text:
# Replacing multiple variables according to the index

I'm currently working on the Polya's Enumeration Theorem implementation in Mathematica. As an example, of what I want to do, here's a formula I'm working with:

$$P_{C_{12}}(x_1,\ldots,x_{12})=\frac{1}{12}\left(x^{12}_1+x^{6}_2+2x^{4}_3+2x^{3}_4+2x^{2}_6+4x^{1}_{12}\right)$$

I'd like to replace $$x_i$$ with another expression that involves $$i$$, $$w^i+b^i$$, and then simplify the whole expression. How could I do that?

The implementation depends on how you define your variables. I have used x[i] as an expression for $$x_i$$.

P = (1/12) (x[1]^12 + x[2]^6 + 2 x[3]^4 + 2 x[4]^3 + 2 x[6]^2 +
4 x[12]^1);

P2 = P /. x[i_] -> w^i + b^i
(* 1/12 ((b + w)^12 + (b^2 + w^2)^6 + 2 (b^3 + w^3)^4 +
2 (b^4 + w^4)^3 + 2 (b^6 + w^6)^2 + 4 (b^12 + w^12)) *)

P2 // Expand
(* b^12 + b^11 w + 6 b^10 w^2 + 19 b^9 w^3 + 43 b^8 w^4 + 66 b^7 w^5 +
80 b^6 w^6 + 66 b^5 w^7 + 43 b^4 w^8 + 19 b^3 w^9 + 6 b^2 w^10 +
b w^11 + w^12 *)

• Thank you, that was exactly what I was looking for. As a side note, is it possible to keep x's with subscripts and do something similar? Jul 26 at 13:41
• You mean like this? P = 1/12 (Subscript[x, 1]^12 + Subscript[x, 2]^6 + 2*Subscript[x, 3]^4 + 2*Subscript[x, 4]^3 + 2*Subscript[x, 6]^2 + 4*Subscript[x, 12]); P2 = P /. Subscript[x, i_] -> w^i + b^i Jul 26 at 13:48


text:
Question

For which problem situation could the equation 45x + 55x = 350 be used?

A 350 mile telephone cable costs $55 per mile for one section and$45 per mile for a different section. What is the total cost of the cable?

A car, traveling 45 mph, leaves Dallas heading for Odessa 350 miles away. At the same time, a truck leaves Odessa for Dallas traveling 55 mph on the same road. When will the two vehicles meet?

A 350 mile telephone cable is cut so the shorter piece is 45/55 of the whole cable. How long is each piece?

Two vehicles start from the same point traveling in the same direction. One vehicle travels 55 miles per hour (mph) and the other 45 mph. When will the two vehicles be 350 miles apart?

1. kimchi2

x=11(13/14)
Step-by-step explanation:
3/5(x-7/10) = -31/4
open the bracket:
3/5 * x-7/10 = -31/4
3/5*-7x/10 =-31/4
3/5 + 31/4 = 7x/10
(12+155)/20 = 7x/10°
167/20 = 7x/10
cross multiply:
140x = 1670
x=1670/140
x=11(130/140)
x = 11(13/14)

format: plain


text:
# History of Topics in Special Relativity/Lorentz transformation (Quaternions)

## Lorentz transformation via quaternions and hyperbolic numbers

### Quaternions

The Lorentz transformations can also be expressed in terms of w:biquaternions: A Minkowskian quaternion (or minquat) q having one real part and one purely imaginary part is multiplied by biquaternion a applied as pre- and postfactor. Using an overline to denote quaternion conjugation and * for complex conjugation, its general form (on the left) and the corresponding boost (on the right) are as follows:[1][2]

{\displaystyle \left.{\begin{matrix}-x_{0}^{\prime 2}+x_{1}^{\prime 2}+x_{2}^{\prime 2}+x_{3}^{\prime 2}=-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\\\hline q'=aq{\bar {a}}^{\ast }\\\hline {\begin{aligned}q&=ix_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}\\q'&=ix_{0}^{\prime }+x_{1}^{\prime }e_{1}+x_{2}^{\prime }e_{2}+x_{3}^{\prime }e_{3}\\a&=\cos \chi +i\sin \chi =e^{i\chi }\end{aligned}}\\\left(a{\bar {a}}=1,\ \chi ={\text{imaginary}}\right)\end{matrix}}\right|{\begin{matrix}\chi ={\frac {1}{2}}i\eta \\\downarrow \\{\begin{aligned}x_{0}^{\prime }&=x_{0}\cosh \eta -x_{1}\sinh \eta \\x_{1}^{\prime }&=-x_{0}\sinh \eta +x_{1}\cosh \eta \\x_{2}^{\prime }&=x_{2},\quad x_{3}^{\prime }=x_{3}\end{aligned}}\end{matrix}}}

(7a)

After the parameter of 3D and 4D rotations were already given by Euler (1770/71), the quaternion transformation describing 3D rotations was given by Hamilton (1844/45) and Cayley (1845) and the corresponding 4D transformation was given by Cayley (1854, 1855). Cox (1882/83) discussed the Lorentz interval in terms of Weierstrass coordinates ${\displaystyle x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=1}$ in the course of adapting w:William Kingdon Clifford's biquaternions a+ωb to hyperbolic geometry by setting ${\displaystyle \omega ^{2}=-1}$ (alternatively, 1 gives elliptic and 0 parabolic geometry). Stephanos (1883) related the imaginary part of w:William Rowan Hamilton's biquaternions to the radius of spheres, and introduced a homography leaving invariant the equations of oriented spheres or oriented planes in terms of w:Lie sphere geometry. Buchheim (1884/85) discussed the Cayley absolute ${\displaystyle x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{2}^{2}=0}$ and adapted Clifford's biquaternions to hyperbolic geometry similar to Cox by using all three values of ${\displaystyle \omega ^{2}}$. Eventually, the modern Lorentz transformation using biquaternions with ${\displaystyle \omega ^{2}=-1}$ as in hyperbolic geometry was given by Klein (1908-11) and Noether (1910) as well as Conway (1911) and Silberstein (1911).

### Hyperbolic numbers

Often connected with quaternionic systems is the w:hyperbolic number ${\displaystyle \varepsilon ^{2}=1}$, which also allows to formulate the Lorentz transformations:[3][4]

{\displaystyle {\begin{aligned}w'&=we^{-\varepsilon \eta }\\&=w(\cosh(-\eta )+\varepsilon \sinh(-\eta ))\\\\w&=w'e^{\varepsilon \eta }\\&=w'(\cosh \eta +\varepsilon \sinh \eta )\end{aligned}}\rightarrow {\begin{aligned}w&=x_{1}+\varepsilon x_{0}\\w'&=x_{1}^{\prime }+\varepsilon x_{0}^{\prime }\end{aligned}}\rightarrow {\begin{aligned}x_{0}^{\prime }&=x_{0}\cosh \eta -x_{1}\sinh \eta \\x_{1}^{\prime }&=-x_{0}\sinh \eta +x_{1}\cosh \eta \\\\x_{0}&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta \\x_{1}&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta \end{aligned}}}

(7b)

After the trigonometric expression ${\displaystyle e^{ix}}$ (w:Euler's formula) was given by Euler (1748), and the hyperbolic analogue ${\displaystyle e^{\varepsilon \eta }}$ as well as hyperbolic numbers by Cockle (1848) in the framework of w:tessarines, it was shown by Cox (1882/83) that one can identify ${\displaystyle ww^{\prime -1}=e^{\varepsilon \eta }}$ with associative quaternion multiplication. Here, ${\displaystyle e^{\varepsilon \eta }}$ is the hyperbolic w:versor with ${\displaystyle \varepsilon ^{2}=1}$, while -1 denotes the elliptic or 0 denotes the parabolic counterpart (not to be confused with the expression ${\displaystyle \omega ^{2}}$ in Clifford's biquaternions also used by Cox, in which -1 is hyperbolic). The hyperbolic versor was also discussed by Macfarlane (1892, 1894, 1900) in terms of w:hyperbolic quaternions. The expression ${\displaystyle \varepsilon ^{2}=1}$ for hyperbolic motions (and -1 for elliptic, 0 for parabolic motions) also appear in "biquaternions" defined by Vahlen (1901/02, 1905).

### Clifford algebra

More extended forms of complex and (bi-)quaternionic systems in terms of w:Clifford algebra can also be used to express the Lorentz transformations. For instance, using a system a of Clifford numbers one can transform the following general quadratic form into itself, in which the individual values of ${\displaystyle i_{1}^{2},i_{2}^{2},\dots }$ can be set to +1 or -1 at will:[5][6]

${\displaystyle {\begin{matrix}i_{1}^{2}x_{1}^{\prime 2}+\cdots +i_{n}^{2}x_{n}^{\prime 2}=i_{1}^{2}x_{1}^{2}+\cdots +i_{n}^{2}x_{n}^{2}\\\hline (1)\ x'=axa^{-1}\\(2)\ x'={\frac {ax+b}{\varepsilon ^{2}bx+a}}\end{matrix}}}$

(7c)

The Lorentz interval follows if the sign of one ${\displaystyle i^{2}}$ differs from all others. The general definite form ${\displaystyle x_{1}^{2}+\cdots +x_{n}^{2}}$ as well as the general indefinite form ${\displaystyle x_{1}^{2}+\cdots +x_{p}^{2}-x_{p+1}^{2}-\cdots -x_{p+q}^{2}}$ and their invariance under transformation (1) was discussed by Lipschitz (1885/86), while hyperbolic motions were discussed by Vahlen (1901/02, 1905) by setting ${\displaystyle \varepsilon ^{2}=1}$ in transformation (2), while elliptic motions follow with -1 and parabolic motions with 0, all of which he also related to biquaternions.

## Historical notation

### Euler (1770/71) – Quaternion rotation parameter

Euler (1770/71) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as w:orthogonal transformation. The transformation in three dimensions was given as:[M 1]

{\displaystyle {\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\\hline {\begin{aligned}X&=Ax+By+Cz\\Y&=Dx+Ey+Fz\\Z&=Gx+Hy+Iz\end{aligned}}{\begin{matrix}\left|{\scriptstyle {\begin{aligned}1&=AA+DD+GG\\1&=BB+EE+HH\\1&=CC+FF+II\\0&=AB+DE+GH\\0&=AG+DF+GI\\0&=BC+EF+HI\end{aligned}}}\right.\end{matrix}}\end{matrix}}}

in which the eight coefficients were related by Euler to four arbitrary parameter p,q,r,s:[M 2]

{\displaystyle {\begin{matrix}{\begin{aligned}A&={\frac {pp+qq-rr-ss}{u}}&B&={\frac {2pq+2ps}{u}}&C&={\frac {2qs-2pr}{u}}\\D&={\frac {2qr-2ps}{u}}&E&={\frac {pp-qq+rr-ss}{u}}&F&={\frac {2pq+2rs}{u}}\\G&={\frac {2qs+2pr}{u}}&H&={\frac {2rs-2pq}{u}}&I&={\frac {pp-qq-rr+ss}{u}}\end{aligned}}\\(u=pp+qq+rr+ss)\end{matrix}}}

which where rediscovered by w:Olinde Rodrigues (1840) who related them to rotation angles.[M 3]

They are equivalent to the parameter of an (improper) quaternion rotation in three dimensions, equivalent to (improper) Euler–Rodrigues parameters.

The case in four dimensions using sixteen coefficients was given by Euler as:[M 4]

{\displaystyle {\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\\hline {\begin{aligned}V&=Av+Bx+Cy+Dz\\X&=Ev+Fx+Gy+Hz\\Y&=Iv+Kx+Ly+Mz\\Z&=Nv+Ox+Py+Qz\end{aligned}}{\begin{matrix}\left|{\scriptstyle {\begin{aligned}1&=AA+RR+II+NN&0&=AB+EF+IK+NO\\1&=BB+FF+KK+OO&0&=AC+EG+IL+NP\\1&=CC+GG+LL+PP&0&=AD+EH+IM+NQ\\1&=DD+HH+MM+QQ&0&=BC+FG+KL+OP\\0&=BD+FH+KM+OQ&0&=CD+FH+LM+PQ\end{aligned}}}\right.\end{matrix}}\end{matrix}}}

Extending the Euler-Rodrigues parameter, he gave the corresponding representation of sixteen coefficients using eight parameters a,b,c,d,p,q,r,s[M 5]

${\displaystyle {\begin{array}{c|c|c|c}+ap+bq+cr+ds&+aq-bp+cs-dr&+ar-bs-cp+dq&+as+br-cq+dp\\\hline +aq-bp-cs+dr&-ap-bq+cr+ds&-as-br-cq-dp&+ar-bs+cp-dq\\\hline +ar+bs-cp-dq&+as-br-cq+dp&-ap+bq-cr+ds&-aq+bp-cs-dr\\\hline +as-br+cq-dp&-ar-bs-cp-dq&+aq+bp+cs+dr&-ap+bq+cr-ds\end{array}}}$
They are equivalent to the parameter of an (improper) quaternion rotation in four dimensions.

