Return-path: Envelope-to: jbovlaste-admin@lojban.org Delivery-date: Mon, 28 Oct 2024 18:47:44 -0700 Received: from [192.168.123.254] (port=38496 helo=jiten.lojban.org) by 4f8f24ca47b9 with smtp (Exim 4.96) (envelope-from ) id 1t5bKf-0003on-29 for jbovlaste-admin@lojban.org; Mon, 28 Oct 2024 18:47:44 -0700 Received: by jiten.lojban.org (sSMTP sendmail emulation); Tue, 29 Oct 2024 01:47:41 +0000 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word mai'u'ei -- By krtisfranks Date: Tue, 29 Oct 2024 01:47:41 +0000 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Message-Id: X-Spam-Score: -1.0 (-) X-Spam_score: -1.0 X-Spam_score_int: -9 X-Spam_bar: - In jbovlaste, the user krtisfranks has edited a definition of "mai'u'ei" in the language "English". Differences: 2,2c2,2 < mekso unary operator: Levi-Civita symbol; for input $n$-tuple $(a_1, a_2,$ ... $, a_n)$, where $n$ is a strictly positive integer, the output is $\varepsilon_{a_1, a_2, ..., a_n}$, where $\varepsilon$ is the Levi-Civita symbol under the convention of mapping $(1, 2,$ ...$, n)$ to $1$.. --- > mekso unary operator: Levi-Civita symbol; for input $n$-tuple $(a_1, a_2,$ ... $, a_n)$, where $n$ is a strictly positive integer, the output is $\varepsilon_{a_1, a_2, ..., a_n}$, where $\varepsilon$ is the Levi-Civita symbol under the convention of mapping $(1, 2,$ ...$, n)$ to $1$. Old Data: Definition: mekso unary operator: Levi-Civita symbol; for input $n$-tuple $(a_1, a_2,$ ... $, a_n)$, where $n$ is a strictly positive integer, the output is $\varepsilon_{a_1, a_2, ..., a_n}$, where $\varepsilon$ is the Levi-Civita symbol under the convention of mapping $(1, 2,$ ...$, n)$ to $1$.. Notes: If any at least two elements of the input tuple are mutually equal (id est: if there exists $i$ and $j$ which are strictly positive integers which do not exceed $n$ such that $a_i = a_j$, where the input tuple is as defined supra), then the output is $0$; else, the output is sgn$((a_1, a_2,$ ... $, a_n))$, where the input to the sgn function there is a permutation denoted in one-line permutation notation. This format is somewhat consistent with index notation in tensor calculus, but treats the indices as ordered entries of a tuple. More generally, the Levi-Civita symbol may be defined for any function $f$ which maps space $S$ to $S$; in such case, the input could be $f$ instead of the aforementioned tuple. Covariant and contravariant Levi-Civita tensors can also be defined. See also: "{mai'u'e}", "{mai'u}". Jargon: Gloss Keywords: Word: Levi-Civita symbol, In Sense: Place Keywords: New Data: Definition: mekso unary operator: Levi-Civita symbol; for input $n$-tuple $(a_1, a_2,$ ... $, a_n)$, where $n$ is a strictly positive integer, the output is $\varepsilon_{a_1, a_2, ..., a_n}$, where $\varepsilon$ is the Levi-Civita symbol under the convention of mapping $(1, 2,$ ...$, n)$ to $1$. Notes: If any at least two elements of the input tuple are mutually equal (id est: if there exists $i$ and $j$ which are strictly positive integers which do not exceed $n$ such that $a_i = a_j$, where the input tuple is as defined supra), then the output is $0$; else, the output is sgn$((a_1, a_2,$ ... $, a_n))$, where the input to the sgn function there is a permutation denoted in one-line permutation notation. This format is somewhat consistent with index notation in tensor calculus, but treats the indices as ordered entries of a tuple. More generally, the Levi-Civita symbol may be defined for any function $f$ which maps space $S$ to $S$; in such case, the input could be $f$ instead of the aforementioned tuple. Covariant and contravariant Levi-Civita tensors can also be defined. See also: "{mai'u'e}", "{mai'u}". Jargon: Gloss Keywords: Word: Levi-Civita symbol, In Sense: Place Keywords: You can go to to see it.