We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$, where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$, $r \geq 2$, and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$-generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$. When $s> {u}/{2}$, it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.

Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.

We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.

Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $. Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic for $\xi $. This is a generalization of a similar theorem by T. Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$, $n\in {\mathbb {Z}}$, $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.

In this paper, we propose a novel approach that employs kinetic equations to describe the collective dynamics emerging from graph-mediated pairwise interactions in multi-agent systems. We formally show that for large graphs and specific classes of interactions a statistical description of the graph topology, given in terms of the degree distribution embedded in a Boltzmann-type kinetic equation, is sufficient to capture the collective trends of networked interacting systems. This proves the validity of a commonly accepted heuristic assumption in statistically structured graph models, namely that the so-called connectivity of the agents is the only relevant parameter to be retained in a statistical description of the graph topology. Then, we validate our results by testing them numerically against real social network data.

Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup of $\textrm{Gal}(K)$, then ${\mathrm{rank}}(A)\le r+1$. Moreover, if $\mathrm{char}(K)=0$, then ${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of $\textrm{Gal}(K)$.

The reduced peripheral system was introduced by Milnor [18] in the 1950s for the study of links up to link-homotopy, that is, up to homotopies leaving distinct components disjoint; this invariant, however, fails to classify links up to link-homotopy for links of four or more components. The purpose of this paper is to show that the topological information which is detected by Milnor’s reduced peripheral system is actually 4-dimensional. The main result gives indeed a complete characterization of links having the same reduced peripheral system, in terms of ribbon solid tori in 4–space up to ribbon link-homotopy. The proof relies on an intermediate characterization given in terms of welded diagrams up to self-virtualization, hence providing a purely topological application of the combinatorial theory of welded links.

Let $X,\,Y$ be Banach spaces and fix a linear operator $T \in \mathcal {L}(X,\,Y)$ and ideals $\mathcal {I},\, \mathcal {J}$ on the nonnegative integers. We obtain Silverman–Toeplitz type theorems on matrices $A=(A_{n,k}: n,\,k \in \omega )$ of linear operators in $\mathcal {L}(X,\,Y)$, so that

\[ \mathcal{J}\text{-}\lim A\boldsymbol{x}=T(\mathcal{I}\text{-}\lim \boldsymbol{x}) \]

$X$

$\boldsymbol {x}=(x_0,\,x_1,\,\ldots )$

$\mathcal {I}$

In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.

Cannibalism is often an extreme interaction in the animal species to quell competition for limited resources. To model this critical factor, we improve the predator–prey model with nonlocal competition effect by incorporating the cannibalism term, and different kernels for competition are considered in this model numerically. We give the critical conditions leading to the double Hopf bifurcation, in which the gestation time delay and the diffusion coefficient were selected as the bifurcation parameters. The innovation of the work lies near the double Hopf bifurcation point, and the stable homogeneous and inhomogeneous periodic solutions can coexist. The theoretical results of the extended centre manifold reduction and normal form method are in good agreement with the numerical simulation.

We prove Abelian and Tauberian theorems for regularized Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularized Cauchy transform.

Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.

In this paper, we prove Kato’s main conjecture for $CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.