### Hamilton (1844/45) – Quaternions

w:William Rowan Hamilton, in an abstract of a lecture held in November 1844 and published 1845/47, showed that spatial rotations can be formulated using his theory of w:quaternions by employing w:versors as pre- and postfactor, with α as unit vector and a as real angle:[M 6]

(1) ${\displaystyle \beta '=\left(\cos {\frac {a}{2}}+\alpha \sin {\frac {a}{2}}\right)\beta \left(\cos {\frac {a}{2}}-\alpha \sin {\frac {a}{2}}\right)}$

In a footnote added before printing, he showed that this is equivalent to Cayley's (1845) rotation formula by setting[M 7]

(2) ${\displaystyle {\begin{matrix}\alpha \tan {\frac {a}{2}}=-\gamma \ \Rightarrow \ \beta '=(1+\gamma )^{-1}\beta (1+\gamma )\\\beta =ix+jy=kz,\ \beta '=ix_{\prime }+jy_{\prime }+kz_{\prime },\ \gamma =i\lambda +j\mu +k\nu \end{matrix}}}$.

Hamilton acknowledged Cayley's independent discovery and priority for first printed (February 1845) publication, but noted that he himself communicated formula (1) already in October 1844 to w:Charles Graves.

Formulas (1) or (2) are role models for Lorentz boost (7a), by replacing versors and quaternions with hyperbolic versors and biquaternions.

### Cayley (1845) – Quaternions

In 1845, w:Arthur Cayley showed that the Euler-Rodrigues parameters in equation E:(Q3) representing rotations can be related to quaternion multiplication by pre- and postfactors (an equivalent rotation formula was also used by Hamilton (1844/45)):[M 8]

{\displaystyle {\begin{matrix}x_{\prime }^{2}+y_{\prime }^{2}+z_{\prime }^{2}=x^{2}+y^{2}+z^{2}\\\hline (1+\lambda i+\mu j+\nu k)^{-1}(ix+jy+kz)(1+\lambda i+\mu j+\nu k)=\\{\scriptstyle ={\frac {1}{1+\lambda ^{2}+\mu ^{2}+\nu ^{2}}}\left\{{\begin{matrix}i\left[x\left(\lambda ^{2}+\mu ^{2}-\nu ^{2}\right)+2y(\mu \nu +\lambda )+2z(\lambda \nu -\mu )\right]\\+j\left[2x\left(\lambda \mu -\nu \right)+y\left(1-\lambda ^{2}+\mu ^{2}-\nu ^{2}\right)+2z(\mu \nu +\lambda )\right]\\+k\left[2x\left(\lambda \nu +\mu \right)+2y\left(\mu \nu -\lambda \right)+z\left(1-\lambda ^{2}-\mu ^{2}+\nu ^{2}\right)\right]\end{matrix}}={\begin{aligned}i(\alpha x+\alpha 'y+\alpha ''z)\\+j(\beta x+\beta 'y+\beta ''z)\\+k(\gamma x+\gamma 'y+\gamma ''z)\end{aligned}}\right\}}\\\downarrow \\{\begin{aligned}x_{\prime }=&i(\alpha x+\alpha 'y+\alpha ''z)\\y_{\prime }=&j(\beta x+\beta 'y+\beta ''z)\\z_{\prime }=&k(\gamma x+\gamma 'y+\gamma ''z)\end{aligned}}\end{matrix}}}

and in 1848 he used the abbreviated form[M 9]

${\displaystyle \Pi _{1}=\Lambda \Pi \Lambda ^{-1}\quad \ \left[{\begin{matrix}\Lambda =1+\lambda i+\mu j+\nu k\\\Pi _{1}=ix_{1}+jy_{1}+kz_{1}\end{matrix}}\right]}$

In 1854 he showed how to transform the sum of four squares into itself:[M 10]

{\displaystyle {\begin{matrix}x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+w_{2}^{2}=x^{2}+y^{2}+z^{2}+w^{2}\\\hline MM'\left(w_{2}+ix_{2}+jy_{2}+kz_{2}\right)=(d+ia+jb+kc)(w+ix+jy+kz)(d'+ia'+jb'+kc')\\\left({\begin{aligned}M^{2}&=a^{2}+b^{2}+c^{2}+d^{2}\\M^{\prime 2}&=a^{\prime 2}+b^{\prime 2}+c^{\prime 2}+d^{\prime 2}\end{aligned}}\right)\end{matrix}}}

or in 1855 (where he also gave credit to Euler (1770)):[M 11]

{\displaystyle {\begin{matrix}\mathrm {x} ^{2}+\mathrm {y} ^{2}+\mathrm {z} ^{2}+\mathrm {w} ^{2}=x^{2}+y^{2}+z^{2}+w^{2}\\\hline \left(\mathrm {x} i+\mathrm {y} j+\mathrm {z} k+\mathrm {w} \right)=-{\frac {1}{\sqrt {MM'}}}(\alpha i+\beta j+\gamma k+\delta )(xi+yj+zk+w)(\alpha 'i+\beta 'j+\gamma 'k+\delta ')\\\left({\begin{aligned}M&=\alpha ^{2}+\beta ^{2}+\gamma ^{2}+\delta ^{2}\\M'&=\alpha ^{\prime 2}+\beta ^{\prime 2}+\gamma ^{\prime 2}+\delta ^{\prime 2}\end{aligned}}\right)\end{matrix}}}
Cayley's quaternion transformation of the sum of four squares, abbreviated QqP, served as a role model for the representation of the Lorentz transformation by Noether (1910), Klein (1910), in which the scalar part is imaginary.

### Cockle (1848) - Tessarines

w:James Cockle (1848) introduced the w:tessarine algebra as follows:[M 12]

${\displaystyle {\begin{matrix}t=w+ix+jy+kz\\t'=w'+ix'+jy'+kz'\\\left[i^{2}=-j^{2}=k^{2}=-1\right]\end{matrix}}}$.

While ${\displaystyle i^{2}=-1}$ is the ordinary imaginary unit, the new unit ${\displaystyle j^{2}=+1}$ led him to formulate the following relation:[M 13]

${\displaystyle \varepsilon ^{jy}=1+jy+{\frac {y^{2}}{2}}+j{\frac {y^{3}}{2.3}}+{\frac {y^{4}}{2.3.4}}+\&c.,\ \left(j^{2}=1\right)}$.
The real tessarine ${\displaystyle w+jy}$ is a w:hyperbolic number, ${\displaystyle \varepsilon ^{jy}}$ a w:hyperbolic versor, and the multiplication ${\displaystyle \varepsilon ^{jy}\left(w+jy\right)}$ leads to Lorentz boost (7b).

### Cox (1882) – Quaternions

w:Homersham Cox (1882/83) described hyperbolic geometry in terms of an analogue to w:quaternions and w:Hermann Grassmann's w:exterior algebra. To that end, he used w:hyperbolic numbers (without mentioning Cockle (1848)) as a means to transfer point P to point Q in the hyperbolic plane, which he wrote in the form:[M 14]

${\displaystyle {\begin{matrix}QP^{-1}=\cosh \theta +\iota \sinh \theta \\QP^{-1}=e^{\iota \theta }\end{matrix}}\left(\iota ^{2}=1\right)}$

In (1882/83a) he showed the equivalence of PQ=-cosh(θ)+ι·sinh(θ) with "quaternion multiplication",[M 15] and in (1882/83b) he described QP−1=cosh(θ)+ι·sinh(θ) as being "associative quaternion multiplication".[M 16] He also showed that the position of point P in the hyperbolic plane may be determined by three quantities in terms of Weierstrass coordinates obeying the relation z2-x2-y2=1.[M 17]

Cox's associative quaternion multiplication using the hyperbolic versor is equivalent to the Lorentz boost (7b) by setting ${\displaystyle Q=x_{1}^{\prime }+\iota x_{0}^{\prime }}$ and ${\displaystyle P=x_{1}+\iota x_{0}}$.

Cox went on to develop an algebra for hyperbolic space analogous to Clifford's w:biquaternions. While Clifford (1873) used biquaternions of the form a+ωb in which ω2=0 denotes parabolic space and ω2=1 elliptic space, Cox discussed hyperbolic space using the imaginary quantity ${\displaystyle {\sqrt {-1}}}$ and therefore ω2=-1.[M 18] He also obtained relations of quaternion multiplication in terms of Weierstrass coordinates:[M 19]

{\displaystyle {\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\\hline {\begin{aligned}PO^{-1}&=\cosh \theta +(li+mj+nk)\sinh \theta \\&=w+xi+yj+zk\\\\OP^{-1}&=\cosh \theta -(li+mj+nk)\sinh \theta \\&=w-xi-yj-zk\end{aligned}}\end{matrix}}}

### Stephanos (1883) – Biquaternions

w:Cyparissos Stephanos (1883)[M 20] showed that Hamilton's biquaternion a0+a1ι1+a2ι2+a3ι3 can be interpreted as an oriented sphere in terms of Lie's sphere geometry (1871), having the vector a1ι1+a2ι2+a3ι3 as its center and the scalar ${\displaystyle a_{0}{\sqrt {-1}}}$ as its radius. Its norm ${\displaystyle a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}}$ is thus equal to the power of a point of the corresponding sphere. In particular, the norm of two quaternions N(Q1-Q2) (the corresponding spheres are in contact with N(Q1-Q2)=0) is equal to the tangential distance between two spheres. The general contact transformation between two spheres then can be given by a w:homography using 4 arbitrary quaternions A,B,C,D and two variable quaternions X,Y:[M 21][7][8]

${\displaystyle XAY+XB+CY+D=0}$ (or ${\displaystyle X=-{\frac {CY+D}{AY+B}}}$).

Stephanos pointed out that the special case A=0 denotes transformations of oriented planes (see Laguerre (1882)).

The Lorentz group SO(1,3) is a subgroup of the conformal group Con(1,3) in terms of Lie's transformations of orientied spheres in which the radius indicates the fourth coordinate. The Lorentz group is isomorphic to the group of Laguerre's transformation of oriented planes.

### Buchheim (1884–85) – Biquaternions

w:Arthur Buchheim (1884, published 1885) applied Clifford's biquaternions and their operator ω to different forms of geometries (Buchheim mentions Cox (1882) as well). He defined the scalar ω2=e2 which in the case -1 denotes hyperbolic space, 1 elliptic space, and 0 parabolic space. He derived the following relations consistent with the Cayley–Klein absolute:[M 22]

{\displaystyle {\begin{matrix}\delta ^{2}+e^{2}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)=0\\\hline P=\delta +\omega \rho =\delta +\omega (\alpha i+\beta j+\gamma k)\\P'=\delta '+\omega \rho '=\delta '+\omega (\alpha 'i+\beta 'j+\gamma 'k)\\{\begin{aligned}PKP'&=(\delta +\omega \rho )(\delta '+\omega \rho ')\\&=\delta \delta '-e^{2}\rho \rho '+\omega (\rho \delta '-\rho '\delta )\\PKP&=\delta ^{2}-e^{2}\rho ^{2}\\\cos(PP')&={\frac {\delta \delta '+e^{2}\left(\alpha \alpha '+\beta \beta '+\gamma \gamma '\right)}{\left\{\delta ^{2}+e^{2}\left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)\right\}^{\frac {1}{2}}\cdot \left\{\delta ^{\prime 2}+e^{2}\left(\alpha ^{\prime 2}+\beta ^{\prime 2}+\gamma ^{\prime 2}\right)\right\}^{\frac {1}{2}}}}\end{aligned}}\end{matrix}}}
By choosing e2=-1 for hyperbolic space, the Cayley absolute becomes the Lorentz interval.

