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## Forum archive 2000-2006

### Lars Jensen - Problem display errors - ww 2.2

by Arnold Pizer -
Number of replies: 0
 Problem display errors - ww 2.2 topic started 5/23/2006; 12:17:06 PMlast post 5/25/2006; 7:17:39 AM
 Lars Jensen - Problem display errors - ww 2.2  5/23/2006; 12:17:06 PM (reads: 832, responses: 16) Hi, There seems to be some difficulties with displaying problems in ww 2.2. The issue does not seem to be related to the particular display mode used. The error message I see again and again is included below. Some problems where I see this error are: File name: asuLibrary/set119LinearSystems/p15.pg File name: asuLibrary/set119LinearSystems/p16.pg File name: asuLibrary/set119LinearSystems/p17.pg Almost all of unionLibrary/setAlgebraAbsoluteValue Almost all of unionLibrary/setAlgebraInequalities Most problems in the dartmouthLibrary/Marsden-Tromba-Weinstein library etc. Any ideas what the error means? Thanks, Lars. ------------------------------------------------------ Warning messages Error messages sourceFilePath is unsafe! at /opt/webwork2/lib/WeBWorK/ContentGenerator/Problem.pm line 462. Call stack The information below can help locate the source of the problem. * in WeBWorK::ContentGenerator::Problem::pre_header_initialize called at line 155 of /opt/webwork2/lib/WeBWorK/ContentGenerator.pm * in WeBWorK::ContentGenerator::go called at line 299 of /opt/webwork2/lib/WeBWorK.pm <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/23/2006; 2:45:06 PM (reads: 879, responses: 1) Hi Lars, This comes from a change in which we insisted that all problems be in subdirectories of the templates directory (or linked to from that directory). This fixes a significant security hole which allowed any readable file on the computer to be retrieved by hacking the URL. (See bug #970 http://bugs.webwork.rochester.edu/show_bug.cgi?id=970 for gory details.) The fix, or at least a partial one, restricts the directories in which problems can be replaced to the templates directory (and subdirectories) and directories linked to from there. I handle this by placing the problem libraries under a webwork directory: webwork | ----------------------------------------------------------- | | | | |webwork2 pg courses rochester_problib union_problib etc. and in the modelCourse/templates directory I place links to each of these problem libraries. When new courses are created using the modelCourse as a template the problem libraries are properly linked in. For existing courses you will need to create the links by hand (if there are few of them) or you can write a script that will place the links there if you have a great many. I'm pretty sure that you will find that you have set the addresses for the problem libraries which are causing trouble to be some directory not under the templates directory. Hope this helps. -- Mike <| Post or View Comments |>

 Lars Jensen - Re: Problem display errors - ww 2.2  5/23/2006; 2:59:54 PM (reads: 940, responses: 0) Hi Mike, Thanks for the answer. How does this explain that the vast majority of problems work? Our rochester_problib problems all seem to show fine. So does most others. The one I listed above are exceptions. Out directory structure is: /opt/webwork2/ /opt/pg/ /opt/wwlib And all the problem libraries are under wwlib. For example, /opt/wwlib/rochester_problib /opt/wwlib/asu_problib Thanks, Lars. <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/23/2006; 3:24:33 PM (reads: 881, responses: 1) Hi Lars, Look at the templates directory for one of the courses where the problem occurs. You might see links such as: rochesterLibrary --> /opt/wwlib/rochester_problib but no link for unionLibrary --> /opt/wwlib/rochester_problib and similarly for asuLibrary you might also see local copies of some of the union library problems which don't cause any problems. Check the url that the problem is trying to reach (particularly the value of sourceFilePath ) and see if it is under the templates directory for that course. -- Mike <| Post or View Comments |>

 Lars Jensen - Re: Problem display errors - ww 2.2  5/23/2006; 5:55:06 PM (reads: 939, responses: 0) Hi Mike, Thanks for the suggestion - however, all the symbolic links are in the templates directory. When I say above that "Almost all of unionLibrary/setAlgebraAbsoluteValue" I mean that almost all of the problems in the set produces the error, but *some do not!* Some of the problems in the same set display fine and the rest produce the error listed above... Any other ideas? Thanks, Lars. <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/23/2006; 7:09:02 PM (reads: 851, responses: 2) Hi, OK. I'm getting a bit puzzled. I can't reproduce this error here. The error message sourceFilePath is unsafe! at /opt/webwork2/lib/WeBWorK/ContentGenerator/Problem.pm line 462. is pretty explicit it happens when the variable &sourceFilePath= .....& is set to something which is not a subset of the templates directory for the course. (this variable will appear in the url, after the ? mark and probably waaaay off to the right in the url that's in the url box when the error is reported. Knowing what sourceFilePath is set too would give us some clue. The other thing is to look at the union setAlgebraAbsoluteValue from the homework editor and look at each of the paths to the problem source files to see if any look very different from the others. I would expect this to fail on a problem by problem basis (the error is in Problem.pm) since the path to each problem is individual, so the fact that some problems in a set work and some don't is not necessarily surprising. Are you looking at these problems from a set that has been built? or from the library browser? Hope this helps. Take care, Mike <| Post or View Comments |>

 Lars Jensen - Re: Problem display errors - ww 2.2  5/24/2006; 11:42:02 AM (reads: 913, responses: 1) Hi Mike, There's more to this problem than I first reported. My first comments was on displaying the problems only: Some problems in the set display fine in the library browser, while others in the *same* set don't. But when I click "Try it" (in the library browser) all problems produce the error message above, no matter how they display. This is also the case for *all* other problems in *all* other libraries, including the rochester library... All symbolic links pointing to the libraries in /opt/wwlib are present in the templates directories of our courses. What could be going on here? Lars. <| Post or View Comments |>

 Lars Jensen - Re: Problem display errors - ww 2.2  5/24/2006; 1:37:27 PM (reads: 977, responses: 0) Hi, This problem is definitely not related to symbolic links because I downloaded the problem library directly into the templates directory of the course, and I still get the same error message.... The &sourceFilePath part of the URL is this: &sourceFilePath=union_problib/setAlgebraAbsoluteValue/p1-abs-a.pg Any ideas? Lars. <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/24/2006; 2:53:57 PM (reads: 851, responses: 0) OK, plan C. It may be that there is a bug in your version of the Problem.pm file (your path for sourceFilePath looks correct by the way, so I don't think that is a problem). Bug #994 addressed a problem with Problem Library: SetMaker uses a relative path for sourceFilePath, which is nice, but which path_is_subdir didn't account for. Added thrid argument, $allow_relative, to that utility which causes$dir to be prepended to $path. Modified Problem.pm to trigger this behavior. Fixed in HEAD: Utils.pm 1.74 Problem.pm 1.198 rel-2-2-dev: Utils.pm 1.71.2.3 Problem.pm 1.191.2.7 can you check your version numbers against those? You can get the latest versions of the files from the CVS. If you are already connected to the CVS (i.e. there CVS directories in among your source files) then cvs update -r 1.71.2.3 Utils.pm will get you the version of Utils.pm connected with rel-2-2-dev. You can also check around line 461 in Problem.pm -- see if you have: die "sourceFilePath is unsafe!" unless path_is_subdir($sourceFilePath, $ce->{courseDirs}->{templates}, 1); # 1==path can be relative to dir If the 1 is not there, then you need to update Problem.pm and also the Utils.pm file. Hopefully this will solve the problem. (Why this problem would cause troubles with some problem paths but not others is a complete mystery to me however.) Good luck. -- Mike <| Post or View Comments |>  Lars Jensen - Re: Problem display errors - ww 2.2 5/24/2006; 4:53:02 PM (reads: 830, responses: 0) Hi Mike, Above instructions - replacing Utils.pm and Problem.pm - definitely helped. Now most of the problems display in the library browser, and "Try it" works for those that do display correctly. Several problems still don't display, but the error message about &sourceFilePath has been replaced by errors like the one below. (The &sourcefilepath error seems to have gone away.) Thanks, Lars. --------------------------------------------------- WeBWorK Error WeBWorK has encountered a software error while attempting to process this problem. It is likely that there is an error in the problem itself. If you are a student, report this error message to your professor to have it corrected. If you are a professor, please consult the error output below for more information. Error messages expected an answer hash at line 2502 of (eval 209) Died within main::trim_whitespace called at line 297 of [PG]/macros/listAnswer.pl from within main::std_list_cmp called at line 33 of [PG]/macros/unionAnswer.pl from within main::num_union_cmp called at line 37 of [TMPL]/unionLibrary/setAlgebraAbsoluteValue/p1-abs-b.pg Error details Problem1 ERROR caught by Translator while processing problem file:unionLibrary/setAlgebraAbsoluteValue/p1-abs-b.pg * expected an answer hash at line 2502 of (eval 209) Died within main::trim_whitespace called at line 297 of [PG]/macros/listAnswer.pl from within main::std_list_cmp called at line 33 of [PG]/macros/unionAnswer.pl from within main::num_union_cmp called at line 37 of [TMPL]/unionLibrary/setAlgebraAbsoluteValue/p1-abs-b.pg * ------Input Read 1 ##DESCRIPTION 2 ## Inequality 3 ##ENDDESCRIPTION 4 ## K. Lesh, Union College, 9/9/03 5 6 DOCUMENT(); # This should be the first executable line in the problem. 7 8 loadMacros( 9 PG.pl, 10 PGbasicmacros.pl, 11 PGchoicemacros.pl, 12 PGanswermacros.pl, 13 PGauxiliaryFunctions.pl, 14 "PGunion.pl", # Union College utilities 15 "unionAnswer.pl", # Load answer-checker appropriate to the problem 16 "PGcourse.pl", # Customization file for the course 17 ); 18 19 TEXT(beginproblem()); 20 21 BEGIN_PROBLEM(); 22 23$a = random(1,8,1); 24 $b = random(2,10,1); 25$c = random(3,15,1); 26 $min=-$a-$b*$c; 27 $max=$b*$c-$a; 28 29 BEGIN_TEXT 30 Solve the following inequality. Enter the answer in interval notation. 31 [ left|frac{x+$a}{$b}right| leq $c ] 32$BR 33 Answer: {ans_rule(35)} 34 $PAR 35$INFINITY_UNION_MESSAGE 36 END_TEXT 37 38 $showPartialCorrectAnswers = 0; 39 ANS(num_union_cmp("[$min,\$max]")); 40 41 END_PROBLEM(); 42 ENDDOCUMENT(); # This should be the last executable line in the problem.; ----- <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/24/2006; 5:52:45 PM (reads: 847, responses: 0) Hi Lars, well that's progress at least. I can reproduce this problem our production server, but not on hosted or on our development server. I haven't yet been able to figure out exactly where this bug was fixed. I'll keep at it. -- MIke <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/24/2006; 6:33:06 PM (reads: 841, responses: 0) Lars, Try updating the union_problib from the CVS -- that solved some of the problema here. Unless there are multiple bugs at your end it will probably work for you as well. -- Mike <| Post or View Comments |>

 Michael Gage - Re: Problem display errors - ww 2.2  5/24/2006; 7:51:07 PM (reads: 839, responses: 0) I solved the remaining bugs here once I removed some old local copies of the macros files which were overriding the new macro files I got from the CVS. I don't think you have local copies of the macro files so once you've updated the union_problib you should be ready to go. Let us know how it turns out. -- Mike <| Post or View Comments |>

