Search

Graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a greater role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e.g. via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Finally, we provide examples of visual certificates for problems in different visual proof complexity classes., Comment: Appears in the Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

The universal relations for spin-$1/2$ fermions with contact interaction in the presence of quenched disorder are discussed. The disorder is modeled by a random external potential with the Gaussian distribution and $\delta$-like two-point correlation function. Utilizing simple scaling arguments, the renormalizability of the theory, and the Hellmann-Feynman theorem we identified the large-momentum tail of particle distribution and analog of Tan's energy relation for many-fermion systems with disorder in arbitrary dimension $d\ge 2$. It is shown that the energy-pressure relation manifests a kind of scale anomaly in two and three dimensions., Comment: 7 pages, 1 figure; Please, feel free to comment and provide relevant references

This paper introduces a novel algorithm for two-player deterministic games with perfect information, which we call PROBS (Predict Results of Beam Search). Unlike existing methods that predominantly rely on Monte Carlo Tree Search (MCTS) for decision processes, our approach leverages a simpler beam search algorithm. We evaluate the performance of our algorithm across a selection of board games, where it consistently demonstrates an increased winning ratio against baseline opponents. A key result of this study is that the PROBS algorithm operates effectively, even when the beam search size is considerably smaller than the average number of turns in the game., Comment: 8 pages, 4 figures

Thermodynamically stable low-temperature phases of the Bose-Fermi mixtures composed of bosons and spinless fermions close to four dimensions are considered. In the regime, where the only boson-fermion two-body interaction is present and tuned to unitary limit, the properties of a system solely depend on the mass and number ratios of constituent atoms. In addition to the phase with the dimers (boson-fermion shallow bound states), we identified one more state of the mixture with the coexistence of fermionic dimers and trimers. The universal physics of these phases, the characteristic feature of is absence of the Bose-Einstein condensate, is discussed., Comment: 7+epsilon pages, 7 figures, comments are welcome

The properties of the one-dimensional $SU(3)$ population-imbalanced fermions are discussed. The system is assumed to be in the two-body resonance where all two-body scattering lengths diverge, and the only interaction between fermions that is taken into account is the short-range three-body one. In particular, we consider the situation when only one `flavor' of fermions is macroscopically occupied, and there are exactly two atoms of two others. This system supports the trimer and the medium-induced dimer states studied here in detail and shows evidence of color superfluidity., Comment: 5 pages, 3 figures; comments welcome

Inspired by the narrow Feshbach resonance in systems with the two-body interaction, we propose the two-channel model of three-component fermions with the three-body interaction that takes into account the finite-range effects in low dimensions. Within this model, the $p$-wave Efimov-like effect in the four-body sector is predicted in fractional dimensions above 1D. The impact of the finite-range interaction on the formation of the four-body bound states in $d=1$ is also discussed in detail., Comment: 5+epsilon pages, 1 table, 5 figures; comments welcome

This paper is dedicated to the fused trend filtering on a general graph, which is a combination of fused estimator and 1-st order trend filtering on a graph. There are two cases of fusion regularisers studied in this work: anisotropic total variation (i.e. fused lasso) and nearly-isotonic restriction. For the trend filtering part we consider general trend filtering on a given graph and Kronecker trend filter for the case of lattice data. We show how these estimators are related to each other and propose a computationally feasible numerical solution with a linear complexity per iteration with respect to the amount of edges in the graph.

The ground state of spin-$1\over 2$ fermions with contact $s$-wave inter- and $p$-wave intra-species interactions is discussed. Particularly, we formulate the mean field scheme for calculating thermodynamic properties of the system in arbitrary dimension $D<2$ and discuss in detail the phase diagram in 1D case. Except clean phases with either singlet or triplet Cooper pairings, we have identified two mixed phases (one stable and another metastable) of the one-dimensional two-component fermions where both pairing mechanisms coexist., Comment: 4+epsilon pages, 2 figures

We formulate a general scheme for calculation of thermodynamic properties of ideal Bose gas with microscopic number of static impurities immersed, when the system is loaded in the harmonic trapping potential with quasi-1D and quasi-2D configurations. The binding energy of a single impurity and a detailed study of the medium-induced Casimir-like forces between two impurities in trapped Bose gas are numerically calculated in wide range of temperatures and interaction strengths., Comment: 6+epsilon pages, 8 figures

