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In our contribution we describe some on-going improvements concerning the Automated Reasoning Tools developed in GeoGebra Discovery, providing different examples of the performance of these new features. We describe the new ShowProof command, that outputs both the sequence of the different steps performed by GeoGebra Discovery to confirm a certain statement, as well as a number intending to grade the difficulty or interest of the assertion. The proposal of this assessment measure, involving the comparison of the expression of the thesis (or conclusion) as a combination of the hypotheses, will be developed., Comment: In Proceedings ADG 2023, arXiv:2401.10725

Ultraviolet spectroscopy is a powerful method to study planetary surface composition through reflectance measurements and atmospheric composition through stellar/solar occultations, transits of other planetary bodies, and direct imaging of airglow and auroral emissions. The next generation of ultraviolet spectrographs (UVS) on board ESA's JUICE (Jupiter Icy Moons Explorer) and NASA's Europa Clipper missions will perform such measurements of Jupiter and its moons in the early 2030's. This work presents a compilation of a detailed UV stellar catalog, named CUBS, of targets with high intensity in the 50-210 nm wavelength range with applications relevant to planetary spectroscopy. These applications include: 1) Planning and simulating occultations, including calibration measurements; 2) Modeling starlight illumination of dark, nightside planetary surfaces primarily lit by the sky; and 3) Studying the origin of diffuse galactic UV light as mapped by existing datasets from Juno-UVS and others. CUBS includes information drawn from resources such as the International Ultraviolet Explorer (IUE) catalog and SIMBAD. We have constructed model spectra at 0.1 nm resolution for almost 90,000 targets using Kurucz models and, when available, IUE spectra. CUBS also includes robust checks for agreement between the Kurucz models and the IUE data. We also present a tool for which our catalog can be used to identify the best candidates for stellar occultation observations, with applications for any UV instrument. We report on our methods for producing CUBS and discuss plans for its implementation during ongoing and upcoming planetary missions., Comment: 13 pages, 4 figures, submitted to JATIS

X ray resonant magnetic scattering (XRMS) was used to characterize the magnetization of 2D arrays of trilayer submicron magnets. The interpretation of the data required the understanding of the morphology of the magnets which was also deduced from the scattered intensity. The magnets consisted of two magnetostatically coupled ferromagnetic layers separated by a non-magnetic spacer. The scattered intensity from the disks resulted to be dependent on the disks surface curvature. This made the collected intensity at each Bragg reflection (BR) to be correlated to the reflected light from locations of the disk with the same angle of curvature. Due to this, quantitative information was obtained, averaged over the disks illuminated by x rays, of the variations in thickness and magnetization across the entire area of the disks. This averaged magnetization mapping of the disks served to study their vortex configuration in each of their magnetic layers, determining the average location of the vortex, the chiral symmetry of its magnetic circulation, and the specific locations where the vortex nucleation starts within the disks. Chiral asymmetry appeared in the disks when the field was oriented at an oblique angle with respect to the easy axis of the array. The local magnetic sensitivity of the technique allowed to identify a non-centrosymmetric distribution of the magnetization of the disks that explains the observed chiral asymmetry. Unexpectedly, the magnetic circulation sense of the vortex was the same in both ferromagnetic layers. In addition, the magnetization of the buried layer was different in the descent branch than in the ascent branch of its hysteresis loops., Comment: 13 pages, 16 figures

In our contribution we will reflect, through a collection of selected examples, on the potential impact of the GeoGebra Discovery application on different social and educational contexts., Comment: In Proceedings ADG 2021, arXiv:2112.14770

Soft X-ray magnetic vector tomography has been used to visualize with unprecedented detail and solely from experimental data the 3D magnetic configuration of a ferrimagnetic Gd12Co88/Nd17Co83/Gd24Co76 multilayer with competing anisotropy, exchange and magnetostatic interactions at different depths. The trilayer displays magnetic stripe domains, arranged in a chevron pattern, which are imprinted from the central Nd17Co83 into the bottom Gd12Co88 layer with a distorted closure domain structure across the thickness. Near the top Gd24Co76 layer, local exchange springs with out-of-plane magnetization reversal, modulated ripple patterns and magnetic vortices and antivortices across the thickness are observed. The detailed analysis of the magnetic tomogram shows that the effective strength of the exchange spring at the NdCo/GdCo interface can be finely tuned by GdxCo1-x composition and anisotropy (determined by sample fabrication) and in-plane stripe orientation (adjustable), demonstrating the capability of 3D magnetic visualization techniques in magnetic engineering research., Comment: 23 pages, 7 figures

