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# Comparison of information structures

## Author

Listed:
• Olivier Gossner

## Abstract

We introduce two ways of comparing information structures, say ${\cal I}$ and ${\cal J}$. First we say that ${\cal I}$ is richer than ${\cal J}$ when for every compact game $G$, all correlated equilibrium distributions of $G$ induced by ${\cal J}$ are also induced by ${\cal I}$. Second, we say that ${\cal J}$ is faithfully reproducable from ${\cal I}$ when all the players can compute from their information in ${\cal I}$ new information'' that they could have received from ${\cal J}$. We prove that ${\cal I}$ is richer than ${\cal J}$ if and only if ${\cal J}$ is faithfully reproducable from ${\cal I}$.

## Suggested Citation

• Olivier Gossner, 1996. "Comparison of information structures," Economics Working Papers 169, Department of Economics and Business, Universitat Pompeu Fabra.
• Handle: RePEc:upf:upfgen:169
as

File URL: https://econ-papers.upf.edu/papers/169.pdf
File Function: Whole Paper

## References listed on IDEAS

as
1. repec:dau:papers:123456789/6244 is not listed on IDEAS
2. Aumann, Robert J, 1987. "Correlated Equilibrium as an Expression of Bayesian Rationality," Econometrica, Econometric Society, vol. 55(1), pages 1-18, January.
3. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
4. Hirshleifer, Jack, 1971. "The Private and Social Value of Information and the Reward to Inventive Activity," American Economic Review, American Economic Association, vol. 61(4), pages 561-574, September.
5. Gossner, Olivier, 1998. "Secure Protocols or How Communication Generates Correlation," Journal of Economic Theory, Elsevier, vol. 83(1), pages 69-89, November.
6. Neyman, Abraham, 1991. "The positive value of information," Games and Economic Behavior, Elsevier, vol. 3(3), pages 350-355, August.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as

Cited by:

1. Gossner, Olivier, 2010. "Ability and knowledge," Games and Economic Behavior, Elsevier, vol. 69(1), pages 95-106, May.
2. Bassan, B. & Gossner, O. & Scarsini, M. & Zamir, S., 1999. "A Class of Games with Positive Value of Information," Papers 99-32, Paris X - Nanterre, U.F.R. de Sc. Ec. Gest. Maths Infor..
3. Liu, Qingmin, 2015. "Correlation and common priors in games with incomplete information," Journal of Economic Theory, Elsevier, vol. 157(C), pages 49-75.
4. Bergemann, Dirk & Morris, Stephen, 2016. "Bayes correlated equilibrium and the comparison of information structures in games," Theoretical Economics, Econometric Society, vol. 11(2), May.
5. Takashi Ui, 2009. "Bayesian potentials and information structures: Team decision problems revisited," International Journal of Economic Theory, The International Society for Economic Theory, vol. 5(3), pages 271-291, September.
6. Cabrales, Antonio & Gossner, Olivier & Serrano, Roberto, 2017. "A normalized value for information purchases," Journal of Economic Theory, Elsevier, vol. 170(C), pages 266-288.
7. Kolotilin, Anton, 2015. "Experimental design to persuade," Games and Economic Behavior, Elsevier, vol. 90(C), pages 215-226.
8. Lehrer, Ehud & Rosenberg, Dinah & Shmaya, Eran, 2010. "Signaling and mediation in games with common interests," Games and Economic Behavior, Elsevier, vol. 68(2), pages 670-682, March.
9. Vida, Péter & Āzacis, Helmuts, 2013. "A detail-free mediator," Games and Economic Behavior, Elsevier, vol. 81(C), pages 101-115.
10. repec:eee:gamebe:v:109:y:2018:i:c:p:126-131 is not listed on IDEAS
11. Olivier Gossner, 1997. "Protocoles de communication robustes," Revue Économique, Programme National Persée, vol. 48(3), pages 685-695.
12. Leal Vizcaíno René & Mekonnen Teddy, 2019. "Bayesian Comparative Statics," Working Papers 2019-03, Banco de México.
13. Lehrer, Ehud & Rosenberg, Dinah, 2006. "What restrictions do Bayesian games impose on the value of information?," Journal of Mathematical Economics, Elsevier, vol. 42(3), pages 343-357, June.
14. Ehud Lehrer & Dinah Rosenberg, 2003. "Information and Its Value in Zero-Sum Repeated Games," Game Theory and Information 0312003, University Library of Munich, Germany.
15. Gossner, Olivier, 1998. "Secure Protocols or How Communication Generates Correlation," Journal of Economic Theory, Elsevier, vol. 83(1), pages 69-89, November.
16. Bruno Bassan & Olivier Gossner & Marco Scarsini & Shmuel Zamir, 2003. "Positive value of information in games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(1), pages 17-31, December.
17. Lehrer, Ehud & Rosenberg, Dinah & Shmaya, Eran, 2013. "Garbling of signals and outcome equivalence," Games and Economic Behavior, Elsevier, vol. 81(C), pages 179-191.
18. repec:kap:compec:v:52:y:2018:i:1:d:10.1007_s10614-017-9646-z is not listed on IDEAS
19. Kloosterman, Andrew, 2015. "Public information in Markov games," Journal of Economic Theory, Elsevier, vol. 157(C), pages 28-48.

### Keywords

Game--theory; information; correlation;

### JEL classification:

• C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
• C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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# Rdpack 2.1.2

• Rdmacros generating references now drop the URL field if a doi field generating the same URL is present. This avoids repetition of the URL since rendering the doi field produces the URL anyway.

• now references with identical authors are sorted by year.

• fixed rebib() to process doi’s properly.

• now checks on github (using github actions) are done on the three major platforms for the following versions of R: release, devel, oldrel, and 3.3.

• now a warning (rather than error) is issued if package ‘testthat’ is not available for tests.

• now bibtex files are parsed with option direct = TRUE when the version of rbibutils is >= ‘2.1.2’. This fixes GeoBosh/rbibutils#3 for Rdpack users (although that needs fixing also for direct = FALSE).

# Rdpack 2.1.1 (CRAN)

• In viewRd, the default for the help type now is options("help_type") (previously the default was "text").

• for roxygen2 users, added a note in README and Rdpack-package.Rd not to indent by exactly four spaces \insertAllCited{} relative to @references when markdown mode is activated. This is to avoid it being translated by roxygen2 as verbatim (\preformatted) text, see issue (#18).

• moved package gbRd to Suggests.

• fixed uses of in several examples (necessitated by a change in R-devel circa February 2021).

# Rdpack 2.1 (CRAN)

• dramatic speed up of processing of bibliography references, most noticeable for users with large number of help pages and large REFERENCES.bib. (prompted by Kisung You, issue #17)

• amended the warning message from bibliography macros to mention duplicated keys.

• added entry for rbibutils in REFERENCES.bib.

• corrected wrong citation of rbibutils.

# Rdpack 2.0 (CRAN)

• removed bibtex from Suggests and elsewhere.

# Rdpack 1.1.0

• imported rbibutils and made suitable adjustments to the code.

• moved bibtex to Suggests (since bibtex has been orphaned for several years).

# Rdpack 1.0.1

• removed a documentation link to a function in gbutils, which is not among the dependencies of Rdpack. This was raising a NOTE on one of the CRAN testing machines.

# Rdpack 1.0.0 (CRAN)

• viewRd() now loads also Rd macros declared by the package to which the rendered Rd file belongs. Previously only macros from Rdpack were loaded but now there is at least one other package, mathjaxr, which defines Rd macros.

• moved grDevices to Suggests.

• edited some documentation files.

# Rdpack 0.11-1 (CRAN)

• in documentation of S4 classes, such as "classname-class.Rd", reprompt() was sometimes inserting entries for slots already listed in the Slots section. The same bug was causing the new slots not to be reported properly to the user.

• in some cases reprompt() failed to process properly \S4method entries in the Usage section of Rd files. Now fixed.

• reprompt() now handles \S4method statements for replacement methods. As for other functions and methods (S3 and S4), it is sufficient to put a declaration with empty argument list in the Usage section and reprompt() will insert the correct formal arguments for the method (they may be different from those of the generic).

• reprompt() now gives a more helpful error message when type is invalid.

• corrected some minor typo’s in the documentation.

• README and the documentation of reprompt() and Rdpack-package now give more details on reprompt()ing replacement functions.

• README.md and README.org now give the correct install.packages instruction for CRAN (pull request #10 from @katrinleinweber).

• some examples were leaving a stray file, dummyfun.Rd, after R CMD check.

# Rdpack 0.11-0 (CRAN)

• Updated the vignette about \insertFig, \printExample and \runExamples, to reflect the lifting in R 3.6.0 of some limitations of Rd processing in previous R versions.

• Vignette ‘Inserting BibTeX references’ now includes a section on bibliography styles. This section was previously only in README.

• The fix in Rdpack 0.10-3 (see below) for an issue introduced in R-devel in Oct 2018 will be made permanent, at least for now. This fix resolves also a similar issue in package pkgdown, see the discussion at https://github.com/GeoBosh/Rdpack/issues/9 for details and further links.

# Rdpack 0.10-3

• fixed issue#9 (reported by by @aravind-j) appearing when a package is built with R-devel (since about Oct 2018) causing references by \insertAllCited{} to appear in a single paragraph in the html rendering of the Rd documentation.

# Rdpack 0.10-2

• added pkgdown site to DESCRIPTION.

• README and the vignette about evaluated examples now state that R-devel no longer gives warnings about \Sexpr not being a top level section. This means that macro runExamples which creates section ‘Examples’ containing code and results of evaluation will be useable in CRAN packages.

• new function Rdo_fetch() gets the Rd object representing a help page from an installed or source package. It works also for packages under devtool’s developer’s mode

• The site created with pkgdown contains fewer errors now. (Note that the documentation builds without error with R’s tools).

• Started moving tests from my local setup to testthat.

# Rdpack 0.10-1 (CRAN)

• removed redundant references from REFERENCES.bib (they were leftovers from testing).

# Rdpack 0.10-0

• updated vignette ‘Inserting_bibtex_references’.

# Rdpack 0.9-1

• now REFERENCES.bib is read-in using the declared encoding for the corresponding package. If there is no declared encoding, “UTF-8” is assumed.

