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Resource allocation in distributed and networked systems such as the Cloud is becoming increasingly flexible, allowing these systems to dynamically adjust toward the workloads they serve, in a demand-aware manner. Online balanced partitioning is a fundamental optimization problem underlying such self-adjusting systems. We are given a set of $\ell$ servers. On each server we can schedule up to $k$ processes simultaneously. The demand is described as a sequence of requests $\sigma_t=\{p_i, p_{j}\}$, which means that the two processes $p_i,p_{j}$ communicate. A process can be migrated from one server to another which costs 1 unit per process move. If the communicating processes are on different servers, it further incurs a communication cost of 1 unit for this request. The objective is to minimize the competitive ratio: the cost of serving such a request sequence compared to the cost incurred by an optimal offline algorithm. Henzinger et al. (at SIGMETRICS 2019) introduced a learning variant of this problem where the cost of an online algorithm is compared to the cost of a static offline algorithm that does not perform any communication, but which simply learns the communication graph and keeps the discovered connected components together. This problem variant was recently also studied at SODA 2021. In this paper, we consider a more general learning model (i.e., stronger adversary), where the offline algorithm is not restricted to keep connected components together. Our main contribution are tight bounds for this problem. In particular, we present two deterministic online algorithms: (1) an online algorithm with competitive ratio $O(\max(\sqrt{k\ell \log k}, \ell \log k))$ and augmentation $1+\epsilon$; (2) an online algorithm with competitive ratio $O(\sqrt{k})$ and augmentation $2+\epsilon$. We further present lower bounds showing optimality of these bounds.

The analysis of Pb+Pb data for net-charge fluctuations at LHC energies within the HYDJET++ model is presented. The strongly intensive quantities $D$ and $\Sigma$ are used to get rid of the effects related to volume fluctuations of the system. We employ two versions of the HYDJET++ for the analysis. The first one is the standard or default version, whereas the second one represents its further modification, which takes into account explicit event-by-event conservation of the electric net-charge of produced particles. Inclusion of the canonical net-charge conservation in the model allows one for better description of the experimental data obtained by the ALICE and the CMS Collaborations. Comparison with calculations of other models is also presented., Comment: LATEX, 8 pages, 8 figures

Using the free fermions technique and bosonization rules we introduce the multicomponent DKP hierarchy as a generating bilinear integral equation for the tau-function. A number of bilinear equations of the Hirota-Miwa type are obtained as its corollaries. We also consider the dispersionless version of the hierarchy as a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the $F$-function). We show that there is an elliptic curve built in the structure of the hierarchy, with the elliptic modulus being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization many dispersionless equations of the Hirota-Miwa type become equivalent to a single equation having a nice form., Comment: 22 pages, no figures

We revisit dispersionless version of the multicomponent KP hierarchy considered previously by Takasaki and Takebe. In contrast to their study, we do not fix any distinguished component treating all of them on equal footing. We obtain nonlinear equations for dispersionless tau-function (the F-function) and represent them using the trigonometric parametrization. In this trigonometric uniformization the equations considerably simplify and acquire a nice form., Comment: 11 pages, no figures

We study the dispersionless limit of the recently introduced Toda lattice hierarchy with constraint of type B (the B-Toda hierarchy) and compare it with that of the DKP and C-Toda hierarchies. The dispersionless limits of the B-Toda and C-Toda hierarchies turn out to be the same., Comment: 8 pages, no figures, typos corrected

We construct quasi-periodic solutions of the universal hierarchy which includes the multi-component KP and Toda hierarchies and show how they fit into the bilinear formalism. The tau-function is expressed in terms of the Riemann theta-function multiplied by exponential function of a quadratic form in the hierarchical times., Comment: 21 pages, no figures. arXiv admin note: text overlap with arXiv:2304.05108

We prove existence of the tau-function for the multi-component CKP hierarchy and find how it is related to the tau-function of the multi-component KP hierarchy., Comment: 14 pages, no figures. arXiv admin note: text overlap with arXiv:2103.07357, revised argument in section 2, results unchanged

The entropy produced by the Unruh radiation is estimated and compared to the entropy of a Schwarzschild black hole. We simulate a spherical system of mass $ M $ by set of Unruh horizons and estimate the total entropy of the outgoing radiation. Dependence on mass and spin of the emitted particles is taken into account. The obtained results can be easily extended to any other intrinsic degrees of freedom of outgoing particles. The ratio of Unruh entropy to the Schwarzschild black hole entropy is derived in exact analytical form. For large black holes this ratio is highly sensitive to quantum numbers of emitted quanta, e.g., spin $ s $, for which it varies from $ 0\% $ for $ s = 0 $ to $ 19\% $ for $ s = 5/2 $., Comment: LATEX, 16 pages, 3 figures

