diff --git a/.codespell-ignorelines b/.codespell-ignorelines index d1ba58d..8e1d679 100644 --- a/.codespell-ignorelines +++ b/.codespell-ignorelines @@ -8,3 +8,10 @@ # lines that codespell should ignore: whitespace matters! +# jmacros.tex: "short implies" not simplifies! +\def\simplies{\;\Rightarrow\;} + +# figMac.tex +\openin\labeLfile=\figdir#1.lbl +\openin\labeLfile=\figdir#2.lbl + diff --git a/extra/CoverPage2018.pdf b/extra/CoverPage2018.pdf new file mode 100644 index 0000000..72be875 Binary files /dev/null and b/extra/CoverPage2018.pdf differ diff --git a/extra/Ex3point3v2.pdf b/extra/Ex3point3v2.pdf new file mode 100644 index 0000000..aed8aa0 Binary files /dev/null and b/extra/Ex3point3v2.pdf differ diff --git a/extra/check.tex b/extra/check.tex new file mode 100644 index 0000000..f452302 --- /dev/null +++ b/extra/check.tex @@ -0,0 +1,114 @@ +\def\ftmagnification{1200} +\def\spacingNumerator{1} +\def\spacingDenominator{1} +\input jmacros +%\input figMac +\def\figdir{fig/} +\def\showintremarks{n} + +\titlea{The Check Column} + +Ordinary arithmetic errors are a big problem when you do row operations +by hand. There is a technique called ``the check column'' (that is modeled +after the ``parity bit'' in computer hardware design) which provides a +very effective way to catch mechanical errors. Here is an example which +illustrates the technique + + + +\example{}{ +The augmented matrix for the system of equations +$$\deqalign{ + 2&x_1&+&&x_2&+3x_3&=\,&&1\cr + 4&x_1&+&5&x_2&+7x_3&=\,&&7\cr + 2&x_1&-&5&x_2&+5x_3&=\,&-&7\cr +}$$ +is +$$ +\left[\left.\matrix{2&1&3\cr 4&5&7\cr 2&-5 &5\cr} +\right|\matrix{1\cr7\cr-7\cr}\right] +$$ +To implement a ``check column'' you tack onto the right hand side of the +augmented matrix an additional column. Each entry in this check column is +the sum of all the entries in the row of the augmented matrix that is to +the left of the check column entry. For example, the top entry in the check +column is $2+1+3+1=7$. +$$ +\left[\left.\matrix{2&1&3\cr 4&5&7\cr 2&-5 &5\cr} +\right|\matrix{1\cr7\cr-7\cr}\right]\matrix{7\cr 23\cr -5} +$$ + To use the check column you just perform the same row +operations on the check column as you do on the augmented matrix. After +each row operation you check that each entry in the check column is +still the sum of all the entries in the corresponding row of the augmented +matrix. + +We now want to eliminate the $x_1$'s from equations (2) and (3). That is, +we want to make the first entries in rows 2 and 3 of the augmented matrix +zero. We can achieve this by subtracting two times row (1) from row (2) and +subtracting row (1) from row (3). +$$ +\matrix{(1)\cr (2)-2(1)\cr (3)-(1)} +\left[\left.\matrix{2&1&3\cr 0&3&1\cr 0&-6 &2\cr} +\right|\matrix{1\cr 5\cr-8\cr}\right]\matrix{7\cr 9\cr -12} +$$ +Observe that the check column entry $9$ is the sum $0+3+1+5$ +of the entries in the second row of the augmented matrix. If this were +not the case, it would mean that we made a mechanical error. Similarly +the check column entry $-12$ is the sum $0-6+2-8$. + +We have now succeeded in eliminating all of the $x_1$'s from equations +(2) and (3). For example, row 2 now stands for the equation +$$ +3x_2+x_3=5 +$$ +We next use equation (2) to eliminate all $x_2$'s from equation +(3). +$$ +\matrix{(1)\cr (2)\cr (3)+2(2)} +\left[\left.\matrix{2&1&3\cr 0&3&1\cr 0&0 &4\cr} +\right|\matrix{1\cr5\cr2\cr}\right]\matrix{7\cr 9\cr 6} +$$ +We can now easily solve (3) for $x_3$, substitute the result back into (2) and +solve for $x_2$ and so on: +$$\meqalign{ +(3)&\implies && 4x_3& =2 && &\implies && x_3&=\half\cr +(2)&\implies && 3x_2+\half&=5 && &\implies && x_2&=\sfrac{3}{2}\cr +(1)&\implies && 2x_1+\sfrac{3}{2}+3\times\half&=1 && &\implies && x_1&=-1\cr +}$$ +This last step is called ``backsolving''. + +Note that there is an easy way to make sure that we have not made any mechanical +errors in deriving this solution --- just substitute the purported solution +$(-1,3/2,1/2)$ back into the original system: +$$\deqalign{ + 2&(-1)&\,+\,&&\sfrac{3}{2}&+3\times\half&=\,&&1\cr + 4&(-1)&\,+&5\times &\sfrac{3}{2}&+7\times\half&=\,&&7\cr + 2&(-1)&\,-&5\times &\sfrac{3}{2}&+5\times\half&=\,&-&7\cr +}$$ +and verify that each left hand side really is equal to its corresponding +right hand side. +} + + + + + + + + + + + + + + + + + + + + + + +\end diff --git a/extra/complex.tex b/extra/complex.tex new file mode 100644 index 0000000..4649da6 --- /dev/null +++ b/extra/complex.tex @@ -0,0 +1,329 @@ +\def\ftmagnification{1000} +\def\spacingNumerator{7} +\def\spacingDenominator{6} +\input jmacros +\def\figdir{fig/} +\input figMac +\def\date{March 4, 2011} +\footline={{\sevenrm\copyright\ Joel Feldman. 2011. All rights reserved.