### Lipschitz (1885/86) – Clifford algebra

w:Rudolf Lipschitz used an even subalgebra of w:Clifford algebra in order to formulate the orthogonal transformation ${\displaystyle \varLambda X=Y\varLambda _{1}}$ of a sum or squares ${\displaystyle x_{1}^{2}+x_{2}^{2}\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}}$ into itself, for which he used real variables and constants, thus Λ becomes a real quaternion for n=3.[M 23] He went further and discussed transformations in which both variables x,y... and constants ${\displaystyle \lambda _{0}\dots }$ are complex, thus Λ becomes a complex quaternion (i.e. biquaternion) for n=3.[M 24] The transformation system for both real and complex quantities has the form:[M 25]

{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}\\\hline \varLambda X=Y\varLambda _{1}\\\left({\begin{aligned}X&=x_{1}+k_{1}k_{2}x_{2}+\dots +k_{1}k_{n}x_{n},\\Y&=y_{1}+k_{1}k_{2}y_{2}+\dots +k_{1}k_{n}y_{n},\\\varLambda &=\lambda _{0}+k_{1}k_{2}\lambda _{12}+\dots +k_{2}k_{3}\lambda _{23}+\dots +k_{1}k_{2}k_{3}k_{4}\lambda _{1234}+\cdots ,\\\varLambda _{1}&=\lambda _{0}-k_{1}k_{2}\lambda _{12}+\dots +k_{2}k_{3}\lambda _{23}+\dots -k_{1}k_{2}k_{3}k_{4}\lambda _{1234}+\cdots ,\end{aligned}}\right)\end{matrix}}}

Lipschitz noticed that this corresponds to the transformations of quadratic forms given by E:Hermite (1854) and E:Cayley (1855). He then modified his equations to discuss the general indefinite quadratic form, by defining some variables and constants as real and some of them as purely imaginary:[M 26]

${\displaystyle {\begin{matrix}{\begin{matrix}x_{1}={\mathfrak {x}}_{1},\dots x_{m}={\mathfrak {x}}_{m},\quad x_{m+1}=-h{\mathfrak {x}}_{m+1},\quad x_{n}=-h{\mathfrak {x}}_{m};\\y_{1}={\mathfrak {y}}_{1},\dots y_{m}={\mathfrak {y}}_{m},\quad y_{m+1}=-h{\mathfrak {y}}_{m+1},\quad y_{n}=-h{\mathfrak {y}}_{m};\end{matrix}}\\\lambda _{0}=\varrho _{0}\\\lambda _{ab}=\varrho _{ab};\quad a\leqq m,\ b\leqq m\\\lambda _{ab}=h\varrho _{ab};\quad a\leqq m,\ b>m\\\lambda _{ab}=h\varrho _{ab};\quad a>m,\ bm,\ b>m\end{matrix}}\ \left(h={\sqrt {-1}}\right)}$

resulting into

{\displaystyle {\begin{matrix}{\mathfrak {x}}_{1}^{2}+\dots +{\mathfrak {x}}_{m}^{2}-{\mathfrak {x}}_{m+1}^{2}-\dots -{\mathfrak {x}}_{n}^{2}={\mathfrak {y}}_{1}^{2}+\dots +{\mathfrak {y}}_{m}^{2}-{\mathfrak {y}}_{m+1}^{2}-\dots -{\mathfrak {y}}_{n}^{2}\\\hline P^{(m+1,\dots n)}{\mathfrak {\bar {X}}}={\mathfrak {\bar {Y}}}P_{1}^{(m+1,\dots n)}\\\left({\begin{aligned}{\mathfrak {\bar {X}}}&={\mathfrak {x}}_{1}+k_{1}k_{2}{\mathfrak {x}}_{2}+\dots +k_{1}k_{m}{\mathfrak {x}}_{m}-k_{1}k_{m+1}h{\mathfrak {x}}_{m+1}-\dots -k_{1}k_{n}h{\mathfrak {x}}_{n}\\{\mathfrak {\bar {Y}}}&={\mathfrak {y}}_{1}+k_{1}k_{2}{\mathfrak {y}}_{2}+\dots +k_{1}k_{m}{\mathfrak {y}}_{m}-k_{1}k_{m+1}h{\mathfrak {y}}_{m+1}-\dots -k_{1}k_{n}h{\mathfrak {y}}_{n}\\P&=\varrho _{0}+k_{1}k_{2}\varrho _{12}+\dots +k_{1}k_{2}k_{3}k_{4}\varrho _{1234}+\cdots ,\end{aligned}}\right)\end{matrix}}}
By setting m=n-1 or n=m+1, the Lorentz interval ${\displaystyle {\mathfrak {x}}_{1}^{2}+\dots +{\mathfrak {x}}_{m}^{2}-{\mathfrak {x}}_{n}^{2}={\mathfrak {y}}_{1}^{2}+\dots +{\mathfrak {y}}_{m}^{2}-{\mathfrak {y}}_{n}^{2}}$ and the Lorentz transformation follows

### Vahlen (1901/02) – Clifford algebra and Möbius transformation

Modifying Lipschitz's (1885/86) variant of Clifford numbers, w:Theodor Vahlen (1901/02) formulated Möbius transformations (which he called vector transformations) and biquaternions in order to discuss motions in n-dimensional non-Euclidean space, leaving the following quadratic form invariant (where j2=1 represents hyperbolic motions, j2=-1 elliptic motions, j2=0 parabolic motions):[M 27]

{\displaystyle {\begin{matrix}x_{0}^{2}-i_{i}^{2}x_{1}^{2}-\dots -i_{p}^{2}x_{p}^{2}=j^{2}\quad \left(i_{\alpha }^{2}=1,\ 0,\ {\text{or}}-1\right)\\\hline ax=ya'\\{\begin{aligned}a&=a_{0}+a_{1}i_{1}+\dots +a_{12}i_{1}i_{2}\dots +a_{12\dots p}i_{1}i_{2}\dots i_{p}\\a'&=a_{0}-a_{1}i_{1}-\dots +a_{12}i_{1}i_{2}\dots +(-1)^{p}a_{12\dots p}i_{1}i_{2}\dots i_{p}\\x&=x_{0}+x_{1}i_{1}+\dots +x_{p}i_{p}\end{aligned}}\\\hline {\text{Vector transformation}}\\y={\frac {ax+b'}{j^{2}bx+a'}}\\\left[z={\frac {cy+d'}{j^{2}dy+c'}},\quad y={\frac {ax+b'}{j^{2}bx+a'}}\right]\rightarrow z={\frac {Ax+B'}{j^{2}Bx+A'}}\\\hline {\text{Biquaternion}}\\(c+d'j)(a+b'j)=A+B'j\quad \left(i_{\alpha }j+ji_{\alpha }=0\right)\end{matrix}}}
The group of hyperbolic motions or the Möbius group are isomorphic to the Lorentz group.

### Klein (1908-11), Noether (1910) – Biquaternions

w:Felix Klein (1908) described Cayley's (1854) 4D quaternion multiplications as "Drehstreckungen" (orthogonal substitutions in terms of rotations leaving invariant a quadratic form up to a factor), and pointed out that the modern principle of relativity as provided by Minkowski is essentially only the consequent application of such Drehstreckungen, even though he didn't provide details.[R 1]

In an appendix to Klein's and Sommerfeld's "Theory of the top" (1910), w:Fritz Noether showed how to formulate hyperbolic rotations using biquaternions with ${\displaystyle \omega ={\sqrt {-1}}}$, which he also related to the speed of light by setting ω2=-c2. He concluded that this is the principal ingredient for a rational representation of the group of Lorentz transformations equivalent to (7a):[R 2]

{\displaystyle {\begin{matrix}V={\frac {Q_{1}vQ_{2}}{T_{1}T_{2}}}\\\hline X^{2}+Y^{2}+Z^{2}+\omega ^{2}S^{2}=x^{2}+y^{2}+z^{2}+\omega ^{2}s^{2}\\\hline {\begin{aligned}V&=Xi+Yj+Zk+\omega S\\v&=xi+yj+zk+\omega s\\Q_{1}&=(+Ai+Bj+Ck+D)+\omega (A'i+B'j+C'k+D')\\Q_{2}&=(-Ai-Bj-Ck+D)+\omega (A'i+B'j+C'k-D')\\T_{1}T_{2}&=T_{1}^{2}=T_{2}^{2}=A^{2}+B^{2}+C^{2}+D^{2}+\omega ^{2}\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\end{aligned}}\end{matrix}}}

Besides citing quaternion related standard works such as Cayley (1854), Noether referred to the entries in Klein's encyclopedia by w:Eduard Study (1899) and the French version by w:Élie Cartan (1908).[9] Cartan's version contains a description of Study's w:dual numbers, Clifford's biquaternions (including the choice ${\displaystyle \omega ={\sqrt {-1}}}$ for hyperbolic geometry), and Clifford algebra, with references to Stephanos (1883), Buchheim (1884/85), Vahlen (1901/02) and others.

Citing Noether, Klein himself published in August 1910 the following quaternion substitutions forming the group of Lorentz transformations:[R 3]

{\displaystyle {\begin{matrix}{\begin{aligned}&\left(i_{1}x'+i_{2}y'+i_{3}z'+ict'\right)\\&\quad -\left(i_{1}x_{0}+i_{2}y_{0}+i_{3}z_{0}+ict_{0}\right)\end{aligned}}={\frac {\left[{\begin{aligned}&\left(i_{1}(A+iA')+i_{2}(B+iB')+i_{3}(C+iC')+i_{4}(D+iD')\right)\\&\quad \cdot \left(i_{1}x+i_{2}y+i_{3}z+ict\right)\\&\quad \quad \cdot \left(i_{1}(A-iA')+i_{2}(B-iB')+i_{3}(C-iC')-(D-iD')\right)\end{aligned}}\right]}{\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)-\left(A^{2}+B^{2}+C^{2}+D^{2}\right)}}\\\hline {\text{where}}\\AA'+BB'+CC'+DD'=0\\A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{matrix}}}

or in March 1911[R 4]

{\displaystyle {\begin{matrix}g'={\frac {pg\pi }{M}}\\\hline {\begin{aligned}g&={\sqrt {-1}}ct+ix+jy+kz\\g'&={\sqrt {-1}}ct'+ix'+jy'+kz'\\p&=(D+{\sqrt {-1}}D')+i(A+{\sqrt {-1}}A')+j(B+{\sqrt {-1}}B')+k(C+{\sqrt {-1}}C')\\\pi &=(D-{\sqrt {-1}}D')-i(A-{\sqrt {-1}}A')-j(B-{\sqrt {-1}}B')-k(C-{\sqrt {-1}}C')\\M&=\left(A^{2}+B^{2}+C^{2}+D^{2}\right)-\left(A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\right)\\&AA'+BB'+CC'+DD'=0\\&A^{2}+B^{2}+C^{2}+D^{2}>A^{\prime 2}+B^{\prime 2}+C^{\prime 2}+D^{\prime 2}\end{aligned}}\end{matrix}}}

### Conway (1911), Silberstein (1911) – Biquaternions

w:Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms of velocity λ:[R 5]

{\displaystyle {\begin{matrix}{\begin{aligned}{\mathtt {D}}&=\mathbf {a} ^{-1}{\mathtt {D}}'\mathbf {a} ^{-1}\\{\mathtt {\sigma }}&=\mathbf {a} {\mathtt {\sigma }}'\mathbf {a} ^{-1}\end{aligned}}\\e=\mathbf {a} ^{-1}e'\mathbf {a} ^{-1}\\\hline a=\left(1-hc^{-1}\lambda \right)^{\frac {1}{2}}\left(1+c^{-2}\lambda ^{2}\right)^{-{\frac {1}{4}}}\end{matrix}}}

Also w:Ludwik Silberstein in November 1911[R 6] as well as in 1914,[10] formulated the Lorentz transformation in terms of velocity v:

{\displaystyle {\begin{matrix}q'=QqQ\\\hline {\begin{aligned}q&=\mathbf {r} +l=xi+yj+zk+\iota ct\\q&'=\mathbf {r} '+l'=x'i+y'j+z'k+\iota ct'\\Q&={\frac {1}{\sqrt {2}}}\left({\sqrt {1+\gamma }}+\mathrm {u} {\sqrt {1-\gamma }}\right)\\&=\cos \alpha +\mathrm {u} \sin \alpha =e^{\alpha \mathrm {u} }\\&\left\{\gamma =\left(1-v^{2}/c^{2}\right)^{-1/2},\ 2\alpha =\operatorname {arctg} \ \left(\iota {\frac {v}{c}}\right)\right\}\end{aligned}}\end{matrix}}}

Silberstein cites Cayley (1854, 1855) and Study's encyclopedia entry (in the extended French version of Cartan in 1908), as well as the appendix of Klein's and Sommerfeld's book.

