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We propose a new general method to study critical points (CP) using the finite-size scaling of Lee-Yang zeros (LYZ). We first study the LYZ in the three-dimensional Ising model on finite lattices. We show that the ratios of multiple LYZ (Lee-Yang-zero ratios: LYZR) have useful scaling properties similar to the Binder cumulants, providing us with a novel method to study CP. In numerical simulations of the Ising model, we confirm that this method works well. We then apply the method to analyze the CP in the three-dimensional three-state Potts model and finite-temperature QCD in heavy-quark region, which are believed to belong to the same universality class as the Ising model. In these models, the partition function at complex parameters can be evaluated by the reweighting method, which allows us to determine the LYZ by varying coupling parameters continuously around the CP. We demonstrate that the LYZR method is powerful in determining the location of the CP in these models., Comment: 10 pages, 5 figures; contribution to the 41th International Symposium on Lattice Field Theory, 2024, Liverpool, UK

Algebraic domains are regions in the plane surrounded by mutually disjoint non-singular real algebraic curves. Poincar'e-Reeb Graphs of them are graphs they naturally collapse: such graphs are formally formulated by Sorea, for example, around 2020. Their studies found that nicely embedded planar graphs are Poincar'e-Reeb graphs of some algebraic domains. These graphs are generic with respect to the projection to the horizontal axis. Problems, methods and results are elementary and natural and they apply natural approximations nicely for example. We present our new approach to extension of the result to a non-generic case and an answer. We first formulate generalized algebraic domains, surrounded by non-singular real algebraic curves which may intersect with normal crossings. Such domains and certain classes of them appear in related studies of graphs and regions surrounded by algebraic curves explicitly., Comment: 10 pages. 3 figures, We have added "Conflict of interest" and "Data availability". Several errors are corrected where essentially our main ingredient is unchanged

We have previously proposed a study of arrangements of small circles which also surround regions in the plane realized as the images of natural real algebraic maps yielding Morse-Bott functions by projections. Among studies of arrangements, families of smooth regular submanifolds in smooth manifolds, this study is fundamental, explicit, and new, surprisingly. We have obtained a complete list of local changes of the graphs the regions naturally collapse to in adding a (generic) small circle to an existing arrangement of the proposed class. Here, we propose a similar and essentially different class of arrangements of circles. The present study also yields real algebraic maps and nice real algebraic functions similarly and we present a similar study. We are interested in topological properties and combinatorics among such arrangements and regions and applications to constructing such real algebraic maps and manifolds explicitly and understanding their global structures., Comment: 16 pages. 7 figures

We present conditions for reconstruction of Morse-Bott functions with prescribed preimages on $3$-dimensional manifolds. The present work strengthens a previous result for the Morse function case by the author and present a related example as another result. This shows a new result on reconstruction of nice smooth functions such that preimages are as prescribed. Such a study has been fundamental, natural, and surprisingly, founded recently, in 2006, by Sharko. Reconstruction of nice smooth functions on closed surfaces has been followed by Masumoto-Saeki, for example, and later, Gelbukh, Marzantowicz, Michalak, and so on, are studying Morse function cases further. The author has started explicit studies for $3$-dimensional cases respecting topologies of preimages of single points and obtained several results. We add another result on this., Comment: 7 pages. Our main result (Theorem 1) extends "Theorem 1 from arXiv:2412.20626" to a certain class of Morse-Bott functions (from Morse functions). After the submission of the preprint to a refereed journal, we hit on an essential idea and this has helped us to present the present preprint

We discuss a necessary and sufficient condition for reconstruction of Morse functions with prescribed preimages on $3$-dimensional manifolds. The present work strengthens a previous result of the author where only sufficient conditions are studied. Our new work is also regarded as a kind of addendum., Comment: 7 pages. A kind of addenda to a published article http://mfat.imath.kiev.ua/article/?id=1911 (arXiv:2108.01300): here we show that the sufficient condition for reconstructing Morse functions there is also a necessary condition. This version is submitted to a refereed journal

