text:
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
To write the given equation in exponential form, we will recall the definition of a logarithm. $\begin{gathered} \log_{{\color{#0000FF}{b}}}({\color{#FF0000}{x}})=\textcolor{darkviolet}{y} \quad \Leftrightarrow \quad {\color{#FF0000}{x}}={\color{#0000FF}{b}}^{\textcolor{darkviolet}{y}} \end{gathered}$ This means that $y$ is the exponent to which $b$ must be raised to get $x.$ In our exercise, $4$ is the exponent to which $5$ must be raised to get $625.$ $\begin{gathered} \log_{{\color{#0000FF}{5}}}({\color{#FF0000}{625}})=\textcolor{darkviolet}{4} \quad \Leftrightarrow \quad {\color{#FF0000}{625}}={\color{#0000FF}{5}}^{\textcolor{darkviolet}{4}} \end{gathered}$


text:
## Duke Mathematical Journal

### On long-range scattering

I. M. Sigal

#### Article information

Source
Duke Math. J., Volume 60, Number 2 (1990), 473-496.

Dates
First available in Project Euclid: 20 February 2004

https://projecteuclid.org/euclid.dmj/1077297301

Digital Object Identifier
doi:10.1215/S0012-7094-90-06019-3

Mathematical Reviews number (MathSciNet)
MR1047762

Zentralblatt MATH identifier
0725.35071

#### Citation

Sigal, I. M. On long-range scattering. Duke Math. J. 60 (1990), no. 2, 473--496. doi:10.1215/S0012-7094-90-06019-3. https://projecteuclid.org/euclid.dmj/1077297301

#### References

• [Com] M. Combescure, Long range scattering for the two-body Schrödinger equation with “Hörmander-like” potentials, Comm. Partial Differential Equations 11 (1986), no. 2, 123–165.
• [En] V. Enss, Quantum scattering theory for two and three-body systems with potentials of short and long range, Schrödinger operators (Como, 1984) ed. S. Graffi, Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 39–176.
• [Hö1] L. Hörmander, The existence of wave operators in scattering theory, Math. Z. 146 (1976), no. 1, 69–91.
• [Hö2] L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985.
• [IK] H. Isozaki and H. Kitada, Modified wave operators with time-independent modifiers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 77–104.
• [K1]1 H. Kitada, Scattering theory for Schrödinger operators with long-range potentials. I: Abstract theory, J. Math. Soc. Japan 29 (1977), no. 4, 665–691.
• [K1]2 H. Kitada, Scattering Theory for Schrödinger operators with long-range potentials. II: Spectral and scattering theory, J. Math. Soc. Japan 30 (1978), no. 4, 603–632.
• [K2] H. Kitada, Scattering theory for Schrödinger equations with time-dependent potentials of long-range type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 353–369.
• [KY]1 H. Kitada and K. Yajima, A scattering theory for time-dependent long range potentials, Duke Math. J. 49 (1982), no. 2, 341–376.
• [KY]2 H. Kitada and K. Yajima, Remarks on our paper: “A scattering theory for time-dependent long-range potentials”, Duke Math. J. 50 (1983), no. 4, 1005–1016.
• [M] Pl. Muthuramalingam, Spectral properties of vaguely elliptic pseudodifferential operators with momentum-dependent long-range potentials using time-dependent scattering theory, J. Math. Phys. 25 (1984), no. 6, 1881–1899, and Bielefeld preprint (1984).
• [Pe] P. Perry, Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. 81 (1981), no. 2, 243–259.
• [RS]1 M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.
• [RS]2 M. Reed and B. Simon, Methods of Modern Mathematical Physics, III, Academic Press, New York, 1979.
• [S1] Y. Saito, On the asymptotic behavior of the solutions of the Schrödinger equation $(-\Delta +Q(y)-k\sp{2})V=F$, Osaka J. Math. 14 (1977), no. 1, 11–35.
• [S2] Y. Saito, Eigenfunction expansions for the Schrödinger operators with long-range potentials$Q(y)=O(\mid y\mid \sp{-\varepsilon })$, $\varepsilon >0$, Osaka J. Math. 14 (1977), no. 1, 37–53.
• [Sig] I. M. Sigal, Lectures on Scattering Theory, Toronto, 1986-1987.
• [SigSof 1] I. M. Sigal and A. Soffer, Local decay and velocity bounds for time-independent and time-dependent Hamiltonians, preprint (Caltech), 1987.
• [SigSof 2] I. M. Sigal and A. Soffer, Asymptotic completeness of $3$- and $4$-body long-range systems, preprint.
• [SigSof 3] I. M. Sigal and A. Soffer, Long-range many-body scattering... Invent. Math. (to appear).
• [St] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, N.J., 1970.

format: plain


text:
Page 1 of 1

Posted: 13 Aug 2011, 19:41
mach1723 wrote: So I am looking for somewhere to start with OpenMW development I figure I will tackle: http://github.com/korslund/openmw/issues#issue/5.

So the question is where should I start?

I have been looking around the code a bit but don't have much of an idea, hopefully this question isn't too general or somesuch.
Star-Demon wrote: Welcome!
mach1723 wrote:
Star-Demon wrote:Welcome!
Thanks.

Also I have been trying to build OpenMW(on Ubuntu Linux 10.04) and I have all of the dependencies and cmake generates a Makefile fine but I get errors about files in extern/caelum trying to include a "Ogre.h" and I look in these files and they are trying to include a "Ogre.h", I don't know if I am missing a file or if these should be including "<OGRE/Ogre.h>".
Zini wrote: First a few comments regarding github:

- You better comment on the issues in Nico's repository. If you add issue comments in your own fork they most likely will get lost.

- It seems something went wrong with your commit. git considers your cpp file a binary.

Regarding the build problem. The code is correct. If it doesn't build you have a setup or configuration problem.

Regarding the issue you want to tackle: I don't know much about the whole resources handling. Nico wrote all the code. I remember though that he recently added a file_finder component, that successfully sorted out most of the problems for sound files.
nicolay wrote: Welcome!

That is correct about file_finder. The problem with path finding is mostly that Unix file systems are case sensitive, while the dataset uses all kinds of cases on filenames it references (and also uses \ dir separators.) The filefinder I wrote browses a given dir tree at startup, and lets you convert file names at runtime. So it converts eg "Fx\item\LongBlad.wav" to "/your/data/dir/Sound/Fx/item/longblad.wav" (if that was an actual file found at startup.) It doesn't do any file reading or anything like that, it's just a pure filename lookup.

It's currently only used in apps/openmw/mwsound/soundmanager.cpp. Another place it would be great to use it is for meshes and textures in components/nifogre/
mach1723 wrote:
Zini wrote:- It seems something went wrong with your commit. git considers your cpp file a binary.
Actually that file is still empty, dunno why github lists that as binary.
nicolay wrote:It's currently only used in apps/openmw/mwsound/soundmanager.cpp. Another place it would be great to use it is for meshes and textures in components/nifogre/
I guess then that having it used for meshes and textures would be the logical first step here then.
mach1723 wrote: Heres some build outputs:

Build output from: make caelum

Code: Select all

[email protected]:~/documents/openmw_build/build$make caelum Scanning dependencies of target caelum [ 0%] Building CXX object extern/caelum/CMakeFiles/caelum.dir/src/UniversalClock.cpp.o In file included from /home/zachary/documents/openmw_build/extern/caelum/src/UniversalClock.cpp:21: /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrecompiled.h:24:18: error: Ogre.h: No such file or directory In file included from /home/zachary/documents/openmw_build/extern/caelum/include/UniversalClock.h:24, from /home/zachary/documents/openmw_build/extern/caelum/src/UniversalClock.cpp:22: /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:114: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:115: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:116: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:117: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:118: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:120: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:121: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:124: error: â??Stringâ?? does not name a type /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:130: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:131: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:132: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:133: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:134: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:135: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:136: error: â??Ogreâ?? has not been declared /home/zachary/documents/openmw_build/extern/caelum/include/CaelumPrerequisites.h:171: error: expected initializer before â??&â?? token In file included from /home/zachary/documents/openmw_build/extern/caelum/src/UniversalClock.cpp:22: /home/zachary/documents/openmw_build/extern/caelum/include/UniversalClock.h:37: error: expected initializer before â??UniversalClockâ?? In file included from /home/zachary/documents/openmw_build/extern/caelum/src/UniversalClock.cpp:23: /home/zachary/documents/openmw_build/extern/caelum/include/Astronomy.h:44: error: expected initializer before â??Astronomyâ?? /home/zachary/documents/openmw_build/extern/caelum/src/UniversalClock.cpp:83: error: expected â??}â?? at end of input make[3]: *** [extern/caelum/CMakeFiles/caelum.dir/src/UniversalClock.cpp.o] Error 1 make[2]: *** [extern/caelum/CMakeFiles/caelum.dir/all] Error 2 make[1]: *** [extern/caelum/CMakeFiles/caelum.dir/rule] Error 2 make: *** [caelum] Error 2 Build output from: make MyGUIOgrePlatform Code: Select all [email protected]:~/documents/openmw_build/build$ make MyGUIOgrePlatform
[  0%] Building CXX object extern/mygui_3.0.1/OgrePlatform/CMakeFiles/MyGUIOgrePlatform.dir/src/MyGUI_OgreDataManager.cpp.o
In file included from /home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:10:
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/include/MyGUI_OgreDataStream.h:14:28: error: OgreDataStream.h: No such file or directory
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:12:18: error: Ogre.h: No such file or directory
In file included from /home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:10:
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/include/MyGUI_OgreDataStream.h:24: error: â??Ogreâ?? has not been declared
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/include/MyGUI_OgreDataStream.h:24: error: expected â??)â?? before â??_streamâ??
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/include/MyGUI_OgreDataStream.h:33: error: â??Ogreâ?? has not been declared
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/include/MyGUI_OgreDataStream.h:33: error: ISO C++ forbids declaration of â??DataStreamPtrâ?? with no type
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/include/MyGUI_OgreDataStream.h:33: error: expected â??;â?? before â??mStreamâ??
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp: In member function â??virtual MyGUI::IDataStream* MyGUI::OgreDataManager::getData(const std::string&)â??:
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:66: error: â??Ogreâ?? has not been declared
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:66: error: expected â??;â?? before â??streamâ??
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:67: error: â??streamâ?? was not declared in this scope
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:71: error: expected type-specifier before â??Ogreâ??
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:71: error: expected â??)â?? before â??::â?? token
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:71: error: expected â??{â?? before â??::â?? token
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:71: error: â??::FileNotFoundExceptionâ?? has not been declared
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:71: error: expected â??;â?? before â??)â?? token
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp: In member function â??const MyGUI::VectorString& MyGUI::OgreDataManager::getDataListNames(const std::string&, bool)â??:
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:94: error: â??Ogreâ?? has not been declared
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:94: error: expected â??;â?? before â??pFileInfoâ??
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:96: error: â??pFileInfoâ?? was not declared in this scope
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:98: error: â??Ogreâ?? has not been declared
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:98: error: expected â??;â?? before â??fiâ??
/home/zachary/documents/openmw_build/extern/mygui_3.0.1/OgrePlatform/src/MyGUI_OgreDataManager.cpp:98: error: â??fiâ?? was not declared in this scope
make[3]: *** [extern/mygui_3.0.1/OgrePlatform/CMakeFiles/MyGUIOgrePlatform.dir/src/MyGUI_OgreDataManager.cpp.o] Error 1
make[2]: *** [extern/mygui_3.0.1/OgrePlatform/CMakeFiles/MyGUIOgrePlatform.dir/all] Error 2
make[1]: *** [extern/mygui_3.0.1/OgrePlatform/CMakeFiles/MyGUIOgrePlatform.dir/rule] Error 2
make: *** [MyGUIOgrePlatform] Error 2

