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Open Access

# On the graph $K_{1,n}$ related configurations of subspaces of a Hilbert space

### Abstract

We study systems of subspaces $H_1,\dots,H_N$ of a complex Hilbert space H that satisfy the following conditions: for every index $k > 1$, the set $\{\theta_{k,1},\ldots,\theta_{k,m_k}\}$ of angles $\theta_{k,i}\in(0,\pi/2)$ between $H_1$ and $H_k$ is fixed; all other pairs $H_k$, $H_j$ are orthogonal. The main tool in the study is a construction of a system of subspaces of a Hilbert space on the basis of its Gram operator (the G-construction).

Key words: System of subspaces, Hilbert space, orthogonal projections, Gram operator.

### Article Information

 Title On the graph $K_{1,n}$ related configurations of subspaces of a Hilbert space Source Methods Funct. Anal. Topology, Vol. 23 (2017), no. 3, 285-300 MathSciNet MR3707523 Milestones Received 23/03/2017 Copyright The Author(s) 2017 (CC BY-SA)

### Authors Information

Alexander Strelets
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine

### Citation Example

Alexander Strelets, On the graph $K_{1,n}$ related configurations of subspaces of a Hilbert space, Methods Funct. Anal. Topology 23 (2017), no. 3, 285-300.

### BibTex

@article {MFAT990,
AUTHOR = {Strelets, Alexander},
TITLE = {On the graph $K_{1,n}$ related configurations of subspaces of a Hilbert space},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {23},
YEAR = {2017},
NUMBER = {3},
PAGES = {285-300},
ISSN = {1029-3531},
MRNUMBER = {MR3707523},
URL = {http://mfat.imath.kiev.ua/article/?id=990},
}

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# Zariski surgically dissecting a flawed proof

The problem of bad science has been in the news recently.  See, for instance this article in the New York Times, the website Retraction Watch, or this retracted article in Science.  In any of the sciences, researchers can get a sense that something is amiss with a published paper just by reading it.  Perhaps some circumstantial fact seems unusual.  A statistical analysis can provide some evidence that there really is a problem, either with the data or with the methods.  But in many cases, a researcher would have to try to recreate the experiment and the data to test the results – a daunting task.  Not so in mathematics.  In a mathematical article, everything is there on the page, open for scrutiny.  If there is a problem, the expert reader need only examine the arguments to discover it.  Oscar Zariski does exactly that in his review of an early attempt to prove resolution of singularities (in characteristic zero).  Zariski was famous for the way he increased the rigor in algebraic geometry by increasing the algebra in algebraic geometry.  In his analysis of Derwidué’s paper, he points out early on that the approach is too geometric to work.  Rather than stopping there, Zariski forges on, zeroing in on the weak points of the argument and giving full explanations of their weaknesses.

The real proof of the resolution of singularities in characteristic zero came thirteen years later, in the famous work of Hironaka

MR0042171 (13,67b)
Derwidué, L.
Le problème de la réduction des singularités d’une variété algébrique. (French)
Math. Ann. 123, (1951). 302–330.

The author presents in this paper an investigation which, he claims, constitutes a complete proof of the famous conjecture that every algebraic variety can be transformed birationally into a variety which is free from singularities. He restricts himself to the case of the complex domain. [In this case, and more generally, in the case of ground fields of characteristic zero, the conjecture has been proved so far only for varieties of dimension $\leq 3$; for the case of dimension 3 see the reviewer’s paper “Reduction of the singularities of algebraic three-dimensional varieties,” [Ann. of Math. (2) 45, 472–542 (1944); MR0011006 (6,102f); this paper will be referred to in this review as RS3].] The author has published a number of other papers on the same subject, in particular a long memoir entitled “Le problème général de la réduction des singularités d’une variété algébrique” [Mém. Soc. Roy. Sci. Liège (4) 9, no. 2 (1949); MR0035487 (11,740c)]. The present paper is described in the introduction as a decisive simplification of the proof given in the above cited memoir. It is stated that the simplification is due chiefly to the author’s recent discovery of what he calls “elementary transformations.” [These transformations, under the name of “monoidal transformations,” have been introduced and fully studied by the reviewer on pages 532–542 of his paper (referred to in the sequel as FBC) “Foundations of a general theory of birational correspondences,” [Trans. Amer. Math. Soc. 53, 490–542 (1943); MR0008468 (5,11b)]; they have been extensively used by the reviewer in RS3.] The monoidal (or “elementary”) transformations are indeed elementary, in more than one sense, and while the reviewer has found them a useful tool for the resolution of the singularities, he finds it rather strange that they should have had such a tremendous effect on the author’s “proof,” for the deeper difficulties of the resolution problem can be affected by such straightforward tools only to a very moderate extent. At any rate, it is stated that his discovery of monoidal transformations has enabled the author to condense “en quelques heures” the 139 pages of his cited memoir to five pages of a note entitled “Méthode simplifiée de réduction des singularités d’une variété algébrique” [Acad. Roy. Belgique. Bull. Cl. Sci. (5) 35, 880–885 (1949); MR0035488 (11,740d)]. According to the author, this short note contained the complete solution of the problem, so much so that he felt that “pour moi, le problèmé était bel et bien résolu, mais il me restait à faire admettre ma solution.” The present more elaborate version of that note has therefore been written chiefly for the purpose of convincing the unbelievers, and the author acknowledges his indebtedness to van der Waerden who, in a series of conferences with the author (“… une dizaine de discussions de deux heures chacune, suivies pour chacun de nous de deux ou trois jours de réflection…”), has closely and critically scrutinized every single detail of the present “proof” (“… et il prit dès lors la peine de passer au crible de son esprit critique bien connu les douze pages du texte que je lui apportais, littéralement ligne par ligne”).

Before describing and discussing the author’s reasoning, a few general remarks about the paper will be in order. Its language is “geometric.” The geometric language, when it is not based on a carefully prepared algebraic basis, is never explicit or convincing in algebraic geometry. On this ground alone the author’s “proof” could be dismissed as incomplete, for in scientific work it is right to hold every author guilty until he proves himself innocent. However, out of consideration for the importance of the problem, and because of the author’s implied belief that his work has not been duly evaluated, this reviewer has reversed his attitude and has read the paper on the assumption that it is the reader who is to be held guilty until he proves himself innocent. In practice that meant that in cases of doubtful points, the reviewer has made an effort to either (a) complete the proof himself or (b) find precisely what is wrong with the proof or (and) (c) find a counterexample. With this approach, the geometric language of the author has the effect of shifting a good deal of the burden from the author to the reader, for in many cases it was at least as difficult to accomplish (a) or (b) or (c) as it was for the author to find his incomplete proofs. However, the problem of resolution of singularities presents not only difficulties of rigor but also difficulties of perception which lie much deeper. It is for this reason that the reviewer was willing to give the author the benefit of the doubt on the formal side of the treatment. He was mainly interested in finding out how the author has dealt with the irreducibly difficult core of the problem. He was forced to conclude that lack of perception of the serious (and not obvious) complications that can (and do) arise in the course of the reduction process have prevented the author from coming to grips with the real difficulties of the problem. And so it happens that the two main pillars of his “proof”—(a) the theorem on first polars on p. 314 (“qui est d’une importance capitale pour la suite”) and (b) the “raisonnement fondamental” on p. 317—are represented by statements of which the first is false and the second is far from having been proved (see theorems ${\rm A}’$ and ${\rm B}’$ below; as a matter of fact also ${\rm B}’$, in the general form stated at the bottom of p. 318, is false). As a consequence, the following can be said about the two other statements (see theorems A and B below) which are made by the author and whose logical sum implies the resolution theorem: A has not been proved because its “proof” is based on ${\rm A}’$; B has not been proved because its “proof” is based on A and ${\rm B}’$. On the whole, the author’s contribution merely scratches the surface of things: it belongs to the prenatal phase of the problem, a phase familiar to anybody who has given this problem any thought. (We pass without comment the author’s astonishing remark that his “methods” can be extended “without difficulties” to perfect fields of characteristic $p\neq 0$, it being quite superficially clear that they could not possibly be so extended.)

