Extended Dimensionality of Interval cmavo: Difference between revisions

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* "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".  I propose that "x mi'i r" is defined 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>.
* "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".  I propose that "x mi'i r" is defined 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>.
** I propose that we adopt additional keywords/glosses/terminology 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".
** I propose that we adopt additional keywords/glosses/terminology 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".
* "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>.
* "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>.
** Terminology can again be updated (id est: added to). The interval should be additionally described as a "n-cell" and "n-orthotope interval"; "rectilinear interval" may additionally be considered.  Both arguments (here denoted by "x" and "y" respectively) should be labelled as "endpoints".  Symmetry between them should be noted in dictionary definitions.
** Terminology can again be updated (id est: added to). The interval should be additionally described as a "n-cell" and "n-orthotope interval"; "rectilinear interval" may additionally be considered.  Both arguments (here denoted by "x" and "y" respectively) should be labelled as "endpoints".  Symmetry between them should be noted in dictionary definitions.
** This extended form of "bi'i" can be obtained via Cartesian products of linear intervals.  We will exploit this fact in the discussion about the endpoint stati (see the section named accordingly).
** This extended form of "bi'i" can be obtained via Cartesian products of linear intervals.  We will exploit this fact in the discussion about the endpoint stati (see the section named accordingly).
* "bi'o" has, to me (lai krtisfranks), no obvious extension since (for example) <math>\mathbb{R}^2</math> cannot be ordered.
* "bi'o" has, to me (lai krtisfranks), no obvious extension since (for example) <math>\mathbb{R}^2</math> cannot be ordered.


_
When <math>F = \mathbb{R}</math>, 1-tuples/1-dimensional endpoints will be isomorphically mapped automatically to the corresponding real numbers. This allows for ease of use and back-compatibility.
When <math>F = \mathbb{R}</math>, 1-tuples/1-dimensional endpoints will be isomorphically mapped automatically to the corresponding real numbers. This allows for ease of use and back-compatibility.
* This is done by establishing the identity/correspondence that <math>\forall (\xi) \in F, (\xi) \leftrightarrow \xi </math>. In other words, the functionality of intervals as defined outside of this whole proposal (which is for the 1-dimensional case) is extended so that endpoints "x" and "y" which are scalars are automatically mapped to (x) and (y) respectively, where the latter are now handled via the extended functionality herein proposed (as a point in (albeit one-dimensional) space and which has coordinates  (well, exactly 1 coördinate)).
* This is done by establishing the identity/correspondence that <math>\forall (\xi) \in F, (\xi) \leftrightarrow \xi </math>. In other words, the functionality of intervals as defined outside of this whole proposal (which is for the 1-dimensional case) is extended so that endpoints "x" and "y" which are scalars are automatically mapped to (x) and (y) respectively, where the latter are now handled via the extended functionality herein proposed (as a point in (albeit one-dimensional) space and which has coordinates  (well, exactly 1 coördinate)).

Revision as of 06:27, 5 March 2016

Presently, the cmavo "mi'i", "bi'i", "bi'o" (which currently constitute all elements of selma'o BIhI) all represent/create one-dimensional intervals. However, in mathematics and even in daily life, there are many instances when higher-dimensional intervals are desired. This functionality should be supported.

Rather than creating new cmavo for this task, the current cmavo (aforementioned) can simply be extended. The proposal described here will have the objective of supporting functionality for description of higher-dimensional intervals via extension only; only mathematical points are being discussed. The result should be back-compatible.

Contributors

The following people have contributed to the writing of this article (or have provided ideas for it):

  • lai .krtisfranks.

Where first-personal pronouns are used, they refer to at least one of these individuals. However, they will be avoided whensoever possible (with explicit mention of the author's name when a personal opinion or insight is conveyed).


Current Functionality

The cmavo of BIhI are nonlogical interval connectives. In mathematics (other options are available), one inputs a real number or possibly a generic endpoint, follows it by a cmavo of BIhI, and then mentions another real number or endpoint. The result is a description of a set of all points belonging to the interval so described. More explicitly:

Let x and y be real numbers or points in geometric space, let r be a nonnegative real number.

  • 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 that is less than (or possibly equal to) r.
  • x bi'i y: generates the interval or 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). "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.
  • 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". Thus "y bi'o x" is backward relative to "x bi'o y".