• Now macros \insertCite and \insertCiteOnly use the correct results=rd instead of results=Rd. This was not catched by Rs building tools but caused errors when processed with pkgdown::build_site(). Fixes issue#8. Also fixes r-lib/pkgdown#784. Thanks to Jay Hesselberth for uncovering this.

• now references produced by the citation macros for BibTeX entries of type @book and @incollection treat field ‘series’ similarly to other BibTeX styles, including JSS (issue#7 raised by Kisung You). Note that even though the underlying base R tools are based on JSS, they treat this field differently.

• Bugfix: now Rd macro \printExample evaluates the expressions in the correct environment.

• help page of get_usage gets a fairly complete Details section with numerous evaluated examples.

• now run_examples() escapes % by default, before returning the text. This is needed for text that is to be included in an Rd file. It can be turned off by setting the new argument escape to FALSE.

• now reprompt() gives a more informative error message if an Rd file describes a non-existent S4 class. This is not captured by Rs tools. It can happen during development if a class is removed.

• now reports printed by reprompt() about methods documented in Usage sections, but no longer existing, are more readable. This is due to a new print method for (the mostly internal) class “f_usage”.

• Many features of Rdpack are best demonstrated on a package. The new package RdpackTester under ./inst/examples now makes this easier.

• now the help page “predefined.Rd” does not print some tables twice in the pdf manual. (This was due to using \if{latex}{}{}' instead of’ for those tables.)

• also in “predefined.Rd”, removed illegal use of vertical bars (in column specifications of tabular environments) from the pure LaTeX code in the \ifelse clause(s) and wrapped them in \out{}.

• in “get_sig_text.Rd”, replace help() with utils::help() to avoid warnings from more stringent R CMD check. Similarly, in “Rdpack-package.Rd” replace packageDescription() with utils::packageDescription(). This may be needed in \Sexpr’s more generally (TODO: check if these would still be needed if the symbols are imported by the package.)

# Rdpack 0.9-0 (CRAN)

• some brush-up of the documentation for the changes since version 0.8-0 of the package.

# Rdpack 0.8-4

• now simple mathematics in BibTeX entries is rendered natively, no need to replace dollars with \eqn{}.

# Rdpack 0.8-3

• new macro \printExample for inclusion of example computations in narrative sections, such as Details. The code is evaluated and printed similarly to R sessions but the code is not prefixed and the output is prefixed with comment symbols, eg.

2+2
##: 4
• new experimental macro \runExamples for use as top level section in an Rd file as a replacement of section \examples. \runExamples{code} evaluates the code and creates section \examples containing the code and the results (similarly to \printExample). So, \runExamples{2 + 2} produces

\examples{
2 + 2
##: 4
}

The generated section examples is processed by R’s documentation tools (almost) as if it was there from the outset.

• new experimental macro \insertFig to create a figure with R code and include it in the documentation. The related macro \makeFig just creates a graphics file, which can be included with the standard Rd command \figure.

• new vignette gives a brief description of the new macros.

# Rdpack 0.8-2

• Now text citations use “et al.” when there are three or more authors. (Issue#6 reported by Timothy P. Bilton)

# Rdpack 0.8-1

• in this file, added backticks to \insertRef and \insertAllCited (see below) - in the rendered News on CRAN the backslashed words had disappeared.

# Rdpack 0.8-0 (CRAN)

• \insertRef and \insertAllCited macros now support bibstyles for formatting references (feature requested by Jamie Halliday, issue#5). Use Rdpack (>= 0.8) in Imports: to use this feature. Currently only long author names are supported but complete support for styles can be added trivially if requested.

• updates to the documentation, in particular the bulk of Rdpack-package.Rd was from 2011!

# Rdpack 0.7-1

• improvements to handling of free form citations in textual mode:

• (bugfix) now the whole citation is not parenthesised in textual mode,

• the handling for textual mode was incomplete in that additional text after the citation was not put inside the parentheses along with the year.

• fix a bug in Rdo_locate_core_section().

# Rdpack 0.7-0 (CRAN)

• consolidated the changes introduced since the previous CRAN release of Rdpack (it was 0.5-5) in preparation for the next release. Users of the new macros for citation can use Rdpack (>=0.7) in the Imports: field of file “DESCRIPTION” to ensure that they are available.

• comprehensive overhaul of handling of errors and warnings during processing of references and citations. In particular, such errors should (in most cases) produce only warnings during R CMD build and R CMD INSTALL, and not prevent the package from being built and installed.

• Unresolved BibTeX keys produce warnings during building and installation of the package, but not errors. Dummy entries are inserted in the documentation explaining what was amiss (currently with ‘author’ A Adummy).

# Rdpack 0.6-x

• new Rd macros for citations

• \insertCite inserts citation(s) for one or more keys and records the keys for \insertAllCited (see below).

• \insertCiteOnly is similar to \insertCite but does not record the keys.

• \insertNoCite records the keys but does not produce a citation.

• \insertAllCited prints a bibliography including all references recorded by \insertCite and \insertNoCite.

• new entries in this file will use markdown syntax.

# Rdpack 0.5-7

• get_bibentries() gets a new argument “everywhere”. When is is TRUE, the default, unescaped percents are escaped in all bibtex fields, otherwise the replacement is done only in field “URL”.

• removed several print() statements which were accidentally left in the code in v. 0.5-6.

• updated help page of get_bibentries() and included examples.

• cleaned up the imports in NAMESPACE.

# Rdpack 0.5-6

• insert_ref() and Rd macro \insertRef should now work ok in the presence of percent encoded symbols in URL field of a BibTeX entry. Closes issue “Unexpected END_OF_INPUT error (URL parsing?)”, see Rdpack issue 3, raised by jdnewmil.

• get_bibentries() now takes care of percent encoded symbols in URLs. It now returns an object from class “bibentryRd”, which inherits from “bibentry” but has its own print method which escapes or unescapes the percent signs in URLs depending on the requested output style. This was also reported by by jdnewmil in issue#3 referenced above.

• get_bibentries() now tries to load the bib file from the development directory of argument “package”, in case it is in development mode under “devtools”. (Even though the search is with system.file(), which devtools replaces with its own version, the bib file still needs to be looked for in “inst/”, i.e. it is not in the root directory as is the case in installed packages.)

# Rdpack 0.5-5 (CRAN)

• Streamlined the help page of reprompt() and Rdpack-package.

• new argument, edit', inreprompt() opens the Rd file after updating it.

• new function ereprompt() (e' for _edit_)* opensreprompt()with suitable defaults to replace the original Rd file and open it in an editor, otherwise equivalent toreprompt().

• now if infile does not exist, reprompt(), and hence ereprompt(), strips the directory part, finds the root of the package directory, and looks for the file under man/ (this uses package rprojroot’). In particular, if the working directory is anywhere under the package root, infile can be simply the name of the Rd file, without path.

• now README.md is generated from README.org. I changed the layout and amended the information in it.

• README.* now get links to georgisemacs for an emacs function to reprompt() the Rd file at point.

• viewRd() now works also when the file is from a package under devtools’ development mode on any platform (e.g. RStudio, Emacs/ESS, Rgui).

# Rdpack 0.5-4.9000

• new RStudio add-in ’Reprompt, contributed by Duncan Murdoch.

If the file being edited in the code editor is an Rd file it calls reprompt(). If the file is as R source file, it looks for the help page of a selected object name and works on it if found, otherwise creates one.

# Rdpack 0.5-4

• added the version of Rdpack to the abstract of the vignette. This seems more informative than giving the compilation date.

• now reprompt() doesn’t give spurious warnings about unknown Rd macros when the Rd file contains user defined Rd macros. These warnings were harmless and rare, but alarming and annoying.

• fixed a bug in inspect_args() which caused the warning “In is.na(x) : is.na() applied to non-(list or vector) of type ‘NULL’” in reprompt(), when the signature of a function in “Usage” section was empty.

# Rdpack 0.5-3 (CRAN)

• The warning message about a missing reference key now appears also in the respective help page.

# Rdpack 0.5-2

• added the github url to DESCRIPTION.

• now give a more informative warning if a key is missing from REFERENCES.bib (thanks to Kisung You for suggesting this).

# Rdpack 0.5-1 (CRAN)

• an example was not restoring the working directory.

# Rdpack 0.5-0

• moved gbRd from Depends to Imports and adjusted some examples to use tools::parse_Rd() rather than parse_Rd() (before this was not necessary since ‘gbRd’ depended on ‘tools’).

# Rdpack 0.4-23

• new function viewRd() parses and shows an Rd file in a source package.
May be particularly helpful for devtools developers.

• new functions makeVignetteReference() and vigbib() generate Bibtex references for vignettes in a package.

# Rdpack 0.4-22 (CRAN)

• Added the requirement to Import Rdpack to the help page of insert_ref(). (It was out of sync with the vignette.)

• Updated Description in DESCRIPTION, which was also out of sync with the vignette.

• Bug fix in the vignette: changed \@references to @references in the roxygen2 example.

# Rdpack 0.4-21 (CRAN)

• Edited and amended vignette “Inserting_bibtex_references”, including:

• additional explanation and an example,

• requirement to import Rdpack and a mention of travis etc.

• changed the URL for parse_Rd.pdf in REFERENCES.bib to https.

# Rdpack 0.4-20 (CRAN)

• cleaned up some minor CRAN issues.

# Rdpack 0.4-19

• new facility for inserting references from BibTeX files - just put the line:

  RdMacros: Rdpack

in the DESCRIPTION file of your package and the references in inst/REFERENCES.bib. Then put \inserRef{key}{yourpackage} in documentation chunks to insert item key from the bib file. This works with both manually created Rd files and roxygen comments, see the documentation for details.

# Rdpack 0.4-18 (CRAN)

• from now on, put (CRAN) next to versions published on CRAN (as above)

• a minor correction in file NEWS.

# Rdpack 0.4-17

• In file DESCRIPTION, changed reprompt and rebib to reprompt() and rebib().

# Rdpack 0.4-16

• don’t export parse_text

• corrected bug in reprompt for S4 classes - new slots were not handled correctly (see slots.R for details).

• reprompt for S4 methods* if the methods documentation describes methods that do not exist, print an informative message for the user. (Such methods are also printed for debugging purposes but in non-user friendly form.)

• included methods in Imports: - around R-devel version 2015-07-01 r68620 not including it triggers warnings)

# Rdpack 0.4-8

• new functions .whichtageq and .whichtagin; replaced most calls to toolsdotdotdotRdTags with these.

