In this paper we confirm a conjecture of Füredi, Jiang, and Seiver, and determine an exact formula for the Turán number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and Füredi for $n\ge 75$ and Csákány and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Turán number, defined as the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$ vertices which is \emph{not} $C^3_3$-free.

First, starting from two hierarchies of autonomous Stäckel ODE's, we reconstruct the hierarchy of KdV stationary systems. Next, we deform considered autonomous Stäckel systems to non-autonomous Painlevé hierarchies of ODE's. Finally, we reconstruct the related non-autonomous KdV stationary hierarchies from respective Painlevé systems.

We propose a method to witness entanglement between two continuous-variable systems in a Gaussian state. Its key ingredient is a local lossy state transfer from the original spatially separated systems onto two spatially separated qubits. The qubits are initially in a pure product state, therefore by detecting entanglement between the qubits we witness entanglement between the two original systems. This method greatly simplifies entanglement witnessing in complex systems.

In this paper we confirm a conjecture of Füredi, Jiang, and Seiver, and determine an exact formula for the Turán number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and Füredi for $n\ge 75$ and Csákány and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Turán number, defined as the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$ vertices which is \emph{not} $C^3_3$-free.

We present a study of the parallel tempering (replica exchange) Monte Carlo method, with special focus on the feedback-optimized parallel tempering algorithm, used for generating an optimal set of simulation temperatures. This method is applied to a lattice simulation of a homopolymer chain undergoing a coil-to-globule transition upon cooling. We select the optimal number of replicas for different chain lengths, N = 25, 50 and 75, using replica's round-trip time in temperature space, in order to determine energy, specific heat, and squared end-to-end distance of the hopolymer chain for the selected temperatures. We also evaluate relative merits of this optimization method.

In this paper we ivestigate Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

Most asteroid families are very homogeneous in physical properties. Some show greater diversity, however. The Flora family is the most intriguing of them. The Flora family is spread widely in the inner main belt, has a rich collisional history, and is one of the most taxonomically diverse regions in the main belt. As a result of its proximity to the asteroid (4) Vesta (the only currently known intact differentiated asteroid) and its family, migration between the two regions is possible. This dynamical path is one of the counter arguments to the hypothesis that there may be traces of a differentiated parent body other than Vesta in the inner main belt region. We here investigate the possibility that some of the V- and A- types (commonly interpreted as basaltoids and dunites - parts of the mantle and crust of differentiated parent bodies) in the Flora dynamical region are not dynamically connected to Vesta.

We present a novel empirical method for correcting asteroid phase curves for rotational and geometrical effects using precomputed spin-and-shape models. Our approach normalizes sparse photometric data to a pole-on geometry, enabling consistent phase-curve fitting across apparitions. We fit both the H,G1,G2 and H,G12 phase functions to the normalized data. We also numerically derive new constraints on parameter ranges that ensure physically meaningful solutions. These constraints are based on the requirement that the reduced magnitude must monotonically decrease with phase angle and remain within plausible slope bounds. Compared to earlier bounds, our new constraints are more permissive. We also compare derivative-based and derivative-free optimization methods, highlighting convergence issues with the HG12 function and offering mitigation strategies. We applied our method to over 25,000 asteroids observed by the ATLAS survey, demonstrating its usability. The new method enables the selection of the preferred spin-and-shape solution based on either statistical phase-curve model selection criteria and/or physically motivated constraints on the phase-curve shape.

Using Monte Carlo simulations we study the dynamics of three-dimensional Ising models with nearest-, next-nearest-, and four-spin (plaquette) interactions. During coarsening, such models develop growing energy barriers, which leads to very slow dynamics at low temperature. As already reported, the model with only the plaquette interaction exhibits some of the features characteristic of ordinary glasses: strong metastability of the supercooled liquid, a weak increase of the characteristic length under cooling, stretched-exponential relaxation and aging. The addition of two-spin interactions, in general, destroys such behaviour: the liquid phase loses metastability and the slow-dynamics regime terminates well below the melting transition, which is presumably related with a certain corner-rounding transition. However, for a particular choice of interaction constants, when the ground state is strongly degenerate, our simulations suggest that the slow-dynamics regime extends up to the melting transition. The analysis of these models leads us to the conjecture that in the four-spin Ising model domain walls lose their tension at the glassy transition and that they are basically tensionless in the glassy phase.

