Strongly Continuous Research Articles

Symbolic extended dynamic mode decomposition.

In this paper, we present a new method of performing extended dynamic mode decomposition (EDMD) on systems, which admit a symbolic representation. EDMD generates estimates of the Koopman operator, K, for a dynamical system by defining a dictionary of observables on the space and producing an estimate, Km, which is restricted to be invariant on the span of this dictionary. A central question for the EDMD is what should the dictionary be? We consider a class of chaotic dynamical systems with a known or estimable generating partition. For these systems, we construct an effective dictionary from indicators of the "cylinder sets," which arise in defining the "symbolic system" from the generating partition. We prove strong operator topology convergence for both the projection onto the span of our dictionary and for Km. We also prove practical finite-step estimation bounds for the projection and Km as well. Finally, we demonstrate some numerical results on eigenspectrum estimation and forecasting applied to the dyadic map and the logistic map.

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.

Convergence of power sequences of operators via their stability

This paper is concerned with the convergence of power sequences and stability of Hilbert space operators, where “convergence” and “stability” are considered with respect to weak, strong and norm topologies. It is proved that an operator has a convergent power sequence if and only if it is a (not necessarily orthogonal) direct sum of an identity operator and a stable operator. This reduces the issue of convergence of the power sequence of an operator T to the study of stability of T. The question of when the limit of the power sequence is an orthogonal projection is investigated. Among operators sharing this property are hyponormal and contractive ones. In particular, a hyponormal or a contractive operator with no identity part is stable if and only if its power sequence is convergent. In turn, a unitary operator has a weakly convergent power sequence if and only if its singular-continuous part is weakly stable and its singular-discrete part is the identity. Characterizations of the convergence of power sequences and stability of subnormal operators are given in terms of semispectral measures.

A Novel Connection Element Method for Multiscale Numerical Simulation of Two-Phase Flow in Fractured Reservoirs

Summary This paper presents a novel approach to the numerical simulation of fractured reservoirs, called the connection element method (CEM), which differs from traditional grid-based methods. The reservoir computational domain is discretized into a series of nodes, and a system of connection elements is constructed based on the given connection lengths and angles. The pressure diffusion term is approximated using generalized finite difference theory. Meanwhile, the transmissibility and volume of the connection elements are determined, and pressure equations are solved discretely to obtain pressure at nodes to approximate the upstream flux along connection elements. Then, we solve the transport equation to obtain oil saturation profiles with low numerical diffusion, utilizing the discontinuous Galerkin (DG) method. Moreover, the flow path tracking algorithm is introduced to quantify the flow allocation factors between wells. In all, the pressure equation can be solved at a global coarse-scale point cloud and the saturation equation is calculated at a local fine-scale connection element. In other words, CEM is of multiscale characteristics relatively. Finally, several numerical examples are implemented to demonstrate that CEM can achieve a relatively better balance between computational accuracy and efficiency compared with embedded discrete fracture modeling (EDFM). Furthermore, CEM adopts flexible meshless nodes instead of grids with strong topology, making it more practical to handle complex reservoir geometry such as fractured reservoirs.

Compactness Criterion for Families of Quantum Operations in the Strong Convergence Topology and Its Applications

Compactness Criterion for Families of Quantum Operations in the Strong Convergence Topology and Its Applications

Extending families of disjoint hypercyclic operators

In the present note, we solve two open questions posed by Salas in [33] about disjoint hypercyclic operators. First, we show that given any family T1,…,TN of disjoint hypercyclic operators, one can always select an operator T such that the extended family T1,…,TN,T of operators remains disjoint hypercyclic. In fact, we prove that the set of operators T which can extend the family of disjoint hypercyclic operators is dense in the strong operator topology in the algebra of bounded operators. Second, we show the existence of two disjoint weakly mixing operators that fail to possess a dense d-hypercyclic manifold. Thus, these operators satisfy the Disjoint Blow-up/Collapse property but fail to satisfy the Strong Disjoint Blow-up/Collapse property, a notion which was introduced by Salas as a sufficient condition for having a dense linear manifold of disjoint hypercyclic vectors.