• removed toolsdotdotdotRdTags

# Rdpack 0.4-7

• removed ::: calls from code; copied some functions from tools to achieve this (see “threedots.R”). Renamed the functions replacing ::: with dotdotdot.

• export by name (not the generic export pattern); preparation for more selective export of functions in the future.

• new functions: Rdo_get_argument_names, Rdo_get_item_labels

# Rdpack 0.4-5

• dependence bibtex becomes “Imports”. tools and gbRd remain (for now) “Depends” since functions from them are used in some examples without require()’s.

• rebib(): now outfile="" can be used to request overwriting infile. (a small convenience; before the change, one could do this by giving the outfile and infile the same values.)

• Bug fix: predefined.Rd contained \tabular environments with vertical bars, |, in the format specification. This is not documented in “Writing R exts” but works for LaTeX and remained unnoticed by me and R CMD check. However, rendering the help page for the objects documented in “predefined.Rd” gave an error in html and text mode. Package installation failed only if html was requested at build time.

• small changes in the documentation

# Rdpack 0.4-1

• new major feature: processing references from Bibtex files. The top user level function is rebib(), which updates section “references” in an Rd file. promptPackageSexpr() has been updated to use this feature. inspect_Rdbib() can be used in programming.

• new auxiliary functions for work with bibentry objects.

• new convenience programming functions for Rd objects.

• some small bug fixes.

• some gaps in the documentation filled.

# Rdpack 0.3-11

• reprompt() was not handling properly S4 replacement methods. Changed the parsing of the arguments to rectify this. Some other functions also needed correction to handle this.

# Rdpack 0.3-10

• Depends field in DESCRIPTION file updated to require R 2.15.0 or later. (because of a few uses of paste0()` in recent code.)

# Rdpack 0.3-8 (CRAN)

• first public version

format: plain


text:
# Surface area problem involving solids of revolution

## Homework Statement

Evaluate the definite integral for the surface area generated by revolving the curve about the y-axis:

## The Attempt at a Solution

Attached

#### Attachments

• image.jpg
38.8 KB · Views: 365

Simon Bridge
Homework Helper
When you write out your maths working - also write some, you know, words - so people know what the equations are supposed to be for.

The curve is y=9-x^2, between 0 < y < 9 and you are rotating about the y axis.

It looks like you have applied a formula instead of understanding it.

i.e. could you find the area dA of the strip on the surface between y and y+dy?

Lastly: you have not asked a question - do you have an issue with what you have done?
This information helps us to direct replies to where you need them.

Sorry, my first question would be, is it correct?

If not could you point out the beginning of my mistake.

Simon Bridge
Homework Helper
I can't really tell without doing it myself - it looks like the right formula.
dx/dy is incorrect, but not in a way that will affect the result.
You should also double-check your algebra in simplifying the integrand.

$$\sqrt{9-y}\cdot \sqrt{1+\left(\frac{1}{2}\frac{1}{\sqrt{9-y}}\right)^2}=\left[1+\frac{1}{4}\frac{1}{9-y}\right]^{1/2}\sqrt{9-y}=\cdots$$... how do you take the √(9-y) inside the other square root?

STEMucator
Homework Helper
Use a vertical slice. The region is bounded between ##-3 ≤ x ≤ 3##.

##r_{in} = x - dx##
##r_{out} = x##
##r_{ave} = \frac{2x - dx}{2}##
##r_{out} - r_{in} = dx##
##h = 9 - x^2##

So your volume element would be:

##dV = 2\pi*r_{ave}*(wall \space thickness)*height##

Simplify that and ignore any ##(dx)^2## terms.

Last edited:
Simon Bridge
Homework Helper
So your volume element would be:...
... to find surface area?

The vertical slice between x and x+dx is a parabola thickness dx
The area of the parabola would be ##\int y(x,z) dz## with apropriate limits - wouldn't it be easier to slice the resulting parabaloid horizontally or use shells?

Anyway: surface area...
What wrong with summing the areas of the annuli between ##y## and ##y+dy## - which is what OP is doing?

STEMucator
Homework Helper
... to find surface area?

The vertical slice between x and x+dx is a parabola thickness dx
The area of the parabola would be ##\int y(x,z) dz## with apropriate limits - wouldn't it be easier to slice the resulting parabaloid horizontally or use shells?

Anyway: surface area...
What wrong with summing the areas of the annuli between ##y## and ##y+dy## - which is what OP is doing?

I just found vertical slices to be convenient. You could also write:

##x = ± \sqrt{9 - y}##, ##0 ≤ y ≤ 9## and use horizontal slices.

Surface area though I didn't see that my bad. Thought it was a volume of revolution problem.

Simon Bridge
Homework Helper
Surface area though I didn't see that my bad. Thought it was a volume of revolution problem.
... well that's what the title says right? ;)
Well sort of - though it is a solid of revolution problem - the task is not to find the volume of the solid.

There's also a niggle: your formula was for cylinder method - not vertical slices.
These details are worth marks.

But good to see someone else weighing in: if you don't speak up you don't learn stuff.

Last edited:
Okay, I made another attempt at this problem but once I start to take the integral things become very ugly.

This is where I am now. http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427e51gddpbmno

If someone could tell me if I'm on the right track and if so which way is the best to integrate and if not then I think I need help setting up the problem. I will be active for the next few hours so we can facilitate the discussion.

 Sorry for using the WolframAlpha tool but I don't know an easier/faster way to post math formulas [/Edit]

SammyS
Staff Emeritus
Homework Helper
Gold Member
Okay, I made another attempt at this problem but once I start to take the integral things become very ugly.

This is where I am now. http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427e51gddpbmno

If someone could tell me if I'm on the right track and if so which way is the best to integrate and if not then I think I need help setting up the problem. I will be active for the next few hours so we can facilitate the discussion.

 Sorry for using the WolframAlpha tool but I don't know an easier/faster way to post math formulas [/Edit]
Early on in this thread, Simon pointed out that you should include words along with your mathematics to explain what it is that you are doing and/or attempting to do.

None of your subsequent posts do that.

I do apologize. I guess I'm not confident in putting math into words.

I'm trying to follow formulas for similar problems.

I rewrote the problem in terms of x which gave me x=sqrt(9-y)

took the derivative which gave (-1/2)(9-y)^(-1/2)

therefore S=∫2∏x ds

with ds=sqrt(1+(1/4(9-y)))

The next step I have is S= 2pi * integral of ((9-y)^(1/2)*(sqrt(1+(1)/(4(9-y))) from there the integral turns really ugly for me so I'm assuming I've made a mistake somewhere

Can anyone spot a/the mistake?

SammyS
Staff Emeritus
Homework Helper
Gold Member
I do apologize. I guess I'm not confident in putting math into words.

I'm trying to follow formulas for similar problems.

I rewrote the problem in terms of x which gave me x=sqrt(9-y)

took the derivative which gave (-1/2)(9-y)^(-1/2)

therefore S=∫2∏x ds

with ds=sqrt(1+(1/4(9-y)))

The next step I have is S= 2pi * integral of ((9-y)^(1/2)*(sqrt(1+(1)/(4(9-y))) from there the integral turns really ugly for me so I'm assuming I've made a mistake somewhere
Use a common denominator inside the radical sqrt(1+(1)/(4(9-y)), then combine the two radical expressions into one radical expression.

I think you can integrate that.

Simon Bridge
Homework Helper
 Sorry for using the WolframAlpha tool but I don't know an easier/faster way to post math formulas [/Edit]
You can use LaTeX here:
https://www.physicsforums.com/showpost.php?p=3977517&postcount=3
... is is extremely worthwhile learning this method.

I do apologize. I guess I'm not confident in putting math into words.
... that's OK, we've all been there. Just give it your best shot and we'll give you pointers.
The only way to get confident is to do it.

I'm trying to follow formulas for similar problems.
... it is better to use maths ;)

I rewrote the problem in terms of x which gave me x=sqrt(9-y)
$$x=\sqrt{9-y}$$
Reasoning: the distance of the surface from the y-axis is x(y).

took the derivative which gave (-1/2)(9-y)^(-1/2)

therefore S=∫2∏x ds

with ds=sqrt(1+(1/(4(9-y))))
Reasoning: the area of the annulus (the surface ring) between y and y+dy is dS=2\pi x ds where ds is the length, along the surface, between x(y) and x(y+dy).
The total surface area is the sum of all the annuli: $$S=2\pi\int_{S}x\; ds$$
... in this case: ##ds = \sqrt{1+(dx/dy)^2}\; dy##, so you need to differentiate x(y). $$\left(\frac{dx}{dy}\right)^2 = \frac{1}{4}\frac{1}{9-y}$$

The next step I have is S= 2pi * integral of ((9-y)^(1/2)*(sqrt(1+(1)/(4(9-y)))
$$S=2\pi\int_0^9 (9-y)^{^{\frac{1}{2}}}\left(1+\frac{1}{4(9-y)}\right)^{\!\!\frac{1}{2}}\;dy$$

... from there the integral turns really ugly for me so I'm assuming I've made a mistake somewhere
... Ugly integrals are normal in real life, though less common in school problems. You've done tricky integrations before, the way forward is to look for a way to simplify it. SammyS suggests:
Use a common denominator inside the radical sqrt(1+(1)/(4(9-y)), then combine the two radical expressions into one radical expression.
... a "radical expression" is anything with a square-root symbol.
You also got a nudge in that direction in post #4.
... how do you take the √(9-y) inside the other square root?
... you should know several techniques for integrating square-root functions.

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Thanks everyone for your help with this one. I talked with my professor last night and her first question was why I set it up as $x = \sqrt{9-y}$ She told me to try it again the way it is already set up.

Honestly I did it the first way because it was my intuition and I didn't question it.

Here is the solution I got after she made that suggestion.

