Reverse engineered Diophantine equations

We answer a question of Samir Siksek, asked at the open problems session of the conference ``Rational Points 2022'', which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set $S$ of perfect integer powers, using Mih\u{a}ilescu's theorem, we construct a polynomial $f_S\in \Z[x]$ such that the set $f_S(\Z)$ contains a perfect integer power if and only if it belongs to $S$. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge's method and Fermat's Last Theorem. Therefore we can construct a polynomial-exponential Diophantine equation whose solutions are described in advance.Comment: 7 page

Variations on the method of Chabauty and Coleman

One of the most important results in Diophantine geometry is the finiteness of the number of rational points on nice curves of genus at least two. However, there are no practical methods to compute the rational points on such curves in general, but the method of Chabauty and Coleman is one of the most successful approaches. We discuss variations on this method.When applicable, the method of Chabauty and Coleman gives an upper bound on the number of rational points. We first study curves that attain this bound. Curves that attain the bound are rare and difficult to find; we construct several new examples.When a curve satisfies a stronger rank condition, the method of Chabauty and Coleman can be extended to study points on symmetric powers of curves, which can tell us the information on points defined over small degree number fields. This method is called Symmetric Chabauty. We generalise the Symmetric Chabauty method so that one can use this extension to compute the cubic or quartic points on certain modular curves.The limitation of the method of Chabauty and Coleman is the rank condition. There is a variation on the method that might be used in this case, called nonabelian Chabauty. When we use p-adic heights to construct a locally analytic function, a special explicit variant is called quadratic Chabauty. We give an algorithm to compute p-adic heights on even degree hyperelliptic curves and to apply quadratic Chabauty to compute integral points on certain even degree hyperelliptic curves

Proposal of the Croatian Microscopy Society for awarding the "Spiridion Brusina" Medal to Professor Miran Čeh, PhD, Jožef Stefan Institute, Ljubljana, Slovenia

Cooperation with Professor Miran Čeh started back in 2007, when a member of the Croatian Microscopy Society (CMS), Andreja Gajović, joined the group of Prof. Miran Čeh for post-doctoral training to Jožef Stefan Institute. Since then, Prof. Čeh collaborated extensively with Croatian scientists through a number of bilateral projects, and in this way participated and still participates in the education of Croatian scientists on high resolution transmission electron microscopy (HRTEM). Moreover, Milivoj Plodinec, a member of CMS, spent a sixmonth TEM training (2011/2012) at the Department for Nanostructured Materials, Jožef Stefan Institute, Ljubljana, Slovenia (Croatian Science Foundation\u27s scholarship for doctoral students, project title "Titanate nanostructures - synthesis and high-resolution transmission electron microscopy (TinaTEM)"). Training was performed under the mentorship of Prof. Čeh in the field of synthesis of nanomaterials and their characterization, using, primarily, basic, and advanced techniques of electron microscopy. Plodinec gathered significant knowledge in scanning electron microscopy (SEM), transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HRTEM), selected area electron diffraction (SAED) and energy dispersive X-ray spectroscopy (EDS) and respected techniques. He transferred the acquired knowledge to the Ruđer Bošković Institute, where he was working on the functionalization of Ti02 nanostructures for various applications

Proposal of the Croatian Microscopy Society for awarding the "Spiridion Brusina" Medal to Professor Miran Čeh, PhD, Jožef Stefan Institute, Ljubljana, Slovenia

Cooperation with Professor Miran Čeh started back in 2007, when a member of the Croatian Microscopy Society (CMS), Andreja Gajović, joined the group of Prof. Miran Čeh for post-doctoral training to Jožef Stefan Institute. Since then, Prof. Čeh collaborated extensively with Croatian scientists through a number of bilateral projects, and in this way participated and still participates in the education of Croatian scientists on high resolution transmission electron microscopy (HRTEM). Moreover, Milivoj Plodinec, a member of CMS, spent a sixmonth TEM training (2011/2012) at the Department for Nanostructured Materials, Jožef Stefan Institute, Ljubljana, Slovenia (Croatian Science Foundation\u27s scholarship for doctoral students, project title "Titanate nanostructures - synthesis and high-resolution transmission electron microscopy (TinaTEM)"). Training was performed under the mentorship of Prof. Čeh in the field of synthesis of nanomaterials and their characterization, using, primarily, basic, and advanced techniques of electron microscopy. Plodinec gathered significant knowledge in scanning electron microscopy (SEM), transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HRTEM), selected area electron diffraction (SAED) and energy dispersive X-ray spectroscopy (EDS) and respected techniques. He transferred the acquired knowledge to the Ruđer Bošković Institute, where he was working on the functionalization of Ti02 nanostructures for various applications

Curves with sharp Chabauty-Coleman bound

We construct curves of each genus $g\geq 2$ for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman's bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman's theorem.Comment: 24 page

Spatial and temporal analysis of fires in Serbia for period 2000-2013

Spatial and temporal analysis of fires in Serbia for period November 2000-August 2013 has been performed to investigate whether spatial relationships exist among fire data. MODIS active fire data were used as fire locations. On such data, different tools of spatial analysis and spatial statistics were used, to determine if there is any spatial relationship. Analysis included data screening, identification of land cover of fire locations, aspect, slope, elevation and solar radiation for each location. Later, different spatial statistics tools were executed against fire locations data, including Getis-Ord Gi* Hot Spots, Global Moran’s I Spatial Autocorrelation, Anselin Local Moran’s I Cluster and Outlier, Ordinary Least Square linear regression and Geographically Weighted Regression. Fire radiative power was used as dependant variable, while terrain morphology and solar radiation were used as explanatory variables. Results shows hot spots of fires in Serbia, and indicates that there is strong relationship between fire radiative power on one side and terrain morphology, land cover, solar radiation and spatial distribution on other side. These analysis have highlighted areas with very intensive fire use associated with land management practices