Build output from 'make openmw' is the same as 'make MyGUIOgrePlatform' because it has to build 'MyGUIOgrePlatform' anyway.

'make MyGUIOgre' and 'make esmtool' both compile fine.

All these builds were attempted against the latest revision here:
http://github.com/korslund/openmw
mach1723 wrote: Fixed my build problem by setting the CPLUS_INCLUDE_PATH environment variable to /usr/include/OGRE/, yay!
mach1723 wrote: An update, I have resource overides loading ingame now, and being used instead of the .bsa contents, I think some of my code is bad at the moment, and an feature that I have not implemented is external resources would only be loaded if they had a newer timestamp than the bsa content.

Git branch: http://github.com/Mach1723/openmw/tree/ ... sitive_bsa

Zini wrote: Sorry, does not work here. I am still getting
I assume you are going through the file finder and the path problems (case and path element separator) should have been addressed. But unfortunately they are not.

Also, the performance is bad. We will have to do something about that, but the correctness is more important now.
Zini wrote: This is only guesswork, but some re-enabled and extended diagnostics indicate, that the problem only shows up with resources, that do not exist in the bsa at all.
mach1723 wrote:
Zini wrote:This is only guesswork, but some re-enabled and extended diagnostics indicate, that the problem only shows up with resources, that do not exist in the bsa at all.
If you have the time it would be helpful if you could uncomment the cout's at line 157-159 of components/bsa/bsa_file.cpp and paste the output somewhere, it would be useful to know what FileFinder is returning to be used as a path.
Zini wrote: That's what I meant with
some re-enabled and extended diagnostics
Little point in posting that though, because this part is working fine. All those files are handled correctly and I don't get any errors for them.

The resources, that are not found, are not listed at all by those couts.
mach1723 wrote:
Little point in posting that though, because this part is working fine. All those files are handled correctly and I don't get any errors for them.

The resources, that are not found, are not listed at all by those couts.
I am not quite understanding the issue at this point.. do the resources have an equivalent in the bsa?
Zini wrote: No, they don't. That exactly is the point.
mach1723 wrote:
Zini wrote:No, they don't. That exactly is the point.
I was planning on adding the loading of non-bsa equivalent resources when we have multiple esp/esm support, since I don't see the point of loading a resource that isn't referenced by an esp/esm/bsa.
Zini wrote: Okay, now I understand. Well, what you have implemented is more like an edge case (but thanks for bringing it up; might have missed it otherwise), while what you are still planning to do is the main issue.

Regarding no point in finishing it up now: Take any total conversion (Redemption for example, but IIRC there is at least another one out there). These usually only use one esm, which might not use a bsa at all.


text:
# Tag Info

13

where does that electricity go? The photons from the sun have energy and momentum, but not "electricity". Essentially, a photon (solar or otherwise) striking the solar panel can create an electron-hole pair (EHP) and, if the EHP is within or near the depletion zone, the pair will be separated by the built-in electric field. This results in a ...

5

Developments as of summer 2013 The hottest kid on the organic-PV block is perovskites: in February 2012, Hardin, Snaith & McGehee published an article in Nature Photonics announcing "The renaissance of dye-sensitized solar cells". Then in May 2013, Oxford Photovoltaics Ltd (a spinoff of the University of Oxford), announced it had hit $15.4\%$ efficiency ...

5

It will vary a lot by site, and on the particular PV technology being proposed. So, you need to know: is it monocrystalline silicon, is it CdTe thin-film, or something else? what's the efficiency of the inverter? what's the guarantee on the kit (5 years, 10 years, 20 years)? Using @Martin Beckett's figure of 2000kWh/y, that means, roughly, that your ...

4

The website is clearly supported by lots of money which doesn't guarantee that it reflects the most accurate scientific information. The very page you quoted says under the picture: The material is graphene, also known as graphite... Well, no. Graphene is not the same thing as graphite. Graphite is a 3-dimensional material used to produce pencils - and ...

4

There are several ways to design the circuit. If it's a Smart system, then when there's surplus power, additional devices will get turned on, to use it: dishwashers, washing machines, or immersion heaters in hot-water storage tanks. If there's still surplus after that, then it's as below. If it's grid-connected, with an inverter, then it's usually designed ...

3

Boden & Bagnall (pdf) looked at this question of reflection, in their paper Bio-Mimetic Subwavelength Surfaces for Near-Zero Reflection Sunrise to Sunset. Their estimate of the proportion of photons reflected, from sunrise to sunset, in a fixed PV system is 20%, rather than your estimate of one-third. The proposed etalon is not an efficient or ...

3

The Juno mission to Jupiter, if successful, will be the furthest that we've managed to get PV work to date. It was launched on 5 August 2011; it is already beyond Mars orbit, and will, all being well, reach Jupiter in July 2016. There isn't a physics limit to the distance at which photovoltaics work. If a photon of suitable wavelength reaches the PV (and we ...

2

Hmm, I think it would depend on the panel circuit. If the panel is not connected, for example, the charge potential would still be created at the leads, but since it's not being drained into a storage device (or otherwise used), the solar medium would saturate at some measurable voltage boundary. Whether it turns to heat at that point depends on whether ...

2

The fundamental mechanism is the same. They are both photovoltaics. The cells on satellites are far more expensive and efficient. Because satellite launch costs dominate any other cost, you might as well pay for a high-efficiency cell. Terrestrial cells are 10-20% efficient, usually made from silicon. The ones in satellites can be double that efficiency, ...

2

Essentially yes they work in the same way. There are slightly different designs, traditionally the space ones had highest performance because money was no object but because it takes years to space qualify a piece of hardware the gap with the latest cheap tech has shrunk. The challenge on most spacecraft is mounting and deploying the panels, which usually ...

2

It sounds like what you're really getting at is a way to make the oft-quoted performance numbers for various electricity generation options (PV included, of course) more understandable to non-experts. Having spent a few years teaching this material, I find that the right approach is to do more than simply quote the fact that X familiar object uses Y power or ...

2

The trouble with an etalon is getting the light into it in the first place. Bearing in mind we want normal inidence to maximise the light intensity our solar panel looks something like: But solar panels aren't transparent (obviously since they reflect two thirds of the incident light) so if you wanted to use a second panel you'd need something like: ...

2

Yes nuclear batteries are reasonably common, it's generally easier to just use the heat given off to drive either a stirling cycle motor or a thermo-electric generator rather than use the energy of the emitted particle directly. They are very useful when you need power for many years without being able to recharge or replace the batteries eg. in space ...

1

You can work through the derivation, but I think you are after a more intuitive answer to the question. Here is the way I think about it. Why exp()? You will know that the I-V curve a resistor is $V=IR$. That is to say that the when you put a voltage across a resistor the current is linearly related to the voltage simply through a constant of ...

1

You're absolutely right that annual mean power is by no means a useful guide to the peak power a panel might generate. You can find places on Earth where annual mean insolation is well below $100~W/m^2$, and places where it's higher than $250~W/m^2$: but at both those places, there will be times when peak insolation is $1000~W/m^2$ or more. And so all PV is ...

1

The short answer is that when an electron in the valance band absorbs energy from a photon to become a conduction band (mobile) electron, both that electron as well as the hole "left behind" in the valence band can participate in an electric current. Thus, it is said, an electron-hole pair is created. Where there was no mobile charge carrier, there are now ...

1

Short answer: out to Mars usually, maybe to the asteroid belt depending on mission details, but not Jupiter unless it's an unusually low-powered spacecraft. Forget Saturn and beyond. PV works great near the Earth, at 1 AU from the Sun, where we receive about 1400 Watts per square meter. (It's more like 1000 on the ground due to our atmosphere in the way.) ...

1

Depends what you mean by "effective". They work quite well at Earth, 150 Million km from the sun. But because the sun's light (or any light) spreads out into a spherical shell - as you double the distance, you only receive 1/4 as much light. So at the orbit of Mars, around 1.5 times as far from the sun, you get only about 1/2 as much sunlight. At the ...