Ratchet devices allow turning an ac input signal into a dc output signal. A ratchet device is set by moving particles driven by zero averages forces on asymmetric potentials. Hybrid nanostructures combining artificially fabricated spin-ice nanomagnet arrays with superconducting films have been identified as a good choice to develop ratchet nanodevices. In our case, the asymmetric potentials are provided by charged N\'eel walls located in the vertices of the magnetic honeycomb array, whereas the role of moving particles is played by superconducting vortices. We have experimentally obtained ratchet effect for different spin ice I configurations and for vortex lattice moving parallel or perpendicular to the magnetic easy axes. Remarkably, the ratchet magnitudes are similar in all the experimental runs; i. e. different spin ice I configurations and in both relevant directions of the vortex lattice motion. We have simulated the interplay between vortex motion directions and a single asymmetric potential. It turns out vortices interact with uneven asymmetric potentials, when they move with trajectories crossing charged N\'eel walls with different orientations. Moreover, the appropriate asymmetric pair potentials which generate the local ratchet effect have been identified. In this rocking ratchet the particles (vortices) on the move are interacting each other (vortex lattice); therefore, the ratchet local effect turns into a global macroscopic effect. In summary, this ratchet device benefits from interacting particles moving in robust and topological protected type I spin ice landscapes., Comment: 21 pages, 13 figures

The mechanisms controlling the growth rate and composition of epitaxial CdTe and CdZnTe films were studied. The films were grown by isothermal closed space configuration technique. A GaAs 100 substrate was exposed sequentially to the elemental sources, Zn, Te, and Cd, in isothermal conditions. While growth of ZnTe followed an atomic layer epitaxy, ALE, regime self regulated at one monolayer per cycle; the CdTe films revealed different growth rates in dependence of the growth parameters,exposure and purge times. Combination of short purge times and larger Cd exposure times led to not self regulated growth regime for CdTe. This is ascribed to large Cd coverages that were dependent on Cd exposure times, following a Brunauer-Emmett and Teller-type adsorption. However, for longer purge times and or short Cd exposure times, an ALE self regulated regime was achieved with 2 ML per cycle. In this sense, the self-regulation of the growth is limited by desorption, instead of absorption, as in the traditional growth technique. Cd atoms substitution by Zn atoms and subsequent evaporation of surface Cd atoms during Zn exposure has been proved. The influence of these facts on the growth and composition of the alloy is discussed.

The sensitivity of Circularly polarized X ray Resonant Magnetic Scattering (CXRMS) to chiral asymmetry has been demonstrated. The study was performed on a 2D array of Permalloy (Py) square nanomagnets of 700 nm lateral size arranged in a chess lattice of 1000 nm lattice parameter. Previous X ray Magnetic Circular Dichroism Photoemission Electron microscopy (XMCD-PEEM) images on this sample showed the formation of vortices at remanence and a preference in their chiral state. The magnetic hysteresis loops of the array along the diagonal axis of the squares indicate a non-negligible and anisotropic interaction between vortices. The intensity of the magnetic scattering using circularly polarized light along one of the diagonal axes of the square magnets becomes asymmetric in intensity in the direction transversal to the incident plane at fields where the vortex states are formed. The asymmetry sign is inverted when the direction of the applied magnetic field is inverted. The result is the expected in the presence of an unbalanced chiral distribution. The effect is observed by CXRMS due to the interference between the charge scattering and the magnetic scattering., Comment: The new revised version corrected some grammatical and orthographic mistakes like "cero" instead of zero and "core vortex" instead of vortex core. Bibliographic entries were modified to have all the same format

The knowledge of how the magnetization looks inside a ferromagnet is often hindered by the limitations of the available experimental methods that are sensitive only to the surface regions or limited in spatial resolution. We report the 3D tomographic reconstruction of the magnetization within a ferromagnetic film of 240 nm in thickness using soft X ray microscopy and magnetic dichroism. The film has periodic magnetic domains forming stripes and closure domains found to be shifted from the stripe array by 1/4 of the period. In addition, the bifurcations of the stripes, which act as inversion nuclei of the magnetization, evidence in 3D meron singularities and Bloch points at the interior of the film. This novel method can be easily extended to magnetic stacks in spintronics applications and other singularities in films., Comment: 13 pages, 4 figures

