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The dynamics of a two-dimensional Bose-Einstein condensate in a presence of quantum fluctuations is studied. The properties of localized density distributions, quantum droplets (QDs), are analyzed by means of the variational approach. [https://www.selleckchem.com/mTOR.html check details] It is demonstrated that the super-Gaussian function gives a good approximation for profiles of fundamental QDs and droplets with nonzero vorticity. The dynamical equations for parameters of QDs are obtained. Fixed points of these equations determine the parameters of stationary QDs. The period of small oscillations of QDs near the stationary state is estimated. It is obtained that periodic modulations of the strength of quantum fluctuations can actuate different processes, including resonance oscillations of the QD parameters, an emission of waves and a splitting of QDs into smaller droplets.In recent decades computer-aided technologies have become prevalent in medicine, however, cancer drugs are often only tested on in vitro cell lines from biopsies. We derive a full three-dimensional model of inhomogeneous -anisotropic diffusion in a tumor region coupled to a binary population model, which simulates in vivo scenarios faster than traditional cell-line tests. The diffusion tensors are acquired using diffusion tensor magnetic resonance imaging from a patient diagnosed with glioblastoma multiform. Then we numerically simulate the full model with finite element methods and produce drug concentration heat maps, apoptosis hotspots, and dose-response curves. Finally, predictions are made about optimal injection locations and volumes, which are presented in a form that can be employed by doctors and oncologists.The two-relaxation-times collision benefits the steady lattice Boltzmann method by yielding viscosity-independent numerical errors. We present in an intuitive way how to incorporate popular force methods into the two-relaxation-times collision. We subsequently rewrite force methods into a generic equation to reveal commonalities and differences. We prove that force methods with a second-order velocity moment of the force break the viscosity independence. A force method with only a first-order velocity moment of the force averts this breakage. We validate our proof numerically.We report the experimental observation of Faraday waves on soft gels. These were obtained using agarose in a mechanically vibrated cylindrical container. Low driving frequencies induce subharmonic standing waves with spatial structure that conforms to the geometry of the container. We report the experimental observation of the first 15 resonant Faraday wave modes that can be defined by the mode number (n,ℓ) pair. We also characterize the shape of the instability tongue and show the complex dependence upon material properties can be understood as an elastocapillary effect.We study an agent-based model of animals marking their territory and evading adversarial territory in one dimension with respect to the distribution of the size of the resulting territories. In particular, we use sophisticated sampling methods to determine it over a large part of territory sizes, including atypically small and large configurations, which occur with probability of less than 10^-30. We find hints for the validity of a large deviation principle, the shape of the rate function for the right tail of the distribution, and insight into the structure of atypical realizations.In this paper, we study the connectivity of a one-dimensional soft random geometric graph (RGG). The graph is generated by placing points at random on a bounded line segment and connecting pairs of points with a probability that depends on the distance between them. We derive bounds on the probability that the graph is fully connected by analyzing key modes of disconnection. In particular, analytic expressions are given for the mean and variance of the number of isolated nodes, and a sharp threshold established for their occurrence. Bounds are also derived for uncrossed gaps, and it is shown analytically that uncrossed gaps have negligible probability in the scaling at which isolated nodes appear. This is in stark contrast to the hard RGG in which uncrossed gaps are the most important factor when considering network connectivity.Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems, while a global statistical approach is overly cursory. Koopman operator technique seems to balance well the two approaches. In this paper we extend Koopman analysis to the study of synchronization of coupled oscillators by extracting important eigenvalues and eigenfunctions from the observed time series. A renormalization group analysis is designed to derive an analytic approximation of the eigenfunction in the case of weak coupling that dominates the oscillation. For moderate or strong couplings, numerical computation further confirms the importance of the average frequencies and the associated eigenfunctions. The synchronization transition points could be located with quite high accuracy by checking the correlation of neighboring eigenfunctions at different coupling strengths, which is readily applied to other nonlinear systems.Quantifying the influence of microscopic details on the dynamics of development of the overall structure of a filamentous network is important in a number of biologically relevant contexts, but it is not obvious what order parameters can be used to adequately describe this complex process. In this paper we investigated the role of multivalent actin-binding proteins (ABPs) in reorganizing actin filaments into higher-order complex networks via a computer model of semiflexible filaments. We characterize the importance of local connectivity among actin filaments, as well as the global features of actomyosin networks. We first map the networks into local graph representations and then, using principles from network-theory order parameters, combine properties from these representations to gain insight into the heterogeneous morphologies of actomyosin networks at a global level. We find that ABPs with a valency greater than 2 promote filament bundles and large filament clusters to a much greater extent than bivalent multilinkers.