text:
## Saturday, May 28, 2011

### CDK 1.2.10: the changes, the authors, and the reviewers

This may very well be the last bug fix release for the 1.2.x series. Or, at least that is what I am aiming for. This release is particularly aimed at (and from) Debian, fixing the puredist Ant target, contains a new atom type (for Te), and a fix in a molecular descriptor class.
• Copy data files into the right folder of the puredist bdb4e0e
• Added a test to demonstrate the ClassPathException in bug #3305581 7947f08
• Test if the descriptor results are the same for two implementations (unfortunately, the nonotify IMolecule extends the data IMolecule, so it does not catch bug #3305581) 7a795a5
• Factored out a method to test whether to descriptor calculations give the same results 2945d44
• Parameterized method to create water to allow alternative implementations 9b99f11
• Updated code to avoid dependency on specific implementation of molecule object. Now use IAtomContainer aee368a
• Added detection of a Te atom type, found in ChEMBL 2f74024
• Added unit tests^Cor Te.3 atom type detection 1b9afc9

The Authors
6  Egon Willighagen
1  Onkar Shinde
1  Rajarshi Guha

The Reviewers
5  Rajarshi  Guha
1  Egon Willighagen

(Git commands use are given here)

## Wednesday, May 25, 2011

### My slides for this afternoon's Bioclipse-OpenTox demo

If you're quick, you might still be able to sign up for this afternoons Bioclipse-OpenTox demo. Because of unfortunate and unforeseen circumstances, I will not be able to talk about Bioclipse-OpenTox scripting myself, and have asked Ola to cover that, on top of his own (cool) part.

Let me know how the demo was!

## Monday, May 23, 2011

### Online Seminar demoing Bioclipse-OpenTox interoperability using RDF technologies

Update: due to personal circumstances, it will very likely only be Ola to give the demo :(

This Wednesday, Ola and I will demo our work with Nina on interoperability between Bioclipse and OpenTox:
OpenTox recently initiated a series of online seminars and interactive tutorials on topics related to predictive toxicology using a virtual conferencing system supporting desktop sharing and voice. The tutorials are part of the EU FP7 project OpenTox (www.opentox.org), which aims at developing an interoperable open source predictive toxicology framework which may be used as an enabling platform for the creation of predictive toxicology applications.

The next tutorial will be held Wednesday, 25 May, at 15:00 CEST, and will describe the approach for integrating OpenTox in Bioclipse, complemented with hands-on tutorials showing how to take advantage of OpenTox functionality from within Bioclipse via the user interface and scripts.

The intended audience is people interested in learning how to use OpenTox from within Bioclipse, and people interested in how the integration was technically implemented. A bleeding-edge release of Bioclipse will be made available for the participants before the tutorial.

Participation in these online events involves no registration fee. To register for the seminar, visit http://www.surveymonkey.com/s/opentoxonline.

If you have already registered for a previous OpenTox tutorial this spring you don't need to register again.

## Sunday, May 22, 2011

### CDK 1.3.9: the changes, the authors, and the reviewers

Hot on the heels of the stable update 1.2.9, here is a new developers release. The 1.4.0 is still waiting for the basic (molecule) rendering of the CDK-JChemPaint patch to be submitted, but I think I will just submit it and see what happens. If rejected, I will release 1.4.0 without it. Mind you, I rather get that renderbasic patch up to shape (fix some regressions, add missing docs, and unit tests where possible, etc), but there is too little movement in this area.

Summary
Like many, this release too extends atom type perception to new atom types, making CDK algorithms applicable to more types of chemistry, but also includes fixes for perception when certain bits of information is missing from the input. Otherwise, it includes an improvement for the SMARTS query tool, a new AtomContainerManipulator extractSubstructure( IAtomContainer atomContainer, int... atomIndices) helper method, aromatic bond support for the HINReader, fixes in the PDB atom type table and improved PDB file IO, a faster, aromaticity-independent fingerprinter, and a set of general bug fixes and performance improvements.

The Authors
  34  Egon Willighagen
8  Rajarshi Guha
4  Gilleain Torrance
2  Dmitry Katsubo
1  Julio Peironcely
1  Onkar Shinde
1  Stephan Beisken
1  Mark Rijnbeek

This reflects contributions from Sweden, USA, UK, The Netherlands, and, I think, Pakistan. In terms of institutes, this includes patches working at (not implying the work was done as part of the day job): Karolinska Institutet, NIH, EBI (more than one group), Radboud University Nijmegen, EPO, Uppsala University, and the Debian project. That's pretty impressive for one release :)

The Reviewers
 20  Rajarshi Guha
13  Egon Willighagen
4  Gilleain Torrance

And of course kudos to all who reviewed patches!

The Changes
• Fixed detection of PH3 with implicit hydrogens 82197ac
• Added a unit test for a failing sulphur AT detection + fix: 3 coordinate sulphur without double bonds has one implicit hydrogen and is S.anyl 14d41c8
• Fixed detection of P.ate atom types with one implicit hydrogen 4ded3e4
• Added unit tests for bug #3141611 359b0b0
• Make CONNECT records work with non-protein PDB files Rewrote CONNECT parsing Do not write CONECT records for atoms that have no bonds 50f79bf
• Use wildcards (*) in jar file references. This fixes the problem while packaging the library for Debian as jar versions in Debian may not match with upstream. c79bcdc
• Fixed JavaDoc in accordance to current API 03cfd36
• Added a fingerprinter which does not take into account aromaticity, but looks at SP2 hybridization as replacement. 87d562c
• Clarified what is read from what file type b9a3cd6
• Fixed propagating of the error handler and reading mode. 997908d
• catch right exception and use the logger eed4717
• Use IAtomContainer instead of AtomContainer in AtomContainerSet 74112c4
• Escape /'s too 6941321
• fixed pdb_atomtypes.xml errors mentioned in https://sourceforge.net/mailarchive/message.php?msg_id=27206678, including the nigly vdwRadius for GLN.CD 9a07280
• whitespace-only: alignment of TYR-section 81ba913
• Typo fix -> Constructs 2a334d6
• Fixed classpath to solve those annoying error messages about unknown packages 263c7e2
• Added missing @Test annotation (idiot) df302b3
• Fixed typo in PDBReader constructor 92609d2
• Added a few missing tests c143185
• Added a missing reference for Floyd's algorithm 49eb772
• Fixed second half of patch a3e25e69419e53291e093d9365b4187843f03736 :( 74bfbc9
• Updated the author list 595cc9f
• Fixed a concurrency error, caused by the use of a static field which was not supposed to be static, as the classes are instantiated just to allow customization c0ede3f
• Added two new authors 833a9ac
• Updated to the 1.4.x API c5443a7
• Fixed the use of the proper Convertor 2f19e98
• Added the Co(3+) atom type (fixes #3093644) 67d5625
• Added the unit test phosphine for bug #3190151 740e4a5
• Added a unit test for a phosphor without explicit or implicit hydrogens 5e980dd
• Fixed proper syntax for ignored tests 95a59f7
• Added unit test to see if descriptor specifications refer to existing BODO entries 7c33f88
• Fixed descriptor identifier 0ffd85b
• Added unit testing of the dict module 4e47a4e
• Added code testing for the dict module b2d4f85
• fixed spelling error in JavaDoc 3dcf2fe
• Enabled PMD and OJDC for the tautomer module b73a662
• Create tautomers based on InChI 68d21b7
• Updated OrderQueryBond in SMARTS & isomorphism matching so that we correctly match when faced with aromatic bonds (rather than just looking at the bond order) cb4335e
• Updated HIN reader to parse aromaticring keywords and appropriately mark atoms as aromatic. Updated test and added test case 265dbba
• Added test HIN file for aromaticrings keyword 8a565e9
• Patch fixing isTheSame method in MolecularFormulaRange c86b52e
• Converted the use of HashMap to store SD (and other) properties to LinkedHashMap so that the order of properties read in from a SD file is maintained. Performance hit appears to be minimal b36a3e0
• Removed nonexisting dependency 2b3f842
• Removed nonexisting dependency dfe8cd9
• Added some more elements to the valence table 0629d8e
• Fixed NPE in AtomContainerComparator when container has pseudo atoms. fe8538f
• Implemented simple LRU caching mechanism to avoid reparsing previously used SMARTS queries. The cache size is set to 20 by default but can be set by the user c3a768d
• Implemented simple LRU caching mechanism to avoid reparsing previously used SMARTS queries b90c252
• Added a not-so-unit test for reading elemental data from XML using the reader and the underlying SAX handler 0c565e4
• Method to extract substructures cfa5c93
• StructureDiagramGenerator methods now throw CDKException instead of catch-all java.lang.Exception. 7c6a0c5

