We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$, where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$. Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$, for $i=1,2$. Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded.

As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: (1) Sampaio’s tangent cone result and (2) links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette.

Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

We construct a flat model structure on the category ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ of additive functors from a small preadditive category $\mathcal {Q}$ satisfying certain conditions to the module category ${_{R}\mathsf {Mod}}$ over an associative ring $R$, whose homotopy category is the $\mathcal {Q}$-shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$, an object in ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal {Q}$, and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.

$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\mathcal {L}_s u=u^{\frac {Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $\mathcal {L}_s$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like $r^{{\mathrm {homdim}}(G) \dim (H)}$. Further, we generalize the concept of acceptance domains to locally compact second countable groups.