Search Results (6)

We study compactness for the complex Green operator ${\mathsf{G}}_q$ associated with the Kohn Laplacian ${\square }_b$ on boundaries of pseudoconvex domains in Stein manifolds. Let $\Omega ⋐X$ be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with $C^1$ boundary. For $1\le q\le n-2$, we first prove a compactness theorem under weak potential-theoretic hypotheses: if $b\Omega$ satisfies weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$, then ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$. This extends known $C^{\infty }$ results in ${\mathbb{C}}^n$ to the minimal regularity $C^1$ and to the Stein setting. On locally convexifiable $C^1$ boundaries, we obtain a full characterization: compactness of ${\mathsf{G}}_q$ is equivalent to simultaneous compactness of ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$, to compactness of the $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ in the interior, to weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$, and to the absence of (germs of) complex varieties of dimensions q and $n-1-q$ on $b\Omega$. A key ingredient is an annulus compactness transfer on ${\Omega }_{+}={\Omega }_2\setminus \overline{{\Omega }_1}$, which yields compactness of ${\mathsf{N}}_q^{{\Omega }_{+}}$ from weak $\left(\mathcal{P}\right)$ near each boundary component and allows us to build compact ${\overline{\partial }}_b$-solution operators via jump formulas. Consequences include the following: compact canonical solution operators for ${\overline{\partial }}_b$, compact resolvent for ${\square }_b$ on the orthogonal complement of its harmonic space (hence discrete spectrum and finite-dimensional harmonic forms), equivalence between compactness and standard compactness estimates, closed range and ${\mathsf{L}}^2$ Hodge decompositions, trace-class heat flow, stability under $C^1$ boundary perturbations, vanishing essential norms, Sobolev mapping (and gains under subellipticity), and compactness of Bergman-type commutators when $q=1$. Full article

This paper studies $(p,q)$-harmonic maps by unified geometric analytic methods. First, we deduce variation formulas of the $(p,q)$-energy functional. Second, we analyze weakly conformal and horizontally conformal $(p,q)$-harmonic maps and prove Liouville results for $(p,q)$-harmonic maps under Hessian and asymptotic conditions on complete Riemannian manifolds. Finally, we define the $(p,q)$-$SSU$ manifold and prove that non-constant stable $(p,q)$-harmonic maps do not exist. Full article

The global transition toward renewable energy and the electrification of transportation is imposing unprecedented power quality (PQ) challenges on modern distribution networks, rendering traditional governance models inadequate. To bridge the existing research gap of the lack of a holistic analytical framework, this review first establishes a systematic diagnostic methodology by introducing the “Triadic Governance Objectives–Scenario Matrix (TGO-SM),” which maps core objectives—harmonic suppression, voltage regulation, and three-phase balancing—against the distinct demands of high-penetration photovoltaic (PV), electric vehicle (EV) charging, and energy storage scenarios. Building upon this problem identification framework, the paper then provides a comprehensive review of advanced mitigation technologies, analyzing the performance and application of key ‘unit operations’ such as static synchronous compensators (STATCOMs), solid-state transformers (SSTs), grid-forming (GFM) inverters, and unified power quality conditioners (UPQCs). Subsequently, the review deconstructs the multi-timescale control conflicts inherent in these systems and proposes the forward-looking paradigm of “Distributed Dynamic Collaborative Governance (DDCG).” This future architecture envisions a fully autonomous grid, integrating edge intelligence, digital twins, and blockchain to shift from reactive compensation to predictive governance. Through this structured approach, the research provides a coherent strategy and a crucial theoretical roadmap for navigating the complexities of modern distribution grids and advancing toward a resilient and autonomous future. Full article

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (${\mathcal{l}}^{\mathtt{q}(·)}\left({\mathtt{L}}^{\mathtt{p}(·)}\right)$). Moreover, it was developed using classical Lebesgue space (${\mathtt{L}}_{\mathtt{p}}$) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not. Full article

It is getting more common every day to install inverters that offer several grid support services in parallel. As these services are provided, a simultaneous need arises to know the combined limit of the inverter for those services. In the present paper, operational limits are addressed based on a utility scale for a real inverter scenario with an energy storage system (ESS) (1.5 MW). The paper begins by explaining how active and reactive power limits are calculated, illustrating the PQ maps depending on the converter rated current and voltage. Then, the effect of the negative sequence injection, the phase shift of compensated harmonics and the transformer de-rating are introduced step-by-step. Finally, inverter limits for active filter applications are summarized, to finally estimate active and reactive power limits along with the harmonic current injection for some example cases. The results show that while the phase shift of the injected negative sequence has a significant effect in the available inverter current, this is not the case for the phase shift of injected harmonics. However, the amplitude of the injected negative sequence and harmonics will directly impact the power capabilities of the inverter and therefore, depending on the grid-side voltage, it might be interesting to design an output transformer with a different de-rating factor to increase the power capabilities. Full article

There are entropic functionals galore, but not simple objective measures to distinguish between them. We remedy this situation here by appeal to Born’s proposal, of almost a hundred years ago, that the square modulus of any wave function ${\left|\psi \right|}^2$ be regarded as a probability distribution P. the usefulness of using information measures like Shannon’s in this pure-state context has been highlighted in [Phys. Lett. A1993, 181, 446]. Here we will apply the notion with the purpose of generating a dual functional [ ${\mathcal{F}}_{{\alpha }_R}:\left\{S_Q\right\}⟶R^{+}$ ], which maps entropic functionals onto positive real numbers. In such an endeavor, we use as standard ingredients the coherent states of the harmonic oscillator (CHO), which are unique in the sense of possessing minimum uncertainty. This use is greatly facilitated by the fact that the CHO can be given analytic, compact closed form as shown in [Rev. Mex. Fis. E 2019, 65, 191]. Rewarding insights are to be obtained regarding the comparison between several standard entropic measures. Full article