Hausdorff’s forgotten proof that almost all numbers are normal

In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in base 2. We generalize this proof to show that almost all numbers are normal. The result is arguably the most elementary proof for this theorem so far and should be accessible to undergraduates in their first year.

Technology factor as society’s information stock: An elegant proof

It has been empirically observed that the income structure of the vast majority of populations in market-economy countries follows an exponential distribution. The empirical evidence has covered more than 66 countries, ranging from Europe to Latin America, North America, and Asia. Here, to further support exponential income distribution as a signature of the well-functioning market economy, we empirically show how the income structure of China evolved towards an exponential distribution after the market-oriented economic reformation. In particular, we strictly prove that, if the income structure of an economy obeys an exponential distribution, the income summation over all households leads to a neoclassical aggregate production function, in which the technology factor is exactly equal to society’s information stock. This finding provides an insight into understanding the underlying implication of technological progress.JEL Classification: O33; D31; D83; C46

Applications of the Bielecki renorming technique

The renorming technique allows one to apply the Banach Contraction Principle for maps which are not contractions with respect to the original metric. This method was invented by Bielecki and manifested in an extremely elegant proof of the Global Existence and Uniqueness Theorem for ODEs. The present paper provides further extensions and applications of Bielecki’s method to problems stemming from functional analysis and from the theory of functional equations.

Colourability and word-representability of near-triangulations

A graph G = (V;E) is word-representable if there is a word w over the alphabet V such that x and y alternate in w if and only if the edge (x; y) is in G. It is known [6] that all 3-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not. There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al. [1] showed that a triangulation of a convex polyomino is word-representable if and only if it is 3-colourable; and Glen and Kitaev [5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed. It was shown in [4] that a near-triangulation is 3-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5].

Minimum eccentric connectivity index for graphs with fixed order and fixed number of pendant vertices

The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product dG(v)eG(v), where dG(v) is the degree of v in G and eG(v) is the maximum distance between v and any other vertex of G. We characterize, with a new elegant proof, those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order n. Then, given two integers n and p with p ? n - 1, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order n with p pendant vertices.

A Refined Approach for Non-Negative Entire Solutions of Δ u + up = 0 with Subcritical Sobolev Growth

Let {N\geq 2} and {1<p<(N+2)/(N-2)_{+}}. Consider the Lane–Emden equation {\Delta u+u^{p}=0} in {\mathbb{R}^{N}} and recall the classical Liouville type theorem: if u is a non-negative classical solution of the Lane–Emden equation, then {u\equiv 0}.The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of Serrin and Zou, originally used for the Lane–Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.