Rigorous justification of the hydrostatic approximation limit of viscous compressible flows

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as “well-prepared” initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to zero.

On the continuity in q of the family of the limit q-Durrmeyer operators

Abstract This study deals with the one-parameter family { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit q q -Durrmeyer operators. The continuity of this family with respect to the parameter q q is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q ∈ [ 0 , 1 ] q\in \left[0,1] . When it comes to the uniform operator topology, the continuity is preserved solely at q = 0 q=0 and fails at all q ∈ ( 0 , 1 ] . q\in \left(0,1]. In addition, a few estimates for the distance between two limit q q -Durrmeyer operators have been derived in the operator norm on C [ 0 , 1 ] C\left[0,1] .

A hybrid topological quantum state in an elemental solid.

Topology1-3and interactions are foundational concepts in the modern understanding of quantum matter. Their nexus yields three important research directions: (1) the competition between distinct interactions, as in several intertwined phases, (2) the interplay between interactions and topology that drives the phenomena in twisted layered materials and topological magnets, and (3) the coalescence of several topological orders to generate distinct novel phases. The first two examples have grown into major areas of research, although the last example remains mostly unexplored, mainly because of the lack of a material platform for experimental studies. Here, using tunnelling microscopy, photoemission spectroscopy and a theoretical analysis, we unveil a 'hybrid' topological phase of matter in the simple elemental-solid arsenic. Through a unique bulk-surface-edge correspondence, we uncover that arsenic features a conjoined strong and higher-order topology that stabilizes a hybrid topological phase. Although momentum-space spectroscopy measurements show signs of topological surface states, real-space microscopy measurements unravel a unique geometry of topologically induced step-edge conduction channels revealed on various natural nanostructures on the surface. Using theoretical models, we show that the existence of gapless step-edge states in arsenic relies on the simultaneous presence of both a non-trivial strong Z2 invariant and a non-trivial higher-order topological invariant, which provide experimental evidence for hybrid topology. Our study highlights pathways for exploring the interplay of different band topologies and harnessing the associated topological conduction channels in engineered quantum or nano-devices.

The hyperbolic cosine transform and its applications to composition operators

In this paper we characterize hyperbolic cosine transforms of (positive) Borel measures ν in terms of exponential convexity (Bernstein's terminology). The case of compactly supported measures ν is also considered. All of this is then applied to (bounded) composition operators CT,ρ:f↦f∘T on L2(Rκ,μρ) with affine symbols T=A+a, where dμρ(x)=ρ(x)dx, ρ(x)=ψ(‖x‖)−1, ψ is a continuous positive real valued function and ‖⋅‖ is the Euclidean norm on Rκ. The main result states that the map Rκ∋a↦CI+a,ρ is continuous in the strong operator topology and has cosubnormal values if and only if ψ is the hyperbolic cosine transform of a compactly supported Borel measure (I is the identity transformation). The case of affine symbols T that are not translations is also discussed.

Emergence of quantum confinement in topological kagome superconductor CsV3Sb5

Quantum confinement is a restriction on the motion of electrons in a material to specific region, resulting in discrete energy levels rather than continuous energy bands. In certain materials, quantum confinement could dramatically reshape the electronic structure and properties of the surface with respect to the bulk. Here, in the recently discovered kagome superconductors CsV3Sb5, we unveil the dominant role of quantum confinement in determining their surface electronic structure. Combining angle-resolved photoemission spectroscopy (ARPES) measurement and density-functional theory simulation, we report the observations of two-dimensional quantum well states due to the confinement of bulk electron pocket and Dirac cone to the nearly isolated surface layer. The theoretical calculations on the slab model also suggest that the ARPES observed spectra are almost entirely contributed by the top two layers. Our results not only explain the disagreement of band structures between the recent experiments and calculations, but also suggest an equally important role played by quantum confinement, together with strong correlation and band topology, in shaping the electronic properties of this material.

