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Graph theoretical problems based on shortest paths are at the core of research due to their theoretical importance and applicability. This paper deals with the geodetic number which is a global measure for simple connected graphs and it belongs to the path covering problems: what is the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Inspired by the exact 0-1 integer linear programming formalism from the recent literature, we propose a new methods to obtain upper bounds for the geodetic number in an algorithmic way. The efficiency of these algorithms are demonstrated on a collection of structurally different graphs.

The spatial distribution of the railway crossings on the river Danube in Hungary is very uneven. There is only one electrified and double-tracked bridge in the country, the Southern Railway Bridge in Budapest. The \'Ujpest bridge in Budapest only provides connection through line 4 which is not electrified and the Baja bridge is not electrified, too, and both of them are single-tracked. The long-planned V0 railway line that is to be cross the Danube approximately halfway between Budapest and Baja would not only help to redistribute the total network flow which currently passes through almost exclusively the Southern bridge but would also provide redundancy for the existing bridges in the case of their disruption. Four of the five proposed V0 path alternatives are analyzed on the basis of these two network properties., Comment: 21 pages, 9 figures, 4 tables

Available alternative routes on which traffic can be rerouted in the case of disruptions are vital for transportation networks. Line sections with less traffic under normal operational conditions but with increased importance in the case of disruptions are identified in the railway network of Hungary by using a weighted directed graph. To describe the goodness of the individual alternative routes the so-called redundancy index is used. The results show that the structure of the network is good, but the lines with the highest redundancy (lines No. 80, 2, 4 and 77 according to the numbering of the national railway operator, M\'AV) are mostly single tracked and in many cases the line speed is low. The building of additional tracks and electrifying these lines while still maintaining the existing diesel locomotives for the case of disruptions of the electric support are the keys to make the performance of the rather dense railway network of Hungary sustainable., Comment: 8 pages, 4 figures. arXiv admin note: text overlap with arXiv:2005.12802

Clustering is an important tool to describe gamma-ray bursts (GRBs). We analyzed the Final BATSE Catalog using Gaussian-mixture-models-based clustering methods for six variables (durations, peak flux, total fluence and spectral hardness ratios) that contain information on clustering. Our analysis found that the five kinds of GRBs previously found by other authors are only the cut groups of the previously well-known three types (short, long and intermediate in duration). The two short and intermediate duration groups differ mostly in the peak flux. Therefore, the reanalysis of the BATSE data finds similar group structures than previously. Because the brightness distribution is asymmetric and not correlated with durations or hardnesses the Gaussian mixture model cuts the Short and the Intermediate duration groups into two subgroups, the dim ones and the bright ones., Comment: 6 pages, 5 figures

The Fermi GBM catalog provides a large database with many measured variables that can be used to explore and verify gamma-ray burst classification results. We have used Principal Component Analysis and statistical clustering techniques to look for clustering in a sample of 801 gamma-ray bursts described by sixteen classification variables. The analysis recovers what appears to be the Short class and two long-duration classes that differ from one another via peak flux, with negligible variations in fluence, duration and spectral hardness. Neither class has properties entirely consistent with the Intermediate GRB class. Spectral hardness has been a critical Intermediate class property. Rather than providing spectral hardness, Fermi GBM provides a range of fitting variables for four different spectral models; it is not intuitive how these variables can be used to support or disprove previous GRB classification results., Comment: accepted

GRB170817A, associated with the LIGO-Virgo GW170817 neutron-star merger event, lacks the short duration and hard spectrum of a Short gamma-ray burst (GRB) expected from long-standing classification models. Correctly identifying the class to which this burst belongs requires comparison with other GRBs detected by the Fermi GBM. The aim of our analysis is to classify Fermi GRBs and to test whether or not GRB170817A belongs - as suggested - to the Short GRB class. The Fermi GBM catalog provides a large database with many measured variables that can be used to explore gamma-ray burst classification. We use statistical techniques to look for clustering in a sample of 1298 gamma-ray bursts described by duration and spectral hardness. Classification of the detected bursts shows that GRB170817A most likely belongs to the Intermediate, rather than the Short GRB class. We discuss this result in light of theoretical neutron-star merger models and existing GRB classification schemes. It appears that GRB classification schemes may not yet be linked to appropriate theoretical models, and that theoretical models may not yet adequately account for known GRB class properties. We conclude that GRB170817A may not fit into a simple phenomenological classification scheme., Comment: accepted in APSS

