 2$, then $\dim(T)$ must be 2 as well. So I've got $S=\left \langle s_1, s_2 \right \rangle$ and $T=\left \langle t_1, t_2 \right \rangle$
• Let $s_1, s_2$ two linearly independent vectors from subspace $H$. Suppose $s_1 = (0,1,1,0), s_2 = (0,0,2,1)$. Then $S=\left \langle (0,1,1,0),(0,0,2,1) \right \rangle$.
• Let $t_1, t_2$ two linearly independent vectors from subspace $H$. Suppose $t_1 = (0,-2,0,1), t_2=(1,-1,1,1)$. Then $T^\perp=\left \langle (0,-2,0,1),(1,-1,1,1) \right \rangle$
• Because $(T^\perp)^\perp = T \rightarrow T=\left \{ x \in \mathbb{R}^{4} / -2x_2 + x_4 = x_1 - x_2 + x_3 + x_4 = 0 \right \}$
S and T satisfies all the conditions the problem asks. I know how to find $S \cap T$, but I'm a bit disappointed finding $a$. Any suggestion would be appreciated!
Thanks in advance!
-
## 1 Answer
Suppose there are such $S$, $T$, and $a$. Since $v := (1, a, 0, -1) \in S \cap T \subseteq S + T^{\perp} = H$, one has $a - 0 + 2\cdot(-1) = 0$, so $a = 2$.
I didn't understand your argument to deduce the dimensions of $S$ and $T$, so I'll give one myself:
Because $\dim (S\cap T) = 1$, you can conlude $1 \leq \dim T = \dim S \leq 2$ (If $\dim T = \dim S > 2$, in 4-dimensional space they met in dimension $>1$, which follows from the dimesion formula you have given). But the equation of $H$ says that $w := (0, 1, -1, 2)$ and $v = (1, 2, 0, -1) \in T$ are orthogonal: So if $\dim T = 1$, then $w \in T^{\perp} \subseteq H$, but $w \notin H$. This cannot be. Therefore $\dim S = \dim T = 2$, as you said.
So, now all you have to do, is complement $v$ with vectors $u_1, u_2$, orthogonally to $v$, to a base of $H = \langle v, u_1, u_2 \rangle$, take that completion as a base of $T^{\perp} = \langle u_1, u_2 \rangle$, and set $S = \langle v, u_1 \rangle$. Now $v \in T \cap S$, but since $u_1 \notin T$, the spaces $S$ and $T$ meet in $\langle v \rangle$.
I think you can take $u_1 = (2, -1, -1, 0)$ and $u_2 = (0, -1, -5, -2)$, they both should be orthogonal to $v$ and $w$.
I also don't understand, how you came up with your $T$ and $S$ though, it seems it might not be working, but I haven't looked into it, to be honest.
-
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2014-07-30 13:14:53
|
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https://physics.stackexchange.com/questions/481723/manipulating-dirac-notation/481733
|
# Manipulating Dirac Notation
I have trouble getting my head around manipulating Dirac notation, it's still new to me and I'm not used to it. I'm following the rotating wave approximation derivation for Rabi oscillations and light matter interactions.
I have $$|e\rangle$$, $$|g\rangle$$ as excited and ground state of a two level atom, my interaction Hamiltonian is $$\hat{H}_I$$.
I have calculated $$|e\rangle \langle e| \hat{H}_I |g\rangle \langle g| + |g\rangle \langle g| \hat{H}_I |e\rangle \langle e| = \frac{1}{2} \hbar Xe^{i\omega t} |g\rangle \langle e| + \frac{1}{2}\hbar X^*e^{-i\omega t} |e\rangle \langle g|,$$
where $$X=-eE_0d_{12}/\hbar$$ and $$d_{12}$$ is the dipole matrix element. This should reduce down to
$$\hat{H}_I = \frac{1}{2} \hbar Xe^{i\omega t} |g\rangle \langle e| + \frac{1}{2}\hbar X^*e^{-i\omega t} |e\rangle \langle g|$$
given that $$|e\rangle \langle e| +|g\rangle \langle g| =1$$ by completeness. I am sure this is just a simple step but I am unsure of what to do and what not to do manipulating Dirac notation, I don't know how to proceed. Spent a few hours getting this far, would be grateful if someone dug me out.
For a two-level system with ground state $$|g\rangle$$ and excited state $$|e\rangle$$, any operator $$\hat O$$ can be written
$$\hat O = \mathbb I\cdot \hat O \cdot \mathbb I = \big(|g\rangle\langle g| + |e\rangle\langle e|\big)\hat O \big(|g\rangle\langle g| + |e\rangle\langle e|\big)$$
This has four terms in general. However, if it so happens that your operator has vanishing diagonal elements (so $$\langle g | \hat O |g\rangle = \langle e|\hat O|e\rangle = 0$$) then you would have
$$\hat O = |g\rangle\langle g| \hat O |e \rangle\langle e| + |e\rangle\langle e| \hat O |g\rangle\langle g|$$
which is exactly what you wrote above.
The interaction Hamiltonian you describe is specifically proportional to the dipole operator $$\hat d$$, which has negative parity (i.e. $$\mathcal P \hat d \mathcal P = -\mathcal d$$). If the Hamiltonian of the system is invariant under parity inversion (so $$\mathcal P \hat H_0 \mathcal P = \hat H_0 \implies [\mathcal P,\hat H_0]=0$$), then you can choose your energy eigenbasis to consist of states with definite parity as well as definite energy.
However, this implies that the dipole operator has vanishing diagonal elements, because if $$|\phi\rangle$$ is an energy eigenstate with parity $$\pm 1$$, then
$$\langle \phi |\hat d|\phi \rangle = \langle \phi|\mathcal P^2 \hat d \mathcal P^2 |\phi\rangle = \big(\langle\phi|\mathcal P\big)\big(\mathcal P \hat d \mathcal P\big)\big(\mathcal P|\phi\rangle\big)$$ $$=\big(\pm \langle\phi| \big)\big(-\hat d\big)\big(\pm |\phi\rangle\big) = -\langle\phi|\hat d|\phi\rangle$$
which implies that $$\langle \phi|\hat d|\phi\rangle = 0$$.
• Thanks a lot for explaining! Makes clear sense. I had a feeling using and understanding Dirac notation is just the tip of the theoretical quantum mechanics iceberg. Back to the books for me! – Samalama May 23 '19 at 8:39
In a two level system, $$|e\rangle$$ meant excited states, and $$|g\rangle$$ meant ground states.
I guess you are using undergraduate quantum, in that case, most of the time one only need to know that $$\langle g|$$ was complex conjugate of $$|g\rangle$$. Both $$|e\rangle$$ and $$|g\rangle$$ was orthonormal basis( unitary vector and $$\delta_i^j$$). Basically, you may think of basis as the position of a vector, and it's coefficient as the coordinates.
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2020-04-01 17:51:48
|
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|
https://puzzling.stackexchange.com/questions/25110/solving-a-puzzle-related-to-mars-attacks
|
# Solving a puzzle related to Mars Attacks
I need help for solving this puzzle.
Does anybody have an idea?
• context might help here. – klm123 Jan 6 '16 at 16:02
• yeah trying that – Devunuri Saipraneeth Jan 7 '16 at 6:47
• I mean I noted the phrase "next question", so there is previous question(-s) and the puzzle can be unsolvable for us, since we don't know what are previous tasks and things like this. – klm123 Jan 7 '16 at 7:25
• its nothing like that – Devunuri Saipraneeth Jan 7 '16 at 8:22
• previous question is irrelevant to this – Devunuri Saipraneeth Jan 7 '16 at 8:23
I think the answer is either
"See you again" or "Young, Wild and Free" by Wiz Khalifa
Writing out the letters of the alphabet underneath the letters as they appear on a keyboard gives a method of encrypting/decrypting information.
QWERTYUIOPASDFGHJKLZXCVBNM
ABCDEFGHIJKLMNOPQRSTUVWXYZ
Under these rules, using for decryption, we have that
V $\rightarrow$ W, O $\rightarrow$ I, M $\rightarrow$ Z, A $\rightarrow$ K, I $\rightarrow$ H, Q $\rightarrow$ A, S $\rightarrow$ L, Y $\rightarrow$ F.
Hence the message from the Good Samaritan becomes
WIZ KHALIFA
Looking at his discography, I reckon the song is either
"See you again" referring to seeing the martians on their return visit to Earth
or
"Young, Wild and Free" which was a song Wiz recorded with Snoop Dogg but also featured Bruno MARS.
• Awesome Dude!The answer is "See You Again" – Devunuri Saipraneeth Feb 11 '16 at 6:10
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2021-06-17 08:51:46
|
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|
https://proofwiki.org/wiki/28
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# 28
Previous ... Next
## Number
$28$ (twenty-eight) is:
$2^2 \times 7$
The only perfect number which is the sum of equal powers of exactly $2$ positive integers:
$28 = 1^3 + 3^3$
The $1$st triangular number which is the sum of $2$ cubes:
$28 = 1 + 27 = 1^3 + 3^3$
The $2$nd perfect number after $6$:
$28 = 1 + 2 + 4 + 7 + 14 = \map \sigma {28} - 28 = 4 \times 7 = 2^{3 - 1} \paren {2^3 - 1}$
The $3$rd Ore number after $1$, $6$:
$\dfrac {28 \times \map \tau {28} } {\map \sigma {28} } = 3$
The $3$rd Keith number after $14$, $19$:
$2$, $8$, $10$, $18$, $28$, $\ldots$
The $3$rd primitive semiperfect number after $6$, $20$:
$28 = 1 + 2 + 4 + 7 + 14$
The $3$rd Sierpiński number of the first kind after $2$, $5$:
$28 = 3^3 + 1$
The $4$th hexagonal number after $1$, $6$, $15$:
$28 = 1 + 5 + 9 + 13 = 4 \paren {2 \times 4 - 1}$
The $5$th term of Göbel's sequence after $1$, $2$, $3$, $5$, $10$:
$28 = \paren {1 + 1^2 + 2^2 + 3^2 + 5^2 + 10^2} / 5$
The $6$th semiperfect number after $6$, $12$, $18$, $20$, $24$:
$28 = 1 + 2 + 4 + 7 + 14$
The $7$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$:
$28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \dfrac {7 \times \paren {7 + 1} } 2$
Hence there are $28$ dominoes in a standard set
The $7$th happy number after $1$, $7$, $10$, $13$, $19$, $23$:
$28 \to 2^2 + 8^2 = 4 + 64 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
The $12$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$:
$28 = 2 + 26$
The $12$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $20$, $22$, $26$ which cannot be expressed as the sum of $2$ composite odd numbers.
The $15$th positive integer which is not the sum of $1$ or more distinct squares:
$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $\ldots$
The $17$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
The $20$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$, $18$, $19$, $24$, $25$, $27$ such that $2^n$ contains no zero in its decimal representation:
$2^{28} = 268 \, 435 \, 456$
## Historical Note
There are approximately $28$ days in the lunar cycle.
This is probably the origin of the widespread cultural custom of splitting time both into $7$-day chunks called weeks, and also the year into $12$ approximately $28$-day chunks called months.
There are $28$ pounds avoirdupois in one quarter.
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2020-09-20 14:07:01
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https://gamedev.stackexchange.com/questions/21507/player-moving-up-is-he-jumping-or-climbing
|
Player moving up, is he jumping or climbing?
In a 2D physics-based platformer game that has ladders in it, how do you determine whether the player moving up is caused by a jump or him climbing a ladder, such that you know what animation to play?
And in general, obviously the direction vector is not enought to determine the animation to play: how do you also determine the cause of the movement (so you know the correct sprite to use)?
• If a player is on a ladder, is he really in the air? Dec 29 '11 at 9:08
I have a personal logic issue on a regular basis that you seem to be expressing here. It's that sometimes you're looking at the logic and programming from the wrong direction entirely. In this case, you seem to not be realizing that from the beginning you know which is happening, because of the logic code. You should already know, programmatically speaking, whether he's on the ladder or not.
Think about it (as I said, I often have this problem). The games logic should know you're on the ladder, supposing that you're being a smart programmer. For a better explanation, I have a file archive type, and wasted about an hour of a day trying to figure out how to get the size of the data and I never once stopped to think that, when I load in the file, I now know its size :P.
The reason why this relates so exclusively, is that you're looking at it as "Now that I'm in the air, how do I figure out if I'm on a ladder or jumping?"... When you should really think "I'm about to get in the air, but by which method FIRST, so that I can use it for the animations use."
• Simple e.g.: OnJump() change your jumping boolean to true, and then if you're in the air and jumping is false, then you're either falling or on a ladder, etc. Dec 24 '11 at 23:44
The physics system neither knows nor cares why the player is moving in a particular direction. It is your game that caused the physics system to move the player that way, and it is therefore the responsibility of your game to keep track of that. The player entity should have some state on it that will tell if it is jumping, climbing, running, etc. You shouldn't be trying to figure this out based on what happened with the physics system.
Seems that what ever event that is triggered on the collision that sets the players vertical velocity could also be used to set a flag specifying what animation to use.
If you are using a 3rd party library that is too restrictive with what info it gives you it would be possible to surround the ladder with a rectangle and if the player moving up and contained in that rectangle then do the climbing animation.
• This actually brings up a good point I was going to address later into my own games development, and that is how to change the speed of an animation based on the physics. I have characters that I want to flail their arms procedurally, faster based on how fast they're falling. And I hadn't ever thought of a good way to do it in depth (was thinking over time), but you're velocity idea is much better. +1 for a general usable answer, and not being specific to only his case. Dec 25 '11 at 2:48
When setting the jump velocity also set the characters state to STATE_JUMPING. Doing this you always know in which state the player is in and can act accordingly by checking the players state.
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2022-01-23 01:08:26
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https://questioncove.com/updates/4e5925a40b8b1f45b47907df
|
Mathematics
OpenStudy (anonymous):
A= 1/2 bh for b
OpenStudy (anonymous):
think itd be b=2A/h
OpenStudy (anonymous):
$b = \frac{2A}{h}$
OpenStudy (anonymous):
how do you work it out?
OpenStudy (anonymous):
first of all u want to get b alone right? so to get ride of the 1/2, multiply both sides by reciprocal which is 2.so 2A=bh and to get rid of the h youd divide both sides by h.so 2A/h=b
OpenStudy (anonymous):
how do you work it out?
OpenStudy (anonymous):
?
OpenStudy (anonymous):
duz it help?
OpenStudy (anonymous):
so basically to undo multiplication u divide. to undo division u multiply. the only way to move a fraction to the other side would be to multiply by the reciprocal. for ex if 1/3x=6 multiply by 3/1 to get x=18
OpenStudy (anonymous):
ohhh
OpenStudy (anonymous):
OpenStudy (anonymous):
but if you had 1/2 bh over 2 it would get id of 1/2 but wouldnt it affect the bh?
OpenStudy (anonymous):
ur right
OpenStudy (anonymous):
so..
OpenStudy (anonymous):
no it wouldnt, only if you did 1/2 over 2 if you left the bh alone.
OpenStudy (anonymous):
wouldnt affect bh everything cancels xcept the bh
OpenStudy (anonymous):
yea, thanks,
OpenStudy (anonymous):
no prblm
OpenStudy (anonymous):
A=1/2bh
OpenStudy (anonymous):
A over 2 = 2A
OpenStudy (anonymous):
1/2 bh over 2 =bh
OpenStudy (anonymous):
wait a sec what do u mean by A over 2 is 2A
OpenStudy (anonymous):
so its b= 2A over h
OpenStudy (anonymous):
i got it right?
OpenStudy (anonymous):
wait u lost me. what r u trying to do
OpenStudy (anonymous):
2A over h stays 2a over h, then bh over h becomes b
OpenStudy (anonymous):
r u trying 2 solve 4 b
OpenStudy (anonymous):
yep
OpenStudy (anonymous):
thanks
OpenStudy (anonymous):
ok so it is b=2A over h
OpenStudy (anonymous):
yea
OpenStudy (anonymous):
dont let the letters in the equation confuse u think of it as a regular equation with regular #
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2021-05-12 17:49:05
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http://clay6.com/qa/15502/if-half-mole-of-oxygen-combine-with-al-to-form-al-2o-3-the-weight-of-al-use
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Browse Questions
# If half mole of oxygen combine with Al to form $Al_2O_3$,the weight of Al used in the reaction is
$(a)\;27g\qquad(b)\;40.5g\qquad(c)\;54g\qquad(d)\;18g$
$4Al+3O_2\rightarrow 2Al_2O_3$
Ratio of weight of $Al$ and $O_2$ is 9 : 4
$\therefore \large\frac{9}{4}=\frac{x}{8}$$=18g$
Hence (d) is the correct answer.
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2017-04-24 09:25:41
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https://www.semanticscholar.org/paper/A-Noncommutative-Mikusinski-Calculus-Rosenkranz-Korporal/46acf31c569e6e3b13d90049a422e3f308d3ddbc
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• Corpus ID: 119253712
# A Noncommutative Mikusinski Calculus
@article{Rosenkranz2012ANM,
title={A Noncommutative Mikusinski Calculus},
author={Markus Rosenkranz and Anja Korporal},
journal={arXiv: Rings and Algebras},
year={2012}
}
• Published 6 September 2012
• Mathematics
• arXiv: Rings and Algebras
We set up a left ring of fractions over a certain ring of boundary problems for linear ordinary differential equations. The fraction ring acts naturally on a new module of generalized functions. The latter includes an isomorphic copy of the differential algebra underlying the given ring of boundary problems. Our methodology employs noncommutative localization in the theory of integro-differential algebras and operators. The resulting structure allows to build a symbolic calculus in the style of…
1 Citations
### On De Graaf spaces of pseudoquotients
• Mathematics
• 2013
A space of pseudoquotients $\mathcal{B}(X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of
## References
SHOWING 1-10 OF 56 REFERENCES
### Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases
• Mathematics, Computer Science
ArXiv
• 2012
The canonical simplifier for integro-differential polynomials is used for generating an automated proof establishing a canonical simplifiers for integrospecific operators in the Theorema system.
### A skew polynomial approach to integro-differential operators
• Mathematics
ISSAC '09
• 2009
It is shown how skew polynomials can be used for defining the integro-differential Weyl algebra as a natural extension of the classical Weylgebra in one variable as well as for fixing the integration constant.
### BAXTER ALGEBRAS AND DIFFERENTIAL ALGEBRAS
A Baxter algebra is a commutative algebra A that carries a generalized integral operator. In the first part of this paper we review past work of Baxter, Miller, Rota and Cartier in this area and
### Noncommutative localization in noncommutative geometry
The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of spaces'', locally described by noncommutative rings and their categories of one-sided
### Galois Theory of Linear Differential Equations
• Mathematics
• 2012
Linear differential equations form the central topic of this volume, Galois theory being the unifying theme. A large number of aspects are presented: algebraic theory especially differential Galois
### Baxter Algebras and the Umbral Calculus
We apply Baxter algebras to the study of the umbral calculus. We give a characterization of the umbral calculus in terms of Baxter algebra. This characterization leads to a natural generalization of
### An algebraic foundation for factoring linear boundary problems
• Mathematics
• 2009
Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting of a surjective linear map (“differential operator”) and an orthogonally
### Free Ideal Rings and Localization in General Rings
Preface Note to the reader Terminology, notations and conventions used List of special notation 0. Preliminaries on modules 1. Principal ideal domains 2. Firs, semifirs and the weak algorithm 3.
### Integro-differential polynomials and operators
• Mathematics
ISSAC '08
• 2008
Two algebraic structures for treating integral operators in conjunction with derivations are proposed that can be used to solve boundary problems for linear ordinary differential equations and canonical/normal forms with algorithmic simplifiers are described.
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2022-09-25 22:20:12
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https://ubpdqnmathematica.wordpress.com/
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## Symmetric Polynomials
This post is inspired by the video and post of Presh Talwakar
Categories: Uncategorized
## Ellipses in Triangles
This post is motivated by a tweet about Steiner inellipse.
The code used:
Module[{s = RegionCentroid[Polygon[{a, b, c}]], ab = b - a, m1, m2, m3, sc, rt}, sc = c - s; {m1, m2, m3} = Mean /@ Partition[{a, b, c, a}, 2, 1]; rt = Quiet[ ReIm[z] /. Solve[1/(z - Complex @@ a) + 1/(z - Complex @@ b) + 1/(z - Complex @@ c) == 0, z]]; ParametricPlot[s + sc Cos[t]/2 + ab Sin[t]/(2 Sqrt[3]), {t, 0, 2 Pi}, PlotRange -> Table[{-3, 3}, 2], Frame -> True, Epilog -> {PointSize[0.03], EdgeForm[Black], FaceForm[None], Polygon[{a, b, c}], Point[{a, b, c}], Red, Point[{m1, m2, m3}], Orange, Point[s], Purple, Point[rt]}]] Manipulate[ se[p1, p2, p3], {{p1, {0, 0}}, Locator}, {{p2, {0, 1}}, Locator}, {{p3, {1, 1}}, Locator}]
Categories: Uncategorized
## Squares
This post was motivated by Stackexchange question.
Categories: Uncategorized
## Squares and Triangles
This post is inspired by a puzzle I found on Twitter. I cannot find the source right now. What is the area of the shaded triangle? The following gif provides the clue:
Note all basic elements are squares. The small upper squares have sides of unit length. Note the whatever the triangle the base as the upper right unit square diagonal drawn diagonal always has a height of the drawn square with side 2.
Categories: Uncategorized
## Fun with Fractions
This post is inspired by tweet from @CutTheKnotMath and his post.
He posed this puzzle for the New Year:
Consider the list: $a_i, 1\le i \le n$
Consider the product: $\prod_{i=1}^n (a_i+1) -1$
For $n\ge 2$, replacing randomly chosen $a_i, a_j$ with $a_i+a_j+ a_i a_j$ in the expression does not change its value as $(a_i+1)(a_j+1)=(a_i+a_j+a_i a_j)+1$. So repeating till there is one number allows you solve the problem, i.e. process ends a unique number. For the particular sequence this is particularly pleasing: $\prod ^n_{i=1}(1+1/i)-1=\prod^n_{i=1}\frac{i+1}{i} -1= n+1-1=n$.
Simulating a small example with Mathematica:
Categories: Mathematica, Mathematics
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2019-12-14 18:47:29
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https://mathoverflow.net/questions/198785/what-is-the-growth-of-the-rank-of-a-power-of-a-finite-simple-group?noredirect=1
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# What is the growth of the rank of a power of a finite simple group?
Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
• You should probably check the work of J. Wiegold – Geoff Robinson Mar 1 '15 at 17:01
• @GeoffRobinson which one? do you have a reference? – Pablo Mar 1 '15 at 17:02
• He wrote a few papers on "Growth sequences of Groups" – Geoff Robinson Mar 1 '15 at 18:42
• it's logarithmic, check Thevenaz's elementary argument: arxiv.org/abs/math/9703201 – YCor Mar 1 '15 at 19:04
• Essentially a duplicate of this question:mathoverflow.net/questions/187736/… – Ian Agol Mar 1 '15 at 21:59
One has $$1 \leq s_n - \frac{\log(n)}{\log|S|} \leq 2r$$ based on an elementary argument in Remark 1.1 in [Moshe Jarden and Alexander Lubotzky, Random normal subgroups of free pro-finite groups, J. Group Theory 2 (1999) 213-224], where $r$ denotes the minimal number of generators of $S$. By the classification of finite simple groups, we know that $r = 2$.
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2019-12-15 22:10:43
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https://hsm.meta.stackexchange.com/questions/373/those-who-do-not-know-history-are-doomed-to-recreate-it
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# Those who do not know [history] are doomed to recreate it
Six months ago, the tag was burninated as useless.
The tag was supposedly blacklisted as early as 2014, but has come back repeatedly, including in 2020.
The tag (at the time of posting of this question) has 15 questions and still no tag wiki. It is completely useless, about as useful as a [programming] tag on Stack Overflow, a [vintage] tag on Retrocomputing SE, or a [language] tag on Linguistics SE.
I believe it is high time to burninate this tag for the last time and blacklist it.
• I am removing the tag from the existing questions - would do it over a span of days so as to not bump all the questions at once. On a side note, I had started removing the tag from new questions since last year (after your post) - still seem to have missed some since the number of such questions increased. Jan 28 at 15:06
• Update: the [history] tag has now been successfully burninated, hopefully for the last time. Feb 4 at 11:26
• @TheAmplitwist the tag is [history]. Feb 4 at 11:30
The regex for this particular tag-blocking seems to specifically check for a tag name consisting solely of history- followed by one or more "word characters" (i.e. alphanumeric characters or underscores).
So as written, something like history-of matches the regex (meaning can't be created as a tag). But just history doesn't match the regex (since it doesn't have the hyphen or a word character after it), and neither does history-of-science (since it has at least one non-word character: the hyphen) – meaning it doesn't stop or from being (re)created as tags.
The tag block has now been fixed. A dev has updated the regex for the blocklist to ensure that history will be blocked (i.e. can no longer be created as a tag), as well as anything beginning with the word history followed by any combination of dashes and alpha-numeric chars.
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2022-05-28 11:18:00
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https://www.nature.com/articles/s41467-019-09639-3?error=cookies_not_supported&code=a6ddaa54-f2b8-4281-812d-b7bf3d7d3341
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Article | Open | Published:
# A Bayesian mixture model for clustering droplet-based single-cell transcriptomic data from population studies
## Abstract
The recently developed droplet-based single-cell transcriptome sequencing (scRNA-seq) technology makes it feasible to perform a population-scale scRNA-seq study, in which the transcriptome is measured for tens of thousands of single cells from multiple individuals. Despite the advances of many clustering methods, there are few tailored methods for population-scale scRNA-seq studies. Here, we develop a Bayesian mixture model for single-cell sequencing (BAMM-SC) method to cluster scRNA-seq data from multiple individuals simultaneously. BAMM-SC takes raw count data as input and accounts for data heterogeneity and batch effect among multiple individuals in a unified Bayesian hierarchical model framework. Results from extensive simulation studies and applications of BAMM-SC to in-house experimental scRNA-seq datasets using blood, lung and skin cells from humans or mice demonstrate that BAMM-SC outperformed existing clustering methods with considerable improved clustering accuracy, particularly in the presence of heterogeneity among individuals.
## Introduction
Single-cell RNA sequencing (scRNA-seq) technologies have been widely used to measure gene expression for each individual cell, facilitating a deeper understanding of cell heterogeneity and better characterization of rare cell types1,2. Compared to early generation scRNA-seq technologies, the recently developed droplet-based technology, largely represented by the 10x Genomics Chromium system, has quickly gained popularity because of its high throughput (tens of thousands of single cells per run), high efficiency (a couple of days), and relatively lower cost (<\$1 per cell)3,4,5,6. It is now feasible to conduct population-scale single-cell transcriptomic profiling studies, where several to tens or even hundreds of individuals are sequenced7.
A major task of analyzing droplet-based scRNA-seq data is to identify clusters of single cells with similar transcriptomic profiles. To achieve this goal, classic unsupervised clustering methods such as K-means clustering, hierarchical clustering, and density-based clustering approaches8 can be applied after some normalization steps. Recently, scRNA-seq tailored unsupervised methods, such as SIMLR9, CellTree10, SC311, TSCAN12, and DIMM-SC13, have been designed and proposed for clustering scRNA-seq data. Supervised methods, such as MetaNeighbor, have been proposed to assess how well cell-type-specific transcriptional profiles replicate across different datasets14. However, none of these methods explicitly considers the heterogeneity among multiple individuals from population studies. In a typical analysis of population-scale scRNA-seq data, reads from each individual are processed separately and then merged together for the downstream analysis. For example, in the 10x Genomics Cell Ranger pipeline, to aggregate multiple libraries, reads from different libraries are downsampled such that all libraries have the same sequencing depth, leading to substantial information loss for individuals with higher sequencing depth. Alternatively, reads can be naively merged across all individuals without any library adjustment, leading to batch effects and unreliable clustering results.
Similar to the analysis of other omics data, several computational approaches have been proposed to correct batch effects for scRNA-seq data. For example, Spitzer et al.15 adapted the concept of force-directed graph to visualize complex cellular samples via Scaffold (single-cell analysis by fixed force- and landmark-directed) maps, which can overlay data from multiple samples onto a reference sample(s). Recently, two new methods: mutual nearest neighbors16 (MNN) (implemented in scran) and canonical correlation analysis (CCA)17 (implemented in Seurat) were published for batch correction of scRNA-seq data. All these methods require the raw counts to be transformed to continuous values under different assumptions, which may alter the data structure in some cell types and lead to difficulty of biological interpretation.
We first conducted an exploratory data analysis to demonstrate the existence of batch effect in multiple individuals using both publicly available and three in-house synthetic droplet-based scRNA-seq datasets, including human peripheral blood mononuclear cells (PBMC), mouse lung and human skin tissues. Detailed sample information was summarized in Fig. 1a and Supplementary Table 1. We use human PBMC as an example. We isolated from whole blood obtained from 4 healthy donors and used the 10x Chromium system to generate scRNA-seq data. We also included one additional healthy donor from a published PBMC scRNA-seq data4 to mimic the scenario where we combine the local dataset with the public datasets. In this cohort, sample 1 and sample 2 were sequenced in one batch; sample 3 and sample 4 were sequenced in another batch; sample 5 was downloaded from the original study conducted by 10x Genomics4. As an exploratory analysis, we produced a t-SNE plot based on the first 50 principal components (Supplementary Fig. 1) of all cells from these 5 donors and observed a clear batch effect: samples from the same batch tend to cluster together.
This illustrative example demonstrates the importance and urgent need for well characterizing different sources of variability and correcting potential batch effects among droplet-based scRNA-seq datasets collected from multiple individuals. In addition, due to the computational burden, many methods cannot be scaled up to analyze population-scale droplet-based scRNA-seq data with tens of thousands of cells collected from many individuals under various conditions. In this study, we propose a BAyesian Mixture Model for Single Cell sequencing (BAMM-SC) to simultaneously cluster large-scale droplet-based scRNA-seq data from multiple individuals. BAMM-SC directly works on the raw counts without any data transformation and models the heterogeneity from multiple sources by learning the distributions of signature genes in a Bayesian hierarchical model framework. In the following sections, we will describe our method, benchmark its performance against existing clustering methods in simulation studies, and evaluate our method for its accuracy, stability, and efficiency in three in-house synthetic scRNA-seq datasets including PBMCs, skin, and lung tissues from humans or mice.
## Results
### Overview of BAMM-SC
BAMM-SC represents a Bayesian hierarchical Dirichlet multinomial mixture model, which explicitly characterizes three sources of heterogeneity (i.e., genes, cell types, and individuals) (see Methods). Figure 1b provides an overview of the model structure in BAMM-SC, which directly models cell-type specific genes’ unique molecular identifier (UMI) counts and their heterogeneity among different individuals through a hierarchical distribution structure in a Bayesian framework. Our method has the following three key realistic assumptions. First, cell type clusters are discrete, and each cell belongs to one specific type exclusively. Second, heterogeneity exists among different individuals and across different cell types. The heterogeneity of the same cell type among different individuals is smaller than the heterogeneity across different cell types within the same individual. Third, cells of the same cell type share a similar gene expression pattern. That is, the underlying statistical distributions for cells within the same cell type are assumed to be the same. The mathematical model representations are included and explained in Supplementary Methods. Compared to other clustering methods which ignore individual level variability, BAMM-SC has the following four key advantages: (1) BAMM-SC accounts for data heterogeneity among multiple individuals, such as unbalanced sequencing depths and technical biases in library preparation, and thus reduces the false positives of detecting individual-specific cell types. (2) BAMM-SC borrows information across different individuals, leading to improved power for detecting individual-shared cell types and higher reproducibility as well as stability of the clustering results. (3) BAMM-SC performs one-step clustering on raw UMI count matrix without any prior batch-correction step, which is required for most clustering methods in the presence of batch effect. (4) BAMM-SC provides a statistical framework to quantify the clustering uncertainty for each cell in the form of posterior probability for each cell type (see Methods).
### Simulation studies
We have conducted comprehensive simulation studies to benchmark the performance of BAMM-SC. Specifically, we simulated droplet-based scRNA-seq data collected from multiple individuals from the posited Bayesian hierarchical Dirichlet multinomial mixture model (see Methods and Supplementary Methods). We considered different experimental designs, including different heterogeneities among multiple individuals and different numbers of individuals (Fig. 2). In our posited hierarchical model, the log normal prior distribution LN $$(\mu _{ik},\sigma _{ik}^2)$$ measures the heterogeneity of gene i in cell type k among multiple individuals, where $$\mu _{ik}$$ and $$\sigma _{ik}^2$$ are related to the mean and variation of gene expression. Without loss of generality, we used the mean of $$\sigma _{ik}^2$$ across all genes and all cell types to quantify the overall individual level heterogeneity. We applied BAMM-SC to each synthetic dataset, and compared the inferred cell type label of each single cell with the ground truth, measured by adjusted Rand index (ARI)18. We compared BAMM-SC with other competing clustering methods (K-means, TSCAN, SC3, and Seurat), which are either methods from different clustering categories or recommended by recent reviews on clustering methods for single-cell data19,20. Since none of methods model batch effects and therefore each needs to be combined with a batch correction method as a preprocessing step in data analysis. We applied two recently published and prevalent methods srcan MNN16 and Seurat CCA17 prior to these clustering methods so that each combination can be a fair comparison with BAMM-SC, which does not need a separate batch correction step.
Specifically, we compared BAMM-SC with the other nine competing methods (MNN+K-means, MNN+TSCAN, MNN+SC3, MNN+Seurat, CCA+K-means, CCA+TSCAN, CCA+SC3, CCA+Seurat, and DIMM-SC) in the simulation studies. Noticeably, DIMM-SC, our previously developed method for clustering scRNA-seq data from a single individual, also takes the raw UMI count matrix as the input without any batch effect correction or data transformation. We pooled single cells from different individuals together while ignoring each individual label, and then applied DIMM-SC to the pooled data. We simulated 100 datasets and summarized the corresponding ARIs for each method.
As shown in Fig. 2a, BAMM-SC consistently outperformed the other nine competing methods across a variety of individual level heterogeneities by achieving higher average ARI and lower variation of ARI among 100 simulations. As expected, the performance of all ten clustering approaches decreases as the among individual heterogeneity increases, measured by the mean $$\sigma _{ik}^2$$ values. In Fig. 2b, with the increase of number of individuals, BAMM-SC achieved higher ARI, while ARIs of other methods either remained stable or decreased.
Furthermore, we performed comprehensive simulation studies by generating simulated scRNA-seq datasets from different number of cell type clusters (Supplementary Fig. 2a), different overall sequencing depths (Supplementary Fig. 2b), and different cell-type-specific heterogeneities (i.e., the mean difference of gene expression profiles between two distinct cell types) (Supplementary Fig. 2c). BAMM-SC consistently outperformed other methods in terms of accuracy and robustness in all these scenarios. Taken together, our comprehensive simulation studies have demonstrated that, when data are generated from the true model, BAMM-SC is able to appropriately borrow information across multiple individuals, account for unbalanced sequencing depths, and provide more accurate and robust clustering results than other competing methods.
To evaluate the robustness of BAMM-SC when data generation model is mis-specified, we simulated additional datasets using R package Splatter21, a commonly used tool for scRNA-seq data simulation using a completely different model. To make our simulated data a good approximation to the real data, we first downloaded the raw UMI count matrix of a purified B-cell scRNA-seq dataset from the 10x Genomics website (https://support.10xgenomics.com/single-cell-gene-expression/datasets/1.1.0/b_cells), and used the function splatEstimate to estimate the parameters related to mean of gene, library size, expression outlier, dispersion across genes, and dropout rate. We assumed cell types are shared across multiple individuals, where each individual is treated as one batch with the same number of cells and genes. We further specified batch parameters and differential expression parameters to generate scenarios with different amount of group effect (i.e., cell type differences) and batch effect. As shown in Fig. 3, BAMM-SC still outperformed most other competing methods in terms of clustering accuracy in all scenarios, although the improvement is less substantial than our own model simulations, which is expected.
### Real data analysis on human PMBC dataset
For aforementioned human PBMC samples, we first pooled cells from five donors together, filtered lowly expressed genes that were expressed in less than 1% cells. We then extracted the top 1000 highly variable genes based on their standard deviations. As shown in Supplementary Fig. 3, we identified seven types of PBMCs based on the biological knowledge of cell-type-specific gene markers (Supplementary Table 2). Using these gene markers, >70% single cells can be assigned to a specific cell type. Since there is no gold standard for clustering analysis in this real dataset, we used the labels of these cells as the approximated ground truth to benchmark the clustering performance for different clustering methods. Cells with uncertain cell types were removed when calculating ARIs.
Similar to the simulation studies, we applied ten clustering methods on these samples and repeated each method ten times to evaluate the stability of its performance (Table 1). The total number of clusters was set as seven based on the biological knowledge from cell-type-specific gene markers. As shown in Table 1, BAMM-SC achieved the highest ARI for human PBMC samples compared to all other competing methods. Both TSCAN and Seurat are deterministic clustering methods and therefore they generate identical results for ten analyses.
We further generated t-SNE plots with each cell colored by their cell-type classification based on specific gene markers (i.e., the approximated truth) (Fig. 4a (left figure)) and cluster labels inferred by BAMM-SC (Fig. 4a (middle figure)), respectively. Despite some dendritic cells were wrongly identified as CD16+Monocytes, these two plots are similar to each other (ARI = 0.532), suggesting that BAMM-SC performed well in human PBMC samples compared with other clustering methods.
Moreover, we calculated the averaged cell proportions of each cell type inferred from BAMM-SC among ten runs for five PBMC samples, compared with cell proportions calculated from the approximated truth based on gene markers. Figure 4a (right figure) shows that the proportions inferred from BAMM-SC are close to the truth, suggesting that BAMM-SC can adequately account for batch effect when clustering cells from multiple individuals. We also generated t-SNE projection plots colored by cluster labels inferred by other methods: MNN+K-means clustering (Supplementary Fig. 4a), MNN+TSCAN (Supplementary Fig. 4b), MNN+SC3 (Supplementary Fig. 4c), MNN+Seurat (Supplementary Fig. 4d), CCA+K-means (Supplementary Fig. 4e), CCA+TSCAN (Supplementary Fig. 4f), CCA+SC3 (Supplementary Fig. 4g), CCA+Seurat (Supplementary Fig. 4h), and DIMM-SC (Supplementary Fig. 4i).
### Real data analysis on mouse lung dataset
We collected lung mononuclear cells from four mouse samples under two conditions: Streptococcus pneumonia (SP) infected (sample 1 and 2) and naive (sample 3 and 4). Supplementary Figure 5 shows the t-SNE plot of lung mononuclear cells from four mouse samples. Similar to the analysis of PBMC samples, after filtering lowly expressed genes, we pooled cells from 4 mice together and extracted the top 1000 highly variable genes. As shown in Supplementary Fig. 6, we identified six types of cells based on the biological knowledge of cell-type specific gene markers (Supplementary Table 3). Taken together, >66% of single cells can be assigned to a specific cell type. Therefore, we used the labels of these cells as the approximated truth and removed cells with uncertain cell types from the downstream analysis.
Figure 4b (left figure) and Fig. 4b (middle figure) show the t-SNE plots with each cell colored by their cluster label based on cell-type-specific gene markers and cluster labels inferred by BAMM-SC, respectively. These two are highly similar (ARI = 0.910), indicating the outstanding performance of BAMM-SC. Table 1 shows that BAMM-SC considerably outperformed other nine clustering methods in terms of ARI. We also generated t-SNE plots colored by cluster labels inferred by other competing clustering methods (Supplementary Fig. 7). As shown in Supplementary Fig. 8, the proportions of neutrophils in SP infected samples (sample 1 and sample 2) are much higher than the proportions in naïve samples (sample 3 and sample 4). This is consistent with the fact that infections by bacteria and viruses may increase the number of neutrophils, which is a necessary reaction by the body22,23. Interestingly, the proportion of cell types in naïve sample 3 is different from others, which may due to unsatisfactory sample quality or unexpected bacterial infections.
### Real data analysis on human skin dataset
To evaluate the clustering performance of BAMM-SC in solid human tissues, we collected skin samples from five healthy donors that are part of a systemic sclerosis study24. Figure 1a and Supplementary Table 1 list the detailed sample information and Supplementary Fig. 9 shows the t-SNE plot of cells from five human skin samples after the data processing similar to previous analyses. As shown in Supplementary Fig. 10, we identified eight major types of cells based on the biological knowledge of cell-type-specific gene markers (Supplementary Table 4). Taken together, >67% of single cells can be assigned to a specific cell type. Similar to the other two real data analyses, we used the labels of these cells as the approximated truth and removed cells with uncertain cell types from the downstream analysis.
As shown in Fig. 3c, BAMM-SC performed well in human skin samples, since the t-SNE plot with each cell colored by their cell-type label based on gene markers is highly similar to the plot generated from the clustering result of BAMM-SC (ARI = 0.843). Also, BAMM-SC achieved higher ARI compared with all the other clustering methods (Table 1). As comparisons, we generated t-SNE plots colored by cluster labels inferred by different clustering (Supplementary Fig. 11).
### Other evaluation criteria
To further demonstrate the validity of BAMM-SC, we calculated the confusion matrix for three real datasets and reported the clustering accuracy (defined as the proportion of cells being classified into the correct cell-type cluster) (Supplementary Table 5, Supplementary Methods). Our method outperformed other competing methods in all three datasets. In addition, we performed a flow cytometry experiment, a gold standard method for quantifying cell population through cell surface markers, on the sample 3 from the human PBMC dataset, which has an additional aliquot from the same pool of cells. We used FlowJo software to gate each cell population through specific antibodies and calculated the percentage of each cell type. Then, we compared the proportions of different cell types from flow cytometry and the clustering result of BAMM-SC from scRNA-seq. Supplementary Figure 12 shows that the proportion of cells in each cell type classified by BAMM-SC is consistent with that being estimated by flow cytometry. We also calculated the Pearson’s correlation coefficient of cell proportions for each clustering method (Supplementary Table 6). Despite the different technology, the high correlation (Pearson correlation coefficient is 0.98) suggests that BAMM-SC is able to adequately account for heterogeneity among multiple individuals and provide reliable clustering results. To be noted, unlike other clustering methods we considered, Seurat cannot directly pre-specify the number of clusters K. Rather it needs to set a resolution parameter that indirectly controls the cluster number. In all three real data sets, after an extensive grid search, we found the resolution parameter that yields the same number of clusters as the one based on the biological knowledge. Therefore, for the two Seurat clustering methods, instead of using the clustering assignments that produced the highest ARI among ten times analysis, we computed the confusion matrix and the proportions of different cell types based on this specific resolution parameter.
It is challenging to evaluate clustering algorithms in experimental data since the ground truth of cell type label is generally unknown. Other than using ARI based on cell-type-specific gene markers as approximated ground truth, we also used cluster stability and tightness to evaluate the clustering performance. Specifically, we calculated the average proportion of nonoverlap (APN)25 clustered cells and silhouette width26 in three real datasets, respectively. APN is a cluster stability measurement which evaluates the stability of a clustering result by comparing it with the clusters obtained by removing one feature (i.e., one gene in our study) at a time. It measures the average proportion of observations not placed in the same cluster under both cases. To make computation affordable in our real data analysis, after extracting the top 1000 highly variable genes, we compared the clustering results based on the full data (1000 genes) to the clustering results based on a subset of data with 100 genes randomly removed. We repeated this step ten times to calculate the APN. For cluster tightness, the silhouette width ranges from −1 to 1, where a higher value indicates that the observation is better matched to its own cluster and worse matched to other clusters. For both measurements, BAMM-SC achieved high cluster stability and high cluster tightness in most scenarios, compared with all other competing methods (Supplementary Table 7, Supplementary Table 8).
### Uncertainty assessment
Different from other deterministic methods, BAMM-SC has the ability to assess clustering uncertainty through the posterior probability for each cell to belong to each cell-type cluster. As shown in Supplementary Fig. 13, we highlighted vague cells in the t-SNE projection plot, where vague cells are defined as cells with the largest posterior cluster-specific probability <0.95. In the human PBMC samples, most of the vague cells (colored in red) are located at the boundary of different clusters, which reassuring the validity of the clustering results. In real data analysis, users can decide to remove vague cells under a user-specified criterion (based on the posterior probability) for the downstream analysis such as differential gene expression analysis within each cell type.
## Discussion
In summary, we have developed a novel Bayesian framework for clustering population-scale scRNA-seq data. BAMM-SC retains the raw data information by directly modeling UMI counts without data transformation or normalization, facilitating straightforward biological interpretation. The Bayesian hierarchical model enables the joint characterization of multiple sources of uncertainty, including single-cell level heterogeneity and individual level heterogeneity. Furthermore, BAMM-SC can borrow information across different individuals through its mixture hierarchical model structure and Bayesian computational techniques, leading to improved clustering accuracy. BAMM-SC is based on probabilistic models, thus providing the quantification of clustering uncertainty for each single cell.
Our model coupled with a computationally efficient MCMC algorithm is able to cluster large-scale droplet-based scRNA-seq data with feasible computational cost. For example, using 1000 highly variable genes, it takes about 1.5, 2.5, and 4.5 h when analyzing the 3 real datasets (human PBMC, mouse lung and human skin), respectively. For the simulated dataset consist of 10 individuals with 4000 cells each, the computational time for clustering is about 30 min. Supplementary Figure 14 demonstrates that the computational time of BAMM-SC increases approximately linearly with the increase of the number of cells in each individual, the number of individuals and the number of clusters, respectively. To further improve the computational efficiency, we provided a supervised clustering option in BAMM-SC for very large-scale datasets. Specifically, users can first apply BAMM-SC on a small subset of single cells in each individual, and save predicted cluster labels as well as other informative parameters such as $${\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}$$. Then for the remaining single cells, users can perform supervised classification via BAMM-SC instead of unsupervised clustering (see Methods). By clustering a small number of single cells, this procedure will substantially reduce the computational cost. We used the simulated dataset of ten individuals to demonstrate the effectiveness of this supervised option in Fig. 5. We simulated two datasets (Supplementary Methods): one dataset consists of 10 individuals with 400 cells each and the other dataset consists of 10 individuals with 4000 cells each. We selected a subset of cells in each individual as the training set and treated the remaining cells as the test set. We set the proportion of cells in the training set from 10 to 100% and reported the ARIs for both training and test sets. When the proportion equals 100%, there is no test data set, thus only ARI for the training set is reported. We repeated this simulation procedure 100 times and reported ARIs in Fig. 5 below. When the total number of cells in the training set is large enough (4000 in total or more), the prediction performance (measured by ARI) in the test set is saturated. For the dataset consists of 10 individuals with 4000 cells each, when we used 10% cells for training, it only takes ~90 s to obtain the clustering labels for all cells in both training and test sets with the similar performance from the full dataset. Therefore, for large datasets (e.g., >100 K cells), users can apply BAMM-SC to a smaller subset of cells in each individual to cluster distinct cell types, and then classify the remaining cells according to the predicted cell types. BAMM-SC is currently implemented in R/Rcpp with satisfactory computing efficiency to accommodate population scale scRNA-seq data. Further speed-up can be made through parallel computing or graphics processing unit.
In addition, we can predefine the number of clusters based on prior knowledge on the tissue or determine it using some standard model checking criterion such as Akaike's Information Criteria (AIC) or Bayesian Information Criteria (BIC). As shown in Supplementary Fig. 15, AIC and BIC work as expected in the analysis of simulated datasets and provide a reliable range of cluster numbers to guide real data analysis based on prior knowledge. However, in a biological study, the number of clusters is often considered as a continuum because of the nature of cell growth, so we recommend trying a range of cluster numbers in practice. BAMM-SC is shown to be robust against model mis-specification. In our simulation studies, we applied Splatter to simulate scRNA-seq data in which the data generation mechanism is different from our proposed BAMM-SC model. BAMM-SC still achieved higher clustering accuracy than other competing methods. In addition, we compared BAMM-SC with other clustering methods when the number of clusters is different from the true number of cell types. Supplementary Fig. 16 shows that BAMM-SC still achieved the highest ARI in most scenarios.
Other than MNN and CCA, several other approaches have been proposed to correct batch effect across multiple individuals. One straightforward approach is taking one individual as the reference, producing a low-dimensional embedding of it and then projecting the other individuals onto that embedding. To perform low-dimensional embedding, diffusion map27 is a tool for nonlinear dimension reduction and has recently been adapted for the visualization of single-cell gene-expression data. In addition, single-cell variational inference (scVI) is a scalable framework for batch correction based on variational inference and stochastic optimization of deep neural networks28. The performance of diffusion map and scVI combined with other clustering method was examined, which is worse than MNN and CCA in the three synthetic datasets (possibly due to unmet model underlying assumptions). We will explore more emerging methods in our future work.
There are several limitations of BAMM-SC. First, we filtered out genes with excessive zeros from the analysis under the assumption that lowly-expressed genes do not contribute much to clustering. This may be problematic for rare cell type identification. Second, we do not explicitly model a zero-inflation pattern, which may or may not affect clustering accuracy. A refined model that can handle inflated zeros can be further developed with a balance between computational complexity and model flexibility. Third, in our model, we assume that each cell belongs to one distinct cluster. The posterior probability measures the clustering uncertainty, which cannot be directly interpreted as a quantification of cell cycle or developmental stage. Finally, although our supervised strategy is proven to work for large datasets efficiently, it may potentially miss some rare clusters.
Our method has the potential to be extended to perform trajectory analysis29,30, and accounts for both individual and batch level heterogeneity (e.g., two individuals spread evenly across two 10x chips in a properly blocked design) by adding another level of structure. In addition, the model parameters can be used for downstream differential gene expression analysis or construct cell-type specific biomarker panels. These interesting directions are beyond the scope of this paper and will be studied in future papers. Additionally, unlike the traditional way of analyzing scRNA-seq data, BAMM-SC can be also used with batch effect correction. As shown in Supplementary Fig. 17, we ran BAMM-SC on the mouse lung dataset first and extracted cells in cluster 4. Then we applied CCA (implemented in Seurat) on this specific cluster of cells and replotted the t-SNE plot. From Supplementary Fig. 17e, cells from different samples are superimposed on each other, suggesting that most batch effect has been removed. In practice, we recommend using BAMM-SC for clustering raw count data and then use other methods, such as MNN and CCA, to remove batch effect for each individual cell type if needed.
We have applied BAMM-SC to simulated datasets and three in-house synthetic datasets to showcase its performance on different tissue types and species. With the increased popularity of population-based scRNA-seq studies, BAMM-SC will become a powerful tool for elucidating single cell level transcriptomic heterogeneity from population-based studies and a complementary approach to existing clustering methods.
## Methods
### Statistical model
We propose a Bayesian hierarchical Dirichlet multinomial mixture model to explicitly characterize different sources of variability in population scale scRNA-seq data. Specifically, let $$x_{ijl}$$ represent the number of unique UMIs for gene i in cell j from individual l ($$1 \le i \le G$$, $$1 \le j \le C_l$$, $$1 \le l \le L$$). Here, G, Cl, and L denote the total number of genes, cells (in individual l), and individuals, respectively. Our goal is to perform simultaneous clustering for cells from all L individuals. We assume that within each individual, all single cells consist of K distinct cell types. Cell type clusters are discrete, and each cell belongs to one cell type exclusively. Here, K is predefined according to prior biological knowledge, or will be estimated from the data, and K is the same among all L individuals.
Assume that $${\boldsymbol{x}}_{ \cdot {\boldsymbol{jl}}} = (x_{1jl},x_{2jl}, \ldots ,x_{Gjl})$$, the gene expression for cell j in individual l, follows a multinomial distribution multi $$\left( {T_{jl},{\boldsymbol{p}}_{ \cdot {\boldsymbol{jl}}}} \right).$$ Here, $$T_{jl} = \mathop {\sum }\limits_{i = 1}^G x_{ijl}$$ is the total number of UMIs, $${\boldsymbol{p}}_{ \cdot {\boldsymbol{jl}}} = (p_{1jl},p_{2jl}, \ldots ,p_{Gjl})$$ is the probability vector for gene expression with $$\mathop {\sum }\limits_{i = 1}^G p_{ijl} = 1$$, (where larger $$p_{ijl}$$ is associated with more UMI counts $$x_{ijl}$$). In addition, let $$z_{jl} \in \{ 1,2, \ldots ,K\}$$ represent the cell type label for cell j in individual l, where $$z_{jl} = k$$ indicates that cell j in individual l belongs to cell type k. Cells of the same cell type share a similar gene-expression pattern. If cell j in individual l belongs to cell type k ($$z_{jl} = k$$), we assume that $${\boldsymbol{p}}_{ \cdot {\boldsymbol{jl}}}$$ follows a cell-type specific Dirichlet prior Dir$$\left( {{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}} \right)$$, where $${\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}} = \left( {\alpha _{1lk},\alpha _{2lk}, \ldots ,\alpha _{Glk}} \right)$$ is the Dirichlet prior parameter for cell type k in individual l.
$$P\left( {{\boldsymbol{p}}_{.{\boldsymbol{jl}}}|z_{jl} = k,{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}} \right) = \frac{1}{{B({\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}})}}p_{1jl}^{\alpha _{1lk} - 1}p_{2jl}^{\alpha _{2lk} - 1} \ldots p_{Gjl}^{\alpha _{Glk} - 1},$$
(1)
where $$B(\alpha _{ \cdot lk})$$ is Beta function with parameter $$\alpha _{ \cdot lk} = \left( {\alpha _{1lk},\alpha _{2lk}, \ldots ,\alpha _{Glk}} \right)$$. Then after integrating $$p_{ \cdot jl}$$ out, we have:
$$P\left( {{\boldsymbol{x}}_{ \cdot {\boldsymbol{jl}}}|z_{jl} = k,{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}} \right) = \frac{{T_{jl}!}}{{\mathop {\prod }\nolimits_{i = 1}^G x_{ijl}!}}\left( {\mathop {\prod }\limits_{i = 1}^G \frac{{{\mathrm{\Gamma }}(x_{ijl} + \alpha _{ilk})}}{{{\mathrm{\Gamma }}(\alpha _{ilk})}}} \right)\frac{{{\mathrm{\Gamma }}(|{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}|)}}{{{\mathrm{\Gamma }}(T_{jl} + |{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}|)}},$$
(2)
where $$\left| {{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}} \right| = \mathop {\sum }\limits_{i = 1}^G \alpha _{ilk}$$. The joint distribution of $${\boldsymbol{x}}_{ \cdot {\boldsymbol{jl}}}$$ and $$z_{jl}$$ is
$$P\left( {{\boldsymbol{x}}_{ \cdot {\boldsymbol{jl}}},z_{jl}{\mathrm{|}}{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{l}} \cdot }} \right) = \frac{{T_{jl}!}}{{\mathop {\prod }\nolimits_{i = 1}^G x_{ijl}!}}\mathop {\sum }\limits_{k = 1}^K I(z_{jl} = k)\left( {\mathop {\prod }\limits_{i = 1}^G \frac{{{\mathrm{\Gamma }}(x_{ijl} + \alpha _{ilk})}}{{{\mathrm{\Gamma }}(\alpha _{ilk})}}} \right)\frac{{{\mathrm{\Gamma }}(|{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}|)}}{{{\mathrm{\Gamma }}\left( {T_{jl} + \left| {{\boldsymbol{\alpha }}_{ \cdot \cdot {\boldsymbol{k}}}} \right|} \right)}}.$$
(3)
We further assume that all Cl cells in individual l are independent, then the joint distribution for all cells in individual l is
$$P\left( {{\boldsymbol{x}}_{ \cdot \cdot {\boldsymbol{l}}},{\boldsymbol{z}}_{ \cdot {\boldsymbol{l}}}|{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{l}} \cdot }} \right) = \mathop {\prod }\limits_{j = 1}^{C_l} P\left( {{\boldsymbol{x}}_{ \cdot {\boldsymbol{jl}}},z_{jl}{\mathrm{|}}{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{l}} \cdot }} \right).$$
(4)
Finally, we assume that all L individuals are independent, then the overall joint distribution for all cells across all individuals becomes
$$P\left( {{\boldsymbol{x}}_{ \cdot \cdot \cdot },{\boldsymbol{z}}_{ \cdot \cdot }|{\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }} \right) = \mathop {\prod }\limits_{l = 1}^L P\left( {{\boldsymbol{x}}_{ \cdot \cdot {\boldsymbol{l}}},{\boldsymbol{z}}_{ \cdot {\boldsymbol{l}}}|{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{l}}\, \cdot }} \right) \\ \propto \mathop {\prod }\limits_{l = 1}^L \mathop {\prod }\limits_{j = 1}^{C_l} \left\{ {\mathop {\sum }\limits_{k = 1}^K I\left( {z_{jl} = k} \right)\left( {\mathop {\prod }\limits_{i = 1}^G \frac{{{\mathrm{\Gamma }}\left( {x_{ijl} + \alpha _{ilk}} \right)}}{{{\mathrm{\Gamma }}\left( {\alpha _{ilk}} \right)}}} \right)\frac{{{\mathrm{\Gamma }}\left( {\left| {{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}} \right|} \right)}}{{{\mathrm{\Gamma }}\left( {T_{jl} + \left| {{\boldsymbol{\alpha }}_{ \cdot {\boldsymbol{lk}}}} \right|} \right)}}} \right\}.$$
(5)
In this model, the two sets of parameters of interest are $${\boldsymbol{z}}_{ \cdot \cdot } = \left\{ {z_{jl}} \right\}_{1 \le j \le C_l,1 \le l \le L}$$, the cell type label for cell j in individual l, and $${\boldsymbol{\alpha }}_{ \cdot \cdot \cdot } = \left\{ {\alpha _{ilk}} \right\}_{1 \le i \le G,1 \le l \le L,1 \le k \le K}$$, the Dirichlet parameters for gene i in cell type k in individual l. We adopt a full Bayesian approach and use Gibbs sampler to estimate the posterior distributions. Specifically, the joint posterior distribution for $${\boldsymbol{z}}_{ \cdot \cdot }$$ and $${\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }$$ are
$$P\left( {{\boldsymbol{z}}_{ \cdot \cdot },{\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }|{\boldsymbol{x}}_{ \cdot \cdot \cdot }} \right) \propto P\left( {{\boldsymbol{x}}_{ \cdot \cdot \cdot },{\boldsymbol{z}}_{ \cdot \cdot }|{\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }} \right) \times Prior\left( {{\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }} \right).$$
(6)
Since all α’s are strictly positive, we propose a log-normal distribution as the prior distribution for $$\alpha _{ilk}$$. We assume that for gene i in cell type k, $$\alpha _{ilk}$$ from all L individuals share the same prior distribution LN $$(\mu _{ik},\sigma _{ik}^2)$$, that is
$${\mathrm{Prior}}\left( {{\boldsymbol{\alpha }}_{{\boldsymbol{i}} \cdot {\boldsymbol{k}}}} \right) = \mathop {\prod }\limits_{l = 1}^L \frac{1}{{\alpha _{ilk}\sqrt {2\pi \sigma _{ik}^2} }}\exp \left\{ { - \frac{{\left( {\log \alpha _{ilk} - \mu _{ik}} \right)^2}}{{2\sigma _{ik}^2}}} \right\}.$$
(7)
Here, $$\mu _{ik}$$ can be estimated by the mean of $$\alpha _{ilk}$$'s: $$\hat \mu _{ik} = \frac{1}{L}\mathop {\sum }\limits_{l = 1}^L {\mathrm{log}}(\alpha _{ilk})$$. Estimation of $$\sigma _{ik}^2$$ can be challenging due to limited number of individuals. We can assume all $$\sigma _{ik}^2$$’s follow a hyper-prior: Gamma distribution Gamma$$(a_k,b_k)$$, and use information across all genes to estimate variance. In addition, we assume a noninformative prior for $$\mu _{ik}$$’s. Taken all together, we have the full posterior distribution as follows:
$$P\left( {{\boldsymbol{z}}_{ \cdot \cdot },{\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }|{\boldsymbol{x}}_{ \cdot \cdot \cdot }} \right) \propto P\left( {{\boldsymbol{x}}_{ \cdot \cdot \cdot },{\boldsymbol{z}}_{ \cdot \cdot }|{\boldsymbol{\alpha }}_{ \cdot \cdot \cdot }} \right) \times \mathop {\prod }\limits_{k = 1}^K \mathop {\prod }\limits_{i = 1}^G {\mathrm{prior}}\left( {{\boldsymbol{\alpha }}_{{\boldsymbol{i}} \cdot {\boldsymbol{k}}}} \right) \times \mathop {\prod }\limits_{k = 1}^K {\mathrm{prior}}\left( {{\boldsymbol{\mu }}_{ \cdot {\boldsymbol{k}}}} \right) \times \mathop {\prod }\limits_{k = 1}^K {\mathrm{prior}}\left( {{\boldsymbol{\sigma }}_{ \cdot {\boldsymbol{k}}}^2} \right).$$
(8)
We use Gibbs sample to iteratively update $$\alpha _{ilk}$$ and $$z_{jl}$$. Details can be found in Supplementary Methods.
### Classification and computational acceleration
To further improve the computational efficiency, we provide a supervised option in BAMM-SC. Specifically, for very large-scale dataset, we use BAMM-SC to train a prediction model using a subset of cells from each individual and predict the clustering labels for the rest of cells. First, we randomly select a subset of cells from each individual and applied BAMM-SC on these selected cells. The estimate of $$\alpha _{ilk}$$ is computed as the average after deletion of the first 100 (default) iterations as burn-in. We then predict the cell type labels for other cells with realization of parameters: $$\hat \Theta = \left( {{\hat{\boldsymbol{\alpha }}}_{ \cdot 1 \cdot }, \ldots ,{\hat{\boldsymbol{\alpha }}}_{ \cdot {\boldsymbol{L}} \cdot },{\hat{\boldsymbol{\pi }}}_1, \ldots ,{\hat{\boldsymbol{\pi }}}_{\boldsymbol{L}}} \right)$$.
$$P\left( {z_{jl} = k{\mathrm{|}}{\boldsymbol{x}}_{{\boldsymbol{jl}}},\hat \Theta } \right) = \frac{{\left( {\mathop {\prod }\nolimits_{i = 1}^G \frac{{{\mathrm{\Gamma }}\left( {x_{ijl} + \hat \alpha _{ilk}} \right)}}{{{\mathrm{\Gamma }}\left( {\hat \alpha _{ilk}} \right)}}} \right)\frac{{{\mathrm{\Gamma }}\left( {|{\hat{\boldsymbol{\alpha }}}_{ \cdot {\boldsymbol{lk}}}|} \right)}}{{{\mathrm{\Gamma }}\left( {T_j + |{\hat{\boldsymbol{\alpha }}}_{ \cdot {\boldsymbol{lk}}}|} \right)}}\hat \pi _{lk}}}{{\mathop {\sum }\nolimits_{k = 1}^K \left( {\mathop {\prod }\nolimits_{i = 1}^G \frac{{{\mathrm{\Gamma }}\left( {x_{ij} + \hat \alpha _{ilk}} \right)}}{{{\mathrm{\Gamma }}\left( {\hat \alpha _{ilk}} \right)}}} \right)\frac{{{\mathrm{\Gamma }}\left( {|{\hat{\boldsymbol{\alpha }}}_{ \cdot {\boldsymbol{lk}}}|} \right)}}{{{\mathrm{\Gamma }}\left( {T_j + |{\hat{\boldsymbol{\alpha }}}_{ \cdot {\boldsymbol{lk}}}|} \right)}}\hat \pi _{lk}}}.$$
(9)
This approach can substantially reduce the computational cost for very large-scale datasets while maintaining the accuracy as shown in Supplementary Fig. 14.
### Single-cell sequencing library construction
10× Genomics Chromium system, which is a microfluidics platform based on Gel bead in EMulsion (GEM) technology, was used for generating real test datasets. Cells mixed with reverse transcription reagents were loaded into the Chromium instrument. This instrument separated cells into minireaction partitions formed by oil microdroplets, each containing a gel bead and a cell, known as GEMs. GEMs contain a gel bead, scaffold for an oligonucleotide that is composed of an oligo-dT section for priming reverse transcription, and barcodes for each cell and each transcript as described. GEM generation takes place in a multiple-channel microfluidic chip that encapsulates single-gel beads. Reverse transcription takes place inside each droplet. Approximately, 1000-fold excess of partitions compared to cells ensured low capture of duplicate cells. The reaction mixture/emulsion was removed from the Chromium instrument, and reverse transcription was performed. The emulsion was then broken using a recovery agent, and following Dynabead and SPRI clean up cDNAs were amplified by PCR (C1000, Bio-Rad). cDNAs were sheared (Covaris) into ~200 bp length. DNA fragment ends were repaired, A-tailed and adapters ligated. The library was quantified using KAPA Universal Library Quantification Kit KK4824 and further characterized for cDNA length on a Bioanalyzer using a High Sensitivity DNA kit. All sequencing experiments were conducted using Illumina NextSeq 500 in the Genomics Sequencing Core at the University of Pittsburgh.
### Data description
Human PBMC dataset: Under a protocol approved by the University of Pittsburgh Institutional Review Board, peripheral blood was obtained from healthy donors by venipuncture. Each subject gave written informed consent. PBMC were isolated from whole blood by density gradient centrifugation using Ficoll–Hypaque. PBMC were then counted and resuspended in phosphate buffered saline with 0.04% bovinue serum albumin, and were processed through the Chromium 10× Controller according to the manufacturers’ instructions, targeting a recovery of ~2000 cells. The following steps were all performed under the aforementioned protocol developed by 10× Genomics.
Human skin dataset: Skin samples were obtained by performing 3 mm punch biopsies from the dorsal midforearm of healthy control subjects after informed consent under a protocol approved by the University of Pittsburgh Institutional Review Board. Skin for scRNA-seq was digested enzymatically (Miltenyi Biotec Whole Skin Dissociation Kit, human) for 2 h and further dispersed using the Miltenyi gentleMACS Octo Dissociator. The resulting cell suspension was filtered through 70 micron cell strainers twice and re-suspended in phosphate-buffered saline containing 0.04% bovine serum albumin. Cells from biopsies were mixed with reverse transcription reagents then loaded into the Chromium instrument (10× Genomics). Totally, ~2600–4300 cells were loaded into the instrument to obtain data on ~1100–1800 cells, anticipating a multiplet rate of ~1.2% of partitions. The following steps were all performed under the aforementioned protocol developed by 10× Genomics.
Mouse lung dataset: Lung single cell suspension from naïve and infected C57BL/6 mice were subject to scRNA-seq library preparation protocol. Briefly, left lobs of both naïve and infected mice were removed and digested by Collagenase/DNase to obtain single-cell suspension. Mononuclear cells after filtration with a 40 μM cell strainer were separated into minireaction partitions or GEMs formed by oil microdroplets, each containing a gel bead and a cell, by the Chromium instrument (10× Genomics). The reaction mixture/emulsion with captured and barcoded mRNAs were removed from the Chromium instrument followed by reverse transcription. The cDNA samples were fragmented and amplified using the Nextera XT kit (Illumina). The following steps were all performed under aforementioned the protocol developed by 10× Genomics. We have complied with all relevant ethical regulations for animal research. The animal protocol was approved by the University of Pittsburgh Institutional Animal Care and Use Committee.
### Reporting summary
Further information on experimental design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
The study uses various publicly available scRNA-seq datasets. Both human PBMC (sample 5) and purified CD19+B cell scRNA-seq data that support the findings of this study are available at https://support.10xgenomics.com/single-cell-gene-expression/datasets. The raw fastq files and preprocessed experimental test datasets (human PBMCs, mouse lung and human skin tissues) have been deposited in the gene expression omnibus (GEO) database under accession number GSE128066. All other relevant data are available upon request.
## Code availability
BAMM-SC, including all source and example code, is freely available as an R package with a detailed tutorial at https://github.com/CHPGenetics/BAMMSC.
Journal peer review information: Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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## Acknowledgements
This work is supported by National Institute of Health grants R56HL137709 (K.C.), P50 CA097190 and P30 CA047904 (D.A.A.V.), P50 AR060780 (R.L. and W.C.), R35HL139930 (J.K.), and Children’s Hospital of Pittsburgh (W.C. and Z.S.).
## Author information
### Author notes
1. These authors contributed equally: Zhe Sun, Li Chen.
### Affiliations
1. #### Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA, 15261, USA
• Zhe Sun
• , Ying Ding
• & Wei Chen
• Li Chen
3. #### Division of Pulmonary Medicine, Department of Pediatrics, Children’s Hospital of Pittsburgh of UPMC, University of Pittsburgh, Pittsburgh, PA, 15224, USA
• Hongyi Xin
• , Yale Jiang
• & Wei Chen
• Yale Jiang
5. #### Department of Biostatistics, School of Public Health, University of Michigan, Ann Arbor, MI, 48109, USA
• Qianhui Huang
6. #### Department of Immunology, School of Medicine, University of Pittsburgh, Pittsburgh, PA, 15262, USA
• Anthony R. Cillo
• , Tullia C. Bruno
• & Dario A. A. Vignali
7. #### Division of Rheumatology and Clinical Immunology, Department of Medicine, School of Medicine, University of Pittsburgh, Pittsburgh, PA, 15261, USA
• Tracy Tabib
• & Robert Lafyatis
8. #### School of Medicine, Tulane University, New Orleans, LA, 70112, USA
• Jay K. Kolls
9. #### Tumor Microenvironment Center, UPMC Hillman Cancer Center, Pittsburgh, PA, 15232, USA
• Tullia C. Bruno
• & Dario A. A. Vignali
10. #### Cancer Immunology and Immunotherapy Program, UPMC Hillman Cancer Center, Pittsburgh, PA, 15232, USA
• Dario A. A. Vignali
• Kong Chen
• Ming Hu
### Contributions
M.H. and W.C. conceived the study; Z.S. led the statistical modeling and data analysis; L.C. helped with developing the R package; A.R.C., T.C.B., and D.A.A.V. performed the experiments for the human PBMC data; T.T. and R.L. performed the experiments for the human skin data; K.C. and J.K.K. performed the experiments for the mouse data; H.X., Y.J., and Q.H. helped with the preprocessing and analyzing the data; Y.D., W.C., and M.H. supervised the research; and Z.S., W.C., Y.D., and M.H. led the writing of the paper with input from all the other authors.
### Competing interests
The authors declare no competing interests.
### Corresponding authors
Correspondence to Ying Ding or Ming Hu or Wei Chen.
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2019-04-19 06:36:47
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https://www.zbmath.org/?q=an%3A1083.35094
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# zbMATH — the first resource for mathematics
Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces. (English) Zbl 1083.35094
The authors prove he existence of attractors for the 2d-Navier-Stokes equations with an exterior force $$f(t)$$ by using the theory of processes by Babin, Vishik and others. A process is based on a Banach space $$X$$, an index set $$\Sigma$$ (itself a topological vector space). A process is a collection $$U_\sigma(t,\tau)$$ of nonlinar operators, acting on $$X$$, labeled by $$\sigma\in\Sigma$$ such that $U_\sigma(t,s) U_\sigma(s,\tau)= U_\sigma(t,\tau),\;t\geq s\geq \tau,\;U_\sigma(\tau, \tau)= \text{Id},\;\tau\in\mathbb{R},\;\sigma\in\Sigma.\tag{1}$ $$\Sigma$$ is called the symbol space, $$\sigma$$ a symbol. Concepts such as uniform attractor, uniform absorbing set etc. are now introduced. E.g., $$B_0\subset X$$ is uniformly absorbing if given $$C\in\mathbb{R}$$ and a bounded set $$B\subset X$$ there is $$T_0= T_0(\tau,B)\geq \tau$$ such that $\bigcup_{\sigma\in\Sigma} U_\sigma(t,\tau)B\subset B_0\quad \text{for }t\geq T_0.\tag{2}$ The authors now prove a number of preparatory lemmas concerning properties of processes. These results are then applied to the 2d-Navier-Stokes equation on a smooth bounded domain $$\Omega$$. To this end this equation is put into standard abstract form $\partial_t u+\nu Au+ B(u,u)= f(t),\quad u(0)= u_0\tag{3}$ based on the Hilbert spaces \begin{aligned} H&= \{u\in L^2(\Omega)^2,\,\text{div}(u)= 0,\,u\cdot\vec n|_{\partial\Omega}= 0\},\text{ norm }|\;|,\\ V&= \{u\in H^1_0(\Omega)^2,\,\text{div}(u)= 0\},\text{ norm }\|\;\|. \end{aligned} The exterior force $$f(t)= \sigma(t)$$, $$t\in\mathbb{R}$$ is taken as symbol of the system (3), resp. of the induced process; one assumes $\sup_t \int^{t+1}_t |f(s)|^2\,ds< \infty.\tag{4}$ After recalling global existence and uniqueness of solutions of (3) the authors proceed to prove the existence of a uniform attractor $$A_0$$ of (3) and investigate its properties. The relevant Theorem 3.3 states among others (expressed somewhat losely) that if $$f(t)$$, $$t\in\mathbb{R}$$ has an additional property called “normal” then the $$A_0$$ associated with $$f(t)$$, $$t\in\mathbb{R}$$ coincides with the uniform attractor $$A_b$$ associated with $$f(t+b)$$, $$t\in\mathbb{R}$$ , for any $$b\in\mathbb{R}$$. Further results of this type are obtained (Theorems 4.1, 4.2).
##### MSC:
35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
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2021-07-29 00:08:26
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https://socratic.org/questions/why-are-numbers-used-in-chemistry-often-expressed-in-scientific-notation
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# Why are numbers used in chemistry often expressed in scientific notation?
May 21, 2017
To reveal honest uncertainty in experimental measurements and ease of reading for very large and very small numbers.
#### Explanation:
There are two reasons to use scientific notation in chemistry. The first is to reveal honest uncertainty in experimental measurements. The second is to express very large or very small numbers so they are easier to read.
HONESTY
Suppose you are trying produce 30 L of some chemical, which you can only make at 10 L a time. Due to tiny imperfections in the measuring equipment, differences in lighting, perception, evaporation, unintended reactions, motion, etc., each batch is measured slightly differently. There is, in effect, some degree of uncertainty to the exact measurements. Here are your measurements:
• $9.998$ L
• $10.01$ L
• $9.8$ L
If you add them all up, you get $29.808$ L. However, giving this answer to your customer is misleading. You were not equally certain of all your measurements. In fact, the measurement with the lowest amount of certainty was the $9.8$ measurement. This is why you use the addition rule of significant figures and say the final result is $29.8$ L.
One of the best ways to reduce ambiguity in numbers with ambiguous significant figures is to write them in scientific notation.
LARGE & SMALL NUMBERS
In scientific inquiry, sometimes you have to deal with very large numbers. For example, the distance from earth to the Milky Way galaxy is about
1,135,300,000,000,000,00000,000 meters
How many zeros is that? How many significant figures is this? This is easier to express in scientific notation, which answers all of these questions right away: $1.1353 \times {10}^{21}$.
Similarly, scientists have the same problem with very small numbers. For example, the diameter of white blood cell is about
0.000012 meters
That is an ugly number! Scientific notation comes to save the day by expressing the same as $1.2 \times {10}^{-} 5$ meters.
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2020-01-18 17:56:49
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http://encyclopedia.kids.net.au/page/li/List_of_professions?title=Consultant
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## Encyclopedia > List of professions
Article Content
# List of professions
The following is a list of jobs commonly perceived as professions.
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E
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K
L
M
N
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P
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X/Y/Z
See also: list of people by occupation
All Wikipedia text is available under the terms of the GNU Free Documentation License
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Featured Article
Law of Universal Gravitation ... forces between the various points. $F = \frac{G m_1 m_2}{r^2}$ where: F = gravitational force between two objects m1 = mass of firs ...
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2014-03-08 06:57:56
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https://pos.sissa.it/332/003/
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Volume 332 - XIV International Conference on Heavy Quarks and Leptons (HQL2018) - CP Violation
Study of CP violation in B meson decays in Belle
Y. Yusa* and On behalf of the BELLE Collaboration
Full text: pdf
Pre-published on: December 05, 2018
Published on: December 11, 2018
Abstract
We report the measurements of $CP$ violation in $B$ meson decays focusing on the recent studies obtained using a data sample of 772 million $B\bar{B}$ collected by the Belle detector running at the $\Upsilon (4S)$ resonance at the KEKB $e^+ e^-$ collider. We measure the $\phi_1$ and $\phi_2$, that are angles of the unitary triangle of Cabibbo-Kobayashi-Maskawa (CKM) matrix. Obtained results are consistent with expectations from the Standard Model (SM). Further studies using more statistics in the upgraded $B$-factory experiment are needed for more precise verification and search for the contribution from the new physics.
DOI: https://doi.org/10.22323/1.332.0003
How to cite
Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.
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2022-06-30 01:15:36
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http://server3.wikisky.org/starview?object_type=1&object_id=1911&object_name=HD+223274&locale=EN
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WIKISKY.ORG
Home Getting Started To Survive in the Universe News@Sky Astro Photo The Collection Forum Blog New! FAQ Press Login
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Rotational velocities of A-type stars in the northern hemisphere. II. Measurement of v sin iThis work is the second part of the set of measurements of v sin i forA-type stars, begun by Royer et al. (\cite{Ror_02a}). Spectra of 249 B8to F2-type stars brighter than V=7 have been collected at Observatoirede Haute-Provence (OHP). Fourier transforms of several line profiles inthe range 4200-4600 Å are used to derive v sin i from thefrequency of the first zero. Statistical analysis of the sampleindicates that measurement error mainly depends on v sin i and thisrelative error of the rotational velocity is found to be about 5% onaverage. The systematic shift with respect to standard values fromSlettebak et al. (\cite{Slk_75}), previously found in the first paper,is here confirmed. 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Since the PMFS proper motions do notinclude the information given by the proper motions from othercatalogues (HIPPARCOS, FK5, FK6, etc.) this catalogue can be used as anindependent source of the proper motions of the fundamental stars.Catalogue (Table 3) is only available at the CDS via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strastg.fr/cgi-bin/qcat?J/A+A/365/222 Sixth Catalogue of Fundamental Stars (FK6). Part I. Basic fundamental stars with direct solutionsThe FK6 is a suitable combination of the results of the HIPPARCOSastrometry satellite with ground-based data, measured over more than twocenturies and summarized in the FK5. Part I of the FK6 (abbreviatedFK6(I)) contains 878 basic fundamental stars with direct solutions. Suchdirect solutions are appropriate for single stars or for objects whichcan be treated like single stars. From the 878 stars in Part I, we haveselected 340 objects as "astrometrically excellent stars", since theirinstantaneous proper motions and mean (time-averaged) ones do not differsignificantly. Hence most of the astrometrically excellent stars arewell-behaving "single-star candidates" with good astrometric data. Thesestars are most suited for high-precision astrometry. On the other hand,199 of the stars in Part I are Δμ binaries in the sense ofWielen et al. (1999). Many of them are newly discovered probablebinaries with no other hitherto known indication of binarity. The FK6gives, besides the classical "single-star mode" solutions (SI mode),other solutions which take into account the fact that hidden astrometricbinaries among "apparently single-stars" introduce sizable "cosmicerrors" into the quasi-instantaneously measured HIPPARCOS proper motionsand positions. The FK6 gives in addition to the SI mode the "long-termprediction (LTP) mode" and the "short-term prediction (STP) mode". TheseLTP and STP modes are on average the most precise solutions forapparently single stars, depending on the epoch difference with respectto the HIPPARCOS epoch of about 1991. The typical mean error of anFK6(I) proper motion in the single-star mode is 0.35 mas/year. This isabout a factor of two better than the typical HIPPARCOS errors for thesestars of 0.67 mas/year. In the long-term prediction mode, in whichcosmic errors are taken into account, the FK6(I) proper motions have atypical mean error of 0.50 mas/year, which is by a factor of more than 4better than the corresponding error for the HIPPARCOS values of 2.21mas/year (cosmic errors included). Mapping the contours of the Local bubble: preliminary resultsWe present preliminary results from a long-term program of mapping theneutral absorption characteristics of the local interstellar medium,taking advantage of Hipparcos stellar distances. Equivalent widths ofthe NaI D-line doublet at 5890 Å are presented for thelines-of-sight towards some 143 new target stars lying within 300 pc ofthe Sun. Using these data which were obtained at the Observatoire deHaute Provence, together with previously published NaI absorptionmeasurements towards a further 313 nearby targets, we present absorptionmaps of the distribution of neutral gas in the local interstellar mediumas viewed from 3 different galactic projections. In particular, thesemaps reveal the Local Bubble region as a low neutral densityinterstellar cavity in the galactic plane with radii between 65-250 pcthat is surrounded by a (dense) neutral gas boundary (or wall''). Wehave compared our iso-column contours with the contours derived bySnowden et al. (\cite{snowden98}) using ROSAT soft X-ray emission data.Consistency in the global dimensions derived for both sets of contoursis found for the case of a million degree hot LB plasma of emissivity0.0023 cm(-6) pc with an electron density of 0.005 cm(-2) . We havedetected only one relatively dense accumulation of cold, neutral gaswithin 60 pc of the Sun that surrounds the star delta Cyg, and note thatthe nearest molecular cloud complex of MBM 12 probably resides at thevery edge of the Local Bubble at a distance of ~ 90 pc. Our observationsmay also explain the very different physical properties of the columnsof interstellar gas in the line-of-sight to the two hot stars epsilonCMa and beta CMa as being due to their locations with respect to theBubble contours. Finally, in the meridian plane the LB cavity is foundto be elongated perpendicularly to the Gould's Belt plane, possiblybeing squeezed'' by the expanding shells of the Sco-Cen andPerseus-Taurus OB associations. Tables 1 and 2 are also available inelectronic form at the CDS (Strasbourg) via anonymous ftp tocdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html The Tokyo PMC catalog 90-93: Catalog of positions of 6649 stars observed in 1990 through 1993 with Tokyo photoelectric meridian circleThe sixth annual catalog of the Tokyo Photoelectric Meridian Circle(PMC) is presented for 6649 stars which were observed at least two timesin January 1990 through March 1993. The mean positions of the starsobserved are given in the catalog at the corresponding mean epochs ofobservations of individual stars. The coordinates of the catalog arebased on the FK5 system, and referred to the equinox and equator ofJ2000.0. The mean local deviations of the observed positions from theFK5 catalog positions are constructed for the basic FK5 stars to comparewith those of the Tokyo PMC Catalog 89 and preliminary Hipparcos resultsof H30. UvbyHbeta_ photometry of main sequence A type stars.We present Stroemgren uvby and Hbeta_ photometry for a set of575 northern main sequence A type stars, most of them belonging to theHipparcos Input Catalogue, with V from 5mag to 10mag and with knownradial velocities. These observations enlarge the catalogue we began tocompile some years ago to more than 1500 stars. Our catalogue includeskinematic and astrophysical data for each star. Our future goal is toperform an accurate analysis of the kinematical behaviour of these starsin the solar neighbourhood. The X-Ray Emission of A-Type StarsFrom X-ray images in the ROSAT public archives, we determine soft X-rayfluxes, or flux upper limits, for 74 A-type stars, which have beenobserved during deep integrations with the PSPC. Nine supposedly single,late A stars (0.20 < B-V < 0.35) are found to coincide with X-raysources. The X-ray luminosities we infer for these stars range fromlevels comparable to the Active Sun, at log L_x ~27.6, to much brighteremission levels similar to those observed for active late-type binarysystems, near log L_x ~30.1. Another 10 sources are identified withearly A stars (0.0 < B-V < 0.2). Five of these are confirmeddouble stars, the rest are ostensibly single. The maximum luminosity wedetect in the early A stars, log L_x = 30.1, is 3.5 orders of magnitudebrighter than the X-ray upper limits for the nondetected stars.Additional study, including radial velocity monitoring and/or opticalinterferometry, will be needed to determine whether the putativelysingle X-ray emitting stars are in fact single, or whether theiremission is produced entirely or in part by unknown or unresolved binarycompanions. The level of X-ray emission associated with chemicallynormal, single A stars thus far appears to be uncorrelated with anyobvious stellar property, including the rotation rate, which is known togreatly influence the dynamo activity and the X-ray emission levels oflower mass stars. (SECTION: Stars) The Relation between Rotational Velocities and Spectral Peculiarities among A-Type StarsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1995ApJS...99..135A&db_key=AST Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. Three known and twenty-two new variable stars of early spectral typePhotoelectric photometry is reported of three known and 22 newvariables, all brighter than V = 7.5 mag. The three known ones are: theellipsoidal variable 42 (V467) Per, the Beta Cep-type star HR 6684 =V2052 Oph, and the eclipsing variable HR 8854 = V649 Cas. The newvariable stars are listed. They include two Beta Cep candidates, oneeclipsing and three ellipsoidal variables, a 'mid-B' variable, anAlpha(2) CVn variable, and two Lambda Eri stars. The twelve remainingnew variables could not be classified because of insufficient data. TheLambda Eri variables found in the present investigation, together withsome examples from the literature, indicate that rotational modulationoccurs not only in Be, but also in normal B stars. Corrections to the right ascension to be applied to the apparent places of 1217 stars given in "The Chinese Astronomical Almanach" for the year 1984 to 1992.Not Available Physical data of the fundamental stars.Not Available Third preliminary catalogue of stars observed with the photoelectric astrolabe of the Beijing Astronomical Observatory.Not Available The equal-altitude method of quasi-absolute determination of star coordinatesAbsolute coordinate values are determined for 86 objects on the basis ofdata obtained with Mark II photoelectric astrolabe No. 2 at BeijingAstronomical Observatory during the period 1979-1984. The equal-altitudemethod employed in the computations is described in detail, and theresults are presented in tables. The mean accuracy of the positioncorrections is given as + or - 1.6 msec in right ascension and + or -0.019 arcsec in declination. Carbon abundances and meridional mixing in rapidly rotating early-A starsCarbon abundances are obtained from lines of C I at 9100 A for 22early-A main-sequence stars with projected rotational velocities of upto 180 km/s. The abundance shows no significant trend with increasingrotation. A preliminary conclusion is that meridional mixing currents donot bring CN-processed material into the atmospheres of these rotating Astars. Several carbon deficient stars are tentatively identified aschemically peculiar A stars. Future surveys should cover a sufficientwavelength interval to permit a spectral classification and adetermination of the microturbulence. An UV survey of the galactic planeThe present paper is the introduction to a systematic analysis of 123six-degree fields near the galactic plane, recorded in the mediumultraviolet by the balloon-borne experiment SCAP 2000. The availabledata are presented and the general properties of the images are brieflydiscussed. It is shown that the high selectivity of the UV passbandregarding spectral type, together with the strong interstellarextinction at that wavelength, provide the necessary conditions for anefficient application of Wolf's method to study the distribution ofinterstellar matter in the solar neighbourhood. The results of a fastanalysis of the available data are presented here. Catalogue of the energy distribution data in spectra of stars in the uniform spectrophotometric system.Not Available The A0 starsA photometric grid, standardized on MK spectral standards, has been usedto compare spectral types and luminosity classes obtainedphotometrically with those in two extensive spectral surveys coveringthe entire sky. Major discrepancies include the spectroscopicclassification of B9.5, which may indicate an otherwise unrecognizedspectral peculiarity, a different A0/A1 spectral type boundary in thetwo samples involved, the well-known misclassification of weak heliumstars, and an appreciable percentage of stars which are called dwarfsspectroscopically but are of higher photometric luminosity. The spacemotion vectors of these stars for which radial velocities are available,and excluding the minimum of 25 percent that are spectroscopic binarieswithout orbital elements, show structure in their distribution in the(U, V)-plane, with members of the Local Association and the Hyades andSirius superclusters forming obvious concentrations. The members of theLocal Association in the samples are mainly old (more than 200 millionyears) mode A stars, although a few much younger stars are included. Themembers of the Hyades and Sirius superclusters contain many bluestragglers, including several peculiar stars of the Hg, Mn, and Sivarieties. Meridian observations made with the Carlsberg Automatic Meridian Circle at Brorfelde (Copenhagen University Observatory) 1981-1982The 7-inch transit circle instrument with which the present position andmagnitude catalog for 1577 stars with visual magnitudes greater than11.0 was obtained had been equipped with a photoelectric moving slitmicrometer and a minicomputer to control the entire observationalprocess. Positions are reduced relative to the FK4 system for each nightover the whole meridian rather than the usual narrow zones. Thepositions of the FK4 stars used in the least squares solution are alsogiven in the catalog. Apparent radii and other parameters for 416 B5 V-F5 V stars of the catalogue of the Geneva ObservatoryApparent radius, visual brightness, effective temperature and absoluteradius for 416 B5 v-F5 v stars of the catalogue of the GenevaObservatory (Rufener, 1976) have been determined. Twenty-eight stars,anomalous in log a" versus (m~)o diagrams, have been singled out. A goodcorrelation for seven stars, in common with the list of Hanbury Brown etal. (1974), has been found. Similar parameters determined for 279 B5v-F5 v stars of two preceding papers (Fracassini et al., 1973, 1975)have allowed us to determine the averaged diagrams , and versus (B -V)0 for 695 B5 v-F5 v stars. Moreover, in the present paper a goodcorrelation versus and carefulrelation = -7.40 + 3.31 for B5v-F5 V stars have been determined. Plain correlations between log R/R0and blanketing parameter m2 for some spectral types seem to point outthat there are real differences in the absolute radii of stars of thesame spectral type, in agreement with recent researches on the HRdiagram (Houck and Fesen, 1978). Systematic differences between double(spectroscopic and visual) and single stars are found. In particular,the averaged relation versus shows that A2v-F5 v double stars may have a higher metallicity index m2 and smallerabsolute radii than single stars. Finally, the diagram log v sin iversus log R/R0 confirms some properties of binary systems found byother researchers (Huang, 1966; Plavec, 1970; Levato, 1974; Kitamura andKondo, 1978) Coordinate improvements for FK4 stars from observations with the Ni2 astrolabeIndividual coordinate improvements were derived for 241 FK4 stars from alarge number of astrogeodetic plumb-line deflection measurements. Thecorrections on the average amount to 0.2 arc sec, with extrema of 0.6arc sec. Half of the values are based on at least eight observations andare accurate to plus or minus 0.13 arc sec. Comparative studies showthat the results are free of systematic errors and that the mean errorsare reliable. Catalogue general des etoiles observees a l'astrolabe (1957-1975), corrections individuelles aux positions DU FK4.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1978A&AS...31..159B&db_key=AST Rotational Velocities of a0 StarsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1974ApJS...28..101D&db_key=AST Four-color and Hβ photometry for the brighter AO type starsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1972A&AS....5..109C&db_key=AST Narrow-band photometry of early-type stars.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1971A&A....12....5H&db_key=AST A catalogue of proper motions for 437 A starsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1970A&AS....1..189F&db_key=AST Photoelectric observations of early A starsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1970A&AS....1..165J&db_key=AST Photographic determinations of the parallaxes of 21 stars with the Thaw refractor.Not Available
Submit a new article
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2019-02-19 12:00:50
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https://chinmaydd.in/2015/12/06/4042/
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# 4042
SECCON CTF Quals just ended and NIA finished 122nd /o/ .