1

A unit of electricity (in most countries) is a kilowatt-hour, ie a 1000 W appliance running for 1hour, or a 100W appliance for 10hours. Typically a regular light bulb (if you still have them) is 100W, a big screen LCD TV is about 250W and an air conditioner or furnace will be between 3000 and 10,000 Watts

1

Yes you can make transparent solar cells. Obviously their efficiency won't be nearly as high as their opaque cousins. http://web.mit.edu/newsoffice/2011/transparent-solar-windows-0415.html There is nothing impossible about a single material that absorbs UV and IR but not visible. Organic semiconductors can do that. Inorganic materials like silicon almost ...

1

Yes, you can have photovoltaic tuned to different areas of the EM Spectrum. No, you can't cover the whole EM spectrum with one detector.[1] In fact, limiting yourself to photovoltaics is throwing out most of the spectrum- you are only looking at materials where EM radiation will cause an electron to jump bands. Is there a particular bandwidth you are ...

1

Let's look at the amount of energy (in Joules) in one second - i.e. in SI units of power, Watts: that way, you can directly compare between them, and scale up the times as you choose. So, for Wh, kWh, MWh, etc - it's each of the activities below, for one hour 2 W average power used in charging a mobile (cell) phone 20 W laptop power consumption, at low ...

1

the injection level is defined as $\delta n/p_0$ where $\delta n$ is the minority carrier (e.g. electrons') density excess at non-equilibrium while $p_0$ is the equilibrium density of the majority carriers (e.g. holes). From the formula which is a ratio of two very similar densities, it's clear that the injection level is dimensionless; "high" and "low" ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Motion of charge in magnetic field with drag force [closed]

Say you have a charged particle in a region that contains a fluid that will produce a drag force that goes as $F=-kv$ where $v$ is the speed and $k$ is some constant. The region also contains a uniform magnetic field. Suppose you give the particle some initial velocity $v_0$ in the plane perpendicular to the magnetic field. What will be the particle's subsequent motion? Please provide semi-quantitative answers. Note that this is not a homework question.

• What have you attempted so far? Dec 25 '13 at 0:00
• My guess is that it is a logarithmic spiral based on the fact that the speed falls off exponentially (from the drag force), although I'm not sure how to show it analytically.
– zeta
Dec 25 '13 at 0:05

Let's find the complete solution of the problem.

A complete solution of the problem would be the solution to the linear ODE,

$m \dot{\mathbf{v}} =q\mathbf{v} \times \mathbf{B}-k \mathbf{v}$

Assume without loss of generality that the magnetic field is pointed along the z-axis,so $\mathbf{B} = B \mathbf{\hat{z}}$. So our equation simplifies to,

$m \dot{\mathbf{v}} =qB\mathbf{v} \times \mathbf{\hat{z}}-k \mathbf{v}$

Dividing both sides of the equation by $m$ and for simplicity in the notation, let $\omega=\frac{qB}{m}$ and $\gamma=\frac{k}{m}$.

So,$\dot{\mathbf{v}} =\omega\mathbf{v} \times \mathbf{\hat{z}}-\gamma \mathbf{v}$

Using $\mathbf{v}=\begin{pmatrix} v_{x}\\v_{y}\\v_{z} \end{pmatrix}$ and writng the given eqaution in matrix form we have,

$\dot{\mathbf{v}} =\begin{pmatrix} -\gamma & \omega & 0 \\ -\omega & -\gamma & 0 \\ 0 & 0 & -\gamma \end{pmatrix} \mathbf{v}=A \mathbf{v}$

This is linear ODE which can be solved using the matrix exponenetial as,

$\mathbf{v}=e^{At} \mathbf{v_0}$ To simply this equation we can find the eigenvaules of A,use a similarity transform to convert it to a diagonal matrix which this greatly simplifies the matrix exponential.

The eigenvalues and the corresponding eigenvectors are,

$\lambda_{1}=-\gamma ,\mathbf{v_1}=\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

$\lambda_{2}=-\gamma-i\omega ,\mathbf{v_2}=\begin{pmatrix} 1 \\ i \\ 0 \end{pmatrix}$

$\lambda_{3}=-\gamma+i\omega ,\mathbf{v_3}=\begin{pmatrix} i \\ 1 \\ 0 \end{pmatrix}$

Using $S=[ \mathbf{v_1} \mathbf{v_2} \mathbf{v_3}]$ and performing a similarity transform on the matrix A,$S^{-1}AS=D$ where D is diagonal matrix with the eigenvalues as the diagonal elements.And here we witness the power of similarity transformations as,

$e^{At}= S \begin{pmatrix} e^{\lambda_{1}t} &0 &0 \\0 & e^{\lambda_{2}t} &0 \\ 0& 0& e^{\lambda_{3}t} \end{pmatrix} S^{-1}$

(After some tedious calculations ans using $e^{ix}=\cos{x}+isin{x}$)

$=\begin{pmatrix} e^{-\gamma t}\cos(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\ -e^{-\gamma t}\sin(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\0 & 0 &e^{-\gamma t}\end{pmatrix}$

Therefore, $\mathbf{v}=\begin{pmatrix} e^{-\gamma t}\cos(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\ -e^{-\gamma t}\sin(\omega t) &e^{-\gamma t}\sin(\omega t) &0\\0 & 0 &e^{-\gamma t}\end{pmatrix} \mathbf{v_0}$

Writing out the components,

$v_{x}=e^{-\gamma t}(v_{x_0} \cos{\omega t}+v_{y_0} \sin{\omega t})$

$v_{x}=e^{-\gamma t}(-v_{x_0} \sin{\omega t}+v_{y_0} \cos{\omega t})$

$v_{x}=e^{-\gamma t} v_{z_0}$

This the just the equation of a helix with both the pitch and radius decreasing exponentially with $\gamma$.However,the angular frequency is the same as that without drag,$\omega$.

Here is a sample trajectory,

The magnetic field does no work, so it does not change the speed of the particle. The drag force results in an exponential decrease in speed. If the drag force is very small, one would expect the particle to move in a circle with radius $r = \frac{mv}{qB}$, where $v = v_0 e^{-\frac{k}{m} t}$, i.e., a spiral.

A precise solution would proceed from

$$m\dot{\mathbf{v}} = q B \, \mathbf{v} \times \hat{\mathbf{z}} - k \,\mathbf{v}$$

This is a set of first-order, linear, coupled ODE's. Not very difficult to solve. In particular, $$\dot{\mathbf{v}} = M \,\mathbf{v}$$ where $M$ is a matrix, so you would start by finding its eigenvectors.


text:
# Find the potential energy of three metal disks

###### Question:

Three charged metal disks are arranged as shown (cutaway view). The disks are held apart by insulating supports not shown in the diagram. Each disk has an area of 2.5 m2 (this is the area of one flat surface of the disk). Use the value 8.85e-12 C2/(N·m2) for e0.The charge Q1 = 3e-8 coulombs, and the charge Q2 = 2e-7 coulombs.
The figure of the disks can be viewed at this website: http://www.webassign.net/mi3/17-070-3plates_D_res.jpg.
What is the potential difference between VC-VB and VD-VF?

#### Similar Solved Questions

##### Castagnola Inc. has provided the following data for the month of January. There were no beginning...
Castagnola Inc. has provided the following data for the month of January. There were no beginning inventories, consequently, th direct materials, direct labor, and manufacturing overhead applied listed below are all for the current month. Direct materials Direct labor Manufacturing overhead applied ...
##### 2. (40 pts) You are a member of the Resistance, tasked with destroying the latest Death...
2. (40 pts) You are a member of the Resistance, tasked with destroying the latest Death Star. After planting a bomb on the Death Star, you escape in your 0.6c spaceship (event E). When you are 1 light-minute from the Death Star, you hit the button to trigger the bomb (event F). The trigger message i...
##### Airs, have been experimentally determined. 5) The following five data points, X-Y pairs, have been experimentary...
airs, have been experimentally determined. 5) The following five data points, X-Y pairs, have been experimentary (1.00, 3.88), (2.00, 8.05), (3.00, 12.01), (4.0, 16.17), (5.00, 19.78) Use a least squares criterion to determine ast squares criterion to determine the equation of the straight line, = b...
##### Room-temperature water boils spontaneously in a vacuum- on the Moon, for example. Could you cook an...
Room-temperature water boils spontaneously in a vacuum- on the Moon, for example. Could you cook an egg in this boiling water? Explain Find what heat in calories (cal) is required to increase the temperature of 60 g water from 0°C to 55°C. The specific heat capacity of water is 1 cal/ g&bull...
##### 5. PROBLEM Compare the mean free path of C-12 to the diffusion length.
5. PROBLEM Compare the mean free path of C-12 to the diffusion length....
##### How do you prove the identity (sin^3 x + cos^3 x)/(sin x +cos x) = 1 - sin x cosx?
How do you prove the identity (sin^3 x + cos^3 x)/(sin x +cos x) = 1 - sin x cosx?...
##### Help please! Show work 5. For the following exothermic reaction, tell how the equilibrium will shift...
Help please! Show work 5. For the following exothermic reaction, tell how the equilibrium will shift in response to... 2H2O2(aq) = 2H20(1) + O2(g) (a) allowing O2 to escape. (b) adding H20 [trick question!). (c) increasing the total pressure. (d) increasing the temperature. (e) justify your answer t...
##### A study regarding the relationship between age and the amount of pressure sales personnel feel in...
A study regarding the relationship between age and the amount of pressure sales personnel feel in relation to their jobs revealed the following sample information. At the 0.02 significance level, is there a relationship between job pressure and age? Age (years) Less than 25 25 up to 40 40 up to 60 6...
##### Your sister owns a small clothing store
Your sister owns a small clothing store. During a conversation at a family dinner, she mentions her frustration with having to manually track and reorder high demand items. She would like an automated system but has a very small budget. Write a 4-5 page paper in which you create a plan for a low-co...
##### Short Answer Calculate the median of the following distribution. State Crime Rate per 1,000 Persons New...
Short Answer Calculate the median of the following distribution. State Crime Rate per 1,000 Persons New Hampshire 1.93 Maine 2.53 Massachusetts 2.82 Rhode Island 2.97 Vermont 2.40 Calculate the mean for the following distribution. State Crime Rate per 1,000 Persons New Hampshire 1.93 Maine 2.53 Mass...
##### Only need part C
Only need part C...
##### The moment generating function of a random variable X is as follows: 1-Xt Find the probability...
The moment generating function of a random variable X is as follows: 1-Xt Find the probability that X is within 0.5 standard deviation from its mean....


text:
## Optimal Evil

This is the main board for discussing philosophy - formal, informal and in between.