$y = 9 - x^2$

Taking the derivative:

$f(x)' = -2x$

Squaring the derivative gives: $4x^2$ (for future use)

Set up the Surface Area equation as:

$2pi \int_0^3 x \sqrt{1 + f(x)'^2}\ dx$

Take integral with u-substitution:

Let $u = 1 + 4x^2$ then $du = 8x$

which gives: $∏/4 \int_0^3 u^{(1/2)} \ du$

Integrate: $∏/4 [{(2/3)} u ^{(3/2)}] = ∏/4 [{(2/3)}(1+4x^2)^{(3/2)}]$ evaluated from [0,3]

Sub in the values: $∏/4[{(2/3)}(37)^{(3/2)}] = 117.84$

This is my first attempt at using the Latex code so please forgive me as parts of it still do not look right. Regardless this is the correct answer. ( At least, according to my professor )

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Simon Bridge
Homework Helper
LaTeX use fine - just a tip: to get lower case Greek letters, spell the name without the capital.

So \Pi gets you ##\Pi## while \pi gets you ##\pi##.

For the math - you still sound like you don't really understand what is going on...
It is much better to understand than to memorize equations.

The surface of a solid of revolution has a rotation axis (in this case the y axis) and it has a radius that varies with position along the rotation axis.

Your method starts by working out the area of the annulus between y and y+dy, which, I agree, is more intuitive. The prof's method is to find the area of the annulus between r and r+dr ... (in this case, r=x).
It is the advantage of using polar coordinates.

The equation of the surface is: y(r)=9-r^2
The annulus between r and r+dr has area ##dS=2\pi r \;dl## where ##dl## is the length along the surface between r and r+dr. So $$dl=\sqrt{dr^2 + (y(r+dr)-y(r))^2}\\ \qquad = \sqrt{1+\left(\frac{y(r+dr)-y(r)}{dr}\right)^2}\;dr$$... notice that the remaining squared term inside the radical is the definition of dy/dr ?

Then you get the formula you used above.

In terms of memorized formula - the surface area of a solid of rotaton is: $$S=2\pi\int r\sqrt{dz^2+dr^2}=2\pi\int_z r\sqrt{1+\left(\frac{dr}{dz}\right)^2}\; dz=2\pi\int_r r\sqrt{1+\left(\frac{dz}{dr}\right)^2}\;dr$$ ... where the z-axis is chosen to correspond to the rotation axis, and ##r^2=x^2+y^2## is the distance from z to the surface.

I really appreciate your help with this, Simon.

When you say:
Your method starts by working out the area of the annulus between y and y+dy, which, I agree, is more intuitive. The prof's method is to find the area of the annulus between r and r+dr ... (in this case, r=x).
It is the advantage of using polar coordinates.

When you say the annulus between y and y+dy I am imagining the small change in the y is accounting for the very small height of the rectangle or rings. This height multiplied by length (2$\pi$r) accounts for the surface area of one ring? I'm trying to understand the concept so please bare with me if I'm off.

Also when you explain that the prof's method is to find the annulus between r and r+dr I am not able to visualize this. Are the rings still horizontal? The part that says "between r and r+dr" is what I'm not able to see.

Simon Bridge
Homework Helper
I really appreciate your help with this, Simon.

When you say:

When you say the annulus between y and y+dy I am imagining the small change in the y is accounting for the very small height of the rectangle or rings. This height multiplied by length (2$\pi$r) accounts for the surface area of one ring? I'm trying to understand the concept so please bare with me if I'm off.
You are doing fine so far.

Also when you explain that the prof's method is to find the annulus between r and r+dr I am not able to visualize this. Are the rings still horizontal? The part that says "between r and r+dr" is what I'm not able to see.
Technically each narrow ring is a truncated cone.
The rings conform to the surface of the solid - it's like when you divide the area under a graph into trapeziums rather than bars.

On your graph y=f(x), the two points (x,f(x)) and (x+dx, f(x+dx)) have a "diagonal" line segment between them. The length of this line segment is the "width" of the annulus.

Some people would just say that dy=f(x+dx)-f(x) right off.

I can definitely see the bigger picture now.

The

$$dl=\sqrt{dr^2 + (y(r+dr)-y(r))^2}\\$$

is the length of the line between (x, f(x)) and (x+dx, f(x+dx) from the Pythagorean theorem.

Then using that as the "height" or "width" of the truncated cone and multiplying it by 2##\pi##r then summing up the cones from [0,9] gives you the surface area!

In the equation above, wouldn't $$(y(r+dr)-y(r))$$ give a negative answer? I know it doesn't matter because it will be squared I am just trying to make sure I understand every part of the equation.

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Simon Bridge
Homework Helper
I can definitely see the bigger picture now.

The

$$dl=\sqrt{dr^2 + (y(r+dr)-y(r))^2}\\$$

is the length of the line between (x, f(x)) and (x+dx, f(x+dx) from the Pythagorean theorem.
Well done.

Then using that as the "height" or "width" of the truncated cone and multiplying it by 2##\pi##r then summing up the cones from [0,9] gives you the surface area!
height and width are not good words for this are they - that's why I was careful to call it the "length along the surface" instead. But that's the idea.
If you are summing along the x axis, the limits have to change though.

In the equation above, wouldn't $$(y(r+dr)-y(r))$$ give a negative answer? I know it doesn't matter because it will be squared I am just trying to make sure I understand every part of the equation.
That is correct - because, for that equation, y decreases with r.
That is perfectly fine and you should have met negative slopes before.

Okay, now I understand this problem and what I learned here has spilled over into other integration problems.

I want to ask though. When you integrate along the x-axis are the rings still in the same position as integrating along the y-axis? When I think about integrating along x my mind automatically turns the rings vertical but then the revolution doesn't work so I'm assuming they stay the same?

Simon Bridge
Homework Helper
When you integrate along the x-axis are the rings still in the same position as integrating along the y-axis?

It doesn't matter.
It is good discipline to think of them as being in different positions and having different widths.
You can see this if you replace the dx and dy terms with deltas instead.

The line between ##\big(x,y(x)\big)## and ##\big(x+\Delta x, y(x+\Delta x)\big)## is not always going to be the same length as the one between ##\big(x(y),y \big)## and ##\big(x(y+\Delta y),y+\Delta y \big)##

The infinitesimal version comes when you take the limit that the delta-thing gets very small.

When I think about integrating along x my mind automatically turns the rings vertical but then the revolution doesn't work so I'm assuming they stay the same?
... the rings are neither vertical nor horizontal, they are "diagonal".

...what I learned here has spilled over into other integration problems...
... which is why I teach it that way: well done :)

When you get used to the process of setting up the infinitesimals that way, you'll find you no longer need to memorize the formulas and you get more confident with the results.

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text:
# ${{\boldsymbol X}^{0}}$ Annihilation Cross Section INSPIRE search

