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diff --git a/Master/texmf-dist/doc/latex/bosisio/sobolev.html b/Master/texmf-dist/doc/latex/bosisio/sobolev.html new file mode 100644 index 00000000000..f354f150347 --- /dev/null +++ b/Master/texmf-dist/doc/latex/bosisio/sobolev.html @@ -0,0 +1,311 @@ +<HTML> + +<!------------------------------------------------------------------------> + +<HEAD> +<TITLE>Package sobolev</TITLE> +</HEAD> +<BODY > + +<!------------------------------------------------------------------------> + +<P> + +<H1 ALIGN="CENTER"> +Package <TT>sobolev</TT><BR> +<SMALL> +This is version 2.2, last revised 1998/02/04; documentation date 1998/02/06. +</SMALL> +</H1> + +<P ALIGN="CENTER"> +<STRONG> +Author: <I>F. Bosisio</I><BR> +<SMALL> +E-mail: <A HREF="MAILTO:bosisio@mate.polimi.it"> + <TT>bosisio@mate.polimi.it</TT> + </A><BR> +Web page: <A HREF="http://ftp.mate.polimi.it/~bosisio/LaTeX"> + <TT>http://ftp.mate.polimi.it/~bosisio/LaTeX</TT> + </A><BR> +CTAN location: <A HREF="http://www.tex.ac.uk/tex-archive/macros/latex/contrib/supported/bosisio/"> + <TT>CTAN:macros/latex/contrib/supported/bosisio/</TT> + </A> +</SMALL> +</STRONG> +</P> + +<P ALIGN="LEFT"></P> + +<!------------------------------------------------------------------------> + +<H3>Abstract:</H3> +<DIV> +Documentation for the package <TT>sobolev</TT>. +</DIV> + +<HR><BR> +<!------------------------------------------------------------------------> + +<H2>Table of contents</H2> + +<P> +<UL> <!-- Beginning of list --> + <LI><A HREF="#SECTION01">Introduction</A> + <LI><A HREF="#SECTION02">The options</A> + <LI><A HREF="#SECTION03">The commands</A> + <UL> <!-- Beginning of sublist --> + <LI><A HREF="#SECTION03a">The <TT>\H</TT> command</A> + <LI><A HREF="#SECTION03b">The <TT>\Hdiv</TT> command</A> + <LI><A HREF="#SECTION03c">The <TT>\L</TT> command</A> + <LI><A HREF="#SECTION03d">The <TT>\W</TT> command</A> + <LI><A HREF="#SECTION03e">The <TT>\D</TT> command</A> + <LI><A HREF="#SECTION03f">The <TT>\Norm</TT> command</A> + <LI><A HREF="#SECTION03g">The <TT>\SemiNorm</TT> command</A> + <LI><A HREF="#SECTION03h">The <TT>\Scalar</TT> command</A> + <LI><A HREF="#SECTION03i">The <TT>\Crochet</TT> command</A> + </UL> <!-- End of sublist --> +</UL> <!-- End of list --> +</P> + +<HR><BR> +<!------------------------------------------------------------------------> + +<H2> +<A NAME="SECTION01">Introduction</A> +</H1> + +<P> +This package provides some commands which are useful when dealing with +Sobolev spaces and their relatives. + +<P> +In particular some commands are redefined, so care should be taken, +expecially when including this package in an already existent LaTeX file. + +<P> +The redefined commands are <TT>\H</TT> and <TT>\L</TT>. +The effect of ``<TT>\H</TT>'' (which is a type of accent) can now be achieved +by the command ``<TT>\HAccent</TT>'', whilst the job of ``<TT>\L</TT>'' +(i.e. print an ``L'' with a superimposed bar) is now done by the command +``<TT>\Lbar</TT>''. + +<H2> +<A NAME="SECTION02">The options</A> +</H2> + +<P> +Two options are available at the moment: <TT>DivInBrackets</TT> and +<TT>DivAsExponent</TT>. +They only affect the output of the ``<TT>\Hdiv</TT>'' command. + +<P> +The firt options (<TT>DivInBrackets</TT>, which is the default) makes +<TT>\Hdiv</TT> behave like ``<TT>H(div;</TT>...