The giant magnetoresistance (GMR) was investigated for electrodeposited Co/Cu multilayers. In order to better understand the formation of individual layers and their influence on GMR, multilayers produced by two different deposition strategies were compared. One series of Co(2 nm)/Cu(tCu) multilayers with tCu ranging from 0.5 nm to 6 nm was produced with the conventional two-pulse plating by using a galvanostatic/potentiostatic (G/P) pulse combination for the magnetic/non-magnetic layer deposition, respectively, whereby the Cu layer deposition was carried out at the electrochemically optimized potential. Another Co(2 nm)/Cu(tCu) multilayer series with the same tCu range was prepared with the help of a G/P/G pulse combination. In this latter case, first a bilayer of Co(2 nm)/Cu(6 nm) was deposited in each cycle as in the G/P mode after which a third G pulse was applied with a small anodic current to dissolve part of the 6 nm thick Cu layer in order to ensure the targeted tCu value. The comparison of the two series revealed that the G/P/G pulse combination yields multilayers for which GMR can be obtained even at such low nominal Cu layer thicknesses where G/P multilayers already exhibit bulk-like anisotropic magnetoresistance only. Surface roughness measurements by atomic force microscopy revealed that the two kinds of pulse combination yield different surface roughness values which correlate with the structural quality of the multilayers as indicated by the absence or presence of multilayer satellite reflections in the X-ray diffraction patterns. The results of multilayer chemical analysis revealed that mainly an increased Cu content of the magnetic layer is responsible for the onset of SPM regions in the form of Co segregations in the G/P/G multilayers with small Cu layer thicknesses., Comment: 32 pages, 10 figures

Although electrodeposition from aqueous media has been widely used to obtain metallic deposits, there are cases where the application of non-aqueous solutions offers advantages over the traditional baths or even represents the only way to electrodeposit some metals. For this reason, a study of the electrodeposition of Ni from various alcoholic solutions was performed. Apart from a detailed cyclic voltammetry study of these solutions, the surface morphology, crystal structure and texture as well magnetic properties of the deposits have also been investigated. The best results were obtained with methanol as solvent, so results on Ni deposits prepared from this solution will be presented in more detail whereas the case of the other alcoholic solutions investigated will be summarized only briefly. Structural and magnetic properties of the deposits obtained have been compared to literature data on Ni samples obtained from various non-aqueous solvents., Comment: 27 pages, 9 figures

The microstructure of electrodeposited Ni films produced without and with organic additives (saccharin and formic acid) was investigated by X-ray diffraction (XRD) line profile analysis and cross-sectional transmission electron microscopy (TEM). Whereas the general effect of these additives on the microstructure (elimination of columnar growth as well as grain refinement) was reproduced, the pronounced intention of this study was to compare the results of various seldom-used high-performance structural characterization methods on identical electrodeposited specimens in order to reveal fine details of structural changes qualitatively not very common in this field. In the film deposited without additives, a columnar structure was observed showing similarities to the T-zone of structure zone models. Both formic acid and saccharin additives resulted in equiaxed grains with reduced size, as well as increased dislocation and twin fault densities in the nanocrystalline films. Moreover, the structure became homogeneous and free of texture within the total film thickness due to the additives. Saccharin yielded smaller grain size and larger defect density than formic acid. A detailed analysis of the grain size and twin boundary spacing distributions was carried out with the complementary application of TEM and XRD, by carefully distinguishing between the TEM and XRD grain sizes., Comment: 26 pages, 11 figures

