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Hadronization is a non-perturbative process, which theoretical description can not be deduced from first principles. Modeling hadron formation requires several assumptions and various phenomenological approaches. Utilizing state-of-the-art Deep Learning algorithms, it is eventually possible to train neural networks to learn non-linear and non-perturbative features of the physical processes. In this study, the prediction results of three trained ResNet networks are presented, by investigating charged particle multiplicities at event-by-event level. The widely used Lund string fragmentation model is applied as a training-baseline at $\sqrt{s}= 7$ TeV proton-proton collisions. We found that neural-networks with $ \gtrsim\mathcal{O}(10^3)$ parameters can predict the event-by-event charged hadron multiplicity values up to $ N_\mathrm{ch}\lesssim 90 $., Comment: 11 pages, 5 figures, proceedings of the 23rd Zimanyi School Winter Workshop on Heavy Ion Physics, Budapest, Hungary, December 4 - 8, 2023, submitted to the International Journal of Modern Physics A, special issue "Zimanyi School 2023"

Machine learning techniques exhibit significant performance in discriminating different phases of matter and provide a new avenue for studying phase transitions. We investigate the phase transitions of three dimensional $q$-state Potts model on cubic lattice by using a transfer learning approach, Domain Adversarial Neural Network (DANN). With the unique neural network architecture, it could evaluate the high-temperature (disordered) and low-temperature (ordered) phases, and identify the first and second order phase transitions. Meanwhile, by training the DANN with a few labeled configurations, the critical points for $q=2,3,4$ and $5$ can be predicted with high accuracy, which are consistent with those of the Monte Carlo simulations. These findings would promote us to learn and explore the properties of phase transitions in high-dimensional systems., Comment: 18pages,36 figures

A transfer learning method, Domain Adversarial Neural Network (DANN), is introduced to study the phase transition of two-dimensional q-state Potts model. With the DANN, we only need to choose a few labeled configurations automatically as input data, then the critical points can be obtained after training the algorithm. By an additional iterative process, the critical points can be captured to comparable accuracy to Monte Carlo simulations as we demonstrate it for q = 3, 4, 5, 7 and 10. The type of phase transition (first or second-order) is also determined at the same time. Meanwhile, for the second-order phase transition at q=3, we can calculate the critical exponent $\nu$ by data collapse. Furthermore, compared to the traditional supervised learning, we found the DANN to be more accurate with lower cost., Comment: 30 pages, 36 figures

Objective. Algorithmic differentiation (AD) can be a useful technique to numerically optimize design and algorithmic parameters by, and quantify uncertainties in, computer simulations. However, the effectiveness of AD depends on how "well-linearizable" the software is. In this study, we assess how promising derivative information of a typical proton computed tomography (pCT) scan computer simulation is for the aforementioned applications. Approach. This study is mainly based on numerical experiments, in which we repeatedly evaluate three representative computational steps with perturbed input values. We support our observations with a review of the algorithmic steps and arithmetic operations performed by the software, using debugging techniques. Main results. The model-based iterative reconstruction (MBIR) subprocedure (at the end of the software pipeline) and the Monte Carlo (MC) simulation (at the beginning) were piecewise differentiable. Jumps in the MBIR function arose from the discrete computation of the set of voxels intersected by a proton path. Jumps in the MC function likely arose from changes in the control flow that affect the amount of consumed random numbers. The tracking algorithm solves an inherently non-differentiable problem. Significance. The MC and MBIR codes are ready for the integration of AD, and further research on surrogate models for the tracking subprocedure is necessary., Comment: 27 pages, 11 figures

The latest advances of statistical physics have shown remarkable performance of machine learning in identifying phase transitions. In this paper, we apply domain adversarial neural network (DANN) based on transfer learning to studying non-equilibrium and equilibrium phase transition models, which are percolation model and directed percolation (DP) model, respectively. With the DANN, only a small fraction of input configurations (2d images) needs to be labeled, which is automatically chosen, in order to capture the critical point. To learn the DP model, the method is refined by an iterative procedure in determining the critical point, which is a prerequisite for the data collapse in calculating the critical exponent $\nu_{\perp}$. We then apply the DANN to a two-dimensional site percolation with configurations filtered to include only the largest cluster which may contain the information related to the order parameter. The DANN learning of both models yields reliable results which are comparable to the ones from Monte Carlo simulations. Our study also shows that the DANN can achieve quite high accuracy at much lower cost, compared to the supervised learning., Comment: 12 pages,10 figures