We characterize $3$-dimensional manifolds represented as connected sums of Lens spaces, copies of $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions. This adds to our previous result around 2024, classifying Morse functions whose preimages containing no singular points are disjoint unions of spheres and tori on $3$-dimensional manifolds represented as connected sums of connected sums of Lens spaces and copies of $S^2 \times S^1$: we have strengthened and explicitized Saeki's result, characterizing the manifolds via such functions, in 2006. We apply similar arguments. However we discuss in a self-contained way essentially., Comment: 13 pages. 5 figures. This is closely related to our previous study arXiv:2411.15943 and this contains new arguments with our new result to be presented in a single form. A new study and result (Theorem 3 and its proof) is added

As a topic of mathematics, "arrangements", systems of hyperplanes, circles, and general (regular) submanifolds, attract us strongly. We present a natural elementary study of arrangements of circles. It is also a kind of new studies. Our study is closely related to geometry and singularity theory of Morse(-Bott) functions. Regions surrounded by circles are regarded as images of real algebraic maps and composing them with projections gives Morse-Bott functions: this observation is natural, and surprisingly, recently presented first, by the author. We present a systematic way of constructing such arrangements by choosing small circles centered at existing circles inductively. We are interested in graphs the regions surrounded by the circles naturally collapse. We have studied local changes of the graphs in adding these circles. These graphs are essentially so-called {\it Reeb graphs} of the previous Morse-Bott functions: they are spaces of all components of preimages of single points for the functions., Comment: 22 pages. 9 figures. Several errors are corrected where readers may understand correctly before the correction

We study the finite-temperature critical point of QCD in the heavy-quark region by a scaling study of the Binder cumulant on large lattices. Extending our previous study at $N_t=4$, we perform simulations on $N_t=6$ and 8 lattices with spatial volumes up to the aspect ratio $LT=N_s/N_t=18$ and 15 ($N_s=108$ and 120), respectively, to determine the critical point in the thermodynamic limit with a high precision. To enable simulations with large spatial volumes, we adopt the hopping parameter expansion combined with a method to effectively incorporate high order terms of the expansion. The reliability of the method is confirmed by examining the effect of high order terms. Using the results of the critical point at $N_t=4$, 6, and 8, we also attempt a preliminary continuum extrapolation of the critical point in physical units., Comment: 9 pages, 5 figures, Contribution to the 41st International Symposium on Lattice Field Theory (LATTICE2024), 28 July - 3 August 2024, Liverpool, UK

Morse functions are important objects and tools in understanding topologies of manifolds since the 20th century. Their classification has been natural and difficult problems, and surprisingly, this is recently developing. Since the 2010's, results for cases of surfaces have been presented by Gelbukh, Marzantowicz and Michalak for example. We have also longed for higher dimensional cases. We present a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one. We concentrate on Morse functions such that preimages of single points containing no singular points are disjoint unions of spheres and tori. Existence of such functions implies that the $3$-dimensional closed and connected manifolds are of such manifolds. This has been shown by Saeki in 2006 and we further study structures of these functions., Comment: 15 pages. 5 figures. Statement of our main result (Theorem 2) is corrected where we do not find essential error in our proof. several small revision

YbCu4Au is a unique material exhibiting multiple quantum fluctuations simultaneously. In this study, we investigated the field-induced criticality in YbCu4Au, based on comprehensive micro and macro measurements, including powder X-ray diffraction (XRD), neutron powder diffraction (NPD), nuclear magnetic resonance, magnetization, resistivity, specific heat, muon spin rotation relaxation (muSR), and X-ray absorption spectroscopy (XAS). Single crystals of YbCu4Au were grown, and their crystal structure was determined using XRD, and NPD measurements. Magnetic successive transitions were observed below 1 T by specific heat, resistivity, NPD, and muSR measurements. XAS measurements further indicate that the valence of Yb ions (+2.93) remained unchanged above 2 T. Moreover, the change in quadrupole frequency observed in the previous study is attributable to the electric quadrupole, as the expected value of the electric quadrupole was finite under magnetic fields [S. Wada et al., Journal of Physics: Condensed Matter, 20, 175201 (2008).]. These experimental results suggest that YbCu4Au exhibited bicritical behavior near 1 T, arising from the competition between RKKY interaction, accounting for the magnetic phases, and the Zeeman effect., Comment: 9 pages, 8 figures