We have designed, fabricated and tested a robust superconducting ratchet device based on topologically frustrated spin-ice nanomagnets. The device is made of a magnetic Co honeycomb array embedded in a superconducting Nb film. This device is based on three simple mechanisms: i) the topology of the Co honeycomb array frustrates in-plane magnetic configurations in the array yielding a distribution of magnetic charges which can be ordered or disordered with in-plane magnetic fields, following spin-ice rules, ii) the local vertex magnetization, which consists of a magnetic half vortex with two charged magnetic N\'eel walls, iii) the interaction between superconducting vortices and the asymmetric potentials provided by the N\'eel walls. The combination of these elements leads to a superconducting ratchet effect. Thus, superconducting vortices driven by alternating forces and moving on magnetic half vortices generate a unidirectional net vortex flow. This ratchet effect is independent of the distribution of magnetic charges in the array., Comment: 17 pages, 7 figures

We have studied two nanomagnet systems with strong (Co/Pd multilayers) and weak (NdCo alloy films) stray magnetic fields by probing the out-of-plane magnetic states with superconducting vortices. The hybrid samples are made of array of nanomagnets embedded in superconducting Nb thin films. The vortex motion detects relevant magnetic state features, since superconducting vortices are able to discriminate between different magnetic stray field strengths and directions. The usual matching effect between the superconducting vortex lattice and the periodic pinning array can be quenched by means of disorder magnetic potentials with strong stray fields at random. Ordered stray fields retrieve the matching effect and yield asymmetry and shift in the vortex dissipation signal. Furthermore vortices can discriminate the sizes of the nanomagnet magnetic domains, detecting magnetic domain sizes as small as 70 nm. In addition, we observe that the vortex cores play the crucial role instead of the supercurrents around the vortex., Comment: 22 pages, 8 figures

We introduce and discuss, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test on the dynamic geometry program GeoGebra., Comment: 18 pages, 5 figures

Magnetic lateral multilayers have been fabricated on weak perpendicular magnetic anisotropy amorphous Nd-Co films in order to perform a systematic study on the conditions for controlled nucleation of topological defects within their magnetic stripe domain pattern. A lateral thickness modulation of period $w$ is defined on the nanostructured samples that, in turn, induces a lateral modulation of both magnetic stripe domain periods $\lambda$ and average in-plane magnetization component $M_{inplane}$. Depending on lateral multilayer period and in-plane applied field, thin and thick regions switch independently during in-plane magnetization reversal and domain walls are created within the in-plane magnetization configuration coupled to variable angle grain boundaries and disclinations within the magnetic stripe domain patterns. This process is mainly driven by the competition between rotatable anisotropy (that couples the magnetic stripe pattern to in-plane magnetization) and in-plane shape anisotropy induced by the periodic thickness modulation. However, as the structural period $w$ becomes comparable to magnetic stripe period $\lambda$, the nucleation of topological defects at the interfaces between thin and thick regions is hindered by a size effect and stripe domains in the different thickness regions become strongly coupled., Comment: 10 pages, 7 figures, submitted to Physical Review B

Perpendicular magnetic anisotropy ferromagnetic/ superconducting (FM/SC) bilayers with a labyrinth domain structure are used to study nucleation of superconductivity on a fractal network, tunable through magnetic history. As clusters of reversed domains appear in the FM layer, the SC film shows a percolative behavior that depends on two independent processes: the arrangement of initial reversed domains and the fractal geometry of expanding clusters. For a full labyrinth structure, the behavior of the upper critical field is typical of confined superconductivity on a fractal network., Comment: 15 pages, 5 figures

Stripe domains are studied in perpendicular magnetic anisotropy films nanostructured with a periodic thickness modulation that induces the lateral modulation of both stripe periods and inplane magnetization. The resulting system is the 2D equivalent of a strained superlattice with properties controlled by interfacial misfit strain within the magnetic stripe structure and shape anisotropy. This allows us to observe, experimentally for the first time, the continuous structural transformation of a grain boundary in this 2D magnetic crystal in the whole angular range. The magnetization reversal process can be tailored through the effect of misfit strain due to the coupling between disclinations in the magnetic stripe pattern and domain walls in the in-plane magnetization configuration.

The motion of a domain wall in a two dimensional medium is studied taking into account the internal elastic degrees of freedom of the wall and geometrical pinning produced both by holes and sample boundaries. This study is used to analyze the geometrical conditions needed for optimizing crossed ratchet effects in periodic rectangular arrays of asymmetric holes, recently observed experimentally in patterned ferromagnetic films. Geometrical calculations and numerical simulations have been used to obtain the anisotropic critical fields for depinning flat and kinked walls in rectangular arrays of triangles. The aim is to show with a generic elastic model for interfaces how to build a rectifier able to display crossed ratchet effects or effective potential landscapes for controlling the motion of interfaces or invasion fronts., Comment: 13 pages, 18 figures