## References

### Historical mathematical sources

1. Euler (1770/71), pp. 75f
2. Euler (1770/71), p. 101
3. Rodrigues (1840), p. 405
4. Euler (1770/71), pp. 89–91
5. Euler (1770/71), p. 102
6. Hamilton (1844/45), p. 13
7. Hamilton (1844/45), p. 14
8. Cayley (1845), p. 142
9. Cayley (1848), p. 196
10. Cayley (1854), p. 211
11. Cayley (1855b), p. 312
12. Cockle (1848), p. 437
13. Cockle (1848), p. 438
14. Cox (1882/83a), pp. 85–86
15. Cox (1882/83a), p. 88
16. Cox (1882/83b), p. 195
17. Cox (1882/83a), p. 97
18. On pp. 104-105 he started using the term v2=-1, on p. 106 he noted that one can simply use ${\displaystyle {\sqrt {-1}}}$ instead of v, and on p. 112 he adopted Clifford's notation by setting ω2=-1.
19. Cox (1882/83a), pp. 108-109
20. Stephanos (1883), p. 590ff
21. Stephanos (1883), p. 592
22. Buchheim (1885), p. 309
23. Lipschitz (1886), pp. 75–79
24. Lipschitz (1886), pp. 134–138
25. Lipschitz (1886), p. 76; p. 137
26. Lipschitz (1886), pp. 145–147
27. Vahlen (1902), pp. 586–587, 590; (1905), p. 282

### Historical relativity sources

1. Klein (1908), p. 165
2. Noether (1910), pp. 939–943
3. Klein (1910), p. 300
4. Klein (1911), pp. 602ff.
5. Conway (1911), p. 8
6. Silberstein (1911/12), p. 793
• Klein, F. (1911). Hellinger, E.. ed. Elementarmethematik vom höheren Standpunkte aus. Teil I (Second Edition). Vorlesung gehalten während des Wintersemesters 1907-08. Leipzig: Teubner.
• Silberstein, L. (1912) [1911], "Quaternionic form of relativity", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23 (137): 790–809, doi:10.1080/14786440508637276

### Secondary sources

1. Synge (1972), pp. 13, 19, 24
2. Girard (1984), pp. 28–29
3. Sobczyk (1995)


text:
finding the jordan canonical with one eigenvalue

Let $$A= \left[ \begin{array}{ c c } -2 & -3 & 1 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{array} \right]$$ Determine the Jordan canonical form of A.

The only eigenvalue I found was $\lambda=-2$

So finding the eigenvector of this is $$\left[ \begin{array}{ c c } 0 & -3 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]. \left[ \begin{array}{ c c } x \\ y \\ z \end{array} \right]=0\implies -3x+z=0\iff z=3x$$ So let $x$ and $y$ be free variables so $x=a$ and $y=b$ with $a,b\in\mathbb{F}$. So $$\left[ \begin{array}{ c c } a \\ b \\ 3b \end{array} \right]=a\left[ \begin{array}{ c c } 1 \\ 0 \\ 0 \end{array} \right]+b\left[ \begin{array}{ c c } 0 \\ 1 \\ 3 \end{array} \right]$$ So we have two eigenvectors that are LI: $$v_1=\left[ \begin{array}{ c c } 1 \\ 0 \\ 0 \end{array} \right], v_2=\left[ \begin{array}{ c c } 0 \\ 1 \\ 3 \end{array} \right]$$

But what would $v_3$ be? Since correct me if I am wrong but our Jordan canonical matrix $J=P^{-1}AP$ where $P$ is the matrix made up of $v_1,v_2,v_3$.

• Your last sentence is only correct if $A$ has a basis of eigenvectors, which this matrix doesn't. Its JCF has a $2\times 2$ block and a $1\times 1$ block. – vadim123 Dec 8 '14 at 22:44
• so what type of block does this one have? – snowman Dec 8 '14 at 22:45
• $J=\left(\begin{smallmatrix}-2&1&0\\0&-2&0\\0&0&-2\end{smallmatrix}\right)$. – vadim123 Dec 8 '14 at 23:36

Hint: You can find a third generalized and linearly independent eigenvector by solving (use Gaussian Elimination):

$$[A-\lambda I]v_3 = [A + 2I]v_3 = v_1$$

You should get:

$$v_3 = \left(0, -\dfrac 13, 0 \right)$$

You can then use the eigenvectors to find (this uses your current eigenvectors) and $v_3$ as:

$$J = P^{-1} A P = \begin{bmatrix}-2 &0&0 \\0 & -2 & 1 \\ 0 & 0 & -2\end{bmatrix}$$

Where $P$ is made up of your column eigenvectors.

• Why is your first equation equal to $v_1$ and not $v_2$? – snowman Dec 9 '14 at 11:36
• that is a question I'd also like to be answered. – etothepitimesi Dec 14 '14 at 3:50
• I understand it would return $v_3=(0,0,0)$. What I'd like to know is if trial and error is the only way to proceed. More generally, I'd like to know if there's a way to find a priori the number of elements of the chain of generalized eigenvectors to which each eigenvector "belongs" :) – etothepitimesi Dec 14 '14 at 4:44
• thanks for the nice reference (and for your time). so, if I understood correctly, the answer to the question "Is there a way to find a priori the number of elements of the chain of generalized eigenvectors to which each eigenvector belongs?" is simply "No", right? – etothepitimesi Dec 14 '14 at 5:23


text:
### Home > PC > Chapter 12 > Lesson 12.2.2 > Problem12-51

12-51.

Earlier we found the matrix associated with the composition of rotating 90° counterclockwise and then reflecting across the x-axis. Now find the matrix associated with reflecting across the x-axis and then rotating 90°. Homework Help ✎

From problem 12-45:
A = matrix associated with 90º counterclockwise rotation
B = martix associated with reflecting across the x-axis

Multiply A and B in the opposite order you did in problem 12-45. That is, find BA.


text:
## donnie1999 one year ago HELP WILL MEDAL!!! simplify the expression 64^1/2

you can convert exponent to root example $\huge\rm x^\frac{ m }{ n } =\sqrt[n]{x^m}$ let m=1 and n=2 n=root number ask if u hve question


text:
Chris Pollett > Publications, Reviews, Master's Student Papers and Talks

# Publications, Reviews, Master's Student Papers, and Talks.

For copyright reasons versions below may be close to but not the final journal version.

## Publications

[25-PDF] [25-PS] Conservative Fragments of S12 and R12. Archive For Mathematical Logic. Vol. 50. Iss. 3. 2011. pp. 367 -- 393.

[24-PDF] [24-PS] Alternating Hierarchies for Time-Space Tradeoffs. (with Eric Miles). arXiv:0801.1307v1. 2008.

[23-PDF] [23-PS] The Weak Pigeonhole Principle for Function Classes in S12. (with Norman Danner). Mathematical Logic Quarterly. Volume 52, Issue 6 (December 2006). pp. 575--584.

[22-PDF] [22-PS] Circuit Principles and Weak Pigeonhole Variants. (with Norman Danner). This paper improves the result of the conference paper below. Theoretical Computer Science. Volume 383. 2007. pp. 115-131.

[21-PDF] [21-PS] Circuit Principles and Weak Pigeonhole Variants. (with Norman Danner). Conferences in Research and Practice in Information Technology -- Computing: The Australasian Theory Symposium (CATS 2005). Eds. M. Atkinson and F. Dehne. Volume 41. Australian Computer Science Communications. Volume 27. Number 4. pp. 31--40.

[20-PDF] [20-PS] On the Computational Power of Probabilistic and Quantum Branching Programs. (with Farid Ablayev, Aida Gainutdinova, Marek Karpinski, and Cristopher Moore). Information and Computation. Dec. 2005. Vol. 203. Iss 2. pp. 145--162.

[19-PDF] [19-PS] Languages to diagonalize against advice classes. Computational Complexity. Vol. 14 Iss 4. pp. 342--361. 2005. An earlier version with many typos appeared as ECCC TR04-014.

[18-PDF] [18-PS] Sk, exp does not prove NP=coNP uniformly. Zapiski Nauchnykh Seminarov. Dom 1. Vol. 304. 2003. pp. 99--120. (Without the uniformity condition the result only implies a restricted form of IOpen +exp cannot prove NP = coNP.) Translated from English to English but with some typos eliminated in Journal of Mathematical Sciences. Vol. 130. No. 2. Oct. 2005. pp.4607--4619.

[17-PDF] [17-PS] A Theory for Logspace and NLIN versus co-NLIN. Journal of Symbolic Logic. Vol. 68 No. 4. 2003. pp. 1082-1090.

[16-PDF] [16-PS] Nepomnjascij's Theorem and Independence Proofs in Bounded Arithmetic. ECCC TR02-051.

[15-PDF] [15-PS] Quantum and Stochastic Branching Programs of Bounded Width. (with Farid Ablayev and Cris Moore) 29th International Colloquium on Automata, Languages, and Programming (ICALP). 2002. p.343--354. Earlier versions can be found at ECCC TR02-013 and LANL quant-ph/020113. The upper bound result in the ICALP paper needed a syntactic restriction. This has been revised in the ECCC and LANL version and in the version on-line here.

[14-PDF] [14-PS] On the Bounded Version of Hilbert's Tenth Problem. Archive for Mathematical Logic. Vol. 42. No. 5. 2003. pp. 469--488.

[13-PDF] [13-PS] Counting, Fanout, and the Complexity of Quantum ACC. (with Fred Green, Steve Homer, and Cris Moore) Quantum Information and Computation. Vol. 2. No. 1. 2002. pp.35--65.

[12-PDF] [12-PS] Minimization and NP multifunctions. (with N. Danner) Theoretical Computer Science. Vol. 318. Iss. 1-2. 2004. pp. 105--119.

[11-PDF] [11-PS] Ordinal Notations and Well-Orderings in Bounded Arithmetic. (with A. Beckmann and S.R. Buss) Annals of Pure and Applied Logic.120 (Issue 1-3) Apr. 2003. pp. 197-223.

[10-PDF] [10-PS] Strengths and Weaknesses of LH Arithmetic. (with R. Pruim) Mathematical Logic Quarterly. 48 (No.2) Feb 2002. pp.221--243.

[9-PDF] [9-PS] On the Complexity of Quantum ACC. (with Fred Green and Steve Homer) Proceedings of the Fifteenth Annual IEEE Conference on Computational Complexity. July 4-7, 2000. pp250--262. Is also Boston University Technical Report BUCS-TR-2000-003 and LANL quant-ph/0002057.

[8-PDF] [8-PS] Multifunction Algebras and the Provability of PH↓. Annals of Pure and Applied Logic. Vol. 104 July 2000. pp. 279--303.

[7-PDF] [7-PS] Translating IΔ0 + exp proofs into weaker systems. Mathematical Logic Quarterly. 46:249-256 (No.2) May 2000.

[6-PDF] [6-PS] On the Δb1-bit comprehension rule. (with Jan Johannsen) Proceedings of Logic Colloquium 1998. edited by S.R. Buss, P. Hajek, P. Pudlak pp.262--270, A.K.Peters and ASL, 2000.

[5-PDF] [5-PS] Structure and Definability in General Bounded Arithmetic Theories. Annals of Pure and Applied Logic. Vol 100. October 1999. pp.189--245.

[4-PDF] [4-PS] On Proofs about Threshold Circuits and Counting Hierarchies (with Jan Johannsen). Proceedings of Thirteenth IEEE Symposium on Logic in Computer Science. pp.444--452. IEEE press. 1998.

[3-PDF] [3-PS] Nonmonotonic reasoning with quantified boolean constraints. (with J. Remmel) Logic Programming and Nonmonotonic Reasoning. 1997. Jurgen Dix, Ulrich Furbach and Anil Nerode (Eds). pp.18--39. LNAI, 1265. Springer. 1997.

[2-PDF] [2-PS] A propositional proof system for Ri2. Proof Complexity and Feasible Arithmetics. Paul W. Beame and Samuel R. Buss eds. pp.253--278. 1997. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science. Vol 39. AMS. 1997.

[1-PDF] Arithmetic Theories with Prenex Normal Form Induction. Ph.D. Thesis. UC San Diego. 1997.

## Reviews

[Rev 13-PDF] [Rev 13-PS] Review of S. Cook and N. Thapen paper: The Strength of Replacement in Weak Arithmetic'' Mathematical Reviews. MR#2264422.

[Rev 12-PDF] [Rev 12-PS] Review of G. Ferreira and I. Oitavem paper: An Interpretation of S12 in Σb2-NIA.'' Mathematical Reviews. MR#2287276.

[Rev 11-PDF] [Rev 11-PS] Review of P. Naumov paper: Upper bounds on complexity of Frege proofs with limited use of certain schemata.'' Mathematical Reviews. MR#2227496.

[Rev 11-PDF] [Rev 11-PS] Review of M. R. Fellows, S. Szeider and G. Wrightson paper: On finding short resolution refutations and small unsatisfiable subsets.'' Mathematical Reviews. MR#2202495.

[Rev 10-PDF] [Rev 10-PS] Review of M. Jarvisalo, T. A. Junttila and I. Niemela paper: Unrestricted vs restricted cut in a tableau method for Boolean circuits.'' Mathematical Reviews. MR#2185702.

[Rev 9-PDF] [Rev 9-PS] Review of J. Krajicek paper: Structured pigeonhole principle, search problems, and hard tautologies.'' Mathematical Reviews. MR#2140049.