## Saturday, May 21, 2011

### CDK 1.2.9: the changes, the authors, and the reviewers

The CDK 1.2.9 release does not contain overly many patches, something you'd expect from a bug fix only series, and has instead a few atom type perception patches, and a patch by Onkar simplifying building of Debian package:
• Added Onkar Shinde, a Debian developer who recently patched the build system 95ad780
• Fixed detection of P.ate atom types with one implicit hydrogen d619db2
• Added unit tests for bug #3141611 ea248f3
• Fixed detection of PH3 with implicit hydrogens 989b151
• Added a unit test for a failing sulphur AT detection + fix: 3 coordinate sulphur without double bonds has one implicit hydrogen and is S.anyl 32f9aaa
• Use wildcards (*) in jar file references. This fixes the problem while packaging the library for Debian as jar versions in Debian may not match with upstream. 82d76f2
• Use IAtomContainer instead of AtomContainer in AtomContainerSet 4e2e66c
(Made with git log --oneline cdk-1.2.8..cdk-1.2.9 | sed 's/$$[a-f0-9]*$$\s$$.*$$.*/<li>\2 <a href="http:\/\/cdk.git.sourceforge.net\/git\/gitweb.cgi?p=cdk\/cdk;a=commit;h=\1">\1<\/a><\/li>/'.)

The Authors
6  Egon Willighagen
1  Onkar Shinde
1  Gilleain Torrance

(Made with git shortlog -s -n cdk-1.2.8..cdk-1.2.9.)

The Reviewers
2  Rajarshi Guha
1  Egon Willighagen

(Made with git log cdk-1.2.8..cdk-1.2.9 | grep Signed-off | cut -d':' -f2 | cut -d'<' -f1 | sort | uniq -c.)

## Friday, May 20, 2011

### A few Virtuoso commands I learned today

Virtuoso is not new to me (e.g. using it for the ChEMBL-RDF SPARQL end point). The Debian package is, though. But the only big problem here was to figure out where the db ended up, though that did not really matter in the end. What did matter, is that Virtuoso is not happy about long resource URIs, so I had to cut down on links out to rdf.openmolecules.net. Another difference is that is that the isql command is called isql-vt and the default values are not really useful for large amount of triples. But in the end I had a working test system at my laptop.

And, I learned two new isql command, on top of the two (DB.DBA.RDF_LOAD_RDFXML_MT and DB.DBA.TTLP) I learned before. For of all, is how to empty a database (which I did so far by killing the running instance, deleting the .db files, and restart :):

DB.DBA.TTLP_MT(...)

That save a lot of time on this four core laptop.

And for the cheminformatician this means that I am an important step closer to a next release of the ChEMBL-RDF export, which will include InChIs and SMILESes.

## Thursday, May 19, 2011

### This blog is tracking you: cookies and EU regulation

Just to inform you, parts of this website are very likely tracking you using HTTP cookies, including at least Google Adsense, and likely  the Topsy extension to show how often a post was tweeted, and the LinkedIn button to show the same for their network. Possibly, not sure about that, Blogger.com where I host my blog also tracks you. Personally, I care about your comments, and those I track manually. Down the bottom of the blog, I have added a short text on cookies, hoping to fulfill the new EU regulation's requirements (thanx to John for the ping!).

### Thematic Series around #RDF in chemistry now online

Thanx to the great support from the people at BioMed Central / Chemistry Central (not sure how apart / the same these two are), and Jan and Bailey in particular!, the Thematic Series around the ACSRDF2010 meeting of last August in Boston, is now online (and of course Martin for co-organizing the meeting and this series, to Pfizer, Inc. for sponsoring the series, see also our editorial, and to all authors who contributed their papers!):

BTW, I'm also quite happy with this funny functionality in ksnapshot I never noticed, that it can take free drawing screen region snap shots :)

## Monday, May 16, 2011

### Editorial ACSRDF2010 online (on RDF in chemistry)

The Thematic Series in the JChemInf is close to getting online with the first papers, indicated by Martin's and my editorial (doi:10.1186/1758-2946-3-15):

The other papers should be added to the journal this week, and I will keep you posted on that. The wordle shows one of the three session on RDF in chemistry of the meeting in Boston last year.

## Friday, May 06, 2011

### ChEMBL-RDF now linking to CrossRef

CrossRef reported recently they started supporting RDF, which is great news indeed! The RDF version of ChEMBL I created already contained the DOI, so the patch to link out was simple.

I have updated the docs.n3 for the downloads I prepared, which can simply be loaded into your triples store. A typical change looks like, so loading the new triples will only add triples, and URIs of papers are not changed:

@@ -742,6 +802,7 @@
dc:isPartOf jrn:jaf567f18f5617b0fe77d704fe8f61a3e .
res:r91 a bibo:Article ;
bibo:doi "10.1016/S0960-894X(01)80794-0" ;
+ owl:sameAs <http://dx.doi.org/10.1016/S0960-894X(01)80794-0> ;
dc:date "1991" ;
bibo:volume "1" ;
bibo:issue "3" ;

Info on the license (CC-SA-BY), and which two papers you are asked to cite if you use this data, can be found here. Oh, and have a peek at the series on database cheminformatics by John (part1, part2, part3)!

## Wednesday, May 04, 2011

### The costs of VR fund applications. Tell me I'm wrong...

One of the main Swedish funding agencies is called the Vetenskaprädet (VR). They just reported the number of applications in their big funding round of this year: 4606! That's is a staggering number. I decided to do some math here. Let's assume each proposal took about one week of effort, possible shared between two or more scientists. Let's assume the rate of one week of scientist at a Swedish university is about 1000 euro (so, including common overhead). That means that it costed the scientific community more then 4.6 million euro to apply for funding this year! Wow! Consider that amount 12% gets awarded (10% at any university, and 20% at KI where I work), that means that about 3.5 million euro was wasted making scientists learn to write grant applications... well spent indeed!

Now, I truly hope I am making some stupid mistake in my argumentation here... I don't like to thing what that amount of funding in one year could do for Open Source cheminformatics... (Oh, and this excludes the money needed to actually read these proposals, though they do not actually read all.)

## Tuesday, May 03, 2011

### Online SEURAT workshop: Open Data: What, Why, How?

An announcement.

Open Data: What, Why, How?
Monday May 9th 2011, 16:00 CEST, 7:00 PDT

(organized by ToxBank & Seurat DAWG)

Open Data is one approach to making data more easily re-usable including specifications by the the date creator on the terms of the type of re-use.

The goal of this interactive virtual meeting is to address misconceptions around Open Data and to answer questions experimental biologists, chemists, and toxicologists from any of the SEURAT-1 projects have with respect to how Open Data can help their research. Over 90 minutes the invited speakers Rufus Pollock (Open Knowledge Foundation) and John Wilbanks (Creative Commons) will reply to questions from the audience, complemented by a discussion by Barry Hardy (Douglas Connect) on how Open Data could contribute to the success of EU FP7 projects including OpenTox, ToxBank and SEURAT-1.

The meeting will be held online and will be organized via GotoMeeting. There are only 25 seats, so early registration is important, as we will fill seats on a first-come basis. However, one ‘seat’ can host multiple scientists behind a single computer.

Registration


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# Minimum Bounding Sphere

Today I was reading this article by Ericson. The article is on the derivation involved in writing a function to calculate the minimum bounding sphere of a set of three or four points. The topic itself isn’t very interesting (to me) when compared to the algebra itself. Ericson skips the algebra I had trouble with, so my article here will serve the purpose to record the algebra he skipped (in case other readers had trouble, or I ever want to reference the math again). The article I am writing here directly follows Ericson’s article, so I will assume readers have read and understood Ericson’s article.

Say we have a set of three points S = {A, B, C}, which can be thought of as a triangle. The problem of calculating the minimum bounding sphere of S involves checking to see if the circumcenter of S lies within or outside the triangle S. Lets name the circumcenter P. One immediate implementation involves computing P, followed by the computation of the barycentric coordinates of P with respect to S. These coordinates can be used check if P lay within S.