Limits are on ${{\mathit \sigma}}{{\mathit v}}$ for ${{\mathit X}^{0}}$ pair annihilation at threshold.
VALUE (cm${}^{3}$s${}^{-1}$) CL% DOCUMENT ID TECN  COMMENT
• • • We do not use the following data for averages, fits, limits, etc. • • •
1
 2019
HAWC DM annihilation to ${{\mathit \gamma}}$s within galactic substructure
$<0.8 \times 10^{-22}$ 95 2
 2019 B
HAWC annihilation/decay to ${{\mathit \gamma}}$ in M31
$<4 \times 10^{-26}$ 95 3
 2019
${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$ and ${{\mathit b}}{{\overline{\mathit b}}}$
$<7 \times 10^{-27}$ 95 4
 2019
FLAT Fermi-LAT M31 and M33
5
 2019
FLAT ${\mathit P}{\mathrm -wave}$ DM; Fermi-LAT
$<2 \times 10^{-26}$ 95 6
 2019 D
FLAT ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit \gamma}}$
$<1 \times 10^{-32}$ 7
 2019
sterile ${{\mathit \nu}}$ decay/annihilation
8
 2019
semi-annihilating DM
$<4 \times 10^{-28}$ 95 9
 2018
HESS ${{\mathit X}^{0}}$ ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}}$ ; galactic halo
$<1 \times 10^{-23}$ 95 10
 2018
MGIC ${{\mathit X}^{0}}$ ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}}$ ; Ursa Major II
$<1 \times 10^{-22}$ 95 11
 2018 B
HAWC ${{\mathit X}^{0}}$ ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}}$ ; Andromeda
$<1 \times 10^{-26}$ 95 12
 2018 A
${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$ $\rightarrow$ ${{\mathit \gamma}}$
13
 2018
THEO Fermi, ${{\mathit \gamma}}$; galaxy groups
14
 2018
FLAT Fermi-LAT CRE data
$<1.2 \times 10^{-23}$ 95 15
 2017 C
ICCB ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ neutrinos
$<1 \times 10^{-23}$ 90 16
 2017 A
ANTR ${{\mathit \nu}}$, DM annihilation
$<1.32 \times 10^{-25}$ 95 17
 2017
VRTS ${{\mathit \gamma}}$ dwarf galaxies
$<7 \times 10^{-21}$ 90 18
 2017
BAIK cosmic ${{\mathit \nu}}$
$<1 \times 10^{-28}$ 19
 2017
MeV DM to ${{\mathit e}^{+}}{{\mathit e}^{-}}$
20
 2016 D
ICCB ${{\mathit \nu}}$, galactic center
$<6 \times 10^{-26}$ 95 21
 2016
HESS Central Galactic Halo
$<1 \times 10^{-27}$ 95 22
 2016 A
HESS WIMP+WIMP $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ ; galactic center
$<3 \times 10^{-26}$ 95 23
 2016
MGFL Satellite galaxy, $\mathit m$(WIMP)=100 GeV
$<1.9 \times 10^{-21}$ 90 24
 2016
BAIK ${{\mathit \nu}}$s from galactic center
$<3 \times 10^{-26}$ 95 25
 2016
FLAT small Magellanic cloud
$<1 \times 10^{-25}$ 95 26
 2016
FLAT Fermi-LAT ${{\mathit \gamma}}$-ray anisotropy
$<5 \times 10^{-27}$ 27
 2016
$<2 \times 10^{-26}$ 95 28
 2016
FLAT dwarf galaxies
$<1 \times 10^{-25}$ 95 29
 2016 A
FLAT Fermi-LAT; M31
$<1 \times 10^{-26}$ 30
 2016
FLAT Fermi-LAT, gamma line
$<1 \times 10^{-25}$ 95 31
 2016
FLAT Fermi-LAT and AMS-02
$<1 \times 10^{-23}$ 95 32
 2016
FLAT extra galactic
33
 2015 C
ICCB ${{\mathit \nu}}$, Galactic halo
34
 2015 E
ICCB ${{\mathit \nu}}$, Galactic center
35
 2015
HESS Galactic center
36
 2015
FLAT monochromatic ${{\mathit \gamma}}$
37
 2015 A
FLAT isotropic ${{\mathit \gamma}}$ background
38
 2015 B
FLAT Satellite galaxy
39
 2015
ANTR ${{\mathit \nu}}$, Galactic center
$<2.90 \times 10^{-26}$ 95 40, 41
 2014
FLAT Satellite galaxy, $\mathit m$ = 10 GeV
$<1.84 \times 10^{-25}$ 95 40, 42
 2014
FLAT Satellite galaxy, $\mathit m$ = 100 GeV
$<1.75 \times 10^{-24}$ 95 40, 42
 2014
FLAT Satellite galaxy, $\mathit m$ = 1 TeV
$<4.52 \times 10^{-24}$ 95 43
 2014
MGIC Segue 1, $\mathit m$ = 1.35 TeV
44
 2013 C
ICCB Galaxies
45
 2013
HESS Central Galactic Halo
46
 2013 A
FLAT Galaxy
47
 2012
HESS Fornax Cluster
48
 2012
FLAT Galaxy
49
 2012
FLAT Galaxy
50
 2012
VRTS Segue 1
$<1 \times 10^{-22}$ 90 51
 2011 C
ICCB Galactic halo, $\mathit m$=1 TeV
$<3 \times 10^{-25}$ 95 52
 2011
HESS Near Galactic center, $\mathit m$=1 TeV
$<1 \times 10^{-26}$ 95 53
 2011
FLAT Satellite galaxy, $\mathit m$=10 GeV
$<1 \times 10^{-25}$ 95 53
 2011
FLAT Satellite galaxy, $\mathit m$=100 GeV
$<1 \times 10^{-24}$ 95 53
 2011
FLAT Satellite galaxy, $\mathit m$=1 TeV
1  ABEYSEKARA 2019 search for ${{\mathit \gamma}}$s from DM annihilation in galactic substructures with HAWC; no signal, limits placed in J$\langle \sigma \cdot{}v\rangle$ vs. declination plane for m(DM) $\sim{}$ $1 - 108$ TeV.
2  ALBERT 2019B search for DM signal from M31 galaxy in ${{\mathit \mu}}$, ${{\mathit \tau}}$, ${{\mathit t}}$, ${{\mathit b}}$, ${{\mathit W}}$ channels using HAWC for m(DM) $\sim{}$ $1 - 100$ TeV; no signal, limits placed in $\langle \sigma \cdot{}v\rangle$ vs. m(DM) plane.
3  CHEUNG 2019 derive model-dependent bounds on $\langle \sigma \cdot{}v\rangle$ from EDGES data: $<$ $4 \times 10^{-26}$ cm${}^{3}$/s for ${{\mathit e}^{+}}{{\mathit e}^{-}}$ and ${{\mathit b}}{{\overline{\mathit b}}}$ for m(${{\mathit \chi}}$) = 100 GeV (including boost factor).
4  DI-MAURO 2019 place limits on WIMP annihilation via Fermi-LAT observation of M31 and M33 galaxies: $\langle \sigma \cdot{}v\rangle$ $<$ $7 \times 10^{-27}$cm${}^{3}$/s for m(${{\mathit \chi}}$) = 20 GeV from M31.
5  JOHNSON 2019 search for ${{\mathit \gamma}}$-rays, $10 - 600$ GeV energy, from ${\mathit P}{\mathrm -wave}$ annihilating DM around SgrA* BH using Fermi-LAT; limits set for various models.
6  LI 2019D search for ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit \gamma}}$ in Fermi-LAT data; no signal, require $\langle \sigma \cdot{}v\rangle$ $<$ $2 \times 10^{-26}$ cm${}^{3}$/s for m(${{\mathit \chi}}$) = 100 GeV.
7  NG 2019 search for X-ray line from sterile ${{\mathit \nu}}$ decay/annihilation using NuStar M-31; no signal: limits placed in m(${{\mathit \nu}}$) vs mixing angle and $\langle \sigma \cdot{}v\rangle$ vs m(${{\mathit \nu}}$).
8  QUEIROZ 2019 examine ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit \chi}}{{\mathit SM}}$ semi-annihilation of DM reaction; limits placed for various assumed ${{\mathit SM}}$ particles in $\langle \sigma \cdot{}v\rangle$ vs. m(${{\mathit \chi}}$) plane.
9  ABDALLAH 2018 search for WIMP WIMP $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}}$ in central galactic halo, 10 years of data; limits placed in $\langle \sigma \cdot{}v\rangle$ vs. m(WIMP) plane for m(WIMP): $0.3 - 70$ TeV.
10  AHNEN 2018 search for WIMP WIMP $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}}$ from Ursa Major II; limits set in $\langle \sigma \cdot{}v\rangle$ vs. m(WIMP) plane for ${{\mathit b}}{{\overline{\mathit b}}}$ , ${{\mathit W}^{+}}{{\mathit W}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ , and ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ annihilation modes.
11  ALBERT 2018B search for TeV-scale WIMPs with WIMP WIMP $\rightarrow$ ${{\mathit \gamma}}{{\mathit X}}$ in Andromeda galaxy using HAWC Observatory; limits set in $\langle \sigma \cdot{}v\rangle$ vs m(WIMP) plane.
12  CHANG 2018A examine ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit b}}{{\overline{\mathit b}}}$ $\rightarrow$ ${{\mathit \gamma}}$ using Fermi Pass 8 data; no signal; require $\langle \sigma \cdot{}v\rangle$ $<$ $10^{-26}$ cm${}^{3}$/s for m(${{\mathit \chi}}$) = 50 GeV.
13  LISANTI 2018 examine Fermi Pass 8 ${{\mathit \gamma}}$-ray data from galaxy groups; report m(WMP) $>$ 30 GeV for annihilation in ${{\mathit b}}{{\overline{\mathit b}}}$ channel.
14  MAZZIOTTA 2018 examine Fermi-LAT electron and positron spectra searching for features originating from DM particles annihilation into ${{\mathit e}^{+}}{{\mathit e}^{-}}$ pairs, from 45 GeV to 2 TeV; no signal found, limits are obtained.
15  AARTSEN 2017C use 1005 days of IceCube data to search for ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ neutrinos via various annihilation channels. Limits set.
16  ALBERT 2017A search for DM annihilation to ${{\mathit \nu}}$s using ANTARES data from $2007 - 2015$. No signal. Limits set in $\langle \sigma \cdot{}v\rangle$ vs. m(DM) plane for m(DM) $\sim{}10 - 10 \times 10^{5}$ GeV. The listed limit is for m(DM) = 100 TeV.
17  ARCHAMBAULT 2017 set limits for WIMP mass between 100 GeV and 1 TeV on $\langle \sigma \cdot{}v\rangle$ for ${{\mathit W}^{+}}{{\mathit W}^{-}}$ , ${{\mathit Z}}{{\mathit Z}}$ , ${{\mathit b}}{{\overline{\mathit b}}}$ , ${{\mathit s}}{{\overline{\mathit s}}}$ , ${{\mathit u}}{{\overline{\mathit u}}}$ , ${{\mathit d}}{{\overline{\mathit d}}}$ , ${{\mathit t}}{{\overline{\mathit t}}}$ , ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit g}}{{\mathit g}}$ , ${{\mathit c}}{{\overline{\mathit c}}}$ , ${{\mathit h}}{{\mathit h}}$ , ${{\mathit \gamma}}{{\mathit \gamma}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ annihilation channels.
18  AVRORIN 2017 find upper limits for the annhilation cross section in various channels for DM particle mass between 30 GeV and 10 TeV. Strongest upper limits coming from the two neutrino channel require $\langle \sigma \cdot{}v\rangle$ $<$ $6 \times 10^{-20}$ cm${}^{3}$/s in dwarf galaxies and $\langle \sigma \cdot{}v\rangle$ $<$ $7 \times 10^{-21}$ cm${}^{3}$/s in LMC for 5 TeV WIMP mass.
19  BOUDAUD 2017 use data from the spacecraft Voyager 1, beyond the heliopause, and from AMS02 on ${{\mathit \chi}}$ ${{\mathit \chi}}$ $\rightarrow$ ${{\mathit e}^{+}}{{\mathit e}^{-}}$ to require $\langle \sigma \cdot{}v\rangle <1. \times 10^{-28}$ cm${}^{3}$/s for m(${{\mathit \chi}}$) = 10 MeV.
20  AARTSEN 2016D search for GeV ${{\mathit \nu}}$s from WIMP annihilation in galaxy; limits set on $\langle \sigma \cdot{}v\rangle$ in Fig. 6, 7.
21  ABDALLAH 2016 require $\langle \sigma \cdot{}v\rangle$ $<$ $6 \times 10^{-26}$ cm${}^{3}$/s for $\mathit m$(WIMP) = 1.5 TeV from 254 hours observation ( ${{\mathit W}}{{\mathit W}}$ channel) and $<$ $2 \times 10^{-26}$ cm${}^{3}$/s for $\mathit m$(WIMP) = 1.0 TeV in ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ channel.
22  ABDALLAH 2016A search for line spectra from WIMP + WIMP $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ in 18 hr HESS data; rule out previous 130 GeV WIMP hint from Fermi-LAT data.
23  AHNEN 2016 require $\langle \sigma \cdot{}v\rangle <3 \times 10^{-26}$ cm${}^{3}$/s for $\mathit m$(WIMP) = 100 GeV ( ${{\mathit W}}{{\mathit W}}$ channel).
24  AVRORIN 2016 require $\langle$s.v$\rangle$ $<$ $1.91 \times 10^{-21}$ cm${}^{3}$/s from WIMP annihilation to ${{\mathit \nu}}$s via ${{\mathit W}}{{\mathit W}}$ channel for $\mathit m$(WIMP) = 1 TeV.
25  CAPUTO 2016 place limits on WIMPs from annihilation to gamma rays in Small Magellanic Cloud using Fermi-LaT data: $\langle \sigma \cdot{}v\rangle <3 \times 10^{-26}$cm${}^{3}$/s for $\mathit m$(WIMP) = 10 GeV.
26  FORNASA 2016 use anisotropies in the ${{\mathit \gamma}}$-ray diffuse emission detected by Fermi-LAT to bound $\langle \sigma \cdot{}v\rangle$ $<$ $10^{-25}$cm${}^{3}$/s for m(WIMP) = 100 GeV in ${{\mathit b}}{{\overline{\mathit b}}}$ channel: see Fig. 28. The limit is driven by dark-matter subhalos in the Milky Way and it refers to their Most Constraining Scenario.
27  LEITE 2016 constrain WIMP annihilation via search for radio emissions from Smith cloud; $\langle \sigma \cdot{}v\rangle$ $<$ $5 \times 10^{-27}$cm${}^{3}$/s in ${{\mathit e}}{{\mathit e}}$ channel for m(WIMP) = 5 GeV.
28  LI 2016 re-analyze Fermi-LAT data on 8 dwarf spheroidals; set limit $\langle \sigma \cdot{}v\rangle <2 \times 10^{-26}$ cm${}^{3}$/s for $\mathit m$(WIMP) = 100 GeV in ${{\mathit b}}{{\overline{\mathit b}}}$ mode with substructures included.
29  LI 2016A constrain $\langle \sigma \cdot{}v\rangle$ $<$ $10^{-25}$cm${}^{3}$/s in ${{\mathit b}}{{\overline{\mathit b}}}$ channel for m(WIMP) = 100 GeV using Fermi-LAT data from M31; see Fig. 6.
30  LIANG 2016 search dwarf spheroidal galaxies, Large Magellanic Cloud, and Small Magellanic Cloud for ${{\mathit \gamma}}$-line in Fermi-LAT data.
31  LU 2016 re-analyze Fermi-LAT and AMS-02 data; require $\langle \sigma \cdot{}v\rangle$ $<10^{-25}$cm${}^{3}$/s for ${\mathit m}_{{{\mathit m}}}$(WIMP) = 1 TeV in ${{\mathit b}}{{\overline{\mathit b}}}$ channel .
32  SHIRASAKI 2016 re-anayze Fermi-LAT extra-galactic data; require $\langle \sigma \cdot{}v\rangle <10^{-23}$cm${}^{3}$/s for $\mathit m$(WIMP) = 1 TeV in ${{\mathit b}}{{\overline{\mathit b}}}$ channel; see Fig. 8.
33  AARTSEN 2015C search for neutrinos from ${{\mathit X}^{0}}$ annihilation in the Galactic halo. See their Figs. 16 and 17, and Table 5 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 100 GeV and 100 TeV.
34  AARTSEN 2015E search for neutrinos from ${{\mathit X}^{0}}$ annihilation in the Galactic center. See their Figs. 7 and 9, and Table 3 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 30 GeV and 10 TeV.
35  ABRAMOWSKI 2015 search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in the Galactic center. See their Fig. 4 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 250 GeV and 10 TeV.
36  ACKERMANN 2015 search for monochromatic ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihlation in the Galactic halo. See their Fig. 8 and Tables $2 - 4$ for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 0.2 GeV and 500 GeV.