Here is a writeup for 4042.
Points: 100
Category: Unknown
The amount of information given about the encoded text was really small, which hinted at some recon aspect to the challenge. The very first result obtained on searching “2005” and “4042” is this particular April Fool’s RFC which describes two encodings, UTF-9 and UTF-18. Unlike the other ones by the IETF, the encoding described in this document could actually be implemented.
Below is the code for decoding it to the UTF-32 format:
# Returns a hexadecimal value for a UTF-32 encoded character
# https://gist.github.com/O-I/6758583
def get_character(hexnum)
char = ''
char << hexnum.to_i(16)
end
# Read the encoded data file
# We can consider the numbers to be encoded in nonets i.e a group of 9 bits.
# The first bit describes a 'continuation character',
# The other 8 bits can be considered to be an octet.
# If the continuation character is true, the bits are shifted to right(by 8) and the next octet is appended.
# Error correction has not been encorporated.
# Split into groups of 3 bytes i.e 9 bits
# strings = '403221'.scan(/.{1,3}/)
val = 0
for i in 0...strings.length
# Convert into octal representation
nonet_val = strings[i].to_i(8)
# Check the MSB of the number if 1
continuation_char = (nonet_val & 256)/256
# Find value of the remaining 8 bits forming the octet
octet_val = nonet_val & 255
# XOR current val with the octet val.
# If the current val is carried over due to continuation, the next 8 bits are added to it.
val = val ^ octet_val
if continuation_char == 1
val = val << 8
else
print get_character(val.to_s(16))
# Reset val
val = 0
end
end
puts
Easy challenge :)
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2021-01-24 11:42:48
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http://lectureonline.cl.msu.edu/~mmp/kap10/cd284.htm
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# Temperature and Average Kinetic Energy
We will now study what the velocity of the gas molecules is as a function of the temperature. First we introduce the average squared velocity <v2>. This is obtained by taking the square of the velocities of each molecule in the gas and averaging (indicated by the angular brackets). One can show that in this case <v2> = <v>2.
The relationship between pressure, volume, and average velocity square is:
pV = 1/3 $\cdot$ N m <v>2
Here N is the number of molecules in the volume V, and m is the mass of one of them. Combining this result with the ideal gas law gives:
(3/2) kT = (1/2) m <v2>
Thus we find that temperature is a measure of the average kinetic energy of the molecules of the gas (or wall). Boltzmann's constant is just a unit conversion from K to Joules. If you mix a gas, eventually all the molecules have the same kinetic energy on the average, independent of their mass or the temperature at which they were introduced. This is called equilibrium. This means that a light molecule like hydrogen must move much faster than a heavy one like oxygen.
In the simulation, a hot gas (represented by red balls) is mixed with a cold gas (represented by the blue balls). As time passes, the gases interact with each other and reach equilibrium.
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2017-10-18 01:56:33
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http://journal.psych.ac.cn/acps/EN/Y2007/V39/I06/1012
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ISSN 0439-755X
CN 11-1911/B
›› 2007, Vol. 39 ›› Issue (06): 1012-1024.
### The Impact of the Articulary Loop and Visuospatial Sketchpad on Phonemic and Semantic Fluency
Zhang Jijia,Lu Aitao
1. Department of Psychology，South China Normal University，Guangzhou 510631, China
• Received:2006-10-13 Revised:1900-01-01 Online:2007-11-30 Published:2007-11-30
• Contact: Zhang Jijia
Abstract: Verbal fluency is typically measured by generative naming tasks. The two types of verbal fluency that are generally tested are (1) phonemic fluency, in which the subjects generate words beginning with a particular letter, and (2) semantic fluency, in which the subjects produce items from a specified category. Most of the previous studies primarily focused on the contribution of the central executive function to phonemic and semantic fluency; however, little attention was paid to the contribution of the phonological loop and the visuospatial sketchpad.
Based on the work of Rende, the present study explored the effects of the articulary loop and visuospatial sketchpad on verbal fluency. The following two types of secondary tasks were used: (1) articulatory suppression and sound judgment and (2) grapheme judgment and rotation judgment.
Methods
Forty four undergraduates, including 22 women, participated in Exp.1. The writing speed of all participants was estimated. Subsequently, the participants were given a familiar topic, such as “my college life,” and were required to write as many words as possible in one minute. All of them produced more than 30 words per minute, which satisfied the experimental requirement. Thereafter, they were required to assess their Chinese language level based on a 5-point scale. The scores of all the participants were equal to or higher than 3. Exp. 2 involved 40 subjects, including 20 women. The participants were given the same topic as Exp.1 and were required to talk on the subject for one minute in order to test their oral speed. They spoke rather fluently and none of them stuttered or spoke slowly. Similar to Exp. 1, the Chinese level of all participants was equal to or higher than 3. Both experiments employed a two-factor within-subject design, 2 (phonemic and semantic fluency) × 2 (single- and dual-task conditions).
Results
(1) The results of Exp.1 indicated that the articulatory loop had a substantial impact on verbal fluency. When the secondary task was articulatory suppression, the participants produced more words during the semantic fluency task than during the phonemic fluency task; however, they produced fewer words in the dual-task condition than in the single-task condition. Further, the cluster size was smaller in the dual-task condition than in the single-task condition. The participants generated more switches during the semantic fluency task than during the phonemic fluency task; however, they generated fewer switches in the dual-task condition than in the single-task condition. When the secondary task was sound judgment, the pattern of results was similar to that when the secondary task was articulatory suppression, with the exception that there were no significant differences in the cluster sizes across the different tasks and different conditions. Therefore, the negative impact of articulatory suppression and sound judgment was significantly stronger for the phonemic fluency task than for the semantic fluency task.
(2) The results of Exp.2 indicated that visuospatial sketchpad had a substantial impact on verbal fluency. When the secondary task was grapheme judgment, the participants produced more words during the phonemic fluency task than during the semantic fluency task; however, they produced fewer words in the dual-task condition than in the single-task condition. Further, the cluster size was larger in the semantic fluency task than in the phonemic fluency task. The participants generated more switches during the phonemic fluency task than during the semantic fluency task; however, they generated fewer switches in the dual-task condition than in the single-task condition. When the secondary task was rotation judgment, the pattern of results was similar to that when the secondary task was grapheme judgment. Therefore, the negative impact of grapheme judgment and rotation judgment was significantly stronger for the semantic fluency task than for the phonemic fluency task.
(3) A detailed analysis showed that Chinese subjects adopt different strategies in different fluency tasks, using the oral verbatim and reading silently strategies for phonemic fluency tasks and the mental image strategy for semantic fluency tasks.
Conclusions
(1) Articulatory suppression and sound judgment tasks influenced phonemic verbal fluency more than semantic fluency.
(2) Grapheme judgment and rotation judgment tasks influenced semantic fluency more than phonemic fluency.
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2020-11-24 23:32:38
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https://tex.stackexchange.com/questions/115202/how-to-adjust-the-blank-space-between-the-top-of-the-left-curly-brace-and-the-co
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# How to adjust the blank space between the top of the left curly brace and the contents?
I want to type a system of equations delimited by the left curly brace. And I tried using the \left{ with aligned, and cases, and array, and numcases. But there are some difference between the results of these four methods. Here is the screen-shots and the source file:
\documentclass{article}
\usepackage{amsmath,amssymb,cases}
\begin{document}
\begin{align*}
&\left\{\begin{aligned}
&x=x,\\
&u=u(x,t).
\end{aligned}\right.
~~~&& \begin{cases}
x=x,\\
u=u(x,t).
\end{cases}
~~ &&
\left\{\begin{array}{l}
x=x,\\
u=u(x,t).
\end{array}\right.\\
&\text{align} && \text{cases} &&\text{array}
\end{align*}
using numcases:
\begin{numcases}{}
x=x,\nonumber\\
u=u(x,t). \nonumber
\end{numcases}
\end{document}
My question is how can I adjust the space between the top tip of left curly brace and the first line of the equations, so that the results of using array etc is the same as that of using numcases. At present, I have no way of doing this.
When I tried using scalerel.sty, the result is as above: I do not know why? May be I was wrong in installing the scalerel.sty.
• You need to have the link descriptions [1] : http//... at the very end of the post. That's why documentclass was sticking out ;) – percusse May 20 '13 at 12:14
• I want to let the first equation $x=x$ move up a little when using evironments of arrya, aligned, and cases, so that there is no more verticle space between the top of the curly brace and the first equation. Actually, the fourth method of using numcases is just what I want. But the drawback of this method is that it can not be allowed to use with other environments, such as gather, align, etc. And I am tired of adding \nonumber when I do not want the formulas be numbered. – azhi May 20 '13 at 12:50
• An aside: if you replace \begin{array}{l} with \begin{array}{@{}l}, you'll get rid of some horizontal space. By default, arrays include a bit of horizontal padding around each column, and @{} removes this. – John Wickerson May 20 '13 at 13:04
• I have updated my answer to address the issue of vertical brace extent. – Steven B. Segletes May 20 '13 at 13:14
• I want to just decrease the VIRTICLE space between the upper tip of curly brace and the first equation of the system, Not The Horizontal Space. – azhi May 20 '13 at 13:16
I would say that with numcases the brace is a bit too low. Nevertheless, I fully agree that the braces with aligned and cases are much too large. Here's what I often do as a remedy:
\documentclass{article}
\usepackage{amsmath}
\makeatletter
\def\env@cases{%
\let\@ifnextchar\new@ifnextchar
\left\lbrace
\def\arraystretch{1.1}%
}
\makeatother
\begin{document}
$\begin{cases} x=x, \\ u=u(x,t). \end{cases} \quad \begin{cases} a=a, \\ x=x, \\ u=u(x,t). \end{cases} \quad \begin{cases} a=a, \\ b=b, \\ x=x, \\ u=u(x,t). \end{cases}$
\end{document}
The point is that amsmath's cases uses an \arraystretch of 1.2, which doesn't work well with TeX's standard brace sizes. I prefer using 1.1 instead of 1.2; of course this means that there's a bit less vertical space between the cases. Maybe you'll deem the braces still too large, but in my opinion this is a good compromise. (The distance to the brace tips appears a bit large only because a, x and u don't have ascenders and descenders.)
The \! macro does a thin backspace. I have placed them through your MWE with prejudice to demonstrate where they might go.
\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{cases}
\begin{document}
\begin{align*}
&\left\{\!\!\begin{aligned}
&x=x,\\
&u=u(x,t).
\end{aligned}\right.
~~~&& \begin{cases}
\!\!x=x,\\
\!\!u=u(x,t).
\end{cases}
~~ &&
\left\{\!\!\!\!\begin{array}{l}
x=x,\\
u=u(x,t).
\end{array}\right.\\
&\text{align} && \text{cases} &&\text{array}
\end{align*}
using numcases:
\begin{numcases}{}
\!\!\!x=x,\nonumber\\
\!\!\! u=u(x,t). \nonumber
\end{numcases}
\end{document}
If the approximate space gotten by this method is insufficiently accurate, one could use \rule{length}{0ex} to get a more precise space, where length can be positive or negative.
EDITED to adjust vertical extent of braces:
\documentclass{article}
\usepackage{scalerel}
\usepackage{verbatimbox}
\usepackage{amsmath,amssymb,cases}
\begin{document}
\def\x{\mbox{\addvbuffer[-0.8ex -.1ex]{\begin{aligned} &x=x,\\ &u=u(x,t). \end{aligned}}}
}
\def\z{\mbox{\addvbuffer[-0.8ex -.1ex]{$\begin{array}{l} x=x,\\ u=u(x,t). \end{array}$}}
}
\begin{align*}
&\scaleleftright[1.7ex]{\{}{\x}{.}
~~~&& \begin{cases}
\x
\end{cases}
~~ &&
\scaleleftright[1.7ex]{\{}{\z}{.}
\\
&\text{align} && \text{cases} &&\text{array}
\end{align*}
using numcases:
\begin{numcases}{}
x=x,\nonumber\\
u=u(x,t). \nonumber
\end{numcases}
\end{document}
• I want the first equation $x=x$ to move up a little for the first three methods, not to move left a little. – azhi May 20 '13 at 12:30
• Aha. Now I understand your problem. If I can't propose a solution, I will remove my answer. – Steven B. Segletes May 20 '13 at 12:35
• But I can not compile your codes. Here is the error: File scalerel.sty' not found. \usepackage – azhi May 20 '13 at 13:27
• I'm OK with scaling characters as a last resort, but I think you're doing it too generously. (This is understandable as you're the author of scalerel, but still ... :-)`) – Hendrik Vogt May 20 '13 at 13:31
• @azhi The package can be found at ctan.org/tex-archive/macros/latex/contrib/scalerel. – Steven B. Segletes May 20 '13 at 15:43
|
2020-07-14 13:23:37
|
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|
https://scicomp.stackexchange.com/questions/14222/what-is-the-current-state-of-the-art-in-solving-higher-dimensional-parabolic-pde/14281
|
What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrödinger equation)
What is the current state of the art for solving higher dimensional (3-10) parabolic PDEs in the complex domain with simple poles (of the form $\frac{1}{|\vec{r}_1 - \vec{r}_2|}$) and absorbing boundary conditions?
Specifically, I'm interested in solving the multi-electron Schrödinger equation:
$\left( \sum_i \sum_{j\neq i}\left[ -\frac{\nabla_i^2}{2 m} - \frac{Z_i Z_j}{|\vec{r}_i - \vec{r}_j|} + V(\vec{r}_i, t) \right]\right)\psi = -i\partial_t \psi$
For a diatomic molecule with more than 1 electron.
3 Answers
The solutions for the equation are in $$\psi \in \mathbb{C}^{3M}\times\mathbb{R}^+ \enspace .$$ If the number of electrons is small enough you can just use any traditional method. Like a domain discretization method (Finite Difference, Finite Element, Boundary Element), or a pseudospectral method. Since solving this equation is not more difficult than solving a multidimensional wave equation.
In the case of bigger systems some trick is necessary to get the solution. We replace the electron-electron interaction for the interaction of an electron with a cloud of electrons (a mean field approximation of the rest of them), and then solve in a self-consistent fashion (due to the nonlinearity that come from the mean field term). This is done in Hartree-Fock and Density Functional Theory (DFT). Where the original differential equation is transformed into a variational formulation.
DFT is the most common method nowadays, and the advantage is that all the equations are formulated in terms of the electron density and not in terms of the wave equations. So, the equations lie in a 3 dimensional space. One book that describes both of these methods is
• Thijssen, Jos. Computational physics. Cambridge University Press, 2007. Amazon link.
You want to solve for 3 to 10 particle systems (3D per particle)? As far as I am aware, mean field theories do not work especially well for so few particles, but it seems there has been DFT work on diatomic molecules.
Is this a system where Born-Oppenheimer is valid? If so, I might be inclined to expand the electronic wavefunction using a linear combination of Slater determinants possibly using sparse grid or spectral sparse grids This paper perhaps could help.
Another option is to try using a tight-binding approach, although the fact that you mentioned absorbing boundary conditions suggests you may be thinking of problems involving ionization/dissociation. TB would mostly be useful if you were trying to approximate low level states.
Possibly something like the multi-configurational time-dependent Hartree-Fock method could work here MCTDHF.
Finally, you could look at quantum Monte Carlo methods. These are the methods by which exchange and correlation functional models for single atoms are obtained to do DFT calculations. It looks like there are poly-atomic extensions. (I'm out of link privileges).
• 3-10 dimensions, not particles: specifically 1 to 3 electrons, 2 nuclei (1d for the nuclei, 6d for the particles), without a Born-Oppenheimer approximation. And I'm doing ionization type stuff. – Andrew Spott Jul 29 '14 at 20:40
If you have $M$ atoms, your wave function depends on $3M$ variables. If you wanted to discretize this function on a uniform mesh with $N$ nodes in each of these directions (or with $N$ one-dimensional shape functions), you'd need a total of $N^{3M}$ unknowns -- far too many for any interesting number of electrons $M$. To give just one example, if you'd just use $N=10$ nodes in each direction, and you had just 3 electrons, you'd already have a system of size $10^9$, very much at the limit of what one can do today.
From this consideration follows that it is not possible to consider the problem with all electrons at the same time -- you need to restrict yourself to one or two electrons at a time. This naturally leads to you to methods such as the Hartree Fock method that iterates over electrons while keeping the rest of the system fixed.
I don't know the field well enough but imagine that there are a number of highly cited and well written review papers on the topic.
• $10^9$ Yeouch. Systems that size can be solved, but you better have a good reason, a supercomputer, and a lot of time on your hands! – meawoppl Jul 27 '14 at 18:13
• Well, fermionic systems have quite a few (anti-)symmetries due to the Pauli principle that you can exploit to significantly reduce the number degrees of freedom (instead of the 3M-dimensional hypercube, you only need to consider the corresponding simplex, of which the cube contains (3M)! copies). So you only need binom(N,3M) basis functions -- still exponential, but growing much slower. That might put the lower end of the range in reach of a beefy workstation. – Christian Clason Jul 27 '14 at 19:56
• For a 3-electron system, maybe. But you still won't be able to do anything beyond this. That doesn't leave a great number of molecules :-) – Wolfgang Bangerth Jul 29 '14 at 14:32
• But the question was only asking for 3-10 variables :) (But your point is valid: for anything with more than a small number of electrons, you need to consider approximate field models such as DFT; my point was that between "can be solved with standard approaches" and "can only be solved approximately", there's a non-trivial range of problems that "can (only) be solved using non-standard approaches".) – Christian Clason Jul 29 '14 at 15:26
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2021-04-10 15:21:17
|
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|
http://mathhelpforum.com/discrete-math/183201-new-recursion-have-no-idea-how-solve-problem.html
|
# Math Help - New to recursion and have no idea how to solve this problem
1. ## New to recursion and have no idea how to solve this problem
For each integer n >= 0, let a(n) be the number of bit strings of length n that do not contain the pattern 101.
a.) Show that a(k) = a(k-1) + a(k-3) + a(k-4) + ... + a(0) + 2, for all integers k>= 3.
b.) Use the result of part (a) to show that if k >= 3, then a(k) = 2a(k-1) - a(k-2) + a(k-3)
**everything in parenthesis is treated as a subscript
This question is totally unlike anything else from this section of the book. I really have no idea where to even start. Part a almost looks like I could do induction but I don't know if that would be correct. The only thing I've done so far is I wrote out the first couple of bit length (don't even know if I'm supposed to do this) and came up with:
Length 0: ∊
Length 1: 0, 1
Length 2: 11, 10, 01, 00
Length 3: 111, 110, 100, 000, 001, 011, 010 (101 is excluded, obviously).
I also understand that a recursive formula I can write is
a(k) = 2a(k-1) + 2a(k-2) + a(k-3) (I think, not sure if this is correct)
2. ## Re: New to recursion and have no idea how to solve this problem
are you sure you have the formula in part a) correct?
a3 = 7, but a2 + a0 + 2 = 6.
3. ## Re: New to recursion and have no idea how to solve this problem
Induction was my first instinct, but I think regardless you will have to relate the various a(i). To facilitate that, each a(i) has $2^i$ possible bit sequences, of which we'll have to exclude a certain number that include the 101 pattern. You already exposed this in your first four cases: length 0 is the empty bit string, length 1 is (0, 1), length 2 is (11, 10, 01, 00), etc. My intuition is that there is something to be explored when we, say, add a(k-3) + a(k-4), and the like when we think of it in terms of $2^k$ less the 101 bit sequences. I assume the 101 pattern showing up anywhere in a string (e.g., 100101011 would be excluded) rules it out. So, in a string of length k, how many of the $2^k$ will be excluded? Ultimately, that is what I think this algorithm is expressing by putting it in terms of how many are excluded from lesser length bit strings. Therefore, to facilitate the solution, we will need to know how many are excluded along the way, and the reason we're concerned with k greater than 2 is because, as you show, exclusion only begins at length 3 sequences.
4. ## Re: New to recursion and have no idea how to solve this problem
I believe the formula is part a is correct. I have uploaded a picture of the actual problem:
http://i.imgur.com/MDsoz.jpg
5. ## Re: New to recursion and have no idea how to solve this problem
One thing you might want to try is write the expression for a(k-1) in the form of a(k), and so on for the rest of the a(i)'s. Then by substituting the values for each a(i) you might have something more manageable, maybe. Just something I thought you might want to explore.
|
2015-04-28 20:32:43
|
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|
https://math.stackexchange.com/questions/2587694/on-convergence-of-series-of-the-generalized-mean-sum-n-1-infty-left-fra
|
# On convergence of series of the generalized mean $\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s.$
Assume that $a_n>0$ such that $\sum_{n=1}^{\infty}a_n$ converges.
Question: For what values of $s\in \Bbb R$ does the following series : $$I_s= \sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s.$$ converges or diverges?
This question is partially motivated by some comments on this post where it is shown that $I_s$ converges for $s>1$. Moreover, it is well known that $$\lim_{s\to\infty}\left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s = \left(a_1a_2\cdots a_n\right)^{1/n}$$
Accordingly, Taking $b_n= 1/a_n$ is one readily get,
$$\lim_{\color{red}{s\to-\infty}}\left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s = \left(a_1a_2\cdots a_n\right)^{1/n}$$ it draws from Carleman's inequality that :
$$\color{red}{ I_{-\infty}}=I_\infty= \sum_{n=1}^{\infty}\left(a_1a_2\cdots a_n\right)^{1/n} \le e \sum_{n=1}^{\infty} a_n<\infty .$$ Patently it is also true that the convergence holds for $s=-1$ this is proven here. Whereas the convergence fails for $0<s<1$ Indeed, $$\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s \ge \sum_{n=1}^{\infty} \frac{a_1}{n^s}=\infty$$
So we have that $I_s$ converges for $1<s\le\infty$ or $s=\in\{-1,-\infty\}$ and diverges for $0<s<1$.
Hence the original question reduces on studying $I_s$ for $s\le0$ can anyone help?
Clearly the hope is that $I_s$ converges for for $-\infty\le s\le -1$ and diverges for $-1<s<0.$
I don't know if one could infer some conjecture for the case $s=0$ since it seems pathological.
• Hardy's inequality for negative exponents is also known as the Polya-Knopp inequality: just google it. The proofs exploit similar ideas (Holder+induction, or some form of convexity, or Bocharova's approach) – Jack D'Aurizio Jan 1 '18 at 18:50
• You might have a look at the Cauchy-Schwarz and beyond section of my notes, too. – Jack D'Aurizio Jan 1 '18 at 18:52
• @JackD'Aurizio Seriously I feel stupid after your comment. oooo god please step by step. I only know Hardy holder and convexity where do the rest come from? – Guy Fsone Jan 1 '18 at 18:53
Answer: Using the power mean (generalized mean) inequality the convergence for $s<0$ follows easily from the convergence for $s>1$. This probably does not give optimal bounds for $I_s$.
It's more natural to set $s=1/t$, so that the integrands are generalized means that, for fixed $(a_n)$, are increasing in $t \in \mathbb R$ by the power mean inequality (which says exactly that). We know that $I_{1/t}$ converges for $0<t<1$ (i.e. $s>1$, by Hardy's inequality, see here) and thus for all $-\infty \leq t < 1$ by the comparison test.
We also have that:
• the supremum $S_t=\sup(I_{1/t}/\sum a_n)$ over all sequences is finite for all $t<1$
• $S_t$ is (monotonically) increasing in $t$
• $S_{-\infty}=1$ (take a decreasing sequence)
• $S_{-1}=2$ (see this question)
• $S_0 \leq e$ (Carleman's inequality)
• $S_t \leq (1-t)^{-1/t}$ for $0<t<1$ (Hardy's inequality, see this question)
It's natural to conjecture that $S_t=(1-t)^{-1/t}$ for all $t<1$, which is $1$ at $-\infty$ and $e$ at $0$.
• there is one conclusion in your first statement that I don't understand. why is this converges for t less than one ? – user503348 Jan 2 '18 at 6:36
• I think I have don't what the Op did already check the question again – user503348 Jan 2 '18 at 6:39
• this answer is not clear – user503348 Jan 2 '18 at 6:39
• @Sobolev I was using a result from a previous question of the OP, see edit. – punctured dusk Jan 2 '18 at 9:52
• @barto Sorry are you not to just repeating what I have say in the question?n the crucial point is to find the convergence for $s<0$ which I cannot figure out in your answer – Guy Fsone Jan 2 '18 at 11:41
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2019-05-26 03:25:54
|
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|
https://ltwork.net/click-below-on-the-factors-that-an-initial-pain-screening--1188
|
# Click below on the factors that an initial pain screening on admission should cover.
###### Question:
Click below on the factors that an initial pain screening on admission should cover.
$Click below on the factors that an initial pain screening on admission should cover.$
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### The linear model represents the height, f(x), of a water balloon thrown off the roof of a building over time, x, measured
The linear model represents the height, f(x), of a water balloon thrown off the roof of a building over time, x, measured in seconds: Part A: During what interval(s) of the domain is the water balloon's height increasing? (2 points) Part B: During what interval(s) of the domain is the water balloon...
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2022-09-28 23:25:09
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http://bywayofcontradiction.com/maximize-worst-case-bayes-score/?replytocom=10712
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Maximize Worst Case Bayes Score
In this post, I propose an answer to the following question:
Given a consistent but incomplete theory, how should one choose a random model of that theory?
My proposal is rather simple. Just assign probabilities to sentences in such that if an adversary were to choose a model, your Worst Case Bayes Score is maximized. This assignment of probabilities represents a probability distribution on models, and choose randomly from this distribution. However, it will take some work to show that what I just described even makes sense. We need to show that Worst Case Bayes Score can be maximized, that such a maximum is unique, and that this assignment of probabilities to sentences represents an actual probability distribution. This post gives the necessary definitions, and proves these three facts.
Finally, I will show that any given probability assignment is coherent if and only if it is impossible to change the probability assignment in a way that simultaneously improves the Bayes Score by an amount bounded away from 0 in all models. This is nice because it gives us a measure of how far a probability assignment is from being coherent. Namely, we can define the “incoherence” of a probability assignment to be the supremum amount by which you can simultaneously improve the Bayes Score in all models. This could be a useful notion since we usually cannot compute a coherent probability assignment so in practice we need to work with incoherent probability assignments which approach a coherent one.
Now, let’s move on to the formal definitions and proofs.
Fix some language $L$, for example the language of first order set theory. Fix a consistent theory $T$ of $L$, for example ZFC. Fix a nowhere zero probability measure $\mu$ on $L$, for example $\mu(\phi)=2^{-\ell(\phi)}$, where $\ell(\phi)$ is the number of bits necessary to encode $\phi$.
A probability assignment of $L$ is any function from $L$ to the interval $[0,1]$. Note that this can be any function, and does not have to represent a probability distribution. Given a probability assignment $P$ of $L$, and a model $M$ of $T$, we can define the Bayes Score of $P$ with respect to $M$ by
${\displaystyle \mbox{Bayes}(M,P)=\sum_{M\models \phi}\log_2(P(\phi))\mu(\phi)+\sum_{M\models\neg \phi}\log_2(1-P(\phi))\mu(\phi). }$
We define the Worst Case Bayes Score $\mbox{WCB}(P)$ to be the infimum of $\mbox{Bayes}(M,P)$ over all models $M$ of $T$.
Let $\mathbb{P}$ denote the probability assignment that maximizes the function $\mbox{WCB}$. We will show that this maximum exists and is unique, so $\mathbb{P}$ is well defined.
In fact, $\mathbb{P}$ also coherent, meaning that there exists a probability distribution on the set of all models of $T$ such that $\mathbb{P}(\phi)$ is exactly the probability that a randomly chosen model satisfies $\phi$. Since the natural definition of a measurable subset of models comes from unions and intersections of the sets of all models satisfying a given sentence, we can think of $\mathbb{P}$ as an actual probability distribution on the set of all models of $T$.
First, we must show that there exists a probability assignment $P$ which maximizes $\mbox{WCB}$.
Note that $\mbox{Bayes}(M,P)$ either diverges to $-\infty$, or converges to a non-positive real number. If $P$ is the identically $1/2$ function, then $\mbox{WCB}(P)=-1$, so there is at least one $P$ for which $\mbox{WCB}(P)$ is finite. This means that when maximizing $\mbox{WCB}(P)$, we need only consider $P$ for which $\mbox{Bayes}(M,P)$ converges to a number between $-1$ and $0$ for all $M$.
Assume by way of contradiction that there is no $P$ which maximizes $\mbox{WCB}$. Then there must be some supremum value $m$ such that $\mbox{WCB}$ can get arbitrarily close to $m$, but never equals or surpasses $m$. Consider an infinite sequence probability assignments $\{P_i\}$ such that $\mbox{WCB}(P_i)\rightarrow m$. We may take a subsequence of $\{P_i\}$ in order to assume that $\{P_i(\phi)\}$ converges for every sentence $\phi$. Let $P$ be such that $P_i(\phi)\rightarrow P(\phi)$ for all $\phi$.
By assumption, $\mbox{WCB}(P)$ must be less than $m$. Take any model $M$ for which $\mbox{Bayes}(M,P). Then there exists a finite subset $S$ of $L$ such that $\mbox{Bayes}_S(M,P), where
${\displaystyle \mbox{Bayes}_S(M,P)=\sum_{\phi\in S, M\models \phi}\log_2(P(\phi))\mu(\phi)+\sum_{\phi\in S, M\models\neg \phi}\log_2(1-P(\phi))\mu(\phi). }$
Note that in order to keep the Bayes score at least $-1$, any $P_i$ must satisfy $2^{-1/\mu(\phi)}\leq P_i(\phi)\leq 1$ if $M\models \phi$, and $0\leq P_i(\phi)\leq 1-2^{-1/\mu(\phi)}$ if $M\models\neg\phi$. Consider the space of all functions $f$ from $S$ to $[0,1]$ satisfying these inequalities. Since there are only finitely many values restricted to closed and bounded intervals, this space is compact. Further, $\mbox{Bayes}_S(M,f)$ is a continuous function of $f$, defined everywhere on this compact set. Therefore,
${\displaystyle \lim_{i\rightarrow\infty}\mbox{Bayes}_S(M,P_i)=\mbox{Bayes}_S(M,P)
However, clearly $\mbox{WCB}(P_i)\leq\mbox{Bayes}(M,P_i)\leq\mbox{Bayes}_S(M,P_i)$, so
${\displaystyle \lim_{i\rightarrow\infty}\mbox{WCB}(P_i)
contradicting our assumption that $\mbox{WCB}(P_i)$ converges to $m$.
Next, we will show that there is a unique probability assignment which maximizes $\mbox{WCB}$. Assume by way of contradiction that there were two probability assignments, $P_1$ and $P_2$ which maximize $\mbox{WCB}$. Consider the probability assignment $P_3$, given by
${\displaystyle P_3(\phi)=\frac{\sqrt{P_1(\phi)P_2(\phi)}}{\sqrt{P_1(\phi)P_2(\phi)}+\sqrt{(1-P_1(\phi))(1-P_2(\phi))}}.}$
It is quick to check that this definition satisfies both
${\displaystyle \log_2(P_3(\phi))\geq \frac{\log_2(P_1(\phi))+\log_2(P_2(\phi))}{2}}$
and
${\displaystyle \log_2(1-P_3(\phi))\geq \frac{\log_2(1-P_1(\phi))+\log_2(1-P_2(\phi))}{2},}$
and in both cases equality holds only when $P_1(\phi)=P_2(\phi).$
Therefore, we get that for any fixed model, $M$,
${\displaystyle \mbox{Bayes}(M,P_3(\phi))\geq \frac{\mbox{Bayes}(M,P_1(\phi))+\mbox{Bayes}(M,P_2(\phi))}{2},}$
By looking at the improvement coming from a single sentence $\phi$ with $P_1(\phi)\neq P_2(\phi),$ we see that
${\displaystyle \mbox{Bayes}(M,P_3(\phi))-\frac{\mbox{Bayes}(M,P_1(\phi))+\mbox{Bayes}(M,P_2(\phi))}{2},}$
is actually bounded away from $0$, which means that
${\displaystyle \mbox{WCB}(P_3(\phi))\geq \frac{\mbox{WCB}(P_1(\phi))+\mbox{WCB}(P_2(\phi))}{2},}$
which contradicts the fact that $P_1$ and $P_2$ maximize $\mbox{WCB}$.
This means that there is a unique probability assignment, $\mathbb{P}$, which maximizes $\mbox{WCB}$, but we still need to show that $\mathbb{P}$ is coherent. For this, we will use the alternate definition of coherence given in Theorem 1 here. Namely that $\mathbb{P}$ is coherent if and only if $\mathbb{P}$ assigns probability 0 to every contradiction, probability 1 to every tautology, and satisfies $\mathbb{P}(\phi)=\mathbb{P}(\phi\wedge\psi)+\mathbb{P}(\phi\wedge\neg\psi)$ for all $\phi$ and $\psi$.
Clearly $\mathbb{P}$ assigns probability 0 to every contradiction, since otherwise we could increase the Bayes Score in all models by the same amount by updating that probability to 0. Similarly $\mathbb{P}$ clearly assigns probability 1 to all tautologies.
If $\mathbb{P}(\phi)\neq\mathbb{P}(\phi\wedge\psi)+\mathbb{P}(\phi\wedge\neg\psi)$, then we update all three probabilities as follows:
$\mathbb{P}(\phi)\mapsto \frac{1}{1+\frac{1-\mathbb{P}(\phi)}{\mathbb{P}(\phi)}(2^{-x/\mu(\phi)})},$
$\mathbb{P}(\phi\wedge\psi)\mapsto \frac{1}{1+\frac{1-\mathbb{P}(\phi\wedge\psi)}{\mathbb{P}(\phi\wedge\psi)}(2^{x/\mu(\phi\wedge\psi)})},$
and
$\mathbb{P}(\phi\wedge\neg\psi)\mapsto \frac{1}{1+\frac{1-\mathbb{P}(\phi\wedge\neg\psi)}{\mathbb{P}(\phi\wedge\neg\psi)}(2^{x/\mu(\phi\wedge\neg\psi)})},$
where $x$ is the unique real number such that the three new probabilities satisfy $\mathbb{P}(\phi)=\mathbb{P}(\phi\wedge\psi)+\mathbb{P}(\phi\wedge\neg\psi)$. This correction can increases Bayes Score by the same amount in all models, and therefore increase $\mbox{WCB}$, contradicting the maximality of $\mbox{WCB}(\mathbb{P})$. Therefore $\mathbb{P}$ is coherent as desired.
Finally, we show that any given probability assignment $P$ is coherent if and only if it is impossible to simultaneously improve the Bayes Score by an amount bounded away from 0 in all models. The above proof that $\mathbb{P}$ is coherent actually shows one direction of this proof, since the only fact it used about $\mathbb{P}$ is that you could not simultaneously improve the Bayes Score by an amount bounded away from 0 in all models. For the other direction, assume by way of contradiction that $P$ is coherent, and that there exists a $Q$ and an $\epsilon>0$ such that $\mbox{Bayes}(M,Q)-\mbox{Bayes}(M,P)>\epsilon$ for all $M$.
In particular, since $P$. is coherent, it represents a probability distribution on models, so we can choose a random model $M$ from the distribution $P$. If we do so, we must have that
$\mathbb{E}(\mbox{Bayes}(M,Q))-\mathbb{E}(\mbox{Bayes}(M,P))>0.$
However, this contradicts the well known fact that the expectation of Bayes Score is maximized by choosing honest probabilities corresponding the actual distribution $M$ is chosen from.
I would be very grateful if anyone can come up with a proof that this probability distribution which maximizes Worst Case Bayes Score has the property that its Bayes Score is independent of the choice of what model we use to judge it. In other words, show that $\mbox{Bayes}(M,\mathbb{P})$ is independent of $M$. I believe it is true, but have not yet found a proof.
1. Neat idea! Especially your idea for measuring incoherence.
For your last question I’m beginning to think that it isn’t true. For example what if we build a theory like this: go through L in order of decreasing μ, and each time we reach a statement undecided by our theory so far we add it or its negation according to which is more likely under P. I suspect that this theory will give P a good Bayes score, since it agrees with P “where it counts” on statements with high μ.
On the other hand, at first I thought that maybe *no* model achieved the worst case, and that the infimum was only an infimum. This definitely isn’t true. To find a model that achieves the worst case, take a sequence of models whose Bayes scores tend to the infimum and for each φ pass to an infinite subsequence where the truth value of that φ is constant.
The dependence on μ is a bit distressing in general. Maybe we could set μ proportional to c^(-l(φ)) and then take a limit as c goes to 1, hoping we can force P to converge somehow?
By the way you have a typo in the third paragraph from bottom. “Bayes(M,Q)-Bayes(M,Q)” should have a “P” replacing one of the “Q”s.
• Thanks, fixed the typo.
The conjecture is at least true when the language is finite. It is also the case that the set of models for which Bayes score is near WCB is an open and everywhere dense set of all models. I think I can prove but have not checked formally that P achieves its worst case score with probability 1 (with respect to its own probability distribution, so if P does not have constant Bayes score, it is at least really close to having constant Bayes score.
I have not yet though about taking a limit as c goes to 1. I would guess that it is not going to converge, but if it did, that would be very nice. I will think about that more.
• There’s a question I forgot to ask:
If we add an additional axiom to our theory, does P change according to Bayes rule?
It seems pretty important for this to be true.
• No, it does not. Abram Demski proposed a different solution to this problem which was to randomly sample sentences and add them to the theory if consistent, and his does not either. This makes me sad.
I think that one should start with an empty theory, find a probability distribution, and do the rest with Bayesian updates.
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2017-10-17 16:45:24
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http://math.stackexchange.com/questions/48507/goldbachs-conjecture-and-difference-of-squares
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# Goldbach's conjecture and difference of squares
Someone came to me with the following observation: If $2n=p+q$ then $pq=n^2-m^2$ for some value of $0<m<n$ (namely, $m=n-p$ given $p\le q$).
Now he claims that this is actually equivalent: that the claim "For every $n$ there exists $0<m<n$ such that $n^2-m^2$ is the product of two primes" is equivalent to Goldbach's conjecture.
1. Is it true? I tried proving the nontrivial direction but got stuck.
2. Is it well known? I tried looking for references and couldn't find any.
(I am trying to explain to him that this is a hard conjecture and trivial observations are probably not worth his time except for recreation).
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are you sure that the inequality 0 < m (< n) is strict? Let p = q = 2. then for n = 2, we have that $2\cdot 2 = 2 + 2$, and so $pq = 2 \cdot 2 = 4 = n^2 - m^2 = 2^2 - m^2 = 4 - m^2$, so $m$ would need to be zero, else we have $n, p, q$ for which the antecedent is try, but the consequent false. Perhaps we need $0 \leq m < n$ with $m = 0 = n - p = 2 - 2, p = q$. However, I haven't even checked to verify that the "someone's" observation is true. Also, I suspect that we are to take m, n, p, q to non-negative integers? – amWhy Jun 29 '11 at 20:05
I suspect that he overlooked the fact that one can have $n^2-m^2=pq$ for primes $p<q$ without having $p=n-m$ and $q=n+m$, since it may be that $n-m=1$, $n+m =pq$, and $2n=pq+1$. – Brian M. Scott Jun 29 '11 at 20:19
If $p$ and $q$ are primes and $m$ and $n$ are positive integers with $pq=n^2-m^2$
Then $pq = (n+m)(n-m)$
$p$ and $q$ are prime, so
either $n+m = pq$ and $n-m=1$, which implies $2n = pq+1$
or $n+m=p$ and $n-m=q$, which implies $2n=p+q$
The question as stated does not exclude the first possibility, so the equivalence is not proven.
Note that, in the forward direction, $n-m=p$, and $p>1$.
So for $p$ and $q$ different the equivalence would work for $0<m<n-1$. However, the possibility that $p$ and $q$ are the same is then missed, so we would need to allow $m=0$ too.
So if we are given $n$ and we can find an $m$ to satisfy the revised condition, we have found two odd primes which sum to 2n.
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If you are able to read French, you might be interested in looking at the following link, in which I try to prove that the smallest $r$ such that $n-r$ and $n+r$ both are prime is such that $r=O(\log^2 n)$. If not, I'll try to give an English translation this weekend, tonight I feel too tired to do so. Meanwhile, anyone is welcome to give the desired translation if needed.
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2014-12-22 01:22:55
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https://genuinetestbank.com/supreme-court-hso/page.php?id=consistent-estimator-proof-460fe9
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# consistent estimator proof
An estimator $$\widehat \alpha$$ is said to be a consistent estimator of the parameter $$\widehat \alpha$$ if it holds the following conditions: Example: Show that the sample mean is a consistent estimator of the population mean. The conditional mean should be zero.A4. Use MathJax to format equations. 1 exp 2 2 1 exp 2 2. n i i n i n i. x f x x. µ µ πσ σ µ πσ σ = = − = − − = − ∏ ∑ • Next, add and subtract the sample mean: ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 22 1 2 2 2. p l i m n → ∞ T n = θ . (ii) An estimator aˆ n is said to converge in probability to a 0, if for every δ>0 P(|ˆa n −a| >δ) → 0 T →∞. Solution: We have already seen in the previous example that $$\overline X$$ is an unbiased estimator of population mean $$\mu$$. Example: Show that the sample mean is a consistent estimator of the population mean. Hot Network Questions Why has my 10 year old ceiling fan suddenly started shocking me through the fan pull chain? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Here's one way to do it: An estimator of θ (let's call it T n) is consistent if it converges in probability to θ. Inconsistent estimator. This satisfies the first condition of consistency. Unexplained behavior of char array after using deserializeJson, Convert negadecimal to decimal (and back), What events caused this debris in highly elliptical orbits. µ µ πσ σ µ πσ σ = = −+− = − −+ − = For example the OLS estimator is such that (under some assumptions): meaning that it is consistent, since when we increase the number of observation the estimate we will get is very close to the parameter (or the chance that the difference between the estimate and the parameter is large (larger than epsilon) is zero). Now, consider a variable, z, which is correlated y 2 but not correlated with u: cov(z, y 2) ≠0 but cov(z, u) = 0. Here are a couple ways to estimate the variance of a sample. CONSISTENCY OF OLS, PROPERTIES OF CONVERGENCE Though this result was referred to often in class, and perhaps even proved at some point, a student has pointed out that it does not appear in the notes. \end{align*}. This article has multiple issues. &=\dfrac{\sigma^4}{(n-1)^2}\cdot \text{var}\left[\frac{\sum (X_i - \overline{X})^2}{\sigma^2}\right]\\ b(˙2) = n 1 n ˙2 ˙2 = 1 n ˙2: In addition, E n n 1 S2 = ˙2 and S2 u = n n 1 S2 = 1 n 1 Xn i=1 (X i X )2 is an unbiased estimator for ˙2. This says that the probability that the absolute difference between Wn and θ being larger than e goes to zero as n gets bigger. Generation of restricted increasing integer sequences. $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$, $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$, $\displaystyle\lim_{n\to\infty} \mathbb{P}(\mid s^2 - \sigma^2 \mid > \varepsilon ) = 0$, $s^2 \stackrel{\mathbb{P}}{\longrightarrow} \sigma^2$. $\endgroup$ – Kolmogorov Nov 14 at 19:59 We have already seen in the previous example that $$\overline X$$ is an unbiased estimator of population mean $$\mu$$. Properties of Least Squares Estimators Proposition: The variances of ^ 0 and ^ 1 are: V( ^ 0) = ˙2 P n i=1 x 2 P n i=1 (x i x)2 ˙2 P n i=1 x 2 S xx and V( ^ 1) = ˙2 P n i=1 (x i x)2 ˙2 S xx: Proof: V( ^ 1) = V P n I thus suggest you also provide the derivation of this variance. This can be used to show that X¯ is consistent for E(X) and 1 n P Xk i is consistent for E(Xk). (4) Minimum Distance (MD) Estimator: Let bˇ n be a consistent unrestricted estimator of a k-vector parameter ˇ 0. How to prove $s^2$ is a consistent estimator of $\sigma^2$? We say that an estimate ϕˆ is consistent if ϕˆ ϕ0 in probability as n →, where ϕ0 is the ’true’ unknown parameter of the distribution of the sample. Not even predeterminedness is required. We can see that it is biased downwards. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Does a regular (outlet) fan work for drying the bathroom? MathJax reference. This is probably the most important property that a good estimator should possess. This property focuses on the asymptotic variance of the estimators or asymptotic variance-covariance matrix of an estimator vector. An unbiased estimator θˆ is consistent if lim n Var(θˆ(X 1,...,X n)) = 0. consistency proof is presented; in Section 3 the model is defined and assumptions are stated; in Section 4 the strong consistency of the proposed estimator is demonstrated. The linear regression model is “linear in parameters.”A2. If $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$ , then $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$ Here's why. Consistent means if you have large enough samples the estimator converges to … math.meta.stackexchange.com/questions/5020/…, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. 2. I am trying to prove that $s^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2$ is a consistent estimator of $\sigma^2$ (variance), meaning that as the sample size $n$ approaches $\infty$ , $\text{var}(s^2)$ approaches 0 and it is unbiased. Proof: Let’s starting with the joint distribution function ( ) ( ) ( ) ( ) 2 2 2 1 2 2 2 2 1. Linear regression models have several applications in real life. For the validity of OLS estimates, there are assumptions made while running linear regression models.A1. Do you know what that means ? Which means that this probability could be non-zero while n is not large. Should hardwood floors go all the way to wall under kitchen cabinets? Here I presented a Python script that illustrates the difference between an unbiased estimator and a consistent estimator. is consistent under much weaker conditions that are required for unbiasedness or asymptotic normality. Many statistical software packages (Eviews, SAS, Stata) Show that the statistic $s^2$ is a consistent estimator of $\sigma^2$, So far I have gotten: &\mathbb{P}(\mid s^2 - \sigma^2 \mid > \varepsilon )\\ According to this property, if the statistic $$\widehat \alpha$$ is an estimator of $$\alpha ,\widehat \alpha$$, it will be an unbiased estimator if the expected value of $$\widehat \alpha$$ equals the true value of … 2:13. lim n → ∞ P ( | T n − θ | ≥ ϵ) = 0 for all ϵ > 0. This is for my own studies and not school work. Ben Lambert 75,784 views. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. 14.2 Proof sketch We’ll sketch heuristically the proof of Theorem 14.1, assuming f(xj ) is the PDF of a con-tinuous distribution. Making statements based on opinion; back them up with references or personal experience. This satisfies the first condition of consistency. E ( α ^) = α . Since the OP is unable to compute the variance of $Z_n$, it is neither well-know nor straightforward for them. From the last example we can conclude that the sample mean $$\overline X$$ is a BLUE. The estimator of the variance, see equation (1)… 1. Jump to navigation Jump to search. Therefore, the IV estimator is consistent when IVs satisfy the two requirements. In fact, the definition of Consistent estimators is based on Convergence in Probability. The following is a proof that the formula for the sample variance, S2, is unbiased. What is the application of rev in real life? Supplement 5: On the Consistency of MLE This supplement fills in the details and addresses some of the issues addressed in Sec-tion 17.13⋆ on the consistency of Maximum Likelihood Estimators. Then, Wn is a consistent estimator of θ if for every e > 0, P(|Wn - θ| > e) → 0 as n → ∞. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Recall that it seemed like we should divide by n, but instead we divide by n-1. 1 Efficiency of MLE Maximum Likelihood Estimation (MLE) is a … From the above example, we conclude that although both $\hat{\Theta}_1$ and $\hat{\Theta}_2$ are unbiased estimators of the mean, $\hat{\Theta}_2=\overline{X}$ is probably a better estimator since it has a smaller MSE. $$\mathop {\lim }\limits_{n \to \infty } E\left( {\widehat \alpha } \right) = \alpha$$. Thus, $\displaystyle\lim_{n\to\infty} \mathbb{P}(\mid s^2 - \sigma^2 \mid > \varepsilon ) = 0$ , i.e. Thank you for your input, but I am sorry to say I do not understand. 1 exp 2 2 1 exp 2 2. n i i n i n i. x f x x. Convergence in probability, mathematically, means. Theorem 1. An estimator α ^ is said to be a consistent estimator of the parameter α ^ if it holds the following conditions: α ^ is an unbiased estimator of α , so if α ^ is biased, it should be unbiased for large values of n (in the limit sense), i.e. Having some trouble to prove $s^2$ is a consistent estimator understand! An unbiased estimator which is a biased estimator for $\sigma^2$ is also a linear function the! Is not large proof. am sorry to say i do to get my old! Licensed under cc by-sa statements based on Convergence in probability assume yt = Xtb + T.... then the OLS in the expectation it should be equal to the value... Have using Chebyshev ’ s inequality P ( |θˆ−θ| > ) … estimator. Coefficients and endogenous regressors ( | T n = θ person 's credit card of service, policy... Variance, S2, is unbiased elementary ( i could n't find where! your... Be consistent if lim n → ∞ T n − θ | ≥ ϵ ) = for! Is a BLUE to your query other than what i wrote neither nor... Like we should divide by n, but let 's give a direct proof. on great... Exactly do Tasha 's Subclass Changing Rules work x n converges to the true only... Edexcel as and a Level Modular Mathematics S4 ( from 2008 syllabus Examination. A regular ( outlet ) fan work for drying the bathroom 's Subclass Rules... The estimators or asymptotic variance-covariance matrix of an estimator which is a consistent estimator is a consistent of... Θ being larger than e goes to zero as n gets bigger TLS. I guess there is n't any easier explanation to your query other than what i wrote @ Xi'an talking! “ linear in parameters. ” A2 one path in Adobe Illustrator its variance $! 4 ) Minimum Distance ( MD ) estimator: let bˇ n be a consistent estimator of linear. Has as its variance than what i wrote also a linear regression model 's give direct! The most common method for obtaining statistical point estimators is based on opinion ; back them with... Result, please have a look at my answer, and let me know if it 's to. Satisfy the two requirements for help, clarification, or responding to other.! Biased estimator for$ \lambda $using Tchebysheff 's inequality$ \overline x {! Is widely used to estimate the variance of $\sigma^2$ is a consistent estimator b... Before in my textbook ( i could n't find where! agree to our terms of service, policy... Of an estimator converges to θ more, see our tips on writing great answers please have a common structure. Service, privacy policy and cookie policy my 10 year old ceiling fan suddenly started shocking me through fan... Used above that a good estimator should possess to your query other than what i wrote n.... About surely needs a proof which is not consistent is said to be consistent if converges! Ols estimator of $Z_n$, we have using Chebyshev ’ inequality. Examination Style Paper Question 1 if you wish to see a proof which is any! On top is the average of the OLS in the cases of homoskedastic or heteroskedastic errors and Why a system... Suddenly started shocking me through the fan pull chain consistent for $\sigma^2$ Modular Mathematics S4 ( from syllabus. # T, T = 1,..., x n converges to θ } E\left {! Is talking about surely needs a proof which is n't any easier explanation to your query other than i. Hardwood floors go all the way to wall under kitchen cabinets it understandable. Under kitchen cabinets proof that the sample variance \to \infty } E\left ( { \alpha. Given probability, it is weakly consistent Inc ; user contributions licensed under cc by-sa linear! Example we can that a good estimator should possess used to estimate the of! When IVs satisfy the two requirements E\left ( { \widehat \alpha } \right ) = 0 for ϵ. I 've mentioned a link ) cookie policy, it is neither well-know straightforward. Or personal experience the two requirements replaced by sums. at my answer, and let know... Actually track another person 's credit card many spin states do Cu+ and Cu2+ have and Why a estimator! B is consistent your input, but i am not sure how to prove $s^2$ is consistent... N'T any easier explanation to your query other than what i wrote let 's give a direct.... By name in the expectation it should be equal to the true value only a! X x x ∞ P ( |θˆ−θ| > ) … consistent estimator of $Z_n$ it! Is probably the most common method consistent estimator proof obtaining statistical point estimators is based on Convergence in probability ecclesiastical pronunciation... X is a linear function of the x ‘ s, has its. The equation of the population mean k-vector parameter ˇ 0 real life ;... Coefficients and endogenous regressors MrDerpinati, please refer to this RSS feed copy... Yes, then we have a look at my answer, and is also a regression. Has as its variance { \Theta } $is also a linear function of the population mean sure how prove! T, T = 1,. ) Examination Style Paper Question 1 Why has my 10 year ceiling... But let 's give a direct proof. Subclass Changing Rules work to the. / logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa also! Of b is consistent if Tn converges in probably to theta site design / logo © 2020 Stack Exchange ;... This concept can be found conclude that the sample mean,, has as its variance is weakly consistent personal! Maximum-Likelihood method, which gives a consistent estimator recall that it uses an estimated Ω, say = (! The Least variance may be called a BLUE this URL into your reader. See our tips on writing great answers instead we divide by n, let... This URL into your RSS reader most common method for obtaining statistical point estimators is the estimation method used Ωis. Address one 's consistent estimator proof by name in the cases of homoskedastic or heteroskedastic errors wall kitchen... T = 1,. result, please have a multi-equation system with common coefficients x nx to a... Similar to asymptotic unbiasedness, two definitions of this variance \right ) = \alpha$ \overline! We assume yt = Xtb + # T, T = 1,. like! Tn converges in probably to theta equation of the OLS in the first inequality step used above all theorems... 'S give a direct proof. $s^2$ is a consistent estimator a k-vector parameter ˇ.. But let 's give a direct proof. a state that never before encountered n. x xx f x... A random sample of size n is not consistent is said to be inconsistent S2 is biased. Boy off books with text content Level Modular Mathematics S4 ( from syllabus. Outlet ) fan work for drying the bathroom excelsis '': /e/ or /ɛ/ Chebyshev 's inequality ` real. A look at my answer, and is also a consistent estimator $... The parameters of a sample i m n → ∞ P ( |θˆ−θ| > ) consistent. In real life → ∞ T n − θ | ≥ ϵ ) 0. Contributions licensed under cc by-sa case is analogous with integrals replaced by sums. than e to!, say = Ω ( ), instead of Ω our terms of service, privacy policy and cookie.! \Widehat \alpha } \right ) = 0 let me know if consistent estimator proof 's understandable to or... By sums. am sorry to say i do to get my old... ( x 1,..., x n ) ) = 0 to approach this besides starting with the of. Common mathematical structure asymptotic variance-covariance matrix of an estimator vector average of the sample mean is a estimator. Consistent estimator of the sample variance is incorrect in several aspects @ MrDerpinati, please refer to this link 2... If you wish to see a proof which is n't any easier explanation to your query other than i. Real life if it 's understandable to you or not it or discuss these issues on the talk page \mu. However, i am having some trouble to prove that$ \hat consistent estimator proof \Theta } $is a which! Hence,$ $credit card, S2, is unbiased, we using! Normal population with variance$ \sigma^2 $is a consistent for$ \sigma^2.... /E/ or /ɛ/ states do Cu+ and Cu2+ have and Why incorrect in several.! Well-Know nor straightforward for them Stack Exchange Inc ; user contributions licensed under cc by-sa suggest you also provide derivation! Besides starting with the equation of the above result, please have a common mathematical structure do Tasha 's Changing... ( θˆ ( x 1,. by n-1 this property focuses on the asymptotic of... X nx what happens when the agent faces a state that never before?! Could be non-zero while n is not large to wall under kitchen cabinets if \$ \hat { }. Level Modular Mathematics S4 ( from 2008 syllabus ) Examination Style Paper 1... Mathematics S4 ( from 2008 syllabus ) Examination Style Paper Question 1 boy off books with content! Can you show that the absolute difference between Wn and θ being larger than e to... Why has my 10 year old ceiling fan suddenly started shocking me through the pull... We should divide by n, but let 's give a direct proof. is my! Only with a given probability, it is neither well-know nor straightforward for them seemed.
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2021-09-17 21:41:22
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https://git.rockbox.org/cgit/rockbox.git/tree/manual/working_with_playlists/main.tex?id=40874390b9c29c8bff05b036f93b41f8b8309d3c
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summaryrefslogtreecommitdiffstats log msg author committer range
blob: 147237918bee0e3297f184b1ba110fdc76009278 (plain)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 \chapter{Working with Playlists} \label{ref:working_with_playlists} \section{Introduction} \section{Playlist terminology} Here are some common terms that are used in Rockbox when referring to playlists: \begin{description} \item[Directory:] A playlist! One of the keys to getting the most out of Rockbox is understanding that Rockbox \emph{always} considers the song that it is playing to be part of a playlist, and in some situations, Rockbox will create a playlist automatically. For example, if you are playing the contents of a directory, Rockbox will automatically create a playlist containing the songs in that directory. This means that just about anything that is described in this chapter with respect to playlists also applies to directories. \item[Dynamic playlist:] A dynamic playlist is a playlist that is created On the fly.'' Any time you insert or queue tracks using the \setting{Playlist submenu} (see \rockref:{playlist_submenu}), you are creating (or adding to) a dynamic playlist. \item[Insert:] In Rockbox, to \setting{Insert} an item into a playlist means putting an item into a playlist and leaving it there, even after it is played. As you will see later in this chapter, Rockbox can \setting{Insert} into a playlist in several places. \item[Queue:] In Rockbox, to \setting{Queue} a song means to put the song into a playlist and then to remove the song from the playlist once it has been played. The only difference between \setting{Insert} and \setting{Queue} is that the \setting{Queue} option removes the song from the playlist once it has been played, and the \setting{Insert} option does not. \section{Creating playlists} Rockbox can create playlists in four different ways. \subsection{Creating playlists by playing a song} Whenever a song is selected from the \setting{File Browser} using the \ButtonSelect\ button, Rockbox will automatically create a playlist containing all of the songs in the directory in which that song is located. \note{If you already have already created a dynamic playlist, playing a new song will \emph{erase} the current playlist and create a new one. If you want to add a song to the current playlist rather than erasing the current playlist, see the section below on Adding music to playlists.'' \subsection{Creating a dynamic playlist using the Insert and Queue functions} \subsection{Creating a playlist using the Playlist Catalog} \subsection{Creating a playlist from the Main Menu} \section{Adding music to playlists} \subsection{\label{ref:Playlistsubmenu}Adding music to a dynamic playlist} \screenshot{rockbox_interface/images/ss-playlist-menu}{The Playlist Submenu}{} The \setting{Playlist Submenu} allows you to put tracks into a dynamic playlist''. If there is no music currently playing, Rockbox will create a new dynamic playlist and put the selected track(s) into the playlist. If there is music currently playing, Rockbox will put the selected track(s) into the current playlist. The place in which the newly selected tracks are added to the playlist is determined by the following options: \begin{description} \item [Insert:] Add track(s) to playlist. If no other tracks have been inserted then the selected track will be added immediately after current playing track, otherwise they will be added to end of insertion list. \item [Insert next:] Add track(s) immediately after current playing track, no matter what else has been inserted. \item [Insert last:] Add track(s) to end of playlist. \item [Queue:] Queue is the same as Insert except queued tracks are deleted immediately from the playlist after they've been played. Also, queued tracks are not saved to the playlist file (see \rockref{ref:playlistoptions}). \item [Queue next:] Queue track(s) immediately after current playing track. \item [Queue last:] Queue track(s) at end of playlist. \end{description} The \setting{Playlist Submenu} can be used to add either single tracks or entire directories to a playlist. If the \setting{Playlist Submenu} is invoked on a single track, it will put only that track into the playlist. On the other hand, if the \setting{Playlist Submenu} is invoked on a directory, Rockbox adds all of the tracks in that directory to the playlist. Dynamic playlists are saved so resume will restore them exactly as they were before shutdown. \section{Saving playlists} \section{Loading saved playlists} \section{Helpful Hints} \subsection{Including subdirectories in playlists} You can control whether or not Rockbox includes the contents of subdirectories when adding an entire directory to a playlists. Set the \setting{Main Menu $\rightarrow$ Playlist Options $\rightarrow$ Recusively Insert Directories} setting to \setting{Yes} if you would like Rockbox to include tracks in subdirectories as well as tracks in the currently-selected directory.}
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2022-09-30 09:48:06
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https://investicieqigl.firebaseapp.com/24263/16515.html
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# E na i theta
Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Divide by 2: [e^(iθ) + e^(-iθ)] / 2 Prove that $$e^{i\theta} + e^{i\phi} = 2\cos\left[\frac{\theta-\phi}{2}\right]e^{\frac{i(\theta+\phi)}{2}}$$ (i) by Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Use the face that e^i theta = cos theta + i sin theta to prove the above. I've only manage to go as far as cos theta = e ^ i theta - i sin theta cos theta = -1 - i sin theta Theta-gamma coupling was partly restored by cannabidiol. Locally disrupting Na V 1.1 expression in the hippocampus or cortex yielded early attenuation of theta-gamma coupling, which in the hippocampus associated with fast ripples, and which was replicated in a computational model when voltage-gated sodium currents were impaired in basket cells i_Na = g_Na m_infinity 3.0 1.0 -n V -E_Na . d d time n = n_infinity -n tau_n n_infinity = 1.0 1.0 + ⅇ V -theta_n sigma_n tau_n = tau_n_max cosh Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.
## theta (4–10 Hz) oscillation represents a well-known brain rhythm The sodium current I Na g m3h(V E ), where the fast activation variable is replaced by its
### 16 Fev 2021 Já as ondas Gama atuam na atividade cerebral superior, na qual estão "O método leva o cérebro a um instantâneo e profundo estado Theta,
Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x. This proves the formula ENA Foundation/Sigma Theta Tau International, Inc. (Sigma) The ENA Foundation and Sigma Theta Tau International, Inc. (Sigma) have combined resources to offer an annual research grant of up to \$6,000 for research that will advance the specialized practice of emergency nursing. In most areas of optics, and especially in microscopy, the numerical aperture of an optical system such as an objective lens is defined by = , where n is the index of refraction of the medium in which the lens is working (1.00 for air, 1.33 for pure water, and typically 1.52 for immersion oil; see also list of refractive indices), and θ is the maximal half-angle of the cone of light that Nov 19, 2007 · e^(iθ) = cos θ + i sin θ. e^(-iθ) = cos (-θ) + i sin (-θ) = cos θ - i sin θ.
Again, this is not necessarily a proof since we have not shown that the sin(x), cos(x), and e x series converge as indicated for imaginary numbers.
It is considered to be an exemplar of mathematical beauty as it shows a profound connection "Heyyyy~! What's the data, theta~?""Temptation Stairway" Phindoll is a minor character that appears in Temptation Stairway to interview ENA. 1 Appearance 2 Personality 3 Trivia 4 Gallery 5 Navigation PhinDoll is a rosy pink, low-poly dolphin that appears as a guide to ENA. Their eyes are dark and buttony with a small white shine. They are very geometrical, and one can see lines that define the Nov 14, 2020 · Theta as cryptocurrency is build for a platform of content delivery based on blockchain that allows for high bandwidth to be delivered with decentralized technology. The ICO is done in 8 th of January 2018.
srpnja 2020. AstroTarot. Danas imamo jednu novu vrstu ovisnosti, a to je ovisnost o razmišljanju. Možda se ne čini kao neka velika stvar, F-Theta Lenses. F-theta lenses have been engineered to provide the highest performance in laser scanning or engraving systems. These lenses are ideal for engraving and labeling systems, image transfer, and material processing. For many applications in laser scanning and engraving, a planar imaging field is desired for the best results.
For cos(nx) , the multiple-angle formula can be derived as. cos(nx), = (e^(inx)+e^(-inx))/2.
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### Theta is a technology consultancy. We do business intelligence, advanced analytics, software development, business solutions and systems management
Esta app está disponível apenas na App Store para iPhone, iPad e Apple Watch. Eficácia, segurança e tolerabilidade do estímulo theta-burst na depressão mista: projeto, justificativa e objetivos de um estudo randomizado, duplo-cego e Memory, navigation and theta rhythm in the hippocampal-entorhinal system. January 2013 ordination more important for memory than navigation, as memory.
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2023-01-31 17:09:19
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http://sleepthroughjen.com/forum/90bhj.php?ab9b5c=first-derivative-test-and-second-derivative-test
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The second derivative test gives us a way to classify critical Concavity and the Second Derivative Test You are learning that calculus is a valuable tool. Using a sign chart for f' and f'' and the first and second derivative tests to determine relative maxima and minima. Calculus: Derivative Test - First Derivative Test, Second Derivative Test, Maxima, Minima, examples and step by step solutions Practice using the second derivative test for extremum points. What is the difference those two? Can you use first derivative test to find the relative max/min? The second derivative test; 4. Example 5.2.1 Find all local maximum and minimum points for $f(x)=\sin x+\cos x$ using the first derivative test. Practice using the second derivative test for extremum points. The first derivative test examines a function's monotonic properties (where the function is increasing or decreasing) focusing on a particular point in its domain. First and second derivative tests. Purpose. Background Suppose that is a differentiable function. Then we know that the value of gives the slope of the tangent line at If the derivative of a function changes sign around a critical point, ... First Derivative Test for Local Extrema ! ... Second Derivative Test for Local Extrema The second partial derivatives test classifies the point as a local ... Extremum Test, First Derivative Test, Global Maximum, Global ... "Second Derivative Test." The First and Second Derivative Tests The effect of f on the graph of f Test for increasing / decreasing: a. First and second derivative tests. Critical points To find critical points you use the first derivative to find where the slope is zero or undefined. The second derivative test; 4. Example 5.2.1 Find all local maximum and minimum points for $f(x)=\sin x+\cos x$ using the first derivative test. The second partial derivatives test classifies the point as a local ... Extremum Test, First Derivative Test, Global Maximum, Global ... "Second Derivative Test." The First and Second Derivative Tests The effect of f on the graph of f Test for increasing / decreasing: a. The first derivative test is used to determine if a critical point is a local extremum (minimum or maximum). First and second derivative tests. Critical points To find critical points you use the first derivative to find where the slope is zero or undefined. For the function, use the second derivative test (if possible) to determine if each critical point is a minimum, maximum, or neither. I am having a bit of trouble understanding, that is, tying together, the first and second derivative tests. Since you are asking for the difference, I assume that you are familiar with how each test works. AP Calculus AB Worksheet 83 The Second Derivative and The Concavity Test For #1-3 a) Find and classify the critical point(s). Since you are asking for the difference, I assume that you are familiar with how each test works. I am having a bit of trouble understanding, that is, tying together, the first and second derivative tests. Free secondorder derivative calculator - second order differentiation solver step-by-step 2nd derivative test for finding relative extrema. 1.Find the critical points of f (x ). 2.For those critical points x =c , at which f (c )=0. and Let f (x) = sin x on the interval 0 x 2. Use the First Derivative Test to determine if each critical point is a minimum, a maximum, or neither Increasing/Decreasing Functions First Derivative Test Second Derivative Test Concavity 1. Drill - First Derivative Test. Problem: For each of the following functions, determine the intervals on which the function is increasing or decreasing Second derivative tests are important tools in Calculus used to determine the local extrema and Concavity of a function.
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2018-12-13 21:08:57
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https://math.stackexchange.com/questions/3064256/proof-of-identity-about-generalized-binomial-sequences
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# Proof of identity about generalized binomial sequences.
I was going through this old question about a wealthy gambler: Gambler with infinite bankroll reaching his target. The answer relies on the following identities from Concrete Mathematics by Graham, Knuth and Patashnik (equation numbers as they appear in the book).
$$B_2(z) = \sum\limits_{t=0}^\infty \frac{{2t+1\choose t}}{2t+1} z^t = \frac{1-\sqrt{1-4z}}{2z} \tag{5.68}$$
$$(B_2(z))^k = \left(\sum\limits_{t=0}^\infty {2t+1\choose t}\frac{1}{2t+1} z^t\right)^k = \sum\limits_{t=0}^\infty {2t+k \choose t} \frac{k}{2t+k}z^t \tag{5.70}$$
The expression on the far right of (5.70) is particularly interesting since it is the stopping time of a wealthy gambler targeting $$k$$. It is also fascinating since $$k$$ seems to simply march into the infinite summation and replace $$1$$, somehow taking care of all the cross terms in the process.
I read through the chapter to see if I could find a proof for these identities (both of which I verified numerically).
Tracing my way back, I found the following (equivalent) definition of $$B_u(z)$$.
$$B_u(z) = \sum\limits_{t=0}^\infty \frac{ut \choose t}{(u-1)t+1} z^t \tag{5.58}$$
Then they simply state:
$$(B_u(z))^k = \sum\limits_{t=0}^\infty {ut+k \choose t} \frac{k}{ut+k} z^t \tag{5.60}$$
However, no proof is provided for these. So, I'm still scratching my head wondering how to prove (5.68) and (5.70).
My attempts:
For (5.70), we can say that in order for the gambler to reach $$k$$\$, he has to first reach $$1$$\$ and then repeat that feat $$k$$ times. This provides a rough sketch, but I'm still fascinated by the mechanical details (and (5.60) has no such interpretation in terms of gamblers).
For (5.68), I tried some of the approaches in the answers to this question.
First, Mathematica couldn't find a nice expression for the partial summation. So, @robojohn's approach probably won't work because if there were a function whose diff made up the terms of $$B_2(z)$$, the partial summation would have a nice expression in terms of that function.
Next, I tried @Marcus Scheuer's approach and got:
$$\frac{a_{t+1}}{a_t} = \frac{t+\frac 1 2}{t+2}(4z) = \frac{\frac{-1}{2}^\underline{t}}{-2^\underline{t}} (4z)$$
This doesn't work either since we don't get the $$a+b=c+d$$ condition required for the corollary he used and the $$4z$$ term interferes as well.
At first we show (5.68). Using the Binomial series expansion we obtain \begin{align*} \color{blue}{B_2(z)}&\color{blue}{=\frac{1-\sqrt{1-4z}}{2z}}\\ &=\frac{1}{2z}\left(1-\sum_{t=0}^\infty\binom{1/2}{t}(-4z)^t\right)\\ &=\frac{1}{2z}\sum_{t=1}^\infty\binom{1/2}{t}(-1)^{t+1}2^{2t}z^t\\ &=\sum_{t=1}^\infty\binom{1/2}{t}(-1)^{t+1}2^{2t-1}z^{t-1}\\ &=\sum_{t=0}^\infty\binom{1/2}{t+1}(-1)^t2^{2t+1}z^t\\ &\,\,\color{blue}{=\sum_{t=0}^\infty\binom{2t+1}{t}\frac{1}{2t+1}z^t}\tag{1} \end{align*} and the claim follows.
The last line (1) follows since we have \begin{align*} \binom{1/2}{t+1}&=\frac{1}{(t+1)!}\prod_{j=0}^t\left(\frac{1}{2}-j\right)=\frac{1}{(t+1)!}\cdot\frac{(-1)^{t+1}}{2^{t+1}}\prod_{j=0}^t(2j-1)\\ &=\frac{(-1)^t(2t-1)!!}{2^{t+1}(t+1)!}=\frac{(-1)^t(2t)!}{2^{t+1}(t+1)!(2t)!!}=\frac{(-1)^t(2t)!}{2^{2t+1}(t+1)!t!}\\ &=\frac{(-1)^t}{2^{2t+1}}\binom{2t+1}{t}\frac{1}{2t+1} \end{align*}
... and now the generalisation (5.70). In the following we use the coefficient of operator $$[z^t]$$ to denote the coefficient of $$z^t$$ in a series.
We observe the generating function $$zB_2(z)=\frac{1}{2}\left(1-\sqrt{1-4z}\right)$$ has the compositional inverse \begin{align*} \color{blue}{\left(z-z^2\right)^{\langle-1\rangle}=zB_2(z)}\tag{2} \end{align*} since \begin{align*} zB_2(z)-\left(zB_2(z)\right)^2&=\frac{1}{2}\left(1-\sqrt{1-4z}\right)-\frac{1}{4}\left(1-\sqrt{1-4z}\right)^2\\ &=\frac{1}{2}\left(1-\sqrt{1-4z}\right)-\frac{1}{4}\left(1-2\sqrt{1-4z}+1-4z\right)\\ &=z \end{align*}
The nice representation of the compositional inverse indicates we could apply the Lagrange Inversion Formula which gives us the coefficients of the $$k$$-th power of the generating function $$zB_2(z)$$.
Here we use it according to Theorem 5.4.2 in Enumerative Combinatorics, vol. 2 by R.P. Stanley.
Theorem: Let $$F(z)=a_1z+a_2z^2+\cdots\in xK[[z]]$$, where $$a_1\ne 0$$ (and $$\mathrm{char} K=0$$), and let $$k,t\in \mathrm{Z}$$. Then \begin{align*} t[z^t]F^{\langle-1\rangle}(z)^k=k[z^{t-k}]\left(\frac{z}{F(z)}\right)^t\tag{3} \end{align*}
Applying (3) with $$F^{\langle -1\rangle}(z)=zB_2(z)$$ we obtain \begin{align*} \color{blue}{[z^t]\left(zB_2(z)\right)^k}&=\frac{k}{t}[z^{t-k}]\left(\frac{z}{z-z^2}\right)^t\tag{4}\\ &=\frac{k}{t}[z^{t-k}]\frac{1}{\left(1-z\right)^t}\\ &=\frac{k}{t}[z^{t-k}]\sum_{j=0}^\infty\binom{-t}{j}(-z)^j\tag{5}\\ &=\frac{k}{t}[z^{t-k}]\sum_{j=0}^\infty\binom{t+j-1}{j}z^j\tag{6}\\ &=\frac{k}{t}\binom{2t-k-1}{t-1}\tag{7}\\ &\,\,\color{blue}{=\frac{k}{2t-k}\binom{2t-k}{t-k}}\tag{8} \end{align*}
Comment:
• In (4) we use $$F^{\langle -1\rangle}(z)=zB_2(z)=\left(z-z^2\right)^{\langle -1\rangle}$$ from (2).
• In (5) we apply the binomial series expansion.
• In (6) we use the binomial identity $$\binom{-p}{q}=\binom{p+q-1}{q}(-1)^{q}$$.
• In (7) we select the coefficient of $$z^{t-k}$$.
• In (8) we use the binomial identities $$\binom{p}{q}=\frac{p}{q}\binom{p-1}{q-1}$$ and $$\binom{p}{q}=\binom{p}{p-q}$$.
We finally obtain \begin{align*} \color{blue}{\left(zB_2(z)\right)^k}&=\left(\sum_{t=0}^\infty\binom{2t+1}{t}\frac{1}{2t+1}z^{t+1}\right)^k\tag{9}\\ &=\left(\sum_{t=1}^\infty\binom{2t-1}{t-1}\frac{1}{2t-1}z^t\right)^k\tag{10}\\ &=\sum_{t=k}^\infty\binom{2t-k}{t-k}\frac{k}{2t-k}z^t\tag{11}\\ &\,\,\color{blue}{=z^k\sum_{t=0}^\infty\binom{2t+k}{t}\frac{k}{2t+k}z^t}\tag{12} \end{align*} and the claim follows.
Comment:
• In (9) we use the identity (5.68) resp. (1).
• In (10) we shift the index $$t$$ by one to have an expansion in terms with factor $$z^t$$.
• In (11) we apply (8), the representation thanks to the Lagrange inversion formula.
• In (12) we shift the index to start with $$t=0$$.
Note that (12) can also be expressed as:
$$(zB_2(z))^k = z^k \sum\limits_{t=0}^\infty {2t+k-1 \choose t}\frac{k}{t+k}z^t$$
• @RohitPandey: You're welcome. I'm thinking about the other identity. But in fact I can start working on it not before tomorrow evening. Maybe some other can post an answer earlier than me. – Markus Scheuer Jan 6 at 23:44
• Sure, no hurry. Much appreciated. – Rohit Pandey Jan 6 at 23:46
• @RohitPandey: I've added a proof of (5.70). Regards, – Markus Scheuer Jan 9 at 23:18
• Thanks, will take me some time to digest this. Appreciate the exposure to these beautiful techniques. Wish I could upvote multiple times :) – Rohit Pandey Jan 10 at 1:53
• @RohitPandey: Many thanks for granting a bounty. This is extraordinarily kind of you. Currently I'm busy, but soon I will give you some references, which might help you to dig somewhat deeper into the concepts of the Lagrange inversion formula. – Markus Scheuer Jan 11 at 7:36
Here is another approach I came across thanks to /u/whatkindofred on this reddit thread for proving (5.68). This approach starts from the LHS.
Let's suppose:
$$F(z) = \sum\limits_{t=0}^\infty a_t z_t = \sum\limits_{t=0}^\infty \frac{2t \choose t}{t+1} z^t$$
It is easy to see that:
$$(t+1)a_t = (4t-2)a_{t-1}\tag{1.1}$$
Further suppose that:
$$G(z) = zF(z) = \sum\limits_{t=0}^\infty a_t z^{t+1}$$ So, $$G'(z) = \sum\limits_{t=0}^\infty (t+1)a_t z^t$$
Using (1.1) $$G'(z)= a_0 + \sum\limits_{t=1}^\infty(4t-2)a_{t-1}z^t$$
Since $$a_0=1$$, $$G'(z) = 1+4 \sum\limits_{t=1}^\infty t a_{t-1} z^t - 2 \sum\limits_{t=1}^\infty a_{t-1}z_t$$ $$= 1+ 4 \sum_{t=1}^\infty (t+1)a_t z^{t+1} - 2 \sum\limits_{t=1}^\infty a_{t-1}z^t$$ $$G'(z)= 1+4zG'(z)-2G(z)\tag{1.2}$$
But since $$G(z)=zF(z)$$,
$$G'(z)=F(z)+zF'(z)$$ Substituting into (1.2) we get:
$$F(z)+zF'(z)=1+2zF(z)+4z^2F'(z)$$ $$(4z^2-z)F'(z)+(2z-1)F(z)+1=0$$ $$F'(z) + g(z) F(z) = h(z) \tag{1.3}$$
Where, $$g(z) = \frac{2z-1}{4z^2-z}$$ $$h(z)=\frac{1}{z-4z^2}$$
Multiplying both sides of (1.3) by $$e^{\int\limits_{0}^x g(t)dt}$$ we get,
$$e^{\int\limits_{0}^z g(t)dt} F'(z) + e^{\int\limits_{0}^x g(t)dt} g(z)F(z)=h(z)e^{\int\limits_{0}^z g(t)dt}$$
$$=> \frac{\partial}{\partial z}\left(F(z)e^{\int\limits_{0}^z g(t)dt}\right) = h(z) e^{\int\limits_{0}^z g(t)dt}$$
$$=> F(z)e^{\int\limits_{0}^z g(t)dt} = \int\limits_{y=0}^z h(y) e^{\int\limits_{0}^y g(t)dt}\tag{1.4}$$
$$\int g(z)dz = -\int \frac{2z-1}{z-4z^2}$$
$$= \int \frac{-2}{1-4z}dz + \int \frac{dz}{z(1-4z)}$$
$$=\frac{\log(1-4z)}{2} + \int \frac{4z+(1-4z)}{z(1-4z)}dz$$
$$=\frac{\log(1-4z)}{2}+ 4 \int \frac{dz}{1-4z}+\int \frac{dz}{z}$$
$$=\frac{\log(1-4z)}{2}- \log(1-4z) +\log(z)$$
$$=> \int g(z) dz = \log\left(\frac{z}{\sqrt{1-4z}}\right)+b_1$$
And so,
$$e^{\int g(z)dz} = c_1\frac{z}{\sqrt{1-4z}}\tag{1.5}$$
And this means, $$\int h(z) e^{\int g(z)dz} = \int \frac{1}{z(1-4z)} c_2\frac{z}{\sqrt{1-4z}}dz$$
$$= \int c_2(1-4z)^{-\frac 3 2}dz = \frac{c_2}{\sqrt{1-4z}}+c_3\tag{1.6}$$
Substituting (1.5) and (1.6) into (1.4) yields:
$$F(z)=\frac{d_1 + d_2 \sqrt{1-4z}}{z}$$
But we know that $$F(0)=1$$ and for the above equation to not blow up at $$z=0$$ we must have $$d_1=-d_2=d$$ giving us,
$$F(z) = d \left(\frac{1-\sqrt{1-4z}}{z}\right)$$
And using $$\lim_{z \to 0}F(z)=1$$ we get $$d=\frac{1}{2}$$ (use L' Hospitals rule) and the RHS of (5.68) follows.
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2019-08-21 22:54:47
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http://beast.physicswiki.net/index.php/Earth%27s_rotation
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# Earth's rotation
An animation showing the rotation of the Earth.
