From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.comp.tex.context/48408 Path: news.gmane.org!not-for-mail From: Mohamed Bana Newsgroups: gmane.comp.tex.context Subject: math problems Date: Sun, 15 Mar 2009 22:42:01 +0000 Message-ID: Reply-To: mailing list for ConTeXt users NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="------------070003060202050806040900" X-Trace: ger.gmane.org 1237157002 27592 80.91.229.12 (15 Mar 2009 22:43:22 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 15 Mar 2009 22:43:22 +0000 (UTC) To: ntg-context@ntg.nl Original-X-From: ntg-context-bounces@ntg.nl Sun Mar 15 23:44:38 2009 Return-path: Envelope-to: gctc-ntg-context-518@m.gmane.org Original-Received: from ronja.vet.uu.nl ([131.211.172.88] helo=ronja.ntg.nl) by lo.gmane.org with esmtp (Exim 4.50) id 1Liz4P-0007mX-AO for gctc-ntg-context-518@m.gmane.org; Sun, 15 Mar 2009 23:44:37 +0100 Original-Received: from localhost (localhost [127.0.0.1]) by ronja.ntg.nl (Postfix) with ESMTP id 4F9FD1FB14; Sun, 15 Mar 2009 23:43:10 +0100 (CET) Original-Received: from ronja.ntg.nl ([127.0.0.1]) by localhost (smtp.ntg.nl [127.0.0.1]) (amavisd-new, port 10024) with LMTP id 27538-04; Sun, 15 Mar 2009 23:42:35 +0100 (CET) Original-Received: from ronja.vet.uu.nl (localhost [127.0.0.1]) by ronja.ntg.nl (Postfix) with ESMTP id 87D5D1FA2B; Sun, 15 Mar 2009 23:42:33 +0100 (CET) Original-Received: from localhost (localhost [127.0.0.1]) by ronja.ntg.nl (Postfix) with ESMTP id 0D9081FA2B for ; Sun, 15 Mar 2009 23:42:30 +0100 (CET) Original-Received: from ronja.ntg.nl ([127.0.0.1]) by localhost (smtp.ntg.nl [127.0.0.1]) (amavisd-new, port 10024) with LMTP id 30471-03-2 for ; Sun, 15 Mar 2009 23:42:21 +0100 (CET) Original-Received: from filter4-til.mf.surf.net (filter4-til.mf.surf.net [194.171.167.220]) by ronja.ntg.nl (Postfix) with ESMTP id 3FA551FA41 for ; Sun, 15 Mar 2009 23:42:20 +0100 (CET) Original-Received: from ciao.gmane.org (main.gmane.org [80.91.229.2]) by filter4-til.mf.surf.net (8.13.8/8.13.8/Debian-3) with ESMTP id n2FMgHvD024957 for ; Sun, 15 Mar 2009 23:42:18 +0100 Original-Received: from list by ciao.gmane.org with local (Exim 4.43) id 1Liz23-0007H1-2S for ntg-context@ntg.nl; Sun, 15 Mar 2009 22:42:11 +0000 Original-Received: from 87-194-191-26.bethere.co.uk ([87.194.191.26]) by main.gmane.org with esmtp (Gmexim 0.1 (Debian)) id 1AlnuQ-0007hv-00 for ; Sun, 15 Mar 2009 22:42:11 +0000 Original-Received: from mbana.lists by 87-194-191-26.bethere.co.uk with local (Gmexim 0.1 (Debian)) id 1AlnuQ-0007hv-00 for ; Sun, 15 Mar 2009 22:42:11 +0000 X-Injected-Via-Gmane: http://gmane.org/ Original-Lines: 498 Original-X-Complaints-To: usenet@ger.gmane.org X-Gmane-NNTP-Posting-Host: 87-194-191-26.bethere.co.uk User-Agent: Thunderbird 2.0.0.19 (X11/20090105) X-Bayes-Prob: 0.0001 (Score 0, tokens from: @@RPTN) X-CanIt-Geo: ip=80.91.229.2; country=NO; region=12; city=Oslo; latitude=59.9167; longitude=10.7500; http://maps.google.com/maps?q=59.9167,10.7500&z=6 X-CanItPRO-Stream: uu:ntg-context@ntg.nl (inherits from uu:default, base:default) X-Canit-Stats-ID: 193881511 - 21b9316a4f1b X-Scanned-By: CanIt (www . roaringpenguin . com) on 194.171.167.220 X-Virus-Scanned: amavisd-new at ntg.nl X-BeenThere: ntg-context@ntg.nl X-Mailman-Version: 2.1.11 Precedence: list List-Id: mailing list for ConTeXt users List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Original-Sender: ntg-context-bounces@ntg.nl Errors-To: ntg-context-bounces@ntg.nl X-Virus-Scanned: amavisd-new at ntg.nl Xref: news.gmane.org gmane.comp.tex.context:48408 Archived-At: This is a multi-part message in MIME format. --------------070003060202050806040900 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit hi guys, i generally use this file to test if math is "working", with the recent update i get a lot of errors such as; ! Missing number, treated as zero. $ \@@dobig ...o #1\bodyfontsize {}\right .\n@space $ }} \@mt@defaultBigl ...\puremathcomm {open}{\Big {#1} } l.99 \pauli = \dirac^2 = \Bigl( \sum_{j=1}^2 \sigma_j\big(-i\PD{}{x^j}-a_j\bi... ? with texlive 2008 it works just fine, i've attached the output of the TL2008. i'm certain that the same file was compiling just fine with (i think) luatex 0.31 (or pre 0.31). the content is from Mikael Persson's thesis. Mohamed --------------070003060202050806040900 Content-Type: text/x-tex; name="mickep-math.tex" Content-Disposition: inline; filename="mickep-math.tex" Content-Transfer-Encoding: quoted-printable \usemodule[bib] \def\mathbb#1{{\blackboard #1}} \def\pauli{\mathfrak{P}} \def\mathfrak#1{{\fraktur #1}} % Matriser av olika typ. % Paranthesis \definemathmatrix [pmatrix] [left=3D{\left(\,},right=3D{\,\right)}] % Brackets \definemathmatrix [bmatrix] [left=3D{\left[\,},right=3D{\,\right]}] % Curly braces \definemathmatrix [Bmatrix] [left=3D{\left\{\,},right=3D{\,\right\}}] % vertical bars \definemathmatrix [vmatrix] [left=3D{\left\vert\,},right=3D{\,\right\vert}] % double vertical bars \definemathmatrix [Vmatrix] [left=3D{\left\Vert\,},right=3D{\,\right\Vert}] \def\PD#1#2{\frac{\partial #1}{\partial #2}} \def\ann{\mathscr{Q}^{\vphantom{*}}} \def\mathscr#1{{\gothic #1}} \def\cre{\mathscr{Q}^*} \def\dirac{{\mathfrak{D}}} \def\pform{\mathfrak{p}} \def\ed{\mfunction{\,d}} \definemathcommand [dom] [nolop] {\mfunction{Dom}} \def\bz{\bar{z}} \def\psip{{\psi_+}} \def\psim{{\psi_-}} % \eqref \definereferenceformat[eqref][left=3D(,right=3D)] \defineenumeration[problem] [text=3DProblem, location=3Dserried, width=3Dfit, indenting=3Dfirst, distance=3D0.5em, way=3Dbysection, ] \subsubsection[ksec:paulitvad]{The Pauli operator in $\mathbb{R}^2$} A charged spin $1/2$ particle is described by the Pauli Hamiltonian, whic= h acts in $L_2(\mathbb{R}^2)\otimes \mathbb{C}^2$, and is formally define= d as \placeformula[keq:pauli] \startformula \pauli =3D=20 \startpmatrix \NC H-\frac{g}{2}B \NC 0 \NR \NC 0 \NC H+\frac{g}{2}B\NR \stoppmatrix. \stopformula Here $H$ is the two-dimensional Schr=C3=B6dinger Hamiltonian $H=3D(-i\nab= la -\vec{a})^2$, $B$ is the magnetic field (In two dimensions we identify= the two-form and the coefficient function), and $g$ is the {\em gyromagn= etic ratio}. We identify the real point $(x^1,x^2)$ with the complex numb= er $z=3Dx^1+ix^2$, and denote a scalar potential