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- Fractal calculus on fractal interpolation functionsPublication . Gowrisankar, Arulprakash; Khalili Golmankhaneh, Alireza; Serpa, CristinaIn this paper, fractal calculus, which is called F α -calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be non differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstrass functions are presented.
- Fractal reconstruction of fiber-reinforced epoxy microstructurePublication . Radovic, Ivona; Stajčić, A.; Mitić, V. V.; Serpa, Cristina; Paunović, V.; Randelović, B.In the past century, the use of polymers and composites with a polymer matrix has expanded to such a level that today it is impossible to imagine life without these materials. Epoxy resin and epoxy-based composites are widely used as construction materials, due to their excellent adhesion, thermal and chemical stability. Fractal nature analysis can provide insight in morphological changes at fiber-matrix interface level, which could give direction for the processing of composites. This mathematical technique can be performed on field emission scanning electron microscopy (FESEM) images, by identifying fiber phase and pores shapes and boundaries, as well as fiber-matrix bonding at the interface. In this study, fiberglass mat was used for the reinforcement of epoxy. FESEM image of enlarged fiber after the composite fracture was used for the reconstruction of data. With the use of affine fractal regression model, software Fractal Real Finder was employed for the reconstruction of fiber shape and the determination of Hausdorff dimension.
- A note on fractal interpolation vs fractal regressionPublication . Serpa, CristinaFractals fascinates both academics and art lovers. They are a form of chaos. A key feature that distinguishes a fractal from other chaotic phenomena is the self-similarity. This is a property that consists of replicating a shape to smaller pieces of the whole. In other words, making zoom in or zoom out gives similar perspectives of the same fractal thing. We may find these shapes everywhere and nature presents many examples of fractal creations. An amazing case is the romanesque cabbage. Mandelbrot is the father of the term fractal and studied various examples (see [3]). Constructing a fractal is a simple task to do, just consider an initial configuration and a replication rule for smaller scales. This is how one gets, for example, the Sierpinski triangle, the dragon curve, or the Koch Snowflake. A simple rule creates complicated shapes with non-classical geometries. Analytically, it is also possible to define fractals as solutions of a system of iterative func tional equations. Barnsley defined such a system in [1]. This non-classical geometric concept has attracted many researchers when they are faced with the need to analyse real data with irregular characteristics.
- Fractal analysis and ferroelectric properties of Nd(Zn-1/Ti-2(1)/(2))O-3(NZT)Publication . Khamoushi, Kouros; Serpa, CristinaThe challenges in productivity of satellite mobile devices are growing rapidly to overcome the question of miniaturization. The intention is to supply the electrical and microwave properties of materials by discovering their outstanding electronic properties. Neodymium Zinc Titanate (NZT) can be a promising ferroelectric material due to its stable dielectric and microwave properties. The grain size and shape of NZT have a strong influence on overall material performances. Therefore, shape, reconstruction and property of the coming compound take an important part and can be predicted before being utilized in the devices. The significant of this research is to define ferroelectric properties of NZT and to characterize it by using Fractal Nature Analysis (FNA). FNA is a powerful mathematical technique that could be applied to improve the grain shape and interface reconstruction. The fractal structure is identified by its self-similarity. The self-similarity of an object means a repetition of shapes in smaller scales. A measure of this structure is computed using the Hausdorff dimension. It is for the first time in this investigation the Fractal analysis method is applied for the microwave materials microstructure reconstruction which makes this research an innovative work and will open the door for Curie-Weiss law fractal correction. In connection to our previous research for dielectric properties fractalization, we had some characterization and reconstruction data which include the Hausdorff dimension (HD).
- Fractal nature of advanced Ni-based superalloys solidified on board the international space stationPublication . Mitic, Vojislav; Serpa, Cristina; Ilic, Ivana; Mohr, Markus; Fecht, Hans-Jorgothers. Therefore, designing, improving and predicting advanced material properties is a crucial necessity. The high temperature creep and corrosion resistance of Ni-based superalloys makes them important materials for turbine blades in aircraft engines and land-based power plants. The investment casting process of turbine blades is costly and time consuming, which makes process simulations a necessity. These simulations require fundamental models for the microstructure formation. In this paper, we present advanced analytical techniques in describing the microstructures obtained experimentally and analyzed on different sample's cross-sectional images. The samples have been processed on board the International Space Station using the MSL-EML device based on electromagnetic levitation principles. We applied several aspects of fractal analysis and obtained important results regarding fractals and Hausdorff dimensions related to the surface and structural characteristics of CMSX-10 samples. Using scanning electron microscopy (SEM), Zeiss LEO 1550, we analyzed the microstructure of samples solidified in space and successfully performed the fractal reconstruction of the sample's morphology. We extended the fractal analysis on the microscopic images based on samples solidified on earth and established new frontiers on the advanced structures prediction.
- Compatibility conditions for systems of iterative functional equations with non-trivial contact setsPublication . Buescu, Jorge; Serpa, CristinaSystems of iterative functional equations with a non-trivial set of contact points are not necessarily solvable, as the resulting intersections may lead to an overdetermination of the system. To obtain existence and uniqueness results additional conditions must be imposed on the system. These are the compatibility conditions, which we define and study in a general setting. An application to the affine and doubly affine cases allows us to solve an open problem in the theory of functional equations. In the last section we consider a special problem in a different perspective, showing that a complex compatibility condition may result in an elegant and simple property of the solution.