Context. Phase curves of small bodies are useful tools to obtain their absolute magnitudes and phase coefficients. The former relates to the object's apparent brightness, while the latter relates to how the light interacts with the surface. Data from multi-wavelength photometric surveys, which usually serendipitously observe small bodies, are becoming the cornerstone of large statistical studies of the Solar System. Nevertheless, to our knowledge, all studies have been carried out in visible wavelengths. Aims. We aim to provide the first catalog of absolute magnitudes in near-infrared filters (Y, J, H, and K). We will study the applicability of a non-linear model to these data and compare it with a simple linear model. Methods. We compute the absolute magnitudes using two photometric models: the HG* 12 and the linear model. We employ a combination of Bayesian inference and Monte Carlo sampling to calculate the probability distributions of the absolute magnitudes and their corresponding phase coefficients. We use the combination of four near-infrared photometric catalogs to create our input database. Results. We produced the first catalog of near-infrared magnitudes. We obtained absolute magnitudes for over 10 000 objects (with at least one absolute magnitude measured), with about 180 objects having four absolute magnitudes. We confirmed that a linear model that fits the phase curves produces accurate results. Since a linear behavior well describes the curves, fitting to a restricted phase angle range (in particular, larger than 9.5 deg) does not substantially affect the results. Finally, we also detect a phase-coloring effect in the near-infrared, as observed in visible wavelengths for asteroids and trans-Neptunian objects.

Bacterial DNA gyrase introduces negative supercoils into chromosomal DNA and relaxes positive supercoils introduced by replication and transiently by transcription. Removal of these positive supercoils is essential for replication fork progression and for the overall unlinking of the two duplex DNA strands, as well as for ongoing transcription. To address how gyrase copes with these topological challenges, we used high-speed single-molecule fluorescence imaging in live Escherichia coli cells. We demonstrate that at least 300 gyrase molecules are stably bound to the chromosome at any time, with ~12 enzymes enriched near each replication fork. Trapping of reaction intermediates with ciprofloxacin revealed complexes undergoing catalysis. Dwell times of ~2 s were observed for the dispersed gyrase molecules, which we propose maintain steady-state levels of negative supercoiling of the chromosome. In contrast, the dwell time of replisome-proximal molecules was ~8 s, consistent with these catalyzing processive positive supercoil relaxation in front of the progressing replisome.

We construct Lax pairs for a wide class of Stäckel systems by applying the multi-parameter Stäckel transform to Lax pairs of a suitably chosen systems from the seed class. For a given Stäckel system, the obtained set of non-equivalent Lax pairs is parametrized by an arbitrary function.

In this paper we construct Lax pairs for Stäckel systems with separation curves from so-called Benenti class. For each system of considered family we present an infinite family of Lax representations, parameterized by smooth functions of spectral parameter.

Current gravitational wave detectors are sensitive to coalescing black holes and neutron stars. However, double white dwarfs (DWDs) have long been recognized as promising sources of gravitational waves, and upcoming detectors like LISA will be able to observe these systems in abundance. DWDs are expected to be the dominant gravitational wave (GW) sources in parts of the LISA frequency range, making it crucial to understand their formation for future detections. The Milky Way contains many white dwarfs (WDs) in both the field and star clusters, promising a rich population of DWDs for LISA. However, the large number of sources may make it difficult to resolve individual binaries, and DWDs in the field and clusters often have similar properties, complicating the identification of their origins from GW signals alone. In this work, we focus on eccentric and tight DWDs, which cannot form in the field, but require dynamical interactions in dense clusters to increase their eccentricity after circularization through mass transfer phases and common-envelope evolution during binary evolution. These binaries may also form in three- and four-body dynamical interactions where a DWD binary may directly form with high eccentricity and low separation. Our results show that we should expect eccentric and tight DWDs in clusters that can provide meaningful GW signal, however, the number is low; with an upper limit of 10-15 in the MW. These can be used to independently obtain distances of their host cluster.

We recently introduced three new parameters that describe the shape of the normalized cumulative radial distribution (nCRD) of the innermost stars in globular clusters and trace the clusters dynamical evolution. Here we extend our previous investigations to the case of a large set of Monte Carlo simulations of globular clusters, started from a broad range of initial conditions. All the models are analyzed at the same age of 13 Gyr, when they have reached different evolutionary phases. The sample of models is well representative of the structural properties of the observed population of Galactic globular clusters. We confirm that the three nCRD parameters are powerful tools to distinguish systems in early stages of dynamical evolution, from those that already experienced core collapse. They might also help disentangle clusters hosting a low-mass intermediate-mass black hole of a few hundred solar masses, from cases with large concentrations of dark remnants in their centers. With respect to other dynamical indicators, the nCRD parameters offer the advantage of being fully empirical and easier to measure from observational data.

We present a micromagnetic approach to exchange bias (EB) in ferromagnetic (FM)/antiferromagnetic (AFM) thin film systems with a small number of irreversible interfacial magnetic moments. We express the exchange bias field $H_{EB}$ in terms of fundamental micromagnetic length scale---the exchange length $l_{ex}$. The benefit from this approach is a better separation of a term related to FM layer from a term related to FM/AFM coupling at interfaces. Using this approach we estimate the highest limit of $H_{EB}$ in real FM/AFM systems.

We present high precision radial velocities (RVs) of double-lined spectroscopic binary stars HD78418, HD123999, HD160922, HD200077 and HD210027. They were obtained based on the high resolution echelle spectra collected with the Keck I/Hires, Shane/CAT/Hamspec and TNG/Sarge telescopes/spectrographs over the years 2003-2008 as a part of TATOOINE search for circumbinary planets. The RVs were computed using our novel iodine cell technique for double-line binary stars. The precision of the RVs is of the order of 1-10 m/s. Our RVs combined with the archival visibility measurements from the Palomar Testbed Interferometer allow us to derive very precise spectroscopic/astrometric orbital and physical parameters of the binaries. In particular, we derive the masses, the absolute K and H band magnitudes and the parallaxes. The masses together with the absolute magnitudes in the K and H bands enable us to estimate the ages of the binaries. These RVs allow us to obtain some of the most accurate mass determinations of binary stars. The fractional accuracy in m*sin(i) only and hence based on the RVs alone ranges from 0.02% to 0.42%. When combined with the PTI astrometry, the fractional accuracy in the masses ranges in the three best cases from 0.06% to 0.5%. Among them, the masses of HD210027 components rival in precision the mass determination of the components of the relativistic double pulsar system PSRJ0737-3039. In the near future, for double-lined eclipsing binary stars we expect to derive masses with a fractional accuracy of the order of up to ~0.001% with our technique. This level of precision is an order of magnitude higher than of the most accurate mass determination for a body outside the Solar System - the double neutron star system PSRB1913+16.

We present a multiparameter generalization of the St\"ackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized St\"ackel transform preserves the Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized St\"ackel transform.

We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.

Biological molecular machines are enzymes that simultaneously catalyze two processes, one donating free energy and second accepting it. Recent studies show that most native protein enzymes have a rich stochastic dynamics that often manifests in fluctuating rates of the catalyzed processes and the presence of short-term memory resulting from transient non-ergodicity. For such dynamics, we prove the generalized fluctuation theorem predicting a possible reduction of energy dissipation at the expense of creating some information stored in memory. The theoretical relationships are verified in computer simulations of random walk on a model critical complex network. The transient utilization of memory may turn out to be crucial for the movement of protein motors and the reason for most protein machines to operate as dimers or higher organized assemblies. From a broader physical point of view, the division of free energy into the operation and organization energy is worth emphasizing. Information can be assigned a physical meaning of a change in the value of both these functions of state.