Dorin Mărghidanu

The purpose of this note is to present a relation between the arithmetic mean, of a finite number of real numbers, and the arithmetic mean of their images through a convex function. Some applications of this inequality are also included.

]]>Marcia R. Pinheiro

In this piece, the paper Deterministic Small-world Communication Networks, a paper from the year of 2000 by Comellas, Ozon, and Peters, is discussed. From the introduction, where some mismatch between sigmatoids, and intended senses is identified, to the conclusion, where the same mismatch appears in the shape of wonder, the findings are of surprising nature: major misunderstandings in the interpretation of the research or teaching or invention of others may lead to great new theories, which may lead to wonderful sets of new, and meaningful mathematical paradigms.

]]>M. R. Pinheiro

In this note, we present a few more important scientific remarks regarding the *S*-convexity phenomenon. This time, we talk about examples. That was one of the first queries Professor Mark Nelson had for us at the ANZIAM meeting that happened this year, in 2017, at the Wollongong University, Information Sciences building. We here talk about a very trivial example. Yet this example will prove a few really old results to be equivocated.

Marcia R. Pinheiro

In this note, we copy the work we presented in *Second* *Note* *on* *the* *Shape* *of* *S-convexity* [1], but apply the reasoning to one of the new limiting lines, limiting lines we presented in *Summary* *and* *Importance* *of* *the* *Results* *Involving* *the* *Definition* *of* *S-Convexity* [2]. That line was called *New* *Positive* *System* in *Third* *Note* *on* *the* *Shape* *of* *S-convexity* [3] because, on that instance, the images of the points of the domain had been replaced with a positive constant, which we called *A*. This is about Possibility 2 of *Summary* *and* *Importance* *of* *the* *Results* *Involving* *the* *Definition* *of* *S-Convexity*. We have called it in *Summary* *and* *Importance* *of* *the* *Results* *Involving* *the* *Definition* *of* *S-Convexity*, and *New* *Positive* *System* in *Third* *Note* *on* *the* *Shape* *of* *S-convexity*. The second part has already been dealt with in *Second* *Note* *on* *the* *Shape* *of* *S-convexity*. In *Second* *Note* *on* *the* *Shape* *of* *S-convexity*, we have already performed the work we performed in *First* *Note* *on* *the* *Shape* *of* *S-convexity* [4] over the case in which the modulus does not equate the function in the system from *Summary* *and* *Importance* *of* *the* *Results* *Involving* *the* *Definition* *of* *S-Convexity*. This paper is about progressing toward the main target: Choosing the best limiting lines amongst our candidates.

J. C. Tiago de Oliveira

Mario Markus, a Chilean scientist and artist from Dortmund Max Planck Institute, has exposed, in (Markus, 2016), a large set of images of Lyapunov exponents for the logistic equation modulated through rhythmic oscillation of parameters. The pictures display features like foreground/background contrast, visualizing superstability, structural instability and, above all, multistability in a way visually analogous to three-dimensional representation (Markus, 2016a). The present paper aims at classifying, through codification of numbers using the unit interval, the ensemble of fractal images thus generated. This is part of a bigger project, which is the classification of style of fractals - a common endeavor to Art and Science.

]]>M. R. Pinheiro

In this note, we introduce some concepts from Graph Theory in the description of the geometry of cybercriminal groups, and we use the work of Broadhurst et al, a piece from 2014, as a foundation of reasoning. We are also worried about suggesting or even creating, if necessary, mathematical jargon, so that also mathematicians, and those who have similar thinking processes, can connect to Broadhurst et al’s work, and create even more ways to deal with cybercrime data. This is a light note, with the sole intent of suggesting ways to go to Broadhurst et al, so that there is even more intersection between their work and ours. What happens with the creation of bridges between Cyber Crime and Mathematics is that we can speak more objectively about things, and, through Mathematics, perhaps optimize the efforts of the computer scientists, or even of the systems analysts, who try to create perfect tools for those who work in such a niche.

]]>Marcia R. Pinheiro

In this paper, we put two concepts together: Shortest paths and starant graphs. We calculate the costs involved in putting two randomly selected individuals in contact in a controlled network. That would be the costs in terms of public health. Disease spread became our main concern in what comes to the starant graphs in the year of 2002 because that is one of the directions the work of Comellas et al. and Watts et al. pointed at, and our work is inspired in theirs. Other factors, such as random, and unexpected, contact between individuals, are disregarded, so that if the individual visits the clinic that belongs to Mister X, his mate, but his usual doctor, Mister Y, is not there, and he is then served by Miss R, we will need new calculations, what means that we go from predictive power to disgrace power, and that frontally opposes our initial intentions with this work.

]]>M. R. Pinheiro

In this note we copy the work we presented on *Second Note on the Shape of S-convexity* [1], but apply the reasoning to one of the new limiting lines, limiting lines we presented on *Summary and Importance of the Results Involving the Definition of S-Convexity* [2]. This is about Possibility 1, second part of the definition, that is, the part that deals with negative real functions. We have called it *S*_{1} in Summary [2]. The first part has already been dealt with in *First Note on the Shape of S-convexity* [3]. This paper is about progressing toward the main target: Choosing the best limiting lines amongst our candidates.

Maria Ribeiro

In this note, we study a possible proof of the Four-colour Theorem, which is the proof contained in (Potapov, 2016), since it is claimed that they prove the equivalent for three colours, and if you can colour a map with three colours, then you can colour it with four, like three starts being the new minimum. We get to prove that this interesting proof, made of terms such as NP-complete, 3-SAT, certificate, hardness, satisfiable, and certifier is at most an attempt to have a proof, and an exotic one, since all that is involved is very different from what we expect to see in a proof of this type.

]]>Maria Ribeiro

The Four Colour Theorem is a relatively old problem (1852 according to our sources). Researcher Gonthier has recently claimed to have proven it in a notice to the American Mathematical Society (Gonthier, 2008). After Dr. Pinheiro found a counter-example to the claims contained in this theorem, however, we succeeded, as expected, in finding flaws in his proof. As we tried to find those in his work, we ended up finding flaws in the work of one of the researchers he mentioned in his paper, and that was Kempe (Kempe, 1879). The wonder is how much to the side of the foundations of the mathematical reasoning the problem we found in their reasoning is. The tools we use here are analysis, comparisons, graphical representation, and synthesis, for instance. Dr. Pinheiro collaborates with us in our research.

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