https://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&feed=atom&action=historyExtended Dimensionality of Interval cmavo - Revision history2024-03-29T05:51:33ZRevision history for this page on the wikiMediaWiki 1.38.4https://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124710&oldid=prevKrtisfranks: Added section on "bi'oi".2022-10-05T07:12:28Z<p>Added section on "bi'oi".</p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Then <math>I = \big( (x_1 - r_1 ,\; x_1 + r_1) \times (x_2 - r_2 ,\; x_2 + r_2) \times [x_3 - r_3 ,\; x_3 + r_3] \big) \setminus (x_1 , x_2 , x_3 ) </math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** Then <math>I = \big( (x_1 - r_1 ,\; x_1 + r_1) \times (x_2 - r_2 ,\; x_2 + r_2) \times [x_3 - r_3 ,\; x_3 + r_3] \big) \setminus (x_1 , x_2 , x_3 ) </math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*** Notice that it is <u>'''NOT'''</u>: <math>I = ((x_1 - r_1 ,\; x_1) \cup (x_1 ,\;x_1 + r_1)) \times ((x_2 - r_2 ,\; x_2) \cup (x_2 ,\;x_2 + r_2)) \times ([x_3 - r_3 ,\; x_3) \cup (x_3 ,\;x_3 + r_3])</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*** Notice that it is <u>'''NOT'''</u>: <math>I = ((x_1 - r_1 ,\; x_1) \cup (x_1 ,\;x_1 + r_1)) \times ((x_2 - r_2 ,\; x_2) \cup (x_2 ,\;x_2 + r_2)) \times ([x_3 - r_3 ,\; x_3) \cup (x_3 ,\;x_3 + r_3])</math>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== "bi'oi" ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">These extensions can also apply to the experimental BIhI cmavo "bi'oi". The definition at https://jbovlaste.lojban.org/dict/bi'oi is designed for the one-dimensional case in a directed space, in order to be consistent with the official definitions of other members BIhI.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The proposal would be for "x bi'oi y" to be extended in terms of dimensionality in the following manner in the context of an n-dimensional connected affine space X. x would belong to X and would remain a point (albeit one which may be specified by an ordered n-tuple); y, then, would be an n-dimensional translation vector with appropriate properties (such as meaning, units, etc.). Then y can be added to x in order to generate another point in X, labelled "x+y". The interval generated by "x bi'oi y" in this case would be the set <math>\{x + \lambda y: 0</math> R<math>_1 \lambda </math> R<math>_2 1 \}</math>, where: relation R<math>_i \in \{'<', '\leq' \} \forall i \in \{1, 2 \}</math>, with each dependent on the clusivity specifications accompanying "bi'oi".</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Further extensions of this word may be possible.</ins></div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Vocabulary/Semantics that have been Introduced ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Vocabulary/Semantics that have been Introduced ==</div></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124709&oldid=prevKrtisfranks: /* Proposed Extension A: "mi'i" */2022-10-05T06:07:35Z<p><span dir="auto"><span class="autocomment">Proposed Extension A: "mi'i"</span></span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Proposed Extension A: "mi'i" ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Proposed Extension A: "mi'i" ===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" can be extended further. Keep the previous definitions and conditions<del style="font-weight: bold; text-decoration: none;">. Now</del>, undefine r. Let <math><del style="font-weight: bold; text-decoration: none;">r_1, r_2, ..., r_n </del>\geq 0</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" can be extended further. <ins style="font-weight: bold; text-decoration: none;">Let X be a vector space or geometric space with an established coördinate system. </ins>Keep the previous definitions and conditions, <ins style="font-weight: bold; text-decoration: none;">except: </ins>undefine r. Let <math><ins style="font-weight: bold; text-decoration: none;">r_i </ins>\geq 0 <ins style="font-weight: bold; text-decoration: none;">\forall i \in \mathbb{Z} \cap [1, n]</ins></math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Then we can express </del>a new formal tuple <math>r = (r_1, ..., r_n)</math> where the order of the entries correspond to similarly labelled coördinates of points in X with respect to the basis established. Note that r does not live in X; it is just a formal n-tuple which has entries ordered in a corresponding manner - in other words, it is just a list of numbers (scalars in the underlying field, more specifically) with the order of presentation fixed by the basis of X and according to the utterer's intention. Notice that r does not technically change if the basis is changed; in such a situation, it may not be possible to describe the n-dimensional interval in simple terms (using only linear combinations of the entries of the new basis) at all and, in any case, the utterer would generally need to supply an entirely different list <math> r\prime </math> in order to convey the same thought.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Define </ins>a new formal tuple <math>r = (r_1, ..., r_n)</math> where the order of the entries correspond to similarly labelled coördinates of points in X with respect to the basis established. Note that r does not live in X; it is just a formal n-tuple which has entries ordered in a corresponding manner - in other words, it is just a list of numbers (scalars in the underlying field, more specifically) with the order of presentation fixed by the basis of X and according to the utterer's intention. Notice that r does not technically change if the basis is changed; in such a situation, it may not be possible to describe the n-dimensional interval in simple terms (using only linear combinations of the entries of the new basis) at all and, in any case, the utterer would generally need to supply an entirely different list <math> r\prime </math> in order to convey the same thought.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* Then we can define "x mi'i r" as <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{<del style="font-weight: bold; text-decoration: none;">N</del>} \cap [1, n]), (d_F (x_i, \alpha_i)</math> ''R'' <math>r_i)) \}</math>. Notice that 'd' is now actually '<math>d_F</math>', id est: the metric on the field F. Here, each <del style="font-weight: bold; text-decoration: none;">coordinate </del>of a point <math>\alpha</math> is being compared to the corresponding coordinate of point x; <del style="font-weight: bold; text-decoration: none;">if </del>they are within the specified distance of one another (given by the corresponding entry in the list r), then that <del style="font-weight: bold; text-decoration: none;">coordinate </del>works out; iff all of the coordinates of the point work out, then the point belongs to the interval so described.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Then we can define "x mi'i r" as <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{<ins style="font-weight: bold; text-decoration: none;">Z</ins>} \cap [1, n]), (d_F (x_i, \alpha_i)</math> ''R'' <math>r_i)) \}</math>. Notice that 'd' is now actually '<math>d_F</math>', id est: the metric on the field F. Here, each <ins style="font-weight: bold; text-decoration: none;">coördinate </ins>of a point <math>\alpha</math> is being compared to the corresponding coordinate of point x; <ins style="font-weight: bold; text-decoration: none;">iff </ins>they are within the specified distance of one another (given by the corresponding entry in the list r), then that <ins style="font-weight: bold; text-decoration: none;">coördinate </ins>works out; iff all of the coordinates of the point work out, then the point belongs to the interval so described.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** This extended form of "mi'i" can be obtained via Cartesian products of linear intervals. We will exploit this fact in the discussion about the endpoint statuses (see the section named accordingly).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** This extended form of "mi'i" can be obtained via Cartesian products of linear intervals. We will exploit this fact in the discussion about the endpoint <ins style="font-weight: bold; text-decoration: none;">clusivity </ins>statuses (see the section named accordingly).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This essentially returns us to the old situation wherein the interval is no longer an n-ball but an n-cell (matching "bi'i"). The <del style="font-weight: bold; text-decoration: none;">side </del>lengths vary (being <math>2 r_i</math> in length, for each <del style="font-weight: bold; text-decoration: none;">side </del>i). The lines which pass through their corresponding/respective midpoints and which are perpendicular to the corresponding hyperfaces will intersect at a single point, videlicet the first argument of "mi'i" <del style="font-weight: bold; text-decoration: none;">constructs </del>(the 'center'; more appropriately: circumcenter), which is the point from which the various perpendicular distances to the boundaries are each measured (being <math> r_i </math>, for the appropriate/corresponding i).