Return-path: Envelope-to: jbovlaste-admin@lojban.org Delivery-date: Fri, 25 Jun 2021 23:53:43 -0700 Received: from [192.168.123.254] (port=41210 helo=web.digitalkingdom.org) by stodi.digitalkingdom.org with smtp (Exim 4.94) (envelope-from ) id 1lx2CC-000SgC-5Y for jbovlaste-admin@lojban.org; Fri, 25 Jun 2021 23:53:43 -0700 Received: by web.digitalkingdom.org (sSMTP sendmail emulation); Sat, 26 Jun 2021 06:53:40 +0000 From: "Apache" To: curtis289@att.net Reply-To: webmaster@lojban.org Subject: [jvsw] Definition Edited At Word ji'i'u -- By krtisfranks Date: Sat, 26 Jun 2021 06:53:40 +0000 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Message-Id: X-Spam-Score: -2.3 (--) X-Spam_score: -2.3 X-Spam_score_int: -22 X-Spam_bar: -- In jbovlaste, the user krtisfranks has edited a definition of "ji'i'u" in the language "English". Differences: 2,2c2,2 < mekso at-most-4-ary operator: a rounding function; ordered input list is $(x,n,m,t)$ and the output is $b^{(-t)} *$ round($b^t * x$), with rounding preference $n$ and where the fractional part of $b^t * x$ being 1/2 causes the function to map $b^t * x$ to the nearest integer of form $2Z+m$, for base b and an integer Z determined by context --- > mekso, at-most-4-ary operator: a rounding function; ordered input list is $(x,n,t,m)$ and the output is sgn$(x) * b^{(-t)} *$ round$_{n} (b^t * $ abs$(x))$, with rounding preference $n$ and where the fractional part of $b^t * x$  being equal to $1/2$ causes the round$_{n} ($•$)$ function to map $b^t * x$ to the nearest integer of form $2Z+m$ for base $b$ and an integer $Z$, each determined by context. 5,5c5,5 < x must be a real number; n must be exactly one of exactly the following: -1, 0, 1; m must be 0 xor 1 if defined at all; t must be an integer if defined at all; the output is a real number. n does not have a contextless default value. m is defined iff n = 0; if m is undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and a third argument would fill the 't' slot under the condition that t is defined). If m is defined, then its contextless default value is m = 0. It first determines the base being used for interpretation of digit strings (determined by context or by explicit specification (JUhAU)) for x; this determination takes place even before inputs are accepted after x; let this base be represented by b throughout this description. If the base is not a positional system wherein each digit represents a corresponding multiple of a fixed natural number raised to the power of its position (as determined relative to the radix point) and wherein the overall number is the sum of these results/representands, then t is defined; if the base is sufficiently bad or unclear, then t is undefined; if t is undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and its input acceptance is terminated by the n or m slot, whichever one is later yet defined) and b = 1 for the purposes of this definition (but not for any digit-to-number interpretation/conversion!). If t is defined, then its contextless default value is t = 0. The rounding function, determined by n, is performed $(b^{t})*x$. If n = 1, then the rounding function is the ceiling function: $(b^{t})*x$ is mapped to the least integer that is greater than or equal to it. If n = -1, then the rounding function is the floor function: $(b^{t})*x$ is mapped to the greatest integer that is less than or equal to it. These integers are both determined by the ordering and metric. If n = 0, then the rounding function maps $(b^{t})*x$ to the integer that minimizes the metric distance between itself and $(b^{t})*x$ if a unique such integer exists (id est: $(b^{t})*x$ is mapped to the nearest integer, where "nearest"-ness is determined according to the order and metric); if no such unique integer exists, then $(b^{t})*x$ is mapped to the unique integer among these aforementioned options for which there exists an integer Z such that 2Z+m is the integer in question; if no unique such integer exists, then the function is undefined. Thus n = 0 produces the commonly used unbiased nearest-integer rounding function. In each of these cases, the output of the rounding function is then multiplied by $b^(-t)$. Thus, it rounds at the $t$th digit, so to speak. The order on, and the operators and metric endowing, the metric space and field of all real numbers is determined by context or by explicit specification. --- > $x$ must be a real number; & $n$ must be exactly one of exactly the following: $-1$, $0$, $1$; & $t$ must be an integer (contextless default: $0$) if defined at all (see infra); & $m$ must be $0$ xor $1$ if it is defined at all (see infra); the output is a real (typically: rational) number. $n$ does not have a contextless default value; it determines the type of rounding which is to be used here (see infra). $m$ is defined iff $n = 0$; if $m$ is/would be undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and the the final argument would fill the '$t$' slot under the condition that three arguments are explicitly used). If $m$ is defined, then its contextless default value is $m = 0$. This function/word first determines the base being used for interpretation of digit strings (determined by context or by explicit specification (JUhAU)) for/as applied to $x$; this determination