The solutions to the operator equation $TXS^* -SX^*T^*=A$ in Hilbert $C^*$-modules

Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids

The Li-Yorke chaotic eigen set of an operator consisting of all λ’s such that T- λI is Li-Yorke chaotic. In this paper, the Li-Yorke chaotic eigen set of the direct sum of linear operators is found to be the union of Li- Yorke chaotic sets of the corresponding operators. Also we discuss about the Li-Yorke chaotic eigen set of compact operators, normal operators and self adjoint operators.

Geophysical Inverse Theory and Regularization Problems

Starting from basic principles, we develop a description of the electromagnetic interactions of a molecular gas in terms of a set of correlation functions which we call the molecular-electromagnetic correlation functions (MECF's). First we use the energy eigenfunctions for an isolated molecule of the species of interest to define a set of molecular creation and annihilation operators. We then derive a closed set of operator equations for these molecular creation and annihilation operators and the electromagnetic vector potential. Explicit definitions of the lowest-order MECF's are given in terms of these operators, and it is shown how the operator equations which have been obtained can be used to derive equations of motion for the MECF's. Finally, we illustrate the use of the MECF's in describing physical properties of the molecular gas and the electromagnetic field. Brief indications are given of the application of the MECF formulation to the semiclassical approximation and to the description of quantum emission of radiation, topics which are treated in greater detail in subsequent papers. The basic MECF formulation described here contains three rather mild approximations: (1) Atomic nuclei are treated as elementary particles; (2) nuclei and electrons are treated nonrelativistically; and (3) the effect of molecular collisions with the container walls on the internal molecular state is neglected. Consequently, the physical description contained in the formulation is rather complete; and the MECF results can be used both to provide a sound basis for some aspects of the usual heuristic models, and to ascertain the ways in which those models are incomplete.

In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from a Hilbert $C^{\ast}$-module $\mathcal{H}$ to itself denoted by $End_{\mathcal{A}}^{\ast}(\mathcal{H}) $. We give some properties relating to some constructions of integral $K$-operator frames and to operators preserving integral $K$-operator frame and we establish some new results.

The work on frame theory has undergone a remarkable evolution over the last century. Several related properties have applications on many fields of mathematics, engineering, signal and image processing, informatics, medecine and probability. In order to search for new results related to the role of operators in frame theory using the characterization of the positive elements in a C∗-algebra, we introduce the concept of positive operator frame, L-positive operator frame, ∗-positive operator frame and ∗-L-positive operator frame for the set of all adjointable operators on a Hilbert C∗-module denoted End∗B(H) where L is a positive operator. Also, we give some new properties.

Syntactic theories relate sentence structure to the details of morphemes, inflections, and word order. Semantic theories relate sentences to the details of formal logic and model theory. But many of the most successful programs for information extraction (IE) are based on domain-dependent templates that ignore the details at the center of attention of the major theories of syntax and semantics. This paper shows that it is possible to find a more primitive set of operations, called the canonical formation rules, which underlie both the template-filling operations of IE and the more formal operations of parsers and theorem provers. These rules are first stated in terms of conceptual graphs and then generalized to any knowledge representation, including predicate calculus, frames, and the IE templates. As a result, the template-filling operations of IE become part of a more general set of operations that can be used in various combinations to process knowledge of any kind,KeywordsKnowledge RepresentationParse TreeDouble NegationCharacter StringGeneralization RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove Korovkin-type theorems in the setting of infinite dimensional Hilbert space operators. The classical Korovkin theorem unified several approximation processes. Also, the non-commutative versions of the theorem were obtained in various settings such as Banach algebras, C^{*}-algebras and lattices etc. The Korovkin-type theorem in the context of preconditioning large linear systems with Toeplitz structure can be found in the recent literature. In this article, we obtain a Korovkin-type theorem on B({mathcal {H}}) which generalizes all such results in the recent literature. As an application of this result, we obtain Korovkin-type approximation for Toeplitz operators acting on various function spaces including Bergman space A^{2}({mathbb {D}}), Fock space F^{2}({mathbb {C}}) etc. These results are closely related to the preconditioning problem for operator equations with Toeplitz structure on the unit disk {mathbb {D}} and on the whole complex plane {mathbb {C}}. It is worthwhile to notice that so far such results are available for Toeplitz operators on circle only. This also establishes the role of Korovkin-type approximation techniques on function spaces with certain oscillation property. To address the function theoretic questions using these operator theory tools will be an interesting area of further research.

