A conventionally cast CoNiFeMn0.5Cr1.5 alloy and two versions with 0.25C&3.7Ta or 0.25C&3.7Hf were tested in oxidation at 1000°C for 50h with thermogravimetric recording of the oxidation kinetic. In all cases the obtained mass gain curves are parabolic. Their parabolic constants are much lower than the Kp previously determined for the original alloys with an equimolar base (CoNiFeMnCr). However, the post–mortem exploitation of the oxidized samples revealed analogous oxidation features on surface and subsurface, with here too external oxide strata on surface with different Mn and Cr contents, and rather great Mn–depletion in addition to a moderate Cr–depletion, in subsurface. Globally, the oxidation behavior is significantly better than earlier observed for the equimolar version of these alloys.

Collatz conjecture (3n+1 problem) is an application of Cantor's isomorphism theorem (Cantor-Bernstein) under recursion. The set of 3n+1 for all odd positive integers n, is an order isomorphism for (odd X, 3X+1). The other (odd X, 3X+1) linear order has been discovered as a bijective order-embedding, with values congruent to powers of four. This is demonstrated using a binomial series as a set rule, then showing the isomorphic structure, mapping, and cardinality of those sets. Collatz conjecture is representative of an order machine for congruence to powers of two. If an initial value is not congruent to a power of two, then the iterative program operates the (odd X, 3X+1) order isomorphism until an embedded value is attained. Since this value is a power of four, repeated division by two tends the sequence to one. Because this same process occurs, regardless of the initial choice for a positive integer, Collatz conjecture is true.

In this paper, we classify the function space of all real-valued functions on R denoted as F(R,R) into 28 distinct blocks. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection, which reveals a special relationship structure between four well-known real-valued functions in real analysis: the Cantor function, Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored.