Lorentz velocity addition

This post follows on the Gamma factor post here. The form of velocity addition based on the Lorentz transformation is related to a combination of additive and harmonic addition. Galilei velocity addition is Lorentz velocity addition is which equals In this way the Lorentz transformation attempts to combine arithmetic and harmonic addition.

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Ancient Greek means

The ancient Greeks defined ten means in terms of the following proportions (see here): Let a > m > b > 0. Then m represents (1) the arithmetic mean of a and b if (2) the geometric mean of a and b if (3) the harmonic mean of a and b if (4) the contraharmonic

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Introduction to logic

This post follows others about logic, such as here. Purpose The purpose of logic is to ensure that one’s discourse makes sense. The most important part of making sense is avoiding contradictions, which would both affirm and deny a proposition. Some propositions may be affirmed in part and denied in part; that is different. The

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N-ary distinctions

The ground of each distinction is an indistinct mass or state or condition, a kind of whole without parts or at least without parts that have been discerned. Every instance of the whole is at first, an instance of one mass or state or condition. A unary distinction is a discernment of something out of

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Moral and civil law

Everyone should understand the distinction between what is moral and what is not moral. I have written briefly about that here. What is legal is not necessarily moral. What is moral is not necessarily legal. What is the relation between the moral law and the civil law? That is something every society must decide for

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Algebra and calculus of ratios

Ratio Algebra Let us define an algebra of ratios. A ratio consists of two numeric expressions separated by a colon, and for clarity enclosed in parentheses, i.e., (a : b) with a, b ∈ ℝ. The expression on the left is the antecedent, and the expression on the right is the consequent. (0 : 0) is

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Complete Galilei Group

The following is based on Lévy-LeBlond’s Galilei Group and Galilean Invariance, §2 (Nuovo Cimento, Jan. 1973). Let Ω be the complete Newtonian space, the points (events) of which we label by their coordinates in some complete Galilean frame, using the notation y = (x(t), z(s)).     (1) The complete proper Galilei group G (or Galilei

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Gamma factor between means

Consider the mean between two quantities: The arithmetic mean is The harmonic mean is where which equals the gamma factor of the Lorentz transformation. The geometric mean is Then the factor γ2 transforms a harmonic mean into an arithmetic mean: The inverse γ factor transforms an arithmetic mean into a geometric mean: so that The

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