The giant magnetoresistance (GMR) and structure was investigated for electrodeposited Co/Cu multilayers prepared by a conventional galvanostatic/potentiostatic pulse combination from a pure sulfate electrolyte with various layer thicknesses, total multilayer thickness and Cu deposition potential. X-ray diffraction (XRD) measurements revealed superlattice satellite reflections for many of the multilayers having sufficiently large thickness (at least 2 nm) of both constituent layers. The bilayer repeats derived from the positions of the visible superlattice reflections were typically 10-20% higher than the nominal values.The observed GMR was found to be dominated by the multilayer-like ferromagnetic (FM) contribution even for multilayers without visible superlattice satellites. There was always also a modest superparamagnetic (SPM) contribution to the GMR and this term was the largest for multilayers with very thin (0.5 nm) magnetic layers containg apparently a small amount of magnetically decoupled SPM regions. No oscillatory GMR behavior with spacer thickness was observed at any magnetic layer thickness. The saturation of the coercivity as measured by the peak position of the MR(H) curves indicated a complete decoupling of magnetic layers for large spacer thicknesses. The GMR increased with total multilayer thickness which could be ascribed to an increasing SPM contribution to the GMR due to an increasing surface roughness, also indicated by the increasing coercivity. For multilayers with Cu layers deposited at more and more positive potentials, the GMRFM term increased and the GMRSPM term decreased. At the same time, a corresponding reduction of surface roughness measured with atomic force microscopy indicated an improvement of the multilayer structural quality which was, however, not accompanied by an increase of the superlattice reflection intensities., Comment: 29 pages, 10 figures

The magnetothermopower and the magnetoresistance of single Co Ni/Cu multilayered nan-owires with various thicknesses of the Cu spacer are investigated. Both kinds of measurement have been performed as a function of temperature (50 K to 325 K) and under applied magnetic fields perpendicular to the nanowire axis, with magnitudes up to 15 % at room temperature. A linear relation between thermopower S and electrical conductivity {\sigma} of the nanowires is found, with the magnetic field as an implicit variable. Combining the linear behavior of the S vs. {\sigma} and the Mott formula, the energy derivative of the resistivity has been determined. In order to extract the true nanowire materials parameters from the measured thermopower, a simple model based on the Mott formula is employed to distinguish the individual thermopower contributions of the sample. By assuming that the non-diffusive thermopower contributions of the nanowire can be neglected, it was found that the magnetic field induced changes of thermopower and resistivity are equivalent. The emphasis in the present paper is on the comparison of the magnetoresistance and magnetothermopower results and it is found that the same correlation is valid between the two sets of data for all samples, irrespective of the relative importance of the giant magnetoresistance or anisotropic magnetoresistance contributions in the various individual nanowires., Comment: 28 pages, 7 figures

A detailed study of the evolution of the magnetoresistance was performed on electrodeposited Co/Cu multilayers with Cu layer thicknesses ranging from 0.5 nm to 4.5 nm. For thin Cu layers (up to 1.5 nm), anisotropic magnetoresistance (AMR) was observed whereas multilayers with thicker Cu layers exhibited clear giant magnetoresistance (GMR) behaviour. The GMR magnitude increased up to about 3.5 to 4 nm Cu layer thickness and slightly decreased afterwards. According to magnetic measurements, all samples exhibited ferromagnetic (FM) behaviour. The relative remanence turned out to be about 0.75 for both AMR and GMR type multilayers. This clearly indicates the absence of an antiferromagnetic (AF) coupling between adjacent magnetic layers for Cu layers even above 1.5 nm where the GMR effect occurs. The AMR behaviour at low spacer thicknesses indicates the presence of strong FM coupling (due to, e.g., pin-holes in the spacer and/or areas of the Cu layer where the layer thickness is very small). With increasing spacer thickness, the pin-hole density reduces and/or the layer thickness uniformity improves which both lead to a weakening of the FM coupling. This improvement in multilayer structure quality results in a better separation of magnetic layers and the weaker coupling (or complete absence of interlayer coupling) enables a more random magnetization orientation of adjacent layers, all this leading to an increase of the GMR. Coercive field and zero-field resistivity measurements as well as the results of a structural study reported earlier on the same multilayers provide independent evidence for the microstructural features established here. The large GMR reported previously on such Co/Cu multilayers at Cu layer thicknesses around 1 nm can be attributed to the presence of a fairly large superparamagnetic (SPM) fraction rather than being due to a strong AF coupling., Comment: 24 pages, 6 figures