37  ACKERMANN 2015A search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation (both Galactic and extragalactic) in the isotropic ${{\mathit \gamma}}$ background. See their Fig. 7 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 10 GeV and 30 TeV.
38  ACKERMANN 2015B search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in 15 dwarf spheroidal satellite galaxies of the Milky Way. See their Figs. 1 and 2 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 2 GeV and 10 TeV.
39  ADRIAN-MARTINEZ 2015 search for neutrinos from ${{\mathit X}^{0}}$ annihilation in the Galactic center. See their Figs. 10 and 11 and Tables 1 and 2 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 25 GeV and 10 TeV.
40  ACKERMANN 2014 search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in 25 dwarf spheroidal satellite galaxies of the Milky Way. See their Tables II--VII for limits assuming annihilation into ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ , ${{\mathit u}}{{\overline{\mathit u}}}$ , ${{\mathit b}}{{\overline{\mathit b}}}$ , and ${{\mathit W}^{+}}{{\mathit W}^{-}}$ , for ${{\mathit X}^{0}}$ mass ranging from 2 GeV to 10 TeV.
41  Limit assuming ${{\mathit X}^{0}}$ pair annihilation into ${{\mathit b}}{{\overline{\mathit b}}}$ .
42  Limit assuming ${{\mathit X}^{0}}$ pair annihilation into ${{\mathit W}^{+}}{{\mathit W}^{-}}$ .
43  ALEKSIC 2014 search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in the dwarf spheroidal galaxy Segue 1. The listed limit assumes annihilation into ${{\mathit W}^{+}}{{\mathit W}^{-}}$ . See their Figs. 6, 7, and 16 for limits on $\sigma \cdot{}$v for annihilation channels ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ , ${{\mathit b}}{{\overline{\mathit b}}}$ , ${{\mathit t}}{{\overline{\mathit t}}}$ , ${{\mathit \gamma}}{{\mathit \gamma}}$ , ${{\mathit \gamma}}{{\mathit Z}}$ , ${{\mathit W}^{+}}{{\mathit W}^{-}}$ , ${{\mathit Z}}{{\mathit Z}}$ for ${{\mathit X}^{0}}$ mass between and $10^{4}$ GeV.
44  AARTSEN 2013C search for neutrinos from ${{\mathit X}^{0}}$ annihilation in nearby galaxies and galaxy clusters. See their Figs. $5 - 7$ for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \nu}}{{\overline{\mathit \nu}}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ , and ${{\mathit W}^{+}}{{\mathit W}^{-}}$ for ${{\mathit X}^{0}}$ mass between 300 GeV and 100 TeV.
45  ABRAMOWSKI 2013 search for monochromatic ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in the Milky Way halo in the central region. Limit on $\sigma \cdot{}$v between $10^{-28}$ and $10^{-25}$ cm${}^{3}$ s${}^{-1}$ (95$\%$ CL) is obtained for ${{\mathit X}^{0}}$ mass between 500 GeV and 20 TeV for ${{\mathit X}^{0}}$ ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ . ${{\mathit X}^{0}}$ density distribution in the Galaxy by Einasto is assumed. See their Fig. 4.
46  ACKERMANN 2013A search for monochromatic ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in the Milky Way. Limit on $\sigma \cdot{}$v for the process ${{\mathit X}^{0}}$ ${{\mathit X}^{0}}$ $\rightarrow$ ${{\mathit \gamma}}{{\mathit \gamma}}$ in the range $10^{-29} - 10^{-27}$ cm${}^{3}$ s${}^{-1}$ (95$\%$ CL) is obtained for ${{\mathit X}^{0}}$ mass between 5 and 300 GeV. The limit depends slightly on the assumed density profile of ${{\mathit X}^{0}}$ in the Galaxy. See their Tables VII$−$X and Fig.10. Supersedes ACKERMANN 2012 .
47  ABRAMOWSKI 2012 search for ${{\mathit \gamma}}$'s from ${{\mathit X}^{0}}$ annihilation in the Fornax galaxy cluster. See their Fig. 7 for limits on $\sigma \cdot{}$v for ${{\mathit X}^{0}}$ mass between 0.1 and 100 TeV for the annihilation channels ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ , ${{\mathit b}}{{\overline{\mathit b}}}$ , and ${{\mathit W}^{+}}{{\mathit W}^{-}}$ .
48  ACKERMANN 2012 search for monochromatic ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in the Milky Way. Limit on $\sigma \cdot{}$v in the range $10^{-28} - 10^{-26}$ cm${}^{3}$s${}^{-1}$ (95$\%$ CL) is obtained for ${{\mathit X}^{0}}$ mass between 7 and 200 GeV if ${{\mathit X}^{0}}$ annihilates into ${{\mathit \gamma}}{{\mathit \gamma}}$ . The limit depends slightly on the assumed density profile of ${{\mathit X}^{0}}$ in the Galaxy. See their Table III and Fig. 15.
49  ACKERMANN 2012 search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in the Milky Way in the diffuse ${{\mathit \gamma}}$ background. Limit on $\sigma \cdot{}$v of $10^{-24}$ cm${}^{3}$s${}^{-1}$ or larger is obtained for ${{\mathit X}^{0}}$ mass between 5 GeV and 10 TeV for various annihilation channels including ${{\mathit W}^{+}}{{\mathit W}^{-}}$ , ${{\mathit b}}{{\overline{\mathit b}}}$ , ${{\mathit g}}{{\mathit g}}$ , ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ . The limit depends slightly on the assumed density profile of ${{\mathit X}^{0}}$ in the Galaxy. See their Figs. $17 - 20$.
50  ALIU 2012 search for ${{\mathit \gamma}}$'s from ${{\mathit X}^{0}}$ annihilation in the dwarf spheroidal galaxy Segue 1. Limit on $\sigma \cdot{}$v in the range $10^{-24} - 10^{-20}$ cm${}^{3}$s${}^{-1}$ (95$\%$ CL) is obtained for ${{\mathit X}^{0}}$ mass between 10 GeV and 2 TeV for annihilation channels ${{\mathit e}^{+}}{{\mathit e}^{-}}$ , ${{\mathit \mu}^{+}}{{\mathit \mu}^{-}}$ , ${{\mathit \tau}^{+}}{{\mathit \tau}^{-}}$ , ${{\mathit b}}{{\overline{\mathit b}}}$ , and ${{\mathit W}^{+}}{{\mathit W}^{-}}$ . See their Fig. 3.
51  ABBASI 2011C search for ${{\mathit \nu}_{{\mu}}}$ from ${{\mathit X}^{0}}$ annihilation in the outer halo of the Milky Way. The limit assumes annihilation into ${{\mathit \nu}}{{\mathit \nu}}$ . See their Fig. 9 for limits with other annihilation channels.
52  ABRAMOWSKI 2011 search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation near the Galactic center. The limit assumes Einasto DM density profile.
53  ACKERMANN 2011 search for ${{\mathit \gamma}}$ from ${{\mathit X}^{0}}$ annihilation in ten dwarf spheroidal satellite galaxies of the Milky Way. The limit for $\mathit m$ = 10 GeV assumes annihilation into ${{\mathit b}}{{\overline{\mathit b}}}$ , the others ${{\mathit W}^{+}}{{\mathit W}^{-}}$ . See their Fig. 2 for limits with other final states. See also GERINGER-SAMETH 2011 for a different analysis of the same data.
References:
 ABEYSEKARA 2019
JCAP 1907 022 Searching for Dark Matter Sub-structure with HAWC
 ALBERT 2019B
JCAP 1904 E01 Search for Dark Matter Gamma-ray Emission from the Andromeda Galaxy with the High-Altitude Water Cherenkov Observatory
 CHEUNG 2019
PL B789 137 The impact of EDGES 21-cm data on dark matter interactions
 DI-MAURO 2019
PR D99 123027 Search for $\gamma$-ray emission from dark matter particle interactions from Andromeda and Triangulum Galaxies with the Fermi Large Area Telescope
 JOHNSON 2019
PR D99 103007 Search for gamma-ray emission from $p$-wave dark matter annihilation in the Galactic Center
 LI 2019D
PR D99 123519 Search for line-like signals in the all-sky Fermi-LAT data
 NG 2019
PR D99 083005 New Constraints on Sterile Neutrino Dark Matter from $NuSTAR$ M31 Observations
 QUEIROZ 2019
JCAP 1904 048 Search for Semi-Annihilating Dark Matter with Fermi-LAT, H.E.S.S., Planck, and the Cherenkov Telescope Array
 ABDALLAH 2018
PRL 120 201101 Search for $\gamma$-Ray Line Signals from Dark Matter Annihilations in the Inner Galactic Halo from 10 Years of Observations with H.E.S.S.
 AHNEN 2018
JCAP 1803 009 Indirect dark matter searches in the dwarf satellite galaxy Ursa Major II with the MAGIC Telescopes
 ALBERT 2018B
JCAP 1806 043 Search for Dark Matter Gamma-ray Emission from the Andromeda Galaxy with the High-Altitude Water Cherenkov Observatory
 CHANG 2018A
PR D98 123004 Search for dark matter annihilation in the Milky Way halo
 LISANTI 2018
PRL 120 101101 Search for Dark Matter Annihilation in Galaxy Groups
 MAZZIOTTA 2018
PR D98 022006 Search for features in the cosmic-ray electron and positron spectrum measured by the Fermi Large Area Telescope
 AARTSEN 2017C
EPJ C77 627 Search for Neutrinos from Dark Matter Self-Annihilations in the Center of the Milky Way with 3 Years of IceCube/DeepCore
 ALBERT 2017A
PL B769 249 Results from the Search for Dark Matter in the Milky Way with 9 Years of Data of the ANTARES Neutrino Telescope
 ARCHAMBAULT 2017
PR D95 082001 Dark Matter Constraints from a Joint Analysis of Dwarf Spheroidal Galaxy Observations with VERITAS
 AVRORIN 2017
JETP 125 80 Dark Matter Constraints from an Observation of dSphs (dwarf spheroidal galaxies) and the LMC (Large Magellanic Cloud) with the Baikal NT200
 BOUDAUD 2017
PRL 119 021103 Novel Cosmic-ray Electron and Positron Constraints on MeV Dark Matter Particles
 AARTSEN 2016D
EPJ C76 531 All-flavour Search for Neutrinos from Dark Matter Annihilations in the Milky Way with IceCube/DeepCore
 ABDALLAH 2016
PRL 117 111301 Search for Dark Matter Annihilations Towards the Inner Galactic Halo from 10 Years of Observations with H.E.S.S
 ABDALLAH 2016A
PRL 117 151302 H.E.S.S. Limits on Linelike Dark Matter Signatures in the 100 GeV to 2 TeV Energy Range Close to the Galactic Center
 AHNEN 2016
JCAP 1602 039 Limits to Dark Matter Annihilation Cross-Section from a Combined Analysis of MAGIC and Fermi-LAT Observations of Dwarf Satellite Galaxies
 AVRORIN 2016
ASP 81 12 A Search for Neutrino Signal from Dark Matter Annihilation in the Center of the Milky Way with Baikal NT200
 CAPUTO 2016
PR D93 062004 Search for Gamma-ray Emission from Dark Matter Annihilation in the Small Magellanic Cloud with the Fermi Large Area Telescope
 FORNASA 2016
PR D94 123005 Angular Power Spectrum of the Diffuse Gamma-Ray Emission as Measured by the Fermi Large Area Telescope and Constraints on its Dark Matter Interpretation
 LEITE 2016
JCAP 1611 021 Synchrotron Emission from Dark Matter in Galactic Subhalos. A Look into the Smith Cloud
 LI 2016
PR D93 043518 Search for Gamma-Ray Emission from Eight dwarf Spheroidal Galaxy Candidates Discovered in Year Two of Dark Energy Survey with Fermi-LAT Data
 LI 2016A
JCAP 1612 028 Constraints on the Dark Matter Annihilation from Fermi-LAT Observation of M31
 LIANG 2016
PR D94 103502 Search for Gamma-Ray Line Features from Milky Way Satellites with Fermi LAT Pass 8 Data
 LU 2016
PR D93 103517 Limits on Dark Matter from AMS-02 Antiproton and Positron Fraction Data
 SHIRASAKI 2016
PR D94 063522 Cosmological Constraints on Dark Matter Annihilation and Decay: Cross-Correlation Analysis of the Extragalactic ${{\mathit \gamma}}$-ray Background and Cosmic Shear
 AARTSEN 2015E
EPJ C75 492 Search for Dark Matter Annihilation in the Galactic Center with IceCube-79
 AARTSEN 2015C
EPJ C75 20 Multipole Analysis of IceCube Data to Search for Dark Matter Accumulated in the Galactic Halo
 ABRAMOWSKI 2015
PRL 114 081301 Constraints on an Annihilation Signal from a Core of Constant Dark Matter Density around the Milky Way Center with H.E.S.S.
 ACKERMANN 2015A
JCAP 1509 008 Limits on Dark Matter Annihilation Signals from the Fermi LAT 4-year Measurement of the Isotropic Gamma-Ray Background
 ACKERMANN 2015B
PRL 115 231301 Searching for Dark Matter Annihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data
 ACKERMANN 2015
PR D91 122002 Updated Search for Spectral Lines from Galactic Dark Matter Interactions with Pass 8 Data from the Fermi Large Area Telescope
JCAP 1510 068 Search of Dark Matter Annihilation in the Galactic Centre using the ANTARES Neutrino Telescope
 ACKERMANN 2014
PR D89 042001 Dark Matter Constraints from Observations of 25 Milky Way Satellite Galaxies with the Fermi Large Area Telescope
 ALEKSIC 2014
JCAP 1402 008 Optimized Dark Matter Searches in Deep Observations of Segue 1 with MAGIC
 AARTSEN 2013C
PR D88 122001 IceCube Search for Dark Matter Annihilation in Nearby Galaxies and Galaxy Clusters
 ABRAMOWSKI 2013
PRL 110 041301 Search for Photon Line-Like Signatures from Dark Matter Annihilations with H.E.S.S
 ACKERMANN 2013A
PR D88 082002 Search for Gamma-Ray Spectral Lines with the Fermi Large Area Telescope and Dark Matter Implications
 ABRAMOWSKI 2012
APJ 750 123 Search for Dark Matter Annihilation Signals from the Fornax Galaxy Cluster with H.E.S.S
 ACKERMANN 2012
PR D86 022002 Fermi LAT (Large Area Telescope) Search for Dark Matter in Gamma-Ray Lines and the Inclusive Photon Spectrum
 ALIU 2012
PR D85 062001 VERITAS Deep Observations of the Dwarf Spheroidal Galaxy Segue 1
 ABBASI 2011C
PR D84 022004 Search for Dark Matter from the Galactic Halo with the IceCube Neutrino Telescope
 ABRAMOWSKI 2011
PRL 106 161301 Search for a Dark Matter Annihilation Signal from the Galactic Center Halo with H.E.S.S.
 ACKERMANN 2011
PRL 107 241302 Constraining Dark Matter Models from a Combined Analysis of Milky Way Satellites with the Fermi Large Area Telescope