Let $V$ be a $k$-dimensional irreducible variety in a projective space $S_q$. To resolve the singularities of $V$ one will begin by applying to $S_q$ a monoidal transformation $T$ whose center is a suitable irreducible nonsingular variety $W$ belonging to the singular variety of $V$. The effect of $T$ is as follows: (a) it transforms birationally $S_q$ into a nonsingular variety $B_q{}’$, immersed in some projective space $S'{}_{n’}$; (b) it transforms birationally $V$ into a variety$V’$ immersed in $B_q{}’$; (c) it blows up $W$ into an irreducible $(q-1)$-dimensional subvariety $\Gamma’$ of $B_q{}’$; this variety $\Gamma’$ is free from singularities and carries a ruling of linear spaces $\Gamma_P{}’$ of dimension $q-1-\rho$ (where $\rho=\dim W$) which correspond to the individual points $P$ of$W$ (see FBC and RS3). In a second step of the resolution process, $V’$ and $B_q{}’$ will replace the original $V$ and $S_q$, and one will apply to $B_q{}’$ a monoidal transformation $T’$ whose center is a suitable irreducible nonsingular variety $W’$ belonging to the singular variety of $V’$ (the steps just described are identical to those used in RS3). We shall therefore assume that we are given, to begin with, a nonsingular $q$-dimensional variety $B\ (q\geq k+1)$ containing $V$and contained in some projective space $S_n\ (n\geq q)$. If $k=n-1$ (whence $B=S_n$) it is well known what is meant by a first polar $F$ of $V$. If $k<n-1$, a first polar of $V$ is defined as a first polar of any hypercone which projects $V$ from an $S_{n-k-2}$. The set $[F]$ of first polars of $V$ is an irreducible algebraic system (linear, if $k=n-1$); its base locus is the singular variety of $V$. The trace of $[F]$ in $B$, i.e., the set of intersections $F\cap B$, is an irreducible algebraic system $[\Phi]$ of $(q-1)$-dimensional subvarieties of $B$, and also the base locus of $[\Phi]$ is the singular variety of $V$. If $V$ is a monoidal transformation of$B$, of the type described above, one defines in an obvious fashion the $T$-transform $[\Phi’]$ of the system $[\Phi]$; this will be then an irreducible algebraic system of $(q-1)$-dimensional subvarieties of $B’\ (=B_q{}’)$ (since $W$ is a base variety of $[\Phi]$, particular members of $[\Phi’]$ may contain $\Gamma’$ as a component, to a suitable multiplicity). This process is to be repeated as long as the successive transforms $V’,V”,\cdots$ of $V$ continue to have singularities. One is thus led to a sequence $\{V^{(i)},B^{(i)},[\Phi^{(i)}]\}$. At each stage, the next, $(i+1)$th member of the sequence depends on the choice of the subvariety$W^{(i)}$ of the singular variety of $V^{(i)}$. For the purposes of this review it must be clearly understood that $[\Phi’]$ no longer has the same relationship to $V’$as $[\Phi]$ has to $V$, i.e., $[\Phi’]$ is not the $B’$-trace of the system of first polars of $V’$ (in relation to the ambient projective space $S'{}_{n’}$ of $B’$). The author then proposes to prove the following two theorems: A. Every singular point of $V^{(i)}$ is a base point of $[\Phi^{(i)}]$. B. By a suitable choice of the centers$W^{(i)}$ of the successive monoidal transformations $T^{(i)}$ it is possible to obtain, after a finite number of steps, a transform $V^{(i)}$ of $V$ such that the system $[\Phi^{(i)}]$ has no base points.

It is clear that A and B together would suffice to establish the theorem of resolution of singularities. [The idea of using the first polars and studying their behavior under quadratic transformations (in the case of algebraic surfaces) goes back to B. Levi, and was used in one form or another in every investigation dealing with theresolution problem. It may be noted that theorem B has been proved by the reviewer in the case $q=3$, for arbitrary systems $[\Phi]$ of surfaces; see RS3, Theorem$7’$, p. 531.] The essential ingredient of the author’s “proof” of A is the following theorem: ${\rm A}’$. The generic first polar $F$ of $V$ has an $(s-1)$-fold point at every (proper or improper) $s$-fold point of $V$ (“improper” means “infinitely near”). Or, in less mysterious language: if a (proper) point $O^{(i)}$ of $V^{(i)}$ is$s$-fold for $V^{(i)}$, then it is $(s-1)$-fold for the generic $\Phi^{(i)}$. The “proof” of ${\rm A}’$ is given only under the following special conditions: $k=n-1$and the monoidal transformations used in the reduction process are all locally quadratic, i.e., their centers are points. The general case is dismissed in two lines by the mere statement that the proof is similar. Now while we shall see in a moment that ${\rm A}’$ is false even under these special conditions, it may be explained now that the general case presents additional difficulties which the author does not perceive. If the center $W$ of $T$ is of positive dimension then it may well happen that$V’$ will contain a generator $\Gamma_P{}’$ of $\Gamma’$ which corresponds to some special point $P$ of $W$. Whatever conclusions one arrives at in the special case of locally quadratic transformations, these conclusions cannot be automatically used, in the general case, for making statements concerning the behavior of $V’$at $\Gamma_P{}’$; they only give information about the behavior of $V’$ at subvarieties $W’$ which correspond to the whole of $W$ (and not to proper subvarieties of $W$). [For a method of dealing with this phenomenon, see RS3.] Furthermore, if $k<n-1$ then the polars $\Phi$ are defined in terms of the general projections of$V$ (in its ambient space $S_n$), and in that case it is not at all clear how the transforms $\Phi’$ of $\Phi$ are related to the polars of the general projections of $V’$(in its ambient space $S'{}_{n’}$) and why this matter should have been dismissed in a few words.

We now come to “theorem” ${\rm A}’$ and accept the special conditions under which the author “proves” it. It is not difficult to give counterexamples, even in the simplest case of plane algebraic curves $(k=1,q=n=2)$, for “theorem” ${\rm A}’$ is “almost always” false. Here is an example. Let $V$ be the plane curve $f(x,y)=0$, where $f(x,y)=y^3-x^5$. Then $V$ has a triple point at the origin $O$ and it has a double point at the improper point $O’$ infinitely near $O$ in the direction $y=0$. By the locally quadratic transformation $T\colon x=x’$, $y=x’y’$, the curve $V$ is transformed into the curve $V’$ given by the equation $x^{\prime 2}-y^{\prime 3}=0$, the improper point $O’$ now being represented by the origin $x’=y’=0$. The generic polar $\Phi$ of $V$ is $uf_x{}’+vf_y{}’+w(xf_x{}’+yf_y{}’-5f)=0$, i.e.,$3vy^2-5ux^4-2wy^3=0$. The $T$-transform of $\Phi$ is $3vy^{\prime 2}-5ux^{\prime 2}-2wx’y^{\prime 3}=0$, and this has at $O’$ a singular point of multiplicity 2 (and not 1, as is claimed in ${\rm A}’$). The error in the author’s proof is the following (we use his notations, pp. 313–314): it is taken for granted that if $P$ is generic with respect to $G$, then $G’$ will be such that also $P’$ will be generic with respect to $G’$. The fact is that in most cases $P’$ will not be generic with respect to $G’$. The author commits here the same error that has been committed once before in the “proof” of the following incorrect statement: “the composition of the singularity of a generic projection of an algebraic curve in $S_3$ is the same as the composition of the corresponding singularity of the space curve itself.” The error in this reasoning has been pointed out long ago by the reviewer (see p. 12 of the reviewer’s book “Algebraic Surfaces” [Springer, Berlin, 1935]). From the moment that $P’$ is not necessarily generic with respect to $G’$ and hence $H_1$ is not necessarily a generic first polar of $G’$, the multiplicity of $H_1$ at$O_1{}’$ may be greater than $s_1-1$, the lemma on pencils of hypersurfaces given on p. 313 is then not applicable, and the entire proof on p. 314 breaks down. What is even more serious is that as soon as the generic $\Phi’$ has at $O_1{}’$ a point of multiplicity $\geq s_1$, then in a second quadratic transformation the fundamental surface $\Pi”$ will detach itself at least $s_1$ times from the transform of $\Phi’$ while it will detach itself precisely $s_1-1$ times from the transform of the generic polar of $G’$. Thus already after two steps we find that further investigation would be necessary before one can make any assertion concerning the multiplicities of $\Phi”$ at the singular points of $G”$, as compared with the multiplicities which the first polars of $G”$ have at these points. What happens after $i$steps is altogether nebulous.

We now pass to theorem B. Part of the “proof” of this theorem, for dimension $k$, presupposes theorem A for all dimensions $\leq k-1$, and hence already on this ground the proof is incomplete. That part of the “proof” which is independent of theorem A is based on a theorem which the author states under the heading: “raisonnement fondamental” (p. 317). We shall explain the gap in the author’s proof, and for simplicity we shall consider the case $k=3$, $q=n=4$. We shall also assume for simplicity that the singular variety of $V$ (and hence also the base locus of $[\Phi]$) is of dimension 1. Let $[\Phi_1]$ be the characteristic system of $[\Phi]$, i.e., the systems of surfaces which are intersections of pairs of $\Phi$‘s. We shall also assume that the characteristic system $[\Phi_2]$ of $[\Phi_1]$ (from which the base curves of $[\Phi_1]$ have been deleted) has no base points (in the notation of the author, we have assumed $\sigma=0$, $k_\sigma=1$). Let $\gamma$ be an irreducible component of the base curve of $[\Phi_1]$. We assume that $\gamma$ is free from singularities. We take now the curve $\gamma$ as the center of our monoidal transformation $T$ and we denote by $[\Phi_1{}’]$ the $T$-transform of the system of surfaces $[\Phi_1]$. The “raisonnement fondamental,” in this special case, is the following assertion: ${\rm B}’$. Let $\gamma’$ be an irreducible curve in $\Gamma’$ which belongs to the base curve of the system $[\Phi_1{}’]$. If the curve $\gamma$ is $s$-fold for the generic $\Phi$ and $t$-fold for the generic $\Phi_1$ and if $m$ is the intersection multiplicity of $\Phi$ and $\Phi_1$ at $\gamma$, then the intersection multiplicity at $\gamma’$ of a generic $\Phi’$ and a generic $\Phi_1{}’$ is $\leq m-st$ (whence, at any rate, $<m$).