No systematic studies have been carried out on the giant magnetoresistance (GMR) of electrodeposited (ED) Fe-Co/Cu multilayers since the elaboration of a method for the optimization of the Cu layer deposition potential. In this paper, we present results on the electrochemical optimization of the Cu layer deposition potential which was found to depend on the relative iron concentration in the bath. An X-ray diffraction study of ED Fe5Co95(1.5 nm)/Cu(dCu) multilayers with dCu ranging from 0.8 nm to 10 nm revealed an fcc structure. For most of the multilayers, a weak superlattice satellite reflection could be identified. The room-temperature magnetoresistance was studied in detail as a function of the individual layer thicknesses. Multilayers with Cu layer thicknesses above about 1.5 nm were found to exhibit a GMR behavior with a maximum GMR of about 5 % and a typical saturation field of 1 kOe. The GMR magnitude decreased with increasing Fe-content in the magnetic layer. The spacer layer thickness evolution of the MR data was established in detail after separating the ferromagnetic and superparamagnetic GMR contributions and no oscillatory GMR was found. A comparison with literature data on both physically deposited and ED Fe-Co/Cu multilayers is also made., Comment: 22 pages, 9 figures

Room-temperature transport properties (the zero-field resistivity, \rho 0, and the GMR) were studied for ED Ni50Co50/Cu multilayers as a function of the individual layer thicknesses and total multilayer thickness. The Cu deposition potential was optimized in order to obtain the preset layer thicknesses. The surface roughness development was studied by AFM, which revealed an exponential roughening with total thickness. The Cu layer thickness strongly influenced the roughness evolution. As expected, \rho 0 decreased with increasing Cu layer thickness whereas it increased strongly for large total multilayer thicknesses that could be ascribed to the observed deposit roughening. All multilayers with Cu layer thicknesses above about 1.5 nm exhibited a GMR behavior with a maximum GMR of about 5 %. The GMR decreased for total multilayer thicknesses above about 300 nm due to the strong increase of \rho 0, the latter caused by the enhanced roughness. The GMR data indicated the appearance of a current at angle to plane type scattering due to the layer undulations. The thickness evolution of the MR data was analyzed in detail after separating the ferromagnetic and superparamagnetic GMR contributions. It could be established that ED Ni-Co/Cu multilayers do not exhibit an oscillatory GMR behavior with spacer thickness., Comment: 27 pages, 11 figures

It has been shown previously for electrodeposited Co/Cu multilayers that the single-bath electrodeposition process can be optimized from an electrochemical point of view in order to avoid unwanted Co dissolution and incorporation of Co in the non-magnetic layer during the Cu deposition pulse. In the present work, electrodeposition of Ni-Co/Cu multilayers has been studied to clarify if the same optimization method is appropriate when two magnetic elements are present and if this potential results in the largest giant magnetoresistance (GMR) for the particular alloy system studied. For this purpose, several Ni-Co/Cu multilayers were prepared by varying the deposition potential of the Cu layer. The composition analysis of the deposits showed that the Ni:Co ratio exhibits a minimum as a function of the Cu deposition potential, which can be explained by considering both the dissolution of Co and the mass transport of the reactants. Both the saturation GMR value and the intensity of the satellite peaks in the X-ray diffractograms were highly correlated with the resulting surface roughness of the deposits which was strongly varying with the Cu deposition potential. Higher GMR values, lower saturation fields and more perfect multilayer structure were observed for sufficiently positive Cu deposition potentials only which enabled a partial Co dissolution resulting in a reduced surface roughness. The results draw attention to the complexity of the optimization procedure of the deposition of multilayers with several alloying components., Comment: 26 pages, 10 figures