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This essentially returns us to the old situation wherein the interval is no longer an n-ball but an n-cell (matching "bi'i"). The <ins style="font-weight: bold; text-decoration: none;">edge </ins>lengths vary (being <math>2 r_i</math> in length, for each <ins style="font-weight: bold; text-decoration: none;">edge which is parallel with basis vector/element or axis <math>e_i \forall </ins>i <ins style="font-weight: bold; text-decoration: none;">\in \mathbb{Z} \cap [1, n]</math></ins>). The lines which pass through their corresponding/respective midpoints and which are perpendicular to the corresponding hyperfaces will intersect at a single point, videlicet the first argument of <ins style="font-weight: bold; text-decoration: none;">the </ins>"mi'i" <ins style="font-weight: bold; text-decoration: none;">construct </ins>(the 'center'; more appropriately: circumcenter), which is the point from which the various perpendicular distances to the boundaries are each measured (being <math> r_i </math>, for the appropriate/corresponding i).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This definition is good for computer science, graphing, and experimental science. It is almost never used in theoretical mathematics<del style="font-weight: bold; text-decoration: none;">. </del>(<del style="font-weight: bold; text-decoration: none;">Literally </del>never in the experience of lai .krtisfranks., at any rate.<del style="font-weight: bold; text-decoration: none;">)</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This definition is good for computer science, graphing, and experimental science. It is almost never used in theoretical mathematics (<ins style="font-weight: bold; text-decoration: none;">literally </ins>never in the experience of lai .krtisfranks., at any rate<ins style="font-weight: bold; text-decoration: none;">) and has no common notation there, but it is a useful concept and resolves the aforementioned lexical gap for uncertainties in measurement</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This additional proposal requires no major update, change, or addition to the glossing/keywords associated with "mi'i" in dictionary definitions, although there would be an implicit understanding of increased generality. If desired, however, "orthotopic interval with given circumcenter" or similar would do nicely.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This additional proposal requires no major update, change, or addition to the glossing/keywords associated with "mi'i" in dictionary definitions, although there would be an implicit understanding of increased generality. If desired, however, "orthotopic interval with given circumcenter" or similar would do nicely.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* Additionally, we could establish the convention-by-definition that: <math>((\exists \rho \geq 0: ((\forall i \in \mathbb{<del style="font-weight: bold; text-decoration: none;">N</del>} \cap [1, n]), (r_i = \rho))) \implies </math> "x mi'i r" = "x mi'i <math>\rho</math>" <math>)</math>; but we would need a way to ensure that the audience recognizes <math>\rho</math> as an n-tuple and not just a scalar. Otherwise, utilization of this convention would be indistinguishable from the previously-mentioned case/proposal wherein the second argument as a single number constitutes the radius of an n-ball.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Additionally, we could establish the convention-by-definition that: <math>((\exists \rho \geq 0: ((\forall i \in \mathbb{<ins style="font-weight: bold; text-decoration: none;">Z</ins>} \cap [1, n]), (r_i = \rho))) \implies </math> "x mi'i r" = "x mi'i <math>\rho</math>" <math>)</math>; but we would need a way to ensure that the audience recognizes <math>\rho</math> as an n-tuple and not just a scalar. Otherwise, utilization of this convention would be indistinguishable from the previously-mentioned case/proposal wherein the second argument as a single number constitutes the radius of an n-ball.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** This complication can be overcome by mentioning "ce'ei'oi" immediately after "<math>\rho</math>" (list sense) in the "mi'i" construct; if this is done, then we are to understand that "<math>\rho</math>" represents - in short-hand form - a formal tuple of identical entries (each being <math>\rho</math> (in the scalar sense)). The <del style="font-weight: bold; text-decoration: none;">elements </del>of this tuple must never be negative.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** This complication can be overcome by mentioning "ce'ei'oi" immediately after "<math>\rho</math>" (list sense) in the "mi'i" construct; if this is done, then we are to understand that "<math>\rho</math>" represents - in short-hand form - a formal tuple of identical entries (each being <math>\rho</math> (in the scalar sense)). The <ins style="font-weight: bold; text-decoration: none;">components/terms </ins>of this tuple must never be negative.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*** If the utterer explicitly defines/declares <math>\rho</math> to be such a formal tuple, then "ce'ei'oi" is not necessary, although it is also not wrong (and may in fact be helpful and encouraged).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*** If the utterer explicitly defines/declares <math>\rho</math> to be such a formal tuple, then "ce'ei'oi" is not necessary, although it is also not wrong (and may in fact be helpful and encouraged).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124708&oldid=prevKrtisfranks: /* Current Functionality */2022-10-05T05:50:04Z<p><span dir="auto"><span class="autocomment">Current Functionality</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 05:50, 5 October 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l33">Line 33:</td>
<td colspan="2" class="diff-lineno">Line 33:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If we accept partial orders, the space X can be all sorts of creatures, including - for example - sets under the strict-containment/proper-subset relation (so that BIhI forms an interval of sets). However, more exotic meanings can be used/intended (although any partial order endowing the space would have to be ignored in context with respect to the meaning of BIhI, which is okay and implicitly possible within the description heretofore provided by the CLL). For example, intervals may just trace out (a possibly ordered/'directed') path between points in X, which may be - for example - the geography of locations on Earth, a network, or a set of sets (which may otherwise but inconsequentially for our purposes be endowed with the proper-subset order). In order to be clear: X need not have an order of any kind endowing it overall; however, if "bi'o" is used, the interval generated does have an ordered endowed on it (alone) which may or may not match the order endowing X, should such an order exist.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If we accept partial orders, the space X can be all sorts of creatures, including - for example - sets under the strict-containment/proper-subset relation (so that BIhI forms an interval of sets). However, more exotic meanings can be used/intended (although any partial order endowing the space would have to be ignored in context with respect to the meaning of BIhI, which is okay and implicitly possible within the description heretofore provided by the CLL). For example, intervals may just trace out (a possibly ordered/'directed') path between points in X, which may be - for example - the geography of locations on Earth, a network, or a set of sets (which may otherwise but inconsequentially for our purposes be endowed with the proper-subset order). In order to be clear: X need not have an order of any kind endowing it overall; however, if "bi'o" is used, the interval generated does have an ordered endowed on it (alone) which may or may not match the order endowing X, should such an order exist.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If the space does 'loop around' in a relevant way (examples: the space is a circle or a modular-arithmetic structure such as the space of fractional parts of real numbers (<math>\mathbb{R} (\mod 1)</math>)), then "x bi'o y" traces out the shortest path from x to y. So, if x and y are points on a circle, then "x bi'o y" is the arc from x to y under an assumed direction of turning/increase in central angle (traditionally in European mathematics: counterclockwise) and "y bi'o x" would be its complement in the circle; in this case, "x bi'i y" should mean the shorted arc which connects x and y. Likewise, in <math>\mathbb{R} (\mod 1)</math>, "x bi'o y" is the interval which is generated by adding all nonnegative values of s to x until y is attained the first time (so, this is counting up from x until y is reached, looping through or ticking back down to 0 if y < x); in other words<del style="font-weight: bold; text-decoration: none;">, </del>if x < y, then this interval is <del style="font-weight: bold; text-decoration: none;">small, </del>and if x > y, then this interval is <del style="font-weight: bold; text-decoration: none;">large </del>(and the complement of the former).