Pseudo-Differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups

Pseudodifferential operators on ultra-modulation spaces

The structure of the global “quantum phase space” is analyzed for the harmonic oscillator perturbed by a monochromatic wave in the limit when the perturbation amplitude is small. Usually, the phenomenon of quantum resonance was studied in nondegenerate [G. M. Zaslavsky, Chaos in Dynamic Systems (Harwood Academic, Chur, 1985)] and degenerate [Demikhovskii, Kamenev, and Luna-Acosta, Phys. Rev. E 52, 3351 (1995)] classically chaotic systems only in the particular regions of the classical phase space, such as the center of the resonance or near the separatrix. The system under consideration is degenerate, and even an infinitely small perturbation generates in the classical phase space an infinite number of the resonant cells which are arranged in the pattern with the axial symmetry of the order 2μ (where μ is the resonance number). We show analytically that the Husimi functions of all Floquet states (the quantum phase space) have the same symmetry as the classical phase space. This correspondence is demonstrated numerically for the Husimi functions of the Floquet states corresponding to the motion near the elliptic stable points (centers of the classical resonance cells). The derived results are valid in the resonance approximation when the perturbation amplitude is small enough, and the stochastic layers in the classical phase space are exponentially thin. The developed approach can be used for studying a global symmetry of more complicated quantum systems with chaotic behavior.

Pseudo-differential operators, Wigner transform and Weyl transform on the affine Poincaré group

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

Kryvokulska N.M. TOOLS FOR IMPROVING THE MANAGEMENT SYSTEM OF ENTERPRISE ENVIRONMENTAL TRANSFORMATION IN THE CONTEXT OF SUSTAINABLE DEVELOPMENT Purpose. The aim of the article is to substantiate the integrated approach and tools for improving the management system of the environmental transformation of the enterprise, taking into account the modern challenges of sustainable development, aimed at increasing its competitiveness and investment attractiveness. Methodology of research. The theoretical and methodological basis of the article is formed by the principles of modern economic science as well as scientific works of domestic and international researchers in the field of sustainable development management and environmental transformation. The methodological basis of the study consists of a systematic approach and systematic analysis, which made it possible to consider the ecological transformation of an enterprise as a multi-component management system; methods of abstraction and logical generalization – to form the author's model of motivational support for the greening of activities; comparative analysis – to compare linear and circular models of economic management. The study employs systemic, structural, and functional analysis to substantiate the management system of enterprise environmental transformation, along with statistical methods to generalize and interpret available quantitative assessments and trends. Findings. From the perspective of a goal-oriented approach, the management system of enterprise environmental transformation is substantiated as an integrated, multi-component, and dynamic managerial system aimed at achieving environmental, economic, and social goals of sustainable development. The key elements of this system are identified, including target, subject–object, institutional–regulatory, organizational–managerial, instrumental, and control–adaptive components, and their functional roles in ensuring the effectiveness of environmental transformation are revealed. It is demonstrated that within environmental transformation, the circular economy should be considered not merely as a set of technological solutions but as a managerial paradigm integrated into the enterprise strategic management system. Originality: for the first time, a multidimensional system of effects of the ecological transformation of an enterprise (ecological, economic, resource, investment, managerial, informational, social and cultural, reputational, strategic and marketing) has been systematized and substantiated, which allows us to consider ecological transformation not as a costly activity, but as a source of creating long-term added value and competitive advantages; It was improved: - methodological approach to the formation of a system for managing the environmental transformation of an enterprise by isolating its structural elements and revealing their functional role in achieving sustainable development goals, which, unlike existing approaches that mainly focus on environmental management as a tool for complying with regulatory requirements or individual management functions, ensures the comprehensive integration of environmental, economic and social goals into a single multi-component and dynamic management system focused on the formation of long-term ecological, economic, managerial and strategic effects; - an approach to substantiation the effectiveness of an enterprise's environmental transformation by establishing a cause-and-effect relationship between the integration of environmental transformation management tools and the formation of a complex ecological, economic and managerial effect, which, unlike existing approaches that mainly record individual results of environmental measures or consider them within the framework of regulatory compliance, allows us to explain the mechanism of the influence of management tools on increasing resource efficiency, reducing environmental risks, increasing investment attractiveness and strengthening the market position of the enterprise. Practical value. The tools proposed in the article for improving the management system of the enterprise's ecological transformation in the context of sustainable development can be used by enterprises of the real sector of the economy as a tool for strategic planning of environmental changes, formation of ESG-oriented management decisions and justification of investments in circular technologies. Their practical implementation creates the basis for increasing resource efficiency, reducing environmental risks, adapting to growing institutional and regulatory requirements, as well as for strengthening the competitive position of the enterprise in domestic and foreign markets in the long term. Key words: sustainable development; management system; enterprise environmental transformation; corporate governance; circular economy; motivational mechanism; investment attractiveness; competitiveness.

The pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ℝ+, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ℝ+ ∋ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.

A finitely generated subgroup${\rm\Gamma}$of a real Lie group$G$is said to be Diophantine if there is${\it\beta}>0$such that non-trivial elements in the word ball$B_{{\rm\Gamma}}(n)$centered at$1\in {\rm\Gamma}$never approach the identity of$G$closer than$|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group$G$is said to be Diophantine if for every$k\geqslant 1$a random$k$-tuple in$G$generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most$5$, or derived length at most$2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

We consider pseudo-differential operators and Gabor frames on locally compact commutative Hausdorff groups. Let [Formula: see text] be an open subgroup of a locally compact commutative Hausdorff group [Formula: see text] and [Formula: see text] be a topological isomorphism, then we define the cross [Formula: see text]-ambiguity function [Formula: see text] and cross [Formula: see text]-Wigner function [Formula: see text] and a pseudo-differentiable operator [Formula: see text] with the symbol [Formula: see text]. Employing properties of [Formula: see text], we establish characteristics of pseudo-differentiable operators. Also, assume that [Formula: see text] and [Formula: see text], then the coefficient operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text]; and similarly, the adjoint operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text].

This paper is the second part of a two-paper series whose aim is to give a detailed description of Connes’ pseudodifferential calculus on noncommutative [Formula: see text]-tori, [Formula: see text]. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension [Formula: see text]. This includes the main results mentioned in [2, 5, 11]. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order [Formula: see text].

This paper is devoted to the study of global controllability of linear systems on higher dimensional Heisenberg groups . Necessary conditions and sufficient ones are provided. They are compared to the known necessary and sufficient conditions on stated in Dath and Jouan [(2016). Controllability of linear systems on low dimensional nilpotent and solvable Lie groups. Journal of Dynamical and Control Systems, 22(2), 207–225]. The notion of decoupled systems is introduced in order to get examples and counter-examples, and more precise controllability criteria are stated for them.

Mapping properties of operator-valued pseudo-differential operators

This book develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. The first chapter studies classes of pseudodifferential operators whose symbols have a limited degree of regularity; the second chapter shows how paradifferential operators yield sharp estimates on the action of various nonlinear operators on function spaces. The third chapter applies this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. The last chapter studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients. Michael Taylor is the author of several well-known books on topics in PDEs and pseudodifferential operators. His Noncommutative Harmonic Analysis, Volume 22 in the Mathematical Surveys and Monographs series published by the AMS, is a good introduction to the use of Lie groups in linear analysis and PDEs. The present book, Tools for PDE, is suitable as a text for advanced graduate students preparing to concentrate in PDE and/or harmonic analysis.

We study unitary representations associated to cocycles of measurable dynamical systems. Our main result establishes conditions on a cocycle, ensuring that ergodicity of the dynamical system under consideration is equivalent to irreducibility of its corresponding unitary representation. This general result is applied to some representations of finite-dimensional nilpotent Lie groups and to some representations of infinite-dimensional Heisenberg groups.

We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach. The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems. We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position. Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space. We present analytical solutions in L2(ℝ) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.

We consider questions related to the following quantization scheme: a classical variable f: Ω → ℝ on a phase space Ω is associated with a unique semispectral measure Ef, such that the kth moment operator of Ef is required to coincide with the operator integral L(fk, E) of fk with respect to a certain fixed phase space semispectral measure E. Mainly, we take the phase space Ω to be a locally compact unimodular group. In the concrete case where Ω = ℝ2 and E is a translation covariant semispectral measure, we determine explicitly the relevant operators L(fk, E) for certain variables f. In addition, we consider the question under what conditions a positive operator measure is projection valued.