text:
Synopsis

# Air showers from ultrahigh-energy cosmic rays

Physics 3, s26
Cascades created by cosmic rays interacting with the atmosphere provide clues about the mass composition of ultrahigh-energy cosmic rays.

The nature and origin of ultrahigh-energy ( $>1018eV$) cosmic rays remain a mystery. Astrophysicists hunt for clues regarding their mass composition, which, along with other properties such as the flux and arrival direction distribution, should help distinguish among the various models of the sources and propagation of cosmic rays. However, because of the low flux of ultrahigh-energy cosmic rays, the mass composition cannot be measured directly. Instead, it is inferred from measurements of the extensive air showers—the cascades of high-energy ions created when incident cosmic rays collide with atoms in the atmosphere. The atmospheric depth at which development of a shower reaches its maximum number of secondary particles depends on the mass and energy of the incident particle. Data on the depth of the maximum thus provide information about the mass composition.

The Pierre Auger Collaboration has presented in Physical Review Letters new measurements of extensive air showers from ultrahigh-energy cosmic rays. Their observations suggest a gradual increase in the average mass of cosmic rays in a region of energies around $1019eV$. The data are the highest statistics measurements to date. – Stanley Brown

## Related Articles

Cosmology

### Dark Matter Detector Proves its Sensitivity

A new sensor provides world-leading sensitivity for distinguishing lightweight dark matter from background noise. Read More »

Cosmology

### Looking for Dark Matter in Neutron Star Light

Radio observations of neutron stars demonstrate a way to search for axions through their expected conversion to electromagnetic waves in a star’s magnetic field. Read More »

Particles and Fields

### Dark Matter Detector Delivers Enigmatic Signal

Are the excess events detected by the XENON1T experiment a harbinger of new physics or a mundane background? Read More »


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# If the elevation above water level of a breaching whale is given by th

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If the elevation above water level of a breaching whale is given by the expression −t^2 + 12t − 35 where t is the time, in seconds, after an observer begins watching, is the whale presently underwater?

(1) t < 8.

(2) t > 5.

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Re: If the elevation above water level of a breaching whale is given by th  [#permalink]

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09 Feb 2017, 03:28
Bunuel wrote:
If the elevation above water level of a breaching wale is given by the expression −t^2 + 12t − 35 where t is the time, in seconds, after an observer begins watching, is the whale presently underwater?

(1) t < 8.

(2) t > 5.

E

when t = 6 value is 1, 7 then value is 0 and 8 then value is -3. I am not sure though
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Re: If the elevation above water level of a breaching whale is given by th  [#permalink]

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16 Feb 2017, 00:47
Bunuel wrote:
If the elevation above water level of a breaching wale is given by the expression $$−t^2 + 12t − 35$$ where t is the time, in seconds, after an observer begins watching, is the whale presently underwater?

