Strongly Continuous Research Articles

Hadamard well-posedness of the gravity water waves system

We consider the system of (pure) gravity water waves in any dimension and in a fluid domain with a general bottom geometry. The unique solvability of this problem was established by Alazard–Burq–Zuily [Invent. Math. 198(1) (2014) 71–163] at a low regularity level where the initial surface is [Formula: see text] in terms of Sobolev embeddings; this result allows the existence of free surfaces with unbounded curvature. Our result states that the solutions obtained in the above work depend continuously on initial data in the strong topology in which the solutions are constructed. This establishes a well-posedness result in the sense of Hadamard.

Global attractor for a strongly damped wave equation with fully supercritical nonlinearities

The paper investigates the existence of global attractor for a strongly damped wave equation with fully supercritical nonlinearities: $ u_{tt}-Δ u- Δu_t+h(u_t)+g(u)=f(x) $. In the case when the nonlinearities $ h(u_t) $ and $ g(u) $ are of fully supercritical growth, which leads to that the weak solutions of the equation lose their uniqueness, by introducing the notion of limit solutions and using the theory on the attractor of the generalized semiflow, we establish the existence of global attractor for the subclass of limit solutions of the equation in natural energy space in the sense of strong topology.

Existence theory for semilinear evolution inclusions involving measures

We provide existence results for semilinear differential inclusions involving measures: urn:x-wiley:0025584X:media:mana201600162:mana201600162-math-0001where A is the infinitesimal generator of a C0‐semigroup of contractions on a separable Banach space X and is a right‐continuous non‐decreasing function. The existence of mild solutions, as well as the compactness of the solution set are obtained via Kakutani–Ky Fan's fixed point theorem in the space of regulated functions endowed with weak, respectively strong topologies. Some examples of special cases covered by our existence results have been included.

Magnetic and nonmagnetic doping dependence of the conducting surface states inSmB6

Kondo insulator $\mathrm{Sm}{\mathrm{B}}_{6}$ has attracted attention because it can realize new topological phenomena driven by the interplay between strong correlation effect and topology. However, its topological nature is still under debate. To examine the topological aspect, we demonstrate the nonmagnetic La and magnetic Ce doping dependence of the resistance of $\mathrm{Sm}{\mathrm{B}}_{6}$. Moreover, the resistance ratios of different thicknesses are analyzed to confirm the surface contribution. Lightly doped La samples show a purely conducting surface region at low temperature, whereas the lightly doped Ce samples do not have any conducting region at low temperature. Furthermore, based on the analysis of the electrical transport data of $\mathrm{S}{\mathrm{m}}_{1\ensuremath{-}x}\mathrm{L}{\mathrm{a}}_{x}{\mathrm{B}}_{6}$ $(0.0\ensuremath{\le}x\ensuremath{\le}1.0)$, an electronic phase diagram was found, composed of four regions: region I $(0.0\ensuremath{\le}x\ensuremath{\le}0.06)$, II $(0.1\ensuremath{\le}x\ensuremath{\le}0.15)$, III $(x\ensuremath{\approx}0.2)$, and IV $(0.25\ensuremath{\le}x\ensuremath{\le}1.0)$. Region I is characterized by the presence of conducting surface states, region II is characterized by the insulating phase due to the $d\ensuremath{-}f$ hybridization gap without the conducting surface state, region III is characterized by the disappearance of the $d\ensuremath{-}f$ hybridization gap and the existence of valence fluctuation, and region IV is a typical metallic state.

Remarks on Measurability of Operator-Valued Functions

The aim of this paper is to consider operator-valued functions that can be approximated in the strong and weak operator topology by simple functions. We relate these notions with the classical formulations of measurability and provide conditions for their coincidence. A number of examples and counterexamples are exhibited.