\hfill\date\hfill Complex Numbers and Exponentials} \hfill\folio} + + +\def\Re{{\rm Re\,}} +\def\Im{{\rm Im\,}} + +\titlea{Complex Numbers and Exponentials} +%%%%%%%%% +\titleb{Definition and Basic Operations} +%%%%%%%%% +A complex number is nothing more than a point in the $xy$--plane. The +sum and product of two complex numbers $\, (x_1,y_1)\, $ and $\, (x_2,y_2) \, $ +is defined by +$$\eqalign{ + (x_1,y_1) + (x_2,y_2) &= (x_1+x_2,y_1+y_2)\cr + (x_1,y_1)\, (x_2,y_2) &= (x_1x_2-y_1y_2,x_1y_2+x_2y_1)\cr +}$$ +respectively. It is conventional to use the notation $x+iy$ +(or in electrical engineering country $x+jy$) +to stand for the complex number $(x,y)$. In other words, it is conventional +to write $x$ in place of $(x,0)$ and $i$ in place of $(0,1)$. In this notation, +the sum and product of two complex numbers $\, z_1=x_1+i y_1 $ and +$\, z_2=x_2 +iy_2 \, $ is given by +$$\eqalign{ +z_1+z_2 &= (x_1+x_2)+i(y_1+y_2)\cr +z_1z_2 &= x_1x_2-y_1y_2+i(x_1y_2+x_2y_1)\cr +}$$ + The complex number $i$ has the special property +$$ +i^2 = (0+1i)(0+1i) = (0\times 0-1\times 1)+i(0\times 1+1\times 0) = -1 +$$ +For example, if $z=1+2i$ and $w=3+4i$, then +$$\deqalign{ +z+w&=(1+2i)+(3+4i)&=4+6i\cr +zw&=(1+2i)(3+4i)&=3+4i+6i+8i^2&=3+4i+6i-8=-5+10i\cr +}$$ +Addition and multiplication of complex numbers obey the familiar algebraic +rules +$$\meqalign{ +z_1+z_2&=z_2+z_1 && z_1z_2&=z_2z_1 \cr +z_1+(z_2+z_3)&=(z_1+z_2)+z_3 && z_1(z_2z_3)&=(z_1z_2)z_3 \cr +0+z_1&=z_1 && 1z_1&=z_1 \cr +z_1(z_2+z_3)&=z_1z_2+z_1z_3 && (z_1+z_2)z_3& = z_1z_3+z_2z_3 \cr +}$$ +The negative of any complex number $z= x+iy$ is defined by $-z=-x+(-y)i$, +and obeys $z+(-z)=0$. +%%%%%%%%% +\goodbreak +\titleb{Other Operations} +%%%%%%%%% +The complex conjugate of $\, z\, $ is denoted $\bar z$ and is +defined to be $\, \bar z=x-i y\, $. That is, to take the complex conjugate, one +replaces every $i$ by $-i$. Note that +$$ +z\bar z=(x+iy)(x-iy)=x^2-ixy+ixy+y^2=x^2+y^2 +$$ +is always a positive real number. In fact, it is the square of the distance +from $x+iy$ (recall that this is the point $(x,y)$ in the $xy$--plane) to $0$ +(which is the point $(0,0)$). The distance from $z=x+iy$ to $0$ is +denoted $\, |z|\, $ and is called the +absolute value, or modulus, of $\, z\, $. It is given by +$$ +|z|\ =\ \sqrt{x^2+y^2}\ =\ \sqrt{z\bar z} +$$ +Since $z_1z_2=(x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1)$, +$$\eqalign{ +|z_1z_2| &=\sqrt{(x_1x_2-y_1y_2)^2+(x_1y_2+x_2y_1)^2} \cr + &=\sqrt{x_1^2x_2^2-2x_1x_2y_1y_2+y_1^2y_2^2 ++x_1^2y_2^2+2x_1y_2x_2y_1+x_2^2y_1^2} \cr + &=\sqrt{x_1^2x_2^2+y_1^2y_2^2+x_1^2y_2^2+x_2^2y_1^2} +=\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)} \cr + &=|z_1||z_2| \cr +}$$ +for all complex numbers $\, z_1,z_2\, $. + +Since $|z|^2=z\bar z$, we have $z\big({ \bar z\over |z|^2}\big)=1$ +for all complex numbers $\, z\ne 0\, $. This says that the multiplicative +inverse, denoted $z^{-1}$ or $\sfrac{1}{z}$, of any nonzero complex number +$z=x+iy$ is +$$ +z^{-1}=\sfrac{\bar z}{|z|^2}=\sfrac{x-iy}{x^2+y^2} +=\sfrac{x}{x^2+y^2}-\sfrac{y}{x^2+y^2}i +$$ +It is easy to divide a complex number by +a real number. For example +$$ +\sfrac{11+2i}{25} = \sfrac{11}{25}+\sfrac{2}{25}i +$$ +In general, there is a trick for rewriting any ratio of complex numbers +as a ratio with a real denominator. For example, suppose that we want to +find $\sfrac{1+2i}{3+4i}$. The trick is to multiply by $1=\sfrac{3-4i}{3-4i}$. +The number $3-4i$ is the complex conjugate of +$3+4i$. Since $(3+4i)(3-4i)=9-12i+12i+16=25$ +$$ +\sfrac{1+2i}{3+4i}=\sfrac{1+2i}{3+4i}\sfrac{3-4i}{3-4i} +=\sfrac{(1+2i)(3-4i)}{25} +=\sfrac{11+2i}{25} += \sfrac{11}{25}+\sfrac{2}{25}i +$$ + +The notations $\Re z$ and $\Im z$ stand for the real and imaginary parts of the +complex number $z$, respectively. If $z=x+ iy$ (with $x$ and $y$ real) they +are defined by +$$ +\Re z=x\qquad \Im z=y +$$ +Note that both $\Re z$ and $\Im z$ are real numbers. +Just subbing in $\bar z=x-iy$ gives +$$ +\Re z=\half(z+\bar z)\qquad \Im z=\sfrac{1}{2i}(z-\bar z) +$$ + +\goodbreak +%%%%%%%%% +\titleb{The Complex Exponential} +%%%%%%%%% +\noindent{\bf Definition and Basic Properties.}\ +For any complex number $z=x+iy$ the exponential $\, e^z\, $, is defined +by +$$ +e^{x+iy}\ =\ e^x\cos y+i e^x\sin y +$$ +In particular, $e^{iy}\ =\ \cos y+i \sin y$. +This definition is not as mysterious as it looks. We could +also define $e^{iy}$ by the subbing $x$ by $iy$ in the Taylor series expansion +$e^x=\sum_{n=0}^\infty \sfrac{x^n}{n!}$. (If you don't know about this +Taylor series expansion, just skip the rest of this paragraph.) +$$ +e^{iy} = 1 +iy +\sfrac{(iy)^2}{2!}+\sfrac{(iy)^3}{3!}+\sfrac{(iy)^4}{4!} ++\sfrac{(iy)^5}{5!}+\sfrac{(iy)^6}{6!}+\cdots +$$ +The even terms in this expansion are +$$ +1 +\sfrac{(iy)^2}{2!}+\sfrac{(iy)^4}{4!}+\sfrac{(iy)^6}{6!}+\cdots +=1 -\sfrac{y^2}{2!}+\sfrac{y^4}{4!}-\sfrac{y^6}{6!}+\cdots +=\cos y +$$ +and the odd terms in this expansion are +$$ + iy +\sfrac{(iy)^3}{3!}+\sfrac{(iy)^5}{5!}+\cdots +=i\Big(y-\sfrac{y^3}{3!}+\sfrac{y^5}{5!}+\cdots\Big) +=i\sin y +$$ + + +For any two complex numbers $z_1$ and $z_2$ +$$\eqalign{ +e^{z_1}e^{z_2} +\ &=\ e^{x_1}(\cos y_1+i \sin y_1)e^{x_2}(\cos y_2+i \sin y_2)\cr +\ &=\ e^{x_1+x_2}(\cos y_1+i \sin y_1)(\cos y_2+i \sin y_2)\cr +\ &=\ e^{x_1+x_2}\left\{(\cos y_1\cos y_2-\sin y_1\sin y_2) ++i (\cos y_1\sin y_2+\cos y_2\sin y_1)\right\}\cr +\ &=\ e^{x_1+x_2}\left\{\cos(y_1+y_2)+i\sin(y_1+y_2)\right\}\cr +\ &=\ e^{(x_1+x_2)+i(y_1+y_2)}\cr +\ &=\ e^{z_1+z_2}\cr +}$$ +so that the familiar multiplication formula also applies to complex exponentials. +For any complex number $c=\al+i\be$ and real number $t$ +$$ +e^{ct}=e^{\al t+i\be t}=e^{\al t}[\cos( \be t)+i\sin(\be t)] +$$ +so that the derivative with respect to $t$ +$$\eqalign{ +\sfrac{d\hfill}{dt} e^{ct} +&=\al e^{\al t}[\cos( \be t)+i\sin(\be t)] ++e^{\al t}[-\be\sin( \be t)+i\be \cos(\be t)]\cr +&=(\al+i\be) e^{\al t}[\cos( \be t)+i\sin(\be t)]\cr +&=ce^{ct}\cr +}$$ +is also the familiar one. +%%%%%%%%%%% +\vskip.2in\noindent +{\bf Relationship with $\sin$ and $\cos$.