Strongly Continuous Research Articles

On the structure of hypergroups with respect to the induced topology

Let H be a hypergroup with left Haar measure and let L1(H) be the complex Lebesgue space associated with it. Let L∞(H) be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locally a.e. It is a standard device to embed L∞(H) into ℬ(L1(H),L∞(H)). We denote the strong operator topology and the weak operator topology on L∞(H) by τso and τwo. Unlike the uniform norm and weak topologies on L∞(H), they depend essentially on the hypergroup structure of H. We derive that the τso-topology is always different from the weak∗-topology whenever H is infinite. We can conclude that for a compact hypergroup H, L1(H) is the dual of (L∞(H),τso). The properties of τso and τwo are then studied further and we pay attention to the τwo-almost periodic elements of L∞(H). Finally we give some further results about bounded linear operators which are τso-τso-continuous.

On the asymptotic behaviour of semigroups for flows in infinite networks

We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures.

On different modes of order convergence and some applications

We give a thorough overview of the different notions for order convergence that are found in the literature and provide a systematic comparison of the associated topologies. As an application of this study we prove a result related to the order topology on von Neumann algebras, complementing the study started in Chetcuti et al. (Stud. Math. 230:95-120, 2015). We show that for every atomic von Neumann algebra (not necessarily \(\sigma \)-finite) the restriction of the order topology to bounded parts of M coincides with the restriction of the \(\sigma \)-strong topology \(s(M,M_*)\). We recall that the methods of Chetcuti et al. (Stud. Math. 230:95-120, 2015) rest heavily on the assumption of \(\sigma \)-finiteness. Furthermore, for a semi-finite measure space, we provide a complete picture of the relations between the topologies on \(L^\infty \) associated with the duality \(\langle L^1, L^\infty \rangle \) and its order topology.

Pseudo almost automorphic mild solutions to non-autonomous differential equations in the “strong topology”

Pseudo almost automorphic mild solutions to non-autonomous differential equations in the “strong topology”

Recent experimental research progress of two-dimensional van der Waals semiconductor moiré superlattices

A moiré superlattice can be formed by overlaying two atomically thin van der Waals materials with a rotation angle or with a lattice mismatch. Since the discovery of correlated insulators and superconductivity in magic angle twisted bilayer graphene, constructing moiré superlattices by various two-dimensional (2D) van der Waals materials and studying their novel properties emerge as a hot topic and research frontier in condensed matter physics. Here we review the recent experimental progress of 2D transition metal dichalcogenide moiré superlattices. In this system, the formation of moiré flat band does not rely on certain magic angles. Experimentally, a series of correlated electron states and topological states have been discovered and confirmed. Further theoretical and experimental studies can find a wealth of emergent phenomena caused by the combined influence of strong correlation and topology in transition metal dichalcogenide moiré superlattice.

Configuration Design and Gait Analysis of Wheel-Legged Mobile Robot Based on the Rubik’s Cube Mechanism

In order to improve the obstacle-crossing ability, motion stability and load-bearing capacity of mobile robots for different terrains, the Rubik’s Cube mechanism (RCM) with strong coupling and variable topology is introduced into the field of mobile robots, and a wheel-legged mobile robot (WLMR) based on RCM is proposed. Aiming at the problems of the classical three-order RCM, such as small internal space, difficult processing and demanding precision, a new type of chute third-order RCM is designed, and its mechanical characteristics analysis and feasibility analysis are carried out. What’s more, a driving configuration analysis method is established according to different driving configuration relationships, and the configuration of WLMR is determined by the configuration stability analysis. Then, a WLMR with polymorphism is designed, and gait planning and gait stability analysis are conducted. Eventually, the co-simulation and prototype experiments are performed to verify the efficiency of the WLMR’s straight motion, in-situ rotation, obstacle-crossing and morphology transformation in complex environments. This research not only provides a reference for the design of polymorphous mobile robots, but also opens up ideas for the application of the RCM in daily production and life.

Quantum anomalous Hall effect from intertwined moiré bands.