Hadron spectral functions carry all the information of hadrons and are encoded in the Euclidean two-point correlation functions. The extraction of hadron spectral functions from the correlator is a typical ill-posed inverse problem and infinite number of solutions to this problem exists. We propose a novel neural network (sVAE) based on the Variation Auto-Encoder (VAE) and Bayesian theorem. Inspired by the maximum entropy method (MEM) we construct the loss function of the neural work such that it includes a Shannon-Jaynes entropy term and a likelihood term. The sVAE is then trained to provide the most probable spectral functions. For the training samples of spectral function we used general spectral functions produced from the Gaussian Mixture Model. After the training is done we performed the mock data tests with input spectral functions consisting 1) only a free continuum, 2) only a resonance peak, 3) a resonance peak plus a free continuum and 4) a NRQCD motivated spectral function. From the mock data test we find that the sVAE in most cases is comparable to the maximum entropy method in the quality of reconstructing spectral functions and even outperforms the MEM in the case where the spectral function has sharp peaks with insufficient number of data points in the correlator. By applying to temporal correlation functions of charmonium in the pseudoscalar channel obtained in the quenched lattice QCD at 0.75 $T_c$ on $128^3\times96$ lattices and $1.5$ $T_c$ on $128^3\times48$ lattices, we find that the resonance peak of $\eta_c$ extracted from both the sVAE and MEM has a substantial dependence on the number of points in the temporal direction ($N_\tau$) adopted in the lattice simulation and $N_\tau$ larger than 48 is needed to resolve the fate of $\eta_c$ at 1.5 $T_c$., Comment: 9 pages, 7 figures. Talk presented at the 38th International Symposium on Lattice Field Theory (Lattice 2021), 26-30 July, 2021, Zoom/Gather@Massachusetts Institute of Technology

Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio $r=N/T$, where $N$ is the number of different assets in the portfolio, and $T$ is the length of the available time series. The critical ratio depends on the confidence level $\alpha$, which means we have a line of critical points on the $\alpha-r$ plane. The large fluctuations in the estimation of ES can be attenuated by the application of regularizers. In this paper, we calculate ES analytically under an $\ell_1$ regularizer by the method of replicas borrowed from the statistical physics of random systems. The ban on short selling, i.e. a constraint rendering all the portfolio weights non-negative, is a special case of an asymmetric $\ell_1$ regularizer. Results are presented for the out-of-sample and the in-sample estimator of the regularized ES, the estimation error, the distribution of the optimal portfolio weights and the density of the assets eliminated from the portfolio by the regularizer. It is shown that the no-short constraint acts as a high volatility cutoff, in the sense that it sets the weights of the high volatility elements to zero with higher probability than those of the low volatility items. This cutoff renormalizes the aspect ratio $r=N/T$, thereby extending the range of the feasibility of optimization. We find that there is a nontrivial mapping between the regularized and unregularized problems, corresponding to a renormalization of the order parameters., Comment: 29 pages, 4 figures

We study the structure of jets in proton-proton collisions at LHC energies using \textsc{Pythia} 8 Monte Carlo simulations. We demonstrate that the radial jet profiles exhibit scaling properties with charged-hadron event multiplicity over a broad transverse-momentum range. We also provide parametrizations of the jet profiles based on different statistically-motivated analytical distributions. Based on this we propose that the scaling behavior stems from fundamental statistical properties of jet fragmentation., Comment: to appear in the Gribov-90 Memorial Volume: Algebraic Methods in QFT

The non-extensive statistical description of the identified final state particles measured in high energy collisions is well-known by its wide range of applicability. However, there are many open questions that need to be answered, including but not limited to, the question of the observed mass scaling of massive hadrons or the size and multiplicity dependence of the model parameters. This latter is especially relevant, since currently the amount of available experimental data with high multiplicity at small systems is very limited. This contribution has two main goals: On the one hand we provide a status report of the ongoing tuning of the soon-to-be-released HIJING++ Monte Carlo event generator. On the other hand, the role of multiplicity dependence of the parameters in the non-extensive hadronization model is investigated with HIJING++ calculations. We present cross-check comparisons of HIJING++ with existing experimental data to verify its validity in our range of interest as well as calculations at high-multiplicity regions where we have insufficient experimental data., Comment: This paper is based on the talk at the 18th Zim\'anyi School, Budapest, Hungary, 3-7 December 2018