In the proof the author takes it for granted that $\gamma’$ meets each generator $\Gamma_P{}’$ of $\Gamma’\ (P\in\gamma)$. In other words, he assumes that $\gamma’$ corresponds to the whole of $\gamma$. He does not consider the possibility that $\gamma’$ may correspond to some special point $P$ of $\gamma$ (in which case $\gamma’\subset\Gamma_P{}’$ and $\gamma’$ does not meet $\Gamma_Q{}’$ if $Q\neq P$). To such a curve $\gamma’$ the author’s reasoning is not applicable. The reviewer has asked himself the following question: May it be that the presence of a base curve $\gamma’$ contained in $\Gamma_P{}’$ implies that $P$ is a base point of $[\Phi_2]$ (contrary to the assumption that $[\Phi_2]$ has no base points)? If that is so, it must be proved. The reviewer doubts that that is so. He knows that the above possibility can actually arise in the case of a linear system of hypersurfaces $[\Phi]$ in $S_4$ whose successive characteristic systems $[\Phi_1]$ and $[\Phi_2]$ satisfy all the conditions (as to their base varieties) stated above; and he does not see why the fact that we are dealing with polar hypersurfaces should make any difference. At any rate, the author correctly remarks that his reasoning makes no use of any special properties of the polars $\Phi$ and that even the fact that the $\Phi$‘s are complete intersections plays no role in the proof. He therefore reformulates his “raisonnement fondamental” as a general theorem concerning arbitrary pairs of varieties $\Phi$ and $\Phi_1$ in B, of dimensions $q-1$ and $q’$ respectively, $q’\leq q-1$ (see end of p. 318). Now this generalization is definitely false, even if $\Phi$ and $\Phi_1$ are complete intersections. Consider, for instance, the following case: $\Phi$ and $\Phi_1$ are respectively a hypersurface and a surface in $S_4$, having a common point $P$ which is an ordinary $n$-fold point of $\Phi$ and an ordinary $\nu$-fold point of $\Phi_1\ (n>1,\nu>1)$; we also assume that $\Phi$ and $\Phi_1$ have in common a line $\gamma$ through $P$ and that the intersection multiplicity at $\gamma$ is 1 (whence $m=s=t=1$). If $\gamma$ is taken as the center of our monoidal transformation $T$, then the following facts can be established in a straightforward manner: (a) the plane $\Gamma_P{}’$ belongs to the three-dimensional variety $\Phi’$, the latter passing through $\Gamma_P{}’$ with multiplicity $\geq n-1$; (b) $\Gamma_P{}’$ meets $\Phi_1{}’$ in a curve $\gamma’$ which is at least $(\nu-1)$-fold for $\Phi_1{}’$. It follows that the intersection multiplicity of $\Phi’$ and$\Phi_1{}’$ at $\gamma’$ is $\geq(n-1)(\nu-1)\geq 1>0=m-st$. This example shows in fact that the intersection multiplicity of $\Phi’$ and $\Phi_1{}’$ at a curve $\gamma’$ belonging to a generator $\Gamma_P{}’$ of $\Gamma’$ may have no particular relationship to either $m-st$ or $m$.

Reviewed by O. Zariski

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I am trying to implement autonomous navigation using 2 lasers and a depth camera. IMU and odometry data will also be available. Basically, I'd like to have the simulated robot spawned in some gazebo world and autonomously drive around it covering all of the floor space.

So far, I've come to the conclusion that using cartographer is best for me because it allows 2 lasers to be directly used without additional tools. However, before trying this I'd played around with gmapping and AMCL, where I'd do the following as a starting point for autonomous exploration:

1. Generate a map by manually moving around
2. Save the map and serve it
3. Localise the robot on the map
4. Provide waypoints to move_base

My question is, what is the order of operations when using cartographer? In my launch file would it just be a cartographer and move_base node and then something else to for frontier exploration? I suppose I'm a bit confused whether I need to replicate the workflow used above with gmapping and AMCL or whether the single cartographer node handles both of these roles in one step.

I'm really getting stuck with some of the tutorials I see online for this as I'm unsure how to adapt them to my use case. I've looked at here, which seems handy but it doesn't work for me as it doesn't do anything when I graphically set a 2DNavGoal in RViz.

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Hi, it seems cartographer can work as SLAM as well as pure localization node. see: Localization only link text

Depending on what your waypoint generating node or usecase works better on, you can either SLAM-navigate, or do mapping and afterwards navigate/plan.

I dont know this tutorial, but when workinng in localization mode one would asume cartographer to demand an initial pose (can be set in RViz or might be served by other means). Some localization nodes dont even start to work before they get an initial pose. In SLAM mode you just start at 0/0 in most cases.

As you run Gabzebo I assume you are running it all on the same computer, otherwise the firewall can hinder proper communication.

Anything else is up and running clean? All depending Jackal packages installed?

more

Thanks for the message! Everything running is on my computer as you suggested. The only bits I'm confident in are the simulated robot in a Gazebo world and a keyboard teleoperation node with successful laser data visualisation in RViz. I can also save .yaml and .pgm map files generated using cartographer and serve them but am unsure whether this is necessary and what the next steps should be without using the AMCL approach before move_base to explore the mapped area.

( 2020-05-27 11:46:17 -0500 )edit

Well, cartographer in pure localization mode as linked in my answer makes cartographer do what amcl does (computationally different but with "the same" outcome). If the map is served to cartographer in this mode (and scans) it should localize and corret odometry, giving move_base a clue where the robot is. In tf, map -> odom would be published by cartographer instead of amcl, and odom -> base_link by odometry as before. If you just droped amcl and launched cartographer its likely that you have some remapping of topics to do, linking node outputs to node inputs and defining frames. By the way dont forget to not! launch the teleop node when move_base comes into play as this would give you publisher conflict on the cmd_vel topic, result is likely nothing happens.

( 2020-05-28 08:27:13 -0500 )edit


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# opening error

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## opening error

 This post was updated on . I installed featool but at first startup I get the following error. I wonder what is the reason.I am using ubuntu 18.04 and matlab 2017a. Starting the FEATool Multiphysics GUI. Loading resources can take a minute . . . . Error using figure uIcontrolfontunits is an invalid class name Error in start_featool_gui Error in start_featool_gui Error in FEAToolMultiphysicsApp/startApp (line 88)                 eval('start_featool_gui'); Error in FEAToolMultiphysicsApp (line 48)             startApp(obj) Error in appinstall.internal.runapp>execute (line 78) out = evalin('caller', [script ';']); Error in appinstall.internal.runapp>runapp13a (line 57) outobj = execute(fullfile(appinstalldir, [wrapperfile 'App.m'])); Error in appinstall.internal.runapp>runcorrectversion (line 36)     appobj = runapp13a(appinstalldir); Error in appinstall.internal.runapp (line 18)         out = runcorrectversion(appmetadata, appentrypoint, addonmetadatadir);   >>
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## Re: opening error

 Administrator This post was updated on . Unfortunately it is not clear what has gone wrong, as this error MURAT wrote Error using figure: uIcontrolfontunits is an invalid class name does not really tell anything about what or where the issue could be. You can possibly try with another Matlab version (or check with the Mathworks about this error). Or you could try and see if starting FEATool manually helps, to do this follow the steps below: 1. Download the archive for the latest release from https://github.com/precise-simulation/featool-multiphysics/releases/latest2. Unpack/unzip it to a folder on your computer (note this folder as path_to_featool_multiphysics_folder). 3. (Re-)start Matlab. 4. Run the following commands in the Matlab terminal (substitute path_to_featool_multiphysics_folder with the actual path to where you have downloaded and extracted the FEATool distribution).  >> cd path_to_featool_multiphysics_folder >> featool Moreover, you can test to check if it is a pure GUI issue or overall installation issue my running the test suite without the GUI (only running the m-script file models). To do this launch the FEATool test suite as  >> featool testm
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## Re: opening error

 Copy the new downloaded version under which folder and open the zip file.
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## Re: opening error

 Warning: Initializing MATLAB Graphics failed. This indicates a potentially serious problem in your MATLAB setup, which should be resolved as soon as possible.  Error detected was: MATLAB:hg:InvalidProperty There is no screenpixelsperinch property on the Root class. > In hgrc (line 163)   In initprefs (line 11)   In matlabrc (line 137) When matlab first opens, I get this error in the command line.
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## Re: opening error

 >> cd path_to_featool_multiphysics_folder Error using cd Cannot CD to path_to_featool_multiphysics_folder (Name is nonexistent or not a directory).
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## Re: opening error

 Administrator This post was updated on . In reply to this post by MURAT MURAT wrote When matlab first opens, I get this error in the command line. Warning: Initializing MATLAB Graphics failed. This indicates a potentially serious problem in your MATLAB setup, which should be resolved as soon as possible.  Error detected was: MATLAB:hg:InvalidProperty This seems to be a Matlab related installation error or issue, in which case you had better contact the Mathworks who develop and license Matlab for support and their advice.
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## Re: opening error

 Administrator In reply to this post by MURAT MURAT wrote >> cd path_to_featool_multiphysics_folder Error using cd Cannot CD to path_to_featool_multiphysics_folder (Name is nonexistent or not a directory). The path_to_featool_multiphysics_folder refers to the folder which you extracted/unzipped the latest "featool-multiphysics.zip" release file to (you must replace this path with the one on your system).
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## Re: Octave install FeaTool

 This post was updated on . hi i will upload for octave. I could not fix the problem in matlab. Can you describe in detail how to install featool into Octave. FEATool-FEniCS, after FEniCS is installed, how to set up featool ???? Can you explain in detail.
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## Re: Octave install FeaTool

 Can I install or run the featools 1.12 .zip package in ocrave.???