takes place even before inputs are accepted after $x$; let this base be represented by $b$ throughout this description. If the base is not a positional system wherein each digit represents a corresponding multiple of a fixed natural number raised to the power of its position (as determined relative to the radix point) and wherein the overall number is the sum of these results/representands, then $t$ is defined; if the base is sufficiently bad or unclear, then $t$ is undefined; if $t$ is undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and its input acceptance is terminated by the '$n$' xor '$m$' slot, whichever one is later yet defined; if '$m$' is defined and explicitly uttered when '$t$' is undefined and the '$x$' and '$n$' values have been explicitly specified, then the third input is '$m$') and $b = 1 =$ "$b^t$" formally for the purposes of this definition (but not for any digit-to-number interpretation/conversion, particularly on $x$!). If $t$ is defined, then its contextless default value is $t = 0$.  The rounding function, determined by $n$, is performed on $(b^{t})*$abs$(x)$. If $n = 1$, then the rounding function is the ceiling function: $(b^{t})*$abs$(x)$ is mapped to the least integer that is greater than or equal to it; in other words, it rounds up.  If $n = -1$, then the rounding function is the floor function: $(b^{t})*$abs$(x)$ is mapped to the greatest integer that is less than or equal to it; in other words, it rounds down. These integers are both determined by the ordering and metric. If $n = 0$, then the rounding function maps $(b^{t})*$abs$(x)$ to the integer that minimizes the metric distance between itself and $(b^{t})*$abs$(x)$ if a unique such integer exists (id est: $(b^{t})*$abs$(x)$ is mapped to the nearest integer, where "nearest"-ness is determined according to the order and metric); if no such unique integer exists, then $(b^{t})*$abs$(x)$ is mapped to the unique integer among these aforementioned options for which there exists an integer $Z$ such that $2Z+m$ is the integer in question; if no unique such integer exists, then the function is undefined. Thus $n = 0$ produces the commonly used unbiased nearest-integer rounding function.  In each of these cases, the output of the rounding function is then multiplied by sgn$(x) * b^{(-t)}$.  Thus, it rounds at the $t$th digit, so to speak. The order on, and the operators and metric endowing, the metric space and field of all real numbers is determined by context or by explicit specification. The absolute value function is denoted as "abs(•)" in the definition, and the signum function sgn is such that sgn$(x) = x/$abs$(x)$ for all real $x: x \neq 0$, and sgn$(0) = 0$. 14a15,17 \n> Word: round up, In Sense: > Word: round down, In Sense: > Word: round to nearest integer, In Sense: Old Data: Definition: mekso at-most-4-ary operator: a rounding function; ordered input list is $(x,n,m,t)$ and the output is $b^{(-t)} *$ round($b^t * x$), with rounding preference $n$ and where the fractional part of $b^t * x$ being 1/2 causes the function to map $b^t * x$ to the nearest integer of form $2Z+m$, for base b and an integer Z determined by context Notes: x must be a real number; n must be exactly one of exactly the following: -1, 0, 1; m must be 0 xor 1 if defined at all; t must be an integer if defined at all; the output is a real number. n does not have a contextless default value. m is defined iff n = 0; if m is undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and a third argument would fill the 't' slot under the condition that t is defined). If m is defined, then its contextless default value is m = 0. It first determines the base being used for interpretation of digit strings (determined by context or by explicit specification (JUhAU)) for x; this determination takes place even before inputs are accepted after x; let this base be represented by b throughout this description. If the base is not a positional system wherein each digit represents a corresponding multiple of a fixed natural number raised to the power of its position (as determined relative to the radix point) and wherein the overall number is the sum of these results/representands, then t is defined; if the base is sufficiently bad or unclear, then t is undefined; if t is undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and its input acceptance is terminated by the n or m slot, whichever one is later yet defined) and b = 1 for the purposes of this definition (but not for any digit-to-number interpretation/conversion!). If t is defined, then its contextless default value is t = 0. The rounding function, determined by n, is performed $(b^{t})*x$. If n = 1, then the rounding function is the ceiling function: $(b^{t})*x$ is mapped to the least integer that is greater than or equal to it. If n = -1, then the rounding function is the floor function: $(b^{t})*x$ is mapped to the greatest integer that is less than or equal to it. These integers are both determined by the ordering and metric. If n = 0, then the rounding function maps $(b^{t})*x$ to the integer that minimizes the metric distance between itself and $(b^{t})*x$ if a unique such integer exists (id est: $(b^{t})*x$ is mapped to the nearest integer, where "nearest"-ness is determined according