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Colloidoscope is a deep learning pipeline employing a 3D residual Unet architecture, designed to enhance the tracking of dense colloidal suspensions through confocal microscopy. This methodology uses a simulated training dataset that reflects a wide array of real-world imaging conditions, specifically targeting high colloid volume fraction and low-contrast scenarios where traditional detection methods struggle. Central to our approach is the use of experimental signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and point-spread-functions (PSFs) to accurately quantify and simulate the experimental data. Our findings reveal that Colloidoscope achieves superior recall in particle detection (finds more particles) compared to conventional heuristic methods. Simultaneously, high precision is maintained (high fraction of true positives.) The model demonstrates a notable robustness to photobleached samples, thereby prolonging the imaging time and number of frames than may be acquired. Furthermore, Colloidoscope maintains small scale resolution sufficient to classify local structural motifs. Evaluated across both simulated and experimental datasets, Colloidoscope brings the advancements in computer vision offered by deep learning to particle tracking at high volume fractions. We offer a promising tool for researchers in the soft matter community, this model is deployed and available to use pretrained: https://github.com/wahabk/colloidoscope., Comment: 20 pages

We study the inclusive and exclusive cross sections of $e^+e^-\to \text{hadrons}$ for center-of-mass energies between 3.70 GeV and 3.83 GeV to infer the mass, width, and couplings of the $\psi(3770)$ resonance. By using a coupled-channel K-matrix approach, we setup our analysis to respect unitarity and the analyticity properties of the underlying scattering amplitudes. We fit several models to the full dataset and identify our nominal results through a statistical model comparison. We find that, accounting for the interplay between the $\psi(2S)$ and the $\psi(3770)$, no further pole is required to describe the $\psi(3770)$ line shape. In particular we derive from the pole location $M_{\psi(3770)} = 3778.8 \pm 0.3$ MeV and $\Gamma_{\psi(3770)} = 25.0 \pm 0.5$ MeV. Moreover, we find the decay to $D^+D^-$ and $D^0\bar{D}^0$ to be consistent with isospin symmetry and derive an upper bound on the branching ratio $\mathcal{B}(\psi(3770) \to \textrm{non-}D\bar{D}) < 6\%$ at $90\%$ probability., Comment: v2: Corrected measured ee->DD cross section, leading to better fit quality and smaller branching ratio for psi(3770)->non-DDbar

Mixing of inertial point particles in a turbulent channel flow at Re{\tau} = 950 is investigated by means of direct numerical simulations. We consider inertial particles, at varying Stokes number, released from pairs of sources located at different positions inside the channel and analyze the rate at which particles come into close proximity to each other. To do so, we employ a Lagrangian framework, which is suitable for the analysis of trajectories and in general for the study of mixing and dispersion problems. By varying the release position of particles along the wall-normal direction we obtain a thorough description of mixing in an anisotropic turbulent flow. Moreover, we analyze the effects of particle inertia and show that these are not univocal but also depend on the position and alignment of the sources, owing in particular to the dependence of the flow timescales on the distance from the wall., Comment: 19 pages, 7 figures

Using dispersive methods, we study the $B \to \gamma^*$ form factors underlying the decay $B^- \to \ell^- \bar{\nu}_\ell \ell^{\prime-} \ell^{\prime+}$. We discuss the ambiguity that arises from a separation of the full $B^- \to \ell^- \bar{\nu}_\ell \ell^{\prime-} \ell^{\prime+}$ amplitude into a hadronic tensor and a final-state-radiation piece, including effects from nonvanishing lepton masses. For the eligibility of a dispersive treatment, we propose a decomposition of the hadronic part that leads to four form factors that are free of kinematic singularities. By establishing a set of dispersion relations, we then relate the $B \to \gamma^*$ form factors to the well-known $B \to V$, $V=\omega(782),\rho(770)$, analogues. Using the combination of a series expansion in a conformal variable and a vector-meson-dominance ansatz to parameterize the $B \to \gamma^*$ form factors, we infer the values of the associated unknown parameters from the available input on $B \to V$. The phenomenological application of our formalism includes the determination of the branching ratios and forward-backward asymmetries of the process $B^- \to \ell^- \bar{\nu}_\ell \ell^{\prime-} \ell^{\prime+}$ for the two lightest leptons., Comment: 23 pages, 6 figures, 5 tables; v2: branching ratios with tau leptons included, version published in Phys. Rev. D

