Let a finite set of points $\{\xi_1,...,\xi_k\}$ be chosen in a metric space $(X,\mathfrak{D})$, and let the squared distance matrix $\mathfrak{D}=(\mathfrak{D}(\xi_i,\xi_j)^2)_{i,j=1}^{k}$ be constructed from them. We propose a geometric approach to studying the spectral properties of squared distance matrices of infinite size, constructed from a countable set of points $\{\xi_k\}_{k\in \mathbb{Z}}$ on Riemannian manifold $(M,g)$. We move from the discrete problem to a continuous one using walk matrices. We describe the structure of the spectrum and study the properties of spectral flows.

The process of writing code and use of features in an integrated development environment (IDE) is a fruitful source of data in computing education research. Existing studies use records of students' actions in the IDE, consecutive code snapshots, compilation events, and others, to gain deep insight into the process of student programming. In this paper, we present a set of tools for collecting and processing data of student activity during problem-solving. The first tool is a plugin for IntelliJ-based IDEs (PyCharm, IntelliJ IDEA, CLion). By capturing snapshots of code and IDE interaction data, it allows to analyze the process of writing code in different languages -- Python, Java, Kotlin, and C++. The second tool is designed for the post-processing of data collected by the plugin and is capable of basic analysis and visualization. To validate and showcase the toolkit, we present a dataset collected by our tools. It consists of records of activity and IDE interaction events during solution of programming tasks by 148 participants of different ages and levels of programming experience. We propose several directions for further exploration of the dataset.

For a given metric space $(P,\phi)$, a tree cover of stretch $t$ is a collection of trees on $P$ such that edges $(x,y)$ of trees receive length $\phi(x,y)$, and such that for any pair of points $u,v\in P$ there is a tree $T$ in the collection such that the induced graph distance in $T$ between $u$ and $v$ is at most $t\phi(u,v).$ In this paper, we show that, for any set of points $P$ on the Euclidean plane, there is a tree cover consisting of two trees and with stretch $O(1).$ Although the problem in higher dimensions remains elusive, we manage to prove that for a slightly stronger variant of a tree cover problem we must have at least $(d+1)/2$ trees in any constant stretch tree cover in $\mathbb R^d$.

The purpose of this work is an attempt to expand the results obtained by A. K. Lakmon and Y. Mensah on embeddings of quantum Sobolev spaces $\mathfrak{H}_\gamma^{s,p}(G,H)$ consisting of Hilbert-Schmidt operators, with $p\neq 2$.

Generalized pseudo-Hamiltonian normal forms (GPHNF) and an effective method of obtaining them are introduced for two-dimensional systems of autonomous ODEs with a Hamiltonian quasi-homogeneous unperturbed part of an arbitrary degree. The terms that can be additionally eliminated in a GPHNF are constructively distinguished, and it is shown that after removing them GPHNF becomes a generalized normal form (GNF). By using the introduced method, all the GNFs are obtained in cases where the unperturbed part of the system is Hamiltonian and has monomial components, which allowed to generalize some results by Takens, Baider and Sanders, as well as Basov et al.

Parametric integration with hyperlogarithms so far has been successfully used in problems of high energy physics (HEP) and critical statics. In this work, for the first time, it is applied to a problem of critical dynamics, namely, a stochastic model of developed turbulence in high-dimensional spaces, which has a propagator that is non-standard with respect to the HEP: $(-i \omega + \nu k^2)^{-1}$. Adaptation of the hyperlogarithm method is carried out by choosing a proper renormalization scheme and considering an effective dimension of the space. Analytical calculation of the renormalization group functions is performed up to the fourth order of the perturbation theory, $\varepsilon$-expansion of the critical exponent $\omega$ responsible for the infrared stability of the fixed point is obtained.