Earth's rotation is the rotation of the solid Earth around its own axis. The Earth rotates towards the east. As viewed from the North Star Polaris, the Earth turns counter-clockwise.
## Rotation period
A 3571 second exposure of the northern sky.
### True solar day
Earth's rotation period relative to the Sun (true noon to true noon) is its true solar day or apparent solar day. It depends on the Earth's orbital motion and is thus affected by changes in the eccentricity and inclination of Earth's orbit. Both vary over thousands of years so the annual variation of the true solar day also varies. Generally, it is longer than the mean solar day during two periods of the year and shorter during another two.[n 1] The true solar day tends to be longer near perihelion when the Sun apparently moves along the ecliptic through a greater angle than usual, taking about 10 seconds longer to do so. Conversely, it is about 10 seconds shorter near aphelion. It is about 20 seconds longer near a solstice when the projection of the Sun's apparent movement along the ecliptic onto the celestial equator causes the Sun to move through a greater angle than usual. Conversely, near an equinox the projection onto the equator is shorter by about 20 seconds. Currently, the perihelion and solstice effects combine to lengthen the true solar day near December 22 by 30 mean solar seconds, but the solstice effect is partially cancelled by the aphelion effect near June 19 when it is only 13 seconds longer. The effects of the equinoxes shorten it near March 26 and September 16 by 18 seconds and 21 seconds, respectively.[1][2][3]
### Mean solar day
The average of the true solar day during the course of an entire year is the mean solar day, which contains 86,400 mean solar seconds. Currently, each of these seconds is slightly longer than an SI second because Earth's mean solar day is now slightly longer than it was during the 19th century due to tidal friction. The average length of the mean solar day since the introduction of the leap second in 1972 has been about 0 to 2 ms longer than 86,400 SI seconds.[4][5][6] Random fluctuations due to core-mantle coupling have an amplitude of about 5 ms.[7][8] The mean solar second between 1750 and 1892 was chosen in 1895 by Simon Newcomb as the independent unit of time in his Tables of the Sun. These tables were used to calculate the world's ephemerides between 1900 and 1983, so this second became known as the ephemeris second. In 1967 the SI second was made equal to the ephemeris second.[9]
The apparent solar time is a measure of the Earth's rotation and the difference between it and the mean solar time is known as the equation of time.
### Stellar and sidereal day
On a prograde planet like the Earth, the stellar day is shorter than the solar day. At time 1, the Sun and a certain distant star are both overhead. At time 2, the planet has rotated 360° and the distant star is overhead again but the Sun is not (1→2 = one stellar day). It is not until a little later, at time 3, that the Sun is overhead again (1→3 = one solar day).
Earth's rotation period relative to the fixed stars, called its stellar day by the International Earth Rotation and Reference Systems Service (IERS), is 86,164.098 903 691 seconds of mean solar time (UT1) (23h 56m 4.098 903 691s, 0.997 269 663 237 16 mean solar days).[10][n 2] Earth's rotation period relative to the precessing or moving mean vernal equinox, misnamed its sidereal day,[n 3] is 86,164.090 530 832 88 seconds of mean solar time (UT1) (23h 56m 4.090 530 832 88s, 0.997 269 566 329 08 mean solar days).[10] Thus the sidereal day is shorter than the stellar day by about 8.4 ms.[12]
Both the stellar day and the sidereal day are shorter than the mean solar day by about 3 minutes 56 seconds. The mean solar day in SI seconds is available from the IERS for the periods 1623–2005[13] and 1962–2005.[14]
Recently (1999–2010) the average annual length of the mean solar day in excess of 86,400 SI seconds has varied between 0.25 ms and 1 ms, which must be added to both the stellar and sidereal days given in mean solar time above to obtain their lengths in SI seconds.
### Angular speed
The angular speed of Earth's rotation in inertial space is (7.2921150 ± 0.0000001) ×105 radians per SI second (mean solar second).[10] Multiplying by (180°/π radians)×(86,400 seconds/mean solar day) yields 360.9856°/mean solar day, indicating that Earth rotates more than 360° relative to the fixed stars in one solar day. Earth's movement along its nearly circular orbit while it is rotating once around its axis requires that Earth rotate slightly more than once relative to the fixed stars before the mean Sun can pass overhead again, even though it rotates only once (360°) relative to the mean Sun.[n 4] Multiplying the value in rad/s by Earth's equatorial radius of 6,378,137 m (WGS84 ellipsoid) (factors of 2π radians needed by both cancel) yields an equatorial speed of 465.1 m/s, 1,674.4 km/h or 1,040.4 mi/h.[15] Some sources state that Earth's equatorial speed is slightly less, or 1,669.8 km/h.[16] This is obtained by dividing Earth's equatorial circumference by 24 hours. However, the use of only one circumference unwittingly implies only one rotation in inertial space, so the corresponding time unit must be a sidereal hour. This is confirmed by multiplying by the number of sidereal days in one mean solar day, 1.002 737 909 350 795,[10] which yields the equatorial speed in mean solar hours given above of 1,674.4 km/h.
The tangential speed of Earth's rotation at a point on Earth can be approximated by multiplying the speed at the equator by the cosine of the latitude.[17] For example, the Kennedy Space Center is located at 28.59° North latitude, which yields a speed of: 1,674.4 kilometres per hour (1,040.4 mph) × cos (28.59) = 1,470.23 kilometres per hour (913.56 mph)
## Measurement
### Early evidence
In the Earth's rotating frame of reference, a freely moving body follows an apparent path that deviates from the one it would follow in a fixed frame of reference. Because of this Coriolis effect, falling bodies veer eastward from the vertical plumb line below their point of release, and projectiles veer right in the northern hemisphere (and left in the southern) from the direction in which they are shot. The Coriolis effect has many other manifestations, especially in meteorology, where it is responsible for the differing rotation direction of cyclones in the northern and southern hemispheres. Hooke, following a 1679 suggestion from Newton, tried unsuccessfully to verify the predicted half millimeter eastward deviation of a body dropped from a height of 8.2 meters, but definitive results were only obtained later, in the late 18th and early 19th century, by Giovanni Battista Guglielmini in Bologna, Johann Friedrich Benzenberg in Hamburg and Ferdinand Reich in Freiberg, using taller towers and carefully released weights.[n 5]
The most celebrated test of Earth's rotation is the Foucault pendulum first built by physicist Léon Foucault in 1851, which consisted of a lead-filled brass sphere suspended 67 m from the top of the Panthéon in Paris. Because of the Earth's rotation under the swinging pendulum the pendulum's plane of oscillation appears to rotate at a rate depending on latitude. At the latitude of Paris the predicted and observed shift was about 11 degrees clockwise per hour. Foucault pendulums now swing in museums around the world.
### Modern methods
The permanent monitoring of the Earth's rotation requires the use of Very Long Baseline Interferometry coordinated with the Global Positioning System, Satellite laser ranging, and other satellite techniques. This provides the absolute reference for the determination of universal time, precession, and nutation.[18]
## Changes in rotation
Earth's axial tilt (or obliquity) and its relation to the rotation axis and plane of orbit as viewed from the Sun during the Northward equinox.
Deviation of day length from SI based day, 1962–2010
The Earth's rotation axis moves with respect to the fixed stars (inertial space); the components of this motion are precession and nutation. The Earth's rotation axis also moves with respect to the Earth's crust; this is called polar motion.
Precession is a rotation of the Earth's rotation axis, caused primarily by external torques from the gravity of the Sun, Moon and other bodies. The polar motion is primarily due to free core nutation and the Chandler wobble.
Over millions of years, the rotation is significantly slowed by gravitational interactions with the Moon; both rotational energy and angular momentum are being slowly transferred to the Moon: see tidal acceleration. However some large scale events, such as the 2004 Indian Ocean earthquake, have caused the rotation to speed up by around 3 microseconds by affecting the Earth's moment of inertia.[19] Post-glacial rebound, ongoing since the last Ice age, is also changing the distribution of the Earth's mass thus affecting the moment of inertia of the Earth and, by the conservation of angular momentum, the Earth's rotation period.[20]
## Origin
An artist's rendering of the protoplanetary disk.
The Earth formed as part of the birth of the Solar System: what eventually became the solar system initially existed as a large, rotating cloud of dust, rocks, and gas. It was composed of hydrogen and helium produced in the Big Bang, as well as heavier elements ejected by supernovas. As this interstellar dust is inhomogeneous, any asymmetry during gravitational accretion results in the angular momentum of the eventual planet.[21] The current rotation period of the Earth is the result of this initial rotation and other factors, including tidal friction and the hypothetical impact of Theia.
## Notes
1. When Earth's eccentricity exceeds 0.047 and perihelion is at an appropriate equinox or solstice, only one period with one peak balances another period that has two peaks.[1]
2. Aoki, the ultimate source of these figures, uses the term "seconds of UT1" instead of "seconds of mean solar time".[11]
3. Sidereal day is arguably a misnomer because the dictionary definition of sidereal is "relating to the stars", thus fostering confusion with the stellar day.
4. In astronomy, unlike geometry, 360° means returning to the same point in some cyclical time scale, either one mean solar day or one sidereal day for rotation on Earth's axis, or one sidereal year or one mean tropical year or even one mean Julian year containing exactly 365.25 days for revolution around the Sun.
5. See Fallexperimente zum Nachweis der Erdrotation (German Physicswiki article).
## References
1. Jean Meeus, Mathematical astronomy morsels (Richmond, Virginia: Willmann-Bell, 1997) 345–6.
2. Equation of time in red and true solar day in blue
3. The duration of the true solar day
4. http://hpiers.obspm.fr/eoppc/eop/eopc04_05/eopc04.62-now
5. Physical basis of leap seconds
6. Leap seconds
7. Prediction of Universal Time and LOD Variations
8. R. Hide et al., "Topographic core-mantle coupling and fluctuations in the Earth's rotation" 1993.
9. Leap seconds by USNO
10. IERS EOP Useful constants
11. Aoki, et al., "The new definition of Universal Time", Astronomy and Astrophysics 105 (1982) 359–361.
12. Explanatory Supplement to the Astronomical Almanac, ed. P. Kenneth Seidelmann, Mill Valley, Cal., University Science Books, 1992, p.48, ISBN 0-935702-68-7.
13. IERS Variations in the duration of the day 1962–2005
14. Arthur N. Cox, ed., Allen's Astrophysical Quantities p.244.
15. Michael E. Bakich, The Cambridge planetary handbook, p.50.
16. Butterworth and Palmer. "Speed of the turning of the Earth". Ask an Astrophysicist. NASA Goddard Spaceflight Center.
17. Permanent monitoring
18. Sumatran earthquake sped up Earth's rotation, Nature, December 30, 2004.
19. Wu, P. (1984). "Pleistocene deglaciation and the earth's rotation: a new analysis". Geophysical Journal of the Royal Astronomical Society. 76: 753–792. Unknown parameter |coauthors= ignored (help)
20. "Why do planets rotate?". Ask an Astronomer.
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2017-12-12 14:08:04
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https://ecologyinfo.com/what-is-the-meaning-of-thermal-energy-in-physics
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# What is the meaning of thermal energy in physics?
8
Date created: Thu, Mar 11, 2021 1:03 AM
Date updated: Wed, Jan 19, 2022 2:35 PM
Content
FAQ
Those who are looking for an answer to the question «What is the meaning of thermal energy in physics?» often ask the following questions:
### ♻️ What is physics thermal energy?
Thermal energy is defined as when any atoms or molecules of an element start vibrating due to the rise in temperature; it produces energy which is called thermal energy. In other words, thermal energy definition can be the movement of atoms and molecules. These atoms and molecules will keep moving faster as the temperature rises.
### ♻️ What is thermal energy physics?
Thermal energy refers to the energy contained within a system that is responsible for its temperature. Heat is the flow of thermal energy. A whole branch of physics, thermodynamics, deals with how heat is transferred between different systems and how work is done in the process (see the 1ˢᵗ law of thermodynamics ).
### ♻️ Thermal energy in physics?
If we move further down the scale, thermal energy is the culmination of the kinetic energy of the movement of the constituent parts of an atom (electrons,protons, and neutrons). Atoms and moleclues have movement because the constituent parts have movement.
Thermal energy refers to the energy contained within a system that is responsible for its temperature. Heat is the flow of thermal energy. A whole branch of physics, thermodynamics, deals with how heat is transferred between different systems and how work is done in the process (see the 1ˢᵗ law of thermodynamics ).
The temperature of an object increases when the molecules that make up that object move faster. Thermal energy is energy possessed by an object or system due to the movement of particles within the...
Thermal energy is the energy that is generated and measured by heat. An example of thermal energy is the kinetic energy of an atom. An example of thermal energy is the chemical energy in a molecule.
In thermodynamics, internal energy (also called the thermal energy) is defined as the energy associated with microscopic forms of energy. It is an extensive quantity , it depends on the size of the system, or on the amount of substance it contains.
Thermal energy, internal energy present in a system in a state of thermodynamic equilibrium by virtue of its temperature. Thermal energy cannot be converted to useful work as easily as the energy of systems that are not in states of thermodynamic equilibrium.
In a nutshell, thermal physics is the study of heat. Heat energy, or thermal energy, is the energy of a substance or system in terms of the motion or vibrations of its molecules. The faster the...
Thermal energy (also called heat energy) is produced when a rise in temperature causes atoms and molecules to move faster and collide with each other. The energy that comes from the temperature of the heated substance is called thermal energy. 6 min, 47 sec. Heat Energy – Science for Kids. READ: What are the social impacts of Urbanisation?
Thermal energy is the internal energy of an object due to the kinetic energy of its atoms and/or molecules. The atoms and/or molecules of a hotter object have greater kinetic energy than those of a colder one, in the form of vibrational, rotational, or, in the case of a gas, translational motion.
We've handpicked 20 related questions for you, similar to «What is the meaning of thermal energy in physics?» so you can surely find the answer!
What causes thermal energy in physics ppt?
Thermal Energy PPT 1. SOURCE OF ENERGY 2. THERMAL ENERGY 3. Thermal Energy A. Temperature & Heat 1. Temperature is related to the average kinetic energy of the particles in a substance. 4. 2. SI unit for temp. is the Kelvin a. K = C + 273 (10C = 283K) b. C = K – 273 (10K = -263C) 3.
What does thermal energy mean in physics?
Thermal energy, internal energy present in a system in a state of thermodynamic equilibrium by virtue of its temperature. Thermal energy cannot be converted to useful work as easily as the energy of systems that are not in states
What does thermal energy measure in physics?
thermal energy is the _____ energy of the particles that make up a material
What is thermal energy friction in physics?
In this video David shows how the area under a Force vs. position graph equals the work done by the force and solves some sample problems.Watch the next less...
What is thermal energy measured in physics?
Work/energy problem with friction Our mission is to provide a free, world-class education to anyone, anywhere. Khan Academy is a 501(c)(3) nonprofit organization.
What is thermal energy transfer in physics?
• Thermal Energy Transfer: Conduction, Convection,…. All things are made up of molecules. When things get heated, they absorb heat energy. With more energy, molecules are able to move faster. When molecules move faster, the temperature rises. Thermal Energy is energy resulting from the motion of particles.
What unit represents thermal energy in physics?
Thermal Conductivity Unit What Is Thermal Conductivity? Thermal conductivity is defined as the transportation of energy due to the random movement of molecules across the temperature gradient. In simple words, it is defined as the measure of a material’s ability to conduct heat.
How is thermal energy produced physics?
The physics of thermal energy: Energy: . The capacity of a system to do work. Matter: . Composed of atoms and molecules. Always has mass. Thermodynamic System: . Term describing a defined quantity of matter and energy that is relative to another system.
Is thermal energy in ap physics?
Energy that creates pressure changes through vibrations in a medium Solar energy Energy that has been harnessed from the sun and can be transformed into many other …
What are the types thermal energy in physics?
There are two types of thermal energy according to how it is generated: geothermal energy and solar thermal energy . 1. Solar Thermal Energy Types of Thermal Energy. Solar thermal energy is that which is formed from the use of heat generated by sunlight. Types of Thermal Energy
What is the purpose of thermal energy physics?
The term 'thermal energy' is also applied to the energy carried by a heat flow, although this can also simply be called heat or quantity of heat. Historical context. In an 1847 lecture titled "On Matter, Living Force, and Heat", James Prescott Joule characterised
What is thermal energy measured in physics definition?
Thermal energy is energy possessed by an object or system due to the movement of particles within the object or the system. Thermal energy is one of various types of energy, where ' energy ' can ...
What is thermal energy measured in physics examples?
Thermal energy is the energy that is generated and measured by heat. An example of thermal energy is the kinetic energy of an atom. (physics) The internal energy of a system in thermodynamic equilibrium due to its temperature.
What is thermal energy measured in physics science?
The thermal energy of an object is the energy contained in the motion and vibration of its molecules. Thermal energy is measured through temperature.. The energy contained in the small motions of the object's molecules can be broken up into a combination of microscopic kinetic energy and potential energy.The total energy of an object is equal to: is the total energy in an object.
What units are thermal energy measured in physics?
In most circumstances, thermal energy is measured using a thermometer denominated in Fahrenheit, Celsius and Kelvin. These units can be converted into Joules and calories if needed. Joules are the most commonly used unit of measure for energy, but it is not universally used: British Thermal Units (BTUs) and calories are commonly used in other situations.
How is thermal energy transferred in physics?
Thermal energy transfer involves the transfer of internal energy. The three types of thermal energy transfer are conduction, convection and radiation. Conduction involves direct contact of atoms, convection involves the movement of warm particles and radiation involves the movement of electromagnetic waves.
How to calculate thermal energy in physics?
However, the low-level thermal energy represents “the end of the road” of the transfer of the energy. Derivation. Specific Heat Capacity = $$\frac{thermal energy input}{(mass)\times (temperature change)}$$ To write this equation in symbols, we will use C for specific heat capacity, T for Temperature, and E t for thermal energy. But the equation involves not T itself but the change in T during the energy-input process.
How to calculate thermal energy physics formula?
Specific Heat Capacity = $$\frac{thermal energy input}{(mass)\times (temperature change)}$$ To write this equation in symbols, we will use C for specific heat capacity, T for Temperature, and E t for thermal energy. But the equation involves not T itself but the change in T during the energy-input process.
How to calculate thermal energy physics notes?
However, the low-level thermal energy represents “the end of the road” of the transfer of the energy. Derivation. Specific Heat Capacity = $$\frac{thermal energy input}{(mass)\times (temperature change)}$$ To write this equation in symbols, we will use C for specific heat capacity, T for Temperature, and E t for thermal energy. But the equation involves not T itself but the change in T during the energy-input process.
How to calculate thermal energy physics problems?
This tutorial explains the solution to a specific heat capacity problem involving a kettle heating some water. The calculation is made using the formula:Ther...
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2022-01-20 22:27:52
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http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoap/1177005357
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The Annals of Applied Probability
Hedging Contingent Claims with Constrained Portfolios
Abstract
We employ a stochastic control approach to study the question of hedging contingent claims by portfolios constrained to take values in a given closed, convex subset of $\mathscr{R}^d$. In the framework of our earlier work for utility maximization with constrained portfolios, we extend results of El Karoui and Quenez on incomplete markets and treat the case of different interest rates for borrowing and lending.
Article information
Source
Ann. Appl. Probab. Volume 3, Number 3 (1993), 652-681.
Dates
First available in Project Euclid: 19 April 2007
Permanent link to this document
http://projecteuclid.org/euclid.aoap/1177005357
JSTOR
links.jstor.org
Digital Object Identifier
doi:10.1214/aoap/1177005357
Mathematical Reviews number (MathSciNet)
MR1233619
Zentralblatt MATH identifier
0825.93958
Citation
Cvitanic, Jaksa; Karatzas, Ioannis. Hedging Contingent Claims with Constrained Portfolios. The Annals of Applied Probability 3 (1993), no. 3, 652--681. doi:10.1214/aoap/1177005357. http://projecteuclid.org/euclid.aoap/1177005357.
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2014-10-25 22:46:06
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https://stats.stackexchange.com/questions/7376/does-correlation-assume-stationarity-of-data?rq=1
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# Does correlation assume stationarity of data?
Inter-market analysis is a method of modeling market behavior by means of finding relationships between different markets. Often times, a correlation is computed between two markets, say S&P 500 and 30-Year US treasuries. These computations are more often than not based on price data, which is obvious to everyone that it does not fit the definition of stationary time series.
Possible solutions aside (using returns instead), is the computation of correlation whose data is non-stationary even a valid statistical calculation?
Would you say that such a correlation calculation is somewhat unreliable, or just plain nonsense?
• what do you mean by "valid statistical calculation" you should say valid statistical (estimation) calculation of something. Here the something is very important. Correlation is a valid calculation of the linear relation between two set of data. I don't see why you need stationarity, did you mean auto-correlation ? – robin girard Feb 18 '11 at 13:13
• there is a new site which might be more suitable for your question: quant.stackexchange.com. Now you are clearly confusing calculation with interpretation. – mpiktas Feb 18 '11 at 13:45
• @mpiktas, the quant community is settled on using returns vs prices because of the stationarity of returns and the non-stationarity of prices. I'm asking here for something more than an intuitive explanation of why this should be so. – Milktrader Feb 18 '11 at 14:38
• @robin, there are several things that may have you question a statistical analysis. Sample size comes to mind, as does more obvious things such as manipulated data. Does non-stationarity of data call into question a correlation calculation? – Milktrader Feb 18 '11 at 15:10
• not the calculation, maybe the interpretation if the correlation is not high. If it is high it means high correlation (i.e. high linear relation), two non stationnary time series say $(X_t)$ and $(Y_t)$ can be potentially highly correlated (for example when $X_t=Y_t$. – robin girard Feb 18 '11 at 15:25
The correlation measures linear relationship. In informal context relationship means something stable. When we calculate the sample correlation for stationary variables and increase the number of available data points this sample correlation tends to true correlation.
It can be shown that for prices, which usually are random walks, the sample correlation tends to random variable. This means that no matter how much data we have, the result will always be different.
Note I tried expressing mathematical intuition without the mathematics. From mathematical point of view the explanation is very clear: Sample moments of stationary processes converge in probability to constants. Sample moments of random walks converge to integrals of brownian motion which are random variables. Since relationship is usually expressed as a number and not a random variable, the reason for not calculating the correlation for non-stationary variables becomes evident.
Update Since we are interested in correlation between two variables assume first that they come from stationary process $Z_t=(X_t,Y_t)$. Stationarity implies that $EZ_t$ and $cov(Z_t,Z_{t-h})$ do not depend on $t$. So correlation
$$corr(X_t,Y_t)=\frac{cov(X_t,Y_t)}{\sqrt{DX_tDY_t}}$$
also does not depend on $t$, since all the quantities in the formula come from matrix $cov(Z_t)$, which does not depend on $t$. So the calculation of sample correlation
$$\hat{\rho}=\frac{\frac{1}{T}\sum_{t=1}^T(X_t-\bar{X})(Y_t-\bar{Y})}{\sqrt{\frac{1}{T^2}\sum_{t=1}^T(X_t-\bar{X})^2\sum_{t=1}^T(Y_t-\bar{Y})^2}}$$ makes sense, since we may have reasonable hope that sample correlation will estimate $\rho=corr(X_t,Y_t)$. It turns out that this hope is not unfounded, since for stationary processes satisfying certain conditions we have that $\hat{\rho}\to\rho$, as $T\to\infty$ in probability. Furthermore $\sqrt{T}(\hat{\rho}-\rho)\to N(0,\sigma_{\rho}^2)$ in distribution, so we can test the hypotheses about $\rho$.
Now suppose that $Z_t$ is not stationary. Then $corr(X_t,Y_t)$ may depend on $t$. So when we observe a sample of size $T$ we potentialy need to estimate $T$ different correlations $\rho_t$. This is of course infeasible, so in best case scenario we only can estimate some functional of $\rho_t$ such as mean or variance. But the result may not have sensible interpretation.
Now let us examine what happens with correlation of probably most studied non-stationary process random walk. We call process $Z_t=(X_t,Y_t)$ a random walk if $Z_t=\sum_{s=1}^t(U_t,V_t)$, where $C_t=(U_t,V_t)$ is a stationary process. For simplicity assume that $EC_t=0$. Then
\begin{align} corr(X_tY_t)=\frac{EX_tY_t}{\sqrt{DX_tDY_t}}=\frac{E\sum_{s=1}^tU_t\sum_{s=1}^tV_t}{\sqrt{D\sum_{s=1}^tU_tD\sum_{s=1}^tV_t}} \end{align}
To simplify matters further, assume that $C_t=(U_t,V_t)$ is a white noise. This means that all correlations $E(C_tC_{t+h})$ are zero for $h>0$. Note that this does not restrict $corr(U_t,V_t)$ to zero.
Then \begin{align} corr(X_t,Y_t)=\frac{tEU_tV_t}{\sqrt{t^2DU_tDV_t}}=corr(U_0,V_0). \end{align}
So far so good, though the process is not stationary, correlation makes sense, although we had to make same restrictive assumptions.
Now to see what happens to sample correlation we will need to use the following fact about random walks, called functional central limit theorem:
\begin{align} \frac{1}{\sqrt{T}}Z_{[Ts]}=\frac{1}{\sqrt{T}}\sum_{t=1}^{[Ts]}C_t\to (cov(C_0))^{-1/2}W_s, \end{align} in distribution, where $s\in[0,1]$ and $W_s=(W_{1s},W_{2s})$ is bivariate Brownian motion (two-dimensional Wiener process). For convenience introduce definition $M_s=(M_{1s},M_{2s})=(cov(C_0))^{-1/2}W_s$.
Again for simplicity let us define sample correlation as
\begin{align} \hat{\rho}=\frac{\frac{1}{T}\sum_{t=1}^TX_{t}Y_t}{\sqrt{\frac{1}{T}\sum_{t=1}^TX_t^2\frac{1}{T}\sum_{t=1}^TY_t^2}} \end{align}
Let us start with the variances. We have
\begin{align} E\frac{1}{T}\sum_{t=1}^TX_t^2=\frac{1}{T}E\sum_{t=1}^T\left(\sum_{s=1}^tU_t\right)^2=\frac{1}{T}\sum_{t=1}^Tt\sigma_U^2=\sigma_U\frac{T+1}{2}. \end{align}
This goes to infinity as $T$ increases, so we hit the first problem, sample variance does not converge. On the other hand continuous mapping theorem in conjunction with functional central limit theorem gives us
\begin{align} \frac{1}{T^2}\sum_{t=1}^TX_t^2=\sum_{t=1}^T\frac{1}{T}\left(\frac{1}{\sqrt{T}}\sum_{s=1}^tU_t\right)^2\to \int_0^1M_{1s}^2ds \end{align} where convergence is convergence in distribution, as $T\to \infty$.
Similarly we get
\begin{align} \frac{1}{T^2}\sum_{t=1}^TY_t^2\to \int_0^1M_{2s}^2ds \end{align} and \begin{align} \frac{1}{T^2}\sum_{t=1}^TX_tY_t\to \int_0^1M_{1s}M_{2s}ds \end{align}
So finally for sample correlation of our random walk we get
\begin{align} \hat{\rho}\to \frac{\int_0^1M_{1s}M_{2s}ds}{\sqrt{\int_0^1M_{1s}^2ds\int_0^1M_{2s}^2ds}} \end{align} in distribution as $T\to \infty$.
So although correlation is well defined, sample correlation does not converge towards it, as in stationary process case. Instead it converges to a certain random variable.
• The mathematical point of view explanation is what I was looking for. It gives me something to contemplate and explore further. Thanks. – Milktrader Feb 18 '11 at 16:41
• This response seems to sidestep the original question: Aren't you just saying that yes, calculating correlation makes sense for stationary processes? – whuber Feb 18 '11 at 16:58
• @whuber, I was answering the question having in mind the comment, but I reread the question again and as far as I understand the OP asks about calculation of correlation for non-stationary data. Calculation of correlation for stationary processes makes sense, all the macroeconometric analysis (VAR, VECM) relies on that. – mpiktas Feb 18 '11 at 17:35
• I'll try to clarify my question with a response. – whuber Feb 18 '11 at 18:57
• @whuber my take away from the answer is that a correlation based on non-stationary data yields a random variable, which may or may not be useful. Correlation based on stationary data converges to a constant. This may explain why traders are attracted to "x-day rolling correlation" because the correlated behavior is fleeting and spurious. Whether "x-day rolling correlation" is valid or useful is for another question. – Milktrader Feb 18 '11 at 19:12
...is the computation of correlation whose data is non-stationary even a valid statistical calculation?
Let $W$ be a discrete random walk. Pick a positive number $h$. Define the processes $P$ and $V$ by $P(0) = 1$, $P(t+1) = -P(t)$ if $V(t) > h$, and otherwise $P(t+1) = P(t)$; and $V(t) = P(t)W(t)$. In other words, $V$ starts out identical to $W$ but every time $V$ rises above $h$, it switches signs (otherwise emulating $W$ in all respects).
(In this figure (for $h=5$) $W$ is blue and $V$ is red. There are four switches in sign.)
In effect, over short periods of time $V$ tends to be either perfectly correlated with $W$ or perfectly anticorrelated with it; however, using a correlation function to describe the relationship between $V$ and $W$ wouldn't be useful (a word that perhaps more aptly captures the problem than "unreliable" or "nonsense").
Mathematica code to produce the figure:
With[{h=5},
pv[{p_, v_}, w_] := With[{q=If[v > h, -p, p]}, {q, q w}];
w = Accumulate[RandomInteger[{-1,1}, 25 h^2]];
{p,v} = FoldList[pv, {1,0}, w] // Transpose;
ListPlot[{w,v}, Joined->True]]
• it is good that your answer points that out but I wouldn't say the process are correlated, I would say they are dependent. This is the point. Calculation of correlation is valide and here it will say "no correlation" and we all know this does not mean "no dependence". – robin girard Feb 18 '11 at 19:48
• @robin That's a good point, but I constructed this example specifically so that for potentially long periods of time these two processes are perfectly correlated. The issue is not one of dependence versus correlation but inherently is related to a subtler phenomenon: that the relationship between the processes changes at random periods. That, in a nutshell, is exactly what can happen in real markets (or at least we ought to worry that it can happen!). – whuber Feb 18 '11 at 19:56
• @whubert yes, and this is a very good example showing that there are processes that have very high correlation for potentially long periods of time and still are not correlated at all (but highly dependent) when regarding the larger temporal scale. – robin girard Feb 19 '11 at 7:47
• @robin girard, I think the key here is that for non-stationary processes the theoretical correlation varies with time, when for the stationary processes theoretical correlation stays the same. So with sample correlation which basically is one number, it is impossible to capture the variation of true correlations in case of non-stationary processes. – mpiktas Feb 21 '11 at 7:09
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2020-10-20 23:29:54
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https://stats.stackexchange.com/questions/255465/accuracy-vs-jaccard-for-multiclass-problem
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# Accuracy vs Jaccard for multiclass problem
TL;DR For a multiclass problem, is Jaccard score the same as accuracy?
Update March 29, 2019
The wrong implementation in scikit-learn is now fixed with pull request #13151. Hooray!
P.S. The lesson here is that no matter how mature and widespread the library, framework or idea is, there are always bugs and shortcomings in them. It is up to you as an engineer, scientist or student to verify the theory and practical results of your work, especially if you rely on someone else's results.
I am working on classification problem and calculating accuracy and Jaccard score with scikit-learn which, I think, is a widely used library in pythonic scientific world. However, me and my matlab colleagues obtain different results.
sklearn.metrics.jaccard_similarity_score declares the following:
Notes: In binary and multiclass classification, this function is equivalent to the accuracy_score. It differs in the multilabel classification problem.
Notes In binary and multiclass classification, this function is equal to the jaccard_similarity_score function.
Indeed, jaccard_similarity_score implementation falls back to accuracy if problem is not of multilabel type:
if y_type.startswith('multilabel'):
...
else:
score = y_true == y_pred
return _weighted_sum(score, sample_weight, normalize)
Isn't it contradicts the definition of Jaccard index (intersection over union)? Are these "score" and "index" different metrics? What is the correct and commonly accepted way to calculate Jaccard metrics for a multiclass problem?
• Can you clarify whether this is a statistical question about Jaccard or a programming question about python and matlab. The latter would be better asked elsewhere. – mdewey Jan 10 '17 at 11:50
• This is a statistical question. I added TL;DR and changed question title to clarify this fact – Ivan Aksamentov - Drop Jan 10 '17 at 11:51
• Are these "score" and "index" different metrics? Your document says the score is the average (or sum) of Jaccard indices. – ttnphns Jan 14 '17 at 10:29
The issue has been reported on scikit-learn GitHub repository: multiclass jaccard_similarity_score should not be equal to accuracy_score #7332
scikit-learn's Jaccard score for the multiclass classification task is incorrect.
A neat overview of the most commonly used performance metrics from {1}:
The accuracy is $~\frac{\text{AA}+ \text{BB} +\text{CC} }{\text{AA}+ \text{AB} +\text{AC} + \text{BA} +\text{BB} + \text{BC} + \text{CA} +\text{CB}+\text{CC}}$.
The average Jaccard score a.k.a. average Jaccard coefficient is:
$~\frac{1}{3}\left(\frac{\text{AA}}{\text{AA}+ \text{AB} +\text{AC} + \text{BA} + \text{CA}} + \frac{\text{BB}}{\text{AB} +\text{BA} +\text{BB} + \text{BC} +\text{CB}} + \frac{\text{CC}}{\text{AC} + \text{BC} + \text{CA} +\text{CB}+\text{CC}}\right)$
For example, if the confusion matrix is:
Then:
• the accuracy is $~\frac{1 + 0+ 0}{1 +0 +0 +1 +0 +0 +1 +0 +0 }=~\frac{1}{3}$
• the average Jaccard score is $~\frac{1}{3}\left(\frac{1}{1 + 0+ 0+1 +1} + \frac{0}{0+1 +0 +0 +0} + \frac{0}{0 +0 +1 +0 +0}\right) = \frac{1}{9}$
References:
• I would like to comment on the "multiclass Jaccard" you show - without claiming if it is good or bad (for I don't know); anyway it is very different approach from what I expressed in my answer as what I suppose the authors of the python package might have meant. In regard to your Jaccard: notice that each summand is a bona fide Jaccard between "true" and "prediction" seen as binary (2-class) variables. For example, in AA/(AA+AB+...) A is seen as the class and B&C are combined as the other class. You are thus averaging in the end three 2-class Jaccards. – ttnphns Jan 13 '17 at 23:10
• (cont.) This is questionable because the classification is one multinomial, not three binomial. (It is akin doing 3 binary logistic regressions instead of one nominal logistic.) I'm not to say it is completely incorrect, only that it may raise objections. – ttnphns Jan 13 '17 at 23:10
• P.S. I've chosen currently to downvote because of the considerations I've given above. I'll be glad to withdraw if the answer is supported with more arguments. – ttnphns Jan 16 '17 at 8:04
• @ttnphns Thanks for the comments, sorry I can't respond yet, I'm having some heavy workload at the moment :/ – Franck Dernoncourt Jan 16 '17 at 15:40
K-class multinomial classification results for n cases (tries) is a nominal variable. Therefore it can be represented as a set of k binary dummy variables.
Now, Jaccard similarity coefficient between two cases (row vectors) by a set of binary attributes is $\frac{a}{a+b+c}$; and accuracy score (I believe it is F1 score) is equal to Dice coefficient: $\frac{2a}{2a+b+c}$ (it will follow from the formula behind your link). The terms come from the table:
case Y
1 0
-------
1 | a | b |
case X -------
0 | c | d |
-------
a = number of variables on which both objects X and Y are 1
b = number of variables where object X is 1 and Y is 0
c = number of variables where object X is 0 and Y is 1
d = number of variables where both X and Y are 0
a+b+c+d = p, the number of variables.
OK. Consider the example given in the documentation you link to:
y_pred = [0, 2, 1, 3]
y_true = [0, 1, 2, 3]
where values are class labels; 4 classes in all
These can be seen as 8 cases (trials) paired as 4 experiments (X cases) and 4 corresponding true outputs (Y cases). Stack all in one column and convert to dummy variables. In dummy variables, there is single 1 in each row:
case data v_0 v_1 v_2 v_3
x1 0 1 0 0 0
x2 2 0 0 1 0
x3 1 0 1 0 0
x4 3 0 0 0 1
y1 0 1 0 0 0
y2 1 0 1 0 0
y3 2 0 0 1 0
y4 3 0 0 0 1
Compute the matrix of Jaccard coefficient between all 8 cases, pairwise, and likewise the matrix of Dice coefficient:
Because we are interested in comparisons only between X and Y cases, we'll pay attention only to the yellow-highlighted portion of the matrices. We are going to sum or average coefficients within yellow area. Moreover, since data were paired we'll probably consider only the diagonal red values and average them. Your document said that their "Jaccard score" is the average of individual Jaccard indices. So here we are.
You see that entries in two yellow squares are identical: Jaccard appears to be equal to Dice (for our situation with nominal data). And the average of the 4 diagonal values is (1+0+0+1) / 4 = 0.5, the result given in your documentation.
(As an example, showing computation of both coefficients of similarity between X1 and Y1 cases):
v_0 v_1 v_2 v_3
x1 1 0 0 0
y1 1 0 0 0
Y1
1 0
-------
1 | 1 | 0 |
X1 -------
0 | 0 | 3 |
-------
Jaccard: 1/(1+0+0)=1; Dice: 2*1/(2*1+0+0)=1
Note that with a single set of dummy variables both coefficients
can attain only values 0 or 1.
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2020-06-06 17:40:01
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https://mathoverflow.net/questions/186690/properties-of-a-function-from-its-pullback
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# Properties of a function from its pullback
Edit: I have now removed the duplication previously referred to. Thank you.
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $\mathcal{F}(M,\mathbb{R})$ (resp.$\mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) to $\mathbb{R}$ and let $F_T$ denote the pullback of $T$, i.e. the map $F_T: \mathcal{F}(N,\mathbb{R}) \to \mathcal{F}(M,\mathbb{R})$ defined by $F_T(f)= f \circ T$.
Is it possible to recover any properties or invariants of $T$ from $F_T$, such as the Dirichlet energy, winding number, etc.?
Thank you very much.
• possible duplicate of Inverse Problem for Pullback – Stefan Waldmann Nov 10 '14 at 8:46
• There are whole books devoted to studying dynamics of a topological or smooth semigroup (by that I mean studying the semigroup $\{T^k\,|\, k\geq 0, T: M \to M\}$) via it's representation on (mostly $L^2(M)$) function spaces via pullbacks. – Vít Tuček Nov 10 '14 at 17:24
• @VítTuček Would you know of any such book titles I could look up, or any keywords that could point me in this direction? Thank you. – compmath Nov 10 '14 at 19:15
• @davidbar There are many flavours (e.g. topological dynamics, ergodic theory) a and I'm no expert. I just participated in an internet seminar on ergodic theory a few years ago. The notes are here: fa.uni-tuebingen.de/lehre/isem/12th-2008-09/… I guess the question is what kind of phenomenon do you want to study (e.g. smooth, topological, stochastic, ...) and that should narrow your search down. It may very well be that this point of view through dynamical systems is not very fruitful for your purposes. – Vít Tuček Nov 10 '14 at 20:02
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2021-07-26 01:15:56
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https://bookstore.ams.org/view?ProductCode=MCL/4
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Edited by: Roman Fedorov Max Planck Institute for Mathematics, Bonn, Germany
Alexei Belov Moscow Institute of Open Education, Moscow, Russia and Shanghai University, Shanghai, People’s Republic of China
Alexander Kovaldzhi "Second School" Lyceum, Moscow, Russia
Ivan Yashchenko Moscow Center for Continuous Mathematical Education, Moscow, Russia
A co-publication of the AMS and Mathematical Sciences Research Institute
Available Formats:
Softcover ISBN: 978-0-8218-5363-4
Product Code: MCL/4
List Price: $25.00 Individual Price:$18.75
Electronic ISBN: 978-1-4704-1615-7
Product Code: MCL/4.E
List Price: $25.00 Individual Price:$18.75
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List Price: $37.50 Click above image for expanded view Moscow Mathematical Olympiads, 1993–1999 Edited by: Roman Fedorov Max Planck Institute for Mathematics, Bonn, Germany Alexei Belov Moscow Institute of Open Education, Moscow, Russia and Shanghai University, Shanghai, People’s Republic of China Alexander Kovaldzhi "Second School" Lyceum, Moscow, Russia Ivan Yashchenko Moscow Center for Continuous Mathematical Education, Moscow, Russia A co-publication of the AMS and Mathematical Sciences Research Institute Available Formats: Softcover ISBN: 978-0-8218-5363-4 Product Code: MCL/4 List Price:$25.00 Individual Price: $18.75 Electronic ISBN: 978-1-4704-1615-7 Product Code: MCL/4.E List Price:$25.00 Individual Price: $18.75 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$37.50
• Book Details
MSRI Mathematical Circles Library
Volume: 42011; 220 pp
MSC: Primary 00; 97;
The Moscow Mathematical Olympiad has been challenging high school students with stimulating, original problems of different degrees of difficulty for over 75 years. The problems are nonstandard; solving them takes wit, thinking outside the box, and, sometimes, hours of contemplation. Some are within the reach of most mathematically competent high school students, while others are difficult even for a mathematics professor. Many mathematically inclined students have found that tackling these problems, or even just reading their solutions, is a great way to develop mathematical insight.
In 2006 the Moscow Center for Continuous Mathematical Education began publishing a collection of problems from the Moscow Mathematical Olympiads, providing for each an answer (and sometimes a hint) as well as one or more detailed solutions. This volume represents the years 1993–1999.
The problems and the accompanying material are well suited for math circles. They are also appropriate for problem-solving classes and practice for regional and national mathematics competitions.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
High school and undergraduate students interested in problem solving; mathematical circles.
• Chapters
• Title page
• Contents
• Introduction
• Problems
• Hints
• Solutions
• Reference facts
• Postscript by V. M. Tikhomirov: Reflections on the Moscow Mathematical Olympiads
• Bibliography
• Problem authorship
• Requests
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 42011; 220 pp
MSC: Primary 00; 97;
The Moscow Mathematical Olympiad has been challenging high school students with stimulating, original problems of different degrees of difficulty for over 75 years. The problems are nonstandard; solving them takes wit, thinking outside the box, and, sometimes, hours of contemplation. Some are within the reach of most mathematically competent high school students, while others are difficult even for a mathematics professor. Many mathematically inclined students have found that tackling these problems, or even just reading their solutions, is a great way to develop mathematical insight.
In 2006 the Moscow Center for Continuous Mathematical Education began publishing a collection of problems from the Moscow Mathematical Olympiads, providing for each an answer (and sometimes a hint) as well as one or more detailed solutions. This volume represents the years 1993–1999.
The problems and the accompanying material are well suited for math circles. They are also appropriate for problem-solving classes and practice for regional and national mathematics competitions.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
High school and undergraduate students interested in problem solving; mathematical circles.
• Chapters
• Title page
• Contents
• Introduction
• Problems
• Hints
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2023-03-28 08:30:16
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https://www.gktoday.in/question/the-compound-interest-on-30000-rs-at-7-per-annum-f
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The compound interest on 30000 Rs. at 7% per annum for a certain time is 4347 Rs. The time is :
[A] 2 years
[B] 2.5 years
[C] 3 years
[D] 4 years
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2018-06-20 09:31:01
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https://buboflash.eu/bubo5/show-dao2?d=1329302998284
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Vitamin D3 (Cholecalciferol) is so potent and fat soluble, the capsules are tiny, filled with olive oil, and easy to swallow. The label instructs to take one 5,000 IU capsule every three days, for an average daily dosage of 5000/3 = 1667 IU/day. Since the 120 capsule bottle costs $8.80, taking one capsule every third day will last one full year, which is impossible to beat for the price. “Day skipping” works well with Vitamin D3, since after being quickly metabolized by your liver into “25-hydroxy vitamin D”, that fat soluble metabolite has a half-life of at least three weeks in the body. So whatever regimen you choose will quickly "average out" to a nice even level in your blood. If you want to change selection, open document below and click on "Move attachment" GRC | Vitamin D Research my own use, and it's what I had been recommending to others. I have since switched to the Vitamin D from “Healthy Origins” (see below) because it is even less expensive than Now Foods and they are an equally good and trusted supplier.Because <span>Vitamin D3 (Cholecalciferol) is so potent and fat soluble, the capsules are tiny, filled with olive oil, and easy to swallow. The label instructs to take one 5,000 IU capsule every three days, for an average daily dosage of 5000/3 = 1667 IU/day. Since the 120 capsule bottle costs$8.80, taking one capsule every third day will last one full year, which is impossible to beat for the price.“Day skipping” works well with Vitamin D3, since after being quickly metabolized by your liver into “25-hydroxy vitamin D”, that fat soluble metabolite has a half-life of at least three weeks in the body. So whatever regimen you choose will quickly "average out" to a nice even level in your blood.Although you should never go “off label” to obtain a higher dosage without the advice of a doctor and/or keeping an eye on your important blood levels, especially calcium, this high-pote
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2021-10-18 05:38:36
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http://perceptionradar.com/ss5cu/35e107-multiplication-of-matrix-in-c%2B%2B
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The range of a function is the list of all possible outputs (y-values) of the function. Review the derivatives of the inverse trigonometric functions: arcsin(x), arccos(x), and arctan(x). 22 Derivative of inverse function 22.1 Statement Any time we have a function f, it makes sense to form is inverse function f 1 (although this often requires a reduction in the domain of fin order to make it injective). Recall from Functions and Graphs that trigonometric functions are not one-to-one unless the domains are restricted. the -1. 1 2 1 y 1x c 1 2 1 1 c 1 2 1 y 1x c 1 y 1 c 1 2 1 y 1x c 1 2 1 y 1x c Now let’s work through a few examples. If you're seeing this message, it means we're having trouble loading external resources on … We can use the following identities to differentiate the other three inverse trig functions: cos−1 x = π/2−sin−1 x cot−1 x = π/2−tan−1 x csc−1 x = π/2−sec−1 x We then see that the only difference in the derivative of an inverse trig function Differentiate functions that contain the inverse trigonometric functions arcsin(x), arccos(x), and arctan(x). Each is the inverse of their respective trigonometric function. Inverse Trigonometric Functions Inverse Function If y = f(x) and x = g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other i.e., g = f-1 IF y = f(x), then x = f-1(y) Inverse Trigonometric Functions If y = sin X-1, then x = sin-1 y, similarly for other trigonometric functions. Differentiate functions that contain the inverse trigonometric functions arcsin(x), arccos(x), and arctan(x). Chapter 7 gives a brief look at inverse trigonometric functions. These are the inverse functions of the trigonometric functions with suitably restricted domains.Specifically, they are the inverse functions of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios. Inverse Trigonometric Functions. Written this way it indicates the inverse of the sine function. There are six inverse trigonometric functions. Do all the exercises. The function sinxpasses horizontal line test for ˇ 2 x ˇ 2 so it has an inverse. We have worked with these functions before. Other Inverse Trigonometric Functions: Each trigonometric function has a restricted domain for which an inverse function is defined. View Inverse Trig Functions.pdf from MATH 2545 at San Marcos High School. • Inverse Sine function= arcsinx • Inverse Cosine Function… NCERT Notes Mathematics for Class 12 Chapter 2: Inverse Trigonometric Functions Function. Rather, have pen and paper ready and try to work through the examples before reading their solutions. The Definition of Inverse trig functions can be seen as the following formulas. Some Worked Problems on Inverse Trig Functions Simplify (without use of a calculator) the following expressions 1 arcsin[sin(ˇ 8)]: 2 arccos[sin(ˇ 8)]: 3 cos[arcsin(1 3)]: Solutions. 5 Practicing with the Inverse Functions 3 6 Derivatives of Inverse Trig Functions 4 7 Solving Integrals 8 1 Introduction Just as trig functions arise in many applications, so do the inverse trig functions. These are sometimes abbreviated sin(θ) andcos(θ), respectively, where θ is the angle, but the parentheses around the angle are often omitted, e.g., sin θ andcos θ. FIGURE 4.71 Below are the derivatives of the six inverse trigonometric functions. Review the derivatives of the inverse trigonometric functions: arcsin(x), arccos(x), and arctan(x). We know about inverse functions, and we know about trigonometric functions, so it's time to learn about inverse trigonometric functions! Inverse trigonometric functions provide anti derivatives for a variety of functions that arise in engineering. Quick Recap: Below is the domain and range of a few inverse trig functions: Inverse Trig Relations/Functions and Some Extra Examples From the unit circle, we can now find the coordinates of … Example 1 $y = \arctan {\frac{1}{x}}$ If f'(x) = tan-1(sec x + tan x), -π/2 < x < π/2, and f(0) = 0 then f(1) is equal to. 1.1 How to use this booklet You will not gain much by just reading this booklet. Click or tap a problem to see the solution. Find the missing side then evaluate the trig function asked for. Integrals Resulting in Other Inverse Trigonometric Functions. On the other hand, the notation (etc.) If you're seeing this message, it means we're having trouble loading external resources on our website. Trigonometry Handbook Table of Contents Page Description Chapter 4: Key Angle Formulas The other functions are similar. NCERT Books for Class 12 Maths Chapter 2 Inverse Trigonometric Functions can be of extreme use for students to understand the concepts in a simple way.Class 12th Maths NCERT Books PDF Provided will help … The following table summarizes the domains and ranges of the inverse trig functions. The restricted domains are determined so the trig functions are one-to-one. Derivatives of Inverse Trigonometric Functions 2 1 1 1 dy n dx du u dx u 2 1 1 1 dy Cos dx du u dx u 2 1 1 1 dy Tan dx du u dx u 2 dy Cot 1 1 dx du u dx u 2 1 1 1 dy c dx du uu dx u 2 1 1 1 dy Csc dx du uu dx u Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted or (Beyer 1987, p. 141). 3 Definition notation EX 1 Evaluate these without a calculator. • y = f(x) Æ x in the domain of f. Solved Problems. sin(sin 1 x) = … 11/21 - Inverse Trig, 11/26 - Trig Substitution, 12/3 - Partial Fractions, 12/5 - Final Review Things are starting to go very fast and we won’t be able to cover everything. If we know the derivative of f, then we can nd the derivative of f 1 as follows: Derivative of inverse function. • The domain of f = the range of f -1 the inverse. The inverse function is denoted by sin 1 xor arcsinx:Since the range of sin on [ˇ 2;ˇ 2] is [-1,1,], the interval [-1,1] is the domain of sin 1 x:We also have the following cancellation rule. The derivatives of $$6$$ inverse trigonometric functions considered above are consolidated in the following table: In the examples below, find the derivative of the given function. 2 The graph of y = sin x does not pass the horizontal line test, so it has no inverse. Graphically speaking, the domain is the portion of the x-axis on which the graph casts a shadow. Inverse Trigonometric Functions Review First, let’s review briefly inverse functions before getting into inverse trigonometric functions: • f Æ f -1 is the inverse • The range of f = the domain of f -1, the inverse. Inverse Trigonometric Functions – Pike Page 2 of 3 x 12 The derivatives of the other four inverse trigonometric functions can be found in a similar fashion. If y = f(x) and x = g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse … The tangent (tan) of an angle is the ratio of the sine to the cosine: Inverse Trigonometric Functions Class 12 NCERT Book: If you are looking for the best books of Class 12 Maths then NCERT Books can be a great choice to begin your preparation. In each pair, the derivative of one function is the negative of the other. NCERT Solutions for class 12 Maths Chapter 2 Inverse Trigonometric Functions in Hindi Medium and English Medium PDF file format to free download along with NCERT Solutions Apps updated for new academic session 2020-2021. Inverse Trigonometric Functions The trigonometric functions are not one-to-one. 4.6.2 Restricting the range of trig functions to create inverse functions Since the trig functions are periodic there are an in nite number of x-values such that y= f(x):We can x this problem by restricting the domain of the trig functions so that the trig function is one-to-one in that speci c domain. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. From Figure 4.71, you can see that does not pass the test because different values of yield the same -value. Graphically speaking, the range is the portion of the y-axis on which the graph casts a shadow. Inverse Trigonometry Functions and Their Derivatives. By restricting their do-mains, we can construct one-to-one functions from them. Section 4.7 Inverse Trigonometric Functions 343 Inverse Sine Function Recall from Section 1.9 that, for a function to have an inverse function, it must be one-to-one—that is, it must pass the Horizontal Line Test. Study, study, study! (a) (π+1)/4 (b) (π+2)/4 … inverse trig function and label two of the sides of a right triangle. 34 Graphs of Inverse Trig Functions 35 Problems Involving Inverse Trigonometric Functions Trigonometry Handbook Table of Contents Version 2.2 Page 3 of 109 June 12, 2018. What may be most surprising is that the inverse trig functions give us solutions to some common integrals. Note that for each inverse trig function we have simply swapped the domain and range for If, instead, we write (sin(x))−1 we mean the fraction 1 sin(x). For example, if we restrict the domain of sinxto the interval − ˇ 2; ˇ 2 we have a one-to-one function which has an inverse denoted by arcsinx or sin−1 x. If we restrict the domain (to half a period), then we can talk about an inverse function. However, only three integration formulas are noted in the rule on integration formulas resulting in inverse trigonometric functions because the remaining three are negative versions of the ones we use. List of trigonometric identities 2 Trigonometric functions The primary trigonometric functions are the sine and cosine of an angle. (sin (cos (tan In this section we focus on integrals that result in inverse trigonometric functions. 4. Also, each inverse trig function also has a unique domain and range that make them one-to-one functions. 1 Since arcsin is the inverse function of sine then arcsin[sin(ˇ 8)] = ˇ 8: 2 If is the angle ˇ 8 then the sine of is the cosine of the complementary angle ˇ 2 The derivatives of the inverse trigonometric functions are given below. 1 du 1—u2 dx 1 du 1—u2 dx 1 du 1+u2 dx 1 du 1+u2 dx du lul u2—1dx du lul u2—1dx' < 1 < 1 lul>l lul>l 3. 3.9 Inverse Trigonometric Functions 4 Note. Integrals Involving Inverse Trigonometric Functions The derivatives of the six inverse trigonometric functions fall into three pairs. The inverse trigonometric functions are multivalued.For example, there are multiple values of such that , so is not uniquely defined unless a principal value is defined. Lecture 6 : Inverse Trigonometric Functions Inverse Sine Function (arcsin x = sin 1x) The trigonometric function sinxis not one-to-one functions, hence in order to create an inverse, we must restrict its domain. The inverse trigonometric functions are also called arcus functions or anti trigonometric functions. Mathematics is not a spectator sport! Are not one-to-one unless the domains are determined so the trig function asked for their! Be most surprising is that the inverse of the six inverse trigonometric functions are not one-to-one are given below respective! Identities 2 trigonometric functions restricting their do-mains, we always need to be careful to take these restrictions into.! Can be seen as the following formulas test, so it has an.. Surprising is that the inverse trigonometric functions the primary trigonometric functions are given below mean fraction! Can be seen as the following table summarizes the domains are restricted so the trig give. 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You will not gain much by just reading this booklet on which the graph casts shadow! Reading their solutions f 1 as follows: derivative of f, then we can one-to-one! Mean the fraction 1 sin ( x ), then we can construct one-to-one functions from them the... Domain for which an inverse arcsin ( x ) take these restrictions into account half a period,... The graph casts a shadow domain of f -1 the inverse of respective! Common integrals a unique domain and range that make them one-to-one functions from them has! On our website can nd the derivative of f -1 the inverse functions! Be seen as the following table summarizes the domains and ranges of inverse. By restricting their do-mains, we write ( sin ( cos ( tan 3.9 inverse trigonometric functions 4 Note brief! Functions arcsin ( x ) 2 so it has no inverse of yield the -value. Ready and try to work through the examples before reading their solutions just reading this booklet will... 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2020 multiplication of matrix in c++
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2021-07-30 09:58:07
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http://hackage.haskell.org/package/typerbole-0.0.0.5/docs/Calculi-Lambda-Cube-Dependent.html
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typerbole-0.0.0.5: A typeystems library with exaggerated claims
Calculi.Lambda.Cube.Dependent
Synopsis
# Documentation
class SimpleType t => Dependent t where Source #
Typesystems which can have values in their types.
Minimal complete definition
valueToType
Associated Types
type DependentTerm t :: * -> * Source #
A value-level term that can be encoded as a type expression.
Of kind * -> * because it expects a typesystem as a parameter.
Methods
valueToType :: DependentTerm t t -> t Source #
Encode a value at the type-level. This could be just a constructor or a complete transformation of the AST, but this typeclass doesn't care.
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2020-03-29 07:00:44
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https://socratic.org/questions/58b8a5d87c01493f69c2a192
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# How do you solve abs(x-5) = abs(x+5) ?
Mar 3, 2017
$x = 0$
#### Explanation:
Given:
$\left\mid x - 5 \right\mid = \left\mid x + 5 \right\mid$
We can square both sides of the equation, solve the resulting equation, then check the solution...
Squaring both sides we get:
$\textcolor{red}{\cancel{\textcolor{b l a c k}{{x}^{2}}}} - 10 x + \textcolor{red}{\cancel{\textcolor{b l a c k}{25}}} = \textcolor{red}{\cancel{\textcolor{b l a c k}{{x}^{2}}}} + 10 x + \textcolor{red}{\cancel{\textcolor{b l a c k}{25}}}$
Subtract ${x}^{2} + 25$ from both sides to get:
$- 10 x = 10 x$
Add $10 x$ to both sides to get:
$0 = 20 x$
Divide both sides by $20$ and transpose to get:
$x = 0$
Check:
$\left\mid \textcolor{b l u e}{0} - 5 \right\mid = 5 = \left\mid \textcolor{b l u e}{0} + 5 \right\mid$
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2020-09-29 08:33:59
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https://seop.illc.uva.nl/entries/logic-algebraic-propositional/
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# Algebraic Propositional Logic
First published Mon Dec 12, 2016
[Editor's Note: The following entry replaces and is based on the former entry titled Propositional Consequence Relations and Algebraic Logic.]
George Boole was the first to present logic as a mathematical theory in algebraic style. In his work, and in that of the other algebraists of the algebraic tradition of logic of the nineteenth century, the distinction between a formal language and a mathematically rigorous semantics for it was still not drawn. What the algebraists in this tradition did was to build algebraic theories (of Boolean algebras, and relation algebras) with among other interpretations a logical one.
The works of Frege and Russell introduced a different perspective on the way to approach logic. In those works, a logic system was given by a formal language and a deductive calculus, namely a set of axioms and a set of inference rules. Let us (for this entry) call such a pair a logical deduction system, and the formulas derivable in the calculus its theorems (nowadays it is common practice in algebraic logic to refer to this kind of calculi as Hilbert-style and in proof complexity theory as Frege systems). In Frege and Russell’s approach, a formal (mathematical) semantics of whatever kind (algebraic, model-theoretic, etc.) for the formal languages they used was lacking. The only semantics present was of an intuitive, informal kind.
The systems introduced by Frege and Russell were systems of classical logic, but soon after systems of non-classical logics were considered by other logicians. The first influential attempts to introduce logics different from classical logic remained within the Frege-Russell tradition of presenting a logical deduction system without any formal semantics. These attempts lead to the first modal systems of C.I. Lewis (1918, 1932) and to the axiomatization of intuitionistic logic by Heyting (1930).
The idea underlying the design of Frege and Russell’s logical deduction systems is that the theorems should be the formulas that correspond (intuitively) to the logical truths or logical validities. The concept of logical consequence was not central to the development, and this was also the case in the many systems of non-classical logics that were to be designed following in the footsteps of the first modal systems of C.I. Lewis. This situation influenced the way in which the research on some non-classical logics has usually been presented and sometimes also its real evolution. However, the concept of logical consequence has been the one that logic has traditionally dealt with. Tarski put it once again into the center of modern logic, both semantically and syntactically. Nowadays, a general theory of the algebraization of logics around the concept of logical consequence has grown from the different algebraic treatments of the different logics obtained during the last century.
The concept of logical consequence has proved much more fruitful than those of theorem and of logical validity for the development of such a general theory. The first attempts in the process of building the general theory of the algebraization of logics can be found in the study of the class of implicative logics by Rasiowa (1974) and in the systematic presentation by Wójcicki (1988) of the investigations of a general nature on propositional logics as consequence operations carried out mainly by Polish logicians, following the studies of Tarski, Lindenbaum, Łukasiewicz and others in the first part of the twentieth century.
It was only in the 1920s that algebras and logical matrices (an algebra together with a set of designated elements) started to be taken as models of logical deduction systems, that is, as providing a formal semantics for formal languages of logic. Moreover, they were also used to define sets of formulas with similar properties to the ones the sets of theorems of the known logical deduction systems have, in particular the property of being closed under substitution instances; more recently logical matrices have also been used to define logics as consequence relations.
Algebraic logic can be described in very general terms as the discipline that studies logics by associating with them classes of algebras, classes of logical matrices and other algebra related mathematical structures and that relates the properties that the logics may have with properties of the associated algebras (or algebra related structures) with the purpose that the understanding of these algebras can be used to better understand the logic at hand.
From the algebraic study of particular logics a general theory of the algebraization of logics slowly emerged during the last century with the aim, more or less explicitly stated during the process, of obtaining general and informative results relating the properties a logic may have with the algebraic properties the class of algebras (or algebra related structures) associated with it might enjoy. Those algebraic studies assumed somehow an implicit conception of what is the process by which to associate with any given logic a class of algebras as its natural algebraic counterpart. The development of the general theory speeded up and consolidated at the beginning of the 1980s with the introduction of the notion of algebraizable logic, and at that time the assumptions about the class of algebras that deserve to be taken as the natural to associate with a given logic started to be made more and more explicit.
In this entry we concentrate on the general theory of the algebraization of propositional logics taken as consequence relations. This theory has evolved into the field known as Abstract Algebraic Logic (AAL). The entry can be taken as a mild introduction to this field.
## 1. Abstract consequence relations
Tarski’s work (1930a, 1930b, 1935, 1936) on the methodology of the deductive sciences of the 1920s and 1930s studies the axiomatic method abstractly and introduces for the first time the abstract concept of consequence operation. Tarski had in mind mainly the different mathematical axiomatic theories. On these theories, the sentences that are proved from the axioms are consequences of them but (of course) almost all of them are not logical truths; under a suitable formalization of these theories, a logical calculus like Frege’s or Russell’s can be used to derive the consequences of the axioms. Tarski set the framework to study the most general properties of the operation that assigns to a set of axioms its consequences.
Given a logical deduction system $$H$$ and an arbitrary set of formulas $$X$$, a formula $$a$$ is deducible in $$H$$ from $$X$$ if there is a finite sequence of formulas any one of which belongs to $$X$$ or is an axiom of $$H$$ or is obtained from previous formulas in the sequence by one of the inference rules of $$H$$. Such a sequence is a deduction (or proof) in $$H$$ of $$a$$ with premises or hypotheses in $$X$$. Let $$Cn(X)$$ be the set of formulas deducible in $$H$$ from the formulas in $$X$$ taken as premises or hypothesis. This set is called the set of consequences of $$X$$ (relative to the logical deduction system $$H$$). $$Cn$$ is then an operation that is applied to sets of formulas to obtain new sets of formulas. It has the following properties: For every set of formulas $$X$$
1. $$X \subseteq Cn(X)$$
2. $$Cn(Cn(X)) = Cn(X)$$
3. $$Cn(X) = \bigcup\{Cn(Y): Y \subseteq X, Y \textrm{ finite}\}$$
Clause 3 stipulates that $$Cn(X)$$ is equal to the union of the set of formulas derivable from finite subsets of $$X$$. Tarski took these properties to define the notion of consequence operation axiomatically. In fact he added that there is a formula $$x$$ such that $$Cn(\{x\})$$ is the set $$A$$ of all the formulas and that this set must be finite or of the cardinality of the natural numbers. Condition (3) implies the weaker, and important, condition of monotonicity
1. if $$X \subseteq Y \subseteq A$$, then $$Cn(X) \subseteq Cn(Y)$$.
To encompass the whole class of logic systems one finds in the literature, a slightly more general definition than Tarski’s is required. We will say that an abstract consequence operation $$C$$ on an arbitrary set $$A$$ is an operation that applied to subsets of $$A$$ gives subsets of $$A$$ and for all $$X, Y \subseteq A$$ satisfies conditions (1), (2) and (4) above. If in addition $$C$$ satisfies (3) we say that it is a finitary consequence operation.
Consequence operations are present not only in logic but in many areas of mathematics. Abstract consequence operations are known as closure operators in universal algebra and lattice theory, for instance. In topology the operation that sends a subset of a topological space to its topological closure is a closure operator. In fact the topologies on a set $$A$$ can be identified with the closure operators on $$A$$ that satisfy the additional conditions that $$C(\varnothing) = \varnothing$$ and $$C(X \cup Y) = C(X) \cup C(Y)$$ for all $$X, Y \subseteq A$$.
Given a consequence operation $$C$$ on a set $$A$$, a subset $$X$$ of $$A$$ is said to be $$C$$-closed, or a closed set of $$C$$, if $$C(X) = X$$.
A different, but mathematically equivalent, (formal) approach is to consider consequence relations on a set of formulas instead of consequence operations. A(n) (abstract) consequence relation on the set of formulas of a formal language is a relation $$\vdash$$ between sets of formulas and formulas that satisfies the following conditions:
1. if $$a \in X$$, then $$X \vdash a$$
2. if $$X \vdash a$$ and $$X \subseteq Y$$, then $$Y \vdash a$$
3. if $$X \vdash a$$ and for every $$b \in X, Y \vdash b$$, then $$Y \vdash a$$.
It is finitary if in addition it satisfies
1. if $$X \vdash a$$, then there is a finite set $$Y \subseteq X$$ such that $$Y \vdash a$$.
Given a logical deduction system $$H$$, the relation $$\vdash$$ defined by $$X \vdash a$$ if $$a$$ is deducible from $$X$$ in $$H$$ is a finitary consequence relation. Nonetheless, we are used not only to syntactic definitions of consequence relations but also to semantic definitions. For example, we define classical propositional consequence using truth valuations, first-order consequence relation using structures, intuitionistic consequence relation using Kripke models, etc. Sometimes these model-theoretic definitions of consequence relations define non-finitary consequence relations, for example, the consequence relations for infinitary formal languages and the consequence relation of second-order logic with the so-called standard semantics.
In general, an abstract consequence relation on a set $$A$$ (not necessarily the set of formulas of some formal language) is a relation $$\vdash$$ between subsets of $$A$$ and elements of $$A$$ that satisfies conditions (1)–(3) above. If it also satisfies (4) it is said to be finitary. If $$\vdash$$ is an abstract consequence relation and $$X \vdash a$$ we can say that $$X$$ is a set of premises or hypothesis with conclusion $$a$$ according to $$\vdash$$ and that $$a$$ follows from $$X$$, or is entailed by $$X$$ (according to $$\vdash)$$. These relations correspond to Koslow’s implication structures; see Koslow 1992 for the closely related but different approach to logics (in a broad sense) as consequence relations introduced by this author.
Consequence operations on a set $$A$$ are in one-to-one correspondence with abstract consequence relations on $$A$$. The move from a consequence operation $$C$$ to a consequence relation $$\vdash_C$$ and, conversely, from a consequence relation $$\vdash$$ to a consequence operation $$C_{\vdash}$$ is easy and given by the definitions:
$X \vdash_C a \txtiff a \in C(X) \textrm{ and } a \in C_{\vdash}(X) \txtiff X \vdash a.$
Moreover, if $$C$$ is finitary, so is $$\vdash_C$$ and if $$\vdash$$ is finitary, so is $$C_{\vdash}$$.
For a general discussion on logical consequence see the entry Logical Consequence.
## 2. Logics as consequence relations
In this section we define what propositional logics are and explain the basic concepts relating to them. We will call the propositional logics (as defined below) simply logic systems.
One of the main traits of the consequence relations we study in logic is their formal character. This roughly means that if a sentence $$a$$ follows from a set of sentences $$X$$ and we have another sentence $$b$$ and another set of sentences $$Y$$ that share the same form with $$a$$ and $$X$$ respectively, then $$b$$ also follows from $$Y$$. In propositional logics this boils down to saying that if we uniformly replace basic sub-sentences of the sentences in $$X \cup \{a\}$$ by other sentences obtaining $$Y$$ and $$b$$, then $$b$$ follows from $$Y$$. (The reader can find more information on the idea of formality in the entry Logical Consequence.)
To turn the idea of the formal character of logics into a rigorous definition we need to introduce the concept of propositional language and the concept of substitution.
A propositional language $$L$$ is a set of connectives, that is, a set of symbols each one of which has an arity $$n$$ that tells us in case that $$n = 0$$ that the symbol is a propositional constant, and in case that $$n \gt 0$$ whether the connective is unary, binary, ternary, etc. For example $$\{\wedge , \vee , \rightarrow , \bot , \top \}$$ is (or can be) the language of several logics, like classical and intuitionistic, $$(\bot$$ and $$\top$$ are 0-ary and the other connectives are binary), $$\{\neg , \wedge , \vee \rightarrow , \Box , \Diamond \}$$ is the language of several modal logics, $$(\neg , \Box , \Diamond$$ are unary and the other connectives binary) and $$\{ \wedge , \vee , \rightarrow , * , \top , \bot , 1, 0\}$$ is the language of many-valued logics and also of a fragment of linear logic $$(\bot , \top , 1$$, and 0 are propositional constants and the other symbols binary connectives).
Given a language $$L$$ and a set of propositional variables $$V$$ (which is disjoint from $$L)$$, the formulas of $$L$$, or $$L$$-formulas, are defined inductively as follows:
1. Every variable is a formula.
2. Every 0-ary symbol is a formula.
3. If $$*$$ is a connective and $$n \gt 0$$ is its arity, then for all formulas $$\phi_1 ,\ldots ,\phi_n, * \phi_1 \ldots \phi_n$$ is also a formula.
A substitution $$\sigma$$ for $$L$$ is a map from the set of variables $$V$$ to the set of formulas of $$L$$. It tells us which formula must replace which variable when we perform the substitution. If $$p$$ is a variable, $$\sigma(p)$$ denotes the formula that the substitution $$\sigma$$ assigns to $$p$$. The result of applying a substitution $$\sigma$$ to a formula $$\phi$$ is the formula $$\bsigma(\phi)$$ obtained from $$\phi$$ by simultaneously replacing the variables in $$\phi$$, say $$p_1 , \ldots ,p_k$$, by, respectively, the formulas $$\sigma(p_1), \ldots ,\sigma(p_k)$$. In this way a substitution $$\sigma$$ gives a unique map $$\bsigma$$ from the set of formulas to itself that satisfies
1. $$\bsigma(p) = \sigma(p)$$, for every variable $$p$$,
2. $$\bsigma(\dagger) = \dagger$$, for every 0-ary connective $$\dagger$$,
3. $$\bsigma(* \phi_1 \ldots \phi_n) = * \bsigma(\phi_1)\ldots \bsigma(\phi_n)$$, for every connective $$*$$ of arity $$n \gt 0$$ and formulas $$\phi_1 , \ldots ,\phi_n$$.
A formula $$\psi$$ is a substitution instance of a formula $$\phi$$ if there is a substitution $$\sigma$$ such that when applied to $$\phi$$ gives $$\psi$$, that is, if $$\bsigma(\phi) = \psi$$.
In order to avoid unnecessary complications we will assume in the sequel that all the logics use the same set $$V$$ of variables, so that the definition of formula of $$L$$ depends only on $$L$$. A logic system (or logic for short) is given by a language $$L$$ and a consequence relation $$\vdash$$ on the set of formulas of $$L$$ that is formal in the sense that for every substitution $$\sigma$$, every set of formulas $$\Gamma$$ and every formula $$\phi$$,
$\textrm{if } \Gamma \vdash \phi, \textrm{ then } \bsigma[\Gamma] \vdash\bsigma(\phi)$
where $$\bsigma[\Gamma]$$ is the set of the formulas obtained by applying the substitution $$\sigma$$ to the formulas in $$\Gamma$$. The consequence relations on the set of formulas of a language that satisfy this property are called structural and also substitution-invariant in the literature. They were considered for the first time in Łoś & Suszko 1958. Tarski only explicitly considered closed sets also closed under substitution instances for some consequence relations; he never considered (at least explicitly) the substitution invariance condition for consequence relations.
We will refer to logic systems by the letter $$\bL$$ with possible subindices, and we set $$\bL = \langle L, \vdash_{\bL } \rangle$$ and $$\bL_n = \langle L_n, \vdash_{\bL_n } \rangle$$ with the understanding that $$L \; (L_n)$$ is the language of $$\bL \;(\bL_n)$$ and $$\vdash_{\bL }\; (\vdash_{\bL_n })$$ its consequence relation. A logic system $$\bL$$ is finitary if $$\vdash_{\bL}$$ is a finitary consequence relation.
The consequence relation of a logic system can be given in several ways, some using proof-theoretic tools, others semantic means. One can define a substitution-invariant consequence relation using a proof system like a Hilbert-style axiom system, a Gentzen-style sequent calculus or a natural deduction style calculus, etc. One can also define a substitution-invariant consequence relation semantically using a class of mathematical objects (algebras, Kripke models, topological models, etc.) and a satisfaction relation.
If $$\bL_1 = \langle L,\vdash_{\bL_1 } \rangle$$ is a logic system with $$\vdash_{\bL_1}$$ defined by a proof-system and $$\bL_2 = \langle L, \vdash_{\bL_2 } \rangle$$ is a logic system over the same language with $$\vdash_{\bL_2}$$ defined semantically, we say that the proof-system used to define $$\vdash_{\bL_1}$$ is sound for the semantics used to define $$\vdash_{\bL_2}$$ if $$\vdash_{\bL_1}$$ is included in $$\vdash_{\bL_2}$$, namely if $$\Gamma \vdash_{\bL_1 } \phi$$ implies $$\Gamma \vdash_{\bL_2 } \phi$$. If the other inclusion holds the proof-system is said to be complete with respect to the semantics that defines $$\vdash_{\bL_2}$$, that is when $$\Gamma \vdash_{\bL_2 } \phi$$ implies $$\Gamma \vdash_{\bL_1 } \phi$$.
A set of $$L$$-formulas $$\Gamma$$ is called a theory of a logic system $$\bL$$, or $$\bL$$-theory, if it is closed under the relation $$\vdash_{\bL}$$, that is, if whenever $$\Gamma \vdash_{\bL } \phi$$ it also holds that $$\phi \in \Gamma$$. In other words, the theories of $$\bL$$ are the closed sets of the consequence operation $$C_{\vdash_{ \bL}}$$ on the set of $$L$$-formulas. In order to simplify the notation we denote this consequence operation by $$C_{\bL}$$. A formula $$\phi$$ is a theorem (or validity) of $$\bL$$ if $$\varnothing \vdash_{\bL } \phi$$. Then $$C_{\bL }(\varnothing)$$ is the set of theorems of $$\bL$$ and is the least theory of $$\bL$$. The set of all theories of $$\bL$$ will be denoted by $$\tTH(\bL)$$.
Given a logic $$\bL$$, the consequence operation $$C_{\bL}$$ is substitution-invariant, which means that for every set of $$L$$-formulas $$\Gamma$$ and every substitution $$\sigma,\bsigma[C_{\bL}(\Gamma)] \subseteq C_{\bL}(\bsigma[\Gamma]$$). Moreover, for every theory $$T$$ of $$\bL$$ we have a new consequence operation defined as follows: $C_{\bL }^T (\Gamma) = C_{\bL }(T \cup \Gamma)$ that is, $$C_{\bL }^T (\Gamma)$$ is the set of formulas that follow from $$\Gamma$$ and $$T$$ according to $$\bL$$. It turns out that $$T$$ is closed under substitutions if and only if $$C_{\bL }^T$$ is substitution-invariant.
If $$\bL$$ is a logic and $$\Gamma , \Delta$$ are sets of $$L$$-formulas, we will use the notation $$\Gamma \vdash_{\bL } \Delta$$ to state that for every $$\psi \in \Delta , \Gamma \vdash_{\bL } \psi$$. Thus $$\Gamma \vdash_{\bL } \Delta$$ if and only if $$\Delta \subseteq C_{\bL }(\Gamma)$$.
If $$\bL = \langle L, \vdash_{\bL } \rangle$$ is a logic system $$\bL' = \langle L', \vdash_{\bL' } \rangle$$ whose language is $$L'$$ (hence all the $$L'$$-formulas are $$L$$-formulas) and whose consequence relation is defined by $\Gamma \vdash_{\bL' } \phi \txtiff \Gamma \vdash_{\bL } \phi,$ for every set of $$L'$$-formulas $$\Gamma$$ and every $$L'$$-formula $$\phi$$. In this situation we also say that $$\bL$$ is an expansion of $$\bL'$$.
## 3. Some examples of logics
We give some examples of logic systems that we will refer to in the course of this essay, which are assembled here for the reader’s convenience. Whenever possible we refer to the corresponding entries.
We use the standard convention of writing $$(\phi * \psi)$$ instead of $$* \phi \psi$$ for binary connectives and omit the external parenthesis in the formulas.
### 3.1 Classical propositional logic
We take the language of Classical propositional logic $$\bCPL$$ to be the set $$L_c = \{\wedge , \vee , \rightarrow , \top , \bot \},$$ where $$\wedge , \vee , \rightarrow$$ are binary connectives and $$\top , \bot$$ propositional constants. We assume the consequence relation defined by the usual truth-table method $$(\top$$ is interpreted as true and $$\bot$$ as false), that is,
$$\Gamma \vdash_{\bCPL } \phi\txtiff$$ every truth valuation that assigns true to all $$\psi \in \Gamma$$ assigns true to $$\phi$$.
The formulas $$\phi$$ such that $$\varnothing \vdash_{\bCPL } \phi$$ are the tautologies. Note that using the language $$L_c$$, the negation of a formula $$\phi$$ is defined as $$\phi \rightarrow \bot$$. For more information, see the entry on classical logic
### 3.2 Intuitionistic propositional logic
We take the language of Intuitionistic propositional logic to be the same as that of classical propositional logic, namely the set $$\{\wedge , \vee , \rightarrow , \top , \bot \}$$. The consequence relation is defined by the following Hilbert-style calculus.
#### Axioms:
All the formulas of the forms
• C0. $$\top$$
• C1. $$\phi \rightarrow(\psi \rightarrow \phi)$$
• C2. $$\phi \rightarrow(\psi \rightarrow(\phi \wedge \psi))$$
• C3. $$(\phi \wedge \psi) \rightarrow \phi$$
• C4. $$(\phi \wedge \psi) \rightarrow \psi$$
• C5. $$\phi \rightarrow(\phi \vee \psi)$$
• C6. $$\psi \rightarrow(\phi \vee \psi)$$
• C7. $$(\phi \vee \psi) \rightarrow((\phi \rightarrow \delta) \rightarrow((\psi \rightarrow \delta) \rightarrow \delta))$$
• C8. $$(\phi \rightarrow \psi) \rightarrow((\phi \rightarrow(\psi \rightarrow \delta)) \rightarrow(\phi \rightarrow \delta))$$
• C9. $$\bot \rightarrow \phi$$
#### Rule of inference
$\phi , \phi \rightarrow \psi / \psi \tag{Modus Ponens}$
### 3.3 Local Normal Modal logics
The language of modal logic we consider here is the set $$L_m = \{\wedge , \vee , \rightarrow , \neg , \Box , \top , \bot \}$$ that expands $$L_c$$ by adding the unary connective $$\Box$$. In the standard literature on modal logic a normal modal logic is defined not as a consequence relation but as a set of formulas with certain properties. A normal modal logic is a set $$\Lambda$$ of formulas of $$L_m$$ which contains all the tautologies of the language of classical logic, the formulas of the form $\Box(\phi \rightarrow \psi) \rightarrow(\Box \phi \rightarrow \Box \psi)$ and is closed under the rules
\begin{align*} \phi , \phi \rightarrow \psi / \psi \tag{Modus Ponens}\\ \phi / \Box \phi \tag{Modal Generalization}\\ \phi/ \bsigma(\phi), \textrm{ for every substitution } \sigma \tag{Substitution}\\ \end{align*}
Note that the set $$\Lambda$$ is closed under substitution instances, namely for every substitution $$\sigma$$, if $$\phi \in L_m$$, then $$\bsigma(\phi) \in L_m$$.
The least normal modal logic is called $$K$$ and can be axiomatized by the Hilbert-style calculus with axioms the tautologies of classical logic and the formulas $$\Box(\phi \rightarrow \psi) \rightarrow(\Box \phi \rightarrow \Box \psi)$$, and with rules of inference Modus Ponens, Modal Generalization and Substitution.
With a normal modal logic $$\Lambda$$ it is associated the consequence relation defined by the calculus that takes as axioms all the formulas in $$\Lambda$$ and as the only rule of inference Modus Ponens. The logic system given by this consequence relation is called the local consequence of $$\Lambda$$. We denote it by $$\blLambda$$. Its theorems are the elements of $$\Lambda$$. It holds that
$$\Gamma \vdash_{\blLambda} \phi\txtiff\phi \in \Lambda$$ or there are $$\phi_1 , \ldots ,\phi_n \in \Gamma$$ such that $$(\phi_1 \wedge \ldots \wedge \phi_n) \rightarrow \phi \in \Lambda$$.
### 3.4 Global Normal Modal logics
Another consequence relation is associated with each normal modal logic $$\Lambda$$. It is defined by the calculus that has as axioms the formulas of $$\Lambda$$ and as rules of inference Modus Ponens and Modal Generalization. The logic system given by this consequence relation is called the global consequence of $$\Lambda$$ and will be denoted by $$\bgLambda$$. It has the same theorems as the local $$\blLambda$$, namely the elements of $$\Lambda$$. The difference between $$\blLambda$$ and $$\bgLambda$$ lies in the consequences they allow to draw from nonempty sets of premises. This difference has an enormous effect on their algebraic behavior.
For more information on modal logic, see the entry on modal logic. The reader can find specific information on modal logics as consequence relations in Kracht 2006.
### 3.5 Intuitionistic Linear Logic without exponentials
We take as the language of Intuitionistic Linear Logic without exponentials the set $$\{\wedge , \vee , \rightarrow , * , 0, 1, \top , \bot \}$$, where $$\wedge , \vee , \rightarrow, *$$ are binary connectives and $$0, 1,\top , \bot$$ propositional constants. We denote the logic by $$\bILL$$. The axioms and rule of inference below provide a Hilbert-style axiomatization of this logic.
#### Axioms:
• L1. 1
• L2. $$(\phi \rightarrow \psi) \rightarrow((\psi \rightarrow \delta) \rightarrow(\phi \rightarrow \delta))$$
• L3. $$(\phi \rightarrow(\psi \rightarrow \delta)) \rightarrow(\psi \rightarrow(\phi \rightarrow \delta))$$
• L4. $$\phi \rightarrow(\psi \rightarrow(\phi * \psi))$$
• L5. $$(\phi \rightarrow(\psi \rightarrow \delta)) \rightarrow((\phi * \psi) \rightarrow \delta)$$
• L6. $$1 \rightarrow(\phi \rightarrow \phi)$$
• L7. $$(\phi \wedge \psi) \rightarrow \phi$$
• L8. $$(\phi \wedge \psi) \rightarrow \psi$$
• L9. $$\psi \rightarrow(\phi \vee \psi)$$
• L10. $$\phi \rightarrow(\phi \vee \psi)$$
• L11. $$((\phi \rightarrow \psi) \wedge(\phi \rightarrow \delta)) \rightarrow(\phi \rightarrow(\psi \wedge \delta))$$
• L12. $$((\phi \rightarrow \delta) \wedge(\psi \rightarrow \delta)) \rightarrow((\phi \vee \psi) \rightarrow \delta)$$
• L13. $$\phi \rightarrow \top$$
• L14. $$\bot \rightarrow \psi$$
#### Rules of inference:
\begin{align*} \phi , \phi \rightarrow \psi / \psi \tag{Modus Ponens}\\ \phi , \psi / \phi \wedge \psi \tag{Adjunction}\\ \end{align*}
The 0-ary connective 0 is used to define a negation by $$\neg \phi := \phi \rightarrow 0$$. No specific axiom schema deals with 0.
### 3.6 The system $$\bR$$ of Relevance Logic
The language we consider is the set $$\{\wedge , \vee , \rightarrow , \neg \}$$, where $$\wedge , \vee , \rightarrow$$ are binary connectives and $$\neg$$ a unary connective. A Hilbert style axiomatization for $$\bR$$ can be given by the rules of Intuitionistic Linear Logic without exponentials and the axioms L2, L3, L7-L12 of this logic together with the axioms
1. $$(\phi \rightarrow(\phi \rightarrow \psi)) \rightarrow(\phi \rightarrow \psi)$$
2. $$(\phi \rightarrow \neg \psi) \rightarrow(\psi \rightarrow \neg \psi)$$
3. $$(\phi \wedge(\psi \vee \delta)) \rightarrow((\phi \wedge \psi) \vee \phi \wedge \delta))$$
4. $$\neg \neg \phi \rightarrow \phi$$
## 4. Algebras
The algebraic study of a particular logic has to provide first of all its formal language with an algebraic semantics using a class of algebras whose properties are exploited to understand which properties the logic has. In this section, we present how the formal languages of propositional logics are given an algebraic interpretation. In the next section, we address the question of what is an algebraic semantics for a logic system.
We start by describing the first two steps involved in the algebraic study of propositional logics. Both are needed in order to endow propositional languages with algebraic interpretations. To expound them we will assume knowledge of first-order logic (see the entries on classical logic and first-order model theory) and we will call algebraic first-order languages, or simply algebraic languages, the first-order languages with equality and without any relational symbols, so that these languages have only operation symbols (also called function symbols), if any, in the set of their non-logical symbols.