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If the space does 'loop around' in a relevant way (examples: the space is a circle or a modular-arithmetic structure such as the space of fractional parts of real numbers (<math>\mathbb{R} (\mod 1)</math>)), then "x bi'o y" traces out the shortest path from x to y. So, if x and y are points on a circle, then "x bi'o y" is the arc from x to y under an assumed direction of turning/increase in central angle (traditionally in European mathematics: counterclockwise) and "y bi'o x" would be its complement in the circle; in this case, "x bi'i y" should mean the shorted arc which connects x and y. Likewise, in <math>\mathbb{R} (\mod 1)</math>, "x bi'o y" is the interval which is generated by adding all nonnegative values of s to x until y is attained the first time (so, this is counting up from x until y is reached, looping through or ticking back down to 0 if y < x); in other words<ins style="font-weight: bold; text-decoration: none;">: </ins>if x < y, then this interval <ins style="font-weight: bold; text-decoration: none;">contains 0 iff x = 0 and x </ins>is <ins style="font-weight: bold; text-decoration: none;">included in the interval; </ins>and if x > y, then this interval is <ins style="font-weight: bold; text-decoration: none;">contains 0 no matter what </ins>(and <ins style="font-weight: bold; text-decoration: none;">is </ins>the complement of the former).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If X is a disconnected space and x and y belong to mutually-disconnected subspaces thereof, then there exists no r such that "x mi'i r" refers to an interval which contains y, and "x bi'i y" (and thus: "x bi'o y", "y bi'i x", and "y bi'o x") would be meaningless (undefined; not even outputting the empty set).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If X is a disconnected space and x and y belong to mutually-disconnected subspaces thereof, then there exists no r such that "x mi'i r" refers to an interval which contains y, and "x bi'i y" (and thus: "x bi'o y", "y bi'i x", and "y bi'o x") would be meaningless (undefined; not even outputting the empty set).</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Notice that, now, "mi'i" and "bi'i" do not have the same "shape" except when the dimensionality involved is 0 or 1. The former is round whereas the latter is rectilinear. This is assuming Euclidean geometry. If other metrics are involved, they may appear to be the same or may actually be the same. For example, in taxicab geometry, a sphere <i>appears</i> to be a cross-polytope of the appropriate dimensionality, yet it is still a sphere (which bounds "mi'i"-intervals). In Chebyshev geometry, a sphere appears as an orthotope of the appropriate dimensionality, yet it is still a sphere (and the boundary of a "mi'i"-interval); in this case, though, it very well may be congruent to the n-cell (orthotope) that "bi'i" produces (under the proper conditions). Regardless of these considerations, in higher-dimensions, both of these intervals have the similarity that they generate subspaces of nonzero Lebesgue measure (such regions of positive volume), assuming that the r > 0 and x <math>\neq</math> y.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Notice that, now, "mi'i" and "bi'i" do not have the same "shape" except when the dimensionality involved is 0 or 1. The former is round whereas the latter is rectilinear. This is assuming Euclidean geometry. If other metrics are involved, they may appear to be the same or may actually be the same. For example, in taxicab geometry, a sphere <i>appears</i> to be a cross-polytope of the appropriate dimensionality, yet it is still a sphere (which bounds "mi'i"-intervals). In Chebyshev geometry, a sphere appears as an orthotope of the appropriate dimensionality, yet it is still a sphere (and the boundary of a "mi'i"-interval); in this case, though, it very well may be congruent to the n-cell (orthotope) that "bi'i" produces (under the proper conditions). Regardless of these considerations, in higher-dimensions, both of these intervals have the similarity that they generate subspaces of nonzero Lebesgue measure (such regions of positive volume), assuming that the r > 0 and x <math>\neq</math> y.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* These distinct definitions are good (utile) and natural in theoretical mathematics.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* These distinct definitions are good (utile) and natural in theoretical mathematics<ins style="font-weight: bold; text-decoration: none;">. They arise in many naturally-defined sets in a theoretical context.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">** Additionally, in this interpretation, they can be used for measurements, forming error/uncertainty bars. Thus, one should not say that a measurement is some value <math>x \pm \delta x</math>, but instead that the measured value belongs to the interval which is generated/referenced by "<math>x</math> mi'i <math>\delta x</math>", assuming that the reported value of <math>x</math> is indeed the midpoint and the error therefrom is uniformly <math>\delta x</math> in all relevant dimensions (under the assumption that all reported errors correspond to, say, the same confidence level); notice that <math>x \pm \delta x</math> suggests that the measurement errors are one-dimensional, but the Lojban solution need not do so under this proposal. If, instead, the error bars form an orthotope, potentially with edges of unequal lengths, then "bi'i" may be used under the assumption of this proposal, although the reported value of <math>x</math> would no longer be expressed via this method (see infra for that). This can be useful when reporting uncertainty in both input and output measurements. Note that <math>x</math> and the uncertainties can be measurement-dimensionful/unitful quantities (such as displacements). For emphasis: a measurement should never really be reported via equality but, instead, via membership as an element of a set (the potentially-multidimensional interval).</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">** The terms can also be used for reporting membership to intervals defined by z-scores in the context of statistics (apart from all of the other obvious aforementioned applications resultant from measure theory, probability, measurement-taking, etc.)</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Alternative #1: Line Segments Unless Specified Otherwise ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Alternative #1: Line Segments Unless Specified Otherwise ===</div></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124707&oldid=prevKrtisfranks: /* Alternative #1: Line Segments Unless Specified Otherwise */2022-10-05T02:55:33Z<p><span dir="auto"><span class="autocomment">Alternative #1: Line Segments Unless Specified Otherwise</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:55, 5 October 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l32">Line 32:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If we accept partial orders, the space X can be all sorts of creatures, including - for example - sets under the strict-containment/proper-subset relation (so that BIhI forms an interval of sets). However, more exotic meanings can be used/intended (although any partial order endowing the space would have to be ignored in context with respect to the meaning of BIhI, which is okay and implicitly possible within the description heretofore provided by the CLL). For example, intervals may just trace out (a possibly ordered/'directed') path between points in X, which may be - for example - the geography of locations on Earth, a network, or a set of sets (which may otherwise but inconsequentially for our purposes be endowed with the proper-subset order). In order to be clear: X need not have an order of any kind endowing it overall; however, if "bi'o" is used, the interval generated does have an ordered endowed on it (alone) which may or may not match the order endowing X, should such an order exist.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If we accept partial orders, the space X can be all sorts of creatures, including - for example - sets under the strict-containment/proper-subset relation (so that BIhI forms an interval of sets). However, more exotic meanings can be used/intended (although any partial order endowing the space would have to be ignored in context with respect to the meaning of BIhI, which is okay and implicitly possible within the description heretofore provided by the CLL). For example, intervals may just trace out (a possibly ordered/'directed') path between points in X, which may be - for example - the geography of locations on Earth, a network, or a set of sets (which may otherwise but inconsequentially for our purposes be endowed with the proper-subset order). In order to be clear: X need not have an order of any kind endowing it overall; however, if "bi'o" is used, the interval generated does have an ordered endowed on it (alone) which may or may not match the order endowing X, should such an order exist.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If the space does 'loop around' in a relevant way (examples: the space is a circle or a modular-arithmetic structure such as the space of fractional parts of real numbers (<math>\mathbb{R} (\mod 1)</math>)), then "x bi'o y" traces out the shortest path from x to y. So, if x and y are points on a circle, then "x bi'o y" is the arc from x to y under an assumed direction of turning/increase in central angle (traditionally in European mathematics: counterclockwise) and "y bi'o x" would be its complement in the circle; in this case, "x bi'i y" should mean the shorted arc which connects x and y. Likewise, in <math>\mathbb{R} (\mod 1)</math>, "x bi'o y" is the interval which is generated by adding all nonnegative values of s to x until y is attained the first time (so, this is counting up from x until y is reached, looping through or ticking back down to 0 if y < x); in other words, if x < y, then this interval is small, and if x > y, then this interval is large (and the complement of the former).