While working on a publication, with an imminent deadline, on a topic I am supposed to be a bit of an expert on, it occurred to me I should confirm the meaning of the word "dimension".

Thank goodness for wikipedia! Simply because I enjoy learning, I used wikipedia since I general find this encyclopedia more educational than a dictionary. It saved me from the embarrassment of using some of my old notes that glibly referred to just 3 physical dimensions and assert that Time should be thought of as the 4th dimension.

Now all I have to do is re-write my drafts, which I hope will be a lot less work that writing responses to the inevitable challenges that would result from being (only a bit, I hope) "out of date".

Amplify’d from en.wikipedia.org

Dimension

From Wikipedia, the free encyclopedia

In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it (for example, to locate a point on the surface of a sphere you need both its latitude and its longitude). The inside of a cube, a cylinder or a sphere is three-dimensional because three co-ordinates are needed to locate a point within these spaces.

[edit] Dimension of a vector space

The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.

[edit] Time

A temporal dimension is a dimension of time. Time is often referred to as the "fourth dimension" for this reason, but that is not to imply that it is a spatial dimension. A temporal dimension is one way to measure physical change. It is perceived differently from the three spatial dimensions in that there is only one of it, and that we cannot move freely in time but subjectively move in one direction.

The equations used in physics to model reality do not treat time in the same way that humans commonly perceive it. The equations of classical mechanics are symmetric with respect to time, and equations of quantum mechanics are typically symmetric if both time and other quantities (such as charge and parity) are reversed. In these models, the perception of time flowing in one direction is an artifact of the laws of thermodynamics (we perceive time as flowing in the direction of increasing entropy).

The best-known treatment of time as a dimension is Poincaré and Einstein's special relativity (and extended to general relativity), which treats perceived space and time as components of a four-dimensional manifold, known as spacetime, and in the special, flat case as Minkowski space. Read more at en.wikipedia.org

I've been collaborating in a geographic mapping project once again and, this time, we've managed to abstract several insights that are likely applicable in part several other contexts I participate in.

For example, the Wikipedia definition of "a hierarchical spatial data structure which subdivides space into buckets of grid shape" can be generalized to apply to other "prime" dimensions, such as Time. A simple example emerges from considering This- ..., Year, Quarter, Month, Fortnight, Week, Day, ... in which arbitrarily named hierarchical intervals are used to subdivide the Time dimension.

Since this is a relatively large Topic, I'll keep this initial posting short and come back to it in the future with additional Comments. Since I'm unsure as to when I might be able to expand these thoughts, I've added a "hashH" tag for anyone that cares enough to pull "alerts". Of course, "interested parties" can also simply add a Comment and request "notification" to monitor this Topic, if they wish.

Amplify’d from en.wikipedia.org

Geohash

From Wikipedia, the free encyclopedia

This article refers to a system for encoding geographic coordinates. Also refers to a location in the sport of geohashing.

Geohash is a latitude/longitude geocode system invented by Gustavo Niemeyer when writing the web service at geohash.org, and put into the public domain. It is a hierarchical spatial data structure which subdivides space into buckets of grid shape.

Geohashes offer properties like arbitrary precision and the possibility of gradually removing characters from the end of the code to reduce its size (and gradually lose precision).

As a consequence of the gradual precision degradation, nearby places will often (but not always) present similar prefixes. On the other side, the longer a shared prefix is, the closer the two places are. Read more at en.wikipedia.org

This information about APL (A Programming Language), a.k.a "Array Programming Language", is useful background for my current work on "theArray".

"theArray itself, is my presumptuous, all encompassing name for a current effort to make use of the 1,100,000+ characters that are possible in the UTF-16 "alphabet", in indexing an "N dimensional" array, that can also be dynamically rotated such that any subset of its cells can be rendered as a two dimensional matrix within the context of a modern browser's AJAX uses.

This is the paragraph (break) or speech (pause) where some of my current associates are frequently heard to advise the un-initiated members of my audience to "Flee! Flee for your very sanity." But I digress...