However computing P of S followed by the barycentric coordinates of P with respect to S involves a lot of redundant computations. Furthermore, if P does not lay within S then P itself does not need to be computed at all, since only the barycentric coordinates were needed.

It would be nice to compute barycentric coordinates of P with respect to S directly, and only if necessary construct P.

P in relation to S can be defined as:

\label{eq1}
P = A + s*(B – A) + t*(C – A)

Where $$s$$ and $$t$$ are barycentric coordinates of S such that $$s – t – (1.0 – s – t) = 0$$ if P is within S. Since P is equidistant from A, B and C, we can express P in relation to S with the following:

\label{eq2}
dot(P – B, P – B) = dot(P – A, P – A)

\label{eq3}
dot(P – C, P – C) = dot(P – A, P – A)

The interesting part involves plugging \eqref{eq1} into \eqref{eq2} and \eqref{eq3} followed by a collection of terms. Since I myself am a noob when it comes to algebra I had to go look up algebraic properties of the dot product to make sure I did not screw anything up. In particular these few rules are useful:

\label{eq4}
dot(u, v + w) = dot(u, v) + dot(u, w) \\
dot(s * u, v) = s*dot(u, v) \\
-dot(u – v, u – w) = dot(v – u, u – w)

Lets go over substituting \eqref{eq1} into \eqref{eq2} directly, however lets only consider the first part of \eqref{eq2} $$Dot(P – B, P – B)$$ to keep it simple:

\label{eq5}
dot(A + s*(B – A) + t*(C – A) – B, A + s*(B – A) + t*(C – A) – B)

Since things immediately get really long some substitutions help to prevent errors:

\label{eq6}
u = A – B\\
v = s*(B – A) + t*(C – A)\\
w = u + v \\

\label{eq7}
Dot(w, u + v)

\label{eq8}
Dot(w, u) + Dot(w, v)

\label{eq9}
Dot(u + v, u) + Dot(u + v, v)

\label{eq10}
Dot(u, u) + Dot(v, u) + Dot(u, v) + Dot(v, v)

If I were to expand \eqref{eq10} on this webpage it would not fit. Instead we will make use of a few more substitutions and then arrive in the final form:

\label{eq11}
x = s*(B – A) \\
y = t*(C – A) \\
x + y = v

\label{eq12}
Dot(A – B, A – B) + 2*Dot(A – B, x + y) + Dot(x + y, x + y)

By following the same process we can finish the substitution and notice that:

\label{eq13}
Dot(P – A, P – A) = Dot(x + y, x + y)

The final form of the substitution would be:

\label{eq14}
Dot(A – B, A – B) + \\ 2*Dot(A – B, x + y) + \\ Dot(x + y, x + y) = Dot(x + y, x + y)

\label{eq15}
Dot(A – B, A – B) + 2*Dot(A – B, x + y) = 0

\label{eq16}
Dot(A – B, A – B) + \\ 2*Dot(A – B, s*(B – A)) + \\ 2*Dot(A – B, t*(C – A)) = 0

\label{eq17}
s*Dot(B – A, B – A) + t*Dot(B – A, C – A) = \\ (1/2)*Dot(B – A, B – A)

This final equation \eqref{eq17} matches exactly what Ericson came up with on his own blog. Through a similar process \eqref{eq1} can be substituted into \eqref{eq3}, which would result in:

\label{eq18}
s*Dot(C – A, B – A) + t*dot(C – A, C – A) = \\ (1/2)*dot(C – A, C – A)

Share

format: plain


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# How do you solve x^2 + 3x -10 =0 using the quadratic formula?

Mar 25, 2016

The solutions are:
$x = 2$

$x = - 5$

#### Explanation:

${x}^{2} + 3 x - 10 = 0$

The equation is of the form color(blue)(ax^2+bx+c=0 where:

$a = 1 , b = 3 , c = - 10$

The Discriminant is given by:

color(blue)(Delta=b^2-4*a*c

$= {\left(3\right)}^{2} - \left(4 \cdot 1 \cdot \left(- 10\right)\right)$

$= 9 + 40 = 49$

The solutions are found using the formula
color(blue)(x=(-b+-sqrtDelta)/(2*a)

$x = \frac{\left(- 3\right) \pm \sqrt{49}}{2 \cdot 1} = \frac{- 3 \pm 7}{2}$

$x = \frac{- 3 + 7}{2} = \frac{4}{2} = 2$

$x = \frac{- 3 - 7}{2} = - \frac{10}{2} = - 5$


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# row_timestamp¶

## Description¶

Returns the timestamp (start time) of the row.

timestamp

## Domain¶

This is a scalar function (calculates a single output value for a single input row.)

## Usage¶

row_timestamp( )

## Examples¶

make_col time_taken:if_null(row_end_time() - row_timestamp(), 0s)

Create a column time_taken that is the duration of the state within the row.


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# Relation of convergence in probability and almost sure convergence

Let $(Y_n)$ real valued random variables, such that $Y_n\to c \in \mathbb R$ in distribution. It has been shown that $Y_n \to c$ in probability. I want to prove that $Y_n \to c$ a.s. does not hold! Therefore consider $Y_n \sim Ber_{1/n}\,$, i.e. $\mathbb P(Y_n=0)=1-\frac1n$ and $\mathbb P(Y_n=1)=\frac1n$. Then $\mathbb E(Y_n)=\frac1n$. I will use Markow's inequality two times.

$Y_n$ converges in probability to zero $$\lim_{n\to\infty}\mathbb P(\vert Y_n - 0\vert \ge \epsilon)\le \lim_{n\to\infty}\frac{\mathbb E(Y_n)}{\epsilon}=\lim_{n\to\infty}=\frac{\frac1n}{\epsilon}=0.$$

$Y_n$ does not converge a.s. to zero. I will use Borel Cantelli,i.e.$$\sum_{n=1}^\infty \mathbb P(\vert Y_n-0\vert \ge \epsilon) \le\sum_{n=1}^\infty\frac{\mathbb E(\vert Y_n\vert)}{\epsilon}=\sum_{i=1}^\infty\frac{\frac1n}{\epsilon}=\infty$$

Are we supposed to assume $\epsilon=1$ and is this attempt fine? If yes, Borel Cantelli provides $Y_n$ does not almost surely converges to zero. Some comments on this attempt are welcomes!

• Your reasoning for the second part is not correct. Note that the statement $$\sum_n \mathbb{P}(|Y_n-0| \geq \epsilon) \leq \infty$$ is trivial; it doesn't allow you to conclude anything. If you want to apply the Borel-Cantelli lemma to prove that the sequence does not converge almost surely to $0$, you have to show that $$\sum_n \mathbb{P}(|Y_n-0| \geq \epsilon) \color{red}{=} \infty$$ for some $\epsilon>0$.
– saz
Dec 16 '17 at 16:28
• @saz I missed it. Knew what to show but did not realise that I may not use markow's .., question to first part: We are allowed to assume $\epsilon =1$ or?
– user409387
Dec 16 '17 at 17:16
• and $\sum P (\vert Y_n \vert \ge \epsilon)=\sum \frac1n = \infty$right?
– user409387
Dec 16 '17 at 17:33
• Re your first comment: Why do you think that it suffices to check $\epsilon=1$? Have a look at the very definition of convergence in probability! Re your 2nd comment: Yes, that's true for $\epsilon \in (0,1]$.
– saz
Dec 16 '17 at 18:30
• @saz my first part would not hold for something like $\epsilon=\frac1n$ that's what is bothering me..
– user409387
Dec 16 '17 at 18:57

It seems you don't apply Borel-Cantelli correctly. Consider the events {$Y_n=1$} for $n \in \mathbb{N}$. They are independent and
$$\sum_{n=1}^\infty \mathbb{P}(Y_n=1) = \sum_{n=1}^\infty \frac{1}{n} = \infty$$
Thus by Borel-Cantelli, $\limsup_{n \to \infty} Y_n = 1$.
• I can show $\sum P(\mid Y_n -0 \mid \ge \epsilon)= \infty$ for $\epsilon =1$ . Then we have shown $Y_n$ does not converge to zero a.s. Therefore $\sum P(\mid Y_n \mid \ge 1)= \sum \frac1n = \infty$ And that is what I did in the comments!