Beyond 2025 we will enter the High-Luminosity era of the LHC, right after the upgrades of the third Long Shutdown of the Large Hadron Collider (LHC). The ongoing state-of-the-art experimental instrument upgrades require high-performance simulation support in the background, that is modern, robust, and comes with long term support. The original FORTRAN based HIJING (Heavy Ion Jet INteraction Generator) has been used intensively since almost three decades by the high energy physics and heavy-ion community. However, it is getting overly challenging to conform to these new requirements. Our novel Monte Carlo event generator, the HIJING++ is the successor of the old FORTRAN version containing all the physics that its predecessor have. Moreover, among others a flexible module handling layer and an analysis interface is also introduced. This latter supports the most popular event container formats such as simple ascii, ROOT and the HepMC format, together with RIVET support. In this paper we compare the pre-release results of HIJING++ with proton-proton experimental data and Pythia8 calculations., Comment: Proceedings from the International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions, 30 September - 5 October 2018 , Aix-Les-Bains, Savoie, France

HIJING++ (Heavy Ion Jet INteraction Generator) is the successor of the widely used original HIJING, developed almost three decades ago. While the old versions (1.x and 2.x) were written in FORTRAN, HIJING++ was completely rewritten in C++. During the development we keep in mind the requirements of the high-energy heavy-ion community: the new Monte Carlo software have a well designed modular framework, therefore any future modifications are much easier to implement. It contains all the physical models that were also present in it's predecessor, but utilizing modern C++ features it also includes native thread based parallelism, an easy-to-use analysis interface and a modular plugin system, which makes room for possible future improvements. In this paper we summarize the results of our performance tests measured on 2 widely used architectures., Comment: Presented at Hot Quarks 2018 -- Workshop for young scientists on the physics of ultrarelativistic nucleus-nucleus collisions, Texel, The Netherlands, September 7-14 2018. Submitted to MDPI Proceedings

We present preliminary results with HIJING++ (3.1.1) for identified hadron production in high-energy heavy ion collisions at LHC energies. The recently developed HIJING++ version is based on the latest version of PYTHIA8 and contains all the nuclear effects that have been included in the HIJING2.552, which will be improved by a new version of the shadowing parametrization and jet quenching module. Here, we summarize the structure and the speed gain due to parallelization of the new program code, also presenting some comparison between experimental data., Comment: proceedings of the 12th International Workshop on High-pT Physics in the RHIC/LHC Era, 2-5 October, 2017, University of Bergen, Bergen, Norway

The optimization of the variance supplemented by a budget constraint and an asymmetric $\ell_1$ regularizer is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions differently, so the present treatment includes the no-short-constrained portfolio optimization problem as a special case. Results are presented for the out-of-sample and the in-sample estimator of the regularized variance, the relative estimation error, the density of the assets eliminated from the portfolio by the regularizer, and the distribution of the optimal portfolio weights. We have studied the dependence of these quantities on the ratio $r$ of the portfolio's dimension $N$ to the sample size $T$, and on the strength of the regularizer. We have checked the analytic results by numerical simulations, and found general agreement. Regularization extends the interval where the optimization can be carried out, and suppresses the large sample fluctuations, but the performance of $\ell_1$ regularization is rather disappointing: if the sample size is large relative to the dimension, i.e. $r$ is small, the regularizer does not play any role, while for $r$'s where the regularizer starts to be felt the estimation error is already so large as to make the whole optimization exercise pointless. We find that the $\ell_1$ regularization can eliminate at most half the assets from the portfolio, corresponding to this there is a critical ratio $r=2$ beyond which the $\ell_1$ regularized variance cannot be optimized: the regularized variance becomes constant over the simplex. These facts do not seem to have been noticed in the literature., Comment: 31 pages, 8 figures

First calculated results with the new HIJING++ are presented for identified hadron production in high-energy heavy ion collisions. The recently developed HIJING++ version is based on the latest version of PYTHIA8 and contains all the nuclear effects has been included in the HIJING2.552, which will be improved by a new version of the shadowing parametrization and jet quenching module. Here, we summarize the major changes of the new program code beside the comparison between experimental data for some specific high-energy nucleus-nucleus collisions., Comment: 4 pages, 3 figures