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#### Volume 11, issue 1 (2007)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
On the automorphism group of generalized Baumslag–Solitar groups

### Gilbert Levitt

Geometry & Topology 11 (2007) 473–515
 arXiv: math.GR/0511083
##### Abstract

A generalized Baumslag–Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that $Out\left(G\right)$ either contains non-abelian free groups or is virtually nilpotent of class $\le$2. It has torsion only at finitely many primes.

One may decide algorithmically whether $Out\left(G\right)$ is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out$\left(G\right)$ virtually nilpotent.

If $G$ is unimodular (virtually ${F}_{n}×ℤ$), then $Out\left(G\right)$ is commensurable with a semi-direct product ${ℤ}^{k}⋊Out\left(H\right)$ with $H$ virtually free.

##### Keywords
Baumslag–Solitar, automorphisms, graphs of groups
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20E08, 20F28
##### Publication
Published: 16 March 2007
Proposed: Martin Bridson
Seconded: Walter Neumann and Wolfgang Lueck
##### Authors
 Gilbert Levitt Laboratoire de Mathématiques Nicolas Oresme UMR 6139 BP 5186 Université de Caen 14032 Caen Cedex France


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Why are diastereomers optically active?

Oct 30, 2014

Many diastereomers are optically active, but many are not.

Explanation:

By definition a diastereomer is any stereoisomer that is not an enantiomer

Consider the possible optical isomers of 2,3-dichlorobutane.

There are two chiral carbons, so there are ${2}^{2} = 4$ possible optical isomers.

However, two of the structures are identical. They are the same meso compound. So there are only three isomers.

Both of the enantiomers are diastereomers.

In each case, the meso compound is not optically active, while its diastereomeric partner is optically active.

It is even possible to have diastereomeric pairs in which neither member is optically active.

Consider the pentose alcohols, ribitol

and xylitol.

They are diastereomers of each other, but they each have an internal plane of symmetry.

They are both meso compounds, and they are both optically inactive.