[Rev 8-PDF] [Rev 8-PS] Review of M. Alekhnovich, E. Ben-Sasson, A. Razborov, and A Wigderson paper: Pseudorandom number generators in propositional proof complexity.'' Mathematical Reviews. MR#2114305.

[Rev 7-PDF] [Rev 7-PS] Review of M. Moniri paper: Polynomial induction and length minimization in intuitionistic bounded arithmetic.'' Mathematical Reviews. MR#2099386.

[Rev 6-PDF] [Rev 6-PS] Review of J. Krajicek paper: Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds .'' Mathematical Reviews. MR#2039361.

[Rev 5-PDF] [Rev 5-PS] Review of A. Beckmann paper: Dynamic ordinal analysis.'' Mathematical Reviews. MR#2018084.

[Rev 4-PDF] [Rev 4-PS] Review of Buresh-Oppenheim, Clegg, Impagliazzo, Pitassi paper: Homogenization and the polynomial calculus.'' Mathematical Reviews. MR#2022043.

[Rev 3-PDF] [Rev 3-PS] Review of P. Pudlak paper: Parallel Strategies.'' Mathematical Reviews. MR#2017351.

[Rev 2-PDF] [Rev 2-PS] Review of M. Moniri paper: Comparing constructive arithmetical theories based on NP-PIND and coNP-PIND.'' Mathematical Reviews. MR#2030203.

[Rev 1-PDF] [Rev 1-PS] Review of A. Beckmann paper: Proving consistency of equational theories in bounded arithmetic.'' Bulletin of Symbolic Logic. Vol. 9. No. 1. 2003. pp.44-45.

## Master's Students Papers

These are some papers I am author on that were mainly dealing with research conducted by master's students who I either supervised or was on their commitee.

[MS 2-PDF] [MS 2-PS] Stealthy Ciphertext. (with Martina Simova and Mark Stamp) Proceeding of the Conference on Internet Computing. 2005. pp. 380--386.

[MS 1-PDF] [MS 1-PS] On using Mouse Movement as a Biometric. (with Shivani Hashia and Mark Stamp) The 3rd International Conference on Computer Science and its Applications. (ICCSA) . 2005. pp. 143--147.

## Talks

This is a partial list of talks I have given at various places, together with overheads if I could find them.

Oct. 9, 2011. On the Finite Axiomatizability of Prenex R12. Talk given at BIRS. Banff, Canada.

April 8. 2008. Weak Definability Notions for Independence Results in Bounded Arithmetic. Talk given at Oberwolfach, Germany.

Feb. 1. 2008. The Surjective Weak Pigeonhole Principle in Bounded Arithmetic. Talk given at VIG 2008, UCLA.

Apr. 10. 2006. When can S12 prove the weak pigeonhole principle? Talk given at Newton Mathematical Institute, Cambridge University. An embarassing glitch appeared in a proof in the original talk on the next to last slide. This is corrected in the version I am posting here.

It is well known result of Krajicek and Pudlak that if S12 could prove the injective weak pigeonhole principle for every polynomial time function then RSA would not be secure. In this talk, we will consider function algebras based on a common characterization of the polynomial time functions where we slightly modify the initial functions and further restrict the amount of allowable recursion. We will then argue that S12 can prove the surjective weak pigeonhole principle for functions in this algebra.

Apr. 4. 2006. Introduction to Quantum Branching Programs. Talk given at Berkeley Quantum Information and Computation Seminar (Chemistry Department, UC Berkeley). (Joint work with Farid Ablayev, Aida Gainutdinova, Marek Karpinski, and Cristopher Moore.)

Branching programs of bounded width are a nonuniform model of computation which can be viewed as generalizing finite automata. As such they are a relatively simple model of computation and quantum versions of branching programs might serve as a good model on which to develop potentially implementable quantum algorithms. In this talk we will introduce quantum branching programs and discuss upper and lower bounds in terms of deterministic and stochastic models of computation on what kind of algorithms are implementable on quantum branching programs.

Dec. 1. 2005. Video Game Engines. Talk given to High School Students at National Hispanic University Charter High School.

Apr. 16. 2005. Circuits Principles and Weak Pigeonhole Principles. Talk given at AMS Sectional Meeting #1007, UC Santa Barbara, Special Session on Complexity of Computation and Algorithms. (Joint work with Norman Danner.) The next four talks listed below are very similar, this talk's set of slides is probably the best. I am retiring this talk at this point.

This talk considers the relational versions of the surjective and multifunction weak pigeonhole principles for various classes of formulas. These principles are interesting because of their close connection to the provability of circuit lower bounds, and hence the P versus NP question, in weak systems of arithmetic. We show that the relational surjective pigeonhole principle for Θb1 in S12 implies a circuit block-recognition principle which in turn implies the surjective weak pigeonhole principle for Σb1 formulas. We introduce a class of predicates corresponding to poly-log length iterates of polynomial-time computable predicates and show that over R12, the multifunction pigeonhole principle for such predicates is equivalent to an "iterative" circuit block-recognition principle. A consequence of this is that if R23 proves this circuit iteration principle then RSA is vulnerable to quasi-polynomial time attacks.

Mar. 22. 2005. Equivalents of the Weak Multifunction Pigeonhole Principle. Talk given at Oberwolfach, Germany. (Joint work with Norman Danner.)

This talk discusses joint work of myself with Norman Danner of Wesleyan University. I began the talk by presenting a recent result of Jerabek on the surjective weak pigeonhole principle for p-time functions. Namely, that over the theory S12 this principle is equivalent to the existence of a string which is hard for any circuit of size nk. This shows that T22, a slightly stronger theory, can prove a predicate exists which is hard for circuits of size nk. Krajicek and Pudlak have shown if the injective weak pigeonhole principle for p-time functions is witnessable from a class C satisfying PTIMEC = C then RSA is insecure against attacks from C. As the multifunction weak pigeonhole principle implies both the injective and surjective principles, it is natural to wonder if there is any circuit class such that the existence of a hard string for this class is equivalent to the multifunction weak pigeonhole principle for the analogous uniform class. We show that for R22, a theory between T22 and S12 in strength, the multifunction weak pigeonhole principle for quasi-log iterated p-time relations is equivalent to circuit lower bounds for quasi-log iterated p-size circuits. Thus, we show if R22 could prove lower bounds for this class of circuits, one can also show RSA is insecure against quasi-polynomial time attacks.

Feb. 1. 2005. Circuit principles and weak pigeonhole variants. Talk given for my CATS 2005 paper, Newcastle, Australia. (Joint work with Norman Danner.)

This paper/talk considers the relational versions of the surjective and multifunction weak pigeonhole principles for PV, Σb1 and Θb2-formulas. We show that the relational surjective pigeonhole principle for Θb2-formulas in S12 implies a circuit block-recognition principle which in turn implies the surjective weak pigeonhole principle for Σb1 formulas. We introduce a class of predicates corresponding to poly-log length iterates of polynomial-time computable predicates and show that over R22, the multifunction pigeonhole principle for such predicates is equivalent to an iterative'' circuit block-recognition principle. A consequence of this is that if R23 proves this circuit iteration principle then RSA is vulnerable to quasi-polynomial time attacks.

Nov. 4. 2004. Circuit Principles, The Weak Pigeonhole Principle, and RSA. SJSU Computer Science Colloquium. Nov 4. 2004. (Joint work with Norman Danner.)

The weak pigeonhole principle for a relation says that the relation does not represent a map of n2 pigeons into n holes such that each pigeon is mapped and each hole only gets one pigeon. This principle for polynomial time relations is closely connected with the RSA cryptographic scheme. In particular, Krajicek and Pudlak have shown that if given any polynomial time relation one can find a polynomial time algorithm which finds for this relation either an unmapped pigeon or two pigeons in the the same hole, then one could break RSA in polynomial time. In this talk we will discuss this result. It is also an open area of research how much mathematics is needed to prove the weak pigeonhole principle. We will show this problem for certain weak systems is connected to whether the system can prove lower bounds on the the size of circuits. After discussing several results of this type from our own research and that of Jerabek, we will finally connect this back to the security of RSA.

Aug. 19, 2003. Weak Arithmetics and Unrelativized Independence Results. Invited talk. Logic Colloquium 2003 (LC 2003). Helsinki, Finland.

In this talk I will survey some techniques which can be used to show unrelativized independence results for questions like NP vs coNP, the collapse of the polynomial hierarchy, or Hilbert's Tenth Problem in weak systems of arithmetic. I will also present some new results concerning the limits of formalizing padding arguments in commonly studied weak arithmetics.

Apr. 15, 2003. On the Power of Classical and Quantum Branching Programs. CS Colloquium. Santa Clara University.

Branching Programs have proven to be a useful model of computation in a variety of domains such as hardware verification, model checking, and other CAD applications. As branching programs are also a very simple model of computation with several easy ways to restrict their power, it is interesting to generalize the branching program model to the quantum setting. In this talk I will survey some of the known results about classical branching programs, discuss (assuming no background in quantum mechanics) how both classical and quantum branching program models work, and describe some of our results concerning the power of the quantum branching program model.

Jan. 25, 2003. Bounded Versions of Hilbert's Tenth Problem and NP = co-NP. Mini-Workshop: Hilbert's Tenth Problem, Mazur's Conjecture and Divisibility Sequences. Mathematisches Forschungsinstitut Oberwolfach. Oberwolfach, Germany.

We discuss the provability of Matijasevich-Robinson-Davis-Putnam (MRDP) result in weak systems of arithmetic. It is a well-known result of Gaifman and Dimitricopoulos IDelta_0+exp proves MRDP. What was shown in their result was that every bounded formula in their language could be rewritten as a formula consisting of an existential block of quantifiers followed by an equation of the form p = q where p and q are polynomials. By Parikh's Theorem, IDelta_0+exp cannot prove the existence of superexponentially fast growing functions. Therefore, one could ask whether if one expanded the language by IDelta_0+exp's access to it, then one could obtain a system unable to prove MRDP in this new language. This is possible because now the bounds on the quantifiers that need to be eliminated are larger than before. In fact, we construct a system that cannot prove MRDP and show as well that it cannot prove NP = co-NP in a certain very uniform way.

Nov. 27, 2002. Nepomnjascij's Theorem and Independence Proofs in Bounded Arithmetic. Logic Colloquium. UC Berkeley.

Many complexity classes in computer science, for example, functions in PTIME, LOGSPACE, NC, AC^0, have characterizations in terms of being the provably "NP-definable" functions of some weak theory of arithmetic. If the complexity class in question is known not to be equal to NP, this can be used to show the weak system of arithmetic cannot prove NP = co-NP. The problem with this kind of result is, of course, you need to be able to show first that the complexity class in question is different from NP. For all but the weakest classes this question has been open for quite some time. In this talk I will survey some of my own work in this area as well as some recent work of Ressayre and Boughattas. Then I will present some new results of mine based on Nepomnjascij's Theorem that get around this barrier in a slightly different setting related to the problem of whether nondeterministic linear time is equal to co-nondeterministic linear time and related to how strong a theory of arithmetic is needed to prove the Matiyasevich-Robinson-Davis-Putnam Theorem.

Aug. 7, 2002. Quantum and Stochastic Branching Programs of Bounded Width. 29th International Colloquium on Automata, Languages, and Programming. (ICALP 2002). Malaga, Spain.

Jun. 29, 2002. Using IDelta_0 in independence proofs. Bounded Arithmetic and Complexity Classes 2002. (BACC 2002). Lisbon, Portugal.

Jun. 7, 2002. On the Bounded Version of Hilbert's Tenth Problem. 21st Days of Weak Arithmetics. (JAF '21) Steklov Institute of Mathematics St.Petersburg, Russia.

Hilbert's Tenth problem concerned the decidability of Diophantine equations over the integers. Its negative solution, Matiyassevich's theorem, amounted to showing that the class of formulas of the form
(exists y)P(x,y)=Q(x,y)
where P, Q are polynomials with natural number coefficients is equivalent to the class of recursively enumerable sets. The bounded form of Hilbert's Tenth problem is whether the NP-predicates are the class D of predicates given by formulas of the form
( exists y)[(\sum_j yj <= 2^{|\sum_i xi|^k}) /\ P(x,y)=Q(x, y)]
where P, Q are polynomials with natural coefficients. This problem is related to the average case completeness of certain NP-problems. In this talk we give lower bounds on the provability of both these problems in weak fragments of arithmetic. We show the theory I^5E cannot prove D=NP. Here I^mE has a finite set of axioms and induction on bounded existential formulas up to m lengths of a number for the language L_2, which has the symbols <= , 0, S, +, x#y, |x|, 2^{|x||y|}, \lfloor x/2^i \rfloor, and limited subtraction. We use the non-provability of D=NP to show that I^5E cannot prove the Matiyassevich's theorem.

