# OBSERVATION

People tie themselves in knots over unstructured time.
keywords: life

# the meaning of life

Life is all about the fact that this world is not our real home, and one’s realization of, and response to, that fact. Your response to that fact is called your legacy. Death consists of moving from the current stage of development of the world to the next stage, and is the complete merging of the concrete and the abstract. The stages of development are determined by the fact that Nature requires us to conduct an increasing percentage of our activities in cyberspace. When the world reaches the next stage, you are reincarnated. Religion is the highly imperfect expression of the realization that this world is not our real home.
“We are not human beings having a spiritual experience. We are spiritual beings having a human experience.”
— Teilhard de Chardin
“He who possess art and science has religion; he who does not possess them, needs religion.”
— Goethe
“Religion is for people who are afraid to go to hell. Spirituality is for those who have already been there.”
— Vine Deloria Jr.
keywords: meaning of life

# RECIPROCITY

The exploits of secret agents are well known.

# Group-theoretic unification of Kaprekar’s constant 6174 and Trigg’s constant 2538

We can use group-theoretic inversion as a mnemonic, or sleight-of-hand, to give Kaprekar’s constant 6174 and Trigg’s constant 2538 a common source. The technique is analogous to what happens in Umbral Calculus. In Umbral Calculus, indices are treated as exponents. Our approach will the the reverse: we will treat exponents as indices. The one and only exponent in question is -1, that is, the multiplicative-inverse operator, but instantiated several times. The key formula that we will use is the well-known fact that the inverse of a product is equal to the reverse product of the inverses, that is, that (ab)^(-1) = (b^(-1))(a^(-1)). So, dropping the exponent, we can say that we have transformed ‘ab’ into ‘ba’. With this mnemonic, or sleight-of-hand, in mind, let us now consider the following two expressions: ((ab)^(-1))(cd)^(-1)) and ((ab)(cd))^(-1). The first expression can be verbalized as ‘the product of the inverses of two products’, and the second one can be verbalized as ‘the inverse of the product of two products’.
To get the iterate for the Kaprekar process, we drop the exponent in the first expression and apply the exponent twice in the second expression, and then drop the exponent. The two resulting expressions are, respectively, abcd and dcba, with the expression
abcd – dcba being the iterate for the Kaprekar process, with the understanding that ‘abcd’ represents the largest number that can be made from the digits of the original number in question.
To get the iterate for the Trigg process, we apply the exponent one time to each side, getting (b^(-1))(a^(-1))(d^(-1))(c^(-1)) and ((cd)^(-1))(ab^(-1)), and then drop the exponent, getting the two expressions badc and cdab, badc – cdab being the iterate for the Trigg process.
So, the Kaprekar process and the Trigg process can be commonly-sourced from Group Theory.
Here is a place in the literature where the Trigg process is described.
keywords: Recreational Mathematics, Number Theory, unification

# lorem ipsum

If everyone would learn Esperanto, it would be an immense economic stimulus and permanent boost for the world economy, when companies can be formed with total liquidity, no longer stopped by the language barrier, or bigoted viewpoints. Esperanto is the original rising tide that lifts all boats. As successful investors will tell you, diversification is king. But nothing supports diversity / diversification like Esperanto does. Esperanto’s star is on the rise.
keywords: prosperity for all, world peace, collaboration, cooperation, teamwork