 Lars Jensen - Re: Problem display errors - ww 2.2  5/25/2006; 1:58:36 AM (reads: 839, responses: 3) Hi Mike, My union_problib was freshly updated already. However, I figured out that one issue is that some Union problems refer to PGauxiliaryfunctions.pl instead of PGauxiliaryFunctions.pl - this obviously produces an error. Another common error is this: Error messages expected an answer hash at line 2502 of (eval 374) Died within main::trim_whitespace called at line 297 of [PG]/macros/listAnswer.pl from within main::std_list_cmp called at line 33 of [PG]/macros/unionAnswer.pl from within main::num_union_cmp called at line 60 of [TMPL]/unionLibrary/setAlgebraInequalities/an_appA_1.pg Error details Problem1 ERROR caught by Translator while processing problem file:unionLibrary/setAlgebraInequalities/an_appA_1.pg * expected an answer hash at line 2502 of (eval 374) Died within main::trim_whitespace called at line 297 of [PG]/macros/listAnswer.pl from within main::std_list_cmp called at line 33 of [PG]/macros/unionAnswer.pl from within main::num_union_cmp called at line 60 of [TMPL]/unionLibrary/setAlgebraInequalities/an_appA_1.pg Lars. <| Post or View Comments |>

 Davide P. Cervone - Re: Problem display errors - ww 2.2  5/25/2006; 6:51:45 AM (reads: 897, responses: 1) Lars: Make sure you update the MACROS that are from the Union library, not just the problem files. Note that the error message indicates that you are getting listAnswer.pl from the pg/macros directory, so unless you have update the copies in pg/macros by hand, you will be getting the old copies. There was a change to the trim_whitespace command last fall that required changes to the code that calls it, and listAnswer.pl was one such place. Now that WeBWorK has a more flexible macro library path variable (which can be set in global.conf), you might want to remove the Union macros from pg/macros and simply have the macrosPath variable include the directory where they can be found (/opt/wwlib/union_problib/macros) so that you don't have to worry about making updates by hand in the future. I'll look for the PGauxiliaryfunctions.pl problem (we're running on a system that does not distinguish upper- and lower-case file names, so that type of error slips by very easily). Davide <| Post or View Comments |>

 Davide P. Cervone - Re: Problem display errors - ww 2.2  5/25/2006; 7:17:39 AM (reads: 908, responses: 0) There were two files in setAnalGeomLines that had the wrong capitalization for PGauxiliaryFunction.pl. I have fixed these in the CVS repository. Thanks for the error report. Davide <| Post or View Comments |>

 Lars Jensen - Re: Problem display errors - ww 2.2  5/26/2006; 1:45:43 AM (reads: 969, responses: 0) Hi Davide, Thanks for the info. Editing the macro path in global.conf does help fix the problems. Lars. <| Post or View Comments |>


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# Trig algebra

1. Dec 8, 2009

### Icelove

1. The problem statement, all variables and given/known data
(1 + tg(x)) * cos^2(x) > 1, solve the inequality

2. Relevant equations
tg(x) = sin(x)/cos(x)
cos(x) * sin(x) = 1/2 sin(2x)

3. The attempt at a solution
I got my final value of:
1-sin^2(x) + 1/2 sin(2x) > 1
but because of the 2 inside the sin
I'm kinda lost so I can't use quadratic formula I presume

2. Dec 8, 2009

### grief

Let me rearrange that a little for you:

1/2sin(2x)>sin^2(x)

which is actually nicer in the following form:

sin(x)cos(x)>sin^2(x).

This should be easier to solve.

3. Dec 8, 2009

### singular

It is important for you to realize where this came from.
If you have notes, cheat sheet, book, or webpage with trig identities, look at the double angle identities.

$$sin2x=2sinxcosx$$

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Documents  Paradan, Paul-Emile | enregistrements trouvés : 6

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Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces Boysal, Arzu | CIRM H

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Research talks;Algebraic and Complex Geometry;Lie Theory and Generalizations;Number Theory

Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.
This is joint work with V. Baldoni and M. Vergne.

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Geometric quantization of toric and semitoric systems Miranda, Eva | CIRM H

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Research talks;Geometry;Mathematical Physics;Topology

One of the many contributions of Kostant is a rare gem which probably has not been sufficiently explored: a sheaf-theoretical model for geometric quantization associated to real polarizations. Kostant’s model works very well for polarizations given by fibrations or fibration-like objects (like integrable systems away from singularities). For toric manifolds where the real polarization is determined by the fibers of the moment map, Kostant’s model yields a representation space whose dimension is the number of integer points inside the corresponding Delzant polytope. We will discuss extensions of this model to consider almost toric manifolds and integrable systems with non-degenerate singularities where “unexpected” infinities can show up even if the manifold is compact. One of the many contributions of Kostant is a rare gem which probably has not been sufficiently explored: a sheaf-theoretical model for geometric quantization associated to real polarizations. Kostant’s model works very well for polarizations given by fibrations or fibration-like objects (like integrable systems away from singularities). For toric manifolds where the real polarization is determined by the fibers of the moment map, Kostant’s ...

53D50

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Quantum footprints of symplectic rigidity Polterovich, Leonid | CIRM H

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Research talks;Geometry;Mathematical Physics

We discuss interactions between quantum mechanics and symplectic topology including a link between symplectic displacement energy, a fundamental notion of symplectic dynamics, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes.
Joint work with Laurent Charles.

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Poisson-Lie duality and Langlands duality via Bohr-Sommerfeld Alekseev, Anton | CIRM H

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Let $G$ be a connected semisimple Lie group with Lie algebra $\mathfrak{g}$. There are two natural duality constructions that assign to it the Langlands dual group $G^\lor$ (associated to the dual root system) and the Poisson-Lie dual group $G^∗$. Cartan subalgebras of $\mathfrak{g}^\lor$ and $\mathfrak{g}^∗$ are isomorphic to each other, but $G^\lor$ is semisimple while $G^∗$ is solvable.
In this talk, we explain the following non-trivial relation between these two dualities: the integral cone defined by the Berenstein-Kazhdan potential on the Borel subgroup $B^\lor \subset G^\lor$ is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on $K^∗ \subset G^∗$ (the Poisson-Lie dual of the compact form $K \subset G$). The first cone parametrizes canonical bases of irreducible $G$-modules. The corresponding points in the second cone belong to integral symplectic leaves of $K^∗$.
The talk is based on a joint work with A. Berenstein, B. Hoffman and Y. Li.
Let $G$ be a connected semisimple Lie group with Lie algebra $\mathfrak{g}$. There are two natural duality constructions that assign to it the Langlands dual group $G^\lor$ (associated to the dual root system) and the Poisson-Lie dual group $G^∗$. Cartan subalgebras of $\mathfrak{g}^\lor$ and $\mathfrak{g}^∗$ are isomorphic to each other, but $G^\lor$ is semisimple while $G^∗$ is solvable.
In this talk, we explain the following non-trivial ...

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Homogeneous vector bundles over abelian varieties Brion, Michel | CIRM H

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The objects of the talk are the translation-invariant vector bundles over an abelian variety. We will present a representation-theoretic description of these vector bundles, which displays a remarkable analogy with finite-dimensional representations of a compact connected Lie group: the weight lattice is replaced with the dual abelian variety, the Weyl group with the Galois group of the ground field...

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Witten non abelian localization for equivariant K-theory, and the $[Q, R] = 0$ theorem Paradan, Paul-Emile ; Vergne , Michele | American Mathematical Society 2019

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ISBN 978-1-4704-3522-6

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# 12 PTS] Suppose you have an income of $200 this year and you expect an income of$110 next year.

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14. -13 points My Notes Ask Your Teacher ] Note: This problem is very similar to your field mapping lab! A given system has the equipotential surfaces shown in the figure below, where Vo = 22.5 V. v-2 y-V -0.00 V Find the electric field in this region in component form, as well as total size and dir...
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In a program helped designed to help patients stop smoking 167 patients were given sustained care, and 85.0% of them were no longer smoking after on month. use a 0.05 significance level to test the claim that 80% of patients stop smoking when giving sustained care 16 of 17 (3) complete) ► ...
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## lunedì 28 agosto 2017

### Final Resume

Summary During the GSoC I worked on different special functions that needed to be improved or implemented from scratch. Discussing with my mentors and the community, we decided that my work should be pushed on a copy of the scource code of Octave on my repository [1] and then I should have work with different bookmarks for each function I had to work on. When different functions happened to be related (e.g. gammainc and gammaincinv), I worked on these on the same bookmark. I present now a summary and the bookmarks related to the functions.

## Incomplete gamma function

bookmark: gammainc
first commit: d1e03faf080b
last commit: 107dc1d24c1b
removed files:/libinterp/corefcn/gammainc.cc, /liboctave/external/slatec-fn/dgami.f, /liboctave/external/slatec-fn/dgamit.f, /liboctave/external/slatec-fn/gami.f, /liboctave/external/slatec-fn/gamit.f, /liboctave/external/slatec-fn/xdgami.f, /liboctave/external/slatec-fn/xdgamit.f, /liboctave/external/slatec-fn/xgmainc.f, /liboctave/external/slatec-fn/xsgmainc.f
modified files: NEWS, /doc/interpreter/arith.txi, /libinterp/corefcn/module.mk, /liboctave/external/slatec-fn/module.mk, /liboctave/numeric/lo-specfun.cc, /scripts/specfun/module.mk

### Summary of the work

On this bookmark I worked on the incomplete gamma function and its inverse.
The incomplete gamma function gammainc had both missing features (it were missed the "scaled" options) and some problem of inaccurate result type (see bug # 47800). Part of the work was already been done by Marco and Nir, I had to finish it. We decided to implement it as a single .m file (gammainc.m) which call (for some inputs) a subfunction written in C++ (__gammainc_lentz__.cc).
The inverse of the incomplete gamma function was missing in Octave (see bug # 48036). I implemented it as a single .m file (gammaincinv.m) which uses a Newton method.

## Bessel functions

bookmark: bessel
first commit: aef0656026cc
last commit: e9468092daf9
modified files: /liboctave/external/amos/README, /liboctave/external/amos/cbesh.f, /liboctave/external/amos/cbesi.f, /liboctave/external/amos/cbesj.f, /liboctave/external/amos/cbesk.f, /liboctave/external/amos/zbesh.f, /liboctave/external/amos/zbesi.f, /liboctave/external/amos/zbesj.f, /liboctave/external/amos/zbesk.f, /liboctave/numeric/lo-specfun.cc, /scripts/specfun/bessel.m

### Summary of the work

On this bookmark I worked on Bessel functions.
There was a bug reporting NaN as output when the argument $x$ was too large in magnitude (see bug # 48316). The problem was given by Amos library, which refuses to compute the output in such cases. I started "unlocking" this library, in such a way to compute the output even when the argument was over the limit setted by the library. Then I compared the results with other libraries (e.g. Cephes [2], Gnu Scientific library [3] and C++ special function library [4]) and some implementations I made. In the end, I discovered that the "unlocked" Amos were the best one to use, so we decided to maintain them (in the "unlocked" form), modifying the error variable to explain the loss of accuracy.

