PROPOSED PROBLEM 2.

Problem 2.

Consider the sequence of real numbers (a_0, a_1, a_2, ...)
where a_m > 0 for every m e IN .

Let f : IR^+ -> IR^+ be a map and let f_m be the restriction
of f to each closed interval [m,m+1] ( m e IN ).

Suppose
f_m: [m,m+1] -> IR^+ , f_m ( x ) = a_m ( x - m ) + B_m
( B_0 =0 and B_m = a_0 + ... + a_(m-1) if m > 0 ).

1.- Prove that f(0) = 0, f continuous in IR^+, f e C^1( [m,m+1] )
with positive derivative in [m,m+1] for every m e IN and as
a consequence f : IR^+ -> f( IR^+) is bijective.

2.- Consider the family of functions

h_k : (0, +oo) -> (0, +oo), h_k (x) = k/x (k real number, k>0).

Prove that the g_k function which determines the graph of
(fxf) (graph h_k)) is g_k( u ) = f ( k/(f^-1(u))
(i.e. g_k = f o h_k o f^-1 )

3.- Prove that g_k is continuous and strictly decreasing.

For w e f ( IR^+) we say that w is f-natural iff there exists
n e IN* such that w = f(n).

4.- Consider (u, v) e f ( IR^+) x f ( IR^+) . Prove that if the
given sequence (a_0, a_1, a_2, ... ) is strictly increasing then:

(u, v) has f-natural coordinates iff in a neighbourhood V of ( u,v)
for every (s,t) belonging to V, g_(f^-1(s) f^-1(t)) is differentiable
at s iff s<>u and t<> v.

Note:

In PROPOSED PROBLEM 1 we called to any function satisfying
the conditions of section 1 in this problem, an " IR^ coding function "
because f transports the usual algebraic structure from IR^+ to
f ( IR^+). Now, it seems adequate to call any function satisfying
the conditions 4. of this problem an "IR^+ prime coding function"
because these functions allow to characterize points of f-natural
coordinates of the transformed hyperbolas i.e., allow to characterize
prime numbers in terms of differenciability. Also, it seems adequate
to call the numbers a_i coefficients of the IR^+ prime coding function.

Fernando Revilla.
.

Relevant Pages

  • PROPOSED PROBLEM 2.
    ... In PROPOSED PROBLEM 1 we called to any function satisfying ... the conditions of section 1 in this problem, an " IR^ coding function " ... it seems adequate to call any function satisfying ... to call the numbers a_i coefficients of the IR^+ prime coding function. ...
    (sci.math)
  • PROPOSED PROBLEM 2.
    ... In PROPOSED PROBLEM 1 we called to any function satisfying ... the conditions of section 1 in this problem, an " IR^ coding function " ... it seems adequate to call any function satisfying ... to call the numbers a_i coefficients of the IR^+ prime coding function. ...
    (sci.math)
  • PROPOSED PROBLEM 2.
    ... In PROPOSED PROBLEM 1 we called to any function satisfying ... the conditions of section 1 in this problem, an " IR^ coding function " ... it seems adequate to call any function satisfying ... to call the numbers a_i coefficients of the IR^+ prime coding function. ...
    (sci.math)
  • Finding out the second derivative of (A_T)^. The essential points.
    ... coefficients of the f R^+ prime coding function. ... 8.- Semi vortex points. ... 13.- Classifying square essential regions. ...
    (sci.math)