Mathematical concepts often have applicability in areas that may have surprised their original developers. This is the case with piecewise isometries (PWIs), which transform an object by cutting it into pieces that are then rearranged to reconstruct the original object, and which also provide a paradigm to study mixing via cutting and shuffling in physical sciences and engineering. Every PWI is characterized by a geometric structure called the exceptional set, $E$, whose complement comprises nonmixing regions in the domain. Varying the parameters that define the PWI changes both the structure of $E$ as well as the degree of mixing the PWI produces, which begs the question of how to determine which parameters produce the best mixing. Motivated by mixing of yield stress materials, for example granular media, in physical systems, we use numerical simulations of PWIs on a hemispherical shell and examine how the fat fractal properties of $E$ relate to the degree of mixing for any particular PWI. We present numerical evidence that the fractional coverage of $E$ negatively correlates with the intensity of segregation, a standard measure for the degree of mixing, which suggests that fundamental properties of $E$ such as fractional coverage can be used to predict the effectiveness of a particular PWI as a mixing mechanism.

Random planar graphs appear in a variety of contexts and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display nontrivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of nontrivial patterns such as central loops. This toy model is made of a star network with $N_b$ branches of size $n$ and links of weight 1, superimposed to a loop at distance $\ell$ from the center and with links of weight $w$. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if $w<w_c$ where the threshold of this transition scales as $w_c\sim n/N_b$. In this regime, there is an optimal position of the loop that scales as ${\ell }_{\mathrm{opt}}\sim N_{b}w/4$. This simple model sheds some light on the organization of these random structures and allows us to discuss the effect of randomness on the centrality of loops. In particular, it suggests that the number and the spatial extension of radial branches are the crucial ingredients that control the existence of central loops.

We propose a method to measure the Hurst exponents of financial time series. The scaling of the absolute moments against the aggregation horizon of real financial processes and of both uniscaling and multiscaling synthetic processes converges asymptotically towards linearity in log-log scale. In light of this we found appropriate a modification of the usual scaling equation via the introduction of a filter function. We devised a measurement procedure which takes into account the presence of the filter function without the need of directly estimating it. We verified that the method is unbiased within the errors by applying it to synthetic time series with known scaling properties. Finally we show an application to empirical financial time series where we fit the measured scaling exponents via a second or a fourth degree polynomial, which, because of theoretical constraints, have respectively only one and two degrees of freedom. We found that on our data set there is not clear preference between the second or fourth degree polynomial. Moreover the study of the filter functions of each time series shows common patterns of convergence depending on the momentum degree.

In this paper we investigate how so-called quorum-sensing networks can be desynchronized. Such networks, which arise in many important application fields, such as systems biology, are characterized by the fact that direct communication between network nodes is superimposed to communication with a shared, environmental variable. In particular, we provide a new sufficient condition ensuring that the trajectories of these quorum-sensing networks diverge from their synchronous evolution. Then, we apply our result to study two applications.

The rearrangement of particles in the final stage of droplet evaporation has been investigated by utilizing differential interference contrast microscopy and the formation mechanism of a network pattern inside a coffee ring has been revealed. A tailored substrate with a circular hydrophilic domain is prepared to obtain thin liquid film containing monolayer particles. Real-time bottom-view images show that the evolution of a dry patch could be divided into three stages: rupture initiation, dry patch expansion, and drying of the residual liquid. A growing number of dry patches will repeat these stages to form the network patterns inside the ringlike stain. It can be shown that the suction effect promotes the rupture of the liquid film and the formation of the dry patch. The particle-assembling process is totally controlled by the liquid film dewetting and dominated by the surface tension of the liquid film, which eventually determine the ultimate deposition patterns.