In this paper we integrate isotonic regression with Stone's cross-validation-based method to estimate a distribution with a general countable support with a partial order relation defined on it. We prove that the estimator is strongly consistent for any underlying distribution, derive its rate of convergence, and in the case of one-dimensional support we obtain Marshal-type inequality for cumulative distribution function of the estimator. Also, we construct the asymptotically correct conservative global confidence band for the estimator. It is shown that, first, the estimator performs good even for small sized data sets, second, the estimator outperforms in the case of non-isotonic underlying distribution, and, third, it performs almost as good as Grenander estimator when the true distribution is isotonic. Therefore, the new estimator provides a trade-off between goodness-of-fit, monotonicity and quality of probabilistic forecast. We apply the estimator to the time-to-onset data of visceral leishmaniasis in Brazil collected from $2007$ to $2014$.

In this paper we introduce and study fused lasso nearly-isotonic signal approximation, which is a combination of fused lasso and generalized nearly-isotonic regression. We show how these three estimators relate to each other, derive solution to the general problem, show that it is computationally feasible and provides a trade-off between piecewise monotonicity, sparsity and goodness-of-fit. Also, we derive an unbiased estimator of the degrees of freedom of the approximator.

We discuss an equilibrium mean-field properties of mixtures consisting of bosons and spin-polarized fermionic atoms with a point-like interaction in an arbitrary dimension $2

We discuss properties of a single impurity atom immersed in the spin-$1/2$ dilute Fermi gas with equal populations of two species in the deep Bose-Einstein condensate (BEC) phase. In this limit, when an almost undepleted BEC of the tightly bound molecules of spin-up and spin-down fermions is formed, we calculate the parameters of an impurity spectrum. It is justified that the leading-order contribution to the impurity energy, while being determined by the two- and three-body scattering processes, is dominated by the former ones., Comment: 6 pages, 7 figures; comments welcome

We present a comprehensive discussion of the ground-state properties of dilute $D$-dimensional Bose gas interacting with a few static impurities. Assuming the short-ranged character of the boson-impurity interaction, we calculate the energy of three- and two-dimensional Bose systems with one and two impurities immersed., Comment: 7 pages, 6 figures

A few-body properties of spinless Bose particles interacting via the contact three-body potential in geometries with fractional dimensions $1

In this paper we consider the stacking of isotonic regression and the method of rearrangement with the empirical estimator to estimate a discrete distribution with an infinite support. The estimators are proved to be strongly consistent with $\sqrt{n}$-rate of convergence. We obtain the asymptotic distributions of the estimators and construct the asymptotically correct conservative global confidence bands. We show that stacked Grenander estimator outperforms the stacked rearrangement estimator. The new estimators behave well even for small sized data sets and provide a trade-off between goodness-of-fit and shape constraints.

A finit periodic $\delta-\delta'$ comb was solved by the help of both classical approach based on a direct solving of a Sr\"{odinger} equation and a quantum wave impedance method. It was demonstrated that the violation of a periodicity leads to the formation of energy levels in band gaps. The expresion for a surface levels (Tamm's levels) were found in this model system. Obtained results allow extending the scope of theoretical models which are applicable for a describing real quantum mechanical systems as well as better understanding the physical properties of systems in which the boundary dependent states are involved, especially the influence of boundary dependent electronic states on the physical properties of low-dimensional systems., Comment: 13 pages, 2 figures

By using simple and efficient method we discuss properties of a single impurity immersed in three-dimensional Bose gas with the interaction between particles tuned to unitary limit. Particularly, adopting the mean-field-like approximation we present the first estimations for the low-momentum parameters of the impurity spectrum, namely, the binding energy, the effective mass and the quasiparticle residue both for repulsive and attractive Bose polarons in the unitary gas., Comment: 6 pages, 9 figures

We analyze zero-temperature universal properties of the simplest Galilean-invariant model of spinless low-dimensional fermions with short-range two-body interactions. In particular, it is shown that after proper renormalization of the coupling constant, even the dilute system possesses rich phase diagram that includes the superfluid state and the metastable `upper branch' behavior., Comment: 5+epsilon pages, 5 figures, comments (criticism) are welcome

The detailed mean-field treatment of the Bose polaron problem in two and three dimensions is presented. Particularly, assuming that impurity is immersed in the dilute Bose gas and interacts with bosons via the hard-sphere two-body potential, we calculate the low-momentum parameters of its spectrum, namely, the binding energy and the effective mass. The limits of applicability of the mean-field approach to a problem of mobile impurity in Bose-Einstein condensates are discussed by comparing our results to the Monte Carlo simulations data., Comment: 9 pages, 9 figures; the discussion of the 2D Bose polaron has been extended; the misleading statement about the average density of bosons has been corrected

