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Purpose: This study aimed to present a preliminary case analysis of the impact of regular and irregular exercise on autonomic regulation and cardiorespiratory performance in young women by comprehensively investigating the nocturnal heart rate variability (HRV) parameters. Methods: Two young female participants were monitored using noncontact ballistocardiography technology to assess their nocturnal HRV daily for 32 weeks. Participant 1 was a 28-year-old woman who engaged in regular running (approximately three times a week, 5km each time), and participant 2 was a 24-year-old woman who participated in irregular running (typically ≤3 times a week, 5km each time). Additionally, cardiorespiratory fitness was evaluated through maximal oxygen uptake (VO₂max), with running data and VO₂max measurements recorded using a wrist bracelet device. Results: During the experiment, the VO₂max value of participant 1 increased by 11.46%, whereas that of participant 2 increased by 3.42%. A correlation was observed between VO₂max and HRV, particularly in the high-frequency (HF) component. The correlation coefficient between ln HF and VO₂max of participant 1 was 0.64, whereas that of participant 2 was 0.28. Additionally, participant 1 exhibited lower HRV complexity than participant 2, with fuzzy entropy values for ln HF of 0.12 and 0.35, respectively. Conclusions: Long-term assessment revealed a correlation between VO₂max and nocturnal HRV in young female exercisers, particularly for the HF index. However, these findings may not apply to other populations, such as men or older individuals.

Let ${ℳ}\subset {ℝ}^n$ be a $C^2$-smooth compact submanifold of dimension $d$. Assume that the volume of ${ℳ}$ is at most $V$ and the reach (i.e. the normal injectivity radius) of ${ℳ}$ is greater than $\tau$. Moreover, let $\mu$ be a probability measure on ${ℳ}$ whose density on ${ℳ}$ is a strictly positive Lipschitz-smooth function. Let $x_j\in {ℳ}$, $j=1,2,\dots ,N$ be $N$ independent random samples from distribution $\mu$. Also, let ${\xi }_j$, $j=1,2,\dots ,N$ be independent random samples from a Gaussian random variable in ${ℝ}^n$ having covariance ${\sigma }^{2}I$, where $\sigma$ is less than a certain specified function of $d,V$ and $\tau$. We assume that we are given the data points $y_j=x_j+{\xi }_j,$$j=1,2,\dots ,N$, modeling random points of ${ℳ}$ with measurement noise. We develop an algorithm which produces from these data, with high probability, a $d$ dimensional submanifold ${{ℳ}}_o\subset {ℝ}^n$ whose Hausdorff distance to ${ℳ}$ is less than $\mathrm{\Delta }$ for $\mathrm{\Delta }>Cd{\sigma }^2/\tau$ and whose reach is greater than $c\tau /d^6$ with universal constants $C,c>0$. The number $N$ of random samples required depends almost linearly on $n$, polynomially on ${\mathrm{\Delta }}^{-1}$ and exponentially on $d$.

An $n$-gram in music is defined as an ordered sequence of $n$ notes of a melodic sequence $m$. $P_m(n)$ is calculated as the average of the occurrence probability without self-matches of all $n$-grams in $m$. Then, $P_m(n)$ is compared to the averages Shuff${}_m(n)$ and Equip${}_m(n)$, calculated from random sequences constructed with the same length and set of symbols in $m$ either by shuffling a given sequence or by distributing the set of symbols equiprobably. For all $n$, both $P_m(n)-{\mathrm{\text{Shuff}}}_m(n)$, $P_m(n)-{\mathrm{\text{Equip}}}_m(n)\ge 0$, and this differences increases with $n$ and the number of notes, which proves that notes in musical melodic sequences are chosen and arranged in very repetitive ways, in contrast to random music. For instance, for $n\le 5$ and for all analyzed genres we found that $1.6<-log(P_m(n))<8.6$, while $1.6<-log({\mathrm{\text{Shuff}}}_m(n))<14.5$ and $1.9<-log({\mathrm{\text{Equip}}}_m(n))<18.3$. $P_m(n)$ of baroque and classical genres are larger than the romantic genre one. $P_m(n)$ vs $n$ is very well fitted to stretched exponentials for all songs. This simple method can be applied to any musical genre and generalized to polyphonic sequences.