We present python libraries for Feynman graphs manipulation. The key feature of these libraries is usage of generalization of graph representation offered by B. G. Nickel et al. In this approach graph is represented in some unique 'canonical' form that depends only on its combinatorial type. The uniqueness of graph representation gives an efficient way for isomorphism finding, searching for subgraphs and other graph manipulation tasks. Though offered libraries were originally designed for Feynman graphs, they might be useful for more general graph problems.

3C 84 is the brightest cluster galaxy in the Perseus Cluster. It is among the closest radio-loud active galaxies and among the very few that can be detected from low frequency radio up to TeV $\gamma$-rays. Here we report on the first X-ray polarization observation of 3C~84 with the Imaging X-ray Polarimetry Explorer, for a total of 2.2 Msec that coincides with a flare in $\gamma$-rays. This is the longest observation for a radio-loud active galaxy that allowed us to reach unprecedented sensitivity, leading to the detection of an X-ray polarization degree of $\rm\Pi_X=4.2\pm1.3\%$ ($\sim3.2\sigma$ confidence) at an X-ray electric vector polarization angle of $\rm \psi_X=163^{\circ}\pm9^{\circ}$, that is aligned with the radio jet direction on the sky. Optical polarization observations show fast variability about the jet axis as well. Our results strongly favor models in which X-rays are produced by Compton scattering from relativistic electrons -- specifically Synchrotron Self-Compton -- that takes places downstream, away from the supermassive black hole.

In this paper, we investigate gravitational waves beyond the linear approximation, focusing on second-order contributions sourced by linearized waves in the transverse-traceless (TT) gauge. A general spacetime metric is constructed, and both timelike and null geodesic congruences are analyzed. For the timelike congruence, a non-vanishing expansion scalar and shear tensor are obtained, while the rotation tensor is found to vanish. In contrast, all these quantities vanish for the null congruence. Using a parallel-transported orthonormal tetrad, we derive the geodesic deviation equations up to second order in the wave amplitude $H$, showing that, as in the linear case, stretching and compression occur in the transverse $x$ and $y$ directions. However, when solving the geodesic equations numerically within the 3+1 formalism, we observe an additional effect: test particles undergo a small, second-order displacement along the direction of wave propagation.

The spin degree of freedom of charge carriers in halide-perovskite semiconductors can be highly useful for information photonics applications. The Faraday effect is known to be the best indicator of paramagnetism of the material and of the spin-light interaction. In this work, the Faraday effect is demonstrated, for the first time, in a hybrid organic-inorganic halide perovskite MAPbI3 (MA+=CH3NH+3). The Faraday rotation and birefringence were measured across the tetragonal-cubic phase transition at 327 K. The Faraday rotation is strongly suppressed below the phase transition temperature due to anisotropy (linear birefringence) of the tetragonal crystal phase. The situation changes drastically above the phase transition temperature, when the crystal becomes optically isotropic. The emerging Faraday rotation obeys the Curie law, demonstrating its population-related paramagnetic nature. This observation opens new prospects for application of these systems and for their investigations using methods of the polarization noise spectroscopy applicable to optically anisotropic materials.

Motivated by recent experimental data on dichloro-tetrakis thiourea-nickel (DTN) [T.A. Soldatov $\textit{et al}$, Phys. Rev. B ${\bf 101}$, 104410 (2020)], a model of antiferromagnet on a tetragonal lattice with single-ion easy-plane anisotropy in the tilted external magnetic field is considered. Using the smallness of the in-plane field component, we analytically address field dependence of the energy gap in ``acoustic'' magnon mode and transverse uniform magnetic susceptibility in the ordered phase. It is shown that the former is non-monotonic due to quantum fluctuations, which was indeed observed experimentally. The latter is essentially dependent on the ``optical'' magnon rate of decay on two magnons. At magnetic fields close to the one which corresponds to the center of the ordered phase, it leads to experimentally observed dynamical diamagnetism phenomenon.

We consider the thin layer quantization with use of only the most elementary notions of differential geometry. We consider this method in higher dimensions and get an explicit formula for quantum potential. For codimension 1 surfaces the quantum potential is presented in terms of principal curvatures, and equivalence with Prokhorov quantization method is proved. It is shown that, in contrast with original da Costa method, Prokhorov quantization can be generalized directly to higher codimensions.