### Optimal Evil

There are things that are generally accepted as wrong, but for which, in practice, the optimal incidence over time is not zero.

A very simple example is disease. In general, we agree that disease is bad, but there are many diseases which it is better that people get over their life time. Chicken pox was a clear case, where getting the disease in childhood was good enough that parents would intentionally expose their kids to the disease. I say "was a clear case" because now we have a vaccine for chicken pox, but vaccines use the same mechanism: we intentionally expose ourselves to small harms. We can also see this in the rate of diseases like asthma in 1st children, where it's likely that too-clean childhoods lead to health problems later in life.

A similar idea can be seen in something as straightforward as pain, where it is better to experience non-zero amounts of pain over the course of one's life. Here, there is a difference, in that not-pain is defined in relation to pain, so that experiencing the evil helps to define the good, as well as to make us aware of it.

Finally, we can look at something broader like the homicide rate. One might think that zero homicides would be the ideal, but in fact a homicide rate of zero is more likely an indicator that we are over-policing, population homogeneity, low population, etc. It remains true that, all else equal, zero murders is better than not zero murders, but in practice all else is not equal: the observation that zero murders are taking place is not necessarily a good observation, and choosing between realistically achievable societies, we will choose one with a non-zero murder rate. More striking, a low-but-non-zero murder rate is likely to be predictive of a better society, i.e. if all you have to go on is murder rate, you should choose a society with a non-zero rate.

These are strange and seemingly difficult cases for consequentialist moral systems. They seem to point to a class of phenomenon which is understood to be negative in every individual case, but when summed across society shows that, though every case is bad, eliminating all cases is worse in practice.
User Control Panel > Board preference > Edit display options > Display signatures: No.
Carleas
Magister Ludi

Posts: 6107
Joined: Wed Feb 02, 2005 8:10 pm
Location: Washington DC, USA

### Re: Optimal Evil

If you put it sharply like this I think moral systems are moot entirely.

If it is all about response patterns, which it is, then every system is a distraction from the eminent fact of it, suggesting that wherever there is a system in place, more evil acts can slip through h cracks of humanity than when there is just pure natural vigilance as in the not so noble not so savage state of frontiers and such.
A moral system offers systemic licence to interpreters of it. You can throw any handful of pertinent terms into a sack and juggle it and you can believe in upholding any of the outcomes on the table, and morality is just that. We know things are important, and we know wed like for these things to go the way we want. Sometimes another wants them to go another way, sometimes another hopes the same as we do, in any case, we would like there to be rules so that things go our way. But what if that is just stupid? What if the case turns out to be that we just need to pay attention?

the whole idea that there are "conflicting goods" is a misconception of the word "good". Who said that if two things are in conflict, they are good?
Who says that the people who wasn't these things deem them "good"? I think no one deems an abortion "good". Except whats that girls name who made the show Girls, who said she hadn't had an abortion but would like one. She thinks they're good for you. But most things that people want are lesser evils.
Conflicting lesser evils isn't any intellectual problem at all, you first try to derive the even lesser evil from the two, and ultimately the least evil. You can call that "best", but that doesn't mean its what anyone would consider to be good.

Goods can also mean resources, in a raw groceries kind of way. Conflicting goods there would mean leaking bottles of water and precious salt-sculptures, or any bad chemistry.

What to do in a world where there are occurrences of bad chemistry?
If we answer his question we are beyond evil, and perhaps a little bit closer to good.
I don't think there is any sense in going beyond good, except as in great or perfect or magnificent.

"Beyond magnificent and disgraceful"
"beyond perfect and flawed"

No one wants to go there. Its where old people go when they're boring or have evil children.

For behold, all acts of love and pleasure are my rituals

Jakob
ILP Legend

Posts: 7193
Joined: Sun Sep 03, 2006 9:23 pm
Location: look at my suit

### Re: Optimal Evil

Carleas wrote:There are things that are generally accepted as wrong, but for which, in practice, the optimal incidence over time is not zero.

I believe that's because we make use of shorthand in the espousal of our moral or ethical convictions.

For example we would say violence is bad... but if you, or better yet a child, is being assaulted, well then the use of violence in defense is not only good but "heroic" even.
One could argue that what was wrong in this telling was to suppose "violence" was ever the problem... that this notion was merely a shorthand for saying try other options first, that violence is among the less preferable methods of resolving conflicts.

Disease is a perfect example of this shorthand... because it is clearly the consequences of certain diseases that we find alarming, consequences that vaccines do not share and in fact help prevent.
Which bleeds into why it's better to experience a non-zero amount of pain...

Often the world forces you to choose the lesser of its many evils... there's not enough food for everyone, someone WILL starve... no one wished this to be the reality, but here it is. What now?
Because of the nature of the world we find ourselves in, learning to endure pain and developing discipline, not only helps us be better prepared but also to better relate to our fellow human beings...
In a different world, under different conditions... this might not be true. We might wish for such a world and even agree that it would be better...
But in our current world, we could get hit hard at any time, so we need to toughen up to minimize the damage it will cause us... a vaccine, if you will.

These are strange and seemingly difficult cases for consequentialist moral systems. They seem to point to a class of phenomenon which is understood to be negative in every individual case, but when summed across society shows that, though every case is bad, eliminating all cases is worse in practice.

You seem to have missed the mark... like with diseases and pain, the vaccine is meant to be the lesser of evils, but if the cure is worse than the disease, it's better to suffer the disease.
There are no stupid questions, just stupid people.

Philosopher

Posts: 2618
Joined: Thu Jan 27, 2005 7:32 pm
Location: Denmark

### Re: Optimal Evil

Jakob wrote:wherever there is a system in place, more evil acts can slip through h cracks

I suspect that this isn't true. Even bad moral systems tend to make things better. Studies have found that putting up the 10 Commandments up in a class where people are taking a test will decrease the rate of cheating. This remains true when the 10 Commandments are replaced by other, novel and non-religious statements of ethical principle. The mere priming of a moral system tends to get everyone to act more morally.

And we should expect this, since moral systems are a coordination game that helps large groups cooperate. Priming for morality is priming for group cooperation. But one takeaway seems to be that almost any system is better than no system (despite that, as you note, systems can also provide clever scoundrels with ways to justify defection as morally permissible).

Jakob wrote:Conflicting lesser evils isn't any intellectual problem at all

I think you're basically right here, if I'm understanding you correctly. But it is weird where the conflict is between a good and itself. So, in the case of abortion, if we should prefer a society with non-zero abortions, we have the evil of abortion conflicting with the evil of the absence of abortion. That strikes me as an intellectual problem.

Mad Man P wrote:For example we would say violence is bad... but if you, or better yet a child, is being assaulted, well then the use of violence in defense is not only good but "heroic" even.

I think this case is more local than I intend. I agree that local trade-offs are straightforward, e.g. using violence to prevent violence. But in the case of murder, we aren't talking about local murder-to-prevent-murder, but about the global issue that non-zero murders across a population is better than zero murders. These murders aren't specifically in exchange for anything, although you may be right that it is ultimately just a trade-off of values, e.g. freedom for murder.

Perhaps mine is just an epistemic point: we can't know everything about a society, so we go off of imperfect proxies (the "shorthand" you note), and that leads to weird outcomes. So, high murder rates can be an indication of social unrest, inequality, distrust, desperation, and very low murder rates can indicate over-policing, low or aging population, or lack of valuable social diversity.

And I also think I may be lumping separate phenomena together. The murder rate point does seem distinct from the pain point; while murder is good as a fuzzy proxy for other good things, pain can be a good in itself (e.g. by giving definition to not-pain).
User Control Panel > Board preference > Edit display options > Display signatures: No.
Carleas
Magister Ludi

Posts: 6107
Joined: Wed Feb 02, 2005 8:10 pm
Location: Washington DC, USA

### Re: Optimal Evil

Carleas wrote:Perhaps mine is just an epistemic point: we can't know everything about a society, so we go off of imperfect proxies (the "shorthand" you note), and that leads to weird outcomes.

I would agree that the big problem that faces consequentialists is when you can't predict the consequences of an action or inaction... how then do you determine the moral act?
But alternatively, if you subscribe to one of the shorthands like "violence is bad" and take it to be a moral edict, you would no longer be a consequentialist... but an idealist.

I would say I'm a utilitarian in my convictions, in that I take human well being and thriving to be the sole concern of morality...
We lack the information and possibly intellect to perfectly determine how best to conduct ourselves to maximise our well-being and minimize suffering... I would agree this is a problem, but the solution can't be to change the objective.

And I also think I may be lumping separate phenomena together. The murder rate point does seem distinct from the pain point; while murder is good as a fuzzy proxy for other good things, pain can be a good in itself (e.g. by giving definition to not-pain).

I don't buy it. The only thing we need to define anything is an alternative... pain need not be part of the equation for us to have a "normal" and "non-normal" state.
Pain is merely how a biological survival instinct manifests... so we don't burn ourselves by sticking our hand into a fire, for example... especially when we're young and stupid.

If you take pain to mean something more broad as say "dislike" well then yes it is a necessary consequence of having any preference... but the extreme discomfort of pain is NOT necessary for preference.
Even if you take pain out of the equation it would not change our preference for bliss over non-bliss.

When people use the term "1st world problems" they are referring to this very phenomenon...
Where in the absence of hardship and pain the things we concern ourselves with and care deeply about have shifted.
There are no stupid questions, just stupid people.