Many models of visual attention have been proposed so far. Traditional bottom-up models, like saliency models, fail to replicate human gaze patterns, and deep gaze prediction models lack biological plausibility due to their reliance on supervised learning. Vision Transformers (ViTs), with their self-attention mechanisms, offer a new approach but often produce dispersed attention patterns if trained with supervised learning. This study explores whether self-supervised DINO (self-DIstillation with NO labels) training enables ViTs to develop attention mechanisms resembling human visual attention. Using video stimuli to capture human gaze dynamics, we found that DINO-trained ViTs closely mimic human attention patterns, while those trained with supervised learning deviate significantly. An analysis of self-attention heads revealed three distinct clusters: one focusing on foreground objects, one on entire objects, and one on the background. DINO-trained ViTs offer insight into how human overt attention and figure-ground separation develop in visual perception.

We present the lattice simulation of the renormalization group flow in the $3$-dimensional $O(N)$ linear sigma model. This model possesses a nontrivial infrared fixed point, called Wilson--Fisher fixed point. Arguing that the parameter space of running coupling constants can be spanned by expectation values of operators evolved by the gradient flow, we exemplify a scaling behavior analysis based on the gradient flow in the large $N$ approximation at criticality. Then, we work out the numerical simulation of the theory with finite $N$. Depicting the renormalization group flow along the gradient flow, we confirm the existence of the Wilson--Fisher fixed point non-perturbatively., Comment: 11 pages, 6 figures, talk presented at the 41st International Symposium on Lattice Field Theory (Lattice2024), July 28th - August 3rd, 2024, The University of Liverpool

We propose a method to numerically determine the location of a critical point in general systems using the finite-size scaling of Lee-Yang zeros. This method makes use of the fact that the ratios of Lee-Yang zeros on various spatial volumes intersect at the critical point. While the method is similar to the Binder-cumulant analysis, it is advantageous in suppressing the finite-volume effects arising from the mixing of variables in general systems. We show that the method works successfully for numerically locating the CP in the three-dimensional three-state Potts model with a nonzero external field., Comment: 6 pages, 3 figures, Minor changes

Density ratio estimation (DRE) is a fundamental machine learning technique for identifying relationships between two probability distributions. $f$-divergence loss functions, derived from variational representations of $f$-divergence, are commonly employed in DRE to achieve state-of-the-art results. This study presents a novel perspective on DRE using $f$-divergence loss functions by deriving the upper and lower bounds on $L_p$ errors. These bounds apply to any estimator within a class of Lipschitz continuous estimators, irrespective of the specific $f$-divergence loss functions utilized. The bounds are formulated as a product of terms that include the data dimension and the expected value of the density ratio raised to the power of $p$. Notably, the lower bound incorporates an exponential term dependent on the Kullback--Leibler divergence, indicating that the $L_p$ error significantly increases with the Kullback--Leibler divergence for $p > 1$, and this increase becomes more pronounced as $p$ increases. Furthermore, these theoretical findings are substantiated through numerical experiments.

We investigate the volume and lifetime of the high baryon-density matter created in heavy-ion collisions and estimate the optimal collision energy to realize the high baryon-density region over a large spacetime volume. We simulate central collisions of gold ions for the center-of-mass energy per nucleon pair $\sqrt{s_{NN}}=2.4 - 19.6\;{\rm GeV}$ with a microscopic transport model JAM. We discover that the optimal collision energy is around $\sqrt{s_{NN}}=3 - 4\;{\rm GeV}$, where a baryon density exceeding three times the normal nuclear density is realized with a substantially large spacetime volume. Higher and lower energies are disfavored due to short lifetime and low density, respectively. We also point out that event-by-event fluctuations of the spacetime density profile are large, indicating the importance of the event selection in the experimental analysis., Comment: 6 pages, 3 figures