</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If X is a disconnected space and x and y belong to mutually-disconnected subspaces thereof, then there exists no r such that "x mi'i r" refers to an interval which contains y, and "x bi'i y" (and thus: "x bi'o y", "y bi'i x", and "y bi'o x") would be meaningless (undefined; not even outputting the empty set).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If X is a disconnected space and x and y belong to mutually-disconnected subspaces thereof, then there exists no r such that "x mi'i r" refers to an interval which contains y, and "x bi'i y" (and thus: "x bi'o y", "y bi'i x", and "y bi'o x") would be meaningless (undefined; not even outputting the empty set).</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Alternative #1: Line Segments Unless Specified Otherwise ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Alternative #1: Line Segments Unless Specified Otherwise ===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>An alternative (which lai .krtisfranks. finds perhaps even better than the previous proposal (Main Proposal #1)) is to have "bi'i" and "bi'o" always default to referencing line segments (generally: geodesics) in any space. That is, regardless of the space (or, in particular, its dimensionality), these two cmavo (but not "mi'i") would 'draw' a line from their first argument to their second one.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>An alternative (which lai .krtisfranks. finds perhaps even better than the previous proposal (Main Proposal #1)) is to have "bi'i" and "bi'o" always default to referencing line segments (generally: geodesics) in any space. That is, regardless of the space (or, in particular, its dimensionality), these two cmavo (but not "mi'i") would 'draw' a line from their first argument to their second one<ins style="font-weight: bold; text-decoration: none;">. Some interpretations of the description provided by the CLL are already supportive of and compatible with, or perhaps even guarantee, this interpretation/so-called 'alternative proposal'</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note: <del style="font-weight: bold; text-decoration: none;">The </del>endpoints (first and second arguments) of "bi'i" and "bi'o" will be points that are specified via multiple coördinates with respect to a basis. <del style="font-weight: bold; text-decoration: none;">They </del>are not merely scalars. They still must live in the same space (X) and thus must have the same number of coördinates. In this situation, the one-dimensional usage which is defined already outside of this whole proposal merely isomorphically maps scalars which are denoted by "x" and "y" to the their corresponding 1-dimensional point specifications "(x)" and "(y)" respectively. (Notice that, without an additional convention, these will never map to "(x,0,0,...)" and "(y,0,0,...)" respectively, despite the isomorphism that may be established. This is meant to avoid the abusive mixing of notation/spaces: there is no interval from (1,2) to 1, for example. <del style="font-weight: bold; text-decoration: none;">We </del>should always specify that the endpoints are higher-dimensional. This note about mapping 1 to (1) is meant solely for the purpose of making this extension back-compatible and natural.)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note: <ins style="font-weight: bold; text-decoration: none;">In many multidimensional formal spaces or, for example, a geometric space, the </ins>endpoints (first and second arguments) of "bi'i" and "bi'o" will be points that are specified via multiple coördinates with respect to a basis. <ins style="font-weight: bold; text-decoration: none;">In such cases, they </ins>are not merely scalars. They still must live in the same space (<ins style="font-weight: bold; text-decoration: none;">using earlier notation/definitions: </ins>X) <ins style="font-weight: bold; text-decoration: none;">as eachother </ins>and thus must have the same number of coördinates <ins style="font-weight: bold; text-decoration: none;">(when applicable)</ins>. In this situation, the one-dimensional usage which is <ins style="font-weight: bold; text-decoration: none;">officially </ins>defined already outside of this whole proposal merely isomorphically maps scalars which are denoted by "x" and "y" to the their corresponding 1-dimensional point specifications "(x)" and "(y)" respectively. (Notice that, without an additional convention, these will never map to "(x,0,0,...)" and "(y,0,0,...)" respectively, despite the isomorphism that may be established. This is meant to avoid the abusive mixing of notation/spaces: there is no interval from <ins style="font-weight: bold; text-decoration: none;">the point </ins>(1,2) to <ins style="font-weight: bold; text-decoration: none;">the point(s) 1 or (</ins>1<ins style="font-weight: bold; text-decoration: none;">)</ins>, for example. <ins style="font-weight: bold; text-decoration: none;">Users </ins>should always specify that the endpoints are higher-dimensional. This note about mapping 1 to (1) is meant solely for the purpose of making this extension back-compatible and natural.)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This would make the default usage automatically compatible with generalized points (see below). Additionally, line segments are generally useful in geometry of any (nontrivial) dimension, so this functionality would be utile.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This <ins style="font-weight: bold; text-decoration: none;">option </ins>would make the default usage automatically compatible with generalized points (see below). Additionally, line segments are generally useful in geometry of any (nontrivial) dimension, so this functionality would be utile.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This also would allow both "bi'i" and "bi'o" to be defined in any decent space (as opposed to only have "bi'i" be defined, which is the case in the aforementioned subproposal).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This <ins style="font-weight: bold; text-decoration: none;">option </ins>also would allow both "bi'i" and "bi'o" to be defined in any decent space (as opposed to only have "bi'i" be defined, which is the case in the aforementioned subproposal).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In this formulation, "x bi'o y" implies that x is in some sense a starting point of reference/of an imaginary journey and y is the corresponding termination point; both are 'endpoints'/terminals, so to speak. If the space does not 'loop around', then the set produced, however, is still exactly equivalent to that produced by "x bi'i y" and any coloring of the connotations is unmathematical (and, thus, should be avoided in the opinion of lai .krtisfranks.),; the latter is generally preferred.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In this formulation, "x bi'o y" implies that x is in some sense a starting point of reference/of an imaginary journey and y is the corresponding termination point; both are 'endpoints'/terminals, so to speak. If the space does not 'loop around', then the set produced, however, is still exactly equivalent to that produced by "x bi'i y" and any coloring of the connotations is unmathematical (and, thus, should be avoided in the opinion of lai .krtisfranks.), <ins style="font-weight: bold; text-decoration: none;">[tagged due to accidental deletion]</ins>; the latter is generally preferred.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Note: There is still no established directionality on the linear interval that is produced by "bi'o". However, as long as it does not conflict with any others, we might be able to assume an established order thereupon. "x bi'o y" does mean that "x < y" (along that line).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Note: There is still no established directionality on the linear interval that is produced by "bi'o". However, as long as it does not conflict with any others, we might be able to assume an established order thereupon. "x bi'o y" does mean that "x < y" (along that line <ins style="font-weight: bold; text-decoration: none;">according at least to the relevant conceptual ordering</ins>).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In this <del style="font-weight: bold; text-decoration: none;">case</del>, <del style="font-weight: bold; text-decoration: none;">we </del>use "ce'ei'oi" (followed by a number larger than 1 if we are being explicit) on either or both points x, y in the constructs "x bi'i y" and "x bi'o y" in order to produce the swept-out higher-dimensional-orthotopal "interval" that was proposed originally. See the 'Handling Generalized Points' section, following, for more details.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In this <ins style="font-weight: bold; text-decoration: none;">alternative proposal</ins>, <ins style="font-weight: bold; text-decoration: none;">one would </ins>use "ce'ei'oi" (followed by a number larger than 1 if we are being explicit) on either or both points x, y in the constructs "x bi'i y" and "x bi'o y" in order to produce the swept-out higher-dimensional-orthotopal "interval" that was proposed originally. See the 'Handling Generalized Points' section, following, for more details.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" will still generalize to a higher-dimensional-ball in the space. (Its functionality, as described previously, and as extended immediately after this section, is unchanged.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" will still generalize to a higher-dimensional-ball in the space. (Its functionality, as described previously, and as extended immediately after this section, is unchanged.)</div></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124706&oldid=prevKrtisfranks: /* Current Functionality */2022-09-30T02:09:22Z<p><span dir="auto"><span class="autocomment">Current Functionality</span></span></p>