In any legal chess position, we define an attacked-square entropy $S$ for either Black or White pieces in terms of the square occupation probability $p_i=m_i∕M$, where $m$ is the number of all possible movements to square $i$ (free or occupied by an opponent’s piece) and $M$ is the total mobility defined as the sum of all possible movements. Thus, each attacked square contributes to the entropy according to its received “firepower” concentration. A simpler nonsquare-dependant equiprobable entropy $S_e$ in terms of equal probabilities $p_e=1∕M$ always yields $S_e-S\ge 0$. On average, the difference $S_e-S$ is very large in the Opening phase and $S$ decreases faster for lower ranked players after move 25. A major cause of the reduction of $S$ during a game is material loss, which is an irreversible process. By game outcome, gaps in average $〈S〉$ among winners, draws and losers are larger for Amateur players than for Elite players, both in the Middlegame and Endgame. Statistically, Elite players exhibit narrower dispersions in $S$. Also, the entropy rates of the Elite level fluctuate much less than the entropies of other levels. Density of attacks in the four-square central zone is very high in the Opening, specially for Elite players.

In this paper, we statistically analyze the density of each chess piece in four square concentric frames for different skill levels (according to the Elo ranking system), phases (opening, middlegame and endgame) and outcomes. The results indicate a tendency for pawns, knights and bishops to move towards the central areas, particularly during the opening phase. Additionally, we identified intrinsic entropy patterns for each piece throughout the entire game, independent of player skill level. We constructed a phase-space from entropy delay time series to provide probabilistic predictions for game outcomes.

Utilizing the gauge framework, software under development at Baylor University, we explicitly construct all layer 1 weakly coupled free fermionic heterotic string (WCFFHS) gauge models up to order 32 in four to ten large spacetime dimensions. These gauge models are well suited to large scale systematic surveys and, while they offer little phenomenologically, are useful for understanding the structure of the WCFFHS region of the string landscape. Herein, we present the gauge groups statistics for this swath of the landscape for both supersymmetric and non-supersymmetric models.

Fractons are anyons classified into equivalence classes and they obey specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the fractional quantum Hall effect (FQHE) and the concept of duality between such classes, defined by shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A β-function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h,ν) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.

Applying the powerful thin film brick-wall model to the general Kerr–Newman black hole, we find out that the entropy calculation result can also satisfy the area theorem. Moreover, the area theorem is not only satisfied for the global black hole, but also for every area cell on its horizon, that means, every cell on the horizon contributes its own part of entropy if we choose a same temperature-related radial cutoff ε'. This new thin film brick-wall model can be used to calculate dynamic black hole which has different temperatures on the horizon. It tells us that the horizon is exactly the statistical origin of a black hole entropy, the total entropy of a black hole is just the sum of all the contributions from every area cell. For a Kerr–Newman black hole, there is also an important difference between the thin film brick-wall model and the original one, that is, we do not need any angular cutoff in the thin film model, and this makes the physical meaning clearer.

The subject of determining the significance of observations in particle physics is discussed. Several dangers are identified, all related to the problem of knowing the sampling probability distribution. Techniques for mitigating these pitfalls are presented.

We give an overview of the recent progress in the field of cosmological model selection. Model selection statistics, such as those based on information theory and on Bayesian statistics are introduced and discussed. In the Bayesian framework, the marginalised model likelihood, or evidence, is the primary model selection statistic. We describe different methods of computing the evidence, and focus in particular on Nested Sampling. We describe the results of applying model selection methods to new cosmological data such as the CMB measurements by WMAP.

We outline a comprehensive and first-principle solution to the wall-crossing problem in D = 4N = 2 Seiberg–Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and of how this allows them to disappear abruptly as parameters or vacuum moduli are continuously changed. This means that the wall-crossing problem is really a bound state formation/dissociation problem. A low energy dynamics for arbitrary collections of dyons is derived, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We discover that the low energy dynamics of such BPS dyons cannot be reduced to one on the classical moduli space, , yet the index can be phrased in terms of . The so-called rational invariant, first seen in Kontsevich–Soibelman formalism of wall-crossing, is shown to incorporate Bose/Fermi statistics automatically. Furthermore, an equivariant version of the index is shown to compute the protected spin character of the underlying D = 4N = 2 theory, where isometry of is identified as a diagonal subgroup of rotation SU(2)_L and R-symmetry SU(2)_R.