In the context of relativistic heavy-ion collisions, we explore the stochastic and dissipative relaxational dynamics of a non-conserved order parameter in a $\lambda\varphi^4$ interaction. The cutoff of the theory is provided by the lattice spacing chosen for our numerical simulations. As a consequence, observables become dependent on that scale. We consider a possible first-order phase transition and an evolution close to a critical point. We demonstrate that using a lattice counterterm restores the expected behavior of the mean, variance and kurtosis: the mean and the variance become lattice spacing independent, and we recover the correct expectation value of the mean, the growth of the variance with the correlation length and the expected minimum in the kurtosis. Our findings hold true in equilibrium and during the dynamical relaxation, and therefore mark an important step towards a fully fluctuating fluid dynamical setup.

Binder-cumulant analysis of the critical point in the heavy-quark region of QCD is performed by Monte-Carlo simulations with the hopping-parameter expansion at $N_t=6$. We extend our previous analysis at $N_t=4$ to finer lattices and perform high-precision analyses on large spatial volumes up to the aspect ratio $LT=N_s/N_t=18$. Higher order terms in the hopping-parameter expansion are incorporated effectively up to 14th order. The numerical results show that the violation of the finite-size scaling becomes more prominent on the finer lattice at a given aspect ratio., Comment: 13 pages, 10 figures

Infra-Red scaling property of inflationary universe is in the same universality class of random walk. The two point correlators of the curvature perturbations are enhanced by the e-folding number N. The distribution function of the curvature perturbation $\rho_t (\zeta)$ satisfies the Fokker Planck equation. The de Sitter universes are dual to the random walk: They belong to the Universality class of dimension two fractal. These boundary and bulk duality are at the heart of holography of quantum gravity. Historically the correspondence of thermodynamics and Einstein's equation are recognized as the first evidence for de Sitter duality .Our de Sitter duality relates the stochastic and geometric point of view. We study two types of the solutions of FP equation in quasi de Sitter space: (1) UV complete spacetime and (2) inflationary spacetime with concave potentials. The maximum entropy principle favors the following scenario: The universe is (a) born with small epsilon and (b) grows by inflation in the concave potential. We predict n_s <0.975(0.97) and r < 0.04(0.03) for N=50(60) at the pivot angle 0.002Mpc^{-1}. We have lowered the upper bound of $r$ by taking account of random walk effect at the boundary. Our predictions are highly consistent with recent observations., Comment: 22 pages. arXiv admin note: text overlap with arXiv:2303.07676

Using SU(3) lattice QCD, we study for the first time the effective potential of the Polyakov loop $\langle P \rangle$ at finite temperature, i.e., the thermodynamic potential, in the field-theoretical way. In the framework of the reweighting method in lattice QCD, we express the effective potential $V_{\rm eff}(\langle P \rangle)$ using the expectation value without a source term. In particular, we consider the most difficult and interesting case of vacuum coexistence at the critical temperature $T_c$. We adopt SU(3) quenched lattice QCD on $48^3 \times 6$ at $\beta$= 5.89379, which corresponds exactly to the critical temperature $T_c$ of the deconfinement phase transition, and use 200,000 Monte Carlo configurations. After categorizing the gauge configurations into one $Z_3$-symmetric and three $Z_3$-broken vacua each, we perform a vacuum-associated reweighting method, using the gauge configurations around each vacuum separately. Finally, we obtain the Polyakov-loop effective potential, which is well depicted around the $Z_3$-symmetric and $Z_3$-broken vacua., Comment: 11 pages, 4 figures, prepared for XIIth Int. Conf. on New Frontiers in Physics (ICNFP2023)

We explore how the electric conductivity and associated relaxation time are modified near the QCD critical point and the phase transition to a color superconducting phase using the two-flavor Nambu-Jona-Lasinio model with finite current quark masses. We give a comprehensive account of the nature of the soft modes associated with these phase transitions and how they affect the photon self-energy when the system approaches these phase transitions in a combined way with an emphasis on the common and different aspects in the two transitions. The formalism developed for describing the paraconductivity in metallic superconductors is used for the analysis of the photon self-energy. We show that the transport coefficients calculated from the self-energy show anomalous enhancements in both cases with different critical exponents for the individual transitions. We briefly discuss the possibility of detecting the enhancements in the relativistic heavy-ion collisions in the present and future facilities., Comment: 29 pages, many figures