<a href="https://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124706&oldid=124705">Show changes</a>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124705&oldid=prevKrtisfranks: /* Current Functionality */2022-09-30T01:22:13Z<p><span dir="auto"><span class="autocomment">Current Functionality</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:22, 30 September 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Let x and y be mutually-distinct real numbers or points in geometric space, let r be a nonnegative real number. Let the space to which x and y belong be X. Further suppose that X does not "loop around" in any sense. Then:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Let x and y be mutually-distinct real numbers or points in geometric space, let r be a nonnegative real number. Let the space to which x and y belong be X. Further suppose that X does not "loop around" in any sense. Then:</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*x mi'i r: generates the interval centered on x which has range r on either side of x; in other words, this is the set of all points that have a distance to x which is less than (or, depending on clusivity or cultural convention for interpretation, possibly equal to) r. x is the midpoint of the interval, and the interval has measure/length 2r. Such a thing is sometimes denoted in a fashion similar to <math> \operatorname{B}_1 (x, r) </math>, where "B" is for "ball" and the subscript - here: "1" - tells the dimensionality of the space; this is also called an r neighborhood of x (sometimes denoted <math> \operatorname{nbhd} (x, r) </math>), where the (one-dimensional) space is inferred from context. In a connected and undirected graph X, this interval could be the set of all nodes which have a graph-geodesic path length to or from x of less than (or, again, possibly equal to) r; if the graph is directed then there is unresolved ambiguity as to whether the "path to x" or "path from x" option is meant (lai .krtisfranks. prefers the latter option, but this not established in any official decision by any Lojbanic authority or by any cultural convention known to him).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*x mi'i r: generates the interval centered on x which has range r on either side of x; in other words, this is the set of all points that have a distance to x which is less than (or, depending on clusivity <ins style="font-weight: bold; text-decoration: none;">of the endpoints </ins>or cultural convention for interpretation, possibly equal to) r. x is the midpoint of the interval, and the interval has measure/length 2r. Such a thing is sometimes denoted in a fashion similar to <math> \operatorname{B}_1 (x, r) </math>, where "B" is for "ball" and the subscript - here: "1" - tells the dimensionality of the space; this is also called an r neighborhood of x (sometimes denoted <math> \operatorname{nbhd} (x, r) </math>), where the (one-dimensional) space is inferred from context. In a connected and undirected graph X, this interval could be the set of all nodes which have a graph-geodesic path length to or from x of less than (or, again, possibly equal to) r; if the graph is directed then there is unresolved ambiguity as to whether the "path to x" or "path from x" option is meant (lai .krtisfranks. prefers the latter option, but this not established in any official decision by any Lojbanic authority or by any cultural convention known to him).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*x bi'i y: generates the interval or unordered line segment with endpoints x and y; in other words, this is the set of all points between x and y (possibly including either, both, or neither of the endpoints) without any implication about whether x < y or y < x (or whether any such ordering exists or is even meaningful). "y bi'i x" is completely equivalent to "x bi'i y"; there is no inherent order to the inputs nor direction to the line segment. This is the generic meaning for "between" when referring to an interval, in normal life or in mathematics; there is no notation for this notion which is both commonly understood and known to lai .krtisfranks. ; supposing an ordering on X, the closest thing would be an interval of form: <math>(\operatorname{min}(\{x,y\}), \operatorname{max}(\{x,y\})) \cup A</math>, where <math>A \subseteq \{x,y\}</math> which is determined by the clusivity of the endpoints. However, if X is a partially ordered space with order relation '<', then we may describe it thusly: let 'R' denote either '<' or its complement/negative '>'; then, if the endpoints are excluded, then "x bi'i y" generates the set <math> \{ \alpha \in X: x </math> R <math> \alpha </math> R <math> y \} </math>. (Notice here that x and y may be presented in either order but for any given selection of presentation order, 'R' is fixed in meaning and present in both relations; if one order of presentation is true, then if the order is switched, then the resulting statement will mean the same thing but the meaning of 'R' will be changed to the other inequality relation in order to preserve the meaning. If an endpoint is to be included, this set will just be united with the singleton set of that endpoint, iterated for each included endpoint. If X cannot be or is not partially ordered, then this present discussion about mathematical representation may be ignored; in such cases, this BIhI construction may still make sense, however - just revert to a more intuitive understanding based on the English description). If X is a connected and undirected graph, then it is not clear what this interval would mean because any node in X would be in a path which connects x to y or vice-versa; however, a natural interpretation would be the set of nodes in X which belong to any geodesic path which connects x to y (or vice-versa), or perhaps any path which connects x to y (or vice-versa) without too much 'back-tracking'. If X is a connected and directed graph, then this latter interpretation would still work (for any path from x to y or any path from y to x; both would be included because there is no preference), but the former interpretation would not generally include all of X because there could be paths from one node to either x or y (or both) but such nodes may not belong to any path which are from x to y or from y to x. For example: In X = "a -> x -> b -> c -> y -> z", nodes 'a' and z would be excluded from the meaning of "x bi'i y" and "y bi'i x" (which are equivalent), which would form a superset of {b, c} and a subset of {x, b, c, y}.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*x bi'i y: generates the interval or unordered line segment with endpoints x and y; in other words, this is the set of all points between x and y (possibly including either, both, or neither of the endpoints) without any implication about whether x < y or y < x (or whether any such ordering exists or is even meaningful). "y bi'i x" is completely equivalent to "x bi'i y"; there is no inherent order to the inputs nor direction to the line segment. This is the generic meaning for "between" when referring to an interval, in normal life or in mathematics; there is no notation for this notion which is both commonly understood and known to lai .krtisfranks. ; supposing an ordering on X, the closest thing would be an interval of form: <math>(\operatorname{min}(\{x,y\}), \operatorname{max}(\{x,y\})) \cup A</math>, where <math>A \subseteq \{x,y\}</math> which is determined by the clusivity of the endpoints. However, if X is a partially ordered space with order relation '<', then we may describe it thusly: let 'R' denote either '<' or its complement/negative '>'; then, if the endpoints are excluded, then "x bi'i y" generates the set <math> \{ \alpha \in X: x </math> R <math> \alpha </math> R <math> y \} </math>. (Notice here that x and y may be presented in either order but for any given selection of presentation order, 'R' is fixed in meaning and present in both relations; if one order of presentation is true, then if the order is switched, then the resulting statement will mean the same thing but the meaning of 'R' will be changed to the other inequality relation in order to preserve the meaning. If an endpoint is to be