Philosopher

Posts: 2618
Joined: Thu Jan 27, 2005 7:32 pm
Location: Denmark

### Re: Optimal Evil

Mad Man P wrote:I would agree that the big problem that faces consequentialists is when you can't predict the consequences of an action or inaction... how then do you determine the moral act?

I think the problem is even worse than that: I think we're fundamentally unable not only to predict the future, but to judge the present. We can't fully know the present state of the world, so we're limited in our ability to say how good or bad it is. On top of that, we don't know the full space of alternatives, so we may have a desirable present that is still less desirable than some alternative that we've thwarted through our actions.

To the latter, compare it to investing money. If we invest money and some time later we have 5% returns, we might think, "Hey great, 5% returns!" But if alternative investments would have gotten 10% returns, and particularly if some straightforward strategy like a fund that tracks the S&P would have gotten 10% returns, we should score our investments as a failure. Despite ending in the black, we've missed an opportunity, and are worse off relative to that alternative.

So too with moral consequences. If we increase policing and prevent a murder, it looks like a good outcome. But if we divert money from health care or education to policing, or if policing itself stifles some type of positive interactions (say, it breaks up gangs, and in doing so it also destroys youth social support networks), we may well make the world worse off than a naive alternative. So even if some act makes the world better at $$t_2$$ than it was at $$t_1$$, it should be scored as a loss if a different would have made the world at $$t_2$$ better still.

Mad Man P wrote:We lack the information and possibly intellect to perfectly determine how best to conduct ourselves to maximise our well-being and minimize suffering... I would agree this is a problem, but the solution can't be to change the objective.

I think I agree, but if I might play devil's advocate: what meaning does a moral system have if it commands us to maximize some unknowable quantity? What would be the difference between a system that says "Maximize happiness" when the quantity of happiness is unknowable, and one that just says, "Maximize X", and doesn't define X? Is a system that seeks to maximize a known and immeasurable thing more prescriptive than one that seeks to maximize an unknown?

Mad Man P wrote:When people use the term "1st world problems" they are referring to this very phenomenon...
Where in the absence of hardship and pain the things we concern ourselves with and care deeply about have shifted.

I think this is a good example of my point. It would be better that people suffer a bit in childhood and become tougher adults who recognize their blessings, than that they have idyllic childhoods free from real suffering, and as adults experience minor inconveniences as world-shattering tragedies.

I will also say that, since having a kid, my perspective on how valuable pain is has changed. Kids are super dumb, it's remarkable how little common sense they have. It's really important to let them hurt themselves to help them understand how the world works. They'll happily crawl of the edge of a couch with no regard for how high it is, but once they've fallen they start to take heights seriously. Since babies are build to be pretty resilient, it's very good to let them experience these small, non-catastrophic falls.

And, further to my previous point, it does give them some perspective. Kids experience silly things as absolutely tragic, and unless they have something to contrast against the comfort of their lives, they'll keep seeing them as tragic.

I take your point that the difference between bliss and 'normal' is real and informative, but if we think of experience of a spectrum between good and bad, only experiencing bliss and normal makes normal seem like it's on the 'bad' end of the spectrum. If we add bad experiences, normal seems like the middle of the spectrum, between good and bad. (This is a bit folk-psychology, but true as a rough model. Too much or too frequent suffering, though, will train the brain to experience suffering.)
User Control Panel > Board preference > Edit display options > Display signatures: No.
Carleas
Magister Ludi

Posts: 6107
Joined: Wed Feb 02, 2005 8:10 pm
Location: Washington DC, USA

### Re: Optimal Evil

Carleas wrote:I think we're fundamentally unable not only to predict the future, but to judge the present. We can't fully know the present state of the world, so we're limited in our ability to say how good or bad it is. On top of that, we don't know the full space of alternatives, so we may have a desirable present that is still less desirable than some alternative that we've thwarted through our actions.

We don't need to "fully" know the present state of the world to judge what we DO know... We don't need an exhaustive list of alternatives to make value judgements between the alternatives we DO know.
Omniscience is not necessary, though it'd be real useful.

Is a system that seeks to maximize a known and immeasurable thing more prescriptive than one that seeks to maximize an unknown?

I'd agree that would make little difference, but I disagree that it applies to our conversation. Well-being is not “immeasurable” it is at worst just measured imprecisely.

Mad Man P wrote:When people use the term "1st world problems" they are referring to this very phenomenon...
Where in the absence of hardship and pain the things we concern ourselves with and care deeply about have shifted.

I think this is a good example of my point. It would be better that people suffer a bit in childhood and become tougher adults who recognize their blessings, than that they have idyllic childhoods free from real suffering, and as adults experience minor inconveniences as world-shattering tragedies.

You seem to have lost sight of my objection... pain cannot be said to be "good in itself" as it's value is contingent.
I agree that we DO happen to live in a world where it has the value of a vaccine, but it's easy to imagine a world where it has no value...
There are no stupid questions, just stupid people.

Philosopher

Posts: 2618
Joined: Thu Jan 27, 2005 7:32 pm
Location: Denmark

### Re: Optimal Evil

Hello Mad Man P. I'm glad You can imagine a world without. a value for pain , because I can not, having never the pleasure of experiencing but a mixture of both.
Meno_
ILP Legend

Posts: 7280
Joined: Tue Dec 08, 2015 2:39 am
Location: Mysterium Tremendum

### Re: Optimal Evil

Meno_ wrote:Hello Mad Man P. I'm glad You can imagine a world without. a value for pain , because I can not, having never the pleasure of experiencing but a mixture of both.

Are you in constant pain?
Do you constantly compare every experience in your life to the worst pain in order to measure your response to it?
I mean this could be true of you... but for me the baseline that I measure my response from is my "normal" state that has nothing to do with pain.

Normal is the state I'm in most of the time... when I experience something NOT normal, I measure it by comparison to normal... that's how I measure pain too.
I can easily imagine a life in which someone only experiences "normal" and "better than normal"...

It's hard to imagine what it would be like to have NEVER known pain... but it's equally hard for me to imagine what it would be like to have never known my father.
That has more to do with my inability to perfectly imagine being someone other than myself...
There are no stupid questions, just stupid people.

Philosopher

Posts: 2618
Joined: Thu Jan 27, 2005 7:32 pm
Location: Denmark


text:
# Help proving Leibniz rule for weak derivatives in Evans' PDEs.

I'm having difficulty proving Leibniz' formula for weak derivatives as presented in Evans PDE. I was hoping that someone could maybe shed some light on it for me? To be honest I'm not entirely comfortable with weak derivative concept. I'm actually an economist by trade hoping to improve my maths. My question is very closely related to the following one posted on this website: Evans's proof of the Leibniz's formula for the weak derivatives in Sobolev spaces however my undrstanding isn't as great as the original poster. I hope this exercise will help others struggling with weak derivatives.

Disclaimer. I have copied the latex code from @Cookie's question as it's exactly the same as Evans' theorem in the book in a bid to save time. Since I'm relatively new to this site I don't know if this is acceptable or not. If not please tell me and I'll rewrite the code.

The theorem and proof is as follows:

• $W^{k,p}(U)$ denotes a Sobolev space that consists of all locally summable functions $u : U \rightarrow \mathbb{R}$ such that for each multiindex $\alpha$ with $|\alpha| \le k$, $D^\alpha u$ exists in the weak sense and belongs to $L^p(U)$.
• $C_c^\infty(U)$ denotes the space of infinitely differentiable functions $\phi : U \rightarrow \mathbb{R}$, with compact support in $U$, and the function $\phi$ is called a test function.

Theorem 1 (Properties of weak derivatives). Assume $u,v, \in W^{k,p}(U), |\alpha| \le k$. Then

$\quad$ (iv) If $\zeta \in C_c^\infty(U)$, then $\zeta u \in W^{k,p}(U)$ and $$D^\alpha (\zeta u)=\sum_{\beta \le \alpha} {\alpha \choose \beta} D^\beta \zeta D^{\alpha - \beta} u \qquad \textit{(Leibniz' formula)} \tag{7}$$ $\quad$ where ${\alpha \choose \beta} = \frac{\alpha!}{\beta!(\alpha-\beta)!}$.

Again, I didn't list properties $\text{(i)-(iii)}$ because I understood them already.

Proof (of property $\text{(iv)}$). We prove $\text{(7)}$ by induction on $|\alpha|$. Suppose first $|\alpha|=1$. Choose any $\phi \in C_c^\infty (U)$. Then \begin{align} \int_U \zeta u D^\alpha \phi \, dx &= \int_U u D^\alpha (\zeta \phi) - u(D^\alpha \zeta) \phi \, dx \\ &= - \int_U (\zeta D^\alpha u + u D^\alpha \zeta) \phi \, dx \end{align} Thus $D^\alpha (\zeta u)=\zeta D^\alpha u + u D^\alpha \zeta$, as required.