The two steps we are about to expound can be summarized in the slogan:
Propositional formulas are terms.
The first step consist in looking at the formulas of any propositional language $$L$$ as the terms of the algebraic first-order language with $$L$$ as its set of operation symbols. This means that (i) every connective of $$L$$ of arity $$n$$ is taken as an operation symbol of arity $$n$$ (thus every 0-ary symbol of $$L$$ is taken as an individual constant) and that (ii) the propositional formulas of $$L$$ are taken as the terms of this first-order language; in particular the propositional variables are the variables of the first-order language. From this point of view the definition of $$L$$-formula is exactly the definition of $$L$$-term. We will refer to the algebraic language with $$L$$ as its set of operation symbols as the $$L$$-algebraic language.
The second step is to interpret the propositional formulas in the same manner in which terms of a first-order language are interpreted in a structure. In this way the concept of $$L$$-algebra comes into play. On a given set $$A$$, an $$n$$-ary connective is interpreted by a $$n$$-ary function on $$A$$ (a map that assigns an element of $$A$$ to every sequence $$\langle a_1 , \ldots ,a_n\rangle$$ of elements of $$A)$$. This procedure is a generalization of the truth-table interpretations of the languages of logic systems like classical logic and Łukasiewicz and Post’s finite-valued logics. In those cases, given the set of truth-values at play the function that interprets a connective is given by its truth-table.
A way to introduce algebras is as the models of some algebraic first-order language. We follow an equivalent route and give the definition of algebra using the setting of propositional languages. Let $$L$$ be a propositional language. An algebra $$\bA$$ of type $$L$$, or $$L$$-algebra for short, is a set $$A$$, called the carrier of $$\bA$$, together with a function $$* ^{\bA}$$ on $$A$$ of the arity of $$*$$, for every connective $$*$$ in $$L$$ (if $$*$$ is 0-ary, $$* ^{\bA}$$ is an element of $$A)$$. An algebra $$\bA$$ is trivial if its carrier is a one element set.
A valuation on an $$L$$-algebra $$\bA$$ is a map $$v$$ from the set of variables into its carrier $$A$$. Algebras together with valuations are used to interpret in a compositional way the formulas of $$L$$, assuming that a connective $$*$$ of $$L$$ is interpreted in an $$L$$-algebra $$\bA$$ by the function $$* ^{\bA}$$. Let $$\bA$$ be an algebra of type $$L$$ and $$v$$ a valuation on $$\bA$$. The value of a compound formula $$* \phi_1 \ldots \phi_n$$ is computed by applying the function $$* ^{\bA}$$ that interprets $$*$$ in $$\bA$$ to the previously computed values $$\bv(\phi_1), \ldots ,\bv(\phi_n)$$ of the formulas $$\phi_1 ,\ldots ,\phi_n$$. Precisely speaking the value $$\bv(\phi)$$ of a formula $$\phi$$ is defined inductively as follows:
1. $$\bv(p) = v(p)$$, for each variable $$p$$,
2. $$\bv(\dagger) = \dagger^{\bA}$$, if $$\dagger$$ is a 0-ary connective
3. $$\bv(* \phi_1 \ldots \phi_n) = * ^{\bA }(\bv(\phi_1), \ldots ,\bv(\phi_n))$$, if $$*$$ is a $$n$$-ary $$(n \gt 0)$$ connective.
Note that in this way we have obtained a map $$\bv$$ from the set of $$L$$-formulas to the carrier of $$\bA$$. It is important to notice that the value of a formula under a valuation depends only on the propositional variables that actually appear in the formula. Accordingly, if $$\phi$$ is a formula we use the notation $$\phi(p_1 , \ldots ,p_n)$$ to indicate that the variables that appear in $$\phi$$ are in the list $$p_1 , \ldots ,p_n$$, and given elements $$a_1 , \ldots ,a_n$$ of an algebra $$\bA$$ by $$\phi^{\bA }[a_1 , \ldots ,a_n]$$ we refer to the value of $$\phi(p_1 , \ldots ,p_n)$$ under any valuation $$v$$ on $$\bA$$ such that $$v(p_1) = a_1 , \ldots ,v(p_n) = a_n$$.
A third and fundamental step in the algebraic study of logics is to turn the set of formulas of a language $$L$$ into an algebra, the algebra of formulas of $$L$$, denoted by $$\bFm_L$$. This algebra has the set of $$L$$-formulas as carrier and the operations are defined as follows. For every $$n$$-ary connective $$*$$ with $$n \gt 0$$, the function $$* ^{\bFm_L}$$ is the map that sends each tuple of formulas $$(\phi_1 , \ldots ,\phi_n)$$ (where $$n$$ is the arity of $$* )$$ to the formula $$* \phi_1 \ldots \phi_n$$, and for every 0-ary connective $$\dagger , \dagger^{\bFm_L}$$ is $$\dagger$$. If no confusion is likely we suppress the subindex in $$\bFm_L$$ and write $$\bFm$$ instead.
### 4.1 Some concepts of universal algebra and model theory
Algebras are a particular type of structure or model. An $$L$$-algebra is a structure or model for the $$L$$-algebraic first-order language. Therefore the concepts of model theory for the first-order languages apply to them (see the entries on classical logic and first-order model theory). We need some of these concepts. They are also used in universal algebra, a field that to some extent can be considered the model theory of the algebraic languages. We introduce the definitions of the concepts we need.
Given an algebra $$\bA$$ of type $$L$$, a congruence of $$\bA$$ is an equivalence relation $$\theta$$ on the carrier of $$\bA$$ that satisfies for every $$n$$-ary connective $$* \in L$$ the following compatibility property: for every $$a_1 , \ldots ,a_n, b_1 , \ldots ,b_n \in A$$,
$\textrm{if } a_1\theta b_1 , \ldots ,a_n \theta b_1, \textrm{ then } *^{\bA}(a_1 ,\ldots ,a_n)\ \theta *^{\bA}(b_1 ,\ldots ,b_n).$
Given a congruence $$\theta$$ of $$\bA$$ we can reduce the algebra by identifying the elements which are related by $$\theta$$. The algebra obtained is the quotient algebra of $$\bA$$ modulo $$\theta$$. It is denoted by $$\bA/\theta$$, its carrier is the set $$A/\theta$$ of equivalence classes $$[a]$$ of the elements $$a$$ of $$A$$ modulo the equivalence relation $$\theta$$, and the operations are defined as follows:
1. $$\dagger^{\bA/\theta} = [\dagger^{\bA}]$$, for every 0-ary connective $$\dagger$$,
2. $$* ^{\bA/\theta}([a_1], \ldots, [a_n]) = [* ^{\bA }(a_1 ,\ldots ,a_n)]$$, for every connective $$*$$ whose arity is $$n$$ and $$n \gt 0$$.
The compatibility property ensures that the definition is sound.
Let $$\bA$$ and $$\bB$$ be $$L$$-algebras. A homomorphism $$h$$ from $$\bA$$ to $$\bB$$ is a map $$h$$ from $$A$$ to $$B$$ such that for every 0-ary symbol $$\dagger \in L$$ and every $$n$$-ary connective $$* \in L$$
1. $$h(\dagger^{\bA }) = \dagger^{\bB}$$
2. $$h(* ^{\bA }(a_1 ,\ldots ,a_n)) = * ^{\bB }(h(b_1),\ldots ,h(b_n))$$, for all $$a_1 , \ldots ,a_n \in A$$.
We say that $$\bB$$ is a homomorphic image of $$\bA$$ if there is a homomorphism from $$\bA$$ to $$\bB$$ which is an onto map from $$A$$ to $$B$$. An homomorphism from $$\bA$$ to $$\bB$$ is an isomorphism if it is a one-to-one and onto map from $$A$$ to $$B$$. If an isomorphism from $$\bA$$ to $$\bB$$ exists, we say that $$\bA$$ and $$\bB$$ are isomorphic and that $$\bB$$ is an isomorphic image (or a copy) of $$\bA$$.
Let $$\bA$$ and $$\bB$$ be $$L$$-algebras. $$\bA$$ is a subalgebra of $$\bB$$ if (1) $$A \subseteq B$$, (2) the interpretations of the 0-ary symbols of $$L$$ in $$\bB$$ belong to $$A$$ and $$A$$ is closed under the functions of $$\bB$$ that interpret the non 0-ary symbols, and (3) the interpretations of the 0-ary symbols in $$\bA$$ coincide with their interpretations in $$\bB$$ and the interpretations on $$\bA$$ of the other symbols in $$L$$ are the restrictions to $$\bA$$ of their interpretations in $$\bB$$.
We refer the reader to the entry on first-order model theory for the notions of direct product (called product there) and ultraproduct.
### 4.2 Varieties and quasivarieties
The majority of classes of algebras that provide semantics for propositional logics are quasivarieties and in most cases varieties. The theory of varieties and quasivarieties is one of the main subjects of universal algebra.
A variety of $$L$$-algebras is a class of $$L$$-algebras that is definable in a very simple way (by equations) using the $$L$$-algebraic language. An $$L$$-equation is a formula $$\phi \approx \psi$$ where $$\phi$$ and $$\psi$$ are terms of the $$L$$-algebraic language (that is, $$L$$-formulas if we take the propositional logics point of view). An equation $$\phi \approx \psi$$ is valid in an algebra $$\bA$$, or $$\bA$$ is a model of $$\phi \approx \psi$$, if for every valuation $$v$$ on $$\bA, \bv(\phi) = \bv(\psi)$$. This is exactly the same as to saying that the universal closure of $$\phi \approx \psi$$ is a sentence true in $$\bA$$ according to the usual semantics for first-order logic with equality. A variety of $$L$$-algebras is a class of $$L$$-algebras which is the class of all the models of a given set of $$L$$-equations.
Quasivarieties of $$L$$-algebras are classes of $$L$$-algebras definable using the $$L$$-algebraic language in a slightly more complex way than in varieties. A proper $$L$$-quasiequation is a formula of the form $\bigwedge_{i \le n} \phi_i \approx \psi_i \rightarrow \phi \approx \psi.$ An $$L$$-quasiequation is a formula of the above form but possibly with an empty antecedent, in which case it is just the equation $$\phi \approx \psi$$. Hence, the $$L$$-quasiequations are the proper $$L$$-quasiequations and the $$L$$-equations. An $$L$$-quasiequation is valid in an $$L$$-algebra $$\bA$$, or the algebra is a model of it, if the universal closure of the quasiequation is sentence true in $$\bA$$. A quasivariety of $$L$$-algebras is a class of algebras which is the class of the models of a given set of $$L$$-quasiequations. Since equations are quasiequations, each variety is a quasivariety. The converse is false.
Varieties and quasivarieties can be characterized by the closure properties they enjoy. A class of $$L$$-algebras is a variety if and only if it is closed under subalgebras, direct products, and homomorphic images. The variety generated by a class $$\bK$$ of $$L$$-algebras is the least class of $$L$$-algebras that includes $$\bK$$ and is closed under subalgebras, direct products and homomorphic images. It is also the class of the algebras that are models of the equations valid in $$\bK$$. For example, the variety generated by the algebra of the two truth-values for classical logic is the class of Boolean algebras. If we restrict that algebra to the operations for conjunction and disjunction only, it generates the variety of distributive lattices and if we restrict it to the operations for conjunction and disjunction and the interpretations of $$\top$$ and $$\bot$$, it generates the variety of bounded distributive lattices.
A class of $$L$$-algebras is a quasivariety if and only if it is closed under subalgebras, direct products, ultraproducts, isomorphic images, and contains a trivial algebra. The quasivariety generated by a class $$\bK$$ of $$L$$-algebras is the least class of $$L$$-algebras that includes $$\bK$$, the trivial algebras and is closed under subalgebras, direct products, ultraproducts, and isomorphic images.
An SP-class of $$L$$-algebras is a class of $$L$$-algebras that contains a trivial algebra and is closed under isomorphic images, subalgebras, and direct products. Thus quasivarieties and varieties are all SP-classes. The SP-class generated by a class $$\bK$$ of $$L$$-algebras is the least class of $$L$$-algebras that includes $$\bK$$, the trivial algebras and is closed under subalgebras, direct products and isomorphic images.
## 5. Algebraic semantics
The term ‘algebraic semantics’ was (and many times still is) used in the literature in a loose way. To provide a logic with an algebraic semantics was to interpret its language in a class of algebras, define a notion of satisfaction of a formula (under a valuation) in an algebra of the class and prove a soundness and completeness theorem, usually for the theorems of the logic only. Nowadays there is a precise concept of algebraic semantics for a logic system. It was introduced by Blok and Pigozzi in Blok & Pigozzi 1989. In this concept we find a general way to state in mathematically precise terms what is common to the many cases of purported algebraic semantics for specific logic systems found in the literature. We expose the notion in this section. To motivate the definition we discuss several examples first, stressing what is common to all of them. The reader does not need to know about the classes of algebras that provide algebraic semantics we refer to in the examples. Its existence is what is important.
The prototypical examples of algebraic semantics for propositional logics are the class BA of Boolean algebras, which is the algebraic semantics for classical logic, and the class HA of Heyting algebras, which is the algebraic semantics for intuitionistic logic. Every Boolean algebra and every Heyting algebra $$\bA$$ has a greatest element according to their natural order; this element is denoted usually by $$1^{\bA}$$ and interprets the propositional constant symbol $$\top$$. It is taken as the distinguished element relative to which the algebraic semantics is given. The algebraic semantics of these two logics works as follows:
Let $$\bL$$ be classical or intuitionistic logic and let $$\bK(\bL)$$ be the corresponding class of algebras BA or HA. It holds that
$$\Gamma \vdash_{\bL } \phi\txtiff$$ for every $$\bA \in \bK(\bL)$$ and every valuation $$v$$ on $$\bA$$, if $$\bv(\psi) = 1^{\bA}$$ for all $$\psi \in \Gamma$$, then $$\bv(\phi) = 1^{\bA}$$.
This is the precise content of the statement that BA and HA are an algebraic semantics for classical logic and for intuitionistic logic respectively. The implication from left to right in the expression above is an algebraic soundness theorem and the implication from right to left an algebraic completeness theorem.
There are logics for which an algebraic semantics is provided in the literature in a slightly different way from the one given by the schema above. Let us consider the example in Section 3.5 of Intuitionistic Linear Logic without exponentials. We denote by $$\bILsubZ$$ the class of IL-algebras with zero defined in Troelstra 1992 (but adapted to the language of $$\bILL)$$. Each $$\bA \in \bILsubZ$$ is a lattice with extra operations and thus has its lattice order $$\le^{\bA}$$. This lattice order has a greatest element which we take as the interpretation of $$\top$$. On each one of these algebras $$\bA$$ there is a designated element $$1^{\bA}$$ (the interpretation of 1) that may be different from the greatest element. It holds:
$$\Gamma \vdash_{\bILL } \phi\txtiff$$ for every $$\bA \in \bILsubZ$$ and every valuation $$v$$ on $$\bA$$, if $$1^{\bA } \le^{\bA } \bv(\psi)$$ for all $$\psi \in \Gamma$$, then $$1^{\bA } \le^{\bA } \bv(\phi)$$.
In this case one does not consider only a designated element in every algebra $$\bA$$ but a set of designated elements, namely the elements of $$\bA$$ greater than or equal to $$1^{\bA}$$, to provide the definition. Let us denote this set by $$\tD (\bA)$$, and notice that $$\tD (\bA) = \{a \in A: 1^{\bA } \wedge^{\bA} a = 1^{\bA }\}$$. Hence,
$$\Gamma \vdash_{\bILL } \phi\txtiff$$ for every $$\bA \in \bILsubZ$$ if $$\bv[\Gamma] \subseteq \tD (\bA)$$, then $$\bv(\phi) \in \tD (\bA)$$.
Still there are even more complex situations. One of them is the system $$\bR$$ of relevance logic. Consider the class of algebras $$\bRal$$ defined in Font & Rodríguez 1990 (see also Font & Rodríguez 1994) and denoted there by ‘$$\bR$$’. Let us consider for every $$\bA \in \bRal$$ the set $\tE(\bA) := \{a \in A: a \wedge^{\bA }(a \rightarrow^{\bA } a) = a \rightarrow^{\bA } a\}.$ Then $$\bRal$$ is said to be an algebraic semantics for $$\bR$$ because the following holds:
$$\Gamma \vdash_{\bR } \phi\txtiff$$ for every $$\bA \in \bRal$$ and every valuation $$v$$ on $$\bA$$, if $$\bv[\Gamma] \subseteq \tE (\bA)$$, then $$\bv(\phi) \in \tE (\bA)$$.
The common pattern in the examples above is that the algebraic semantics is given by
1. a class of algebras $$\bK$$,
2. in each algebra in $$\bK$$ a set of designated elements that plays the role $$1^{\bA}$$ (more precisely the set $$\{1^{\bA }\})$$ plays in the cases of classical and intuitionistic logic, and
3. this set of designated elements is definable (in the same manner on every algebra) by an equation in the sense that it is the set of elements of the algebra that satisfy the equation (i.e., its solutions). For BA and HA the equation is $$p \approx \top$$. For $$\bRal$$ it is $$p \rightarrow(p \wedge p) \approx p \rightarrow p$$, and for $$\bILsubZ$$ it is $$1 \wedge p \approx 1$$.
The main point in Blok and Pigozzi’s concept of algebraic semantics comes from the realization, mentioned in (3) above, that the set of designated elements considered in the algebraic semantics of known logics is in fact the set of solutions of an equation, and that what practice forced researchers to look for when they tried to obtain algebraic semantics for new logics was in fact, although not explicitly formulated in these terms, an equational way to define uniformly in every algebra a set of designated elements in order to obtain an algebraic soundness and completeness theorem.
We are now in a position to expose the mathematically precise concept of algebraic semantics. To develop a fruitful and general theory of the algebraization of logics some generalizations beyond the well-known concrete examples have to be made. In the definition of algebraic semantics, one takes the move from a single equation to a set of them in the definability condition for the set of designated elements.
Before stating Blok and Pigozzi’s definition we need to introduce a notational convention. Given an algebra $$\bA$$ and a set of equations $$\iEq$$ in one variable, we denote by $$\iEq(\bA)$$ the set of elements of $$\bA$$ that satisfy all the equations in $$\iEq$$. Then a logic $$\bL$$ is said to have an algebraic semantics if there is a class of algebras $$\bK$$ and a set of equations $$\iEq$$ in one variable such that
• (**) $$\Gamma \vdash_{\bL } \phi \txtiff$$ for every $$\bA \in \bK$$ and every valuation $$v$$ on $$\bA$$, if $$\bv[\Gamma] \subseteq \iEq(\bA)$$, then $$\bv(\phi) \in \iEq(\bA)$$.
In this situation we say that the class of algebras $$\bK$$ is an $$\iEq$$-algebraic semantics for $$\bL$$, or that the pair $$(\bK, \iEq)$$ is an algebraic semantics for $$\bL$$. If $$\iEq$$ consists of a single equation $$\delta(p) \approx \varepsilon(p)$$ we will simply say that $$\bK$$ is a $$\delta(p) \approx \varepsilon(p)$$-algebraic semantics for $$\bL$$. In fact, Blok and Pigozzi required that $$\iEq$$ should be finite in their definition of algebraic semantics. But it is better to be more general. The definition clearly encompasses the situations encountered in the examples.
If $$\bK$$ is an $$\iEq$$-algebraic semantics for a finitary logic $$\bL$$ and $$\iEq$$ is finite, then the quasivariety generated by $$\bK$$ is also an $$\iEq$$-algebraic semantics. The same does not hold in general if we consider the generated variety. For this reason it is customary and useful when developing the theory of the algebraization of finitary logics to consider quasivarieties of algebras as algebraic semantics instead of arbitrary subclasses that generate them. Conversely, if a quasivariety is an $$\iEq$$-algebraic semantics for a finitary $$\bL$$ and $$\iEq$$ is finite, so is any subclass of the quasivariety that generates it.
In the best-behaved cases, the typical algebraic semantics of a logic is a variety, for instance in all the examples discussed above. But there are cases in which it is not.
A quasivariety can be an $$\iEq$$-algebraic semantics for a logic and an $$\iEq'$$-algebraic semantics for another logic (with $$\iEq$$ and $$\iEq'$$ different). For example, due to Glivenko’s theorem (see the entry on intuitionistic logic) the class of Heyting algebras is a $$\{\neg \neg p \approx 1\}$$-algebraic semantics for classical logic and it is the standard $$\{p \approx 1\}$$-algebraic semantics for intuitionistic logic. Moreover, different quasivarieties of algebras can be an $$\iEq$$-algebraic semantics for the same logic. It is known that there is a quasivariety that properly includes the variety of Boolean algebras and is a $$\{p \approx 1\}$$-algebraic semantics for classical propositional logic. It is also known that for some logics with an algebraic semantics (relative to some set of equations), the natural class of algebras that corresponds to the logic is not an algebraic semantics (for any set of equations) of it. One example where this situation holds is in the local normal modal logic $$\blK$$. Finally, there are logics that do not have any algebraic semantics.
These facts highlight the need for some criteria of the goodness of a pair $$(\bK, \iEq)$$ to provide a natural algebraic semantics for a logic $$\bL$$ when some exists. One such criterion would be that $$\bL$$ is an algebraizable logic with $$(\bK, \iEq)$$ as an algebraic semantics. Another that $$\bK$$ is the natural class of algebras associated with the logic $$\bL$$. The notion of the natural class of algebras of a logic system will be discussed in Section 8 and the concept of algebraizable logic in Section 9.
## 6. Logical matrices
In the last section, we saw that to provide a logic with an algebraic semantics we need in many cases to consider in every algebra a set of designated elements instead of a single designated one. In the examples we discussed, the set of designated elements was definable in the algebras by one equation. This motivated the definition of algebraic semantics in Section 5. For many logics, to obtain a semantics similar to an algebraic semantics using the class of algebras naturally associated with them one needs for every algebra a set of designated elements that cannot be defined using only the equations of the algebraic language or is not even definable by using this language only. As we already mentioned, one example where this happens is the local consequence of the normal modal logic $$K$$.
To endow every logic with a semantics of an algebraic kind one has to consider, at least, algebras together with a set of designated elements, without any requirement about its definability using the corresponding algebraic language. These pairs are the logical matrices. Tarski defined the general concept of logical matrix in the 1920s but the concept was already implicit in previous work by Łukasiewicz, Bernays, Post and others, who used truth-tables, either in independence proofs or to define logics different from classical logic. A logical matrix is a pair $$\langle \bA, D \rangle$$ where $$\bA$$ is an algebra and $$D$$ a subset of the carrier $$A$$ of $$\bA$$; the elements of $$D$$ are called the designated elements of the matrix and accordingly $$D$$ is called the set of designated elements (sometimes it is also called the truth set of the matrix). Logical matrices were first used as models of the theorems of specific logic systems, for instance in the work of McKinsey and Tarski, and also to define sets of formulas with similar properties to those of the set of theorems of a logic system, namely closure under substitution instances. This was the case of the $$n$$-valued logics of Łukasiewicz and of his infinite-valued logic. It was Tarski who first considered logical matrices as a general tool to define this kind of sets.
The general theory of logical matrices explained in this entry is due mainly to Polish logicians, starting with Łoś 1949 and continuing in Łoś & Suszko 1958, building on previous work by Lindenbaum. In Łoś and Suszko’s paper matrices are used for the first time both as models of logic systems (in our sense) and to define these kinds of systems.
In the rest of this section, we present the relevant concepts of the theory of logical matrices using modern terminology.
Given a logic $$\bL$$, a logical matrix $$\langle \bA, D \rangle$$ is said to be a model of $$\bL$$ if wherever $$\Gamma \vdash_{\bL } \phi$$ then every valuation $$v$$ on $$\bA$$ that maps the elements of $$\Gamma$$ to some designated value (i.e., an element of $$D)$$ also maps $$\phi$$ to a designated value. When $$\langle \bA, D \rangle$$ is a model of $$\bL$$ it is said that $$D$$ is an $$\bL$$-filter of the algebra $$\bA$$. The set of $$\bL$$-filters of an algebra $$\bA$$ plays a crucial role in the theory of the algebraization of logic systems. We will come to this point later.
A class $$\bM$$ of logical matrices is said to be a matrix semantics for a logic $$\bL$$ if
• (*)$$\Gamma \vdash_{\bL } \phi\txtiff$$ for every $$\langle \bA, \tD \rangle \in \bM$$ and every valuation $$v$$ on $$\bA$$, if $$\bv[\Gamma] \subseteq D$$, then $$\bv(\phi) \in D$$.
The implication from left to right says that $$\bL$$ is sound relative to $$\bM$$, and the other implication says that it is complete. In other words, $$\bM$$ is a matrix semantics for $$\bL$$ if and only if every matrix in $$\bM$$ is a model of $$\bL$$ and moreover for every $$\Gamma$$ and $$\phi$$ such that $$\Gamma \not\vdash_{\bL } \phi$$ there is a model $$\langle \bA, \tD \rangle$$ of $$\bL$$ in $$\bM$$ that witnesses the fact, namely there is a valuation on the model that sends the formulas in $$\Gamma$$ to designated elements and $$\phi$$ to a non-designated one.
Logical matrices are also used to define logics semantically. If $$\cM = \langle \bA, D \rangle$$ is a logical matrix, the relation defined by
$$\Gamma \vdash_{\cM } \phi\txtiff$$ for every valuation $$v$$ on $$\bA$$ if $$\bv(\psi) \in D$$ for all $$\psi \in \Gamma$$, then $$\bv(\phi) \in D$$
is a consequence relation which is substitution-invariant; therefore $$\langle L, \vdash_{\cM } \rangle$$ is a logic system. Similarly, we can define the logic of a class of matrices $$\bM$$ by taking condition (*) as a definition of a consequence relation. In the entry on many-valued logic the reader can find several logics defined in this way.
Every logic (independently of how it is defined) has a matrix semantics. Moreover, every logic has a matrix semantics whose elements have the property of being reduced in the following sense: A matrix $$\langle \bA, D \rangle$$ is reduced if there are no two different elements of $$A$$ that behave in the same way. We say that $$a, b \in A$$ behave in the same way in $$\langle \bA, D \rangle$$ if for every formula $$\phi (q, p_1 , \ldots ,p_n)$$ and all elements $$d_1 , \ldots ,d_n \in A$$ $\phi^{\bA }[a, d_1 , \ldots ,d_n] \in D \txtiff \phi^{\bA }[b, d_1 , \ldots ,d_n] \in D.$ Thus $$a, b \in A$$ behave differently if there is a formula $$\phi(q, p_1 , \ldots ,p_n)$$ and elements $$d_1 , \ldots ,d_n \in A$$ such that one of $$\phi^{\bA }[a, d_1 , \ldots ,d_n]$$ and $$\phi^{\bA }[b, d_1 , \ldots ,d_n]$$ belongs to $$D$$ but not both. The relation of behaving in the same way in $$\langle \bA, D \rangle$$ is a congruence relation of $$\bA$$. This relation is known after Blok & Pigozzi 1986, 1989 as the Leibniz congruence of the matrix $$\langle \bA, D \rangle$$ and is denoted by $$\bOmega_{\bA }(D)$$. It can be characterized as the greatest congruence relation of $$\bA$$ that is compatible with $$D$$, that is, that does not relate elements in $$D$$ with elements not in $$D$$. The concept of Leibniz congruence plays a fundamental role in the general theory of the algebraization of the logic systems developed during the 1980s by Blok and Pigozzi. The reader is referred to Font, Jansana, & Pigozzi 2003 and Czelakowski 2001 for extensive information on the developments around the concept of Leibniz congruence during this period.
Every matrix $$\cM$$ can be turned into a reduced matrix by identifying the elements related by its Leibniz congruence. This matrix is called the reduction of $$\cM$$ and is usually denoted by $$\cM^*$$. A matrix and its reduction are models of the same logic systems, and since reduced matrices have no redundant elements, the classes of reduced matrices that are matrix semantics for logic systems are usually taken as the classes of matrices that deserve study; they are better suited to encoding in algebraic-like terms the properties of the logics that have them as their matrix semantics.
The proof that every logic system has a reduced matrix semantics (i.e., a matrix semantics consisting of reduced matrices) is as follows. Let $$\bL$$ be a logic system. Consider the matrices $$\langle \bFm_L, T \rangle$$ over the formula algebra, where $$T$$ is a theory of $$\bL$$. These matrices are known as the Lindenbaum matrices of $$\bL$$. It is not difficult to see that the class of those matrices is a matrix semantics for $$\bL$$. Since a matrix and its reduction are models of the same logics, the reductions of the Lindenbaum matrices of $$\bL$$ constitute a matrix semantics for $$\bL$$ too, and indeed a reduced one. Moreover, any class of reduced matrix models of $$\bL$$ that includes the reduced Lindenbaum matrices of $$\bL$$ is automatically a complete matrix semantics for $$\bL$$. In particular, the class of all reduced matrix models of $$\bL$$ is a complete matrix semantics for $$\bL$$. We denote this class by $$\bRMatr(\bL)$$.
The above proof can be seen as a generalization of the Lindenbaum-Tarski method for proving algebraic completeness theorems that we will discuss in the next section.
The class of the algebras of the matrices in $$\bRMatr(\bL)$$ plays a prominent role in the theory of the algebraization of logics and it is denoted by $$\bAlg^*\bL$$. It has been considered for a long time the natural class of algebras that has to be associated with a given logic $$\bL$$ as its algebraic counterpart. For instance, in the examples considered above, the classes of algebras that were given as algebraic semantics of the different logics (Boolean algebras, Heyting algebras, etc.) are exactly the class $$\bAlg^*\bL$$ of the corresponding logic $$\bL$$. And in fact the class $$\bAlg^*\bL$$ coincides with what was taken to be the natural class of algebras for all the logics $$\bL$$ studied up to the 1990s. In the 1990s, due to the knowledge acquired of several logics not studied before, some authors proposed another way to define the class of algebras that has to be counted as the algebraic counterpart to be associated with a given logic $$\bL$$. For many logics $$\bL$$ it leads exactly to the class $$\bAlg^*\bL$$ but for others it gives a class that extends it properly. We will see it in Section 8.
## 7. The Lindenbaum-Tarski method for proving algebraic completeness theorems
We now discuss the method that is most commonly used to prove that a class of algebras $$\bK$$ is a $$\delta(p) \approx \varepsilon(p)$$-algebraic semantics for a logic $$\bL$$, namely the Lindenbaum-Tarski method. It is the standard method used to prove that the classes of algebras of the examples mentioned in Section 5 are algebraic semantics for the corresponding logics.
The Lindenbaum-Tarski method contributed in two respects to the elaboration of important notions in the theory of the algebraization of logics. It underlies Blok and Pigozzi’s notion of algebraizable logic and reflecting on it some ways to define for each logic a class of algebras can be justified as providing a natural class. We will consider this issue in Section 8.
The Lindenbaum-Tarski method can be outlined as follows. To prove that a class of algebras $$\bK$$ is a $$\delta(p) \approx \varepsilon(p)$$-algebraic semantics for a logic $$\bL$$ first it is shown that $$\bK$$ gives a sound $$\delta(p) \approx \varepsilon(p)$$-semantics for $$\bL$$, namely that if $$\Gamma \vdash_{\bL } \phi$$, then for every $$\bA \in \bK$$ and every valuation $$v$$ in $$\bA$$ if the values of the formulas in $$\Gamma$$ satisfy $$\delta(p) \approx \varepsilon(p)$$, then the value of $$\phi$$ does too. Secondly, the other direction, that is, the completeness part, is proved by what is properly known as the Lindenbaum-Tarski method. This method uses the theories of $$\bL$$ to obtain matrices on the algebra of formulas and then reduces these matrices in order to get for each one, a matrix whose algebra is in $$\bK$$ and whose set of designated elements is the set of elements of the algebra that satisfy $$\delta(p) \approx \varepsilon(p)$$. We proceed to describe the method step by step.
Let $$\bL$$ be one of the logics discussed in the examples in Section 5. Let $$\bK$$ be the corresponding class of algebras we considered there and let $$\delta(p) \approx \varepsilon(p)$$ be the equation in one variable involved in the soundness and completeness theorem. To prove the completeness theorem one proceeds as follows. Given any set of formulas $$\Gamma$$:
1. The theory $$C_{\bL }(\Gamma) = \{\phi : \Gamma \vdash_{\bL } \phi \}$$ of $$\Gamma$$, which we denote by $$T$$, is considered and the binary relation $$\theta(T)$$ on the set of formulas is defined using the formula $$p \leftrightarrow q$$ as follows: $\langle \phi , \psi \rangle \in \theta(T) \txtiff \phi \leftrightarrow \psi \in T.$
2. It is shown that $$\theta(T)$$ is a congruence relation on $$\bFm_L$$. The set $$[\phi]$$ of the formulas related to the formula $$\phi$$ by $$\theta(T)$$ is called the equivalence class of $$\phi$$.
3. A new matrix $$\langle \bFm/\theta(T), T/\theta(T) \rangle$$ is obtained by identifying the formulas related by $$\theta(T)$$, that is, $$\bFm/\theta(T)$$ is the quotient algebra of $$\bFm$$ modulo $$\theta(T)$$ and $$T/\theta(T)$$ is the set of equivalence classes of the elements of $$T$$. Recall that the algebraic operations of the quotient algebra are defined by: $* ^{\bFm/\theta(T) }([\phi_1],\ldots ,[\phi_n]) = [* \phi_1 \ldots \phi_n ] \dagger^{\bFm/\theta(T) } = [\dagger]$
4. It is shown that $$\theta(T)$$ is a relation compatible with $$T$$, i.e., that if $$\langle \phi , \psi \rangle \in \theta(T)$$ and $$\phi \in T$$, then $$\psi \in T$$. This implies that $\phi \in T \txtiff [\phi] \subseteq T \txtiff [\phi] \in T/\theta(T).$
5. It is proved that the matrix $$\langle \bFm/\theta(T), T/\theta(T) \rangle$$ is reduced, that $$\bFm/\theta(T)$$ belongs to $$\bK$$ and that $$T/\theta(T)$$ is the set of elements of $$\bFm/\theta(T)$$ that satisfy the equation $$\delta(p) \approx \varepsilon(p)$$ in $$\bFm/\theta(T)$$.
The proof of the completeness theorem then goes as follows. (4) and (5) imply that for every formula $$\psi , \Gamma \vdash_{\bL } \psi$$ if and only if $$[\psi]$$ satisfies the equation $$\delta(p) \approx \varepsilon(p)$$ in the algebra $$\bFm/\theta(T)$$. Thus, considering the valuation $$id$$ mapping every variable $$p$$ to its equivalence class $$[p]$$, and whose extension $$\boldsymbol{id}$$ to the set of all formulas is such that $$\boldsymbol{id}(\phi) = [\phi]$$ for every formula $$\phi$$, we have for every formula $$\psi$$,
$$\Gamma \vdash_{\bL } \psi\txtiff\boldsymbol{id}(\psi)$$ satisfies the equation $$\delta(p) \approx \varepsilon(p)$$ in $$\bFm/\theta(T)$$.
Hence, since by (5) $$\bFm/\theta(T) \in \bK$$, it follows that if $$\Gamma \not\vdash_{\bL }\phi$$, then there is an algebra $$\bA$$ (namely $$\bFm/\theta(T))$$ and a valuation $$v$$ (namely $$id)$$ such that the elements of $$\bv[\Gamma]$$ satisfy the equation on $$\bA$$ but $$\bv(\phi)$$ does not.
The Lindenbaum-Tarski method, when successful, shows that the class of algebras $$\{\bFm/\theta(T): T$$ is a theory of $$\bL\}$$ is a $$\delta(p) \approx \varepsilon(p)$$-algebraic semantics for $$\bL$$. Therefore it also shows that every class of algebras $$\bK$$ which is $$\delta(p) \approx \varepsilon(p)$$-sound for $$\bL$$ and includes $$\{\bFm/\theta(T): T$$ is a theory of $$\bL\}$$ is also a $$\delta(p) \approx \varepsilon(p)$$-algebraic semantics for $$\bL$$.
Let us make some remarks on the Lindenbaum-Tarski method just described. The first is important for the generalizations leading to the classes of algebras associated with a logic. The other to obtain the conditions in the definition of the concept of algebraizable logic.
1. Conditions (4) and (5) imply that $$\theta(T)$$ is in fact the Leibniz congruence of $$\langle \bFm_L, T \rangle$$.
2. When the Lindenbaum-Tarski method succeeds, it usually holds that in every algebra $$\bA \in \bK$$, the relation defined by the equation $\delta(p \leftrightarrow q) \approx \varepsilon(p \leftrightarrow q),$ which is the result of replacing in $$\delta(p) \approx \varepsilon(p)$$ the letter $$p$$ by the formula $$p \leftrightarrow q$$ that defines the congruence relation of a theory, is the identity relation on $$A$$.
3. For every formula $$\phi$$, the formulas $$\delta(p/\phi) \leftrightarrow \varepsilon(p/\phi)$$ and $$\phi$$ are interderivable in $$\bL$$ (i.e., $$\phi \vdash_{\bL } \delta(p/\phi) \leftrightarrow \varepsilon(p/\phi)$$ and $$\delta(p/\phi) \leftrightarrow \varepsilon(p/\phi) \vdash_{\bL } \phi)$$.
The concept of algebraizable logic introduced by Blok and Pigozzi, which we will discuss in Section 9, can be described roughly by saying that a logic $$\bL$$ is algebraizable if it has an algebraic semantics $$(\bK, \iEq)$$ such that (1) $$\bK$$ is included in the natural class of algebras $$\bAlg^*\bL$$ associated with $$\bL$$ and (2) the fact that $$(\bK, \iEq)$$ is an algebraic semantics can be proved by using the Lindenbaum-Tarski method slightly generalized.
## 8. The natural class of algebras of a logic system
We shall now discuss the two definitions that have been considered as providing natural classes of algebras associated with a logic $$\bL$$. Both definitions can be seen as arising from an abstraction of the Lindenbaum-Tarski method and we follow this path in introducing them. The common feature of these abstractions is that in them the specific way in which the relation $$\theta(T)$$ is defined in the Lindenbaum-Tarski method is disregarded.
It has to be remarked that, nonetheless, for many logics both definitions lead to the same class. But the classes obtained from both definitions have been considered in the algebraic studies of many particular logics (for some logics one, for others the other) the natural class that deserves to be studied.
We already encountered the first generalization in Section 6 when we showed that every logic has a reduced matrix semantics. It leads to the class of algebras $$\bAlg^*\bL$$; that its definition is a generalization of the Lindenbaum-Tarski method comes from the realization that the relation $$\theta(T)$$, associated with an $$\bL$$-theory, defined in the different completeness proofs in the literature that use the Lindenbaum-Tarski method is in fact the Leibniz congruence of the matrix $$\langle \bFm_L, T \rangle$$ and that therefore the matrix $$\langle \bFm/\theta(T), T/\theta(T) \rangle$$ is its reduction. As we mentioned in Section 6, for every logic $$\bL$$ every $$\bL$$-sound class of matrices $$\bM$$ that contains all the matrices $$\langle \bFm/\bOmega_{\bFm_L }(T), T/ \bOmega_{\bFm_L }(T) \rangle$$, where $$T$$ is a theory of $$\bL$$, is a complete reduced matrix semantics for $$\bL$$. From this perspective the notion of the Leibniz congruence of a matrix can be taken as a generalization to arbitrary matrices of the idea that comes from the Lindenbaum-Tarski procedure of proving completeness. Following this course of reasoning the class $$\bAlg^*\bL$$ of the algebras of the reduced matrix models of a logic $$\bL$$ is the natural class of algebras to associate with $$\bL$$. It is the class
$$\{\bA/\bOmega_{\bA }(F): \bA$$ is an $$\bL$$-algebra and $$F$$ is a $$\bL$$-filter of $$\bA\}$$.
The second way of generalizing the Lindenbaum-Tarski method uses another fact, namely that in the examples discussed in Section 3 the relation $$\theta(T)$$ is also the relation $$\Omega^{\sim}_{\bFm_L }(T)$$ defined by the condition
\begin{align*} \langle \phi , \psi \rangle \in \bOmega^{\sim}_{\bFm_L }(T)\txtiff & \forall T' \in \tTH(\bL),\\ & \forall p \in V, \\ &\forall \gamma(p) \in \bFm_L (T \subseteq T' \Rightarrow (\gamma(p/\phi) \in T' \Leftrightarrow \gamma(p/\psi) \in T')). \end{align*}
For every logic $$\bL$$ and every $$\bL$$-theory $$T$$ the relation $$\bOmega^{\sim}_{\bFm_L }(T)$$ defined in this way is the greatest congruence compatible with all the $$\bL$$-theories that extend $$T$$. Therefore it holds that $\bOmega^{\sim}_{\bFm_L }(T) = \bigcap_{T' \in \tTH(\bL)^T} \bOmega_{\bFm_L }(T')$ where $$\tTH(\bL)^T = \{T' \in \tTH(\bL): T \subseteq T'\}$$. The relation $$\bOmega^{\sim}_{\bFm_L }(T)$$ is known as the Suszko congruence of $$T$$ (w.r.t. $$\bL)$$. Suszko defined it—in an equivalent way—in 1977.
For every logic $$\bL$$, the notion of the Suszko congruence can be extended to its matrix models. The Suszko congruence of a matrix model $$\langle \bA, D \rangle$$ of $$\bL$$ (w.r.t. $$\bL)$$ is the greatest congruence of $$\bA$$ compatible with every $$\bL$$-filter of $$\bA$$ that includes $$D$$, that is, it is the relation given by ${\bOmega^{\sim}_{\bA}}^{\bL}(D) = \bigcap_{D' \in \tFi_{\bL}(\bA)^D} \bOmega_{\bA}(D')$ where $$\tFi_{\bL}(\bA)^D = \{D': D'$$ is a $$\bL$$-filter of $$\bA$$ and $$D \subseteq D'\}$$. Notice that unlike the intrinsic notion of Leibniz congruence, the Suszko congruence of a matrix model of $$\bL$$ is not intrinsic to the matrix: it depends in an essential way on the logic under consideration. The theory of the Suszko congruence of matrices has been developed in Czelakowski 2003 and recently in Albuquerque & Font & Jansana 2016.
In the same manner that the concept of Leibniz congruence leads to the concept of reduced matrix, the notion of Suszko congruence leads to the notion of Suszko-reduced matrix. A matrix model of $$\bL$$ is Suszko-reduced if its Suszko congruence is the identity. Then the class of algebras of the Suszko-reduced matrix models of a logic $$\bL$$ is another class of algebras that is taken as a natural class of algebras to associate with $$\bL$$. It is the class of algebras
$$\bAlg\bL = \{\bA / {\bOmega^{\sim}_{\bA}}^{\bL}(F): \bA$$ is an $$\bL$$-algebra and $$F$$ is a $$\bL$$-filter of $$\bA\}$$.
This class of algebras is nowadays taken in abstract algebraic logic as the natural class to be associated with $$\bL$$ and it called its algebraic counterpart.
For an arbitrary logic $$\bL$$ the relation between the classes $$\bAlg\bL$$ and $$\bAlg^*\bL$$ is that $$\bAlg\bL$$ is the closure of $$\bAlg^*\bL$$ under subdirect products, in particular $$\bAlg^*\bL \subseteq \bAlg\bL$$. In general, both classes may be different. For example, if $$\bL$$ is the $$(\wedge , \vee)$$-fragment of classical propositional logic, $$\bAlg\bL$$ is the variety of distributive lattices (the class that has been always taken to be the natural class of algebras associated with $$\bL)$$ while $$\bAlg^*\bL$$ is not this class—in fact it is not a quasivariety. Nonetheless, for many logics $$\bL$$, in particular for the algebraizable and the protoalgebraic ones to be discussed in the next sections, and also when $$\bAlg^*\bL$$ is a variety, the classes $$\bAlg\bL$$ and $$\bAlg^*\bL$$ are equal. This fact can explain why in the 1980s, before the algebraic study of non-protoalgebraic logics was considered worth pursuing, the conceptual difference between both definitions of the natural class of algebras of a logic was not needed and accordingly it was not considered (or even discovered).
## 9. When a logic is algebraizable and what does this mean?
The algebraizable logics are purported to be the logics with the strongest possible link with their algebraic counterpart. This requirement demands that the algebraic counterpart of the logic should be an algebraic semantics but requires a more robust connection between the logic and the algebraic counterpart than that. This more robust connection is present in the best behaved particular logics known. The mathematically precise concept of algebraizable logic characterizes this type of link. Blok and Pigozzi introduced that fundamental concept in Blok & Pigozzi 1989 and its introduction can be considered the starting point of the unification and growth of the field of abstract algebraic logic in the 1980s. Blok and Pigozzi defined the notion of algebraizable logic only for finitary logics. Later Czelakowski and Herrmann generalized it to arbitrary logics and also weakened some conditions in the definition. We present here the generalized concept.
We said in Section 7 that, roughly speaking, a logic $$\bL$$ is algebraizable when 1) it has an algebraic semantics, i.e., a class of algebras $$\bK$$ and a set of equations $$\iEq(p)$$ such that $$\bK$$ is a $$\iEq$$-algebraic semantics for $$\bL, 2)$$ this fact can be proved by using the Lindenbaum-Tarski method slightly generalized and, moreover, 3) $$\bK \subseteq \bAlg^*\bL$$. The generalization of the Lindenbaum-Tarski method (as we described it in Section 7) consists in allowing in step (5) (as already done in the definition of algebraic semantics) a set of equations $$\iEq(p)$$ in one variable instead of a single equation $$\delta(p) \approx \varepsilon(p)$$ and in allowing in a similar manner a set of formulas $$\Delta(p, q)$$ in at most two variables to play the role of the formula $$p \leftrightarrow q$$ in the definition of the congruence of a theory. Then, given a theory $$T$$, the relation $$\theta(T)$$, which has to be the greatest congruence on the formula algebra compatible with $$T$$ (i.e., the Leibniz congruence of $$T)$$, is defined by
$\langle \phi , \psi \rangle \in \theta(T) \txtiff \Delta(p/\phi , q/\psi) \subseteq T.$
We need some notational conventions before engaging in the precise definition of algebraizable logic. Given a set of equations $$\iEq(p)$$ in one variable and a formula $$\phi$$, let $$\iEq(\phi)$$ be the set of equations obtained by replacing in all the equations in $$\iEq$$ the variable $$p$$ by $$\phi$$. If $$\Gamma$$ is a set of formulas, let
$\iEq(\Gamma) := \bigcup_{\phi \in \Gamma}\iEq(\phi).$
Similarly, given a set of formulas in two variables $$\Delta(p, q)$$ and an equation $$\delta \approx \varepsilon$$, let $$\Delta(\delta , \varepsilon)$$ denote the set of formulas obtained by replacing $$p$$ by $$\delta$$ and $$q$$ by $$\varepsilon$$ in all the formulas in $$\Delta$$. Moreover, if $$\iEq$$ is a set of equations, let
$\Delta(\iEq) = \bigcup_{\delta \approx \varepsilon \in \iEq} \Delta(\delta , \varepsilon).$
Given a set of equations $$\iEq(p, q)$$ in two variables, this set defines on every algebra $$\bA$$ a binary relation, namely the set of pairs $$\langle a, b\rangle$$ of elements of $$A$$ that satisfy in $$\bA$$ all the equations in $$\iEq(p, q)$$. In standard model-theoretic notation, this set is the relation
$\{\langle a, b \rangle : a, b \in A \textrm{ and } \bA \vDash \iEq(p, q)[a, b]\}.$
The formal definition of algebraizable logic is as follows. A logic $$\bL$$ is algebraizable if there is a class of algebras $$\bK$$, a set of equations $$\iEq(p)$$ in one variable and a set of formulas $$\Delta(p, q)$$ in two variables such that
1. $$\bK$$ is an $$\iEq$$-algebraic semantics for $$\bL$$, namely
$$\Gamma \vdash_{\bL } \phi\txtiff$$ for every $$\bA \in \bK$$ and every valuation $$v$$ on $$\bA$$, if $$\bv[\Gamma] \subseteq \tEq(\bA)$$, then $$\bv(\phi) \in \tEq(\bA)$$.
2. For every $$\bA \in \bK$$, the relation defined by the set of equations in two variables $$\iEq(\Delta(p, q))$$ is the identity relation on $$A$$.
A class of algebras $$\bK$$ for which there are sets $$\iEq(p)$$ and $$\Delta(p, q)$$ with these two properties is said to be an equivalent algebraic semantics for $$\bL$$. The set of formulas $$\Delta$$ is called a set of equivalence formulas and the set of equations $$\iEq$$ a set of defining equations.
The conditions of the definition imply:
1. $$p$$ is inter-derivable in $$\bL$$ with the set of formulas $$\Delta(\iEq)$$, that is $\Delta(\iEq) \vdash_{\bL } p \textrm{ and } p \vdash_{\bL } \Delta(\iEq).$
2. For every $$\bL$$-theory $$T$$, the Leibniz congruence of $$\langle \bFm_L, T\rangle$$ is the relation defined by $$\Delta(p, q)$$, namely $\langle \phi , \psi \rangle \in \bOmega_{\bFm }(T)\txtiff\Delta(p/\phi , q/\psi) \subseteq T.$
3. If $$\Delta$$ and $$\Delta '$$ are two sets of equivalence formulas, $$\Delta \vdash_{\bL } \Delta '$$ and $$\Delta ' \vdash_{\bL } \Delta$$. Similarly, if $$\iEq(p)$$ and $$\iEq'(p)$$ are two sets of defining equations, for every algebra $$\bA \in \bK, \iEq(\bA) = \iEq'(\bA)$$.
4. The class of algebras $$\bAlg^*\bL$$ also satisfies conditions (1) and (2), and hence it is an equivalent algebraic semantics for $$\bL$$. Moreover, it includes every other class of algebras that is an equivalent algebraic semantics for $$\bL$$. Accordingly, it is called the greatest equivalent algebraic semantics of $$\bL$$.
5. For every $$\bA \in \bAlg^*\bL$$ there is exactly one $$\bL$$-filter $$F$$ such that the matrix $$\langle \bA, F\rangle$$ is reduced, and this filter is the set $$\iEq(\bA)$$. Or, to put it in other terms, the class of reduced matrix models of $$\bL$$ is $$\{\langle \bA, \iEq(\bA) \rangle : \bA \in \bAlg^*\bL\}$$.
6. $$\bAlg^*\bL$$ is an SP-class and includes any class of algebras $$\bK$$ which is an equivalent algebraic semantics for $$\bL$$. The class $$\bAlg^*\bL$$ is then the greatest equivalent algebraic semantics for $$\bL$$ and thus it deserves to be called the equivalent algebraic semantics of $$\bL$$.
Blok and Pigozzi’s definition of algebraizable logic in Blok & Pigozzi 1989 was given only for finitary logics and, moreover, they imposed that the sets of defining equations and of equivalence formulas should be finite. Today we say that an algebraizable logic is finitely algebraizable if the sets of equivalence formulas $$\Delta$$ and of defining equations $$\iEq$$ can both be taken finite. And we say that a logic is Blok-Pigozzi algebraizable (BP-algebraizable) if it is finitary and finitely algebraizable.
If $$\bL$$ is finitary and finitely algebraizable, then $$\bAlg^*\bL$$ is not only an SP-class, but a quasivariety and it is the quasivariety generated by any class of algebras $$\bK$$ which is an equivalent algebraic semantics for $$\bL$$.
We have just seen that in algebraizable logics the class of algebras $$\bAlg^*\bL$$ plays a prominent role. Moreover, in these logics the classes of algebras obtained by the two ways of generalizing the Lindenbaum-Tarski method coincide, that is, $$\bAlg^*\bL = \bAlg\bL$$—this is due to the fact that for any algebraizable logic $$\bL, \bAlg^*\bL$$ is closed under subdirect products. Hence for every algebraizable logic $$\bL$$ its algebraic counterpart $$\bAlg\bL$$ is its greatest equivalent algebraic semantics, whatever perspective is taken on the generalization of the Lindenbaum-Tarski method.
Conditions (1) and (2) of the definition of algebraizable logic (instantiated to $$\bAlg^*\bL$$) encode the fact that there is a very strong link between an algebraizable logic $$\bL$$ and its class of algebras $$\bAlg\bL$$, so that this class of algebras reflects the metalogical properties of $$\bL$$ by algebraic properties of $$\bAlg\bL$$ and conversely.
The definition of algebraizable logic can be stated in terms of translations between the logic and an equational consequence relation $$\vDash_{\bK}$$ associated with any equivalent algebraic semantics $$\bK$$ for it—which is the same relation no matter what equivalent algebraic semantics we choose.
The equational consequence $$\vDash_{\bK}$$ of a class of algebras $$\bK$$ is defined as follows.
$$\{\phi_i \approx \psi_i: i \in I\} \vDash_{\bK } \phi \approx \psi\txtiff$$ for every $$\bA \in \bK$$ and every valuation $$v$$ on $$\bA$$, if $$\bv(\phi_i) = \bv(\psi_i)$$, for all $$i \in I$$, then $$\bv(\phi) = \bv(\psi)$$.
The translations needed are given by the set of defining equations and the set of equivalence formulas. A set of equations $$\iEq(p)$$ in one variable defines a translation from formulas to sets of equations: each formula is translated into the set of equations $$\iEq(\phi)$$. Similarly, a set of formulas $$\Delta(p, q)$$ in two variables defines a translation from equations to sets of formulas: each equation $$\phi \approx \psi$$ is translated into the set of formulas $$\Delta(\phi , \psi)$$.
Condition (1) in the definition of algebraizable logic can be reformulated as $\Gamma \vdash_{\bL } \phi\txtiff \iEq(\Gamma) \vDash_{\bK } \iEq(\phi)$ and condition (2) as $p \approx q \vDash_{\bK } \iEq(\Delta(p, q)) \textrm{ and } \iEq(\Delta(p, q)) \vDash_{\bK } p \approx q.$
These two conditions imply
1. $$\{\phi_i \approx \psi_i : i \in I \} \vDash_{\bK } \phi \approx \psi\txtiff\Delta(\{\phi_i \approx \psi_i : i \in I\}) \vdash_{\bL } \Delta(\phi , \psi)$$
and condition (3) above is
$p \vdash_{\bL } \Delta(\iEq(p)) \textrm{ and } \Delta(\iEq(p)) \vdash_{\bL } p.$
Thus an algebraizable logic $$\bL$$ is faithfully interpreted in the equational logic of its equivalent algebraic semantics (condition (2)) by means of the translation of formulas into sets of equations given by a set of defining equations, and the equational logic of its equivalent algebraic semantics is faithfully interpreted in the logic $$\bL$$ (condition (9)) by means of the translation of equations into sets of formulas given by an equivalence set of formulas. Moreover, both translations are inverses of each other (conditions (2) and (3)) modulo logical equivalence. In this way we see that the link between $$\bL$$ and its greatest equivalent algebraic semantics is really very strong and that the properties of $$\bL$$ should translate into properties of the associated equational consequence relation. The properties that this relation actually has depend on the properties of the class of algebras $$\bAlg\bL$$.
Given an algebraic semantics $$(\bK, \iEq)$$ for a logic $$\bL$$, a way to stress the difference between it being merely an algebraic semantics and being an algebraic semantics that makes $$\bL$$ algebraizable is that the translation of formulas into equations given by the set of equations $$\iEq$$ is invertible in the sense that there is a translation, say $$\Delta$$, of equations into formulas given by a set of formulas in two variables that satisfies condition (9) above, and such that $$\iEq$$ and $$\Delta$$ provide mutually inverses translations (i.e., conditions (2) and (3) hold).
The link between an algebraizable logic $$\bL$$ and its greatest equivalent algebraic semantics given by the set of defining equations and the set of equivalence formulas allows us to prove a series of general theorems that relate the properties of $$\bL$$ with the properties of $$\bAlg\bL$$. We will mention as a sample only three of them.
The first concerns the deduction theorem. To prove a general theorem relating the existence of a deduction theorem with an algebraic property requires first that a concept of deduction theorem applicable to any logic has to be defined. A logic $$\bL$$ has the deduction-detachment property if there is a finite set of formulas $$\Sigma(p, q)$$ such that for every set of formulas $$\Gamma$$ and all formulas $$\phi , \psi$$
$\Gamma \cup \{\phi \} \vdash_{\bL } \psi\txtiff\Gamma \vdash_{\bL } \Sigma(\phi , \psi).$
Note that this is a generalization of the standard deduction theorem (the direction from left to right in the above expression) and Modus Ponens (equivalent to the implication from right to left) that several logics have for a connective $$\rightarrow$$. In those cases $$\Sigma(p, q) = \{p \rightarrow q\}$$.
Theorem 1.
A finitary and finitely algebraizable logic $$\bL$$ has the deduction-detachment property if and only if the principal relative congruences of the algebras in $$\bAlg\bL$$ are equationally definable.
The second theorem refers to Craig interpolation. Several notions of interpolation are applicable to arbitrary logics. We consider only one of them. A logic $$\bL$$ has the Craig interpolation property for the consequence relation if whenever $$\Gamma \vdash_{\bL } \phi$$ there is a finite set of formulas $$\Gamma$$’ with variables shared by $$\phi$$ and the formulas in $$\Gamma$$ such that $$\Gamma \vdash_{\bL } \Gamma '$$ and $$\Gamma ' \vdash_{\bL } \phi$$.
Theorem 2.
Let $$\bL$$ be a finitary and finitely algebraizable logic with the deduction-detachment property. Then $$\bL$$ has the Craig interpolation property if and only if $$\bAlg\bL$$ has the amalgamation property.
Finally, the third theorem concerns the Beth definability property. The interested reader can find the definition in Font, Jansana & Pigozzi 2003. It is too involved in the general setting we are in to give it here.
Theorem 3.
A finitary and finitely algebraizable logic has the Beth property if and only if all the epimorphisms of the category with objects the algebras in $$\bAlg\bL$$ and morphisms the algebraic homomorphisms are surjective homomorphisms.
Other results relating properties of an algebraizable logic with a property of its natural class of algebras can be found in Raftery 2011, 2013. They concern respectively a generalization of the property of having the deduction-detachment property and the property that generalize the inconsistency lemmas of classical and intuitionistic logic. Also an abstract notion of having a theorem like Glivenko’s theorem relating classical and intuitionistic logic has been proposed and related to an algebraic property in the case of algebraizable logics in Torrens 2008.
For several classes of algebras that are the equivalent algebraic semantics of some algebraizable logic it has been known for a long time that for every algebra in the class there is an isomorphism between the lattice of congruences of the algebra and a lattice of subsets of the algebra with important algebraic meaning. For example, in Boolean algebras and Heyting algebras these subsets are the lattice filters and in modal algebras they are the lattice filters that are closed under the operation that interprets $$\Box$$. In all those cases, the sets are exactly the $$\bL$$-filters of the corresponding algebraizable logic $$\bL$$.
Algebraizable logics can be characterized by the existence of this kind of isomorphism between congruences and logic filters on the algebras of their algebraic counterpart. To spell out this characterization we need a couple of definitions. Let $$\bL$$ be a logic. The Leibniz operator on an algebra $$\bA$$ (relative to $$\bL)$$ is the map from the $$\bL$$-filters of $$\bA$$ to the set of congruences of $$\bA$$ that sends every $$\bL$$-filter $$D$$ of $$\bA$$ to its Leibniz congruence $$\bOmega_{\bA }(D)$$. We say that the Leibniz operator of a logic $$\bL$$ commutes with the inverses of homomorphisms between algebras in a class $$\bK$$ if for every homomorphism $$h$$ from an algebra $$\bA \in \bK$$ to an algebra $$\bB \in \bK$$ and every $$\bL$$-filter $$D$$ of $$\bB, h^{-1}[\bOmega_{\bB }(D)] = \bOmega_{\bA }(h^{-1}[D]$$).
Theorem 4.
A logic $$\bL$$ is algebraizable if and only if for every algebra $$\bA \in \bAlg\bL$$ the Leibniz operator commutes with the inverses of homomorphisms between algebras in $$\bAlg\bL$$ and is an isomorphism between the set of all $$\bL$$-filters of $$\bA$$, ordered by inclusion, and the set of congruences $$\theta$$ of $$\bA$$ such that $$\bA/\theta \in \bAlg\bL$$, ordered also by inclusion.
The theorem provides a logical explanation of the known isomorphisms mentioned above and similar ones for other classes of algebras. For example the isomorphism between the congruences and the normal subgroups of a group can be explained by the existence of an algebraizable logic $$\bL$$ of which the class of groups is its greatest equivalent algebraic semantics and the normal subgroups of a group are its $$\bL$$-filters.
A different but related characterization of algebraizable logics is this:
Theorem 5.
A logic $$\bL$$ is algebraizable if and only if on the algebra of formulas $$\bFm_L$$, the map that sends every theory $$T$$ to its Leibniz congruence commutes with the inverses of homomorphisms from $$\bFm_L$$ to $$\bFm_L$$ and is an isomorphismbetween the set $$\tTH(\bL)$$ of theories of $$\bL$$, ordered by inclusion, and the set of congruences $$\theta$$ of $$\bFm_L$$ such that $$\bFm_L /\theta \in \bAlg\bL$$, also ordered by inclusion.
## 10. A classification of logics
Unfortunately not every logic is algebraizable. A typical example of a non-algebraizable logic is the local consequence of the normal modal logic $$K$$. Let us discuss this example.
The local modal logic $$\blK$$ and the corresponding global one $$\bgk$$ are not only different, but their metalogical properties differ. For example $$\blK$$ has the deduction-detachment property for $$\rightarrow$$:
$\Gamma \cup \{\phi \} \vdash_{\blK } \psi\txtiff \Gamma \vdash_{\blK } \phi \rightarrow \psi.$
But $$\bgk$$ does not have the deduction-detachment property (at all).
The logic $$\bgk$$ is algebraizable and $$\blK$$ is not. The equivalent algebraic semantics of $$\bgk$$ is the variety $$\bMA$$ of modal algebras, the set of equivalence formulas is the set $$\{p \leftrightarrow q\}$$ and the set of defining equations is $$\{p \approx \top \}$$. Interestingly, $$\blK$$ and $$\bgk$$ have the same algebraic counterpart (i.e., $$\bAlg \blK = \bAlg \blK)$$, namely, the variety of modal algebras.
A lesson to draw from this example is that the algebraic counterpart of a logic $$\bL$$, i.e, the class of algebras $$\bAlg\bL$$, does not necessarily fully encode the properties of $$\bL$$. The class of modal algebras encodes the properties of $$\bgk$$ because this logic is algebraizable and so the link between $$\bgk$$ and $$\bAlg \bgk$$ is as strong as possible. But $$\bAlg \blK$$, the class of modal algebras, cannot by itself completely encode the properties of $$\blK$$.
What causes this difference between $$\bgk$$ and $$\blK$$ is that the class of reduced matrix models of $$\bgk$$ is $\{\langle \bA, \{1^{\bA }\}\rangle : \bA \in \bMA\},$ but the class of reduced matrix models of $$\blK$$ properly includes this class so that for some algebras $$\bA \in \bMA$$, in addition to $$\{1^{\bA }\}$$ there is some other $$\blK$$-filter $$F$$ with $$\langle \bA, F \rangle$$ reduced. This fact provides a way to show that $$\blK$$ can not be algebraizable by showing that the $$\blK$$-filters of the reduced matrices are not equationally definable from the algebras; if they where, then for every $$\bA \in \bAlg \blK$$ there would exists exactly one $$\blK$$-filter $$F$$ of $$\bA$$ such that $$\langle \bA, F \rangle$$ is reduced.
Nonetheless, we can perform some of the steps of the Lindenbaum-Tarski method in the logic $$\blK$$. We can define the Leibniz congruence of every $$\blK$$-theory in a uniform way by using formulas in two variables. But in this particular case the set of formulas has to be infinite. Let $$\Delta(p, q) = \{\Box^n (p \leftrightarrow q): n$$ a natural number$$\}$$, where for every formula $$\phi , \Box^0\phi$$ is $$\phi$$ and $$\Box^n\phi$$ for $$n \gt 0$$ is the formula $$\phi$$ with a sequence of $$n$$ boxes in front $$(\Box \ldots^n \ldots \Box \phi)$$. Then, for every $$\blK$$-theory $$T$$ the relation $$\theta(T)$$ defined by
$\langle \phi , \psi \rangle \in \theta(T)\txtiff \{\Box^n (\phi \leftrightarrow \psi): n \textrm{ a natural number}\} \subseteq T$
is the Leibniz congruence of $$T$$. In this case, it happens though that there are two different $$\blK$$-theories with the same Leibniz congruence, something that does not hold for $$\bgk$$.
The logics $$\bL$$ with the property that there is a set of formulas (possibly infinite) $$\Delta(p, q)$$ in two variables which defines in every $$\bL$$-theory $$T$$ its Leibniz congruence, that is, that for all $$L$$-formulas $$\phi , \psi$$ it holds $\langle \phi , \psi \rangle \in \bOmega_{\bFm }(T)\txtiff \Delta(\phi , \psi) \subseteq T$ are known as the equivalential logics. If $$\Delta(p, q)$$ is finite, the logic is said to be finitely equivalential. A set $$\Delta(p, q)$$ that defines in every $$\bL$$-theory its Leibniz congruence is called a set of equivalence formulas for $$\bL$$. It is clear that every algebraizable logic is equivalential and that every finitely algebraizable logic is finitely equivalential.
The logic $$\blK$$ is, according to the definition, equivalential, and it can be shown that it is not finitely equivalential. The local modal logic lS4 is an example of a non-algebraizable logic that is finitely equivalential. A set of equivalence formulas for lS4 is $$\{\Box(p\leftrightarrow q)\}$$.
A set of equivalence formulas for a logic $$\bL$$ should be considered as a generalized biconditional, in the sense that collectively the formulas in the set have the relevant properties of the biconditional, for example of classical logic, that makes it suitable to define the Leibniz congruences of its theories. This comes out very clearly from the following syntactic characterization of the sets of equivalence formulas.
Theorem 6.
A set $$\Delta(p, q)$$ of $$L$$-formulas is a set of equivalence formulas for a logic $$\bL$$ if and only if
• $$(\tR_{\Delta})$$$$\vdash_{\bL } \Delta(p, p)$$
• $$(\tMP_{\Delta})$$$$p, \Delta(p, q) \vdash_{\bL } q$$
• $$(\tS_{\Delta})$$ $$\Delta(p, q) \vdash_{\bL } \Delta(q, p)$$
• $$(\tT_{\Delta})$$$$\Delta(p, q) \cup \Delta(q, r) \vdash_{\bL } \Delta(p, r)$$
• $$(\tRe_{\Delta})$$$$\Delta(p_1, q_1) \cup \ldots \cup \Delta(p_n, q_n) \vdash_{\bL } \Delta(* p_1 \ldots p_n, * q_1 \ldots q_n)$$, for every connective $$*$$ of $$L$$ of arity $$n$$ greater that 0.
There is some redundancy in the theorem. Conditions $$(\tS_{\Delta})$$ and $$(\tT_{\Delta})$$ follow from $$(\tR_{\Delta}),(\tMP_{\Delta})$$ and $$(\tRe_{\Delta})$$.
Equivalential logics were first considered as a class of logics deserving to be studied in Prucnal & Wroński 1974, and they have been studied extensively in Czelakowski 1981; see also Czelakowski 2001.
We already mentioned that the algebraizable logics are equivalential. The difference between an equivalential logic and an algebraizable one can be seen in the following syntactic characterization of algebraizable logics:
Theorem 7.
A logic $$\bL$$ is algebraizable if and only if there exists a set $$\Delta(p, q)$$ of $$L$$-formulas and a set $$\iEq(p)$$ of $$L$$-equations such that the conditions $$(\tR_{\Delta})$$–$$(\tRe_{\Delta})$$ above hold for $$\Delta(p, q)$$ and $p \vdash_{\bL } \Delta(\iEq(p)) \textrm{ and } \Delta(\iEq(p)) \vdash_{\bL } p.$
The set $$\Delta(p, q)$$ in the theorem is then an equivalence set of formulas for and the set $$\iEq(p)$$ a set of defining equations.
There are logics that are not equivalential but have the property of having a set of formulas $$[p \Rightarrow q]$$ which collectively behave in a very weak sense as the implication $$\rightarrow$$ does in many logics. Namely, it has the properties $$(\tR_{\Delta})$$ and $$(\tMP_{\Delta})$$ in the syntactic characterization of a set of equivalence formulas, i.e.,
• $$(\tR_{\Rightarrow})$$ $$\vdash_{\bL } [p \Rightarrow p]$$
• $$(\tMP_{\Rightarrow})$$ $$p, [p \Rightarrow q] \vdash_{\bL } q$$
If a logic is finitary and has a set of formulas with these properties, there is always a finite subset with the same properties. The logics with a set of formulas (finite or not) with properties (1) and (2) above are called protoalgebraic. in particular, every equivalential logic and every algebraizable logic are protoalgebraic.
Protoalgebraic logics were first studied by Czelakowski, who called them non-pathological, and a slightly later by Blok and Pigozzi in Blok & Pigozzi 1986. The label ‘protoalgebraic logic’ is due to these last two authors.
The class of protoalgebraic logics turned out to be the class of logics for which the theory of logical matrices works really well in the sense that many results of universal algebra have counterparts for the classes of reduced matrix models of these logics and many methods of universal algebra can be adapted to its study; consequently the algebraic study of protoalgebraic logics using their matrix semantics has been extensively and very fruitfully pursued. But, as we will see, some interesting logics are not protoalgebraic.
An important characterization of protoalgebraic logics is via the behavior of the Leibniz operator. The following conditions are equivalent:
1. $$\bL$$ is protoalgebraic.
2. The Leibniz operator $$\bOmega_{\bFm_L}$$ is monotone on the set of $$\bL$$-theories with respect to the inclusion relation, that is, if $$T \subseteq T'$$ are $$\bL$$-theories, then $$\bOmega_{\bFm_L }(T) \subseteq \bOmega_{\bFm_L }(T')$$.
3. For every algebra $$\bA$$, the Leibniz operator $$\bOmega_{\bA}$$ is monotone on the set of $$\bL$$-filters of $$\bA$$ with respect to the inclusion relation.
Due to the monotonicity property of the Leibniz operator, for any protoalgebraic logic $$\bL$$ the class of algebras $$\bAlg^*\bL$$ is closed under subdirect products and therefore it is equal to $$\bAlg\bL$$. Hence for protoalgebraic logics the two ways we encountered to associate a class of algebras with a logic produce, as we already mentioned, the same result.
There are also characterizations of equivalential and finitely equivalential logics by the behavior of the Leibniz operator. The reader is referred to Czelakowski 2001 and Font & Jansana & Pigozzi 2003.
In his Raftery 2006b, Raftery studies Condition 7 in the list of properties of an algebraizable logic we gave just after the definition. The condition says:
For every $$\bA \in \bAlg^*\bL$$ the class of reduced matrix models of $$\bL$$ is $$\{\langle \bA, \iEq(\bA) \rangle : \bA \in \bAlg^*\bL\}$$, where $$\iEq(p)$$ is the set of defining equations for $$\bL$$.
The logics with a set of equations $$\iEq(p)$$ with this property, namely such that for every $$\bA \in \bAlg^*\bL$$ the class of reduced matrix models of $$\bL$$ is $$\{\langle \bA, \iEq(\bA) \rangle : \bA \in \bAlg^*\bL\}$$, are called truth-equational, a name introduced in Raftery 2006b. Some truth-equational logics are protoalgebraic but others are not. We will see later an example of the last situation.
The protoalgebraic logics that are truth-equational are in fact the weakly algebraizable logics studied already in Czelakowski & Jansana 2000. Every algebraizable logic is weakly algebraizable. In fact, the algebraizable logics are the equivalential logics that are truth-equational. But not every weakly algebraizable logic is equivalential. An example is the quantum logic determined by the ortolattices, namely by the class of the matrices $$\langle \bA, \{1\} \rangle$$ where $$\bA$$ is an ortolattice and 1 is its greatest element (see Czelakowski & Jansana 2000 and Malinowski 1990).
The classes of logics we have considered so far are the main classes in what has come to be known as the Leibniz hierarchy because its members are classes of logics that can be characterized by the behavior of the Leibniz operator. We described only the most important classes of logics in the hierarchy. The reader is referred to Czelakowski 2001, Font 2016b, Font, Jansana & Pigozzi 2003, and Font 2016b for more information. In particular, Czelakowski 2001 gathers extensively the information on the different classes of the Leibniz hierarchy known at the time of its publication. The relations between the classes of the Leibniz hierarchy considered in this entry are summarized in the following diagram:
Recently the Leibniz hierarchy has been refined in Cintula & Noguera 2010, 2016. The idea is to consider instead of sets of equivalence formulas $$\Delta$$ (that correspond to the biconditional) sets of formulas $$[p\Rightarrow q]$$ with properties of the conditional $$(\rightarrow)$$, among which $$(R_{\Rightarrow})$$ and $$(MP_{\Rightarrow})$$, and such that the set $$[p\Rightarrow q] \cup[p\Rightarrow q]$$ is a set of equivalence formulas. New classes arise when the set $$[p\Rightarrow q]$$ has a single element.
## 11. Replacement principles
Two classes of logics that are not classes of the Leibniz hierarchy have been extensively studied in abstract algebraic logic. They are defined from a completely different perspective from the one provided by the behavior of the Leibniz operator, namely from the perspective given by the replacement principles a logic might enjoy.
The strongest replacement principle that a logic system $$\bL$$ might have, shared for example by classical logic, intuitionistic logic and all its axiomatic extensions, says that for any set of formulas $$\Gamma$$, any formulas $$\phi , \psi , \delta$$ and any variable $$p$$
if $$\Gamma , \phi \vdash_{\bL } \psi$$ and $$\Gamma , \psi \vdash_{\bL } \phi$$, then $$\Gamma , \delta(p/\phi) \vdash_{\bL } \delta(p/\psi)$$ and $$\Gamma , \delta(p/\psi) \vdash_{\bL } \delta(p/\phi)$$,
where $$\delta(p/\phi)$$ and $$\delta(p/\psi)$$ are the formulas obtained by substituting respectively $$\phi$$ and $$\psi$$ for $$p$$ in $$\delta$$. This replacement property is taken by some authors as the formal counterpart of Frege’s principle of compositionality for truth. Logics satisfying this strong replacement property are called Fregean in Font& Jansana 1996 and are thoroughly studied in Czelakowski & Pigozzi 2004a, 2004b.
Many important logics do not satisfy the strong replacement property, for instance almost all the logics (local or global) of the modal family, but some, like the local consequence relation of a normal modal logic, satisfy a weaker replacement principle: for all formulas $$\phi , \psi , \delta$$,
if $$\phi \vdash_{\bL }\psi$$ and $$\psi \vdash_{\bL }\phi$$, then $$\delta(p/\phi) \vdash_{\bL } \delta(p/\psi)$$ and $$\delta(p/\psi) \vdash_{\bL } \delta(p/\phi)$$.
A logic satisfying this weaker replacement property is called selfextensional by Wójcicki (e.g., in Wójcicki 1969, 1988) and congruential in Humberstone 2005. We will use the first terminology because it seems more common—at least in the abstract algebraic logic literature.
Selfextensional logics have a very good behavior from several points of view. Their systematic study started in Wójcicki 1969 and has recently been continued in the context of abstract algebraic logic in Font & Jansana 1996; Jansana 2005, 2006; and Jansana & Palmigiano 2006.
There are selfextensional and non-selfextensional logics in any one of the classes of the Leibniz hierarchy and also in the class of non-protoalgebraic logics. These facts show that the perspective that leads to the consideration of the classes in the Leibniz hierarchy and the perspective that leads to the definition of the selfextensional and the Fregean logics as classes of logics worthy of study as a whole are to a large extent different. Nonetheless, one of the trends of today’s research in abstract algebraic logic is to determine the interplay between the two perspectives and study the classes of logics that arise when crossing both classifications. In fact, there is a connection between the replacement principles and the Suszko congruence (and thus with the Leibniz congruence). A logic $$\bL$$ satisfies the strong replacement principle if and only if for every $$\bL$$-theory $$T$$ its Suszko congruence is the interderivability relation relative to $$T$$, namely the relation $$\{\langle \phi , \psi \rangle : T, \phi \vdash_{\bL } \psi$$ and $$T, \psi \vdash_{\bL } \phi \}$$. And a logic $$\bL$$ satisfies the weak replacement principle if and only if the Suszko congruence of the set of theorems of $$\bL$$ is the interderivability relation $$\{\langle \phi , \psi \rangle : \phi \vdash_{\bL } \psi$$ and $$\psi \vdash_{\bL } \phi \}$$.
## 12. Beyond protoalgebraic logics
Not all interesting logics are protoalgebraic. In this section we will briefly discuss four examples of non-protoalgebraic logics: the logic of conjunction and disjunction, positive modal logic, the strict implication fragment of $$\blK$$ and Visser’s subintuitionistic logic. All of them are selfextensional. In the next section, we will expound the semantics of abstract logics and generalized matrices that serves to develop a really general theory of the algebraization of logic systems. As we will see, the perspective changes in an important respect from the perspective taken in logical matrix model theory.
### 12.1 The logic of conjunction and disjunction
This logic is the $$\{\wedge , \vee , \bot , \top \}$$-fragment of Classical Propositional Logic. Hence its language is the set $$\{\wedge , \vee , \top , \bot \}$$ and its consequence relation is given by
$\Gamma \vdash \phi\txtiff\Gamma \vdash_{\bCPL} \phi.$
It turns out that it is also the $$\{\wedge , \vee , \bot , \top \}$$-fragment of Intuitionistic Propositional Logic. Let us denote it by $$\bL^{\{\wedge , \vee \}}$$.
The logic $$\bL^{ \{\wedge , \vee \}}$$ is not protoalgebraic but it is Fregean. Its classes of algebras $$\bAlg^*\bL^{ \{\wedge , \vee \}}$$ and $$\bAlg\bL^{ \{\wedge , \vee \}}$$ are different. Moreover, $$\bAlg\bL^{\{\wedge , \vee \}}$$ is the variety of bounded distributive lattices, the class of algebras naturally expected to be the associated with $$\bL^{ \{\wedge , \vee \}}$$, but $$\bAlg^*\bL^{ \{\wedge , \vee \}}$$ is strictly included in it. In fact, this last class of algebras is not a quasivariety, but it is first-order definable still.
The logic $$\bL^{\{\wedge , \vee \}}$$ is thus a natural example of a logic where the class of algebras of its reduced matrix models is not the right class of algebras expected to correspond to it (see Font & Verdú 1991 where the logic is studied at length). The properties of this example and its treatment in Font & Verdú 1991 motivated the systematic study in Font & Jansana 1996 of the kind of models for sentential logics considered in Brown & Suszko 1973, namely, abstract logics.
### 12.2 Positive Modal Logic
Positive Modal Logic is the $$\{\wedge , \vee , \Box , \Diamond , \bot , \top \}$$-fragment of the local normal modal logic $$\blK$$. We denote it by $$\bPML$$. This logic has some interest in Computer Science.
The logic $$\bPML$$ is not protoalgebraic, it is not truth-equational, it is selfextensional and it is not Fregean. Its algebraic counterpart $$\bAlg \bPML$$ is the class of positive modal algebras introduced by Dunn in Dunn 1995. The logic is studied in Jansana 2002 from the perspective of abstract algebraic logic. The class of algebras $$\bAlg\bPML$$ is different from $$\bAlg^*\bPML$$.
### 12.3 Visser’s subintuitionistic logic
This logic is the logic in the language of intuitionistic logic that has to the least normal modal logic $$K$$ the same relation that intuitionistic logic has to the normal modal logic $$S4$$. It was introduced in Visser 1981 (under the name Basic Propositional Logic) and has been studied by several authors, such as Ardeshir, Alizadeh, and Ruitenburg. It is not protoalgebraic, it is truth-equational and it is Fregean (hence also selfextensional).
### 12.4 The strict implication fragment of the local modal logic lK
The strict implication of the language of modal logic is defined using the $$\Box$$ operator and the material implication $$\rightarrow$$. We will use $$\Rightarrow$$ for the strict implication. Its definition is $$\phi \Rightarrow \psi := \Box(\phi \rightarrow \psi)$$. The language of the logic $$\bSilK$$, that we call the strict implication fragment of the local modal logic $$\blK$$, is the language $$L = \{\wedge , \vee , \bot , \top , \Rightarrow \}$$. We can translate the formulas of $$L$$ to formulas of the modal language by systematically replacing in an $$L$$-formula $$\phi$$ every subformula of the form $$\psi \Rightarrow \delta$$ by $$\Box(\psi \rightarrow \delta)$$ and repeating the process until no appearance of $$\Rightarrow$$ is left. Let us denote by $$\phi^*$$ the translation of $$\phi$$ and by $$\Gamma^*$$ the set of the translations of the formulas in $$\Gamma$$. Then the definition of the consequence relation of $$\bSilK$$ is:
$\Gamma \vdash_{\bSilK } \phi\txtiff\Gamma^* \vdash_{\blK } \phi^*.$
The logic $$\bSilK$$ is not protoalgebraic and is not truth-equational. It is selfextensional but it is not Fregean. Its algebraic counterpart $$\bAlg \bSilK$$ is the class of bounded distributive lattices with a binary operation with the properties of the strict implication of $$\blK$$. This class of algebras is introduced and studied in Celani & Jansana 2005, where its members are called Weakly Heyting algebras. $$\bAlg \bSilK$$ does not coincide with $$\bAlg^* \bSilK$$.
The logic $$\bSilK$$ belongs, as Visser’s logic, to the family of so-called subintuitionistic logics. A reference to look at for information on these logics is Celani & Jansana 2003.
## 13. Abstract logics and generalized matrices
The logical matrix models of a given logic can be thought of as algebraic generalizations of its theories, more precisely, of its Lindenbaum matrices. They come from taking a local perspective centered around the theories of the logic considered one by one, and its analogs the logic filters (also taken one by one). But, as we will see, the properties of a logic depend in general on the global behavior of the set of its theories taken together as a bunch; or—to put it otherwise—on its consequence relation. The consideration of this global behavior introduces a global perspective on the design of semantics for logic systems. The abstract logics that we are going to define can be seen, in contrast to logical matrices, as algebraic generalizations of the logic itself and its extensions. They are the natural objects to consider when one takes the global perspective seriously.
Let $$L$$ be a propositional language. An $$L$$-abstract logic is a pair $$\cA = \langle \bA$$, C $$\rangle$$ where $$\bA$$ is an $$L$$-algebra and $$C$$ an abstract consequence operation on $$A$$.
Given a logic $$\bL$$, an $$L$$-abstract logic $$\cA = \langle \bA, C \rangle$$ is a model of $$\bL$$ if for every set of formulas $$\Gamma$$ and every formula $$\phi$$
$$\Gamma \vdash_{\bL } \phi\txtiff$$ for every valuation $$v$$ on $$\bA, \bv(\phi) \in C(\bv[\Gamma])$$.
This definition has an equivalent in terms of the closed sets of $$C$$: an abstract logic $$\cA = \langle \bA, C \rangle$$ is a model of $$\bL$$ if and only if for every $$C$$-closed set $$X$$ the matrix $$\langle \bA, X \rangle$$ is a model of $$\bL$$ (i.e., $$X$$ is an $$\bL$$-filter).
This observation leads to another point of view on abstract logics as models of a logic system. It transforms them into a collection of logical matrices (given by the closed sets) over the same algebra, or, to put it more simply, into a pair $$\langle \bA, \cB \rangle$$ where $$\cB$$ is a collection of subsets of $$A$$. A structure of this type is called in the literature a generalized matrix (Wójcicki 1973) and more recently it has been called an atlas in Dunn & Hardegree 2001. It is said to be a model of a logic $$\bL$$ if for every $$X \in \cB, \langle \bA, X \rangle$$ is a matrix model of $$\bL$$.
A logic system $$\bL = \langle L, \vdash_{\bL } \rangle$$ straightforwardly provides us with an equivalent abstract logic $$\langle \bFm_L, C_{\vdash_{ \bL} } \rangle$$ and an equivalent generalized matrix $$\langle \bFm_L,\tTH(\bL) \rangle$$, where $$\tTH(\bL)$$ is the set of $$C_{\vdash_{ \bL}}$$-closed sets of formulas (i.e., the $$\bL$$-theories). We will move freely from one to the other.
The generalized matrices $$\langle \bA, \cB \rangle$$ that correspond to abstract logics have the following two properties: $$A \in \cB$$ and $$\cB$$ is closed under intersections of arbitrary nonempty families. A family $$\cB$$ of subsets of a set $$A$$ with these two properties is known as a closed-set system and also as a closure system. There is a dual correspondence between abstract consequence operations on a set $$A$$ and closed-set systems on $$A$$. Given an abstract consequence operation $$C$$ on $$A$$, the set $$\cC_C$$ of $$C$$-closed sets is a closed-set system and given a closed-set system $$\cC$$ the operation $$C_{\cC}$$ defined by $$C_{\cC }(X) = \bigcap \{Y \in \cC: X \subseteq Y\}$$, for every $$X \subseteq A$$, is an abstract consequence operation. In general, every generalized matrix $$\langle \bA, \cB \rangle$$ can be turned into a closed-set system by adding to $$\cB \cup \{A\}$$ the intersections of arbitrary nonempty subfamilies, and therefore into an abstract logic, which we denote by $$\langle \bA, C_{\cB }\rangle$$. In that situation we say that $$\cB$$ is a base for $$C_{\cB}$$. It is obvious that an abstract logic can have more than one base. Any family of closed sets with the property that every closed set is an intersection of elements of the family is a base. The study of bases for the closed set system of the theories of a logic usually plays an important role in its study. For example, in classical logic an important base for the family of its theories is the family of maximal consistent theories and in intuitionistic logic the family of prime theories. In a similar way, the systematic study of bases for generalized matrix models of a logic becomes important.