In this work, the notion of a quantum inverse semigroup is introduced as a linearized generalization of inverse semigroups. Beyond the algebra of an inverse semigroup, which is the natural example of a quantum inverse semigroup, several other examples of this new structure are presented in different contexts, those are related to Hopf algebras, weak Hopf algebras, partial actions and Hopf categories. Finally, a generalized notion of local bisections are defined for commutative Hopf algebroids over a commutative base algebra giving rise to new examples of quantum inverse semigroups associated to Hopf algebroids in the same sense that inverse semigroups are related to groupoids., Comment: 1 figure. Revised version

The dispersion and mixing of passive particles in a turbulent channel flow is investigated by means of a network-based representation of their motion. We employ direct numerical simulations at five different Reynolds numbers, from $\mathrm{Re}_{\tau}$ = 180 up to $\mathrm{Re}_{\tau}$ = 950, and obtain sets of particle trajectories via numerical integration. By dividing the channel domain into wall-normal levels, the motion of particles across these levels is used to build a time-varying complex network, which is able to capture the transient phase of the wall-normal mixing process and its dependence on the Reynolds number, $\mathrm{Re}_{\tau}$. Using network metrics, we observe that the dispersion of clouds of tracers depends highly on both their wall-normal starting position and the time elapsed from their release. We identify two main mechanisms that contribute to the long lasting inhibition of the dispersion of particles released near the walls. We also show how the relative importance of these mechanisms varies with the Reynolds number. In particular, at low $\mathrm{Re}_{\tau}$ the weaker velocity fluctuations appear dominant in inhibiting dispersion, while at higher Reynolds numbers a larger role is played by cyclic patterns of motion. At the higher Reynolds numbers employed in this work, we find that most network properties are Reynolds-independent when scaled with outer flow variables. Instead, at lower $\mathrm{Re}_{\tau}$, the aforementioned scaling is not observed. We explore the meaning of the emergence of this scaling in relation to the features of dispersion and to the network definition., Comment: 18 pages, 7 figures

EOS is an open-source software for a variety of computational tasks in flavor physics. Its use cases include theory predictions within and beyond the Standard Model of particle physics, Bayesian inference of theory parameters from experimental and theoretical likelihoods, and simulation of pseudo events for a number of signal processes. EOS ensures high-performance computations through a C++ back-end and ease of usability through a Python front-end. To achieve this flexibility, EOS enables the user to select from a variety of implementations of the relevant decay processes and hadronic matrix elements at run time. In this article, we describe the general structure of the software framework and provide basic examples. Further details and in-depth interactive examples are provided as part of the EOS online documentation., Comment: 32 pages, 8 figures, 4 ancillary Jupyter example notebooks

In this paper we intend to complement the identification given in Kuerten & Fernandes-Silva [1] (Mod. Phys. Lett. A. Vol. 33, No. 16) which relates the axion to a metric spinor phase by means of Maxwell's theory in the Infeld-van der Waerden's {\gamma}-formalism. Thus, we obtain two alternative identifications: the first focuses on Dirac's theory so that when obtaining an axion-like phase-fermion coupling, we achieve the first identification, and the last one investigates the phase behavior under Peccei-Quinn rotations in order to show that the phase changes as an axion pseudoparticle. With the formal aspects established, we also study the semiclassical fermion-photon system to demonstrate that the magnetic monopole current defined in [1] has dyon charge in flat universe and acquires a Witten effect form when there is a demand for chiral symmetry., Comment: 13 pages

Atomically sharp domain walls (DWs) in ferroelectrics are considered as an ideal platform to realize easy-to-reconfigure nanoelectronic building blocks, created, manipulated and erased by external fields. However, conductive DWs have been exclusively observed in oxides, where DW mobility and conductivity is largely influenced by stoichiometry and defects. In contrast, we here report on conductive DWs in the non-oxide ferroelectric GaV$_4$S$_8$, where charge carriers are provided intrinsically by multivalent V$_4$ molecular clusters. We show that this new mechanism gives rise to DWs composed of nanoscale stripes with alternating electron and hole conduction, unimaginable in oxides. By exerting magnetic control on these segments we promote the mobile and effectively 2D DWs into dominating the 3D conductance, triggering abrupt conductance changes as large as eight orders of magnitude. The flexible valency, as origin of these novel hybrid DWs with giant conductivity, demonstrates that non-oxide ferroelectrics can be the source of novel phenomena beyond the realm of oxide electronics., Comment: 8 pages, 4 figures