It should be noted that, unless is defined (in a way which lai .krtisfranks. has not cared to figure out) and true in a given case, , "x bi'o r" refers to/forms a bounded interval or the whole space; no semi-infinite (that is: bounded from above/positive xor below/negative) can be produced. These proposals (where '' no refers to any point which has a distance of from the origin) will (or, at least, thus far have) not changed this fact. In both the current (non-proposed) functionality and the herein proposed functionality, the other two intervals (namely, those which are produced by "bi'i" and "bi'o") can be bounded, semi-infinite (in any number of directions (so long as that number is a nonnegative integer less than or equal to the dimensionality of the space)), or infinite in both positive and negative directions (doubly-infinite/infinite; in any number of directions (so long as that number is a nonnegative integer less than or equal to the dimensionality of the space)).

Proposed Functionality

Any commentary in this article is meant to be taken as part of the "whole proposal". What follows are specific details which, broken into labelled sections for the sake of reference.

Main Proposal #1

Fix a space X which is endowed with a metric d and defined over an ordered field F which is also endowed with a compatible metric ; fix a basis B thereof. Let r be a nonnegative real number. Let x and y live in the same space X. Define the dimensionality of X to be dim(X) = n, where n is any nonnegative integer or (for simplicity: countable) infinity. Define B = {}. Then there exists . Let "R" denote an 'ordering' relation on F or the ordered field of real numbers (as appropriate) which may be either the "less than" relation (denoted "<") or, as appropriate (determined by GAhO; generically, elliptical), the "less than or equal to" relation (denoted "").

We let the dimensionality of our space (which is and can be inferred from the dimensionality of x and/or y, which should match) determine the nature of our intervals.

  • "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". I propose that "x mi'i r" is defined 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 R .
    • I propose that we adopt additional keywords/glosses/terminology 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".
  • "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 R R .
    • Terminology can again be updated (id est: added to). The interval should be additionally described as a "n-cell" and "n-orthotope interval"; "rectilinear interval" may additionally be considered. Both arguments (here denoted by "x" and "y" respectively) should be labelled as "endpoints". Symmetry between them should be noted in dictionary definitions.
    • This extended form of "bi'i" can be obtained via Cartesian products of linear intervals. We will exploit this fact in the discussion about the endpoint stati (see the section named accordingly).
  • "bi'o" has, to me (lai krtisfranks), no obvious extension since (for example) cannot be ordered.

_ When , 1-tuples/1-dimensional endpoints will be isomorphically mapped automatically to the corresponding real numbers. This allows for ease of use and back-compatibility.

  • This is done by establishing the identity/correspondence that . In other words, the functionality of intervals as defined outside of this whole proposal (which is for the 1-dimensional case) is extended so that endpoints "x" and "y" which are scalars are automatically mapped to (x) and (y) respectively, where the latter are now handled via the extended functionality herein proposed (as a point in (albeit one-dimensional) space and which has coordinates (well, exactly 1 coördinate)).

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.

  • These distinct definitions are good (utile) and natural in theoretical mathematics.

Alternative #1: Line Segments Unless Specified Otherwise

An alternative (which lai .krtisfranks. finds perhaps even better than the previous proposal) is to have "bi'i" and "bi'o" always default to referencing line segments (generally: geodesics) in any space.

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.)

This would make the default usage automatically compatible with generalized points (see below). Additionally, line segments are generally useful in geometry of any dimension.

This also would allow both to be defined in any decent space (also opposed to only have "bi'i" be defined).

In this formulation, "x bi'o y" implies that x is in some sense a starting point of reference/imaginary journey and y is the corresponding termination point; both are 'endpoints'/terminals, so to speak. 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.

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).

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'of y" in order to produce the swept-out higher-dimensional-orthotopal "interval" that was proposed originally.

"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.)

Further Extension A: "mi'i"

"mi'i" can be extended further. keep the previous definitions and conditions; undefine r. Let .

Then we can express a new formal tuple where the order of the entries correspond to similarly labelled coordinates of points in X with respect to the basis established. Note that r does not live in X; it is just an n-tuple which has entries ordered in a corresponding manner.

  • Then we can define "x mi'i r" as R . Notice that 'd' is now the metric on the field F; each coordinate of a point is being compared to the corresponding point of x; if they are within the specified distance of one another, then that coordinate works out; if all of the coordinates of the point work out, then the point belongs to the interval.
    • 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 stati (see the section named accordingly).

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 for side i) and the lines passing through their corresponding/respective midpoints will intersect at a single point, videlicet the first argument (the 'center') which is the point from which the boundaries are 'measured'.

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.)