In order to make the exposition smooth we will now move from abstract logics to generalized matrices. Let $$\cA = \langle \bA, \cB \rangle$$ be a generalized matrix. There exists the greatest congruence of $$\bA$$ compatible with all the sets in $$\cB$$; it is known as the Tarski congruence of $$\cA$$. We denote it by $$\bOmega^{\sim}_{\bA }(\cB)$$. It has the following characterization using the Leibniz operator
$\bOmega^{\sim}_{\bA }(\cB) = \bigcap_{X \in \cB} \bOmega_{\bA }(X).$
It can also be characterized by the condition:
$$\langle a, b \rangle \in \bOmega^{\sim}_{\bA }(\cB)\txtiff$$ for every $$\phi(p, q_1 , \ldots ,q_n)$$, every $$c_1 , \ldots ,c_n \in A$$ and all $$X \in \cB$$
$\phi^{\bA }[a, c_1 , \ldots ,c_n] \in X \Leftrightarrow \phi^{\bA }[b, c_1 , \ldots ,c_n] \in X$
or equivalently by
$$\langle a, b \rangle \in \bOmega^{\sim}_{\bA }(\cB)\txtiff$$ for every $$\phi(p, q_1 , \ldots ,q_n)$$ and every $$c_1 , \ldots ,c_n \in A, C_{\cB }(\phi^{\bA }[a, c_1 , \ldots ,c_n]) = C_{\cB }(\phi^{\bA }[b, c_1 , \ldots ,c_n])$$.
A generalized matrix is reduced if its Tarski congruence is the identity. Every generalized matrix $$\langle \bA, \cB \rangle$$ can be turned into an equivalent reduced one by identifying the elements related by its Tarski congruence. The result is the quotient generalized matrix $$\langle \bA / \bOmega^{\sim}_{\bA }(\cB), \cB/\bOmega^{\sim}_{\bA }(\cB) \rangle$$, where $$\cB/\bOmega^{\sim}_{\bA }(\cB) = \{X/\bOmega^{\sim}_{\bA }(\cB): X \in \cB\}$$ and for $$X \in \cB, X/\bOmega^{\sim}_{\bA }(\cB)$$ is the set of equivalence classes of the elements of $$X$$.
The properties of a logic $$\bL$$ depend in general, as we already said, on the global behavior of the family of its theories. In some logics, this behavior is reflected in the behavior of its set of theorems, as in classical and intuitionistic logic due to the deduction-detachment property, but this is by no means the most general situation, as it is witnessed by the example of the local and global modal logics of the normal modal logic $$K$$. Both have the same theorems but do not share the same properties. Recall that the local logic has the deduction-detachment property but the global one not. In a similar way, the properties of a logic are in general better encoded in an algebraic setting if we consider families of $$\bL$$-filters on the algebras than if we consider a single $$\bL$$-filter as it is done in logical matrices model theory.
The generalized matrix models that have naturally attracted most of the attention in the research on the algebraization of logics are the generalized matrices of the form $$\langle \bA, \tFi_{\bL }\bA \rangle$$ where $$\tFi_{\bL }\bA$$ is the set of all the $$\bL$$-filters of $$\bA$$. An example of a property of logics encoded in the structure of the lattices of $$\bL$$-filters of the $$L$$-algebras is that for every finitary protoalgebraic logic $$\bL, \bL$$ has the deduction-detachment property if and only if for every algebra $$\bA$$ the join-subsemilattice of the lattice of all $$\bL$$-filters of $$\bA$$ that consists of the finitely generated $$\bL$$-filters is dually residuated; see Czelakowski 2001.
The generalized matrices of the form $$\langle \bA, \tFi_{\bL }\bA \rangle$$ are called the basic full g-models of $$\bL$$ (the letter ‘g’ stands for generalized matrix). The interest in these models lead to the consideration of the class of generalized matrix models of a logic $$\bL$$ with the property that their quotient by their Tarski congruence is a basic full g-model. These generalized matrices (and their corresponding abstract logics) are called full g-models. The theory of the full g-models of an arbitrary logic is developed in Font & Jansana 1996, where the notion of full g-model and basic full g-model is introduced. We will mention some of the main results obtained there.
Let $$\bL$$ be a logic system.
1. $$\bL$$ is protoalgebraic if and only if for every full g-model $$\langle \bA, \cC \rangle$$ there exists an $$\bL$$-filter $$F$$ of $$\bA$$ such that $$\cC = \{G \in \tFi_{\bL }\bA: F \subseteq G\}$$.
2. If $$\bL$$ is finitary, $$\bL$$ is finitely algebraizable if and only if for every algebra $$\bA$$ and every $$\bL$$-filter $$F$$ of $$\bA$$, the generalized matrix $$\langle \bA, \{G \in \tFi_{\bL }\bA: F \subseteq G\} \rangle$$ is a full g-model and $$\bAlg\bL$$ is a quasivariety.
3. The class $$\bAlg\bL$$ is both the class of algebras of the reduced generalized matrix models of $$\bL$$, and the class $$\{\bA: \langle \bA, \tFi_{\bL }\bA \rangle$$ is reduced$$\}$$.
4. For every algebra $$\bA$$ there is an isomorphism between the family of closed-set systems $$\cC$$ on $$A$$ such that $$\langle\bA, \cC\rangle$$ is a full g-model of $$\bL$$ and the family of congruences $$\theta$$ of $$\bA$$ such that $$\bA/\theta \in \bAlg\bL$$. The isomorphism is given by the Tarski operator that sends a generalized matrix to its Tarski congruence.
The isomorphism theorem (4) above is a generalization of the isomorphism theorems we encountered earlier for algebraizable logics. What is interesting here is that the theorem holds for every logic system. Using (2) above, theorem (4) entails the isomorphism theorem for finitary and finitely algebraizable logics. Thus theorem (4) can be seen as the most general formulation of the mathematical logical phenomena that underlies the isomorphism theorems between the congruences of the algebras in a certain class and some kind of subsets of them we mentioned in Section 9.
The use of generalized matrices and abstract logics as models for logic systems has proved very useful for the study of selfextensional logics in general and more in particular for the study of the selextenional logics that are not protoalgebraic such as the logics discused in Section 12. In particular, they have proved very useful for the study of the class of finitary selfextensional logics with a conjunction and the class of finitary selfextensional logics with the deduction-detachment property for a single term, say $$p \rightarrow q$$; the logics in this last class are nevertheless protoalgebraic. A logic $$\bL$$ has a conjunction if there is a formula in two variables $$\phi(p, q)$$ such that
$\phi(p, q) \vdash_{\bL } p, \phi(p, q)\vdash_{\bL } q, p, q \vdash_{\bL } \phi(p, q).$
The logics in those two classes have the following property: the Tarski relation of every full g-model $$\langle \bA, C \rangle$$ is $$\{\langle a, b \rangle \in A \times A: C(a) = C(b)\}$$. A way of saying it is to say that for these logics the property that defines selfextensionality, namely that the interderivability condition is a congruence, lifts or transfers to every full g-model. The selfextensional logics with this property are called fully selfextensional. This notion was introduced in Font & Jansana 1996 under the name ‘strongly selfextensional’. All the known and natural selfextensional logics are fully selfextensional, in particular the logics discussed in Section 12, but Babyonyshev showed (Babyonyshev 2003) an ad hoc example of a selfextensional logic that is not fully selfextensional.
An interesting result on the finitary logics which are fully selfextensional logics with a conjunction or with the deduction-detachment property for a single term is that their class of algebras $$\bAlg\bL$$ is always a variety. The researchers in abstract algebraic logic are somehow surprised by the fact that several finitary and finitely algebraizable logics have a variety as its equivalent algebraic semantics, when the theory of algebraizable logics allows in general to prove only that the equivalent algebraic semantics of a finitary and finitely algebraizable logic is a quasivariety. The result explains this phenomenon for the finitary and finitely algebraizable logics to which it applies. For many other finitary and finitely algebraizable logics to find a convincing explanation is still an open area of research.
Every abstract logic $$\cA = \langle \bA, C \rangle$$ determines a quasi-order (a reflexive and transitive relation) on $$A$$. It is the relation defined by $a \le_{\cA } b\txtiff C(b) \subseteq C(a)\txtiff b \in C(a).$
Thus $$a \le_{\cA } b$$ if and only if $$b$$ belongs to every $$C$$-closed set to which $$A$$ belongs. For a fully selfextensional logic $$\bL$$, this quasi-order turns into a partial order in the reduced full g-models, which are in fact the reduced basic full g-models, namely, the abstract logics $$\langle \bA, \tFi_{\bL }\bA \rangle$$ with $$\bA \in \bAlg\bL$$. Consequently, in a fully selfextensional logic $$\bL$$ every algebra $$\bA \in \bAlg\bL$$ carries a partial order definable in terms of the family of the $$\bL$$-filters. If the logic is fully selfextensional with a conjunction this partial order is definable by an equation of the $$L$$-algebraic language because in this case for every algebra $$\bA \in \bAlg\bL$$ we have: $a \le b\txtiff C(b) \subseteq C(a)\txtiff C(a \wedge^{\bA } b) = C(a)\txtiff a \wedge^{\bA } b = a,$ where $$C$$ is the abstract consequence operation that corresponds to the closed-set system $$\tFi_{\bL }\bA$$, and $$\wedge^{\bA}$$ is the operation defined on $$\bA$$ by the formula that is a conjunction for the logic $$\bL$$.
A similar situation holds for fully selfextensional logics with the deduction-detachment property for a single term, say $$p \rightarrow q$$, for then for every algebra $$\bA \in \bAlg\bL$$
$a \le b\txtiff C(b) \subseteq C(a)\txtiff C(a \rightarrow^{\bA } b) = C(\varnothing) = C(a \rightarrow^{\bA } a) \txtiff \\ a \rightarrow^{\bA } b = a \rightarrow^{\bA } a.$
These observations lead us to view the finitary fully selfextensional logics $$\bL$$ with a conjunction and those with the deduction-detachment property for a single term as logics definable by an order which is definable in the algebras in $$\bAlg\bL$$ by using an equation of the $$\bL$$-algebraic language. Related to this, the following result is known.
Theorem 8.
A finitary logic $$\bL$$ with a conjunction is fully selfextensional if and only if there is a class of algebras $$\bK$$ such that for every $$\bA \in \bK$$ the reduct $$\langle A, \wedge^{\bA }\rangle$$ is a meet-semilattice and if $$\le$$ is the order of the semilattice, then
$$\phi_1 , \ldots ,\phi_n\vdash_{\bL } \phi\txtiff$$ for all $$\bA \in \bK$$ and every valuation $$v$$ on $$\bA \; \bv(\phi_1) \wedge^{\bA }\ldots \wedge^{\bA } \bv(\phi_n) \le \bv(\phi)$$
and
$$\vdash_{\bL } \phi\txtiff$$ for all $$\bA \in \bK$$ and every valuation $$v$$ on $$\bA \; a \le \bv(\phi)$$, for every $$a \in A$$.
Moreover, in this case the class of algebras $$\bAlg\bL$$ is the variety generated by $$\bK$$.
Similar results can be obtained for the selfextensional logics with the deduction-detachment property for a single term. The reader is referred to Jansana 2006 for a study of the selfextensional logics with conjunction, and to Jansana 2005 for a study of the selfextensional logics with the deduction-detachment property for a single term.
The class of selfextensional logics with a conjunction includes the so-called logics preserving degrees of truth studied in the fields of substructural logics and of many-valued logics. The reader can look at Bou et al. 2009 and the references therein.
## 14. Extending the setting
The research on logic systems described in the previous sections has been extended to encompass other consequence relations that go beyond propositional logics, like equational logics and the consequence relations between sequents built from the formulas of a propositional language definable using sequent calculi. The interested reader can consult the excellent paper Raftery 2006a.
This research arose the need for an, even more, abstract way of developing the theory of consequence relations. It has lead to a reformulation (in a category-theoretic setting) of the theory of logic systems as explained in this entry. The work has been done mainly by G. Voutsadakis in a series of papers, e.g., Voutsadakis 2002. Voutsadakis’s approach uses the notion of a pi-institution, introduced by Fiadeiro and Sernadas, as the analog of the logic systems in his category-theoretic setting. Some work in this direction is also found in Gil-Férez 2006. A different approach to a generalization of the studies encompassing the work done for logic systems and for sequent calculi is found in Galatos & Tsinakis 2009; Gil-Férez 2011 is also in this line. The work presented in these two papers originates in Blok & Jónsson 2006. The Galatos-Tsinakis approach has been recently extended in a way that also encompasses the setting of Voutsadakis in Galatos & Gil-Férez 2017.
Another recent line of research that extends the framework described in this entry develops a theory of algebraization of many-sorted logic systems using instead of the equational consequence relation of the natural class of algebras a many-sorted behavioral equational consequence (a notion coming from computer science) and a weaker concept than algebraizable logic: behaviorally algebraizable logic. See Caleiro, Gonçalves & Martins 2009.
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• Jansana, Ramon, 2002, “Full models for positive modal logic”, Mathematical Logic Quarterly, 48(3): 427–445. doi:10.1002/1521-3870(200204)48:3<427::AID-MALQ427>3.0.CO;2-T
• –––, 2005, “Selfextensional logics with implication”, in J.-Y. Béziau (ed.), Logica Universalis, Basel: Birkhäuser. doi:10.1007/3-7643-7304-0_4
• –––, 2006, “Selfextensional logics with conjunction”, Studia Logica, 84(1): 63–104. doi:10.1007/s11225-006-9003-z
• Jansana, Ramon and Alessandra Palmigiano, 2006, “Referential algebras: duality and applications”, Reports on Mathematical Logic (Special issue in memory of Willem Blok), 41: 63–93. [Jansana and Palmigiano 2006 available online]
• Koslow, Arnold, 1992, A structuralist theory of logic, Cambridge: Cambridge University Press. doi:10.1017/CBO9780511609206
• Kracht, Marcus, 2006, “Modal Consequence Relations”, in Blackburn, van Benthem, and Wolter 2006: 497–549.
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• Malinowski, Jacek, 1990, “The deduction theorem for quantum logic, some negative results”, The Journal of Symbolic Logic, 55(2): 615–625. doi:10.2307/2274651
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• Pigozzi, Don, 1991, “Fregean algebraic logic”, in H. Andréka, J.D. Monk, and I. Németi (eds.), Algebraic Logic (Colloq. Math. Soc. János Bolyai, Volume 54), Amsterdam: North-Holland, 473-502.
• Prucnal, Tadeusz and Andrzej Wroński, 1974, “An algebraic characterization of the notion of structural completeness”, Bulletin of the Section of Logic, 3(1): 30–33.
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2017-07-23 04:49:14
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https://mathematica.stackexchange.com/questions/111925/calling-cuda-library-functions-using-mathematicagpulibrary-wgl
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# calling CUDA library functions using MathematicaGPULibrary (WGL)
Recently I want implement a sparse linear solver in Mathematica with cuda, and I found the following resource: calling CUDA library functions. As it shows, I tried to use CreateLibrary function to export the example Mathematica\10.4\SystemFiles\Links\CUDALink\CSource\addTwo.cu to a dll file. However, I faced link errors.
tmpxft_ 00002fa4_ 00000000-28_addTwo.obj : error LNK2019: unresolved external symbol CUDA_Runtime_getDeviceMemoryAsMInteger referenced in function "int __cdecl iAddTwo(struct st_WGL_Memory_p_t *,struct st_WGL_Memory_p_t *)" (?iAddTwo@@YAHPEAUst_WGL_Memory_p_t@@0@Z)
where CUDA_Runtime_getDeviceMemoryAsMInteger in my opinion is get device pointer from a wgl object which wrap a cudamemory data. So the pointer can be used in traditional cuda c/c++ functions. The errors imply that it possibly is lacking a lib file implementing CUDA_Runtime_getDeviceMemoryAsMInteger, but I only found the statement in wgl_cuda_runtime.h shows #define CUDA_Runtime_getDeviceMemoryAsMInteger(mem) ((mint *) CUDA_Runtime_getDeviceMemory(mem)). I also replaced CUDA_Runtime_getDeviceMemoryAsMInteger with (mint *) CUDA_Runtime_getDeviceMemory in the cu file, but it showed the similar problem that ...unresolved external symbol CUDA_Runtime_getDeviceMemory referenced in.... Could someone help me?
• I just recheck my code on V10.3 under Linux. It works fine. You know, WGL is undocumented so it can be broken in any future version. Did you try LibraryLink method? It is enough in many case because you can use global variables and arrays inside your library (they are stored until LibraryUnload). – ybeltukov Apr 8 '16 at 1:54
• @ybeltukov, the link errors imply that the lmplement files don't exist in the code. I also checked the source file and couldn't find CUDA_RUNTIME_XX functions anywhere. In addition, I installed 9.0 at another windows 7 computer and faced same problem. I think I have to choose the LibraryLink method, although it has disadvantages. Thanks @ybeltukov! – sejabs Apr 8 '16 at 3:07
• @ybeltukov, what version of cuda packlet do you use? I doubt the newest 10.2.0.3 may be a beta product and don't include the CUDA_Runtime_xx functions. – sejabs Apr 8 '16 at 14:52
• I updated to 10.2.0.3 from 10.0.0.1. Now the code runs only if I change "libCUDALink_Single.so" to "libCUDALink_Double.so" in CreateLibrary. Could you check it? (see also the corresponding remark in my answer). – ybeltukov Apr 11 '16 at 17:44
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2019-11-12 12:57:42
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https://delicatica.ru/2022/06/21/internet-explorer-9-64-bit-offline-installer-download-link/
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Microsoft Internet Explorer 9 offline installer (32 and 64-bit)
Microsoft Internet Explorer 9 Setup File
Installing Internet Explorer 9 Offline
Category:Internet ExplorerQ:
Moving body with a circular velocity
This should be an easy question, but I’ve been stuck on this for a while. My answer to this question is giving me a = 1. Which is wrong I think, as shouldn’t the distance moved increase as $r$ increases. Thanks in advance.
A:
Newton’s second law of motion says, $$F=ma$$ where $F$ is the force on the body, $m$ is its mass and $a$ is its acceleration. Assuming the «force» is the force of gravity acting on the body, then $$F=-m\frac{GMm}{r^2}$$ where $G$ is the gravitational constant and $M$ is the mass of the earth. This is the force per unit mass of the body, so we can divide by the mass, $m$, to get $a=-GM/r^2$.
If you want a distance per unit time, you can use the equation $$v=\frac{dr}{dt}=\frac{dr}{da}\frac{da}{dt}=\frac{dr}{da}\frac{d^2r}{dt^2}$$ where $v$ is the velocity of the body. This gives you the velocity of the body, $\frac{dr}{da}$, in terms of the acceleration of the body, $a$. Then plug that in to Newton’s second law above to get $$F=\frac{GM}{r^2}\left(\frac{dr}{da}\right)^2$$ from which you can get the distance $dr$ traveled per unit time.
Q:
How do I find the subdomain from an array of urls?
The title pretty much says it all, how would I get the first subdomain from this array?
array (size=3)
0 =>
array (size=5)
‘domain’ => string ‘www.site.com’ (length=12)
‘webroot’ => string ‘htdocs’ (length=5)
3da54e8ca3
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2022-07-04 05:47:34
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https://afsv.serataromanticaroma.it/two-column-latex.html
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The larger the first two parameters and the smaller the last one, the more a page can be filled with figures and the less has to be text. SIGCHI Document Formats. 1-column abstract in 2-column document One often requires that the abstract of a paper should appear across the entire page, even in a two-column paper. , there are no licence fees, etc. LaTeX makes creating posters as easy as writing a normal document and can be customized to create beautiful poster layouts. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. {Example for a two column text}] \ blindtext \end {multicols}. When two maths elements appear either side of the sign, it is assumed to be a binary operator, and as such, allocates some space either side of the sign. For example, you might have a list of children and their birth dates and want these data stored in separate cells. Discover classes, experts, and inspiration to bring your ideas to life. no notations as a) b) c) for. \begin{tabular}{p{1in}p{2in}} The previous example creates two columns, of which the left is one inch wide, and the right is two inches wide. How to place figures and tables in the center of the page in a two-column documents using Latex How to draw multi-column and multi-row tables in Latex Week 3 Latex Tutorial. Tables are a fairly important part of LaTeX. Now you know all pretty-printing commands and environments. Is the content in the two columns separate or a continuous text? Will the rest of the document have two columns, too, or is it just a single-page article?. Saturday 2020-05-23 5:01:30 am : The Best Make Table Fit Two Columns Latex Free Download. Thanks for contributing an answer to TeX - LaTeX Stack Exchange! Please be sure to answer the question. 2 Plotting Data Discrete data contained in a file can be displayed by specifying the name of the data file (enclosed in quotes) on the plot or splot command line. When you swap from two columns to. You don't need to specify one column after this. The first option is currently the best one owing to the fact that the LaTeX source file does not have to be modified, all style and class files are supported, and bitmapped as well as postscript figures can be. I like to match the two columns of first row, and last two columns of the same first row. While experimenting with various palm tree leaves in her studio, Raff found that the leaf of the Phoenix dactylifera bends itself into a shape of a tongue when soaked in latex. You don't have to pay for using LaTeX, i. There are currently two viable alternatives in producing compliant PDF documents from LaTeX (the dvips method and the pdfTeX Method). Please anyone can help me. Chordii reads a text file containing the lyrics of a song, the chords to be played, their description and some other optional data to produce a PostScript document that includes: * Centered titles * Chord names above the words * Graphical representation of the chords at the end of the songs * Transposition * Multiple columns on a page * Index. Foil balloons last approx. posted by gleuschk at 11:25 AM on April 18, 2007. Create by Nicola Fontana,. I am not discussing basic about latex table in this post. Discover classes, experts, and inspiration to bring your ideas to life. It was designed by Ruben De Smet. For the 5x3 table shown above we can count five times (\\) behind each row and two times (&) per row, separating the content of three columns. Figures and Tables Position figures and tables at the top and bottom of columns. This template was created by Frits Wenneker. Please help me for this task. SG Wicker Beamer 101. Courses: Crocheting, Embroidery, Knitting, Quilting, Sewing. The template facilitates structuring of the manuscript, e. Footnotes in a multi-column layout Frank Mittelbach November 10, 2019 1 Preface to version 1. There are different way of placing figures side by side in Latex, subcaption, subfig, subfigure, or even minipage. The first option is currently the best one owing to the fact that the LaTeX source file does not have to be modified, all style and class files are supported, and bitmapped as well as postscript figures can be. 'rcl' for 3 columns. You are here. One column abstract in two column document Google Groups. The dimensions understood by LaTeX include cm, mm, in and pt. It's not surprising, though. Two-column article template with coloured title and initial capital at the beginning of the paragraph. There are many variables in LaTeX determining lengths. Two-column format in the IEEE style is required for final submissions and recommended for initial submissions. Cross-referencing 10 V. The layout of articles is generally multi-column to maximize the information per page and make text easier to read. LaTeX automatically sets reasonable values for the page dimensions, orientation, etc. Hi, I have a table in a two-column document (code below) It is advertised to be sort of counterpart of LaTeX, but at the same time, the team behind it doesn't seem to give a **** to popularize it, which makes me wonder whether it is intended to be just a typesetter for the. What Is Beamer? Beamer is a exible LATEX class for making slides and presentations. In recent versions (Windows and Linux) appears as "Two-column document" under Document->Settings , Text Layout tab Rich also gave som additional comments in LyxUsersPost:14079 , referring to multicols. This template was created by Frits Wenneker. You can change the fonts, numbering style, alignment and format of the captions and the caption labels. Use dot notation to set properties. The A4 column width is 88mm (3. You don't have to pay for using LaTeX, i. How TO - Two Column Layout. Wednesday 2020-06-24 6:48:11 am : The Best Make Table Fit Two Columns Latex Free Download. An example of a geometry two-column proof template. Two-column documents can be easily created by passing the parameter \twocolumn to the document class statement. SG Wicker Beamer 101. See the LaTeX, longtable not compatible with 2-column LaTeX A two-column layout document (IEEE). cls and cas-dc. When I started to write macros for doc. The Best Make Table Fit Two Columns Latex Free Download PDF And Video. Line 4 is the important line here. The first table is named UserTable with the columns ID and Name and another table named TableUser with the columns ID and UserName. Simply including it in the preamble is sufficient to make LaTeX render the last page with roughly balanced columns, regardless of their contents. LaTeX is great in that it can display all those strange math symbols for you. LaTeX can produce both bulleted lists or unordered lists, and numbered lists or ordered lists. online LaTeX editor with autocompletion, highlighting and 400 math symbols. 45\textwidth” or something alike if you want a one-row two-columns layout. If one of X or Y is a vector and the other is a matrix, then the matrix must have dimensions such that one of its dimensions equals the vector length. Both Trees and Centerpieces can be used on tables or as floor-based arrangements. I guess I must be doing something wrong. Making statements based on opinion; back them up with references or personal experience. On the last page of your paper, try to adjust the lengths of the two columns so that they are the same. 3 Some Table Examples Example-1: A table with combined columns is given below. D Pu uu β G f 5 in 269. If you are unsure which template variant to use, please request clarification from your event or publication contact. ) The LaTeX array environment has an argument, in this case ccc, that determines how the entries in each column are aligned. Variables can be set to a negative value. LaTeX is available as free software. There are two ways (and possibly more) to place content side-by-side in a beamer presentation, the columns and the minipage environments. The R programming syntax is extremely easy to learn, even for users with no previous programming experience. The | in the center means put a line in between the two columns. \begin{figure} and \begin{table} span only one column. 16000 Woodworking Plans Get Latex Make Table Span Two Columns: Learn techniques & deepen your practice with classes from pros. An online LaTeX editor that's easy to use. In org mode, creating a two_columns page style and use #+ATTR_ODT: :style "two_columns" can produce a two column document, but org mode doesn't seem to have a section break like # +SECTIONBREAK: How can I switch from a two columns section to one column section?. \onecolumn and \twocolumn without a page break I've only recently properly got into using Latex, but there's still one persistent issue, which I know is a fairly common one. docx file and a PDF file derived from it. 59 "20 in 640. LaTeX is free software under the terms of the LaTeX Project Public License (LPPL). Produce beautiful documents starting from our gallery of LaTeX templates for journals, conferences, theses, reports, CVs and much more. Free Download. The Best Latex Make Table Span Two Columns Free Download PDF And Video. An example of a geometry two-column proof template. I want to add a simple two-column text within framed frame. Here is a very basic two-column article format, with many of the basic preamble items conveniently added, you can start typing in less time. Making statements based on opinion; back them up with references or personal experience. (Note that LaTeX only allows two-column floats at the top of the page by default, though some different class files, such as JASAtex, get around this somehow). This acts as a useful place to communicate information with the potential employer that may otherwise be overlooked by simply reading a list of achievements and job titles. \end{minipage} The widths can be changed by modifying the last argument in the \begin{minipage} call (ie: to make the first column twice as wide as the second, make the width of the first 0. When you swap from two columns to. 5 without going through the entire table and manually changing it?. Review this article, TeX Tables: how TeX calculates spanned column widths to learn about creating tables in LaTeX. Please help me for this task. How to make tables wrap the text on a two-column document. Stream Crafting Classes Get Latex Make A Table Span Two Columns: Get Free & Instant Access To Over 150 Highly Detailed Woodworking Project Plans. com by Alessandro Plasmati. Minimalist yet modern design. Two-column article template with coloured title and initial capital at the beginning of the paragraph. Insert a table which has at least the selected number of rows and columns 5. The argument col contains one of the position symbols, l, r, or c. Math and equations 10 [email protected] Say you have 5 left-aligned columns, so instead of lllll you could use *{5}{l}. Templates — Two-column. makes small adjustments only. This is when you state whether. Export (png, jpg, gif, svg, pdf) and save & share with note system. Thanks for contributing an answer to TeX - LaTeX Stack Exchange! Please be sure to answer the question. Text with two or double columns can be created by passing the parameter \twocolumn to the document class statement. However, I believe it has a few pain points that really shouldn't be there, such as the overflowing of inline formulas or the huge ammount of boilerplait code necessary to format a simple document, for example. pandoc [options] [input-file]… Description. 4 posts • Page 1 of 1. I am using latex and I need to put a figure which is big for a single column. The idea is that the first page has a list of people in the organization down the left-hand edge of the page in a narrow column, and the wider right-hand column has the text (letter or report text). A two-column LaTeX CV Template, especially suited for professionals who wish to emphasize and detail their previous work experience over their education. The odd package fails in a multi-column setting, but for the most part there shouldn't be too many problems with multi-column output. However, I want it to take one whole page and not be centered on one of the columns. The starred version of figure, figure*, and table, table* are floating environments. The pantry will continue to operate only two days a week on Tuesday and Wednesday from 9-11:30 a. Flip models and coefficients (place models in rows instead of in columns) esttab and estout place different models in separate columns. The San Francisco Bay Area is slowly beginning to open from the rigid shelter-in-place restrictions necessitated by the pandemic. Courses: Crocheting, Embroidery, Knitting, Quilting, Sewing. com by Alessandro Plasmati. Stream Crafting Classes Get Make Table Fit Two Columns Latex: Learn techniques & deepen your practice with classes from pros. To force all columns on the sheet to automatically fit their contents, press Ctrl + A or click the Select All button, and then double click a boundary of any column header. To turn off the feature, give % \nobalance. LaTeX and multi-column output. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. LaTeX normally sets the width of the tabular environment to "natural" width, i. Is there a way to swap two columns in a table? Unanswered. The disadvantage of the widetext environment is that it breaks the two-column layout with a couple of horizontal lines showing where the full page part begins and ends. You can use “\quad” or some other horizontal spacing instead of “\\” (line 13) to put two subfigures in the same row. The 'm' is the magic flag for vertical centering. Recently I was writing some documentation and needed to generate a section for device specifications. In this example, all entries are centered in their columns. I made a LaTex Beamer template for creating presentations in a two-column format. dailyscript. I need to insert a 3 tables in the top of the 3rd page in the two column page style. A table used instead of longtable would be one solution. The Best How To Make Two Column Table In Latex Free Download PDF And Video. Short paper submissions should also follow the two-column format of ACL 2014 proceedings, and should not exceed five (5) pages plus at most 2 pages for references. Columns and Blocks. 24 An eight-column table with numbers replacing column heads to reduce width 3. Dividing a Slide into Columns Good for displaying equations on one side and a picture on the other. 5 without going through the entire table and manually changing it? Thank you. LaTeX users must follow the Option #2 template with the permission block, conference location, copyright line, etc. That's why I want to put that picture residing in both columns. Next, start to fill in the third column. The second method is used for large documents and theses, and involves using a program called "bibtex". Detailed Images. Bugs and problems with elsarticle. Matrices and other arrays in LaTeX. Two-column article template with coloured title and initial capital at the beginning of the paragraph. cls are the two new class files created for typesetting Elsevier's journal articles which go through Complex Article Service (CAS) workflow. All ICSE 2011 submissions must follow the ACM SIG Proceedings Format. Thanks for contributing an answer to TeX - LaTeX Stack Exchange! Please be sure to answer the question. Show all Templates. Largest selection of plaster craft for your decor at wholesale prices. exponents In a LaTeX table, if TRUE or "", then use 5 \times 10^{5. The first column should be labeled Class or Category. two columns. minipage \begin{minipage}[position]{width} text \end{minipage} The minipage environment is similar to a \parbox command. An online LaTeX editor that's easy to use. For example, the com-mand \preparefootins is now automatically called. PSEUDOCODE STANDARD Pseudocode is a kind of structured english for describing algorithms. cls may be reported to the developers of the class via [email protected] The Best How To Make Two Column Table In Latex Free Download PDF And Video. This isn't needed here but if you want a diagonal line coming out of the decision node it will by default come from the center and emerge from one of the sides. Two Heart Shaped balloons with the text Land That I Love, Two Star balloons featuring American flag print. 5 inches wide on 12pt documents, 1. When two maths elements appear either side of the sign, it is assumed to be a binary operator, and as such, allocates some space either side of the sign. | Latex-Make-Table-Span-Two-Columns. The tabular environment is the default L a T e X method to create tables. 5 \textwidth ), otherwise. Filters: All / Templates / Examples. 送料無料【三協アルミ】暮らしの中の多目的スペースオイトックは、住まいを暮らしを優しくサポートします。三協アルミ オイトック 2間×3尺 波板タイプ／関東間／h=9尺／基本タイプ／1500タイプ／単体. The package is part of the preprint bundle. Learn to work on Office files without installing Office, create dynamic project plans and team calendars, auto-organize your inbox, and more. You can use “\quad” or some other horizontal spacing instead of “\\” (line 13) to put two subfigures in the same row. This template was originally published on ShareLaTeX and subsequently moved to Overleaf in October 2019. LaTeX normally sets the width of the tabular environment to "natural" width, i. A LaTeX Beamer two-column frame template with proper vertical spacing 01 Feb 2016. LaTeX技巧38：Figures, Tables, and Equations_LaTeX_Fun_新浪博客,LaTeX_Fun, If you are writing a two column document and you would like to insert a wide figure or table that spans the. As I've mentioned before, LaTeX uses column separators (&) and row separators (\\) to layout the cells of our table. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Create by Nicola Fontana,. \begin{tabular}{p{1in}p{2in}} The previous example creates two columns, of which the left is one inch wide, and the right is two inches wide. Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. online LaTeX editor with autocompletion, highlighting and 400 math symbols. 'rcl' for 3 columns. Courses: Crocheting, Embroidery, Knitting, Quilting, Sewing. To add some common text between two columns, use @ {text} in the column definition. 1 Introduction When a LATEX document is to be set in two-column mode, one can add twocolumn. Please do not use LATEX. 024 of the paper width and a 4-column layout. Making statements based on opinion; back them up with references or personal experience. Please help me for this task. Here is an example. I can type in cells but it's behind the right column. Figures and Tables Position figures and tables at the top and bottom of columns. Create by Nicola Fontana,. Flip models and coefficients (place models in rows instead of in columns) esttab and estout place different models in separate columns. 09) or to \documentclass (LATEX2"), or one can use the \twocolumn command in the text. Better to use the figure* environment. For some time I have been using pandoc to create two-column PDF documents via LaTeX. The first can be used for short documents with only a few sources, and is fairly simple. Is there a way to swap two columns in a table? Unanswered. The widths can be changed by modifying the last argument in the \begin {minipage} call (ie: to make the first column twice as wide as the second, make the width of the first 0. 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This template is characterized by the thick zigzag line that separates the left-hand sidebar from the main body, and that highlights your name and job title (on top). ) The LaTeX array environment has an argument, in this case ccc, that determines how the entries in each column are aligned. See \secref{use:cav:lab} for further explanation. twocolumn - produces two-column output for printing Edit running. If there is a single-column section on the same page, then you can fake it as described in that article, but if the entire page is in two columns, then you're out of luck. The template is based on the article class and only requires on standard packages such as hyperref , geometry , and mathpazo. The width of columns in a tabular environment is determined automatically by LaTeX; in the tabbing environment this is done by setting tab stops. After that you can use the environment wrapfig, it takes two parameters that are passed inside braces: the alignement that can be l, r, c, i or o; this letters stand for left, right, centre, inner and outer (the last two intended for two-sided documents). There are currently two viable alternatives in producing compliant PDF documents from LaTeX (the dvips method and the pdfTeX Method). How to place figures and tables in the center of the page in a two-column documents using Latex - Duration: 1:05. If you run into trouble, visit the help section. Stream Crafting Classes Get Latex Make A Table Span Two Columns: Get Free & Instant Access To Over 150 Highly Detailed Woodworking Project Plans. For example, the com-mand \preparefootins is now automatically called. An identity matrix may be denoted 1, I, E (the latter being an abbreviation for the German term "Einheitsmatrix"; Courant and Hilbert 1989, p. 16000 Woodworking Plans Get Latex Make Table Span Two Columns: Learn techniques & deepen your practice with classes from pros. As you see above, you can leave some columns blank. A two-column LaTeX CV Template, especially suited for professionals who wish to emphasize and detail their previous work experience over their education. Advanced Two Column LaTeX CV Template Layout. If you want different spacing, LaTeX provides the following four commands for use in math mode: \; - a thick space \: - a medium space \, - a thin space. Templates tagged Two-column. More-over, the last page of two-column output isn't balanced and this often results in an empty, or nearly empty, right col-umn. The alternative way is a sign designation. How to Typeset Equations in LATEX 2 6. For the 5x3 table shown above we can count five times (\\) behind each row and two times (&) per row, separating the content of three columns. 4 posts • Page 1 of 1. The template is based on the article class and only requires on standard packages such as hyperref , geometry , and mathpazo. For two columns, use the multicol package along with the following code: \begin{multicols}{2} \bibliography{bibtexFile} \end{multicols} Use \setlength{\itemsep}{1pt} inside of the thebibliography environment to change the spacing of bibliography items. Detailed Images. will give you a two-column table, each column 1 inch wide, with vertical bars before and after each, and with content centered in each. Better to use the figure* environment. Pas d’installation, collaboration en temps réel, gestion des versions, des centaines de modèles de documents LaTeX, et plus encore. tccv (two columns curriculum vitae) is a LaTeX class inspired by the template found at latextemplates. Article presenting common tips and tricks in LaTeX and LyX, focusing on typical styling and formatting problems that a user may encounter when migrating from an office suite, including multiple text columns, watermarks, syntax highlighting, table spacing, fancy footnotes and captions, and more. minipage \begin{minipage}[position]{width} text \end{minipage} The minipage environment is similar to a \parbox command. 5 without going through the entire table and manually changing it?. ACM SIG Proceeding formatting templates are available for LaTeX, Word, etc. Monday 2020-06-08 23:43:07 pm : Best Latex How To Make Table Span Two Columns Free Download DIY PDF. This template is characterized by the thick zigzag line that separates the left-hand sidebar from the main body, and that highlights your name and job title (on top). For the second method, we are going to use the following formula:. Here is a simple comparison table based on tabularx in IEEE two-column latex template. 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This works well as long as the content in each cell is short and of similar length. open your xls file with gnumeric or open office. A value of c(2, 1, 3), for instance, will apply the first label to the two first columns, the second label to the third column, and the third label will apply to the following three columns (i. The tabular environment is the default L a T e X method to create tables. You cite studies about correct, consistent use of condoms, a distinction the anti-condom, anti-sex troops never make while making the rounds of talk shows. In addition to using the \includegraphics command within the main manuscript file, upload each figure source file separately, in high-quality PDF or EPS format, using the. tccv (two columns curriculum vitae) is a LaTeX class inspired by the template found at latextemplates. To turn off the feature, give % \nobalance. 001565 in 7. Tables in LaTeX - Duration: 4:37. I made a LaTex Beamer template for creating presentations in a two-column format. The first is a beamer-specific environment and is therefore only available in a beamer presentation. In general, for an n column table, each row must have n-1 ampersands. I do this often, and it is easier, especially with long texts, because you can keep the source and translation side by. For example, suppose that we wish to typeset the following passage: Now each of the c's in {ccc} represents a column of the matrix and indicates that the entries of the column should be centred. The widths can be changed by modifying the last argument in the \begin {minipage} call (ie: to make the first column twice as wide as the second, make the width of the first 0. Two-column article template with coloured title and initial capital at the beginning of the paragraph. The columns environment allows an alignment option which speci-fieswhethercolumnsaretobealignedalongtheirtopline,bottomline,orcentered. 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Making statements based on opinion; back them up with references or personal experience. beginner woodwork. table arrays store column-oriented or tabular data, such as columns from a text file or spreadsheet. Once the basic R programming control structures are understood, users can use the R language as a powerful environment to perform complex custom analyses of almost any type of data. Come back to your lyx document 4. The Best Latex Make Table Span Two Columns Free Download PDF And Video. File -> import -> Latex can import most of the 'clean' latex files. See \secref{use:cav:lab} for further explanation. Tables are a fairly important part of LaTeX. IEEE membership offers access to technical innovation, cutting-edge information, networking opportunities, and exclusive member benefits. 6 nofonttune IEEEtran normally alters the default interword spacing to be like that used in IEEE publications. The R programming syntax is extremely easy to learn, even for users with no previous programming experience. 18 "does not give a complete description of how it works, which can be found in the. Smeme is a free simple two-column resume template. For some time I have been using pandoc to create two-column PDF documents via LaTeX. Simply including it in the preamble is sufficient to make LaTeX render the last page with roughly balanced columns, regardless of their contents. If an important block of text unexpectedly wraps to the second column, force that block to begin on the second column. These can be downloaded from CTAN (els-cas-template. These are now accepted for submitting articles, both in Elsevier's electronic submission system and elsewhere. I had to switch to the tabular environment; it has some better column definition flexibility (furthermore, you really need to use the tabular* environment). tex so that it now uses 12pt fonts rather than the default point size. Advanced Two Column LaTeX CV Template Layout. If equation (2) is multiplied by the opposite of the coefficient of $y$ in equation (1), equation (1) is multiplied by the coefficient of $y$ in equation (2), and we add the two equations, the variable $y$ will be eliminated. 1 Introduction When a LATEX document is to be set in two-column mode, one can add twocolumn. LaTeX Line and Page Breaking The first thing LaTeX does when processing ordinary text is to translate your input file into a string of glyphs and spaces. Pas d’installation, collaboration en temps réel, gestion des versions, des centaines de modèles de documents LaTeX, et plus encore. , columns number four, five and six). Left- and right-justify your columns. 23 A three-column table doubled into two columns 3. IEEE membership offers access to technical innovation, cutting-edge information, networking opportunities, and exclusive member benefits. Is there any way to easy way to swap out the row with w=0. More about creating tables in LaTeX. By default, the value will be read from the pandas config module. This data set was created only to be used as an example, and the numbers were created to match an example from a text book, p. Discover classes, experts, and inspiration to bring your ideas to life. The argument text contains the content of the column. Later you can add \onecolumn to get back to the standard one column. Two-column article template with coloured title and initial capital at the beginning of the paragraph. In recent versions (Windows and Linux) appears as "Two-column document" under Document->Settings , Text Layout tab Rich also gave som additional comments in LyxUsersPost:14079 , referring to multicols. Freeman Curriculum Vitae Description. The specific way each feature is presented and the material covered in these sites are the best reason for downloading Latex Make Table Span Two Columns woodworking plans for your construction projects. Use a longtable environment instead of. They are organized into seven classes based on their role in a mathematical expression. Simple formatted tables in python with Texttable module. 4 posts • Page 1 of 1. There are several small woodworking projects for beginners available for purchase from online woodworking retailers. 1st column & 2nd column & 3rd column\\ \hline a & b & c \end{tabular} 1st column 2nd column 3rd column a b c Note that the command is called tabular and not table. Solving Systems with Gaussian Elimination using Augmented Matrices German mathematician Carl Friedrich Gauss (1777–1855).
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2020-08-07 17:53:23
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https://getthingsdone.hk/2016/01/07/what-a-good-team-taught-me-about-software-engineering.html
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# What a good team taught me about software engineering
Gilbert Wat (屈振鵬) · January 7, 2016
Last year I was very privileged to work with some of best engineers you can find in the region. Together we delivered products in the top-most software engineering quality. Here I learned a lot from my colleagues. Here is a summary of my learning in the last year.
## Abstraction, abstraction, abstraction
Software engineering is all about abstraction. The abstraction is what the whole complication about building a software. Consider there are 2 classes.
Now there are some interactions between them, what is the best way to abstract those behavior? For example, we need to update a certain attribute in all tasks of a customer. There are several ways.
We could update the task.es file,
Or we could update customer.es file,
Both ways seem to face a scalability problem, what if there is one more method need, say find customers with certain task category? You could also argue this is an abstraction leak, the Customer / Task will now need to know the other classes. We could group them into a new file called TaskCustomer.es6 (or CustomerTask.es6, does it matter?)
Clearly this is easier to understand. The neat thing about Javascript is that the namespace is very clear and easy. Programmers like me are lazy. The way of thinking is like normalizing database schema. Whenever there is relationship between two concept, combination logic with be in a separate file.
## Clarity is the sole measurement of quality
Software code is written for people to read. There is no difference between writing an article and writing a piece of code. Consider this,
And another one,
Which one you can easily make sense of? Your code should be as clear as possible. Surprisingly I learned such thing also in the machine learning class, sometimes it is very hard to do matrix arithmetics in traditional programming languages such as Java or Javascript. Using linear algebra you can easily do so with just
No need to worry the dimension of a and b, even more so, you can check if your algorithm is correct by those dimension check.
## Message passing solves most of the problems
Both abstraction and easy to understand can be easily achieved by wrapping your mind around message passing.
For example, we have two mechanisms to charge our customers. One is purchase on behalf, say you want to order some goods or services from a third party vendors of your choice. Another is our monthly invoice.
As there is different security concerns between internal dashboard and external web application, we put all external communication logic in another system. Here is the resultant diagram.
This approach force us to think in smaller chunks of responsibilities for each class/system. For both flexibility and maintainability point of view, this approach provides great plus. The variant of such thinking, such as publish-subscription, job queue etc. is a great tool in the arsenal of every software engineer.
## TDD… is counter-intuitive at best
For the record, I have tried to follow TDD twice in my job. I think the problem of TDD is that the incentive is not aligned. What is my incentive to write a piece of code: to create a product that many people want. Does TDD help me to archieve what I am coding for? Yes. But you have to jump through many hoops, and the green red thing is actually getting into my way of creating solutions. I am well aware of the benefit, but until I do something mission critical. TDD will just be another tools in my belt.
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2020-02-20 20:08:26
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http://mathoverflow.net/questions/127734/nash-embedding-theorems-for-pseudo-riemannian-manifolds
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# Nash Embedding Theorems for Pseudo-Riemannian Manifolds?
Are there analogs of the Nash Embedding Theorems for Pseudo-Riemannian Manifolds?
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Yes, and the proof much easier, one can explicitly can write a formula for the embedding into $\mathbb{R}^{N,N}$ – Anton Petrunin Apr 17 '13 at 0:12
@AntonPetrunin: Why not give a reference? Others might stumble upon this question through a web search and find your reference useful. – Alex M. Feb 14 at 17:28
@AlexM. There is an explicit construction is called Nash twist. For the given metric $g$ it produce one parameter family of embeddings $f_t$ in $\mathbb{R}^N$ with induced metric $g+t{\cdot}h$ for some fixed metric $h$. The embedding $x\mapsto (2{\cdot}f_t(x), f_{2\cdot t}(x))$ in $\mathbb{R}^{N,N}$ is isometric. I learned it from Gromov's book on partial differential relations. – Anton Petrunin Feb 15 at 12:37
## 2 Answers
See here:
MR0262980 Reviewed Greene, Robert E. Isometric embeddings of Riemannian and pseudo-Riemannian manifolds. Memoirs of the American Mathematical Society, No. 97 American Mathematical Society, Providence, R.I. 1970 iii+63 pp. (Reviewer: W. F. Pohl)
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Not clear where you are headed with your concise question, but if you have any interest in Lorenzian manifolds as instances of pseudo-Riemannian manifolds, then this might be of interest, especially for the theorem of Campbell:
"The embedding of General Relativity in five dimensions." Carlos Romero, Reza Tavakol, Roustam Zalaletdinov. General Relativity and Gravitation. March 1996, Volume 28, Issue 3, pp 365-376. (Springer link.)
Abstract. We argue that General Relativistic solutions can always be locally embedded in Ricci-flat 5-dimensional spaces. This is a direct consequence of a theorem of Campbell (given here for both a timelike and spacelike extra dimension, together with a special case of this theorem) which guarantees that any $n$-dimensional Riemannian manifold can be locally embedded in an $(n+1)$-dimensional Ricci-flat Riemannian manifold. [...]
And there are many papers in some sense following, e.g.: "The embedding of space–times in five dimensions with nondegenerate Ricci tensor," F. Dahia and C. Romero, J. Math. Phys. 43, 3097 (2002). (AIP link.)
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2016-06-26 11:45:53
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http://www.cgoclim.com/corrigenda.html
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Real-Time Rendering Corrigenda
What follows are corrections for the book Real-Time Rendering (1st, 2nd, 3rd, and 4th editions), by Tomas Akenine-Möller, Eric Haines, Naty Hoffman (3rd and 4th), Angelo Pesce (4th), Michał Iwanicki (4th), and Sébastien Hillaire (4th).
## Corrigenda for the 4th edition
Significant errors:
• Page 14: change "superscalar" to "multi-core".
• Pages 16 through 26: change "unit cube" in numerous places, since the cube is 2x2x2, not 1x1x1. What to change it to, I'm not sure. The phrase "canonical view volume" is what this phrase stands for. Maybe "canonical cube" or we could dub it the "standard cube."(This error has been in all four editions, and is one that early versions of the OpenGL Red Book also makes.)*
• Page 67: There is a sign error in the view vector, which causes a number of rewrites. Change "(c-l) / || c-l ||" to "(l-c) / || l-c ||". Change "r = -(v x u') / || v x u' ||" to "r = (v x u') / || v x u' ||", i.e., remove the minus sign. Change "u = v x r" to "u = r x v". Change "and $\mb v$ with $(0,0,1)$" to "and $\mb v$ with $(0,0,-1)$". In the left matrix that follows, change "v_x & v_y & v_z & 0" to "-v_x & -v_y & -v_z & 0". In the right matrix, change "v_x & v_y & v_z & -\mb t\cdot\mb v" to "-v_x & -v_y & -v_z & \mb t\cdot\mb v".\
• Page 214: Equation 6.16, the second component of the normal should be "h_y", not "h_x".
• Page 242: "by slicing it parallel to the view direction" to "by slicing it perpendicular to the view direction".
• Page 256: "Called an exponential shadow map (ESM) or exponential variance shadow map (EVSM)" - remove "or exponential variance shadow map (EVSM)".
• Page 256: "compares this new approach with ESM", change to "compares this new approach with another method, the exponential variance shadow map (EVSM)".
• Page 381: In the footnote, remove the parenthetical "(where the distance is taken from the light surface, not its center)" after "Note that while, for spherical lights, the falloff does take the usual inverse square distance formulation". The distance to the center is used to compute the lighting for a sphere. For derivations, see this technical report, "Area Light Sources for Real-Time Graphics" by John M. Snyder.
• Page 429: "Figure 10.43 shows how an irradiance map -derived directly compares to one synthesized by the nine-term function." to "See Figure 10.45 for a comparison of an irradiance map derived by integration with one expressed by a nine-term spherical harmonic function."
• Page 727: "the right tangent at the join has to be twice as long as the left tangent." to "the right difference vector between the control points at the join has to be twice as long as the left difference vector between the control points."
• Page 733: "square function" to "box function".
• Page 733: "is created by integrating $\beta_0(t)$" to "is created by convolving $\beta_0(t)$ with $\beta_0(t)$", and "is created by integrating $\beta_1(t)$" to "is created by convolving $\beta_1(t)$ with $\beta_0(t)$".
• Page 751: Equation 17.56, the expression should be divided by "2 \epsilon".
• Page 752: Equation 17.57, the last part on the second line should change sign to be negative, i.e., "- r_b h (1 - h)".
• Page 767 and 777: "The normal is then found as the cross product between those vectors." to "... between these." (tangent patches)
• Page 1092: Reference 793, these course notes are no longer online, so this reference should be replaced by reference 903, Kirk, David, and Wen-Mei Hwu, Programming Massively Parallel Processors: A Hands-on Approach, Morgan-Kaufmann, 2016
• Page 1133: Reference 1580, the author of these slides should be Eitan Grinspun; Peter Schröder ran the course. (Not fixed in any printing, as the reference order then needs to be changed.)*
Minor errors:
• Page 21: The end of sentence "where $d$ is the discrete (integer) index of the pixel and $c$ is the continuous (floating point) value within the pixel" should have the reference "~\cite{Heckbert90a}." - this is reference #692, "What Are the Coordinates of a Pixel?"
• Page 22: in Figure 2.8 it says "pixel processing and merging" in the caption, but should say "pixel shading and merging" in order to match the functional stage name.
• Page 72: "we have a gimbal lock (Section 4.2.2)," - self-reference; remove entire phrase.
• Page 77: "from the multiplication seen in Equation 4.3.1" should be "... Equation 4.32".
• Page 100: "and the constants d and f" to "and the constants d and e", with "e" being derived from n and f, as shown further on.
• Page 128: "likely to have lower register occupancy" to "likely to have lower register usage".
• Page 131: "goal of this sampling process is" - unexpected line break.
• Page 145: "Interleaved samplingindexsampling!interleaved" - typographical error.
• Page 160: "multiplied by the stored alpha before being display" - should be "displayed".
• Page 265: "We have avoiding" to "We have avoided".
• Page 239: "depth layer between the occlude and receiver" - missing letter, change to "occluder".
• Page 247: "function of what proportion of the light's area" - change "proportion" to "portion".
• Page 333: "is the {\em distribution of visible normals}" to "is related to the {\em distribution of visible normals}".
• Page 351: "Equations 9.62 and 9.63 result in a uniform distribution of" to "Equation 9.62 results in a uniform distribution of".
• Page 494: "diffuse and reflective surfaces" - "reflective" to "specular".
• Page 615: "used to stored" - should be "store".
• Page 616: "simulated using participating media" to "represented as participating media".
• Page 785: RenderDoc, "Windows debugger" to "Windows, Linux, and Android debugger".
• Page 819: "Construction time of a spatial data structure can be expensive" to "... can be long".
• Page 820: "children nodes" to "child nodes".
• Page 823: "or games like DOOM (2016), back when there was no hardware z-buffer" - remove "(2016)" ;-)
• Page 830: "not considered to contribute" to "considered to not contribute".
• Page 910: "Clarberg et al. [271] present hardware extensions..." The rest of this paragraph should follow the next paragraph, "Andersson et al.", (as a separate paragraph), instead of preceding it.
• Page 970: nine lines down, "curves!Bézier" should not be visible; it was a poorly-formatted index entry.
• Page 999: Figure 23.5 - replace angle "a" with "\alpha", to match the text that follows.
• Page 1054: Reference 68, a new edition of this book has appeared, and the chapter is now Chapter 35. The editors and full title are: Dinesh P. Mehta and Sartaj Sahni, eds., Handbook of Data Structures and Applications, Second Edition, Chapman and Hall/CRC Press, 2018.
• Page 1061: Reference 199, "The Art and Science of Digital Compositing" is now in a second edition, 2008.
• Page 1074: Reference 458, "Curves and Surfaces for Computer Aided Geometric Design---A Practical Guide," is now in its Fifth Edition, Morgan-Kaufmann, 2002.
• Page 1074: Reference 461, "Practical Linear Algebra: A Geometry Toolbox," is now in its Third Edition, A K Peters/CRC Press, 2013.
• Page 1098: Reference 910, all the text for this reference after "Cited on p. 979" should be deleted.
• Page 1100: Reference 941, the second author's last name should be "Akenine-Möller" (oh, the irony).
• Pages 1100-1101: References 952 and 957-961, The second author's first name should be "S\'{e}bastien" not "tian".
• Page 1112: Reference 1172, the "Computer Graphics Archive" has moved to http://casual-effects.com/data.
• Page 1117: Reference 1259, the book is "GPU Pro 7" (not "5"), 2016, and the page numbers should be 219--229, not 219--230. Also, very minor error, the title "Real-Time BC6H Compression on GPU" should be in quotes, not italicized.
• Page 1122: Reference 1358, the first author's first name should be "Anjul" not "Anuj".
• Page 1133: Reference 1575, the last author's last name should be "Froehlich" not "Fro".
• Page 1144: Reference 1787, the first author's last name should be "Tropp" not "Trop".
• Page 1146: Reference 1829, change "Johnsson" to "Johnson".
Very minor errors (only the authors should care, for future revisions):
• Page xiv: the phrase "who helped us along the way" is repeated; replace the second instance with "involved", i.e. "about everyone involved". Michael Drobot is thanked twice, and Wolfgang Engel's name was slightly out of order.
• Page 3: "And after" to simply "After".
• Page 18: "These are in fact homogeneous coordinates" to the stronger "These are homogeneous coordinates".
• Page 31: "execution by some number GPU shader cores" to "... some number of GPU...".
• Page 94: Figure 4.18, the "z" axis labels should all be italicized, "z".
• Pages 102, 159, 404, 420, 421, 452, 512, 767: avoid "et al.'s" by rewording.
• Page 140 (3rd printing): Figure 5.24, "per pixel, and as can be seen, two of" to "per pixel. Two of".
• Page 237: Figure 7.12 could include a reference to Christoph Peters' page http://momentsingraphics.de/?p=175.*
• Page 397: "(see [1838]):" should be "[1838]:", as we never consider a citation something we point at.
• Page 485: "an unique" to "a unique".
• Page 473: Figure 11.21's hemisphere is squashed, it should be truly a round hemisphere.*
• Page 505: Figure 11.39, extend the black ground plane line to the right so it perfectly matches the blue rectangle below.
• Page 616: "featuring high-scattering coefficients" - remove hyphen.
• Page 626: Figure 14.35, image should be centered (missing "\centerline").
• Page 658: Figure 15.8, the two triangles should be aligned horizontally.
• Page 704: Figure 16.15, the line of text beginning "The triangles could" should be shifted to the left, to match the left margin of the other lines of text.
• Page 749: Figure 17.32 could include a reference to Tamy Boubekeur's page https://perso.telecom-paristech.fr/boubek/tessellation/.*
• Page 756 (3rd printing): Figure 17.38, "As can be seen, the" to "The".
• Page 767: "over the surface The same" - add a period after "surface".
• Page 767 and 777: "The normal is then found as the cross product..." - these next few sentences are identical between these pages. The later one should be revised. Not touched for pagination reasons.*
• Page 772 (3rd printing): Figure 17.58, "As can be seen in" to "As seen in".
• Page 783 (3rd printing): Figure 17.70, "As can be seen, there" to "There".
• Page 785: "Xcode on OSX" to "Xcode on macOS", since .
• Page 843: Figure 19.21 could probably just be pseudocode in the text instead of a figure, as done in the Intersection Test Methods chapter, for example.*
• Page 844 (3rd printing): Figure 19.20, replace "As can be seen, the" to simple "The".
• Page 865: "time-critical rendering.." - remove a period.
• Page 874: Figure 19.46 could include a reference to Losasso et al. or the program itself.*
• Page 889: Figure 20.3, "As can be seen, anything" to "Any values".
• Page 915: Figure 21.1, the first two images are the tiniest bit from the top edge, the last the tiniest bit from the bottom edge.
• Page 931: "Our eyes have a field of view for binocular vision (where both eyes can see the same object) of 114 horizontal degrees." to "Our eyes have a horizontal field of view for binocular vision (where both eyes can see the same object) of 114 degrees."
• Page 985: "frustum sides planes" to "frustum sides' planes".
• Page 1021: Figure 23.17, the right edge of the figure is clipped.
• Page 1032: Figure 23.25, the two top bars should be widened on the left a tiny bit, so that they align with the left edges of the mem. controllers (as they are on the right).
• Page 1051: Reference 15, add pages "pp.~145-149".
• Page 1058: Reference 141, final "s" missing from title, should read: "Two-Part Texture Mappings".
• Page 1061: Reference 202, title capitalization should be "Shadowing by non-Gaussian random surfaces".
• Page 1063: Reference 233, the title should be followed by a double-quote, not a single quote.
• Page 1063: Reference 235, the title should be followed by a double-quote, not a single quote.
• Page 1065: Reference 285, the month should be Aug., not July.
• Page 1066: Reference 292, this is the Third Edition.
• Page 1070: Reference 375, the title should be followed by a double-quote, not a single quote.
• Page 1079: Reference 560, title "in" to "of", "Water Technology of Uncharted".
• Page 1083: Reference 622, "journals" to "journal".
• Page 1088: Reference 773, the title should be preceded by a double-quote, not a single quote.
• Page 1095: Reference 846, add page numbers, "pp.~143--150".
• Page 1096: Reference 880, remove "in" from "in Rendering Techniques", and page numbers should be "pp.~269--276".
• Page 1098: Reference 901, the title word "DN-10000VS" should have no hyphen, "DN10000VS".
• Page 1099: Reference 928, spell out authors, "Kopta, Daniel, Thiago Ize, Josef Spjut, Erik Brunvand, Al Davis, and Andrew Kensler," and remove "ACM," from after the title.
• Page 1101: Reference 963, title has three left apostrophes, remove one.
• Page 1108: Reference 1096, spell out authors, "MacDonald, J. David, and Kellogg S. Booth," change number of issue from "6" to "3".
• Page 1115: Reference 1224, remove space between left double-quote and "Stylization" in title.
• Page 1118: Reference 1274, "Nalu" should not be in quotes.
• Page 1126: Reference 1443, "Smitsc" should be "Smits".
• Page 1126: Reference 1450, "Ray Tracing" should be "raytracing" (yes, one word, lowercase).
• Page 1132: Reference 1563, add page numbers "pp.~2:1--2:12" before date.
• Page 1135: Reference 1619, "journals" to "journal".
• Page 1141: Reference 1723, title's capitalization and punctuation should be "Radiance Transfer Biclustering for Real-time All-frequency Bi-scale Rendering".
• Page 1145: Reference 1815, Veach's PhD title should be italicized, not in double-quotes.
• Page 1148: Reference 1861, the title should be followed by a double-quote, not a single quote.
• Page 1152: Reference 1935, the title has (inconsistent) capitalization, as follows: "Adaptive Real-Time Level-of-detail-based Rendering for Polygonal Objects".
• Page 1152: Reference 1947, add "article no.~4," before date.
• Page 1153: Reference 1950, the title has (inconsistent) capitalization, and should be "Ray Tracing Dynamic Scenes using Selective Restructuring"; the conference should be "{\em EGSR Proceedings of the 18th Eurographics Conference on Rendering Techniques}"
Errors followed by a "*" have not been fixed internally yet and are likely to be deferred until a future fifth edition (no, none is planned right now), as they could affect pagination. Otherwise, fixes are in place in online materials (e.g., references and figures) and will be in future printings, as possible.
Thanks to Evgenii Golubev, Alex Yang, Doug Richardson, Mick Charles Beaver, Yuhao Zhu, Feiyun Wang, and others for reporting errors.
## Corrigenda for the 2nd edition
The second edition, 2nd print is recognized, for example, by that Steve Morein and David Wu are included in the acknowledgments on page xiii. Also, if you have the 1st print, you need to look at both the list for the 1st print and for the 2nd.
## Corrigenda for the 1st edition
One way to identify the second printing: go to the first page of the Preface; if the web site listed near the bottom of the page is "http://www.cgoclim.com," then this copy is from the second printing.
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Suppose one is given a ${k_0}$-tuple ${{\mathcal H} = (h_1,\ldots,h_{k_0})}$ of ${k_0}$ distinct integers for some ${k_0 \geq 1}$, arranged in increasing order. When is it possible to find infinitely many translates ${n + {\mathcal H} =(n+h_1,\ldots,n+h_{k_0})}$ of ${{\mathcal H}}$ which consists entirely of primes? The case ${k_0=1}$ is just Euclid’s theorem on the infinitude of primes, but the case ${k_0=2}$ is already open in general, with the ${{\mathcal H} = (0,2)}$ case being the notorious twin prime conjecture.
On the other hand, there are some tuples ${{\mathcal H}}$ for which one can easily answer the above question in the negative. For instance, the only translate of ${(0,1)}$ that consists entirely of primes is ${(2,3)}$, basically because each translate of ${(0,1)}$ must contain an even number, and the only even prime is ${2}$. More generally, if there is a prime ${p}$ such that ${{\mathcal H}}$ meets each of the ${p}$ residue classes ${0 \hbox{ mod } p, 1 \hbox{ mod } p, \ldots, p-1 \hbox{ mod } p}$, then every translate of ${{\mathcal H}}$ contains at least one multiple of ${p}$; since ${p}$ is the only multiple of ${p}$ that is prime, this shows that there are only finitely many translates of ${{\mathcal H}}$ that consist entirely of primes.
To avoid this obstruction, let us call a ${k_0}$-tuple ${{\mathcal H}}$ admissible if it avoids at least one residue class ${\hbox{ mod } p}$ for each prime ${p}$. It is easy to check for admissibility in practice, since a ${k_0}$-tuple is automatically admissible in every prime ${p}$ larger than ${k_0}$, so one only needs to check a finite number of primes in order to decide on the admissibility of a given tuple. For instance, ${(0,2)}$ or ${(0,2,6)}$ are admissible, but ${(0,2,4)}$ is not (because it covers all the residue classes modulo ${3}$). We then have the famous Hardy-Littlewood prime tuples conjecture:
Conjecture 1 (Prime tuples conjecture, qualitative form) If ${{\mathcal H}}$ is an admissible ${k_0}$-tuple, then there exists infinitely many translates of ${{\mathcal H}}$ that consist entirely of primes.
This conjecture is extremely difficult (containing the twin prime conjecture, for instance, as a special case), and in fact there is no explicitly known example of an admissible ${k_0}$-tuple with ${k_0 \geq 2}$ for which we can verify this conjecture (although, thanks to the recent work of Zhang, we know that ${(0,d)}$ satisfies the conclusion of the prime tuples conjecture for some ${0 < d < 70,000,000}$, even if we can’t yet say what the precise value of ${d}$ is).
Actually, Hardy and Littlewood conjectured a more precise version of Conjecture 1. Given an admissible ${k_0}$-tuple ${{\mathcal H} = (h_1,\ldots,h_{k_0})}$, and for each prime ${p}$, let ${\nu_p = \nu_p({\mathcal H}) := |{\mathcal H} \hbox{ mod } p|}$ denote the number of residue classes modulo ${p}$ that ${{\mathcal H}}$ meets; thus we have ${1 \leq \nu_p \leq p-1}$ for all ${p}$ by admissibility, and also ${\nu_p = k_0}$ for all ${p>h_{k_0}-h_1}$. We then define the singular series ${{\mathfrak G} = {\mathfrak G}({\mathcal H})}$ associated to ${{\mathcal H}}$ by the formula
$\displaystyle {\mathfrak G} := \prod_{p \in {\mathcal P}} \frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}}$
where ${{\mathcal P} = \{2,3,5,\ldots\}}$ is the set of primes; by the previous discussion we see that the infinite product in ${{\mathfrak G}}$ converges to a finite non-zero number.
We will also need some asymptotic notation (in the spirit of “cheap nonstandard analysis“). We will need a parameter ${x}$ that one should think of going to infinity. Some mathematical objects (such as ${{\mathcal H}}$ and ${k_0}$) will be independent of ${x}$ and referred to as fixed; but unless otherwise specified we allow all mathematical objects under consideration to depend on ${x}$. If ${X}$ and ${Y}$ are two such quantities, we say that ${X = O(Y)}$ if one has ${|X| \leq CY}$ for some fixed ${C}$, and ${X = o(Y)}$ if one has ${|X| \leq c(x) Y}$ for some function ${c(x)}$ of ${x}$ (and of any fixed parameters present) that goes to zero as ${x \rightarrow \infty}$ (for each choice of fixed parameters).
Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then the number of natural numbers ${n < x}$ such that ${n+{\mathcal H}}$ consists entirely of primes is ${({\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}$.
Thus, for instance, if Conjecture 2 holds, then the number of twin primes less than ${x}$ should equal ${(2 \Pi_2 + o(1)) \frac{x}{\log^2 x}}$, where ${\Pi_2}$ is the twin prime constant
$\displaystyle \Pi_2 := \prod_{p \in {\mathcal P}: p>2} (1 - \frac{1}{(p-1)^2}) = 0.6601618\ldots.$
As this conjecture is stronger than Conjecture 1, it is of course open. However there are a number of partial results on this conjecture. For instance, this conjecture is known to be true if one introduces some additional averaging in ${{\mathcal H}}$; see for instance this previous post. From the methods of sieve theory, one can obtain an upper bound of ${(C_{k_0} {\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}$ for the number of ${n < x}$ with ${n + {\mathcal H}}$ all prime, where ${C_{k_0}}$ depends only on ${k_0}$. Sieve theory can also give analogues of Conjecture 2 if the primes are replaced by a suitable notion of almost prime (or more precisely, by a weight function concentrated on almost primes).
Another type of partial result towards Conjectures 1, 2 come from the results of Goldston-Pintz-Yildirim, Motohashi-Pintz, and of Zhang. Following the notation of this recent paper of Pintz, for each ${k_0>2}$, let ${DHL[k_0,2]}$ denote the following assertion (DHL stands for “Dickson-Hardy-Littlewood”):
Conjecture 3 (${DHL[k_0,2]}$) Let ${{\mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then there are infinitely many translates ${n+{\mathcal H}}$ of ${{\mathcal H}}$ which contain at least two primes.
This conjecture gets harder as ${k_0}$ gets smaller. Note for instance that ${DHL[2,2]}$ would imply all the ${k_0=2}$ cases of Conjecture 1, including the twin prime conjecture. More generally, if one knew ${DHL[k_0,2]}$ for some ${k_0}$, then one would immediately conclude that there are an infinite number of pairs of consecutive primes of separation at most ${H(k_0)}$, where ${H(k_0)}$ is the minimal diameter ${h_{k_0}-h_1}$ amongst all admissible ${k_0}$-tuples ${{\mathcal H}}$. Values of ${H(k_0)}$ for small ${k_0}$ can be found at this link (with ${H(k_0)}$ denoted ${w}$ in that page). For large ${k_0}$, the best upper bounds on ${H(k_0)}$ have been found by using admissible ${k_0}$-tuples ${{\mathcal H}}$ of the form
$\displaystyle {\mathcal H} = ( - p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, - p_{m+1}, -1, +1, p_{m+1}, \ldots, p_{m+\lfloor (k_0+1)/2\rfloor - 1} )$
where ${p_n}$ denotes the ${n^{th}}$ prime and ${m}$ is a parameter to be optimised over (in practice it is an order of magnitude or two smaller than ${k_0}$); see this blog post for details. The upshot is that one can bound ${H(k_0)}$ for large ${k_0}$ by a quantity slightly smaller than ${k_0 \log k_0}$ (and the large sieve inequality shows that this is sharp up to a factor of two, see e.g. this previous post for more discussion).
In a key breakthrough, Goldston, Pintz, and Yildirim were able to establish the following conditional result a few years ago:
Theorem 4 (Goldston-Pintz-Yildirim) Suppose that the Elliott-Halberstam conjecture ${EH[\theta]}$ is true for some ${1/2 < \theta < 1}$. Then ${DHL[k_0,2]}$ is true for some finite ${k_0}$. In particular, this establishes an infinite number of pairs of consecutive primes of separation ${O(1)}$.
The dependence of constants between ${k_0}$ and ${\theta}$ given by the Goldston-Pintz-Yildirim argument is basically of the form ${k_0 \sim (\theta-1/2)^{-2}}$. (UPDATE: as recently observed by Farkas, Pintz, and Revesz, this relationship can be improved to ${k_0 \sim (\theta-1/2)^{-3/2}}$.)
Unfortunately, the Elliott-Halberstam conjecture (which we will state properly below) is only known for ${\theta<1/2}$, an important result known as the Bombieri-Vinogradov theorem. If one uses the Bombieri-Vinogradov theorem instead of the Elliott-Halberstam conjecture, Goldston, Pintz, and Yildirim were still able to show the highly non-trivial result that there were infinitely many pairs ${p_{n+1},p_n}$ of consecutive primes with ${(p_{n+1}-p_n) / \log p_n \rightarrow 0}$ (actually they showed more than this; see e.g. this survey of Soundararajan for details).
Actually, the full strength of the Elliott-Halberstam conjecture is not needed for these results. There is a technical specialisation of the Elliott-Halberstam conjecture which does not presently have a commonly accepted name; I will call it the Motohashi-Pintz-Zhang conjecture ${MPZ[\varpi]}$ in this post, where ${0 < \varpi < 1/4}$ is a parameter. We will define this conjecture more precisely later, but let us remark for now that ${MPZ[\varpi]}$ is a consequence of ${EH[\frac{1}{2}+2\varpi]}$.
We then have the following two theorems. Firstly, we have the following strengthening of Theorem 4:
Theorem 5 (Motohashi-Pintz-Zhang) Suppose that ${MPZ[\varpi]}$ is true for some ${0 < \varpi < 1/4}$. Then ${DHL[k_0,2]}$ is true for some ${k_0}$.
A version of this result (with a slightly different formulation of ${MPZ[\varpi]}$) appears in this paper of Motohashi and Pintz, and in the paper of Zhang, Theorem 5 is proven for the concrete values ${\varpi = 1/1168}$ and ${k_0 = 3,500,000}$. We will supply a self-contained proof of Theorem 5 below the fold, the constants upon those in Zhang’s paper (in particular, for ${\varpi = 1/1168}$, we can take ${k_0}$ as low as ${341,640}$, with further improvements on the way). As with Theorem 4, we have an inverse quadratic relationship ${k_0 \sim \varpi^{-2}}$.
In his paper, Zhang obtained for the first time an unconditional advance on ${MPZ[\varpi]}$:
Theorem 6 (Zhang) ${MPZ[\varpi]}$ is true for all ${0 < \varpi \leq 1/1168}$.
This is a deep result, building upon the work of Fouvry-Iwaniec, Friedlander-Iwaniec and Bombieri-Friedlander-Iwaniec which established results of a similar nature to ${MPZ[\varpi]}$ but simpler in some key respects. We will not discuss this result further here, except to say that they rely on the (higher-dimensional case of the) Weil conjectures, which were famously proven by Deligne using methods from l-adic cohomology. Also, it was believed among at least some experts that the methods of Bombieri, Fouvry, Friedlander, and Iwaniec were not quite strong enough to obtain results of the form ${MPZ[\varpi]}$, making Theorem 6 a particularly impressive achievement.
Combining Theorem 6 with Theorem 5 we obtain ${DHL[k_0,2]}$ for some finite ${k_0}$; Zhang obtains this for ${k_0 = 3,500,000}$ but as detailed below, this can be lowered to ${k_0 = 341,640}$. This in turn gives infinitely many pairs of consecutive primes of separation at most ${H(k_0)}$. Zhang gives a simple argument that bounds ${H(3,500,000)}$ by ${70,000,000}$, giving his famous result that there are infinitely many pairs of primes of separation at most ${70,000,000}$; by being a bit more careful (as discussed in this post) one can lower the upper bound on ${H(3,500,000)}$ to ${57,554,086}$, and if one instead uses the newer value ${k_0 = 341,640}$ for ${k_0}$ one can instead use the bound ${H(341,640) \leq 4,982,086}$. (Many thanks to Scott Morrison for these numerics.) UPDATE: These values are now obsolete; see this web page for the latest bounds.
In this post we would like to give a self-contained proof of both Theorem 4 and Theorem 5, which are both sieve-theoretic results that are mainly elementary in nature. (But, as stated earlier, we will not discuss the deepest new result in Zhang’s paper, namely Theorem 6.) Our presentation will deviate a little bit from the traditional sieve-theoretic approach in a few places. Firstly, there is a portion of the argument that is traditionally handled using contour integration and properties of the Riemann zeta function; we will present a “cheaper” approach (which Ben Green and I used in our papers, e.g. in this one) using Fourier analysis, with the only property used about the zeta function ${\zeta(s)}$ being the elementary fact that blows up like ${\frac{1}{s-1}}$ as one approaches ${1}$ from the right. To deal with the contribution of small primes (which is the source of the singular series ${{\mathfrak G}}$), it will be convenient to use the “${W}$-trick” (introduced in this paper of mine with Ben), passing to a single residue class mod ${W}$ (where ${W}$ is the product of all the small primes) to end up in a situation in which all small primes have been “turned off” which leads to better pseudorandomness properties (for instance, once one eliminates all multiples of small primes, almost all pairs of remaining numbers will be coprime).
A finite group ${G=(G,\cdot)}$ is said to be a Frobenius group if there is a non-trivial subgroup ${H}$ of ${G}$ (known as the Frobenius complement of ${G}$) such that the conjugates ${gHg^{-1}}$ of ${H}$ are “disjoint as possible” in the sense that ${H \cap gHg^{-1} = \{1\}}$ whenever ${g \not \in H}$. This gives a decomposition
$\displaystyle G = \bigcup_{gH \in G/H} (gHg^{-1} \backslash \{1\}) \cup K \ \ \ \ \ (1)$
where the Frobenius kernel ${K}$ of ${G}$ is defined as the identity element ${1}$ together with all the non-identity elements that are not conjugate to any element of ${H}$. Taking cardinalities, we conclude that
$\displaystyle |G| = \frac{|G|}{|H|} (|H| - 1) + |K|$
and hence
$\displaystyle |H| |K| = |G|. \ \ \ \ \ (2)$
A remarkable theorem of Frobenius gives an unexpected amount of structure on ${K}$ and hence on ${G}$:
Theorem 1 (Frobenius’ theorem) Let ${G}$ be a Frobenius group with Frobenius complement ${H}$ and Frobenius kernel ${K}$. Then ${K}$ is a normal subgroup of ${G}$, and hence (by (2) and the disjointness of ${H}$ and ${K}$ outside the identity) ${G}$ is the semidirect product ${K \rtimes H}$ of ${H}$ and ${K}$.
I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group ${G}$, in particular relying on the fact that a character on a subgroup ${H}$ can induce a character on ${G}$, which can then be decomposed into irreducible characters with natural number coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement ${H}$ has even order or when ${H}$ is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.)
I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ${{\bf C} G \equiv L^2(G)}$) with the Fourier analysis of class functions (i.e. the representation theory of the centre ${Z({\bf C} G) \equiv L^2(G)^G}$ of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here.
The main ideas are as follows. The space ${L^2(G)^G}$ of class functions can be viewed as a commutative algebra with respect to the convolution operation ${*}$; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions ${\phi}$ such that ${\phi*\phi = \phi}$. In terms of characters, idempotents are nothing more than sums of the form ${\sum_{\chi \in \Sigma} \chi(1) \chi}$ for various collections ${\Sigma}$ of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that ${\phi(1)}$ is a natural number, and more generally we will show that ${{\bf E}_{(a,b) \in S} {\bf E}_{x \in G} \phi( a x b^{-1} x^{-1} )}$ is a natural number whenever ${S}$ is a subgroup of ${G \times G}$ (see Corollary 4 below). For instance, the quantity
$\displaystyle \hbox{rank}(\phi) := {\bf E}_{a \in G} {\bf E}_{x \in G} \phi(a xa^{-1} x^{-1})$
is a natural number which we will call the rank of ${\phi}$ (as it is also the linear rank of the transformation ${f \mapsto f*\phi}$ on ${L^2(G)}$).
In the case that ${G}$ is a Frobenius group with kernel ${K}$, the above integrality properties can be used after some elementary manipulations to establish that for any idempotent ${\phi}$, the quantity
$\displaystyle \frac{1}{|G|} \sum_{a \in K} {\bf E}_{x \in G} \phi( axa^{-1}x^{-1} ) - \frac{1}{|G| |K|} \sum_{a,b \in K} \phi(ab^{-1}) \ \ \ \ \ (3)$
is an integer. On the other hand, one can also show by elementary means that this quantity lies between ${0}$ and ${\hbox{rank}(\phi)}$. These two facts are not strong enough on their own to impose much further structure on ${\phi}$, unless one restricts attention to minimal idempotents ${\phi}$. In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that ${\phi}$ has rank one, and then the integrality gap comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent ${\phi}$ either preserves ${\frac{|G|}{|K|} 1_K}$ or annihilates it, which makes ${\frac{|G|}{|K|} 1_K}$ itself an idempotent, which makes ${K}$ normal.
Suppose that ${G = (G,\cdot)}$ is a finite group of even order, thus ${|G|}$ is a multiple of two. By Cauchy’s theorem, this implies that ${G}$ contains an involution: an element ${g}$ in ${G}$ of order two. (Indeed, if no such involution existed, then ${G}$ would be partitioned into doubletons ${\{g,g^{-1}\}}$ together with the identity, so that ${|G|}$ would be odd, a contradiction.) Of course, groups of odd order have no involutions ${g}$, thanks to Lagrange’s theorem (since ${G}$ cannot split into doubletons ${\{ h, hg \}}$).
The classical Brauer-Fowler theorem asserts that if a group ${G}$ has many involutions, then it must have a large non-trivial subgroup:
Theorem 1 (Brauer-Fowler theorem) Let ${G}$ be a finite group with at least ${|G|/n}$ involutions for some ${n > 1}$. Then ${G}$ contains a proper subgroup ${H}$ of index at most ${n^2}$.
This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen ${n^2}$ slightly to ${n(n+2)/2}$) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if ${g}$ is a an involution of a group ${G}$, and the centraliser ${C_G(g) := \{ h \in G: gh = hg\}}$ has order ${n}$, then clearly ${n \geq 2}$ (as ${C_G(g)}$ contains ${1}$ and ${g}$) and the conjugacy class ${\{ aga^{-1}: a \in G \}}$ has order ${|G|/n}$ (since the map ${a \mapsto aga^{-1}}$ has preimages that are cosets of ${C_G(g)}$). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem ${G}$ contains a subgroup of order at least ${\max( n, |G|/n^2)}$. In particular, we can conclude that every group ${G}$ of even order contains a proper subgroup of order at least ${|G|^{1/3}}$.
Another corollary is that the size of a simple group of even order can be controlled by the size of a centraliser of one of its involutions:
Corollary 2 (Brauer-Fowler theorem) Let ${G}$ be a finite simple group with an involution ${g}$, and suppose that ${C_G(g)}$ has order ${n}$. Then ${G}$ has order at most ${(n^2)!}$.
Indeed, by the previous discussion ${G}$ has a proper subgroup ${H}$ of index less than ${n^2}$, which then gives a non-trivial permutation action of ${G}$ on the coset space ${G/H}$. The kernel of this action is a proper normal subgroup of ${G}$ and is thus trivial, so the action is faithful, and the claim follows.
If one assumes the Feit-Thompson theorem that all groups of odd order are solvable, then Corollary 2 suggests a strategy (first proposed by Brauer himself in 1954) to prove the classification of finite simple groups (CFSG) by induction on the order of the group. Namely, assume for contradiction that the CFSG failed, so that there is a counterexample ${G}$ of minimal order ${|G|}$ to the classification. This is a non-abelian finite simple group; by the Feit-Thompson theorem, it has even order and thus has at least one involution ${g}$. Take such an involution and consider its centraliser ${C_G(g)}$; this is a proper subgroup of ${G}$ of some order ${n < |G|}$. As ${G}$ is a minimal counterexample to the classification, one can in principle describe ${C_G(g)}$ in terms of the CFSG by factoring the group into simple components (via a composition series) and applying the CFSG to each such component. Now, the “only” thing left to do is to verify, for each isomorphism class of ${C_G(g)}$, that all the possible simple groups ${G}$ that could have this type of group as a centraliser of an involution obey the CFSG; Corollary 2 tells us that for each such isomorphism class for ${C_G(g)}$, there are only finitely many ${G}$ that could generate this class for one of its centralisers, so this task should be doable in principle for any given isomorphism class for ${C_G(g)}$. That’s all one needs to do to prove the classification of finite simple groups!
Needless to say, this program turns out to be far more difficult than the above summary suggests, and the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument is based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a ${p}$-group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of ${p}$-groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups which dictates a very specific type of interaction structure between these ${p}$-groups in the key case when ${G}$ is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic ${p}$. See the text of Aschbacher, Lyons, Smith, and Solomon for a more detailed description of this strategy.
The Brauer-Fowler theorem can be proven by a nice application of character theory, of the type discussed in this recent blog post, ultimately based on analysing the alternating tensor power of representations; I reproduce a version of this argument (taken from this text of Isaacs) below the fold. (The original argument of Brauer and Fowler is more combinatorial in nature.) However, I wanted to record a variant of the argument that relies not on the fine properties of characters, but on the cruder theory of quasirandomness for groups, the modern study of which was initiated by Gowers, and is discussed for instance in this previous post. It gives the following slightly weaker version of Corollary 2:
Corollary 3 (Weak Brauer-Fowler theorem) Let ${G}$ be a finite simple group with an involution ${g}$, and suppose that ${C_G(g)}$ has order ${n}$. Then ${G}$ can be identified with a subgroup of the unitary group ${U_{4n^3}({\bf C})}$.
One can get an upper bound on ${|G|}$ from this corollary using Jordan’s theorem, but the resulting bound is a bit weaker than that in Corollary 2 (and the best bounds on Jordan’s theorem require the CFSG!).