Electron correlation and topology are two central threads of modern condensed matter physics. Semiconductor moiré materials provide a highly tuneable platform for studies of electron correlation1-12. Correlation-driven phenomena, including the Mott insulator2-5, generalized Wigner crystals2,6,9, stripe phases10 and continuous Mott transition11,12, have been demonstrated. However, non-trivial band topology has remained unclear. Here we report the observation of a quantum anomalous Hall effect in AB-stacked MoTe2 /WSe2 moiré heterobilayers. Unlike in the AA-stacked heterobilayers11, an out-of-plane electric field not only controls the bandwidth but also the band topology by intertwining moiré bands centred at different layers. At half band filling, corresponding to one particle per moiré unit cell, we observe quantized Hall resistance, h/e2 (with h and e denoting the Planck's constant and electron charge, respectively), and vanishing longitudinal resistance at zero magnetic field. The electric-field-induced topological phase transition from a Mott insulator to a quantum anomalous Hall insulator precedes an insulator-to-metal transition. Contrary to most known topological phase transitions13, it is not accompanied by a bulk charge gap closure. Our study paves the way for discoveryof emergent phenomena arising from the combined influence of strong correlation and topology in semiconductor moiré materials.

Infinite sum relations on universal C*-algebras

We extend the usual theory of universal C*-algebras from generators and relations in order to allow some relations to be described using the strong operator topology. In particular, we can allow some infinite sum relations. We prove a universal property for the algebras we define and we show how the Cuntz algebra of infinite isometries as well as the Exel-Laca algebras can be described using infinite sum relations. Finally, we give some sufficient conditions for when a C*-algebra generated by projections and partial isometries is a universal C*-algebra using only norm relations, in case one still wants to avoid using relations with respect to the strong operator topology.

Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a “lagrangian formulation” of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the BV regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a BV-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property.

Interplay between strong correlations and electronic topology in the underlying kagome lattice of Na2/3CoO2

Electronic topology in metallic kagome compounds is under intense scrutiny. We present transport experiments in Na2/3CoO2 in which the Na order differentiates a Co kagome sub-lattice in the triangular CoO2 layers. Hall and magnetoresistance (MR) data under high fields give evidence for the coexistence of light and heavy carriers. At low temperatures, the dominant light carrier conductivity at zero field is suppressed by a B-linear MR suggesting Dirac like quasiparticles. Lifshitz transitions induced at large B and T unveil the lower mobility carriers. They display a negative B^2 MR due to scattering from magnetic moments likely pertaining to a flat band. We underline an analogy with heavy Fermion physics.

Theoretical study of the crystal structure, stability, and properties of phases in the V-N system

Stable compounds in the V-N binary system are systematically investigated and four new phases are found: $Pbam\text{\ensuremath{-}}{\mathrm{V}}_{5}{\mathrm{N}}_{2}, Pnma\text{\ensuremath{-}}{\mathrm{V}}_{2}\mathrm{N}, P\overline{3}m1\text{\ensuremath{-}}{\mathrm{V}}_{2}{\mathrm{N}}_{3}$, and $I4/mcm\text{\ensuremath{-}}{\mathrm{VN}}_{2}$. All the predicted high-pressure vanadium nitrides are dynamically stable at ambient pressure. Moreover, the thermodynamic stability of vanadium nitrides in the temperature range of 0--1500 K at different pressures (0, 20, 40, 60, and 120 GPa) was also evaluated within the harmonic approximation. The sequence of phases of ${\mathrm{V}}_{2}\mathrm{N}$ under pressure is $\ensuremath{\varepsilon}{\text{-Fe}}_{2}\text{N}\phantom{\rule{0.16em}{0ex}}\text{type}\ensuremath{\rightarrow}\ensuremath{\zeta}{\text{-Fe}}_{2}\text{N}\phantom{\rule{0.16em}{0ex}}\text{type}\ensuremath{\rightarrow}{\text{Fe}}_{2}\text{C}\phantom{\rule{0.16em}{0ex}}\text{type}\ensuremath{\rightarrow}Pnma{\text{-V}}_{2}\text{N}$. In addition, relative stability and lattice dynamics properties of several vanadium mononitrides are systematically calculated and discussed. Structural features, mechanical properties, electronic structures, and chemical bonding of all the V-N compounds are analyzed at 0 GPa. Among these vanadium nitrides, WC-type VN has the highest Vickers hardness ($\ensuremath{\sim}37\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$) and superior fracture toughness (4.3--6.1 MPa ${\mathrm{m}}^{1/2}$), which mainly originate from its strong V-N bonding as well as its strong three-dimensional V-N covalent bond network. The configuration of the strong and short N-N covalent bonds enables the new phase $I4/mcm\text{\ensuremath{-}}{\mathrm{VN}}_{2}$ to exhibit good mechanical properties. Our results also reveal that the formation of a strong covalent-bond network topology in a crystal is a fundamental principle for designing a hard or superhard structure.

