# OBSERVATION (which one trumps: mls or gls)

In ordinary life mls (momentary life situation) trumps gls (general life situation). War is when gls trumps mls. Rebellious spirits want gls to trump mls, and war is God’s way of saying to them, as C.S. Lewis noted, “Okay, have it your way.”
keywords:
[priority]
[priorities]
[spirituality]
[Thy will be done.]

# QUIP (capitalism versus communism)

“The difference between capitalism and communism is that under capitalism the priority is personal growth-opportunities, whereas under communism it’s the reverse: the priority is personal-growth opportunities.”
keywords:
[politics]
[psychology]
[priorities]

# (generalized) backward difference operator

(suggested new terminology)
I suggest ‘f(x) – f(red(x)’ as the (generalized) backward difference operator, where ‘red’ is short for ‘reduction of’. (This is a ‘horizontal’ generalization, as opposed to a ‘vertical’ generalization as might be needed in heavy-duty numerical analysis.) Two common forms of reduction of x are -x and x – 1. Actual use of the backward difference operator might occur in a kind of spread-out stepwise manner that disguises the fact that the backward difference operator is at work. For example, at a certain juncture in the derivation of the computational formula for the logarithm that Tom Apostol provides in volume 1 of his 2-volume Calculus textbook, there is an instance of the form ‘red(x) = -x’. A more directly visible occurrence of the backward difference operator is in specifying the general term in the summation of the average speed over members of a partition of an interval of time: (p(t(k)) – p(t(k – 1))/(t(k) – t(k – 1)), where p is the position function, the denominator of this expression being a ‘red(x) = x – 1’ instance of the backward difference operator. Another occurrence of the backward difference operator where ‘red(x)’ has the form ‘x – 1’ is the backward difference operator f(n) – f(n – 1) defined by Michael Z. Spivey in his article ‘The Humble Sum of Remainders Function’. (The Farlex Dictionary offers a generalized version as f(x) – f(x – h) – “where h is a constant denoting the difference between successive points of interpolation or calculation” – others calling h the ‘step size’.)
Another occurrence of the backward difference operator is in defining the Kaprekar process. The definition of the Kaprekar process can be worded as follows: “For a given 4-digit positive integer n (expressed in base 10, leading 0’s allowed) containing at least 2 digits, let f(n) be the largest number expressible using the digits of n, and let f(-n) be the smallest number expressible using the digits of n, and let k(1) = f(n) – f(-n). Then apply the preceding to k(1) as the ‘new n’, obtaining k(2), and so on.” (This process reaches the number 6174 in at most 7 steps.) Another occurrence of the backward difference operator is the probability formula P(A) = 1 – P(A’), which can be written as P(A) = P(S) – P(red(S)), where red(S) = S – A. In a similar way, the expression 1 – 1/p, which can be written as i(p/p) – i(red(p/p), where red(p/p) = 1/p (i being the identity function) can be considered an occurrence of the the backward difference operator, which Euler put to good use in evaluating ζ(2). Another occurrence of the backward difference operator is the definition of the interquartile range (IQR), namely, Q(3) – Q(1), which can be written in the form Q(3) – Q(red(3)), where red(3) = 1.
Just as group theory can be use additive notation (typically, for abelian groups) or multiplicative notation, so also the ‘red(x) = -x’ version of the backward difference operator can be stated in multiplicative form as well: f(x)/(f(1/x).
keywords:
[Mathematics]
[gbdo]
[subtraction]
[subtracting]
[backward difference formula]
[f(x) – f(-x)]
[f(x) minus f(-x)]
[f(x) minus f(reduction of(x))]
[f(x) minus f(red(x))]

