arXiv subjects

John Ryan

Publications and source records attributed to John Ryan.

At least 19 recordsLinked to original sources

arXiv2024

Bosonic Laplacians in higher spin Clifford analysis

In this article, we firstly introduce higher spin Clifford analysis, which are considered as generalizations of classical Clifford analysis by considering functions taking values in irreducible representations of the spin group. Then, we introduce a type of second order conformally invariant differential operators, named as bosonic Laplacians, in the higher spin Clifford analysis. In particular, we will show their close connections to classical Maxwell equations. At the end, we will introduce a new perspective to define bosonic Laplacians, which simplifies the connection between bosonic Laplacians and Rarita-Schwinger type operators obtained before. Moreover, a matrix type Rarita-Schwinger operator is obtained and some results related to this new first order matrix type operator are provided.

math-ph↗

arXiv2020

Polynomial Null Solutions to Bosonic Laplacians, Bosonic Bergman and Hardy Spaces

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

math.CV↗

arXiv2020

Green's Formulas and Poisson's Equation for Bosonic Laplacians

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given $L^2$ inner product on certain compact supported function spaces.

math.AP↗

arXiv2020

Boundary value problems in Euclidean space for Bosonic Laplacians

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.

math-ph↗

arXiv2016

Some Properties of the Higher Spin Laplace Operator

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformally invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.

math.CV↗

arXiv2016

Harassment detection: a benchmark on the #HackHarassment dataset

Online harassment has been a problem to a greater or lesser extent since the early days of the internet. Previous work has applied anti-spam techniques like machine-learning based text classification (Reynolds, 2011) to detecting harassing messages. However, existing public datasets are limited in size, with labels of varying quality. The #HackHarassment initiative (an alliance of 1 tech companies and NGOs devoted to fighting bullying on the internet) has begun to address this issue by creating a new dataset superior to its predecssors in terms of both size and quality. As we (#HackHarassment) complete further rounds of labelling, later iterations of this dataset will increase the available samples by at least an order of magnitude, enabling corresponding improvements in the quality of machine learning models for harassment detection. In this paper, we introduce the first models built on the #HackHarassment dataset v1.0 (a new open dataset, which we are delighted to share with any interested researcherss) as a benchmark for future research.

cs.CL↗

arXiv2016

Spherical $\Pi$-type Operators in Clifford Analysis and Applications

The $\Pi$-operator (Ahlfors-Beurling transform) plays an important role in solving the Beltrami equation. In this paper we define two $\Pi$-operators on the n-sphere. The first spherical $\Pi$-operator is shown to be an $L^2$ isometry up to isomorphism. To improve this, with the help of the spectrum of the spherical Dirac operator, the second spherical $\Pi$ operator is constructed as an isometric $L^2$ operator over the sphere. Some analogous properties for both $\Pi$-operators are also developed. We also study the applications of both spherical $\Pi$-operators to the solution of the spherical Beltrami equations.

math.CV↗

arXiv2016

Construction of Arbitrary Order Conformally Invariant Operators in Higher Spin Spaces

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slov\'{a}k has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for arbitrary order conformally invariant differential operators, where by conformally invariant we mean equivariant with respect to the conformal group of $\mathbb{S}^m$ acting in Euclidean space $\mathbb{R}^m$. We name these the fermionic operators when the order is odd and bosonic operators when the order is even. Our approach explicitly uses convolution type operators to construct conformally invariant differential operators. These convolution type operators are examples of Knapp-Stein operators and they can be considered as the inverses of the corresponding differential operators. Intertwining operators of these convolution type operators are provided and intertwining operators of differential operators follow immediately. This reveals that our convolution type operators and differential operators are all conformally invariant. This also gives us a class of conformally invariant convolution type operators in higher spin spaces. Their inverses, when they exist, are conformally invariant pseudo-differential operators. Further we use Stein Weiss gradient operators and representation theory for the Spin group to naturally motivate the Rarita-Schwinger operators.

math.DG↗

arXiv2015

Higher Order Fermionic and Bosonic Operators

This paper studies a particular class of higher order conformally invariant dif- ferential operators and related integral operators acting on functions taking values in particular finite dimensional irreducible representations of the Spin group. The differential operators can be seen as a generalization to higher spin spaces of kth- powers of the Euclidean Dirac operator. To construct these operators, we use the framework of higher spin theory in Clifford analysis, in which irreducible represen- tations of the Spin group are realized as polynomial spaces satisfying a particular system of differential equations. As a consequence, these operators act on functions taking values in the space of homogeneous harmonic or monogenic polynomials de- pending on the order. Moreover, we classify these operators in analogy with the quantization of angular momentum in quantum mechanics to unify the terminology used in studying higher order higher spin conformally invariant operators: for integer and half-integer spin, these are respectively bosonic and fermionic operators. Fundamental solutions and their conformal invariance are presented here.