Weighted Composition Semigroups on Some Banach Spaces

We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis (Linear Algeb Appl 84:359–371, 1986) and König (Michigan Math J 37:469–476, 1990) and answers a question of Siskakis, A. G. proposed in Siskakis (Contemp Math 213:229–252, 1998). We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator \(M_z\).

Kagome Flatbands for Coherent Exciton-Polariton Lasing

Kagome lattices supporting Dirac cone and flatband dispersions are well known as a highly frustrated, two-dimensional lattice system. Particularly the flatbands therein are attracting continuous interest based on their link to topological order, correlations and frustration. In this work, we realize coupled microcavity implementations of Kagome lattices hosting exciton-polariton quantum fluids of light. We demonstrate precise control over the dispersiveness of the flatband as well as selective condensation of exciton-polaritons into the flatband. Subsequently, we focus on the spatial and temporal coherence properties of the laser-like emission from these polariton condensates that are closely connected to the flatband nature of the system. Notably, we find a drastic increase in coherence time due to the localization of flatband condensates. Our work illustrates the outstanding suitability of the exciton-polariton system for detailed studies of flatband states as a platform for microlaser arrays in compact localized states, including strong interactions, topology and non-linearity.

Dislocation non-Hermitian skin effect

We demonstrate that crystal defects can act as a probe of intrinsic non-Hermitian topology. In particular, in point-gapped systems with periodic boundary conditions, a pair of dislocations may induce a non-Hermitian skin effect, where an extensive number of Hamiltonian eigenstates localize at only one of the two dislocations. An example of such a phase are two-dimensional systems exhibiting weak non-Hermitian topology, which are adiabatically related to a decoupled stack of Hatano-Nelson chains. Moreover, we show that strong two-dimensional point-gap topology may also result in a dislocation response, even when there is no skin effect present with open boundary conditions. For both cases, we directly relate their bulk topology to a stable dislocation non-Hermitian skin effect. Finally, and in stark contrast to the Hermitian case, we find that gapless non-Hermitian systems hosting bulk exceptional points also give rise to a well-localized dislocation response.

Unconventional sequence of correlated Chern insulators in magic-angle twisted bilayer graphene

The interplay between strong electron-electron interactions and band topology can lead to novel electronic states that spontaneously break symmetries. The discovery of flat bands in magic-angle twisted bilayer graphene (MATBG) with nontrivial topology has provided a unique platform in which to search for new symmetry-broken phases. Recent scanning tunneling microscopy and transport experiments have revealed a sequence of topological insulating phases in MATBG with Chern numbers $C=\pm 3, \, \pm 2, \, \pm 1$ near moir\'e band filling factors $\nu = \pm 1, \, \pm 2, \, \pm 3$, corresponding to a simple pattern of flavor-symmetry-breaking Chern insulators. Here, we report high-resolution local compressibility measurements of MATBG with a scanning single electron transistor that reveal a new sequence of incompressible states with unexpected Chern numbers observed down to zero magnetic field. We find that the Chern numbers for eight of the observed incompressible states are incompatible with the simple picture in which the $C= \pm 1$ bands are sequentially filled. We show that the emergence of these unusual incompressible phases can be understood as a consequence of broken translation symmetry that doubles the moir\'e unit cell and splits each $C=\pm 1$ band into a $C=\pm 1$ band and a $C=0$ band. Our findings significantly expand the known phase diagram of MATBG, and shed light onto the origin of the close competition between different correlated phases in the system.