The giant magnetoresistance (GMR) effect has been widely investigated on electrodeposited ferromagnetic/non-magnetic (FM/NM) multilayers generally containing a large number of bilayers. In most applications of the GMR effect, layered structures consisting of a relatively small number of consecutive FM and NM layers are used. It is of great interest, therefore, to investigate the initial stages of GMR multilayer film growth by electrodeposition. In the present work we have extended our previous studies on ED GMR multilayers to layered structures with a total thickness ranging from a few nanometers up to 70 nm. The evolution of the surface roughness and electrical transport properties of such ultrathin ED Co/Cu layered structures was investigated. Various layer combinations were produced including both Co and Cu either as starting or top layers in order (i) to see differences in the nucleation of the first layer and (ii) to trace out the effect of the so called exchange reaction. Special attention was paid to measure the field dependence of the magnetoresistance, MR(H) in order to derive information for the appearance of superparamagnetic regions in the magnetic layers. This proved to be helpful for monitoring the evolution of the layer microstructure at each step of the deposition sequence., Comment: 29 pages, 12 figures

The electrical resistivity and the anisotropic magnetoresistance (AMR) was investigated for Ni Co alloys at and below room temperature. The Ni Co alloy layers having a thickness of about 2 um were prepared by electrodeposition on Si wafers with evaporated Cr and Cu underlayers. The alloy composition was varied in the whole concentration range by varying the ratio of Ni sulfate and Co sulfate in the electrolyte. The Ni Co alloy deposits were investigated first in the as deposited state on the substrates and then, by mechanically stripping them from the substrates, as self supporting layers both without and after annealing. According to an X ray diffraction study, a strongly textured face centered cubic (fcc) structure was formed in the as deposited state with an average grain size of about 10 nm. Upon annealing, the crystal structure was retained whereas the grain size increased by a factor of 3 to 5, depending on alloy composition. The zero field resistivity decreased strongly by annealing due to the increased grain size. The annealing hardly changed the AMR below 50 at.% Co but strongly decreased it above this concentration. The composition dependence of the resistivity and the AMR of the annealed Ni Co alloy deposits was in good quantitative agreement with the available literature data both at 13 K and at room temperature. Both transport parameters were found to exhibit a pronounced maximum in the composition range between 20 and 30 at.% Co and the data of the Ni Co alloys fit well to the limiting values of the pure component metals (fcc Ni and fcc Co). The only theoretical calculation reported formerly on fcc Ni Co alloys yielded at T=0K a resistivity value smaller by a factor of 5 and an AMR value larger by a factor of about 2 than the corresponding low temperature experimental data, although the theoretical results properly reproduced the composition dependence of both quantities., Comment: 27 pages, 10 figures

Decades ago two classes of gamma-ray bursts were identified and delineated as having durations shorter and longer than about 2 s. Subsequently indications also supported the existence of a third class. Using maximum likelihood estimation we analyze the duration distribution of 888 Swift BAT bursts observed before October 2015. Fitting three log-normal functions to the duration distribution of the bursts provides a better fit than two log-normal distributions, with 99.9999% significance. Similarly to earlier results, we found that a fourth component is not needed. The relative frequencies of the distribution of the groups are 8% for short, 35% for intermediate and 57% for long bursts which correspond to our previous results. We analyse the redshift distribution for the 269 GRBs of the 888 GRBs with known redshift. We find no evidence for the previously suggested difference between the long and intermediate GRBs' redshift distribution. The observed redshift distribution of the 20 short GRBs differs with high significance from the distributions of the other groups., Comment: accepted