## Incomplete beta function

bookmark: betainc
first commit: 712a069d2860
last commit: e0c0dd40f096
removed files: /libinterp/corefcn/betainc.cc, /liboctave/external/slatec-fn/betai.f, /liboctave/external/slatec-fn/dbetai.f, /liboctave/external/slatec-fn/xbetai.f, /liboctave/external/slatec-fn/xdbetai.f
modified files: /libinterp/corefcn/module.mk, /liboctave/external/slatec-fn/module.mk, /liboctave/numeric/lo-specfun.cc, /liboctave/numeric/lo-specfun.h, /scripts/specfun/module.mk, /scripts/statistics/distributions/betainv.m, /scripts/statistics/distributions/binocdf.m

### Summary of the work

On this bookmark I worked on the incomplete beta function and its inverse.
The incomplete beta function missed the "upper" version and had reported bugs on input validation (see bug # 34405) and inaccurate result (see bug # 51157). We decided to rewrite it from scratch. It is now implemented ad a single .m file (betainc.m) which make the input validation part, then the output is computed using a continued fraction evaluation, done by a C++ function (__betainc_lentz__.cc).
The inverse was present in Octave but missed the "upper" version (since it was missing also in betainc itself). The function is now written as a single .m file (betaincinv.m) which implement a Newton method where the initial guess is computed by few steps of bisection method.

## Integral functions

bookmark: expint
first commit: 61d533c7d2d8
last commit: d5222cffb1a5
modified files: /doc/interpreter/arith.txi, /libinterp/corefcn/module.mk, /scripts/specfun/expint.m, /scripts/specfun/module.mk

### Summary of the work

On this bookmark I worked on exponential integral, sine integral and cosine integral. I already rewrote the exponential integral before the GSoC. Here I just moved the Lentz algorithm to an external C++ function (__expint_lentz__.cc), accordingly to gammainc and betainc. I've also modified the exit criterion for the asymptotic expansion using [5] (pages 1 -- 4) as reference.
The functions sinint and cosint were present only in the symbolic package of Octave but was missing a numerical implementation in the core. I wrote them as .m files (sinint.m and cosint.m). Both codes use the series expansion near the origin and relations with expint for the other values.

## To do

There is still room for improvement for some of the functions I wrote. In particular, gammainc can be improved in accuracy for certain couple of values, and I would like to make a template version for the various Lentz algorithms in C++ so to avoid code duplication in the functions.
In October I will start a PhD in Computer Science, still here in Verona. This will permit me to remain in contact with my mentor Marco Caliari, so that we will work on these aspects.

[1] https://bitbucket.org/M_Ginesi/octave
[2] http://www.netlib.org/cephes/
[3] https://www.gnu.org/software/gsl/
[4] http://en.cppreference.com/w/cpp/numeric/special_math
[5] N. Bleistein and R.A. Handelsman, "Asymptotic Expansions of Integrals", Dover Publications, 1986.

## sabato 19 agosto 2017

### Integral functions

Integral functions During the last week I made few modifications to expint.m and wrote sinint.m and cosint.m from scratch. All the work done can be found on the bookmark expint of my repository.

## expint

As I mentioned here I rewrote expint.m from scratch before the GSoC. During the last week I moved the Lentz algorithm to a .cc function (in order to remain coherent with the implementations of gammainc and betainc) and added few tests.

## sinint

The sinint function is present in the symbolic package, but is not present a numerical implementation in the core.
The sine integral is defined as $$\text{Si} (z) = \int_0^z \frac{\sin(t)}{t}\,dt.$$ To compute it we use the series expansion $$\text{Si}(z) = \sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{(2n+1)(2n+1)!}$$ when the module of the argument is smaller than 2. For bigger values we use the following relation with the exponential integral $$\text{Si} = \frac{1}{2i} (E_1(iz)-E_1(-iz)) + \frac{\pi}{2},\quad |\text{arg}(z)| < \frac{\pi}{2}$$ and the following simmetry relations $$\text{Si}(-z) = -\text{Si}(z),$$ $$\text{Si}(\bar{z}) = \overline {\text{Si}(z)}.$$ The function is write as a single .m file.

## cosint

As the sinint function, also cosint is present in the symbolic package, but there is not a numerical implementation in the core.
The cosine integral is defined as $$\text{Ci} (z) = -\int_z^\infty \frac{\cos(t)}{t}\,dt.$$ An equivalent definition is $$\text{Ci} (z) = \gamma + \log z + \int_0^z \frac{\cos t - 1}{t}\,dt.$$ To compute it we use the series expansion $$\text{Ci}(z) = \gamma + \log z + \sum_{n=1}^\infty \frac{(-1)^n z^{2n}}{(2n)(2n)!}$$ when the module of the argument is smaller than 2. For bigger values we use the following relation with the exponential integral $$\text{Ci} = -\frac{1}{2} (E_1(iz)+E_1(-iz)),\quad |\text{arg}(z)| < \frac{\pi}{2}$$ and the following simmetry relations $$\text{Ci}(-z) = \text{Ci}(z) -i\pi,\quad 0<\text{arg}(z)<\pi,$$ $$\text{Ci}(\bar{z}) = \overline{\text{Ci}(z)} .$$ As for sinint, also cosint is written as a single .m file.

## sabato 12 agosto 2017

### betaincinv

betaincinv The inverse of the incomplete beta function was present in Octave, but without the "upper" option (since it was missing in betainc itself). We decided to rewrite it from scratch using Newton method, as for gammaincinv (see my post on it if you are interested).
To make the code numerically more accurate, we decide which version ("lower" or "upper") invert depending on the inputs.
At first we compute the trivial values (0 and 1). Then the remaining terms are divided in two sets: those that will be inverted with the "lower" version, and those that will be inverted with the "upper" one. For both cases, we perform 10 iterations of bisection method and then we perform a Newton method.
The implementation (together with the new implementation of betainc) can be found on my repository, bookmark "betainc".

## mercoledì 2 agosto 2017

### betainc

betainc The betainc function has two bugs reported: #34405 on the input validation and #51157 on inaccurate result. Moreover, it is missing the "upper" version, which is present in MATLAB.

# The function

The incomplete beta function ratio is defined as $$I_x(a,b) = \dfrac{B_x(a,b)}{B(a,b)},\quad 0\le x \le 1,\,a>0,\,b>0,$$ where $B(a,b)$ is the classical beta function and $$B_x(a,b)=\int_0^x t^{a-1}(1-t)^{b-1}\,dt.$$ In the "upper" version the integral goes from $x$ to $1$. To compute this we will use the fact that $$\begin{array}{rcl} I_x(a,b) + I_x^U(a,b) &=& \dfrac{1}{B(a,b)}\left( \int_0^x t^{a-1}(1-t)^{b-1}\,dt + \int_x^1 t^{a-1}(1-t)^{b-1}\,dt\right)\\ &=&\dfrac{1}{B(a,b)}\int_0^1 t^{a-1}(1-t)^{b-1}\,dt\\ &=&\dfrac{B(a,b)}{B(a,b)}\\ &=&1 \end{array}$$ and the relation $$I_x(a,b) + I_{1-x}(b,a) = 1$$ so that $$I_x^U(a,b) = I_{1-x}(b,a).$$

# The implementation

Even if it is possible to obtain a Taylor series representation of the incomplete beta function, it seems to not be used. Indeed the MATLAB help cite only the continuous fraction representation present in "Handbook of Mathematical Functions" by Abramowitz and Stegun: $$I_x(a,b) = \dfrac{x^a(1-x)^b}{aB(a,b)}\left(\dfrac{1}{1+} \dfrac{d_1}{1+} \dfrac{d_2}{1+}\ldots\right)$$ with $$d_{2m+1} = -\dfrac{(a+m)(a+b+m)}{(a+2m)(a+2m+1)}x$$ and $$d_{2m} = \dfrac{m(b-m)}{(a+2m-1)(a+2m)}x$$ which seems to be the same strategy used by GSL. To be more precise, this continued fraction is computed directly when $$x<\dfrac{a-1}{a+b-2}$$ otherwise, the computed fraction is used to compute $I_{1-x}(b,a)$ and then it is used the fact that $$I_x(a,b) = 1-I_{1-x}(b,a).$$ In my implementation I use a continued fraction present in "Handboob of Continued Fractions for Special Functions" by Cuyt, Petersen, Verdonk, Waadeland and Jones, which is more complicated but converges in fewer steps: $$\dfrac{B(a,b)I_x(a,b)}{x^a(1-x)^b} = \mathop{\huge{\text{K}}}_{m=1}^\infty \left(\dfrac{\alpha_m(x)}{\beta_m(x)}\right),$$ where $$\begin{array}{rcl} \alpha_1(x) &=&1,\\ \alpha_{m+1}(x) &=&\dfrac{(a+m-1)(a+b+m-1)(b-m)m}{(a+2m-1)^2}x^2,\quad m\geq 1,\\ \beta_{m+1}(x) &=&a + 2m + \left( \dfrac{m(b-m)}{a+2m-1} - \dfrac{(a+m)(a+b+m)}{a+2m+1} \right)x,\quad m\geq 0. \end{array}$$ This is most useful when $$x\leq\dfrac{a}{a+b},$$ thus, the continued fraction is computed directly when this condition is satisfied, while it is used to evaluate $I_{1-x}(b,a)$ otherwise.
The function is now written as a .m file, which check the validity of the inputs and divide the same in the values which need to be rescaled and in those wo doesn't need it. Then the continued fraction is computed by an external .c function. Finally, the .m file explicit $I_x(a,b)$.

# betaincinv

Next step will be to write the inverse. It was already present in Octave, but is missing the upper version, so it has to be rewritten.

## lunedì 24 luglio 2017

### Second period resume

Second period resume Here I present a brief resume of the work done in this second month.

## Bessel functions

The topic of this month were Bessel functions. On the bug tracker it is reported only a problem regarding the "J" one (see bug 48316), but the same problem is present in every type of Bessel function (they return NaN + NaNi when the argument is too big).