A large portfolio of independent returns is optimized under the variance risk measure with a ban on short positions. The no-short selling constraint acts as an asymmetric $\ell_1$ regularizer, setting some of the portfolio weights to zero and keeping the out of sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case. However, the susceptibility, i.e. the sensitivity of the optimal portfolio weights to changes in the returns, diverges at a critical value $r=2$. This means that a ban on short positions does not prevent the phase transition in the optimization problem, it merely shifts the critical point from its non-regularized value of $r=1$ to $2$. At $r=2$ the out of sample estimator for the portfolio variance stays finite and the estimated in-sample variance vanishes. We have performed numerical simulations to support the analytic results and found perfect agreement for $N/T<2$. Numerical experiments on finite size samples of symmetrically distributed returns show that above this critical point the probability of finding solutions with zero in-sample variance increases rapidly with increasing $N$, becoming one in the large $N$ limit. However, these are not legitimate solutions of the optimization problem, as they are infinitely sensitive to any change in the input parameters, in particular they will wildly fluctuate from sample to sample. We also calculate the distribution of the optimal weights over the random samples and show that the regularizer preferentially removes the assets with large variances, in accord with one's natural expectation., Comment: 27 pages, 2 figures

The optimization of a large random portfolio under the Expected Shortfall risk measure with an $\ell_2$ regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number $N$ of different assets in the portfolio is much less than the length $T$ of the available time series, the regularizer plays a negligible role even if its strength $\eta$ is large, while in the opposite limit, where the size of samples is comparable to, or even smaller than the number of assets, the optimum is almost entirely determined by the regularizer. We construct the contour map of estimation error on the $N/T$ vs. $\eta$ plane and find that for a given value of the estimation error the gain in $N/T$ due to the regularizer can reach a factor of about 4 for a sufficiently strong regularizer., Comment: 14 pages, 8 figures

The contour maps of the error of historical resp. parametric estimates for large random portfolios optimized under the risk measure Expected Shortfall (ES) are constructed. Similar maps for the sensitivity of the portfolio weights to small changes in the returns as well as the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-the-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the optimization for a given number of different assets in the portfolio, at a given confidence level and a given level of relative estimation error. The necessary sample sizes invariably turn out to be unrealistically large for any reasonable choice of the number of assets and the confidence level. These results are obtained via analytical calculations based on methods borrowed from the statistical physics of random systems, supported by numerical simulations., Comment: 47 pages, 14 figures

The contour map of estimation error of Expected Shortfall (ES) is constructed. It allows one to quantitatively determine the sample size (the length of the time series) required by the optimization under ES of large institutional portfolios for a given size of the portfolio, at a given confidence level and a given estimation error., Comment: 5 pages, 1 figure

Correlations and other collective phenomena in a schematic model of heterogeneous binary agents (individual spin-glass samples) are considered on the complete graph and also on 2d and 3d regular lattices. The system's stochastic dynamics is studied by numerical simulations. The dynamics is so slow that one can meaningfully speak of quasi-equilibrium states. Performing measurements of correlations in such a quasi-equilibrium state we find that they are random both as to their sign and absolute value, but on average they fall off very slowly with distance in all instances that we have studied. This means that the system is essentially non-local, small changes at one end may have a strong impact at the other. Correlations and other local quantities are extremely sensitive to the boundary conditions all across the system, although this sensitivity disappears upon averaging over the samples or partially averaging over the agents. The strong, random correlations tend to organize a large fraction of the agents into strongly correlated clusters that act together. If we think about this model as a distant metaphor of economic agents or bank networks, the systemic risk implications of this tendency are clear: any impact on even a single strongly correlated agent will spread, in an unforeseeable manner, to the whole system via the strong random correlations., Comment: 28 pages, 17 figures

We study di-hadron correlations in proton-proton collisions at $\sqrt{s} = 200$ GeV and interpret experimental data in terms of a fragmentation width and a momentum imbalance. A fragmentation width of $580 \pm 50$ GeV/c is obtained, and the measured momentum imbalance gives an `intrinsic' transverse momentum width of partons in the proton of $2.6 \pm 0.2$ GeV/c. Consequences to heavy ion collisions are discussed., Comment: 6 pages, 4 figures, QM2006 proceeding