}\ +When $\th$ is a real number +$$\eqalign{ +e^{i \th} &= \cos \th+i \sin \th\cr +e^{-i \th} &= \cos \th-i \sin \th=\overline{e^{i\th}}\cr +}$$ +are complex numbers of modulus one. +Solving for $\cos\th$ and $\sin\th$ (by adding and subtracting the two equations) +$$\deqalign{ +\cos\th&=\sfrac{1}{2}(e^{i\th}+e^{-i\th})&=\Re e^{i\th}\cr +\sin\th&=\sfrac{1}{2i}(e^{i\th}-e^{-i\th})&=\Im e^{i\th}\cr +}$$ +These formulae make it easy derive trig identities. For example +$$\eqalign{ +\cos\th\cos\phi &= \sfrac{1}{4}(e^{i\th}+e^{-i\th})(e^{i\phi}+e^{-i\phi})\cr + &= \sfrac{1}{4}(e^{i(\th+\phi)}+e^{i(\th-\phi)} ++e^{i(-\th+\phi)}+e^{-i(\th+\phi)})\cr +&= \sfrac{1}{4}(e^{i(\th+\phi)}+e^{-i(\th+\phi)}+e^{i(\th-\phi)} ++e^{i(-\th+\phi)})\cr +&= \sfrac{1}{2}\big(\cos(\th+\phi)+\cos(\th-\phi)\big)\cr +}$$ +and, using $(a+b)^3=a^3+3a^2b+3ab^2+b^3$, +$$\eqalign{ +\sin^3\th&=-\sfrac{1}{8i}\big(e^{i\th}-e^{-i\th}\big)^3\cr +&=-\sfrac{1}{8i}\big(e^{i3\th}-3e^{i\th}+3e^{-i\th}-e^{-i3\th}\big)\cr +&=\sfrac{3}{4}\sfrac{1}{2i}\big(e^{i\th}-e^{-i\th}\big) +-\sfrac{1}{4}\sfrac{1}{2i}\big(e^{i3\th}-e^{-i3\th}\big)\cr +&=\sfrac{3}{4}\sin\th-\sfrac{1}{4}\sin(3\th)\cr +}$$ +and +$$\eqalign{ +\cos(2\th)&=\Re e^{i2\th}=\Re \big(e^{i\th}\big)^2\cr +&=\Re \big(\cos \th+i\sin\th\big)^2\cr +&=\Re \big(\cos^2 \th+2i\sin\th\cos\th-\sin^2\th\big)\cr +&=\cos^2\th-\sin^2\th\cr +}$$ +%%%%%%%%%%% +\vskip.2in\noindent +{\bf Polar Coordinates.}\ +Let $z=x+iy$ be any complex number. Writing $(x,y)$ in polar coordinates +in the usual way gives $x=r\cos\th$, $y=r\sin\th$ and +$$ +x+iy=r\cos\theta+ir\sin\theta=re^{i\theta}\qquad +\figplace{polar}{0 in}{-.4 in} +$$ +In particular +$$ +\hbox{\figplace{polar2}{-.3 in}{-.4 in} +\vbox{\halign{\hfil$#$ &\hfil$#\ =\ $\hfil &\hfil$#$\hfil &\hfil$#\ =\ $\hfil +&\hfil$#$\hfil &\hfil$#\ =\ $\hfil &\hfil$#$\hfil &#\hskip.2in for $k=0,\pm 1,\pm2,\cdots$\hfil \cr +1 && e^{i0} && e^{2\pi i} && e^{2k\pi i} &\cr +-1 && e^{i\pi} && e^{3\pi i} && e^{(1+2k)\pi i} &\cr +i && e^{i\pi/2}&& e^{{5\over 2}\pi i}&& e^{({1\over 2}+2k)\pi i}&\cr +-i && e^{-i\pi/2}&&e^{{3\over 2}\pi i}&& e^{(-{1\over 2}+2k)\pi i}&\cr +}}}$$ +The polar coordinate $\th=\tan^{-1}\sfrac{y}{x}$ associated with the +complex number $z=x+iy$ is also called the argument of $z$. + +The polar coordinate representation makes it easy to find square roots, +third roots and so on. Fix any positive integer $n$. The $n^{\rm th}$ roots +of unity are, by definition, all solutions $z$ of +$$ +z^n\ =\ 1 +$$ +Writing $z=re^{i\th}$ +$$ +r^ne^{n\th i}\ =\ 1e^{0i} +$$ +The polar coordinates $(r,\th)$ and $(r',\th')$ represent the same point +in the $xy$--plane if and only if $r=r'$ and $\th=\th'+2k\pi$ for some +integer $k$. So $z^n=1$ if and only if $r^n=1$, i.e. $r=1$, and $n\th =2 +k\pi$ for some integer $k$. The $n^{\rm th}$ roots of unity are all complex numbers +$e^{2\pi i{k\over n}}$ with $k$ integer. There are precisely $n$ distinct +$n^{\rm th}$ roots of unity because $e^{2\pi i{k\over n}}=e^{2\pi i{k'\over n}}$ +if and only if $2\pi {k\over n}-2\pi i{k'\over n}=2\pi {k-k'\over n}$ is +an integer multiple of $2\pi$. That is, if and only if $k-k'$ is an integer +multiple of $n$. The $\, n\, $ distinct nth roots of unity are +$$ +1\ ,\ e^{2\pi i{1\over n}}\ ,\ e^{2\pi i{2\over n}} +\ ,\ e^{2\pi i{3\over n}}\ ,\ \cdots\ ,\ e^{2\pi i{n-1\over n}} +\figplace{polar3}{.8 in}{-.7 in} +$$ + + +%%%%%%%%%%% +\vskip.2in\noindent +{\bf Phasors and Phasor Diagrams.}\ +Algebraic expressions involving complex numbers may be evaluated +geometrically by exploiting the following two observations. +\item{$\circ$}(Addition and subtraction) A complex number is nothing more +than a point in the $xy$--plane. So we may identify the complex number +$A=a+ib$ with the vector whose tail is at the origin and whose head is at +the point $(a,b)$. Similarly, we may identify the complex number +$C=c+id$ with the vector whose tail is at the origin and whose head is at +the point $(c,d)$. Those two vectors form two sides of a parallelogram. +The vector for the sum $A+C=(a+c)+i(b+d)$ is that from the origin to +the diagonally opposite corner of the parallelogram. +The vector for the difference $A-C=(a-c)+i(b-d)$ has its tail at $C$ and +its head at $A$. + +\centerline{\figput{sum}\hskip1.0in\figput{diff}} + +\item{$\circ$}(Multiplication and Division) To multiply or divide two +complex numbers, write them in their polar coordinate forms $A= re^{i\th}$, +$C=\rho e^{i\varphi}$. So $r$ and $\rho$ are the lengths of $A$ and $C$, +respectively, and $\th$ and $\varphi$ are the angles from the positive $x$--axis +to $A$ and $C$, respectively. Then $AC=r\rho e^{i(\th+\varphi)}$. This vector +has length equal to the product of the lengths of $A$ and $C$. The angle +from the positive $x$--axis to $AC$ is the sum of the angles $\th$ and +$\varphi$. And $\sfrac{A}{C}=\sfrac{r}{\rho} e^{i(\th-\varphi)}$. This vector +has length equal to the ratio of the lengths of $A$ and $C$. The angle from the +positive $x$--axis to $AC$ is the difference of the angles $\th$ and $\varphi$. + +\centerline{\figput{mult}\hskip1.0in\figput{div}} + + + +\noindent +Complex numbers are also called ``phasors'' by some electrical engineers. +They call the diagrams resulting from the geometric evaluation, as above, +of algebraic expressions involving complex numbers ``phasor diagrams''. +For example, suppose that an AC signal of frequency $\om$ is applied to +the left hand end of the parallel circuit + +\centerline{\figput{parallel}} + +\noindent Then the impedances across the three circuit elements are +$$ +Z_R=R\qquad +Z_L=i\om L\qquad +Z_C=\sfrac{1}{i\om C} +$$ +and the impedance, $Z$, of the parallel circuit as a whole is determined +by +$$ +\sfrac{1}{Z}=\sfrac{1}{Z_R}+\sfrac{1}{Z_C}+\sfrac{1}{Z_L} +=\sfrac{1}{R}+i\om C-\sfrac{i}{\om L} +$$ +To evaluate $Z$ geometrically, we +\item{$\circ$}add $Z_L^{-1}=-\sfrac{i}{\om L}$ to $Z_C^{-1}=i\om C$ +(In the phasor diagram, below, I am considering the case that +$\sfrac{1}{\om L}>\om C> 0$.) +\item{$\circ$}add $Z_R^{-1}=\sfrac{1}{R}$ to the result to give +$\sfrac{1}{Z}$ and finally +\item{$\circ$}invert the result, using that $\sfrac{1}{re^{i\th}}=\sfrac{1}{r}e^{-i\th}$ + +\centerline{\figput{phasordia}} + +\end + diff --git a/extra/figMac.tex b/extra/figMac.tex new file mode 100644 index 0000000..5956532 --- /dev/null +++ b/extra/figMac.tex @@ -0,0 +1,213 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%% Macros for figure insertion +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%% +%%%%%%% The two main figure insertion macros are +%%%%%%% +% \figput{} +% \figplace{}{}{} +%%%%%%% +%%%%%%% The first just inserts the figure at the current location. The +%%%%%%% second inserts the figure at the current location but then shifts +%%%%%%% horizontally by the second argument and vertically by the third. +%%%%%%% +%%%%%%% Some typical TeX commands for inserting figures are +%%%%%%% \centerline{\figput{}} +%%%%%%% \vadjust{\centerline{\figput{}}} +%%%%%%% \midinsert\centerline{\figput{}}\endinsert +%%%%%%% \topinsert\centerline{\figput{}}\endinsert + + + + +%%%%%%% +%%%%%%% TO SET A FIGURE DIRECTORY INSERT, FOR EXAMPLE, +%%%%%%% \def\figdir{figures/} +%%%%%%% IN YOUR SOURCE FILE. REMEMBER THE TAILING / +%%%%%%% + + + +%%%%%%% +%%%%%%% SELECT (a) YOUR POSTSCRIPT FILE SUFFIX AND (b) YOUR SYSTEM NOW! +%%%%%%% +\def\suffix{ps} +\newcount\system +%\global\system=1 % for textures +%\global\system=2 % for msdos +\global\system=3 % for unix(dvips) +%\global\system=4 % for unix(dvips) scaled by a factor of 1.2 +%\global\system=6 % for xdvik + + + +\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax} +\ifundefined{figdir}\def\figdir{}\fi +% +% Now for the definitions and main macro for figure inclusion. +% +\newcount\firstline +\newdimen\pswidth \newdimen\xleft +\newdimen\psheight \newdimen\ytop \newdimen\ybot +\newcount\justx \newcount\justy +\global\justx=0 \global\justy=0 +\newdimen\vpos \newtoks\labeL +\newread\labeLfile \newdimen\xcoord \newdimen\ycoord +\newif\ifdoit +\newbox\labox +% variables for use with xdvik +\newdimen\xdvikwid +\newdimen\xdvikht +\newdimen\pspoints +\newdimen\rwi +\pspoints=1bp +% +\newcount\temp +\def\readdim#1{\global\read\labeLfile to \temp +\global #1=\temp pt} +% +% +% figcrop{} treats the first two labels as marking +% the upper left and lower right corners of the figure. This is for +% positioning purposes only. The figure may extend beyond the corners. +% The corner markers are not printed. +% +% +\def\figcrop#1{\par% #1=filename +\openin\labeLfile=\figdir#1.lbl +\global\read\labeLfile to\firstline\message{#1} +\global\read\labeLfile to\temp%read overall dimensions +\readdim{\ybot} +\readdim{\xleft}% read upper left point +\readdim{\ytop} +\global\read\labeLfile to\justx%ignore +\global\read\labeLfile to\justy%ignore +\global\read\labeLfile to\labeL%ignore +\readdim{\pswidth}% read lower right point +\global\advance\pswidth by -\xleft +\readdim{\psheight} +\global\advance\ybot by -\psheight +\global\advance\psheight by -\ytop +\global\read\labeLfile to\justx%ignore +\global\read\labeLfile to\justy%ignore +\global\read\labeLfile to\labeL%ignore +\vbox to\psheight{\vfill +%%% +%%% NOTE: next line may have to be changed for your DVIPS driver %%% +\ifnum\system=1\special{postscript grestore newpath gsave}\fi %textures +\ifnum\system=2\special{postscript grestore newpath gsave}\fi %msdos +\ifnum\system=3 + %% \special{" grestore newpath gsave} + \fi %%unix:dvips +\ifnum\system=4\special{" grestore newpath gsave}\fi %%unix:dvips,scaled +\ifnum\system=1 +\hbox to \pswidth{\kern-\xleft\special{postscriptfile \figdir#1.\suffix }\hfil}\fi + %textures +\ifnum\system=2 +\hbox to \pswidth{\kern-\xleft\special{ps: plotfile \figdir#1.\suffix }\hfil}\fi + %mdos +\ifnum\system=3 +\hbox to \pswidth{\kern-\xleft\special{psfile=\figdir#1.\suffix }\hfil}\fi + %unix:dvips +\ifnum\system=4 +\hbox to \pswidth{\kern-\xleft\special{psfile=\figdir#1.\suffix hscale=120 vscale=120}\hfil}\fi + %unix:dvips,scaled +\ifnum\system=5 +\hbox to \pswidth{\kern-\xleft\special{psfile="\figdir#1.\suffix"}\hfil}\fi %orphee +\ifnum\system=6 + \xdvikwid=\pswidth + \xdvikht=\psheight + {\global\divide\xdvikwid by \pspoints} + {\global\divide\xdvikht by \pspoints} + \rwi=\xdvikwid + {\global\multiply\rwi by 10} +\hbox to \pswidth{\kern-\xleft\special{psfile=\figdir#1.