By way of explanation for this post...

What I am hoping to achieve, using Amplify, is to capture the isolated points that may appeal to a few very specific audiences, tag them of subsequent retrieval and contextual association, and ultimately weave them into a comprehensive, meaningful thesis.

Amplify’d from www.chilton.com

APL On The Internet

Keywords A fundamental problem with discussing APL on the Internet is APL symbols, which can't be included directly in e-mail, news postings, or web pages. One common solution is to represent symbols with keywords such as {rho} and {iota}. In small quantities, the translation between symbols and keywords can be done by hand. For larger jobs, the APLASCII workspace can be used to automate the process. About {keywords}: APL-ASCII Transliteration   A quick overview of the process, including links to the APLASCII workspace (which is available for most modern APL systems), and two technical papers that describe the process in detail. Transferring Workspaces   Moving programs and data between different APL systems can be quite a chore. This describes an easy way to get the job done using the APLASCII workspace. Read more at www.chilton.com

Some worthy points regarding the need(s) for improvements to Global Positioning Systems.

Amplify’d from mikepearsonnz.wordpress.com

Mike Pearson

Helping people work better with technology

Posted by: mikepearsonnz | August 24, 2010 19 billion reasons why we need a better GPS

Should a Location Standard be a public good?

$19 billion of buried treasure

Everyone has trouble accurately locating those underground hidden assets. There is no overall view of the shared underground space.  There is uncertainty as to what is under the ground, and where exactly it is, for several reasons:

Broadband is going to change the financial impact of disruption. As we bury more broadband, the value of our underground assets is no longer the commodity price of water or electricity.  Rather it is the economic value of the information flowing through the underground fiber.  The financial impact for anyone interrupting that service will run into millions of dollars.  Unless we invest in better location information, there will be more hand shoveling in the future. Read more at mikepearsonnz.wordpress.com

These few paragraphs have been excerpted from a summary of some R&D undertaken by the AdManCa.PIL (... Product Initiatives Laboratory) in 2006. The illustrate that...

* Canada's "ANANAN" postal codes can be thought of as a tag for a "block face" geoCode in most urban areas.

* These Tags can be used as an index into an EnDimensionalArray that consists of all of the data accessible via the internet (a.k.a. TheArray ; theArray ; ...)

* Tuples can be used to "nest" references to any of the Cells of TheArray created by its intersecting Vectors vectors.

* Numeral systems can be used to automate the calculation of Tags that correspond to any specific Cell of TheArray (and that this can have a preferred, personal, myTagName "alias" that can easily be equated to the calculated positional Tag).

Caution:
The referenced publication contains a trivial mathematical error that will be corrected in the next Version, when it is published.

Amplify’d from janus.datafix.com

Data Tags

Storage and subsequent retrieval of data on the Internet can be particularly difficult. Stored data is increasingly impossible to navigate by traditional means. Witness the success of search engines like Google with their simple syntax for searching for keywords on the World Wide Web.

We can use these "Tag" values as efficient aliases and indices.

·        A tag can be thought of as the name of a cell in the two dimensional Array (e.g. ‘B3’ for column 2, row 3).

·        This can be easily extended to a three dimensional array by considering the Excel’ worksheet as a third dimension of ‘pages’ or ... ‘Sheets’.

·        To those with a sufficient background in mathematics, this is obviously an example of a ‘bounded’ (preferable terminology to ‘infinite’) EnDimensionalArray, which extends the concepts of...

o       An (Entity; Property) or (Name; Value) tuple which is a two element Vector, that can be one axis of a Matrix (‘table’ or ‘Spread sheet’) resulting from rendering two (orthogonal) vectors as a Matrix.

Again, is it a mathematical fact that this is an index into a bounded array of cells with a maximum number of:

Z9Z 9Z9

This clearly establishes the fact that a tag value can be an index into an EnDmensionalArray.