$\quad$ Next assume $l < k$ and formula $\text{(7)}$ is valid for all $|\alpha| \le l$ and all functions $\zeta$. Choose a multiindex $\alpha$ with $|\alpha| = l+1$. Then $\alpha = \beta + \gamma$ for some $|\beta|=l, |\gamma| = 1$. Then for $\phi$ as above,

\begin{align} \int_U \zeta u D^\alpha \phi \, dx &= \int_U \zeta u D^\beta (D^\gamma \phi) \, dx \\ &= (-1)^{|\beta|} \int_U \sum_{\sigma \le \beta} {\beta \choose \sigma} D^\sigma \zeta D^{\beta - \sigma} u D^\gamma \phi \, dx \tag{A} \\ &= (-1)^{|\beta|+|\gamma|} \int_U \sum_{\sigma \le \beta} {\beta \choose \sigma} D^\gamma(D^\sigma \zeta D^{\beta - \sigma} u ) \phi \, dx \tag{B} \\ &= (-1)^{|\alpha|} \int_U \sum_{\sigma\le \beta} {\beta \choose \sigma} [D^\rho \zeta D^{\alpha - \rho} u + D^\sigma \zeta D^{\alpha - \sigma} u] \phi \, dx \tag{C} \\ &= (-1)^{|\alpha|} \int_u \left[\sum_{\sigma \le \alpha} {\alpha \choose \sigma} D^\sigma \zeta D^{\alpha - \sigma} u \right] \phi \, dx. \tag{D} \end{align}

Just to make sure I am understanding this correctly, the first part of the proof follows because

\begin{align} \int_U u D^\alpha (\zeta \phi) - u(D^\alpha \zeta) \phi \, dx &=\int_U u \zeta D^\alpha \phi\, dx + \int_U u \phi D^\alpha \zeta\,dx - \int_U u(D^\alpha \zeta) \phi \, dx \\ &=\int_U \zeta u D^\alpha \phi \, dx \end{align} and we have since $\zeta \phi \in C^\infty_c(U)$ \begin{align*} \int_U \zeta u D^\alpha \phi \, dx=\int_U u D^\alpha(\zeta \phi)\, dx - \int_U u(D^\alpha \zeta)\phi \,dx&=-\int_U D^\alpha u (\zeta \phi)\,dx-\int_U u(D^\alpha \zeta)\phi \,dx \\ &= -\int(\zeta D^\alpha u + u D^\alpha \zeta)\phi\,dx \end{align*}

I also understand how to get (A). My main problem is that I don't see how to get from (A) to (B), (B) to (C) and (C) to (D). Many thanks.

(A) to (B): Usual integration by parts formula ($|\gamma|=1$).

(B) to (C): $|\alpha|=|\beta|+|\gamma|,$ and $\rho=\sigma+\gamma,$ so $\beta-\sigma=(\beta+\gamma)-(\sigma+\gamma)=\alpha-\rho$.

(C) to (D): Let us call the terms in (C) with $\rho$ "sum 1" and the terms without "sum 2." Observe that $\binom{\alpha}{\sigma}=\binom{\alpha_{1}}{\sigma_{1}}\binom{\alpha_{2}}{\sigma_{2}}\cdots\binom{\alpha_{n}}{\sigma_{n}},$ so if $\gamma=e_{j},$ we can see that (when $\sigma\geq \gamma$): $$\binom{\beta}{\sigma}+\binom{\beta}{\sigma-\gamma}=\left(\binom{\beta_{j}}{\sigma_{j}}+\binom{\beta_{j}}{\sigma_{j}-1}\right)\prod_{i=1,i\neq j}^{n}\binom{\beta_{i}}{\sigma_{i}}=\binom{\beta_{j}+1}{\sigma_{j}}\prod_{i=1,i\neq j}^{n}\binom{\beta_{i}}{\sigma_{i}}=\binom{\alpha}{\sigma}.$$ Similarly, when $\sigma+\gamma\leq\beta,$ $\binom{\beta}{\sigma}+\binom{\beta}{\sigma+\gamma}=\binom{\alpha}{\sigma+\gamma}$ (just let $\sigma'=\sigma+\gamma$ and use the above).

We have three cases to consider. Either $\rho\leq\beta,$ so there is a match between a term in sum 1 and a term in sum 2; $\rho\not\leq\beta,$ so there is a term in sum 1 with no match; or $\sigma\leq\beta$ is such that there is no corresponding $\rho=\sigma,$ so there is a term in sum 2 with no match. For case 1, if $\rho\leq \beta,$ applying the equality above for $\binom{\beta}{\sigma}D^{\rho}\zeta D^{\alpha-\rho}u+\binom{\beta}{\sigma+\gamma}D^{\sigma+\gamma}\zeta D^{\alpha-\sigma-\gamma}u$ (where the first term comes from sum 1, and the second term comes from sum 2), we get $\binom{\alpha}{\sigma+\gamma}D^{\sigma+\gamma}\zeta D^{\alpha-\sigma-\gamma}u$ (note that the derivatives are exactly the same, so we're just using the formula for these binomial coefficients). For case 2, if $\rho\not\leq\beta,$ then $\sigma_{j}=\beta_{j},$ so $\binom{\beta_{j}}{\sigma_{j}}=1=\binom{\alpha_{j}}{\sigma_{j}+1},$ which gives us $\binom{\beta}{\sigma}=\binom{\alpha}{\sigma+\gamma},$ since none of the other factors are affected by adding $\gamma.$ This leaves case 3, where $\sigma\leq\beta$ is not of the form $\sigma'+\gamma$ for some $\sigma'\leq\beta.$ But this means that $\sigma_{j}=0,$ so again $\binom{\beta_{j}}{\sigma_{j}}=1=\binom{\alpha_{j}}{\sigma_{j}},$ which implies that $\binom{\beta}{\sigma}=\binom{\alpha}{\sigma}$ for such $\sigma.$ It should be clear that each $\sigma\leq \alpha$ appears as one of the three cases above, so this completes the proof.

• Many thanks for your answer. I don't suppose you could go through in slightly more detail the left hand equality with the sum of the multinomial coefficients. What do you mean by $\gamma=e_j$? Cheers. – mark Jun 27 '17 at 13:30
• Sorry, I meant that $\gamma$ is the $j$th standard unit vector, so equal to one in the $j$th component and 0 in all others. – RideTheWavelet Jun 27 '17 at 13:58
• Is the equality $\binom{\alpha}{\sigma}=\prod_{i=1}^{n}\binom{\alpha_{i}}{\sigma_{i}}$ clear? I'm just using this on both terms and factoring out all but the $j$th factor. – RideTheWavelet Jun 27 '17 at 14:01
• I'll have to have another look as I'm still not getting it. Yes, I thought that's what $e_j$ meant – mark Jun 27 '17 at 14:37
• I certainly understand most of the argument now. I think I'm being confused by the multi-indices. I find the terminology $\sum_{\sigma \leq \beta}$ confusing.I presume this just means $\sum_{\sigma_1 \leq \beta_1} \sum_{\sigma_2 \leq \beta_2} \cdots \sum_{\sigma_n \leq \beta _n}$ I also find the notation $\sum_{|\alpha|\leq k}$ is rather confusing. – mark Jun 27 '17 at 19:31


text:
# 21.0: Prelude to Circuits and DC Instruments

Electric circuits are commonplace. Some are simple, such as those in flashlights. Others, such as those used in supercomputers, are extremely complex.

This collection of modules takes the topic of electric circuits a step beyond simple circuits. When the circuit is purely resistive, everything in this module applies to both DC and AC. Matters become more complex when capacitance is involved. We do consider what happens when capacitors are connected to DC voltage sources, but the interaction of capacitors and other nonresistive devices with AC is left for a later chapter. Finally, a number of important DC instruments, such as meters that measure voltage and current, are covered in this chapter.

This page titled 21.0: Prelude to Circuits and DC Instruments is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


text:
# Sufficient optimality condition for a non-smooth quasiconvex problem

The result of relaxing to an integer program is the following optimization problem: $$\min_{\textbf{x}} \sum_{i=1}^n \alpha_i h(x_i)\quad subject \; to \quad A\textbf{x} = \textbf{0}$$ where $\textbf{x}=(x_1,\cdots, x_n)^T$, $\alpha_i \in \mathbb{R}$, $A\in\mathbb{R}^{m\times n}$ with $m<n$ ($A$ is a fat random Gaussian matrix). The function $h$ is the Heaviside function $$h(x) = \left\{ \begin{array}{lr} 1 & x \geq 0\\ 0 & x <0 \end{array}. \right.$$ Given a point $\textbf{x}^*$, what is a rather general sufficient condition (in terms of $A$ and $\alpha_i$) for $\textbf{x}^*$ to be a global minimizer to the program above? For instance when $\textbf{x}^*$ is in the feasible subspace and $sign(x_i^*)= - sign(\alpha_i)$, then $\textbf{x}^*$ is a global minimizer. What happens when there are components of $\mathbf{x}$ for which $sign(x_i^*)\neq - sign(\alpha_i)$?

In case there is no immediate response to this, are there any well-known techniques (e.g., formulating everything in terms of a convex problem) to handle this? Are there any useful literature that I can refer to?

-Thanks

• You say your problem is a relaxation: can you include the original problem you are relaxing in this way, and why you choose such a relaxation? I don't know any conditions for global minima for nonconvex programming, but maybe your original problem can be relaxed to a convex program. – Cristóbal Guzmán Apr 28 '14 at 22:20
• Thanks Cristobal. Actually the initial problem is not very easy to explain (I will email you the details). I already have a convex relaxation to that problem. The problem above is a better approximation (than the convex form) that under some conditions on the $\alpha_i$ values becomes equivalent to the combinatorial cost. The ultimate goal is to use this program as an intermediate program to show the convex formulation and the combinatorial form coincide under certain criteria. – Ali Apr 28 '14 at 23:50


text:
# Difference between revisions of "Research:Thrown weapons"

## Judging thrown weapons

Here's some data Frazyl and I got for the damage that thrown weapons of different types do. This was with chest focus, and a fighting.range.thrown bonus of 260. They were judged with a 224 ad.ev.we bonus. Each number is an amount of damage done by being hit with it when it was thrown (misses were ignored).

dart ("pathetic", thrown weapon only):

• 0 several times, 1, can poison but more often doesn't

tlinga ("very poor", thrown weapon only):

• 87, 87, 88, 88, 89

throwing spike ("reasonable", thrown weapon only):

• 109, 115, 116, 119, 119, 119,

throwing star ("reasonable", also misc (misc bonus 204)):

• 163, 164, 165, 168, 169, 170, 172, 181

small throwing hatchet ("quite good", also axe (axe bonus 177)):

• 268, 280, 283, 283, 285, 289

large throwing hatchet ("good", also axe (axe bonus 177)):

• 265, 271, 278, 281, 282, 282

medium throwing hatchet ("good", also axe (axe bonus 177)):

• 303, 303, 304, 306, 308, 308, 310, 310, 313, 315

no. 5 throwing knife ("good", also dagger (dagger bonus 200)):

• 283, 293, 295, 299, 299, 313, 314, 322, 322

javelin ("powerful", also polearm (polearm bonus 245)):

• 545, 570, 572, 579, 584, 606

Those were all the ctf weapons available that had any judge as thrown weapons... so it seems that pathetic < very poor < reasonable < quite good <? good < powerful. Skill may be a factor... the throwing hatchets in particular interest me, as the large one did a bit worse than the small one despite, I think, judging better.