included, this set will just be united with the singleton set of that endpoint, iterated for each included endpoint. If X cannot be or is not partially ordered, then this present discussion about mathematical representation may be ignored; in such cases, this BIhI construction may still make sense, however - just revert to a more intuitive understanding based on the English description). If X is a connected and undirected graph, then it is not clear what this interval would mean because any node in X would be in a path which connects x to y or vice-versa; however, a natural interpretation would be the set of nodes in X which belong to any geodesic path which connects x to y (or vice-versa), or perhaps any path which connects x to y (or vice-versa) without too much 'back-tracking'. If X is a connected and directed graph, then this latter interpretation would still work (for any path from x to y or any path from y to x; both would be included because there is no preference), but the former interpretation would not generally include all of X because there could be paths from one node to either x or y (or both) but such nodes may not belong to any path which are from x to y or from y to x. For example: In X = "a -> x -> b -> c -> y -> z", nodes 'a' and z would be excluded from the meaning of "x bi'i y" and "y bi'i x" (which are equivalent), which would form a superset of {b, c} and a subset of {x, b, c, y}.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*x bi'o y: generates the interval or 'directed' line segment with endpoints x and y in that order (starting from x and going to y); otherwise, it is equivalent to "bi'i". This is the typical meaning of intervals of form [x, y) and the like. Thus "y bi'o x" is backward relative to "x bi'o y". Continuing the discussion in the immediately previous point about "bi'i" which was concerned with mathematical representation of the construct formed, and supposing the same conditions and notation, then "x bi'o y" is exactly the same (and the same commentary applies) except that it demands that 'R' represent '<'. (Notice additionally that, in this case, x < y must be true. However be careful to avoid reading "<" as the symbol representing "less than" in the intuitive sense for real numbers; it could just as easily be any other partial order, including the "greater than" relation). In other words, if the endpoints are excluded, then "x bi'o y" generates the set <math> \{ \alpha \in X: x < \alpha < y \} </math>; but, in this case, "y bi'o x" would either be malformed or would, in some sense, be the negative or complement interval of that which is generated by "x bi'o y". (Note: The "complement" interpretation should not be adopted in the context of X being guaranteed to not loop around. The "negative" interpretation is meant as in the context of integration in which reversing the direction of the integral produces the same answer in absolute value but negates it in signum; this concept may generalize or be useful in other contexts, including for notational simplicity). If X is a connected and undirected graph, then this word might have the same meaning (and have the same ambiguities or issues) as "bi'i" but could be distinguished from it by a further (as-yet-undetermined) convention for the sake of utility. If the graph X is connected and directed, then the interval would be as in the case for "bi'i" but would be required to follow only those directed paths which are from x (the first term) to y (the second term). Thus, in X = "a -> x -> b -> c -> y -> z", "x bi'o y" would mean the same thing as "x bi'i y" (for same endpoint clusivity statuses), but "y bi'o x" would be the empty set.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*x bi'o y: generates the interval or 'directed' line segment with endpoints x and y in that order (starting from x and going to y); otherwise, it is equivalent to "bi'i". This is the typical meaning of intervals of form [x, y) and the like. Thus "y bi'o x" is backward relative to "x bi'o y". Continuing the discussion in the immediately previous point about "bi'i" which was concerned with mathematical representation of the construct formed, and supposing the same conditions and notation, then "x bi'o y" is exactly the same (and the same commentary applies) except that it demands that 'R' represent '<'. (Notice additionally that, in this case, x < y must be true. However be careful to avoid reading "<" as the symbol representing "less than" in the intuitive sense for real numbers; it could just as easily be any other partial order, including the "greater than" relation). In other words, if the endpoints are excluded, then "x bi'o y" generates the set <math> \{ \alpha \in X: x < \alpha < y \} </math>; but, in this case, "y bi'o x" would either be malformed or would, in some sense, be the negative or complement interval of that which is generated by "x bi'o y". (Note: The "complement" interpretation should not be adopted in the context of X being guaranteed to not loop around. The "negative" interpretation is meant as in the context of integration in which reversing the direction of the integral produces the same answer in absolute value but negates it in signum; this concept may generalize or be useful in other contexts, including for notational simplicity). If X is a connected and undirected graph, then this word might have the same meaning (and have the same ambiguities or issues) as "bi'i" but could be distinguished from it by a further (as-yet-undetermined) convention for the sake of utility. If the graph X is connected and directed, then the interval would be as in the case for "bi'i" but would be required to follow only those directed paths which are from x (the first term) to y (the second term). Thus, in X = "a -> x -> b -> c -> y -> z", "x bi'o y" would mean the same thing as "x bi'i y" (for same endpoint clusivity statuses), but "y bi'o x" would be the empty set.</div></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124704&oldid=prevKrtisfranks: /* Main Proposal #1 */ -- Copy-edit for improper insertion of text.2022-09-24T02:25:31Z<p><span dir="auto"><span class="autocomment">Main Proposal #1: </span> -- Copy-edit for improper insertion of text.</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:25, 24 September 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* "mi'i" is the easiest to redefine. In fact, the previous description needs no reworking, so long as we understand the space to be potentially larger than a line and loosen our notion of "interval". The proposal is that "x mi'i r" be defined so as to be/describe the n-dimensional hyperball (or, possibly, the closure thereof) which is centered on/at x and which has radius r. Notationally, it is <math>\{ z \in X: d(x,z)</math> ''R'' <math>r \}</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* "mi'i" is the easiest to redefine. In fact, the previous description needs no reworking, so long as we understand the space to be potentially larger than a line and loosen our notion of "interval". The proposal is that "x mi'i r" be defined so as to be/describe the n-dimensional hyperball (or, possibly, the closure thereof) which is centered on/at x and which has radius r. Notationally, it is <math>\{ z \in X: d(x,z)</math> ''R'' <math>r \}</math>.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** interval is a neighborhood in the space; that is to say, it is a region of the space which is bounded by and internal to a sphere (but which may possibly include the boundary and/or exclude <del style="font-weight: bold; text-decoration: none;">thexternal e </del>center). This region is called a "ball"<del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** interval is a neighborhood in the space; that is to say, it is a region of the space which is bounded by and internal to a sphere (but which may possibly include the <ins style="font-weight: bold; text-decoration: none;">external </ins>boundary and/or exclude <ins style="font-weight: bold; text-decoration: none;">the </ins>center). This region is called a "ball"; it is a closed ball or ball-with-sphere if the external boundary be included; it is punctured if the center be <ins style="font-weight: bold; text-decoration: none;">excluded.</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">** Further, the proposal is also that the Lojban community ado</del>; it is a closed ball or ball-with-sphere if the external boundary be included; it is <del style="font-weight: bold; text-decoration: none;">a </del>punctured if the center be <del style="font-weight: bold; text-decoration: none;">excludedpt </del>additional keywords/glosses/terminology or similar for "mi'i". "mi'i" should be given the keyword/gloss "centered interval"; it might also deserve the keyword/gloss "n-ball". The second argument (here denoted by "r") should be called the "radius" (in addition to "range"). The first argument (here denoted by "x") can remain with the sole label of "center".</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">** Further, the proposal is also that the Lojban community adopt </ins>additional keywords/glosses/terminology or similar for "mi'i". "mi'i" should be given the keyword/gloss "centered interval"; it might also deserve the keyword/gloss "n-ball". The second argument (here denoted by "r") should be called the "radius" (in addition to "range"). The first argument (here denoted by "x") can remain with the sole label of "center".