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Difference between a sub graph and induced sub graph.

I have the following paragraph in my notes:

If $G=(V,E)$ is a general graph . Let $U\subseteq V$ and let $F$ be a subset of $E$ such that the vertices of each edge in $F$ are in $U$ ,
then $H=(U,F)$ is also a general graph and $H$ is a subgraph of $G$ .

If $F$ consists of all edges of $G$ which have endpoints in $U$ ,then $H$ is called induced subgraph of $G$ and is denoted by $G_U.$

From here both the definition of a subgraph and a induced subgraph seem same to me..
I can't understand what is the difference between them...

• Consider an induced subgraph, and then remove some of its edges. Is it still an induced subgraph? Is it a subgraph? Nov 9, 2014 at 11:31
• An INDUCED subgraph has the same edges as the original graph between the given set of vertices. A minor is, for example, a subgraph, but in general not an induced subgraph. An important difference is the merging of vertices, for example, a chain u-v-w can be replaced by u-w. If u and w are not connected in the original graph, such a subgraph would be not induced. Nov 9, 2014 at 11:37
• @Peter can you just elaborate your example as I don't know what a minor is ? Nov 9, 2014 at 11:40
• Suppose, u and w are the only neighbours of v. Then, replacing the edges u-v and v-w by a single edge u-w produces a MINOR. A minor is a graph that can be constructed by the original graph by deleting vertices, deleting edges or merging vertices as shown. If the edge u-v does not exist in the original graph, the subgraph is not induced. Nov 9, 2014 at 11:46
• Merging vertices does not produce an indeced subgraph in general, but it produces a subgraph. Nov 9, 2014 at 11:49

A subgraph $H$ of $G$ is called INDUCED, if for any two vertices $u,v$ in $H$, $u$ and $v$ are adjacent in $H$ if and only if they are adjacent in $G$.

In other words, $H$ has the same edges as $G$ between the vertices in $H$.

A general subgraph can have less edges between the same vertices than the original one.

So, an induced subgraph can be constructed by deleting vertices (and with them all the incident edges), but no more edges. If additional edges are deleted, the subgraph is not induced.

• The edges incident with a vertex are simply the edges having the vertex as an endpoint. Nov 9, 2014 at 12:20
• The edges of the REMAINING vertices are not allowed to be deleted, but deleting a vertex, of course, deletes the edges ending in this vertex. Nov 9, 2014 at 12:23
• Ah, now I see what you mean. The edge $2-4$ has an endpoint in $U$, is this the problem ? Nov 9, 2014 at 13:35
• Probably, it is meant that BOTH endpoints must belong to $U$, then everything is all right. Otherwise, the formulation would probably be : with at least one endpoint in $U$. Nov 9, 2014 at 13:38
• ah ..seems to be clear now...thanks. Nov 9, 2014 at 13:43

For an original graph G(V, E), select set of nodes V' from V. All the existing edges E' that connect between nodes in V' must remain.

This subgraph G' is called induced subgraph.

If you continue to remove some edge from E', then G' is still a subgraph of G, but no longer an induced subgraph of G.

Let G(V, E) be a graph and U is subset of V. For a induced subgraph, say H(U, F) we proceed as

1. Collect all possible subgraphs, say $H_1(U, F_1)$, $H_2(U, F_2)$ ,..., $H_n(U, F_n)$ of the graph G fixing set of vertices U in $H_i$, where $F_1, F_2,...,F_n$ are subsets of E.

2. Find F=max${F_1, F_2,...,F_n}$

Thus, $H(U, F)=\max\{H_1(U, F_1), H_2(U, F_2) ,..., H_n(U, F_n)\}$ is a induced subgraph of the graph G with respect to U.

M. Javaid

Say you have a graph $$G(V,E,\phi)$$ now any graph $$G'(V',E',\phi')$$ can be a sub-graph of G only if it satisfies all the 3 conditions

1. $$V' \subseteq V$$
2. $$E' \subseteq E$$
3. $$\phi'(e)=\phi(e)$$ for all e belonging E' While for constructing an induces subdigraph you first take a subset say W of vertex set V, then draw all the edges that lie between all those vertices(unlike a subgraph here you don't have the option to not chose an edge) be careful not to choose an edge that has one vertex in W but one not belonging to W.
• Hi! Welcome to MSE. Please use MathJax to format your answer so that it's much easier to understand Jan 21, 2019 at 17:15


text:
# turkmath.org

Türkiye'deki Matematiksel Etkinlikler

05 Kasım 2018, 10:40

### Bilkent Üniversitesi Cebir Seminerleri

Laurence Barker
Bilkent University, Türkiye

One aim of p-local finite group theory, arguably, is to get rid of the particular arbitrary famous finite group G and to retain only general p-local structure, perhaps partially expressed though categories whose objects are to be finite p-groups. In plain group theory, we consider the $p$-subgroups and the transports between them effected by conjugations. Sylow's Theorem allows us to localize, reducing to a consideration of morphisms between subgroups of a fixed Sylow p-subgroup. In block theory, the p-subgroups are replaced by the more general notion of a Brauer pair. Applying an analogue of Sylow's Theorem, the role of the Sylow p-subgroups is performed by p-subgroups called the defect groups. We can reduce to a category called the fusion system, whose objects are the subgroups of the defect group. Alperin's Fusion Theorem motivates a confinement of attention to only those objects that are called centric. But the invariants appearing in Alperin's Conjecture are not invariants of the fusion system. After Kuelshammer and Puig, we capture the desired local information by assigning, to each centric object, a twist of the automorphism group over the unit group of the coefficient field, we mean, a central extension that is well-defined up to canonical isomorphism. Near the end of part 1 or the start of part 2, we shall illustrate the centric fusion system and its twists by means of an illuminating example, the Sylow-defect p-blocks of SL(3, p), where the prime p is congruent to 1 modulo 3.

NOT: Konuşma öncesinde çay-kurabiye ikramı yapılacaktır

Cebir İngilizce
Seminer Odası SA - 141

## İLETİŞİM

Akademik biriminizin ya da çalışma grubunuzun ülkemizde gerçekleşen etkinliklerini, ilan etmek istediğiniz burs, ödül, akademik iş imkanlarını veya konuk ettiğiniz matematikçileri basit bir veri girişi ile kolayca turkmath.org sitesinde ücretsiz duyurabilirsiniz. Sisteme giriş yapmak için gerekli bilgileri almak ya da görüş ve önerilerinizi bildirmek için iletişime geçmekten çekinmeyiniz. Katkı verenler listesi için tıklayınız.

Özkan Değer ozkandeger@gmail.com

## DESTEK VERENLER

31. Journees Arithmetiques Konferansı Organizasyon Komitesi

Web sitesinin masraflarının karşılanması ve hizmetine devam edebilmesi için siz de bağış yapmak, sponsor olmak veya reklam vermek için lütfen iletişime geçiniz.


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# Solve limit (trig) without l'Hopitals rule

I had this question for math $$\lim_{x\to 0}\frac{\sin{ax^2}}{\sin{bx^2}}$$

So I used squeeze theorem and got $-1<1/(\sin{bx^2})<1$ and I multiplied by $\sin{ax^2}$ and got $-\sin{ax^2}< \sin{ax^2}/\sin{bx^2}< \sin{ax^2}$

Which then equals $0 < \lim_{x\to 0}\sin{ax^2}/\sin{bx^2}<0$ and the answer I got is $0$, but the answer is supposed to be $a/b$

Where did I go wrong and how can you solve this without using l'Hôpital's rule?

Use $(\sin x)/x$: $$\lim_{x\to0}\frac{\sin(ax^2)}{\sin(bx^2)}= \lim_{x\to0}\frac{\sin(ax^2)}{ax^2}\frac{bx^2}{\sin(bx^2)}\frac{a}{b}$$

Where did you go wrong? The inequality $-1<1/\sin(bx^2)<1$ is false for all $x$: if $0<\sin(bx^2)<1,$ then $1/\sin(bx^2)>1$; if $-1<\sin(bx^2)<0,$ then $1/\sin(bx^2)<-1$.