### Amos, Cephes, C++ and GSL

Actually, Bessel functions in Octave are computed via the Amos library, written in Fortran. Studying the implementation I discovered that the reported bug follows from the fact that if the input is too large in module, the function zbesj.f set IERR to 4 (IERR is a variable which describe how the algorithm has terminate) and set the output to zero, then lo-specfunc.cc return NaN when IERR=4. Obviusly, the same happen for other Bessel functions.
What I initially did was to "unlock" these .f files in such a way to give still IERR=4, but computing anyway the output, and modify lo-specfun.cc in order to return the value even if IERR is 4. Then I tested the accuracy, together with other libraries.
On the bug report where suggested the Cephes library, so I tested also them, the C++ Special mathematical functions library, and the GSL (Gnu Scientific library). Unfortunately, both these alternatives work worse than Amos. I also tried to study and implement some asymptotic expansions by myself to use in the cases which give inaccurate results, unfortunately without success.
For completeness, in the following table there are some results of the tests (ERR in Cephes refers to cos total loss of precision error):
1e09 1e10 1e11 1e12 1e13
Amos 1.6257e-16 0.0000e+00 0.0000e+00 1.3379e-16 1.1905e-16
Cephes 2.825060e-08 ERR ERR ERR ERR
GSL 4.8770e-16 2.2068e-16 4.2553e-16 1.3379e-16 1.1905e-16
C++ 2.82506e-08 2.68591e-07 1.55655e-05 8.58396e-07 0.000389545

1e15 1e20 1e25 1e30
Amos 1.3522e-16 1.6256e-16 15.22810 2.04092
GSL 1.3522e-16 0 15.22810 2.04092

### The problem with the double precision

As I explained in my last post the problem seems to be that there are no efficient algorithms in double precision for arguments so large. In fact, the error done by Amos is quite small if compared with the value computed with SageMath (which I'm using as reference one), but only if we use the double precision also in Sage: using more digits, one can see that even the first digit change. Here the tests:
sage: bessel_J(1,10^40).n()
7.95201138653707e-21
sage: bessel_J(1,10^40).n(digits=16)
-7.522035532561300e-21
sage: bessel_J(1,10^40).n(digits=20)
-5.7641641376196142462e-21
sage: bessel_J(1,10^40).n(digits=25)
1.620655257788838811378880e-21
sage: bessel_J(1,10^40).n(digits=30)
1.42325562284462027823158499095e-21
sage: bessel_J(1,10^40).n(digits=35)
1.4232556228446202782315849909515310e-21
sage: bessel_J(1,10^40).n(digits=40)
1.423255622844620278231584990951531026335e-21
The value stabilizes only when we use more than 30 digits (two times the digits used in double precision).

### The decision

Even if we are aware of the fact that the result is not always accurate, for MATLAB compatibility we decided to unlock the Amos, since they are still the most accurate and, even more important, the type of the inputs is the same as in MATLAB (while, for example, GSL doesn't accept complex $x$ value). Moreover, in Octave is possible to obtain in output the value of IERR, thing that is not possible in MATLAB.
You can find the work on the bookmark "bessel" of my repository.

## Betainc

During these last days I also started to implement betainc from scratch, I think it will be ready for the first days of August. Then, it will be necessary to rewrite also betaincinv, since the actual version doesn't have the "upper" version. This should not be too difficult. I think we can use a simple Newton method (as for gammaincinv), the only problem will be, as for gammaincinv, to find good initial guesses.

## Bessel functions

During the second part of GSoC I have to work on Bessel functions. The bug is related in particular on the Bessel function of first kind and regard the fact that the function is not computed if the $x$ argument is too big.
During this week I studied the actual implementation i.e. the Amos library, written in Fortran. The problem is related to the fact that zbesj.f simply refuse to compute the result when the input argument is too large (the same problem happens for all bessel functions, both for real and complex arguments). What I did was to "unlock" the amos in such a way to compute the value in any case (which seems to be the strategy used by MATLAB). Then I compared the relative errors with the relative errors done by the Gnu Scientific Library (GSL).
At first, I would precise some limitation present in GSL:
• The parameter $x$ must be real, while it can be complex in Amos and MATLAB. The same holds for the parameter $\alpha$.
• The class of the output does not adapt to the class of the input, returning always a double.
Doing some tests, it seems that the amos work better in terms of accuracy (even if we are talking about errors which do not differ in the order). I concentrated on values in which Amos usually refuse to compute, since in every other zone of the $x-\alpha$ plane it is known the error is in the order of 1e-15. For $\alpha\in\{-1,0,1\}$ they return the same result, while for other values of $\alpha$, Amos are in general more accurate.
Anyway, I would remark the fact that there are not, as far as I know, accurate algorithms in double precision for very large magnitude. In fact, for such arguments, both Amos and GSL make a relative error of the order of the unity. This problem is evident when using SageMath to compute accurate values, e.g.
sage: bessel_J(1,10^40).n()
7.95201138653707e-21
sage: bessel_J(1,10^40).n(digits=16)
-7.522035532561300e-21
sage: bessel_J(1,10^40).n(digits=20)
-5.7641641376196142462e-21
sage: bessel_J(1,10^40).n(digits=25)
1.620655257788838811378880e-21
sage: bessel_J(1,10^40).n(digits=30)
1.42325562284462027823158499095e-21
sage: bessel_J(1,10^40).n(digits=35)
1.4232556228446202782315849909515310e-21
sage: bessel_J(1,10^40).n(digits=40)
1.423255622844620278231584990951531026335e-21
The values "stabilize" only when the number of digits is bigger than 30, far away from double precision.

## Incomplete gamma function

I also gave a look on how to use GSL to eventually improve the incomplete gamma function. It is not possible, however, to use only GSL functions gsl_sf_gamma_inc_P and gsl_sf_gamma_inc_Q due to their limitations:
• There is no the "scaled" option.
• The value of $x$ must be real. This may be not a problem since even MATLAB does not accept complex value (while the version I worked on does).
• The parameter $x$ must be positive. This is actually a problem of MATLAB compatibility.
• The class of the output does not adapt to the class of the input, returning always a double.
I tested the accuracy of the GSL and they work better when $a<1$ and $x\ll1$ so I think I will fix gammainc.m using the algorithm of gsl_sf_gamma_inc_P and gsl_sf_gamma_inc_Q for that values of the input arguments.

## Incomplete Beta function

The actual incomplete beta function needs to be replaced. I've already studied the continued fraction exapnsion (which seems to be the best one to use). In GSL is implemented in a good way but still present two limitations:
• Does not adapt to the input class (it always work in double precision).
• There is no "upper" version. This is missing also in the current betainc function, but it is present in MATLAB. Unfortunately, it is not sufficient to compute it as $1 - I_x(a,b)$, it is necessary to find an accurate way to compute directly it.
So I will take inspiration to GSL version of the function but I think I will write beatinc as a single .m file.

## sabato 1 luglio 2017

### First period resume

First period resume

# First period resume

Here I present a brief resume of the work done in this first month.

## 1st week: Gammainc

The incomplete gamma function presented a series of inaccurate results in the previous implementation, so it has been decided to rewrite it from scratch. A big part of the work was done by Marco and Nir (here the discussion). I gave my contribution by fixing some problems in the implementation and by making a functioning commit (thanks to the suggestions given by Carne during the OctConf in Geneve).

## 2nd and 3rd week: Gammaincinv

The inverse of the incomplete gamma function was completely missing in Octave (and this was a problem of compatibility with Matlab). Now the function is present, written as a single .m file. The implementation consist in a simple, but efficient, Newton's method.

## Last week: Betainc

When I first submitted my timeline, there was only a bug related to betainc, involving the input validation. During the first period of GSoC emerged a new bug of "incorrect result" type. From this, me and my mentor decided that it is necessary to rewrite the function, so I used this last week to study efficient algorithms to evaluate it. In particular, seems that the most efficient way is to use continude fractions, maybe after a rescale of the function. I'm already working on it, but I will complete it during the first week of august, after finishing the work on Bessel functions.


text:
# How do you find dy/dx by implicit differentiation of xcosy=1 and evaluate at point (2, pi/3)?

Mar 1, 2017

${\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]}_{2 , \frac{\pi}{3}} = \frac{1}{2 \sqrt{3}}$

#### Explanation:

Differentiate the equation with respect to $x$:

$\frac{d}{\mathrm{dx}} \left(x \cos y\right) = 0$

$\cos y - x \sin y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\cos y = x \sin y \frac{\mathrm{dy}}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x t g y}$

In the point $\left(2 , \frac{\pi}{3}\right)$ the value is then:

${\left[\frac{\mathrm{dy}}{\mathrm{dx}}\right]}_{2 , \frac{\pi}{3}} = \frac{1}{2 t g \left(\frac{\pi}{3}\right)} = \frac{1}{2 \sqrt{3}}$


text:
Odd characterizations of almost simple groups - 3-local and character theoretic methods

Astill, Sarah (2010). Odd characterizations of almost simple groups - 3-local and character theoretic methods. University of Birmingham. Ph.D.

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Astill09PhD.pdf
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Abstract

In this PhD thesis we discuss methods of recognizing finite groups by the structure of normalizers of certain 3-subgroups. We explain a method for characterizing groups using character theoretic and block theoretic methods and we use these methods to characterize Alt(9). Furthermore, we describe a particular hypothesis related to the 3-local structure of finite groups of local characteristic 3 and characterize two almost simple proper extensions of $$\Omega$$$$_8$$ (2) as examples of the local approach to group recognition. We also give a 3-local characterization of the sporadic simple group HN using local methods whilst applying a character theoretic result.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Parker, ChristopherUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Mathematics
Funders: None/not applicable
Subjects: Q Science > QA Mathematics
URI: http://etheses.bham.ac.uk/id/eprint/520

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text:
### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Watch the Clock

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

### Walk and Ride

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

# The Hare and the Tortoise

##### Age 7 to 11 Challenge Level:

Most of you will know the story of the hare and the tortoise with the moral tag "slow and steady wins the race".

In this version

• The race is $10$ km (kilometres) long.
• The hare runs at $10$ times the speed of the tortoise.
• The tortoise take $2$ hours and $30$ minutes to complete the race.
• The hare arrives at the finish $30$ seconds after the tortoise.

For how long does the hare sleep?


text:
/ Math 36

### HW Assignments

• Daily Work: Just as in any other subject, developing proficiency in mathematical modeling requires sustained, consistent effort. Between in-class assignments and brief writing assignments. These problems will mostly be short written responses or computational in nature and provide opportunities for you to review that material from that day's lecture.
• Essays: The main assessments in this course will be in the form of 2-3 page essays, one for each of the large topic segments in the class. These assignments will generally require you to perform an analysis using relevant techniques from the section and draw conclusions based on your experiments. The associated essay will describe your efforts, analysis, and conclusions (specific instructions will be presented for each assignment).
• Quizzes: Each of the topic segments will conclude with a short quiz covering the computational methods covered. These quizzes will occur approximately once every two weeks.

#### Homework Problems

Week Date Sections Daily Work
Essays and Quizzes

1 9/11 1.1

2. Write one paragraph describing your personal goals for this course. In particular, if there are specific types of data or social frameworks that would be particularly interesting to you, this would be a great place to describe them.
9/13 1.2 and MatLab

No Homework

9/14 Individual Student Meetings
9/15 1.3

Find population data for at least 5 years for a city that is important to you and plot the population against time. Write one sentence explaining why you chose this city, one sentence describing the data (units, time interval, source, etc.), and one sentence explaining which of the models that we looked at in class best matches your data or why it doesn't match any of the models.

2 9/18 1.3

1. Read Models of Human Population Growth'' (linked on the resources page). Choose one of the arguments that he makes for the inadmissability of the logistic model and describe whether you think it is important or relevant.
2. Using the logistic model MatLab code on the resources page, find a set of parameters (P_0,k,n,m) that displays chaotic behavior.
Due: 10/2

Goal: Write a 2-3 page essay describing a modification of the logistic equation for population growth.

In class we have discussed some of the properties of the model P_{n+1)=P_n+k(M-P_n)P_n as well as its predictions and limitations. Your task is to identify a single term that you think should be added to this difference equation and discuss the expected consequences of modifying the model. You should explain why you believe the term you have added is important, what type of data you would need to estimate the new effects, and how you believe the addition will make the model more accurate/useful.