In this paper we describe RooFitUnfold, an extension of the RooFit statistical software package to treat unfolding problems, and which includes most of the unfolding methods that commonly used in particle physics. The package provides a common interface to these algorithms as well as common uniform methods to evaluate their performance in terms of bias, variance and coverage. In this paper we exploit this common interface of RooFitUnfold to compare the performance of unfolding with the Richardson–Lucy, Iterative Dynamically Stabilized, Tikhonov, Gaussian Process, bin-by-bin and inversion methods on several example problems.

We continue the study of Confined Vortex Surfaces (CVS) that we introduced in the previous paper. We classify the solutions of the CVS equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation $\left|y\right||x{|}^{\mu }=\text{const}$ in each quadrant of the tube cross-section ($xy$ plane).

We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We vindicate this assumption by the scaling laws for the surface shrinking to zero in the extreme turbulent limit. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate.

We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the $xy$ plane. This phenomenon naturally leads to the “multifractal” scaling of the moments of velocity difference $v(r_1)-v(r_2)$. More precisely, these moments have a nontrivial dependence of $n$, $log\mathrm{\Delta }r$, approximating power laws with effective index $\zeta (n,log\mathrm{\Delta }r)$. We derive some general formulas for the moments containing multidimensional integrals. The rough estimate of resulting moments shows the log–log derivative $\zeta (n,log\mathrm{\Delta }r)$ which is approximately linear in $n$ and slowly depends on $log\mathrm{\Delta }r$. However, the value of effective index is wrong, which leads us to conclude that some other solution of the CVS equations must be found. We argue that the approximate phenomenological relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the CVS theory. We reinterpret their renormalization parameter $\alpha \approx 0.95$ in the Bernoulli law $p=-\frac{1}{2}\alpha v^2$ as a probability to find no vortex surface at a random point in space.

Shortly after his escape, in 1968, from the DDR to the Western world, Harald Fritzsch made a fast career in physics while investigating the strong force in elementary particles. Inspired by Gell-Mann, Yang and Mills and many others, he contributed to the rise of the Standard Model, and in particular the developments of theories for the color confinement mechanism.

The statistical modeling was used in the analysis of structures containing heterogeneous masses. The statistical probability of a system of coupled oscillators was found taking into account the mass dependence of energy levels. It is essentially a new approach. Both the low as well as high temperature expressions for diffusion coefficient were found. Comparison with experimental data gave satisfactory agreement.

In a recent paper several species of octagonal patterns have been introduced with the help of a construction which allows us to derive them by means of inflation rules. Non-deterministic patterns can be generated by composition of the inflation rules. In this paper we show how a similar construction produces patterns with hexagonal symmetry. The non-deterministic rhombus–triangle tilings are obtained by local rearrangements of tiles which are included in the inflation rules. This property allows to compute the configurational entropy.

The ubiquitous presence of complexity in nature makes it necessary to seek new mathematical tools which can probe physical systems beyond linear or perturbative approximations. The random matrix theory is one such tool in which the statistical behavior of a system is modeled by an ensemble of its replicas. This paper is an attempt to review the basic aspects of the theory in a simplified language, aimed at students from diverse areas of physics.

An interpretation of Icosahedral Danzer tilings in terms of algebraic substitutions is used in order to study the Fourier transform of suitable mass distributions. Numerical results are obtained for a mass distribution placed on vertex positions.

Classical statistical inference is nonmonotonic in nature. We show how it can be formalized in the default logic framework. The structure of statistical inference is the same as that represented by default rules. In particular, the prerequisite corresponds to the sample statistics, the justifications require that we do not have any reason to believe that the sample is misleading, and the consequence corresponds to the conclusion sanctioned by the statistical test.

The chaotic motion of a ray path in a deep water acoustic waveguide with internal-wave-induced fluctuations of the sound speed is investigated. A statistical approach for the description of chaotic rays is discussed. The behavior of ray trajectories is studied using Hamiltonian formalism expressed in terms of action-angle variables. It is shown that the range dependence of the action variable of chaotic ray can be approximated by a random Wiener process. On the basis of this result, analytical expressions for probability density functions of ray parameters are derived. Distributions of coordinates, momenta (grazing angles), and actions of sound rays are evaluated. Numerical simulation shows that statistical characteristics of ray parameters weakly depend on a particular realization of random perturbation giving rise to ray chaos.