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula., Comment: 12 pages, no figures

The performance of many large-scale and data-intensive distributed systems critically depends on the capacity of the interconnecting network. This paper is motivated by the vision of self-adjusting infrastructures whose resources can be adjusted according to the workload they currently serve, in a demand-aware manner. Such dynamic adjustments can be exploited to improve network utilization and hence performance, by dynamically moving frequently interacting communication partners closer, e.g., collocating them in the same server or datacenter rack. In particular, we revisit the online balanced graph partitioning problem which captures the fundamental tradeoff between the benefits and costs of dynamically collocating communication partners. The demand is modelled as a sequence $\sigma$ (revealed in an online manner) of communication requests between $n$ processes, each of which is running on one of the $\ell$ servers. Each server has capacity $k=n/\ell$, hence, the processes have to be scheduled in a balanced manner across the servers. A request incurs cost $1$, if the requested processes are located on different servers, otherwise the cost is 0. A process can be migrated to a different server at cost $1$. This paper presents the first online algorithm for online balanced graph partitioning achieving a polylogarithmic competitive ratio for the fundamental case of ring communication patterns. Specifically, our main contribution is a $O(\log^3 n)$-competitive randomized online algorithm for this problem. We further present a randomized online algorithm which is $O(\log^2 n)$-competitive when compared to a static optimal solution. Our two results rely on different algorithms and techniques and hence are of independent interest.

This is a short review of the construction of quasi-periodic (algebraic-geometrical) solutions to hierarchies of nonlinear integrable equations. As is well known, the solutions are expressed through Riemann's theta-functions associated with algebraic curves. It is explained how solutions from this class can be treated within the framework of the approach to the integrable hierarchies developed by the Kyoto school. Three representative examples are considered in detail: the Kadomtsev-Petviashvili hierarchy, the 2D Toda lattice hierarchy and the B-version of the Kadomtsev-Petviashvili hierarchy., Comment: 29 pages, no figures

We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B) which can be regarded as a discretization of the BKP hierarchy. We introduce the tau-function of the B-Toda hierarchy and obtain the bilinear equations for it. Examples of soliton tau-functions are presented in the explicit form., Comment: 18 pages, no figures

We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars-Schneider system. We find its commutation representation in the form of the Manakov triple and study properties of the spectral curve. By studying more general elliptic solutions (elliptic families), we also suggest an extension of the deformed Ruijsenaars-Schneider system to a field theory., Comment: 23 pages, no figures

We obtain B\"acklund transformations and integrable time discretization of the recently introduced deformed Ruijsenaars-Schneider many-body system which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. We also show that the deformed Ruijsenaars-Schneider system in discrete time is the dynamical system for poles of elliptic solutions to the fully discrete Kadomtsev-Petviashvili equation of type B. Besides, we suggest a field analogue of the deformed Ruijsenaars-Schneider system on a space-time lattice., Comment: 18 pages, no figures

We find integrals of motion for the recently introduced deformed Ruijsenaars-Schneider many-body system which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. Our method is based on the fact that equations of motion for this system coincide with those for pairs of Ruijsenaars-Schneider particles which stick together preserving a special fixed distance between the particles., Comment: Section 5 added, 23 pages, 1 figure

We further develop the approach to many-body systems based on finding conditions of existence of meromorphic solutions to certain linear partial differential and difference equations which serve as auxiliary linear problems for nonlinear integrable equations such as KP, BKP, CKP and different versions of the Toda lattice. These conditions imply equations of the time evolution for poles of singular solutions to the nonlinear equations which are equations of motion for integrable many-body systems of Calogero-Moser and Ruijsenaars-Schneider type. A new many-body system is introduced, which governs dynamics of poles of elliptic solutions to the Toda lattice of type B., Comment: 32 pages, no figures

Data from the Large Hadron Collider on the charge balance function in Pb+Pb collisions at center-of-mass energy 2.76~TeV per nucleon pair are analyzed and interpreted within the framework of the \hydjet++ model. This model allows us to qualitatively reproduce the experimentally observed centrality dependence of the balance function widths at relatively low transverse momentum intervals due to the different charge creation mechanisms in soft and hard processes. However, a fully adequate description of the balance function in these intervals implies an essential modification of the model by including exact charge conservation via the canonical rather than the grand canonical ensemble. A procedure is proposed for introducing charge correlations into the thermal model without changing other model parameters. With increasing transverse momenta, the default model results describe the experimental data much better because the contribution of the soft component of the model is significantly reduced in these transverse momentum intervals. In practical terms, there is a transition to a single source of charge correlations, namely, charge correlations in jets in which exact charge conservation holds at each stage., Comment: Matches published version

We define the Dyson diffusion process on a curved smooth closed contour in the plane and derive the Fokker-Planck equation for probability density. Its stationary solution is shown to be the Boltzmann weight for the logarithmic gas confined on the contour., Comment: 8 pages, no figures