We exactly calculate the full temperature dependence of Casimir-like forces appearing between two and three static impurities loaded in the ideal Bose gas below the Bose-Einstein condensation transition point. Assuming the short-ranged character of the boson-impurity interaction, the calculation procedure presented here can be easily extended on a Bose system with an arbitrary number of impurities immersed., Comment: 9+epsilon pages, 8 figures; the manuscript was substantially extended

The two-component mixture of Bose particles with short-range pairwise interaction at finite temperatures in three dimensions is considered. Particularly we examine, by means of the large-$N$ expansion technique, the stability of mixed state below the Bose-Einstein transition point and the temperature dependence of the condensate density for symmetric mixture of Bose gases. The presented analysis reveals the importance of finite-temperature excitations of the non-condensed particles in formation of the phase diagram of two-component Bose systems., Comment: 6 pages, 5 figures

The full momentum dependence of spectrum of a point-like impurity immersed in a dilute one-dimensional Bose gas is calculated on the mean-field level. In particular we elaborate, to the finite-momentum Bose polaron, the path-integral approach whose semi-classical approximation leads to the conventional mean-field treatment of the problem while quantum corrections can be easily accounted by standard loop expansion techniques. The extracted low-energy parameters of impurity spectrum, namely, the binding energy and the effective mass of particle, are shown to be in qualitative agreement with the results of quantum Monte Carlo simulations., Comment: 8+epsilon pages, 7 figures

We consider a system of interacting bosons in one dimension at a two-body resonance. This system, which is weakly interacting, is known to give rise to effective three-particle interactions, whose dynamics is similar to that of a two-dimensional Bose gas with two-body interactions, and exhibits an identical scale anomaly. We consider the experimentally relevant scenario of a harmonically trapped system. We solve the three-body problem exactly and evaluate the shifts in the frequency of the lowest compressional mode with respect to the dipole mode, and find that the effect of the anomaly is to increase the mode's frequency. We also consider the weak-coupling regime of the trapped many-boson problem and find, within the local density approximation, that the frequency of the lowest compressional mode is also shifted upwards in this limit. Moreover, the anomalous frequency shifts are enhanced by the higher particle number to values that should be observable experimentally., Comment: 8 pages, 2 figures

We perform the comprehensive comparison of properties of the condensate and superfluid densities for the $N$-component three-dimensional Bose gas with the symmetric inter- and intraspecies short-range interaction between particles. In particular, based on the large-$N$ expansion approach for many-boson systems we obtain general expression for density of the superfluid component that at very low temperatures reproduce the well-know Landau's formula and non-trivially includes the thermal fluctuations in the finite-temperature region, and compare it to the condensate density calculated previously. The numerically evaluated temperature dependencies are in a qualitatively good agreement with the results of Monte Carlo simulations., Comment: 6 pages, 3 figures; false statement about the nature of critical exponent for the superfluid density was clarified

We exactly analyze, on the mean-field level, the low-momentum properties of a single impurity atom loaded in the dilute one-dimensional Bose gas with two- and three-body short-range interactions. Particularly the Bose polaron binding energy and the quasiparticle residue are calculated for the considered system in the broad region of parameters change. We also explore the generic mean-field formula for the polaron effective mass which was shown to depend on the density profile of bath particles with a motionless impurity immersed., Comment: 6+epsilon pages, 6 figures

In this paper we propose estimators of the distribution of events of different kinds in a multimode Poisson process. We give the explicit solution for the maximum likelihood estimator, and derive its strong consistency and asymptotic normality. We also provide an order restricted estimator and derive its consistency and asymptotic distribution. We discuss the application of the estimator to the detection of neutrons in a novel detector being developed at the European Spallation Source in Lund, Sweden. The inference problem gives rise to Sylvester-Ramanujan system of equations.

