Modeling cell differentiation from omics data is an essential problem in systems biology research. Although many algorithms have been established to analyze scRNA-seq data, approaches to infer the pseudo-time of cells or quantify their potency have not yet been satisfactorily solved. Here, we propose the Landscape of Differentiation Dynamics (LDD) method, which calculates cell potentials and constructs their differentiation landscape by a continuous birth-death process from scRNA-seq data. From the viewpoint of stochastic dynamics, we exploited the features of the differentiation process and quantified the differentiation landscape based on the source-sink diffusion process. In comparison with other scRNA-seq methods in seven benchmark datasets, we found that LDD could accurately and efficiently build the evolution tree of cells with pseudo-time, in particular quantifying their differentiation landscape in terms of potency. This study provides not only a computational tool to quantify cell potency or the Waddington potential landscape based on scRNA-seq data, but also novel insights to understand the cell differentiation process from a dynamic perspective.

Using the tri-hamiltonian splitting method, the authors of [Anco and Mobasheramini, Physica D, 355: 1--23, 2017] derived two U (1) -invariant nonlinear PDEs that arise from the hierarchy of the nonlinear Schrdinger equation and admit peakons ($ non-smooth\solitons $). In the present paper, these two peakon PDEs are generalized to a family of U (1) -invariant peakon PDEs parametrized by the real projective line U (1) . All equations in this family are shown to posses $ conservative\peakon\solutions $(whose Sobolev U (1) norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon ODEs are shown to be Hamiltonian with respect to several Poisson structures. It is shown that the resulting Hamilonian peakon flows in the case of the two peakon equations derived in [Anco and Mobasheramini, Physica D, 355: 1--23, 2017] form orthogonal families, while in general the Hamiltonian peakon flows for two different equations in the general family intersect at a fixed angle equal to the angle between two lines in U (1) parametrizing those two equations. Moreover, it is shown that inverse spectral methods allow one to solve explicitly the dynamics of conservative peakons using explicit solutions to a certain interpolation problem. The graphs of multipeakon solutions confirm the existence of multipeakon breathers as well as asymptotic formation of pairs of two peakon bound states in the non-periodic time domain.

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet’s theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner’s theorem on measure rigidity for unipotent flows, and linearization technique.

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$.

The Novikov equation is an integrable analogue of the CamassaHolm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (B-Toda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians.