On adjacency operators of locally finite graphs

A graph $\Gamma$ is called locally finite if, for each vertex $v$ of $\Gamma$, the set $\Gamma(v)$ of all neighbors of $v$ in $\Gamma$ is finite. For any locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and for any field $F$, let $F^{V(\Gamma)}$ be the vector space over $F$ of all functions $V(\Gamma) \to F$ (with natural componentwise operations) and let $A^{({\rm alg})}_{\Gamma,F}$ be the linear operator $F^{V(\Gamma)} \to F^{V(\Gamma)}$ defined by $(A^{({\rm alg})}_{\Gamma,F}(f))(v) = \sum_{u \in \Gamma(v)}f(u)$ for all $f \in F^{V(\Gamma)}$, $v \in V(\Gamma)$. In the case of finite graph $\Gamma$ the mapping $A^{({\rm alg})}_{\Gamma,F}$ is the well known operator defined by the adjacency matrix of $\Gamma$ (over $F$), and the theory of eigenvalues and eigenfunctions of such operator is a well-developed (at least in the case $F = \mathbb{C}$) part of the theory of finite graphs. In this paper we develope a theory of eigenvalues and eigenfunctions of $A^{({\rm alg})}_{\Gamma,F}$ for arbitrary infinite locally finite graphs $\Gamma$ (although a few results may be of interest for finite graphs) and fields $F$ with a special emphasis on the case when $\Gamma$ is connected with uniformly bounded vertex degrees and $F = \mathbb{C}$. By the author opinion, previous attempts in this direction were not quite satisfactory since were limited by consideration of rather special eigenfunctions and corresponding eigenvalues.Comment: in Russia

Systems of Hess-Appel'rot Type and Zhukovskii Property

We start with a review of a class of systems with invariant relations, so called {\it systems of Hess--Appel'rot type} that generalizes the classical Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the {\it Zhukovskii property}: these are Hamiltonian systems with invariant relations, such that partially reduced systems are completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rote type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type - the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian $Gr^+(4,2)$ has natural interpretation within Zhukovskii property and it is equivalent to a partial reduction of certain system of Hess-Appel'rot type. We perform a classical and an algebro-geometric integration of the system, as an example of an isoholomorphic system. The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems.Comment: 42 page

Magnetic and structural properties of barium hexaferrite BaFe12O19 from various growth techniques

Barium hexaferrite powder samples with grains in the m-range were obtained from solid-state sintering, and crystals with sizes up to 5 mm grown from PbO, Na2CO3, and BaB2O4 fluxes, respectively. Carbonate and borate fluxes provide the largest and structurally best crystals at significantly lower growth temperatures of 1533 K compared to flux-free synthesis (1623 K). The maximum synthesis temperature can be further reduced by the application of PbO-containing fluxes (down to 1223 K upon use of 80 at % PbO), however, Pb-substituted crystals Ba1-xPbxFe12O19 with Pb contents in the range of 0.23(2) x 0.80(2) form, depending on growth temperature and flux PbO content. The degree of Pb-substitution has only a minor influence on unit cell and magnetic parameters, although the values for Curie temperature, saturation magnetization, as well as the coercivity of these samples are significantly reduced in comparison with those from samples obtained from the other fluxes. Due to the lowest level of impurities, the samples from carbonate flux show superior quality compared to materials obtained using other methods

MUSiC : a model-unspecific search for new physics in proton-proton collisions at root s=13TeV

Results of the Model Unspecific Search in CMS (MUSiC), using proton-proton collision data recorded at the LHC at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb(-1), are presented. The MUSiC analysis searches for anomalies that could be signatures of physics beyond the standard model. The analysis is based on the comparison of observed data with the standard model prediction, as determined from simulation, in several hundred final states and multiple kinematic distributions. Events containing at least one electron or muon are classified based on their final state topology, and an automated search algorithm surveys the observed data for deviations from the prediction. The sensitivity of the search is validated using multiple methods. No significant deviations from the predictions have been observed. For a wide range of final state topologies, agreement is found between the data and the standard model simulation. This analysis complements dedicated search analyses by significantly expanding the range of final states covered using a model independent approach with the largest data set to date to probe phase space regions beyond the reach of previous general searches.Peer reviewe

Development and validation of HERWIG 7 tunes from CMS underlying-event measurements

This paper presents new sets of parameters (“tunes”) for the underlying-event model of the HERWIG7 event generator. These parameters control the description of multiple-parton interactions (MPI) and colour reconnection in HERWIG7, and are obtained from a fit to minimum-bias data collected by the CMS experiment at s=0.9, 7, and 13Te. The tunes are based on the NNPDF 3.1 next-to-next-to-leading-order parton distribution function (PDF) set for the parton shower, and either a leading-order or next-to-next-to-leading-order PDF set for the simulation of MPI and the beam remnants. Predictions utilizing the tunes are produced for event shape observables in electron-positron collisions, and for minimum-bias, inclusive jet, top quark pair, and Z and W boson events in proton-proton collisions, and are compared with data. Each of the new tunes describes the data at a reasonable level, and the tunes using a leading-order PDF for the simulation of MPI provide the best description of the dat

Fabrication of High-Entropy Alloys Using a Combination of Detonation Spraying and Spark Plasma Sintering: A Case Study Using the Al-Fe-Co-Ni-Cu System

The use of pre-alloyed powders as high-entropy alloy (HEA) coating precursors ensures a predetermined (unaltered) elemental composition of the coating with regard to the feedstock powder. At the same time, it is interesting to tackle a more challenging task: to form alloy coatings from powder blends (not previously alloyed). The powder-blend-based route of coating formation eliminates the need to use atomization or ball milling equipment for powder preparation and allows for the introduction of additives into the material in a flexible manner. In this work, for the first time, a HEA was obtained using detonation spraying (DS) followed by spark plasma sintering (SPS). A powder mixture with a nominal composition of 10Al-22.5Fe-22.5Co-22.5Ni-22.5Cu (at.%) was detonation-sprayed to form a multicomponent metallic coating on a steel substrate. The elemental composition of the deposited layer was (9 ± 1)Al-(10 ± 1)Fe-(20 ± 1)Co-(34 ± 1)Ni-(27 ± 1)Cu (at.%), which is different from that of the feedstock powder because of the differences in the deposition efficiencies of the metals during DS. Despite the compositional deviations, the deposited layer was still suitable as a precursor for a HEA with a configurational entropy of ~1.5R, where R is the universal gas constant. The subsequent SPS treatment of the substrate/coating assembly was carried out at 800–1000 °C at a uniaxial pressure of 40 MPa. The SPS treatment of the deposited layer at 1000 °C for 20 min was sufficient to produce an alloy with a single-phase face-centered cubic structure and a porosity of 0.3). The hardness of the coatings measured in two perpendicular directions did not differ significantly. The features of the DS–SPS route of the formation of HEA coatings and its potential applications are discussed