Once you have settled on a modification, your writing should be motivated by the following question: How will the earth's population grow over time?'' In the essay, show your analysis and results and discuss their implications to this question. Be sure to discuss the strengths and limitations of the model and their impact on your confidence in the conclusions you draw.

To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.
1. What are the meaning of the terms of the model? What assumptions to they imply? How are they interpreted?
2. What are reasonable magnitudes for the constants? Why are values outside this range inappropriate?
3. Can the population ever exceed the carrying capacity? If so, where are the boundary cases (i.e. what combinations of constants create one situation or the other?)?
4. Under what conditions is the model appropriate/applicable? Are these conditions extant today? Do we expect them to continue in the future?
5. How do your modifications affect the arguments made in Overpopulation is not the problem'' and Models of Human Population Growth''?
9/30 1.4

Read Overpopulation is not the problem'' (linked on the resources page). Write at most a paragraph comparing and constrasting this piece to Monday's reading. In particular, how do the viewpoints of the authors differ?

9/22 Chapter 2

Describe the modificiation to the logistic model that you are proposing for your writing assignment.

3 9/25 Chapter 3

Do problem 2 in Section 3.3 of your textbook (typed below). The numerical answers are in the back of your textbook and you should be able to check your answers in MatLab as well. Note that the textbook uses $m$ for the number of data points.

Use the normal equations $$a=\dfrac{m\left(\sum_{i=1}^mx_iy_i\right) - \left(\sum_{i=1}^n x_i\right)\left(\sum_{i=1}^n y_i\right)}{m\left(\sum_{i=1}^nx_i^2\right)-\left(\sum_{i=1}^mx_i\right)^2}$$ and $$b=\dfrac{\left(\sum_{i=1}^mx_i^2\right)\left(\sum_{i=1}^my_i\right)-\left(\sum_{i=1}^n x_iy_i\right)\left(\sum_{i=1}^n x_i\right)}{m\left(\sum_{i=1}^nx_i^2\right)-\left(\sum_{i=1}^mx_i\right)^2}$$ for the following data sets:

1. $\{(1.0,3.6),(2.3,3.0),(3.7,3.2),(4.2,5.1),(6.1,5.3),(7.0,6.8)\}$
2. $\{(29.1,.0493),(48.2,.0821),(72.7,.123),(92.0,.154),(118,.197),(140,.234),(165,.274),(199,.328) \}$
3. $\{(2.5,4.32),(3.0,4.83),(3.5,5.27),(4.0,5.74),(4.5,6.26),(5.0,6.79),(5.5,7.23) \}$
9/27 Data Analysis Day

What are the equilibrium states for the following system: $$a_{n+1}= a_n + 3a_n+4b_n -2$$ $$b_{n+1}=b_n+ 2a_n-3b_n +1$$

9/29 Preference Systems

Read an article on the ranked choice voting controversy in Maine (a couple are linked on the resources page). Think about how mathematical models could inform this debate (you don't have to hand anything in).

4 10/2 Lecture Notes

1. Construct a preference graph on 5 options where each option has an equal number of wins and losses
2. Explain why no such graph exists for 6 options.
3. Construct a preference graph for 6 options where three options have three wins and three options have three losses.
10/4 Lecture Notes

Consider the preference system: $$\begin{array}{|c|c|c|c|} \hline 1&2&3&1\\ \hline \hline A&B&D&A\\ \hline B&C&A&C\\ \hline C&A&C&B\\ \hline D&D&B&D\\ \hline \end{array}$$ Determine the winners under the plurality, Borda, and Condorcet methods? Which option do you think should win a fair election?

Due: 10/16

1. What are the meaning of the specific matrix entries? What assumptions do they imply? How are they interpreted in this specific case?
2. How do the differences in magnitudes between the options of a single player affect the analysis? What about the differences between the values available to the two players?
3. Does your game match in structure or outcome one of the stylized models we analyzed in class? What are the differences between your model and that one?
4. How does our discussion of preference rankings and voting systems affect your analysis? What models would make the most sense if the members of one or both of the parties had to rank the options and vote for the strategy to select?

10/6 Lecture Notes

Assume that the preference system: $$\begin{array}{|c|c|c|} \hline 4&3&5\\ \hline B&A&C\\ \hline A&B&A\\ \hline C&C&B\\ \hline\end{array}$$ is known to all parties and that candidates $A$ and $B$ are allowed to play a game to determine the final voting method, choosing between options plurality, Borda (with 10 points for first, 2 for second, 1 for third), or Condorcet. Each candidate would prefer to win the election and $A$ and $B$ would both prefer that $C$ win rather than the other player. If the candidates select the same method that method is chosen, otherwise one of the three methods is selected uniformly at random. Write a paragraph describing which method you believe each of $A$ and $B$ should select and justifying your selection. Does one of the players have more leverage than the other? Why or why not?

5 10/9 Lecture Notes

1. Construct a normal form matrix where player 1 has a dominated strategy but player 2 does not.
2. Determine the values in a normal form matrix for the game consisting of two '21s running around the homecoming fire. Their options are to keep running laps like normal or try to touch the fire. Is your formulation a dilemma?

10/11 Lecture Notes

1. Use IESDS to find reduce the following game: $$\begin{array}{|c|c|c|} \hline 6,8&2,6&8,2\\ \hline 8,2&4,4&9,5\\ \hline 8,10&4,6&6,7\\ \hline \end{array}$$
2. What is the Nash equilibrium for: $$\begin{array}{|c|c|c|} \hline 4,3&5,1&6,2\\ \hline 2,1&8,4&3,6\\ \hline 3,0&9,6&2,8\\ \hline \end{array}$$

10/13 Lecture Notes

Input some text you have written (perhaps an essay for this class) to the Markov text generator here: link and look at the output. Would the output of the computer program get an A in this class? (you don't have to hand anything in)

This is not required, but if you are interested in the practical aspects of working with data, the following description of the process to construct a data set for generating Markov comics is an interesting read. Notice how much effort is required to format the data in a useful fashion before any actual mathematics can be applied. This is becoming an increasingly important feature of data-drive'' jobs, that is rarely discussed in the classroom.

6 10/16 6.1
In Hanover, we have four types of weather, sunny, cloudy, snowy, and rainy. If it rains or snows then the next day has a 40% chance of the other type of precipitation, a 30% chance of the same type of precipitation, a 20% chance of clouds, and a 10% chance of sun. When it is sunny, the next day is sunny 65% of the time, cloudy 25% of the time, and each type of precipitation happens 5% of the time. Finally, when it is cloudy, the next day any of the options could happen with equal probability. Formulate a Markov chain for this situation and determine the associated steady state. What is the probability that the weather for this schoolweek is exactly, sunny, cloudy, cloudy, snowy, sunny if it rained on Sunday? What is the most likely weather on Saturday if it snows on Thursday?
10/18 Lecture Notes
Modify the chain from Monday's assignment under the winter assumption'' that if it snows one day then it will continue to snow, every day, forever :( If it is sunny today, how many days is it expected to take until it begins snowing and how many of those days are likely to be cloudy?

Due: 10/30

Goal: Write a 2-3 page essay analyzing the use of Markov chains for text analysis.

Details: The purpose of this assignment is to explore the use of Markov models and random processes. In class, we looked at some simple examples of generating text using different order Markov models but in this assignment we will also be analyzing the behavior as the underlying text changes. I have provided transition matrices and labels for 1--4 order Markov chains for several input texts and the associated MatLab code to generate samples from each chain. There are two components to this assignment, one focusing on comparing the properties of specific authors and another on the sample outputs.

Both Markov and Shannon were interested in the individual transition probabilities between letters as a method for analyzing interesting features of specific authors. Using properties of the first order Markov chains, determine a criterion for attempting to distinguish between works of different authors and justify your decision. The goal is to formulate a rule that allows you to determine whether a particular piece of writing was authored by one of the authors in our set, or not. You may consider any aspects of the first order chain, including its steady state behavior, convergence time, or some of the specific matrix entries.

For the longer chains, the task is to analyze how increasing the order of the Markov chain affects the structure of the output strings. For each of the chains, you should generate several example texts of varying lengths and discuss your observations. Discuss how the features of the text, such as length, genre, or time period, relate to the outputs that you observe. Pay particular attention to the repetitions that occur as you generate longer text examples and discuss what aspects of the original text are responsible for them.

To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.

1. Can you determine anything about the average length of the words from the first order chain? Are there any differences in vowel distribution? What types of letters are most likely to display useful anomalies for distinguishing between authors?
2. How does the prevalence of particular words or proper nouns affect the observed behavior? Does the length of the text matter?
3. Does the type of writing, fiction vs. non--fiction, prose vs. poetry, etc. change the expected behavior? What pieces of the analysis is this consideration most relevant for?
4. Are there phrases that appear multiple times in the output for some of the chains? Is it always true that larger values of $n$ lead to more realistic outputs? How are these questions related?
10/20 No Class
No Class.

7 10/23 Lecture Notes

Construct the transition matrix for the random Tower of Hanoi game with two rings and three pegs.

10/25 Lecture Notes

Consider the following time series: $\{-0.0475,-0.0185,-0.3178,-0.2189,0.0164,0.4838,0.3183,-0.3729,0.1841, 0.4489\}$. What are the distributions of patterns of length 2, 3, and 4 that appear? Do these numbers appear to have been drawn from a random distribution?

10/26 X-hour
10/27 Lecture Notes

No homework for Monday.

No quiz this week.

8 10/30 8.1
Construct your ego network (with at least 5 nodes) and compute at least two of the local measures that we discussed in class (code is linked on the resources page). Any interesting observations?
11/1 8.2

No Homework for Friday.

Due: 11/13

Goal: Write a 2-3 page essay analyzing the fission in a social network.

Details: Zachary's karate club is a standard example of a social network originally analyzed in the 1970's, usually in the context of fission in social groups. The nodes represent members of a university Karate club and the edges represent social relationships between them. The MatLab file containing the adjacency matrix is attached here and you can find Zachary's original research paper at: this link (don't read the paper yet).

Your task has two components. First, you will perform an analysis of the network, using the tools that we have discussed in class, focusing on how the network could split into groups and who the leaders of those groups might be. Secondly, you will analyze your results in light of the contents of Zachary's original research. Your essay should explain your original hypotheses and your reaction to and analysis of the methods and conclusions presented by Zachary. Critically, you should complete the first part before actually reading the paper, so as not to prejudice your initial work.

To help you along the way, here are some questions that may help guide your analysis. Discussion of these need not appear in the essay you turn in but will likely help you perform a thorough analysis on which to base your writing.

1. Are some nodes more important than others? How can we tell? How do the different centrality measures we discussed in class impact the community structure? We have discussed both local and global measures for centrality, how do these differences impact the analysis?
2. How does this network compare to the null models that we studied in class? Can you estimate reasonable values for the models to generate examples that look like this network? What does this tell you? Does modularity or assortativity tell you anything interesting?
3. What is the problem Zacharyis trying to address? What is the anthropological model behind the question? What is his network model and how does it reflect the anthropological model? What assumptions does he make?
4. What is the network analytic method that Zachary uses? How does it differ from your methods? How does your analysis compare to the author's? What are the positive and negative aspects of the different methodologies?
11/3 8.3

Calculate the assortativity of the dolphin and otter networks.