\suffix\space + llx=0\space lly=0\space + urx=\number\xdvikwid\space ury=\number\xdvikht\space + rwi=\number\rwi}\hfil}\fi %xdvik +%%% +\vskip -\baselineskip +\vskip -\ybot +\vskip-\psheight % +\hbox to\pswidth {\hss}% +\parindent=0pt\offinterlineskip +\vpos=0 pt% +\loop\readdim{\xcoord} +\ifdim \xcoord < -999pt \doitfalse\else\doittrue\fi +\ifdoit \advance \xcoord by -\xleft +\readdim{\ycoord} +\advance \ycoord by -\ytop +\global\read\labeLfile to\justx +\global\read\labeLfile to\justy +\global\read\labeLfile to\labeL +\global\setbox\labox=\hbox{\labeL\hskip-0.3em}% +\advance\vpos by-\ycoord +\vskip-\vpos \vpos=\ycoord +\hbox to\pswidth{\hskip\xcoord % +\hbox to 0pt{\ifnum\justx>0\hss\fi% +\vbox to0pt{% +\ifnum\justy<2\vss\fi% +\copy\labox\kern0pt% +\ifnum\justy>0\vss\fi}% +\ifnum\justx<2\hss\fi}% +\hss}% +\repeat% +\advance\vpos by-\psheight% +\vskip-\vpos % +}\closein\labeLfile} +% +% +% \figplace{}{}{} +% moves to the right by and down by +% and then applies \figcrop +% +\def\figplace#1#2#3{ +\openin\labeLfile=\figdir#1.lbl +\ifeof \labeLfile + \immediate\write16{***Can't find \figdir#1.lbl; Skipping it.***} +\else \closein\labeLfile + \null\hskip#2\raise #3 \hbox{\figcrop{#1}} +\fi +} +% +% +% \figput{} +% +% just applies \figcrop +% +\def\figput#1{ +\openin\labeLfile=\figdir#1.lbl +\ifeof \labeLfile + \immediate\write16{***Can't find \figdir#1.lbl; Skipping it.***} +\else \closein\labeLfile + \hbox{\figcrop{#1}} +\fi +} + +% +% +% \rfig{text}{}{caption} +% +% Macro for text on left side of page and figure on right side. +% Figure is captioned below. +% +\newdimen\textsize +\long\def\rfig#1#2#3{\par\smallskip +\openin\labeLfile=\figdir#2.lbl +\ifeof\labeLfile + \immediate\write16{***Can't find \figdir#2.lbl; Skipping it.***} +\else + \global\read\labeLfile to\headerinfo + \readdim{\pswidth} + \closein\labeLfile + + \textsize=\hsize + \advance\textsize by-\pswidth + \advance\textsize by-1pc + \line{\vtop{\hsize=\textsize{#1}\vfil} + \hfill\vtop{\hsize=\pswidth \vskip0pt + \figcrop{#2}\smallskip + \line{\hfil#3\hfil}\smallskip}} +\fi +} + diff --git a/extra/figures/diff.eps b/extra/figures/diff.eps new file mode 100644 index 0000000..5018e39 --- /dev/null +++ b/extra/figures/diff.eps @@ -0,0 +1,715 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Creator: dvips(k) 5.98 Copyright 2009 Radical Eye Software +%%Title: figure.dvi +%%CreationDate: Sat May 30 17:23:54 2015 +%%BoundingBox: 262 524 352 614 +%%DocumentFonts: CMMI10 CMSY10 +%%EndComments +%DVIPSWebPage: (www.radicaleye.com) +%DVIPSCommandLine: dvips -E -o diff.eps figure +%DVIPSParameters: dpi=600 +%DVIPSSource: TeX output 2015.05.30:1723 +%%BeginProcSet: tex.pro 0 0 +%! 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+%%Date: Wed Jul 17 09:30:43 1996 +%%EndComments + +gsave + + +%% ellipse library + +/ellipsedict 8 dict def %% arguments: xcentre ycentre xradius yradius +ellipsedict /emtrx matrix put %% startangle endangle + +/ellipse + {ellipsedict begin + /endangle exch def /startangle exch def + /yrad exch def /xrad exch def + /y exch def /x exch def + /savematrix emtrx currentmatrix def + x y strans + xrad xscale mul yrad yscale mul scale + newpath + 0 0 1 startangle endangle arc + savematrix setmatrix + end + } def + + +%% scr2ps dictionary + +/scr2psDict 30 dict def +scr2psDict begin + /iscp false def + {pop pop /iscp true def} {pop pop} {pop pop pop pop pop pop} {} pathforall + iscp {currentpoint translate} if + /np {newpath} bind def + /mt {moveto} bind def + /li {lineto} bind def + /ls {lineto stroke} bind def + /setxscale {/xscale exch def} bind def + /setyscale {/yscale exch def} bind def + /smt {yscale mul exch xscale mul exch moveto} bind def + /sli {yscale mul exch xscale mul exch lineto} bind def + /sls {yscale mul exch xscale mul exch lineto stroke} bind def + /strans {yscale mul exch xscale mul exch translate} bind def + /sct {yscale mul /y3 exch def + xscale mul /x3 exch def + yscale mul /y2 exch def + xscale mul /x2 exch def + yscale mul exch xscale mul exch x2 y2 x3 y3 curveto} bind def + /setpsfont {exch findfont exch scalefont setfont} bind def + /stringsize {/temp exch def gsave newpath 0 0 moveto + temp true charpath flattenpath pathbbox + /strh exch def /strw exch def pop pop grestore} bind def + +1 setxscale +1 setyscale + +np -1.0 -11.8 mt 123.4 -11.8 li 123.4 112.6 li -1.0 112.6 li closepath clip + 0.0 -10.8 translate + + 0.0 setlinewidth + + 1.020 setxscale + 60.000 0 strans + + 1.020 setyscale + 0 60.000 strans + +%% draw line + np -50.000 0.000 smt 50.000 0.000 sls + +%% draw line + np 0.000 -50.000 smt 0.000 50.000 sls + + 0.000 0.000 10.000 10.000 0.0 90.0 ellipse stroke + 0.000 0.000 12.000 12.000 -90.0 0.0 ellipse stroke + 0.000 0.000 8.000 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600 (figure.dvi) +@start /Fa 212[77 43[{}1 99.6264 /CMR10 rf /Fb 188[71 +1[75 65[{}2 99.6264 /CMMI10 rf end +%%EndProlog +%%BeginSetup +%%Feature: *Resolution 600dpi +TeXDict begin + end +%%EndSetup +TeXDict begin 1 0 bop 1607 1622 a @beginspecial @setspecial +%%BeginDocument: sum.ps +%!PS-Adobe-3.0 +%%BoundingBox: 0.0 0.0 72.0 72.0 +%%Title: +%%Created by: Figure.java +%%Date: Fri Dec 31 23:15:19 PST 2004 +%%EndComments + +gsave + +%% ellipse library + +/ellipsedict 10 dict def %% arguments: xcentre ycentre xradius yradius +ellipsedict /emtrx matrix put %% startangle endangle closetype + +/ellipse %% closetype: 0 for open, 1 for chord, 2 for wedge + {ellipsedict begin + /closetype exch def + /endangle exch def /startangle exch def + /yrad exch def /xrad exch def + /y exch def /x exch def + /savematrix emtrx currentmatrix def + x y translate + xrad yrad scale + newpath + closetype 2 eq {0 0 moveto} if + 0 0 1 startangle endangle arc + closetype 1 eq {closepath} if + closetype 2 eq {closepath} if + savematrix setmatrix + end + } def + +%% Figure2ps dictionary + +/Figure2psDict 30 dict def +Figure2psDict begin + /iscp false def + {pop pop /iscp true def} {pop pop} {pop pop pop pop pop pop} {} pathforall + iscp {currentpoint translate} if + /setpsfont {exch findfont exch scalefont setfont} bind def + /stringsize {/temp exch def gsave newpath 0 0 moveto + temp true charpath flattenpath pathbbox + /strh exch def /strw exch def pop pop grestore} bind def + +newpath -1.0 -1.0 moveto 73.0 -1.0 lineto 73.0 73.0 lineto -1.0 73.0 lineto closepath clip +/Helvetica 10 setpsfont +0.4 setlinewidth +newpath +0.0 12.0 moveto +72.0 12.0 lineto stroke +newpath +12.0 0.0 moveto +12.0 72.0 lineto stroke +newpath +48.0 24.0 moveto +60.0 60.0 lineto stroke +newpath +24.0 48.0 moveto +60.0 60.0 lineto stroke +0.6 setlinewidth +newpath +12.0 12.0 moveto +48.0 24.0 lineto stroke +gsave + 0.0 setgray +newpath 48.0 24.0 moveto 41.675444679663244 24.0 lineto 43.730925158772685 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differ diff --git a/extra/jmacros.tex b/extra/jmacros.tex new file mode 100644 index 0000000..35644c5 --- /dev/null +++ b/extra/jmacros.tex @@ -0,0 +1,468 @@ +%%% +%%% jmacros.tex +%%% Version of 27/12/11 +%%% Joel Feldman feldman@math.ubc.ca +%%% +%%%%% +%%%%% To produce the jmacros documentation: +%%%%% 1. tex jmacrosdoc +%%%%% 2. xdvi jmacrosdoc +%%%%% + +% This is an interlock so the macros get loaded only once +% (see the TeXbook p. 383): +\ifx\jmacsin\relax\endinput\else\let\jmacsin=\relax\fi + +\def\ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax} +\ifundefined{ftmagnification} \def\ftmagnification{1200} \fi +\ifundefined{spacingNumerator} \def\spacingNumerator{3} \fi +\ifundefined{spacingDenominator} \def\spacingDenominator{2} \fi + + +%%%%%%% VARIOUS SETTINGS +\magnification\ftmagnification +\tolerance=10000 +\hsize=6.5truein\vsize=8.9truein +\def\letterPaper{\hsize=6.5truein\vsize=8.9truein} +\def\afourPaper{\hsize=17truecm\vsize=23truecm} +\parindent=20pt +\mathsurround=0pt +\ifundefined{singlespace} + \multiply\baselineskip by \spacingNumerator + \divide \baselineskip by \spacingDenominator \else \fi +\def\showintremarks{y} %%% choose this line to include internal remarks +%\def\showintremarks{n} %%% choose this line to exclude internal remarks +\long\def\intremark#1{\if\showintremarks y + \removelastskip\vskip.1in\hrule\penalty500\vskip.05in\noindent + {\bf Internal Remark}\vskip.01in\penalty500 + #1 + %\hfill\break{\bf End Internal Remark} + \vskip.05in\penalty500\hrule\vskip.1in\penalty-500 + \fi} + +% +% HEADERS +% +\def\today{\ifcase\month\or January\or February\or March\or April\or + May\or June\or July\or August\or September\or October\or November\or + December\fi\space\number\day, \number\year} +\def\getArchdate{\openin7=\jobname.arx + \ifeof7 \def\Archdate{\today} + \else \read7 to \Archdate\fi + \closein7} +\def\header#1{\getArchdate + \rm\nopagenumbers \hfil\underbar{#1} \hfil\Archdate\bigskip + \headline={\rm\ifnum\pageno>1 #1\hfil\Archdate\hfil Page \folio\else\hfil\fi}} +% +% STYLE SHORT FORMS +% +\def\dst{\displaystyle} +\def\sst{\scriptstyle} +\def\tst{\textstyle} +\def\ssst{\scriptscriptstyle} +% +% EQUATIONS +% +\def\frac#1#2{\dst {#1\over#2}} % fractions in displaystyle +\def\sfrac#1#2{{\mathchoice{{\tst{#1\over#2}}}{{\tst{#1\over#2}}} + {{#1\over#2}}{{#1\over#2}}}} + +\def\deqalign#1{\vcenter{\openup1\jot \mathsurround=0pt \ialign{ + \strut\hfil$\displaystyle{##}$&&$\displaystyle{{}##}$\hfil + \crcr + #1\crcr}}} % double eqalign + +\def\meqalign#1{\vcenter{\openup1\jot \mathsurround=0pt \ialign{ + &\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil& + \quad$##$\crcr + #1\crcr}}} % multiple eqalign +\def\leftdisplay#1{\leftline{$\displaystyle{#1}$}} +\def\leftrightdisplay#1#2{\line{$\displaystyle{#1}$\hfil$\displaystyle{#2}$}} + +\def\eqn#1{\eqno{({\rm #1})}} +% +% GREEK LETTERS +% +\def\al{\alpha} +\def\be{\beta} +\def\ga{\gamma} +\def\de{\delta} +\def\ep{\epsilon} +\def\ze{\zeta} +\def\et{\eta} +\def\th{\theta} +\def\ka{\kappa} +\def\la{\lambda} +\def\rh{\rho} +\def\si{\sigma} +\def\ta{\tau} +\def\up{\upsilon} +\def\ph{\phi} +\def\ch{\chi} +\def\ps{\psi} +\def\om{\omega} +\def\veps{\varepsilon} +\def\Ga{\Gamma} +\def\De{\Delta} +\def\Th{\Theta} +\def\La{\Lambda} +\def\Si{\Sigma} +\def\Up{\Upsilon} +\def\Ph{\Phi} +\def\Ps{\Psi} +\def\Om{\Omega} +% +% BOLD FACE, SCRIPT and ROMAN CHARACTERS +% +\def\pmb#1{\setbox0=\hbox{#1} % generate bold face + \kern-.025em\copy0\kern-\wd0 + \kern.05em\copy0\kern-\wd0 + \kern-.025em\box0} %Knuth puts in \raise.0433em before box0 +\def\0{{\bf 0}} +\def\k{{\bf k}} +\def\p{{\bf p}} +\def\q{{\bf q}} +\def\x{{\bf x}} +\def\y{{\bf y}} +\def\z{{\bf z}} +\def\bL{{\bf L}} +\def\rF{{\rm F}} +\def\bDe{{\bf \De}} +\def\bphi{{\bf \Phi}} +\def\bpsi{{\bf \Psi}} +\def\bPhi{{\bf \Phi}} +\def\bPsi{{\bf \Psi}} +\def\bSi{{\bf \Si}} +\def\bC{{\bf C}} +\def\bS{{\bf S}} +\def\cA{{\cal A}} +\def\cB{{\cal B}} +\def\cE{{\cal E}} +\def\cF{{\cal F}} +\def\cG{{\cal G}} +\def\cH{{\cal H}} +\def\cI{{\cal I}} +\def\cJ{{\cal J}} +\def\cL{{\cal L}} +\def\cO{{\cal O}} +\def\cS{{\cal S}} +\def\cT{{\cal T}} +\def\cY{{\cal Y}} +% +% FONT FAMILIES +% +\font\tenfrak = eufm10 +\font\sevenfrak = eufm7 +\font\fivefrak = eufb5 +\newfam\frakfam + \textfont\frakfam=\tenfrak + \scriptfont\frakfam=\sevenfrak + \scriptscriptfont\frakfam=\fivefrak +\def\frak{\fam\frakfam\tenfrak} +% +\font \sixsans = cmss10 at 6pt +\font \ninesans = cmss10 at 9pt +\font \tensans = cmss10 +\font \fivesans = cmss10 at 5pt +\font \sevensans = cmss10 at 7pt +\font \twelvesans = cmss12 +\newfam\sansfam + \textfont\sansfam=\tensans + \scriptfont\sansfam=\sevensans + \scriptscriptfont\sansfam=\fivesans +\def\sans{\fam\sansfam\tensans} +% +%\font \smallescriptscriptfont = cmr5 +%\font \smallescriptfont = cmr5 at 7pt +%\font \smalletextfont = cmr5 at 10pt +% +\font\tenmsb =msbm10 +\font\sevenmsb =msbm7 +\font\fivemsb =msbm5 +\newfam\msbfam + \textfont\msbfam=\tenmsb + \scriptfont\msbfam=\sevenmsb + \scriptscriptfont\msbfam=\fivemsb +\def\msbfamno{\ifcase \msbfam 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or + 9\or A\or B\or C\or D\or E\or F\fi} +\def\bbb{\fam\msbfam\tenmsb} +% +\font\tenmsa =msam10 +\font\sevenmsa =msam7 +\font\fivemsa =msam5 +\newfam\msafam + \textfont\msafam=\tenmsa + \scriptfont\msafam=\fivemsa %NOTE small scriptsize + \scriptscriptfont\msafam=\fivemsa +\def\msafamno{\ifcase \msafam 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or + 9\or A\or B\or C\or D\or E\or F\fi} + +\font\sixrm=cmr6 \font\eightrm=cmr8 +\font\sixi=cmmi6 \font\eighti=cmmi8 +\font\sixsy=cmsy6 \font\eightsy=cmsy8 +\font\sixbf=cmbx6 \font\eightbf=cmbx8 + \font\eightit=cmti8 + \font\eightsl=cmsl8 + \font\eighttt=cmtt8 + \font\eightfrak=eufm7 at 8pt +% +\def\eightpoint{\def\rm{\fam0\eightrm}% switch to 8-point type + \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm + \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei + \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy + \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex + \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% + \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% + \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% + \textfont\frakfam=\eightfrak \def\frak{\fam\frakfam\tenfrak}% + \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf + \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% + \normalbaselineskip=9pt + \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% + \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm} +% +\font\twelverm=cmr12 +\font\twelvei=cmmi12 +\font\twelvesy=cmsy10 at 12pt +\font\twelvebf=cmbx12 +\font\twelveit=cmti12 +\font\twelvesl=cmsl12 +\font\twelvett=cmtt12 +% +\def\twelvepoint{\def\rm{\fam0\twelverm}% switch to 12-point type + \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm + \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi + \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy + \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex + \textfont\itfam=\twelveit \def\it{\fam\itfam\twelveit}% + \textfont\slfam=\twelvesl \def\sl{\fam\slfam\twelvesl}% + \textfont\ttfam=\twelvett \def\tt{\fam\ttfam\twelvett}% + \textfont\bffam=\twelvebf \scriptfont\bffam=\eightbf + \scriptscriptfont\bffam=\sixbf \def\bf{\fam\bffam\twelvebf}% + \normalbaselineskip=14pt + \setbox\strutbox=\hbox{\vrule height10pt depth4pt width0pt}% + \let\sc=\tenrm \let\big=\twelvebig \normalbaselines\rm} + +\font\tenBigrm=cmr10 scaled\magstep2 +\font\tenBigi=cmmi10 scaled\magstep2 +\font\tenBigsy=cmsy10 scaled\magstep2 +\font\tenBigbf=cmbx10 scaled\magstep2 +\font\tenBigit=cmti10 scaled\magstep2 +\font\tenBigsl=cmsl10 scaled\magstep2 +\font\tenBigtt=cmtt10 scaled\magstep2 +% +\def\tenBigpoint{\def\rm{\fam0\tenBigrm}% switch to 10x1.44-point type + \textfont0=\tenBigrm \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm + \textfont1=\tenBigi \scriptfont1=\teni \scriptscriptfont1=\seveni + \textfont2=\tenBigsy \scriptfont2=\tensy \scriptscriptfont2=\sevensy + \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex + \textfont\itfam=\tenBigit \def\it{\fam\itfam\tenBigit}% + \textfont\slfam=\tenBigsl \def\sl{\fam\slfam\tenBigsl}% + \textfont\ttfam=\tenBigtt \def\tt{\fam\ttfam\tenBigtt}% + \textfont\bffam=\tenBigbf \scriptfont\bffam=\tenbf + \scriptscriptfont\bffam=\sevenbf \def\bf{\fam\bffam\tenBigbf}% + \normalbaselineskip=17.3pt + \setbox\strutbox=\hbox{\vrule height12.2pt depth5pt width0pt}% + \let\sc=\twelverm \let\big=\tenBigbig \normalbaselines\rm} + +% +% Footnotes +% +\catcode`@=11 +\def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf + \insert\footins\bgroup\eightpoint + \interlinepenalty100 \let\par=\endgraf + \leftskip=0pt \rightskip=0pt + \splittopskip=10pt plus 1pt minus 1pt \floatingpenalty=20000 + \smallskip\item{#1}\bgroup\strut\aftergroup\@foot\let\next} +\skip\footins=12pt plus 2pt minus 4pt +\dimen\footins=30pc +\catcode`@=12 + + +% +% Blackboard characters +% +\def\bbbr{{\rm I\!R}} +\def\bbbn{{\rm I\!N}} +\def\bbbm{{\rm I\!M}} +\def\bbbh{{\rm I\!H}} +\def\bbbf{{\rm I\!F}} +\def\bbbk{{\rm I\!K}} +\def\bbbp{{\rm I\!P}} +\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} %\bbbone +{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}} +\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox %\bbbc +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox +to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} +\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm %\bbbq +Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise +0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} +\def\bbbz{{\mathchoice {\hbox{$\sans\textstyle Z\kern-0.4em Z$}} %\bbbz +{\hbox{$\sans\textstyle Z\kern-0.4em Z$}} +{\hbox{$\sans\scriptstyle Z\kern-0.3em Z$}} +{\hbox{$\sans\scriptscriptstyle Z\kern-0.2em Z$}}}} +\def\bbbt{{\bbb T}} %\bbbt +\def\bbbs{{\mathchoice %\bbbs +{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox +to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\hbox +to0pt{\kern0.55\wd0\vrule height0.5\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.35\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox +to0pt{\kern0.5\wd0\vrule height0.45\ht0\hss}\box0}} +{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox +to0pt{\kern0.4\wd0\vrule height0.45\ht0\hss}\raise0.05\ht0\hbox +to0pt{\kern0.55\wd0\vrule height0.45\ht0\hss}\box0}}}} +% +% SYMBOLS +% +\def\const{{\rm const}\,} +\def\cst#1#2{{\rm const}^{#1}_{#2}\,} +\def\sgn{{\rm sgn}} +\def\half{\sfrac{1}{2}} +\def\tr{{\rm tr\, }} +\def\Tr{{\rm Tr\, }} +\def\optbar#1{\vbox{\ialign{##\crcr\hfil${\scriptscriptstyle(}\mkern -1mu + \vrule height 1.2pt width 3pt depth -.8pt + {\scriptscriptstyle)}$\hfil\crcr + \noalign{\kern-1pt\nointerlineskip}$\hfil\displaystyle{#1}\hfil$\crcr}}} +\def\<{\left<} +\def\>{\right>} +\def\dbar{{\mkern6mu\mathchar'26\mkern-12mud}} +\def\smprod{\mathop{\textstyle\prod}} +\def\smsum{\mathop{\textstyle\sum}} +\def\set#1#2{\big\{ \ #1\ \big|\ #2\ \big\}} +\def\eval#1{\big|\lower4pt\hbox{$\displaystyle\sst #1$}} +\def\slim{\mathop{\,\hbox{\rm s-lim}\,}} +\def\cartprod{\mathop{\lower2pt\hbox{{\twelvesans X}}}} +\def\eqdef{{\buildrel \rm def \over =}} +\def\v{\pmb{\big\vert}} +\def\bbar#1{\setbox0=\hbox{$#1$}% + \copy0\kern-\wd0 + \raise.0433em\box0} +\def\sbar#1{\vbox{\ialign{##\crcr + \noalign{\kern1pt\nointerlineskip} + \hfil$ \mkern -1mu\vrule height 1.2pt width 3pt depth -.9pt$\hfil\crcr + \noalign{\kern0pt\nointerlineskip} + $\hfil{\sst #1}\hfil$\crcr}}} +\mathchardef\lle="3\msafamno36 +\mathchardef\gge="3\msafamno3E +\def\llt{{\scriptscriptstyle <}} +\def\ggt{{\scriptscriptstyle >}} +\mathchardef\evall="2\msafamno16 +\mathchardef\smsetminus="2\msbfamno72 +\def\implies{\;\Longrightarrow\;} +\def\simplies{\;\Rightarrow\;} +% +% SECTION TITLES +% +\def\titlea#1{\centerline{\tenBigpoint\bf #1 }\vskip.5truein} +%\def\titlea#1{\centerline{\tenBigpoint\bf #1 }\vskip.75truein} +%\def\titleb#1{\noindent{\twelvepoint\bf #1 }\vskip.25truein} +\def\titleb#1{\removelastskip\vskip.3truein% +\noindent{\twelvepoint\bf #1 }\vskip.25truein} +%\def\titlec#1{\noindent{\bf #1 }\vskip.1truein} +\def\titlec#1{\removelastskip\vskip.15truein\noindent{\bf #1 }\vskip.1truein} +%\def\sec{\mathhexbox278} +% +% THEOREM etc. +% +\def\newenvironment#1#2#3#4{\long\def#1##1##2{% +\removelastskip\penalty-100\vskip\baselineskip% +\noindent{#3#2\if!##1!.\else\unskip\ \ignorespaces +##1\unskip\fi\ }{#4\ignorespaces##2\vskip\baselineskip}}} +% +\newenvironment\lemma{Lemma}{\bf}{\it} +\newenvironment\proposition{Proposition}{\bf}{\it} +\newenvironment\theorem{Theorem}{\bf}{\it} +\newenvironment\corollary{Corollary}{\bf}{\it} +\newenvironment\example{Example}{\bf}{\rm} +\newenvironment\problem{Problem}{\bf}{\rm} +\newenvironment\definition{Definition}{\bf}{\rm} +\newenvironment\remark{Remark}{\bf}{\rm} +\newenvironment\hypothesis{Hypothesis}{\bf}{\it} +\newenvironment\convention{Convention}{\bf}{\it} + +\def\Item{\vskip.1in\noindent} + +% +% PROOF, QED +% +\long\def\proof#1{\removelastskip\penalty-100\vskip\baselineskip\noindent{\bf + Proof\if!#1!\else\ \ignorespaces#1\fi:\ }\ \ \ignorespaces} +\long\def\prf{\removelastskip\penalty-100\vskip\baselineskip\noindent{\bf + Proof:\ }\ \ \ignorespaces} +\def\endproof{\hfill\vrule height .6em width .6em depth 0pt\goodbreak\vskip.25in } +\def\sq{\hbox{\rlap{$\sqcap$}$\sqcup$}} +\def\qed{\ifmmode\sq\else{\unskip\nobreak\hfil + \penalty50\hskip1em\null\nobreak\hfil\sq + \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi} + +% +% ISSUES +% +%\def\showissues{y} %%% choose this line to include "issue" internal remarks +\def\showissues{n} %%% choose this line to exclude "issue" internal remarks +\edef\Raum{ } +\newcount\issuesD +\newcount\issuesND +\long\def\issue#1{\if\showissues y \global\advance\issuesD by 1% +% {\bf [ISSUE[} +% #1 +% {\bf ]END ISSUE] } + {\noindent\bf ??? #1 ??? } + \else \global\advance\issuesND by 1\fi} +\def\printIssueCount{ + \immediate\write16{\Raum\Raum\number\issuesD\Raum Issues displayed.} + \immediate\write16{\Raum\Raum\number\issuesND\Raum Issues not displayed.}} + + + + +\newdimen\rulesz \newdimen\bordersz \newdimen\boxshift +% +% +% +% MACRO FOR DRAWING BOXES +% +% boxes drawn will have rules 1 point wide surrounding a 2 point wide border +% +% You can change these measurements by redefining \rulesz +% and \bordersz in your TeX source. +% +% to show a box without the box affecting other spacing set\rulesz=.1pt +% \bordersz=-\rulesz +% +% WARNING: removing the %'s below or adding blank spaces will affect box positioning +% +\rulesz=1pt \bordersz=2pt +\def\shbox#1{\setbox111=\hbox{#1\hss}% +\boxshift=\dp111\advance\boxshift by \rulesz\advance\boxshift by \bordersz% +\lower\boxshift\vbox{\hrule height\rulesz% +\hbox{\vrule width\rulesz\kern\bordersz% +\vbox{\kern\bordersz\copy111\kern\bordersz}\kern\bordersz% +\vrule width\rulesz}\hrule height\rulesz}}% +%\def\shbox#1{\lower.4pt\vbox{\hrule\hbox{\vrule\hbox{#1\hss}\vrule}\hrule}} % box it +% + + + + + + + + diff --git a/extra/readme.txt b/extra/readme.txt new file mode 100644 index 0000000..97a5000 --- /dev/null +++ b/extra/readme.txt @@ -0,0 +1,3 @@ +Various files, mostly in the "new/" directory from the zip file from Brian Wetton. + + - Colin Macdonald, 2025-06-14