I had a 198 dodge bonus for this, although my actual skill at dodging varied a bit as the weapons ended up in my inventory and burdened me to varying degrees. Hopefully that doesn't matter, though... there only seemed to be two messages for them hitting:

• "hits you in the chest": hatchets, tlinga, throwing spike, bo
• "pierces your chest": javelin, throwing star, throwing knife

The messages don't seem to relate to damage, since the throwing star was worse than the hatchets... probably just depends on whether it's piercing damage or sharp/blunt.

--Ilde 02:43, 15 April 2010 (UTC)

It should be noted that I lacked the infinite patience and time to judge all the ctf weapons. Those I did didn't have a thrown judge message.
In case some other skill matters, my fighting skills were the following (I had a couple tms while testing but the only bonus that increased because of them was unarmed.)

=======SKILLS=======Level/Bonus=====================================================================
fighting............  138  216   | | misc............  117  204   | | parrying........  100  200
| melee.............  108  200   | range.............   81  193   | | blocking........  100  206
| | dagger..........  103  200   | | thrown..........  204  260   | | dodging.........  250  271
| | sword...........  251  285   | | bow.............   20  110   | special...........  179  243
| | heavy-sword.....  100  202   | | fired...........   20  104   | | weapon..........  235  291
| | axe.............   57  177   | unarmed...........  131  222   | | unarmed.........  168  247
| | mace............   20   99   | | striking........   52  172   | | tactics.........  295  322
| | flail...........   20  103   | | grappling.......  211  265   | | mounted.........   20   94
| | polearm.........  201  245   | defence...........  150  233   | points............  181  228

In particular, the large hatchets doing less damage could be a sign that axe plays a part in throwing axes. Maybe! --Frazyl 04:55, 15 April 2010 (UTC)


text:
Author
AbstractA thesis frontpage for Bachelor students studying Mathematics at the University of Groningen


text:
# Query about cubic roots and the discriminant

I have an equation a little like this;

$$2c^3 - 3oc^2 + oG = 0$$

In my equations, $o$ is always $> G$ . I'm trying to find a general solution rather than constantly estimating it graphically, and I got onto the Wikipedia article on cubic equations. If I've done my algebra correctly, the discriminant is $\operatorname{Dis} = 108(G^2\cdot o^2)(o^2 - G^2)$ which is always real and positive, meaning there should be three real solutions.

My problem is this however; there is a square root term that is $-27(2)\cdot \operatorname{Dis}$ ; this will always be negative, and always give a pair of complex roots; worse, I then have to take the cubic root of this at least twice; this is throwing me and I don't know how to explicitly solve the equation, or find a general form?

Can anyone help or point out any mistakes I'm making ? I'd be most grateful!

Cheers!

-


text:
## Which countries are research giants?

On its news website, the journal Nature recently published a graphic illustrating scientific research activity around the world as measured by the number of papers published from January to October 2011. Nature obligingly arranged the data by country.

A glance at the graphic tells you that the U.S., China, the U.K. and certain European countries are the big producers of scientific research. But does that mean that these countries are doing a good job educating their citizens and encouraging research? A closer examination of the data says no.

### Here are the data

I’ve sorted it in descending number of papers.

I’ve put them in descending order. True, the United States, China, Britain and Germany as the big contributors to scientific progress. Those countries lead the pack, while small, developing countries, such as Thailand, trail.

### But there’s more

But there’s more here than meets the eye. Of course, the US tops the list: as well as being developed economically, it’s got a substantial population, and consequently a large number of scientists. As you might expect, countries flush with people, like India and China, are also near the top.

But are those top publishers really pulling their weight in terms of production per person?

One way of adjusting for population size is to compute the number of papers per hundred thousand inhabitants. Taking Wikipedia as the source of population, we get the following. (I used Excel to compute papers per hundred thousand and then sorted the data in descending order of this ratio.

Now Switzerland heads the list, followed by the Scandinavian four and the Netherlands. Of these, only the Netherlands has more than 10 million inhabitants. The US has been topped by 16 other countries, including the United Kingdom and three British Commonwealth countries, Canada, Australia and New Zealand. China and India have plunged.

### But there’s still more

Is there a benefit to being a small country? It’s worth plotting the data on a graph with the population of the country as the horizontal axis and the “papers per 100K” on the vertical axis.

Those two outliers, India and China, with their large populations, mask any relationships hiding in the rest of the data. The US is a bit of an outlier, too.

In order to pick apart the data, I’ll delete China and India as special cases. This corrupts the generalization of the analysis. So noted. Here’s the new graph.

Now a relationship is beginning to emerge. Small countries do have an advantage. But the US — that point out to the right — is still anomalous; and the  plot is not linear — it’s got an L-shape.

### Time for some statistical pyrotechnics

Skip over this section if you are uncomfortable with logarithms.

In a situation like this, statisticians usually reach for a transformation or two. After fiddling about with the data, I chose two.

I replaced the “papers per 100K” with the logarithm of that variable; and I did the same with the population variable. (“What kind of logarithm?” you may ask. I used natural logs (base e), but base 10 works just as well, so long as you remain consistent.)

I then fit a regression line (the best fitting straight line) to the data.

The statistical sophisticate will note the heteroskedasticity here; I’m ignoring it.  The plot looks nice and linear. The US is still an outlier.

The regression equation, with w = Log(papers per 100k) and z = Log(Population) is

$w = -0.5719z + 6.0707$

The F-value is 25.04 (1,36) df. Significance p < 0.0001.

(Log base 10 transformations would have yielded different coefficients, but identical F and p statistics.)

It is informative to transform the equation back into terms of x and y where y = papers per 100k and x = population in millions. We have a = -0.5719 and b = 6.0707. So our equation is

$w = az + b$

$\log(y) = a\log(x) + b$

Taking exponents (base e) of both sides leads to

$y= \exp(a\log(x)+b) = x^a e^b$

Putting in values for a and b leads to

$y = \frac{e^{6.0707}}{x^{0.5719}}$

The numerator is $e^{6.0707} = 433.0$, rounded to 1 decimal place.

### Math phobes, you may return

The relationship between the “papers per 100k” (designated by y) and “population in millions” (x) is summarized in the equation

$y = \frac{433.0}{x^{0.57}}$

We can plot this as a line on the graph we looked at before.

### Now, which countries are producing above their weight?

The dots above the line represent the countries whose output exceeds what is explained by the size of their populations. They are:  Switzerland, Denmark, Sweden, Norway, the Netherlands,  Finland,  Australia,  Belgium, UK, Canada,
Israel, Austria, Taiwan, Germany,  US, Spain, France, South Korea, Italy and Japan.

Now all these are well-developed nations with relatively high education standards. If we wanted to delve into the data more deeply, we could include, along with population size, such factors as public education, economics, health and standard of living.

But I’ve said and done enough.

### Caveat

Unhappily, Nature News does not say how it classified the papers, but presumably that is based on the location of the lead author of each paper. Happily, the graphic includes real data. Unhappily, Nature News does not give a source for the data.


text:
# Journey from Somewhere to Elsewhere

Alice and Bob are both residents of Somewhere, and they are both coworkers in a company located in Elsewhere. Every weekday morning and evening, Alice and Bob commute between these two towns, which lie on adjacent exits on a stretch of Highway 773. They make their commutes at roughly the same time each day.

One day, at the water cooler, Bob complained to Alice about the other cars on Highway 773. During his commutes, Bob would see many more cars that violated the speed limit than those that adhered to it. Alice, in response, stated that in her experience, the opposite was true; she had seen more cars following the speed limit during her commutes than she had seen violating it.

In spite of this apparent contradiction, both Alice and Bob were honest in stating their own observations. How can this be so?

Alice is speeding, and Bob is not.

Assume 100 cars are spaced evenly and do the speed limit of 60 mph
100 additional cars are doing 70 mph and spaced evenly

Bob will only see the 60 mph car in front and in back
Bob will see the 70 mph cars pass him
Alice will only see the 70 mph car in front and in back
Alice will see the 60 mph cars she passes

I think that the answer is:

Alice is speeding, and Bob isn't.

My reasoning for this is:

I'll use a scenario to illustrate my point. Let's say that the speed limit on Highway 773 is 100km/h (or 60mph, depending on what system Somewhere uses), and that every hour, 12 cars use the highway. In each hour, half of these cars (so 6) drive lawfully, and 6 drive above the speed limit. Let's use the assumption that speeders drive 50% faster than they should.

Hence, it takes speeders two-thirds of the time to travel from Somewhere to Elsewhere, compared to lawful drivers. Hence, there would be, on average, $1\div\frac{2}{3}=\frac{3}{2}$, or 50% more drivers who are speeding past Bob, even though the amount of speeders and lawful drivers on the road are equivalent.

Whereas for Alice, who is speeding, in the time it takes her to get from Somewhere to Elsewhere, she will pass all the lawful drivers that started with her in Somewhere, but she will also pass the lawful drivers who passed Somewhere $\frac{1}{3}x$ before she did, where $x$ is the amount of time it takes her to drive from Somewhere to Elsewhere. Hence, she will see a disproportionate skew of lawful drivers to speeding drivers during her journey.