Arrays of Ni nanodots embedded in Nb superconducting films have been fabricated by sputtering and electron beam lithography techniques. The arrays are periodic triangular lattices of circular Ni dots arranged in a kagom\'e-like pattern with broken reflection symmetry. Relevant behaviors are found in the vortex lattice dynamics : i) At values lower than the first integer matching field, several fractional matching fields are present when the vortex lattice moves parallel or perpendicular to the reflection symmetry axis of the array showing a clear anisotropic character in the magnetoresistance curves, ii) injecting an ac current perpendicular to the reflection symmetry axis of the array yields an unidirectional motion of the vortex lattice (ratchet effect) as a result of the interaction between the whole vortex lattice and the asymmetric lattice of dots, iii) increasing the input current amplitudes the ratchet effect changes polarity independently of matching field values. These experimental results can be explained taking into account the vortex lattice density., Comment: 9 pages, 4 figures, 1 table

We study both experimentally and theoretically the driven motion of domain walls in extended amorphous magnetic films patterned with a periodic array of asymmetric holes. We find two crossed ratchet effects of opposite sign that change the preferred sense for domain wall propagation, depending on whether a flat or a kinked wall is moving. By solving numerically a simple $\phi^4$-model we show that the essential physical ingredients for this effect are quite generic and could be realized in other experimental systems involving elastic interfaces moving in multidimensional ratchet potentials., Comment: 4 pages, 4 figures, submitted to PRL

Driven vortex lattices have been studied in a material with strong pinning, such as Nb films. Samples in which natural random pinning coexists with artificial ordered arrays of defects (submicrometric Ni dots) have been fabricated with different geometries (square, triangular and rectangular). Three different dynamic regimes are found: for low vortex velocities, there is a plastic regime in which random defects frustrate the effect of the ordered array; then, for vortex velocities in the range 1-100 m/s, there is a sudden increase in the interaction between the vortex lattice and the ordered dot array, independent on the geometry. This effect is associated to the onset of quasi long range order in the vortex lattice leading to an increase in the overlap between the vortex lattice and the magnetic dots array. Finally, at larger velocities the ordered array-vortex lattice interaction is suppresed again, in agreement with the behavior found in numerical simulations., Comment: 8 text pages + 4 figures

Pinning interactions between superconducting vortices in Nb and magnetic Ni dots were studied as a function of current and temperature to clarify the nature of pinning mechanisms. A strong current dependence is found for a square array of dots, with a temperature dependent optimum current for the observation of periodic pinning, that decreases with temperature as (1-T/Tc)3/2. This same temperature dependence is found for the critical current at the first matching field with a rectangular array of dots. The analysis of these results allows to narrow the possible pinning mechanisms to a combination of two: the interaction between the vortex and the magnetic moment of the dot and the proximity effect. Moreover, for the rectangular dot array, the temperature dependence of the crossover between the low field regime with a rectangular vortex lattice to the high field regime with a square configuration has been studied. It is found that the crossover field increases with decreasing temperature. This dependence indicates a change in the balance between elastic and pinning energies, associated with dynamical effects of the vortex lattice in the high field range., Comment: 12 text pages (revtex), 6 figures (1st jpeg, 2nd-6th postscript) accepted in Physical Review B

The flux flow regime of high-T$_{\rm c}$ samples of different normal state resistivities is studied in the temperature range where the sign of the Hall effect is reversed. The scaling of the vortex viscosity with normal state resistivity is consistent with the Bardeen-Stephen theory. Estimates of the influence of possible mechanisms suggested for the sign reversal of the Hall effect are also given., Comment: 3 pages. 4 figures upon request

We give a constructive procedure to check basicness of open (or closed) semialgebraic sets in a compact, non singular, real algebraic surface $X$. It is rather clear that if a semialgebraic set $S$ can be separated from each connected component of $X\setminus(S\cup\frz S)$ (when $\frz S$ stands for the Zariski closure of $(\ol S\setminus{\rm Int}(S))\cap{\rm Reg}(X)$), then $S$ is basic. This leads to associate to $S$ a finite family of sign distributions on $X\setminus\frz S$; we prove the equivalence between basicness and two properties of these distributions, which can be tested by an algorithm. There is a close relation between these two properties and the behaviour of fans in the algebraic functions field of $X$ associated to a real prime divisor, which gives an easy proof, for a general surface $X$, of the well known 4-elements fan's criterion for basicness (Brocker, Andradas-Ruiz). Furthermore, if the criterion fails, using the description of fans in dimension 2, we find an algorithmic method to exhibit the failure. Finally, exploiting this thecnics of sign distribution we give one improvement of the 4-elements fan's criterion of Brocker to check if a semialgebraic set is principal., Comment: 23 pages, amslatex (+bezier.sty) report: 1.89.(766) october 1993