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Kenneth Chan

Publications and source records attributed to Kenneth Chan.

At least 19 recordsLinked to original sources

Log-ozone groups and centers of polynomial Poisson algebras

In previous work, the authors introduced the ozone group of an associative algebra as the subgroup of automorphisms which fix the center pointwise. The authors studied PI skew polynomial algebras, using the ozone group to understand their centers and to characterize them among graded algebras. In this work, we introduce and study the log-ozone group of a Poisson algebra over a field of positive characteristic. The log-ozone group is then used to characterize polynomial Poisson algebras with skew symmetric structure. We prove that unimodular Poisson algebras with skew symmetric structure have Gorenstein centers. A related result is proved for graded polynomial Poisson algebras of dimension three.

math.RA

On Prior Confidence and Belief Updating

We experimentally investigate how confidence over multiple priors affects belief updating. Theory predicts that the average Bayesian posterior is unaffected by confidence over multiple priors if average priors are the same. We manipulate confidence by varying the time subjects view a black-and-white grid, the proportion representing the prior in a Bernoulli distribution. We find that when subjects view the grid for a longer duration, they have more confidence, under-update more, and place more (less) weight on priors (signals). Overall, confidence over multiple priors matters when it should not, while confidence in prior beliefs does not matter when it should.

econ.GN

Ozone groups of Artin--Schelter regular algebras satisfying a polynomial identity

We study the ozone group of noetherian Artin--Schelter regular algebras satisfying a polynomial identity (or PI for short). The ozone group was shown in previous work by the authors to be an important invariant in the study of PI skew polynomial rings and their centers. In this paper, we show that skew polynomial rings are in fact characterized as those algebras with maximal rank ozone groups. We also classify those with trivial ozone groups, which must necessarily be Calabi--Yau. This class includes most three-dimensional PI Sklyanin algebras. Further examples and applications are given, including applications to the Zariski Cancellation Problem.

math.RA

Structurally Different Neural Network Blocks for the Segmentation of Atrial and Aortic Perivascular Adipose Tissue in Multi-centre CT Angiography Scans

Since the emergence of convolutional neural networks (CNNs) and, later, vision transformers (ViTs), deep learning architectures have predominantly relied on identical block types with varying hyperparameters. We propose a novel block alternation strategy to leverage the complementary strengths of different architectural designs, assembling structurally distinct components similar to Lego blocks. We introduce LegoNet, a deep learning framework that alternates CNN-based and SwinViT-based blocks to enhance feature learning for medical image segmentation. We investigate three variations of LegoNet and apply this concept to a previously unexplored clinical problem: the segmentation of the internal mammary artery (IMA), aorta, and perivascular adipose tissue (PVAT) from computed tomography angiography (CTA) scans. These PVAT regions have been shown to possess prognostic value in assessing cardiovascular risk and primary clinical outcomes. We evaluate LegoNet on large datasets, achieving superior performance to other leading architectures. Furthermore, we assess the model's generalizability on external testing cohorts, where an expert clinician corrects the model's segmentations, achieving DSC > 0.90 across various external, international, and public cohorts. To further validate the model's clinical reliability, we perform intra- and inter-observer variability analysis, demonstrating strong agreement with human annotations. The proposed methodology has significant implications for diagnostic cardiovascular management and early prognosis, offering a robust, automated solution for vascular and perivascular segmentation and risk assessment in clinical practice, paving the way for personalised medicine.

eess.IV

Ozone groups and centers of skew polynomial rings

We introduce the ozone group of a noncommutative algebra $A$, defined as the group of automorphisms of $A$ which fix every element of its center. In order to initiate the study of ozone groups, we study PI skew polynomial rings, which have long proved to be a fertile testing ground in noncommutative algebra. Using the ozone group and other invariants defined herein, we give explicit conditions for the center of a PI skew polynomial to be Gorenstein (resp. regular) in low dimension.

math.RA

Reflexive hull discriminants and applications

We introduce the reflexive hull discriminant as a tool to study noncommutative algebras that are finitely generated, but not necessarily free, over their centers. As an example, we compute the reflexive hull discriminants for quantum generalized Weyl algebras and use them to determine automorphism groups and other properties, recovering results of Su{\'a}rez-Alvarez, Vivas, and others.

math.RA

Noncommutative cyclic isolated singularities

The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He and the third-named author and results of Gaddis, Kirkman, Moore and Won.

math.RA

The minimal model program for b-log canonical divisors and applications

We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b- log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or "non-commutative algebraic varieties which are finite over their centres". The case of toric Brauer pairs is discussed in further detail.

math.AG

McKay Correspondence for semisimple Hopf actions on regular graded algebras, II

We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A^H, over A\#H, and over \End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A^H.

math.RA

McKay Correspondence for semisimple Hopf actions on regular graded algebras, I

In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.

math.RA

Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations

Let $kE$ denote the group algebra of an elementary abelian $p$-group of rank $r$ over an algebraically closed field of characteristic $p$. We investigate the functors $\mathcal{F}_i$ from $kE$-modules of constant Jordan type to vector bundles on $\mathbb{P}^{r-1}(k)$, constructed by Benson and Pevtsova. For a $kE$-module $M$ of constant Jordan type, we show that restricting the sheaf $\mathcal{F}_i(M)$ to a dimension $s-1$ linear subvariety of $\mathbb{P}^{r-1}(k)$ is equivalent to restricting $M$ along a corresponding rank $s$ shifted subgroup of $kE$ and then applying $\mathcal{F}_i$. In the case $r=2$, we examine the generic kernel filtration of $M$ in order to show that $\mathcal{F}_i(M)$ may be computed on certain subquotients of $M$ whose Loewy lengths are bounded in terms of $i$. More precise information is obtained by applying similar techniques to the $n$th power generic kernel filtration of $M$. The latter approach also allows us to generalise our results to higher ranks $r$.

math.RT

Quantum binary polyhedral groups and their actions on quantum planes

We classify quantum analogues of actions of finite subgroups G of SL_2(k) on commutative polynomial rings k[u,v]. More precisely, we produce a classification of pairs (H,R), where H is a finite dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin-Schelter regular algebra R of global dimension two. Remarkably, the corresponding invariant rings R^H share similar regularity and Gorenstein properties as the invariant rings k[u,v]^G in the classic setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.

math.RA

Hopf actions on filtered regular algebras

We study finite dimensional Hopf algebra actions on so-called filtered Artin-Schelter regular algebras of dimension n, particularly on those of dimension 2. The first Weyl algebra is an example of such on algebra with n=2, for instance. Results on the Gorenstein condition and on the global dimension of the corresponding fixed subrings are also provided.

math.RA

Hopf actions and Nakayama automorphisms

Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin-Schelter regular algebra. We study connections between the Nakayama automorphism of A and S^2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given.

math.RA

Rational curves and ruled orders on surfaces

We study ruled orders. These arise naturally in the Mori program for orders on projective surfaces and morally speaking are orders on a ruled surface ramified on a bisection and possibly some fibres. We describe fibres of a ruled order and show they are in some sense rational. We also determine the Hilbert scheme of rational curves and hence the corresponding non-commutative Mori contraction. This gives strong evidence that ruled orders are examples of the non-commutative ruled surfaces introduced by Van den Bergh.

math.RA