Return-path: Envelope-to: jbovlaste-admin@lojban.org Delivery-date: Thu, 09 Mar 2023 13:37:44 -0800 Received: from [192.168.123.254] (port=47316 helo=jiten.lojban.org) by 8612a944938c with smtp (Exim 4.94.2) (envelope-from ) id 1paNxF-0005gT-6v for jbovlaste-admin@lojban.org; Thu, 09 Mar 2023 13:37:43 -0800 Received: by jiten.lojban.org (sSMTP sendmail emulation); Thu, 09 Mar 2023 21:37:41 +0000 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word cpolinomi'a -- By krtisfranks Date: Thu, 9 Mar 2023 21:37:41 +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 "cpolinomi'a" in the language "English". Differences: 2,2c2,2 < $x_1$ is a formal polynomial with coefficients $x2$ (ordered list) of degree $x_3$ (li; nonnegative integer) over structure/ring $x_4$ (to which coefficients $x_2$ all belong) and in indeterminant $x_5$ --- > $x_1$ is a formal polynomial with coefficients $x2$ (ordered list) of degree $x_3$ (li; nonnegative integer) over structure/ring $x_4$ (to which coefficients $x_2$ all belong) and in indeterminant $x_5$. 5,5c5,5 < $x_3$ must be greater than or equal to the number of entries in $x_2$; if these two values are not equal, then the explicitly mentioned entries of $x_2$ are the values of the coefficients as will be described next, starting with the most important one; all the following coefficients (which are not explicitly mentioned) are {xo'ei} (taking appropriate values) until and including once the constant term's coefficient (when understood as a function) is reached. If $x_2$ is presented as an ordered list, the entries represent the 'coefficients' of the particular polynomial and are specified in the order such that the $i$th entry/term is the $(n-i+1)$th 'coefficient', for all natural numbers $i$ between $1$ and $n+1$ inclusively, where the ordering of 'coefficients' is determined by the exponent of the indeterminate associated therewith (when treated as a function); thus, the last entry is the constant term (when treated as a function), the penultimate term is the coefficient of the argument of $x_5$ (when treated as a function), and the first term is the coefficient of the argument of $x_5$ exponentiated by $n$ (which is the degree of the polynomial). See also: {tefsujme'o} (polynomial function) --- > $x_3$ must be greater than or equal to the number of entries in $x_2$; if these two values are not equal, then the explicitly mentioned entries of $x_2$ are the values of the coefficients as will be described next, starting with the most important one; all the following coefficients (which are not explicitly mentioned) are {xo'ei} (taking appropriate values) until and including once the constant term's coefficient (when understood as a function) is reached. If $x_2$ is presented as an ordered list, the entries represent the 'coefficients' of the particular polynomial and are specified in the order such that the $i$th entry/term is the $(n-i+1)$th 'coefficient', for all positive integers $i$ which are less than or equal to $n+1$, where the ordering of 'coefficients' is determined by the exponent of the indeterminate associated therewith (when treated as a function); thus, the last entry is the constant term (when treated as a function), the penultimate term is the coefficient of the argument of $x_5$ (when treated as a function), and the first term is the coefficient of the argument of $x_5$ exponentiated by $n$ (which is the degree of the polynomial); in other words, the $i$th entry of $x_2$ (where indexing of the ordered list starts at $1$) is the coefficient of $x^{(n-i+1)}$, where $x_5 = x$ is the indeterminate. See also: {tefsujme'o} (polynomial function) Old Data: Definition: $x_1$ is a formal polynomial with coefficients $x2$ (ordered list) of degree $x_3$ (li; nonnegative integer) over structure/ring $x_4$ (to which coefficients $x_2$ all belong) and in indeterminant $x_5$ Notes: $x_3$ must be greater than or equal to the number of entries in $x_2$; if these two values are not equal, then the explicitly mentioned entries of $x_2$ are the values of the coefficients as will be described next, starting with the most important one; all the following coefficients (which are not explicitly mentioned) are {xo'ei} (taking appropriate values) until and including once the constant term's coefficient (when understood as a function) is reached. If $x_2$ is presented as an ordered list, the entries represent the 'coefficients' of the particular polynomial and are specified in the order such that the $i$th entry/term is the $(n-i+1)$th 'coefficient', for all natural numbers $i$ between $1$ and $n+1$ inclusively, where the ordering of 'coefficients' is determined by the exponent of the indeterminate associated therewith (when treated as a function); thus, the last entry is the constant term (when treated as a function), the penultimate term is the coefficient of the argument of $x_5$ (when treated as a function), and the first term is the coefficient of the argument of $x_5$ exponentiated by $n$ (which is the degree of the polynomial). See also: {tefsujme'o} (polynomial function) Jargon: Gloss Keywords: Word: coefficient, In Sense: polynomial Word: coefficient ring, In Sense: of polynomial Word: indeterminant, In Sense: of formal polynomial Word: polynomial, In Sense: formal, ring element (not as: a function or a funcion evaluated at a particular input value) Place Keywords: Word: degree, In Sense: polynomial, For Place: 1 New Data: Definition: $x_1$ is a formal polynomial with coefficients $x2$ (ordered list) of degree $x_3$ (li; nonnegative integer) over structure/ring $x_4$ (to which coefficients $x_2$ all belong) and in indeterminant $x_5$. Notes: $x_3$ must be greater than or equal to the number of entries in $x_2$; if these two values are not equal, then the explicitly mentioned entries of $x_2$ are the values of the coefficients as will be described next, starting with the most important one; all the following coefficients (which are not explicitly mentioned) are {xo'ei} (taking appropriate values) until and including once the constant term's coefficient (when understood as a function) is reached. If $x_2$ is presented as an ordered list, the entries represent the 'coefficients' of the particular polynomial and are specified in the order such that the $i$th entry/term is the $(n-i+1)$th 'coefficient', for all positive integers $i$ which are less than or equal to $n+1$, where the ordering of 'coefficients' is determined by the exponent of the indeterminate associated therewith (when treated as a function); thus, the last entry is the constant term (when treated as a function), the penultimate term is the coefficient of the argument of $x_5$ (when treated as a function), and the first term is the coefficient of the argument of $x_5$ exponentiated by $n$ (which is the degree of the polynomial); in other words, the $i$th entry of $x_2$ (where indexing of the ordered list starts at $1$) is the coefficient of $x^{(n-i+1)}$, where $x_5 = x$ is the indeterminate. See also: {tefsujme'o} (polynomial function) Jargon: Gloss Keywords: Word: coefficient, In Sense: polynomial Word: coefficient ring, In Sense: of polynomial Word: indeterminant, In Sense: of formal polynomial Word: polynomial, In Sense: formal, ring element (not as: a function or a funcion evaluated at a particular input value) Place Keywords: Word: degree, In Sense: polynomial, For Place: 1 You can go to to see it.