of $B$ by $W$, \startformula -\Delta W =3D B. \stopformula We set $\Pi_j =3D -i\PD{}{x^j}-a_j$ and \startformula \ann =3D \Pi_1 - i \Pi_2,\quad \cre =3D \Pi_1 + i\Pi_2, \stopformula and note that \placeformula[keq:komm] \startformula \ann\cre=3D\cre\ann+2B,\quad H=3D\cre\ann+B=3D\ann\cre-B. \stopformula From~\eqref[keq:pauli] and~\eqref[keq:komm] we get \placeformula[keq:paulig] \startformula \pauli =3D=20 \startpmatrix \NC \cre\ann-\frac{g-2}{2}B \NC 0 \NR \NC 0 \NC \ann\cre+\frac{g-2}{2}B\NR \stoppmatrix. \stopformula The number $\frac{g-2}{2}$ is called the {\em anomaly factor} of the magn= etic moment. Experiments give an anomaly factor of $0.00159$ for the elec= tron~\cite[bovo]. We assume that $g=3D2$, which is the simplest case. Thu= s, the Pauli Hamiltonian we study in this thesis is formally defined by \placeformula[keq:pauliw] \startformula \pauli =3D=20 \startpmatrix \NC \cre\ann \NC 0 \NR \NC 0 \NC \ann\cre\NR \stoppmatrix. \stopformula The Pauli operator can be written as the square of the Dirac operator \placeformula[keq:paulia] \startformula \pauli =3D \dirac^2 =3D \Bigl(\sum_{j=3D1}^2 \sigma_j\big(-i\PD{}{x^j}-a_= j\big)\Bigr)^2 =3D=20 \startpmatrix \NC 0 \NC \cre \NR \NC \ann \NC 0\NR \stoppmatrix^2 \stopformula from which it follows that it is a non-negative operator. Now let us be m= ore precise about the domains. As in the case of the Schr=C3=B6dinger Ham= iltonian there is a problem in defining the Pauli Hamiltonian if the magn= etic field is too singular. The quadratic form corresponding to $\pauli$ = is given by \placeformula[keq:pform] \startformula \pform(\psi,\psi)=3D \int_{\mathbb{R}^2} \Big|\sum_{j=3D1}^2 \sigma_j\big= (-i\PD{}{x^j}-a_j\big)\psi\Big|^2\ed m(x). \stopformula If $\vec{a}\in L_{2,\text{loc}}(\mathbb{R}^2)\otimes \mathbb{R}^2$ then $= \pform(\psi,\psi)$ makes sense for $\psi\in C_0^\infty(\mathbb{R}^2)\otim= es \mathbb{C}^2$. We define the {\em minimal} Pauli form $\pform_{\text{m= in}}$ as \startformula \startalign \NC \dom(\pform_{\text{min}}) \NC =3D C_0^\infty(\mathbb{R}^2)\otimes \ma= thbb{C}^2;\NR \NC \pform_{\text{min}}(\psi,\psi) \NC =3D \pform(\psi,\psi),\quad \psi\i= n\dom(\pform_{\text{min}}).\NR \stopalign \stopformula It is closable and thus a self-adjoint operator $\pauli_{\text{min}}$ can= be defined. We also define the {\em maximal} Pauli form $\pform_{\text{m= ax}}$ as \placeformula \startformula \startalign \NC \dom(\pform_{\text{max}}) \NC =3D \bigl\{\,\psi\in L_2(\mathbb{R}^2)\= otimes \mathbb{C}^2\bigm| \pform(\psi,\psi)<\infty\,\bigr\};\NR[keq:pform= max] \NC \pform_{\text{max}}(\psi,\psi) \NC =3D \pform(\psi,\psi),\quad\psi\in= \dom(\pform_{\text{max}}).\NR \stopalign \stopformula In the presence of AB solenoids, $\vec{a}$ does not belong to $L_{2,\text= {loc}}(\mathbb{R}^2)\otimes \mathbb{R}^2$. It was proved in~\cite[so] tha= t the Pauli form can not be defined on smooth compactly supported $\psi$ = via~\eqref[keq:pform] in this case. The way out of this is to redefine th= e Pauli form $\pform$ by an expression that makes sense even in this more= singular case. This is done in~\cite[ervo] by writing the operators $\an= n$ and $\cre$ as \placeformula[keq:anncre] \startformula \ann =3D-2i e^{W}\PD{}{\bar{z}} e^{-W}\quad\text{and}\quad \cre =3D -2i e= ^{-W}\PD{}{z} e^{W}, \stopformula and noting that the quadratic form \placeformula[keq:pformw] \startformula \pform(\psi,\psi) =3D 4\int_{\mathbb{R}^2} \Big|\PD{}{\bz}\left(e^{-W}\ps= ip\right)\Big|^2e^{2W}+\Big|\PD{}{z}\left(e^{W}\psim\right)\Big|^2 e^{-2W= }\ed m(x), \stopformula $\psi=3D(\psip,\psim)^t$ makes sense even with this more singular field. = If $\pform$ is defined on a maximal domain in the same way as in~\eqref[k= eq:pformmax], it yields a self-adjoint operator even with this singular f= ield, usually called the {\em maximal} Pauli operator. The forms in~\eqre= f[keq:pform] and~\eqref[keq:pformw] coincides for more regular fields. \page \placeformula[keq:pauli] \startformula \pauli =3D=20 \startpmatrix \NC H-\frac{g}{2}B \NC 0 \NR \NC 0 \NC H+\frac{g}{2}B\NR \stoppmatrix. \stopformula \placeformula \startformula -\Delta W =3D B. \stopformula We set $\Pi_j =3D -i\PD{}{x^j}-a_j$ and \startformula \ann =3D \Pi_1 - i \Pi_2,\quad \cre =3D \Pi_1 + i\Pi_2, \stopformula \placeformula[keq:komm] \startformula \ann\cre=3D\cre\ann+2B,\quad H=3D\cre\ann+B=3D\ann\cre-B. \stopformula \placeformula[keq:paulig] \startformula \pauli =3D=20 \startpmatrix \NC \cre\ann-\frac{g-2}{2}B \NC 0 \NR \NC 0 \NC \ann\cre+\frac{g-2}{2}B\NR \stoppmatrix. \stopformula \placeformula[keq:pauliw] \startformula \pauli =3D=20 \startpmatrix \NC \cre\ann \NC 0 \NR \NC 0 \NC \ann\cre\NR \stoppmatrix. \stopformula \placeformula[keq:paulia] \startformula \pauli =3D \dirac^2 =3D \Bigl(\sum_{j=3D1}^2 \sigma_j\big(-i\PD{}{x^j}-a_= j\big)\Bigr)^2 =3D=20 \startpmatrix \NC 0 \NC \cre \NR \NC \ann \NC 0\NR \stoppmatrix^2 \stopformula \placeformula[keq:pform] \startformula \pform(\psi,\psi)=3D \int_{\mathbb{R}^2} \Big|\sum_{j=3D1}^2 \sigma_j\big= (-i\PD{}{x^j}-a_j\big)\psi\Big|^2\ed m(x). \stopformula \startformula \startalign \NC \dom(\pform_{\text{min}}) \NC =3D C_0^\infty(\mathbb{R}^2)\otimes \ma= thbb{C}^2;\NR \NC \pform_{\text{min}}(\psi,\psi) \NC =3D \pform(\psi,\psi),\quad \psi\i= n\dom(\pform_{\text{min}}).\NR \stopalign \stopformula \placeformula \startformula \startalign \NC \dom(\pform_{\text{max}}) \NC =3D \bigl\{\,\psi\in L_2(\mathbb{R}^2)\= otimes \mathbb{C}^2\bigm| \pform(\psi,\psi)<\infty\,\bigr\};\NR[keq:pform= max] \NC \pform_{\text{max}}(\psi,\psi) \NC =3D \pform(\psi,\psi),\quad\psi\in= \dom(\pform_{\text{max}}).\NR \stopalign \stopformula \placeformula[keq:anncre] \startformula \ann =3D-2i e^{W}\PD{}{\bar{z}} e^{-W}\quad\text{and}\quad \cre =3D -2i e= ^{-W}\PD{}{z} e^{W}, \stopformula \placeformula[keq:pformw] \startformula \pform(\psi,\psi) =3D 4\int_{\mathbb{R}^2} \Big|\PD{}{\bz}\left(e^{-W}\ps= ip\right)\Big|^2e^{2W}+\Big|\PD{}{z}\left(e^{W}\psim\right)\Big|^2 e^{-2W= }\ed m(x), \stopformula \placeformula \startformula H =3D (-i\nabla-\vec{a})^2. \stopformula It is an unbounded operator, so one should be more specific about the dom= ain $\dom(H)$. To be able to define $H$ on $C^\infty_0(\mathbb{R}^{n})$ i= t is sufficient that \placeformula[keq:areq] \startformula \vec{a}\in L_{4,\text{loc}}(\mathbb{R}^{n})\otimes \mathbb{R}^{n},\quad \= div\vec{a}\in L_{2,\text{loc}}(\mathbb{R}^{n}). \stopformula This follows by expanding $H$ as \startformula H =3D -\Delta + i\div \vec{a}+2i\vec{a}\cdot\nabla+\vec{a}\cdot\vec{a}. \stopformula \placeformula[keq:sform] \startformula h(u,u)=3D\int_{\mathbb{R}^n} \left|(-i\nabla-\vec{a})u\right|^2\ed m(x). \stopformula Assuming that $\vec{a}\in L_{2,\text{loc}}(\mathbb{R}^n)\otimes \mathbb{R= }^n$, we can define two forms $h_{\min}$ and $h_{\max}$ as \startformula \startalign \NC \dom(h_{\text{min}}) \NC =3D C_0^\infty(\mathbb{R}^n);\NR \NC h_{\text{min}}(u,u)\NC =3Dh(u,u),\quad u\in\dom(h_{\text{min}});\NR \stopalign \stopformula \placeformula \startformula \startalign \NC \dom(h_{\text{max}}) \NC =3D \bigl\{\,u\in L_2(\mathbb{R}^n)\bigm| h(= u,u)<\infty\,\bigr\}\NR \NC h_{\text{max}}(u,u)\NC =3Dh(u,u),\quad u\in\dom(h_{\text{max}}).\NR \stopalign \stopformula \placeformula \startformula e^{if}(-i\nabla-\vec{a}_1)^2 e^{-if} =3D (-i\nabla -\vec{a}_2)^2. \stopformula \placeformula \startformula B(x)=3D2\pi\alpha \delta(x) \ed x^1\wedge \ed x^2. \stopformula \placeformula \startformula \vec{a}(x)=3D\frac{\alpha}{|x|^2}\bigl(-x^2,x^1\bigr). \stopformula \placeformula \startformula \startalign \NC h(u,u) \NC =3D\int_{\mathbb{R}^2} |(-i\nabla-\vec{a})u|^2\ed m(x) \ge= q \int_{|x|<1}|(-i\nabla-\vec{a})u|^2\ed m(x)\NR \NC \NC =3D \int_{|x|<1}|\vec{a}|^2\ed m(x) =3D \int_{|x|<1}\frac{\alpha^= 2}{|x|^2}\ed m(x) =3D +\infty\NR \stopalign \stopformula \placeformula \startformula u(re^{i\theta})\sim c_{-\alpha}r^{-\alpha}+c_{\alpha-1}r^{\alpha-1}e^{-i\= theta}+c_{\alpha}r^{\alpha}+c_{1-\alpha}r^{1-\alpha}e^{-i\theta},\quad r\= searrow 0,=20 \stopformula \placeformula \startformula N\bigl(\Lambda_q\pm\lambda,\mu_{\pm},H(\vec{a}_0,\pm V)\bigr) \sim \frac{= |\log\lambda|}{\log|\log\lambda|},\quad\lambda \searrow 0, \stopformula Moreover, if $W_\infty=3D0$, then they are able to prove a similar formul= a for the higher Landau levels, \startformula \lim_{j\to\infty} \bigl(\pm j!(\lambda_{j,q}^{\pm}-\Lambda_q\bigr)^{1/j} = =3D \frac{B_0}{2}\kap(K)^2,\quad q=3D0,1,\ldots. \stopformula \placeformula \startformula \lambda_{1,q}^+\geq\lambda_{2,q}^+\geq=20 \cdots,\qquad \lambda_{1,q}^-\leq\lambda_{2,q}^-\leq \cdots \stopformula be the eigenvalues of $H(\vec{a},\pm V)$ in $(\Lambda_{q},\Lambda_{q+1})$= (for~$+$) and $(\Lambda_{q-1},\Lambda_q)$ (for~$-$), and let $\kap(K)$ b= e the logarithmic capacity of the set $K$, see~\cite[lan]. Then, if the m= agnetic scalar potential can be written as $W=3D-\frac{B_0}4|z|^2+W_\inft= y$, where $W_\infty$ is bounded, and $V$ has support in a compact $K$, $V= \geq c>0$ on $K$, it holds that \startformula \lim_{j\to\infty} \bigl(\pm j!(\lambda_{j,0}^{\pm}-\Lambda_0)\bigr)^{1/j}= =3D \frac{B_0}{2}\kap(K)^2. \stopformula Moreover, if $W_\infty=3D0$, then they are able to prove a similar formul= a for the higher Landau levels, \startformula \lim_{j\to\infty} \bigl(\pm j!