Strong topologies for spaces of smooth maps with infinite-dimensional target

In this article we study two “strong” topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modelled on a locally convex space. Namely, we construct Whitney type topologies for these spaces and a certain refinement corresponding to Michor’s FD-topology. Then we establish the continuity of certain mappings between spaces of smooth mappings, e.g. the continuity of the joint composition map. As a first application we prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group (with respect to these topologies).For the reader’s convenience the article includes also a proof of the folklore fact that the Whitney topologies defined via jet bundles coincide with the ones defined via local charts.

On the optimal estimation of probability measures in weak and strong topologies

Given random samples drawn i.i.d. from a probability measure $\mathbb{P}$ (defined on say, $\mathbb{R}^d$), it is well-known that the empirical estimator is an optimal estimator of $\mathbb{P}$ in weak topology but not even a consistent estimator of its density (if it exists) in the strong topology (induced by the total variation distance). On the other hand, various popular density estimators such as kernel and wavelet density estimators are optimal in the strong topology in the sense of achieving the minimax rate over all estimators for a Sobolev ball of densities. Recently, it has been shown in a series of papers by Gin\'{e} and Nickl that these density estimators on $\mathbb{R}$ that are optimal in strong topology are also optimal in $\|\cdot\|_{\mathcal{F}}$ for certain choices of $\mathcal{F}$ such that $\|\cdot\|_{\mathcal{F}}$ metrizes the weak topology, where $\|\mathbb{P}\|_{\mathcal{F}}:=\sup\{\int f\,\mathrm{d}\mathbb{P}: f\in\mathcal{F}\}$. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing $\mathcal{F}$ to be a unit ball in a reproducing kernel Hilbert space (say $\mathcal{F}_H$ defined over $\mathbb{R}^d$), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that $\|\cdot\|_{\mathcal{F}_H}$ metrizes the weak topology and the kernel density estimator (with $L^1$ optimal bandwidth) estimates $\mathbb{P}$ at dimension independent optimal rate of $n^{-1/2}$ in $\|\cdot\|_{\mathcal{F}_H}$ along with providing a uniform central limit theorem for the kernel density estimator.

Topological properties of spaces of projective unitary representations

<div data-canvas-width="10.602647058244418">Sea G un grupo de Lie compacto y conexo y PU(H) el grupo de operadores proyectivos e unitarios en un espacio de Hilbert separable e infinito dimensional H, provisto de la topología fuerte de operadores. Estudiamos el espacio homst(G,PU(H)) de homomorfismos continuos desde G a PU (H) que son estables, es decir homomorfismos cuyas representaciones inducidas contienen cada representación irreducible un número infinito de veces. Demostramos que las componentes conexas del espacio homst(G,PU(H)) están parametrizadas por las clases de isomorfía de extensiones centrales de</div><div>G por el grupo S1, y que cada componente conexa tiene por grupo fundamental al grupo hom(G,S1) y sus grupos de homotopía superiores son triviales. Estudiamos la aplicación conjugación PU(H)→homst(G,PU(H)),F→FαF−1 , demostramos que no tiene secciones locales y demostramos que para cualquier aplicación continua B→homst(G,PU(H)) con B paracompacto de dimensión paracompacta finita, los levantamientos locales a PU (H) sí existen. © Acad. Colomb. Cienc. Ex. Fis. Nat. 2016.</div>

Conditional Lp-spaces and the duality of modules over f-algebras

Motivated by dynamic asset pricing, we extend the dual pairs' theory of Dieudonné (1942) and Mackey (1945) to pairs of modules over a Dedekind complete f-algebra with multiplicative unit. The main tools are:•a Hahn–Banach Theorem for modules of this kind;•a topology on the f-algebra that has the special feature of coinciding with the norm topology when the algebra is a Banach algebra and with the strong order topology of Filipovic, Kupper, and Vogelpoth (2009), when the algebra of all random variables on a probability space (Ω,G,P) is considered. As a leading example, we study in some detail the duality of conditional Lp-spaces.