Apr. 8, 2002. Implicit Complexity and Bounded Arithmetic Theories. Mathematische Logik. Mathematisches Forschungsinstitut Oberwolfach. Oberwolfach, Germany.

Bounded arithmetic theories are weak fragments of arithmetic useful in the study of computational complexity classes. In this talk we will discuss known techniques for doing independence proofs in these theories. We will then consider the question of classifying the Sigma^b_1≠definable multifunctions of a particular bounded arithmetic theory, S_2. Such a classification might prove useful in proving new independence results. We will then indicate why an implicit complexity approach to this characterization might be reasonable.

Jun., 2001. Computational complexity and independence results. Meeting on Discrete Mathematics and Mathematical Cybernetics organized by Moscow State University. Dubna, Russia.

Jul. 30, 2000. Strengths and Weaknesses of LH Arithmetic. Contributed paper and talk. Logic Colloquium 2000 (LC 2000). Paris, France.

In Pollett~\cite{cpollett00} a bounded arithmetic theory $Z$ was shown not to be able to prove the collapse of the polynomial hierarchy. This theory also had the property that if $Z \subseteq S^i_2$ for any $i\geq 1$ then the polynomial hierarchy collapses. Here $S^i_2$ are the theories of Buss~\cite{bus86a}. Unfortunately, despite this property $Z$ seemed too weak a theory to formalize many of the arguments that have been used in computational complexity. In this talk, we give a new arithmetic characterization of the levels of $\log$-time hierarchy. Using this characterization, we propose a new variant of the theory $TAC^0$ of Clote and Takeuti~\cite{clotak95}. This variant has nice deductive fragments which in some sense correspond to the levels of the log-time hierarchy. We show that this theory (like $Z$) cannot prove the collapse of the polynomial hierarchy. Furthermore, we give some evidence that this theory may be strong enough to prove that the log-time hierarchy is infinite, so unlike $Z$ it can carry out useful complexity arguments.

Jul., 2000. On the Complexity of Quantum ACC. Computational Complexity 2000. Florence, Italy. The linked slides have some the content of the original talk; however, some slides are from a job interview at SJSU and page 13 has been lost.

Nov. 5, 1999. Strengths and Weaknesses of LH Arithmetic. Logic Colloquium, UCLA.

One way to quantify the difficulty of P = NP problem would be to exhibit a logical theory which is capable of formalizing current attempts at answering this question and yet which is not powerful enough to prove or disprove this equality. Razborov has argued that most current circuit lower bound techniques can be formalized in a fragment of arithmetic which roughly has induction on NE-predicates. Nevertheless, exhibiting any fragment of arithmetic which one can demonstrate cannot prove the collapse of the polynomial hierarchy is nontrivial. In this talk we will consider a much weaker theory than Razborov's, but which we argue can still do some interesting mathematics. Namely, it can "reason" about all the functions in the log-time hierarchy, it can prove the log-time hierarchy differs from NP, and, in fact, we give some evidence it might be able to show the log-time hierarchy is infinite. (In the real world this is known to be true.) On the other hand, we show this theory cannot prove the polynomial hierarchy collapses. So, in particular, it cannot prove P = NP, and it follows that any proof that the polynomial hierarchy collapses, if such a proof exists, must be formalized in a stronger fragment than ours.

Mar. 21, 1999. Using translation to separate bounded arithmetic theories. ASL Annual Meeting. UC San Diego.

Aug. 3-7, 1998. Multifunction Algebras and Provability of the Collapse of PH. Proof Theory and Complexity 1998. (PTAC 1998). BRICS, Aarhus, Denmark.

In this talk I will introduce some multifunctions algebras which for i > 1 correspond to functions computable in polynomial time with a limited number of witnessing queries to an oracle at the i - 1 level of the hierarchy. We then consider two subtheories of the well-studied bounded arithmetic theory S_2 of Buss. Actually, one of our theories is contained in the other. Using our algebras (mainly the i = 1 variants on our algebras) we establish the following properties for these theories: (1) Neither theory can prove the polynomial hierarchy collapses.
(2) If either theory is contained in S^i_2
for some i then the polynomial hierarchy collapses.
(3) If either theory proves the polynomial hierarchy is infinite then for all i, S^i_2
can separate the ith level of the hierarchy.
(4) There is an interesting initial segment of any model of the weaker theory that satisfies all of IDelta_0 + exp. (Postscript: statement 4 in restrospect was not quite correct, although it did lead to me write a paper about IDelta_0 +exp presented in San Diego.)

Apr. 17, 1998. Bounded query classes and Bounded Arithmetic. Adelphia University.

In trying to solve the P=NP question people have developed lower bounds techniques such as Hastad's Switching Lemma, Razborov-Smolensky, etc. These techniques can often be formalized in very weak theories of arithmetic. One such theory which can prove the Switching Lemma is S_2(\alpha). A natural question is "can one show that the P=NP problem is independent of such theories?" If so, one can rule out certain lower bounds methods as a means of solving this problem. In this talk we will consider some complexity questions related to fragments of S_2(\alpha) and bounded query classes. We will use our results to derive a weak relativized independence result.

Mar. 5, 1998. Bounded query classes and bounded arithmetic. Logic, Computability and Complexity 1998. (LCC 1998). Amsterdam, Netherlands.

There is a well-known result of Krajicek which connects the bounded query class P^{\Sigma^p_i}(log) with the Delta^b_{i+1}-predicates of S^i_2. In this talk we discuss a generalization of this result which was motivated by trying to show the Delta^b_{i+1}-predicates of R^i_2 are P^{Sigma^p_i}(loglog). We also discuss a general condition result which can be used to show on bounded arithmetic theory is conservative over another. We then use these results together with recent results about bounded query classes to derive tighter collapses of the polynomial hierarchy under the assumption that various bounded arithmetic theories are equal. We finally discuss new relativized seperations of bounded arithmetic theories and a weak relativized independence result.

Oct.(?), 1997. Structure and Definability in Bounded Arithmetic Theories. Logic Seminar. Massachusetts Institute of Technology.

Jul. 28, 1997. Nonmonotonic Reasoning with Quantified Boolean Constraints. The Fourth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR '97). Schloss Dagstuhl, Germany.

We define and investigate the complexity of several nonmonotonic logics with quantified Boolean formulas as constraints. We give quantified constraint versions of the constraint programming formalism of Marek, Nerode, and Remmel [15] and of the natural extension of their theory to default logic. We also introduce a new formalism which adds constraints to circumscription. We show that standard complexity results for each of these formalisms generalize in the quantified constraint case. Gogic, Kautz, Papadimitriou, and Selman [8] have introduced a new method for measuring the strengths of reasoning formalisms based on succinctness of model representation. We show a natural hierarchy based on this measure exists between our versions of logic programming, circumscription, and default logic. Finally, we discuss some results about the relative succinctness of our reasoning formalisms versus any formalism for which model checking can be done somewhere in the polynomial time hierarchy.

May 27, 1997. My thesis defense talk. UC San Diego.


text:
# Does Rolle's Theorem imply Dedekind completeness?

I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually a solution to a Monthly problem) in which M. J. Pelling claims to construct a non-Archimedean ordered field in which Rolle's Theorem holds; see http://www.jstor.org/pss/2321145 .

Since I'm working on an article for the Monthly ( http://jamespropp.org/reverse.pdf ) that claims to prove that the Mean Value Theorem (which follows from Rolle's Theorem) implies Dedekind completeness, I'd like to know who's right!

First, I'll show that the Mean Value Theorem implies the Constant Value Theorem. Then I'll show that the latter implies the Cut Property. Then I'll show that the Cut Property implies Dedekind completeness. And then one of you will tell me where my mistake is, or where Pelling's mistake is.

Step 1: Suppose $f$ is a continuous function on $[0,1]$ that is differentiable on $(0,1)$, with $f'(x)=0$ for all $x$ in $(0,1)$. We'd like to show that $f$ is constant. Suppose not; consider any $a > 0$ with $f(a) \neq f(0)$. Then (by the Mean Value Theorem) there must exist $c$ in $(0,a)$ with $f'(c) = (f(a)-f(0))/(a-0) \neq 0$, which is a contradiction. Hence the Constant Value Theorem holds.

Step 2: Suppose $A$ and $B$ are disjoint sets whose union is the whole ordered field, such that $a < b$ for all $a \in A$, $b \in B$. We'd like to show that there exists $c$ in the ordered field such that $a \leq c$ for all $a \in A$ and $b \geq c$ for all $b \in B$. Suppose not. Then for all $a \in A$ there exists $a' \in A$ with $a' > a$, and for all $b \in B$ there exists $b' \in B$ with $b' < b$, from which it follows that the indicator function of the set $A$ is continuous and differentiable everywhere, with derivative constantly 0. This contradicts the Constant Value Theorem. Hence the Cut Property holds.

Step 3: Let $S$ be a bounded non-empty subset of the ordered field. We'd like to show that $S$ has a least upper bound. Let $B$ be the set of upper bounds of $S$, and let $A$ be the complement of $B$. Then $A$ and $B$ satisfy the hypotheses of the Cut Property, so there exists $c$ such that $a \leq c$ for all $a \in A$ and $b \geq c$ for all $b \in B$. It is easy to check that $c$ is a least upper bound for $S$.

This argument seems fairly straightforward to me, so I'm inclined to think that the error must be in Pelling's more complicated argument (which I won't pretend to understand). Though perhaps it will turn out that Pelling and I mean different things when we say than an ordered field satisfies Rolle's theorem.

Can anyone shed light on this?

-
This is just a random thought--I have not read through your proof--but the exactly what is meant by a "continuous" function might be an important point here. In particular, a non-Archimidean field often comes with its own topology, which I can imagine might not agree with the order topology if the field happens to be ordered. –  Charles Staats Feb 7 '12 at 23:48

It turns out that Pelling and I mean different things by "Rolle's Theorem": his form of Rolle's Theorem treats only polynomials. I figured that out when I came across the following passage: "... Then $F$ is not real-closed since, e.g., $z^{1/2} \in L - F$, and it remains to show that Rolle's theorem is valid in $F$. By linear changes of variable if necessary it suffices to prove that if $p(x)$ is a polynomial [emphasis mine] over $F$ such that $p(0)=p(1)=0$ then $p'(x)$ has a root in the open interval $(0,1)$ in $F$." Case closed (I think).


text:
# How would you arrange the ions N^-3, O^-2, Mg2+, Na+, and F- in order of increasing ionic radius?

May 13, 2016

$\text{_12Mg^"-3"< ""_11Na^"+"<""_9F^"-"<""_8O^"-2"<""_7N^"-3}$

#### Explanation:

If the ions derived from different atoms are isoelectronic species, then they all have same number of electrons in their electronic shells and will have got same electronic configuration but their nuclear charge will differ because of their difference in number of protons in the nucleus. With increase in number of protons in the nucleus the electrons are more attracted towards nucleus thereby causing the decrease in ionic radius. On this principle our problem will be solved

The given ions are

• $\text{_7N^"-3"->"no. of proton"=7 and" no of electron} = 10$
• $\text{_8O^"-2"->"no. of proton"=8 and" no of electron} = 10$
• $\text{_9F^"-"->"no. of proton"=9 and" no of electron} = 10$
• $\text{_11Na^"+"->"no. of proton"=11 and" no of electron} = 10$
• $\text{_12Mg^"-3"->"no. of proton"=12 and" no of electron} = 10$

Hence the increasing order of ionic radius is

$\text{_12Mg^"-3"< ""_11Na^"+"<""_9F^"-"<""_8O^"-2"<""_7N^"-3}$

To rmember ->For isoelectronic species lower the nuclear charge higher the radius


text:
Coin toss

A fair coin is tossed 8 times, find the probability that the resulting sequences of heads and tails looks same when viewed from the beginning or from the end.

If answer is in the form $$\dfrac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers, then find $$a+b$$.

×


text:
# How do you solve y+6=2x & 4x-10y=4?

May 25, 2017

$x = \frac{7}{2}$ and $y = 1$

#### Explanation:

Strategy: Solve for $y$ in terms of $x$. Plug in that $y$ equation into the other equation to solve for $x$. With the solution for $x$, you can back track and find the solution for $y$.

