Consider a commutative ring $R$ and a simplicial map, $X\mathop {\longrightarrow }\limits ^{\pi }K,$ of finite simplicial complexes. The simplicial cochain complex of $X$ with $R$ coefficients, $\Delta ^*X,$ then has the structure of an $(R,K)$ chain complex, in the sense of Ranicki . Therefore it has a Ranicki-dual $(R,K)$ chain complex, $T \Delta ^*X$. This (contravariant) duality functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$ was defined algebraically on the category of $(R,K)$ chain complexes and $(R,K)$ chain maps.

Our main theorem, 8.1, provides a natural $(R,K)$ chain isomorphism:

\[ T\Delta^*X\cong C(X_K) \]

$C(X_K)$

$X_K$

$X_K$

$X$

$(R,K)$

$C(X_K)$

This paper is focused on spreading dynamics for a discrete Nicholson's blowflies model with time convolution kernel. This problem arises in the invasive activity of blowflies scattered in discrete spatial environment and has distributed maturated age. We found that for a general convolution kernel, the model can exhibit travelling wave phenomena in a discrete spatial habitat. In particular, we determine the minimal wave speed of travelling waves by deriving the non-existence of travelling waves, and we demonstrate that the minimal wave speed can determine the long time behaviour of solutions with compact initial function. Moreover, we prove that all travelling waves are strictly increasing, which implies that the waveforms remain monotone in the propagation process. Some numerical simulations are also presented to confirm the analytical results.

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.

We study the asymptotic behaviour, as $p\to 1^+$, of the solutions of the following inhomogeneous Robin boundary value problem:P

\begin{equation*} \begin{cases} \displaystyle -\Delta_p u_p = f & \text{ in }\Omega,\\ \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g & \text{ on } \partial\Omega, \end{cases} \end{equation*}

$\Omega$

$\mathbb {R}^{N}$

$\nu$

$\Delta _p v$

$p$

$p>1$

$f\in L^{N,\infty }(\Omega )$

$\lambda,\,g$

$\partial \Omega$

$\lambda \ge 0$

$p$

$1$

$p\to 1^+$

We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from Conti and Schweizer (Commun. Pure Appl. Math. 59 (2006), 830–868) and Knüpfer and Kohn (Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 467 (2011), 695–717), we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we fill that gap for the Liouville property, by proving its stability by quasi-isometries for every Riemann surface (and even Riemannian surfaces with pinched negative curvature). Also, a key result characterizes Riemannian surfaces which are quasi-isometric to $\mathbb {R}$.

For an odd prime $p$, we consider free actions of $(\mathbb {Z}_{/{p}})^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb {Z}_{/{p}})^2$ on $\mathbb {R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the $k$-invariants and the Pontrjagin classes from the rotation numbers.

We analyse the $\Gamma$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.

In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

We develop a method based on the Burau matrix to detect conditions on the linking numbers of braid strands. Our main application is to iterated exchanged braids. Unless the braid permutation fixes both braid edge strands, we establish under some fairly generic conditions on the linking numbers a ‘subsymmetry’ property; in particular at most two such braids can be mutually conjugate. As an addition, we prove that the Burau kernel is contained in the commutator subgroup of the pure braid group. We discuss also some properties of the Burau image.

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

Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.

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 paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$, and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0< L_*\leq L^*$, and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$, we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$), then the protection zone should start from somewhere away from $0$, and as far away from $0$ as possible.

We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below $C^2$. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class $C^1$ and that the converse holds for $C^{1,1}$-metrics under an additional convergence condition on regularizations of the metric.

The main purpose of this paper is to study weight-semi-greedy Markushevich bases, and in particular, find conditions under which such bases are weight-almost greedy. In this context, we prove that, for a large class of weights, the two notions are equivalent. We also show that all weight semi-greedy bases are truncation quasi-greedy and weight-superdemocratic. In all of the above cases, we also bring to the context of weights the weak greedy and Chebyshev greedy algorithms—which are frequently studied in the literature on greedy approximation. In the course of our work, a new property arises naturally and its relation with squeeze symmetric and bidemocratic bases is given. In addition, we study some parameters involving the weak thresholding and Chebyshevian greedy algorithms. Finally, we give examples of conditional bases with some of the weighted greedy-type conditions we study.

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.

We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential $V$, (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential $V$ is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials $V$, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant $V$, our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More precisely, we investigate some $L^2$-estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$ utilizing the group Fourier transform on $G$. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot )\|_{L^2(G)}$ by further assuming the $L^1(G)$-regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$