In simple practical terms, the analogy to a spreadsheet discussed during the second review meeting may illustrate this more clearly since it is quite common to use terminology like ‘B3’ to identify the cell at the intersection of the second column and the third row. In this example, the worksheet can be thought of as an extension to a third dimension of an array, which might extend its comparable Tag value to (S4) B3.

Proof of the validity of this ‘theoretical’ but fundamentally important realization was the motivation for much of the research in to the frequency, scope, and acceptability of tags in a RepresentativeSample(s) of very large GroupFormingNetworks of electronic ‘communities’ such as flickr, DiiGO and ‘DelIcioUs’ documented inRead more at janus.datafix.com

APL and its adherents have "reshaped" my thinking since I my introduction in 1972-05 at IPSA. Now, I find myself adapting many of its symbols and methods in the AdManCa.PIL (Product Initiatives Laboratory) work on theArray.

Amplify’d from en.wikipedia.org

APL (programming language)

APL (named after the book A Programming Language)[5] is an interactive array-oriented language and integrated development environment which is available from a number of commercial and non-commercial vendors[6] and for most computer platforms.[7] It is based on a mathematical notation developed by Kenneth E. Iverson.

APL has a combination of unique and relatively uncommon features that appeal to programmers and make it a productive programming language:[8]

It is concise, using symbols rather than words and applying functions to entire arrays without using explicit loops.

It is solution focused, emphasizing the expression of algorithms independently of machine architecture or operating system.

It has just one simple, consistent, and recursive precedence rule: the right argument of a function is the result of the entire expression to its right.

It facilitates problem solving at a high level of abstraction. Read more at en.wikipedia.org

An over-arching concept of my on-line collaborations is my view of "TheArray". Recently, my efforts to explain some of these concepts have...

* ...referenced some of the supporting mathematical explanations at Wikipedia.

* ... involved the use of the GreekAlphabet's letters as Tags that name "dimensions" (of TheArray).

This post should be helpful to several of collaborators since...

* ... It illustrates the use of "Delta" as a symbol for "change", for the benefit of the math folks who are working on the problems of computing a unique address for each and every internet destination, using a UTF-16 symbol set.

* ... It cautions the math folks (who understand "dot product" notations) and the javascript programmers (who understand the "." operator and OOPs nameConventions that they need to be aware of the risks of not distinguishing the different contexts within which they use these symbols.)

And it reminds me to make more use of wikipedia as the wonderful resource that it is.

Finally, this post is starting to make use of some of the Tags that I append to my 5,000+ DelIcioUs bookmarks that I use to index the on-line material of the various Groups I participate in.

Amplify’d from en.wikipedia.org

Work (physics)

If the force and the displacement are parallel and in the same direction, the mechanical work is positive. If the force and the displacement are parallel but in opposite directions (i.e. antiparallel), the mechanical work is negative.

However, if the force and the displacement act perpendicular to each other, zero work is done by the force:[4]

Read more at en.wikipedia.org

Occasionally, I find I need to go back to basics and review the foundations on which I'm trying to build something. Currently, the work on to build a PDA (Personal Digital Assistant) is predicated on using an Inference Engine to assess "theArray" of all content that can be accessed via the internet.

I sure hope this works, 'cause I'm falling further and further behind in my reading. :-)

Amplify’d from en.wikipedia.org

Associative array

From Wikipedia, the free encyclopedia

An associative array (also associative container, map, mapping, dictionary, finite map, and in query-processing an index or index file) is an abstract data type composed of a collection of unique keys and a collection of values, where each key is associated with one value (or set of values). The operation of finding the value associated with a key is called a lookup or indexing, and this is the most important operation supported by an associative array. The relationship between a key and its value is sometimes called a mapping or binding. For example, if the value associated with the key "bob" is 7, we say that our array maps "bob" to 7. Associative arrays are very closely related to the mathematical concept of a function with a finite domain. As a consequence, a common and important use of associative arrays is in memoization. Read more at en.wikipedia.org