9 11/6 Lecture Notes

Read the first four pages of this paper: link. Don't hand anything in.

11/8 Chapter 11

Read the beginning of Chapter 11 of your textbook until things stop making sense.

10/26 X-hour
11/10 Lecture Notes

No homework for Monday.

Quiz solutions will go here.

10 11/13 12

You can download the final exam here: link. Your solutions are due on 11/21 at 11am. The exam is open book and open notes (and open MatLab) but the Honor Principle requires that you neither give nor receive aid on this exam. In particular, you may not discuss the problems or solutions with anyone else, although you may ask me questions for clarification.

Daryl DeFord
Last updated November 09, 2017


text:
### Project 13: "Additive and combinatorial number theory"

This project is running since 2015.

Principal investigator: Christian Elsholtz
Graz University of Technology, Austria.
Mentor for: Ddamulira, Wiegel, Technau; Iaco, Kreso, Smertnig.

### DK Student

• Second phase of the doctoral program:
• Stefan Planitzer (Austria; since July 2015)
Email: stefan.planitzer@student.tugraz.at
Mentors: Robert Tichy, Daniel Krenn.
• Thesis Title: "Sums of unit fractions, Romanov type problems and sequences with property P.".
PhD Defense: August 27, 2018.
Chair: A. Geroldinger.
Referees: C. Elsholtz, J. Schlage-Puchta (Rostock), Greg Martin (Vancouver).
Examiners: C. Elsholtz, J. Schlage-Puchta (Rostock).

### Project description

I am particularly interested in the interplay of additive and multiplicative questions. The methods used are often a combination of tools from combinatorics and analytic number theory.

Topics that were studied in this project, in collaboration with Stefan Planitzer, Niclas Technau, Rainer Dietmann, Florian Luca, Jan-Christoph Schlage-Puchta, Igor Shparlinski, Marc Technau and Robert Tichy include:

1. Number of solutions of Diophantine equation, e.g. the equation 1=1/x_1+ ... + 1/x_k, in integers x_i, (see arxiv:1805.02945)
2. Questions what happens if one adds prime numbers to a thin sequence, such as powers of 2? (On Romanov's constant and Romanov type problem)
3. Constructing large sets of integers such that no a_i divides a sum a_j+a_k, (arxiv:1609.07935)
4. Regularity of distribution of prime numbers (arxiv:1602.04317, arxiv:1702.00289)
5. Maximum order of iterated multiplicative functions (arxiv:1709.04799)
6. Subsetsums which are square-free or similar, (see for example arxiv:1510.05260, arxiv:1601.04754)
Future work can be related to the topics above, but can also include a variety of topics in additive combinatorics or combinatorial number theory, for example:
• Solutions of diophantine equations.
• Sets of integers or lattice points with (or without) specified patterns, such as arithmetic progressions, (e.g. Roth-type results, cap sets etc).
• Prime numbers, sieve methods.
• Size of sum and product sets.


text:
# zbMATH — the first resource for mathematics

Phase retrievable projective representation frames for finite abelian groups. (English) Zbl 1440.42156
Summary: We consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering for the phase-retrieval problem. For a given multiplier $$\mu$$ of a finite abelian group $$G$$, we show that the representation dimension of any irreducible $$\mu$$-projective representation of $$G$$ is exactly the rank of the symmetric multiplier matrix associated with $$\mu$$. With the help of this result we are able to prove that every irreducible $$\mu$$-projective representation of a finite abelian group $$G$$ admits a frame vector with the maximal span property, and obtain a complete characterization for all such frame vectors. Consequently the complement of the set of all the maximal span frame vectors for any projective unitary representation of any finite abelian group is Zariski-closed. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements.

##### MSC:
 42C15 General harmonic expansions, frames 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 20C25 Projective representations and multipliers 20K01 Finite abelian groups
##### Keywords:
phase retrieval; frames; group representations
PhaseLift
Full Text:
##### References:
 [1] Alexeev, B.; Bandeira, AS; Fickus, M.; Mixon, DG, Phase retrieval with polarization, SIAM J. Imaging Sci., 7, 35-66, (2014) · Zbl 1304.49071 [2] Backhouse, NB; Bradley, CJ, Projective representations of Abelian groups, Proc. Am. Math. Soc., 36, 260-266, (1972) · Zbl 0256.22009 [3] Balan, R.: Stability of phase retrievable frames. In: Proceedings of SPIE, Wavelets and Sparsity XV, 88580H (2013) [4] Balan, R.; Wang, Y., Invertibility and robustness of phaseless reconstruction, Appl. Comput. Harmon. Anal., 38, 469-488, (2015) · Zbl 1337.42032 [5] Balan, R.; Zou, D., On Lipschitz analysis and Lipschitz synthesis for the phase retrieval problem, Linear Algebra Appl., 496, 152-181, (2016) · Zbl 1332.15042 [6] Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction from the absolute value of the frame coefficients. In: Proceedings of SPIE, vol. 5914, pp. 1-8 (2005) [7] Balan, R.; Casazza, P.; Edidin, D., On signal reconstruction without phase, Appl. Comput. Harmon. Anal., 20, 345-356, (2006) · Zbl 1090.94006 [8] Balan, R.; Casazza, PG; Edidin, D., Equivalence of reconstruction from the absolute value of the frame coefficients to a sparse representation problem, IEEE Signal Process. Lett., 14, 341-345, (2007) [9] Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Fast algorithms for signal reconstruction without phase. In: Proceedings of SPIE-Wavelets XII, San Diego, vol. 6701, pp. 670111920-670111932 (2007) [10] Balan, R.; Bodmann, BG; Casazza, PG; Edidin, D., Painless reconstruction from magnitudes of frame vectors, J. Fourier Anal. Appl., 15, 488-501, (2009) · Zbl 1181.42032 [11] Bandeira, AS; Cahill, J.; Mixon, DG; Nelson, AA, Saving phase: injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal., 37, 106-125, (2014) · Zbl 1305.90330 [12] Bandeira, AS; Chen, Y.; Mixon, DG, Phase retrieval from power spectra of masked signals, Inf. Inference, 3, 83-102, (2014) · Zbl 1308.94031 [13] Bendory, T., Eldar, Y.C.: A least squares approach for stable phase retrieval from short-time Fourier transform magnitude, preprint arXiv:1510.00920 (2015) [14] Bodmann, BG; Hammen, N., Stable phase retrieval with low-redundancy frames, Adv. Comput. Math., 41, 317-33, (2015) · Zbl 1316.42035 [15] Bodmann, B.G., Casazza, P.G., Edidin, D., Balan, R.: Frames for Linear Reconstruction Without Phase. CISS Meeting, Princeton, NJ (2008) · Zbl 1181.42032 [16] Bojarovska, I.; Flinth, A., Phase retrieval from Gabor measurements, J. Fourier Anal. Appl., 22, 542-567, (2016) · Zbl 1341.42049 [17] Candès, EJ; Li, X., Solving quadratic equations via PhaseLift when there are about as many equations as unknowns, Found. Comput. Math., 14, 1017-1026, (2014) · Zbl 1312.90054 [18] Candès, EJ; Eldar, YC; Strohmer, T.; Voroninski, V., Phase retrieval via matrix completion, SIAM J. Imaging Sci., 6, 199-225, (2013) · Zbl 1280.49052 [19] Candès, EJ; Strohmer, T.; Voroninski, V., PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming, Commun. Pure Appl. Math., 66, 1241-1274, (2013) · Zbl 1335.94013 [20] Cheng, C., A character theory for projective representations of finite groups, Linear Algebra Appl., 469, 230-242, (2015) · Zbl 1312.20005 [21] Cheng, C., Fu, J.: On the rings of projective characters of Abelian groups and Dihedral groups, preprint (2016) [22] Conca, A.; Edidin, D.; Hering, M.; Vinzant, C., An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal., 38, 346-356, (2015) · Zbl 1354.42003 [23] Eldar, YC; Sidorenko, P.; Mixon, DG; Barel, S.; Cohen, O., Sparse phase retrieval from short-time Fourier measurements, IEEE Signal Process. Lett., 22, 638-642, (2015) [24] Eldar, Y.C., Hammen, N., Mixon, D.: Recent advances in phase retrieval. In: IEEE Signal Processing Magazine, pp. 158-162 (September 2016) [25] Fickus, M.; Mixon, DG; Nelson, AA; Wang, Y., Phase retrieval from very few measurements, Linear Algebra Appl., 449, 475-499, (2014) · Zbl 1286.42043 [26] Jaganathan, K., Eldar, Y.C., Hassibi, B.: Phase retrieval: an overview of recent developments. arXiv:1510.07713 [27] Karpilovsky, G.: The Schur Multiplier. London Mathematical Society Monographs. Clarendon Press, Oxford (1987) · Zbl 0619.20001 [28] Li, L.; Cheng, C.; Han, D.; Sun, Q.; Shi, G., Phase retrieval from multiple-window short-time Fourier measurements, IEEE Signal Process. Lett., 24, 372-376, (2017) [29] Nawab, SH; Quatieri, TF; Lim, JS, Signal reconstruction from short-time Fourier transform magnitude, IEEE Trans. Acoust. Speech Signal Process., 31, 986-998, (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
, 24.01.2020jo17miapierre

# Simplify the expression (2x−9)+(x−7)

2x²+3x-54

Step-by-step explanation:

To simplify the expression

(2x − 9)(x + 6)

We will multiply both linear equations together to get a quadratic equation of second degree.

(2x − 9)(x + 6)

= 2x²+12x-9x-54

= 2x²+3x-54

This gives the required quadratic expression after simplification.

13x +5

Step-by-step explanation:

(2x + 9) + (11x-4)

Combine like terms

2x+11x +9-4

13x +5

Step-by-step explanation:

The given expression is

Distributive property:

or

By distributive property the above expression will be written as :

Hence,

Step-by-step explanation:

The given expression is

Distributive property:

or

By distributive property the above expression will be written as :

Hence,

Step-by-step explanation:

The given expression is

Distributive property:

or

By distributive property the above expression will be written as :

Hence,

2x^2+12x-9x-54

2x^2+3x-54
(2x-9)•(x+6)

2x•x+2x•6-9•x-9•6

2x²+12x-9x-54
2x²+(12-9)•x -54
2x²+3x-54
(2x-9)(x+6)
2x(x+6)-9(x+6)
2x^2+6x-9x-54
2x^2+3x-54=c
It belongs to option C
(2x - 9)(x + 6) = 2x(x + 6) - 9(x + 6) = 2x² + 12x - 9x - 54 = 2x² + 3x - 54
( 2x - 9 ) ( x + 6 )

( 2x ) ( x ) + ( 2x ) ( 6 ) + ( - 9 ) ( x ) + ( - 9 ) ( 6 )

2x² + 12x - 9x - 54

2x² + 3x - 54.

