The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

This paper focuses on the vanishing limit problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) system, which is commonly used to describe the dynamic properties of polymeric fluids. Our purpose is to show the relation of the PTT system to the well-known Oldroyd-B system (with or without damping mechanism). The suitable a priori estimates and global existence of strong solutions are established for the PTT system with small initial data. Taking advantage of uniform energy and decay estimates for the PTT system with respect to time $t$ and coefficients $a$ and $b$, then allows us to justify in particular the vanishing limit for all time. More precisely, we prove that the solution $(u,\,\tau )$ of PTT system with $0\leq b\leq Ca$ converges globally in time to some limit $(\widetilde {u},\,\widetilde {\tau })$ in a suitable Sobolev space when $a$ and $b$ go to zero simultaneously (or, only $b$ goes to zero). We may check that $(\widetilde {u},\,\widetilde {\tau })$ is indeed a global solution of the corresponding Oldroyd-B system. In addition, a rate of convergence involving explicit norm will be obtained. As a byproduct, similar results are also true for the local a priori estimates in large norm.

Let ${\mathcal {D}}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\mathbb {R}}^n$. Then ${\mathcal {D}}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\mathcal {D}}$.

The one-dimensional transient solidification of a binary alloy undergoing shrinkage is well-known as an invaluable benchmark for the testing of numerical codes that model macrosegregation. Here, recent work that considered the small-time behaviour of this problem is extended until complete solidification, thereby determining the solute profile across the entire solidified domain. The small-time solution is used as the initial condition for the numerical integration of a problem having three moving boundaries. Of particular significance is the so-called inverse segregation that is observed at the start of solidification, and the extreme segregation that is observed at the end; in the case of the example presented, which is for the often-cited Al–Cu system, the macrosegregation is found to be positive or negative, depending on whether Scheil’s equation or the lever rule is assumed at the microscale, respectively. The relevance of these results for the modelling of steady-state continuous casting processes – in particular, the phenomenon of centreline segregation – is also discussed.

Using totally symmetric sets, Chudnovsky–Kordek–Li–Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.

In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are $C^2$ diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.

In this paper, we consider a general single population model with delay and patch structure, which could model the population loss during the dispersal. It is shown that the model admits a unique positive equilibrium when the dispersal rate is smaller than a critical value. The stability of the positive equilibrium and associated Hopf bifurcation are investigated when the dispersal rate is small or near the critical value. Moreover, we show the effect of network topology on Hopf bifurcation values for a delayed logistic population model.

The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.

This paper concerns the monostable cooperative system with nonlocal diffusion and free boundaries, which has recently been discussed by Du and Ni [J. Differential equations 308(2021) 369-420 and arXiv:2010.01244]. We here aim at four aspects: the first is to give more accurate estimates for the longtime behaviours of the solution; the second is to discuss the limits of solution pair of a semi-wave problem; the third is to investigate the asymptotic behaviours of the corresponding Cauchy problem; the last is to study the limiting profiles of the solution as one of the expanding rates of free boundaries converges to $\infty$. Moreover, some epidemic models are given to illustrate their own rich longtime behaviours, which are quite different from those of the relevant existing works.

We prove a stability theorem for spaces of smooth concordance embeddings. From it we derive various applications to spaces of concordance diffeomorphisms and homeomorphisms.

In this note, we prove that the moduli stack of vector bundles on a curve with a fixed determinant is ${\mathbb A}^1$-connected. We obtain this result by classifying vector bundles on a curve up to ${\mathbb A}^1$-concordance. Consequently, we classify ${\mathbb P}^n$-bundles on a curve up to ${\mathbb A}^1$-weak equivalence, extending a result in [3] of Asok-Morel. We also give an explicit example of a variety which is ${\mathbb A}^1$-h-cobordant to a projective bundle over ${\mathbb P}^2$ but does not have the structure of a projective bundle over ${\mathbb P}^2$, thus answering a question of Asok-Kebekus-Wendt [2].

We establish a priori bounds, existence and qualitative behaviour of positive radial solutions in annuli for a class of nonlinear systems driven by Pucci extremal operators and Lane-Emden coupling in the superlinear regime. Our approach is purely nonvariational. It is based on the shooting method, energy functionals, spectral properties, and on a suitable criteria for locating critical points in annular domains through the moving planes method that we also prove.

In this paper, we give a new method to classify all simple cuspidal modules for the $\mathbb {Z}$-graded and $1/2\mathbb {Z}$-graded Ovsienko–Roger superalgebras. Using this result, we classify all simple Harish–Chandra modules over some related Lie superalgebras, including the $N=1$ BMS$_3$ superalgebra, the super $W(2,2)$, etc.

In this short note, we deal with complete noncompact expanding and steady Ricci solitons of dimension $n\geq 3.$ More precisely, under an integrability assumption, we obtain a characterization for the generalized cigar Ricci soliton and the Gaussian Ricci soliton.

In this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set

\begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*}

$ \alpha\in[0,1] $

$ \alpha $

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles–Giga functional. We introduce a nonlinear $\operatorname {\mathrm {curl}}$ operator for such unoriented vector fields as well as a family of even entropies which we call ‘trigonometric entropies’. Using these tools, we show two main theorems which parallel some results in the literature on the classical Aviles–Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg–Landau energy. Our methods provide alternative proofs in the classical Aviles–Giga context.

We categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ andtheir infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

This work is devoted to the study of the sub-critical case of an anisotropic fully parabolic Keller–Segel chemotaxis system. We prove the existence of nonnegative weak solutions of (1.1) without restriction on the size of the initial data.

For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.

The aim of this paper is to study the dimension reduction analysis of an elastic plate with small thickness reinforced with increasing number of thin ribbons developing fractal geometry. We prove the $\Gamma $-convergence of the energy functionals to a two-dimensional effective energy including singular terms supported within the Sierpinski carpet.