We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

We consider densities $D_\Sigma(A)$, $\overline{D}_\Sigma(A)$ and $\underline{D}_\Sigma(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $\Sigma$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic, weakly mixing and strongly mixing $\times p$-invariant measures on the unit circle $\mathbb{T}$. Combining these, we prove the following measure rigidity results: on $\mathbb{T}$, the Lebesgue measure is the only non-atomic $\times p$-invariant measure satisfying one of the following: (1) $\mu$ is ergodic and there exist a F\o lner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $D_\Sigma(A)=1$; (2) $\mu$ is weakly mixing and there exist a F\o lner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $\overline{D}_\Sigma(A)>0$; (3) $\mu$ is strongly mixing and there exists a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for infinitely many $j$. Moreover, a $\times p$-invariant measure satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure. As an application we prove that for every increasing function $\tau$ defined on positive integers with $\lim_{n\to\infty}\tau(n)=\infty$, there exists a multiplicative semigroup $S_\tau$ of $\mathbb{Z}^+$ containing $p$ such that $|S_\tau\cap[1,n]|\leq (\log_p n)^{\tau(n)}$ and the Lebesgue measure is the only non-atomic ergodic $\times p$-invariant measure which is $\times q$-invariant for all $q$ in $S_\tau$.

In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.

A vast class of exponential functions are shown to be deterministic. This class includes functions whose exponents are polynomial-like or ``piece-wise'' close to polynomials after differentiation. Many of these functions are proved to be disjoint from the M\"obius function.

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in $${\mathbb{R}^{3}}$$ . More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in $${\mathbb{R}^{3}}$$ , we show that they can be transformed with a$C$^{$m$}-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted thin vortex tubes can be traced back to Lord Kelvin.

This paper presents a 2×2 dynamical system to study the cyclic appearance and disappearance of the gas phase in a two-component (CO2, H2O), two phase (gas, liquid) flow in a single pore. Depending on injection rate, linearization of the dynamical system around equilibrium shows that the gas phase can exhibit two behaviors, either cyclically vanishing and appearing, or approaching a steady-state volume value. Numerical simulations were also run on the fully non-linear dynamical system to verify the results of the linearized model.

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We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

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.

Skew-orthogonal polynomials (SOPs) arise in the study of the <i>n</i>-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in Forrester and Kieburg (Commun Math Phys 342(1):151187, 2016), we propose the concept of <i>partial-skew-orthogonal polynomials</i> (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the multiple integral representations. Among these integrable lattices, some of them are

To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.

For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

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We show that theMöbius function is disjoint to every analytic skew product dynamical system on T2 over a rotation of the circle.

Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence, and a system related to so-called A_1 A_1-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Backlund transformation of the Somos-4 recurrence. Related topics about Somos polynomials are also studied.

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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 $C(X)\rtimes \Gamma$.

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.

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Firstly, a formal correspondence is established between the CamassaHolm (CH) equation and a two-component modified CH (or called SQQ) equation according to the method of <i>moment modification</i> for multipeakon formulae. Secondly, based on the generalized nonisospectral CH equation in Chang et al. (2014) [14] and the interlacing multipeakons of the two-component modified CH equation in Chang et al. (2016) [15], we propose a new generalized two-component modified CH equation with two parameters, which possesses a nonisospectral Lax pair. The proposed equation still admits multipeakon solutions of explicit and closed form. Sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented. Last of all, as a by-product, a generalized nonisospectral modified CH equation is deduced, together with its Lax pair.

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.

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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.

<i>Peakons</i> are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified CamassaHolm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified CamassaHolm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem, the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes continued fractions and multi-point Pad approximations. We propose

We prove the dynamical Mordell-Lang conjecture for birational polynomial morphisms on A^2.