The dynamic scaling of the island size distribution (ISD) in the submonolayer growth regime of low-dimensional nanostructured systems is a long standing problem in epitaxial growth. In this study, kinetic Monte Carlo simulations of a realistic atomistic lattice-gas model describing the one-dimensional nucleation and growth of Al on Si(100):$2\times 1$ were performed to investigate the scaling behavior under varied growth conditions. Consistent with previous predictions, our results show that the shape of the scaled island size distribution can be altered by controlling the temperature and the $C$-defect density. The shifts in the scaled ISD are opposite to each other with temperature depending on the density of $C$-defects. For low $C$-defect density, a shift from a monomodal to a monotonically decreasing distribution as temperature increases is observed. We attribute the monomodal distribution to enhanced nucleation and aggregation whereas a monotonically decreasing distribution is attributed to restricted aggregation with defects playing only a minor role. At higher $C$-defect density, we show that the scaled ISD shift is from a monotonically decreasing to a monomodal distribution with increasing temperature. We argue that the reversal of the shift is due to competing effects introduced by high $C$-defect concentration. In addition, results show that the ISD is generally insensitive to flux variations and that at a high coverage regime the shift in the scaling behavior vanishes. Lastly, we posit that the shift in the scaled ISD indicates the departure of the island density's temperature dependence from predictions of classical nucleation theory.

The elastoplastic behavior of sodium silicate glasses is studied at different scales as a function of composition and pressure, with the help of quasistatic atomistic simulations. The samples are first compressed and then sheared at constant pressure to calculate yield strength and permanent plastic deformations. Changes occurring in the global response are then compared to the analysis of local plastic rearrangements and strain heterogeneities. It is shown that the plastic response results from the succession of well-identified localized irreversible deformations occurring in a nanometer-size area. The size and the number of these local rearrangements, as well as the amount of internal deviatoric and volumetric plastic deformation, are sensitive to the composition and to the pressure. In the early stages of the deformation, plastic rearrangements are driven by sodium mobility. Consequently, the elastic yield strength decreases when the sodium content increases, and the same when pressure increases. Finally, good correlation was found between global and local stress-strain relationships, reinforcing again the role of sodium ions as local initiators of the plastic behavior observed at larger scales.

Microfluidic flow-focusing devices offer excellent control over fluid flow, enabling formation of drops with a narrow size distribution. However, the throughput of microfluidic flow-focusing devices is limited and scale-up through operation of multiple drop makers in parallel often compromises the robustness of their operation. We demonstrate that parallelization is facilitated if the outer phase is injected from the direction opposite to that of the inner phase, because the fluid injection flow rate, where the drop formation transitions from the squeezing into the dripping regime, is shifted towards higher values.

We perform first-principles path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) calculations to explore warm dense matter states of LiF. Our simulations cover a wide density-temperature range of $2.08–15.70\mathrm{g}{\mathrm{cm}}^{-3}$ and ${10}^4–{10}^9$ K. Since PIMC and DFT-MD accurately treat effects of atomic shell structure, we find a pronounced compression maximum and a shoulder on the principal Hugoniot curve attributed to K-shell and L-shell ionization. The results provide a benchmark for widely used EOS tables, such as SESAME, LEOS, and models. In addition, we compute pair-correlation functions that reveal an evolving plasma structure and ionization process that is driven by thermal and pressure ionization. Finally, we compute electronic density of states of liquid LiF from DFT-MD simulations and find that the electronic gap can remain open with increasing density and temperature to at least 15.7 $\mathrm{g}{\mathrm{cm}}^{-3}$.

Noise in a low-current Townsend discharge in argon and nitrogen is studied. Experiments were performed in a “semiconductor–gas-discharge” structure constituted by a short plane-parallel discharge gap and a high-resistivity photosensitive semiconductor electrode. The noise of optical emission from gas excited by the discharge was investigated with a photomultiplier. Experiments were done at the room temperature of the discharge structure. According to the obtained data, the noise of discharge glow in Ar is astonishingly low—similar to that earlier found for the cryogenic discharge in this gas. At equal time-average photon fluxes emitted by discharges in both gases, the noise of gas glow in nitrogen substantially exceeds the corresponding value for argon. A statistical processing of signals of glow brightness for the two gases revealed the origin of this difference: While the power spectral density of noise in nitrogen is specified by a band of frequencies with the increased density, in argon it is nearly constant in an extended frequency range. At the same time, intensity of noise is practically the same for both gases in the ranges of low and high frequencies. The relationship of the increased noise in nitrogen with a low critical current density for oscillatory instability of the Townsend discharge in this gas is considered.

We present a general framework for classifying partially observed dynamical systems based on the idea of learning in the model space. In contrast to the existing approaches using point estimates of model parameters to represent individual data items, we employ posterior distributions over model parameters, thus taking into account in a principled manner the uncertainty due to both the generative (observational and/or dynamic noise) and observation (sampling in time) processes. We evaluate the framework on two test beds: a biological pathway model and a stochastic double-well system. Crucially, we show that the classification performance is not impaired when the model structure used for inferring posterior distributions is much more simple than the observation-generating model structure, provided the reduced-complexity inferential model structure captures the essential characteristics needed for the given classification task.

In this work, we propose a general approach for modeling mass transfer and reaction of dilute solute(s) in incompressible three-phase flows by introducing a collision operator in lattice Boltzmann (LB) method. An LB equation was used to simulate the solute dynamics among three different fluids, in which the newly expanded collision operator was used to depict the interface behavior of dilute solute(s). The multiscale analysis showed that the presented model can recover the macroscopic transport equations derived from the Maxwell-Stefan equation for dilute solutes in three-phase systems. Compared with the analytical equation of state of solute and dynamic behavior, these results are proven to constitute a generalized framework to simulate solute distributions in three-phase flows, including compound soluble in one phase, compound adsorbed on single-interface, compound in two phases, and solute soluble in three phases. Moreover, numerical simulations of benchmark cases, such as phase decomposition, multilayered planar interfaces, and liquid lens, were performed to test the stability and efficiency of the model. Finally, the multiphase mass transfer and reaction in Janus droplet transport in a straight microchannel were well reproduced.

The slow-bond problem is a long-standing question about the minimal strength ${\epsilon }_{\mathrm{c}}$ of a local defect with global effects on the Kardar-Parisi-Zhang (KPZ) universality class. A consensus on the issue has been delayed due to the discrepancy between various analytical predictions claiming ${\epsilon }_{\mathrm{c}}=0$ and numerical observations claiming ${\epsilon }_{\mathrm{c}}>0$. We revisit the problem via finite-size scaling analyses of the slow-bond effects, which are tested for different boundary conditions through extensive Monte Carlo simulations. Our results provide evidence that the previously reported nonzero ${\epsilon }_{\mathrm{c}}$ is an artifact of a crossover phenomenon which logarithmically converges to zero as the system size goes to infinity.

We investigate the scaling properties of the order parameter and the largest nonvanishing Lyapunov exponent for the fully locked state in the Kuramoto model with a finite number $N$ of oscillators. We show that, for any finite value of $N$, both quantities scale as ${(K-K_L)}^{1/2}$ with the coupling strength $K$ sufficiently close to the locking threshold $K_L$. We confirm numerically these predictions for oscillator frequencies evenly spaced in the interval $[-1,1]$ and additionally find that the coupling range $\delta K$ over which this scaling is valid shrinks like $\delta K\sim N^{-\alpha }$ with $\alpha \approx 1.5$ as $N\to \infty$. Away from this interval, the order parameter exhibits the infinite-$N$ behavior $r-r_L\sim {(K-K_L)}^{2/3}$ proposed by Pazó [Phys. Rev. E 72, 046211 (2005)]. We argue that the crossover between the two behaviors occurs because at the locking threshold, the upper bound of the continuous part of the spectrum of the fully locked state approaches zero as $N$ increases. Our results clarify the convergence to the $N\to \infty$ limit in the Kuramoto model.

In the statistical learning language, samples are snapshots of random vectors drawn from some unknown distribution. Such vectors usually reside in a high-dimensional Euclidean space, and thus the “curse of dimensionality” often undermines the power of learning methods, including community detection and clustering algorithms, that rely on Euclidean geometry. This paper presents the idea of effective dissimilarity transformation (EDT) on empirical dissimilarity hyperspheres and studies its effects using synthetic and gene expression data sets. Iterating the EDT turns a static data distribution into a dynamical process purely driven by the empirical data set geometry and adaptively ameliorates the curse of dimensionality, partly through changing the topology of a Euclidean feature space ${\mathbb{R}}^n$ into a compact hypersphere $S^n$. The EDT often improves the performance of hierarchical clustering via the automatic grouping information emerging from global interactions of data points. The EDT is not restricted to hierarchical clustering, and other learning methods based on pairwise dissimilarity should also benefit from the many desirable properties of EDT.

