Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

f:I->R; I is an interval on the real line; f is differentiable
(1) f is convex
(2) f' is increasing
Prove that (1) => (2)
Hint:
x<c<y from I, (f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c)
f:I->R; I is an interval on the real line; f is differentiable
(1) f is convex
(2) f' is increasing
Prove that (2) => (1)
Use Lagrange's mean-value theorem.
page 119 in palace
f, g, F : I -> R; I is an interval
F = max{f,g}
f, g are convex, does F have to be convex?
Yes.
f, g, F : I -> R; I is an interval
F = min{f,g}
f, g are convex, does F have to be convex?
No.
f(x)=7
g(x)=x
Prove Jensen's inequality for a convex function.
page 122 in palace
Prove that
(a1 * a2 * ... * an)^(1/n) <= (a1 + a2 + ... + an)/n
Use Jensen's inequality for the minus logarithmic function, which is convex.
Prove that
n / (1/a1 + 1/a2 + ... + 1/an) <= (a1 * a2 * ... * an)^(1/n)
Use: (a1 * a2 * ... * an)^(1/n) <= (a1 + a2 + ... + an)/n
(1) f:(0;oo)->|R
(2) f is convex
(3) f is strictly increasing
Prove that lim f(x) = infinity, as x approaches infinity.
Choose 0<a<b. Put k = (f(b) - f(a)) / (b-a). By (3) k>0.
Since f is convex conclude that
for all x > b, kx + (f(b)-kb) <= f(x).
And hence conclude the thesis.
f,g : I -> R; I is an interval on the real line;
f,g are convex; a,b are positive numbers
Prove that af+bg is convex.
page 124 in palace
(1) f : [0,oo) -> |R
(2) f is convex
(3) f(0) = 0
(4) g : (0,oo) -> |R , g(x)=f(x)/x
What can we conclude about function g?
g is increasing
Hint: If f is convex, then for all x<c<y
(f(c) - f(x)) / (c-x) <= (f(y) - f(c)) / (y-c).
Use f(0)=0.
page 124 in the palace notebook