text:
usable graphing calculator: Try the Free Math Solver or Scroll down to Resources! Displaying top 8 worksheets found for - Simplifying Radicals With Fractions. ]. Q2_Wk5_127_-_1211__Simplifying_Radicals - Q2 Wk5 12\/7 12\/11 Simplifying Radicals 1 4 9 16 25 36 49,64 81 100 121 144 169 \u2026 U2.1 Simplifying In beginning algebra, we typically assume that all variable expressions within the radical are positive. Simplify: √252. IXL Learning Learning. Sign in. Found worksheet you are looking for? Improve your math knowledge with free questions in "Simplify radical expressions with variables I" and thousands of other math skills. sample lesson plan for radical expressions. graphing linear equations with a domain and range. Many algebraic expressions contain radicals. Sign in now. Since a negative number times a negative number is always a positive number, you need to remember when taking a square root that the answer will be both a positive and a negative number or expression. Algrebra online, Problem 1 : Simplify : √(16u 4 v 3) Problem 2 : Simplify : √(147m 3 n 3) Problem 3 : Simplify : 3 √(125p 6 q 3) Problem 4 : Simplify : 4 √(x 4 /16) Problem 5 : Simplify : 6√(72y 2) Problem 6 : Simplify : √ (196a 6 b Simplifying radical expressions: three variables. National curriculum . Simplify the expressions both inside and outside the radical by multiplying. , you have to take one term out of fourth root for every four same terms multiplied inside the radical. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. They will use their answers to solve the joke/riddle. factors to , so you can take a out of the radical. ], In this section, you will learn how to simplify radical expressions with variables. Come to Mathfraction.com and master radical, common factor and lots of additional math subjects Year 10 V.4 Simplify radical expressions with variables . Use the following rules to enter expressions into the calculator. The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial. , you have to take one term out of the square root for every two same terms multiplied inside the radical. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Improve your maths skills by practising free problems in 'Simplify radical expressions with variables' and thousands of other practice lessons. Improve your math knowledge with free questions in "Simplify radical expressions with variables I" and thousands of other math skills. Like Radicals : The radicals which are having same number inside the root and same index is called like radicals. For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero. The radicand contains no fractions. We just have to work with variables as well as numbers. For , there are pairs of 's, so goes outside of the radical, and one remains underneath the radical. Simplifying Square Roots that Contain Variables. Simplifying Radical Expressions with Variables. Simplifying Radical Expressions with Variables When radicals (square roots) include variables, they are still simplified the same way. No radicals appear in the denominator. Displaying top 8 worksheets found for - Simplifying Radicals Without Variables. In this lesson, we are going to take it one step further, and simplify square roots that contain variables. Combine the radical terms using mathematical operations. Explanation: . solve radical expression [ Def: A mathematical phrase made up of numbers and/or variables, connected with operational signs. We can add and subtract like radicals only. From simplifying expressions with integers to equations in two variables, we have every aspect included. My students loved this activity as it's a fun twist on an important concep . Simplifying rational exponent expressions: ... Let's do another one of these. Simplifying Radical Expressions A radical expression is composed of three parts: a radical symbol, a radicand, and an index In this tutorial, the primary focus is on simplifying radical expressions with an index of 2. Simplifying Square Roots (Review) Let's review the steps involved in simplifying square roots: Factor the number inside the square root sign. I would start by doing a factor tree for , so you can see if there are any pairs of numbers that you can take out. , you have to take one term out of cube root for every three same terms multiplied inside the radical. Simplifying radicals with variables is a bit different than when the radical terms contain just numbers. Start off by having students list the perfect squares on their paper by taking each natural number and squaring it. Simplifying radical expressions This calculator simplifies ANY radical expressions. Awards . If a radical is fully simplified, there is no factor of the radicand that can be written as a power greater than or equal to the index, there are no fractions under the radical sign, and there are no radicals in the denominator. Take a look at the following radical expressions. Simplifying radicals with variables is a bit different than when the radical terms contain just numbers. If you have fourth root (4â), you have to take one term out of fourth root for every four same terms multiplied inside the radical. The radicals which are having same number inside the root and same index is called like radicals. Simplifying Radical Expressions with Variables Worksheet - Concept ... Variables and constants. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. Word problems on mixed fractrions. Simplifying Radicals with Variables. Simplifying square roots of fractions. ]. But for radical expressions, any variables outside the radical should go in front of the radical, as shown above. how to factor by grouping polynomials , Grades: 9 th, 10 th, 11 th, 12 th. When radicals (square roots) include variables, they are still simplified the same way. Radical Expressions Simplifying Radicals Worksheets - No Variables Simplifying Radicals Worksheets Radical Form to Exponential Form Worksheets Exponential Form to Radical Form Worksheets Adding Subtracting Multiplying Radicals Worksheets Dividing Radicals Worksheets Algebra 1 Algebra 2 Square Roots Radical Expressions Introduction Topics: We present examples on how to simplify complex fractions including variables along with their detailed solutions. Example 1. Simplify radical expressions containing variables. This type of radical is commonly known as the square root. Some of the worksheets for this concept are Grade 9 simplifying radical expressions, Grade 5 fractions work, Radical workshop index or root radicand, Dividing radical, Radical expressions radical notation for the n, Simplifying radical expressions date period, Reducing fractions work 2, Simplifying rational expressions. Radical Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Scientific Notation Arithmetics Free Download of IT Companies Aptitude Test Papers, Special care must be taken when simplifying radicals containing variables. Simplify by multiplication of all variables both inside and outside the radical. â18 + â8  =  â(3 â 3 â 2) + â(2 â 2 â 2), â(16u4v3)  =  â(4 â 4 â u2 â u2 â v â v â v), â(147m3n3)  =  â(7 â 7 â 3 â m â m â m â n â n â n), 3â(125p6q3)  =  3â(5 â 5 â 5 â p2 â p2 â p2 â q â q â q), 4â(x4/16)  =  4â(x â x â x â x) / 4â(2 â 2 â 2 â 2), 6â(72y2)  =  6â(6 â 6 â 2 â y â y), â(196a6b8c10)  =  â(14 â 14 â a3 â a3 â b4 â b4 â c5 â c5). example. Problem. Improve your skills with free problems in 'Simplify radical expressions with variables' and thousands of other practice lessons. Find the prime factors of the number inside the radical. 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Writing and evaluating expressions. If a factor appears twice, cross out both and write the factor one time to the left of the square root sign. Teach your students everything they need to know about Simplifying Radicals through this Simplifying Radical Expressions with Variables: Investigation, Notes, and Practice resource.This resource includes everything you need to give your students a thorough understanding of Simplifying Radical Expressions with Variables with an investigation, several examples, and practice problems. Example 1. A radical expression is considered simplified when there are no perfect root factors left in the radical. mathematics trivia, Here are the steps required for Simplifying Radicals: Step 1: Find the prime factorization of the number inside the radical. Radical expressions are written in simplest terms when. Free Algebraic Calculator Online, Factor the number into its prime factors and expand the variable (s). I use this lesson as part of an algebra 1 u Decompose the number inside the radical into prime factors. Recommendations Recs . 1) 125 n 5 5n 2) 216 v 6 6v 3) 512 k2 16 k 2 4) 512 m3 16 m 2m 5) 216 k4 6k2 6 6) 100 v3 10 v v 7) 80 p3 4p 5p 8) 45 p2 3p 5 9) 147 m3n3 7m ⋅ n 3mn 10) 200 m4n 10 m2 2n 11) 75 x2y 5x 3y 12) 64 m3n3 8m ⋅ n mn 13) 16 u4v3 4u2 ⋅ v v 14) 28 x3y3 2x ⋅ y 7xy-1-©x 32w0y1 j2f 1K Ruztoa X mSqo 0fvt Kwnayr GeF DLuL ZCI. Simplify square roots that contain variables in them, like √(8x³) If you're seeing this message, it means we're having trouble loading external resources on our website. Simplifying Radical Expressions with Variables Worksheet - Concept - Problems with step by step explanation. Simplifying Radical Expressions Date_____ Period____ Simplify. We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . How do you do this? solving a second order differential equation [ Def: A mathematical sentence containing an equal sign. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Some of the worksheets for this concept are 1 simplifying square roots, Simplifying square roots work, Simplifying square roots examples, Radicals, Square roots and other radicals, Math 6 notes name, Radicals, Simplifying variable expressions. Sign in Remember. If you are looking to simplify square roots that contain numerals as the radicand, then visit our page on how to simplify square roots.. One step equation word problems. In this example, we simplify √(60x²y)/√(48x). And this one is a little more interesting, 'cause we are starting to involve a variable, we have the variable w, but it's really going to be somewhat the same process. Look at the two examples that follow. How to Write a Decimal as a Mixed Number [ Def: A number written as a whole number and a fraction. SIMPLIFYING RADICAL EXPRESSIONS WITH VARIABLES WORKSHEET . Some of the worksheets for this concept are Grade 9 simplifying radical expressions, Radical workshop index or root radicand, Simplifying variable expressions, Simplifying radical expressions date period, Algebra 1 common core, Radicals, Unit 4 packetmplg, Radical expressions radical notation for the n. Writing out the complete factorization would be a bore, so I'll just use what I know about powers. Variables. If you have square root (â), you have to take one term out of the square root for every two same terms multiplied inside the radical. Simplifying radicals containing variables. Unlike Radicals : Unlike radicals don't have same number inside the radical sign or index may not be same. Find the slop caculator, free 7th grade english worksheets, free printouts for teachers maths sheets, online rational calculator, factor polynomials online calculator, TI 84 roms. Thew following steps will be useful to simplify any radical expressions. Improve your math knowledge with free questions in "Simplify radical expressions involving fractions" and thousands of other math skills. To simplify radicals, rather than looking for perfect squares or perfect cubes within a number or a variable the way it is shown in most books, I choose to do the problems a different way, and here is how. Some of these radicals can be simplified prior to simplifying the entire algebraic expression. We just have to work with variables as well as numbers You can use the Mathway widget below to practice simplifying fractions containing radicals (or radicals containing fractions). Simplify expressions with square roots that contain variables; Square Roots and the Order of Operations. Key Takeaways Key Points. We can add and subtract like radicals only. Example 1: to simplify $(\sqrt{2}-1)(\sqrt{2}+1)$ type (r2 - 1)(r2 + 1) . The same general rules and approach still applies, such as looking to factor where possible, but a bit more attention often needs to be paid. ]. These radicals may contain numbers and/or variables. These perfect squares will be key to successfully mastering this concept. I know it has something to do with absolute value because x might be negative and you don't know if it is and you can't have a negative radicand, but I don't know how you DO it. Then click the button and select "Simplify" to compare your answer to Mathway's. We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . The same general rules and approach still applies, such as looking to factor where possible, but a bit more attention often needs to be paid. Simplifying square roots of fractions. Subjects: Math, PreCalculus, Algebra 2. To simplify radicals, I like to approach each term separately. Improve your maths skills by practising free problems in 'Simplify radical expressions with variables' and thousands of other practice lessons. ]. 252 = 2 x 2 x 3 x 3 x 7. ], Convert Decimal To Fraction [ Def: A number that names a part of a whole or a part of a group. I would start by doing a factor tree for , so you can see if there are any pairs of numbers that you can take out. Simplifying Radical Expressions. Simplifying Radicals with Variables - Google Form & Video Lesson! ... Word problems on fractions. This allows us to focus on simplifying radicals without the technical issues associated with the principal nth root. Simplifying Radical Expressions with Variables When you need to simplify a radical expression that has variables under the radical sign, first see if you can factor out a square. Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction. Simplifying Complex Fractions With Variables Worksheets Posted on January 15, 2020 January 15, 2020 by Myrl Simplifying Complex Fractions Worksheet & algebra 2 worksheets dynamically created algebra 2 worksheets. 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If you are looking to simplify square roots that contain numerals as the radicand, then visit our page on how to simplify square roots. In this section, you will learn how to simplify radical expressions with variables. ]. Always put everything you take out of the radical in front of that radical (if anything is left inside it). how to do cubic root on TI-84 silver plus. Solving linear equations using elimination method. Simplifying Exponential Expressions. Example 1 Add the fractions: $$\dfrac{2}{x} + \dfrac{3}{5}$$ Solution to Example 1 Simplify . Learning; Diagnostic; Analytics; Membership. Exponents are supported on variables using the ^ (caret) symbol. When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. word problems about coin problem with exaples. English. finding the scale factor [ Def: The ratio of a distance measured on a scale drawing to the corresponding distance measured on the actual object. Displaying top 8 worksheets found for - Simplifying Radicals With Variables. Simplify square roots (radicals) that have fractions In these lessons, we will look at some examples of simplifying fractions within a square root (or radical). Some of the worksheets for this concept are grade 9 simplifying radical expressions radical workshop index or root radicand simplifying variable expressions simplifying radical expressions date period algebra 1 common core radicals unit 4 packetmplg radical expressions radical notation for the n. 25 scaffolded questions that start out relatively easy and end with some real challenges. Convert Decimal To Fraction [ Def: A number that names a part of a whole or a part of a group. If you have any feedback about our math content, please mail us : You can also visit the following web pages on different stuff in math. 1000 aptitude questions with solutions download. Example. Move only variables that make groups of 2 or 3 from inside to outside radicals. Exponents. Explanation: . I'm not so fluent in arithmetic. Simplifying Radical Expressions: Students are asked to simplifying 18 radical expressions, some containing variables and negative numbers (there are 3 imaginary numbers). Rationalizing Radicals; Simplifying Exponential ... we are assuming that variables in radicals are non-negative, and denominators ... And most teachers will want you to rationalize radical fractions, which means getting rid of radicals in the denominator. Combination rules +algerbra, simplifying square roots fractions, finding the points of intersection algebraically absolute value functions. Maths. We simplify any expressions under the radical sign before performing other operations. Some of the worksheets below are Simplifying Radical Expressions Worksheet, Steps to Simplify Radical, Combining Radicals, Simplify radical algebraic expressions, multiply radical expressions, divide radical expressions, Solving Radical Equations, Graphing Radicals, … Once you find your worksheet(s), you can either click on the pop-out icon or download button to print or download … maths homework cheater for functions, https://study.com/.../simplifying-radical-expressions-with-variables.html Solution. Simplifying Radicals with Variables. Special care must be taken when simplifying radicals containing variables. Content Continues Below. If you have cube root (3â), you have to take one term out of cube root for every three same terms multiplied inside the radical. Simplifying Radical Expressions with Variables When you need to simplify a radical expression that has variables under the radical sign, first see if you can factor out a square. Join now More. The index is as small as possible. To simplify radical expressions, look for factors of the radicand with powers that match the index. Number that names a part of a series teaching radicals to students before they learn right. Simplify by multiplication of all examples and then the expression is simplified we have every part covered website... If anything is left inside it ) simplify expressions with variables skills with free questions in  ''! 'S a fun twist on an important concep are still simplified the way! That the domains *.kastatic.org and *.kasandbox.org are unblocked series teaching radicals to students before learn! This product includes: ( 1 ) factor the number inside the square root sign are supported on variables the! Into its prime factors of the radical by multiplying sure that the domains *.kastatic.org and *.kasandbox.org unblocked! And Write the factor one time to the left of the number inside the.... And outside the radical some radicals needs to be estimated in order to the. Enter expressions into the calculator whole or a part of a group the inside... To compare your answer to Mathway 's come to Mathfraction.com and master radical, and one underneath..., so you can take a out of the examples below, are... Compare your answer to Mathway 's roots and the order of operations to simplify any expressions under radical! Both and Write the factor one time to the left of the radical, common factor and lots of math... 'Simplify radical expressions involving fractions '' and thousands of other math skills radical if! Of numbers and/or variables, connected with operational signs Decimal as a variable in it performing operations! Right triangles in geometry the radicand ( the numbers/variables inside the root and same is. Simplifying square roots with variables I '' and thousands of other practice.! A mathematical phrase made up of numbers and/or variables, connected with signs. 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text:
## By Pavel Panchekha, Jeffrey Prouty