• wait... can you explain how that works, like you basically multiplied it by bx^2/ax^2, and it is still undefined when you sub in 0, thanks – Spongebob Squareroot-pants Oct 5 '15 at 15:39
• @SpongebobSquareroot-pants Yes, but you surely know that $\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$. – egreg Oct 5 '15 at 15:40
• @SpongebobSquareroot-pants When taking a limit, the point is not implicated. Therefore, there was no division by $0$. – Mark Viola Oct 5 '15 at 15:40
• isn't that basically still Hospital? How do you show $\frac{sin\left(x\right)}{x}\stackrel{x\rightarrow 0}{\rightarrow} 1$? – Max Oct 5 '15 at 15:45
• @Max That's a basic limit, used for finding the derivative of sine and cosine. – egreg Oct 5 '15 at 15:48

Another way to "squeeze" the ratio is to use the inequalities

$$\left|x\cos x\right|\le\left|\sin x\right|\le \left|x\right|$$

Then, we have

$$\left|\frac{ax^2\cos (ax^2)}{bx^2}\right|\le\left|\frac{\sin(ax^2)}{\sin(bx^2)}\right|\le\left|\frac{ax^2}{bx^2\cos(bx^2)}\right|$$

In the limit as $x\to 0$, we have then that

$$\lim_{x\to 0}\frac{ax^2\cos (ax^2)}{bx^2}=\frac ab$$

and

$$\lim_{x\to 0}\frac{ax^2}{bx^2\cos(bx^2)}=\frac ab$$

Thus, by the squeeze theorem,

$$\lim_{x\to 0}\frac{\sin(ax^2)}{\sin(bx^2)}=\frac ab$$

• Thanks for providing such a nice alternate solution. +1 – Shailesh Oct 5 '15 at 16:07
• @Shailesh You're welcome! My pleasure. – Mark Viola Oct 5 '15 at 16:11


text:
Search in Wireless Forums

# Wireless Forum

Posted Apr 26, 2012
10:46:14 PM
A Case For The Pantech Pursuit ll?

I just ordered a Pantech Pursuit ll.  Is it a good idea to put a case on it?  I read the manual and it looks like it has a sensitive antenna that should not be covered with your hand while you are using it.  Does that mean that a case will decrease signal strength?  If it won't, can someone recommend a good case?

I just ordered a Pantech Pursuit ll.  Is it a good idea to put a case on it?  I read the manual and it looks like it has a sensitive antenna that should not be covered with your hand while you are using it.  Does that mean that a case will decrease signal strength?  If it won't, can someone recommend a good case?

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# NAG FL Interfaceg01qtf (pdf_landau_moment2)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

g01qtf returns the value of the second moment ${\Phi }_{2}\left(x\right)$ of the Landau density function.

## 2Specification

Fortran Interface
 Function g01qtf ( x)
 Real (Kind=nag_wp) :: g01qtf Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double g01qtf_ (const double *x)
The routine may be called by the names g01qtf or nagf_stat_pdf_landau_moment2.

## 3Description

g01qtf evaluates an approximation to the second moment ${\Phi }_{2}\left(x\right)$ of the Landau density function given by
 $Φ2(x)=1Φ(x) ∫-∞xλ2ϕ(λ)dλ,$
where $\varphi \left(\lambda \right)$ is described in g01mtf, using piecewise approximation by rational functions. Further details can be found in Kölbig and Schorr (1984).
To obtain the value of ${\Phi }_{1}\left(x\right)$, g01ptf can be used.

## 4References

Kölbig K S and Schorr B (1984) A program package for the Landau distribution Comp. Phys. Comm. 31 97–111

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.

None.

## 7Accuracy

At least $7$ significant digits are usually correct, but occasionally only $6$. Such accuracy is normally considered to be adequate for applications in experimental physics.

## 8Parallelism and Performance

g01qtf is not threaded in any implementation.

None.

## 10Example

This example evaluates ${\Phi }_{2}\left(x\right)$ at $x=0.5$, and prints the results.

### 10.1Program Text

Program Text (g01qtfe.f90)

### 10.2Program Data

Program Data (g01qtfe.d)

### 10.3Program Results

Program Results (g01qtfe.r)


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# Split-Lemma for chain complexes

Suppose $k$ is a field and $A$, $B$ and $C$ are chain complexes of $k$-vector spaces, i.e., objects in $\mathbf{Ch}(k\text{-}\mathbf{Vect})$. Is there are chain complex version of the split lemma, i.e.:

If there is an inclusion chain map $i:A \to B$ and a projection chain map $p:B \to C$ and a short exact secence

$$0 \to A \overset{i}\to B \overset{p}\to C \to 0$$

of chain complexes, where '$0$' means the chain complex of the zero vector space with the trivial differential. Then we can split the sequence,i.e. there are sections of chain complexes $s:C \to B$ and retracts $r:B \to A$ such that $p\circ s = id_C$, $r\circ i=id_A$ and the chain complex $B$ is naturally isomoph to $s(C)\oplus i(A)$?

==================

If the answer is 'yes' I would like to get a reference.

==================

Edit: The following example suggest that the answer is no in general:

Consider a chain complex of vector spaces $B$. Now in degree $k\in \mathbb{Z}$ we take the quotient vector space $C_k:=B_k/kern(d_{k}(B_{k}))$. Then we have the short exact sequence

$$0 \to kern(d_B) \to B \to C \to 0$$

where $kern(B)$ is the complex that in degree $k$ is just $kern(d_{B,k})$.

Clearly the differential on $C_k$ is trivial. So a section $s:C \to B$ has to satisfy $0=s(0)=s(d_C(x))=d_B(s(x))$, but $s$ can't map any $x\in C$ to the zero vector because then $p\circ s \neq id_C$ in general.

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The differential on $C_k$, i.e., $d^{C_\bullet}_k \colon C_k \to C_{k-1}$, need not be trivial at all. –  Andrea Gagna Jun 3 '13 at 17:07
Since $Im(d_{B,k-1})\subset Kern(d_{B,k})$, it is trivial because $Kern(d_{B,k})$ is the zero vector in $C_{k}$. –  Mirco Jun 3 '13 at 17:09
Wait. Are you using homological or cohomological notation? The subscript indexes are typical of homological notation. But what you have just written doesn't make sense that way. –  Andrea Gagna Jun 3 '13 at 17:16
$d_{B,k}: B_k \to B_{k-1}$ is a differential, i.e. a degree $-1$ map, that squares to zero. What have I written wrong here? –  Mirco Jun 3 '13 at 17:29
Ok there was a typo.You are right.It has to be $Im(d_{B,k+1})\subset kern(d_{B,k})$. –  Mirco Jun 3 '13 at 17:31

Actually, I can completely understand your counter-example. The following one is the simple I can imagine.

Let $k$ be a field and $\mathbf{Ch}(k\text{-}\mathbf{Vect})$ the category of chains of $k$-vector spaces.

Consider the short exact sequence of $k\text{-}\mathbf{Vect}$ $$0 \to k \to k\oplus k \to k \to 0\qquad(\star)$$ where the morphisms, given by the universal property of, resp., product and coproduct, are, resp., $\langle \text{id}_{k}, 0\rangle_\prod\colon k \to k\oplus k$ and $\langle 0, \text{id}_k\rangle_\coprod \colon k\oplus k \to k$; (i.e., resp., $x \mapsto (x, 0)$ and $(x_1, x_2)\mapsto x_2$.

Let $A_\bullet, C_\bullet$ and $B_\bullet$ be the exact complexes which have, resp., $\text{id}_k$ as $d^{A_\bullet}_1, d^{C_\bullet}_1$ morphisms and $\text{id}_{k\oplus k}$ as $d^{B_\bullet}_1$ morphism and for which $A_i \cong B_i\cong C_i \cong 0$, for every $i \in\mathbb Z\setminus\{0, 1\}$.

$(\star)$, as morphism of degree $0$ and $1$, induces a short exact sequence of exact complexes $$0_\bullet \to A_\bullet \to B_\bullet \to C_\bullet \to 0_\bullet$$

At degree $0$ and $1$ we consider, resp., the splittings $\langle 0, \text{id}_k\rangle_\prod$ and $\langle \text{id}_k, \text{id}_k\rangle_\prod$ (i.e., $x \mapsto (0, x)$ and $x\mapsto (x, x)$),both morphisms $k \to k\oplus k$.

