Composition Operators Research Articles

Difference of composition operators on the weighted Bergman spaces over the half-plane

Difference of composition operators on the weighted Bergman spaces over the half-plane

Composition operators and generalized primes

We study composition operators on the Hardy space H 2 \mathcal {H}^2 of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on H 2 \mathcal {H}^2 . To do that we extend our notions to a Hardy space H Λ 2 \mathcal {H}_{\Lambda }^2 of generalized Dirichlet series, induced in a natural way by a sequence of Beurling’s primes.

SMOTE-CD: SMOTE for compositional data.

Compositional data are a special kind of data, represented as a proportion carrying relative information. Although this type of data is widely spread, no solution exists to deal with the cases where the classes are not well balanced. After describing compositional data imbalance, this paper proposes an adaptation of the original Synthetic Minority Oversampling TEchnique (SMOTE) to deal with compositional data imbalance. The new approach, called SMOTE for Compositional Data (SMOTE-CD), generates synthetic examples by computing a linear combination of selected existing data points, using compositional data operations. The performance of the SMOTE-CD is tested with three different regressors (Gradient Boosting tree, Neural Networks, Dirichlet regressor) applied to two real datasets and to synthetic generated data, and the performance is evaluated using accuracy, cross-entropy, F1-score, R2 score and RMSE. The results show improvements across all metrics, but the impact of oversampling on performance varies depending on the model and the data. In some cases, oversampling may lead to a decrease in performance for the majority class. However, for the real data, the best performance across all models is achieved when oversampling is used. Notably, the F1-score is consistently increased with oversampling. Unlike the original technique, the performance is not improved when combining oversampling of the minority classes and undersampling of the majority class. The Python package smote-cd implements the method and is available online.

C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches

Let [Formula: see text] be a compact metric space and let [Formula: see text] be continuous. We study a [Formula: see text]-algebra [Formula: see text] generated by all multiplication operators by continuous functions on [Formula: see text] and a composition operator [Formula: see text] induced by [Formula: see text] on a certain [Formula: see text] space. Let [Formula: see text] be a system of proper contractions on [Formula: see text]. Suppose that [Formula: see text] are inverse branches of [Formula: see text] and [Formula: see text] is self-similar. We consider the Hutchinson measure [Formula: see text] of [Formula: see text] and the [Formula: see text] space [Formula: see text]. Then we show that the [Formula: see text]-algebra [Formula: see text] is isomorphic to the [Formula: see text]-algebra [Formula: see text] associated with [Formula: see text] under some conditions.

Composition operators on Dirichlet spaces over the half-plane

ABSTRACT As a continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which have a dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane.

The Compound Forest–Medicinal Plant System Enhances Soil Carbon Utilization

The sensible use of forest resources and the sound management of forests have become increasingly important throughout the years. In keeping with the trend, a composite forestry operation model has emerged. Traditional Chinese culture and forest management are particularly intertwined in China. Thus, use of the forest–medicine compound management model is recommended. The majority of research on the management of forest–medicine compounds has focused on how to grow more effective medicinal plants, ignoring the effects of the chemicals used on the soil environment, particularly the soil micro-environment. A forest–medicine system was established in South China to investigate the impacts of planting Aspidistra elatior on the variety of rhizospheric microorganisms and their ability to use carbon sources. In the plots with or without A. elatior, three dominant plants (Castanopsis hystrix, Psychotria rubra, and Ficus hirta) grew soil rhizosphere microbes, which were analyzed using Biolog EcoPlates. The study found that planting medicinal plants in the understory improved the soil’s nutritional content, increased the inter-root microbial communities of other medicinal plants, and enhanced the microbes’ ability to use soil carbon sources. The forest–medicine complex model, which rationalizes the use of forest clearings and generates economic and ecological benefits, can significantly increase the quantity of dominant microorganisms and enhance the enrichment of other species, resulting in a positive impact on the soil environment. These findings suggest that the forest–medicine compound management model can improve the use of soil carbon sources throughout the forest system.

The inverse Mellin transform via analytic continuation

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors.

Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball

Unitary weighted composition operators on Bergman-Besov and Hardy Hilbert spaces on the ball

Composition operators between Bergman spaces on infinite type convex domains in ℂ2

ABSTRACT We provide characterizations of boundedness and compactness for composition operators between Bergman spaces of smoothly bounded convex domains in C 2 admitting the so-called geometric F-type. Such geometric type contains all domains of finite type and many cases of infinite type in the sense of Range.