Turbulent mixing is studied in the Lagrangian framework with an approach based on the complex network formalism. We consider the motion of passive, noninertial particles inside a turbulent channel simulated at $Re_{\tau} = 950$. The time-dependent network is built to evaluate the transfer of tracers between thin wall-parallel layers which partition the channel in the wall-normal direction. By doing so, we are able to assess the spatial and temporal complexities arising from turbulence dynamics and their influence on the mixing process. This approach highlights the effects of small-scale features of turbulent flow structures and also the larger scale effects determined by wall-induced anisotropy. Complex networks, coupled to the Lagrangian description of turbulence, are effective in providing novel insights into inhomogeneous turbulence and mixing., Comment: 13 pages, 6 figures

Improper ferroelectrics are described by two order parameters: a primary one, driving a transition to long-range distortive, magnetic or otherwise non-electric order, and the electric polarization, which is induced by the primary order parameter as a secondary, complementary effect. Using low-temperature scanning probe microscopy, we show that improper ferroelectric domains in YMnO$_3$ can be locally switched by electric field poling. However, subsequent temperature changes restore the as-grown domain structure as determined by the primary lattice distortion. The backswitching is explained by uncompensated bound charges occuring at the newly written domain walls due to the lack of mobile screening charges at low temperature. Thus, the polarization of improper ferroelectrics is in many ways subject to the same electrostatics as in their proper counterparts, yet complemented by additional functionalities arising from the primary order parameter. Tailoring the complex interplay between primary order parameter, polarization, and electrostatics is therefore likely to result in novel functionalities specific to improper ferroelectrics.

A temporal complex network-based approach is proposed as a novel formulation to investigate turbulent mixing from a Lagrangian viewpoint. By exploiting a spatial proximity criterion, the dynamics of a set of fluid particles is geometrized into a time-varying weighted network. Specifically, a numerically solved turbulent channel flow is employed as an exemplifying case. We show that the time-varying network is able to clearly describe the particle swarm dynamics, in a parametrically robust and computationally inexpensive way. The network formalism enables to straightforwardly identify transient and long-term flow regimes, the interplay between turbulent mixing and mean flow advection, and the occurrence of proximity events among particles. Thanks to their versatility and ability to highlight significant flow features, complex networks represent a suitable tool for Lagrangian investigations of turbulence mixing. The present application of complex networks offers a powerful resource for Lagrangian analysis of turbulent flows, thus providing a further step in building bridges between turbulence research and network science.

A network-based analysis of a turbulent channel flow numerically solved at $Re_\tau=180$ is proposed as an innovative perspective for the spatial characterization of the flow field. Two spatial networks corresponding to the streamwise and wall-normal velocity components are built, where nodes represent portions of volume of the physical domain. For each network, links are active if the correlation coefficient of the corresponding velocity component between pairs of nodes is sufficiently high, thus unveiling the strongest kinematic relations. Several network measures are studied in order to explore the interrelations between nodes and their neighbors. Specifically, long-range links are localized between near-wall regions and associated with the temporal persistence of coherent patterns, namely high and low speed streaks. Furthermore, long-range links play a crucial role as intermediary for the kinematic information flow, as emerges from the analysis of indirect connections between nodes. The proposed approach provides a framework to investigate spatial structures of the turbulent dynamics, showing the full potential of complex networks. Although the network analysis is based on the two-point correlation, it is able to advance the level of information, by exploiting the texture created by active links in all directions. Based on the observed findings, the current approach can pave the way for an enhanced spatial interpretation of the turbulence dynamics.

