Received: from 173-13-139-235-sfba.hfc.comcastbusiness.net ([173.13.139.235]:40724 helo=jukni.digitalkingdom.org) by stodi.digitalkingdom.org with smtp (Exim 4.91) (envelope-from ) id 1fp4aP-0001n1-Ep for jbovlaste-admin@lojban.org; Sun, 12 Aug 2018 21:36:11 -0700 Received: by jukni.digitalkingdom.org (sSMTP sendmail emulation); Sun, 12 Aug 2018 21:36:09 -0700 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word cnanfadi -- By rlpowell Date: Sun, 12 Aug 2018 21:36:09 -0700 MIME-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Message-Id: X-Spam-Score: -0.9 (/) X-Spam_score: -0.9 X-Spam_score_int: -8 X-Spam_bar: / In jbovlaste, the user rlpowell has edited a definition of "cnanfadi" in the language "English". Differences: 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 complex-valued, single-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the $p$th-power function ($z$^${p}$) for some nonzero $p$ (note that it need not be positive or an integer) and indeterminate/variable/input $z$, 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 weight-sum-scaled 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 weight-sum-scaled 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 with each term equal identically to $1/n$, 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-power function, but instead the $p$th-power function left-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 $p$th-power function as before iff $p$ is nonzero real, the $max$ or $min$ as before if $p$ is infinite (according to its signum as before), 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. --- > 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 complex-valued, single-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the $p$th-power function ($z^p$) for some nonzero $p$ (note that it need not be positive or an integer) and indeterminate/variable/input $z$, 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 weight-sum-scaled 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 weight-sum-scaled 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 with each term equal identically to $1/n$, 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-power function, but instead the $p$th-power function left-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 $p$th-power function as before iff $p$ is nonzero real, the $max$ or $min$ as before if $p$ is infinite (according to its signum as before), 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; 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 complex-valued, single-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the $p$th-power function ($z$^${p}$) for some nonzero $p$ (note that it need not be positive or an integer) and indeterminate/variable/input $z$, 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 weight-sum-scaled 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 weight-sum-scaled 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 with each term equal identically to $1/n$, 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-power function, but instead the $p$th-power function left-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 $p$th-power function as before iff $p$ is nonzero real, the $max$ or $min$ as before if $p$ is infinite (according to its signum as before), 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: 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 complex-valued, single-valued function which is defined and continuous on $x_2$ and which is injective; it defaults to the $p$th-power function ($z^p$) for some nonzero $p$ (note that it need not be positive or an integer) and indeterminate/variable/input $z$, 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 weight-sum-scaled 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 weight-sum-scaled 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 with each term equal identically to $1/n$, 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-power function, but instead the $p$th-power function left-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 $p$th-power function as before iff $p$ is nonzero real, the $max$ or $min$ as before if $p$ is infinite (according to its signum as before), 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.