### Other questions on the subject: Mathematics

Mathematics, 21.06.2019, nuneybaby
Scale factor is always (new/original) so it depends which rectangle came first. if it started as large and was dilated to the smaller one, then the ratio is (.5/2.5) which simplifi...Read More
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# Prediction interval for log transformed variable in Stata [duplicate]

I want to predict $$y$$ with $$x_1$$ and $$x_2$$, including an out of sample prediction interval. However, $$y$$ has large outliers, so I log transform $$y$$ and estimate $$\log(y) = a + b_1 x_1 + b_2 x_2 + e$$, where $$e$$ is my normally distributed error term. Here are some data.

. clear

. set obs 2001
number of observations (_N) was 0, now 2,001

. global X x_1 x_2

.
. generate x_1 = runiform()

. generate x_2 = runiform()

. generate y = exp(5*x_1 + 5*x_2 + rnormal())

. replace x_1 = 1 if (_n == 1)

. replace x_2 = 1 if (_n == 1)

. replace y = exp(5*x_1 + 5*x_2 + 0) if (_n == 1)

. generate logy = log(y)

In Stata, normally, I would use predict to estimate $$\hat y$$ and predict, stdf to estimate prediction error $$SE$$, then $$\hat y \pm 2 \times SE$$ to generate upper and lower bounds for my prediction interval.

. regress y $X if (_n > 1) Source | SS df MS Number of obs = 2,000 -------------+---------------------------------- F(2, 1997) = 47.60 Model | 1.9090e+10 2 9.5451e+09 Prob > F = 0.0000 Residual | 4.0041e+11 1,997 200507843 R-squared = 0.0455 -------------+---------------------------------- Adj R-squared = 0.0446 Total | 4.1950e+11 1,999 209857103 Root MSE = 14160 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x_1 | 7837.173 1104.463 7.10 0.000 5671.154 10003.19 x_2 | 7390.569 1084.627 6.81 0.000 5263.45 9517.687 _cons | -5763.221 849.7131 -6.78 0.000 -7429.638 -4096.804 ------------------------------------------------------------------------------ . predict yhat_1, xb . predict yhat_1_se, stdf . generate yhat_1_lb = yhat_1 - 2*yhat_1_se . generate yhat_1_ub = yhat_1 + 2*yhat_1_se . list y yhat_1* if (_n == 1) +------------------------------------------------------+ | y yhat_1 yhat_1~e yhat_~lb yhat_~ub | |------------------------------------------------------| 1. | 22026.46 9464.521 14184.66 -18904.8 37833.84 | +------------------------------------------------------+  However, because I log transformed $$y$$, my code above is not correct and I can't apply this technique to estimate a $$\hat y$$ prediction interval. I can use predictnl to estimate $$\hat y$$. But there is no , stdf option for predictnl. How can I adjust predictnl's confidence interval to generate a prediction error? Is there a manual solution? . regress logy$X if (_n > 1)

Source |       SS           df       MS      Number of obs   =     2,000
-------------+----------------------------------   F(2, 1997)      =   4246.62
Model |  8975.08597         2  4487.54298   Prob > F        =    0.0000
Residual |  2110.29548     1,997  1.05673284   R-squared       =    0.8096
Total |  11085.3814     1,999  5.54546346   Root MSE        =     1.028

------------------------------------------------------------------------------
logy |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
x_1 |   5.237723   .0801803    65.32   0.000     5.080477    5.394969
x_2 |   5.203039   .0787403    66.08   0.000     5.048617    5.357461
_cons |   -.200541   .0616864    -3.25   0.001    -.3215174   -.0795646
------------------------------------------------------------------------------

. predictnl yhat_2 = exp(xb()), ci(yhat_2_lb yhat_2_ub) level(95)
note: confidence intervals calculated using Z critical values

. list y yhat_2* if (_n == 1)

+-------------------------------------------+
|        y     yhat_2   yha~2_lb   yha~2_ub |
|-------------------------------------------|
1. | 22026.46   28007.31   24680.86   31333.77 |
+-------------------------------------------+

How can I adjust predictnl's confidence interval to generate a prediction error? Is there a manual solution?

• Although your question asks about a prediction interval, the Stata code seems to be computing confidence intervals. The former are easy to deal with--you just back-transform them--while the latter are not. It matters very much, then, that you clarify what you are trying to ask. Could you do that for us? – whuber Nov 30 '18 at 21:47
• @whuber I would like a prediction interval from predictnl. But, you're right, my code generates confidence interval because I don't know how to get prediction interval from predictnl. – Richard Herron Nov 30 '18 at 21:50
• @whuber In other words... I have to estimate my regression in logs. But I want my predicted value and prediction interval in levels, not logs. – Richard Herron Nov 30 '18 at 21:55


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# Is the velocity with which molecules of the medium oscillate same as velocity of the sound in that medium?

I wanted to know that under sound field applied molecules of the medium oscillate with some velocity in the direction of sound propagation. Is this velocity same as velocity of the sound in that medium? I am interested in liquid medium.

-
This is a good question and while I don't know the answer, I strongly suspect the two velocities are related. It would be interesting to know by what relationship and if it had anything to do with the frequency of the sound. – Dom Mar 16 at 10:31
If a molecule is oscillating there is not a unique velocity vector, by definition. Are you talking about an instantaneous velocity in the direction of sound propagation? that at best will be enlarging and diminishing? an average speed? it is not a well defined quantity – anna v Mar 16 at 11:58
Do you mean average kinetic energy in the bulk flow rest frame (formally the second velocity moment of the distribution function) or something like the quantum vibrational modes of an atom? They are very different phenomena as some answers have eluded to below. – honeste_vivere Mar 16 at 12:33

It can't be, for the celerity of "vibrating" fluid parcels / groups of molecules (for volume or surface waves) also depends on the amplitude of the wave. Moreover $c_\phi = \omega/k$ ( and $c_g = \partial\omega/\partial k$ if dispersive) while the fastest parcels on the fastest moment of the cycle move at $A\omega$, without a dependency on k.

Now at the true molecular scale, even without sounds wave around, the thermal agitation of molecules (not parcels) makes a distribution a velocity around 0 (if no ground motion like wind... or sound waves :-) ) with magnitude peaking around the order of magnitude of sound waves, at least for perfect gas. But it's a wide distribution (responsible for many macroscopic effect, e.g. evaporation/condensation, etc).

-
Remarkably enough though, sound speed in a gas is about the same as the mean speed of thermal motion of the particles! This follows from the equipartition theorem. (It's not in fact the same speed; for a monoatomic gas it's about 75%.) – leftaroundabout Mar 16 at 23:21
and it's the (quadratic) mean. I.e. there is a whole large distribution of speeds around. – Fabrice NEYRET Mar 17 at 2:23

I am interested in liquid medium.

The answer is probably not because the speed of sound in most liquids decreases with increasing temperature and yet the molecules in a liquid would have more kinetic energy if the temperature increased. Source: Kaye & Laby Properties of sound in liquids

As ever water is anomalous and shows an increase in speed reach a maximum at about 75 $^\circ$C. Kaye & Laby suggest the following reason "mainly because of the temperature dependence of the adiabatic compressibility of the water molecule itself"

-
It is true that as temperature rises molecules of the liquid will have greater kinetic energy and hence higher velocity but I am interested to the velocity with which molecules of the medium oscillate under applied sound field and not its thermal motion which is random and increases with increase of temperature. – Vishal Mar 16 at 11:18
So, does increased kinetic energy of a medium cause increased amplitude of "brownian motion", or increased velocity, or both, or other? – Todd Wilcox Mar 16 at 14:26

I can elaborate on this question with respects to crystalline solids. Usually atoms in solids posses many possible vibrational modes. Information of all of them are collected in the phonon spectrum (see Fig. below).

Those vibrations are not oscillations of individual atoms, but their collective correlated motion. Atoms can vibrate in traverse or longitudinal directions relative to the direction of the wave propagation. These vibrations correspond to optical and acoustic phonons respectively. However, the sound waves are longitudinal waves only. Therefore, from many possible collective oscillations of atoms, only some of them are responsible for the propagation of sound.

-

No. For linear acoustics, the speed of sound, $c$, is given by $$c^{2} = \left(\frac{\partial \rho}{\partial p}\right)_{\!s} \, ,$$ and therefore is related to the thermodynamic equation of state. (The subscript $s$ indicates constant entropy.) This is the speed at which the wave propagates through the medium. A typical value for water, for example, is $c \approx 1500 \; \mathrm{ m/s}$.

The amplitude of the velocity oscillation of the molecules is known as the particle velocity, $v$. For a plane wave, the particle velocity is proportional to the acoustic pressure $p$ as $$v = \frac{p}{\rho c}.$$ The denominator, $\rho c$, is called the characteristic impedance. Again for water, $\rho \approx 1000 \; \mathrm{ kg/m^{3} }$, so $\rho c \approx 1.5 \times 10^{6} \; \mathrm{ kg / (m^{2} \, s )}$. A modest acoustic pressure of 1 Pa would then have an associated particle velocity of $v \approx 6.7 \times 10^{-7} \; \mathrm{ m/s }$— Quite different from the sound speed!

You can refer to Chapter 1 of Acoustics by Allan Pierce for derivations and details.

-
So, How to now acoustic pressure acting on the fluid? – Vishal Mar 17 at 10:44
Ideally, this is what you measure as a waveform with a hydrophone or transducer. Alternatively, if you know your sound pressure level in dB, you can invert the formula to get the RMS pressure. Make sure to use 1 micropascal as your reference pressure if in water. This all assumes plane waves, though, and there are modifications for wavefront spreading. Do you have a particular geometry or experiment in mind? – Greg Lyons Mar 17 at 14:13
Yes, I wanted to know how to calculate pressure density from the probe sonicator immersed in suspension. From there I will be able to calculate the velocity of the fluid as you suggested earlier. – Vishal Mar 17 at 16:27
I've used sonicators, but I don't know how I'd calculate the pressure field. I'd measure the SPL with an appropriate ultrasonic transducer. The formula I gave is not going to be correct for that geometry: You'll have spreading effects and reflections from the vessel walls. This is becoming significantly different from your original question, though. – Greg Lyons Mar 18 at 1:48

## protected by Qmechanic♦Mar 16 at 20:01

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).


text:
pgf scatter not assigning classes styles

I'm trying to generate a scatter plot using the pgfplots package. However, I'm having troubles with the scatter/classes and scatter srcoptions. For some reason, LaTeX keeps sending me the following error:

Package pgfplots Warning: scatter/classes: can't find class for

At the end, LaTeX generates the chart but it does not assign any style to the coordinates:

Here's a minimal example of my code:

\documentclass[border = 1cm]{standalone}

\usepackage{tikz} % https://www.ctan.org/pkg/pgf
\usepackage{pgfplots} % https://www.ctan.org/pkg/pgfplots
% ->
\usetikzlibrary{plotmarks}

\begin{filecontents}{data.dat}
c1  c2  c3
1   1   a
2   4   b
7   2   c
\end{filecontents}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
scatter/classes = {
a = {mark = *, draw = green},
b = {mark = *, draw = blue},
c = {mark = *, draw = red}
}]
scatter,
only marks,
scatter src = explicit symbolic]
table[
x = c1,
y = c2,
meta = c3,
col sep = space
]{data.dat};
\end{axis}
\end{tikzpicture}

\end{document}

• Tip: Instead of linking to some file for download, you could add \pgfplotstableread[row sep=\\]{c1 c2 c3 \\ 0 0 a \\ 1 0 b \\ 2 0 c\\}\datatable to the preamble, and use \datatable instead of data.csv in the \addplot. Makes the example self contained as well. (You can also use actual linebreaks instead of \\, but that doesn't work well in comments.) Oct 31 '19 at 22:54
• I almost completely agree with @TorbjørnT. but would actually suggest to add the table via filecontents, the reason being that preloading the table and plotting it from the file are not completely equivalent. I will add this to my answer.
– user194703
Oct 31 '19 at 23:01
• Thank you. I edited my question to be self-contained with filecontents. Oct 31 '19 at 23:12

You only need to replace the spaces between a and =, and similarly for b and c. Otherwise the names of the classes will contain these spaces.