(\lambda_{j,q}^{\pm}-\Lambda_q\bigr)^{1/j} = =3D \frac{B_0}{2}\kap(K)^2,\quad q=3D0,1,\ldots. \stopformula \placeformula \startformula \startalign \NC N(-\infty,\Lambda_0,H)\NC \leq 2n,\NR \NC N(\Lambda_q,\Lambda_{q+1},H)\NC \leq qn,\quad q=3D0,1,\ldots.\NR \stopalign \stopformula \placeformula \startformula \lim_{j\to\infty} \bigl(j!(\lambda_{j,q}^{+}-\Lambda_q)\bigr)^{1/j} =3D \= frac{B_0}{2}\kap(K)^2,\quad q=3D0,1,\ldots. \stopformula \placeformula \startformula \lim_{j\to\infty} \bigl(j!(\Lambda_q-\lambda_{j,q}^{-})\bigr)^{1/j} =3D \= frac{B_0}{2}\kap(K)^2,\quad q=3D0,1,\ldots. \stopformula \placeformula[keq:gpauli] \startformula \pauli =3D=20 \startpmatrix \NC \vphantom{\frac{g-2}{2}B}\cre\ann \NC 0 \NR \NC 0 \NC \vphantom{\frac{g-2}{2}B}\ann\cre\NR \stoppmatrix + \startpmatrix \NC -\frac{g-2}{2}B \NC 0 \NR \NC 0 \NC \frac{g-2}{2}B\NR \stoppmatrix, \stopformula \placeformula \startproblem (This problem was proposed to me by Prof. Ari Laptev in pri= vate communication) Study Schr=C3=B6dinger operators with singular potent= ials in higher dimensions. For example, one could try to describe the sel= f-adjoint extensions of the Schr=C3=B6dinger operator in $\mathbb{R}^{2d}= $ with magnetic one-form \startformula a(x) =3D \sum_{j\neq k} \Phi_{j,k}\frac{x^{2k}-x^{2j}}{|z^j-z^k|^2}\ed x^= {2j-1} + \Phi_{j,k} \frac{x^{2j-1}-x^{2k-1}}{|z^j-z^k|^2}\ed x^{2j} \stopformula initially defined on $C_0^\infty\bigl(\mathbb{R}^{2d}\setminus \{z^j=3Dz^= k\}_{j\neq k}\bigr)$. The main difficulty is that the vector potential is singular, not just in= one point as in two dimensions, but in all hyperplanes $z^j=3Dz^k$. In t= wo dimensions the extensions were described by certain singular boundary = terms at the singular points.=20 \stopproblem --------------070003060202050806040900 Content-Type: text/x-log; name="mickep-math.log" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mickep-math.log" This is pdfTeX, Version 3.1415926-1.40.9 (Web2C 7.5.7) (format=cont-en 2009.3.11) 13 MAR 2009 21:37 entering extended mode \write18 enabled. (/home/mbana/context-minimals/tex/texmf-context/web2c/natural.tcx) **mickep-math.tex \emergencyend (./mickep-math.tex ConTeXt ver: 2009.03.11 09:45 MKII fmt: 2009.3.11 int: english/english system : cont-new loaded (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/cont-new.tex systems : beware: some patches loaded from cont-new.tex (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/cont-new.mkii) (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/cont-mtx.tex)) system : cont-fil loaded (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/cont-fil.tex loading : Context File Synonyms ) system : cont-sys.rme loaded (/home/mbana/context-minimals/tex/texmf-context/tex/context/user/cont-sys.rme (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/type-tmf.tex) (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/type-siz.tex) (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/type-one.tex) (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/type-akb.tex)) system : mickep-math.top loaded (./mickep-math.top) bodyfont : 12pt rm is loaded language : patterns nl->texnansi:texnansi->1->2:2 nl->ec:ec->2->2:2 fr-> texnansi:texnansi->3->2:2 fr->ec:ec->4->2:2 de->texnansi:texnansi->5->2:2 