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This refers to ODE models for the parachute problem by Douglas B. Meade. On page four the author says:

When $k$ is piecewise constant, the snatch force has a jump discontinuity at the instant of deployment: $$[j(t_d)]:=j(t_d^+)-j(t_d^-)=-\frac{[k(t_d)a(t_d)]}{m}$$

I have two questions. What is the mathematical meaning of the brackets?

What is the mathematical meaning of the colon equals (i.e. :=)?

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The symbol $:=$ indicates that the left hand side is being defined to be equal to the right-hand side. The author has just made the definition that square brackets around something like $f(x)$ mean the jump of $f$ at $x$, that is $[f(x)]=f(x^+)-f(x^-)$.

And then this definition is immediately used: the expression $[k(t_d)a(t_d)]$ appearing on the right should be understood as $$k(t_d^+)a(t_d^+)-k(t_d^-)a(t_d^-)$$

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