  • Additionally, we could establish the convention-by-definition that: "x mi'i r" = "x mi'i " ; but we would need a way to ensure that the audience recognizes 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 ""; if this is done, then we are to understand that "" represents - in short-hand form - a formal tuple of identical entries (each being ). The elements of this tuple must never be negative.
      • If the utterer explicitly defines/declares 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).

Handling Generalized Points

If the input (x or y) are generalized points (such as towns), then they likely live in at least two-dimensional space, as is the case on Earth. However, they are being treated as distinct points (assigned real numbers isomorphically). According to this proposal, it is not possible for the interpretation of an interval with these arguments to be anything except one-dimensional. But maybe we want to discuss the 'rectangular' (more on this immediately later/below) area of the globe between Olympia, Washington, U.S.A. (defined as x) and Tallahassee, Florida, U.S.A. (defined as y). (This rectangle covers much of the continental/contiguous U.S.) "x bi'i y" would, presumably, give the quickest route for a roadtrip from Olympia to Tallahassee (the geodesic line segment (more on this immediately later/below)) or reverse; it would not yield the 'rectangle' that we want. The word "ce'ei'oi" fixes this issue. If it is used on a generalized point (rather than a formal tuple - see immediately previous/above ("mi'i" discussion)), then it indicates that that endpoint (and, consequently, all others) is to be treated multi-dimensionally (unless the argument of "ce'ei'oi" is identically and exactly equal to 1). It is not necessary if the points are already defined to belong to a well-described space (of known dimensionality) or are decomposed in terms of their basis/represented as a tuple. If we do this, with the former definitions, then "x ce'ei'oi bi'i y" will suddenly indicate not the line(ar interval) between x and y, but instead the 'area between them'. (This area may be visualized thusly: Imagine a map with x and y on it, on a computer. Click on one of these points, drag the cursor to the other. In many programs, a (possibly degenerate) rectangle is swept out, usually with a dashed or dotted outline. The opposite corners of this rectangle are x and y; the remaining corners are given by one coordinate of x and the other coordinate of y, as appropriate. The space highlighted (within this rectangle) is the interval formed.)

The space in which this interval exists is determined by context. This is a problem even in the unextended (one-dimensional) version of these words in this usage. Is the unordered interval from Paris to London along the Earth's surface (geodesic) or is it a straight line through space (intersecting the Earth's surface at some point)? Is the (un)ordered interval, if geodesic, following the shorter segment of the great-circle connecting the two cities, or the longer one?

Note that in any case x and y need to live in the same space. So, if one is a generalized point, then the other must be. It makes no sense to discuss the interval from 1 to you or from Olympia to my imaginary friend. Moreover, they must have the same dimensionality; placing "ce'ei'oi" on one of them determines the nature of the other (and so is unnecessary); however, the argument of "ce'ei'oi" does need to be compatible with both x and y if present (for "bi'i"; this is not the case for its being used upon the first argument of "mi'i", but is the case in its being used on the second argument of "mi'i").

Endpoint Stati (Inclusion/Exclusion)

(To be addressed shortly)

Vocabulary/Semantics that have been Introduced

Mostly, old vocabulary has been expanded in functionality.


New vocabulary:

  • ce'ei'oi
  • vau'e'oi
  • vau'o'oi

Miscellany

"mi'i" is really good for error bars in the sciences. In English, scientists often say stuff like "g is 9.85 plus-or-minus .05 meters per second per second". This is abusive. The only options for the value of g in such a case would be 9.80 m/s2 or 9.90 m/s2. What they mean is that the value of g is between these two values (possibly including either of them). Following the format of the example quote, we have "mi'i" being the intention, with 9.85 functioning as the center and .05 as the radius. In describing a data set, one should make sure to say that the variable (usually dependent) belongs to this set, rather than it being this set. This variable will be the one with error bars in the graph. If a single variable is described in such a manner, the error bars graphically are parallel to only one axis: the actual value (as measured, within standard deviation/error) can shift in this direction (so long as it remains within the bars) but cannot shift in any other. If the error bars are applied to the point, rather than the variable (which acts as a coordinate of the former), though, then the error bar will (under this proposal) envelop the point in a ball of the given radius; the actual value (as measured, within standard deviation/error) could thus shift in any direction within n-space so long as it remains within the provided radius of the given (measured, central) value. Adopting the further extension, each coordinate can be individually and independently assigned/associated with an error; the error bars will graphically be parallel to each axis (or will be 0); the actual value (as measured, within standard deviation/error for each measurement/variable/coordinate) can shift relative to the data point along each axis so long as it stays within the axis-appropriate radius of the data point.