to the order and metric); if no such unique integer exists, then $(b^{t})*x$ is mapped to the unique integer among these aforementioned options for which there exists an integer Z such that 2Z+m is the integer in question; if no unique such integer exists, then the function is undefined. Thus n = 0 produces the commonly used unbiased nearest-integer rounding function. In each of these cases, the output of the rounding function is then multiplied by $b^(-t)$. Thus, it rounds at the $t$th digit, so to speak. The order on, and the operators and metric endowing, the metric space and field of all real numbers is determined by context or by explicit specification. Jargon: Gloss Keywords: Word: ceiling function, In Sense: Word: floor function, In Sense: Word: nearest integer function, In Sense: Word: rounding function, In Sense: rounds at given digit Place Keywords: New Data: Definition: mekso, at-most-4-ary operator: a rounding function; ordered input list is $(x,n,t,m)$ and the output is sgn$(x) * b^{(-t)} *$ round$_{n} (b^t * $ abs$(x))$, with rounding preference $n$ and where the fractional part of $b^t * x$  being equal to $1/2$ causes the round$_{n} ($•$)$ function to map $b^t * x$ to the nearest integer of form $2Z+m$ for base $b$ and an integer $Z$, each determined by context. Notes: $x$ must be a real number; & $n$ must be exactly one of exactly the following: $-1$, $0$, $1$; & $t$ must be an integer (contextless default: $0$) if defined at all (see infra); & $m$ must be $0$ xor $1$ if it is defined at all (see infra); the output is a real (typically: rational) number. $n$ does not have a contextless default value; it determines the type of rounding which is to be used here (see infra). $m$ is defined iff $n = 0$; if $m$ is/would be undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and the the final argument would fill the '$t$' slot under the condition that three arguments are explicitly used). If $m$ is defined, then its contextless default value is $m = 0$. This function/word first determines the base being used for interpretation of digit strings (determined by context or by explicit specification (JUhAU)) for/as applied to $x$; this determination takes place even before inputs are accepted after $x$; let this base be represented by $b$ throughout this description. If the base is not a positional system wherein each digit represents a corresponding multiple of a fixed natural number raised to the power of its position (as determined relative to the radix point) and wherein the overall number is the sum of these results/representands, then $t$ is defined; if the base is sufficiently bad or unclear, then $t$ is undefined; if $t$ is undefined, then its slot is automatically and implicitly deleted from this word in that context (so the operator is at-most-3-ary and its input acceptance is terminated by the '$n$' xor '$m$' slot, whichever one is later yet defined; if '$m$' is defined and explicitly uttered when '$t$' is undefined and the '$x$' and '$n$' values have been explicitly specified, then the third input is '$m$') and $b = 1 =$ "$b^t$" formally for the purposes of this definition (but not for any digit-to-number interpretation/conversion, particularly on $x$!). If $t$ is defined, then its contextless default value is $t = 0$.  The rounding function, determined by $n$, is performed on $(b^{t})*$abs$(x)$. If $n = 1$, then the rounding function is the ceiling function: $(b^{t})*$abs$(x)$ is mapped to the least integer that is greater than or equal to it; in other words, it rounds up.  If $n = -1$, then the rounding function is the floor function: $(b^{t})*$abs$(x)$ is mapped to the greatest integer that is less than or equal to it; in other words, it rounds down. These integers are both determined by the ordering and metric. If $n = 0$, then the rounding function maps $(b^{t})*$abs$(x)$ to the integer that minimizes the metric distance between itself and $(b^{t})*$abs$(x)$ if a unique such integer exists (id est: $(b^{t})*$abs$(x)$ is mapped to the nearest integer, where "nearest"-ness is determined according to the order and metric); if no such unique integer exists, then $(b^{t})*$abs$(x)$ is mapped to the unique integer among these aforementioned options for which there exists an integer $Z$ such that $2Z+m$ is the integer in question; if no unique such integer exists, then the function is undefined. Thus $n = 0$ produces the commonly used unbiased nearest-integer rounding function.  In each of these cases, the output of the rounding function is then multiplied by sgn$(x) * b^{(-t)}$.  Thus, it rounds at the $t$th digit, so to speak. The order on, and the operators and metric endowing, the metric space and field of all real numbers is determined by context or by explicit specification. The absolute value function is denoted as "abs(•)" in the definition, and the signum function sgn is such that sgn$(x) = x/$abs$(x)$ for all real $x: x \neq 0$, and sgn$(0) = 0$. Jargon: Gloss Keywords: Word: ceiling function, In Sense: Word: floor function, In Sense: Word: nearest integer function, In Sense: Word: rounding function, In Sense: rounds at given digit Word: round up, In Sense: Word: round down, In Sense: Word: round to nearest integer, In Sense: Place Keywords: You can go to to see it.