We determined perturbatively the low-energy universal thermodynamics of dilute one-dimensional bosons with the three-body repulsive forces. The final results are presented for the limit of vanishing potential range in terms of three-particle scattering length. An analogue of Tan`s energy theorem for considered system is derived in generic case without assuming weakness of the interparticle interaction. We also obtained an exact identity relating the three-body contact to the energy density., Comment: 4+epsilon pages, 2 figures

We rigorously discuss the large-$N$ thermodynamics of a Bose gas with a short-range two-body potential. Considering the system as a mixture of $N$ identical components with symmetrical interaction we calculated numerically the temperature dependence of the leading-order corrections to the depletion of Bose-Einstein condensate and to the isothermal compressibility., Comment: 5 pages, 4 figures

We revisit the problem of the calculation of low-temperature properties for the dilute two-dimensional Bose gas. By using Popov's hydrodynamic approach and perturbation theory on the one-loop level we recover not only the known expansion for the ground-state energy but also calculate for the first time the condensate density and Tan's contact., Comment: 10+epsilon pages, 1 figure; comments welcome

The properties of a dilute Bose gas with the non-Gaussian quenched disorder are analysed. Being more specific we have considered a system of bosons immersed in the classical bath consisting of the non-interacting particles with infinite mass. Making use of perturbation theory up to second order we have studied the impact of environment on the ground-state thermodynamic and superfluid characteristics of the Bose component., Comment: almost 7 pages; comments welcome

The properties of a Bose polaron immersed in a dilute two-dimensional medium at finite temperatures are discussed. Assuming that the impurity is weakly-coupled to the bath particles we have perturbatively calculated the polaron energy, effective mass, quasiparticle residue and damping rate. The parameters of impurity spectrum are found to be well-defined in the whole temperature region whereas the pole structure of the impurity Green's function is visible only at absolute zero. At any finite temperatures the quasiparticle residue is logarithmically divergent signalling of the branch-cut behavior of the polaron propagator., Comment: 4+epsilon pages, 3 figures

The properties of impurity immersed in the dilute $D$-dimensional Bose gas at temperatures close to the second-order phase transition point are considered. Particularly by means of the $1/N$-expansion we calculated the leading-order polaron energy and the damping rate in the limit of vanishing boson-boson interaction. It is show that the perturbative effective mass and the quasiparticle residue diverge logarithmically in the long-length limit signalling the non-analytic behavior of impurity spectrum and a non-pole structure of a polaron Green's function in the infrared region, respectively., Comment: 10+epsilon pages, 5 figures

We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator's asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.

The thermodynamic and superfluid properties of the dilute two-dimensional binary Bose mixture at low temperatures are discussed. We also considered the problem of the emergence of the long-range order in these systems. All calculations are performed by means of celebrated Popov's path-integral approach for the Bose gas with a short-range interparticle potential., Comment: 5+epsilon pages, 2 figures

The detailed study of the low-energy spectrum for a mobile impurity in the one-dimensional bosonic enviroment is performed. Particularly we have considered only two analytically accessible limits, namely, the case of an impurity immersed in a dilute Bose gas, where one can use many-body perturbation techniques for low-dimensional bosonic systemsm and the case of the Tonks-Girardeau gas, for which the usual fermionic diagrammatic expansion up to the second order is applied., Comment: 7 pages, 8 figures, references added, misprints corrected

We proposed the non-perturbative scheme for calculation of the impurity spectrum in the Bose system at zero temperature. The method is based on the path-integral formulation and describes an impurity as a zero-density ideal Fermi gas interacting with Bose system for which the action is written in terms of density fluctuations. On the example of the $^3$He atom immersed in the liquid helium-4 a good consistency with experimental data and results of Monte Carlo simulations is shown., Comment: 5 pages, 4 figures

We revised the large-$N$ expansion for a three-dimensional Bose system with short-range repulsion in normal phase. Particularly, for the model potential that is characterised only by the $s$-wave scattering length $a$ the full numerical calculations of the critical temperature in the $1/N$-approximation as a function of the gas parameter $an^{1/3}$ are performed. Additionally to the well-known result in the dilute limit we estimated analytically the leading-order strong-coupling behavior of the Bose-Einstein condensation transition temperature. It is shown that the critical temperature shift of the non-ideal Bose gas grows at small $an^{1/3}$, reaches some maximal value and then falls down becoming negative., Comment: 6 pages, 3 figures

We studied the properties of a single impurity atom immersed in a dilute Bose condensate at low temperatures. In particular, we perturbatively obtained the momentum dependence of the impurity spectrum and damping. By means of the Brillouin-Wigner perturbation theory we also calculated the self-energy both for attractive and repulsive polaron in the long-wavelength limit. The stability problem of the impurity atom in a weakly-interacting Bose gas is also examined., Comment: 11 pages, 4 figures