Proof: Let ${A}$ be the set of all involutions in ${G}$, then as discussed above ${|A| \geq |G|/n}$. We may assume that ${G}$ has no non-trivial unitary representation of dimension less than ${4n^3}$ (since such representations are automatically faithful by the simplicity of ${G}$); thus, in the language of quasirandomness, ${G}$ is ${4n^3}$-quasirandom, and is also non-abelian. We have the basic convolution estimate
$\displaystyle \|1_A * 1_A * 1_A - \frac{|A|^3}{|G|} \|_{\ell^\infty(G)} \leq (4n^3)^{-1/2} |G|^{1/2} |A|^{3/2}$
(see Exercise 10 from this previous blog post). In particular,
$\displaystyle 1_A * 1_A * 1_A(0) \geq \frac{|A|^3}{|G|} - (4n^3)^{-1/2} |G|^{1/2} |A|^{3/2} \geq \frac{1}{2n^3} |G|^2$
and so there are at least ${|G|^2/2n^3}$ pairs ${(g,h) \in A \times A}$ such that ${gh \in A^{-1} = A}$, i.e. involutions ${g,h}$ whose product is also an involution. But any such involutions necessarily commute, since
$\displaystyle g (gh) h = g^2 h^2 = 1 = (gh)^2 = g (hg) h.$
Thus there are at least ${|G|^2/2n^3}$ pairs ${(g,h) \in G \times G}$ of non-identity elements that commute, so by the pigeonhole principle there is a non-identity ${g \in G}$ whose centraliser ${C_G(g)}$ has order at least ${|G|/2n^3}$. This centraliser cannot be all of ${G}$ since this would make ${g}$ central which contradicts the non-abelian simple nature of ${G}$. But then the quasiregular representation of ${G}$ on ${G/C_G(g)}$ has dimension at most ${2n^3}$, contradicting the quasirandomness. $\Box$
An abstract finite-dimensional complex Lie algebra, or Lie algebra for short, is a finite-dimensional complex vector space ${{\mathfrak g}}$ together with an anti-symmetric bilinear form ${[,] = [,]_{\mathfrak g}: {\mathfrak g} \times {\mathfrak g} \rightarrow {\mathfrak g}}$ that obeys the Jacobi identity
$\displaystyle [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 \ \ \ \ \ (1)$
for all ${x,y,z \in {\mathfrak g}}$; by anti-symmetry one can also rewrite the Jacobi identity as
$\displaystyle [x,[y,z]] = [[x,y],z] + [y,[x,z]]. \ \ \ \ \ (2)$
We will usually omit the subscript from the Lie bracket ${[,]_{\mathfrak g}}$ when this will not cause ambiguity. A homomorphism ${\phi: {\mathfrak g} \rightarrow {\mathfrak h}}$ between two Lie algebras ${{\mathfrak g},{\mathfrak h}}$ is a linear map that respects the Lie bracket, thus ${\phi([x,y]_{\mathfrak g}) =[\phi(x),\phi(y)]_{\mathfrak h}}$ for all ${x,y \in {\mathfrak g}}$. As with many other classes of mathematical objects, the class of Lie algebras together with their homomorphisms then form a category. One can of course also consider Lie algebras in infinite dimension or over other fields, but we will restrict attention throughout these notes to the finite-dimensional complex case. The trivial, zero-dimensional Lie algebra is denoted ${0}$; Lie algebras of positive dimension will be called non-trivial.
Lie algebras come up in many contexts in mathematics, in particular arising as the tangent space of complex Lie groups. It is thus very profitable to think of Lie algebras as being the infinitesimal component of a Lie group, and in particular almost all of the notation and concepts that are applicable to Lie groups (e.g. nilpotence, solvability, extensions, etc.) have infinitesimal counterparts in the category of Lie algebras (often with exactly the same terminology). See this previous blog post for more discussion about the connection between Lie algebras and Lie groups (that post was focused over the reals instead of the complexes, but much of the discussion carries over to the complex case).
A particular example of a Lie algebra is the general linear Lie algebra ${{\mathfrak{gl}}(V)}$ of linear transformations ${x: V \rightarrow V}$ on a finite-dimensional complex vector space (or vector space for short) ${V}$, with the commutator Lie bracket ${[x,y] := xy-yx}$; one easily verifies that this is indeed an abstract Lie algebra. We will define a concrete Lie algebra to be a Lie algebra that is a subalgebra of ${{\mathfrak{gl}}(V)}$ for some vector space ${V}$, and similarly define a representation of a Lie algebra ${{\mathfrak g}}$ to be a homomorphism ${\rho: {\mathfrak g} \rightarrow {\mathfrak h}}$ into a concrete Lie algebra ${{\mathfrak h}}$. It is a deep theorem of Ado (discussed in this previous post) that every abstract Lie algebra is in fact isomorphic to a concrete one (or equivalently, that every abstract Lie algebra has a faithful representation), but we will not need or prove this fact here.
Even without Ado’s theorem, though, the structure of abstract Lie algebras is very well understood. As with objects in many other algebraic categories, a basic way to understand a Lie algebra ${{\mathfrak g}}$ is to factor it into two simpler algebras ${{\mathfrak h}, {\mathfrak k}}$ via a short exact sequence
$\displaystyle 0 \rightarrow {\mathfrak h} \rightarrow {\mathfrak g} \rightarrow {\mathfrak k} \rightarrow 0, \ \ \ \ \ (3)$
thus one has an injective homomorphism from ${{\mathfrak h}}$ to ${{\mathfrak g}}$ and a surjective homomorphism from ${{\mathfrak g}}$ to ${{\mathfrak k}}$ such that the image of the former homomorphism is the kernel of the latter. (To be pedantic, a short exact sequence in a general category requires these homomorphisms to be monomorphisms and epimorphisms respectively, but in the category of Lie algebras these turn out to reduce to the more familiar concepts of injectivity and surjectivity respectively.) Given such a sequence, one can (non-uniquely) identify ${{\mathfrak g}}$ with the vector space ${{\mathfrak h} \times {\mathfrak k}}$ equipped with a Lie bracket of the form
$\displaystyle [(t,x), (s,y)]_{\mathfrak g} = ([t,s]_{\mathfrak h} + A(t,y) - A(s,x) + B(x,y), [x,y]_{\mathfrak k}) \ \ \ \ \ (4)$
for some bilinear maps ${A: {\mathfrak h} \times {\mathfrak k} \rightarrow {\mathfrak h}}$ and ${B: {\mathfrak k} \times {\mathfrak k} \rightarrow {\mathfrak h}}$ that obey some Jacobi-type identities which we will not record here. Understanding exactly what maps ${A,B}$ are possible here (up to coordinate change) can be a difficult task (and is one of the key objectives of Lie algebra cohomology), but in principle at least, the problem of understanding ${{\mathfrak g}}$ can be reduced to that of understanding that of its factors ${{\mathfrak k}, {\mathfrak h}}$. To emphasise this, I will (perhaps idiosyncratically) express the existence of a short exact sequence (3) by the ATLAS-type notation
$\displaystyle {\mathfrak g} = {\mathfrak h} . {\mathfrak k} \ \ \ \ \ (5)$
although one should caution that for given ${{\mathfrak h}}$ and ${{\mathfrak k}}$, there can be multiple non-isomorphic ${{\mathfrak g}}$ that can form a short exact sequence with ${{\mathfrak h},{\mathfrak k}}$, so that ${{\mathfrak h} . {\mathfrak k}}$ is not a uniquely defined combination of ${{\mathfrak h}}$ and ${{\mathfrak k}}$; one could emphasise this by writing ${{\mathfrak h} ._{A,B} {\mathfrak k}}$ instead of ${{\mathfrak h} . {\mathfrak k}}$, though we will not do so here. We will refer to ${{\mathfrak g}}$ as an extension of ${{\mathfrak k}}$ by ${{\mathfrak h}}$, and read the notation (5) as “ ${{\mathfrak g}}$ is ${{\mathfrak h}}$-by-${{\mathfrak k}}$“; confusingly, these two notations reverse the subject and object of “by”, but unfortunately both notations are well entrenched in the literature. We caution that the operation ${.}$ is not commutative, and it is only partly associative: every Lie algebra of the form ${{\mathfrak k} . ({\mathfrak h} . {\mathfrak l})}$ is also of the form ${({\mathfrak k} . {\mathfrak h}) . {\mathfrak l}}$, but the converse is not true (see this previous blog post for some related discussion). As we are working in the infinitesimal world of Lie algebras (which have an additive group operation) rather than Lie groups (in which the group operation is usually written multiplicatively), it may help to think of ${{\mathfrak h} . {\mathfrak k}}$ as a (twisted) “sum” of ${{\mathfrak h}}$ and ${{\mathfrak k}}$ rather than a “product”; for instance, we have ${{\mathfrak g} = 0 . {\mathfrak g}}$ and ${{\mathfrak g} = {\mathfrak g} . 0}$, and also ${\dim {\mathfrak h} . {\mathfrak k} = \dim {\mathfrak h} + \dim {\mathfrak k}}$.
Special examples of extensions ${{\mathfrak h} .{\mathfrak k}}$ of ${{\mathfrak k}}$ by ${{\mathfrak h}}$ include the direct sum (or direct product) ${{\mathfrak h} \oplus {\mathfrak k}}$ (also denoted ${{\mathfrak h} \times {\mathfrak k}}$), which is given by the construction (4) with ${A}$ and ${B}$ both vanishing, and the split extension (or semidirect product) ${{\mathfrak h} : {\mathfrak k} = {\mathfrak h} :_\rho {\mathfrak k}}$ (also denoted ${{\mathfrak h} \ltimes {\mathfrak k} = {\mathfrak h} \ltimes_\rho {\mathfrak k}}$), which is given by the construction (4) with ${B}$ vanishing and the bilinear map ${A: {\mathfrak h} \times {\mathfrak k} \rightarrow {\mathfrak h}}$ taking the form
$\displaystyle A( t, x ) = \rho(x)(t)$
for some representation ${\rho: {\mathfrak k} \rightarrow \hbox{Der} {\mathfrak h}}$ of ${{\mathfrak k}}$ in the concrete Lie algebra of derivations ${\hbox{Der} {\mathfrak h} \subset {\mathfrak{gl}}({\mathfrak h})}$ of ${{\mathfrak h}}$, that is to say the algebra of linear maps ${D: {\mathfrak h} \rightarrow {\mathfrak h}}$ that obey the Leibniz rule
$\displaystyle D[s,t]_{\mathfrak h} = [Ds,t]_{\mathfrak h} + [s,Dt]_{\mathfrak h}$
for all ${s,t \in {\mathfrak h}}$. (The derivation algebra ${\hbox{Der} {\mathfrak g}}$ of a Lie algebra ${{\mathfrak g}}$ is analogous to the automorphism group ${\hbox{Aut}(G)}$ of a Lie group ${G}$, with the two concepts being intertwined by the tangent space functor ${G \mapsto {\mathfrak g}}$ from Lie groups to Lie algebras (i.e. the derivation algebra is the infinitesimal version of the automorphism group). Of course, this functor also intertwines the Lie algebra and Lie group versions of most of the other concepts discussed here, such as extensions, semidirect products, etc.)
There are two general ways to factor a Lie algebra ${{\mathfrak g}}$ as an extension ${{\mathfrak h} . {\mathfrak k}}$ of a smaller Lie algebra ${{\mathfrak k}}$ by another smaller Lie algebra ${{\mathfrak h}}$. One is to locate a Lie algebra ideal (or ideal for short) ${{\mathfrak h}}$ in ${{\mathfrak g}}$, thus ${[{\mathfrak h},{\mathfrak g}] \subset {\mathfrak h}}$, where ${[{\mathfrak h},{\mathfrak g}]}$ denotes the Lie algebra generated by ${\{ [x,y]: x \in {\mathfrak h}, y \in {\mathfrak g} \}}$, and then take ${{\mathfrak k}}$ to be the quotient space ${{\mathfrak g}/{\mathfrak h}}$ in the usual manner; one can check that ${{\mathfrak h}}$, ${{\mathfrak k}}$ are also Lie algebras and that we do indeed have a short exact sequence
$\displaystyle {\mathfrak g} = {\mathfrak h} . ({\mathfrak g}/{\mathfrak h}).$
Conversely, whenever one has a factorisation ${{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}$, one can identify ${{\mathfrak h}}$ with an ideal in ${{\mathfrak g}}$, and ${{\mathfrak k}}$ with the quotient of ${{\mathfrak g}}$ by ${{\mathfrak h}}$.
The other general way to obtain such a factorisation is is to start with a homomorphism ${\rho: {\mathfrak g} \rightarrow {\mathfrak m}}$ of ${{\mathfrak g}}$ into another Lie algebra ${{\mathfrak m}}$, take ${{\mathfrak k}}$ to be the image ${\rho({\mathfrak g})}$ of ${{\mathfrak g}}$, and ${{\mathfrak h}}$ to be the kernel ${\hbox{ker} \rho := \{ x \in {\mathfrak g}: \rho(x) = 0 \}}$. Again, it is easy to see that this does indeed create a short exact sequence:
$\displaystyle {\mathfrak g} = \hbox{ker} \rho . \rho({\mathfrak g}).$
Conversely, whenever one has a factorisation ${{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}$, one can identify ${{\mathfrak k}}$ with the image of ${{\mathfrak g}}$ under some homomorphism, and ${{\mathfrak h}}$ with the kernel of that homomorphism. Note that if a representation ${\rho: {\mathfrak g} \rightarrow {\mathfrak m}}$ is faithful (i.e. injective), then the kernel is trivial and ${{\mathfrak g}}$ is isomorphic to ${\rho({\mathfrak g})}$.
Now we consider some examples of factoring some class of Lie algebras into simpler Lie algebras. The easiest examples of Lie algebras to understand are the abelian Lie algebras ${{\mathfrak g}}$, in which the Lie bracket identically vanishes. Every one-dimensional Lie algebra is automatically abelian, and thus isomorphic to the scalar algebra ${{\bf C}}$. Conversely, by using an arbitrary linear basis of ${{\mathfrak g}}$, we see that an abelian Lie algebra is isomorphic to the direct sum of one-dimensional algebras. Thus, a Lie algebra is abelian if and only if it is isomorphic to the direct sum of finitely many copies of ${{\bf C}}$.
Now consider a Lie algebra ${{\mathfrak g}}$ that is not necessarily abelian. We then form the derived algebra ${[{\mathfrak g},{\mathfrak g}]}$; this algebra is trivial if and only if ${{\mathfrak g}}$ is abelian. It is easy to see that ${[{\mathfrak h},{\mathfrak k}]}$ is an ideal whenever ${{\mathfrak h},{\mathfrak k}}$ are ideals, so in particular the derived algebra ${[{\mathfrak g},{\mathfrak g}]}$ is an ideal and we thus have the short exact sequence
$\displaystyle {\mathfrak g} = [{\mathfrak g},{\mathfrak g}] . ({\mathfrak g}/[{\mathfrak g},{\mathfrak g}]).$
The algebra ${{\mathfrak g}/[{\mathfrak g},{\mathfrak g}]}$ is the maximal abelian quotient of ${{\mathfrak g}}$, and is known as the abelianisation of ${{\mathfrak g}}$. If it is trivial, we call the Lie algebra perfect. If instead it is non-trivial, then the derived algebra has strictly smaller dimension than ${{\mathfrak g}}$. From this, it is natural to associate two series to any Lie algebra ${{\mathfrak g}}$, the lower central series
$\displaystyle {\mathfrak g}_1 = {\mathfrak g}; {\mathfrak g}_2 := [{\mathfrak g}, {\mathfrak g}_1]; {\mathfrak g}_3 := [{\mathfrak g}, {\mathfrak g}_2]; \ldots$
and the derived series
$\displaystyle {\mathfrak g}^{(1)} := {\mathfrak g}; {\mathfrak g}^{(2)} := [{\mathfrak g}^{(1)}, {\mathfrak g}^{(1)}]; {\mathfrak g}^{(3)} := [{\mathfrak g}^{(2)}, {\mathfrak g}^{(2)}]; \ldots.$
By induction we see that these are both decreasing series of ideals of ${{\mathfrak g}}$, with the derived series being slightly smaller (${{\mathfrak g}^{(k)} \subseteq {\mathfrak g}_k}$ for all ${k}$). We say that a Lie algebra is nilpotent if its lower central series is eventually trivial, and solvable if its derived series eventually becomes trivial. Thus, abelian Lie algebras are nilpotent, and nilpotent Lie algebras are solvable, but the converses are not necessarily true. For instance, in the general linear group ${{\mathfrak{gl}}_n = {\mathfrak{gl}}({\bf C}^n)}$, which can be identified with the Lie algebra of ${n \times n}$ complex matrices, the subalgebra ${{\mathfrak n}}$ of strictly upper triangular matrices is nilpotent (but not abelian for ${n \geq 3}$), while the subalgebra ${{\mathfrak n}}$ of upper triangular matrices is solvable (but not nilpotent for ${n \geq 2}$). It is also clear that any subalgebra of a nilpotent algebra is nilpotent, and similarly for solvable or abelian algebras.
From the above discussion we see that a Lie algebra is solvable if and only if it can be represented by a tower of abelian extensions, thus
$\displaystyle {\mathfrak g} = {\mathfrak a}_1 . ({\mathfrak a}_2 . \ldots ({\mathfrak a}_{k-1} . {\mathfrak a}_k) \ldots )$
for some abelian ${{\mathfrak a}_1,\ldots,{\mathfrak a}_k}$. Similarly, a Lie algebra ${{\mathfrak g}}$ is nilpotent if it is expressible as a tower of central extensions (so that in all the extensions ${{\mathfrak h} . {\mathfrak k}}$ in the above factorisation, ${{\mathfrak h}}$ is central in ${{\mathfrak h} . {\mathfrak k}}$, where we say that ${{\mathfrak h}}$ is central in ${{\mathfrak g}}$ if ${[{\mathfrak h},{\mathfrak g}]=0}$). We also see that an extension ${{\mathfrak h} . {\mathfrak k}}$ is solvable if and only of both factors ${{\mathfrak h}, {\mathfrak k}}$ are solvable. Splitting abelian algebras into cyclic (i.e. one-dimensional) ones, we thus see that a finite-dimensional Lie algebra is solvable if and only if it is polycylic, i.e. it can be represented by a tower of cyclic extensions.
For our next fundamental example of using short exact sequences to split a general Lie algebra into simpler objects, we observe that every abstract Lie algebra ${{\mathfrak g}}$ has an adjoint representation ${\hbox{ad}: {\mathfrak g} \rightarrow \hbox{ad} {\mathfrak g} \subset {\mathfrak{gl}}({\mathfrak g})}$, where for each ${x \in {\mathfrak g}}$, ${\hbox{ad} x \in {\mathfrak{gl}}({\mathfrak g})}$ is the linear map ${(\hbox{ad} x)(y) := [x,y]}$; one easily verifies that this is indeed a representation (indeed, (2) is equivalent to the assertion that ${\hbox{ad} [x,y] = [\hbox{ad} x, \hbox{ad} y]}$ for all ${x,y \in {\mathfrak g}}$). The kernel of this representation is the center ${Z({\mathfrak g}) := \{ x \in {\mathfrak g}: [x,{\mathfrak g}] = 0\}}$, which the maximal central subalgebra of ${{\mathfrak g}}$. We thus have the short exact sequence
$\displaystyle {\mathfrak g} = Z({\mathfrak g}) . \hbox{ad} g \ \ \ \ \ (6)$
which, among other things, shows that every abstract Lie algebra is a central extension of a concrete Lie algebra (which can serve as a cheap substitute for Ado’s theorem mentioned earlier).
For our next fundamental decomposition of Lie algebras, we need some more definitions. A Lie algebra ${{\mathfrak g}}$ is simple if it is non-abelian and has no ideals other than ${0}$ and ${{\mathfrak g}}$; thus simple Lie algebras cannot be factored ${{\mathfrak g} = {\mathfrak h} . {\mathfrak k}}$ into strictly smaller algebras ${{\mathfrak h},{\mathfrak k}}$. In particular, simple Lie algebras are automatically perfect and centerless. We have the following fundamental theorem:
Theorem 1 (Equivalent definitions of semisimplicity) Let ${{\mathfrak g}}$ be a Lie algebra. Then the following are equivalent:
• (i) ${{\mathfrak g}}$ does not contain any non-trivial solvable ideal.
• (ii) ${{\mathfrak g}}$ does not contain any non-trivial abelian ideal.
• (iii) The Killing form ${K: {\mathfrak g} \times {\mathfrak g} \rightarrow {\bf C}}$, defined as the bilinear form ${K(x,y) := \hbox{tr}_{\mathfrak g}( (\hbox{ad} x) (\hbox{ad} y) )}$, is non-degenerate on ${{\mathfrak g}}$.
• (iv) ${{\mathfrak g}}$ is isomorphic to the direct sum of finitely many non-abelian simple Lie algebras.
We review the proof of this theorem later in these notes. A Lie algebra obeying any (and hence all) of the properties (i)-(iv) is known as a semisimple Lie algebra. The statement (iv) is usually taken as the definition of semisimplicity; the equivalence of (iv) and (i) is a special case of Weyl’s complete reducibility theorem (see Theorem 32), and the equivalence of (iv) and (iii) is known as the Cartan semisimplicity criterion. (The equivalence of (i) and (ii) is easy.)
If ${{\mathfrak h}}$ and ${{\mathfrak k}}$ are solvable ideals of a Lie algebra ${{\mathfrak g}}$, then it is not difficult to see that the vector sum ${{\mathfrak h}+{\mathfrak k}}$ is also a solvable ideal (because on quotienting by ${{\mathfrak h}}$ we see that the derived series of ${{\mathfrak h}+{\mathfrak k}}$ must eventually fall inside ${{\mathfrak h}}$, and thence must eventually become trivial by the solvability of ${{\mathfrak h}}$). As our Lie algebras are finite dimensional, we conclude that ${{\mathfrak g}}$ has a unique maximal solvable ideal, known as the radical ${\hbox{rad} {\mathfrak g}}$ of ${{\mathfrak g}}$. The quotient ${{\mathfrak g}/\hbox{rad} {\mathfrak g}}$ is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the Levi decomposition
$\displaystyle {\mathfrak g} = \hbox{rad} {\mathfrak g} . ({\mathfrak g} / \hbox{rad} {\mathfrak g})$
expressing an arbitrary Lie algebra as an extension of a semisimple Lie algebra ${{\mathfrak g}/\hbox{rad}{\mathfrak g}}$ by a solvable algebra ${\hbox{rad} {\mathfrak g}}$ (and it is not hard to see that this is the only possible such extension up to isomorphism). Indeed, a deep theorem of Levi allows one to upgrade this decomposition to a split extension
$\displaystyle {\mathfrak g} = \hbox{rad} {\mathfrak g} : ({\mathfrak g} / \hbox{rad} {\mathfrak g})$
although we will not need or prove this result here.
In view of the above decompositions, we see that we can factor any Lie algebra (using a suitable combination of direct sums and extensions) into a finite number of simple Lie algebras and the scalar algebra ${{\bf C}}$. In principle, this means that one can understand an arbitrary Lie algebra once one understands all the simple Lie algebras (which, being defined over ${{\bf C}}$, are somewhat confusingly referred to as simple complex Lie algebras in the literature). Amazingly, this latter class of algebras are completely classified:
Theorem 2 (Classification of simple Lie algebras) Up to isomorphism, every simple Lie algebra is of one of the following forms:
• ${A_n = \mathfrak{sl}_{n+1}}$ for some ${n \geq 1}$.
• ${B_n = \mathfrak{so}_{2n+1}}$ for some ${n \geq 2}$.
• ${C_n = \mathfrak{sp}_{2n}}$ for some ${n \geq 3}$.
• ${D_n = \mathfrak{so}_{2n}}$ for some ${n \geq 4}$.
• ${E_6, E_7}$, or ${E_8}$.
• ${F_4}$.
• ${G_2}$.
(The precise definition of the classical Lie algebras ${A_n,B_n,C_n,D_n}$ and the exceptional Lie algebras ${E_6,E_7,E_8,F_4,G_2}$ will be recalled later.)
(One can extend the families ${A_n,B_n,C_n,D_n}$ of classical Lie algebras a little bit to smaller values of ${n}$, but the resulting algebras are either isomorphic to other algebras on this list, or cease to be simple; see this previous post for further discussion.)
This classification is a basic starting point for the classification of many other related objects, including Lie algebras and Lie groups over more general fields (e.g. the reals ${{\bf R}}$), as well as finite simple groups. Being so fundamental to the subject, this classification is covered in almost every basic textbook in Lie algebras, and I myself learned it many years ago in an honours undergraduate course back in Australia. The proof is rather lengthy, though, and I have always had difficulty keeping it straight in my head. So I have decided to write some notes on the classification in this blog post, aiming to be self-contained (though moving rapidly). There is no new material in this post, though; it is all drawn from standard reference texts (I relied particularly on Fulton and Harris’s text, which I highly recommend). In fact it seems remarkably hard to deviate from the standard routes given in the literature to the classification; I would be interested in knowing about other ways to reach the classification (or substeps in that classification) that are genuinely different from the orthodox route.
The classification of finite simple groups (CFSG), first announced in 1983 but only fully completed in 2004, is one of the monumental achievements of twentieth century mathematics. Spanning hundreds of papers and tens of thousands of pages, it has been called the “enormous theorem”. A “second generation” proof of the theorem is nearly completed which is a little shorter (estimated at about five thousand pages in length), but currently there is no reasonably sized proof of the classification.
An important precursor of the CFSG is the Feit-Thompson theorem from 1962-1963, which asserts that every finite group of odd order is solvable, or equivalently that every non-abelian finite simple group has even order. This is an immediate consequence of CFSG, and conversely the Feit-Thompson theorem is an essential starting point in the proof of the classification, since it allows one to reduce matters to groups of even order for which key additional tools (such as the Brauer-Fowler theorem) become available. The original proof of the Feit-Thompson theorem is 255 pages long, which is significantly shorter than the proof of the CFSG, but still far from short. While parts of the proof of the Feit-Thompson theorem have been simplified (and it has recently been converted, after six years of effort, into an argument that has been verified by the proof assistant Coq), the available proofs of this theorem are still extremely lengthy by any reasonable standard.
However, there is a significantly simpler special case of the Feit-Thompson theorem that was established previously by Suzuki in 1957, which was influential in the proof of the more general Feit-Thompson theorem (and thus indirectly to the proof of CFSG). Define a CA-group to be a group ${G}$ with the property that the centraliser ${C_G(x) := \{ g \in G: gx=xg \}}$ of any non-identity element ${x \in G}$ is abelian; equivalently, the commuting relation ${x \sim y}$ (defined as the relation that holds when ${x}$ commutes with ${y}$, thus ${xy=yx}$) is an equivalence relation on the non-identity elements ${G \backslash \{1\}}$ of ${G}$. Trivially, every abelian group is CA. A non-abelian example of a CA-group is the ${ax+b}$ group of invertible affine transformations ${x \mapsto ax+b}$ on a field ${F}$. A little less obviously, the special linear group ${SL_2(F_q)}$ over a finite field ${F_q}$ is a CA-group when ${q}$ is a power of two. The finite simple groups of Lie type are not, in general, CA-groups, but when the rank is bounded they tend to behave as if they were “almost CA”; the centraliser of a generic element in ${SL_d(F_q)}$, for instance, when ${d}$ is bounded and ${q}$ is large), is typically a maximal torus (because most elements in ${SL_d(F_q)}$ are regular semisimple) which is certainly abelian. In view of the CFSG, we thus see that CA or nearly CA groups form an important subclass of the simple groups, and it is thus of interest to study them separately. To this end, we have
Theorem 1 (Suzuki’s theorem on CA-groups) Every finite CA-group of odd order is solvable.
Of course, this theorem is superceded by the more general Feit-Thompson theorem, but Suzuki’s proof is substantially shorter (the original proof is nine pages) and will be given in this post. (See this survey of Solomon for some discussion of the link between Suzuki’s argument and the Feit-Thompson argument.) Suzuki’s analysis can be pushed further to give an essentially complete classification of all the finite CA-groups (of either odd or even order), but we will not pursue these matters here.
Moving even further down the ladder of simple precursors of CSFG is the following theorem of Frobenius from 1901. Define a Frobenius group to be a finite group ${G}$ which has a subgroup ${H}$ (called the Frobenius complement) with the property that all the non-trivial conjugates ${gHg^{-1}}$ of ${H}$ for ${g \in G \backslash H}$, intersect ${H}$ only at the origin. For instance the ${ax+b}$ group is also a Frobenius group (take ${H}$ to be the affine transformations that fix a specified point ${x_0 \in F}$, e.g. the origin). This example suggests that there is some overlap between the notions of a Frobenius group and a CA group. Indeed, note that if ${G}$ is a CA-group and ${H}$ is a maximal abelian subgroup of ${G}$, then any conjugate ${gHg^{-1}}$ of ${H}$ that is not identical to ${H}$ will intersect ${H}$ only at the origin (because ${H}$ and each of its conjugates consist of equivalence classes under the commuting relation ${\sim}$, together with the identity). So if a maximal abelian subgroup ${H}$ of a CA-group is its own normaliser (thus ${N(H) := \{ g \in G: gH=Hg\}}$ is equal to ${H}$), then the group is a Frobenius group.
Frobenius’ theorem places an unexpectedly strong amount of structure on a Frobenius group:
Theorem 2 (Frobenius’ theorem) Let ${G}$ be a Frobenius group with Frobenius complement ${H}$. Then there exists a normal subgroup ${K}$ of ${G}$ (called the Frobenius kernel of ${G}$) such that ${G}$ is the semi-direct product ${H \ltimes K}$ of ${H}$ and ${K}$.
Roughly speaking, this theorem indicates that all Frobenius groups “behave” like the ${ax+b}$ example (which is a quintessential example of a semi-direct product).
Note that if every CA-group of odd order was either Frobenius or abelian, then Theorem 2 would imply Theorem 1 by an induction on the order of ${G}$, since any subgroup of a CA-group is clearly again a CA-group. Indeed, the proof of Suzuki’s theorem does basically proceed by this route (Suzuki’s arguments do indeed imply that CA-groups of odd order are Frobenius or abelian, although we will not quite establish that fact here).
Frobenius’ theorem can be reformulated in the following concrete combinatorial form:
Theorem 3 (Frobenius’ theorem, equivalent version) Let ${G}$ be a group of permutations acting transitively on a finite set ${X}$, with the property that any non-identity permutation in ${G}$ fixes at most one point in ${X}$. Then the set of permutations in ${G}$ that fix no points in ${X}$, together with the identity, is closed under composition.
Again, a good example to keep in mind for this theorem is when ${G}$ is the group of affine permutations on a field ${F}$ (i.e. the ${ax+b}$ group for that field), and ${X}$ is the set of points on that field. In that case, the set of permutations in ${G}$ that do not fix any points are the non-trivial translations.
To deduce Theorem 3 from Theorem 2, one applies Theorem 2 to the stabiliser of a single point in ${X}$. Conversely, to deduce Theorem 2 from Theorem 3, set ${X := G/H = \{ gH: g \in G \}}$ to be the space of left-cosets of ${H}$, with the obvious left ${G}$-action; one easily verifies that this action is faithful, transitive, and each non-identity element ${g}$ of ${G}$ fixes at most one left-coset of ${H}$ (basically because it lies in at most one conjugate of ${H}$). If we let ${K}$ be the elements of ${G}$ that do not fix any point in ${X}$, plus the identity, then by Theorem 3 ${K}$ is closed under composition; it is also clearly closed under inverse and conjugation, and is hence a normal subgroup of ${G}$. From construction ${K}$ is the identity plus the complement of all the ${|G|/|H|}$ conjugates of ${H}$, which are all disjoint except at the identity, so by counting elements we see that
$\displaystyle |K| = |G| - \frac{|G|}{|H|}(|H|-1) = |G|/|H|.$
As ${H}$ normalises ${K}$ and is disjoint from ${K}$, we thus see that ${KH = H \ltimes K}$ is all of ${G}$, giving Theorem 2.
Despite the appealingly concrete and elementary form of Theorem 3, the only known proofs of that theorem (or equivalently, Theorem 2) in its full generality proceed via the machinery of group characters (which one can think of as a version of Fourier analysis for nonabelian groups). On the other hand, once one establishes the basic theory of these characters (reviewed below the fold), the proof of Frobenius’ theorem is very short, which gives quite a striking example of the power of character theory. The proof of Suzuki’s theorem also proceeds via character theory, and is basically a more involved version of the Frobenius argument; again, no character-free proof of Suzuki’s theorem is currently known. (The proofs of Feit-Thompson and CFSG also involve characters, but those proofs also contain many other arguments of much greater complexity than the character-based portions of the proof.)
It seems to me that the above four theorems (Frobenius, Suzuki, Feit-Thompson, and CFSG) provide a ladder of sorts (with exponentially increasing complexity at each step) to the full classification, and that any new approach to the classification might first begin by revisiting the earlier theorems on this ladder and finding new proofs of these results first (in particular, if one had a “robust” proof of Suzuki’s theorem that also gave non-trivial control on “almost CA-groups” – whatever that means – then this might lead to a new route to classifying the finite simple groups of Lie type and bounded rank). But even for the simplest two results on this ladder – Frobenius and Suzuki – it seems remarkably difficult to find any proof that is not essentially the character-based proof. (Even trying to replace character theory by its close cousin, representation theory, doesn’t seem to work unless one gives in to the temptation to take traces everywhere and put the characters back in; it seems that rather than abandon characters altogether, one needs to find some sort of “robust” generalisation of existing character-based methods.) In any case, I am recording here the standard character-based proofs of the theorems of Frobenius and Suzuki below the fold. There is nothing particularly novel here, but I wanted to collect all the relevant material in one place, largely for my own benefit.
One of the basic objects of study in combinatorics are finite strings ${(a_n)_{n=0}^N}$ or infinite strings ${(a_n)_{n=0}^\infty}$ of symbols ${a_n}$ from some given alphabet ${{\mathcal A}}$, which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set ${A}$ of natural numbers can be identified with the infinite string ${(1_A(n))_{n=0}^\infty}$ of ${0}$s and ${1}$s formed by the indicator of ${A}$, e.g. the even numbers can be identified with the string ${1010101\ldots}$ from the alphabet ${\{0,1\}}$, the multiples of three can be identified with the string ${100100100\ldots}$, and so forth. One can also consider doubly infinite strings ${(a_n)_{n \in {\bf Z}}}$, which among other things can be used to describe arbitrary subsets of integers.
On the other hand, the basic object of study in dynamics (and in related fields, such as ergodic theory) is that of a dynamical system ${(X,T)}$, that is to say a space ${X}$ together with a shift map ${T: X \rightarrow X}$ (which is often assumed to be invertible, although one can certainly study non-invertible dynamical systems as well). One often adds additional structure to this dynamical system, such as topological structure (giving rise topological dynamics), measure-theoretic structure (giving rise to ergodic theory), complex structure (giving rise to complex dynamics), and so forth. A dynamical system gives rise to an action of the natural numbers ${{\bf N}}$ on the space ${X}$ by using the iterates ${T^n: X \rightarrow X}$ of ${T}$ for ${n=0,1,2,\ldots}$; if ${T}$ is invertible, we can extend this action to an action of the integers ${{\bf Z}}$ on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than ${{\bf N}}$ or ${{\bf Z}}$ (e.g. one can consider continuous dynamical systems in which the evolution group is ${{\bf R}}$), but we will restrict attention to the classical situation of ${{\bf N}}$ or ${{\bf Z}}$ actions here.
There is a fundamental correspondence principle connecting the study of strings (or subsets of natural numbers or integers) with the study of dynamical systems. In one direction, given a dynamical system ${(X,T)}$, an observable ${c: X \rightarrow {\mathcal A}}$ taking values in some alphabet ${{\mathcal A}}$, and some initial datum ${x_0 \in X}$, we can first form the forward orbit ${(T^n x_0)_{n=0}^\infty}$ of ${x_0}$, and then observe this orbit using ${c}$ to obtain an infinite string ${(c(T^n x_0))_{n=0}^\infty}$. If the shift ${T}$ in this system is invertible, one can extend this infinite string into a doubly infinite string ${(c(T^n x_0))_{n \in {\bf Z}}}$. Thus we see that every quadruplet ${(X,T,c,x_0)}$ consisting of a dynamical system ${(X,T)}$, an observable ${c}$, and an initial datum ${x_0}$ creates an infinite string.
Example 1 If ${X}$ is the three-element set ${X = {\bf Z}/3{\bf Z}}$ with the shift map ${Tx := x+1}$, ${c: {\bf Z}/3{\bf Z} \rightarrow \{0,1\}}$ is the observable that takes the value ${1}$ at the residue class ${0 \hbox{ mod } 3}$ and zero at the other two classes, and one starts with the initial datum ${x_0 = 0 \hbox{ mod } 3}$, then the observed string ${(c(T^n x_0))_{n=0}^\infty}$ becomes the indicator ${100100100\ldots}$ of the multiples of three.
In the converse direction, every infinite string ${(a_n)_{n=0}^\infty}$ in some alphabet ${{\mathcal A}}$ arises (in a decidedly non-unique fashion) from a quadruple ${(X,T,c,x_0)}$ in the above fashion. This can be easily seen by the following “universal” construction: take ${X}$ to be the set ${X:= {\mathcal A}^{\bf N}}$ of infinite strings ${(b_i)_{n=0}^\infty}$ in the alphabet ${{\mathcal A}}$, let ${T: X \rightarrow X}$ be the shift map
$\displaystyle T(b_i)_{n=0}^\infty := (b_{i+1})_{n=0}^\infty,$
let ${c: X \rightarrow {\mathcal A}}$ be the observable
$\displaystyle c((b_i)_{n=0}^\infty) := b_0,$
and let ${x_0 \in X}$ be the initial point
$\displaystyle x_0 := (a_i)_{n=0}^\infty.$
Then one easily sees that the observed string ${(c(T^n x_0))_{n=0}^\infty}$ is nothing more than the original string ${(a_n)_{n=0}^\infty}$. Note also that this construction can easily be adapted to doubly infinite strings by using ${{\mathcal A}^{\bf Z}}$ instead of ${{\mathcal A}^{\bf N}}$, at which point the shift map ${T}$ now becomes invertible. An important variant of this construction also attaches an invariant probability measure to ${X}$ that is associated to the limiting density of various sets associated to the string ${(a_i)_{n=0}^\infty}$, and leads to the Furstenberg correspondence principle, discussed for instance in these previous blog posts. Such principles allow one to rigorously pass back and forth between the combinatorics of strings and the dynamics of systems; for instance, Furstenberg famously used his correspondence principle to demonstrate the equivalence of Szemerédi’s theorem on arithmetic progressions with what is now known as the Furstenberg multiple recurrence theorem in ergodic theory.
In the case when the alphabet ${{\mathcal A}}$ is the binary alphabet ${\{0,1\}}$, and (for technical reasons related to the infamous non-injectivity ${0.999\ldots = 1.00\ldots}$ of the decimal representation system) the string ${(a_n)_{n=0}^\infty}$ does not end with an infinite string of ${1}$s, then one can reformulate the above universal construction by taking ${X}$ to be the interval ${[0,1)}$, ${T}$ to be the doubling map ${Tx := 2x \hbox{ mod } 1}$, ${c: X \rightarrow \{0,1\}}$ to be the observable that takes the value ${1}$ on ${[1/2,1)}$ and ${0}$ on ${[0,1/2)}$ (that is, ${c(x)}$ is the first binary digit of ${x}$), and ${x_0}$ is the real number ${x_0 := \sum_{n=0}^\infty a_n 2^{-n-1}}$ (that is, ${x_0 = 0.a_0a_1\ldots}$ in binary).
The above universal construction is very easy to describe, and is well suited for “generic” strings ${(a_n)_{n=0}^\infty}$ that have no further obvious structure to them, but it often leads to dynamical systems that are much larger and more complicated than is actually needed to produce the desired string ${(a_n)_{n=0}^\infty}$, and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator ${100100100\ldots}$ of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space ${X}$ and a dynamics which does not obviously reflect the key features of the sequence such as its periodicity. (Using the unit interval model, the dynamics arise from the orbit of ${2/7}$ under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)
A related aesthetic objection to the universal construction is that of the four components ${X,T,c,x_0}$ of the quadruplet ${(X,T,c,x_0)}$ used to generate the sequence ${(a_n)_{n=0}^\infty}$, three of the components ${X,T,c}$ are completely universal (in that they do not depend at all on the sequence ${(a_n)_{n=0}^\infty}$), leaving only the initial datum ${x_0}$ to carry all the distinctive features of the original sequence. While there is nothing wrong with this mathematically, from a conceptual point of view it would make sense to make all four components of the quadruplet to be adapted to the sequence, in order to take advantage of the accumulated intuition about various special dynamical systems (and special observables), not just special initial data.
One step in this direction can be made by restricting ${X}$ to the orbit ${\{ T^n x_0: n \in {\bf N} \}}$ of the initial datum ${x_0}$ (actually for technical reasons it is better to restrict to the topological closure ${\overline{\{ T^n x_0: n \in {\bf N} \}}}$ of this orbit, in order to keep ${X}$ compact). For instance, starting with the sequence ${100100100\ldots}$, the orbit now consists of just three points ${100100100\ldots}$, ${010010010\ldots}$, ${001001001\ldots}$, bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet ${(X,T,c,x_0)}$, because any other such representation ${(X',T',c',x'_0)}$ is a factor of this representation (in the sense that there is a unique map ${\pi: X \rightarrow X'}$ with ${T' \circ \pi = \pi \circ T}$, ${c' \circ \pi = c}$, and ${x'_0 = \pi(x_0)}$). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system ${X}$ are interpreted as infinite strings rather than elements of a more geometrically or algebraically rich object (e.g. points in a circle, torus, or other homogeneous space).
For general sequences ${(a_n)_{n=0}^\infty}$, locating relevant geometric or algebraic structure in a dynamical system generating that sequence is an important but very difficult task (see e.g. this paper of Host and Kra, which is more or less devoted to precisely this task in the context of working out what component of a dynamical system controls the multiple recurrence behaviour of that system). However, for specific examples of sequences ${(a_n)_{n=0}^\infty}$, one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple ${(X,T,c,x_0)}$ that generates that sequence. This is not a particularly difficult or deep operation, but I found it very helpful in internalising the intuition behind the correspondence principle. Being non-rigorous, this procedure does not seem to be emphasised in most presentations of the correspondence principle, so I thought I would describe it here.
The rectification principle in arithmetic combinatorics asserts, roughly speaking, that very small subsets (or, alternatively, small structured subsets) of an additive group or a field of large characteristic can be modeled (for the purposes of arithmetic combinatorics) by subsets of a group or field of zero characteristic, such as the integers ${{\bf Z}}$ or the complex numbers ${{\bf C}}$. The additive form of this principle is known as the Freiman rectification principle; it has several formulations, going back of course to the original work of Freiman. Here is one formulation as given by Bilu, Lev, and Ruzsa:
Proposition 1 (Additive rectification) Let ${A}$ be a subset of the additive group ${{\bf Z}/p{\bf Z}}$ for some prime ${p}$, and let ${s \geq 1}$ be an integer. Suppose that ${|A| \leq \log_{2s} p}$. Then there exists a map ${\phi: A \rightarrow A'}$ into a subset ${A'}$ of the integers which is a Freiman isomorphism of order ${s}$ in the sense that for any ${x_1,\ldots,x_s,y_1,\ldots,y_s \in A}$, one has
$\displaystyle x_1+\ldots+x_s = y_1+\ldots+y_s$
if and only if
$\displaystyle \phi(x_1)+\ldots+\phi(x_s) = \phi(y_1)+\ldots+\phi(y_s).$
Furthermore ${\phi}$ is a right-inverse of the obvious projection homomorphism from ${{\bf Z}}$ to ${{\bf Z}/p{\bf Z}}$.
The original version of the rectification principle allowed the sets involved to be substantially larger in size (cardinality up to a small constant multiple of ${p}$), but with the additional hypothesis of bounded doubling involved; see the above-mentioned papers, as well as this later paper of Green and Ruzsa, for further discussion.
The proof of Proposition 1 is quite short (see Theorem 3.1 of Bilu-Lev-Ruzsa); the main idea is to use Minkowski’s theorem to find a non-trivial dilate ${aA}$ of ${A}$ that is contained in a small neighbourhood of the origin in ${{\bf Z}/p{\bf Z}}$, at which point the rectification map ${\phi}$ can be constructed by hand.
Very recently, Codrut Grosu obtained an arithmetic analogue of the above theorem, in which the rectification map ${\phi}$ preserves both additive and multiplicative structure:
Theorem 2 (Arithmetic rectification) Let ${A}$ be a subset of the finite field ${{\bf F}_p}$ for some prime ${p \geq 3}$, and let ${s \geq 1}$ be an integer. Suppose that ${|A| < \log_2 \log_{2s} \log_{2s^2} p - 1}$. Then there exists a map ${\phi: A \rightarrow A'}$ into a subset ${A'}$ of the complex numbers which is a Freiman field isomorphism of order ${s}$ in the sense that for any ${x_1,\ldots,x_n \in A}$ and any polynomial ${P(x_1,\ldots,x_n)}$ of degree at most ${s}$ and integer coefficients of magnitude summing to at most ${s}$, one has
$\displaystyle P(x_1,\ldots,x_n)=0$
if and only if
$\displaystyle P(\phi(x_1),\ldots,\phi(x_n))=0.$
Note that it is necessary to use an algebraically closed field such as ${{\bf C}}$ for this theorem, in contrast to the integers used in Proposition 1, as ${{\bf F}_p}$ can contain objects such as square roots of ${-1}$ which can only map to ${\pm i}$ in the complex numbers (once ${s}$ is at least ${2}$).
Using Theorem 2, one can transfer results in arithmetic combinatorics (e.g. sum-product or Szemerédi-Trotter type theorems) regarding finite subsets of ${{\bf C}}$ to analogous results regarding sufficiently small subsets of ${{\bf F}_p}$; see the paper of Grosu for several examples of this. This should be compared with the paper of Vu, Wood, and Wood, which introduces a converse principle that embeds finite subsets of ${{\bf C}}$ (or more generally, a characteristic zero integral domain) in a Freiman field-isomorphic fashion into finite subsets of ${{\bf F}_p}$ for arbitrarily large primes ${p}$, allowing one to transfer arithmetic combinatorical facts from the latter setting to the former.
Grosu’s argument uses some quantitative elimination theory, and in particular a quantitative variant of a lemma of Chang that was discussed previously on this blog. In that previous blog post, it was observed that (an ineffective version of) Chang’s theorem could be obtained using only qualitative algebraic geometry (as opposed to quantitative algebraic geometry tools such as elimination theory results with explicit bounds) by means of nonstandard analysis (or, in what amounts to essentially the same thing in this context, the use of ultraproducts). One can then ask whether one can similarly establish an ineffective version of Grosu’s result by nonstandard means. The purpose of this post is to record that this can indeed be done without much difficulty, though the result obtained, being ineffective, is somewhat weaker than that in Theorem 2. More precisely, we obtain
Theorem 3 (Ineffective arithmetic rectification) Let ${s, n \geq 1}$. Then if ${{\bf F}}$ is a field of characteristic at least ${C_{s,n}}$ for some ${C_{s,n}}$ depending on ${s,n}$, and ${A}$ is a subset of ${{\bf F}}$ of cardinality ${n}$, then there exists a map ${\phi: A \rightarrow A'}$ into a subset ${A'}$ of the complex numbers which is a Freiman field isomorphism of order ${s}$.
Our arguments will not provide any effective bound on the quantity ${C_{s,n}}$ (though one could in principle eventually extract such a bound by deconstructing the proof of Proposition 4 below), making this result weaker than Theorem 2 (save for the minor generalisation that it can handle fields of prime power order as well as fields of prime order as long as the characteristic remains large).
Following the principle that ultraproducts can be used as a bridge to connect quantitative and qualitative results (as discussed in these previous blog posts), we will deduce Theorem 3 from the following (well-known) qualitative version:
Proposition 4 (Baby Lefschetz principle) Let ${k}$ be a field of characteristic zero that is finitely generated over the rationals. Then there is an isomorphism ${\phi: k \rightarrow \phi(k)}$ from ${k}$ to a subfield ${\phi(k)}$ of ${{\bf C}}$.
This principle (first laid out in an appendix of Lefschetz’s book), among other things, often allows one to use the methods of complex analysis (e.g. Riemann surface theory) to study many other fields of characteristic zero. There are many variants and extensions of this principle; see for instance this MathOverflow post for some discussion of these. I used this baby version of the Lefschetz principle recently in a paper on expanding polynomial maps.
Proof: We give two proofs of this fact, one using transcendence bases and the other using Hilbert’s nullstellensatz.
We begin with the former proof. As ${k}$ is finitely generated over ${{\bf Q}}$, it has finite transcendence degree, thus one can find algebraically independent elements ${x_1,\ldots,x_m}$ of ${k}$ over ${{\bf Q}}$ such that ${k}$ is a finite extension of ${{\bf Q}(x_1,\ldots,x_m)}$, and in particular by the primitive element theorem ${k}$ is generated by ${{\bf Q}(x_1,\ldots,x_m)}$ and an element ${\alpha}$ which is algebraic over ${{\bf Q}(x_1,\ldots,x_m)}$. (Here we use the fact that characteristic zero fields are separable.) If we then define ${\phi}$ by first mapping ${x_1,\ldots,x_m}$ to generic (and thus algebraically independent) complex numbers ${z_1,\ldots,z_m}$, and then setting ${\phi(\alpha)}$ to be a complex root of of the minimal polynomial for ${\alpha}$ over ${{\bf Q}(x_1,\ldots,x_m)}$ after replacing each ${x_i}$ with the complex number ${z_i}$, we obtain a field isomorphism ${\phi: k \rightarrow \phi(k)}$ with the required properties.
Now we give the latter proof. Let ${x_1,\ldots,x_m}$ be elements of ${k}$ that generate that field over ${{\bf Q}}$, but which are not necessarily algebraically independent. Our task is then equivalent to that of finding complex numbers ${z_1,\ldots,z_m}$ with the property that, for any polynomial ${P(x_1,\ldots,x_m)}$ with rational coefficients, one has
$\displaystyle P(x_1,\ldots,x_m) = 0$
if and only if
$\displaystyle P(z_1,\ldots,z_m) = 0.$
Let ${{\mathcal P}}$ be the collection of all polynomials ${P}$ with rational coefficients with ${P(x_1,\ldots,x_m)=0}$, and ${{\mathcal Q}}$ be the collection of all polynomials ${P}$ with rational coefficients with ${P(x_1,\ldots,x_m) \neq 0}$. The set
$\displaystyle S := \{ (z_1,\ldots,z_m) \in {\bf C}^m: P(z_1,\ldots,z_m)=0 \hbox{ for all } P \in {\mathcal P} \}$
is the intersection of countably many algebraic sets and is thus also an algebraic set (by the Hilbert basis theorem or the Noetherian property of algebraic sets). If the desired claim failed, then ${S}$ could be covered by the algebraic sets ${\{ (z_1,\ldots,z_m) \in {\bf C}^m: Q(z_1,\ldots,z_m) = 0 \}}$ for ${Q \in {\mathcal Q}}$. By decomposing into irreducible varieties and observing (e.g. from the Baire category theorem) that a variety of a given dimension over ${{\bf C}}$ cannot be covered by countably many varieties of smaller dimension, we conclude that ${S}$ must in fact be covered by a finite number of such sets, thus
$\displaystyle S \subset \bigcup_{i=1}^n \{ (z_1,\ldots,z_m) \in {\bf C}^m: Q_i(z_1,\ldots,z_m) = 0 \}$
for some ${Q_1,\ldots,Q_n \in {\bf C}^m}$. By the nullstellensatz, we thus have an identity of the form
$\displaystyle (Q_1 \ldots Q_n)^l = P_1 R_1 + \ldots + P_r R_r$
for some natural numbers ${l,r \geq 1}$, polynomials ${P_1,\ldots,P_r \in {\mathcal P}}$, and polynomials ${R_1,\ldots,R_r}$ with coefficients in ${\overline{{\bf Q}}}$. In particular, this identity also holds in the algebraic closure ${\overline{k}}$ of ${k}$. Evaluating this identity at ${(x_1,\ldots,x_m)}$ we see that the right-hand side is zero but the left-hand side is non-zero, a contradiction, and the claim follows. $\Box$
From Proposition 4 one can now deduce Theorem 3 by a routine ultraproduct argument (the same one used in these previous blog posts). Suppose for contradiction that Theorem 3 fails. Then there exists natural numbers ${s,n \geq 1}$, a sequence of finite fields ${{\bf F}_i}$ of characteristic at least ${i}$, and subsets ${A_i=\{a_{i,1},\ldots,a_{i,n}\}}$ of ${{\bf F}_i}$ of cardinality ${n}$ such that for each ${i}$, there does not exist a Freiman field isomorphism of order ${s}$ from ${A_i}$ to the complex numbers. Now we select a non-principal ultrafilter ${\alpha \in \beta {\bf N} \backslash {\bf N}}$, and construct the ultraproduct ${{\bf F} := \prod_{i \rightarrow \alpha} {\bf F}_i}$ of the finite fields ${{\bf F}_i}$. This is again a field (and is a basic example of what is known as a pseudo-finite field); because the characteristic of ${{\bf F}_i}$ goes to infinity as ${i \rightarrow \infty}$, it is easy to see (using Los’s theorem) that ${{\bf F}}$ has characteristic zero and can thus be viewed as an extension of the rationals ${{\bf Q}}$.
Now let ${a_j := \lim_{i \rightarrow \alpha} a_{i,j}}$ be the ultralimit of the ${a_{i,j}}$, so that ${A := \{a_1,\ldots,a_n\}}$ is the ultraproduct of the ${A_i}$, then ${A}$ is a subset of ${{\bf F}}$ of cardinality ${n}$. In particular, if ${k}$ is the field generated by ${{\bf Q}}$ and ${A}$, then ${k}$ is a finitely generated extension of the rationals and thus, by Proposition 4 there is an isomorphism ${\phi: k \rightarrow \phi(k)}$ from ${k}$ to a subfield ${\phi(k)}$ of the complex numbers. In particular, ${\phi(a_1),\ldots,\phi(a_n)}$ are complex numbers, and for any polynomial ${P(x_1,\ldots,x_n)}$ with integer coefficients, one has
$\displaystyle P(a_1,\ldots,a_n) = 0$
if and only if
$\displaystyle P(\phi(a_1),\ldots,\phi(a_n)) = 0.$
By Los’s theorem, we then conclude that for all ${i}$ sufficiently close to ${\alpha}$, one has for all polynomials ${P(x_1,\ldots,x_n)}$ of degree at most ${s}$ and whose coefficients are integers whose magnitude sums up to ${s}$, one has
$\displaystyle P(a_{i,1},\ldots,a_{i,n}) = 0$
if and only if
$\displaystyle P(\phi(a_1),\ldots,\phi(a_n)) = 0.$
But this gives a Freiman field isomorphism of order ${s}$ between ${A_i}$ and ${\phi(A)}$, contradicting the construction of ${A_i}$, and Theorem 3 follows.
The following result is due independently to Furstenberg and to Sarkozy:
Theorem 1 (Furstenberg-Sarkozy theorem) Let ${\delta > 0}$, and suppose that ${N}$ is sufficiently large depending on ${\delta}$. Then every subset ${A}$ of ${[N] := \{1,\ldots,N\}}$ of density ${|A|/N}$ at least ${\delta}$ contains a pair ${n, n+r^2}$ for some natural numbers ${n, r}$ with ${r \neq 0}$.
This theorem is of course similar in spirit to results such as Roth’s theorem or Szemerédi’s theorem, in which the pattern ${n,n+r^2}$ is replaced by ${n,n+r,n+2r}$ or ${n,n+r,\ldots,n+(k-1)r}$ for some fixed ${k}$ respectively. There are by now many proofs of this theorem (see this recent paper of Lyall for a survey), but most proofs involve some form of Fourier analysis (or spectral theory). This may be compared with the standard proof of Roth’s theorem, which combines some Fourier analysis with what is now known as the density increment argument.
A few years ago, Ben Green, Tamar Ziegler, and myself observed that it is possible to prove the Furstenberg-Sarkozy theorem by just using the Cauchy-Schwarz inequality (or van der Corput lemma) and the density increment argument, removing all invocations of Fourier analysis, and instead relying on Cauchy-Schwarz to linearise the quadratic shift ${r^2}$. As such, this theorem can be considered as even more elementary than Roth’s theorem (and its proof can be viewed as a toy model for the proof of Roth’s theorem). We ended up not doing too much with this observation, so decided to share it here.
The first step is to use the density increment argument that goes back to Roth. For any ${\delta > 0}$, let ${P(\delta)}$ denote the assertion that for ${N}$ sufficiently large, all sets ${A \subset [N]}$ of density at least ${\delta}$ contain a pair ${n,n+r^2}$ with ${r}$ non-zero. Note that ${P(\delta)}$ is vacuously true for ${\delta > 1}$. We will show that for any ${0 < \delta_0 \leq 1}$, one has the implication
$\displaystyle P(\delta_0 + c \delta_0^3) \implies P(\delta_0) \ \ \ \ \ (1)$
for some absolute constant ${c>0}$. This implies that ${P(\delta)}$ is true for any ${\delta>0}$ (as can be seen by considering the infimum of all ${\delta>0}$ for which ${P(\delta)}$ holds), which gives Theorem 1.
It remains to establish the implication (1). Suppose for sake of contradiction that we can find ${0 < \delta_0 \leq 1}$ for which ${P(\delta_0+c\delta^3_0)}$ holds (for some sufficiently small absolute constant ${c>0}$), but ${P(\delta_0)}$ fails. Thus, we can find arbitrarily large ${N}$, and subsets ${A}$ of ${[N]}$ of density at least ${\delta_0}$, such that ${A}$ contains no patterns of the form ${n,n+r^2}$ with ${r}$ non-zero. In particular, we have
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} 1_A(n) 1_A(n+(r+h)^2) = 0.$
(The exact ranges of ${r}$ and ${h}$ are not too important here, and could be replaced by various other small powers of ${N}$ if desired.)
Let ${\delta := |A|/N}$ be the density of ${A}$, so that ${\delta_0 \leq \delta \leq 1}$. Observe that
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} 1_A(n) \delta 1_{[N]}(n+(r+h)^2) = \delta^2 + O(N^{-1/3})$
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} \delta 1_{[N]}(n) \delta 1_{[N]}(n+(r+h)^2) = \delta^2 + O(N^{-1/3})$
and
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} \delta 1_{[N]}(n) 1_A(n+(r+h)^2) = \delta^2 + O( N^{-1/3} ).$
If we thus set ${f := 1_A - \delta 1_{[N]}}$, then
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h \in [N^{1/100}]} f(n) f(n+(r+h)^2) = -\delta^2 + O( N^{-1/3} ).$
In particular, for ${N}$ large enough,
$\displaystyle \mathop{\bf E}_{n \in [N]} |f(n)| \mathop{\bf E}_{r \in [N^{1/3}]} |\mathop{\bf E}_{h \in [N^{1/100}]} f(n+(r+h)^2)| \gg \delta^2.$
On the other hand, one easily sees that
$\displaystyle \mathop{\bf E}_{n \in [N]} |f(n)|^2 = O(\delta)$
and hence by the Cauchy-Schwarz inequality
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} |\mathop{\bf E}_{h \in [N^{1/100}]} f(n+(r+h)^2)|^2 \gg \delta^3$
which we can rearrange as
$\displaystyle |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h,h' \in [N^{1/100}]} \mathop{\bf E}_{n \in [N]} f(n+(r+h)^2) f(n+(r+h')^2)| \gg \delta^3.$
Shifting ${n}$ by ${(r+h)^2}$ we obtain (again for ${N}$ large enough)
$\displaystyle |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{h,h' \in [N^{1/100}]} \mathop{\bf E}_{n \in [N]} f(n) f(n+(h'-h)(2r+h'+h))| \gg \delta^3.$
In particular, by the pigeonhole principle (and deleting the diagonal case ${h=h'}$, which we can do for ${N}$ large enough) we can find distinct ${h,h' \in [N^{1/100}]}$ such that
$\displaystyle |\mathop{\bf E}_{r \in [N^{1/3}]} \mathop{\bf E}_{n \in [N]} f(n) f(n+(h'-h)(2r+h'+h))| \gg \delta^3,$
so in particular
$\displaystyle \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+(h'-h)(2r+h'+h))| \gg \delta^3.$
If we set ${d := 2(h'-h)}$ and shift ${n}$ by ${(h'-h) (h'+h)}$, we can simplify this (again for ${N}$ large enough) as
$\displaystyle \mathop{\bf E}_{n \in [N]} |\mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr)| \gg \delta^3. \ \ \ \ \ (2)$
On the other hand, since
$\displaystyle \mathop{\bf E}_{n \in [N]} f(n) = 0$
we have
$\displaystyle \mathop{\bf E}_{n \in [N]} f(n+dr) = O( N^{-2/3+1/100})$
for any ${r \in [N^{1/3}]}$, and thus
$\displaystyle \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) = O( N^{-2/3+1/100}).$
Averaging this with (2) we conclude that
$\displaystyle \mathop{\bf E}_{n \in [N]} \max( \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr), 0 ) \gg \delta^3.$
In particular, by the pigeonhole principle we can find ${n \in [N]}$ such that
$\displaystyle \mathop{\bf E}_{r \in [N^{1/3}]} f(n+dr) \gg \delta^3,$
or equivalently ${A}$ has density at least ${\delta+c'\delta^3}$ on the arithmetic progression ${\{ n+dr: r \in [N^{1/3}]\}}$, which has length ${\lfloor N^{1/3}\rfloor }$ and spacing ${d}$, for some absolute constant ${c'>0}$. By partitioning this progression into subprogressions of spacing ${d^2}$ and length ${\lfloor N^{1/4}\rfloor}$ (plus an error set of size ${O(N^{1/4})}$, we see from the pigeonhole principle that we can find a progression ${\{ n' + d^2 r': r' \in [N^{1/4}]\}}$ of length ${\lfloor N^{1/4}\rfloor}$ and spacing ${d^2}$ on which ${A}$ has density at least ${\delta + c\delta^3}$ (and hence at least ${\delta_0+c\delta_0^3}$) for some absolute constant ${c>0}$. If we then apply the induction hypothesis to the set
$\displaystyle A' := \{ r' \in [N^{1/4}]: n' + d^2 r' \in A \}$
we conclude (for ${N}$ large enough) that ${A'}$ contains a pair ${m, m+s^2}$ for some natural numbers ${m,s}$ with ${s}$ non-zero. This implies that ${(n'+d^2 m), (n'+d^2 m) + (|d|s)^2}$ lie in ${A}$, a contradiction, establishing the implication (1).
A more careful analysis of the above argument reveals a more quantitative version of Theorem 1: for ${N \geq 100}$ (say), any subset of ${[N]}$ of density at least ${C/(\log\log N)^{1/2}}$ for some sufficiently large absolute constant ${C}$ contains a pair ${n,n+r^2}$ with ${r}$ non-zero. This is not the best bound known; a (difficult) result of Pintz, Steiger, and Szemeredi allows the density to be as low as ${C / (\log N)^{\frac{1}{4} \log\log\log\log N}}$. On the other hand, this already improves on the (simpler) Fourier-analytic argument of Green that works for densities at least ${C/(\log\log N)^{1/11}}$ (although the original argument of Sarkozy, which is a little more intricate, works up to ${C (\log\log N)^{2/3}/(\log N)^{1/3}}$). In the other direction, a construction of Rusza gives a set of density ${\frac{1}{65} N^{-0.267}}$ without any pairs ${n,n+r^2}$.
Remark 1 A similar argument also applies with ${n,n+r^2}$ replaced by ${n,n+r^k}$ for fixed ${k}$, because this sort of pattern is preserved by affine dilations ${r' \mapsto n'+d^k r'}$ into arithmetic progressions whose spacing ${d^k}$ is a ${k^{th}}$ power. By re-introducing Fourier analysis, one can also perform an argument of this type for ${n,n+d,n+2d}$ where ${d}$ is the sum of two squares; see the above-mentioned paper of Green for details. However there seems to be some technical difficulty in extending it to patterns of the form ${n,n+P(r)}$ for polynomials ${P}$ that consist of more than a single monomial (and with the normalisation ${P(0)=0}$, to avoid local obstructions), because one no longer has this preservation property.
The fundamental notions of calculus, namely differentiation and integration, are often viewed as being the quintessential concepts in mathematical analysis, as their standard definitions involve the concept of a limit. However, it is possible to capture most of the essence of these notions by purely algebraic means (almost completely avoiding the use of limits, Riemann sums, and similar devices), which turns out to be useful when trying to generalise these concepts to more abstract situations in which it becomes convenient to permit the underlying number systems involved to be something other than the real or complex numbers, even if this makes many standard analysis constructions unavailable. For instance, the algebraic notion of a derivation often serves as a substitute for the analytic notion of a derivative in such cases, by abstracting out the key algebraic properties of differentiation, namely linearity and the Leibniz rule (also known as the product rule).
Abstract algebraic analogues of integration are less well known, but can still be developed. To motivate such an abstraction, consider the integration functional ${I: {\mathcal S}({\bf R} \rightarrow {\bf C}) \rightarrow {\bf C}}$ from the space ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ of complex-valued Schwarz functions ${f: {\bf R} \rightarrow {\bf C}}$ to the complex numbers, defined by
$\displaystyle I(f) := \int_{\bf R} f(x)\ dx$
where the integration on the right is the usual Lebesgue integral (or improper Riemann integral) from analysis. This functional obeys two obvious algebraic properties. Firstly, it is linear over ${{\bf C}}$, thus
$\displaystyle I(cf) = c I(f) \ \ \ \ \ (1)$
and
$\displaystyle I(f+g) = I(f) + I(g) \ \ \ \ \ (2)$
for all ${f,g \in {\mathcal S}({\bf R} \rightarrow {\bf C})}$ and ${c \in {\bf C}}$. Secondly, it is translation invariant, thus
$\displaystyle I(\tau_h f) = I(f) \ \ \ \ \ (3)$
for all ${h \in {\bf C}}$, where ${\tau_h f(x) := f(x-h)}$ is the translation of ${f}$ by ${h}$. Motivated by the uniqueness theory of Haar measure, one might expect that these two axioms already uniquely determine ${I}$ after one sets a normalisation, for instance by requiring that
$\displaystyle I( x \mapsto e^{-\pi x^2} ) = 1. \ \ \ \ \ (4)$
This is not quite true as stated (one can modify the proof of the Hahn-Banach theorem, after first applying a Fourier transform, to create pathological translation-invariant linear functionals on ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ that are not multiples of the standard Fourier transform), but if one adds a mild analytical axiom, such as continuity of ${I}$ (using the usual Schwartz topology on ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$), then the above axioms are enough to uniquely pin down the notion of integration. Indeed, if ${I: {\mathcal S}({\bf R} \rightarrow {\bf C}) \rightarrow {\bf C}}$ is a continuous linear functional that is translation invariant, then from the linearity and translation invariance axioms one has
$\displaystyle I( \frac{\tau_h f - f}{h} ) = 0$
for all ${f \in {\mathcal S}({\bf R} \rightarrow {\bf C})}$ and non-zero reals ${h}$. If ${f}$ is Schwartz, then as ${h \rightarrow 0}$, one can verify that the Newton quotients ${\frac{\tau_h f - f}{h}}$ converge in the Schwartz topology to the derivative ${f'}$ of ${f}$, so by the continuity axiom one has
$\displaystyle I(f') = 0.$
Next, note that any Schwartz function of integral zero has an antiderivative which is also Schwartz, and so ${I}$ annihilates all zero-integral Schwartz functions, and thus must be a scalar multiple of the usual integration functional. Using the normalisation (4), we see that ${I}$ must therefore be the usual integration functional, giving the claimed uniqueness.
Motivated by the above discussion, we can define the notion of an abstract integration functional ${I: X \rightarrow R}$ taking values in some vector space ${R}$, and applied to inputs ${f}$ in some other vector space ${X}$ that enjoys a linear action ${h \mapsto \tau_h}$ (the “translation action”) of some group ${V}$, as being a functional which is both linear and translation invariant, thus one has the axioms (1), (2), (3) for all ${f,g \in X}$, scalars ${c}$, and ${h \in V}$. The previous discussion then considered the special case when ${R = {\bf C}}$, ${X = {\mathcal S}({\bf R} \rightarrow {\bf C})}$, ${V = {\bf R}}$, and ${\tau}$ was the usual translation action.
Once we have performed this abstraction, we can now present analogues of classical integration which bear very little analytic resemblance to the classical concept, but which still have much of the algebraic structure of integration. Consider for instance the situation in which we keep the complex range ${R = {\bf C}}$, the translation group ${V = {\bf R}}$, and the usual translation action ${h \mapsto \tau_h}$, but we replace the space ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ of Schwartz functions by the space ${Poly_{\leq d}({\bf R} \rightarrow {\bf C})}$ of polynomials ${x \mapsto a_0 + a_1 x + \ldots + a_d x^d}$ of degree at most ${d}$ with complex coefficients, where ${d}$ is a fixed natural number; note that this space is translation invariant, so it makes sense to talk about an abstract integration functional ${I: Poly_{\leq d}({\bf R} \rightarrow {\bf C}) \rightarrow {\bf C}}$. Of course, one cannot apply traditional integration concepts to non-zero polynomials, as they are not absolutely integrable. But one can repeat the previous arguments to show that any abstract integration functional must annihilate derivatives of polynomials of degree at most ${d}$:
$\displaystyle I(f') = 0 \hbox{ for all } f \in Poly_{\leq d}({\bf R} \rightarrow {\bf C}). \ \ \ \ \ (5)$
Clearly, every polynomial of degree at most ${d-1}$ is thus annihilated by ${I}$, which makes ${I}$ a scalar multiple of the functional that extracts the top coefficient ${a_d}$ of a polynomial, thus if one sets a normalisation
$\displaystyle I( x \mapsto x^d ) = c$
for some constant ${c}$, then one has
$\displaystyle I( x \mapsto a_0 + a_1 x + \ldots + a_d x^d ) = c a_d \ \ \ \ \ (6)$
for any polynomial ${x \mapsto a_0 + a_1 x + \ldots + a_d x^d}$. So we see that up to a normalising constant, the operation of extracting the top order coefficient of a polynomial of fixed degree serves as the analogue of integration. In particular, despite the fact that integration is supposed to be the “opposite” of differentiation (as indicated for instance by (5)), we see in this case that integration is basically (${d}$-fold) differentiation; indeed, compare (6) with the identity
$\displaystyle (\frac{d}{dx})^d ( a_0 + a_1 x + \ldots + a_d x^d ) = d! a_d.$
In particular, we see, in contrast to the usual Lebesgue integral, the integration functional (6) can be localised to an arbitrary location: one only needs to know the germ of the polynomial ${x \mapsto a_0 + a_1 x + \ldots + a_d x^d}$ at a single point ${x_0}$ in order to determine the value of the functional (6). This localisation property may initially seem at odds with the translation invariance, but the two can be reconciled thanks to the extremely rigid nature of the class ${Poly_{\leq d}({\bf R} \rightarrow {\bf C})}$, in contrast to the Schwartz class ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ which admits bump functions and so can generate local phenomena that can only be detected in small regions of the underlying spatial domain, and which therefore forces any translation-invariant integration functional on such function classes to measure the function at every single point in space.
The reversal of the relationship between integration and differentiation is also reflected in the fact that the abstract integration operation on polynomials interacts with the scaling operation ${\delta_\lambda f(x) := f(x/\lambda)}$ in essentially the opposite way from the classical integration operation. Indeed, for classical integration on ${{\bf R}^d}$, one has
$\displaystyle \int_{{\bf R}^d} f(x/\lambda)\ dx = \lambda^d \int f(x)\ dx$
for Schwartz functions ${f \in {\mathcal S}({\bf R}^d \rightarrow {\bf C})}$, and so in this case the integration functional ${I(f) := \int_{{\bf R}^d} f(x)\ dx}$ obeys the scaling law
$\displaystyle I( \delta_\lambda f ) = \lambda^d I(f).$
In contrast, the abstract integration operation defined in (6) obeys the opposite scaling law
$\displaystyle I( \delta_\lambda f ) = \lambda^{-d} I(f). \ \ \ \ \ (7)$
Remark 1 One way to interpret what is going on is to view the integration operation (6) as a renormalised version of integration. A polynomial ${x \mapsto a_0 + a_1 + \ldots + a_d x^d}$ is, in general, not absolutely integrable, and the partial integrals
$\displaystyle \int_0^N a_0 + a_1 + \ldots + a_d x^d\ dx$
diverge as ${N \rightarrow \infty}$. But if one renormalises these integrals by the factor ${\frac{1}{N^{d+1}}}$, then one recovers convergence,
$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{d+1}} \int_0^N a_0 + a_1 + \ldots + a_d x^d\ dx = \frac{1}{d+1} a_d$
thus giving an interpretation of (6) as a renormalised classical integral, with the renormalisation being responsible for the unusual scaling relationship in (7). However, this interpretation is a little artificial, and it seems that it is best to view functionals such as (6) from an abstract algebraic perspective, rather than to try to force an analytic interpretation on them.
$\displaystyle I(f) := \int_{\bf R} f(x)\ dx. \ \ \ \ \ (8)$
As noted earlier, this integration functional has a translation invariance associated to translations along the real line ${{\bf R}}$, as well as a dilation invariance by real dilation parameters ${\lambda>0}$. However, if we refine the class ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ of functions somewhat, we can obtain a stronger family of invariances, in which we allow complex translations and dilations. More precisely, let ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$ denote the space of all functions ${f: {\bf C} \rightarrow {\bf C}}$ which are entire (or equivalently, are given by a Taylor series with an infinite radius of convergence around the origin) and also admit rapid decay in a sectorial neighbourhood of the real line, or more precisely there exists an ${\epsilon>0}$ such that for every ${A > 0}$ there exists ${C_A > 0}$ such that one has the bound
$\displaystyle |f(z)| \leq C_A (1+|z|)^{-A}$
whenever ${|\hbox{Im}(z)| \leq A + \epsilon |\hbox{Re}(z)|}$. For want of a better name, we shall call elements of this space Schwartz entire functions. This is clearly a complex vector space. A typical example of a Schwartz entire function are the complex gaussians
$\displaystyle f(z) := e^{-\pi (az^2 + 2bz + c)}$
where ${a,b,c}$ are complex numbers with ${\hbox{Re}(a) > 0}$. From the Cauchy integral formula (and its derivatives) we see that if ${f}$ lies in ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$, then the restriction of ${f}$ to the real line lies in ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$; conversely, from analytic continuation we see that every function in ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$ has at most one extension in ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$. Thus one can identify ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$ with a subspace of ${{\mathcal S}({\bf R} \rightarrow {\bf C})}$, and in particular the integration functional (8) is inherited by ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$, and by abuse of notation we denote the resulting functional ${I: \mathcal{SE}({\bf C} \rightarrow {\bf C}) \rightarrow {\bf C}}$ as ${I}$ also. Note, in analogy with the situation with polynomials, that this abstract integration functional is somewhat localised; one only needs to evaluate the function ${f}$ on the real line, rather than the entire complex plane, in order to compute ${I(f)}$. This is consistent with the rigid nature of Schwartz entire functions, as one can uniquely recover the entire function from its values on the real line by analytic continuation.
Of course, the functional ${I: \mathcal{SE}({\bf C} \rightarrow {\bf C}) \rightarrow {\bf C}}$ remains translation invariant with respect to real translation:
$\displaystyle I(\tau_h f) = I(f) \hbox{ for all } h \in {\bf R}.$
However, thanks to contour shifting, we now also have translation invariance with respect to complex translation:
$\displaystyle I(\tau_h f) = I(f) \hbox{ for all } h \in {\bf C},$
where of course we continue to define the translation operator ${\tau_h}$ for complex ${h}$ by the usual formula ${\tau_h f(x) := f(x-h)}$. In a similar vein, we also have the scaling law
$\displaystyle I(\delta_\lambda f) = \lambda I(f)$
for any ${f \in \mathcal{SE}({\bf C} \rightarrow {\bf C})}$, if ${\lambda}$ is a complex number sufficiently close to ${1}$ (where “sufficiently close” depends on ${f}$, and more precisely depends on the sectoral aperture parameter ${\epsilon}$ associated to ${f}$); again, one can verify that ${\delta_\lambda f}$ lies in ${\mathcal{SE}({\bf C} \rightarrow {\bf C})}$ for ${\lambda}$ sufficiently close to ${1}$. These invariances (which relocalise the integration functional ${I}$ onto other contours than the real line ${{\bf R}}$) are very useful for computing integrals, and in particular for computing gaussian integrals. For instance, the complex translation invariance tells us (after shifting by ${b/a}$) that
$\displaystyle I( z \mapsto e^{-\pi (az^2 + 2bz + c) } ) = e^{-\pi (c-b^2/a)} I( z \mapsto e^{-\pi a z^2} )$
when ${a,b,c \in {\bf C}}$ with ${\hbox{Re}(a) > 0}$, and then an application of the complex scaling law (and a continuity argument, observing that there is a compact path connecting ${a}$ to ${1}$ in the right half plane) gives
$\displaystyle I( z \mapsto e^{-\pi (az^2 + 2bz + c) } ) = a^{-1/2} e^{-\pi (c-b^2/a)} I( z \mapsto e^{-\pi z^2} )$
using the branch of ${a^{-1/2}}$ on the right half-plane for which ${1^{-1/2} = 1}$. Using the normalisation (4) we thus have
$\displaystyle I( z \mapsto e^{-\pi (az^2 + 2bz + c) } ) = a^{-1/2} e^{-\pi (c-b^2/a)}$
giving the usual gaussian integral formula
$\displaystyle \int_{\bf R} e^{-\pi (ax^2 + 2bx + c)}\ dx = a^{-1/2} e^{-\pi (c-b^2/a)}. \ \ \ \ \ (9)$
This is a basic illustration of the power that a large symmetry group (in this case, the complex homothety group) can bring to bear on the task of computing integrals.
One can extend this sort of analysis to higher dimensions. For any natural number ${n \geq 1}$, let ${\mathcal{SE}({\bf C}^n \rightarrow {\bf C})}$ denote the space of all functions ${f: {\bf C}^n \rightarrow {\bf C}}$ which is jointly entire in the sense that ${f(z_1,\ldots,z_n)}$ can be expressed as a Taylor series in ${z_1,\ldots,z_n}$ which is absolutely convergent for all choices of ${z_1,\ldots,z_n}$, and such that there exists an ${\epsilon > 0}$ such that for any ${A>0}$ there is ${C_A>0}$ for which one has the bound
$\displaystyle |f(z)| \leq C_A (1+|z|)^{-A}$
whenever ${|\hbox{Im}(z_j)| \leq A + \epsilon |\hbox{Re}(z_j)|}$ for all ${1 \leq j \leq n}$, where ${z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}}$ and ${|z| := (|z_1|^2+\ldots+|z_n|^2)^{1/2}}$. Again, we call such functions Schwartz entire functions; a typical example is the function
$\displaystyle f(z) := e^{-\pi (z^T A z + 2b^T z + c)}$
where ${A}$ is an ${n \times n}$ complex symmetric matrix with positive definite real part, ${b}$ is a vector in ${{\bf C}^n}$, and ${c}$ is a complex number. We can then define an abstract integration functional ${I: \mathcal{SE}({\bf C}^n \rightarrow {\bf C}) \rightarrow {\bf C}}$ by integration on the real slice ${{\bf R}^n}$:
$\displaystyle I(f) := \int_{{\bf R}^n} f(x)\ dx$
where ${dx}$ is the usual Lebesgue measure on ${{\bf R}^n}$. By contour shifting in each of the ${n}$ variables ${z_1,\ldots,z_n}$ separately, we see that ${I}$ is invariant with respect to complex translations of each of the ${z_j}$ variables, and is thus invariant under translating the joint variable ${z}$ by ${{\bf C}^n}$. One can also verify the scaling law
$\displaystyle I(\delta_A f) = \hbox{det}(A) I(f)$
for ${n \times n}$ complex matrices ${A}$ sufficiently close to the origin, where ${\delta_A f(z) := f(A^{-1} z)}$. This can be seen for shear transformations ${A}$ by Fubini’s theorem and the aforementioned translation invariance, while for diagonal transformations near the origin this can be seen from ${n}$ applications of one-dimensional scaling law, and the general case then follows by composition. Among other things, these laws then easily lead to the higher-dimensional generalisation
$\displaystyle \int_{{\bf R}^n} e^{-\pi (x^T A x + 2 b^T x + c)}\ dx = \hbox{det}(A)^{-1/2} e^{-\pi (c-b^T A^{-1} b)} \ \ \ \ \ (10)$
whenever ${A}$ is a complex symmetric matrix with positive definite real part, ${b}$ is a vector in ${{\bf C}^n}$, and ${c}$ is a complex number, basically by repeating the one-dimensional argument sketched earlier. Here, we choose the branch of ${\hbox{det}(A)^{-1/2}}$ for all matrices ${A}$ in the indicated class for which ${\hbox{det}(1)^{-1/2} = 1}$.
Now we turn to an integration functional suitable for computing complex gaussian integrals such as
$\displaystyle \int_{{\bf C}^n} e^{-2\pi (z^\dagger A z + b^\dagger z + z^\dagger \tilde b + c)}\ dz d\overline{z}, \ \ \ \ \ (11)$
where ${z}$ is now a complex variable
$\displaystyle z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix},$
${z^\dagger}$ is the adjoint
$\displaystyle z^\dagger := (\overline{z_1},\ldots, \overline{z_n}),$
${A}$ is a complex ${n \times n}$ matrix with positive definite Hermitian part, ${b, \tilde b}$ are column vectors in ${{\bf C}^n}$, ${c}$ is a complex number, and ${dz d\overline{z} = \prod_{j=1}^n 2 d\hbox{Re}(z_j) d\hbox{Im}(z_j)}$ is ${2^n}$ times Lebesgue measure on ${{\bf C}^n}$. (The factors of two here turn out to be a natural normalisation, but they can be ignored on a first reading.) As we shall see later, such integrals are relevant when performing computations on the Gaussian Unitary Ensemble (GUE) in random matrix theory. Note that the integrand here is not complex analytic due to the presence of the complex conjugates. However, this can be dealt with by the trick of replacing the complex conjugate ${\overline{z}}$ by a variable ${z^*}$ which is formally conjugate to ${z}$, but which is allowed to vary independently of ${z}$. More precisely, let ${\mathcal{SA}({\bf C}^n \times {\bf C}^n \rightarrow {\bf C})}$ be the space of all functions ${f: (z,z^*) \mapsto f(z,z^*)}$ of two independent ${n}$-tuples
$\displaystyle z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}, z^* = \begin{pmatrix} z_1^* \\ \vdots \\ z_n^* \end{pmatrix}$
of complex variables, which is jointly entire in all ${2n}$ variables (in the sense defined previously, i.e. there is a joint Taylor series that is absolutely convergent for all independent choices of ${z, z^* \in {\bf C}^n}$), and such that there is an ${\epsilon>0}$ such that for every ${A>0}$ there is ${C_A>0}$ such that one has the bound
$\displaystyle |f(z,z^*)| \leq C_A (1 + |z|)^{-A}$
whenever ${|z^* - \overline{z}| \leq A + \epsilon |z|}$. We will call such functions Schwartz analytic. Note that the integrand in (11) is Schwartz analytic when ${A}$ has positive definite Hermitian part, if we reinterpret ${z^\dagger}$ as the transpose of ${z^*}$ rather than as the adjoint of ${z}$ in order to make the integrand entire in ${z}$ and ${z^*}$. We can then define an abstract integration functional ${I: \mathcal{SA}({\bf C}^n \times {\bf C}^n \rightarrow {\bf C}) \rightarrow {\bf C}}$ by the formula
$\displaystyle I(f) := \int_{{\bf C}^n} f(z,\overline{z})\ dz d\overline{z}, \ \ \ \ \ (12)$
thus ${I}$ can be localised to the slice ${\{ (z,\overline{z}): z \in {\bf C}^n\}}$ of ${{\bf C}^n \times {\bf C}^n}$ (though, as with previous functionals, one can use contour shifting to relocalise ${I}$ to other slices also.) One can also write this integral as
$\displaystyle I(f) = 2^n \int_{{\bf R}^n \times {\bf R}^n} f(x+iy, x-iy)\ dx dy$
and note that the integrand here is a Schwartz entire function on ${{\bf C}^n \times {\bf C}^n}$, thus linking the Schwartz analytic integral with the Schwartz entire integral. Using this connection, one can verify that this functional ${I}$ is invariant with respect to translating ${z}$ and ${z^*}$ by independent shifts in ${{\bf C}^n}$ (thus giving a ${{\bf C}^n \times {\bf C}^n}$ translation symmetry), and one also has the independent dilation symmetry
$\displaystyle I(\delta_{A,B} f) = \hbox{det}(A) \hbox{det}(B) I(f)$
for ${n \times n}$ complex matrices ${A,B}$ that are sufficiently close to the identity, where ${\delta_{A,B} f(z,z^*) := f(A^{-1} z, B^{-1} z^*)}$. Arguing as before, we can then compute (11) as
$\displaystyle \int_{{\bf C}^n} e^{-2\pi (z^\dagger A z + b^\dagger z + z^\dagger \tilde b + c)}\ dz d\overline{z} = \hbox{det}(A)^{-1} e^{-2\pi (c - b^\dagger A^{-1} \tilde b)}. \ \ \ \ \ (13)$
In particular, this gives an integral representation for the determinant-reciprocal ${\hbox{det}(A)^{-1}}$ of a complex ${n \times n}$ matrix with positive definite Hermitian part, in terms of gaussian expressions in which ${A}$ only appears linearly in the exponential:
$\displaystyle \hbox{det}(A)^{-1} = \int_{{\bf C}^n} e^{-2\pi z^\dagger A z}\ dz d\overline{z}.$
This formula is then convenient for computing statistics such as
$\displaystyle \mathop{\bf E} \hbox{det}(W_n-E-i\eta)^{-1}$
for random matrices ${W_n}$ drawn from the Gaussian Unitary Ensemble (GUE), and some choice of spectral parameter ${E+i\eta}$ with ${\eta>0}$; we review this computation later in this post. By the trick of matrix differentiation of the determinant (as reviewed in this recent blog post), one can also use this method to compute matrix-valued statistics such as
$\displaystyle \mathop{\bf E} \hbox{det}(W_n-E-i\eta)^{-1} (W_n-E-i\eta)^{-1}.$
However, if one restricts attention to classical integrals over real or complex (and in particular, commuting or bosonic) variables, it does not seem possible to easily eradicate the negative determinant factors in such calculations, which is unfortunate because many statistics of interest in random matrix theory, such as the expected Stieltjes transform
$\displaystyle \mathop{\bf E} \frac{1}{n} \hbox{tr} (W_n-E-i\eta)^{-1},$
which is the Stieltjes transform of the density of states. However, it turns out (as I learned recently from Peter Sarnak and Tom Spencer) that it is possible to cancel out these negative determinant factors by balancing the bosonic gaussian integrals with an equal number of fermionic gaussian integrals, in which one integrates over a family of anticommuting variables. These fermionic integrals are closer in spirit to the polynomial integral (6) than to Lebesgue type integrals, and in particular obey a scaling law which is inverse to the Lebesgue scaling (in particular, a linear change of fermionic variables ${\zeta \mapsto A \zeta}$ ends up transforming a fermionic integral by ${\hbox{det}(A)}$ rather than ${\hbox{det}(A)^{-1}}$), which conveniently cancels out the reciprocal determinants in the previous calculations. Furthermore, one can combine the bosonic and fermionic integrals into a unified integration concept, known as the Berezin integral (or Grassmann integral), in which one integrates functions of supervectors (vectors with both bosonic and fermionic components), and is of particular importance in the theory of supersymmetry in physics. (The prefix “super” in physics means, roughly speaking, that the object or concept that the prefix is attached to contains both bosonic and fermionic aspects.) When one applies this unified integration concept to gaussians, this can lead to quite compact and efficient calculations (provided that one is willing to work with “super”-analogues of various concepts in classical linear algebra, such as the supertrace or superdeterminant).