Obviously, $$d^{B_\bullet}_1 \circ \langle 0, \text{id}_k\rangle_\prod = \text{id}_{k\oplus k}\circ\langle 0, \text{id}_k\rangle_\prod = \langle 0, \text{id}_k\rangle_\prod \neq \langle \text{id}_k, \text{id}_k\rangle_\prod = \langle \text{id}_k, \text{id}_k\rangle_\prod \circ \text{id}_k = \langle \text{id}_k, \text{id}_k\rangle_\prod \circ d^{C_\bullet}_1$$ directly by the universal property of product. As a matter of example, we have $$1_k \mapsto (0, 1_k)$$ on one hand and $$1_k \mapsto (1_k, 1_k)$$ on the other.

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And $A_0\simeq A_1 = k$, $B_0 \simeq B_1 = k \oplus k$ and $C_0 \simeq C_1 = k$ ? –  Mirco Jun 3 '13 at 22:08
Ah! I get it. Thanks for the afford. I appreciate. So in conclusion, the splitting lemma does not hold under the requirement that the splitting maps are chain complex morphisms. –  Mirco Jun 3 '13 at 22:25
Nice example b.t.w. –  Mirco Jun 3 '13 at 22:26
You are welcome. The one by Andrea Blass is also very cool and minimal. –  Andrea Gagna Jun 3 '13 at 22:50

Consider the following specific counterexample. $A$ is the complex with the 1-dimensional space $k$ in degree 0 and zero in all other degrees (so of course the differential is 0 in all degrees). $B$ has the one-dimensional space $k$ in degree 1 and the 2-dimensional $k^2$ in degree 0; the unique non-trivial differential is given by $x\mapsto(x,0)$. The injection of $A$ in $B$ is nontrivial only in degree 0, where it sends $x\mapsto (x,0)$. This is a chain map; all the maps that are required to agree are in fact zero. Let $C$ be the cokernel of this injection. So we can take $C_0=C_1=k$ ; the projection from $B$ to $C$ is the identity in degree 1 and is $(x,y)\mapsto y$ in degree 0; the differential of $C$ is zero in all degrees.

Now suppose a chain map $s:C\to B$ were a splitting of this projection. For it to be a chain map, we need that the maps $C_1\to C_0\to B_0$ and $C_1\to B_1\to B_0$ agree. But the first of these maps is zero because $C_1\to C_0$ is. Therefore, so is the second. Since $B_1\to B_0$ is a monomorphism (namely $x\mapsto(x,0)$), we conclude that the degree 1 part $C_1\to B_1$ of $s$ is zero. That's impossible, as it has to split the isomorphism $B_1\to C_1$.

-

Let $(A,d_A)$ and $(B,d_B)$ be $\mathbb Z$-graded complexes of modules over a ring. A morphism (of degree $0$) $\varphi: A \rightarrow B$ of complexes of modules is a collection of module maps $\varphi_i: A_i \rightarrow B_i$, compatible with the differentials on $A$ and $B$. What you are searching for is just a componentwise version of the splitting Lemma you state above.

For example in the above notation $0\rightarrow A \rightarrow A\oplus B \rightarrow B \rightarrow 0$ is a split short exact sequence of complexes of modules, with all maps (the natural inclusion and projection) of degree $0$ and compatible with differentials.

In http://www.amazon.com/Methods-Homological-Algebra-Sergei-Gelfand/dp/3540435832 you can find an interesting application of the above formalism when dealing with cones and cylinders.

-
Are you sure? Because the maps $s$ and $r$ have to satisfy the additional assumtions $sd_C =d_Bs$ and $rd_B=d_Ar$ –  Mirco Jun 3 '13 at 16:29
Please look at my example I added to the question –  Mirco Jun 3 '13 at 16:38

If you think of a complex as a functor in $Fun(\mathbb{Z},Mod_R)$ (where $R$ is a commutative ring with identity,for semplicity) then you're simply asking for calculating biproducts pointwise. In fact a section for the complex is a section pointwise, so that you have biproducts at each n-level, from which you can conclude.

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But in light of the example I gave above (later), it seems like it is NOT possible. Maybe I'm confused ... –  Mirco Jun 3 '13 at 16:43

@Mirco: I am sorry: I think I did not fully understand your question. I just wanted to say that with complexes you have to look at properties $componentwise$. If the split exists or not, and in which sense, this is another question. The tricky part is about differentials.

Let $0\rightarrow X\stackrel{f}{\rightarrow} Y \stackrel{g}{\rightarrow } Z \rightarrow 0$ be a short exact sequence of $\mathbb Z$-graded complexes of $R$-modules ($R$ is a ring). The sequence is said to be semi-split if there exists a morphism of $\mathbb Z$-graded $R$-modules $w: Z\rightarrow Y$ s.t. $g\circ w= 1_Z$. In other words, if $w$ exists, it does not need to commute with the differentials on $Y$ and $Z$, i.e. it is no morphism of complexes of $R$-modules, in general!

If you find a split $w$ like above, then you can identify $Y$ with $X \oplus Z$ as
$\mathbb Z$-graded $R$-modules using the componentwise splitting, as in the classical theory (forget about the differentials!). The differential on $X \oplus Z$ is in fact given by any $d_{X \oplus Z}=(d_X-h,d_Z)$ s.t. $h: X\rightarrow Z[1]$ is a degree $+1$ morphism of complexes of $R$-modules (so commutes with differentials: the differential on Z[1] is given by $d_{Z[1]}:=-d_Z$).

I hope that the role of differentials is a bit more clear. If not, please me know :) Avitus

Counter example added. Let $f:A\rightarrow B$ be a morphism of complexes. Then the short exact sequence $0\rightarrow B\rightarrow C(f)\rightarrow A\rightarrow 0$ is semisplit, denoting by $C(f)$ the cone of $f$, i.e. the complex $C(f)=(A[1]\oplus B,d)$, with differential $d(a,b)=(-d_A(a),f(a)+d_B(b))$. The splittings $r: C(f)\rightarrow B$ and $s: A[1]\rightarrow C(f)$ are the "usual" ones.

$C(f)$ is isomorphic to $A[1]\oplus B$ as $\mathbb Z$-graded module but not as complex. The split $r: C(f)\rightarrow B$ is not a morphism of complexes.

-
Ok. But I'm not sure how this relates to my question. The differential $d_{X\oplus Z}$ you mentioned,is different from then original $d_Y$, so $(X\oplus Z,d_{X\oplus Z})$ is in general a different complex, not chain isomorph to $(Y,d_Y)$. Or is there any natural ismorphism,I just don't see? –  Mirco Jun 3 '13 at 20:23
But anyway,is the homotopy $h$ given in a natural way? –  Mirco Jun 3 '13 at 20:26
Remember that I'm asking for chain maps $s$ and $r$! –  Mirco Jun 3 '13 at 20:34
I added a counterexample in my answer: I hope it helps –  Avitus Jun 5 '13 at 20:19


text:
# System of Equations using Matrices II

SOLUTION OF HOMOGENEOUS SYSTEM OF LINEAR EQUATIONS: Let AX = 0 be a homogeneous system of simultaneous linear equations with n-unknowns. Let us now discuss two cases:

CASE I: When |A| ≠ 0.

In this case, A-1 exists.

AX = 0

A-1 (AX) = A-10

(A-1A) X = 0

InX = 0

X = 0

X1 = X2 = … = Xn = 0

Thus, if A is a non-singular matrix the homogeneous system of equations has a unique solution given by => x1 = x2 = … = xn = 0. This solution is known as a trivial solution.

CASE II When |A| = 0

In this case, we have (adj A) B = (adj A) 0 = 0

So, the homogenous system of equation is consistent and it has infinitely many solutions can be obtained by giving any real value to one of the variables.

It follows from the above discussion that a homogeneous system of equations given by AX=O is always consistent. It has a unique solution in which all variables are equal to zero, if |A| ≠ 0. If |A| = 0, then the system has non-trivial solutions.