Competing itinerant and local spin interactions in kagome metal FeGe

The combination of a geometrically frustrated lattice, and similar energy scales between degrees of freedom endows two-dimensional Kagome metals with a rich array of quantum phases and renders them ideal for studying strong electron correlations and band topology. The Kagome metal, FeGe is a noted example of this, exhibiting A-type collinear antiferromagnetic (AFM) order at TN ≈ 400 K, then establishes a charge density wave (CDW) phase coupled with AFM ordered moment below TCDW ≈ 110 K, and finally forms a c-axis double cone AFM structure around TCanting ≈ 60 K. Here we use neutron scattering to demonstrate the presence of gapless incommensurate spin excitations associated with the double cone AFM structure of FeGe at temperatures well above TCanting and TCDW that merge into gapped commensurate spin waves from the A-type AFM order. Commensurate spin waves follow the Bose factor and fit the Heisenberg Hamiltonian, while the incommensurate spin excitations, emerging below TN where AFM order is commensurate, start to deviate from the Bose factor around TCDW, and peaks at TCanting. This is consistent with a critical scattering of a second order magnetic phase transition with decreasing temperature. By comparing these results with density functional theory calculations, we conclude that the incommensurate magnetic structure arises from the nested Fermi surfaces of itinerant electrons and the formation of a spin density wave order.

The solvability of inhomogeneous boundary-value problems in Sobolev spaces

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.

Representative Functions, Variational Convergence and Almost Convexity

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

Stability Analysis of Two-Dimensional Ideal Flows With Applications to Viscous Fluids and Plasmas

Abstract We are interested in the stability analysis of two-dimensional incompressible inviscid fluids. Specifically, we revisit a known recent result on the stability of Yudovich’s solutions to the incompressible Euler equations in $L^{\infty }([0,T];H^{1})$ by providing a new approach to its proof based on the idea of compactness extrapolation and by extending it to the whole plane. This new method of proof is robust and, when applied to viscous models, leads to a remarkable logarithmic improvement on the rate of convergence in the vanishing viscosity limit of two-dimensional fluids. Loosely speaking, this logarithmic gain is the result of the fact that, in appropriate high-regularity settings, the smoothness of solutions to the Euler equations at times $t\in [0,T)$ is strictly higher than their regularity at time $t=T$. This “memory effect” seems to be a general principle that is not exclusive to fluid mechanics. It is therefore likely to be observed in other setting and deserves further investigation. Finally, we also apply the stability results on Euler systems to the study of two-dimensional ideal plasmas and establish their convergence, in strong topologies, to solutions of magnetohydrodynamic systems, when the speed of light tends to infinity. The crux of this asymptotic analysis relies on a fine understanding of Maxwell’s system.

Comparison Theorems for Weak Topologies (3)

Weak topology on a nonempty set X is defined as the smallest or weakest topology on X with respect to which a given (fixed) family of functions on X is continuous. In an effort to construct and analyze strictly stronger weak topologies, only four weak topologies (namely, the Arens topology, the Mackey topology, the weakened topology, and the strong topology) have been compared before now. Also, all these weak topologies were constructed with an eye on only the polars of linear spaces (i.e. with a focus on only linear functionals on linear spaces). To that extent, the study of strictly stronger weak topologies before now can be described as a study of linear topological spaces. Here in Part 3 of our Comparison Theorems for Weak Topologies: We established and clarified the place of our comparison theorems in the context of the weakened topology, the Arens topology, the Mackey topology, and the strong topology that have up to now been compared. We showed that there are many other weak topologies between the four already compared weak topologies—constructible even by the use of polars. In particular, we showed that we can we find a weak topology stronger than the one hitherto known and referred to as the strong topology. Then we showed that if two weak topologies, generated by one family of functions on a set, are strictly comparable, then there exist in a range space two strictly comparable topologies which induce the weak topologies. The rather simplistic view that” The weak topology is Hausdorff” has been held by many authors for very long time. We changed that narrative here, as we stated and proved (with examples) that:” Some weak topologies are Hausdorff while others are not.”