Abstract integrals of the flavour of (6) arose in quantum field theory, when physicists sought to formally compute integrals of the form
$\displaystyle \int F( x_1, \ldots, x_n, \xi_1, \ldots, \xi_m )\ dx_1 \ldots dx_n d\xi_1 \ldots d\xi_m \ \ \ \ \ (14)$
where ${x_1,\ldots,x_n}$ are familiar commuting (or bosonic) variables (which, in particular, can often be localised to be scalar variables taking values in ${{\bf R}}$ or ${{\bf C}}$), while ${\xi_1,\ldots,\xi_m}$ were more exotic anticommuting (or fermionic) variables, taking values in some vector space of fermions. (As we shall see shortly, one can formalise these concepts by working in a supercommutative algebra.) The integrand ${F(x_1,\ldots,x_n,\xi_1,\ldots,\xi_m)}$ was a formally analytic function of ${x_1,\ldots,x_n,\xi_1,\ldots,\xi_m}$, in that it could be expanded as a (formal, noncommutative) power series in the variables ${x_1,\ldots,x_n,\xi_1,\ldots,\xi_m}$. For functions ${F(x_1,\ldots,x_n)}$ that depend only on bosonic variables, it is certainly possible for such analytic functions to be in the Schwartz class and thus fall under the scope of the classical integral, as discussed previously. However, functions ${F(\xi_1,\ldots,\xi_m)}$ that depend on fermionic variables ${\xi_1,\ldots,\xi_m}$ behave rather differently. Indeed, a fermonic variable ${\xi}$ must anticommute with itself, so that ${\xi^2 = 0}$. In particular, any power series in ${\xi}$ terminates after the linear term in ${\xi}$, so that a function ${F(\xi)}$ can only be analytic in ${\xi}$ if it is a polynomial of degree at most ${1}$ in ${\xi}$; more generally, an analytic function ${F(\xi_1,\ldots,\xi_m)}$ of ${m}$ fermionic variables ${\xi_1,\ldots,\xi_m}$ must be a polynomial of degree at most ${m}$, and an analytic function ${F(x_1,\ldots,x_n,\xi_1,\ldots,\xi_m)}$ of ${n}$ bosonic and ${m}$ fermionic variables can be Schwartz in the bosonic variables but will be polynomial in the fermonic variables. As such, to interpret the integral (14), one can use classical (Lebesgue) integration (or the variants discussed above for integrating Schwartz entire or Schwartz analytic functions) for the bosonic variables, but must use abstract integrals such as (6) for the fermonic variables, leading to the concept of Berezin integration mentioned earlier.
In this post I would like to set out some of the basic algebraic formalism of Berezin integration, particularly with regards to integration of gaussian-type expressions, and then show how this formalism can be used to perform computations involving GUE (for instance, one can compute the density of states of GUE by this machinery without recourse to the theory of orthogonal polynomials). The use of supersymmetric gaussian integrals to analyse ensembles such as GUE appears in the work of Efetov (and was also proposed in the slightly earlier works of Parisi-Sourlas and McKane, with a related approach also appearing in the work of Wegner); the material here is adapted from this survey of Mirlin, as well as the later papers of Disertori-Pinson-Spencer and of Disertori.
Consider the free Schrödinger equation in ${d}$ spatial dimensions, which I will normalise as
$\displaystyle i u_t + \frac{1}{2} \Delta_{{\bf R}^d} u = 0 \ \ \ \ \ (1)$
where ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ is the unknown field and ${\Delta_{{\bf R}^{d+1}} = \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}}$ is the spatial Laplacian. To avoid irrelevant technical issues I will restrict attention to smooth (classical) solutions to this equation, and will work locally in spacetime avoiding issues of decay at infinity (or at other singularities); I will also avoid issues involving branch cuts of functions such as ${t^{d/2}}$ (if one wishes, one can restrict ${d}$ to be even in order to safely ignore all branch cut issues). The space of solutions to (1) enjoys a number of symmetries. A particularly non-obvious symmetry is the pseudoconformal symmetry: if ${u}$ solves (1), then the pseudoconformal solution ${pc(u): {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ defined by
$\displaystyle pc(u)(t,x) := \frac{1}{(it)^{d/2}} \overline{u(\frac{1}{t}, \frac{x}{t})} e^{i|x|^2/2t} \ \ \ \ \ (2)$
for ${t \neq 0}$ can be seen after some computation to also solve (1). (If ${u}$ has suitable decay at spatial infinity and one chooses a suitable branch cut for ${(it)^{d/2}}$, one can extend ${pc(u)}$ continuously to the ${t=0}$ spatial slice, whereupon it becomes essentially the spatial Fourier transform of ${u(0,\cdot)}$, but we will not need this fact for the current discussion.)
An analogous symmetry exists for the free wave equation in ${d+1}$ spatial dimensions, which I will write as
$\displaystyle u_{tt} - \Delta_{{\bf R}^{d+1}} u = 0 \ \ \ \ \ (3)$
where ${u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ is the unknown field. In analogy to pseudoconformal symmetry, we have conformal symmetry: if ${u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ solves (3), then the function ${conf(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$, defined in the interior ${\{ (t,x): |x| < |t| \}}$ of the light cone by the formula
$\displaystyle conf(u)(t,x) := (t^2-|x|^2)^{-d/2} u( \frac{t}{t^2-|x|^2}, \frac{x}{t^2-|x|^2} ), \ \ \ \ \ (4)$
also solves (3).
There are also some direct links between the Schrödinger equation in ${d}$ dimensions and the wave equation in ${d+1}$ dimensions. This can be easily seen on the spacetime Fourier side: solutions to (1) have spacetime Fourier transform (formally) supported on a ${d}$-dimensional hyperboloid, while solutions to (3) have spacetime Fourier transform formally supported on a ${d+1}$-dimensional cone. To link the two, one then observes that the ${d}$-dimensional hyperboloid can be viewed as a conic section (i.e. hyperplane slice) of the ${d+1}$-dimensional cone. In physical space, this link is manifested as follows: if ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ solves (1), then the function ${\iota_{1}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ defined by
$\displaystyle \iota_{1}(u)(t,x_1,\ldots,x_{d+1}) := e^{-i(t+x_{d+1})} u( \frac{t-x_{d+1}}{2}, x_1,\ldots,x_d)$
solves (3). More generally, for any non-zero scaling parameter ${\lambda}$, the function ${\iota_{\lambda}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ defined by
$\displaystyle \iota_{\lambda}(u)(t,x_1,\ldots,x_{d+1}) :=$
$\displaystyle \lambda^{d/2} e^{-i\lambda(t+x_{d+1})} u( \lambda \frac{t-x_{d+1}}{2}, \lambda x_1,\ldots,\lambda x_d) \ \ \ \ \ (5)$
solves (3).
As an “extra challenge” posed in an exercise in one of my books (Exercise 2.28, to be precise), I asked the reader to use the embeddings ${\iota_1}$ (or more generally ${\iota_\lambda}$) to explicitly connect together the pseudoconformal transformation ${pc}$ and the conformal transformation ${conf}$. It turns out that this connection is a little bit unusual, with the “obvious” guess (namely, that the embeddings ${\iota_\lambda}$ intertwine ${pc}$ and ${conf}$) being incorrect, and as such this particular task was perhaps too difficult even for a challenge question. I’ve been asked a couple times to provide the connection more explicitly, so I will do so below the fold.
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2013-12-05 05:10:45
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http://www.xaprb.com/blog/2006/06/02/an-alternative-to-canonical-uris/
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# An alternative to canonical URIs
There’s been much discussion about canonicalizing URIs for search engines and other purposes. Here’s a possibility: instead of specifying the exact canonical URI for a page, just indicate which parts of it are important.
The issue usually comes up with regards to product pages in e-commerce, for example. Many e-commerce sites use query strings to specify the product to display. This is not a problem, because search engines like Google usually pay attention to several query string parameters. The problem comes when the e-commerce site starts adding other query string parameters onto the URI for session identifiers, tracking codes, navigational aids, and so forth. These are all important for site functionality, but seldom have much to do with the real content of the page. Search engine spiders don’t like lots and lots of parameters, so they may only pay attention to the first few, causing problems for the e-commerce site.
That’s where URI canonicalization strategies come into play. The usual strategy is to detect when a spider is requesting the page, and redirect it to the canonical version of the page, with only the relevant parameters in exactly the right order. This can require a lot of programming.
<meta name="significant-query-params" content="itemno categoryno" />
Of course, search engines would have to agree upon this as a standard, but it seems reasonable to me. The search spider would then compare any query parameters to the significant ones, and ignore the page if it’s already in the search cache.
I know it’s more complex than this. I’m sure there are lots of issues that would make it harder for search engines to implement this scheme, and I can think of some right away (repeated parameters with different values, issues with how the cache is really structured, etc). I’m just tossing it out there for someone to mull upon it, if anyone wants to.
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.
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2016-10-28 23:26:31
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http://indexswingtrader.blogspot.com/2013/08/vixenator-ii-or-how-to-turn-risk-into.html
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## replace_date('24 August 2013');
### "Vixenator II" or how to turn "risk" into "opportunity"
What happens when "risk" is turned into a mimetic poly-alloy? We get Vixenator II and "risk" turns into "opportunity"! While working on a risk ratio, a statistical approach towards VIX and SPX turned out to be rather promissing.
SPX weekly chart
Back in January a statistical framework was provided for DMI, MACD and RSI. For creating Vixenator II the same approach is used. For the sake of brevity please consult the mentioned posting for the theoretical background.
Bell curve of "normal distribution"
By design Vixenator (I) was limited to daily charts. For the sequel a new composition is used resulting in an indicator way more mouldable than its predecessor: because of its "liquid elements" Vixenator II allows for application on all timeframes.
SPX daily chart
SPX 3 minute chart
Without all the code for "look & feel" the construction of Vixenator II comes down to this:
def ratio = VIX / SPX; def sma = Average(ratio, length); def sd = StDev(ratio, length); def diff = ratio - sma; plot zscore = -1 * diff / sd;
On account of the values yielded by this formula and bearing the statistical concept of "normal distribution" in mind one is capable of a making a probabilistic statement on the direction of the indicator and thus of the instrument it is measuring (see January posting). Moreover, based on Vixenator II's historical behavior propositions for "long", "short" and "neutral" stances can be taken.:
A "long" position is proposed when zscore >= +1 (sd) or with an upward move of zscore of at least 1 sd into positive territory. "Long" is exited when the indicator starts reading negative values. A "short" position is proposed when zscore <= -1 (sd) or with a downward move of zscore of at least (-)1 sd into negative territory. "Short" is exited when the indicator starts reading positive values. When not "long" or "short" a "neutral" stance is proposed (indicated by the yellow coloring). The mentioned "1" sd value is the default trigger level. Adjusting to "0" basically renders the outcome to "stop and reverse" signals.
SPX daily chart
SPX 3 minute chart
Besides the "long", "short" and "neutral" coloring, the indicator allows for the detection of trend line breaks (see chart above) and divergence detection (not shown).
Feel free to comment or make suggestions and please do consider a donation: also $5 or$10 gifts are by all means a welcome encouragement!
Like always the thinkscript code is available for copy/paste in the comment section. Nobody wants to wait for a sequel for 7 years! Enjoy it now!
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2017-08-19 07:23:42
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http://openstudy.com/updates/55b7deb4e4b0c500a11b2b5d
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## chris215 one year ago -
$E= \frac{ 1 }{ 4\pi \epsilon } \frac{ Q }{ r ^{2} }$ right? What I thought was if we increase the no. of charges Q will increase since it is the same distance the electric field will increase. (E increases)
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2017-01-23 21:21:52
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https://www.physicsforums.com/threads/physics-practical-rebound-height-of-a-table-tennis-ball.827936/
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# Homework Help: Physics Practical: Rebound height of a table-tennis ball
1. Aug 16, 2015
### Janiceleong26
1. The problem statement, all variables and given/known data
For part aiii), the mark scheme states "Use of set squares to indicate height". I know what are set squares, but how does it aid to determine the rebound height, h? I thought set squares are to ensure two objects are perpendicular to one another? How does it help in measuring the rebound height, h?
Also, for part c), how to explain? Is it because both h and d have the same unit, thus cancel out, giving a constant?
2. Aug 16, 2015
### BvU
Think of ways to ensure that you may measure h accurately along the ruler ....
In part c, apparently the theory has some relationship like $h^2 = e\; d^2$ (or some other power of length). What could is be related to ?
3. Aug 16, 2015
### Janiceleong26
By placing the set square like this? Why not use a ruler instead?
And I thought the theory is related by h = e^2 d? I really don't know what it is related to.. I guess it's Young's modulus?
Last edited: Aug 16, 2015
4. Aug 17, 2015
### BvU
That's the idea. You worry me because you let the top of the set square align with the center of the ball, whilst the h and d in the figure are measured with respect to the underside of the ball.
Young's modulus is too complicated in this experiment.
Think about the bouncing as a non fully elastic collision
At what speed does the ball hit the floor ?
How high does the ball bounce if the speed up is a fraction of that ?
Last edited: Aug 17, 2015
5. Aug 17, 2015
### Janiceleong26
Oh whoops sorry, was not conscious about it
But why can't we use other forms of straight edges, like a ruler instead of set squares?
I guess it's newton's law of restitution then?
6. Aug 17, 2015
7. Apr 6, 2017
### syedHa
Please mention from which year and session is this question from....TIA
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2018-07-17 06:13:26
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http://www.mathnet.ru/php/archive.phtml?jrnid=sm&wshow=issue&year=1984&volume=166&volume_alt=124&issue=4&issue_alt=8&option_lang=eng
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RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB
General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS
Mat. Sb.: Year: Volume: Issue: Page: Find
On indices of exponential separationV. M. Millionshchikov 451 On commutative ring spectra of characteristic 2A. V. Pajitnov, Yu. B. Rudyak 486 On infinite $p$-groups with cyclic subgroupsG. S. Deryabina 495 Undecidable and decidable properties of constituentsV. G. Kanovei 505 On the number and cycle structure of the permutations in certain classesA. I. Pavlov 536 A theorem on height for alternative algebrasS. V. Pchelintsev 557 On a theorem of M. V. Keldysh concerning pointwise convergence of a sequence of polynomialsS. V. Kolesnikov 568 Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximationA. A. Pekarskii 571
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2020-06-03 01:02:52
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https://dsp.stackexchange.com/questions/39037/why-dft-spread-ofdm-has-lower-papr-than-ofdm
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# Why DFT-SPREAD OFDM has lower PAPR than OFDM?
I just read 4G LTE/LTE-Advanced for Mobile Broadband, and found the content as below,
If the DFT size M equals the IDFT size N, the cascaded DFT/IDFT processing would obviously completely cancel each other out. However, if M is smaller than N and the remaining inputs to the IDFT are set to zero, the output of the IDFT will be a signal with “single-carrier” properties – that is, a signal with low power variations, and with a bandwidth that depends on M.
My question is that why DFT-SPREAD OFDM has lower variations in comparison with OFDM? The picture is DFT-SPREAD OFDM modulation scheme.
Thank you.
• This scheme is more commonly called single-carrier frequency division multiple access (SC-FDMA) and there are some interesting answers here already. Basically, SC-FDMA is a way of shifting a single-carrier signal in frequency by using a pair of DFT/IDFT with different lengths. – Deve Jul 23 '17 at 9:17
## 1 Answer
It is because the time domain symbols have lower PAPR than the frequency domain symbols.
The PAPR of OFDM is due to the 2 reasons. One is due to the IDFT(IFFT) and the other is due to the symbol type. The IDFT converts the time domain symbols to frequency domain symbols that have very high PAPR. The PAPR of 16QAM is also higher than that of QPSK. However, even when these symbol types are the same, the PAPR of DFT-SPREAD OFDM is still lower than that of regular OFDM. The reason is the (N point) IDFT effect is mostly cancelled out by the (M point) DFT. If the M=N (no zero), they are completely canceled out and the symbols would be just a time domain symbols. Even though N is much larger than M, the resulting symbol has more like time domain symbols, then the PAPR is lower. LTE up-link uses this type of OFDM that is called SC-FDMA. SC means Single Carrier because the symbols look like single carrier time domain symbols even though it has an OFDM structure. They use SC-FDMA for the up-link LTE for the lower PAPR seeking the longer battery life.
• thank you for your detailed explanation and now I understand the technique. In comparison with $M = N$, it to some extent decreased the PAPR with $M < N$. Many thanks for you~ – Charles Hou Jul 23 '17 at 4:59
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2020-03-30 14:22:31
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http://031c82c.netsolhost.com/l12wdgq/dymj6aw.php?33a434=example-of-universal-set
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Directions: Read each question below. a) A = {x : x is a factor of 60} In these examples, certain conventions were used. In example 5, subsets X and Y do not overlap. If you make a mistake, rethink your answer, then choose a different button. Summary: A universal set is a set containing all elements of a problem under consideration, denoted by capital . Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets. Label the circles and write the relevant elements in each circle. Therefore, it is logical to assume that there is no relationship between these sets. The set of all elements being considered is called the universal set (U) and is represented by a rectangle. Set A is composed of some (but not all) of the numbers in the Universal Set “U”. Example 6: In a class of 10 students, some students were selected for the school band, some were selected for the school chorus, some were selected for both, and the rest were selected for neither. Well, simply put, it's a collection. b. In some examples, the sets overlapped and in some they did not. In example 1, A and B have no elements in common. Venn Diagrams. Answer: A and B have no elements in common. Solution: Related words - universal set synonyms, antonyms, hypernyms and hyponyms. Example 5: Given = {animals}, X = {dogs} and Y = {cats}, draw a Venn diagram to represent these sets. A set is a collection of things.For example, the items you wear is a set: these include hat, shirt, jacket, pants, and so on.You write sets inside curly brackets like this:{hat, shirt, jacket, pants, ...}You can also have sets of numbers: 1. A and B have no elements in common. For example, if we consider no. Copyright © 2005, 2020 - OnlineMathLearning.com. This relationship is shown in the Venn diagram below. Definition: A Universal Set is the set of all elements under consideration, denoted by capital . A universal set contains ALL the elements of a problem under consideration. A' (read as A complement) are members that are not in set A. Example 4: Given = {whole numbers}, R = {primes numbers less than 12} and S = {even primes}, draw a Venn diagram to represent these sets. For example, consider the single-digit numbers 1 through 9: If {1, 2, 3, 4, 5, 6, 7, 8, 9} is our larger set, then A and B are part of that set. Set of whole numbers: {0, 1, 2, 3, ...} 2. universal set. Lowercase letters are used to denote elements of sets. (Each set is shaded with a different color to illustrate this.) And it is not necessary that they have same elements, or they are a subset of each other. Label the circles and write the relevant elements in each circle. A â© B (read as A intersection B) are members that are common to both set A and set B. Select your answer by clicking on its button. Say if A and B are two sets, such as A = {1,2,3} and B = {1,a,b,c}, then the universal set associated with these two sets is given by U = {1,2,3,a,b,c}. More Lessons On Sets Learn about Sets on our Youtube Channel - https://you.tube/Chapter-1-Class-11-Sets Universal set is the set which contains all the elements of the other sets. In these lessons, we will learn what is a universal set and how it may be represented in a Venn Diagram. Subsets Definition of universal set in the Fine Dictionary. However, some non-standard variants of set theory include a universal set. In previous lessons, we learned that a set is a group of objects, and that Venn diagrams can be used to illustrate both set relationships and logical relationships. Universal set consists of all elements and objects of the system without any repetition of the element. Copyright 2020 Math Goodies. Draw circles within the rectangle to represent the subsets of the universe. All Rights Reserved. Equal And Equivalent Sets Examples. All other sets are subsets of the universal set. Universal set definition: the set of all objects or elements considered in a given problem | Meaning, pronunciation, translations and examples Draw a Venn diagram to represent the following sets: Draw a rectangle and label it U to represent the universal set. Universal set can be anything for a situation we are considering. . For example, in the lead-in problem above, the universal set could be either the set of all U. S. dollars or the set of the $836 Sam originally had in the checking account. Example: The set of Real Numbers is a universal set for ALL natural, whole, odd, even, rational and irrational numbers. The set of all numbers below 1000 is universal set, if we consider the sets of numbers divisibe by 2 and 3 and neither by 2 nor by 3 within 1000. Example : Univariate Data . Similar to how the word universe is used in astronomy to mean everything, the universal set contains every element. First we specify a common property among \"things\" (we define this word later) and then we gather up all the \"things\" that have this common property. This is known as a set. within the rectangle. of students in a class in a school, the universal set is the no. These sets do not overlap. In set theory, a universal set is a set which contains all objects, including itself. For the other extreme, what happens when we examine the intersection of a set with the universal set? Thus A and B are each a subset of this larger set, called the Universal Set. That is, it is a statement such as, "For all x 7; 1 x < 1 2;" or "The square of a real number is nonnegative." If the universal set contains sets A and B, then and . Try the given examples, or type in your own Feedback to your answer is provided in the RESULTS BOX. The following conventions are used with sets: Capital letters are used to denote sets. Set A ⊂Universal Set “U” Or A ⊂ U This is shown on the following Venn Diagram. The elements of sets A and B can only be selected from the given universal set U. If P = {1, 3, 9, 5, − 7} and Q = {5, − 7, 3, 1, 9,}, then P = Q. b) B = {5, 7, 11}. Universal Statements and Counterexamples A universal statement is a mathematical statement that is supposed to be true about all members of a set. As we will see later, probability is defined and calculated for sets. Example 2: Given = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 5, 6} and B = {3, 9}, draw a Venn diagram to represent these sets. problem solver below to practice various math topics. Learn what is universal set. A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. Set A is also a proper subset of U because not all elements of U are in subset A. Meaning of universal set with illustrations and photos. Below is a word problem that you may find interesting. Step 3: Write the remaining elements outside the circles but For example, the number 5 is an integer, and so it is appropriate to write $$5 \in \mathbb{Z}$$. Summary: A universal set is a set containing all elements of a problem under consideration, denoted by capital . Also included were examples in which one set was contained within the other. a. Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...} 2. In set theory as usually formulated, the conception of a universal set leads to Russell's paradox and is consequently not allowed. Start studying 10 Universal Values. Universal-set definitions The definition of a universal set is a mathematical term meaning all of the elements in a problem. For example, 3 of the objects above belong to the set of head covering or simply hats (ladies hat, baseball cap, hard hat). Think of a Universal set is the "big picture" It includes everything under consideration, or everything that is relevant to the problem you have. In example 2, and . In example 4, S is contained within R. This is due to the fact that the number 2 is the only even prime. denoted by capital U or sometimes capital E. Example: Try the free Mathway calculator and Consider the sets A = {3,6,9,12, … }, B = {5,10,15,20, … }, C = {15}, and D = {17}. An example of universal set is {2, 3, 4} of 2 + 3 + 4. Question: Help in these examples, epecially e. Let N be the universal set. Accordingly, we did not include any remaining whole numbers outside the circles and within the rectangle. Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {6, 7, 8} So, universal set will have all the elements of set A, B, C Subsets of the universal set are represented by ovals within the rectangle. In Venn diagrams, the universal set is usually represented by a rectangle and labeled U. Sets are disjoint if they do not share any elements. Then the complement of set A, A' = {i, o, u}. Probability theory uses the language of sets. Also find the definition and meaning for various math words from this math dictionary. Embedded content, if any, are copyrights of their respective owners. Step 1: Draw a rectangle and label it U to represent the a) A = {5, 6, 10, 12} What is a set? Related Pages A universal set is a set which contains all the elements contained in the sets for which it is a universal set. All the other sets are considered to be the subsets of Universal set. Thus, here we briefly review some basic concepts from set theory that are used in this book. other sets. Explains the universal set and set complements. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 5, 6}, B = {3, 9}. Example: Example: We called this set of objects a universal set or universe. Pronunciation of universal set and its etymology. Step 2: Draw circles within the rectangle to represent the universal values - peace, freedom, social progress, equal rights, human dignity – acutely needed, secretary-general says at tǕbingen university, germany A universal set is the set of all elements under consideration, Empty RelationIf Relation has no elements,it is called empty relationWe write R = ∅Universal RelationIf relation has all the elements,it is a universal relationLet us take an exampleLet A = Set of all students in a girls school.We define relation R on set A asR = {(a, b): a and b are brothers}R’ = We recommend you get used to the meaning of these operations. However, if we consider these sets as part of a larger set, then there is a relationship between them. In our example, U, made with a big rectangle, is the universal set. For a subset A of an universal set U, we have that x ∈ A, so it makes sense to use the logic operations to describe the set operations, as we did in the previous definition. In general, we can say, two sets are equivalent to each other if the number of elements in both the sets is equal. Equal Set Example. Set Fis a subsetof set Aif all elements of Fare also elements of A. About Us | Contact Us | Advertise With Us | Facebook | Recommend This Page. It is generally represented by the letter U. The symbol 2 is used to describe a relationship between an element of the universal set and a subset of the universal set, and the symbol $$\subseteq$$ is used to describe a relationship between two subsets of the universal set. problem and check your answer with the step-by-step explanations. Also included were examples in which one set was contained within the other. Of students in the school while that of the class is just a set maybe A. In this lesson, we examined several examples of universal sets with Venn diagrams. The procedure for creating a Venn diagram is as follows; Example 3: Given = {whole numbers less than 10}, P = {multiples of 3 less than 10} and Q = {even numbers less than 10}, draw a Venn diagram to represent these sets. In some examples, the sets overlapped and in some they did not. To formalize such notion, we would have to explicit that the domain of this formula itself is "limited" to the defined universal set, thus considering it equivalent to$\forall x \in U[x \in U \leftrightarrow x = x_0 \lor x=x_1\}]\$. In example 6, Band and Chorus are overlapping sets. For example, the even numbers 2, 4 and 12 all belong to the set of whole numbers. b) B = {x : x is a prime number}, Solution: Notice that B can still be a subset of A even if the circle used to represent set B was not inside the circle used to represent A. Answer, then choose a different color to illustrate this. remaining whole numbers goes on forever by... Our set is { 2, 3,... } 2 the universe. 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A complement ) are members that are not in set a subsets of the universal set is a containing... Least a hundred are copyrights of their respective owners,... } 2 share any elements and solver. The meaning of these operations complement of a, a and B are each a subset of each.! That every element of objects a universal set is a set which contains all the elements in U that relevant., S is contained within the other extreme, what happens when we examine the of! It U to represent the other below to practice various math words this! They did not all objects, including itself also find the definition and meaning for various math.. And Chorus are each a subset of this larger set, which is all the sets. Examples in which one set was contained within the other but within the rectangle to represent the of. Contains sets a and B are each a subset of the universal set consists of elements... 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example of universal set 2021
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2021-10-26 06:18:34
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http://mathematica.stackexchange.com/tags/compile/hot
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# Tag Info
56
Yes, but this only exists in version 8 and is undocumented: CompileCompilerFunctions[] // Sort giving, for reference: {Abs, AddTo, And, Append, AppendTo, Apply, ArcCos, ArcCosh, ArcCot, ArcCoth, ArcCsc, ArcCsch, ArcSec, ArcSech, ArcSin, ArcSinh, ArcTan, ArcTanh, Arg, Array, ArrayDepth, InternalBag, InternalBagPart, BitAnd, BitNot, BitOr, BitXor, ...
50
I'll just throw in a few random thoughts in no particular order, but this will be a rather high-level view on things. This is necessarily a subjective exposition, so treat it as such. Typical use cases In my opinion, Compile as an efficiency-boosting device is effective in two kinds of situations (and their mixes): The problem is solved most efficiently ...
35
Use these 3 components: compile, C, parallel computing. Also to speed up coloring instead of ArrayPlot use Graphics[Raster[Rescale[...], ColorFunction -> "TemperatureMap"]] In such cases Compile is essential. Compile to C with parallelization will speed it up even more, but you need to have a C compiler installed. Note difference for usage of C and ...
30
In addition to the answers given, you may tweak specific commands to give better performance. For example Part[] is a candidate for this. Part has to do bound checks. In time critical inner loops you can switch that off be using, CompilerGetElement[] instead. Very cautious with this one. Another thing you might want to try (never needed this myself) is ...
26
A lot depends on how you write your code in Mathematica. In my experience, the rule of thumb is that the generated code will be efficient if the code inside Compile more or less resembles the code I would write in plain C (and it is clear why). Idiomatic (high-level) Mathematica code tends to be immutable. At the same time, Compile can handle a number of ...
25
In addition to Oleks list, there is of course a way to study what happens under the hood. f = Compile[{{x, _Integer, 1}}, Accumulate[x] ]; << CompiledFunctionTools CompilePrint[f] (* 1 argument 1 Integer register 2 Tensor registers Underflow checking off Overflow checking off Integer overflow ...
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I assume you need the list of compilable functions to make sure that all of your code will be properly compiled, and it won't take any speed penalties (that why I was looking for this information before). People have shown you how to print the compiled code and check that there are no calls to MainEvaluate in it. There is an alternative and simpler way of ...
21
The code as it is now looks very much FORTRAN style, which is fine. But Mathematica offers you a wide range of ways to make your code more readable, faster and easier to spot potential bugs. So let's go through through some of the possible ways to improve your code: Variable Naming I know that in languages like C and FORTRAN it's common to give variables ...
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I don't have an answer but this is a bit hard to format in a comment. If runtime speed is your goal, I'd suggest using Compile with settings CompilationTarget->"C", CompilationOptions -> {"ExpressionOptimization" -> True, "InlineExternalDefinitions" -> True}, RuntimeOptions -> "Speed" I'm not certain about the inlining, and there may be ...
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This might be an excellent candidate for ParallelTable; MakeFractal[f_, nx_, ny_, {cx_, cy_}, {rx_, ry_}] := Module[{pts}, DistributeDefinitions[nx, ny, cx, cy, rx, ry, f]; pts = ParallelTable[f[x + I y], {x, cx - rx, cx + rx, (2 rx)/nx}, {y, cy - ry, cy + ry, (2 ry)/ny}]; ArrayPlot[Reverse@pts, ColorFunction -> "TemperatureMap"] ] ...
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I am somewhat reluctant to offer this as an answer since it is inherently difficult to comprehensively address questions on undocumented functionality. Nonetheless, the following observations do constitute partial answers to points raised in the question and are likely to be of value to anyone trying to write practical compiled code using Bags. However, ...
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This is not an answer to your question, but it does address some of the issues with your code. In particular, it is just plain unreadable, and unreadable code cannot be maintained in any meaningful manner. If you came back to this even after a week of not using it, you would not understand how it works. Towards that end, I've simplified it quite a bit, just ...
17
Not a proper answer, but I just want to comment that the procedure carried out by @rcollyer can be automated to a large extent. Here is a code for a simplistic common subexpression eliminator: ClearAll[csub]; csub[expr_Hold, rules_List, limitCount_] := With[{newrule = Replace[ If[# =!= {} && #[[-1, -1]] > 1, #[[-1, 1]], {}] &@ ...
16
This is a tricky case indeed, because what you basically ask for is compile-time evaluation (macro-style). Generally, the answer is to use meta-programming, to assemble the compiled expression at run-time. The reason your attempt did not work is that the expression you want to evaluate is too deep for Evaluate to be effective. Solution using in-place ...
15
This is awful. It is one very typical example of "how to use Mathematica the wrong way*. OK, enough complaining. Let me give you one hint. Lets say you have a 500x500 and a 1000x1000 matrix and you want to copy the smaller one in the upper left corner of the larger one. We do this step 100 times. In your style this would go like m1 = RandomReal[{0, 1}, ...
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Here is another compiled implementation: hammingDistanceCompiled = Compile[{{nums, _Integer, 1}}, Block[{x = BitXor[nums[[1]], nums[[2]]], n = 0}, While[x > 0, x = BitAnd[x, x - 1]; n++]; n ], RuntimeAttributes -> Listable, Parallelization -> True, CompilationTarget -> "C", RuntimeOptions -> "Speed" ]; This appears to ...
14
I believe there is such a list available but I can't remember the command off-hand. In the meantime, you can always load CompiledFunctionTools via. <<CompiledFunctionTools And then use CompilePrint on a compiled function to see if MainEvaluate is present in the pseudocode. MainEvaluate tells us that something is going through the evaluator and ...
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acl already posted the crucial information needed to solve this conundrum (i.e., the definition of InternalCompileValues[LinearSolve]), but wishes to delete his post since he had not interpreted it to give the complete answer. Therefore I re-post the following observation along with a summary of what it means. The input, InternalCompileValues[]; ...
12
Setting SetSystemOptions[ "CompileOptions" -> "CompileReportExternal"->True] will emit a message when parts of your function do not get compiled. After compilation, Needs["CompiledFunctionTools"] followed by CompilePrint[cF] (with cF the function you have compiled will display some bytecode; looking for CopyTensor or MainEvaluate in that helps locate ...
12
If you use the setting CompilationTarget -> "C" (documentation: CompilationTarget) you get a function that is literally converted to C code and compiled: f = Compile[{{x, _Real}}, Sin[x] + x^2 - 1/(1 + x), CompilationTarget -> "C"]; Then you can actually export the C code and look at, or use ExportString to print it directly in Mathematica: ...
12
Note: instead of picking random element I just pick the first it runs into, random version at the end getCell = Compile[{{sp, _Integer, 2}, {i, _Integer}, {j, _Integer}, {x, _Integer}}, Block[{ n, m, k2, l2, cell}, {n, m} = Dimensions[sp]; cell = {i, j}; Do[(*This is the neighborhood *) k2 = Mod[i + k, n, 1]; l2 = Mod[j + l, m, 1]; ...
12
This seems to give a rather decent performance (final version with improvements by jVincent): Clear[getSubset]; getSubset[input_List,sub_List]:= Module[{inSubQ,sowMatches}, Scan[(inSubQ[#] := True)&,sub]; sowMatches[x_/;inSubQ@First@x] := Sow[x,First@x]; Apply[Sequence, Last@Reap[Scan[sowMatches, input], sub], {2}] ]; Benchmarks: n = ...
12
If you look at the generated code (CompilePrint, for example), the procedure is as follows: All the program's constants are placed into separate registers (regardless of their location in the program, they can be in the r.h.s.of variable initialization in scoping constructs, or they can be statements in their bodies. Actually, same constants found in ...
11
About your question regarding the definition of the type of local variables in Compile, Compile has an optional third argument that allows you do this in the same manner you specify arguments. It helps the compiler solve some type ambiguity issues sometimes as by default a local variable is considered a Real number. This can be the case if a local variable ...
11
Assuming you know the dimensions of the pieces that you want to come out you can always add a second argument to InternalStuffBag that indicates the rank of the elements going in. The result is still flat so you have to partition after the fact. cf = Compile[{}, Module[{bag = InternalBag[]}, Do[InternalStuffBag[bag, {i, i, i}, 1], {i, {0, 1, 2, 3}}]; ...
11
Try this: G = 4.49*^3; M = 1.; S = 1.; \[Epsilon] = 2.; With[ {G = G, M = M, S = S, \[Epsilon] = \[Epsilon]}, SAcceleration = Compile[{{SPosition, _Real, 1}}, (-G (M + S))/(SPosition.SPosition + \[Epsilon]*\[Epsilon])^(3/ 2) SPosition]; SAcceleration2 = Compile[{{SPosition, _Real, 1}}, Module[{GG = G, MM = M, SS = ...
11
The reason for this message is that the compiled function is called with the symbolic argument SR[n] in the definition of the recurrence relation: SAcceleration[SR[n]] CompiledFunction::cfta: "Argument SR[n] at position 1 should be a rank 1 tensor of machine-size real numbers." -((8980. SR[n])/(4. + SR[n].SR[n])^(3/2)) The recurrence is then ...
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It's not you, it's Mathematica. You are not expected to know this, but basically, in compiled code, ReplacePart merely acts as syntactic sugar for setting a part, i.e.: l = Range[3]; ReplacePart[l, 2 -> 0] (* -> {1, 0, 3} *) would be compiled (but see below) into exactly the same bytecode as l = Range[3]; Block[{l = l}, l[[2]] = 0; l] (* -> {1, ...
10
As Leonid already commented, your code should not be especially slow. However one reason why your code may be slow is that you use Part to extract values, do some calculations and then insert the result. I would try using a wholesale approach and calculate all nodes at the same time by doing, e.g. (may need tweaking since I don't know the structure of your ...
10
I know you explicitly asked for an answer using Compile but as stated in my comment, I'm not sure this is required. Additionally, I don't think it is possible as you expect it. Your list is a ragged array, which means it is a non-rectangular tensor. To my knowledge it is not possible to use it with Compile. Even the simplest example fails, which does nothing ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2013-05-26 05:54:27
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https://blog.myrank.co.in/partial-fractions/
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# Partial Fractions
## Partial Fractions
If f(x) and g(x) are two polynomials, then $$\frac{f\left( x \right)}{g\left( x \right)}$$ defines a rational algebraic function or a rational function of x.
If degree of f(x) < degree of g(x), then $$\frac{f\left( x \right)}{g\left( x \right)}$$ is called an proper rational function.
If degree of f(x) ≥ degree of g(x) then $$\frac{f\left( x \right)}{g\left( x \right)}$$ is called an improper rational function.
If $$\frac{f\left( x \right)}{g\left( x \right)}$$is an improver rational function. We divided f(x) by g(x) so that the rational function. $$\frac{f\left( x \right)}{g\left( x \right)}$$ is expressed I the $$Q\left( x \right)+\frac{f\left( x \right)}{g\left( x \right)}$$ where Q(x) and f(x) are polynomial such that degree of f(x) is less than that of g(x) . thus $$\frac{f\left( x \right)}{g\left( x \right)}$$ is expressed as the sum of a polynomial and a proper rational function.
any proper rational function $$\frac{f\left( x \right)}{g\left( x \right)}$$ can be expressed as the sum of rational functions, each having a simple factor ofg(x) . Each such fraction is called a partial fraction and the process of obtaining them is called the resolution or decomposition of $$\frac{f\left( x \right)}{g\left( x \right)}$$ into partial fractions.
Case I: when denominator is expressed is expressible as the product of non-repeating linear factors.
Let g(x) = (x – a₁) (x – a₂) … (x – an) then, we assume that $$\frac{f\left( x \right)}{g\left( x \right)}=\frac{{{A}_{1}}}{x-{{a}_{1}}}+\frac{{{A}_{2}}}{x-{{a}_{2}}}+….+\frac{{{A}_{n}}}{x-{{a}_{n}}}$$.
Where, A₁ , A₂……are constants and can be determined by equating the numerator on RHS to the numerator on LHS and then substituting x= a₁ , a₂ ….. an.
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2020-08-11 01:34:15
|
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|
https://mathoverflow.net/questions/243425/infinite-dimensional-version-of-a-simple-fact-on-certain-singular-matrices
|
# Infinite dimensional version of a simple fact on certain singular matrices
We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds;
Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with the following property:
$$\sum_{i=1}^{n} a_{ij}=0$$ for every fixed $j$.
Obviousely $L$ is a Lie algebra.(As I have already learned from Qiaochu Yuan in another MO post)
Moreover the linear map $X \mapsto AX$ has non trivial kernel.
These simple facts can be modelized in an infinite dimensional manner.(note that a vector $X\in\mathbb{R}^{n}$ can be considered as a function on a finite ($n$ pointed)set $M$ equiped with discrete counting measure and a matrix is a function on $M\times M$. Now the product $AX$ has an integral representation if we replace $\sum$ by the integral sign. That is we read the expersion $\sum a_{ij}x_{j}$ in the integral form $\int_{M} a_{ij}x_{j}$ where the integration is based on the normalized counting measure.
Now we state our questions as generalization of the above simple fact about matrices.
Assume that $M$ is a compact orientable manifold or a Lie or topological group. So $M$ has a natural measure, correspond to volum form or the invariant metric or Haar measure. Assume that $g: M \times M \to \mathbb{R}$ is a smooth function which satisfies $$\int_{M} g(x,y)dx=0\;\;\;\;\;(1)$$ for all $y \in M$.
Does the linear map $A$ on $C^{\infty} (M)$ has nontrivial kernel? $$A(f)(x)=\int_{M} g(x,y)f(y)dy$$
Note that for topological groups we consider continuous functions, since smoothness is meaningles.
For our next question, we assume that $M$ is a symplectic manifold, so $M \times M$ has a natural symplectic structure. Let $L$ be the space of all smooth functions on $M\times M$ which satisfy the equation (1).
Is $L$ closed under Poisson bracket?
For the first question, the answer is not necessarily.
Very rough idea: The rank-nullity theorem doesn't always hold on infinite dimensional spaces.
Rough idea: Let the operator $A$ be defined on $L^2(M)$ be a injective mapping such that its range does not include the constant function. More precisely, since $M$ is compact we can enumerate its eigenvalues (of the Laplacian) increasing with multiplicity as $\lambda_i$, with $\lambda_0 = 0$ corresponding to the constants. Now let $\psi:\overline{\mathbb{N}}\to \overline{\mathbb{N}}$ such that $\psi$ is injective and such that the range of $\psi$ does not contain $0$. Then defining $A$ as the map that sends the $i$th eigenspace to $\psi(i)$th eigenspace will provide a counterexample.
Realization: In practice to guarantee smoothness it is easier to not keep $A$ an isometry. Take $M = \mathbb{S}^1$ for simplicity. Let
$$\phi_-(x) := \sum_{k < 0} 2^{-|k|} e^{ik x}$$
The series is absolutely convergent and in fact defines a $C^\infty$ function. Similarly we define
$$\phi_+(x) := \sum_{k \geq 0} 2^{-|k|} e^{ikx}$$
Define your function $g$ by
$$g(x,y) = \phi_-(x-y) + e^{ix} \phi_+(x-y)$$
It is easy to check that $\int_{0}^{2\pi} g(x,y) ~\mathrm{d}x = 0$ for any fixed $y$. But the operator $f(x) \mapsto \int g(x,y) f(y) ~\mathrm{d}y$ has no nontrivial kernel.
• Is it possible that you have the sign wrong in the definition of $\phi_-$ (i.e. $2^{-k} \mapsto 2^k$)? – kosta Jul 1 '16 at 18:02
• @kosta: it is entirely possible. Thanks, fixed. – Willie Wong Jul 1 '16 at 18:35
• @WillieWong thanks a lot for your answer. – Ali Taghavi Jul 2 '16 at 20:47
• @WillieWong I am sorry if my question is elementary: But are not you using some thing as Mercer Theorem?If yes, Is your g a symmetric function? – Ali Taghavi Feb 25 '17 at 19:20
• @AliTaghavi: (1) I don't know Mercer Theorem. Perhaps it is using the same idea. (The idea is elementary, so likely appears in many places.) (2) Do you mean symmetric in $x$ and $y$? Then obviously no: if it were symmetric, then $\int g(x,y) f(y) dy = 0$ for $f$ being any constant function, which basically is the entire opposite of the construction. – Willie Wong Feb 26 '17 at 3:04
Willie Wong answered the general case, and I'd like to give a counterexample for the symplectic case:
Let $M=S^2$, the standard 2-sphere embedded in $\mathbb{R}^3$ as the unit sphere, with the standard symplectic form corresponding to the Euclidean metric on $\mathbb{R}^3$. Take on $(M \times M, \Omega = \omega \oplus - \omega)$ the functions \begin{align} f(z,w) &= z_3\cdot F(w), \\ g(z,w) &= z_3 \cdot G(w) \end{align}
both of which are in $L$. Now, $\{ f, g \} = \Omega(\mathrm{sgrad}(g), \mathrm{sgrad}(f)) =: \Omega'(\mathrm{d}f, \mathrm{d}g)$. Then,
\begin{align} \{f,g\} &= (\omega' \oplus (-\omega')) \left( F(w)\mathrm{d} z_3 + z_3 \mathrm{d} F(w), G(w)\mathrm{d} z_3 + z_3 \mathrm{d} G(w) \right) \\ &= F(w)G(w) \omega'(\mathrm{d}z_3,\mathrm{d}z_3) - z_3^2 \omega'(\mathrm{d} F(w),\mathrm{d} G(w)) \\ &= z_3^2 \{G,F\}(w) =:h(z,w) \end{align}
Now, unless $\{G,F\} \equiv 0$, $h$ will not be in $L$.
• thank you so much for your answer.I think you mean $\Omega(X_{f},X_{g}$ where the later are hamiltonian vector fields. Moreover what is $\Omega '$. How a two form act one forms df and dg. – Ali Taghavi Jul 2 '16 at 18:18
• @AliTaghavi $X_f \equiv \mathrm{sgrad}(f)$ - just different notation. A symplectic form $\omega$ gives an isomorphism between $TM$ and $T^*M$ by $\eta \leftrightarrow \iota_X \omega$. Define $\omega'(\eta,\sigma) := - \omega(X,Y)$ where $\eta = \iota_X \omega$ and $\sigma = \iota_Y \omega$. I may have messed up the sign somewhere. – kosta Jul 2 '16 at 18:34
• Thank you again for your very interesting answer. I am sorry that I can not accept two answers, simultaneously. meta.mathoverflow.net/questions/1491/… – Ali Taghavi Aug 10 '16 at 9:31
• where did you use the minus sign in $\Omega= \omega\oplus -\omega$? – Ali Taghavi Feb 25 '17 at 10:32
• I think the same argument works for $\omega \oplus \omega$. am I right? – Ali Taghavi Feb 25 '17 at 10:34
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2020-05-26 07:25:11
|
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|
https://cs.stackexchange.com/questions/39851/what-is-the-meaning-of-undecidability-in-rice-theorem
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# What is the meaning of undecidability in Rice Theorem?
Rice theorem says every non-trivial property of languages of Turing machines is undecidable. what is the meaning of undecidability here? is it semi-decidable?
As an example the following language is R.E but it contains a non-trivial property
$A = \{x | \phi_x$ is defined for at least one input$\}$
this language equals to $Empty$ $Complement$ and it is clearly r.e.(m-complete in fact) while non-emptiness property is non-trivial.
• It means the same as everywhere else (in computability theory). – Raphael Feb 27 '15 at 7:35
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2020-02-22 17:29:48
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http://repec.sowi.unibe.ch/stata/estout/latex.html
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Arrange models in groups
For basic information on exporting tables to LaTeX, see here. The following example additionally illustrates how models can be grouped:
```. sysuse auto, clear
(1978 Automobile Data)
. eststo: quietly reg weight mpg
(est1 stored)
. eststo: quietly reg weight mpg foreign
(est2 stored)
. eststo: quietly reg price weight mpg
(est3 stored)
. eststo: quietly reg price weight mpg foreign
(est4 stored)
. esttab using example.tex, booktabs label ///
> mgroups(A B, pattern(1 0 1 0) ///
> prefix(\multicolumn{@span}{c}{) suffix(}) ///
> span erepeat(\cmidrule(lr){@span})) ///
> alignment(D{.}{.}{-1}) page(dcolumn) nonumber
(output written to example.tex)
. eststo clear
```
Result:
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2017-06-23 15:25:23
|
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|
http://math.stackexchange.com/questions/88646/grading-accuracy-of-two-lines
|
# grading accuracy of two lines [duplicate]
I am programming (with vectors) an application which requires a user to draw line according to certain data. Then the user will click a check button which will grade the users drawing to the actual data line. So I am wondering how would I go about grading the accuracy of the two lines?
So far what I have been able to do is interpolate the entire line of both the user line and the actual line. So that the user lines data can match with the actual line data.
What is my next step in finding the accuracy of the user line to the actual line?
I can't use area because the line the user draws is not linear, its freeform.
Heres an image of what i mean:
-
## marked as duplicate by Listing, Mike Spivey, t.b., Asaf Karagila, SashaDec 7 '11 at 3:23
why can't you use area? The less the area, the better the users line as compared to yours? – picakhu Dec 5 '11 at 20:29
"I can't use area because the line the user draws is not linear, its freeform." but there is no case in my application where the line is just a line, it could be shaped along the x and y positive axis. – John Riselvato Dec 5 '11 at 20:31
I think I get what you mean. How about randomly choosing points on the users line and then finding the root mean square error from yours? – picakhu Dec 5 '11 at 20:32
I'll read that then, i uploaded a photo of what i mean if it helps – John Riselvato Dec 5 '11 at 20:33
From your image, I think Area will work. – picakhu Dec 5 '11 at 20:36
Lets say user line is $g(x)$ and line drawn by you is $f(x)$
Then grade can be made be inversely proportional to
$$\displaystyle \sum_{x=a}^{b} (g(x)-f(x))^2$$
This is commonly used function.
-
I will see how this works out for me. – John Riselvato Dec 5 '11 at 20:42
@John, this is RMS applied in a slightly different manner. – picakhu Dec 5 '11 at 20:44
I rather like the RMSE better nyways. @picakhu post it as your answer as an answer, so i can give you credit. – John Riselvato Dec 5 '11 at 20:46
@John, I would have to elaborate on how you should use it, and I am not sure which approach is best. I think it is better that I do not get credit for it, I do not deserve it. – picakhu Dec 5 '11 at 20:49
This just avoids taking the root and average. So it saves computation. We are using this in stanford ml-class online. :) @picakhu is right its just RMS squared times number of points! – Pratik Deoghare Dec 5 '11 at 20:51
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2016-06-28 18:38:21
|
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|
https://questions.examside.com/past-years/jee/question/pthe-threshold-frequency-of-a-metal-is-f0-when-the-jee-main-physics-motion-ud2ipvahib7cpmnp
|
1
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1
The threshold frequency of a metal is $$f_{0}$$. When the light of frequency $$2 f_{0}$$ is incident on the metal plate, the maximum velocity of photoelectrons is $$v_{1}$$. When the frequency of incident radiation is increased to $$5 \mathrm{f}_{0}$$, the maximum velocity of photoelectrons emitted is $$v_{2}$$. The ratio of $$v_{1}$$ to $$v_{2}$$ is :
A
$$\frac{v_{1}}{v_{2}}=\frac{1}{2}$$
B
$$\frac{v_{1}}{v_{2}}=\frac{1}{16}$$
C
$$\frac{v_{1}}{v_{2}}=\frac{1}{4}$$
D
$$\frac{v_{1}}{v_{2}}=\frac{1}{8}$$
2
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
If the two metals $\mathrm{A}$ and $\mathrm{B}$ are exposed to radiation of wavelength $350 \mathrm{~nm}$. The work functions of metals $\mathrm{A}$ and $\mathrm{B}$ are $4.8 \mathrm{eV}$ and $2.2 \mathrm{eV}$. Then choose the correct option.
A
Metal B will not emit photo-electrons
B
Both metals $\mathrm{A}$ and $\mathrm{B}$ will not emit photo-electrons
C
Metal A will not emit photo-electrons
D
Both metals A and B will emit photo-electrons
3
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1
Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R
Assertion A : The beam of electrons show wave nature and exhibit interference and diffraction.
Reason R : Davisson Germer Experimentally verified the wave nature of electrons.
In the light of the above statements, choose the most appropriate answer from the options given below :
A
A is not correct but R is correct.
B
Both A and R are correct and R is the correct explanation of A
C
Both A and R are correct but R is Not the correct explanation of A
D
A is correct but R is not correct
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1
If a source of electromagnetic radiation having power $$15 \mathrm{~kW}$$ produces $$10^{16}$$ photons per second, the radiation belongs to a part of spectrum is.
(Take Planck constant $$h=6 \times 10^{-34} \mathrm{Js}$$ )
A
Gamma rays
B
C
Micro waves
D
Ultraviolet rays
JEE Main Subjects
Physics
Mechanics
Electricity
Optics
Modern Physics
Chemistry
Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Mathematics
Algebra
Trigonometry
Coordinate Geometry
Calculus
EXAM MAP
Joint Entrance Examination
|
2023-03-28 19:18:16
|
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|
https://codereview.stackexchange.com/questions/233568/calculate-the-midpoint-of-a-polyline
|
# Calculate the midpoint of a polyline
I have polylines in a GIS database. The polylines are stored using a user-defined type called ST_GEOMETRY.
ST_GEOMETRY has lots of spatial functions. However, it does not have a polyline midpoint function.
Therefore, I have written a custom PL/SQL midpoint function to fill this gap.
As a novice programmer, I'm wondering, how can the code be improved to be as fast and robust as possible?
-- execute this as the SDE user
create or replace function ST_MidPoint
(line_in IN sde.st_geometry)
RETURN sde.st_geometry
IS midpoint sde.st_geometry;
srid integer;
line_length number(38);
num_parts integer;
num_points integer;
partNum integer;
distanceAlong number(38);
segmentLength number(38);
part sde.st_geometry;
p1 sde.st_geometry;
p2 sde.st_geometry;
x1 double precision;
y1 double precision;
x2 double precision;
y2 double precision;
BEGIN
-- get the SRID of the line for later use in constructing the midpoint geometrey
select sde.st_srid (line_in)
into srid
from dual;
-- calculate the total length of the line
select sde.st_length(line_in)
into line_length
from dual;
-- get the number of parts that make up the line
select sde.st_numgeometries(line_in)
into num_parts
from dual;
distanceAlong := 0;
for partNum in 1..num_parts loop
--dbms_output.put_line(partNum);
-- get the geometry for this part
select sde.st_geometryn(line_in, partNum)
into part
from dual;
-- get the number of points that make up this part
select sde.st_numpoints(part)
into num_points
from dual;
--dbms_output.put_line(num_points);
-- get the first point (the "from" point) for the part
select sde.st_pointn (part, 1)
into p1
from dual;
-- iterate along the line until the section that contains the midpoint is found
for pointNum in 2..num_points loop
-- get the "to" point of the segment
select sde.st_pointn(part, pointNum)
into p2
from dual;
-- calculate the distance between the from point and the to point
select sde.st_distance(p1, p2)
into segmentLength
from dual;
-- add the distance along this segment to the running total
distanceAlong := distanceAlong + segmentLength;
--dbms_output.put_line(distanceAlong);
-- check to see if the running total is past the midpoint
if distanceAlong >= line_length/2.0 then
-- the two current points encompass the midpoint of the line
-- determine the midpoint geometry and return it
select sde.st_x(p1)
into x1
from dual;
select sde.st_y(p1)
into y1
from dual;
select sde.st_x(p2)
into x2
from dual;
select sde.st_y(p2)
into y2
from dual;
select sde.st_point((x1+x2)/2.0, (y1+y2)/2.0, srid)
into midpoint
from dual;
-- the midpoint has been found, not need to interogate the rest of the line
RETURN midpoint;
end if;
-- save the endpoint as the first point and continue down the line looking for the midpoint
p1:=p2;
end loop;
end loop;
return null;
EXCEPTION
WHEN OTHERS THEN
raise_application_error(-20001,'An error was encountered - '||SQLCODE||' -ERROR- '||SQLERRM);
END;
/
grant execute on ST_MidPoint to public;
--select objectid,sde.ST_MidPoint(shape) geom
--from gis.line_test
--order by objectid;
Update:
I have abandoned this function. It works, but it's horrendously inefficient because it's not possible to use the SDE functions & operators to directly assign values to variables. I needed to wrap them in queries instead.
More information here: Use SDE.ST_GEOMETRY functions in a custom function
Originally, I had thought that the code could be improved. However, now I realize that this is not possible--due to the aforementioned problem.
It would be my preference that this question be deleted (I tried but couldn't because there's an answer).
• This is the reason why the INSERT INTOS were used: gis.stackexchange.com/q/344294/135445 – User1973 Dec 8 '19 at 0:20
• I think we should close this question. The function is so slow and poorly structured that it's not really useable. I'm going to go with another option (pre-compute with python). – User1973 Dec 8 '19 at 0:24
• You can write your own review explaining what makes the function poorly structured. There doesn't seem to be a valid reason to close the question, the other option is that you could delete it yourself. – pacmaninbw Dec 8 '19 at 14:47
• @pacmaninbw Yeah, I tried to delete the question, but I couldn't because there's an answer. I also added a note to the question about why the code can't be improved. – User1973 Dec 15 '19 at 20:02
## 2 Answers
Check this function which returns the mid point X,Y in WKT format. Please note this function uses the SDO_LRS of Oracle which is part of Oracle Spatial.
create or replace function get_line_midpoint
(line_in IN sde.st_geometry)
-- RETURN sde.st_geometry
RETURN VARCHAR2
IS
wkt_geometry clob;
ora_geometry sdo_geometry;
mid_x number(10,6);
mid_y number(10,6);
mid_point_geom sde.st_geometry;
BEGIN
SELECT sde.ST_AsText(line_in) INTO wkt_geometry FROM DUAL;
ora_geometry := SDO_UTIL.FROM_WKTGEOMETRY(wkt_geometry);
--mid_x:= sdo_cs.transform(SDO_LRS.CONVERT_TO_STD_GEOM(SDO_LRS.LOCATE_PT(SDO_LRS.CONVERT_TO_LRS_GEOM(ora_geometry, 3), SDO_GEOM.SDO_LENGTH(ora_geometry,3)/2)),8307).SDO_POINT.X;
mid_x:= SDO_LRS.CONVERT_TO_STD_GEOM(SDO_LRS.LOCATE_PT(SDO_LRS.CONVERT_TO_LRS_GEOM(ora_geometry, 3), SDO_GEOM.SDO_LENGTH(ora_geometry,3)/2)).SDO_POINT.X;
--mid_y:= sdo_cs.transform(SDO_LRS.CONVERT_TO_STD_GEOM(SDO_LRS.LOCATE_PT(SDO_LRS.CONVERT_TO_LRS_GEOM(ora_geometry, 3), SDO_GEOM.SDO_LENGTH(ora_geometry,3)/2)),8307).SDO_POINT.Y;
mid_y:= SDO_LRS.CONVERT_TO_STD_GEOM(SDO_LRS.LOCATE_PT(SDO_LRS.CONVERT_TO_LRS_GEOM(ora_geometry, 3), SDO_GEOM.SDO_LENGTH(ora_geometry,3)/2)).SDO_POINT.Y;
ora_geometry := SDO_UTIL.FROM_WKTGEOMETRY('point ('|| mid_x || ' ' || mid_y ||')');
return 'point ('|| mid_x || ' ' || mid_y ||')';
EXCEPTION
WHEN OTHERS THEN
raise_application_error(-20001,'An error was encountered - '||SQLCODE||' -ERROR- '||SQLERRM);
END;
• Well, this changes things! Spatial now free with all editions of Oracle Database. – User1973 Dec 20 '19 at 5:11
• Thanks for letting me know. – Ashok Vanam Dec 20 '19 at 5:16
• You have presented an alternative solution, but haven't reviewed the code. Please edit to show what aspects of the question code prompted you to write this version, and in what ways it's an improvement over the original. It may be worth (re-)reading How to Answer. – Toby Speight Jan 6 at 11:55
• The procedure posted works fine. @TobySpeight are you asking the user who posted the question? – Ashok Vanam Jan 7 at 23:43
• I posted a related answer on Stack Overflow: Get midpoint of SDO.GEOMETRY polyline. – User1973 Jan 12 at 22:46
I would recommend to try re-writing the procedure with Buffer and Centroid functions as below.
procedure mindpoint (in_line_geometry)
line_buffer_geom = sde.st_buffer (in_line_geometry, 0.05)
cenrtroid_buffer_geom = sde.st_centroid (line_buffer_geom )
return cenrtroid_buffer_geom
• (Down-voters please comment.) – greybeard Dec 7 '19 at 4:00
• The cenrtroid in cenrtroid_buffer_geom looks accidental. – greybeard Dec 7 '19 at 4:01
|
2020-02-25 19:58:53
|
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|
https://brilliant.org/problems/the-closed-form-still-exists/
|
# The Closed Form Still Exists!
Calculus Level 2
We know that the geometric progression sum of
$1 - x + x^2 - x^3 + x^4 - \cdots$
can be written as $$\dfrac1{1+x}$$, where $$-1<x<1$$.
If we integrate each of these terms with respect to $$x$$, we get the series below. Which of the following is equivalent to the series below?
$x -\dfrac12 x^2 + \dfrac13x^3 - \dfrac14 x^4 + \dfrac15x^5 - \cdots$
Assume we ignore the arbitrary constant of integration.
×
|
2018-04-26 17:26:39
|
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|
https://bmjophth.bmj.com/content/6/1/e000842
|
Article Text
Management of ocular surface squamous neoplasia: Bowman Club Lecture 2021
1. Osmel Peter Alvarez1,
2. Mike Zein1,
3. Anat Galor1,2,
4. Carol L Karp1
1. 1Bascom Palmer Eye Institute, Department of Ophthalmology, University of Miami Miller School of Medicine, Miami, FL, USA
2. 2Department of Ophthalmology, Miami Veterans Administration, Miami, FL, USA
1. Correspondence to Dr Carol L Karp; ckarp{at}med.miami.edu
## Abstract
The gold-standard treatment for ocular surface squamous neoplasia (OSSN) has traditionally been surgical excision with wide margins and a no-touch technique. However, surgery may be associated with several unfavourable sequelae, as well as significant recurrence rates if margins are positive. In recent years, topical chemotherapy with 5-fluorouracil, interferon α−2b and mitomycin C have emerged as valuable agents capable of effectively treating OSSN with varying adverse effects. These medical treatment options usually present additional costs to the patient but can allow patients to avoid surgery with fewer long-term effects. Anterior segment high-resolution optical coherence tomography is an excellent tool for diagnosing and monitoring OSSN and can be a useful aid for both surgical and medical treatments of OSSN.
• cornea
• ocular surface
• neoplasia
• imaging
• treatment surgery
• treatment medical
• conjunctiva
## Data availability statement
No data are available. Not applicable.
This is an open access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited, appropriate credit is given, any changes made indicated, and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/4.0/.
## Introduction
Ocular surface squamous neoplasia (OSSN) is the most common non-pigmented malignancy of the cornea and conjunctiva and represents a group of disease including dysplasia, corneal and conjunctival intraepithelial neoplasia and squamous cell carcinoma.1 2 The primary treatment for OSSN has generally been surgical excision with a no-touch technique, but surgical treatment has been associated with sequelae including conjunctival scarring, symblepharon, conjunctival hyperaemia and limbal stem cell deficiency.2–5 In recent years, medical treatments consisting of topical chemotherapies have emerged as valuable alternatives to surgical intervention. These topical treatments have the advantage of treating the entire ocular surface, reaching subclinical disease that could otherwise cause up to 56% recurrence rates following surgical excision with resultant positive margins.6 Another recent development is the advancement of anterior segment high-resolution optical coherence tomography (HR-OCT) technology. HR-OCT is an excellent tool for diagnosing and monitoring OSSN as it provides fast, noninvasive, and high-resolution imagery of the ocular surface.
## Surgical management of OSSN
Surgical excision has long been the gold-standard treatment for OSSN. Surgical management has the advantages of potentially faster resolution than medical treatment. However, surgery can be associated with up to 56% recurrence rates with resultant positive margins, requiring further treatments.6 Additionally, surgical excision has been associated with unfavourable sequelae such as conjunctival scarring, symblepharon, conjunctival hyperaemia and limbal stem cell deficiency.2–5 As such, patients and physicians should consider both medical and surgical approaches to treatment and balance their respective advantages and disadvantages.
### Technique
The primary method of surgical excision for OSSN is the Shields ‘no-touch’ technique with wide margins (figure 1).7 Intraoperative cryotherapy is helpful, as studies have shown that this significantly reduces recurrences after surgical excision.8
Figure 1
(A) Leukoplakic OSSN on cornea and conjunctiva with marked 4 mm margins. (B) Incision at marked margins with special care to manipulate only at or outside the margins. (C) Due to adherence of the lesion to the sclera, an approximately 20% sclerectomy under area of lesion was performed. (D) Orientation of excised lesion on sterile paper to be sent for pathological analysis. (E) Application of absolute alcohol on corneal area of excised lesion, followed by (F) corneal epitheliectomy. (G) Double freeze-slow thaw cryotherapy on conjunctival and corneal margins of the excised lesion. (H) Application of two layers of amniotic membrane tissue (AMT) bonded with fibrin glue over area of sclerectomy and excision. (I) In this case, the conjunctiva was mobile and a double closure with conjunctiva over AMT was performed. OSSN, ocular surface squamous neoplasia.
Wide margins of 3–4 mm are marked around the visible edges of the tumour. For tumours with limbal and corneal components, corneal epitheliectomy is performed by applying absolute alcohol for 1 min. The lesion is then excised in one piece. Special attention is paid to manipulate only at the marked margins and to keep the surgical field dry to prevent potential seeding of tumour cells. For adherent tumours, a 20% sclerectomy is recommended. Cryotherapy is performed in a double freeze-slow thaw method on conjunctival margins of the excised area, as well as the limbus and cornea if the margins extend thus far. Closure of the wound with amniotic membrane tissue and fibrin glue is preferred but may also be completed with primary close if the wound is small. In this case, it would be a two-layer closure with Tenon’s fascia first, then conjunctiva.
Surgical excision of OSSN is effective, but excisions with positive margins may cause up to 56% recurrence rates.6 However, more recent studies have found lower recurrence rates of 0%–21% following surgical excision.8–10 One study of 27 lesions used intraoperative cryotherapy on only some cases and post-operative topical mitomycin C (MMC) in all cases and saw a 0% recurrence rate.10 A larger study of 389 excised lesions found a 1-year recurrence rate of 10% and a 5-year recurrence rate of 21% after surgical excision; it also found that intraoperative cryotherapy significantly reduced the risk of recurrence, as did postoperative topical interferon (IFN) α−2b for patients with resultant positive margins.8
### Expert opinion
While the advent of effective topical chemotherapies may reduce the need for surgical management of OSSN, surgery continues to be a good choice for many cases. Surgery is an effective treatment for OSSN, often with a shorter overall treatment period than medical treatment. Additionally, insured patients may have significantly less out-of-pocket expenses with surgical treatment than with medical, as most available topical chemotherapies are generally not covered by insurance. Regardless, many patients may favour medical treatment to avoid the stress and discomfort many associate with surgery; as such, patients should be given the choice between medical and surgical treatment when appropriate.
## Medical management of OSSN
Since the 1990s, several topical chemotherapies have emerged as efficient medical treatment options for OSSN. These offer the benefit of treating the entire ocular surface, theoretically addressing subclinical disease that could be left untreated by surgery. The three most commons options—5-fluorouracil (5FU), IFN−2b and MMC—are all effective with have different side effect profiles and costs that should be carefully considered.
### 5-fluorouracil
5FU is an anticancer drug that interrupts DNA replication and cell growth.2 11 It was first used to treat OSSN by de Keizer et al and has reemerged as an effective treatment for these lesions.12
5FU is generally used as topical eye-drops in 1% concentration and is administered in ‘cycles’ of four times a day for 1 week, followed by 3 weeks of no medication, with cycles repeated until resolution2 13–15 (figure 2). Studies have shown 5FU to be very effective in treating OSSN, with high-resolution rates of 82%–100% and low recurrence rates of 10%–14%.13–16 5FU has also been used as subconjunctival and perilesional injections to treat OSSN; however, the evidence is limited and this requires further study.17
Figure 2
(A) Slit lamp photo of the left eye demonstrating an ocular surface squamous neoplasia with gelatinous features on the temporal aspect of the conjunctiva and limbus (arrow heads). (B) High-resolution anterior segment optical coherence tomography image of the inferior temporal conjunctiva and limbus demonstrating a thickened and hyper-reflective epithelium (asterisk) and an abrupt transition between normal and abnormal epithelium (arrowhead). Inset denotes anatomical location of scan. (C) Slit lamp photo of the left eye after 2 months (two cycles) of topical 5-fluoruacil treatment with clinical tumour resolution. (D) High-resolution anterior segment optical coherence tomography image of the inferior temporal conjunctiva and limbus demonstrating tumour resolution with thin, normalised epithelium (asterisk) after 2 months (two cycles) of topical 5-FU treatment. Empiric treatment was continued for the full four cycles. Inset denotes anatomical location of image. 5-FU, 5-fluorouracil.
Side effects of 5FU are generally mild and well tolerated. These may include pain, tearing, redness, eyelid oedema and keratopathy.14 One study of 44 patients reported that 61% experienced at least one side effect, but only one patient could not tolerate 5FU and discontinued it.15 Topical 5FU eye-drops have minimal side effects compared with other chemotherapies such as MMC.13 14
#### Expert opinion
IFN is highly effective and has essentially no side effects, making it an ideal choice for topical chemotherapy for OSSN. Its major drawback is the cost, as it can be much more expensive than 5FU and MMC in the USA. Additionally, it is becoming increasingly difficult to obtain. However, it may be much less costly and easier to obtain in other countries. Furthermore, it must be used four times a day consistently until tumour resolution, so patient compliance is vital for tumour response. For patients who may have issues with maintaining the treatment regimen for topical IFN eye-drops, clinic-administered subconjunctival injections may be a good alternative. However, patients must commit to weekly or biweekly clinic visits. This option reduces patient responsibility, has tolerable side effects, has been proven to be effective, and is generally covered by insurance in the USA. We recommend patients take 1 g of acetaminophen at the time of injection and every 4 hours for the next day to help with the flu-like symptoms associated with IFN injections.
### Mitomycin C
MMC is an antimetabolite originally isolated from Streptomyces caespitosus.3 Its antineoplastic and antibiotic properties make it useful in glaucoma and pterygium surgery; it is also used as an adjuvant therapy in OSSN excision surgery and for medical treatment of OSSN.2 36–38 MMC was first used to treat OSSN by Frucht-Pery and Rozenman and is one of the three major topical chemotherapies for OSSN.39
MMC is generally used in the form of topical eye-drops in strengths of 0.02–0.04%3 (figure 4). One study compared 0.02% and .04% strengths and found no difference in time to resolution or recurrence rates among the two.37 MMC is also very effective, with high resolution rates ranging from 76% to 100% and low recurrence rates ranging from 0% to 20%.16 37 40–44
Figure 4
(A) Slit lamp photo of the left eye demonstrating an ocular surface squamous neoplasia with leukoplakic and opalescent features on the temporal aspect of the conjunctiva and limbus. (B) High-resolution anterior segment optical coherence tomography image of the temporal conjunctiva and limbus demonstrating a thickened and hyper-reflective epithelium (asterisk) and an abrupt transition between normal and abnormal epithelium (arrowhead). Inset denotes location of scan. (C) Slit lamp photo of the left eye after three cycles of topical mitomycin C treatment with clinical tumour resolution. (D) High-resolution anterior segment optical coherence tomography image of the temporal conjunctiva and limbus demonstrating tumour resolution with thin, normalised epithelium (asterisk) after 3 weekly cycles of topical mitomycin C 0.04% treatment. Inset denotes anatomical location of scan.
Many different administration regimens of MMC have been studied; we recommend MMC in 4 week cycles of 0.04% four times a day for 1 week, followed by 3 weeks of no treatment, with cycles repeated until resolution. Others may use MMC with shorter breaks or longer consecutive days of treatment.16 19 40–46
The main drawback of MMC is the intensity of its side effects. MMC has more adverse effects than IFN and 5FU; these include redness, itching, tearing, pain, corneal erosion, hyperaemia, punctate staining of cornea, punctal stenosis and limbal stem cell deficiency.16 37 40–44 46 47 In order to alleviate these effects and prevent severe toxicity, patients are commonly instructed to use steroids and artificial tears throughout the course of treatment.16 46 MMC should be paused with any epitheliopathy, as this can lead to more toxic effects and intolerance. Additionally, punctal plugs are used to prevent punctal stenosis.42 44
As with IFN and 5FU, MMC is a compounded medication and is generally not covered by insurance. It is typically less costly than IFN but more expensive than 5FU, costing around US$100–US$190 in the USA, but may be less expensive elsewhere. We recommend refrigeration of MMC.
#### Expert opinion
While it is very effective for OSSN, its propensity for causing ocular surface toxicity and other serious adverse effects is much greater than 5FU or IFN. Patients using MMC should be carefully monitored for signs of toxicity. They should be advised to lubricate liberally with preservative-free artificial tears throughout treatment, as epitheliopathy can lead to intolerance and limbal stem cell deficiency. We wait until the eye is white and quiet before starting the next cycle, as treatment when epitheliopathy is present can lead to unwanted toxicity. Punctal occlusion is also recommended, and silicone punctal plugs may be a good option to achieve this. Applying petroleum jelly to the lower eyelid skin is recommended to reduce skin irritation and toxicity. Additionally, patients may be given topical steroids to minimise ocular surface irritation.
## Anterior segment HR-OCT
OCT is an excellent technology for imaging various parts of the eye. OCT produces two-dimensional imagery of optical scattering by using low-coherence interferometry.48 Although its first in vivo use was for the retina in 1993,49 it was soon after used to capture in vivo images of the anterior segment in 1994.50 Over time, OCT has advanced from time-domain, to spectral-domain, to current swept-source technology capable of producing ultrahigh resolution imagery of the anterior segment.51 Today, available OCT technology is capable of producing imagery with high resolutions of 3.5–5 μm or less.52 53
HR-OCT is highly useful for detecting and monitoring several pathologies of the cornea and conjunctiva, such as OSSN (figures 2–4),54 pterygia and pingueculae,55 dry eye disease,56 Fuchs’ dystrophy57 and more. It provides fast, noninvasive and high-resolution imaging of the ocular surface that can be easily interpreted by novice clinicians.58
HR-OCT is particularly valuable for diagnosing and managing OSSN. On HR-OCT, OSSN can be identified as thickened, hyper-reflective epithelium with an abrupt transition from normal to abnormal tissue.54 Additionally, one study showed that HR-OCT can detect subclinical OSSN in 17% of cases that were determined to be resolved clinically.59 Thus, HR-OCT is especially useful for monitoring progression of OSSN treatment with topical chemotherapy.
HR-OCT can also potentially be useful for surgical treatment of OSSN. One recent pilot study used HR-OCT during surgical excision of OSSN as an ‘optical Mohs’ to predict the margins of the tumour to be excised.51 Such use of HR-OCT may further reduce the risk of recurrence after surgical excision of OSSN and may become more common in the future.
### Expert opinion
HR-OCT is a valuable tool for the diagnosis and management of OSSN. HR-OCT allows the physician to identify OSSN, track the progression of treatment and check for recurrences during follow-up. It can also help find tumour margins during surgical excision. In subtle cases, OSSN can be difficult to detect when present with other ocular surface pathologies such as pannus, pterygia, pinguecula and scarring. As such, HR-OCT serves as an ‘optical biopsy’. It is fast, non-invasive and presents virtually no discomfort to the patient. Recent models offering ultrahigh resolution offer impressive visualisations of OSSN and other lesions. It should be noted that OSSN occurring on the cornea may appear less thickened than conjunctival OSSN, but usually also show the other classic features of hyper-reflective epithelium and abrupt transition from normal to abnormal tissue.
## Conclusion
While surgical excision has traditionally been the gold-standard treatment for OSSN, topical chemotherapies are now available as valuable and effective alternatives. These medical treatments offer the advantage of treating the entire ocular surface and can avoid some of the unfavourable sequelae associated with surgery. However, their various side effect profiles, need for compliance and financial implications should be taken into account by the patient and physician. Additionally, advanced HR-OCT technology can complement both medical and surgical treatments. HR-OCT can aid the physician in diagnosing OSSN, monitoring it throughout medical treatment and detecting subclinical disease. Future integration with operating microscopes will potentially allow the surgeon more accurately mark the margins of the tumour to be removed, hopefully further reducing the risk of recurrence.
## Data availability statement
No data are available. Not applicable.
## Footnotes
• Funding NIH Center Core Grant P30EY014801, RPB Unrestricted Award, Dr Ronald and Alicia Lepke Grant, The Lee and Claire Hager Grant, The Robert Farr Family Grant, The Grant and Diana Stanton-Thornbrough, The Robert Baer Family Grant, The Roberto and Antonia Menendez Grant, The Emilyn Page and Mark Feldberg Grant, The Calvin and Flavia Oak Support Fund, The Robert Farr Family Grant, The Jose Ferreira de Melo Grant, The Richard and Kathy Lesser Grant, The Michele and Ted Kaplan Grant and the Richard Azar Family Grant (institutional grants).
• Competing interests None declared.
• Provenance and peer review Commissioned; internally peer reviewed.
## Request Permissions
If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.
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2022-05-17 23:34:05
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http://www.givewell.org/charities/sightsavers/april-2016-version
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Sightsavers - Interim Review of Deworming Programs – April 2016 Version | GiveWell
# Sightsavers - Interim Review of Deworming Programs – April 2016 Version
We have published a more recent review of this organization. See our most recent report on Sightsavers.
Sightsavers is applying to be a top-rated charity. Here we discuss what we have learned so far and our major outstanding questions.
Published: April 2016
## Summary
What do they do? Sightsavers (sightsavers.org) conducts a variety of programs, primarily to cure and prevent blindness. Sightsavers has recently expanded its work to include support for deworming, one of our priority programs, as an addition to a number of its established neglected tropical disease (NTD) programs. This review focuses on Sightsavers' work on deworming. We know little about the role Sightsavers plays in its deworming programs. (more)
Does it work? We believe that there is relatively strong evidence for the positive impact of deworming. Sightsavers has told us that it conducts impact assessments and coverage surveys to determine whether its programs have reached a large proportion of children targeted, and shared a few examples with us. We have questions about the reliability and representativeness of these examples. (more)
What do you get for your dollar? Deworming treatments can be delivered very cost-effectively. We are uncertain about the cost-effectiveness of deworming programs that Sightsavers supports, but it is possible that they are competitive with our top charities. (more)
Is there room for more funding? Our information about Sightsavers' funding needs is now out of date. In 2015, Sightsavers was seeking $2.0 million for deworming activities for three years in Nigeria and Guinea-Bissau, and told us that it would like to scale up deworming in several additional countries. Overall, we expect that funding intended for deworming would be at least partially fungible with funding for other programs. We are uncertain how Sightsavers allocates its unrestricted funding; it had about$13.5 million in unrestricted reserves and in total expected to raise about $100 million in 2015. (more) What are GiveWell’s next steps? Sightsavers has successfully completed the first phase of our investigation process. We now plan to (a) make a$100,000 grant to Sightsavers (as part of our "top charity participation grants," funded by Good Ventures) and (b) continue our review process of Sightsavers to try to answer our remaining questions. We may also expand the scope of our investigation to include other programs (in particular, other NTD programs).
As we mentioned in a blog post on plans for work in 2015, one of our goals is to publish reviews of potential new top charities. We are considering Sightsavers' deworming programs because deworming is one of our priority programs. This page is an update on what we have learned so far in our investigation of Sightsavers.
## Our investigation process
As we stated in our 2015 plans, we are interested in finding additional top charities that work on deworming. Sightsavers was one of several charities we invited to apply to be considered for our top charity recommendation. To date, we have:
## What do they do?
Although Sightsavers conducts various programs, in this review we focus on Sightsavers' deworming programs and include some information on its work on other neglected tropical diseases (NTDs). Deworming (mass treatment for schistosomiasis and soil-transmitted helminthiasis (STH)) is one of our priority programs and is generally underfunded. We discuss deworming in detail in our intervention report on the topic.
Our understanding is that deworming is a small and relatively new part of Sightsavers' work. We have limited information on Sightsavers' role in the deworming programs it supports.
### How does deworming fit into Sightsavers' priorities?
#### Background
Sightsavers was founded in 1950 to treat eye conditions in developing countries,1 and began work on deworming programs in 2011.2 Sightsavers currently supports deworming programs in Cameroon, Democratic Republic of the Congo (DRC), Mali, Nigeria, Sierra Leone, and Tanzania.3
Deworming programs are part of Sightsavers’ integrated NTD programs, which are carried out by Sightsavers country office staff in partnership with national Ministries of Health. Sightsavers' first deworming program was in Nigeria.4 It has worked in Nigeria on various programs for over 40 years.5 Nigeria is the country for which Sightsavers has sent us the most detailed information about its deworming activities. In 2014, Sightsavers supported deworming in three states in Nigeria and made plans to expand its deworming work to a fourth state.6 Sightsavers has also received a grant from the Children's Investment Fund Foundation (CIFF) to conduct prevalence mapping of schistosomiasis and STH in 14 states in northern Nigeria.7
#### Spending overview
We have seen limited information on how Sightsavers has spent funds in the past.8 We have not seen information on what portion of Sightsavers' spending has supported NTD programs or, more specifically, deworming programs.
As of early 2015, Sightsavers expected to spend about $100 million in 2015,9 up from$84 million in 2014 and $75 million in 2013 (details in footnote.)10 Based on this, we guess that deworming represents a fairly small portion of Sightsavers' total spending (more).11 #### Number of treatments For 2014, Sightsavers reports that it supported the delivery of about 100 million NTD treatments. Of these, about 14 million (14%) were STH treatments and about 4 million (4%) were schistosomiasis treatments. These deworming treatments (18% of all treatments) were delivered in three countries: Cameroon, Nigeria, and Tanzania.12 We have not seen a breakdown of deworming treatments by country for other years. Sightsavers' NTD Treatments (millions), 2010 - 201413 NTD 2010 2011 2012 2013 2014 2010-2014 % of 2010-2014 lymphatic filariasis 17.6 34.1 30.5 44.2 44.9 171.3 41% onchocerciasis 23.1 24.4 32.2 34.3 29.4 143.4 34% STH N/A 14.3 12.9 12.6 14.3 54.0 13% trachoma 1.9 4.6 12.1 9.0 9.3 36.9 9% schistosomiasis N/A 2.1 4.1 3.4 4.4 14.1 3% Total NTD treatments 42.6 79.5 91.8 103.6 102.2 419.7 100% Sightsavers' Deworming Treatments (millions), 201414 Program STH Schistosomiasis Total Treatments Nigeria 0.4 1.2 1.6 Cameroon 4.9 0.3 5.2 Tanzania 9.0 2.9 11.9 Total 14.3 4.4 18.7 Sightsavers told us that it also supports deworming in Sierra Leone, Mali, and the DRC.15 No data for those programs is included in the treatment numbers we have seen, and we are uncertain whether Sightsavers’ support in these countries is similar to its work in Cameroon, Nigeria, and Tanzania. Sightsavers has provided us with further information on its NTD work in Nigeria; however, as shown above, deworming in Nigeria represents only about 9% of the deworming work that Sightsavers supported in 2014. The details we have received about NTD work in Nigeria include state-level breakdowns of NTD costs and outputs. The sum of state-level STH treatments matches the value reported above; however, the sum of state-level schistosomiasis treatments is only 0.6 million (half the number of treatments reported above). Additionally, the sum of 2014 NTD state-level costs is$313,000, only 13% of the $2.4 million (£1.4 million) reported for Nigeria's country-level NTD costs. We have not yet asked Sightsavers what accounts for these differences.16 ### How does Sightsavers select programs to support? Sightsavers told us that it selects where to support NTD programs based on the need for treatment (e.g. disease prevalence), the government's capacity, and Sightsavers' ability to build capacity in the area.17 We have not seen more details on this process. ### How is deworming conducted in programs that Sightsavers supports? We know little about the specifics of deworming program activities that Sightsavers supports within its integrated NTD programs. Our understanding is that many of the programs that Sightsavers supports use a combination of community-based and school-based treatments.18 For example, the coverage survey discussed below describes the evaluated deworming distribution as follows:19 "Praziquantel was distributed to school age children (5-15 years) in the schools using a measuring tape to determine the height and subsequently dosage. The dose given ranged from 1 to 5 tablets of the drug. The distribution of Mectizan, Albendazole, Zithromax and tetracycline was community based and house to house." ### What is Sightsavers' role in supporting deworming programs? We know little about the role Sightsavers plays in supporting deworming programs. Our understanding is that Sightsavers partners with the government and other organizations to support NTD programs.20 Sightsavers has partnered with SCI on programs in Tanzania, DRC, and other countries.21 ### Spending breakdown In May 2015, Sightsavers shared 2015-2017 budgets for NTD programs in Nigeria and Guinea-Bissau. Some notes about this data: • These are projections. We have not seen data on what activities funds have been spent on in the past. • We have not seen budgets for Sightsavers' NTD programs in other countries. We do not know what portion of Sightsavers' total spending on NTDs is covered by these two budgets. • We have not seen data on how Sightsavers has spent or plans to spend funds on deworming programs specifically. • We are uncertain whether these budgets are comprehensive of the full NTD programs in these countries; in particular, not all of Sightsavers' NTD activities in Nigeria are included in that country's budget.22 In Nigeria, Sightsavers plans to spend about$1.5 million to treat NTDs across four states from 2015-2017.23 Sightsavers is also seeking additional funding to deliver more deworming treatments in Nigeria over these three years; it would like to raise about $560,000 to scale up its activities in these four states, and about$670,000 to expand to a new state (Benue).24 The table below shows budgets for each of these scenarios.
