Testing - J Wiki

This page is for testing. The contents can be expunged at any time.

-- BillLam 2010-02-10 15:22:34

E
0 1 5 2 3 4 6

E
0 1 5 2 3 4 6

E
0 1 5 2 3 4 6

column 0	column 1	column 2
0	zero	zilch
1	one	eins
2	two	zwei

Notation as a Tool of Thought, Figure 3.

$\sum_{j=1}^{n} j \cdot 2^{-j}$

$1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots \textit{n terms} \leftrightarrow \frac{1}{4} n (n+1) (n+2) (n+3)$

$1 \cdot 2 \cdot 3 \cdot 4 + 2 \cdot 3 \cdot 4 \cdot 5 + \dots \textit{n terms} \leftrightarrow \frac{1}{5} n (n+1) (n+2) (n+3) (n+4)$

$\frac{[\frac{x-a}{N}]^{-q}}{\Gamma(-q)}\sum_{j=0}^{N-1} \frac{\Gamma(j-q)}{\Gamma(j+1)} f\bigg(x- j \bigg[\frac{x-a}{N}\bigg] \bigg)$

$x \bmod y = x - y \lfloor x/y \rfloor, \ \ \ \ for \ y \ne 0.$

Let $P$ be the set of primes $p$ such that $p^2\leq n$ , and let $m=\left| P\right|$ . Then the number of semiprimes less than or equal to $n$ is given by
$\sum_{p\in P} \left(\pi(n/p)-\pi(p)\right)=\sum_{p\in P} \pi(n/p)- \frac{m(m-1)}2$

Something is still amiss in the formula. For

$\sum_{p\in P} \left(\pi(n/p)-\pi(p)\right)=\sum_{p\in P} \pi(n/p)- \frac{m(m-1)}2$

to be true, it must be that

$\sum_{p\in P}\pi(p)=\frac{m(m-1)}2$

But:

   pi=: _1&p: + 1&p:
   ] p=: p: i.m=: 5
2 3 5 7 11
   pi p
1 2 3 4 5
   +/ pi p
15
   2 %~ m*m-1
10

The following is generated by <<RandomQuote(Doc/A Programming Language)>>

No quotes on Doc/A Programming Language.

-- RogerHui 2007-10-28 03:35:43

a b c d

$\sum_{i=0}^n a_i x^i$

$++/ (M=\iota^1 \,\, {}_+^\circ \,\, \iota^1) // M$

$+/ (M=\iota^1(\mu(M)) \,\, {}_+^\circ \,\, \iota^1(\nu(M))) / M$

$\iota^k(n)$	`k`-origin indexing; `k+i.n` in J
${}_+^\circ$	outer product; the dyad `+/` in J
$\mu(M)$	the number of rows in matrix ; `#M` in J
$\nu(M)$	the number of columns in matrix ; `{:$M` in J
	equivalent to `,B#"1 M` in J
	equivalent to `,B#"1&\|:M` in J

The formula $\sqrt{l^2 + w^2}$ computes the diagonal of a rectangle with sides $$l$$ and $$w$$

Tip: The system replaces @SIG@ in the text by your UserName and current timestamp.

User Suggestions

This is very cool Roger. Embedded latex. This Wiki stuff is deeper than I thought. JohnBaker

Actually this whole page is for just testing. Do you recognize the expressions? RogerHui

No. JohnBaker

The story behind $++/ (M=\iota^1 \,\, {}_+^\circ \,\, \iota^1) // M$ is recounted on page 25 of A Source Book of APL. In the Q&A session after Ken Iverson's presentation of Formalism in Programming Languages at the Working Conference on Mechanical Language Structures in 1963, Dijkstra asked, "How would you represent a more complex operation, for example, the sum of all elements of a matrix which are equal to the sum of the corresponding row and column indices?" The above formula was Ken's one-liner answer. It was in Iverson notation (APL did not yet exist). According to the notation in A Programming Language, a valid answer would be $+/ (M=\iota^1(\mu(M)) \,\, {}_+^\circ \,\, \iota^1(\nu(M))) / M$ (Two slashes instead of one in front of the final $M$ would also work.) I think a conversant APL or J user would agree that Dijkstra's question was very simple, so I don't think Ken would have had any trouble, and attribute the first version to clerical errors. Signed with "at SIG at" as a test. -- RogerHui 2005-10-14 13:35:30

I'd define dijkstra =: $ (] +/@:= (+/@#: i.@:#)) , in J.

Very very cool - practically cryogenic. Suggest changing Tex & LaTex to the officially approved TeX & LaTeX, including modifying the name of the nascent JWiki LaTex page. EwartShaw

Done. -- RogerHui 2005-10-14 15:51:35

Still can't get the hang of when LaTeX goes to a new line and when it doesn't. -- RogerHui 2005-10-14 16:28:52

See new page for a $\LaTeX$ how to -- ChrisBurke 2005-10-16 08:49:32