The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$.

In this note, we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.

Dans cet article, nous étudions la cohomologie de de Rham du premier revêtement de la tour de Drinfel’d. En particulier, nous obtenons une preuve purement locale du fait que la partie supercuspidale réalise la correspondance de Jacquet-Langlands locale pour $\mathrm {GL}_n$ en la comparant à la cohomologie rigide de certaines variétés de Deligne-Lusztig. Les représentations obtenues sont analogues à celles qui apparaissent dans la cohomologie $\ell $-adique lorsqu’on oublie l’action du groupe de Weil. La preuve repose sur une généralisation d’un résultat d’excision de Grosse-Klönne et de la description explicite du premier revêtement en tant que revêtement cyclique obtenu par l’auteur dans un travail précédent.

In this paper, we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.

We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

There are several factors that can cause the excessive accumulation of biofluid in human tissue, such as pregnancy, local traumas, allergic responses or the use of certain therapeutic medications. This study aims to further investigate the shear-dependent peristaltic flow of Phan–Thien–Tanner (PTT) fluid within a planar channel by incorporating the phenomenon of electro-osmosis. This research is driven by the potential biomedical applications of this knowledge. The non-Newtonian fluid features of the PTT fluid model are considered as physiological fluid in a symmetric planar channel. This study is significant, as it demonstrates that the chyme in the small intestine can be modelled as a PTT fluid. The governing equations for the flow of the ionic liquid, thermal radiation and heat transfer, along with the Poisson–Boltzmann equation within the electrical double layer, are discussed. The long-wavelength ($\delta \ll 1$) and low-Reynolds-number approximations ($Re \to 0$) are used to simplify the simultaneous equations. The solutions analyse the Debye electronic length parameter, Helmholtz–Smoluchowski velocity, Prandtl number and thermal radiation. Additionally, streamlines are used to examine the phenomenon of entrapment. Graphs are used to explain the influence of different parameters on the flow and temperature. The findings of the current model have practical implications in the design of microfluidic devices for different particle transport phenomena at the micro level. Additionally, the noteworthy results highlight the advantages of electro-osmosis in controlling both flow and heat transfer. Ultimately, our objective is to use these findings as a guide for the advancement of lab-on-a-chip systems.

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

Bedford and Smillie [A symbolic characterization of the horseshoe locus in the Hénon family. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1389–1412] classified the dynamics of the Hénon map $f_{a, b} : (x, y)\mapsto (x^2-a-by, x)$ defined on $\mathbb {R}^2$ in terms of a symbolic dynamics when $(a, b)$ is close to the boundary of the horseshoe locus. The purpose of the current article is to generalize their results for all $b\ne 0$ (including the case $b < 0$ as well). The method of the proof is first to regard $f_{a, b}$ as a complex dynamical system in $\mathbb {C}^2$ and second to introduce the new Markov-like partition in $\mathbb {R}^2$ constructed by us [On parameter loci of the Hénon family. Comm. Math. Phys. 361(2) (2018), 343–414].