Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

Review of the Water Footprint Project within Geographically Delineated Area

During the last decade, there has been an intensive research activity concerning the concept of the Water Footprint （WF） approach, which was firstly introduced by Arjen Hoekstra in 2002. WF is an indicator of direct and indirect freshwater use of a consumer or producer that takes into account water consumption in every step （intermediate and final） along the production chain and services. The concept can be implemented in various levels such as products, consumers, producers, nations and river basins etc.. The water footprint within a geographically delineated area equals the sum of the process water footprints of all processes taking place in the area. The aim of current research is a review of the most important WF studies, with a special focus on applications within regional, basin and administrative unit level. National and global scales are not included in the current paper. The article presents the most widespread methodologies and approaches that attempt to evaluate water footprints of specific defined areas and highlights their recent advances as well as shortcomings in the constantly evolving research efforts.

Creating a Geographical Observatory of Atmospheric Pollution （GEOAP） for the Greek Territory

The main purpose of this paper is to analyze the necessity of a Geographical Observatory of Atmospheric Pollution （GEOAP） in the Greek territory. The analysis performed is mainly focused on the benefits of the future function of the GEOAP to the environmental planning of the country and it could also provide an environmental management tool for the whole region. Measuring and mapping the pollution data and at the same time performing the geographical analysis of the complexity and the characteristics of natural and human environment can be useful tool in observation, management, and planning of the environmental policy of the country.

Harmony Search and Cellular Automata in Spatial Optimization

The combined optimization problem of resource production and allocation is considered. The spatial character of the problem is emphasized and cellular modeling is introduced. First a new enhanced harmony search algorithm is applied combined with cellular concepts. Then another new approach is presented involving a cellular automaton combined with harmony search. This second approach renders solutions with greater compactness, a desirable characteristic in spatial optimization. The two algorithms are compared and discussed.

Underwater Recording and 3D Modeling during a Dive Training Course： A Case Study at Panormos, Skopelos, Greece

In the field of underwater survey, there is a variety of methods which differ from each other in terms of the operating environment and the purpose that each method is used. Hence, some of the state-of-the-art methods, that have many applications in the various scientific fields nowadays, are introduced in this paper. Additionally, the application of the procedures for an underwater survey in shallow depths is analyzed in accordance with the training standards of the PADI Underwater Survey Diver distinctive specialty. The main goal is to demonstrate not only the theoretical background of underwater surveys but also their operational issues, in order to facilitate the development of knowledge and skills during a training course. Finally, a case study for the recording and 3D modeling of the seabeds＇ morphology in shallow water is presented as it has been organized and accomplished by the participants of such a training course. By this way, it is expected that the reader will understand explicitly the application of the procedures prior, during and after the dive for an underwater survey in shallow depths.

Extending the decision-making process during yellow phase from human drivers to autonomous vehicles:A microsimulation study with safety considerations

One of the main factors affecting the safety of signalised intersections is the stop/go behaviour during the yellow interval.Although previous research has exhaustively examined drivers’stop/go decision-making,the expected autonomous vehicles’(AVs’)stop/go behaviour has not yet been thoroughly investigated.Through a series of simulation experiments developed for conventional and autonomous vehicles using different carfollowing,lane-changing,lateral placement and stop/go model parameter values,we examine here whether the default VISSIM stop/go parameter values can adequately replicate the observed drivers’behaviour at the considered intersection and assess the suitability of using the currently available options,albeit referring to human drivers,to simulate the expected stop/go behaviour of AVs.We also propose a policy framework for determining the desired behaviour of AVs in yellow interval,which is integrated into an AVs logic and achieved in the last simulation to explore the effect of automation on the stop/go outcome and,hence,on the safety level of signalised intersections.Several data analysis and modeling techniques were used for the formulation of certain scenarios,including binary choice models.The default stop/go parameter values were found unfit to replicate the observed stop/go behaviour and subjected to calibration.Compared to the currently available options,the proposed AVs logic proved to produce the most accurate results,in terms of the stop/go simulation outcome.Regarding the impact of automation on the stop/go outcome,the simulation experiments showed that AVs preferred a more conservative behaviour in favor of road safety,as indicated by the significant reduction(≈15%)in the number of vehicles crossing the stop line during the yellow light and zero instances of red light violation.However,compared to the conservative drivers represented by the default stop/go parameter values,AVs preferred a more rational behaviour in favor of intersection capacity without compromising road safety.

Spatial resource allocation via extremal optimization enhanced by cell-based local search

A new treatment is presented for land use planning problems by means of extremal optimization(EO)in conjunction to cell-based neighborhood local search.EO,inspired by self-organized critical models of evolution has been applied mainly to the solution of classical combinatorial optimization problems.Cell-based local search has been employed by the author elsewhere in problems of spatial resource allocation in combination with genetic algorithms and simulated annealing.In this paper,it complements EO in order to enhance its capacity for a spatial optimization problem.The hybrid method thus formed is compared to methods of the literature on a specific resource allocation problem by taking into account both the development and the transportation cost.It yields better results both in terms of objective function values and in terms of compactness.The latter is an important quantity for spatial planning and its meaning is discussed.The appearance of significant compactness values as emergent results is investigated.