Share under CC-BY-SA.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

# Problem Set 9 (Given 01/08/12)

## Level I

1. Find the three partial derivatives $$\frac{\partial}{\partial x}$$, $$\frac{\partial}{\partial y}$$, and $$\frac{\partial}{\partial z}$$ of $$f(x, y, z) = xyz + x^2 y + y^2 z + z^2 x - xy - yz - xz$$.
2. Find the path integral along the unit circle in the $$xy$$ plane of $$F(x, y, z) = (x + 3) {\hat x} + (y + 2) {\hat y} + (z - 1) {\hat z}$$.
3. Convince yourself that work is the path-integral of force. Formalize what this means.

## Level II

1. Find the electric field inside and outside a shell of charge. This is often referred to as the "conducting sphere" case — "conducting" objects only have charge on their boundary.
2. Find the electric field inside a plane of finite thickness (not infinitely thin).
3. We discussed in class that when charge is uniformly distributed throughout a sphere, the electric field strength increases linearly as distance from the center increases (while you are within the sphere). Could you, and if yes how could you, distribute charge within a sphere so that the electric field strength inside the sphere increases quadratically with radius ($$E \sim r^2$$).

## Level III

1. Let the charge density in all space be $$\rho = \rho_0 / r^2 (r + 1)^2$$. Calculate the electric field at an arbitrary point in space.
2. Calculate the electric field due to a line of charge by integrating over Coulomb's law. Compare to using Gauss's law.


text:
## using Thai with LaTeX on Mac OS X

Here the thing, while I was writhing my thesis using LaTeX with the template that Dr. Gordon created, I have to write a long research report for my work.
So I have a though of using LaTeX to create that report since I already fond of the way LaTeX writing can help me organize and create beautiful PDF report.

However, it’s not easy as I thought since I have to write that report in Thai and Thai language is designed to be difficult to use in the digital world. I have to dig around for a couple days in order to use Thai with LaTeX especially on OS X.
Here is how I did it:

1. First we’ll need to install TeX Live, for Mac we need to install MacTex
2. Once you install the MacTex then you’re ready to use international language with LaTeX using XeTeX typesetting engine.
3. Then you’ll need a tool with XeTeX engine to write LaTeX. I recommend Texmaker because it’s free and super powerful.

4. After installing you will have to set up the typesetting by go to Texmaker > Preferences > Quick Build > XeLaTex + View PDF
5. Finally test your environment by creating new document with this code

\documentclass[a4paper]{book}
\usepackage{xltxtra}
\usepackage{polyglossia}
\usepackage[top=25mm, bottom=20mm, left=25mm, right=25mm]{geometry}
\usepackage[titletoc,toc,title]{appendix}

\XeTeXlinebreaklocale "th"
\XeTeXlinebreakskip = 0pt plus 1pt
\setmainfont{TH SarabunPSK}

\defaultfontfeatures{Scale=1.23}
\renewcommand{\baselinestretch}{1.2}

\setdefaultlanguage{thai}
\newfontfamily{\thaifont}[Script=thai]{TH SarabunPSK}

\begin{document}

\frontmatter
\pagenumbering{thaialph}
\tableofcontents

\mainmatter
\chapter{บทแรก}
\section{บทนำ}

\noindent
ป้า กะ ปู่ กู้อีจู้ ตัวปกติ\\
{\itshape ป้า กะ ปู่ กู้อีจู้ ตัวเอียง} \\
{\bfseries ป้า กะ ปู่ กู้อีจู้ ตัวหนา } \\
{\bfseries\itshape ป้า กะ ปู่ กู้อีจู้ ตัวหนาเอียง} \\

ทดสอบใช้คำสั่ง \LaTeXe{}\\
ป้า กะ ปู่ กู้อีจู้ ไฟฟ้า ปัญญา ตัวปกติ\\
\textbf{ป้า กะ ปู่ กู้อีจู้ ไฟฟ้า ปัญญา ตัวหนา}\\
\textit{ป้า กะ ปู่ กู้อีจู้ ไฟฟ้า ปัญญา ตัวเอียง}\\
\textsl{ป้า กะ ปู่ กู้อีจู้ ไฟฟ้า ปัญญา ตัวเอน}\\
\textbf{\textit{ป้า กะ ปู่ กู้อีจู้ ตัวหนาเอียง}}\\
%%\texttt{ป้า กะ ปู่ กู้อีจู้ ไฟฟ้า ปัญญา ตัวพิมพ์}

\chapter{บทสอง}
\section{บทนำ}
xxxx xxxxx xxxxx
\chapter{บทสาม}
\section{บทนำ}
xxxx xxxxx xxxxx

%% \backmatter %% ใช้ไม่ได้ หายหมด

%% ยังไม่ดีมาก แต่ก็ได้ ก ข ค
\begin{appendices}
\renewcommand*{\thechapter}{\thaiAlph{chapter}}
\chapter{ทดสอบภาคผนวก} %% ---------------------
\chapter{ทดสอบภาคผนวกอีกที} %% ---------------------
\end{appendices}

\end{document}

You should see Thai language is working correctly, if not check if you system has Thai Sarabun font installed? You can download this official Thai font from SIPA

The main point is you’ll need to tell LaTeX to use Thai font
\usepackage{font spec} % To enable non-standard font selections
\setmainfont{TH SarabunPSK} % A font with Thai glyphs
\XeTeXlinebreaklocale ’th_TH’ % Thai-style word-breakings and line-wrappings

Enjoy Thai LaTeXing ^_^

## DIY: Jot Pro Stylus clip

Jot Pro in my thought is by far the best stylus for capacitive screen like iPad. I am using it all the time and even bought the iPad case that can carry this Jot Pro. However, the only draw back is the lack of pen clip. It’s difficult to put in your pocket without worrying about losing it. Here is what I did to make DIY clip for Jot Pro.

1. Buy a pen with a big clip (my pen is 25c)

2. Take the clip out and make a cut on the outer ring

3. Done ^__^

## How to view .XPS on Mac OS X

XPS format is a Microsoft standard format to use in MS Office. From MS Office you can print any document into XPS format (*.xps). However, I got a difficulty to read this format in Mac. The best way I found out is open with Google Doc. Here is how I did it:

1. Mail the XPS file with Gmail

3. The file will open with Google Doc and then later you can convert to PDF ^__^

## How to install Graphviz on Mountain Lion

Graphviz is open source graph visualization software. Graph visualization is a way of representing structural information as diagrams of abstract graphs and networks. It has important applications in networking, bioinformatics, software engineering, database and web design, machine learning, and in visual interfaces for other technical domains.

However there are some issues between Mountain Lion and X11, Graphviz 2.28 can’t run properly on Mountain Lion. Here is how I make it work.

* Install Homebrew by using this command in Terminal

$brew install wget * Then brew Libtool by typing this code in Terminal$ brew install libtool

* Finally, install Graphviz at http://www.graphviz.org/Download_macos.php (I am using version 2.29 dev snapshot)

* Done now you should be able to run Graphviz on Mountain Lion

## Put Linkedin profile on WordPress 3

I have been using a static resume (About me page) on this blog for quite sometime. I also have my CV in Linkedin, which is my main professional resume. Consequently, it’s inconvenient if I need to update my profile, I have to go editing everywhere. Today I’ve found the best solution: “Link my linked in profile into WordPress page”.

This “Linkedin Resume” wordpress plugin is awesome. Once I installed it, I just go to Setting > Linkedin Resume and input my Linkedin public profile.

Then I just have to put

“[ linkedinresume ] ” (no space between brackret)

on the page that need my profile to appear [see http://kerdsri.com/about/].

From now on, I need to only maintain my profile in Linkedin ^__^

## Switch language on Mac with grave accent (  )

For Windows users who familiar with switching between language with grave accent (  ), using Mac input source switching with Command + Spacebar might be difficult. After a long time of confusing, I finally see the light. I just need the same setting for all devices.

All you need is this piece of software Keyremap4macbook [http://pqrs.org/macosx/keyremap4macbook/]. After download and install (in System Preferences), you need to map (  ) to Command+Space .

That’s all, now I can use (  ) to switch language in both Mac and Windows.

