In this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field $K$. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to $\mathbb{Z}[1/p]$-coefficients.

In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

In this paper the authors consider the weighted estimates for the Calderón commutator defined by

\mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,

$L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$

$L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$

$\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$

$\mathcal {C}_{m+1, A}$

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

Let ${\mathcal{X}}$ be a regular variety, flat and proper over a complete regular curve over a finite field such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of ${\mathcal{X}}$ is finite if and only Tate’s conjecture for divisors on $X$ holds and the Tate–Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the known formula for a surface.

In this paper, we study the existence of weak solutions of the quasilinear equation

\begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}

This paper is concerned with the fractional system

\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}

$p \leqslant \frac {n+\alpha +2a}{n-\beta }$

$q \leqslant \frac {n+\beta +2b}{n-\alpha }$

$p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$

We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

For certain pairs of Lie groups (G, H) and primes p, Harris showed a relation of the p-localized homotopy groups of G and H. This is reinterpreted as a p-local homotopy equivalence G ≃ (p)H × G/H, and so there is a projection G(p) → H(p). We show how much this projection preserves the higher homotopy associativity.

The origin of this special issue took place at the 9th European Nonlinear Dynamics Conference (ENOC 2017) in Budapest, Hungary. Specifically, the mini-symposium on Random Dynamical Systems – Recent Advances and New Directions brought together novel perspectives on analyzing stochastic dynamics with applications including biology, structural dynamics, control, energy and mechanics. The expanded use of stochasticity in more realistic models exposes questions related to bifurcations, meta-stability, tipping and early warning signals, multiscale dynamics, and connections between chaos and stochastic dynamics. The observed phenomena in applications drive new methodologies and analyses, needed to understand the interplay between different sources of stochastic effects and nonlinearities, network structure, multi-mode and multi-scale behavior, non-smooth dynamics, energy transfer, and spatio-temporal phenomena. Of course, a single issue cannot hope to cover all of the new topics in stochastic analysis for applications. Nevertheless, we hope that the collection of applications and stochastic models presented in this issue illustrates some of the exciting advances and perspectives relevant for broad classes of stochastic models and demonstrates the need in advancing the theory of stochastic processes.

A credit default swap (CDS) is an exchange of premium payments for a compensation for the occurrence of a credit event. Counterparty risks refer to defaults of parties holding CDS contracts. In this paper we develop a valuation/pricing model for a CDS subject to counterparty risks. Using the Cox–Ingersoll–Ross (CIR) model for interest rate and first arrival times of Poisson processes with variable intensities for the occurrences of credit default and counterparty defaults, we derive a mathematical formulation and make a full theoretical investigation. In addition, we develop a full theory for the corresponding infinite horizon problem and establish its connection with the asymptotic long expiry behaviour of finite horizon problem. Furthermore, we establish a connection between two major frameworks for default times: the structure model approach and the intensity model approach. We show that a solution of the structure model can be obtained as the limit of a sequence of solutions of intensity models. Regarded as an important theoretical development, we remove a constraint typically imposed on the parameters of the CIR model; that is, the well-posedness (existence, uniqueness and continuous dependence of parameters) of the mathematical model holds for any empirically calibrated parameters for the CIR model.

We correct an error in our paper published in volume 56 of this journal.

Steady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder.

We consider two-dimensional mass transport to a finite absorbing strip in a uniform shear flow as a model of surface-based biosensors. The quantity of interest is the Sherwood number Sh, namely the dimensionless net flux onto the strip. Considering early-time absorption, it is a function of the Péclet number Pe and the Damköhler number Da, which, respectively, represent the characteristic magnitude of advection and reaction relative to diffusion. With a view towards modelling nanoscale biosensors, we consider the limit Pe«1. This singular limit is handled using matched asymptotic expansions, with an inner region on the scale of the strip, where mass transport is diffusively dominated, and an outer region at distances that scale as Pe-1/2, where advection enters the dominant balance. At the inner region, the mass concentration possesses a point-sink behaviour at large distances, proportional to Sh. A rescaled concentration, normalised using that number, thus possesses a universal logarithmic divergence; its leading-order correction represents a uniform background concentration. At the outer region, where advection by the shear flow enters the leading-order balance, the strip appears as a point singularity. Asymptotic matching with the concentration field in that region provides the Sherwood number as

$${\rm{Sh}} = {\pi \over {\ln (2/{\rm{P}}{{\rm{e}}^{1/2}}) + 1.0559 + \beta }},$$