Equation 3 · Humanity Has Always Been a Human–Technology System

What does this equation mean?

\dot{x}_i = x_i(f_i-\bar f) + \sum_j M_{ji}x_j - \sum_j M_{ij}x_i,

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsx_i(f_i-bar f) + sum_j M_jix_j - sum_j M_ijx_i

Result or conditionẋ_i

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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\dot{x}_i

Symbol ẋ_i

ẋ_i has a dot, marking the rate of change of the underlying indexed quantity with respect to the article’s time variable.

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x_i

Symbol x_i

the frequency.

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f_i

Symbol f_i

context-dependent cultural fitness.

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\bar f

Symbol bar f

bar f is one of the signed contributions combined to compute the quantity on the left.

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j

Symbol j

j appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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M_{ji}

Symbol M_ji

$M_j$ i is one of the signed contributions combined to compute the quantity on the left.

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x_j

Symbol x_j

$x_j$ is one of the signed contributions combined to compute the quantity on the left.

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M_{ij}

Symbol M_ij

transformation or transmission from variant i to j.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

Add the term after the plus sign to the term or group before it.

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subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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j

Starting index or lower bound: j

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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j

Starting index or lower bound: j

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

A minimal cultural-selection model can represent the frequency $x_i$ of a variant i as $\dot{x}_i = x_i(f_i-\bar f) + \sum_j M_{ji}x_j - \sum_j M_{ij}x_i$ . where $f_i$ is context-dependent cultural fitness and $M_{ij}$ represents transformation or transmission from variant i to j . The equation is illustrative, not a claim that beliefs have fixed genes or that cultural change is reducible to one replicator law. Unlike alleles, cultural variants can be intentionally modified, horizontally transmitted, blended, externally stored, and adopted because of prestige, conformity, coercion, or institutional mandate.

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Sources cited in the article section

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