We introduce a new integrable hierarchy of nonlinear differential-difference equations which is a subhierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. The 2D Toda lattice with the constraint can be regarded as a discretization of the BKP hierarchy. We construct its algebraic-geometrical solutions in terms of Riemann and Prym theta-functions., Comment: 24 pages, 1 figure

LHC data on the correlations of the elliptic flow $v_2$ of particles at low and high transverse momenta $p_T$ from Pb+Pb collisions at center-of-mass energy per nucleon pair $\sqrt{s_{NN}} = 5.02$ TeV are analyzed in the framework of the HYDJET++ model. This model includes soft and hard components which allows to describe the region of both low and high transverse momenta. The origin of $v_2$ values in different $p_T$ regions is investigated at different centralities. It is shown that the experimentally observed correlations between $v_2$ at low and high $p_T$ in peripheral lead-lead collisions is due to correlation of particles in jets.

We study the dispersionless version of the recently introduced constrained Toda hierarchy. Like the Toda lattice itself, it admits three equivalent formulations: the formulation in terms of Lax equations, the formulation of the Zakharov-Shabat type and the formulation through the generating equation for the dispersionless limit of logarithm of the tau-function. We show that the dispersionless constrained Toda hierarchy describes conformal maps of reflection-symmetric planar domains to the exterior of the unit disc. We also find finite-dimensional reductions of the hierarchy and show that they are characterized by a differential equation of the L\"owner type which we call the symmetric radial L\"owner equation. It is also shown that solutions to the symmetric radial L\"owner equation are conformal maps of the exterior of the unit circle with two symmetric slits to the exterior of the unit circle., Comment: 24 pages, 5 figures; revised

The Nuclotron-base Ion Collider fAcility (NICA) is under construction at the Joint Institute for Nuclear Research (JINR), with commissioning of the facility expected in late 2022. The Multi-Purpose Detector (MPD) has been designed to operate at NICA and its components are currently in production. The detector is expected to be ready for data taking with the first beams from NICA. This document provides an overview of the landscape of the investigation of the QCD phase diagram in the region of maximum baryonic density, where NICA and MPD will be able to provide significant and unique input. It also provides a detailed description of the MPD set-up, including its various subsystems as well as its support and computing infrastructures. Selected performance studies for particular physics measurements at MPD are presented and discussed in the context of existing data and theoretical expectations., Comment: 53 pages, 68 figures, submitted as a Review article to EPJA

We study elliptic families of solutions to the recently introduced constrained Toda hierarchy, i.e., solutions which are elliptic functions of some linear combination of the hierarchical times. Equations of motion for poles of such solutions are obtained., Comment: 9 pages, no figures. arXiv admin note: text overlap with arXiv:2107.01697

We introduce and study a model of a logarithmic gas with inverse temperature $\beta$ on an arbitrary smooth closed contour in the plane. This model generalizes Dyson's gas (the $\beta$-ensemble) on the unit circle. We compute the non-vanishing terms of the large $N$ expansion of the free energy ($N$ is the number of particles) by iterating the "loop equation" that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann-Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour., Comment: 35 pages, no figures

We present new results on the thermal production yield of a hypothetical state made of six quarks $uuddss$ assuming its production in heavy-ion collisions at the CERN LHC. A state with this quark content and mass low enough to be stable against decay in timescales of the order of the age of the Universe, has been hypothesized by one of us (G.F.) and has been discussed as a possible dark matter candidate. In this work we address for the first time the thermal production rate that can be expected for this state in heavy-ion collisions at colliders. For this estimate we use a thermal model which has been shown to describe accurately the production of hadrons and nuclei in heavy-ion collisions at LHC energy. This estimate is of great relevance for sexaquark searches at colliders as well as for its consideration as a dark matter candidate and for the composition of neutron stars., Comment: 11 pages, 2 figures

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^{\dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities., Comment: 24 pages, no figures

We suggest to explore an entirely new method to experimentally and theoretically study the phase diagram of strongly interacting matter based on the triple nuclear collisions (TNC). We simulated the TNC using the UrQMD 3.4 model at the beam center-of-mass collision energies $\sqrt{s_{NN}} = 200$ GeV and $\sqrt{s_{NN}} = 2.76$ TeV. It is found that in the most central and simultaneous TNC the initial baryonic charge density is about 3 times higher than the one achieved in the usual binary nuclear collisions at the same energies. As a consequence, the production of protons and $\Lambda$-hyperons is increased by a factor of 2 and 1.5, respectively. Using the MIT Bag model equation we study the evolution of the central cell in TNC and demonstrate that for the top RHIC energy of collision the baryonic chemical potential is 2-2.5 times larger than the one achieved in the binary nuclear collision at the same type of reaction. Based on these estimates, we show that TNC offers an entirely new possibility to study the QCD phase diagram at very high baryonic charge densities., Comment: Summary of the talk given at the Online Conference "Strangeness in Quark Matter 2021"