Step 1. Solve for $y$ in terms of $x$.
$y + 6 = 2 x$ given
$y = 2 x - 6$ subtract 6 from both sides

Step 2. Plug this $y$ equation into the other equation.
$4 x - 10 \textcolor{red}{y} = 4$ ; given
$4 x - 10 \left(\textcolor{red}{2 x - 6}\right) = 4$ ; replace variable $y$ with $2 x - 6$
$4 x - 20 x + 60 = 4$ ; distribute $- 10$ through
$- 16 x = - 56$ ; combine $x$ terms and subtract $60$ both sides
$x = \frac{7}{2}$ ; divide both sides by $- 16 \mathmr{and} red u c e$

Step 3. Plug this solution back into the equation of step 1.
$y = 2 \textcolor{red}{x} - 6$ ; final part of step 1
$y = 2 \left(\textcolor{red}{\frac{7}{2}}\right) - 6$ ; solution for $x$ in step 2
$y = 7 - 6 = 1$

So your solution is $x = \frac{7}{2}$ and $y = 1$

May 25, 2017

$\left(\frac{7}{2} , 1\right)$

#### Explanation:

$\textcolor{red}{y} + 6 = 2 x \to \left(1\right)$

$4 x - 10 \textcolor{red}{y} = 4 \to \left(2\right)$

$\text{note that in " (1)" y can be expressed in terms of x}$

$\Rightarrow \textcolor{red}{y} = 2 x - 6 \to \left(3\right)$

$\text{substitute into } \left(2\right)$

$\Rightarrow 4 x - 10 \left(2 x - 6\right) = 4$

$\Rightarrow 4 x - 20 x + 60 = 4 \leftarrow \text{ distributing}$

$\Rightarrow - 16 x + 60 = 4 \leftarrow \text{ simplifying left side}$

$\text{subtract 60 from both sides}$

$- 16 x \cancel{+ 60} \cancel{- 60} = 4 - 60$

$\Rightarrow - 16 x = - 56$

$\text{divide both sides by - 16}$

$\frac{\cancel{- 16} x}{\cancel{- 16}} = \frac{- 56}{- 16}$

$\Rightarrow x = \frac{56}{16} = \frac{7}{2}$

$\text{substitute this value in " (3)" and evaluate for y}$

$y = \left(2 \times \frac{7}{2}\right) - 6 = 7 - 6 = 1$

$\left(\frac{7}{2} , 1\right) \text{ is the point of intersection of the 2 equations}$
graph{(y-2x+6)(y-2/5x+2/5)=0 [-10, 10, -5, 5]}


text:
Bessel's Inequality for Inner Product Spaces

## Bessel's Inequality for Inner Product Spaces

Recall from the The Pythagorean Identity for Inner Product Spaces page that if $H$ is an inner product space and $\{ x_1, x_2, ..., x_n \}$ is an orthonormal subset of $H$ then for all $c_1, c_2, ..., c_n \in \mathbb{C}$ we have that:

(1)
\begin{align} \quad \biggr \| \sum_{k=1}^{n} c_kx_k \biggr \|^2 = \sum_{k=1}^{n} |c_k|^2 \end{align}

We now use the Pythagorean identity to prove the very important Bessel's inequality.

 Theorem 1 (Bessel's Inequality for Inner Product Spaces): Let $H$ be an inner product space and let $(x_n)_{n=1}^{\infty}$ be an orthonormal sequence in $H$. Then for every $y \in H$ we have that $\displaystyle{\sum_{n=1}^{\infty} |\langle y, x_n \rangle|^2 \leq \| y \|^2}$.
• Proof: For each $N \in \mathbb{N}$ let:
(2)
\begin{align} \quad y_N = \sum_{n=1}^{N} \langle y, x_n \rangle x_n \end{align}
• Since $\{ x_1, x_2, ..., x_N \}$ is an orthonormal subset of $H$, by the Pythagorean identity for inner product spaces we have that:
(3)
\begin{align} \quad \| y_N \|^2 &= \biggr \| \sum_{n=1}^{N} \langle y, x_n \rangle x_n \biggr \|^2 \\ &= \sum_{n=1}^{N} |\langle y, x_n \rangle|^2 \end{align}
• Therefore:
(4)
\begin{align} \quad 0 &\leq \| y - y_N \|^2 \\ & \leq \langle y - y_N, y - y_N \rangle \\ & \leq \| y \|^2 - 2 \mathrm{Re} \langle y, y_N \rangle + \| y_N \|^2 \\ & \leq \| y \|^2 - 2 \mathrm{Re} \langle y, y_N \rangle + \sum_{n=1}^{N} |\langle y, x_n \rangle|^2 \quad (*) \end{align}
• Observe that:
(5)
\begin{align} \quad \mathrm{Re} \langle y, y_N \rangle &= \mathrm{Re} \sum_{n=1}^{N} \overline{\langle y, x_n \rangle} \langle y, x_n \rangle \\ &= \mathrm{Re} \sum_{n=1}^{N} |\langle y, x_n \rangle|^2 \\ &= \sum_{n=1}^{N} |\langle y, x_n \rangle|^2 \end{align}
• So from $(*)$ we see that:
(6)
\begin{align} \quad 0 \leq \| y \|^2 - \sum_{n=1}^{N} |\langle y, x_n \rangle|^2 \end{align}
• So for each $N \in \mathbb{N}$
(7)
\begin{align} \quad \sum_{n=1}^{N} |\langle y, x_n \rangle|^2 \leq \| y \|^2 \end{align}
• Taking the limit as $N \to \infty$ gives us that:
(8)
\begin{align} \quad \sum_{n=1}^{\infty} |\langle y, x_n \rangle|^2 \leq \| y \|^2 \end{align}


text:
# 2D arrays in c++

This topic is 3989 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Im making a console c++ program for a personal assignment, and a part of it needs operations on an array.
    A B C D E F G H I...Z

A   a b c d e f g h i...z
B   b c d e f g h i...z a
C   c d e f g h i...z a b
D   d e f g h i...z a b c
...
Z z
(capital letters denote a referencing row)

Operation - 1. User picks two letters from the reference rows (eg 'C' and 'D') 2. Program shows the alphabet which occours in the same row and column as both of the reference letters (here that'd be 'f', for 'B' and 'G' that'd be 'h', etc) Could anyone please give me the code fragment for the two given operations? -regards, BlueBolt [Edited by - BlueBolt on September 17, 2007 8:16:39 AM]

##### Share on other sites
Do you have an actual question?

##### Share on other sites
You don't need operations on arrays for this. The structure of values inside the arrays is so regular that you can generate it on the fly using two '+' and one '%'.

##### Share on other sites
@SiCrane - Of course...just asked one.
Quote:
 Original post by ToohrVykYou don't need operations on arrays for this. The structure of values inside the arrays is so regular that you can generate it on the fly using two '+' and one '%'.

I tried to use the ASCII of a character to generate the entire English alphabet (as a starting step to implement your advice)...it just spewed me with junk values...
#include<iostream>using namespace std;int main(){    char alpha;    alpha=1; //or maybe alpha=a;    cout<<alpha;    cin>>alpha;    for(int i=0; i<=26; alpha++)            cout<<alpha;    return 0;}

I have a gist that if I knew how to represent alphabets in their ASCII code in the program, I would be able to make the alphabetic sequences (a matter of an increment sign, as ToohrVyk said...dunno where the modules pops in)

-regards,
BlueBolt

##### Share on other sites
char alpha = 'a';

##### Share on other sites
Quote:
 Could anyone please give me the code fragment for the two given operations?

But, given that the entire structure of the OP is very much like a homework assignment, most would be unwilling to do so.

Quote:
 for(int i=0; i<=26; alpha++) cout<

Letters are represented as ASCII table. But since you can treat them as numbers in your particular case, then this becomes valid:

for (char c = 'a'; c <= 'z'; c++) std::cout << c;

The other way, to obtain n-th character, can be obatined using:
int index = 14;std::cout << ( 'a' + index );

Obviously, it goes the other way as well. Which number is the letter x?
char x = 'f';std::cout << (x - 'a');

And no, there is no need for a table, everything can be calculated using the above operations.

1. 1
2. 2
3. 3
4. 4
frob
14
5. 5

• 16
• 12
• 20
• 12
• 19
• ### Forum Statistics

• Total Topics
632167
• Total Posts
3004528

×


text:
# Law of sines

### Law of sines

In this section, we will learn about the Law of Sines, also known as the Sines Rule. The Law of Sines is a formula that models the relationship between the sides and the angles of any triangle, be it a right-angled triangle, an obtuse triangle, or an acute triangle. In order to use the Law of Sines, we need to satisfy the "one pair, one additional information" condition (i.e. Angle-Angle-Side abbreviated as AAS, and Angle-Side-Angle abbreviated as ASA). We will also explore the concept of the Ambiguous Case of the Law of Sines.

#### Lessons

Law of Sine
For any $\triangle$ ABC,

$\frac{a}{\sin(A)}$ $=\frac{b}{\sin(B)}$ $=\frac{c}{\sin(C)}$
and,
$\frac{\sin(A)}{a}$ $=\frac{\sin(B)}{b}$ $=\frac{\sin(C)}{c}$

Use the Law of Sine when given a pair!

Ambiguous case
Ambiguous case of the Law of Sine arises when given SSA (side-side-angle)

Step 1) Use the given angle to find the height of the triangle: $h=b \sin (A)$

Step 2) Check if,
$Side\;a$ < $h$,
then no triangles
$Side\;a=h$,
then 1 triangle
$Side\;a$ > $h$,
then 1 triangle
$h$ < $Side\;a$ < $Side\;b$,
then 2 triangles

Step 3) Solve the triangle(s)!
• Introduction

• 1.
Given the following triangle $\triangle ABC$,

a)
Solve for$\angle C$

b)
Solve for $a$

• 2.
Solve for side $x$

• 3.
Ambiguous case: SSA triangles
In $\triangle DEF$, $DE=21cm$, $\angle$$F=45$°, and $EF=24cm$; find $DF$.


text:
$$\mathbb {S}^{1}$$ can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme." /> $$\mathbb {S}^{1}$$ can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme." /> $$\mathbb {S}^{1}$$ can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme." /> Shape Analysis of Planar Multiply-Connected Objects Using Conformal Welding
The Community for Technology Leaders
Subscribe
Issue No.07 - July (2014 vol.36)
pp: 1384-1401
Xianfeng Gu , Department of Computer Science, State University of New York at Stony Brook, Stony Brook, New York, NY, USA
ABSTRACT
Shape analysis is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial but difficult. It usually involves an efficient representation of 2D shape space with a metric, so that its mathematical structure can be used for further analysis. Although the study of 2D simply-connected shapes has been subject to a corpus of literatures, the analysis of multiply-connected shapes is comparatively less studied. In this work, we propose a representation for general 2D multiply-connected domains with arbitrary topologies using conformal welding. A metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains (unit disk with several inner disks removed), using holomorphic 1-forms. A set of diffeomorphisms of the unit circle $$\mathbb {S}^{1}$$ can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme.
INDEX TERMS
Shape, Welding, Educational institutions, Equations, Shape measurement, Electronic mail,Computational models of vision, Numerical algorithms, Applications, Vision and Scene Understanding, Representations, data structures, and transforms, Image Processing and Computer Vision, Reconstruction, Shape
CITATION
Xianfeng Gu, "Shape Analysis of Planar Multiply-Connected Objects Using Conformal Welding", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.36, no. 7, pp. 1384-1401, July 2014, doi:10.1109/TPAMI.2013.215


text:
# Timeline graph with ggplot2

July 7, 2012
By

(This article was first published on FishyOperationsCategory Archives: r, and kindly contributed to R-bloggers)

This post shows how to create a timeline graph by using ggplot2.