# (generalized) backward difference operator

(suggested new terminology)
I suggest ‘f(x) – f(red(x)’ as the (generalized) backward difference operator, where ‘red’ is short for ‘reduction of’. (This is a ‘horizontal’ generalization, as opposed to a ‘vertical’ generalization as might be needed in heavy-duty numerical analysis.) Two common forms of reduction of x are -x and x – 1. Actual use of the backward difference operator might occur in a kind of spread-out stepwise manner that disguises the fact that the backward difference operator is at work. For example, at a certain juncture in the derivation of the computational formula for the logarithm that Tom Apostol provides in volume 1 of his 2-volume Calculus textbook, there is an instance of the form ‘red(x) = -x’. A more directly visible occurrence of the backward difference operator is in specifying the general term in the summation of the average speed over members of a partition of an interval of time: (p(t(k)) – p(t(k – 1))/(t(k) – t(k – 1)), where p is the position function, the denominator of this expression being a ‘red(x) = x – 1’ instance of the backward difference operator. Another occurrence of the backward difference operator where ‘red(x)’ has the form ‘x – 1’ is the backward difference operator f(n) – f(n – 1) defined by Michael Z. Spivey in his article ‘The Humble Sum of Remainders Function’. (The Farlex Dictionary offers a generalized version as f(x) – f(x – h) – “where h is a constant denoting the difference between successive points of interpolation or calculation” – others calling h the ‘step size’.)
Another occurrence of the backward difference operator is in defining the Kaprekar process. The definition of the Kaprekar process can be worded as follows: “For a given 4-digit positive integer n (expressed in base 10, leading 0’s allowed) containing at least 2 digits, let f(n) be the largest number expressible using the digits of n, and let f(-n) be the smallest number expressible using the digits of n, and let k(1) = f(n) – f(-n). Then apply the preceding to k(1) as the ‘new n’, obtaining k(2), and so on.” (This process reaches the number 6174 in at most 7 steps.) Another occurrence of the backward difference operator is the probability formula P(A) = 1 – P(A’),
which can be written as P(A) = P(S) – P(red(S)), where red(S) = S – A. In a similar way, the expression 1 – 1/p, which can be written as i(p/p) – i(red(p/p), where red(p/p) = 1/p (i being the identity function) can be considered an occurrence of the the backward difference operator, which Euler put to good use in evaluating ζ(2). Another occurrence of the backward difference operator is the definition of the interquartile range (IQR), namely, Q(3) – Q(1), which can be written in the form Q(3) – Q(red(3)), where red(3) = 1.
Just as group theory can be use additive notation (typically, for abelian groups) or multiplicative notation, so also the ‘red(x) = -x’ version of the backward difference operator can be stated in multiplicative form as well: f(x)/(f(1/x).
keywords:
[Mathematics]
[gbdo]
[subtraction]
[subtracting]
[backward difference formula]
[f(x) – f(-x)]
[f(x) minus f(-x)]
[f(x) minus f(reduction of(x))]
[f(x) minus f(red(x))]

# La Velŝipo Magdalena

(kanzono en Esperanto)
Jen la ligilo.

# diagram of the limit of sin(x)/x as x goes to 0

keywords:
[Mathematics]
[Geometry]
[Trigonometry]
[sin(x) / x]
[sin x / x]
[limit is 1]
[limit is unity]
[fundamental inequality]
[inequalities]

# Reuleaux triangles

A Reuleaux triangle is a curve of constant width other than a circle.
Here is the link to the Wikipedia article on this topic.
keywords:
[Mathematics]
[Geometry]
[counter-intuitive results]
[counterintuitive results]
[arc of a circle]
[equilateral triangle]

Advertising is invitations to the hoi polloi.
keywords:
[politics]
[journalism]
[entertainment]

# OBSERVATION (teaching)

In an ‘I-don’t-care.’ environment, proactive explanations are useless.
keywords:
[education]
[training]
[educators]
[curriculum]
[lack of motivation]

# OBSERVATION (synchronization)

The amount of synchronization in your life is constant. It consists of a component of passive synchronization and a component of active synchronization. Passive synchronization is the result of lack of buffering. Active synchronization consists of buffering. No synchronization is more active than the act of gifting.
keywords:
[life]
[General Systems Theory]