Recently, an interesting gravitational model was proposed in order to mimic the effect of Dark Matter. Chamseddine and Mukhanov in the arXiv preprint 1308.5410 have separated the conformal mode of a physical metric in the form of a squared gradient of an auxiliary scalar field. Notably, the variational principle has given a more general equation of motion than that of purely Einsteinian relativity theory, with a possibility of reproducing an effective Dark Matter. In this short paper, we explain the nature of this phenomenon in terms of the class of functions on which the variation takes place. Then we give a more transparent equivalent formulation of the model without an auxiliary metric. Finally, we speculate a bit about possible extensions.

In the age of big data, sorting is an indispensable operation for DBMSes and similar systems. Having data sorted can help produce query plans with significantly lower run times. It also can provide other benefits like having non-blocking operators which will produce data steadily (without bursts), or operators with reduced memory footprint. Sorting may be required on any step of query processing, i.e., be it source data or intermediate results. At the same time, the data to be sorted may not fit into main memory. In this case, an external sort operator, which writes intermediate results to disk, should be used. In this paper we consider an external sort operator of the comparison-based sort type. We discuss its implementation and describe related design decisions. Our aim is to study the impact on performance of a data structure used on the merge step. For this, we have experimentally evaluated three data structures implemented inside a DBMS. Results have shown that it is worthwhile to make an effort to implement an efficient data structure for run merging, even on modern commodity computers which are usually disk-bound. Moreover, we demonstrated that using a loser tree is a more efficient approach than both the naive approach and the heap-based one.

We advance in constructing a bottom-up holographic theory of linear meson Regge trajectories that generalizes and unites into one logical framework various bottom-up holographic approaches proposed in the past and scattered in the literature. The starting point of the theory is a quadratic in fields holographic five-dimensional action in which the Poincar\'{e} invariance along the holographic coordinate is violated in the most general way compatible with the linear Regge behavior of the discrete spectrum in four dimensions. It is further demonstrated how different Soft Wall (SW) like holographic models existing in the literature plus some new ones emerge from our general setup. Various interrelations between the emerging models are studied. These models include the known SW models with different sign in the exponential background, the SW models with certain generalized backgrounds, with modified metrics, and No Wall models with 5D mass depending on the holographic coordinate in a simple polynomial way. We argue that this dependence allows to describe the effects caused by the main non-local phenomena of strongly coupled 4D gauge theory, the confinement and chiral symmetry breaking, in terms of a local 5D dual field theory in the AdS space. We provide a detailed comparison of our approach with the Light Front holographic QCD, with the spectroscopic predictions of the dual Veneziano like amplitudes, and with the experimental Regge phenomenology. We apply our general approach to a holographic study of confinement, chiral symmetry breaking, and the pion form factor.

The advancement of the Event Horizon Telescope has enabled the study of relativistic jets in active galactic nuclei down to sub-parsec linear scales even at high redshift. Quasi-simultaneous multifrequency observations provide insights into the physical conditions in compact regions and allow testing accretion theories. Initially we aimed at measuring the magnetic field strength close to the central supermassive black hole in NRAO 530 (1730-130) by studying frequency-dependent opacity of the jet matter, Faraday rotation and the spectral index in the mm-radio bands. NRAO 530 was observed quasi-simultaneously at 15, 22, 43, 86, and 227 GHz at four different very long baseline interferometer (VLBI) networks. By the means of imaging and model-fitting, we aligned the images, taken at different frequencies. We explored opacity along the jet and distribution of the linearly polarized emission in it. Our findings reveal that the jet of NRAO 530 at 86 and 227 GHz is transparent down to its origin, with 70 mJy emission detected at 227 GHz potentially originating from the accretion disk. The magnetic field strength near the black hole, estimated at $5r_\mathrm{g}$, is $3\times10^3-3\times10^4$ G (depending on the central black hole mass). These values represent some of the highest magnetic field strengths reported for active galaxies. We also report the first ever VLBI measurement of the Faraday rotation at 43-227 GHz, which reveals rotation measure values as high as -48000 rad/m2 consistent with higher particle density and stronger magnetic fields at the jet's outset. The complex shape of the jet in NRAO 530 is in line with the expected behavior of a precessing jet, with a period estimated to be around $6\pm4$~years.