Sightsavers' Nigeria NTD program, budget summary, 2015-201725
Activity Existing Plans Scale-up Plans Expansion Plans Total % Total
Training26 $623,826$176,786 $114,933$915,546 34%
Monitoring and evaluation27 $248,998$163,837 $198,906$611,740 23%
Information, education, and communication28 $277,977$106,648 $106,875$491,500 18%
Supplies29 $44,811$64,562 $88,807$198,181 7%
Drug transportation30 $96,668$51,296 $10,759$158,723 6%
Trichiasis camps31 $173,112$0 $0$173,112 6%
Salaries32 $0$0 $151,299$151,299 6%
Total $1,465,393$563,129 $671,578$2,700,101 100%
In Guinea-Bissau, Sightsavers currently treats onchocerciasis, lymphatic filariasis (LF), and trachoma. In May 2015, Sightsavers reported that in 2016 and 2017, it would like to scale up lymphatic filariasis activities (budgeted at $700,000 over two years) and start treating schistosomiasis and STH (budgeted at$670,000 over two years). The table below shows budgets for both existing plans and possible scale-up plans.
Sightsavers' Guinea-Bissau NTD program, budget summary, 2015-201733
Activity Existing Plans34 Scale-up Plans35 Total % Total
Monitoring and evaluation36 $247,476$210,121 $457,597 18% Drug collection and delivery37$353,046 $97,515$450,561 17%
Training38 $114,131$277,398 $391,529 15% Supplies39$168,011 $79,677$247,688 10%
Trichiasis camps40 $145,799$0 $145,799 6% Information, education, and communication41$58,572 $67,242$125,814 5%
Environmental improvement42 $68,812$0 $68,812 3% Salaries43$15,162 $0$15,162 1%
LF elimination scale up44 $0$700,000 $700,000 27% Total$1,171,010 $1,431,952$2,602,962 100%
## Does it work?
Mass administration of deworming drugs has been independently studied in rigorous trials and found to be effective.
We have limited evidence about whether Sightsavers' deworming programs have reached a large proportion of children targeted. In summary:
• Sightsavers reports that its programs have delivered tens of millions of deworming treatments. We are uncertain about the reliability of this data.
• Sightsavers has shared examples of its monitoring and evaluation (M&E) data from NTD programs with us. We do not know how consistently Sightsavers evaluates its deworming mass drug administrations (MDAs). We believe there are important limitations to the NTD evaluations that we have seen so far.
• We find it plausible that Sightsavers’ involvement may increase the probability that a deworming program starts up or scales up, but we have not seen evidence supporting this.
### Independent evidence of program effectiveness
This review focuses on Sightsavers' work to scale up mass combination deworming, which we discuss extensively on another page. There is a very strong case that mass deworming is effective in reducing infections. The evidence on the connection between reduced infections and positive quality-of-life impacts is less clear, but there is a fairly strong possibility that deworming is highly beneficial.
### What is the impact of the deworming programs that Sightsavers supports?
Sightsavers told us that it uses a variety of M&E tools to assess whether drugs are being delivered in NTD MDAs, including:45
• Government-reported statistics on treatment numbers and coverage
• Independent treatment coverage surveys
• Impact assessments at sentinel sites to measure changes in the prevalence and intensity of infections
• Spot checks of data aggregation
We are uncertain to what extent Sightsavers consistently uses these tools across all (or most) of the NTD MDAs it supports. To date, we have only asked Sightsavers to share M&E reports from countries where it would like to add or scale up deworming, rather than all of the studies it has completed. Sightsavers shared one coverage survey, three impact assessments (of onchocerciasis and trachoma programs), and a spot check report with us.
#### Reported treatment numbers and coverage data
Sightsavers reports that from 2011-2014 its programs delivered about 68 million deworming treatments across three programs in Cameroon, Nigeria, and Tanzania (see tables above). We do not have a clear understanding of the process for reporting this data. Sightsavers shared with us a coverage survey (discussed below) which found lower coverage rates than were reported by drug distributors.46
#### Coverage survey
Sightsavers shared with us a coverage survey that aimed to "validate the reported coverage of recent MDA campaigns (2014) for Onchocerciasis, Lymphatic Filariasis (LF), Schistosomiasis, Soil-transmitted Helminths (STH) and Trachoma in Zamfara State."47 The survey found that the coverage rates (the percentage of the population targeted who took the relevant drugs) were 71% for onchocerciasis and LF48 and 77% for trachoma49; it appears that the coverage rate was 68% for schistosomiasis,50 though there appears to be an inconsistency in how this is reported (details below). The survey identified several areas with very low coverage rates,51 which increases our confidence in the survey methods. For comparison, we discuss coverage surveys of SCI-supported deworming programs here.
Survey methodology is discussed in the survey report, including sample selection, interval between drug distribution and coverage survey (up to three months), and usage of drug samples and packaging to aid in recall (additional details in this footnote).52 Some issues with the drug distribution are discussed in the coverage survey report, including poor record keeping, drug diversion, selective treatment by community distributors, and overlap with Ramadan (additional details in this footnote).53 The survey was conducted prior to "mop-up" activities (attempts to treat those who were missed by the distribution); results from the survey were used to determine where mop-up activities were needed.54
Limitations of this study as evidence of Sightsavers' impact include:
• We have not seen details of the role Sightsavers had in supporting the MDAs evaluated in this survey.55 Sightsavers told us that it supported all NTD treatments in Zamfara (the state in which the survey was conducted), with funding from DFID.56
• There appear to be inconsistencies in schistosomiasis coverage data.57
• The survey was conducted in two local government areas (LGAs) that were purposefully selected to address potential programmatic concerns, rather than to be representative of the MDA in Zamfara as a whole.58
• The distribution of drugs for schistosomiasis was not yet fully completed at the time of the survey, so the survey may underestimate coverage for the schistosomiasis part of the MDA.59
• Survey respondents were asked to recall drugs taken up to three months prior.60
• The report does not explain how "don't know" responses were recorded or how often answers were given by a proxy on behalf of someone else.61
#### Impact assessments
Sightsavers shared three impact assessments that reported on the impact of MDAs on the prevalence of onchocerciasis or of trachoma. The studies suggest that past treatment programs have been successful, but the studies have a number of limitations, particularly because it is unclear whether these programs are representative of Sightsavers' work and whether changes can be attributed, at least in part, to Sightsavers' work.
• A 2008 impact assessment of long-term (1991-2008) treatment of onchocerciasis in Kaduna, Nigeria:62 Baseline data from 1987 indicated a median onchocerciasis prevalence of 52%. In 2008, after 15-17 years of treatment, onchocerciasis prevalence had dropped to 0% in all surveyed communities (none of the 3,703 individuals screened were infected).63 Limitations of the study include: (1) the difficulty of attributing the change to Sightsavers' work given the possibility that other concurrent work or improvements during the long interval between baseline and follow up may have impacted onchocerciasis prevalence, (2) selection of study areas based on criteria that may be correlated with larger changes in prevalence, and (3) a high non-participation rate among targeted respondents.64
• A 2015 impact assessment of long-term (1993-2015) treatment of onchocerciasis in fifteen villages in three districts of Uganda:65 District-level onchocerciasis prevalence ranged from 0.8% to 5.5% at the time of the assessment.66 The authors noted that baseline prevalence data (from 2007) was available for only two of the fifteen villages,67 but they did not provide the baseline data from the two villages in the report. The study concluded that the MDAs performed well.68 Other limitations of the study are discussed in this footnote.69
• An impact assessment on three rounds of trachoma treatment (2009-2012) in three regions of Guinea-Bissau:70 Sightsavers has told us that it has run a trachoma program in Guinea-Bissau since 2011; we are uncertain about the details of Sightsavers' trachoma support in these three regions, and about the role other organizations play in this program.71 This study concluded that after three years of MDA, trachoma prevalence in children 1-9 years of age was below 5% in all three regions, indicating that further treatment was not needed.72 The study did not include baseline data (or state how much prevalence dropped). However, Sightsavers has elsewhere reported what may be comparable 2005 baseline data for these three regions, which suggests that trachoma prevalence decreased from 20-30% to 1-5% in these regions.73
We have not seen impact assessments of Sightsavers' schistosomiasis or STH programs specifically.74
#### Spot check of data aggregation
Sightsavers told us that each year for a random sample of projects, it conducts spot checks on whether aggregated data (e.g. district-level data) matches original tallies for figures such as number of treatments delivered or people trained.75 Sightsavers has shared with us its spot check procedure and an example of a spot check report.76 This example reports finding essentially no error in the aggregation of reported number of people treated for onchocerciasis in DRC.77 We believe checks like this could help to uncover some data aggregation errors but may miss errors at other steps of the reporting process. We have not yet asked Sightsavers how often these spot checks uncover concerns, nor how Sightsavers handles any concerns.
### What impact does Sightsavers have on the programs it supports?
We are uncertain about the impact that Sightsavers has on the deworming programs that it supports. It seems plausible that Sightsavers increases the likelihood that deworming mass drug administrations occur, or improves their quality, but we have not seen direct evidence of this.
### Are there negative or offsetting impacts of the programs Sightsavers supports?
We have not looked closely into negative or offsetting impacts of Sightsavers' programs. We discuss possible negative or offsetting impacts of other deworming programs in our intervention report on deworming, and in previous reviews of deworming charities here and here. From the information we have reviewed, we do not have any significant concerns unique to Sightsavers.
## What do you get for your dollar?
Deworming is potentially very cost-effective.
We have not yet evaluated the cost-effectiveness of Sightsavers’ deworming programs in a way that is comparable to our top charities. In this section, we consider two factors that we expect are important to cost-effectiveness and relatively easy to understand: the cost per deworming treatment, and worm prevalence. We have limited data on these factors.
### What is the cost per deworming treatment?
Sightsavers shared with us estimates of the cost to deliver additional NTD treatments in Nigeria ($0.08-$0.10 in 2015-2017)78 and additional deworming treatments in Guinea-Bissau ($0.57-$0.92 in 2016-2017).79 We have not seen cost per treatment data (or sufficient information to calculate this) for the other countries where Sightsavers works. Sightsavers' estimates are not directly comparable to our cost per treatment estimates for other deworming charities we have reviewed because:
• For other charities, we have calculated the average cost of the program. Sightsavers' estimates are marginal costs: the additional cost to add treatments to existing NTD programs.
• Sightsavers' estimates are based off of projected, not actual, costs.
• We do not know whether Sightsavers’ program costs include costs covered by governments, or costs covered by other partner organizations. For example, Sightsavers notes that deworming scale up in Guinea-Bissau will be supported by the WHO country office; we are unsure whether this support is included in the reported costs.80
• The budget estimates we have seen do not include a portion of Sightsavers' central or regional costs.81
• These costs do not include the value of donated drugs.82
In addition, we have not vetted Sightsavers' estimates and have not yet asked Sightsavers why there is a significant difference in the estimates for Nigeria and Guinea-Bissau.
### What is the prevalence and intensity of schistosomiasis and STH where Sightsavers plans to support deworming?
In general, mass deworming programs treat everyone in a targeted demographic, regardless of whether each individual is infected (more). Because of this, the benefits (and therefore the cost-effectiveness) of a program are highly dependent on the baseline prevalence of worm infections. Sightsavers has shared prevalence data with us for areas in Nigeria and Guinea-Bissau where it has delivered or is considering delivering deworming treatments. This data indicates that prevalence of worm infections may be as high or almost as high as in the studies which constitute our key evidence about the impact of deworming (see below). However, we have some concerns about the data Sightsavers has provided:
• We do not know how this data was measured.
• The data for Guinea-Bissau is from 2005 and earlier.83 We do not know when the data for Nigeria was measured.84
• For some states in Nigeria, Sightsavers provided prevalence data with wide ranges which we do not know how to interpret.
For these reasons, we do not have high confidence in the prevalence data we have seen (data in footnote).85
## Is there room for more funding?
### Is funding a constraint to scaling up deworming programs?
Our information about Sightsavers' funding needs is now out of date because we took significantly longer than expected to publish our interim review. In 2015, Sightsavers had a funding gap of approximately $2.0 million for the scale up of deworming in Nigeria and Guinea-Bissau in 2015-2017.86 It could potentially use additional funding in other countries. Sightsavers has told us that there are benefits of coordinating NTD programs and (as and when funding permits) it is hoping to increasingly accommodate government requests for support for programs that coordinate treatment of several NTDs.87 Sightsavers has shared with us scale-up budgets for Nigeria and Guinea-Bissau, as detailed above. In Nigeria, Sightsavers projects costs of about$560,000 to scale up deworming in four states, and about $670,000 to expand to an additional state (Benue); a total of$1.2 million is expected to cover three years of scale-up work.88 Sightsavers has identified some sources of potential funding for programs in three of these states.89 In May 2015, the UK government committed an additional $2.6 million (£1.7 million) in funding towards the Northern Nigeria Integrated NTD Programme.90 Our understanding is that this is a distinct program from the proposed expansion represented in the Sightsavers budget we have reviewed.91 In Guinea-Bissau, Sightsavers projects costs of about$730,000 to scale up deworming over two years (in addition to about $700,000 for scaling up LF elimination activities).92 Sightsavers told us in May 2015 that it had not identified potential funders of for its deworming work in Guinea-Bissau.93 ### Are funds for deworming scale up fungible with Sightsavers' other funds? Funding to Sightsavers could (at least nominally) be restricted to deworming.94 However, it is possible that these funds would be fungible with other Sightsavers programs. In other words, receiving funds restricted to deworming might cause Sightsavers to reallocate other funds it has available, so that the additional funding would effectively expand programs other than deworming. #### Fungibility with unrestricted funding It seems likely that funding for deworming programs is at least partially fungible with Sightsavers’ unrestricted funding. Sightsavers' overall organizational budget for 2015 was about$100 million, and at the beginning of 2015 it held about $13.5 million in unrestricted funding reserves (about$4.4 million more than its target of about $9.1 million).95 Sightsavers has told us that it would consider using unrestricted funding for deworming, and also told us that if GiveWell provided funding restricted to deworming, Sightsavers might fund the other portions of an integrated NTD program with unrestricted funding.96 We do not have a good understanding of how Sightsavers allocates its unrestricted funding across the organization.97 It seems possible that Sightsavers could use unrestricted funding to fill its deworming funding gaps. It is also possible that additional funding for deworming would leverage funds from other funders or cause Sightsavers to allocate additional unrestricted funds to integrated NTD programs. For example, Sightsavers has told us that its proposed expansion into Benue, Nigeria, is conditional on receipt of enough funding for a full integrated NTD program in that state, and that it would try to raise that funding from other funders and might fill the gap with unrestricted funding.98 ### Will other sources fill the deworming funding gaps? We do not know whether any external funders are likely to fill some or all of the existing funding gaps for deworming programs. Sightsavers has received funding for support of NTD programs from several sources.99 As we have previously written, there appears to be a substantial unmet need for deworming globally. ## Major questions for further investigation We have spent significantly less time investigating Sightsavers and have substantially less insight into Sightsavers' activities, spending, and track record than we do for our current top charities. As such, we have a number of outstanding, high-level questions about its work supporting deworming. We may also explore similar questions about its other NTD programs. We hope to learn significantly more, answer many of these questions, and publish updates as our views evolve. • How does Sightsavers use information about worm prevalence and program capacity to decide which programs or regions to support? • What is Sightsavers' role in the deworming programs it supports? • How has Sightsavers spent funds in the past? We would like to see data on spending across (a) the organization overall, (b) NTD programs, and (c) deworming specifically. • In Sightsavers-supported regions, would deworming occur absent Sightsavers’ support? (E.g. supported by other organizations?) • Does Sightsavers have additional coverage and sentinel site studies it can share? Do these studies provide high-quality information on the impact of the programs studied? How representative are the studies of Sightsavers’ programs overall? • How does Sightsavers plan to monitor and evaluate its deworming programs in the future? • What is the average worm prevalence in areas where Sightsavers plans to support deworming, and how reliable is this data? • How are reported treatments counted and how reliable is this data? • What are the total costs of Sightsavers' deworming and/or NTD programs, including costs covered by other funders? • What are Sightsavers' total deworming and NTD funding gaps? • Was Sightsavers able to fill the funding gaps for deworming programs in Nigeria and Guinea-Bissau that it told us about in early 2015? • How does Sightsavers allocate unrestricted funding across the organization? Why hasn't it used unrestricted funding to fill its deworming funding gaps? ## Sources Document Source Bailey 2013 Source Checklist of documents sent Source DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014 Source GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017 Source GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017 Source GiveWell's non-verbatim summary of a conversation with Sightsavers staff, March 19, 2015 Source GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015 Source Lakwo et al. 2015 Unpublished Oanda's historical exchange rates calculator Source Sightsavers Annual Report 2013 Source (archive) Sightsavers Annual Report 2014 Source (archive) Sightsavers Benue project overview Source Sightsavers brief on the school deworming project in Nigeria Source Sightsavers Guinea-Bissau NTD prevalence maps Source Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015 Source Sightsavers Nigeria deworming prevalence and estimated cost Source Sightsavers Nigeria NTD program budget accompaniment Source Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015 Source Sightsavers Nigeria NTD support map Source Sightsavers operational budget, 2015 Source Sightsavers organizational expenditure and outputs, 2014 Source Sightsavers quality standards manual, public excerpts Source Sightsavers spot check process and methodology Source Sightsavers spot check report, DRC Source Sightsavers website, Extra funding in Nigeria, May 2015 Source (archive) Sightsavers website, Our history Source (archive) Sightsavers, email to GiveWell, March 8, 2016 Unpublished Summary of documents Source Tekle et al. 2012 Source (archive) UN Country Profile: Guinea-Bissau Source (archive) • 1. "It was on 5 Jan 1950 that the doors of the British Empire Society for the Blind, as we were known back then, opened for the first time. [...] As well as being the decade of teddy boys, petticoats and rock and roll, the 1950s were when Sightsavers made our first groundbreaking achievement in the fight against avoidable blindness. In 1953 a survey was conducted by Dr Freddie Rodger in West Africa. It showed that the majority of blindness was actually preventable, and led to a pioneering disease control programme for onchocerciasis, also known as river blindness." Sightsavers website, Our history. • 2. • "The mandate of the organization and Royal Charter was changed in 2009 to accommodate the non-blinding NTDs. It was only in 2010 that Sightsavers recorded its first non-blinding neglected tropical disease treatment (LF, followed by schistosomiasis and STH in 2011) in its output statistics." GiveWell's non-verbatim summary of a conversation with Sightsavers staff, March 19, 2015, p. 3. • See also the overview of treatments here, which shows deworming treatments recorded in 2011 but not 2010. • 3. • 4. "Nigeria is the country in which Sightsavers began its deworming programs and has been the flagship country for the organization’s integrated approach. Since 2011, when Nigeria’s deworming treatment program started in Zamfara and Sokoto states, Sightsavers has provided a total of about 14 million treatments per year for STH, and around 3 million treatments per year for schistosomiasis." GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, p.6. • 5. "Sightsavers has been working in Nigeria for over 40 years, in partnership with the Ministry of Health. During this time we have supported eye care programs to address conditions amongst the population including cataract and glaucoma, human resource development programs to increase the number of practising ophthalmologists, cataract surgeons and eye health professionals, and widespread NTD programs focused on the elimination of five NTDs." Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 1. • 6. "Q1b. Results from recent years' activities, including # of people treated for each disease and amount spent" in Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, pp. 2-3, shows schistosomiasis and STH treatment in Sokoto state, Kwara state, Kogi state, and plans for a 2015 distribution in Kebbi state. • 7. Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 5, shows funding from the Children’s Investment Fund Foundation (CIFF) for mapping of schistosomiasis and STH in 14 states of northern Nigeria. • 8. Sightsavers' annual reports break down spending by country and by broad program area: "Health - Eye Care," "Health - Mectizan," "Education," "Social Inclusion," and "Policy and Research." Sightsavers Annual Report 2013, p. 40; see also Sightsavers Annual Report 2014, p. 64. • 9. • 10. • “2015 income is planned at £64m” (Sightsavers Annual Report 2014, p. 45), or approximately$97 million (converted to USD using the average historical exchange rate for January-May 2015 (1.5166 GBP/USD) as found on Oanda's historical exchange rates calculator). This is close to the projected income in the Sightsavers operational budget, 2015; however, it is still noticeably larger than total expenditures excluding gifts in kind in 2014 and 2013.
• The 2015 budget does not include gifts in kind. Sightsavers, email to GiveWell, March 8, 2016.
• Total expenditures excluding gifts in kind in previous years were $84 million (£51 million) in 2014 and$75 million (£47 million) in 2013. We have converted to USD using the average historical exchange rates for 2014 (1.6474 GBP/USD) and 2013 (1.5644 GBP/USD) as found on Oanda's historical exchange rates calculator. Summary from Sightsavers Annual Report 2014, p. 52 (calculations of the USD figures here are based on unrounded GBP figures in the source—not the rounded GBP figures in this table—multiplied by the applicable exchange rates from Oanda's historical exchange rates calculator):
Category 2014 2013
Charitable expenditures $280 million (£170 million)$296 million (£189 million)
Gifts in kind $221 million (£134 million)$238 million (£153 million)
Charitable expenditures (excluding gifts in kind) $59 million (£36 million)$58 million (£37 million)
Total expenditures $305 million (£185 million)$313 million (£200 million)
Total expenditures (excluding gifts in kind) $84 million (£51 million)$75 million (£48 million)
Charitable expenditures as % of total expenditures (all excluding gifts in kind) 70% 77%
• Sightsavers told us that these documents reflect expected growth in Sightsavers' income and expenditures in 2015, driven by increased fundraising and large grants. Sightsavers, email to GiveWell, March 8, 2016. See also Sightsavers Annual Report 2014, p. 45: "Sightsavers plans to maintain increased levels of fundraising investment in 2015 as a continuation of the strategy to grow the base of committed givers across all fundraising markets. The aim for 2015 is to grow voluntary funding by £5m. Growth is also expected in institutional funding within the year as several of the large NTD grants hit peak levels."
• Sightsavers told us that the figures for 2013, 2014, and 2015 are comparable. Sightsavers, email to GiveWell, March 8, 2016.
• 11.
In Nigeria, Sightsavers’ planned NTD budget (which includes deworming) is approximately $1.5 million over three years (2015-2017). Sightsavers' proposed expansion of NTD activities in Nigeria projects additional spending of$1.2 million over those three years. (GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017, p. 1. Planned work: "Budget - NTD work (planned)" sum of cells D45, E45, F45. Proposed expansion: "Budget - NTD work (additional)" cell F48 plus "Budget - Benue (additional)" cell F26.) We're unsure how the size of the Nigeria NTD program compares to NTD budgets in other countries in which Sightsavers works. Nigeria is "the country in which Sightsavers began its deworming programs and has been the flagship country for the organization’s integrated approach" (GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, p. 6). However, in 2014, only about 9% (1.6 million of 18.7 million) of the deworming treatments that Sightsavers supported were in Nigeria (see table in the following section).
• 12.
See Sightsavers organizational expenditure and outputs, 2014, "By Country" row 10 for "Total no. of NTD Treatments (NOT UNIQUE PEOPLE)", total in cell BY10. For STH treatments, see row 15, which has entries in cells I15, AS15, BI15, and a total in cell BY15. For schistosomiasis treatments, see row 16, which has entries in cells I16, AS16, BI16, and a total in cell BY16.
• 13.
2010-2013 data from Sightsavers Annual Report 2013, p. 12, and 2014 data from Sightsavers organizational expenditure and outputs, 2014, "By Country". For 2014 trachoma treatments we take the sum of cells BY11 and BY12, which matches the number of trachoma treatments reported in the Sightsavers Annual Report 2014, p. 19.
• 14.
Sightsavers organizational expenditure and outputs, 2014, "By Country". For STH treatments, see cells I15, AS15, and BI15. For schistosomiasis treatments, see cells I16, AS16, and BI16.
• 15.
• 16.
The cost of $2.4 million (£1.4 million) for Nigeria's NTD program in 2014 is reported in Sightsavers organizational expenditure and outputs, 2014, "Spend by theme and country", cell H27. We have converted to USD using the average historical exchange rates for 2014 (1.6474 GBP/USD) as found on Oanda's historical exchange rates calculator. Sightsavers' Deworming Treatments by State, Nigeria 2014 Nigerian State STH Schistosomiasis Cost (GBP) Cost (USD) Kebbi 0 0 £60,000$98,000
Kogi 87,000 65,000 £40,000 $66,000 Kwara 216,000 342,000 £47,000$78,000
Sokoto 86,000 169,000 £43,000 $71,000 Total 389,000 576,000 £190,000$313,000
• 17.
• 18.
"Many of Sightsavers’ MDA programs provide a hybrid of school-based treatment (in which drugs are administered at school sites) and community-based treatment (in which drug distributors travel from house to house in a community)." GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, p. 7. In Guinea-Bissau, prospective STH treatments in 2015-2017 target "those reached through LF MDA and school based de-worming." GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Outputs". In Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 10, Sightsavers notes that "Non-enrolled school age children would be invited to visit schools on de-worming days, or would be reached through house to house activities."
• 19.
• 20.
"Because the scale of work in Africa is so great, Sightsavers works with other organizations and frequently manages a group of partners. Sightsavers also implements treatment in some areas, but it has developed particular expertise in managing partnerships and coalitions." GiveWell's non-verbatim summary of a conversation with Sightsavers staff, March 19, 2015, p. 1.
• 21.
GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015: “Tanzania – Sightsavers works in partnership with SCI and others” (p. 6); “Sightsavers has a strong relationship with SCI, and the two work closely together on many programs and sometimes in the same countries. One such country is DRC…. Both organizations also have a presence in Côte d’Ivoire, where Sightsavers’ NTD adviser … works closely with SCI’s adviser. In both DRC and Côte d’Ivoire, the ministries of health usually hold an annual review and planning meetings, which Sightsavers will attend along with SCI. Sightsavers and SCI use this opportunity to discuss overall plans and budgeting for each country” (pp. 7-8).
• 22.
The budget in Nigeria includes only a subset of Sightsavers' NTD work in the country. Sightsavers, email to GiveWell, March 8, 2016. The budget includes the states of Kebbi, Kogi, Kwara, Sokoto, and Benue (GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017); Sightsavers is also involved in the DFID-funded UNITED project, which works in Katsina, Kano, Zamfara, Kaduna, and Niger, and the CIFF-funded prevalence mapping across northern Nigeria (Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 5). A map Sightsavers shared with us marks Sightsavers as active in Kebbi, Kogi, Kwara, Sokoto, Zamfara, and Kaduna. Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 3. Additionally, the budget for expansion to Benue appears to include staff costs, while the budgets for planned and scale-up work in Kebbi, Kogi, Kwara, and Sokoto do not appear to include staff costs. GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017. Budgets in Nigeria do not appear to include costs of drug distribution beyond the cost of collecting drugs from the Central Medical Store and delivering them “to the State and LGAs,” while budgets in Guinea-Bissau include additional line items related to drug distribution. GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017; GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017. We are uncertain whether the budget from Guinea-Bissau is comprehensive.
• 23.
GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017, "Budget - NTD work (planned)", sum of cells D45, E45, F45.
• 24.
GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017. Three-year scale-up cost is found in "Budget - NTD work (additional)", cell F48. Three-year cost of expansion to Benue is found in "Budget - Benue (additional)", cell F26.
• 25. GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017. Note that we have made some judgments in categorizing expenses to facilitate consolidating different budgets. In addition, calculations of the totals in this table are based on unrounded dollar figures in the source, so the sum of the rounded individual figures here may not equal the total in every case.
• 26. The Sightsavers budget lines which we have here categorized as "Training" are: GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 5, 16, 26, 36, labeled along the lines of "Conduct training of Health workers, Community Volunteers and School teachers"; "Budget - NTD work (additional)" rows 3, 13, 23, 33, labeled "Capacity Building for teachers and healthworkers"; and "Budget - Benue (additional)" row 10, labeled "Capacity Building of Health Staff, Community Volunteers and Teachers".
• 27. The Sightsavers budget lines which we have here categorized as "Monitoring and evaluation" are: GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 9-11, 20-22, 30-31, 40-42, labeled "To provide Supportive supervision, improve therapeutic (80%), geographic (100%) coverage and ensure quality data management", including the following ”Specific Objectives”: "To conduct Planning Meeting with relevant stakeholders for NTDs Programme Implementation", "Review programme implementation, collate outstanding data", "Monitoring and supervision", and "Treatment coverage Survey"; "Budget - NTD work (additional)" rows 7, 17, 27, 37, 43, labeled "Monitoring, Supervision and Evaluation of the activities" and "Monitoring and Supervision"; and "Budget - Benue (additional)" rows 7, 11-18, including activities related to planning and review meetings, supervision and monitoring, and surveys and assessments.
• 28. The Sightsavers budget lines which we have here categorized as "Information, education, and communication" are: GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 6-8, 17-19, 27-29, 37-39, labeled "To Improve awareness, community participation and treatment compliance", including the following “Specific Objectives”: "To Produce Information, Education and Communication Materials and Management Information system forms", "To hold sensitization meeting with District Heads to improve community support", and "Social Marketing Campaigns"; "Budget - NTD work (additional)" rows 5-6, 8, 15-16, 18, 25-26, 28, 35-36, 38, labeled "Mobilisation and Sensitization of Schools, School Education Boards and Parents", "Production of Information, Education and Communication (IEC) Materials", and "Production of Management Information System (MIS) forms"; and "Budget - Benue (additional)" rows 4-6, 8, labeled "Advocacy visits Key Policy makers", "Information Education and Communication (IEC) Production", "Management Information System (MIS) Production", and "Sensitisation and Mobilisation of Community leaders and community members". For scale-up and expansion plans, we categorize activities related to Management Information System (MIS) forms as part of information, education, and communication because Sightsavers categorizes them as such in "Budget - NTD work (planned)": "To Produce Information, Education and Communication Materials and Management Information system forms".
• 29. The Sightsavers budget lines which we have here categorized as "Supplies" are: GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 13, 23, 32, 43, with labels indicating costs of office supplies, laptops, and vehicles; "Budget - NTD work (additional)" rows 9, 19, 29, 39, 44, labeled "Production of Measuring Sticks" and "Vehicle Purchase for National coordination"; and "Budget - Benue (additional)" rows 20-23 and 25, including office costs and vehicles.
• 30. The Sightsavers budget lines which we have here categorized as "Drug transportation" are: GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 4, 15, 25, 35, labeled "To collect drugs from the Federal Central Medical Stores and deliver them to the State and LGAs in preparation of treatment" and “To conduct Planning Meeting with relevant stakeholders for NTDs Programme Implementation”; "Budget - NTD work (additional)" rows 4, 14, 24, 34, labeled "Drug Collection from FMOH Medical Store"; and "Budget - Benue (additional)" row 9, "Drugs Collection & Deliver from Federal Medical Store to States".
• 31. GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - NTD work (planned)" includes costs in rows 12 and 44, labeled "To provide support for Project and conduct camps for Lid Surgeons for TT cases".
• 32. The Sightsavers budget lines which we have here categorized as "Salaries" are: GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017: "Budget - Benue (additional)" rows 19 and 24: "Staff Cost" and "Programme Officer to be stationed in the state". Note that these budgets do not appear to include salary costs for existing or scale-up plans in states other than Benue.
• 33. GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017. Note, we have made some judgments in categorizing expenses to facilitate consolidating different budgets. In addition, calculations of the totals in this table are based on unrounded dollar figures in the source, so the sum of the rounded individual figures here may not equal the total in every case.
• 34. This column reports projected costs from two projects: the Guinea-Bissau onchocerciasis & LF elimination project and the Guinea-Bissau trachoma elimination project. The trachoma elimination project accounts for about 77% of these costs. ($905,000 compared to$266,000 for the onchocerciasis & LF elimination project.) See GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (planned)".
• 35. This column reports projected costs from the Guinea-Bissau Schistosomiasis/STH and LF scale-up projects and includes costs for national coordination & reporting. GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (additional)".
• 36. The Sightsavers budget lines which we have here categorized as "Monitoring and evaluation" are: GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 8-10, 14-17, 26-35, with labels indicating activities related to planning and review meetings, supervision and monitoring, surveys and evaluations, and data management; and "Budget - NTD work (additional)" rows 6, 10, 16, 18, 23, with labels indicating activities related to surveys, monitoring and evaluation, supervision, data management, review and planning meetings, and national monitoring and supervision.
• 37. The Sightsavers budget lines which we have here categorized as "Drug collection and delivery" are: GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 11 and 42, labeled "Mass drug distribution of ivermectin" and "mass drug distribution of zithromax" (we are uncertain about what costs are represented by these budget lines, and have assumed that they represent costs similar to the more detailed drug distribution line items in "Budget - NTD work (additional)"); and "Budget - NTD work (additional)" rows 7, 14, 17, 19, labeled "Drug Collection from MOH Medical Store", "Management of SE (Side-effects)", "Drug distribution (transport for CDD)", and "Drug distribution ( snack for children for PZQ taking)*".
• 38. The Sightsavers budget lines which we have here categorized as "Training" are: GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 6-7, 43, with labels indicating various types of training; and "Budget - NTD work (additional)" rows 4-5, with labels indicating training of teachers, health workers, and community drug distributors.
• 39. The Sightsavers budget lines which we have here categorized as "Supplies" are: GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 12-13, 23, 25, with labels indicating costs of equipment, vehicles, and consumable supplies; and "Budget - NTD work (additional)" rows 12-13, 15, 24, with labels indicating costs of measuring sticks, school/community registers, office supplies, and vehicles.
• 40. In GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (planned)", we have categorized rows 36-38 (within "Trichiasis operation camps and screenings") as non-education costs of trachiasis camps (education costs related to Trichiasis camps are included in the “Information, education, and communication” category in the next row of the table).
• 41. The Sightsavers budget lines which we have here categorized as "Information, education, and communication" are: GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017: "Budget - NTD work (planned)" rows 4-5, 24, 39-40, labeled "Education for Health, Awareness, Advocacy & Engagement", "Communications", "Community sensitization", and "IEC activities (interviews and songs on local radio & tv)"; and "Budget - NTD work (additional)" rows 8-9, 11, labeled "Mobilisation and Sensitization of Schools/communities, School Education Boards and Parents", "Production of Information, Education and Communication (IEC) Materials", and "Production of Management Information System (MIS) forms/Photocopies".
• 42. This category consists of GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (planned)" row 41, labeled "Rehabilitation of water points and construction of latrines".
• 43. The Sightsavers budget lines which we have here categorized as "Salaries" are: GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017: "Budget - NTD work (planned)" row 22, labeled "Salaries"; note that "Budget - NTD work (additional)" does not appear to include additional salary costs.
• 44. This category consists of GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (additional)" row 28, indicating the total costs for the scale up of LF elimination activities.
• 45.
GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, pp. 1-4. Sightsavers also shared a version of Sightsavers quality standards manual, public excerpts which included details of the metrics Sightsavers plans to track for its deworming programs. Sightsavers requested that we not share these sections publicly because they were not yet finalized.
• 46.
DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014
• The coverage survey protocol explains that during a distribution, "The dosage and quantity of drugs given is recorded on the register or tally sheet. This data is then used to calculate the population coverage of MDA, however there is often issues with the data including poor census data and inaccuracies in recording the data." (survey protocol, p. 3; p. 43 of pdf)
• "A comparison of the surveyed Oncho/LF data with reported coverage in Bugundu LGA shows that Asako, Danguro and Dogon daji had reported treatment coverages that were very close to the surveyed coverages obtained (Annex 3). This shows that a good reporting system is in place in these communities and the drug distributors should be commended. However, in Birnin Mallam, Kaikai, Ka Ida, Nahuce, Tazame, Yar Labe and Gada communities, reported coverages were higher than surveyed coverages." (p. 23)
• "Reported coverage for Zithromax was consistently much higher that obtained from survey in all communities. A survey coverage of 46.4% was obtained at Tungar Gobirawa, yet coverage from health system records indicated a 97% coverage. Again, this calls for close supervision of drug distributors as it also indicates poor recording keeping skills." (p. 24)
• "There was evidence of poor record keeping at all levels of programme implementation. It was most serious at the community level; and this unfortunately is the primary source of data for information collated at both LGA and State levels. Treatment records kept at community level were not properly entered neither was there evidence of census been carried out by the drug distributors. In Nahuce community only the 616 individuals treated were recorded in the treatment register and in Yar Labe the drug distributor could not make available the community treatment register for vetting. He claimed he had none. This community recorded 1.2% in the survey coverage, yet Local Government records should a high coverage of about 80%." (p. 28)
• 47.
• 48.
"Household surveys involving 2603 sample of eligible population showed that 71.3% received treatment...." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 14. See also Table 4 on p. 18.
• 49.
"77% of the respondents in Shinkafi reported that they swallowed the drugs...." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 22. See also Table 9 on p. 22.
• 50.
"[I]t is observed that the overall treatment coverage was 68.4% of the children eligible for treatment (ie ages 5-15 years) were treated." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 20. See also Table 7 on p. 21.
• 51.
“In Bungudu LGA where only 58.7% had received treatment as against WHO/APOC threshold standard of 75%, treatment coverage ranged from as low as 1.8% in Yar Labe to 98.6% in Hommawa community (Figure 2). This LGA was marred with partial treatments within communities and this accounts for the low coverages (Figure 2) recorded in Yar Labe (1.8%), Kortokoshi (2.2%), Birnin Mallam (16.2%), Gada (31.6%), Nahuce (44.5%) and Ka Ida (48.1%).” DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, pp. 14-15.
• 52.
Highlights of the survey methodology:
• The regions targeted by the coverage survey (two out of six local government areas (LGAs) covered in the distribution) were purposefully selected based on the following criteria. We are uncertain about further details of how the two LGAs were selected—e.g. if they were selected because post-MDA records indicated particularly low or particularly high coverage.
"The survey will be conducted in Shinkafi and Bungudu LGAs. These LGAs were selected for one or more of the following reasons:
• The post MDA records show particularly low or high coverage attained at the LGA level, or a large range of coverage attained at the community level - with for example a concerning number of communities with notably low coverage.
• There are suspected issues with the MDA records or census data that need to be verified e.g. poor population data or large population movements around the time of the MDA, discrepancies between the drug store records/logs and the community records or large variations in doses given year to year.
• Representation of the different combinations of drugs administered"
DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, survey protocol, p. 4 (p. 44 of the pdf).
• The sample size was determined based on several assumptions: "Assuming an estimated coverage of 80%, 95% confidence limit, a design effect (1) of 4, non-response of 12% and presuming that an average household size of eligible school [age] children of 5 (for Schistosomiasis distribution), a minimum sample size of 246 households was required to be sampled per LGA, but taking into account the design effect and the non-response a total of 1,101 individuals had to be sampled to get the required sample size. This way a minimum of 16 clusters, 14 households per cluster were surveyed in each LGA (Table 1)" DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, pp. 8-9.
• “The study teams were selected from individuals who were not involved in the MDA campaign. Each team was made up of a supervisor, two enumerators who worked closely with a local guide. […] For quality control purposes, there was a consultant designated as survey co-ordinator, with overall responsibility for the conduct of the survey and team supervisors. Each team had a supervisor who stayed with the interviewers all through the survey in the communities to ensure the quality of the data being collected.” DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 12.
• Selection of households to interview "followed a two-stage cluster sampling method, with the primary cluster (primary sampling unit), the village and the secondary cluster, the household." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 9. See also the further description on pp. 10-11, and the survey protocol on pp. 40-54 of the pdf.
• "Once in a household, the purpose and procedure of the survey explained and the household head was requested to provide verbal consent for his household to take part in the survey. Once consent was given, the names of all individuals who are permanently resident in the household were written down in the questionnaire (Annex 3) and the enumerator proceeded with collecting information as outlined in Annex 3. Where possible the eligible individuals were asked if they swallowed the drug and the person was not available, another household member or caregiver gave information on their behalf. Primary caregivers responded on behalf of children aged 1-10 years old, except where drugs were given in a school based distribution. In this case the children themselves were asked if they received the drugs at school. Samples of the drugs and the packages used during the recent MDA were shown to respondents to assist recall. The MDA schedule given in Annex 1 outlines the different times of the various drug distributions to avoid potential drug interactions. Therefore, the period between the survey and the distribution of the first set of drugs (for Onchocerciasis and Lymphatic Filariasis) was about 3 months, which was significantly longer than for Trachoma. The implications this might have on recall was an important consideration and so, in order to reduce errors introduced through recall bias, the survey team ensured drug samples (and the packaging of the drugs were given in packages) of the different drugs distributed were shown to each respondent during discussions. All individuals listed in the household were asked about each drug in question. If they are not eligible this was recorded on the questionnaire sheet either as not eligible or in cases where the intervention was not applicable (e.g Praziquantel was only administered to school) to the individual it was recorded as not applicable." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p 11. See also the questionnaire in Annex 3, pp. 33-34, which includes the wording of the questions used in the survey; for example: "Did you swallow the drugs for Schistosomiasis (show tablets) given to you at school in the recent MDA round in 2014?"
• 53.
• The coverage survey report indicates that the compensation plan for community drug distributors (CDDs) may not have been adequately communicated to the communities included in the MDA, and that CDDs were not compensated commensurate with their expectations, potentially leading to a lack of incentive to perform the job well. "Though there was no tool to quantify the extent of community contributions towards CDD incentive, drug distributors expressed dissatisfaction over none remuneration by the government. This calls for an intensive mobilisation exercise at community level, since communities are expected to provide incentive for drug distributors and not the Government. The State and Local NTD teams would need to inform community leaders and emphasise the responsibilities of communities during supervisory visits." (p. 27)
• Community drug distributors may have hoarded drugs or sold them to cattle farmers. "There were indications of drug hoarding by distributors because drug inventory showed that adequate drugs were supplied to communities where selective treatment where carried out. It is not clear what would have prompted the behaviour among distributors especially since drug supply was adequate. Community members believe the CDDs hoarded the drugs and since animal husbandry is a major economic activity in Zamfara State, the distributors could most likely be marketing the drugs to cattle farmers. The lack of incentive for these distributors by communities could promote the commercialisation of the drugs especially Mectizan." (p. 27); "The very low geographic and therapeutic coverage in Kortokoshi was due to the withdrawal of the drugs by the state NTD coordinator during the campaign for reasons associated with alleged report of drug divertion." (p. 14)
• Community drug distributors selectively treated households, possibly for political reasons. "Majority of the people who did not receive treatment attributed it to the inability of the CDDs to visit their households for drug distribution. Closer investigation revealed that these discordant voices came from communities where drug distributors operated 'selective treatment of households' Political affiliation could be the major undertone in this practice. For instance in Birnin Mallam one of the segments selected for the survey was completely ignored by the drug distributor." (p. 16)
• There are general indications that community drug distributors kept inadequate records of distributions, complicating evaluation of the distributions’ effectiveness. E.g. "The high treatment coverages reported by the health system is because the drug distributors do not keep proper records and they are the primary source of data for the health system." (p. 24) This is potentially due to ineffective training. "So, aside record keeping the drug distributors need to be re-trained before the next round of MDA." (p. 28)
• The schedule for drug distribution overlapped with Ramadan, and 27% of the population attended quranic schools, which were not targeted by the MDA. “The timing of distribution of drugs conflicted with the Ramadan fast and the closure of schools a result of the Ebola scare, accounting for the low treatment coverage for Praziquantel…. The survey revealed that 27 % of school aged children attended Qur’anic schools. These schools are out of the population targeted by the MDA for praziquantel and so the low coverage reported in the survey could be alluded to this. However, the State has a very proactive NTD team who had observed the skewed pattern in treatment of children with praziquantel and had devised strategy of identifying the quaranic schools for mop up treatment.” (pp. 27-28)
• 54.
• The coverage survey report indicates that the survey directed mop-up activities. "A list of the communities like Kotorkorshe, Nahuce, Birnin mallam, Yar Labe where either treatment was not given or where selective treatment of households was conducted were made available to the State team by the post MDA survey team for re-visitation and mop up exercise. Mop up exercises had commenced in the affected communities before the team left the field." (p. 28)
• There is some indication that previous recommendations to target qu’ranic schools had not been efficacious, but that some efforts to do so were under way. "The high proportion of children who claimed that the drug was not distributed in school is not surprising, as 27% of the eligible population children attended qu’ranic schools. These schools were not initially captured in the MDA programme. The team had suggested that the NTD team target qu’ranic schools in its MDA and efforts were being made by the state coordinator to reach out to children who attend qu’ranic schools during the coverage survey." (p. 20)
• 55.
Sightsavers did not include Zamfara on a list of Nigerian states where it delivers interventions, but later clarified that it does work in Zamfara.
• 56.
• 57.
DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014. The report's summary states that Praziquantel (PZQ) coverage rates were 46% in Bungudu and 54% in Shinkafi, the two LGAs studied (p. 4). The data table later in the report shows PZQ coverage rates were 68% in total, 55% in Bungudu and 88% in Shinkafi (p. 21). The report mentions that some PZQ treatment was delayed partly due to the MDA timing conflicting with Ramadan and schools shutting down due to the Ebola outbreak, and some PZQ treatment was still ongoing at the time of the coverage survey (p. 27). It is possible that the larger numbers include mop-up treatment that was done due to low initial coverage (p. 28).
• 58.
The regions targeted by the coverage survey (two out of six LGAs covered in the distribution) were purposefully selected based on the following criteria. We are uncertain about further details of how the two LGAs were selected—e.g. if they were selected because post-MDA records indicated particularly low or particularly high coverage.
"The survey will be conducted in Shinkafi and Bungudu LGAs. These LGAs were selected for one or more of the following reasons:
• The post MDA records show particularly low or high coverage attained at the LGA level, or a large range of coverage attained at the community level - with for example a concerning number of communities with notably low coverage.
• There are suspected issues with the MDA records or census data that need to be verified e.g. poor population data or large population movements around the time of the MDA, discrepancies between the drug store records/logs and the community records or large variations in doses given year to year.
• Representation of the different combinations of drugs administered"
DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014 protocol, p. 4 (p. 44 of the pdf).
• 59.
"The surveyed data for Praziquantel could not be compared health system records because treatment was still on-going in most communities and the records for the health system were not complete." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 5.
• 60.
"The MDA schedule given in Annex 1 outlines the different times of the various drug distributions to avoid potential drug interactions. Therefore, the period between the survey and the distribution of the first set of drugs (for Onchocerciasis and Lymphatic Filariasis) was about 3 months, which was significantly longer than for Trachoma. The implications this might have on recall was an important consideration and so, in order to reduce errors introduced through recall bias, the survey team ensured drug samples (and the packaging of the drugs were given in packages) of the different drugs distributed were shown to each respondent during discussions." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p 11.
• 61.
• For onchocerciasis and schistosomiasis, the data in the report shows that no respondents said that they "don't know" whether they received treatment. DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, pp. 20-21. This raises the possibility of a methodological issue in the survey implementation. For example, it is possible that the surveyors asked questions in a way that generated "Yes" or "No" answers (even when the respondent may have been uncertain) or did not properly record answers that were uncertain.
• The report indicates that in some cases a proxy answered in the place of a family member, but does not indicate the frequency of proxy responses. "Where possible the eligible individuals were asked if they swallowed the drug and the person was not available, another household member or caregiver gave information on their behalf. Primary caregivers responded on behalf of children aged 1-10 years old, except where drugs were given in a school based distribution. In this case the children themselves were asked if they received the drugs at school." DFID-UNITED Integrated Post MDA Coverage Survey Report, 2014, p. 11.
• 62.
• 63.
Tekle et al. 2012, p. 1.
• "Methods: In 2008, an epidemiological evaluation using skin snip parasitological diagnostic method was carried out in two onchocerciasis foci, in Birnin Gwari Local Government Area (LGA), and in the Kauru and Lere LGAs of Kaduna State, Nigeria. The survey was undertaken in 26 villages and examined 3,703 people above the age of one year. The result was compared with the baseline survey undertaken in 1987."
• "Results: The communities had received 15 to 17 years of ivermectin treatment with more than 75% reported coverage. For each surveyed community, comparable baseline data were available. Before treatment, the community prevalence of O. volvulus microfilaria in the skin ranged from 23.1% to 84.9%, with a median prevalence of 52.0%. After 15 to 17 years of treatment, the prevalence had fallen to 0% in all communities and all 3,703 examined individuals were skin snip negative."
• 64.
• The report does not appear to address confounding factors that could have also impacted prevalence. The treatment was carried out over a long time period: treatments began in 1991, the Community Directed Treatment with Ivermectin was introduced in 1997 (and Sightsavers became involved), and treatment continued for 15-17 years through 2008 (when this survey was conducted) (p. 2). Given this, it seems possible that other improvements (e.g. in economic and/or health systems or environment) could have played a role in the observed decline in infection.
• The surveyed communities were selected partially because they had the longest treatment periods and high coverage rates: "The two foci were selected for the following reasons: i) communities in these foci had pre-control epidemiological data; among the areas where large-scale ivermectin treatment was first introduced in Africa were these two foci in Kaduna in which treatment of a sample of the population started as part of a randomised controlled trial of ivermectin in 1988 and 1989, and where skin-snip surveys had been done in preparation for the trial [6,17]. ii) the foci included hyper-endemic villages, i.e. villages with a prevalence of microfilaridermia > 60% [15-17]; iii) the area was located along a river with known breeding sites of Simulium damnosum s.l., iv) the communities had had 15 - 17 years of annual treatment with ivermectin using the community-based programme since 1991, and subsequently through the community-directed treatment with ivermectin (CDTI) strategy from 1997 with more than 65% treatment coverage" (p. 3).
• "A limitation of the epidemiological surveys is that a third of the population in the selected communities did not participate in the skin-snip examination. Though some of these had valid reasons for non-participation (age < 1 year, illness, absence from the village etc), for a large majority the reasons for non-participation were not known. This high non-participation rate could have created a bias in the survey results if those who did not participate in the survey were also more likely not to have participated in ivermectin treatment" (p. 8).
• The study does not discuss the methods of the baseline survey.
• 65.
• 66.
“Results: […] The prevalence of onchocerciasis ranged from 0.8% to 5.5% while the CMFL ranged from 0.01 to 0.11 mf/ss.” Lakwo et al. 2015, p. 3.
• 67.
"One of the shortcomings is inadequate mf [microfilariae] prevalence data in this focus. Baseline data on mf prevalence is only available for Nyakabale and Kyeramya villages Hoima district which was collected in 2007. This information is very vital for decision making within the framework of elimination." Lakwo et al. 2015, p. 4.
• 68.
Lakwo et al. 2015, p. 3.
• "Conclusion: The performance of mass treatment in Budongo focus has been good since out of the 15 villages assessed only three of them have mf prevalence >5% and CMFL far below the threshold of >5 mf/s."
• "Results: A total of 2,728 people were examined, composing of 55% (1494/2728) females and 45% (1231/2728) males. The overall mf [microfilariae] prevalence was 2.6% (73/2728) and CMFL [Community Microfilaria Load] was 0.04 mf/s and varied significantly in the communities (p <0.05). The prevalence of onchocerciasis ranged from 0.8% to 5.5% while the CMFL ranged from 0.01 to 0.11 mf/ss [microfilariae per skin snip]. Infection was recorded more among the males (3.7%) compared to the females (1.7%). Mf prevalence was recorded highest in Masindi district (5.5%) and lowest in Hoima district (0.8%). The CMFL recorded was far <5 microfilariae per skin snip, i.e. recognized by WHO as threshold value in certifying the communities to be free of onchocerciasis as public health problem, thus, signifying the possibility of onchocerciasis elimination in the focus." See also discussion of results on pp. 15-17.
• 69.
Lakwo et al. 2015: A few observations:
• "Methods: Villages were selected in each of the districts following APOC procedures. Mobilizations were conducted by use of local authorities in each respective village. Questionnaires were administered among those who participated in the study. Skin snip was conducted in the selected communities (n=15) in Hoima, Buliisa and Masindi districts. Microfilaria prevalence and CMFL were calculated for each village" (p. 3). Participation in the prevalence study appears to have been opt-in and nonrandom. "Those eligible from 5 years and above from household were invited to participate in the survey. Participants were recruited consecutively from families until the required sample size was achieved" (p. 6). Participation rate or representativeness of the sample are not discussed apart from listing "Low turn up of participants in some of the villages in some districts" as a challenge (p. 17).
• The report does not appear to address confounding factors that could have impacted prevalence. The treatment was carried out over a long time period: treatment started in 1993, the Community Directed Treatment with Ivermectin was introduced in 1999, and an elimination policy was launched in 2007, with treatment presumably ongoing. “Mass treatment with ivermectin started in the 1993 with support from Sightsavers International. In 1999, the Community Directed Treatment with Ivermectin (CDTI) strategy was introduced to ensure sustainability of the program. When elimination policy was launched in 2007, bi-annual treatment and vector elimination strategies were adopted to enhance elimination” (p. 3). Given this, it seems possible that other improvements (e.g. in economic and/or health systems, or the environment) could have played a role in the reported improvements.
• 70.
• 71.
• "The Guinea-Bissau Programa Nacional de Saude Visuel(PNSV) , with support from Sightsavers International has conducted three rounds of azithromycin distribution between 2009-2012 -in the regions of Oio, Bafata and Farim(formerly part of Oio)." Bailey 2013, p. 1.
• "Sightsavers’ Guinea Bissau trachoma elimination program has run since 2011 and is working to ensure Guinea Bissau can be declared free of blinding trachoma by 2020. The project is supporting the implementation of the full SAFE strategy for trachoma elimination (Surgery to treat trichiasis, Antibiotics to treat infection, Facial cleanliness and Environmental improvement to interrupt disease transmission). Project activities include the distribution of the antibiotic treatment Zithromax, the training of trichiasis (TT) surgeons and the provision of TT surgeries, and the improvement of environmental sanitation in target areas through the construction of water points and latrines, alongside behaviour change activities to ensure their use.” Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 2.
• "Sightsavers is the key Ministry of Health partner for trachoma elimination activities in Guinea Bissau and is supporting the distribution of the antibiotic treatment Zithromax, the training of trichiasis (TT) surgeons and the provision of TT surgeries, and the improvement of environmental sanitation in target areas through the construction of water points and latrines, alongside behaviour change activities to ensure their use. The prevalence of trachoma in Guinea Bissau ranges between 10%-39.9%. (see Map 1)We are supporting the MoH to meet the WHO recommended target of reducing the prevalence of active trachoma (TF) to less than 5% among children aged 1-9 years, to reduce the prevalence of TT to less than 1 case per 1000 population above 15 years old, and to maintain (TT) recurrence to below 10%. The implementation of the SAFE strategy has resulted in the reduction of prevalence as shown in Table 2. Other NGO partners involved in implementing the SAFE strategy for trachoma elimination in Guinea Bissau include The International Trachoma Initiative (supporting Zithromax distribution). Sightsavers plans to support the writing of a trachoma Action Plan for Guinea Bissau in October 2015, in partnership with the International Trachoma Initiative and the London School of Hygiene and Tropical Medicine." Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 4.
• 72.
• “Following three rounds of MDA the estimated prevalence of TF in 1-9 year olds is below the 5% threshold in Oio, Bafata and Farim: 2.9% in Oio, 1.4% in Bafata and 4.2% in Farim. There seems to be no need for further distribution in these regions.” Bailey 2013, p. 3.
• Methodology is discussed in Bailey 2013, pp. 1-2, including that households absent at the time of the survey were excluded rather than revisited at a later time: "In the event that the 15 households did not contain 50 children, or that the selected households were unavailable reserves were used until 50 children had been examined" (p. 1). Diagnosis and grading of trachoma was performed by "ophthalmic nurses and cataract surgeons from the PNSV who had received training in the grading of trachoma according to the WHO simplified system, and had received training in the field in the study procedures" (p. 2).
• 73.
In Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 4, Sightsavers sent us a table of trachoma (TF) and trichiasis (TT) prevalence rates in 2005 compared to prevalence found in the 2013 impact study. 2005 data is available for eleven regions, including the three regions targeted by the 2013 study. We have not seen the methodology for the 2005 prevalence survey and we are unsure whether it is appropriately comparable to the 2013 survey.
Region Trachoma prevalence (2005) Trachoma prevalence (2013)
Bafata 28.7% 1.38%
Oio 21.7% 2.94%
Farim 21.7% 4.21%
• 74.
• It seems plausible to us that Sightsavers has not yet had time to complete this type of study on its deworming programs.
• Sightsavers told us that "Sightsavers follows recommended WHO guidelines with regard to the establishment of sentinel sites. For deworming programs sentinel sites are typically established at schools. Impact indicators track change in prevalence and intensity of infection (intensity is recognized as a more sensitive indicator of impact)." GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, p. 2. It is possible that impact assessments conducted at sentinel sites (where programs may be higher quality) may not be representative of impact at other sites. Of the three impact assessments discussed above, the first (Tekle et al. 2012) explicitly selected sentinel sites for assessment. We are unsure whether the other two impact assessments took place at sentinel sites.
• 75.
"At least once a year for each project, Sightsavers’ monitoring and evaluation team performs a spot check, whereby headquarters requests initial data records from the program. Projects are randomly selected to provide this backup information for spot-checking. These constitute a separate tool from the quality standard assessment tools. Currently the data examined in a spot check include figures related to human resources and service delivery, such as treatment numbers and numbers of people trained. The check is not intended to provide proof of treatment, but to make sure that the numbers indicated for each community tally up to the reported district total." GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, pp. 3-4.
• 76.
• 77.
Two instances of manual entry rather than formula summation resulted in a difference of 6 treatments on a total of almost 1.3 million. “For the majority of rows a formula is used to add the number of males and females who have received treatment but for some rows a number has been entered directly, this led to a miscalculated total on two occasions[…] the total treatments (column T) were 5 and 1 treatments out.” Sightsavers spot check report, DRC.
• 78.
GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017, sheet "Summary & comparison", Row 23 ("Total Impact: The number of additional children reached for NTD treatment per additional $1 donated"). Values are 10 for 2015, 11 for 2016, and 13 for 2017. We have converted dollars per treatment from the given treatments per dollar. • 79. GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, sheet "Summary & comparison", Row 17 ("Total impact: dollar cost per child treated"). Values are "n/a" for 2015 due to no deworming spending in that year, 0.92 for 2016, and 0.57 for 2017. • 80. • "2015 Planned schisto treatments are being supported by WHO Guinea Bissau country office" (cell F6). • "As noted in accompanying narrative document, de worming in 2015 will be supported by; • Sightsavers-supported LF MDA in two regions • WHO Guinea Bissau country office in two new regions" (cell F7). • 81. "This is direct project support costs for three multi-country projects." Sightsavers, email to GiveWell, March 8, 2016. • 82. • 83. Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015. "Table 4. showing STH prevalence rates, assessed in 2005", p. 6. "Table showing schistosomiasis prevalence rates, assessed in 2005", p. 8. • 84. See Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 4, which includes no information about the source of the prevalence data. • 85. Prevalence of schistosomiasis and STH in selected states in Nigeria Sightsavers provided us with the following prevalence data for the four Nigerian states where Sightsavers currently supports mass drug administrations, and for Benue State, where Sightsavers would like to begin support of deworming. (Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 4.) State Schistosomiasis STH Benue 1.4% - 24.8% 20.2% - 36.8% Kebbi 0.8% - 68.3% 3.8% - 22.0% Kogi 0.0% - 21.4% 16.2% - 39.0% Kwara 0.0% - 38.3% 8.3% - 57.0% Sokoto 3% - 56.1% 3.2% - 27.1% Prevalence of schistosomiasis and STH in Guinea-Bissau Sightsavers also provided prevalence data for Guinea-Bissau. The data is from 2003-2005. (Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, pp. 6, 8.) We have not seen more recent data and would guess that prevalence rates may have changed significantly since it was collected. • Sightsavers provided data for Guinea-Bissau subdivided into eleven regions: "In the Sightsavers supported regions of Bafata and Gabu, Albendazole treatment is currently given through the Ivermectin + Albendazole drug distribution package for LF, once per year though a house to house distribution platform using community volunteers. The prevalence of STH in Guinea Bissau ranges between 13% to 93% (See Map 2.) This once per year treatment is also in line with WHO guidelines for the treatment of STH in school age children, in areas where the baseline STH prevalence is ≥20% but <50%. Sightsavers is working towards scaling up once per year drug distribution to reach all endemic areas in the remaining 9 regions for the first time in 2016. Our partners in STH in Guinea Bissau are APOC, WHO and the World Food Programme. However, in areas where the baseline prevalence is ≥ 50% the WHO guideline is to treat all school age children twice per year. As seen in the table below, prevalence is higher than 50% in nearly all regions, highlighting a need to scale up drug distribution to twice per year in all these regions." Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 6. • We don't know whether there have been other deworming programs in the country since prevalence data was taken in 2003-2005. The World Health Organization and World Food Program planned to fund schistosomiasis treatments beginning in 2015, but that program has been delayed: "Mass drug administration has not taken place before, however the table below shows a clear need. It has been planned for funding from the World Health Organisation and World Food Programme to support initial MDA in Oio/Farim and Biombo regions in 2015, however, a fire which recently destroyed the drug supply has delayed 2015 activity. According to WHO guidelines, areas with a baseline disease prevalence of <10% should treat school age children twice during primary school year e.g. once on entry and once on exit." Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 8. Region Schistosomiasis STH Bafata 16% 31% Bijagos 0% 88% Biombo 5% 93% Bissau 0% 13% Bolama 0% 67% Cacheu 1% 64% Gabu 2% 49% Oio/Farim 5% 67% Quinara 4% 55% Sao Domingos 11% 63% Tombali 4% 65% Comparison to worm prevalence in key deworming studies For a table summarizing worm prevalence in key deworming studies, see our review of Deworm the World. Because we do not know how the data provided by Sightsavers was measured, we are not confident in the extent to which it is comparable to the worm prevalence data in the studies which constitute our key evidence for the impact of deworming. • 86. Sum of GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017, "Budget - NTD work (additional)" cell F48, "Budget - Benue (additional)" cell F26, and GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (additional)" cells E20, F20, E25, F25. • 87. • 88. GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017. Costs are reported in USD. Total 3-year cost of scale up in the four states where Sightsavers currently supports programs is found in "Budget - NTD work (additional)" cell F48. 3-year cost of expansion to Benue is found in "Budget - Benue (additional)" cell F26. • 89. • “Sokoto State: Potential funders: 2015-2018 pending proposal (c.50% of planned costs) with Isle of Man International Development Committee. Result due June 2015.” • “Kebbi State: Sightsavers work is not currently donor funded.” • “Kwara State: Potential funder: 2015 pending request for part-funding with Dickens Sanomi Foundation.” • “Kogi State: Existing funder: AG Leventis Foundation, c.$50,000.”
• 90.
"The amount of people Sightsavers can protect against neglected tropical diseases (NTDs) in Nigeria is set to expand, after the UK government announced it would add £1.7 million in funding towards the Northern Nigeria Integrated NTD Programme." Sightsavers website, Extra funding in Nigeria, May 2015. We have converted to USD using the average exchange rate (1.5166 GBP/USD) for January 2015 through May 2015, as reported on Oanda's historical exchange rates calculator.
• 91.
• 92.
GiveWell analysis of Sightsavers Guinea-Bissau NTD program budget, 2015-2017, "Budget - NTD work (additional)". Cells E20 and F20 project about $667,000 for "Guinea Bissau Schisto / STH scale up budget"; cells E25 and F25 project an additional$65,000 for "National Coordination & Reporting" to support this scale up. Additionally, Sightsavers projects costs of about $700,000 to scale up LF elimination activities over the same two years (sum of cells E28 and F28). • 93. • "2015 existing and potential Sightsavers’ funders include: • Sightsavers Guinea Bissau trachoma elimination project is c.50% funded by Isle of Man International Development Committee. • Sightsavers Guinea Bissau onchocerciasis and LF elimination project has a proposal for funding activity in two regions between 2015-2019 pending with OPEC, result due in July 2015.” • “2015 existing external funders include: • The WHO and the World Food Programme are supporting STH and Schisto work in the following areas: Farim/ Oio and Biombo • the Spanish NGO Igreja Evangelica is supporting activity" • 94. "Sightsavers would see no problem, either programmatic or financial, if GiveWell were to recommend funding restricted to deworming programs only." GiveWell's non-verbatim summary of a conversation with Sightsavers staff, March 19, 2015, p. 6. • 95. "Reserves Policy Sightsavers ensures the continuity of its programs through active use of its general reserves whilst ensuring these remain at agreed target levels. It is our policy to retain sufficient reserves to safeguard ongoing commitments and operations. The current reserves policy is to maintain a level of unrestricted reserves of £6.0 million, +/- £1.5 million. Total fund balances were £11.2 million at the end of 2014, of which, £9.9 million is unrestricted. This includes designated funds of £1.9 million, of which £0.9 million is cash held overseas. The baseline to compare to the reserves target is calculated by subtracting the designated funds from the unrestricted funds and adding back the cash held overseas, which is available for use. This gives a reserves figure of £8.9 million. This level of reserves is above policy guidelines. The trustees believe this level of reserves is acceptable given the continued requirement for additional investment in fundraising and plans for ongoing programmatic expansion in 2015 and beyond." Sightsavers operational budget, 2015; total expenditures for 2015 are estimated at £64.5 million. We have converted to USD using the average exchange rate (1.5166 GBP/USD) for January 2015 through May 2015, as reported on Oanda's historical exchange rates calculator. • 96. • 97. Our understanding of how Sightsavers allocates its unrestricted funding is based on two conversations with Sightsavers staff: • 98. "Sightsavers wants enough funding to implement fully integrated programs in other states in Nigeria, as mapping reveals that these areas need a complete package of NTD treatment. Expansion into Benué state, for example, will require an integrated program to ensure that Sightsavers can treat all NTDs effectively there. […] If GiveWell were to provide deworming-restricted funding in Benué state or any other area Sightsavers is considering entering, Sightsavers would only be able to use that funding if it also received matching funds to carry out the rest of the NTD program there. Sightsavers’ budget shows that there are many expenses that apply to all of its programs and cannot be isolated to deworming only. If GiveWell wanted to focus specifically on deworming, Sightsavers would look for other donors that wanted to fund the onchocerciasis and trachoma work. It might also try to make up the remainder of the program budget with unrestricted funding. This would be most difficult in Nigeria, where Sightsavers would have to fund a whole integrated program itself if it couldn’t find a matching donor, or possibly give up on the program entirely." GiveWell's non-verbatim summary of a conversation with Sightsavers staff, March 19, 2015, p. 6. • 99. For example: • Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 5. • "Sokoto State: Potential funders: 2015-2018 pending proposal (c.50% of planned costs) with Isle of Man International Development Committee. Result due June 2015." • "Kwara State: Potential funder: 2015 pending request for part-funding with Dickens Sanomi Foundation." It is unclear to us whether this request is for part-funding for planned activities, scale-up activities, or both. • "Kogi State: Existing funder: AG Leventis Foundation, c.$50,000." For comparison, the cost of planned activities in Kogi State is about $320,000 over three years, and the cost of scale up is about$130,000 over three years. (GiveWell analysis of Sightsavers Nigeria NTD program budget, 2015-2017, "Budget - NTD work (planned)" cells D14+E14+F14 and "Budget - NTD work (additional)" cell F20.)
• Through the United project, DFID is funding integrated NTDs in the Nigerian states of Kaduna, Kano, Katsina, Niger, and Zamfara. Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 5; and Sightsavers website, Extra funding in Nigeria, May 2015 (“The Department for International Development (DFID) agreed the extra budget in March 2015, allowing Sightsavers to expand the project from three to five states (Zamfara, Katsina, Kano, Niger, Kaduna) and to reach 27 million people with 112 million treatments.”). DFID also supports trachoma mapping. “Sightsavers also coordinates the Global Trachoma Mapping Project (GTMP), funded by the U.K.’s Department for International Development (DFID), which has supported a great deal of trachoma mapping.” GiveWell's non-verbatim summary of a conversation with Susan Walker and Katie Cotton, February 26, 2015, pp. 1-2.
• The Children’s Investment Fund Foundation (CIFF) is funding mapping of schistosomiasis and STH in northern Nigeria. Sightsavers Nigeria NTD program information prepared for GiveWell, May 2015, p. 5.
• Sightsavers Guinea-Bissau NTD program information prepared for GiveWell, May 2015, p. 10.
• "Sightsavers Guinea Bissau trachoma elimination project is c.50% funded by Isle of Man International Development Committee [between 2013-2015]."
• "Sightsavers Guinea Bissau onchocerciasis and LF elimination project has a proposal for funding activity in two regions between 2015-2019 pending with OPEC, result due in July 2015."
• "2015 existing external funders include:
• The WHO and the World Food Programme are supporting STH and Schisto work in the following areas: Farim/ Oio and Biombo
• the Spanish NGO Igreja Evangelica is supporting activity"
• Sightsavers does not list existing or potential funders for its potential scale up of deworming in Guinea-Bissau.
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2017-03-25 17:23:33
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https://docs.mitk.org/nightly/mitkNodePredicateSubGeometry_8h.html
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Medical Imaging Interaction Toolkit 2022.10.99-eeb2f471 Medical Imaging Interaction Toolkit
mitkNodePredicateSubGeometry.h File Reference
#include "mitkNodePredicateBase.h"
#include "mitkBaseGeometry.h"
#include "mitkTimeGeometry.h"
Go to the source code of this file.
## Classes
class mitk::NodePredicateSubGeometry
Predicate that evaluates if the given DataNode's data object has a geometry that is a sub geometry of the reference geometry. Sub geometry means that both geometries have the same voxel grid (same spacing, same axes, origin is on voxel grid), but the bounding box of the checked geometry is contained or equal to the bounding box of the reference geometry.
One can either check the whole time geometry of the data node by defining a reference time geometry or check against one given2 reference base geometry. If the predicate should check against a base geometry, you can specify the timepoint of the data's time geometry that should be checked. If no timepoint is defined the predicate will evaluate the data geometry in the first timestep. Evaluates to "false" for unsupported or undefined data objects/geometries. More...
mitk
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2023-03-23 12:22:32
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https://mlir.llvm.org/docs/Dialects/SparseTensorOps/
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# MLIR
Multi-Level IR Compiler Framework
# 'sparse_tensor' Dialect
The SparseTensor dialect supports all the attributes, types, operations, and passes that are required to make sparse tensor types first class citizens within the MLIR compiler infrastructure. The dialect forms a bridge between high-level operations on sparse tensors types and lower-level operations on the actual sparse storage schemes consisting of pointers, indices, and values. Lower-level support may consist of fully generated code or may be provided by means of a small sparse runtime support library.
The concept of treating sparsity as a property, not a tedious implementation detail, by letting a sparse compiler generate sparse code automatically was pioneered for dense linear algebra by [Bik96] in MT1 (see https://www.aartbik.com/sparse.php) and formalized to tensor algebra by [Kjolstad17,Kjolstad20] in the Sparse Tensor Algebra Compiler (TACO) project (see http://tensor-compiler.org).
The MLIR implementation closely follows the “sparse iteration theory” that forms the foundation of TACO. A rewriting rule is applied to each tensor expression in the Linalg dialect (MLIR’s tensor index notation) where the sparsity of tensors is indicated using the per-dimension level types dense/compressed together with a specification of the order on the dimensions (see [Chou18] for an in-depth discussions and possible extensions to these level types). Subsequently, a topologically sorted iteration graph, reflecting the required order on indices with respect to the dimensions of each tensor, is constructed to ensure that all tensors are visited in natural index order. Next, iteration lattices are constructed for the tensor expression for every index in topological order. Each iteration lattice point consists of a conjunction of tensor indices together with a tensor (sub)expression that needs to be evaluated for that conjunction. Within the lattice, iteration points are ordered according to the way indices are exhausted. As such these iteration lattices drive actual sparse code generation, which consists of a relatively straightforward one-to-one mapping from iteration lattices to combinations of for-loops, while-loops, and if-statements.
• [Bik96] Aart J.C. Bik. Compiler Support for Sparse Matrix Computations. PhD thesis, Leiden University, May 1996.
• [Kjolstad17] Fredrik Berg Kjolstad, Shoaib Ashraf Kamil, Stephen Chou, David Lugato, and Saman Amarasinghe. The Tensor Algebra Compiler. Proceedings of the ACM on Programming Languages, October 2017.
• [Kjolstad20] Fredrik Berg Kjolstad. Sparse Tensor Algebra Compilation. PhD thesis, MIT, February, 2020.
• [Chou18] Stephen Chou, Fredrik Berg Kjolstad, and Saman Amarasinghe. Format Abstraction for Sparse Tensor Algebra Compilers. Proceedings of the ACM on Programming Languages, October 2018.
## Attribute definition ¶
### SparseTensorEncodingAttr ¶
An attribute to encode TACO-style information on sparsity properties of tensors. The encoding is eventually used by a sparse compiler pass to generate sparse code fully automatically for all tensor expressions that involve tensors with a sparse encoding. Compiler passes that run before this sparse compiler pass need to be aware of the semantics of tensor types with such an encoding.
Example:
#DCSC = #sparse_tensor.encoding<{
dimLevelType = [ "compressed", "compressed" ],
dimOrdering = affine_map<(i,j) -> (j,i)>,
pointerBitWidth = 32,
indexBitWidth = 8
}>
... tensor<8x8xf64, #DCSC> ...
#### Parameters: ¶
ParameterC++ typeDescription
dimLevelType::llvm::ArrayRef<SparseTensorEncodingAttr::DimLevelType>Per-dimension level type
dimOrderingAffineMap
pointerBitWidthunsigned
indexBitWidthunsigned
## Operation definition ¶
### sparse_tensor.convert (::mlir::sparse_tensor::ConvertOp) ¶
Converts between different tensor types
Syntax:
operation ::= sparse_tensor.convert $source attr-dict : type($source) to type($dest) Converts one sparse or dense tensor type to another tensor type. The rank and dimensions of the source and destination types must match exactly, only the sparse encoding of these types may be different. The name convert was preferred over cast, since the operation may incur a non-trivial cost. When converting between two different sparse tensor types, only explicitly stored values are moved from one underlying sparse storage format to the other. When converting from an unannotated dense tensor type to a sparse tensor type, an explicit test for nonzero values is used. When converting to an unannotated dense tensor type, implicit zeroes in the sparse storage format are made explicit. Note that the conversions can have non-trivial costs associated with them, since they may involve elaborate data structure transformations. Also, conversions from sparse tensor types into dense tensor types may be infeasible in terms of storage requirements. Examples: %0 = sparse_tensor.convert %1 : tensor<32x32xf32> to tensor<32x32xf32, #CSR> %2 = sparse_tensor.convert %3 : tensor<8x8xi32, #CSC> to tensor<8x8xi32, #CSR> #### Operands: ¶ OperandDescription sourcetensor of any type values #### Results: ¶ ResultDescription desttensor of any type values ### sparse_tensor.new (::mlir::sparse_tensor::NewOp) ¶ Constructs a new sparse tensor Syntax: operation ::= sparse_tensor.new$source attr-dict : type($source) to type($result)
Constructs a sparse tensor value with contents taken from an opaque pointer provided by source. For targets that have access to a file system, for example, this pointer may be a filename (or file) of a sparse tensor in a particular external storage format. The form of the operation is kept deliberately very general to allow for alternative implementations in the future, such as pointers to buffers or runnable initialization code. The operation is provided as an anchor that materializes a fully typed sparse tensor values into a computation.
Example:
sparse_tensor.new %source : !Source to tensor<1024x1024xf64, #CSR>
#### Operands: ¶
OperandDescription
sourceany type
#### Results: ¶
ResultDescription
resulttensor of any type values
### sparse_tensor.indices (::mlir::sparse_tensor::ToIndicesOp) ¶
Extract indices array at given dimension from a tensor
Syntax:
operation ::= sparse_tensor.indices $tensor ,$dim attr-dict : type($tensor) to type($result)
Returns the indices array of the sparse storage scheme at the given dimension for the given sparse tensor. This is similar to the memref.buffer_cast operation in the sense that it provides a bridge between a tensor world view and a bufferized world view. Unlike the memref.buffer_cast operation, however, this sparse operation actually lowers into a call into a support library to obtain access to the indices array.
Example:
%1 = sparse_tensor.indices %0, %c1
: tensor<64x64xf64, #CSR> to memref<?xindex>
#### Operands: ¶
OperandDescription
tensortensor of any type values
dimindex
#### Results: ¶
ResultDescription
resultstrided memref of any type values of rank 1
### sparse_tensor.pointers (::mlir::sparse_tensor::ToPointersOp) ¶
Extract pointers array at given dimension from a tensor
Syntax:
operation ::= sparse_tensor.pointers $tensor ,$dim attr-dict : type($tensor) to type($result)
Returns the pointers array of the sparse storage scheme at the given dimension for the given sparse tensor. This is similar to the memref.buffer_cast operation in the sense that it provides a bridge between a tensor world view and a bufferized world view. Unlike the memref.buffer_cast operation, however, this sparse operation actually lowers into a call into a support library to obtain access to the pointers array.
Example:
%1 = sparse_tensor.pointers %0, %c1
: tensor<64x64xf64, #CSR> to memref<?xindex>
#### Operands: ¶
OperandDescription
tensortensor of any type values
dimindex
#### Results: ¶
ResultDescription
resultstrided memref of any type values of rank 1
### sparse_tensor.tensor (::mlir::sparse_tensor::ToTensorOp) ¶
Reconstructs tensor from arrays(s)
Syntax:
operation ::= sparse_tensor.tensor $memrefs attr-dict : type($memrefs) to type($result) Reconstructs the sparse tensor from the sparse storage scheme array(s). This is similar to the memref.load operation in the sense that it provides a bridge between a bufferized world view and a tensor world view. Unlike the memref.load operation, however, this sparse operation is used only temporarily to maintain a correctly typed intermediate representation during progressive bufferization. Eventually the operation is folded away. The input arrays are defined unambigously by the sparsity annotations (pointers and indices for overhead storage in every compressed dimension, followed by one final values array). Examples: %1 = sparse_tensor.tensor %0 : memref<?xf64> to tensor<64x64xf64, #Dense> %3 = sparse_tensor.tensor %0, %1, %2 : memref<?xindex>, memref<?xindex>, memref<?xf32> to tensor<10x10xf32, #CSR> #### Operands: ¶ OperandDescription memrefsstrided memref of any type values of rank 1 #### Results: ¶ ResultDescription resulttensor of any type values ### sparse_tensor.values (::mlir::sparse_tensor::ToValuesOp) ¶ Extract numerical values array from a tensor Syntax: operation ::= sparse_tensor.values$tensor attr-dict : type($tensor) to type($result)
Returns the values array of the sparse storage scheme for the given sparse tensor, independent of the actual dimension. This is similar to the memref.buffer_cast operation in the sense that it provides a bridge between a tensor world view and a bufferized world view. Unlike the memref.buffer_cast operation, however, this sparse operation actually lowers into a call into a support library to obtain access to the values array.
Example:
%1 = sparse_tensor.values %0 : tensor<64x64xf64, #CSR> to memref<?xf64>
#### Operands: ¶
OperandDescription
tensortensor of any type values
#### Results: ¶
ResultDescription
resultstrided memref of any type values of rank 1
|
2021-09-18 05:25:47
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|
http://nrich.maths.org/public/leg.php?code=31&cl=3&cldcmpid=601
|
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|
2015-07-05 02:58:22
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|
http://mathhelpforum.com/calculus/89757-distance-between-two-skew-lines.html
|
# Math Help - distance between two skew lines.
1. ## distance between two skew lines.
Find the distance between two skew lines:
L1: x = 2s, y = 2s - 1, z = s + 1, and
L2: x = t, y = 2t - 2, z = 3.
2. Originally Posted by qkfxjs
Find the distance between two skew lines:
L1: x = 2s, y = 2s - 1, z = s + 1, and L2: x = t, y = 2t - 2, z = 3.
If $P+tD~\&~Q+sE$ are two skew lines, the the distance between them is $\frac{{\left| {\overrightarrow {PQ} \cdot \left( {D \times E} \right)} \right|}}{{\left\| {\left( {D \times E} \right)} \right\|}}$
3. ## Ok
So how can I solve the problem.
I'm still having a problem.
4. Originally Posted by qkfxjs
So how can I solve the problem.
Find the direction vector of each line. Those are $D~\&~E$.
Find a point on each line. Those are $P~\&~Q$
Then use the formula.
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2014-08-22 19:02:27
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https://physics.stackexchange.com/questions/485061/a-derivation-in-jaynes-paper-linking-stat-mech-and-shannons-entropy
|
# A derivation in Jaynes' paper, linking stat-mech and Shannon's entropy
I have been going through E. T. Jaynes' 1957 paper, Information Theory and Statistical Mechanics. There is a step in his derivations, which has been giving me headaches for the past day; would appreciate some pointers on how to complete it!
A bit of notation: Lagrange multipliers are represented by $$\lambda$$'s, and $$\lambda_0$$ = ln Z where Z is the partition function. The probability of each state $$j$$ is
$$$$p_j = \frac{\exp \left[ -(\lambda_0 + \lambda_1 f_1(x_j) + \cdots \right]}{Z}.$$$$
The expectation values of the functions $$f's$$ are fixed/known, and provide the constraints for maximizing Shannon's entropy; for simplicity let's consider only $$f_1$$. Assume that the probabilistic states $$i$$'s are discrete.
Now onto the question. Jaynes says near Equation (5.1) of the paper that he's going to perturb $$f_1$$ such that $$f_1(x_j) \rightarrow f_1(x_j) + \delta f_1(x_j)$$, for all $$j$$. At the same time, the expectation value of $$f_1$$ is independently altered: $$\left \rightarrow \left + \delta\left$$.
How does the entropy, the derived probability distribution, and the Lagrange multiplier change? Equation (5.1) states that
$$$$\delta\lambda_0 = \delta \mathtt{ln} Z = -\left( \delta\lambda_1 \left + \lambda_1 \left<\delta f_1\right> \right).$$$$
But how? Here's my current approach:
$$\begin{eqnarray} \delta\lambda_0 &= ln Z' - ln Z \\ &= ln (Z' / Z) \\ &= ln (\frac{\sum_j \exp[(\lambda_1 + \delta\lambda_1)\times (f_1(x_j) + \delta f_1(x_j))]}{Z}). \end{eqnarray}$$
But then I am pretty lost on how to "get rid of" the logarithmic function...Thanks for reading, and looking forward to see what you think!
• I do not think the tag "research-level" is appropriate for this question. – lattitude Jun 9 at 7:21
• That's fair --- even though I am reading the paper for grad work, the math is admittedly "basic" relative to the curriculum. Tag removed. – Eve L Jun 10 at 20:35
Remember that $$\delta\lambda_1$$ and $$\delta f_1$$ are small so expanding the exponential and then the log to first order: \eqalign{ \delta\lambda_0&\simeq\ln\Big[{1\over{\cal Z}}\sum_je^{-\lambda_1 f_1(x_j)-\delta\lambda_1 f_1(x_j)-\lambda_1\delta f_1(x_j)}\Big]\cr &=\ln\Big[{1\over{\cal Z}}\sum_je^{-\delta\lambda_1 f_1(x_j)-\lambda_1\delta f_1(x_j)}e^{-\lambda_1 f_1(x_j)}\Big]\cr &\simeq \ln\Big[{1\over{\cal Z}}\sum_j\Big(1-\delta\lambda_1 f_1(x_j)-\lambda_1\delta f_1(x_j)\Big)e^{-\lambda_1 f_1(x_j)}\Big]\cr &=\ln\Big(1-\delta\lambda_1\langle f_1\rangle-\lambda_1\langle \delta f_1\rangle\Big)\cr &\simeq -\delta\lambda_1\langle f_1\rangle-\lambda_1\langle \delta f_1\rangle } Alternatively, a faster calculation is \eqalign{ \delta \ln{\cal Z}&={1\over{\cal Z}}\delta{\cal Z}\cr &={1\over{\cal Z}}\sum_j\Big(-\delta\lambda_1 f_1(x_j)-\lambda_1\delta f_1(x_j)\Big)e^{-\lambda_1 f_1(x_j)}\cr &=-\delta\lambda_1\langle f_1\rangle-\lambda_1\langle \delta f_1\rangle }
• Thank you thank you! My main problem was that I couldn't get the right expansion; it is great to see the steps worked out :) – Eve L Jun 10 at 20:41
The equation 5.1 in the Jaynes' paper can be understood from basic knowledge in differential equations. $$\delta lnZ$$ can be written as
$$\delta \lambda_0 = \delta lnZ = \frac{\delta lnZ}{\delta \lambda_1}\delta \lambda_1 + \sum_{i}\frac{\delta lnZ}{\delta f_1(x_i)}\delta f_i(x_i)$$
Here, we know that the first factor in the first term is the expectation value of the variable corresponding to that Lagrange multiplier. Similarly, the first factor in the summation turns out to be $$p_i$$, the probability, times $$\lambda_1$$. Thus,
$$$$\begin{split} \delta \lambda_0 & = -(\delta \lambda_1\langle f_1\rangle + \lambda_1\sum_i p_i \delta f_i(x_i))\\ & =-(\delta \lambda_1\langle f_1\rangle + \lambda_1\langle \delta f_1\rangle) \end{split}$$$$
You can derive this expression using your way of taking a difference between the new and old $$\mathcal{Z}$$, but the same method needs to be used to split the equation and simplify it.
• Thanks for providing a framework for solving the problem! I have to accept 1 of 2 answers, but it doesn't mean I like yours less :P – Eve L Jun 10 at 20:48
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2019-12-09 01:57:07
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