Metric spinor phase of the Infeld-van der Waerden {\gamma}-formalism and axion field were identified in Ref. [1], by using Maxwell theory. Since axion couples with fermions, we will investigate Dirac theory to extend the work given in [1], showing that is possible again to identify this phase with the axion. By searching an exact identification, we will work yet the spin transformations to adapt chiral rotations and so to show that the metric spinor phase behaves exactly as required for the axion field. We will study also the Dirac-Maxwell system which provides a minimal Lagrangian approach for the axion classical sector and, finally, we will obtain an explicit 2-spinor description of magnetic monopole and its charge., Comment: 13 pages

In this article we will explore the spacetime phase structure inside the Infeld-van der Waerden formalisms, showing that Maxwell's theory in this scenario leads to an axion-like phase electrodynamics, which suggests that this phase may originate the axion field., Comment: 15 pages

The magnetic properties of a system of three interacting magnetic elliptical discs are examined. For the various levels of uniaxial anisotropy investigated a complicated series of phase transitions exist. These are marked by the critical lines of stability (CLS) that are demonstrated in an applied magnetic field plane diagram., Comment: 12 pages, 4 figures

The Greek aperitif Ouzo is not only famous for its specific anise-flavored taste, but also for its ability to turn from a transparent miscible liquid to a milky-white colored emulsion when water is added. Recently, it has been shown that this so-called Ouzo effect, i.e. the spontaneous emulsification of oil microdroplets, can also be triggered by the preferential evaporation of ethanol in an evaporating sessile Ouzo drop, leading to an amazingly rich drying process with multiple phase transitions [H. Tan et al., Proc. Natl. Acad. Sci. USA 113(31) (2016) 8642]. Due to the enhanced evaporation near the contact line, the nucleation of oil droplets starts at the rim which results in an oil ring encircling the drop. Furthermore, the oil droplets are advected through the Ouzo drop by a fast solutal Marangoni flow. In this article, we investigate the evaporation of mixture droplets in more detail, by successively increasing the mixture complexity from pure water over a binary water-ethanol mixture to the ternary Ouzo mixture (water, ethanol and anise oil). In particular, axisymmetric and full three-dimensional finite element method simulations have been performed on these droplets to discuss thermal effects and the complicated flow in the droplet driven by an interplay of preferential evaporation, evaporative cooling and solutal and thermal Marangoni flow. By using image analysis techniques and micro-PIV measurements, we are able to compare the numerically predicted volume evolutions and velocity fields with experimental data. The Ouzo droplet is furthermore investigated by confocal microscopy. It is shown that the oil ring predominantly emerges due to coalescence.

Evaporating liquid droplets are omnipresent in nature and technology, such as in inkjet printing, coating, deposition of materials, medical diagnostics, agriculture, food industry, cosmetics, or spills of liquids. While the evaporation of pure liquids, liquids with dispersed particles, or even liquid mixtures has intensively been studied over the last two decades, the evaporation of ternary mixtures of liquids with different volatilities and mutual solubilities has not yet been explored. Here we show that the evaporation of such ternary mixtures can trigger a phase transition and the nucleation of microdroplets of one of the components of the mixture. As model system we pick a sessile Ouzo droplet (as known from daily life - a transparent mixture of water, ethanol, and anise oil) and reveal and theoretically explain its four life phases: In phase I, the spherical cap-shaped droplet remains transparent, while the more volatile ethanol is evaporating, preferentially at the rim of the drop due to the singularity there. This leads to a local ethanol concentration reduction and correspondingly to oil droplet nucleation there. This is the beginning of phase II, in which oil microdroplets quickly nucleate in the whole drop, leading to its milky color which typifies the so-called 'Ouzo-effect'. Once all ethanol has evaporated, the drop, which now has a characteristic non-spherical-cap shape, has become clear again, with a water drop sitting on an oil-ring (phase III), finalizing the phase inversion. Finally, in phase IV, also all water has evaporated, leaving behind a tiny spherical cap-shaped oil drop., Comment: 40 pages, 12 figures