de->e c:ec->6->2:2 it->texnansi:texnansi->7->2:2 it->ec:ec->8->2:2 pt->texnansi:texna nsi->9->2:2 pt->ec:ec->10->2:2 hr->ec:ec->11->2:2 pl->pl0:pl0->12->2:2 pl->ec:e c->13->2:2 pl->qx:qx->14->2:2 cs->il2:il2->15->2:2 cs->ec:ec->16->2:2 sk->il2:i l2->17->2:2 sk->ec:ec->18->2:2 sl->ec:ec->19->2:2 ru->t2a:t2a->21->2:2 gb->ec:e c->22->2:2 us->ec:ec->23->2:2 agr->agr:agr->24->2:2 da->ec:ec->25->2:2 sv->ec:e c->26->2:2 af->ec:ec->27->2:2 nb->ec:ec->28->2:2 nn->ec:ec->29->2:2 deo->ec:ec- >30->2:2 es->ec:ec->35->2:2 ca->ec:ec->36->2:2 la->ec:ec->37->2:2 ro->ec:ec->38 ->2:2 tr->ec:ec->39->2:2 fi->ec:ec->41->2:2 hu->ec:ec->42->2:2 loaded specials : dvips loaded \openout3 = `mickep-math.tui'. \openout0 = `mickep-math-mpgraph.mp'. \openout0 = `mpgraph.mp'. systems : system commands are enabled language : language en is active specials : loading definition file tpd (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/spec-tpd.tex specials : loading definition file fdf (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/spec-fdf.tex (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/spec-fdf.mkii) ) (/home/mbana/context-minimals/tex/texmf-context/tex/context/base/spec-tpd.mkii) ) system : module bib loaded (/home/mbana/context-minimals/tex/texmf-context/tex/context/bib/t-bib.tex publications : loading formatting style from bibl-apa (/home/mbana/context-minimals/tex/texmf-context/tex/context/bib/bibl-apa.tex)) subsubsection : 1 The Pauli operator in $\mathbb {R}^2$ (./mickep-math.tuo system : new version of utility file, second pass needed ) references : unknown reference [][keq:pauli] references : unknown reference [][keq:komm] publications : warning: cite argument bovo is unknown on 170 ! Missing number, treated as zero. $ \@@dobig ...o #1\bodyfontsize {}\right .\n@space $ }} \@mt@defaultBigl ...\puremathcomm {open}{\Big {#1} } l.99 \pauli = \dirac^2 = \Bigl( \sum_{j=1}^2 \sigma_j\big(-i\PD{}{x^j}-a_j\bi... ? X Here is how much of TeX's memory you used: 2089 strings out of 257043 34980 string characters out of 1271064 2692006 words of memory out of 4125384 41891 multiletter control sequences out of 10000+100000 166255 words of font info for 80 fonts, out of 2000000 for 5000 486 hyphenation exceptions out of 8191 35i,10n,33p,409b,619s stack positions out of 10000i,500n,10000p,4000000b,50000s No pages of output. PDF statistics: 0 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 131072) 1 words of extra memory for PDF output out of 10000 (max. 10000000) --------------070003060202050806040900 Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Disposition: inline ___________________________________________________________________________________ If your question is of interest to others as well, please add an entry to the Wiki! maillist : ntg-context@ntg.nl / http://www.ntg.nl/mailman/listinfo/ntg-context webpage : http://www.pragma-ade.nl / http://tex.aanhet.net archive : https://foundry.supelec.fr/projects/contextrev/ wiki : http://contextgarden.net ___________________________________________________________________________________ --------------070003060202050806040900--