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of $L$-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing $\eta$, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit $\eta \to 0$ is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed., Comment: 46 pages, typos corrected

A simple geometrical model with event-by-event fluctuations is suggested to study elliptical and triangular eccentricities in the initial state of relativistic heavy-ion collisions. This model describes rather well the ALICE and ATLAS data for Pb+Pb collisions at center-of-mass energy $\sqrt{s_{NN}} = 5.02$~TeV per nucleon pair, assuming that the second, $v_2$, and third, $v_3$, harmonics of the anisotropic flow are simply linearly proportional to the eccentricities $\varepsilon_2$ and $\varepsilon_3$, respectively. We show that the eccentricity $\varepsilon_3$ has a pure fluctuation origin and is substantially dependent on the size of the overlap area only, while the eccentricity $\varepsilon_2$ is mainly related to the average collision geometry. Elliptic flow, therefore, is weakly dependent on the event-by-event fluctuations everywhere except of the very central collisions 0--2%, whereas triangular flow is mostly determined by the fluctuations. The scaling dependence of the magnitude of the flow harmonics on atomic number, $v_n \propto A^{-1/3}$, is predicted for this centrality interval., Comment: Further development and extension of ideas discussed in arXiv:2007.04850. Accepted for publication in J. of Physics G: Nucl. and Part. Physics

An analytical formula that accurately accounts for the Lorentz contraction of the excluded volume of two relativistic hadrons with hard-core repulsion is worked out. Using the obtained expression we heuristically derive the equation of state of Boltzmann particles with relativistic excluded volumes in terms of system pressure and its surface and curvature tension coefficients. The behavior of effective excluded volumes of lightest baryons and mesons is studied at very high temperatures (particle number densities) and for very large values of degeneracy factors. Several parameterizations of the obtained equation of state demonstrate a universal asymptotics of the effective excluded volume at high particle number densities. It is peculiar, that the found maximal packing fraction $\eta \simeq 0.75$ of Lorentz contracted particles is very close to the dense packing limit of classical hard spheres of same radius $\eta_{exc} \approx 0.74$. We show that the developed equation of state is the grand canonical formulation of the morphological thermodynamics approach applied to Lorentz contracted rigid spheres., Comment: 15 pages, 3 figures

Here we develop an original approach to investigate the grand canonical partition function of the multicomponent mixtures of Boltzmann particles with hard-core interaction in finite and even small systems of the volumes above 20 fm$^3$. The derived expressions of the induced surface tension equation of state are analyzed in details. It is shown that the metastable states, which can emerge in the finite systems with realistic interaction, appear at very high pressures at which the hadron resonance gas, most probably, is not applicable at all. It is shown how and under what conditions the obtained results for finite systems can be generalized to include into a formalism the equation for curvature tension. The applicability range of the obtained equations of induced surface and curvature tensions for finite systems is discussed and their close relations to the equations of the morphological thermodynamics are established. The hadron resonance gas model on the basis of the obtained advanced equation of state is worked out. Also, this model is applied to analyze the chemical freeze-out of hadrons and light nuclei with the number of (anti-)baryons not exceeding 4, including the most problematic ratios of hyper-triton and its antiparticle. Their multiplicities were measured by the ALICE Collaboration in the central lead-lead collisions at the center-of-mass energy $\sqrt{s_{\rm NN}} =$ 2.76 TeV., Comment: 24 pages, 3 figures and 1 table

We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. It is known that poles $x_i$ and matrix residues at the poles $\rho_i^{\alpha \beta}=a_i^{\alpha}b_i^{\beta}$ of such solutions as functions of the time $t_2$ move as particles of spin generalization of the elliptic Calogero-Moser model (elliptic Gibbons-Hermsen model). In this paper we establish the correspondence with the spin elliptic Calogero-Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the $k$-th hierarchical time of the matrix KP hierarchy is Hamiltonian with the Hamiltonian $H_k$ obtained via an expansion of the spectral curve near the marked points. The Hamiltonians are identified with the Hamiltonians of the elliptic spin Calogero-Moser system with coordinates $x_i$ and spin degrees of freedom $a_i^{\alpha}, \, b_i^{\beta}$., Comment: 25 pages, no figures. arXiv admin note: text overlap with arXiv:1910.00434

We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians which govern the dynamics of poles with respect to the m-th hierarchical times t_m and \bar t_m of the 2D Toda lattice hierarchy are obtained from expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points., Comment: 27 pages, no figures. arXiv admin note: text overlap with arXiv:1907.06621, arXiv:2102.03784