Jump to: navigation, search

Extended Dimensionality of Interval cmavo

77 bytes added, 3 months ago
Proposed Extension A: "mi'i": -- typos
"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>.
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 coordinates 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 isnjust is just a list of numbers (scalars in thebunderlying 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.
* 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.
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).
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.)
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.
* 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.
** 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 elements of this tuple must never be negative.*** 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 helpfuland encouraged).
== Handling Generalized Points ==

Navigation menu