Let’s start by loading the ggplot2 library.

library(ggplot2)

Next let’s create a dataset which we will use to feed the graph. In the last column (y), I create random positive values for the first three rows (which will be shown above the timeline) and random negative values for the last three rows (to be shown below the timeline).

timeset<-data.frame(year=c(1986,1995,2011,1990,1998,2010),text=c('I was born','Had a nice icecream','Spotted a dodo','First swim','Crashed my bicycle','Bought a helmet'),y=c(runif(3,.5,1),runif(3,-1,-.5)))

Now comes the tricky part, creating the graph.

plot<-ggplot(timeset,aes(x=year,y=0))

Let’s add the lines which will run from the timeline arrow to the text label.

plot<-plot+geom_segment(aes(y=0,yend=y,xend=year))

The text is added as follows;

plot<-geom_text(aes(y=ytext,label=text),size=2.5,vjust=-1)

Now to fancy it up a bit with some points at the end of the line segments;

plot<-plot+geom_point(aes(y=y))

Putting some limits on the y-axis.

scale_y_continuous(limits=c(-2,2))

As the y-values are, in this case, non-informative, let’s hide ‘em.

plot<-plot+scale_y_continuous(limits=c(-2,2))

Let’s draw the timeline arrow . I agree this, is fiddling, and these values can possibly be extracted from the x-axis.

plot<-plot+geom_hline(y=0,size=1,color='purple')   #draw a vertical line
plot<-plot+geom_segment(x=2011.4,xend=2012.2,y=.2,yend=0,color='purple',size=1) + geom_segment(x=2011.4,xend=2012.2,y=-.2,yend=0,color='purple',size=1) #drawing the actual arrow

Now to finish it up, let’s set some options.

plot<-plot+opts(axis.text.y =theme_blank(),title='My timeline')+ylab('')+xlab('')

Let’s have a look at the result!

print(plot)

The post Timeline graph with ggplot2 appeared first on FishyOperations.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Tags: , , , ,


text:
Properties

 Label 1620.2.a.j Level $1620$ Weight $2$ Character orbit 1620.a Self dual yes Analytic conductor $12.936$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 180) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + ( 1 - \beta_{1} ) q^{7} +O(q^{10})$$ $$q + q^{5} + ( 1 - \beta_{1} ) q^{7} + \beta_{2} q^{11} + ( 2 + \beta_{2} ) q^{13} -\beta_{2} q^{17} + ( 2 - \beta_{2} ) q^{19} + ( 1 + \beta_{1} ) q^{23} + q^{25} + ( 1 - 2 \beta_{1} ) q^{29} + ( 2 - \beta_{2} ) q^{31} + ( 1 - \beta_{1} ) q^{35} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 + \beta_{2} ) q^{43} + ( -5 + \beta_{1} - \beta_{2} ) q^{47} + ( 6 + \beta_{2} ) q^{49} + ( 2 + 2 \beta_{1} ) q^{53} + \beta_{2} q^{55} + ( -2 - 2 \beta_{1} ) q^{59} + ( 7 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( 2 + \beta_{2} ) q^{65} + ( 3 + \beta_{1} + \beta_{2} ) q^{67} + ( -8 - 2 \beta_{1} + \beta_{2} ) q^{71} + 8 q^{73} + ( -2 - 2 \beta_{1} + 3 \beta_{2} ) q^{77} + 2 q^{79} + ( 7 + \beta_{1} ) q^{83} -\beta_{2} q^{85} + 3 q^{89} + ( -4 \beta_{1} + 3 \beta_{2} ) q^{91} + ( 2 - \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{5} + 3q^{7} + O(q^{10})$$ $$3q + 3q^{5} + 3q^{7} + 6q^{13} + 6q^{19} + 3q^{23} + 3q^{25} + 3q^{29} + 6q^{31} + 3q^{35} + 12q^{37} - 3q^{41} + 6q^{43} - 15q^{47} + 18q^{49} + 6q^{53} - 6q^{59} + 21q^{61} + 6q^{65} + 9q^{67} - 24q^{71} + 24q^{73} - 6q^{77} + 6q^{79} + 21q^{83} + 9q^{89} + 6q^{95} + 18q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{2} + 2 \beta_{1} + 11$$$$)/3$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.51414 −2.08613 0.571993
0 0 0 1.00000 0 −3.83502 0 0 0
1.2 0 0 0 1.00000 0 2.73419 0 0 0
1.3 0 0 0 1.00000 0 4.10083 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.a.j 3
3.b odd 2 1 1620.2.a.i 3
4.b odd 2 1 6480.2.a.bw 3
5.b even 2 1 8100.2.a.u 3
5.c odd 4 2 8100.2.d.o 6
9.c even 3 2 540.2.i.b 6
9.d odd 6 2 180.2.i.b 6
12.b even 2 1 6480.2.a.bt 3
15.d odd 2 1 8100.2.a.v 3
15.e even 4 2 8100.2.d.p 6
36.f odd 6 2 2160.2.q.i 6
36.h even 6 2 720.2.q.k 6
45.h odd 6 2 900.2.i.c 6
45.j even 6 2 2700.2.i.c 6
45.k odd 12 4 2700.2.s.c 12
45.l even 12 4 900.2.s.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 9.d odd 6 2
540.2.i.b 6 9.c even 3 2
720.2.q.k 6 36.h even 6 2
900.2.i.c 6 45.h odd 6 2
900.2.s.c 12 45.l even 12 4
1620.2.a.i 3 3.b odd 2 1
1620.2.a.j 3 1.a even 1 1 trivial
2160.2.q.i 6 36.f odd 6 2
2700.2.i.c 6 45.j even 6 2
2700.2.s.c 12 45.k odd 12 4
6480.2.a.bt 3 12.b even 2 1
6480.2.a.bw 3 4.b odd 2 1
8100.2.a.u 3 5.b even 2 1
8100.2.a.v 3 15.d odd 2 1
8100.2.d.o 6 5.c odd 4 2
8100.2.d.p 6 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1620))$$:

 $$T_{7}^{3} - 3 T_{7}^{2} - 15 T_{7} + 43$$ $$T_{11}^{3} - 24 T_{11} + 36$$ $$T_{17}^{3} - 24 T_{17} - 36$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$43 - 15 T - 3 T^{2} + T^{3}$$
$11$ $$36 - 24 T + T^{3}$$
$13$ $$76 - 12 T - 6 T^{2} + T^{3}$$
$17$ $$-36 - 24 T + T^{3}$$
$19$ $$4 - 12 T - 6 T^{2} + T^{3}$$
$23$ $$-9 - 15 T - 3 T^{2} + T^{3}$$
$29$ $$279 - 69 T - 3 T^{2} + T^{3}$$
$31$ $$4 - 12 T - 6 T^{2} + T^{3}$$
$37$ $$436 - 36 T - 12 T^{2} + T^{3}$$
$41$ $$81 - 81 T + 3 T^{2} + T^{3}$$
$43$ $$76 - 12 T - 6 T^{2} + T^{3}$$
$47$ $$27 + 39 T + 15 T^{2} + T^{3}$$
$53$ $$-72 - 60 T - 6 T^{2} + T^{3}$$
$59$ $$72 - 60 T + 6 T^{2} + T^{3}$$
$61$ $$409 + 63 T - 21 T^{2} + T^{3}$$
$67$ $$151 - 21 T - 9 T^{2} + T^{3}$$
$71$ $$-324 + 108 T + 24 T^{2} + T^{3}$$
$73$ $$( -8 + T )^{3}$$
$79$ $$( -2 + T )^{3}$$
$83$ $$-243 + 129 T - 21 T^{2} + T^{3}$$
$89$ $$( -3 + T )^{3}$$
$97$ $$424 + 36 T - 18 T^{2} + T^{3}$$


text:
Is this a high pass or low pass active filter?

Homework Statement

Here is the problem with the solution:

The Attempt at a Solution

I don't understand why the filter is low pass. How I solve this usually is I plug in ω=0 and ω=∞. When I plug in 0 I get the gain of a non inverting op amp. When I plug in ∞ I get that the transfer function (gain) is equal to 1. Since it is equal to 1 doesn't that mean it lets high frequencies pass? I remember in passive filters when you get 1 for ω=∞ it means it is a high pass filter. HOWEVER, the tricky thing about active filters is that when ω=0, the gain is not 0, instead it is 1+R1/R2 which I think is why I am confused.

berkeman
Mentor

Homework Statement

Here is the problem with the solution:

The Attempt at a Solution

I don't understand why the filter is low pass. How I solve this usually is I plug in ω=0 and ω=∞. When I plug in 0 I get the gain of a non inverting op amp. When I plug in ∞ I get that the transfer function (gain) is equal to 1. Since it is equal to 1 doesn't that mean it lets high frequencies pass? I remember in passive filters when you get 1 for ω=∞ it means it is a high pass filter. HOWEVER, the tricky thing about active filters is that when ω=0, the gain is not 0, instead it is 1+R1/R2 which I think is why I am confused.

That gain equation looks wrong. First of all, it's an inverting amp, so the gain has to be negative. Second, the feedback impedance goes to zero at high frequencies, so the gain should go toward -0 at high frequencies...

That gain equation looks wrong. First of all, it's an inverting amp, so the gain has to be negative. Second, the feedback impedance goes to zero at high frequencies, so the gain should go toward -0 at high frequencies...

Hi,

I compared it with wikipedia and it looks like a non inverting amp to me? :S

http://en.wikipedia.org/wiki/Operational_amplifier#Non-inverting_amplifier

I also solved it on my own, as you can see, when I plug in a high frequency I do not get 0. Instead I get 1.

That gain equation looks wrong. First of all, it's an inverting amp, so the gain has to be negative.

Have another look.. it is not inverting :) Vi is applied to the + t-erminal and will also appear at the - terminal. This causes current to flow from the top of R2 to ground, so the op amp will be pushing current from the output toward ground through R2 and R1||C, making the output at higher voltage than the voltage at the - terminal which is Vi. So Vo > Vi.

thebeast, what you did was right. You just need to be more careful about your conclusion.

At w->0, the capacitor has infinite impedance so it is out of the circuit. The output is Vo = (1+R1/(R1+R2))*Vi.

At w->∞, the capacitor is a short so R1 is out of the circuit. Now Vo=Vi.

The output at low frequencies is higher than the output at high frequencies so this is a low pass filter. It's a little different because the gain does not approach 0 as w→∞

You can also see this from the suggested solution. There is both a zero and a pole. The pole occurs at a lower frequency so a bode plot will show the gain flat, then head down at -20db/decade at the low pass corner frequency (the pole) and then become constant when the high pass corner frequency is met (the zero adds a rising slope of 20db/decade). This is what you found -- higher gain at 0 and lower but non-zero gain at infinity.

Have another look.. it is not inverting :) Vi is applied to the + t-erminal and will also appear at the - terminal. This causes current to flow from the top of R2 to ground, so the op amp will be pushing current from the output toward ground through R2 and R1||C, making the output at higher voltage than the voltage at the - terminal which is Vi. So Vo > Vi.

thebeast, what you did was right. You just need to be more careful.

At w->0, the capacitor has infinite impedance so it is out of the circuit. The output is Vo = (1+R1/(R1+R2))*Vi.

At w->∞, the capacitor is a short so R1 is out of the circuit. Now Vo=Vi.

The output at low frequencies is higher than the output at high frequencies so this is a low pass filter. It's a little different because the gain does not approach 0 as w→∞

You can also see this from the suggested solution. There is both a zero and a pole. The zero occurs at a lower frequency so a bode plot will show the gain flat, then head down at -20db/decade at the low pass corner frequency (the zero) and then stay constant when the high pass corner frequency is met (the pole adds a rising slope of 20db/decade). This is what you found -- higher gain at 0 and lower but non-zero gain at infinity.

Thank you! Your first paragraph really helped me in understanding how the non inverting op amp works.

Also... So whichever one is higher is the frequency that is allowed to pass? Unfortunately our professor did not teach us bode plots so I think that is the reason to why I am having difficulty in understanding this. I will definitely check it out though!

berkeman
Mentor
Have another look.. it is not inverting :)

Oops! You are correct, my mistake. I usually draw my opamp circuits with the inverting terminal on top, but that's still not a good excuse for missing the non-inverting configuration.

Unfortunately you quoted me before I finished an edit :) I had zero and pole swapped in my last paragraph. My post is corrected but the quoted text still has it wrong.

Unfortunately our professor did not teach us bode plots so I think that is the reason to why I am having difficulty in understanding this. I will definitely check it out though!

Alright well the solution is depending on a bode plot argument so that may be a problem for you to follow.

Also... So whichever one is higher is the frequency that is allowed to pass?

All frequencies are allowed to pass since there is no w where the gain is zero; it's just that some frequencies are emphasized and some are not, making them smaller in comparison. You found the low frequencies get high gain and the high frequencies get (relatively) low gain.


text:
# How do you find the x and y intercepts of y=2/3x-4?

Mar 1, 2017

See the entire solution process below:

#### Explanation:

To find the y-intercept by substitution, substitute $0$ for $x$ and calculate $y$:

$y = \left(\frac{2}{3} \times 0\right) - 4$

$y = 0 - 4$

$y = - 4$ Therefore the y-intercept is $- 4$ or $\left(0 , - 4\right)$

To find the x-intercept by substitution, substitute $0$ for $y$ and solve for $x$:

$0 = \frac{2}{3} x - 4$

$0 + \textcolor{red}{4} = \frac{2}{3} x - 4 + \textcolor{red}{4}$

$4 = \frac{2}{3} x - 0$

$4 = \frac{2}{3} x$

$\frac{\textcolor{red}{3}}{\textcolor{b l u e}{2}} \times 4 = \frac{\textcolor{red}{3}}{\textcolor{b l u e}{2}} \times \frac{2}{3} x$

$\frac{12}{\textcolor{b l u e}{2}} = \frac{\cancel{\textcolor{red}{3}}}{\cancel{\textcolor{b l u e}{2}}} \times \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{2}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}} x$

$6 = x$

$x = 6$ Therefore the x-intercept is $6$ or $\left(6 , 0\right)$