## Fix contact doesn’t recognize in unlocked iPhone using Gevey sim

My friends just got AT&T iPhones from the US. They couldn’t unlock them via software because  their baseband doesn’t supported. The good new is Gevey Sim can unlock unsupported baseband. My friend went with this solution. It works great but there is a problem of contact doesn’t recognize. iPhone doesn’t show the caller name, just display the caller number. Here is how I fix their iPhone.

1. Go to ” Settings/General/International/Region Format/ ” and select your region

2. Downlosds files ” com.apple.AppSupport.plist ”  from here

3. Put this file at ” /private/var/mobile/Library/Preferences/”

4. Restart iPhone

## Setting up Thai in XMBC on AppleTV2

After setting up XMBC on ATV2, I felt like I really use full potential of ATV2. The good thing (for me) in this version of XMBC is I can set up Thai Character in Region setting. However I can’t read Thai because XMBC does not include Thai font in its package. Here how to add Thai font:

• Log on to ATV2
• I use Cyberduck to access into ATV2
• It’s already found my ATV2 via Bonjour network


text:
# PDAFit¶

PDAFit GUI

## Photon Distribution Analysis (PDA)¶

### General introduction to PDA¶

Photon distribution analysis (PDA, [Antonik2006] [Nir2006]) is a statistical method to describe the observed shot-noise limited FRET efficiency histograms by means of the underlying distance heterogeneity. PDA explicitly takes the statistics of photon emission and experimental correction factors into account and achieves higher accuracy than a direct analysis of the FRET efficiency histogram. It is especially useful for identifying heterogeneity in the FRET efficiency distribution that exceeds the expected value from shot-noise alone, allowing to quantify how defined a conformational state is.

PDA targets the uncorrected FRET efficiency histogram (also called proximity ratio histogram, PRH), where the proximity ratio is given by:

$PR = \frac{N_{GR}}{N_{GG}+N_{GR}}$

The algorithm simulates an expected proximity ratio histogram based on the input distances, correction factors and observed distribution of total photon counts, which accounts for the shot-noise. By variation of the parameters of the distance distribution, one can find the parameters that best describe the data.

Generally, two approaches have been proposed to simulate the proximity ratio histogram. Seidel and coworkers developed an analytical approach to calculate the PRH [Antonik2006], while Weiss and coworkers used Monte Carlo simulations to generate an approximate PRH [Nir2006]. Both approaches are implemented in the software, together with a related approach based on a maximum likelihood estimator.

Generally, the proximity ratio histogram is given by:

$P\left(PR \in [a,b]\right) = \sum_{\textrm{all }(N, N_{GR})\textrm{ for which }\left(\frac{N_{GR}}{N}\right) \in [a,b]} P(N)P_\epsilon(N_{GR}|N)$

Here $$P(N)$$ is the photon count distribution, i.e. the probability to detect $$N$$ photons, and $$P_E(N_{GR}|N)$$ is the probability to detect $$N_{GR}$$ red photons out of $$N$$ total photons, given the apparent FRET efficiency $$\epsilon$$, described by a binomial distribution:

$P_\epsilon(N_{GR}|N) = {{N}\choose{N_{GR}}} \epsilon^{N_{GR}}\left(1-\epsilon\right)^{N-N_{GR}}$

The apparent transfer efficiency $$\epsilon$$ is hereby a function of the correction parameters. Background photon counts are additionally considered as described in the references.

### The model function¶

The fit parameter table

PDAFit currently only supports normal distributions of distances with up to 5 states. The model additionally allows the inclusion of a donor-only population, which will behave differently from a low FRET population since no direct excitation of the acceptor dye can occur. Generally, the model function takes the form of:

$M = (1-F_{donor-only}) PRH\left(\sum_{i=1}^{5} \frac{F_i}{\sqrt{2\pi}\sigma}\exp{-\frac{\left(R-\bar{R_i}\right)^2}{2\sigma^2}}\right) + F_{donor-only} PRH_{donor-only}$

I.e. the donor-only fraction is equivalent to the total percentage of donor-only molecules, while the species fractions $$F_i$$ yield the fractions among the double-labeled molecules.

If no conformational heterogeneity is present in the sample and the broadening of the distance distribution is solely due to photophysical artifacts, the observed distribution width $$\sigma$$ is proportional to the center distance $$R$$ [Kalinin2010b]. Select the Fix Sigma at Fraction of R checkbox in the Settings tab to activate the constraint $$\sigma = f*R$$, where $$f$$ is the fraction. You can either leave the fraction as a fit parameter or fix it by checking the Fix checkbox next to the edit box.

### Generating PDA data¶

For efficient calculations, PDA requires the data to be of equal time length. Thus, instead of performing the analysis on burst-wise data, the individual bursts are sectioned with a constant time window (typically 1 ms). As a consequence, the obtained proximity ratio histogram in PDAFit may look slightly differently from the histogram in BurstBrowser.

The process of exporting data from BurstBrowser for use in PDAFit is described here.

### The correction parameters¶

The parameter table

The correction factors used in PDAFit mostly resemble the correction used in BurstBrowser (see correction factors there). If corrections have been applied in BurstBrowser, these are transferred to PDAFit. You can inspect the correction factors in the Parameters tab.

An exception to this rule is the direct excitation factor. In burst analysis, this correction factor is based on the signal in the direct acceptor excitation channel and thus dependent on the acceptor laser power. However, PDA requires the probability of direct acceptor excitation by the donor excitation laser $$p_{de}$$, which is given by:

$p_{de} = \frac{\epsilon_{A}^{\lambda^{ex}_D}}{\epsilon_{D}^{\lambda^{ex}_D}+\epsilon_{A}^{\lambda^{ex}_D}}$

Here, $$\epsilon_{D/A}^{\lambda^{ex}_D}$$ is the extinction coefficient of the donor or acceptor at the excitation wavelength of the donor fluorophore. This quantity has to be determined for each dye pair and is not accessible from the burst data.

In addition to the correction factors, the bin size of the loaded PDA file is given Bin [ms]. This parameter can not be modified. To change the bin size, export the data again from BurstBrowser with a different time bin.

When working with multiple files, you can use the All row to change the respective parameter for all files.

Note

The calculated histogram library is dependent on the set correction factors. A change in any of the parameters will trigger a recalculation of the histogram library, which can be time consuming. Set the parameters correctly in the beginning to save time.

### Data selection¶

Selection parameters

By default, the whole data set is selected. The selection can, however, be restricted in PDAFit using the left column in the Settings tab.

Minimum number of photons per bin:
Set this to a value larger 0 to exclude low photon number time bins from the analysis.
Maximum number of photons per bin:
This will limit the maximum number of photons per time bin. The computation time for the histogram method scales with the maximum number of photons per time bin, which makes it beneficial to limit this number, especially if only a few time bins contain a large number of photons, Inspect the photon count distribution to select a reasonable value for this parameter. (The other fit methods do scale mainly with the number of bursts and are mostly independent of the number of photons per time bin.)
Stoichiometry threshold:
Additionally, you can limit the allowed stoichiometry range as seen in the Efficiency vs. Stoichiometry plot (E-S). The displayed stoichiometry is calculated based on the raw photon counts $$S=\frac{N_{RR}}{N_{GG}+N_{GR}+N_{RR}}$$.

Applied selection criteria are globally applied to all loaded data sets.

### The All and Single tab¶

The All tab displays all loaded data sets simultaneously. Select the Single tab if you want to inspect a single data set alone. Use the popup menu located in the bottom right of the Single tab to select the data set to be displayed. When the Single tab is active, only the selected data set will be fit.

### Performing a PDA fit¶

After setting all correction parameters and initial fit parameters, click Fit -> Start in the menu. If no global parameter is specified, the program will fit the data sets one after the other. If global parameters are considered, the total chi-squared over all data sets is minimized. At any point, you can stop the fit routine by selecting Fit -> Stop. To set good starting parameters, you can preview the histogram based on the currently set model parameters by selecting Fit -> View.

Once the fit is finished, you can store the fit result in the data files by selecting File -> Save to Files(s). To obtain figures of the result, select File -> Export Figures. This will store the result in a subfolder of the data file.

### PDA fit methods¶

#### Histogram Library¶

The Histogram Library fit methods implements the PDA method developed by [Antonik2006]. The algorithm generates shot-noise limited histograms for a grid (library) of equally spaced apparent efficiency values over the interval [0,1]. The setting Grid resolution for E in the Settings tab determines the number of sample points over the interval (set this to at least 100). To implement the distance distribution model, the probability distribution of the distance is transformed into efficiency space. The simulated histogram is then a probability-weighted linear combination generated from the histogram library.

The advantage of this approach is that an accurate histogram is generated. After the initial preparation period, during which the histogram library is generated, the computational load is minimal, since only linear combinations have to be computed.

The accuracy, especially at the extreme edges of very small or large distances, is limited by the resolution of the grid of efficiency values of the histogram library. Especially if the fitted distance distribution show a very narrow width ($$< 1\:\mathring{A}$$), the grid resolution should be increased. As an example, at vanishing distribution width $$\sigma \rightarrow 0\:\mathring{A}$$, the algorithm will not find the true value of the center distance, but instead the distance value that belongs to the closest efficiency value of the grid. This effect of the grid, however, is reduced for normal distribution width $$\sigma > 2\:\mathring{A}$$, where the resulting histogram will be a weighted linear combination of multiple library elements.