We investigate the thermodynamics and phase structure of $SU(3)$ Yang-Mills theory on $\mathbb{T}^2\times\mathbb{R}^2$ in Euclidean spacetime in an effective-model approach. The model incorporates two Polyakov loops along two compactified directions as dynamical variables, and is constructed to reproduce thermodynamics on $\mathbb{T}^2\times\mathbb{R}^2$ measured on the lattice. The model analysis indicates the existence of a novel first-order phase transition on $\mathbb{T}^2\times\mathbb{R}^2$ in the deconfined phase, which terminates at critical points that should belong to the two-dimensional $Z_2$ universality class. We argue that the interplay of the Polyakov loops induced by their cross term in the Polyakov-loop potential is responsible for the manifestation of the first-order transition., Comment: 16 pages, 15 figures

The values of Hubble constant H0 by direct measurements with standard distance ladder are typically larger than those obtained from the observation of cosmic microwave background and the galaxy survey with inverse distance ladder. On the other hand, although the errors are still large, various determinations of the value of matter density parameter Omega_m are consistent with each other. Therefore, it is possible that the difference in Hubble constant is translated to the difference of physical matter energy density omega_m = Omega_m h^2, where h = H0 / (100 Km/s/Mpc). In this article it is examined the possibility of an increase of the physical dark matter energy density (comoving energy density without the effect of expansion of the universe) by a fast transition at a certain value of redshift as a possible resolution of the Hubble tension. A phenomenological fluid model of the dark sector, which is the modification of a so-called unified dark matter model, is introduced to concretely realize such a fast transition in the physical dark matter energy density., Comment: 17 pages, 3 figures

Density ratio estimation (DRE) is a fundamental machine learning technique for capturing relationships between two probability distributions. State-of-the-art DRE methods estimate the density ratio using neural networks trained with loss functions derived from variational representations of $f$-divergence. However, existing methods face optimization challenges, such as overfitting due to lower-unbounded loss functions, biased mini-batch gradients, vanishing training loss gradients, and high sample requirements for Kullback-Leibler (KL) divergence loss functions. To address these issues, we focus on $\alpha$-divergence, which provides a suitable variational representation of $f$-divergence. Subsequently, a novel loss function for DRE, the $\alpha$-divergence loss function ($\alpha$-Div), is derived. $\alpha$-Div is concise but offers stable and effective optimization for DRE. The boundedness of $\alpha$-divergence provides the potential for successful DRE with data exhibiting high KL-divergence. Our numerical experiments demonstrate the effectiveness in optimization using $\alpha$-Div. However, the experiments also show that the proposed loss function offers no significant advantage over the KL-divergence loss function in terms of RMSE for DRE. This indicates that the accuracy of DRE is primarily determined by the amount of KL-divergence in the data and is less dependent on $\alpha$-divergence., Comment: $\mathcal{T}_{\text{Lip}}$ in Theorem 7.1 (Theorem B.15.) was changed to the set of all locally Lipschitz continuous functions. In the previous version, $\mathcal{T}_{\text{Lip}}$ was defined as the set of all Lipschitz continuous functions, which is unsuitable for the statement of case (ii) in the theorem

We perform a finite-size scaling analysis of the critical point in the heavy-quark region of QCD at nonzero temperature. Our previous analysis on the Binder cumulant at $N_t=4$ is extended to finer lattices with $N_t=6$ and $8$. The aspect ratio is also extended up to $15$ to suppress the non-singular contribution. High-precision analysis of the Binder cumulant is realized by an efficient Monte-Carlo simulation with the hopping-parameter expansion (HPE). Effects of higher-order terms in the HPE are incorporated by the reweighting method., Comment: 7 pages, 8 figures, proceedings of Lattice 2023