</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* "bi'i" requires a little more work. I propose that "x bi'i y" generates/describes the n-cell/n-orthotope which has opposite vertices at points x and y. This is <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{N} \cap [1, n]), (x_i</math> ''R'' <math> \alpha_i </math> ''R'' <math> y_i)) \}</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* "bi'i" requires a little more work. I propose that "x bi'i y" generates/describes the n-cell/n-orthotope which has opposite vertices at points x and y. This is <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{N} \cap [1, n]), (x_i</math> ''R'' <math> \alpha_i </math> ''R'' <math> y_i)) \}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** This is similar to the rectangle made on a computer by clicking the mouse at one endpoint and holding-with-dragging the cursor to the other endpoint. Note that there are as many ways to generate the same 'rectangle' as there are vertices on/of the 'rectangle' (where this jumber scales with dimensionality of the 'recrangle'). It need not be two-dimensional, though.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** This is similar to the rectangle made on a computer by clicking the mouse at one endpoint and holding-with-dragging the cursor to the other endpoint. Note that there are as many ways to generate the same 'rectangle' as there are vertices on/of the 'rectangle' (where this jumber scales with dimensionality of the 'recrangle'). It need not be two-dimensional, though.</div></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124703&oldid=prevKrtisfranks: Spelling changes, corrections, and additional info.2022-09-24T02:05:01Z<p>Spelling changes, corrections, and additional info.</p>
<a href="https://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=124703&oldid=123222">Show changes</a>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=123222&oldid=prevKrtisfranks: /* Proposed Extension A: "mi'i" */ -- typos2019-05-23T17:21:13Z<p><span dir="auto"><span class="autocomment">Proposed Extension A: "mi'i": </span> -- typos</span></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" can be extended further. Keep the previous definitions and conditions. Now, undefine r. Let <math>r_1, r_2, ..., r_n \geq 0</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" can be extended further. Keep the previous definitions and conditions. Now, undefine r. Let <math>r_1, r_2, ..., r_n \geq 0</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Then we can express a new formal tuple <math>r = (r_1, ..., r_n)</math> where the order of the entries correspond to similarly labelled <del style="font-weight: bold; text-decoration: none;">coordinates </del>of points in X with respect to the basis established. Note that r does not live in X; it is just a formal n-tuple which has entries ordered in a corresponding manner - in other words, it <del style="font-weight: bold; text-decoration: none;">isnjust </del>a list of numbers (scalars in <del style="font-weight: bold; text-decoration: none;">thebunderlying </del>field, more specifically) with the order of presentation fixed by the basis of X and according to the utterer's intention. Notice that r does not technically change if the basis is changed; in such a situation, it may not be possible to describe the n-dimensional interval in simple terms (using only linear combinations of the entries of the new basis) at all and, in any case, the utterer would generally need to supply an entirely different list <math> r\prime </math> in order to convey the same thought.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Then we can express a new formal tuple <math>r = (r_1, ..., r_n)</math> where the order of the entries correspond to similarly labelled <ins style="font-weight: bold; text-decoration: none;">coördinates </ins>of points in X with respect to the basis established. Note that r does not live in X; it is just a formal n-tuple which has entries ordered in a corresponding manner - in other words, it <ins style="font-weight: bold; text-decoration: none;">is just </ins>a list of numbers (scalars in <ins style="font-weight: bold; text-decoration: none;">the underlying </ins>field, more specifically) with the order of presentation fixed by the basis of X and according to the utterer's intention. Notice that r does not technically change if the basis is changed; in such a situation, it may not be possible to describe the n-dimensional interval in simple terms (using only linear combinations of the entries of the new basis) at all and, in any case, the utterer would generally need to supply an entirely different list <math> r\prime </math> in order to convey the same thought.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Then we can define "x mi'i r" as <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{N} \cap [1, n]), (d_F (x_i, \alpha_i)</math> ''R'' <math>r_i)) \}</math>. Notice that 'd' is now actually '<math>d_F</math>', id est: the metric on the field F. Here, each coordinate of a point <math>\alpha</math> is being compared to the corresponding coordinate of point x; if they are within the specified distance of one another (given by the corresponding entry in the list r), then that coordinate works out; iff all of the coordinates of the point work out, then the point belongs to the interval so described.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Then we can define "x mi'i r" as <math>\{ \alpha = \alpha_1 e_1 +...+ \alpha_n e_n = (\alpha_1, ..., \alpha_n) \in X: ((\forall i \in \mathbb{N} \cap [1, n]), (d_F (x_i, \alpha_i)</math> ''R'' <math>r_i)) \}</math>. Notice that 'd' is now actually '<math>d_F</math>', id est: the metric on the field F. Here, each coordinate of a point <math>\alpha</math> is being compared to the corresponding coordinate of point x; if they are within the specified distance of one another (given by the corresponding entry in the list r), then that coordinate works out; iff all of the coordinates of the point work out, then the point belongs to the interval so described.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l95">Line 95:</td>
<td colspan="2" class="diff-lineno">Line 95:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This essentially returns us to the old situation wherein the interval is no longer an n-ball but an n-cell (matching "bi'i"). The side lengths vary (being <math>2 r_i</math> in length, for each side i). The lines which pass through their corresponding/respective midpoints and which are perpendicular to the corresponding hyperfaces will intersect at a single point, videlicet the first argument of "mi'i" constructs (the 'center'; more appropriately: circumcenter), which is the point from which the various perpendicular distances to the boundaries are each measured (being <math> r_i </math>, for the appropriate/corresponding i).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This essentially returns us to the old situation wherein the interval is no longer an n-ball but an n-cell (matching "bi'i"). The side lengths vary (being <math>2 r_i</math> in length, for each side i). The lines which pass through their corresponding/respective midpoints and which are perpendicular to the corresponding hyperfaces will intersect at a single point, videlicet the first argument of "mi'i" constructs (the 'center'; more appropriately: circumcenter), which is the point from which the various perpendicular distances to the boundaries are each measured (being <math> r_i </math>, for the appropriate/corresponding i).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This definition is good for computer science, graphing, and experimental science. It is almost never used in theoretical mathematics. (Literally never in the experience of lai krtisfranks, at any rate.)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This definition is good for computer science, graphing, and experimental science. It is almost never used in theoretical mathematics. (Literally never in the experience of lai <ins style="font-weight: bold; text-decoration: none;">.</ins>krtisfranks<ins style="font-weight: bold; text-decoration: none;">.</ins>, at any rate.)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This additional proposal requires no major update, change, or addition to the glossing/keywords associated with "mi'i" in dictionary definitions, although there would be an implicit understanding of increased generality. If desired, however, "orthotopic interval with given circumcenter" or similar would do nicely.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This additional proposal requires no major update, change, or addition to the glossing/keywords associated with "mi'i" in dictionary definitions, although there would be an implicit understanding of increased generality. If desired, however, "orthotopic interval with given circumcenter" or similar would do nicely.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Additionally, we could establish the convention-by-definition that: <math>((\exists \rho \geq 0: ((\forall i \in \mathbb{N} \cap [1, n]), (r_i = \rho))) \implies </math> "x mi'i r" = "x mi'i <math>\rho</math>" <math>)</math>; but we would need a way to ensure that the audience recognizes <math>\rho</math> as an n-tuple and not just a scalar. Otherwise, utilization of this convention would be indistinguishable from the previously-mentioned case/proposal wherein the second argument as a single number constitutes the radius of an n-ball.