The sparsity of the isotope Helium-3, ongoing since 2009, has initiated a new generation of neutron detectors. One particularly promising development line for detectors is the multilayer gaseous detector. In this paper, a stochastic process approach is used to determine the neutron's energy from the additional data afforded by the multilayer nature of these novel detectors. The data from a multi-layer detector consists of counts of the number of absorbed neutrons along the sequence of the detector's layers, in which the neutron absorption probability is unknown. We study the Maximum Likelihood estimator for the intensity and absorption probability, show its consistency and asymptotic normality, as the experiment time (or the number of incoming neutrons) goes to infinity. We combine these results with known results on the relation between the absorption probability and the wavelength to derive an estimator of the wavelength and to show consistency and asymptotic normality for the estimator.

We propose a new estimator of a discrete monotone probability mass function with known flat regions. We analyse its asymptotic properties and compare its performance to the Grenander estimator and to the monotone rearrangement estimator., Comment: 16 pages, 1 figure

We rigorously analyze the low-temperature properties of homogeneous three-dimensional two-component Bose mixture with dipole-dipole interaction. For such a system the effective hydrodynamic action that governs the behavior of low-energy excitations is derived. The infrared structure of the exact single-particle Green's functions is obtained in terms of macroscopic parameters, namely the inverse compressibility and the superfluid density matrices. Within one-loop approximation we calculate the anisotropic superfluid and condensate densities and give the beyond mean-field stability condition for the binary dipolar Bose gas. A brief variational derivation of the coupled equations that describe macroscopic hydrodynamics of the system in the external non-uniform potential at zero temperature is presented., Comment: 6 pages, 2 figures

We rigorously discuss the infrared behavior of the uniform three dimensional dipolar Bose systems. In particular, it is shown that low-temperature physics of the system is controlled by two parameters, namely isothermal compressibility and intensity of the dipole-dipole interaction. By using hydrodynamic approach we calculate the spectrum and damping of low-lying excitations and analyze infrared behavior of the one-particle Green's function. The low-temperature corrections to the anisotropic superfluid density as well as condensate depletion are found. Additionally we derive equations of the two-fluid hydrodynamics for dipolar Bose systems and calculate velocities of first and second sound., Comment: 8 pages, 3 figures, references added, misprints corrected

We consider a simple model of the quasi-two-dimensional dipolar Bose gas confined in the one-dimensional square well potential. All dipoles are assumed to be oriented along the confining axis. By means of hydrodynamic approach it is shown that the general structure of the low-lying excitations can be analyzed exactly. We demonstrate that the problem significantly simplifies in the high-density limit for which the density profile in the confined direction as well as the leading-order contribution to the ground-state energy and spectrum of elementary excitations are calculated. The low-temperature result for the damping rate of the phonon mode is also presented., Comment: 7 pages, 4 figures; references added, typos corrected

The formula for the temperature dependence of the effective mass of a $^{4}% $He atom in the superfluid and normal phases is obtained.\,\,This expression for the effective mass allows one to eliminate infra-red divergences, being applicable at all temperatures, except for a narrow fluctuation region 0.97~$\lesssim T/T_{\rm c}\leq1$.\,\,In the high and low temperature limits, as well as in the interactionless limit, the obtained expression reproduces the well known results.\,\,The temperature dependence of the heat capacity and the phase transition temperature $T_{\rm c}\approx$~2.18~K are calculated, by using the formula obtained for the effective mass.\,\,In the framework of the approach proposed in this work, the small critical index $\eta$ is determined in the random phase approximation.\,\,The obtained value corresponds to the well known result., Comment: 9 pages, 4 figures

In this paper, with the corresponding formula for internal energy obtained in Ref. [J. Phys. Stud. {\bf 11}, 259 (2007)], combined with a simple calculation of the effective mass of interacting Bose particles, the behavior of the heat capacity of liquid $^4$He is analyzed numerically for the entire temperature range. The results agree quite well with experimental data., Comment: 15 pages, 2 figures

We calculate the gravitational correction to the phase difference between neutrino mass eigenstates. There is a number of papers, where this correction was calculated for the spherically symmetric gravitational field described by the Schwarzschild metric. The results of these papers differ from each other. Our result is close to that of Bhattachary, Habib and Mottola (see ref. [3,6]) and differs only by a coefficient: our correction is twice smaller., Comment: 6 pages