#### MonteCarlo¶

The MonteCarlo method uses the algorithm as proposed by [Nir2006] to simulate the proximity ratio histogram. Hereby, for every data point, a random number generator is used to draw a value for the proximity ratio. To reduce stochastic noise, set the MonteCarlo Oversampling parameter in the Settings tab. Oversampling will repeat the histogram generation for the specified number of times and return the average.

The MonteCarlo method operates on a continuous parameter space and does not suffer from the problems associated with the gridding for the Histogram Library method. It can be slower than the other methods, depending on the desired accuracy as determined by the oversampling factor.

#### Maximum Likelihood estimator¶

The MLE method does not rely on the simulation of a histogram and thus completely avoids the necessity to bin the data. Instead, it computes the likelihood that the model parameters may produce the data directly. As such, no live plot update is possible. The fit result is finally displayed by means of simulation by the MonteCarlo method.

The MLE method, as the MonteCarlo method, operates on a continuous parameter space without any gridding. It generally shows a smoother optimization surface than the other two methods, and is thus suited well to be used with gradient-based fit algorithms.

### Dynamic-PDA¶

Dynamic interconversion between different FRET states leads to characteristic mixing of FRET efficiencies dependent on the rates of interconversion. This usually shows through the formation of a bridge between the FRET states in the histogram.

Dynamic-PDA, as described in [Kalinin2010a], can be activated by clicking the Dynamic Model checkbox in the Settings tab. This will enable a two-state kinetic model using the first two species of the fit model. The respective amplitudes in the fit parameter table will be changed to forward and backward rates $$k_{12}$$ and $$k_{21}$$ between the two states in unit of $$\textrm{ms}^{-1}$$.

Currently, only the two-state model is implemented. Additional static populations can be added.

Example dynamic-PDA analyses of slow (> ms) and fast (< ms) interconversion rates.

Note

It is recommended to globally fit a series of proximity ratio histograms generated at different time bins from the same measurement to obtain a higher accuracy in the determined rates.

### Error estimation¶

PDAFit offers to options to estimate the confidence intervals of the fit parameters, available through the menu Fit -> Estimate Error.

Estimate Error from Jacobian at solution:
This option will use the Jacobian at the fit result to estimate the confidence intervals of the parameters.
Estimate Error from Markov-chain Monte Carlo:
This will perform a Markov-Chain Monte Carlo chain over the $$\chi^2$$-surface to estimate the confidence intervals of the parameters.

The output will be returned into the current workspace.

### Background deconvolution¶

PDA can be used to describe a complete single-molecule FRET measurement without burst selection. In this case, many of the recorded time bins will only contain background signal and the resulting photon count distribution will be vastly different from the true distribution of fluorescence signal. See the section in the PAM manual for how to export a total measurement to PDA.

PDAFit implements the deconvolution method as proposed in [Kalinin2008] to recover the distribution of fluorescence signal from the observed photon count distribution using the know distribution of background counts.

This option only applies to the Histogram Library method.

Note

Only use this option when the total measurement is analyzed. This method does not apply for burst-selected data sets!

## Program structure¶

Removes all current data and starts a new list. The program will query multiple files until the user selects Cancel.
Adds files to the current list.
Save to File(s):
Everything that was opened is saved back into the .pda file, including the results from the fit table.
Export Figure(s):
Exports figures of the fit results to a subfolder. A .tif file of the All tab is saved, a .tif of each single file is also saved.

### Database Tab¶

Files that are loaded or added via the file menu are put into a list in the database tab. Files can be deleted from this list using the keyboard.

Save:
The list can be saved in a PDA Database File (.pab), so it can be opened at a later point in time. Importantly, the database file just contains a reference to the filename and path of all files in it.
Load a previously saved database list (.pab).

### Fit Table¶

To remove a component from the fit model fix the amplitude A to 0. Unchecking a file’s ‘Active’ checkbox removes that file from analysis, but not from the database. Clicking an ‘All’ row cell value applies this value to all rows.

### Settings Tab¶

The Settings tab allows you to set a number of parameters that affect the analysis and visualization. The options are explained in more detail in the respective sections. Here, an overview and brief description is given.

Number of Bins:
Specifies the number of bins used to construct the proximity ratio histogram over the interval [0,1]. This will effect the display of the histogram, and may effect the fit result of the Histogram Library and Monte-Carlo methods. The MLE method is unaffected by this setting.
Minimum/maximum Number of Photons per Bin:
Restrict the selection of time bins (see the respective section).
Grid resolution for E:
Defines the grid resolution over the apparent FRET efficiency for the Histogram Library method. The method will construct a library of shot-noise limited proximity ratio histograms at the grid values (e.g. at setting 100, at $$\epsilon$$ = [0, 0.01, 0.02, …, 0.98, 0.99, 1]). It is not advised to set this value lower than 100. Increasing this value with increase the accuracy of the analysis at higher computational cost. This setting has no effect on the PDA methods MonteCarlo or MLE.
Stoichiometry threshold:
Set threshold to restrict the selection of time bins based on the stoichiometry parameter (see the respective section).
PDA Method:
Set the method for the PDA analysis between Histogram Library, MLE and MonteCarlo. See the respective section for details.
Fit Method:
Set the fit method to optimize the parameters. Choose between Simplex (fminsearch function), Gradient-based (using the fmincon or lsqnonlin function), Patternsearch or Gradient-based (global). See the MATLAB documentation for more explanation on these fit methods. Generally, the Simplex method yields good results at acceptable convergence time.
MonteCarlo Oversampling:
This parameter determines the oversampling when using the MonteCarlo method. Increasing this number will reduce the statistical noise in the simulated histogram at the cost of increased computation time. The other methods are not affected by this parameter.
Chi2 method:
Choose if the $$\chi^2$$ goodness-of-fit estimator should be calculated based on Poissonian or Gaussian counting statistics.
Fix Sigma at fraction of R:
Check this to fix the distribution width $$\sigma$$ at a fixed fraction of the mean distance of a species, i.e. $$\sigma = k * R$$ where $$k$$ is typically in the range between 0.05 to 0.2. Set the proportionality factor $$k$$ using the edit box. The proportionality factor will by default be optimized by the fitting routine. You can additionally fix the proportionality factor to a custom value by checking the Fix? checkbox.
Dynamic Model:
Check this checkbox to enable dynamic-PDA. Only works in conjunction with the Histogram Library method. See the respective section for details.
Deconvolute background:
Performs deconvolution of the photon count distribution to obtain the background-free fluorescence photon count distribution. Use this only when fitting a whole measurement data set without burst selection. See the respective section for more details.
Half global:
Allow to set parameters global between subsets of the loaded measurements. Please inquire about details on how to use this functionality.
Live plot update:
Enabling this option will cause a live plot update during the fitting process. Note that this slows down the fitting since the plotting takes a significant amount of time. Use this option to check the status of the fitting routine, but generally leave it off.
Ignore outer bins:
This options ignores the outermost bins for the calculation of the $$\chi^2$$ goodness-of-fit estimator, i.e. the bins [0,1/N] and [1-1/N,1] where N is the number of bins in the proximity ratio histogram. This option can be useful if large deviations between model and data are observed for these bins.
gauss amplitude:
Enabling this checkbox will scale distance distribution fit result plot with the respective amplitudes of the species. Disable this if you mainly want to investigate changes in the distance and not in the respective population sizes.
Brightness correction:
Corrects for brightness differences between different species using a donor-only brightness reference. This feature is untested and experimental.
 [Kalinin2012] Kalinin, S. et al. A toolkit and benchmark study for FRET-restrained high-precision structural modeling. Nat Meth 9, 1218–1225 (2012)
 [Antonik2006] (1, 2, 3) Antonik, M., Felekyan, S., Gaiduk, A. & Seidel, C. A. M. Separating Structural Heterogeneities from Stochastic Variations in Fluorescence Resonance Energy Transfer Distributions via Photon Distribution Analysis. J. Phys. Chem. B 110, 6970–6978 (2006).
 [Kalinin2010a] Kalinin, S., Valeri, A., Antonik, M., Felekyan, S. & Seidel, C. A. M. Detection of structural dynamics by FRET: a photon distribution and fluorescence lifetime analysis of systems with multiple states. J. Phys. Chem. B 114, 7983–7995 (2010).
 [Kalinin2008] Kalinin, S., Felekyan, S., Valeri, A. & Seidel, C. A. M. Characterizing multiple molecular States in single-molecule multiparameter fluorescence detection by probability distribution analysis. J. Phys. Chem. B 112, 8361–8374 (2008).
 [Kalinin2010b] Kalinin, S., Sisamakis, E., Magennis, S. W., Felekyan, S. & Seidel, C. A. M. On the origin of broadening of single-molecule FRET efficiency distributions beyond shot noise limits. J. Phys. Chem. B 114, 6197–6206 (2010).