Class Of Symbols Research Articles

On Compactness of Products of Toeplitz Operators

AbstractWe study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols f and g, we show that $$T_fT_g$$ T f T g is compact if and only if fg vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of fg on the whole polydisc might not imply the compactness of $$T_fT_g$$ T f T g . On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

Pseudo‐differential operators associated with Gyrator transform on Sobolev space

The main aim of this article is to focus on the fundamental properties of the gyrator transform on the Schwartz spaces. A more generalization Hörmander symbol class is introduced and studied the boundedness properties of pseudo‐differential operators associated with the gyrator transform. Integral representation and kernel of the pseudo‐differential operators are derived. It is shown that pseudo‐differential operators satisfies certain norm inequalities on Sobolev space. As an application, we have successfully applied the gyrator transform to solve heat equation and Fredholm integral equation.

Few-Shot Symbol Detection in Engineering Drawings

ABSTRACT Recently, there has been significant interest in digitizing engineering drawings due to their complexity and practical benefits. Symbol digitization, a critical aspect in this field, is challenging as utilizing Deep Learning-based methods to recognize symbols of interest requires a large number of training instances for each class of symbols. Acquiring and annotating sufficient diagrams is difficult due to concerns about confidentiality and availability. The conventional manual annotation process is time-consuming, costly, and prone to human error. Additionally, obtaining an adequate number of samples for rare classes proves to be exceptionally challenging. This paper introduces a few-shot framework to address these challenges. Several experiments with fewer than ten, and sometimes just one, training instance per class using complex engineering drawings from industry sources were carried out. The results suggest that our method not only significantly improves symbol detection performance compared to other state-of-the-art methods but also decreases the necessary number of training instances.

Ap weights and an application to Hankel operators on Fock spaces with variable exponents on [formula omitted]

We characterize Ap weights for p>1 and via extrapolation we characterize boundedness and compactness of Hankel operators between Fock spaces of variable exponent and the Lebesgue spaces of variable exponents. We also give some characterizations of the symbol class which is some BMO-type spaces with variable exponent on the complex plane Cn.

Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} consists on those Dirichlet series convergent in the right half-plane C+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}_+$$\\end{document} and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols Φ:C+→C+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi :{{\\mathbb {C}}}_+\\rightarrow {{\\mathbb {C}}}_+$$\\end{document} giving rise to bounded composition operators CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document} in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} and denote this class by GA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}_{\\mathcal {A}}$$\\end{document}. We also characterise when the operator CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document} is compact in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document}. As a byproduct, we show that the weak compactness is equivalent to the compactness for CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document}. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions {Φt}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Phi _t\\}$$\\end{document} in the class GA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}_{\\mathcal {A}}$$\\end{document} and strongly continuous semigroups of composition operators {Tt}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{T_t\\}$$\\end{document}, Ttf=f∘Φt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_tf=f\\circ \\Phi _t$$\\end{document}, f∈A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\in \\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document}. We conclude providing examples showing the differences between the symbols of bounded composition operators in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} and the Hardy spaces of Dirichlet series Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^p$$\\end{document} and H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^{\\infty }$$\\end{document}.

Tinned: A symbolic library for response theory and high-order derivatives.

A symbolic C++ library-Tinned-has been developed for symbolic differentiation and manipulation in response theory. By recognizing different key building blocks in the density matrix-based (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) and coupled-cluster response theories, we have implemented their corresponding C++ symbolic classes, including but not limited to one- and two-electron operators, exchange-correlation energy and potential, and coupled-cluster operator. Formulas of response theory can be well expressed in terms of the symbolic classes in the library Tinned. Their high-order perturbation-strength derivatives can be straightforwardly computed and extracted afterwards for numerical evaluation. The library Tinned will greatly facilitate the development work of response theory and may lead to a unified framework for response theory at different levels of electronic structure theory.

Weighted estimates for Hermite pseudo-multipliers with rough symbols

We introduce a class of rough symbols for pseudo-multipliers for Hermite expansions and obtain Lp and weighted Lp estimates. These symbols generalise the class of rough symbols introduced by Kenig–Staubach.

Animating Cartographic Meaning: Unveiling the Impact of Pictorial Symbol Motion Speed in Preattentive Processing

The primary objective of this study is to assess how the motion of dynamic point symbols impacts preattentive processing on a map. Specifically, it involves identifying the motion velocity parameters for cartographic animated pictorial symbols that contribute to the preattentive perception of the target symbols. We created five pictorial symbols, each accompanied by a unique animation tailored to convey the meaning associated with each symbol. The animation dynamics of symbols on the administrative map were distributed across arithmetic, logarithmic, and exponential scales. Eye-tracking technology was utilized to explain the user’s visual attention. The key findings reveal that, although movement does not uniformly hold the same designation in cartographic communication, it could guide user attention to identify the value peaks in quantitative map visualization. Motion velocity enhances the salience of animated symbols, making them stand out, not only against static elements but also against other animated distractors. Additionally, motion distributions between symbol classes based on exponential or arithmetic scales were identified as the most successful. Nevertheless, certain types of motion, such as rotational, do not perform well with pictorial symbols, even on the most effective motion distribution scale.

A CLASS OF SYMBOLS THAT INDUCE BOUNDED COMPOSITION OPERATORS FOR DIRICHLET-TYPE SPACES ON THE DISC

Abstract We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $\varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.