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Additionally, we could establish the convention-by-definition that: <math>((\exists \rho \geq 0: ((\forall i \in \mathbb{N} \cap [1, n]), (r_i = \rho))) \implies </math> "x mi'i r" = "x mi'i <math>\rho</math>" <math>)</math>; but we would need a way to ensure that the audience recognizes <math>\rho</math> as an n-tuple and not just a scalar. Otherwise, utilization of this convention would be indistinguishable from the previously-mentioned case/proposal wherein the second argument as a single number constitutes the radius of an n-ball.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>** This complication can be overcome by mentioning "ce'ei'oi" immediately after "<math>\rho</math>"; if this is done, then we are to understand that "<math>\rho</math>" represents - in short-hand form - a formal tuple of identical entries (each being <math>\rho</math>). The elements of this tuple must never be negative.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>** This complication can be overcome by mentioning "ce'ei'oi" immediately after "<math>\rho</math>" <ins style="font-weight: bold; text-decoration: none;">(list sense) in the "mi'i" construct</ins>; if this is done, then we are to understand that "<math>\rho</math>" represents - in short-hand form - a formal tuple of identical entries (each being <math>\rho</math> <ins style="font-weight: bold; text-decoration: none;">(in the scalar sense)</ins>). The elements of this tuple must never be negative.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*** If the utterer explicitly defines/declares <math>\rho</math> to be such a formal tuple, then "ce'ei'oi" is not necessary, although it is also not wrong (and may in fact be helpful).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*** If the utterer explicitly defines/declares <math>\rho</math> to be such a formal tuple, then "ce'ei'oi" is not necessary, although it is also not wrong (and may in fact be helpful <ins style="font-weight: bold; text-decoration: none;">and encouraged</ins>).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Handling Generalized Points ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Handling Generalized Points ==</div></td></tr>
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</table>Krtisfrankshttps://mw-live.lojban.org/index.php?title=Extended_Dimensionality_of_Interval_cmavo&diff=123221&oldid=prevKrtisfranks: /* Alternative #1: Line Segments Unless Specified Otherwise */ - minor corrections2019-05-23T15:57:22Z<p><span dir="auto"><span class="autocomment">Alternative #1: Line Segments Unless Specified Otherwise: </span> - minor corrections</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:57, 23 May 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l72">Line 72:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note: The endpoints (first and second arguments) of "bi'i" and "bi'o" will be points that are specified via multiple coördinates with respect to a basis. They are not merely scalars. They still must live in the same space (X) and thus must have the same number of coördinates. In this situation, the one-dimensional usage which is defined already outside of this whole proposal merely isomorphically maps scalars which are denoted by "x" and "y" to the their corresponding 1-dimensional point specifications "(x)" and "(y)" respectively. (Notice that, without an additional convention, these will never map to "(x,0,0,...)" and "(y,0,0,...)" respectively, despite the isomorphism that may be established. This is meant to avoid the abusive mixing of notation/spaces: there is no interval from (1,2) to 1, for example. We should always specify that the endpoints are higher-dimensional. This note about mapping 1 to (1) is meant solely for the purpose of making this extension back-compatible and natural.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note: The endpoints (first and second arguments) of "bi'i" and "bi'o" will be points that are specified via multiple coördinates with respect to a basis. They are not merely scalars. They still must live in the same space (X) and thus must have the same number of coördinates. In this situation, the one-dimensional usage which is defined already outside of this whole proposal merely isomorphically maps scalars which are denoted by "x" and "y" to the their corresponding 1-dimensional point specifications "(x)" and "(y)" respectively. (Notice that, without an additional convention, these will never map to "(x,0,0,...)" and "(y,0,0,...)" respectively, despite the isomorphism that may be established. This is meant to avoid the abusive mixing of notation/spaces: there is no interval from (1,2) to 1, for example. We should always specify that the endpoints are higher-dimensional. This note about mapping 1 to (1) is meant solely for the purpose of making this extension back-compatible and natural.)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This would make the default usage automatically compatible with generalized points (see below). Additionally, line segments are generally useful in geometry of any dimension.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This would make the default usage automatically compatible with generalized points (see below). Additionally, line segments are generally useful in geometry of any <ins style="font-weight: bold; text-decoration: none;">(nontrivial) </ins>dimension<ins style="font-weight: bold; text-decoration: none;">, so this functionality would be utile</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This also would allow both "bi'i" and "bi'o" to be defined in any decent space (as opposed to only have "bi'i" be defined, which is the case in the aforementioned subproposal).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This also would allow both "bi'i" and "bi'o" to be defined in any decent space (as opposed to only have "bi'i" be defined, which is the case in the aforementioned subproposal).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In this formulation, "x bi'o y" implies that x is in some sense a starting point of reference/of an imaginary journey and y is the corresponding termination point; both are 'endpoints'/terminals, so to speak. <del style="font-weight: bold; text-decoration: none;">The </del>set produced, however, is still exactly equivalent to that produced by "x bi'i y" and any coloring of the connotations is unmathematical (and, thus, should be avoided in the opinion of lai .krtisfranks.); the latter is generally preferred.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In this formulation, "x bi'o y" implies that x is in some sense a starting point of reference/of an imaginary journey and y is the corresponding termination point; both are 'endpoints'/terminals, so to speak. <ins style="font-weight: bold; text-decoration: none;">If the space does not 'loop around', then the </ins>set produced, however, is still exactly equivalent to that produced by "x bi'i y" and any coloring of the connotations is unmathematical (and, thus, should be avoided in the opinion of lai .krtisfranks.)<ins style="font-weight: bold; text-decoration: none;">,</ins>; the latter is generally preferred.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note: There is still no established directionality on the linear interval that is produced by "bi'o". However, as long as it does not conflict with any others, we might be able to assume an established order thereupon. "x bi'o y" does mean that "x < y" (along that line).</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Note: There is still no established directionality on the linear interval that is produced by "bi'o". However, as long as it does not conflict with any others, we might be able to assume an established order thereupon. "x bi'o y" does mean that "x < y" (along that line).</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In this case, we use "ce'ei'oi" (followed by a number larger than 1 if we are being explicit) on either or both points x, y in the constructs "x bi'i y" and "x bi'o y" in order to produce the swept-out higher-dimensional-orthotopal "interval" that was proposed originally.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In this case, we use "ce'ei'oi" (followed by a number larger than 1 if we are being explicit) on either or both points x, y in the constructs "x bi'i y" and "x bi'o y" in order to produce the swept-out higher-dimensional-orthotopal "interval" that was proposed originally<ins style="font-weight: bold; text-decoration: none;">. See the 'Handling Generalized Points' section, following, for more details</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" will still generalize to a higher-dimensional-ball in the space. (Its functionality, as described previously, and as extended immediately after this section, is unchanged.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>"mi'i" will still generalize to a higher-dimensional-ball in the space. (Its functionality, as described previously, and as extended immediately after this section, is unchanged.)</div></td></tr>
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</table>Krtisfranks