Let $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

A dynamic multilevel compositional solver (C-ADM) is introduced for fully(and sequentially-) implicit systems arising from compositional displacements in natural porous media. The fully (or sequential) implicit system is first described at a fine-scale resolution, where phases are allowed to consist of different components (based on thermodynamics equilibrium). In addition, heterogeneous capillary functions (defined based on Leverett's J-function) and gravitational effects are both considered, adding significantly to the non-linear complexity of the processes. Given this complex fine-scale system for a heterogeneous reservoir, C-ADM defines a dynamic multilevel system, based on an error criterion, where the grid resolution is defined based on the physics of the process as well as geological complexities and location of wells. Once this multilevel grid is defined, sequences of prolongation and restriction operators are employed to obtain an accurate and efficient multilevel system. CADM allows for a general set of prolongation operators, e.g., constant, bilinear (or polynomial), and multiscale basis functions. The restriction operators, however, are constructed based on a mass-conservative finite-volume formulation at all levels. For several challenging test cases it is shown that C-ADM employs only a small fraction of the fine-scale grids to provide an accurate description of the process. C-ADM casts a promising approach in the application of dynamic grid refinement methods for real-field applications.

For each \(1\le q<p<\infty \) we study the sharp versions of the \(L^{p,\infty }\rightarrow L^q\) estimates for the dyadic maximal operator on \(\mathbb {R}^n\). Actually, this is done in the more general setting of maximal operators associated with a tree-like structure. The proof rests on a novel combination of the Bellman function technique and optimization arguments.

This article comprises two parts. In the first, we study |$L^p$| to |$L^q$| bounds for spectral multipliers and Bochner–Riesz means with negative index in the general setting of abstract self-adjoint operators. In the second, we obtain the uniform Sobolev estimates for constant coefficients higher order elliptic operators |$P(D)-z$| and all |$z\in {\mathbb C}\backslash [0, \infty)$|⁠, which give an extension of the second order results of Kenig–Ruiz–Sogge [42]. Next we use perturbation techniques to prove the uniform Sobolev estimates for Schrödingier operators |$P(D)+V$| with small integrable potentials |$V$|⁠. Finally, we deduce spectral multiplier applications for all these operators, including sharp Bochner–Riesz summability results.

We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [Formula: see text]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

We present a mathematical framework for viscoelastic full waveform inversion (FWI) in vertically transverse isotropic media. FWI can be formulated as the nonlinear inverse problem of identifying parameters in the underlying attenuating anisotropic wave equation given partial wave field measurements (seismograms). From a mathematical point of view, one has to solve an operator equation for the full waveform forward operator, which is the corresponding parameter-to-state map. We give a rigorous definition of this operator, show its Fréchet differentiability, and explicitly characterize the adjoint operator of its Fréchet derivative. Thus, we provide the main ingredients to implement Newton-type/gradient-based regularization schemes for FWI. Our approach can be directly applied to other concepts of anisotropy.

A necessary and sufficient condition for the existence of the general common positive solution to equations A1X = C1, XB2 = C2, A3XA � = C3, A4XA � = C4 for operators between Hilbert C � -modules is established, and an expression for the common positive solution to the equations is derived when the solvability conditions are satisfied. As an application, a new necessary and sufficient condition for the system of adjointable operator equations over Hilbert C � -modules to have a common Re-positive solution is proved. Moreover, an expression of the general Re-positive solution is derived when the consistent conditions are met. The results of this paper extend some known results in the literature.