(1) $$t < 8$$.

(2) $$t > 5$$.

Official solution from Veritas Prep.
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Re: If the elevation above water level of a breaching whale is given by th  [#permalink]

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16 Feb 2017, 09:41
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Bunuel wrote:
If the elevation above water level of a breaching wale is given by the expression −t^2 + 12t − 35 where t is the time, in seconds, after an observer begins watching, is the whale presently underwater?

(1) t < 8.

(2) t > 5.

A breaching whale is a whale leaping out of the water.

Elevation = −t^2 + 12t − 35

(1) t < 8
Let's notice what happens at exactly t = 8
Elevation = -(8)^2 + 12*8 - 35 = -3
It means the whale is under water at this time.

Put t = 7
Elevation = -(7)^2 + 12*7 - 35 = 0
So the whale is exactly at the surface at this time.

Put t = 6
Elevation = -(6)^2 + 12*6 - 35 = 1
The whale has a positive elevation at this time.

Let's find out what happens at t = 5
Elevation = -(5)^2 + 12*5 - 35 = 0
So the whale is exactly at the surface at this time.

We see what is happening here. The whale is exactly at the surface at t = 5, is out of water and then back in the water at t = 7.

Given that t < 8, the whale is underwater if t > 7 or t < 5. Not sufficient.

(2) t > 5
If t = 6, the whale is out of water but if t > 7, it is underwater. Not sufficient.

Using both, If t = 6, the whale is out of water but if t > 7, the whale is underwater. Not sufficient.

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text:
1. ## Quick homework help...

Could some explain how i could i start these problems?

and
solving this by factoring

Thanks for the help

thind06

2. Originally Posted by thind06
Could some explain how i could i start these problems?
To learn how to get started, try studying some online lessons on solving radical equations and factoring quadratics.

Once you've learned the basic terms and techniques, you can attempt the exercises. For the first one, a good start would be to subtract the x to the right-hand side, and then square both sides, keeping in mind that (8 - x)^2 means (8 - x)(8 - x).

For the second one, I'm not sure if they want you to square first, and then solve by factoring, or else rearrange as follows:

. . . . . $\left(\sqrt{x}\right)^2\, -\, 1\sqrt{x}\, -\, 6\, =\, 0$


text:
Mathematics

# Euclid's division lemma states that for two positive integers a and b, there exists unique integers q and rr such that $a=bq+r$, where r must satisfy $0\le r\le b$ is this statement true?

##### SOLUTION
required conditions
$0\le r<b$
$a$ is the quotient, $b$ is the divisor $a$ is divided and $r$ is remainder
Remainder cannot be more than divisor

You're just one step away

Subjective Medium Published on 09th 09, 2020
Questions 120418
Subjects 10
Chapters 88
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#### Realted Questions

Q1 Single Correct Medium
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.
(iii) Things which are double of the same thing are equal to one another.
Check whether the given system of axioms is consistent or inconsistent
• A. inconsistent
• B. Only (i) & (ii) are consistent
• C. Only  (iii) is consistent
• D. consistent

Asked in: Mathematics - Introduction to Euclid’s Geometry

1 Verified Answer | Published on 09th 09, 2020

Q2 TRUE/FALSE Medium
State TRUE or FALSE
There exist at least three points that are not on the same line.
• A. False
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The Euclidean geometry is valid only for figures in the plane.
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According to Euclid : The whole is greater than the part .State whether that this is true or false.
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Q5 Subjective Medium
Give an example for the following axioms from your experience:
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text:
# Is this the correct way to obtain $<f|i>$ term in $\phi^4$ interaction theory? [closed]

Lets first write the expectation value of the fields in the interaction picture;

$$<\Omega|T\phi(x_1)\phi(x_2)\phi(x_1')\phi(x_2')|\Omega>\\=\frac{i\lambda}{4!}<0|T\phi_I(x_1)\phi_I(x_2)\phi_I(x_1')\phi_I(x_2')\int d^4y \phi_I^4(y)|0>\\+\frac{1}{2}(\frac{i\lambda}{4!})^2<0|T\phi_I(x_1)\phi_I(x_2)\phi_I(x_1')\phi_I(x_2')\int d^4y \phi_I^4(y)\int d^4z \phi_I^4(z)|0>+O(\lambda^3)\\ = \frac{i\lambda}{4!}\int d^4y 4\Delta(x_1-y)3\Delta(x_2-y)2\Delta(x_1'-y)\Delta(x_2'-y) \\+\frac{1}{2}(\frac{i\lambda}{4!})^2\int d^4y\int d^4z 16\Delta(y-z)9\Delta(y-z)4\Delta(x_1-y)\{2\Delta(x_2-y)\Delta(x_1'-z)\Delta(x_2'-z)+\Delta(x_2-z)2\Delta(x_1'-y)\Delta(x_2'-z)+\Delta(x_2-z)\Delta(x_1'-z)2\Delta(x_2'-y)\}+O(\lambda^3)\\ = i\lambda\int d^4y \Delta(x_1-y)\Delta(x_2-y)\Delta(x_1'-y)\Delta(x_2'-y) \\+(i\lambda)^2\int d^4y\int d^4z \Delta(y-z)\Delta(y-z)\{\Delta(x_1-y)\Delta(x_2-y)\Delta(x_1'-z)\Delta(x_2'-z)+\Delta(x_1-y)\Delta(x_2-z)\Delta(x_1'-y)\Delta(x_2'-z)+\Delta(x_1-y)\Delta(x_2-z)\Delta(x_1'-z)\Delta(x_2'-y)\}+O(\lambda^3)\\$$

Then we need to write $<f|i>$ such as;

$$<f|i>=(i)^4\int d^4x_1 \int d^4x_2\int d^4x_3\int d^4x_4(-\partial_1^2+m^2)(-\partial_1^2+m^2)(-\partial_1'^2+m^2)(-\partial_2'^2+m^2)<\Omega|T\phi(x_1)\phi(x_2)\phi(x_1')\phi(x_2')|\Omega>$$

My question is that; is the following part right? $$...16\Delta(y-z)9\Delta(y-z)...$$ I mean if I investigate $\phi^5$ theory then this term would contained 3 $\Delta(y-z)$ such as,

$$...25\Delta(y-z)16\Delta(y-z)9\Delta(y-z)...$$

right?

## closed as off-topic by ACuriousMind♦, HDE 226868, Gert, Kyle Kanos, user36790 Nov 3 '15 at 9:08

• This question does not appear to be about physics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.


text:
# American Institute of Mathematical Sciences

October  2020, 13(10): 2905-2915. doi: 10.3934/dcdss.2020221

## Space-time kernel based numerical method for generalized Black-Scholes equation

 Department of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan

* Corresponding author: Marjan Uddin

Received  February 2019 Revised  June 2019 Published  December 2019

Fund Project: This work is supported by HEC Pakistan

In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

Citation: Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221
##### References:

show all references

##### References:
A typical centers arrangements in global space-time domain as well as in a local sub-domain, and sparsity of descretized operator of problem 1, where $m = 100$, $n = 10$
The exact solution versus the approximate solution in space-time domain corresponding to problem 1, when $m = 1600$ and $n = 10$ in domain $(x,t)\in (-2,2)\times(0,1)$
The numerical solution and error in space-time domain, corresponding to problem 2 when $m = 400$ and $n = 10$ in the domain $(x,t)\in (0,1)\times(0,1)$
Double barrier option prices obtained by space-time local kernel method
Call option prices obtained by space-time local kernel method
Put option prices obtained by space-time local kernel method
Sapce-time (ST) solution of problem 1 for different total collocation points $m$ and stencil size $n$, and time integration (TI) solution in domain $(x,t)\in (-2,2)\times(0,1)$
 $m$ $n$ $L_{\infty}$ ST method (C.time) TI method (C.time) 100 10 1.23E-02 0.6783 3.2891 400 9.49E-02 0.7123 6.2821 1600 1.08E-04 0.9129 10.2370 2500 1.26E-04 1.1234 15.2950 100 15 2.37E-02 0.8234 4.3491 400 1.45E-03 10.2356 9.2371 1600 2.45E-03 11.2916 13.9820 2500 3.45E-04 13.1087 20.3582 100 20 2.22E-02 9.1835 10.10491 400 2.89E-03 11.2349 12.7146 1600 9.88E-03 14.8679 21.3812 2500 7.23E-04 16.1955 25.0492
 $m$ $n$ $L_{\infty}$ ST method (C.time) TI method (C.time) 100 10 1.23E-02 0.6783 3.2891 400 9.49E-02 0.7123 6.2821 1600 1.08E-04 0.9129 10.2370 2500 1.26E-04 1.1234 15.2950 100 15 2.37E-02 0.8234 4.3491 400 1.45E-03 10.2356 9.2371 1600 2.45E-03 11.2916 13.9820 2500 3.45E-04 13.1087 20.3582 100 20 2.22E-02 9.1835 10.10491 400 2.89E-03 11.2349 12.7146 1600 9.88E-03 14.8679 21.3812 2500 7.23E-04 16.1955 25.0492
Observed maximum absolute error for example 1 in Reza [20] and Kadabajoo [13] for different $\theta$ in domain $(x,t)\in (-2,2)\times(0,1)$
 $M = N$ 10 20 40 80 160 [20] for $\theta=1$ 0.0724 0.0312 0.0139 0.0608 0.000271 [20] for $\theta=\frac{1}{2}$ 0.00112 0.0208 3.91e-05 7.19e-06 1.31e-06 [13] for $\theta=1$ 0.0744 0.0394 0.0202 0.0102 0.00516 [13] for $\theta=\frac{1}{2}$ 0.00589 0.00146 0.000364 9.1e-05 2.27e-05
 $M = N$ 10 20 40 80 160 [20] for $\theta=1$ 0.0724 0.0312 0.0139 0.0608 0.000271 [20] for $\theta=\frac{1}{2}$ 0.00112 0.0208 3.91e-05 7.19e-06 1.31e-06 [13] for $\theta=1$ 0.0744 0.0394 0.0202 0.0102 0.00516 [13] for $\theta=\frac{1}{2}$ 0.00589 0.00146 0.000364 9.1e-05 2.27e-05
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