We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

We consider the parabolic one-dimensional Allen–Cahn equation

The steady state connects, as a ‘transition layer’, the stable phases –1 and +1. We construct a solution u with any given number k of transition layers between –1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t → –∞. More precisely, we find

where the functions ξj (t) satisfy a first-order Toda-type system. They are given by

for certain explicit constants γjk.

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

We obtain intertwining dilation theorems for non-commutative regular domains 𝒟f and non-commutative varieties 𝒱J in B(𝓗)n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain 𝒟f (respectively, variety 𝒱J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety 𝒱J. We provide Andô-type dilations and inequalities for bi-domains 𝒟f ×c 𝒟g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ 𝒟f and Y := (Y1,…, Yn2) ∊ 𝒟g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱J1×c 𝒱J2, where 𝒱J1 and 𝒱J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.

We describe completely the link invariants constructed using Markov traces on the Yokonuma–Hecke algebras in terms of the linking matrix and the Hoste–Ocneanu–Millett–Freyd–Lickorish–Yetter–Przytycki–Traczyk (HOMFLY-PT) polynomials of sublinks.

This review paper is concerned with the stability analysis of the continuity equation in the DiPerna–Lions setting in which the advecting velocity field is Sobolev regular. Quantitative estimates for the equation were derived only recently, but optimality was not discussed. We revisit the results from our 2017 paper, compare the new estimates with previously known estimates for Lagrangian flows and demonstrate how these can be applied to produce optimal bounds in applications from physics, engineering and numerical analysis.

We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation polynomials. Based on the non-vanishing property of these determinants, we apply the functional equations to get an infinite sequence of approximations that is used to produce the linear independence measures.

We consider the finite-time blow-up of solutions for the following two kinds of nonlinear wave equation in de Sitter spacetime:

This proof is based on a new blow-up criterion, which generalizes that by Sideris. Furthermore, we give the lifespan estimate of solutions for the problems.