We study the critical behavior of a continuous opinion model, driven by kinetic exchanges in a fully connected population. Opinions range in the real interval $[-1,1]$, representing the different shades of opinions against and for an issue under debate. Individuals' opinions evolve through pairwise interactions, with couplings that are typically positive, but a fraction $p$ of negative ones is allowed. Moreover, a social temperature parameter $T$ controls the tendency of the individual responses toward neutrality. Depending on $p$ and $T$, different collective states emerge: symmetry broken (one side wins), symmetric (tie of opposite sides), and absorbing neutral (indecision wins). We find the critical points and exponents that characterize the phase transitions between them. The symmetry breaking transition belongs to the usual Ising mean-field universality class, but the absorbing-phase transitions, with $\beta =0.5$, are out of the paradigmatic directed percolation class. Moreover, ordered phases can emerge by increasing social temperature.

We present an analysis of citations accrued over time by patents granted by the United States Patent and Trademark Office in 1998. In contrast to previous studies, a disaggregation by technology category is performed, and exogenously caused citation-number growth is controlled for. Our approach reveals an intrinsic citation rate that clearly separates into an—in the long run, exponentially time-dependent—aging function and a completely time-independent preferential-attachment-type growth kernel. For the general case of such a separable citation rate, we obtain the time-dependent citation distribution analytically in a form that is valid for any functional form of its aging and growth parts. Good agreement between theory and long-time characteristics of patent-citation data establishes our work as a useful framework for addressing still open questions about knowledge-propagation dynamics, such as the observed excess of citations at short times.

A modified test-chain self-consistent field theory (SCFT) is presented to study the intra- and intermolecular correlations of linear and branched polymers in various solutions and melts. The key to the test-chain SCFT is to break the the translational symmetry by fixing a monomer at the origin of a coordinate. This theory successfully describes the crossover from self-avoiding walk at short distances to screened random walk at long distances in a semidilute solution or melt. The calculations indicated that branching enhances the swelling of polymers in melts and influences stretching at short distances. The test-chain SCFT calculations show good agreement with experiments and classic polymer theories. We highlight that the theory presented here provides a solution to interpret the polymer conformation and behavior under various conditions within the framework of one theory.

The pressure and internal energy of an ultracold plasma in a state of quasiequilibrium are evaluated using classical molecular dynamics simulations. Coulomb collapse is avoided by modeling electron-ion interactions using an attractive Coulomb potential with a repulsive core. We present a method to separate the contribution of classical bound states, which form due to recombination, from the contribution of free charges when evaluating these thermodynamic state variables. It is found that the contribution from free charges is independent of the choice of repulsive core length scale when it is sufficiently short-ranged. The partial pressure associated with the free charges is found to closely follow that of the one-component plasma model, reaching negative values at strong coupling, while the total system pressure remains positive. This pseudopotential model is also applied to Debye-Hückel theory to describe the weakly coupled regime.

We present both exact and numerical results for the behavior of the Casimir force in $O\left(n\right)$ systems with a finite extension $L$ in one direction when the system is subjected to surface fields that induce helicity in the order parameter. We show that for such systems, the Casimir force in certain temperature ranges is of the order of $L^{-2}$, both above and below the critical temperature, $T_c$, of the bulk system. An example of such a system would be one with chemically modulated bounding surfaces, in which the modulation couples directly to the system's order parameter. We demonstrate that, depending on the parameters of the system, the Casimir force can be either attractive or repulsive. The exact calculations presented are for the one-dimensional $XY$ and Heisenberg models under twisted boundary conditions resulting from finite surface fields that differ in direction by a specified angle, and the three-dimensional Gaussian model with surface fields in the form of plane waves that are shifted in phase with respect to each other. Additionally, we present exact and numerical results for the mean-field version of the three-dimensional $O\left(2\right)$ model with finite surface fields on the bounding surfaces. We find that all significant results are consistent with the expectations of finite-size scaling.