sequence

=Sequence=

toc Read here about sequence, a term that is very useful to discussing and understanding relationships in scale studies, both intra-scale and inter-scale.

=Introduction= We borrow from naïve set theory to develop a set of names describing a super-scale survey. Imagine a line L calibrated with regularly spaced scales to be surveyed listed in stepped sequence A, B, C. Each step bears a signifier or label. Signifier A signifies a set of phenomena A associated with the scale at step A, likewise signifiers B, C and so on.

There is an operation that can be performed on set A: e.g. aggregation or disaggregation, construction or deconstruction, or motion along a path. The operation modifies set A to become set A’. Repeat the operation to generate A’’, then A’’’, until at some point the signifier A is no longer sufficient to signify the set of phenomena A’’’’, and so new signifier B is needed.

We thus have a sequence of A, A', A, A', B, B', etc. =Mathematical Sequence= In seeking a deeper understanding of the nature of this sequence, we turn to mathematics for a precise definition.

In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence is included in most notions of sequence, but may be excluded depending on the context.

Subsequence
A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.

Monotone
If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function. A monotone function is also called isotone, or order-preserving. The dual notion is often called antitone, anti-monotone, or order-reversing.

Contrast between Linear and Multiplicative Sequence
Exponential expands by repeatedly multiplying by a constant. Linear expands by repeatedly adding a constant. The linear vs. exponential dialectic has also been called arithmetic vs. geometric. Only exponential sequences have a similar proportion between successive members. That is, 1+1 is doubling, 1,000,000 + 1 is barely any change at all; in contrast 1*10 yields a new entity 10x the value of the previous, and 1,000,000 * 10 has the same effect.

=See Also= Embodied arithmetic, succession, set theory.

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