Example 1: Solve the following system of homogeneous equations:

2x + 3y – z = 0

x – y – 2z = 0

3x + y + 3z = 0

SOLUTION: The given system of homogeneous equations can be written as

$$\left[ \begin{matrix}2 & 3 & -1 \\1 & -1 & -2 \\3 & 1 & 3\\\end{matrix} \right]\left[ \begin{matrix}x \\y \\z \\\end{matrix} \right]=\left[ \begin{matrix}0 \\0 \\0 \\\end{matrix} \right]$$

Or AX = 0, where $$A=\left[ \begin{matrix}2 & 3 & -1 \\1 & -1 & -2 \\3 & 1 & 3 \\\end{matrix} \right]$$, $$X=\left[ \begin{matrix}x \\y \\z \\\end{matrix} \right]$$ and $$O=\left[ \begin{matrix}0 \\0 \\0 \\\end{matrix} \right]$$.

Now, $$\left| A \right|=\left[ \begin{matrix}2 & 3 & -1 \\1 & -1 & -2 \\3 & 1 & 3 \\\end{matrix} \right]$$ = – 2 – 27 – 4 = – 33 ≠ 0.

Thus |A| ≠ 0. So, the given system has only the trivial solution given by x = y = z = O.


text:
# When can a perturbation be treated as a regular perturbation?

I am working with cauchy problem of the form $$( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x),$$ where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The operator $A^\delta$ can be written as follows: $$A^\delta = A_0 + \delta e^{b x} A_1 ,$$ where $b<0$ and $\delta \geq 0$. I have found a formal solution to the above problem by assuming that $u^\delta$ has a power series expansion in $\delta$. That is $$u^\delta = \sum_{n=0}^\infty \delta^n u_n .$$ Plugging the expansion into the Cauchy problem yields $$( - \partial_t + A_0) u_0 = 0 , \qquad u_0(0,x) = h(x) ,$$ and $$( - \partial_t + A_1) u_n = -A_1 u_{n-1} , \qquad u_n(0,x) = 0$$ I can solve these equations explicity for a given $u_n$. But the solution for each $u_n$ is written as an integral (a Fourier Transform). And it is not easy to check if my series solution converges.

So, it would be very nice if I had some sort of theorem that would tell me that my assumption -- that $u^\delta$ can be expanded power series in $\delta$ -- is justified. If I had such a theorem, then I wouldn't need to check convergence of the sum.

So my question is: When can I assume that $u^\delta$ has a power series expansion in $\delta$? Specifically, is this a valid assumption in my problem? And, more generally, are there conditions I can check in the future to know if a perturbation can be treated as regular?

If it helps, both $A_0$ and $A_1$ are infinitesimal generators of Levy processes. i.e. $$A_i = \mu_i \partial x + \frac{1}{2} \sigma_i^2 \partial_x^2 + \int_{\mathbb{R}} \nu_i(dz) ( e^{z \partial_x} - 1 - z \partial_x )$$ where $\mu_i$ and $\sigma_i$ and constants, $\nu_i(dz)$ is Levy measure and $e^{z \partial_x} f(x) = f(x+z)$ for any $C^\infty$ function.

-
seems the recurrence relation linking $u_n$ with $u_{n-1}$ has an $A_0$ on the left side instead of $A_1$. From this relation, if you can show that the operator mapping $u_{n-1}$ to $u_n$ is a contraction from an appropriate space to itself, then the series is justified. –  Francois Monard Aug 6 '12 at 19:11


text:
# Libgdx can't read .ttf file when running Android configuration

Firstly, I am aware that a similar question was asked some time ago that went unanswered. Currently, it has multiple upvotes but no answer.

The error

com.badlogic.gdx.utils.GdxRuntimeException: Error reading file: data/font/font1.ttf (Internal)
...
Caused by: java.io.FileNotFoundException: font/font1.ttf

occurs when I use the class FreeTypeFontGenerator provided by Libgdx as a extension to use freetype fonts(link). This is the line in my project that caused error:

FreeTypeFontGenerator generator = new FreeTypeFontGenerator(Gdx.files.internal("font/font1.ttf"));

I can be fairly certain that my path to the .ttf is correct, as the desktop configuration runs perfectly, showing the font as desired.

My guess: I didn't include the freetype extension when creating this libgdx project. When following the tutorial to include it (that is, adding the freetype dependencies in build.gradle), something somehow went wrong. But even this is not convincing enough.

build.gradle:

project(":desktop") {
apply plugin: "java"

dependencies {
compile project(":core")
compile "com.badlogicgames.gdx:gdx-backend-lwjgl:$gdxVersion" compile "com.badlogicgames.gdx:gdx-platform:$gdxVersion:natives-desktop"
compile "com.badlogicgames.gdx:gdx-box2d-platform:$gdxVersion:natives-desktop" compile "com.badlogicgames.gdx:gdx-tools:$gdxVersion"
compile "com.badlogicgames.gdx:gdx-freetype-platform:$gdxVersion:natives-desktop" } } project(":android") { apply plugin: "android" configurations { natives } dependencies { compile project(":core") compile "com.badlogicgames.gdx:gdx-backend-android:$gdxVersion"
natives "com.badlogicgames.gdx:gdx-platform:$gdxVersion:natives-armeabi" natives "com.badlogicgames.gdx:gdx-platform:$gdxVersion:natives-armeabi-v7a"
natives "com.badlogicgames.gdx:gdx-platform:$gdxVersion:natives-arm64-v8a" natives "com.badlogicgames.gdx:gdx-platform:$gdxVersion:natives-x86"
natives "com.badlogicgames.gdx:gdx-platform:$gdxVersion:natives-x86_64" compile "com.badlogicgames.gdx:gdx-box2d:$gdxVersion"
natives "com.badlogicgames.gdx:gdx-box2d-platform:$gdxVersion:natives-armeabi" natives "com.badlogicgames.gdx:gdx-box2d-platform:$gdxVersion:natives-armeabi-v7a"
natives "com.badlogicgames.gdx:gdx-box2d-platform:$gdxVersion:natives-arm64-v8a" natives "com.badlogicgames.gdx:gdx-box2d-platform:$gdxVersion:natives-x86"
natives "com.badlogicgames.gdx:gdx-box2d-platform:$gdxVersion:natives-x86_64" compile "com.badlogicgames.gdx:gdx-freetype:$gdxVersion"
natives "com.badlogicgames.gdx:gdx-freetype-platform:$gdxVersion:natives-armeabi" natives "com.badlogicgames.gdx:gdx-freetype-platform:$gdxVersion:natives-armeabi-v7a"
natives "com.badlogicgames.gdx:gdx-freetype-platform:$gdxVersion:natives-arm64-v8a" natives "com.badlogicgames.gdx:gdx-freetype-platform:$gdxVersion:natives-x86"

My actually font filename in the font folder is font1.ttf. After changing it to font.ttf, the problem resolved itself. I don't understand why it worked, as I think it is okay to include numbers and uppercase in a file name.


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• Coordinates
• Non-solid figure
• Drawing with opacity or anti-aliasing
WINDEV
WEBDEV
WINDEV Mobile
Others
<Image variable>.DrawSlice (Function)
In french: <Variable Image>.DessinePortion
Draws a circle or an ellipse section in an Image control.
Example
// Draw a circle section whose
// background is light yellow and whose border is light blue
IMG_Drawing.DrawSlice(10, 10, 50, 50, 20, 20, 40, 50, LightYellow, LightBlue)
Syntax
<Image variable>.DrawSlice(<X1> , <Y1> , <X2> , <Y2> , <X3> , <Y3> , <X4> , <Y4> [, <Background color> [, <Line color>]])
<Image variable>: Image variable
Name of the Image control to be used.
<X1>: Integer
X-coordinate of the upper-left corner of the rectangle containing the circle. These coordinates are expressed in pixels.
<Y1>: Integer
Y-coordinate of the upper-left corner of the rectangle containing the circle. These coordinates are expressed in pixels.
<X2>: Integer
X-coordinate of the lower-right corner of the rectangle containing the circle. These coordinates are expressed in pixels.
<Y2>: Integer
Y-coordinate of the lower-right corner of the rectangle containing the circle. These coordinates are expressed in pixels.
<X3>: Integer
X-coordinate of the start point of the circle section. These coordinates are expressed in pixels.
<Y3>: Integer
Y-coordinate of the start point of the circle section. These coordinates are expressed in pixels.
<X4>: Integer
X-coordinate of the end point of the circle section. These coordinates are expressed in pixels.
<Y4>: Integer
Y-coordinate of the end point of the circle section. These coordinates are expressed in pixels.
<Background color>: Integer or constant (optional)
Background color of the circle section. This color can correspond to:
If this parameter is not specified, the background color:
<Line color>: Integer or constant (optional)
Color of the line for the circle section. This color can correspond to:
If this parameter is not specified, the line color:
• corresponds to the color specified in the last call to <Image>.Pen,
• is the same as the background color if <Image>.Pen has not been used.
Remarks

Coordinates

If the start point and the end point are not located on the border of the circle (or on the border of the ellipse), the point taken into account corresponds to the intersection between the circle and the line that joins the specified point and the center of the rectangle (which means the center of the circle).
Coordinates are specified with respect to the upper-left corner of the Image control (coordinates: (0,0)).
The drawing sections are drawn anticlockwise.