<TT>)</TT>'', while the +second one (<TT>DivAsExponent</TT>) makes <TT>\Hdiv</TT> expand to +``<TT>H^{div}(</TT>...<TT>)</TT>''. + +<H1> +<A NAME="SECTION03">The commands</A> +</H1> + +<P> +Most of the subsequent space-generating commands have mandatory arguments to +indicate the type of the space. +Often this argument consists of a single digit: in this case it is <EM>not</EM> +necessary to enclose it in brackets, since in LaTeX the names of commands +consists of letters only, and so a digit following it is certainly an argument. +This saves a lot of typing and is the only reason that makes these commans +useful (if you always had to type the brackets, then it would have been simpler +to type the expansion of the command than the command itself !). +In other words, you can think as if several commands exist (like <TT>\H</TT>, +<TT>\H1</TT>, <TT>\H10</TT>, etc.), the ones with the digit beeing a sort of +abbreviation for the general one. + +<H3> +<A NAME="SECTION03a">The <TT>\H</TT> command</A> +</H3> + +<P> +The <TT>\H</TT> command is used to generate the symbol of sobolev spaces. +It takes a mandatory argument, which is used as a superscript, and an optional +argument, which is used as a subscript. + +<P> +As explained above, if the mandatory argument is a digit, it need not be +enclosed in brackets. +Moreover, if the optional argument is the digit ``0'', it can be typed without +the square brackets. + +<P> +Here are some examples (whith the <TT>\DefaultSet</TT> set to its default +value <TT>\Omega</TT>): +<PRE> + \H2 ==> H^2(\Omega) + \H10 ==> H^1_0(\Omega) + \H1[\Gamma_D] ==> H^1_{\Gamma_D}(\Omega) + \H^{-1/2} ==> H^{-1/2}(\Omega) +</PRE> + +<H3> +<A NAME="SECTION03b">The <TT>\Hdiv</TT> command</A> +</H3> + +<P> +The <TT>\Hdiv</TT> command is used to generate the sobolev space +called ``H div''. +It takes only an optional argument, which is used as a subscript and which +need not to be surrounded by the square brackets if it is the digit ``0''. + +<P> +If the (default) option <TT>DivInBrackets</TT> is in effect, it differs from +the command <TT>\H</TT> in that the word ``div'' is printed (in roman type) +inside brackets, before the set. +If, instead, the option <TT>DivAsExponent</TT> is active, then it is simply +an abbreviation for <TT>\Hdiv</TT>. + +<P> +Here are some examples: +<PRE> + DivInBrackets DivAsExponent + \Hdiv ==> H(\mathrm{div};\Omega) H^{\mathrm{div}}(\Omega) + \Hdiv0 ==> H_0(\mathrm{div};\Omega) H^{\mathrm{div}}_0(\Omega) + \Hdiv[\Gamma_D] ==> H_{\Gamma_D}(\mathrm{div};\Omega) H^{\mathrm{div}}_{\Gamma_D}(\Omega) +</PRE> + +<H3> +<A NAME="SECTION03c">The <TT>\L</TT> command</A> +</H3> + +<P> +The <TT>\L</TT> command is used to generate the symbol of Lebesgue-measurable +functions. +It has one argument which is the exponent of the L-space. +Again, if this argument is a digit (or a single symbol, +like ``<TT>\infty</TT>'') the surrounding braces are optional. +Like for the <TT>\H</TT> command, the output of <TT>\DefaultSet</TT> +is appended. + +<P> +Here are some examples: +<PRE> + \L2 ==> L^2(\Omega) + \L10 ==> L^{10}(\Omega) + \L ==> L^\infty(\Omega) +</PRE> + +<H3> +<A NAME="SECTION03d">The <TT>\W</TT> command</A> +</H3> + +<P> +The <TT>\W</TT> command is completly analogous, except that it prints a ``W'' +insted af an ``L'' and that it has two argument, both printed as a supercript, +separated by a comma. +It is used for the generalized Sobolev spaces. + +<P> +Here is an example of how it is used: +<PRE> + \W{k}{p} ==> W^{k,p}(\Omega) + \W1 ==> W^{1,\infty}(\Omega) +</PRE> + +<H3> +<A NAME="SECTION03e">The <TT>\D</TT> command</A> +</H3> + +<P> +The <TT>\D</TT> command is used in the theory of distributions: +it prints the space of distributions over the <TT>\DefaultSet</TT> if +followed by a prime symbol, or its dual space, otherwise. +<PRE> + \D ==> \mathcal{D}(\Omega) + \D' ==> \mathcal{D}'(\Omega) +</PRE> + +<H3> +<A NAME="SECTION03f">The <TT>\Norm</TT> command</A> +</H3> + +<P> +The <TT>\Norm</TT> command has a mandatory and an optional argument; +it generates the norm of the mandatory argument, with the optional argument, +if present, as a whole subscript, to denote the space within which the norm +is taken. + +<P> +Some examples: +<PRE> + \Normf(x) ==> \left\Vert f(x)\right\Vert + \Normg[L^2] ==> \left\Vert g\right\Vert _{L^2} +</PRE> + +<H3> +<A NAME="SECTION03g">The <TT>\SemiNorm</TT> command</A> +</H3> + +<P> +The <TT>\SemiNorm</TT> command is completly analoguos, but generates the +semi-norm instead of the norm. + +<P> +Some examples: +<PRE> + \SemiNorm{f(x)} ==> \left\Vert f(x)\right\Vert + \SemiNormg[H^1] ==> \left\vert g\right\vert _{H^1} +</PRE> + +<H3> +<A NAME="SECTION03h">The <TT>\Scalar</TT> command</A> +</H3> + +<P> +The <TT>\Scalar</TT> command has two arguments; a third optional argument +(which is used as a whole subscript) may follow inside square brackets. +The output consists of the two arguments separated by a comma and enclosed +in a pair of adjustable-size brackets, with the optional argument placed as +a subscript (to denote the space inside which the scalr product is taken). + +<P> +Some examples: +<PRE> + \Scalar{f}{g} ==> \left(f,g\right) + \Scalar{u}{v}[L^2] ==> left(u,v\right)_{L^2} +</PRE> + +<H3> +<A NAME="SECTION03i">The <TT>\Crochet</TT> command</A> +</H3> + +<P> +The <TT>\Crochet</TT> command has two arguments; a third optional argument +(which is used as a whole subscript) may follow inside square brackets. +The output consists of the two arguments separated by a comma and enclosed +in a pair of adjustable-size angular-parenthesys, with the optional argument +placed as a subscript (to denote the space inside which the duality is taken). + +<P> +Some examples: +<PRE> + \Crochet{f}{g} ==> \left\langle f,g\right\rangle + \Crochet{u}{v}[D] ==> \left\langle u,v\right\rangle_D +</PRE> + +<HR><BR> +<!------------------------------------------------------------------------> + +<ADDRESS> +<I>Francesco Bosisio</I><BR> +E-mail: <A HREF="MAILTO:bosisio@mate.polimi.it"> + <TT>bosisio@mate.polimi.it</TT> + </A><BR> +Web page: <A HREF="http://ftp.mate.polimi.it/~bosisio/LaTeX"> + <TT>http://ftp.mate.polimi.it/~bosisio/LaTeX</TT> + </A> +</ADDRESS> + +<!------------------------------------------------------------------------> + +</BODY> +</HTML> |