We report on the quantum criticality of YbCu$_4$Ni as revealed by our combined micro- and macro-measurements. We determine the crystal structure of YbCu$_4$Ni with site mixing by neutron diffraction measurements, which suggests the possible presence of Kondo disorder. However, decreasing the local spin susceptibility distribution and the development of spin fluctuations below 10 K at ambient pressure by muon spin rotation and relaxation measurements suggests that YbCu4Ni exhibits quantum criticality. Therefore, our experimental results indicate that YbCu4Ni is a new material that exhibits quantum criticality under a zero magnetic field and ambient pressure., Comment: 8 pages, 5 figures

Morse functions with exactly two singular points on homotopy spheres and canonical projections of spheres are generalized as special generic maps. A special generic map is, roughly, a smooth map represented as the composition of a smooth surjection onto a manifold whose preimages are diffeomorphic to a unit sphere in the interior of the manifold and single point sets on the boundary with a smooth immersion of codimension $0$. This paper constructs real algebraic maps topologically special generic maps whose images are smoothly embedded manifolds. We are also interested in construction of explicit and meaningful smooth maps in differential topology and recently ones in real algebraic geometry. This has been an important and difficult problem. In such stories, we have previously constructed real algebraic maps topologically regarded as special generic maps. This paper is a kind of additional short remark on such maps., Comment: 17 pages, the title changed, In Main Theorem 3 complicated arguments and statements are changed into simplified ones, some arguments are revised with more precise exposition, several presentations on real algebraic geometry such as Hilbert's 16th problem are added, small errors are corrected, this is submitted to a refereed journal

We study how the dilepton production rates and electric conductivity are affected by the phase transition to color superconductivity and the QCD critical point. Effects of the soft modes associated with these phase transitions are incorporated through the photon self-energy called the Aslamazov-Larkin, Maki-Thompson, and density-of-states terms, which are responsible for the paraconductivity in metallic superconductors. We show that anomalous enhancements of the production rate in the low energy/momentum region and the conductivity occur around the respective critical points.

We determine the location of the critical point where the first-order deconfining transition in the heavy-quark region turns into a crossover in finite-temperature and density lattice QCD with 2+1 flavors of Wilson quarks. Combining a hopping parameter expansion (HPE) of the quark determinant with a reweighting method, we evaluate the chemical potential dependence of the critical point. By systematically calculating the coefficients of the hopping parameter expansion up to a high order of HPE at finite chemical potential, we find that the higher order terms are strongly correlated with the Polyakov loop, which is the leading-order term, on each configuration. Moreover, their complex phases themselves, which are important at finite density, are also found to be strongly correlated with the complex phase of the Polyakov loop. Using this property, we develop a method for estimating the critical point incorporating high-order terms from calculations with only low-order terms. We report that the first-order phase transition region in the heavy-quark region becomes narrower exponentially with increasing the chemical potential. Since the hopping parameter of the critical point decreases exponentially as the density increases, the sign problem does not become serious even when the density increases, and critical points can be evaluated up to high densities., Comment: 7 pages, 6 figures, Proceedings of the 40th International Symposium on Lattice Field Theory (Lattice 2023), July 31st - August 4th, 2023, Fermi National Accelerator Laboratory

The Hubble constant problem is that the values of Hubble constant from the observation of cosmic microwave background assuming the LambdaCDM model disagrees with the values from direct measurements. This problem suggests some new physics beyond the LambdaCDM model. Typically there are two ways of reconciliation: one is the realization of smaller value of sound horizon at recombination, and the other is the modification of the way of expansion of the universe after recombination. In this letter we examine the latter possibility by comparing two typical phenomenological dark energy models with the distance-redshift relation provided by Pantheon catalogue of supernova observations and galaxy surveys by BOSS and eBOSS collaborations. Though these phenomenological dark energy models globally fit observations better than the LambdaCDM model, they are strongly disfavored by the distance-redshift relation as almost the same level as the LambdaCDM model defined by cosmic microwave background observations. The distance-redshift relation strongly suggests some new physics which realize smaller value of sound horizon at recombination., Comment: 16 pages, 5 figures