Sparks of New Metaphysics and the Limits of Explanatory Abstractions

Abstract Physical reality as an explanatory model is an abstraction of the mind. Every perceptual system is a user interface, like the dashboard of an aeroplane or the desktop of a computer. We do not see or otherwise perceive reality but only interface with reality. The user interface concept is a starting point for a critical dialogue with those epistemic theories that present themselves as veridical and take explanatory abstractions as ontological primitives. At the heart of any scientific model are assumptions about which things exist, how they are related, and how we can know them. Scientific models take our knowledge beyond ordinary experience toward explanatory abstractions. The main problem with veridical models lies in why we cannot express our theories and the explanatory abstractions associated with them other than through classical perceptual interface symbols. This study analyses the limits, possibilities and constraints of explanatory abstractions.

Multilinear Fourier integral operators on modulation spaces

Abstract In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely 𝐒𝐆 {\mathbf{SG}} -class.

Semi-supervised symbol detection for piping and instrumentation drawings

Current computer vision methods for symbol detection in piping and instrumentation diagrams (P&IDs) face limitations due to the manual data annotation resources they require. This paper introduces a versatile two-stage symbol detection pipeline that optimizes efficiency by (1) labeling only data samples with minimal cumulative informational redundancy, (2) restricting annotation to the minimal effective training dataset size, and (3) expanding the training dataset using pseudo-labels. In Stage-1, the method performs generic symbol detection, while Stage-2 focuses on symbol differentiation through metric learning. To enhance robustness and generalizability, the model is trained on a diverse dataset collected from both industry sources and web scraping. The achieved Top-1 accuracy is 85.39%, with a Top-5 accuracy of 95.19% on a test dataset containing 102 symbol classes. These results suggest the potential for a shift from resource-intensive supervised learning approaches to a more efficient semi-supervised paradigm.

Offline handwritten mathematical expression recognition with graph encoder and transformer decoder

Handwritten mathematical expression recognition (HMER) has attracted extensive attention. Despite the significant progress achieved in recent years attributed to the development of deep learning approaches, HMER remains a challenge due to the complex spatial structure and variable writing styles. Encoder–decoder models with attention mechanism, which treats HMER as an image-to-sequence (i.e. LaTeX) generation task, have boosted the accuracy, but suffer from low interpretability in that the symbols are not segmented explicitly. Symbol segmentation is desired for facilitating post-processing and human interaction in real applications. In this paper, we formulate the mathematical expression as a graph and propose a Graph-Encoder-Transformer-Decoder (GETD) approach for HMER. For constructing the graph from input image, candidate symbols are first detected using an object detector and represented as the nodes of a graph, called symbol graph, and the edges of the graph encodes the between-symbol relationship. The spatial information is aggregated in a graph neural network (GNN), and a Transformer-based decoder is used to identify the symbol classes and structure from the graph. Experiments on public datasets demonstrate that our GETD model achieves competitive expression recognition performance while offering good interpretability compared with previous methods.

Lipschitz continuity of spectra of pseudodifferential operators in a weighted Sjöstrand class and Gabor frame bounds

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions.

On the rôle of singular functions in extending the probabilistic symbol to its most general class

The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving Hunt behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle.

Truncation quantization in the edge calculus

Pseudodifferential operators on the half-space associated with classical symbols of order zero without transmission property are shown to belong to the so-called edge algebra.

Emergency Floor Plan Digitization Using Machine Learning.

An increasing number of special-use and high-rise buildings have presented challenges for efficient evacuations, particularly in fire emergencies. At the same time, however, the use of autonomous vehicles within indoor environments has received only limited attention for emergency scenarios. To address these issues, we developed a method that classifies emergency symbols and determines their location on emergency floor plans. The method incorporates color filtering, clustering and object detection techniques to extract walls, which were used in combination to generate clean, digitized plans. By integrating the geometric and semantic data digitized with our method, existing building information modeling (BIM) based evacuation tools can be enhanced, improving their capabilities for path planning and decision making. We collected a dataset of 403 German emergency floor plans and created a synthetic dataset comprising 5000 plans. Both datasets were used to train two distinct faster region-based convolutional neural networks (Faster R-CNNs). The models were evaluated and compared using 83 floor plan images. The results show that the synthetic model outperformed the standard model for rare symbols, correctly identifying symbol classes that were not detected by the standard model. The presented framework offers a valuable tool for digitizing emergency floor plans and enhancing digital evacuation applications.

Positivity results for Weyl’s pseudo-differential calculus on the Wiener space

This paper deals with positivity properties for a pseudo-differential calculus, generalizing Weyl’s classical quantization, and set on an infinite dimensional configuration space, the Wiener space. In this frame, we show that a positive symbol does not, in general, give a positive operator. In order to measure the nonpositivity, we establish a Gårding’s inequality, which holds for the symbol classes at hand. Nevertheless, for symbols with radial aspects, additional assumptions ensure the positivity of the associated operator.

Microlocal analysis for Gelfand–Shilov spaces

We introduce an anisotropic global wave front set of Gelfand–Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand–Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.

A hyperbolic analogue of the Rademacher symbol

One of the most famous results of Dedekind is the transformation law of $$\log \Delta (z)$$ . After a half-century, Rademacher modified Dedekind’s result and introduced an $${{\,\textrm{SL}\,}}_2({\mathbb {Z}})$$ -conjugacy class invariant (integer-valued) function $$\Psi (\gamma )$$ called the Rademacher symbol. Inspired by Ghys’ work on modular knots, Duke–Imamoḡlu–Tóth (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol $$\Psi _\gamma (\sigma )$$ and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.