arXiv subjects
Publications and source records attributed to Yingzi Yang.
We will construct solvable lattice models whose partition functions are Demazure characters. We will construct a crystal structure on the states of the model and prove that the states of the closed model form a Demazure crystal.
Every positive integer may be written uniquely as a base-$β$ decomposition--that is a legal sum of powers of $β$--where $β$ is the dominating root of a non-increasing positive linear recurrence sequence. Guided by earlier work on a two-player game which produces the Zeckendorf Decomposition of an integer (see [Bai+19]), we define a broad class of two-player games played on an infinite tuple of non-negative integers which decompose a positive integer into its base-$β$ expansion. We call this game the Generalized Bergman Game. We prove that the longest possible Generalized Bergman game on an initial state $S$ with $n$ summands terminates in $Θ(n^2)$ time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between $Ω(n)$ and $O(n^2)$ time. We also show a linear bound on the maximum length of the tuple used throughout the game.
We have studied the conformational and scaling behaviors of a flexible dendrimer immersed in athermal or good solvents. A self-consistent field theory combined with a pre-averaged excluded volume potential representing the two-body short-ranged interaction between the segments, was adopted to calculate the density profile of various generations and branch points thoroughly. Our calculation results support the "dense-core" model. We find the conformation of the dendrimer is strongly stretched in the dense central region, but much weakly stretched in the outer region where the segment density profile is shoulder shaped. Both our self-consistent field theory calculation and the Flory mean-field theory calculation give the same scaling law R proportional to (GP)^0.2N^0.4, where G is the generation number of the dendrimer, P is the spacer segment number, and N is the total segment number. If we fix G, the scaling law is simplified to R proportional to P^0.6 in good solvent.
Self-propelled point-like particles move along circular trajectories when their translocation velocity is constant and the angular velocity related to their orientation vector is also constant. We investigate the collective behavior of ensembles of such circle swimmers by Brownian dynamics simulations. If the particles interact via a "velocity-trajectory coordination" rule within neighboring particles, a self-organized vortex pattern emerges. This vortex pattern is characterized by its particle-density correlation function $G_ρ$, the density correlation function $G_c$ of trajectory centers, and an order parameter $S$ representing the degree of the aggregation of the particles. Here, we systematically vary the system parameters, such as the particle density and the interaction range, in order to reveal the transition of the system from a light-vortex-dominated to heavy-vortex-dominated state, where vortices contain mainly a single and many self-propelled particles, respectively. We also study a semi-dilute solution of curved, sinusoidal-beating flagella, as an example of circling self-propelled particles with explicit propulsion mechanism and excluded-volume interactions. Our simulation results are compared with previous experimental results for the vortices in sea-urchin sperm solutions near a wall. The properties of the vortices in simulations and experiments are found to agree quantitatively.
Systems of self-propelled particles are known for their tendency to aggregate and to display swarm behavior. We investigate two model systems, self-propelled rods interacting via volume exclusion, and sinusoidally-beating flagella embedded in a fluid with hydrodynamic interactions. In the flagella system, beating frequencies are Gaussian distributed with a non-zero average. These systems are studied by Brownian-dynamics simulations and by mesoscale hydrodynamics simulations, respectively. The clustering behavior is analyzed as the particle density and the environmental or internal noise are varied. By distinguishing three types of cluster-size probability density functions, we obtain a phase diagram of different swarm behaviors. The properties of clusters, such as their configuration, lifetime and average size are analyzed. We find that the swarm behavior of the two systems, characterized by several effective power laws, is very similar. However, a more careful analysis reveals several differences. Clusters of self-propelled rods form due to partially blocked forward motion, and are therefore typically wedge-shaped. At higher rod density and low noise, a giant mobile cluster appears, in which most rods are mostly oriented towards the center. In contrast, flagella become hydrodynamically synchronized and attract each other; their clusters are therefore more elongated. Furthermore, the lifetime of flagella clusters decays more quickly with cluster size than of rod clusters.
We consider an inextensible, semiflexible polymer or worm-like chain, with persistence length $P$ and contour length $L$, fluctuating in a cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$, corresponding to a long, tightly confined polymer, the average length of the channel $ $ occupied by the polymer and the mean square deviation from the average vary as $ =[1-α_\circ(D/P)^{2/3}]L$ and $<ΔR_\parallel^{\thinspace 2}\thinspace>=β_\circ(D^2/P)L$, respectively, where $α_\circ$ and $β_\circ$ are dimensionless amplitudes. In earlier work we determined $α_\circ$ and the analogous amplitude $α_\Box$ for a channel with a rectangular cross section from simulations of very long chains. In this paper we estimate $β_\circ$ and $β_\Box$ from the simulations. The estimates are compared with exact analytical results for a semiflexible polymer confined in the transverse direction by a parabolic potential instead of a channel and with a recent experiment. For the parabolic confining potential we also obtain a simple analytic result for the distribution of $R_\parallel$ or radial distribution function, which is asymptotically exact for large $L$ and has the skewed shape seen experimentally.
Sperm swimming at low Reynolds number have strong hydrodynamic interactions when their concentration is high in vivo or near substrates in vitro. The beating tails not only propel the sperm through a fluid, but also create flow fields through which sperm interact with each other. We study the hydrodynamic interaction and cooperation of sperm embedded in a two-dimensional fluid by using a particle-based mesoscopic simulation method, multi-particle collision dynamics (MPC). We analyze the sperm behavior by investigating the relationship between the beating-phase difference and the relative sperm position, as well as the energy consumption. Two effects of hydrodynamic interaction are found, synchronization and attraction. With these hydrodynamic effects, a multi-sperm system shows swarm behavior with a power-law dependence of the average cluster size on the width of the distribution of beating frequencies.
We consider a long, semiflexible polymer, with persistence length $P$ and contour length $L$, fluctuating in a narrow cylindrical channel of diameter $D$. In the regime $D\ll P\ll L$ the free energy of confinement $ΔF$ and the length of the channel $R_\parallel$ occupied by the polymer are given by Odijk's relations $ΔF/R_\parallel=A_\circ k_BTP^{-1/3}D^{-2/3}$ and $R_\parallel=L[1-α_\circ(D/P)^{2/3}]$, where $A_\circ$ and $α_\circ$ are dimensionless amplitudes. Using a simulation algorithm inspired by PERM (Pruned Enriched Rosenbluth Method), which yields results for very long polymers, we determine $A_\circ$ and $α_\circ$ and the analogous amplitudes for a channel with a rectangular cross section. For a semiflexible polymer confined to the surface of a cylinder, the corresponding amplitudes are derived with an exact analytic approach. The results are relevant for interpreting experiments on biopolymers in microchannels or microfluidic devices.