\documentclass[border = 1cm]{standalone}
\usepackage{filecontents}
\begin{filecontents*}{data.csv}
c1;c2;c3
1;1;a
2;4;b
7;2;c
\end{filecontents*}

\usepackage{tikz} % https://www.ctan.org/pkg/pgf
\usepackage{pgfplots} % https://www.ctan.org/pkg/pgfplots
% ->
\usetikzlibrary{plotmarks}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
scatter/classes = {
a={mark = *, draw = green},
b={mark = *, draw = blue},
c={mark = *, draw = red}
}]
scatter,
only marks,
scatter src=explicit symbolic]
table[
x = c1,
y = c2,
meta = c3,
col sep = semicolon
]{data.csv};
\end{axis}
\end{tikzpicture}
\end{document}

Now the marks have a (hardly visible) halo of the respective color, if you want them also filled in these colors, use

\documentclass[border = 1cm]{standalone}
\usepackage{filecontents}
\begin{filecontents*}{data.csv}
c1;c2;c3
1;1;a
2;4;b
7;2;c
\end{filecontents*}

\usepackage{tikz} % https://www.ctan.org/pkg/pgf
\usepackage{pgfplots} % https://www.ctan.org/pkg/pgfplots
% ->
\usetikzlibrary{plotmarks}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
scatter/classes = {
a={mark = *, green},
b={mark = *, blue},
c={mark = *, red}
}]
scatter,
only marks,
scatter src=explicit symbolic]
table[
x = c1,
y = c2,
meta = c3,
col sep = semicolon
]{data.csv};
\end{axis}
\end{tikzpicture}
\end{document}

• Thank you very much. I didn't know that aspect of the scatter/classes names. Oct 31 '19 at 22:42


text:
# general form to standard form

• November 26th 2009, 03:12 PM
katt
general form to standard form
i need to change = -3x^2+x-2 to standard form( a(x-p)^2 +q)
can anyone show me the algebraic steps through completing the square?
• November 26th 2009, 03:28 PM
Raoh
$ax^{2}+bx+c=a\left [ \left ( x+\frac{b}{2a}\right )^{2}-\frac{\Delta}{4a^{2}}\right ]$
• November 26th 2009, 03:48 PM
katt
• November 26th 2009, 05:56 PM
dedust
this is the details

$ax^{2}+bx+c=0$

$x^{2}+\frac{b}{a}x+\frac{c}{a}=0$

$\left( x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}\right) +\frac{c}{a}%
-\frac{b^{2}}{4a^{2}}=0$

$\left( x+\frac{b}{2a}\right) ^{2}+\frac{c}{a}-\frac{b^{2}}{4a^{2}}=0$

$\left( x+\frac{b}{2a}\right) ^{2}+\frac{4ac-b^{2}}{4a^{2}}=0$
• November 27th 2009, 01:04 AM
Raoh
• November 29th 2009, 03:43 AM
Raoh
okay,
$-3x^2+x-2$ = $-3\left ( x^2-\frac{1}{3}x+\frac{2}{3} \right )=-3\left ( (x-\frac{1}{6})^2 -\frac{1}{36}+\frac{2}{3}\right )$= $-3\left ( (x-\frac{1}{6})^2+\frac{23}{36} \right )$
(Happy)


text:
# how to show sinx uni converges

• October 26th 2008, 05:12 AM
szpengchao
how to show sinx uni converges
$y_{n}(x)=\sin(x), 0\leq |x| \leq 2\pi n$ and

$y_{n}(x)=0, 2\pi n< |x| <\infty$
• October 27th 2008, 02:42 AM
CaptainBlack
Quote:

Originally Posted by szpengchao
$y_{n}(x)=\sin(x), 0\leq |x| \leq 2\pi n$ and

$y_{n}(x)=0, 2\pi n< |x| <\infty$

$y_n(x)$ converges pointwise to $\sin(x)$ ,but for all $n$ there exists $x_n$ such that $|y_n(x_n)-\sin(x_n)|=1$.

CB


text:
# Normalizing Hermite Polynomials

1. May 5, 2017

### rmiller70015

1. The problem statement, all variables and given/known data
Evaluate the normalization integral in (22.15). Hint: Use (22.12) for one of the $H_n(x)$ factors, integrate by parts, and use (22.17a); then use your result repeatedly.

2. Relevant equations
(22.15) $\int_{-\infty}^{\infty}e^{-x^2}H_n(x)H_m(x)dx = \sqrt{\pi}2^nn!$ when $n=m$
(22.12) $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$
(22.17a) $H'_n(x)=2nH_{n-1}(x)$

3. The attempt at a solution
I started out by writing my integral from (22.15) as:
$\int_{-\infty}^{\infty}e^{-x^2}[H_n(x)]^2dx$
Expanding one of the $H_n(x)$ terms gives:
$\int_{-\infty}^{\infty}e^{-x^2}H_n(x)e^{x^2}(-1)^n\frac{d^n}{dx^n}e^{-x^2}dx$
$=(-1)^n\int_{-\infty}^{\infty}H_n(x)\frac{d^n}{dx^n}e^{-x^2}dx$
Integration by parts and using equation 22.17a gives:
$(-1)^n[H_n(x)e^{-x^2}|^{\infty}_{-\infty}-\int^{\infty}_{-\infty}2nH_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}e^{-x^2}dx$
The first term goes to zero and leaves:
$(-1)^n\int^{\infty}_{-\infty}2nH_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}e^{-x^2}dx$
IBP n-1 more times gives:
$(-1)^n2^nn!\int^{\infty}_{-\infty}H_0(x)e^{-x^2}dx=(-1)^n2^nn!\int^{\infty}_{-\infty}e^{-x^2}dx = (-1)^n2^nn!\sqrt{\pi}$

This is correct except I have this extra factor of $(-1)^n$ that shouldn't be there, I'm not sure how to get rid of it.

2. May 5, 2017

### I like Serena

Hi rmiller,

Let's take a closer look at the IBP:
Shouldn't that be:
$$(-1)^n\left[H_n(x)e^{-x^2}|^{\infty}_{-\infty}-\int^{\infty}_{-\infty}2nH_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}e^{-x^2}dx\right] \\ =(-1)^n \cdot \left[ -\int^{\infty}_{-\infty}2nH_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}e^{-x^2}dx \right] \\ =(-1)^{n-1} \int^{\infty}_{-\infty}2nH_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}e^{-x^2}dx$$
?

3. May 8, 2017

### rmiller70015

For future people, you get a factor of -2 each time you do ibp and in the end you're left with (-1)^n (-2)^n when you combine them it gives 2^n.


text:
# Joule free expansion and differentials in irreversible processes

I am studying thermodynamics (undergraduate level) and I am not sure I understand what happens in irreversible processes.

I thought I understood but then I got confused in the Joule free expansion (not Joule-Thomson). This is my thought process:

I know it's essentially an adiabatic expansion into vacuum. Adiabatic means $$q = 0$$ and free expansion - vacuum means external pressure $$p_{ext} = 0$$, therefore for pV work, $$w = 0$$. So, $$ΔU = q + w = 0$$.

From my lecture notes, I see that: $$dU = 0 = C_V dT_U + \left(\frac{\partial U}{\partial V} \right)_{T} dV_U$$, therefore $$\left(\frac{\partial U}{\partial V} \right)_{T} = -C_V \left(\frac{\partial T}{\partial V} \right)_{U}$$, which makes sense considering for ideal gas it's $$0$$.

But my problem is that since it's a free expansion, it's also an irreversible process. What is the meaning of the differential $$dU$$ in an irreversible process? Isn't the whole problem in irreversible processes that it is not defined since the state is not in equilibrium? Why can we even use data from this experiment in the general expression for $$dU$$?

From my understanding, the only states in equilibrium are the initial and the final, so $$ΔU = 0$$ is correct. But why is $$dU$$ equal to 0?

Also, I am still on the first law and although I know about entropy in general and that $$ΔS > 0$$ in irreversible processes, I would prefer to understand this without involving entropy if it's possible.

• For a tutorial on how to determine the change in entropy for a system that experiences an irreversible process, see the following link:chemistry.stackexchange.com/questions/109654/…. This link also has a worked example for this exact problem. – Chet Miller Feb 19 at 2:01

For the system you describe in which no heat is transferred across the system boundary (insulated) Q=0 and no work is done on the rigid boundary surrounding the contents of the chamber W = 0, the first law of thermodynamics tells us that the change in internal energy of the system is zero: $$\Delta U=0$$For an ideal gas, the internal energy is a function only of temperature (and not volume) U=U(T), so this means that the change in temperature of the gas is zero: $$\Delta T=0$$ For a tutorial on how to determine the change in entropy for a system that experiences an irreversible process, see the following link: https://chemistry.stackexchange.com/questions/109654/entropy-change-of-isothermal-irreversible-expansion-of-ideal-gas


text:
chapter  2
Organizational Choices
Pages 50

Rationality in Organizational Design Before discussing in detail the variety of

organizational possibilities, however, they need a setting. Pfeffer , introducing his study, says, "A critical dimension distinguishing among theories of organizations is the perspective on action adopted, either explicitly or implicitly, by each theory. The three perspectives seen in the literature are: action seen as purposive, boundedly or intendedly rational, and prospective or goal directed; action seen as externally constrained or situationally determined;

and action seen as being somewhat more random and dependent on an emergent, unfolding process, with rationality in the second and third perspectives being constructed after the fact to make sense of behaviours that have already occurred."


text:
Mathematics
Easy

Question

# In an inverse variation, x = 6 and y = . What is the equation that represents the inverse variation?

Hint:

## The correct answer is: y =

### Here, we are given that in an inverse variation, x = 6 and y = 1/2.We have to find the relation between x and y.We have, y = =>  = k/6=> k = 3.Equation: y = .Hence, the correct option is B.

We can also take the relation as xy = k instead of y = k/x, where k is a constant.


text:
## Lesson goal: Do some computerized drawing

Previous: Drawing with text | Home | Next: The Cartesian Coordinate System

After you can draw pixels, lines, and circles, why don't you see if you can put them together and draw something? How about a stick figure? Car? House?

To get this to work, you have to visualize your drawing in terms of $(x,y)$ points on the grid, with $(0,0)$ at the center. You can move the mouse pointer over the plotting area (to map out needed $(x,y)$ points) as you create your work. Have fun!