• This is the right train of thought, but the reasoning is incorrect. – redyoshi49q Jul 15 '17 at 1:09
• @redyoshi49q I'm sorry, could you please explain what you mean by this being the right train of thought? – Lucas Jul 15 '17 at 1:12
• The crux of the apparent paradox is a result of Alice and Bob driving at different speeds, and while there isn't enough information to deternine how fast each person drives, there is enough to determine that one drives faster than the other (and who that is). However, your answer is not quite correct. – redyoshi49q Jul 15 '17 at 1:23
• I can give a further hint if such is desired. – redyoshi49q Jul 15 '17 at 1:26
• @redyoshi49q Ahhh :) did I get it now? – Lucas Jul 15 '17 at 2:42


text:
# Weird behavior of HypergeometricPFQ

Consider the following function:

fun = HypergeometricPFQ[{1, x1-y1, 1-x1-y1-2u}, {x2-y1, 3-x2-y1-2u}, 1];

Let us try and evaluate this function on the following specific values of the arguments:

sub = {x1 -> 3, x2 -> 2, y1 -> 7, y2 -> 4, u -> 51/10};
fun/.sub

43/11

Alternatively, we can functionally expand fun first to obtain

funEx = FunctionExpand[fun, Element[x1 | y1 | x2 | y2, Integers] && Element[u, Rationals] && y1 >= x1 && y1 >= x2 && y2 >= x2 && y1 >= y2 && x1 >= x2]

-((2 (-1 + x2 - y1))/((-1 - x1 + x2) (-2 + 2 u + x1 + x2))) + ( 2 u (-1 + x2 - y1))/((-1 - x1 + x2) (-2 + 2 u + x1 + x2)) + ( x2 (-1 + x2 - y1))/((-1 - x1 + x2) (-2 + 2 u + x1 + x2)) + ((-1 + x2 - y1) y1)/((-1 - x1 + x2) (-2 + 2 u + x1 + x2)) - ( Gamma[3 - 2 u - x2 - y1] Gamma[ x2 - y1])/((1 + x1 - x2) (-2 + 2 u + x1 + x2) Gamma[ 1 - 2 u - x1 - y1] Gamma[x1 - y1])

First of all we see that even though FunctionExpand was explicitly told that y1>=x1 and y1>=x2 and x1,x2,y1 are integers, it still produced the ill behaved ratio Gamma[x2 - y1]/Gamma[x1 - y1], such that trying to evaluate funEx/.sub leads to the error:

Indeterminate

What is worse, even if we try to regularize by setting y1->7+e and expanding around small e, we find a discrepancy with the original evaluation above:

Series[funEx /. y1 -> 7 + e /. sub, {e, 0, 0}] // FullSimplify// Normal

-(5203/125)

Which is clearly different from 43/11. Why does this function behave this way? I need to simplify the function analytically and I am only evaluating it to check if the functional expansion was correct. Clearly it is not correct and I am at a loss on how to proceed. Any suggestions on how to obtain the correct simplification?

EDIT:

J. M. suggested to use the Thomae transformation. In the above case this leads to

subv={a -> 1, b -> x1 - y1, c -> 1 - x1 - y1 - 2 u, e -> x2 - y1, f -> 3 - x2 - y1 - 2 u};
s = e + f - a - b - c;
fun2=(Gamma[e] Gamma[f] Gamma[s])/(Gamma[a] Gamma[s + b] Gamma[s + c])HypergeometricPFQ[{s, e - a, f - a}, {s + b, s + c}, 1]/.subv

Now, trying to functionally expand this as before, gives

funEx2=FunctionExpand[fun2, Element[x1 | y1 | x2 | y2, Integers] && Element[u, Rationals] && y1 >= x1 && y1 >= x2 && y2 >= x2 && y1 >= y2 && x1 >= x2] //FullSimplify

-(((-1 + x2 - y1) (-2 + 2 u + x2 + y1) + (Gamma[3 - 2 u - x2 - y1] Gamma[x2 - y1])/(Gamma[1 - 2 u - x1 - y1] Gamma[x1 - y1]))/((1 + x1 - x2) (-2 + 2 u + x1 + x2)))

which still produces the same ill behaved ratio of gamma functions.

Resolution:

Just trying to evaluate the function fun a few times on some different y1 and x1 values like

fun/. y1 -> 3 /. x1 -> 2 // Simplify

((-4 + x2) (1 + 2 u + x2))/((-3 + x2) (2 u + x2))

One can infer that the whole hypergeometric function actually in general reduces to just

((-1 + x2 - y1) (-2 + 2 u + x2 + y1))/((-1 + x2 - x1) (-2 + 2 u + x1 + x2))

This is basically what both the functional expansions above produced, but with all the gamma functions dropped. Even though I am almost certain now that this is the correct expansion, this way to do things is highly unsatisfying. Mathematica should do better than this.

• Have you tried Thomae, by any chance? Jun 2, 2016 at 1:03
• Thank you for the suggestion, I will try and then update the question afterwards. Jun 2, 2016 at 1:45
• Updated the question, and posted my ad hoc solution. Jun 4, 2016 at 15:44
• I was actually hoping you'd try to track the other ${}_3 F_2$ formulae for unit argument, like Saalschutz's, if Thomae didn't work. Although, I probably should have pointed out Bühring's paper to begin with. Jun 4, 2016 at 15:49
• That would be the digamma function, considering the context. (Note that Abramowitz and Stegun was referred to by Bühring, and similar expressions show up in the sections he pointed out.) Jun 4, 2016 at 16:53

Just trying to evaluate the function fun a few times on some different y1 and x1 values like

fun/. y1 -> 3 /. x1 -> 2 // Simplify

((-4 + x2) (1 + 2 u + x2))/((-3 + x2) (2 u + x2))

One can infer that the whole hypergeometric function actually in general reduces to just

((-1 + x2 - y1) (-2 + 2 u + x2 + y1))/((-1 + x2 - x1) (-2 + 2 u + x1 + x2))

This is basically what both the functional expansions in the question above produced, but with all the gamma functions dropped. Even though I am almost certain now that this is the correct expansion, this way to do things is highly unsatisfying. Mathematica should do better than this.

EDIT

With a little bit of hindsight, it is possible to interpret mathematica output a bit better. As already mentioned, the first part which is independent of the gamma functions:

((-1 + x2 - y1) (-2 + 2 u + x2 + y1))/((1 + x1 - x2) (2 - 2 u - x1 - x2))

is the result we are after. But why do we get some extra terms with gamma functions - is something going wrong or does it have an interpretation? Well, it turns out that if we regularize the gamma functions, we get the extra contribution:

(-1)^(x1 - x2) (Gamma[3 - 2 u - x2 - y1] Gamma[y1 - x1 + 1] )/( Gamma[1 - 2 u - x1 - y1] Gamma[ y1 - x2 + 1] (1 + x1 - x2) (2 - 2 u - x1 - x2))

This contribution makes up for the difference in results when the arguments are non-integers. Having a negative integer for one of the first arguments in the hypergeometric function makes the sum terminate, which gives back just the first part of the result (and which we are interested in here). If this one argument is even slightly different from integer, the summation continues until infinity - and picks up exactly the second part of the result involving the gamma functions. From this perspective it is also clear why the gamma functions needed to be regularized - they only arise in the case when the argument actually is not integer so that the sum does not terminate.

• As you probably already know, Mathematica returns results that are only generically correct, in that they work for most complex values except at some distinguished points (the integers are a frequent point of failure). Jun 7, 2016 at 2:54


text:
# Formulation of Two Photon and Two Gluon Decays of Pseudoscalar $\eta$ Mesons in a Relativistic Bound State Calculation

Authors
Publication Date
Jan 01, 1987
Identifiers
DOI: 10.1103/PhysRevD.36.3392
OAI: oai:inspirehep.net:250614
Source
INSPIRE-HEP
Keywords
Unknown

## Abstract

We formulate a relativistic bound-state calculation of the decay of η (0−+) mesons into two gluons or two photons. The decays of ηc, ηc’, ηb, and ηb’ are calculated and a value of αs deduced from the total width of ηc. The calculated value for ηc→2γ is in agreement with experiment. The decays of η→2γ and η’→2γ are calculated and by comparison with experiment and the bound on ι→2γ, restrictions are placed on the Xη, Yη, Xη’, and Yη’ mixing coefficients of Rosner. The comparison with the Schrödinger-equation calculations shows that the inclusion of relativistic propagators spreads out the annihilations so that they do not occur at r=0. This suppresses the 2S and higher-level decays, with the greatest suppression for the lightest quarks.

Seen <100 times


text:
# How do you find a hole in a rational function graph?

## Question:

How do you find a hole in a rational function graph?

## Rational Functions:

A rational function is a function that is a ratio of two polynomials, i.e. of the form

{eq}\begin{align*} f (x) ****= \frac{P (x)}{Q (x)} \end{align*} {/eq}

where {eq}P {/eq} and {eq}Q {/eq} are both polynomials. Note that this function is undefined whenever {eq}Q = 0 {/eq} and in those special places where part of the denominator cancels part of the numerator, we will find holes in the graph.

It is important to know that when we graph a function, holes remain hidden! This is because the hole is infinitesimally small, and is easily glossed over by our graphing utilities. For this reason it is important to check for holes. To do so for a rational function we need to factor the numerator and denominator see if it anything can "cancel out" (we use quotes because we need to be very careful when we simplify). If it does, then we have a hole in the function. The easiest function to see this in is probably {eq}f (x) = \frac{x^2}x {/eq}. Note that {eq}x \neq 0 {/eq}. And if it does not, then {eq}f (x) = x {/eq}. And so the function {eq}f (x) = \frac{x^2}x {/eq} is the line {eq}f (x) = x {/eq} with a hole in it at {eq}x = 0 {/eq}.

More generally if we have a rational function of the form

{eq}\begin{align*} f (x) &= \frac{ax^2 + bx + c}{x+r} \end{align*} {/eq}

where we can write {eq}ax^2 + bx + c = (x+r)(x+s) {/eq}, then

{eq}\begin{align*} f (x) &= \frac{ax^2 + bx + c}{x+r} \end{align*} {/eq}

is actually the line

{eq}\begin{align*} f (x) &= \frac{ax^2 + bx + c}{x+r} \\ &= \frac{ (x+r)(x+s) }{x+r} \\ &= x+s \end{align*} {/eq}

with a hole at {eq}x = r {/eq}.

This can basic idea can be extended to any degree polynomial to find all holes.