Stability of strong attractors for the extensible beam equation with gentle dissipation

The paper investigates the stability of the strong global and exponential attractors for the extensible beam equation with gentle dissipation: utt−βM(‖∇u‖2)Δu+Δ2u+(−Δ)2αut+f(u)=g(x), where the perturbed parameter α∈(0,1/2] is a dissipative index and β∈[0,1] is an extensibility parameter. We show that when the nonlinearity f(u) is of the optimally subcritical growth depending on α: 1≤p<pα:=N+8α(N−4)+, (i) the strong (H,H2)-global attractors obtained in [15] for this model are upper semicontinuous with respect to α in H2-topology and with respect to β in stronger V4×V2+α1-topology, with α1∈(0,α), respectively; (ii) for any fixed β∈[0,1] and α0∈(0,1/2], the related dynamical system (Sα(t),H) has a family of strong (H,H2)-exponential attractors Eexpα which are continuous with respect to α at α0 in H2-topology; (iii) for any fixed α∈(0,1/2], the related dynamical system (Sβ(t),H) has a family of strong (H,V4×V2+2α1)-exponential attractors Eexpβ which are Hölder continuous with respect to β at any point β0∈[0,1] in V4×V2+2α1-topology. All these strong global and exponential attractors are also the standard global and exponential attractors of optimal regularity. The method used here allows to obtain the stability of the strong attractors with respect to the perturbed parameters in strong topology instead of only standard energy topology.

Almost Everywhere Ergodicity in Quantum Lattice Models

We rigorously examine, in generality, the ergodic properties of quantum lattice models with short range interactions, in the C^* algebra formulation of statistical mechanics. Ergodicity results, in the context of group actions on C^* algebras, assume that the algebra is asymptotically abelian, which is not generically the case for time evolution. The Lieb-Robinson bound tells us that, in a precise sense, the spatial extent of any time-evolved local operator grows linearly with time. This means that the algebra of observables is asymptotically abelian in a space-like region, and implies a form of ergodicity outside the light-cone. But what happens within it? We show that the long-time limit of the n-th moment of a ray-averaged observable, along space-time rays of almost every speed, converges to the n-th power of its expectation in the state (i.e. its ensemble average). Thus ray averages do not fluctuate in the long time limit. This is a statement of ergodicity, and holds in any state that is invariant under space-time translations and that satisfies weak clustering properties in space. The ray averages can be performed in a way that accounts for oscillations, showing that ray-averaged observables cannot sustainably oscillate in the long time limit. We also show that in the GNS representation of the algebra of observables, for any KMS state with the above properties, the long-time limit of the ray average of any observable converges (in the strong operator topology) to the ensemble average times the identity, again along space-time rays of almost every speed. This is a strong version of ergodicity, and indicates that, as operators, observables get “thinner” almost everywhere within the light-cone. A similar statement holds under oscillatory averaging.

Nonlocal Wasserstein distance: metric and asymptotic properties

The seminal result of Benamou and Brenier provides a characterization of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is the kinetic energy. Here we consider a fundamental modification of the framework where the paths are solutions of nonlocal (jump) continuity equations and the action is a nonlocal kinetic energy. The resulting nonlocal Wasserstein distances are relevant to fractional diffusions and Wasserstein distances on graphs. We characterize the basic properties of the distance and obtain sharp conditions on the (jump) kernel specifying the nonlocal transport that determine whether the topology metrized is the weak or the strong topology. A key result of the paper are the quantitative comparisons between the nonlocal and local Wasserstein distance.

Quasi-monotone convergence of plurisubharmonic functions

The complex Monge-Ampère operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded quasi-plurisubharmonic functions by various authors. As this operator is not continuous for the L1-topology, several stronger topologies have been introduced over the last decades to remedy this, while maintaining efficient compactness criteria. The purpose of this note is to show that these stronger topologies are essentially equivalent to the natural quasi-monotone topology that we introduce and study here.