Received: from 173-13-139-235-sfba.hfc.comcastbusiness.net ([173.13.139.235]:44252 helo=jukni.digitalkingdom.org) by stodi.digitalkingdom.org with smtp (Exim 4.89) (envelope-from ) id 1f2SaQ-0000mT-0W for jbovlaste-admin@lojban.org; Sat, 31 Mar 2018 19:19:16 -0700 Received: by jukni.digitalkingdom.org (sSMTP sendmail emulation); Sat, 31 Mar 2018 19:19:13 -0700 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word cnanfadi -- By krtisfranks Date: Sat, 31 Mar 2018 19:19:13 -0700 MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Message-Id: X-Spam-Score: 3.1 (+++) X-Spam_score: 3.1 X-Spam_score_int: 31 X-Spam_bar: +++ X-Spam-Report: Spam detection software, running on the system "stodi.digitalkingdom.org", has NOT identified this incoming email as spam. The original message has been attached to this so you can view it or label similar future email. If you have any questions, see the administrator of that system for details. Content preview: In jbovlaste, the user krtisfranks has edited a definition of "cnanfadi" in the language "English". Differences: 2,2c2,2 < $x_1$ (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data $x_2$ (completely specified ordered multiset/list) using function $x_3$ (defaults according to [...] Content analysis details: (3.1 points, 5.0 required) pts rule name description ---- ---------------------- -------------------------------------------------- 1.4 RCVD_IN_BRBL_LASTEXT RBL: No description available. [173.13.139.235 listed in bb.barracudacentral.org] -1.9 BAYES_00 BODY: Bayes spam probability is 0 to 1% [score: 0.0000] 1.0 RDNS_DYNAMIC Delivered to internal network by host with dynamic-looking rDNS 2.6 TO_NO_BRKTS_DYNIP To: lacks brackets and dynamic rDNS In jbovlaste, the user krtisfranks has edited a definition of "cnanfadi" in the language "English". Differences: 2,2c2,2 < $x_1$ (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data $x_2$ (completely specified ordered multiset/list) using function $x_3$ (defaults according to the notes) with weights $x_4$ (completely specified ordered multiset/list with same cardinality/length as $x_2$; defaults according to Notes). --- > $x_1$ (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data $x_2$ (completely specified ordered multiset/list) using function $x_3$ (defaults according to the notes; if it is an extended-real number, then it has a particular interpretation according to the Notes) with weights $x_4$ (completely specified ordered multiset/list with same cardinality/length as $x_2$; defaults according to Notes). 5,5c5,5 < Potentially dimensionful. Make sure to convert $x_3$ from an operator to a sumti; $x_3$ is the 'f' in "f-mean" and must be a real-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the monic, single-term polynomial represented by "$z^p$" (of degree $p$) for some nonzero $p$ and indeterminate/variable $z$, or $log$, or $exp$ (as functions); culture or context can further constrain the default. If $x_2$ is set to "$z^{+ \infty .}$" (the exponent is positive infinity, given by "{ma'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term polynomial as the degree increases without bound), then the result ($x_1$) is the maximum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if $x_2$ is set to "$z^{- \infty .}$" (the exponent is negative infinity, given by "{ni'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term reciprocal-polynomial as the reciprocal-degree increases without bound (or the degree decreases without bound)), then the result ($x_1$) is the minimum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of $x_4$ is the ordered set of $n$ terms, each of which is $1/n$, where the cardinality of $x_2$ is $n$. Let "f" denote the sumti in $x_3$, "$y_i$" denote the $i$th term in $x_2$ for all $i$, $n$ denote the cardinality of $x_2$ (thus also $x_4$), and $w_i$ denote the $i$th term in $x_4$ for any $i$; then the result, $x_1$, is equal to: $f^{(-1)}($Sum$(w_i f(y_i), i$ in Set$(1,...,n)) / $Sum$(w_i, i$ in Set$(1,...,n)))$. Note that if the weights are all $1$ and $x_2$ is set equal to not the $p$th monic, single-term polynomial, but instead that same polynomial composed with the absolute value function (or the forward difference function), then the result is the $p$-norm on $x_2$ scaled by $n^{(-1/p)}$ for integer $n$ being the cardinality of $x_2$. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow $x_2$ to be the image (set) of the function whose average is desired over the entire relevant subset of its domain - notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily $x_3$. --- > Potentially dimensionful. Make sure to convert $x_3$ from an operator to a sumti; $x_3$ is the 'f' in "f-mean" and must be a real-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the monic, single-term formal polynomial represented by "$z^p$" (of degree $p$) for some nonzero $p$ (note that it need not be positive or an integer) and indeterminate/variable $z$, or $log$, or $exp$ (as functions); culture or context can further constrain the default. If $x_2$ is set to "$z^{+ \infty .