math.DG↗

arXiv2015

Higher order fermionic and bosonic operators on cylinders and Hopf manifolds

Higher order higher spin operators are generalizations of $kth$-powers of the Dirac operator. In this paper, we study higher order higher spin operators defined on some conformally flat manifolds, namely cylinders and Hopf manifolds. We will also construct the kernels of these operators on these manifolds.

math.DG↗

arXiv2015

On Some Conformally Invariant Operators in Euclidean Space

The aim of this paper is to correct a mistake in earlier work on the conformal invariance of Rarita-Schwinger operators and use the method of correction to develop properties of some conformally invariant operators in the Rarita-Schwinger setting. We also study properties of some other Rarita-Schwinger type operators, for instance, twistor operators and dual twistor operators. This work is also intended as an attempt to motivate the study of Rarita-Schwinger operators via some representation theory. This calls for a review of earlier work by Stein and Weiss.

math.CV↗

arXiv2012

Some Rarita-Schwinger Type Operators

In this paper we study a generalization of the classical Rarita-Schwinger type operators and construct their fundamental solutions. We give some basic integral formulas related to these operators. We also establish that the projection operators appearing in the Rarita-Schwinger operators and the Rarita-Schwinger equations are conformally invariant. We further obtain the intertwining operators for other operators related to the Rarita-Schwinger operators under actions of the conformal group.

math.CV↗

arXiv2012

Rarita-Schwinger Type Operators on Spheres and Real Projective Space

In this paper we deal with Rarita-Schwinger type operators on spheres and real projective space. First we define the spherical Rarita-Schwinger type operators and construct their fundamental solutions. Then we establish that the projection operators appearing in the spherical Rarita-Schwinger type operators and the spherical Rarita-Schwinger type equations are conformally invariant under the Cayley transformation. Further, we obtain some basic integral formulas related to the spherical Rarita-Schwinger type operators. Second, we define the Rarita-Schwinger type operators on the real projective space and construct their kernels and Cauchy integral formulas.

math.CV↗

arXiv2011

Rarita-Schwinger Type operators on Cylinders

Here we define Rarita-Schwinger operators on cylinders and construct their fundamental solutions. Further the fundamental solutions to the cylindrical Rarita-Schwinger type operators are achieved by applying translation groups. In turn, a Borel-Pompeiu Formula, Cauchy Integral Formula and a Cauchy Transform are presented for the cylinders. Moreover we show a construction of a number of conformally inequivalent spinor bundles on these cylinders. Again we construct Rarita-Schwinger operators and their fundamental solutions in this setting. Finally we study the remaining Rarita-Schwinger type operators on cylinders.

math.AP↗

arXiv2011

Some Operators Associated to Rarita-Schwinger Type Operators

In this paper we study some operators associated to the Rarita-Schwinger operators. They arise from the difference between the Dirac operator and the Rarita-Schwinger operators. These operators are called remaining operators. They are based on the Dirac operator and projection operators $I-P_k.$ The fundamental solutions of these operators are harmonic polynomials, homogeneous of degree $k$. First we study the remaining operators and their representation theory in Euclidean space. Second, we can extend the remaining operators in Euclidean space to the sphere under the Cayley transformation.

math.CV↗

arXiv2011

A new class of hypercomplex analytic cusp forms

In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator $D Δ^{k/2}$ for some even $k \in {\mathbb{Z}}$. They will be called $k$-holomorphic Cliffordian automorphic forms. $k$-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions have also the property that they are solutions to the homogeneous and inhomogeneous Weinstein equation. This function class includes the set of $k$-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing $k$-hypermonogenic cusp forms for $k \neq 0$, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms we turn to the construction of Poincaré series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of $k$-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of $k$-hypermonogenic Eisenstein series and of $k$-holomorphic Cliffordian cusp forms.

math.NT↗

arXiv2008

$p$-Dirac Operators

We introduce non-linear Dirac operators in $\mathbb{R}^{n}$ associated to the $p$-harmonic equation and we extend to other contexts including spin manifolds and the sphere.

math.CV↗

arXiv2007

Some Sharp L^2 Inequalities for Dirac Type Operators

We use the spectra of Dirac type operators on the sphere $S^{n}$ to produce sharp $L^{2}$ inequalities on the sphere. These operators include the Dirac operator on $S^{n}$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in $\mathbb{R}^{n}$.

math-ph↗