Ramp Function

A Ramp Function is an Unary Function defined as [math]\displaystyle{ R(x) := \max(x,0) }[/math]

• AKA: Max Function.
• Example(s):
  • [math]\displaystyle{ R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x\lt 0 \end{cases} }[/math]
  • [math]\displaystyle{ R(x) := \frac{x+|x|}{2} }[/math]
• Counter-Example(s)
  • Exponential Function,
  • Linear Threshold Function,
  • Boxcar Function.
• See: Unary Function, Piecewise Function, Real Function, Graph of a Function, Scaling And Shifting.

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References

2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Ramp_function Retrieved:2018-2-18.
  • The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for other functions obtained by scaling and shifting, and the function in this article is the unit ramp function (slope 1, starting at 0).

    This function has numerous applications in mathematics and engineering, and goes by various names, depending on the context.

    Definitions

    The ramp function (R(x) : ℝ → ℝ₀⁺) may be defined analytically in several ways. Possible definitions are:

    • A piecewise function:
      • [math]\displaystyle{ R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x\lt 0 \end{cases} }[/math]
    • The max function:
      • [math]\displaystyle{ R(x) := \max(x,0) }[/math]
    • The mean of an independent variable and its absolute value (a straight line with unity gradient and its modulus):
      • [math]\displaystyle{ R(x) := \frac{x+|x|}{2} }[/math]

this can be derived by noting the following definition of max(a,b),

[math]\displaystyle{ \max(a,b) = \frac{a+b+|a-b|}{2} }[/math]

for which a = x and b = 0

• The Heaviside step function multiplied by a straight line with unity gradient:

• [math]\displaystyle{ R\left( x \right) := xH(x) }[/math]
• The convolution of the Heaviside step function with itself:
• [math]\displaystyle{ R\left( x \right) := H(x) * H(x) }[/math]

• The integral of the Heaviside step function:

• [math]\displaystyle{ R(x) := \int_{-\infty}^{x} H(\xi)\,d\xi }[/math]

• Macaulay brackets:

• [math]\displaystyle{ R(x) := \langle x\rangle }[/math]

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