Anisotropic phonon-mediated electronic transport in chiral Weyl semimetals

Discovery and observations of exotic, quantized optical and electrical responses have sparked renewed interest in nonmagnetic chiral crystals. Within this class of materials, six group V transition metal ditetrelides, that is, XY$_2$ (X = V, Nb, Ta and Y = Si, Ge), host composite Weyl nodes on high-symmetry lines, with Kramers-Weyl fermions at time-reversal invariant momenta. In addition, at least two of these materials, NbGe$_2$ and NbSi$_2$, exhibit superconducting transitions at low temperatures. The interplay of strong electron-phonon interaction and complex Fermi surface topology present an opportunity to study both superconductivity and hydrodynamic electron transport in these systems. Towards this broader question, we present an ab initio theoretical study of the electronic transport and electron-phonon scattering in this family of materials, with a particular focus on NbGe$_2$ vs. NbSi$_2$, and the other group V ditetrelides. We shed light on the microscopic origin of NbGe$_2$'s large and anisotropic room temperature resistivity and contextualize its strong electron-phonon scattering with a presentation of other relevant scattering lifetimes, both momentum-relaxing and momentum-conserving. Our work explores the intriguing possibility of observing hydrodynamic electron transport in these chiral Weyl semimetals.

Operator algebras of countable range multiplicity

In this paper, the spectrum of operators commuting with operator algebras of countable range multiplicity is studied. It is shown that if the commutant of a set which does not contain any scalar operator has countable range multiplicity then it has a non trivial invariant subspace. If the range multiplicity of an operator algebra is one then it is shown that the strong and uniform topologies coincide on the commutant of the algebra and also each collection of mutually orthogonal projections in the commutant is finite. In addition, if the operator algebra is self adjoint also then it is shown that the underline Hilbert space has a finite orthogonal decomposition such that each of its components reduces the algebra.

Convergence of a generalized penalty and regularization method for quasi–variational–hemivariational inequalities

In the paper an elliptic quasi–variational–hemivariational inequality with constraints in a Banach space is studied. First, we apply the Minty technique, the KKM principle and the theory of nonsmooth analysis to establish the solvability of the inequality problem. Then, we employ a generalized penalty and regularization method for the inequality and introduce a family of penalized and regularized problems with no constraints and with Gâteaux differentiable potentials. Through a limit procedure, we prove that the Kuratowski upper limit with respect to the weak topology of the solution sets to penalized and regularized problems, is a nonempty subset of the solution set to the original inequality problem. Next, if a set-valued operator in the inequality has (S)+-property, then the Kuratowski upper limits with respect to the weak and strong topologies for the solution sets coincide. Finally, we illustrate our results by examining a nonlinear elliptic inclusion with the subgradient term of a locally Lipschitz function, mixed boundary conditions and an obstacle unilateral constraint which appears in a semipermeability problem.

Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology

In this paper, the concept of probabilistic g-metric space with degree l, which is a generalization of probabilistic G-metric space, is introduced. Then, by endowing strong topology, the definition of l-dimensional asymptotic density of a subset A of mathbb{N}^{l} is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.

Does a typical ℓ_{𝑝}-space contraction have a non-trivial invariant subspace?

Given a Polish topology τ \tau on B 1 ( X ) \mathcal {B}_{1}(X) , the set of all contraction operators on X = ℓ p X=\ell _p , 1 ≤ p > ∞ 1\le p>\infty or X = c 0 X=c_0 , we prove several results related to the following question: does a typical T ∈ B 1 ( X ) T\in \mathcal {B}_{1}(X) in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense G δ G_\delta set G ⊆ ( B 1 ( X ) , τ ) \mathcal G\subseteq (\mathcal {B}_{1}(X),\tau ) such that every T ∈ G T\in \mathcal G has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong ∗ ^* Operator Topology.