Recently, Borzou et al. (BSSY) generalized the Shiromizu-Maeda-Sasaki (SMS) formulation to f(R)-bulks. BSSY brane projected equation carries an additional stress tensor, besides SMS correction for Einstein's theory on the brane. If we change this perspective, by requiring BSSY tensor in the geometrical side acting as f(R)-brane generator, it is possible to relate f(R)-brane/bulk theories, by using curvature dynamical constraint (CDC), a concept that we developed. Since that brane and bulk are f(R), 5/4D scalar curvatures also play a dynamical role and, thus, a dynamical version to Gauss equation trace, or CDC, must be offered. We will work yet in a specific case to obtain f(R)-unimodular gravity, formally identical with obtained by Nojiri et al. (NOO). Therefore, two applications which consider cosmological scenarios in f(R)-unimodular gravity with dark radiation correction will be offered., Comment: 20 pages

We identified quasiparticle states at well-defined energies inside the superconducting gap of the electron system at the LaAlO$_3$-SrTiO$_3$ interface using tunneling spectroscopy. The states are found only in a number of samples and depend upon the thermal-cycling history of the samples. The states consist of a peak at zero energy and other peaks at finite energies, symmetrically placed around zero energy. These peaks disappear, together with the superconducting gap, with increasing temperature and magnetic field. We discuss the likelihood of various physical mechanisms that are known to cause in-gap states in superconductors and conclude that none of these mechanisms can easily explain the results. The conceivable scenarios are the formation of Majorana bound states, Andreev bound states, or the presence of an odd-frequency spin triplet component in the superconducting order parameter., Comment: 11 pages, 5 figures

We develop the concept of fractal metamaterials which consist of arrays of nano and micron sized rings containing Josephson junctions which play the role of "atoms" in such artificial materials. We show that if some of the junctions have $\pi$-shifts in the Josephson phases that the "atoms" become magnetic and their arrays can have tuned positive or negative permeabilty. Each individual "$\pi$-ring" - the Josephson ring with one $\pi$-junction - can be in one of two energetically degenerate magnetic states in which the supercurrent flows in the clockwise or counter-clockwise direction. This results in magnetic moments that point downwards or upwards, respectively. The value of the total magnetization of such a metamaterial may display fractal features. We describe the magnetic properties of such superconducting metamaterials, including the magnetic field distribution in them (i.e. in the network that is made up of these rings). We also describe the way that the magnetic flux penetrates into the Josephson network and how it is strongly dependent on the geometry of the system., Comment: 10 pages, 9 figures

For atmospheric sulfuric acid (SA) concentrations the presence of dimethylamine (DMA) at mixing ratios of several parts per trillion by volume can explain observed boundary layer new particle formation rates. However, the concentration and molecular composition of the neutral (uncharged) clusters have not been reported so far due to the lack of suitable instrumentation. Here we report on experiments from the Cosmics Leaving Outdoor Droplets chamber at the European Organization for Nuclear Research revealing the formation of neutral particles containing up to 14 SA and 16 DMA molecules, corresponding to a mobility diameter of about 2 nm, under atmospherically relevant conditions. These measurements bridge the gap between the molecular and particle perspectives of nucleation, revealing the fundamental processes involved in particle formation and growth. The neutral clusters are found to form at or close to the kinetic limit where particle formation is limited only by the collision rate of SA molecules. Even though the neutral particles are stable against evaporation from the SA dimer onward, the formation rates of particles at 1.7-nm size, which contain about 10 SA molecules, are up to 4 orders of magnitude smaller comparedwith those of the dimer due to coagulation and wall loss of particles before they reach 1.7 nm in diameter. This demonstrates that neither the atmospheric particle formation rate nor its dependence on SA can simply be interpreted in terms of cluster evaporation or the molecular composition of a critical nucleus., Comment: Main text plus SI