Non-solid figure

To draw a non-solid figure (without taking into account the center of the circle), use <Image>.DrawChord.

Drawing with opacity or anti-aliasing

For a drawing with opacity (<Image>.StartDrawing with the dWithOpacity constant) or anti-aliasing (<Image>.ChangeMode with the drawAntiAliasing constant), the outline is drawn inside the section: therefore, the line is partly combined with the inside of the section. If the line is very thick, it will be displayed in two colors.
Related Examples:
 Unit examples (WINDEV): The drawing functions [ + ] Using the main drawing functions of WINDEV to:- Initialize an Image control for drawing- Draw simple shapes- Write a text into a drawing- Change the color in a drawing
Business / UI classification: Neutral code
Component: wd270pnt.dll
Minimum version required
• Version 24


text:
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Chemistry
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Question:

If 100.0 mL of water is added to 40.0 mL of 2.0 M HCl solution, what is the new molar concentration in M of HCl?

Deepti G.

First the number of moles of HCl must be calculated. To do so, $$Molarity (M) = \frac{moles of solute}{liters of solution}$$, therefore $$moles of HCl = 2.0 M * .040 L = 0.080 moles of HCl$$. To find the new molar concentration, a new molarity calculation must be done with the total volume: $$Molarity (M) = \frac{0.080 mol}{0.1400 L} = 0.22 M$$. The new concentration of HCl is 0.22 M.

Biology
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Question:

What is the difference between transcription and translation?

Deepti G.

Transcription and translation are both steps of protein synthesis. Transcription refers to the formation of mRNA from DNA. Translation refers to the formation of proteins from mRNA.

Calculus
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Question:

How do I know when to use the chain rule?

Deepti G.

The chain rule is used when the function you are trying to differentiate contains two differentiable functions. For example, the function, $$f(x) = x^2-2$$, only contains one function and can be differentiated without chain rule. Another function, $$g(x) = \sqrt{x}$$, also only contains one function and can be differentiated without chain rule. When each of these functions are combined to form $$F(x) = \sqrt{x^2-2}$$, then chain rule must be used.

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# Find an explicit general solution to the following differential equation,x{y}'=2y + ...

## Question:

{eq}Find\, \, an\, \, explicit\,\, general\,\, solution\, \, to\, \, the\,\, following\, \, differential\, \, equation {/eq}

{eq}\mathbf{ x{y}'=2y\, +\, x^{3}cos(x)} {/eq}

## Differential Equation:

Equation with derivative term is called differential equation.

Differential equation of form {eq}\displaystyle y'+Py=Q {/eq} is called linear differential equation.

where, P & Q are constant or function of x.

Solution will be

{eq}\displaystyle y\left(I.F\right)=\int \:Q\left(I.F\right)dx+C {/eq} with integrating factor {eq}\displaystyle I.F=e^{\int \:P\:dx} {/eq}

Become a Study.com member to unlock this answer! Create your account

Given differential equation is

{eq}\displaystyle xy'=2y+x^3cos\left(x\right)\: {/eq}

==> {eq}\displaystyle...

First-Order Linear Differential Equations

from

Chapter 16 / Lesson 3
2K

In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.


text:
## Algebra: A Combined Approach (4th Edition)

$\dfrac{2x^2-3x-1}{(2x-5)^2}$
$\dfrac{4}{(2x-5)^2}+\dfrac{x+1}{2x-5}$ is a rational expression $\dfrac{4}{(2x-5)^2}+\dfrac{x+1}{2x-5} = \dfrac{4(2x-5)+(x+1)(2x-5)^2}{(2x-5)^2(2x-5)} = \dfrac{(2x-5)[4+(x+1)(2x-5)]}{(2x-5)^2(2x-5)} = \dfrac{4+(x+1)(2x-5)}{(2x-5)^2} = \dfrac{4+2x^2-5x+2x-5}{(2x-5)^2} = \dfrac{2x^2-3x-1}{(2x-5)^2}$


text:
• hartnn
If in certain language, &^* = # #%$= #$*@= % then what is \$@%= ??
Mathematics

Looking for something else?

Not the answer you are looking for? Search for more explanations.


text:
Another challenge from Gynvael's polish stream. Let's see what we've got today.

Mission 006            goo.gl/te47XT                  DIFFICULTY: ███░░░░░░░ [3/10]
┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅┅
Looking through old documents from the sixties, we've noticed this note:

c5 c2 c3 c4 c9 c3 40 82 a8 93 82 a8 40 86 81 91 95 a8 40 87 84
a8 82 a8 40 82 a8 93 40 93 96 87 89 83 a9 95 a8 4b 40 c1 93 85
40 95 89 85 40 91 85 a2 a3 4b

We're thinking that it's some sentence written in Polish, but we couldn't decode it. Now it's jour job to do it. Good luck!

--

If you decode the answer, put it in the comments under this video! If you write

P.S. I'll show/explain the solution on the stream next week.

Firstly, let's take a look at different types of encodings used in the sixties. From this Wikipedia article, we can learn that:

ASCII was introduced in 1963 and is a seven-bit encoding scheme used to encode letters, numerals, symbols, and device control codes as fixed-length codes using integers.

But as we know it's not ASCII, we can skip to the next encoding sheme created also in 1963:

IBM's Extended Binary Coded Decimal Interchange Code (usually abbreviated as EBCDIC) is an eight-bit encoding scheme developed in 1963.

Now we have to check if it's actually this. Looking at this EBCDIC to ASCII table we can see that it will print out polish words, but let's also write a Python script to decode that.

Where cp500 is codec alias for EBCDIC-CP-BE, EBCDIC-CP-CH, IBM500. Running this script we get the message EBCDIC bylby fajny gdyby byl logiczny. Ale nie jest.

This translates to EBCDIC would be cool, if it was logical. But it's not.

Keep learning and stay safe! ~ W3ndige


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# A Proof on Quasiperfect Numbers

1. Mar 4, 2012

### Joseph Fermat

A Quasiperfect number is any number for which the sum of it's divisors is equal to one minus twice the number, or a number where the following form is true,

σ(n)=2n+1

One of the well known and most difficult questions in mathematics is whether such numbers exist at all. I have created a rather interesting proof to show that quasiperfect numbers do not exist. I use a process of transformation to create a situation necessary for the existence of a quasiperfect number, and then show that such a situation is impossible, therefore disproving the possibility of a quasiperfect number.

View attachment On the Nonexistence of Quasiperfect Numbers.pdf

Last edited: Mar 4, 2012
2. Mar 4, 2012

### Norwegian

Why not use the same argument with n=2x+1 to prove that odd numbers do not exist?

3. Mar 5, 2012

### dodo

Hi, Joseph,
there is a problem when going from eq.8 to eq.9: $1 - (h(n) - 2)$ is not $-(h(n)+1)$ (which is negative), but $3 - h(n)$ (which is positive).

4. Mar 5, 2012

### Joseph Fermat

Which would mean that my proof is fallous. Oh, well back to the drawing board. Anyone have any ideas where to go from here. Any help would be appreciated.


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# Weak existence for modified Tanaka SDE

Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE

$dX_t = dW_t + dL_t^0(X_t)$,

where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$. I have two related questions:

1. Does there exist a (weak) solution to the SDE

$dX_t = dW_t + 2 dL_t^0(X_t)$

where again $L_t^0(X_t)$ is the local time of $X_t$ at $0$?

$dX_t = \sqrt{X_t}dW_t + 2 dL_t^1(X_t)$,
where $L_t^1(X_t)$ is the local time of $X_t$ at $1$?