The strong ultrafilter topology on spaces of ideals

The patch/constructible refinement of the Zariski topology on the prime spectrum of a commutative ring is well known and well studied. Recently, Fontana and Loper gave an equivalent definition of this topology using ultrafilters. In this note we distinguish between two different types of ultrafilter convergence and use them to define two new topologies on the prime spectrum of a ring. We study various properties of these topologies. As applications we use the ultrafilters to classify all the compact subsets of a spectral space in the Zariski topology and we classify Grothendieck's retrocompact spaces again using ultrafilters.

A van der Corput-type lemma for power bounded operators

We prove a van der Corput-type lemma for power bounded Hilbert space operators. As a corollary we show that \(N^{-1}\sum _{n=1}^N T^{p(n)}\) converges in the strong operator topology for all power bounded Hilbert space operators T and all polynomials p satisfying \(p(\mathbb {N}_0)\subset \mathbb {N}_0\). This generalizes known results for Hilbert space contractions. Similar results are true also for bounded strongly continuous semigroups of operators.

On Douady–Earle extension and the contractibility of the VMO-Teichmüller space

Douady–Earle extension provides a quasiconformal extension of a quasisymmetric homeomorphism to the unit disk and induces a continuous self-map σ of Beltrami coefficients. In this note, we show that σ is also continuous (under a stronger topology) on those Beltrami coefficients which induce vanishing Carleson measures and thus project to the VMO-Teichmüller space. An immediate consequence is that the VMO-Teichmüller space is contractible.

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

Abstract The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

Continuity of the sequential product of sequential quantum effect algebras

In order to study quantum measurement theory, sequential product defined by A∘B = A1/2BA1/2 for any two quantum effects A, B has been introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A∘B = A1/2BA1/2 with respect to other important topologies, such as norm topology, weak operator topology, order topology, and interval topology.

An Agler-type model theorem forC0-semigroups of Hilbert space contractions

We investigate suitable conditions for a C0-semigroup (T(t))t≥0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup (S∗γ(t))t≥0 to an invariant subspace of the standard weighted Bergman space Aγ−2(C+,K). It turns out that (T(t))t≥0 admits a model by (S∗γ(t))t≥0 if and only if its cogenerator is γ-hypercontractive and limt→0T(t)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ-hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization. (Less)

Strong convergence in the weighted setting of operator-valued Fourier series defined by the Marcinkiewicz multipliers

Suppose that 1<p<∞ and let w be a bilateral weight sequence satisfying the discrete Muckenhoupt Ap weight condition. We show that every Marcinkiewicz multiplier ψ:T→C has an associated operator-valued Fourier series which serves as an analogue in B(ℓp(w)) of the usual Fourier series of ψ, and this operator-valued Fourier series is everywhere convergent in the strong operator topology. In particular, we deduce that the partial sums of the usual Fourier series of ψ are uniformly bounded in the Banach algebra of Fourier multipliers for ℓp(w). These results transfer to the framework of invertible, modulus mean-bounded operators acting on Lp spaces of sigma-finite measures.

Non-commutative functional calculus in finite type I von Neumann algebras

A certain class of matrix-valued Borel matrix functions is introduced. It is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f1, f2,… of functions that converge pointwise to 0 transform into sequences f1[T], f2[T],… of operators in T that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T that acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all functions from the aforementioned class. Some conclusions are drawn concerning so-called operator-spectra of such operators and a new variation of the spectral theorem is formulated.

Non-commutative functional calculus in finite type I von Neumann algebras

A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.

Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator.

Operator means of probability measures and generalized Karcher equations

In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a large part of Kubo–Ando theory to arbitrary many variables, in fact, to the case of probability measures with bounded support on the cone of positive definite operators. This framework characterizes each operator mean extrinsically as unique solutions of generalized Karcher equations which are obtained by exchanging the matrix logarithm function in the Karcher equation to arbitrary operator monotone functions over the positive real half-line. If the underlying Hilbert space is finite dimensional, then these generalized Karcher equations are Riemannian gradients of convex combinations of strictly geodesically convex log-determinant divergence functions, hence these new means are their global minimizers, in analogy to the case of the Karcher mean as pointed out. Our framework is based on fundamental contraction results with respect to the Thompson metric, which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above.