Smooth Transition Functions
The function
has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval . To have the smooth transition in the real interval with a < b, consider the function
For real numbers a < b < c < d, the smooth function
equals 1 on the closed interval and vanishes outside the open interval (a,d).
Read more about this topic: Non-analytic Smooth Function
Famous quotes containing the words smooth, transition and/or functions:
“Beautiful women seldom want to act. They are afraid of emotion and they do not try to extract anything from a character that they are portraying, because in expressing emotion they may encourage crow’s feet and laughing wrinkles. They avoid anything that will disturb their placidity of countenance, for placidity of countenance insures a smooth skin.”
—Laurette Taylor (1887–1946)
“The most remarkable aspect of the transition we are living through is not so much the passage from want to affluence as the passage from labor to leisure.... Leisure contains the future, it is the new horizon.... The prospect then is one of unremitting labor to bequeath to future generations a chance of founding a society of leisure that will overcome the demands and compulsions of productive labor so that time may be devoted to creative activities or simply to pleasure and happiness.”
—Henri Lefebvre (b. 1901)
“In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)