}$" (the exponent is positive infinity, given by "{ma'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term polynomial as the degree increases without bound), then the result ($x_1$) is the maximum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if $x_2$ is set to "$z^{- \infty .}$" (the exponent is negative infinity, given by "{ni'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term reciprocal-polynomial as the reciprocal-degree increases without bound (or the degree decreases without bound)), then the result ($x_1$) is the minimum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of $x_4$ is the ordered set of $n$ terms, each of which is $1/n$, where the cardinality of $x_2$ is $n$. Let "f" denote the sumti in $x_3$, "$y_i$" denote the $i$th term in $x_2$ for all $i$, $n$ denote the cardinality of $x_2$ (thus also $x_4$), and $w_i$ denote the $i$th term in $x_4$ for any $i$; then the result, $x_1$, is equal to: $f^{(-1)}($Sum$(w_i f(y_i), i$ in Set$(1,...,n)) / $Sum$(w_i, i$ in Set$(1,...,n)))$. Note that if the weights are all $1$ and $x_2$ is set equal to not the $p$th monic, single-term polynomial, but instead that same polynomial composed with the absolute value function (or the forward difference function), then the result is the $p$-norm on $x_2$ scaled by $n^{(-1/p)}$ for integer $n$ being the cardinality of $x_2$. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow $x_2$ to be the image (set) of the function whose average is desired over the entire relevant subset of its domain - notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily $x_3$. If $x_3$ is a single extended-real number $p$ (not a function), then this word refers to the weighted power-mean and it is equivalent to letting $x_2$ equal the monic, single-term formal polynomial of $p$th degree as before if $p$ is nonzero real, the $max$ or $min$ as before if $p$ is infinity (according to signum), and $log$ if $p=0$ (thus making the overall mean refer to the geometric mean); this overloading is for convenience of usage and will not cause confusion because constant functions are very much so not injective. Old Data: Definition: $x_1$ (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data $x_2$ (completely specified ordered multiset/list) using function $x_3$ (defaults according to the notes) with weights $x_4$ (completely specified ordered multiset/list with same cardinality/length as $x_2$; defaults according to Notes). Notes: Potentially dimensionful. Make sure to convert $x_3$ from an operator to a sumti; $x_3$ is the 'f' in "f-mean" and must be a real-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the monic, single-term polynomial represented by "$z^p$" (of degree $p$) for some nonzero $p$ and indeterminate/variable $z$, or $log$, or $exp$ (as functions); culture or context can further constrain the default. If $x_2$ is set to "$z^{+ \infty .}$" (the exponent is positive infinity, given by "{ma'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term polynomial as the degree increases without bound), then the result ($x_1$) is the maximum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if $x_2$ is set to "$z^{- \infty .}$" (the exponent is negative infinity, given by "{ni'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term reciprocal-polynomial as the reciprocal-degree increases without bound (or the degree decreases without bound)), then the result ($x_1$) is the minimum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of $x_4$ is the ordered set of $n$ terms, each of which is $1/n$, where the cardinality of $x_2$ is $n$. Let "f" denote the sumti in $x_3$, "$y_i$" denote the $i$th term in $x_2$ for all $i$, $n$ denote the cardinality of $x_2$ (thus also $x_4$), and $w_i$ denote the $i$th term in $x_4$ for any $i$; then the result, $x_1$, is equal to: $f^{(-1)}($Sum$(w_i f(y_i), i$ in Set$(1,...,n)) / $Sum$(w_i, i$ in Set$(1,...,n)))$. Note that if the weights are all $1$ and $x_2$ is set equal to not the $p$th monic, single-term polynomial, but instead that same polynomial composed with the absolute value function (or the forward difference function), then the result is the $p$-norm on $x_2$ scaled by $n^{(-1/p)}$ for integer $n$ being the cardinality of $x_2$. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow $x_2$ to be the image (set) of the function whose average is desired over the entire relevant subset of its domain - notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily $x_3$. Jargon: Gloss Keywords: Word: arithmetic mean, In Sense: Word: average, In Sense: generalized f-mean Word: generalized f-mean, In Sense: Word: generalized mean, In Sense: Word: geometric mean, In Sense: Word: harmonic mean, In Sense: Word: log-sum-exponential, In Sense: Word: LSE, In Sense: log-sum-exponential Word: max, In Sense: Word: mean, In Sense: generalized f-mean Word: mean value, In Sense: Word: min, In Sense: Word: norm, In Sense: average, typical, or 'normal' value Word: norm, In Sense: math terminology, specifically p-norm Word: p-norm, In Sense: Word: power mean, In Sense: Word: quasi-arithmetic mean, In Sense: Word: RMS, In Sense: root-mean-square Word: root-mean-square, In Sense: Word: typical, In Sense: average value Place Keywords: New Data: Definition: $x_1$ (li; number/quantity) is the weighted quasi-arithmetic mean/generalized f-mean of/on data $x_2$ (completely specified ordered multiset/list) using function $x_3$ (defaults according to the notes; if it is an extended-real number, then it has a particular interpretation according to the Notes) with weights $x_4$ (completely specified ordered multiset/list with same cardinality/length as $x_2$; defaults according to Notes). Notes: Potentially dimensionful. Make sure to convert $x_3$ from an operator to a sumti; $x_3$ is the 'f' in "f-mean" and must be a real-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the monic, single-term formal polynomial represented by "$z^p$" (of degree $p$) for some nonzero $p$ (note that it need not be positive or an integer) and indeterminate/variable $z$, or $log$, or $exp$ (as functions); culture or context can further constrain the default. If $x_2$ is set to "$z^{+ \infty .}$" (the exponent is positive infinity, given by "{ma'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term polynomial as the degree increases without bound), then the result ($x_1$) is the maximum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering; likewise, if $x_2$ is set to "$z^{- \infty .}$" (the exponent is negative infinity, given by "{ni'uci'i}") for indeterminate/variable $z$ (the function is the functional limit of the monic, single-term reciprocal-polynomial as the reciprocal-degree increases without bound (or the degree decreases without bound)), then the result ($x_1$) is the minimum of the products of the data (terms of $x_2$) with their corresponding weights (terms of $x_4$) according to the standard ordering on the set of all real numbers or possibly some other specified or assumed ordering. The default of $x_4$ is the ordered set of $n$ terms, each of which is $1/n$, where the cardinality of $x_2$ is $n$. Let "f" denote the sumti in $x_3$, "$y_i$" denote the $i$th term in $x_2$ for all $i$, $n$ denote the cardinality of $x_2$ (thus also $x_4$), and $w_i$ denote the $i$th term in $x_4$ for any $i$; then the result, $x_1$, is equal to: $f^{(-1)}($Sum$(w_i f(y_i), i$ in Set$(1,...,n)) / $Sum$(w_i, i$ in Set$(1,...,n)))$. Note that if the weights are all $1$ and $x_2$ is set equal to not the $p$th monic, single-term polynomial, but instead that same polynomial composed with the absolute value function (or the forward difference function), then the result is the $p$-norm on $x_2$ scaled by $n^{(-1/p)}$ for integer $n$ being the cardinality of $x_2$. This should typically not refer to the mean of a function ( https://en.wikipedia.org/wiki/Mean_of_a_function ), although it generalizes easily; alternatively, with appropriate weighting, allow $x_2$ to be the image (set) of the function whose average is desired over the entire relevant subset of its domain - notice that the weights will have to themselves be functions of the data or the domain of the function; in this context, the function is not necessarily $x_3$. If $x_3$ is a single extended-real number $p$ (not a function), then this word refers to the weighted power-mean and it is equivalent to letting $x_2$ equal the monic, single-term formal polynomial of $p$th degree as before if $p$ is nonzero real, the $max$ or $min$ as before if $p$ is infinity (according to signum), and $log$ if $p=0$ (thus making the overall mean refer to the geometric mean); this overloading is for convenience of usage and will not cause confusion because constant functions are very much so not injective. Jargon: Gloss Keywords: Word: arithmetic mean, In Sense: Word: average, In Sense: generalized f-mean Word: generalized f-mean, In Sense: Word: generalized mean, In Sense: Word: geometric mean, In Sense: Word: harmonic mean, In Sense: Word: log-sum-exponential, In Sense: Word: LSE, In Sense: log-sum-exponential Word: max, In Sense: Word: mean, In Sense: generalized f-mean Word: mean value, In Sense: Word: min, In Sense: Word: norm, In Sense: average, typical, or 'normal' value Word: norm, In Sense: math terminology, specifically p-norm Word: p-norm, In Sense: Word: power mean, In Sense: Word: quasi-arithmetic mean, In Sense: Word: RMS, In Sense: root-mean-square Word: root-mean-square, In Sense: Word: typical, In Sense: average value Place Keywords: You can go to to see it.