We have measured direct photons for p_T<5~GeV/$c$ in minimum bias and 0\%--40\% most central events at midrapidity for Cu$+$Cu collisions at $\sqrt{s_{_{NN}}}=200$ GeV. The $e^{+}e^{-}$ contribution from quasi-real direct virtual photons has been determined as an excess over the known hadronic contributions in the $e^{+}e^{-}$ mass distribution. A clear enhancement of photons over the binary scaled $p$$+$$p$ fit is observed for p_T<4 GeV/$c$ in Cu$+$Cu data. The $p_T$ spectra are consistent with the Au$+$Au data covering a similar number of participants. The inverse slopes of the exponential fits to the excess after subtraction of the $p$$+$$p$ baseline are 285$\pm$53(stat)$\pm$57(syst)~MeV/$c$ and 333$\pm$72(stat)$\pm$45(syst)~MeV/$c$ for minimum bias and 0\%--40\% most central events, respectively. The rapidity density, $dN/dy$, of photons demonstrates the same power law as a function of $dN_{\rm ch}/d\eta$ observed in Au$+$Au at the same collision energy.

We fit various colour-magnitude diagrams (CMDs) of the high-latitude Galactic globular clusters NGC\,5024 (M53), NGC\,5053, NGC\,5272 (M3), NGC\,5466, and NGC\,7099 (M30) by isochrones from the Dartmouth Stellar Evolution Database and Bag of Stellar Tracks and Isochrones for $\alpha$-enrichment [$\alpha$/Fe]$=+0.4$. For the CMDs, we use data sets from {\it Hubble Space Telescope}, {\it Gaia}, and other sources utilizing, at least, 25 photometric filters for each cluster. We obtain the following characteristics with their statistic uncertainties for NGC\,5024, NGC\,5053, NGC\,5272, NGC\,5466, and NGC\,7099, respectively: metallicities [Fe/H]$=-1.93\pm0.02$, $-2.08\pm0.03$, $-1.60\pm0.02$, $-1.95\pm0.02$, and $-2.07\pm0.04$ dex with their systematic uncertainty 0.1 dex; ages $13.00\pm0.11$, $12.70\pm0.11$, $11.63\pm0.07$, $12.15\pm0.11$, and $12.80\pm0.17$ Gyr with their systematic uncertainty 0.8 Gyr; distances (systematic uncertainty added) $18.22\pm0.06\pm0.60$, $16.99\pm0.06\pm0.56$, $10.08\pm0.04\pm0.33$, $15.59\pm0.03\pm0.51$, and $8.29\pm0.03\pm0.27$ kpc; reddenings $E(B-V)=0.023\pm0.004$, $0.017\pm0.004$, $0.023\pm0.004$, $0.023\pm0.003$, and $0.045\pm0.002$ mag with their systematic uncertainty 0.01 mag; extinctions $A_\mathrm{V}=0.08\pm0.01$, $0.06\pm0.01$, $0.08\pm0.01$, $0.08\pm0.01$, and $0.16\pm0.01$ mag with their systematic uncertainty 0.03 mag, which suggest the total Galactic extinction $A_\mathrm{V}=0.08$ across the whole Galactic dust to extragalactic objects at the North Galactic pole. The horizontal branch morphology difference of these clusters is explained by their different metallicity, age, mass-loss efficiency, and loss of low-mass members in the evolution of the core-collapse cluster NGC\,7099 and loose clusters NGC\,5053 and NGC\,5466.