We compare eigenvalue densities of Wigner random matrices whose elements are independent identically distributed (iid) random numbers with a Levy distribution and maximally random matrices with a rotationally invariant measure exhibiting a power law spectrum given by stable laws of free random variables. We compute the eigenvalue density of Wigner-Levy (WL) matrices using (and correcting) the method by Bouchaud and Cizeau (BC), and of free random Levy (FRL) rotationally invariant matrices by adapting results of free probability calculus. We compare the two types of eigenvalue spectra. Both ensembles are spectrally stable with respect to the matrix addition. The discussed ensemble of FRL matrices is maximally random in the sense that it maximizes Shannon's entropy. We find a perfect agreement between the numerically sampled spectra and the analytical results already for matrices of dimension N=100. The numerical spectra show very weak dependence on the matrix size N as can be noticed by comparing spectra for N=400. After a pertinent rescaling spectra of Wigner-Levy matrices and of symmetric FRL matrices have the same tail behavior. As we discuss towards the end of the paper the correlations of large eigenvalues in the two ensembles are however different. We illustrate the relation between the two types of stability and show that the addition of many randomly rotated Wigner-Levy matrices leads by a matrix central limit theorem to FRL spectra, providing an explicit realization of the maximal randomness principle., Comment: 18 pages, 14 figures, added: comparison of BC's and our equations; discussion on spectral stability and maximal entropy principle, fig.6 changed

We study empirical covariance matrices in finance. Due to the limited amount of available input information, these objects incorporate a huge amount of noise, so their naive use in optimization procedures, such as portfolio selection, may be misleading. In this paper we investigate a recently introduced filtering procedure, and demonstrate the applicability of this method in a controlled, simulation environment., Comment: 9 pages with 3 EPS figures

We study the performance of Weibull and scale free Internet-like networks and compare them to a classical random graph based network. The scaling of the traffic load with the nodal degree is established, and confimed in a numerical simulation of the TCP traffic. The scaling allows us to estimate the link capacity upgrade required making and extra connection to an existing node., Comment: 6 pages, 7 figures Added references

In this talk we present a perturbative QCD improved parton model calculation for light meson production in high energy heavy ion collisions. In order to describe the experimental data properly, one needs to augment the standard pQCD model by the transverse momentum distribution of partons ("intrinsic k_T"). Proton-nucleus data indicate the presence of nuclear shadowing and multiscattering effects. Further corrections are needed in nucleus-nucleus collisions to explain the observed reduction of the cross section. We introduce the idea of proton dissociation and compare our calculations with the SPS and RHIC experimental data., Comment: Talk presented by G. Papp at Zakopane 2001 School, Zakopane, 2001 June; 10 pages with 3 EPS figures

The intrinsic transverse momentum distribution of partons in the nucleon can be used to explain a large amount of high-$p_T$ hadron and photon production data in high-energy nucleon-nucleon collisions at energies $\sqrt{s} \approx 20$ to 1800 GeV. However, proton-nucleus experiments at energies $\approx 30$ GeV show an extra enhancement (Cronin effect) in the yield of photons and mesons compared to a simple extrapolation of the proton-proton data. This enhancement is due to the effect of the dense hadronic matter encountered by the projectile proton in the nuclear environment. We discuss the origin and the properties of the nuclear enhancement., Comment: 5 pages, LaTeX with one EPS figure; Contribution to the 'International Workshop XXVIII on Gross Properties of Nuclei and Nuclear Excitations', Hirschegg, Austria, 16-22.01.2000

We model the effects of a large number of zero and near-zero modes in the QCD partition function by using sparse chiral matrix models with an emphasis on the quenched topological susceptibility in the choice of the measure. At finite temperature, the zero modes are not affected by temperature but are allowed to pair into topologically neutral near-zero modes which are gapped at high temperature. In equilibrium, chiral and U(1) symmetry are simultaneously restored for total pairing, evading mean-field arguments. We analyze a number of susceptibilities versus the light quark masses. At the transition point the topological susceptibility vanishes, and the dependence on the vacuum angle $\theta$ drops out. Our results are briefly contrasted with recent lattice simulations., Comment: 15 pages with 1 epf figure included; Proc of the Light-Cone Workshop "New Directions in QCD", June 1999, Kyungju, Korea