INTL. SYMP. ON SUSTAINABLE MATHEMATICS APPLICATIONS

To find an analytically solution of a problem involving a system of partial differential equation is a challenging tusk. So, we use iterative methods to obtain an approximate solution. In inverse scattering the transmission eigenvalue problem is important do determine data for the scatterer. From the complexity of the domain (scatterer) we use the finite element method because we can obtain the best approximation of the required zone. The problem we solve is nonlinear and non-selfadjoint. Using variational method and Fredholm alternative we transform it in order to be discretize. Colton and Cakoni give inferior and superior of the refractive index. This information is used in an inequality given by Colton and Haddar to determine a boundary for the eigenvalues involving the first Dirichlet eigenvalue as well. We use an algorithm to find the first eigenvalue. We have the refractive index n also The algorithm used is a combination of finite element method with GMRES algorithm.

Forecasting energy production by hydropower plants (HPP) is a challenge because of their correlation with many exogenous variables such as precipitations, water inflow, temperature, the minimum and maximum level of the HPP, etc. Albania has a favorite geographical position which makes the electrical energy the main source of energy produced in the country. In our work, we try to analyze hourly and daily data of energy produced in the main cascade of the country which produces the main amount of energy consumption. Our focus is on analyzing the situation of energy demand on a 24 hour period and also on a weekly /monthly period. We analyze the seasonality patterns of the energy demand and try to fit different models to predict the upcoming season (hours or days). Several modeling strategies among hierarchical forecasting, neural network, multistage forecasting, econometric forecasting models were tested and the best was selected looking at the performance obtained on the testing period. Our goal is to use the proposed models to obtain the forecast for electric energy demand in the country which will help the Albanian Power Corporation (KESH) to build various scenarios on optimizing the country demand and production capacities on HPP cascade.

Structures made from high strength metal alloys (aluminum, steel, and titanium) are being used as structural/load bearing members in many areas including transportation, civil infrastructure, offshore structures, water distribution networks, and nuclear industries. Such systems operate in harsh and uncertain environments and exhibit a strong propensity to pitting and stress corrosion cracking. Pitting corrosion-fatigue is recognized to be one of the major potential degradation mechanisms in aging infrastructures [1-4], so much so that such a combination, when left unchecked, can lead to catastrophic failures. Although pits can initiate from both physical and chemical heterogeneities on the surface, the role of inclusions and second-phase particles (constituent particles) in inducing pitting/stress corrosion cracking in aluminum metal alloys is more common.
Corrosion-fatigue in metal alloys generally involves the formation of pits, pit growth, nucleation of cracks from pits, and the eventual crack propagation to failure. Corrosion pits seem to significantly shorten the fatigue crack initiation, decrease the threshold stress intensity by 50% or more, and lower the fatigue strength by about 40%. Even though the corrosion and fatigue mechanisms have been studied well individually, the coupled effects of corrosion and fatigue have not been studied in detail [5]. With corrosion-fatigue being generally recognized by the structural integrity community as a potential cause for failure in many infrastructural structures and materials, and with replacement of such components being unlikely due to excessive costs, the need for predictive methodologies and models cannot be overstated. In order to continue operating structural systems worldwide in a reliable and sustainable manner, however, better prediction methods based on additional knowledge of the mechanisms associated with corrosion and fatigue are required. This, in turn, would also reduce repair and maintenance costs.
Quantitative analysis through multiscale discrete models for corrosion through computational simulations and imaging data that correlates to pitting and cracking is being investigated. These analysis models will be presented and discussed at the conference.

Although nanoindentation has been used for more than 40 years for calculating elastic constants (modulus and hardness) of materials at the nanoscale, the test still lacks a concrete theoretical framework. The main problems of the current conceptual framework pertaining nanoindentation include: the one-dimensional consideration of a three-dimensional problem, the calculation of elastic constants after strong local plasticity, and the dependence of the calculated elastic properties on the maximum penetration depth or maximum load. Another problem is that within the current theoretical framework, the measurements acquired through the use of instruments of increasing accuracy are theoretically interpreted by semi-empirical methods, involving many assumptions.
The proposed framework in which nanoindentation is considered to be an inhomogeneous compression is due to the tip pyramidal geometry (Berkovich, Vickers) which tries to provide solutions to the aforementioned problems. In the proposed framework, the effect of the tip geometry is modeled in a way to be deducted from the calculation of the modulus of elasticity and hardness. Preliminary results in this direction indicate that the use of gradient theory can actually eliminate the effect of tip geometry by providing values for both the elastic modulus [1] and hardness [2]. These are independent of the maximum indentation depth or load, i.e. proving that the so-called indentation size effect (ISE) is just an artifact of the specific tip geometry.

Though a wide variety of mathematical techniques are used in physics and its applications, much of this stems from complexity rather than the underlying fundamental theories. At the most fundamental level, Nature appears to use mathematics in a strikingly minimal way, basing itself on only a few simple principles. Nature, in its most basic aspect, does not appear to require sophisticated mathematics, which seemingly arises only from the entanglement of multiple simple systems. Mathematics at the most fundamental level is almost an accidental result of a very few basic symmetries (principally, duality and anticommutativity, based on the numbers 2 and 3) and a requirement to maintain totality zero. These generate a particular double space algebra which can be seen as the basis of quantum mechanics, classical mechanics, particle physics and even aspects of chemistry and biology, in particular the genetic code.

The evolutionary methods are optimization methods that converge to the global solution. There are many optimization techniques nowadays used and the one we are working is the evolutionary method PSO. Many authors have proposed various modifications of the basic PSO parameters with the goal to obtain a variant of PSO with best performance algorithm complexity. In our case, first, we present a modified PSO algorithm. Then we analyze the convergence of the proposed algorithm using differential equations. More precisely we relate a difference equation with a differential equation, and study the behavior of its solution. The solution brings results for the parameters of PSO, specifically for the coefficients of acceleration. Since the PSO results depend on its parameters, we propose new parametersthem based on the convergence study. We give an application in the energetic field in Albanian case, in the main three hydropower cascades of the country, which consist of three hydro power plants.