Within the braneworld paradigm the tachyonic scalar field has been used to generate models that attempt to solve some of the open problems that physics faces nowadays, both in cosmology and high energy physics as well. When these field configurations are produced by the interplay of higher dimensional warped gravity with some matter content, braneworld models must prove to be {\it stable} under the whole set of small fluctuations of the gravitational and matter fields background, among other consistency tests. Here we present a complete proof of the stability under scalar perturbations of tachyonic thick braneworlds with an embedded maximally symmetric 4D space-time, revealing its physical consistency. This family of models contains a recently reported tachyonic de Sitter thick braneworld which possesses a series of appealing properties. These features encompass complete regularity, asymptotic flatness (instead of being asymptotically dS or AdS) even when it contains a negative bulk cosmological constant, a relevant 3-brane with dS metric which naturally arises from the full set of field equations of the 5D background (it is not imposed), qualitatively describing the inflationary epochs of our Universe, and a graviton spectrum with a single zero mode bound state that accounts for the 4D graviton localised on the brane and is separated from the continuum of Kaluza-Klein massive graviton excitations by a mass gap. Gauge vector fields with a single massless bound state in its mass spectrum are also localised on this braneworld model a fact that allows us to recover the Coulomb's law of our 4D world. All these properties of the above referred tachyonic braneworld together with the positive stability analysis provided in this work, constitute a firm step towards the construction of realistic cosmological models within the braneworld paradigm., Comment: 25 pages

From the dynamics of a brane-world with matter fields present in the bulk, the bulk metric and the black string solution near the brane are generalized, when both the dynamics of inhomogeneous dust/generalized dark radiation on the brane-world and inhomogeneous dark radiation in the bulk as well are considered -- as exact dynamical collapse solutions. Based on the analysis on the inhomogeneous static exterior of a collapsing sphere of homogeneous dark radiation on the brane, the associated black string warped horizon is studied, as well as the 5D bulk metric near the brane. Moreover, the black string and the bulk are shown to be more regular upon time evolution, for suitable values for the dark radiation parameter in the model, by analyzing the physical soft singularities., Comment: 26 pages, 14 figures

The perihelion precession, the deflection of light, and the radar echo delay are classical tests of General Relativity here used to probe brane world topologically charged black holes in a f(R) bulk and to constrain the parameter that arises from the Shiromizu-Maeda-Sasaki procedure applied to a f(R) bulk as well. The existing Solar system observational data constrain the possible values of the tidal charge parameter and the effective cosmological constant including f(R) brane world effects. We show that the observational/experimental data for both perihelion precession and radar echo delay make the black hole space of parameters to be more strict than the ones for the Dadhich, Maartens, Papadopoulos and Rezania (DMPR) black hole geometry. Furthermore, the deflection of light constrains the tidal charge parameter similarly as the DMPR black holes due to a peculiarity in the equation of motion., Comment: 14 pages, 2 figures, to appear in Gen. Relat. Grav

This paper aims to evince the corrections on the black string warped horizon in the braneworld paradigm, and their drastic physical consequences, as well as to provide subsequent applications in astrophysics. Our analysis concerning black holes on the brane departs from the Schwarzschild case, where the black string is unstable to large-scale perturbation. The cognizable measurability of the black string horizon corrections due to braneworld effects is investigated, as well as their applications in the variation of quasars luminosity. We delve into the case wherein two solutions of Einstein's equations proposed by Casadio, Fabbri, and Mazzacurati, regarding black hole metrics presenting a post-Newtonian parameter measured on the brane. In this scenario, it is possible to analyze purely the braneworld corrected variation in quasars luminosity, by an appropriate choice of the post-Newtonian parameter that precludes Hawking radiation on the brane: the variation in quasars luminosity is uniquely provided by pure braneworld effects, as the Hawking radiation on the brane is suppressed., Comment: 13 pages, 4 figures (to appear in Class. Quant. Grav.)

In this paper, the results of an international collaborative test case relative to the production of a Direct Numerical Simulation and Lagrangian Particle Tracking database for turbulent particle dispersion in channel flow at low Reynolds number are presented. The objective of this test case is to establish a homogeneous source of data relevant to the general problem of particle dispersion in wall-bounded turbulence. Different numerical approaches and computational codes have been used to simulate the particle-laden flow and calculations have been carried on long enough to achieve a statistically-steady condition for particle distribution. In such stationary regime, a comprehensive database including both post-processed statistics and raw data for the fluid and for the particles has been obtained. The complete datasets can be downloaded from the web at http://cfd.cineca.it/cfd/repository/. In this paper, the most relevant velocity statistics (for both phases) and particle distribution statistics are discussed and benchmarked by direct comparison between the different numerical predictions.

We discuss stationary solutions of the discrete nonlinear Schr\"odinger equation (DNSE) with a potential of the $\phi^{4}$ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a $1D$ chain of atoms --- the (Rashba)--Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.