Stationary phase method, powers of functions, and applications to functional analysis

The utility of the (weighted) van der Corput inequalities or of the stationary phase method is illustrated with various examples borrowed from: differentiability issues (Riemann’s function and related); functional analysis on Banach spaces or algebras of analytic functions (composition operators); and local Banach space geometry (Schäffer’s problem).

Gradient-flowed order parameter for spontaneous gauge symmetry breaking

The gauge-invariant two-point function of the Higgs field at the same spacetime point can make a natural gauge-invariant order parameter for spontaneous gauge symmetry breaking. However, this composite operator is ultraviolet divergent and is not well defined. We propose using a gradient flow to cure the divergence from putting the fields at the same spacetime point. As a first step, we compute it for the Abelian Higgs model with a positive mass squared at the one-loop order in the continuum theory using the saddle-point method to estimate the finite part. The order parameter consistently goes to zero in the infrared limit of the infinite flow time.

Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Abstract The gradient flow exact renormalization group (GFERG) is an idea that incorporates gauge-invariant gradient flows into the formalism of the exact renormalization group (ERG). GFERG introduces a Wilson action with a cutoff while keeping the vector gauge invariance manifestly. The details of the formalism are still to be worked out. In this paper, we apply GFERG to construct the Wilson action of massless Dirac fermions under the background chiral gauge fields. By formulating the chiral anomaly as a “composite operator”, we make the scale invariance of the anomaly manifest. We argue that the same result extends to QCD.

Composition Operators on Sobolev Spaces and Q-Homeomorphisms

Composition Operators on Sobolev Spaces and Q-Homeomorphisms

Schramm spaces and composition operators

Schramm spaces and composition operators

Weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces

Abstract A family of Zygmund-type spaces, called Dirichlet–Zygmund-type spaces, are introduced. The boundedness, compactness and the essential norm of weighted composition operators from Dirichlet–Zygmund-type spaces into Stević-type spaces are also investigated in this paper.

Polynomial differentiation composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball

Polynomial differentiation composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball

Revealing topology in metals using experimental protocols inspired by K-theory

Topological metals are conducting materials with gapless band structures and nontrivial edge-localized resonances. Their discovery has proven elusive because traditional topological classification methods require band gaps to define topological robustness. Inspired by recent theoretical developments that leverage techniques from the field of C∗-algebras to identify topological metals, here, we directly observe topological phenomena in gapless acoustic crystals and realize a general experimental technique to demonstrate their topology. Specifically, we not only observe robust boundary-localized states in a topological acoustic metal, but also re-interpret a composite operator—mathematically derived from the K-theory of the problem—as a new Hamiltonian whose physical implementation allows us to directly observe a topological spectral flow and measure the topological invariants. Our observations and experimental protocols may offer insights for discovering topological behaviour across a wide array of artificial and natural materials that lack bulk band gaps.

Correction to: Semigroups of composition operators on Hardy spaces of the half-plane

Correction to: Semigroups of composition operators on Hardy spaces of the half-plane

Composition operators on weighted Hilbert spaces of Dirichlet series

AbstractWe study composition operators of characteristic zero on weighted Hilbert spaces of Dirichlet series. For this purpose, we demonstrate the existence of weighted mean counting functions associated with the Dirichlet series symbol, and provide a corresponding change of variables formula for the composition operator. This leads to natural necessary conditions for the boundedness and compactness. For Bergman‐type spaces, we are able to show that the compactness condition is also sufficient, by employing a Schwarz‐type lemma for Dirichlet series.

Boundedness of composition operators on higher order Besov spaces in one dimension

This paper aims to characterize boundedness of composition operators on Besov spaces $$B^s_{p,q}$$ of higher order derivatives $$s>1+1/p$$ on the one-dimensional Euclidean space. In contrast to the lower order case $$0<s<1$$ , there were a few results on the boundedness of composition operators for $$s>1$$ . We prove a relation between the composition operators and pointwise multipliers of Besov spaces, and effectively use the characterizations of the pointwise multipliers. As a result, we obtain necessary and sufficient conditions for the boundedness of composition operators for general p, q, and s such that $$1<p\le \infty $$ , $$0<q\le \infty $$ , and $$s>1+1/p$$ . In this paper, we treat, as a map that induces the composition operator, not only a homeomorphism on the real line but also a continuous map whose number of elements of inverse images at any one point is bounded above. We also show a similar characterization of the boundedness of composition operators on Sobolev spaces.