Pointwise convergence of noncommutative Fourier series

This paper is devoted to the study of pointwise convergence of Fourier series for compact groups, group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this work studies the refined counterpart of pointwise convergence of these Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to $1$ pointwise, so that the associated Fourier multipliers on noncommutative $L_p$-spaces satisfy the pointwise convergence for all $1<p<\infty$. In a similar fashion, we also obtain results for a large subclass of groups (as well as discrete quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fej\'{e}r means and Bochner-Riesz means in the noncommutative setting. Even back to the Fourier series of $L_p$-functions on Euclidean spaces and non-abelian compact groups, our results seem novel and yield new problems. On the other hand, we obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.Comment: v3: 83 pages; this version contains some corrections. v2: 74 pages; new results are added in Section 4, Section 5 and Section 6.

Characteristic polynomials and finitely dimensional representations of simple Lie Algebras

In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed. Furthermore, the characteristic polynomials of sl(2, C) on some classical simple Lie algebras through adjoint representations are studied, and we present some results of Borel subalgebras and parabolic subalgebras of simple Lie algebras through characteristic polynomials.Comment: 1

A Systematic Review for Transformer-based Long-term Series Forecasting

The emergence of deep learning has yielded noteworthy advancements in time series forecasting (TSF). Transformer architectures, in particular, have witnessed broad utilization and adoption in TSF tasks. Transformers have proven to be the most successful solution to extract the semantic correlations among the elements within a long sequence. Various variants have enabled transformer architecture to effectively handle long-term time series forecasting (LTSF) tasks. In this article, we first present a comprehensive overview of transformer architectures and their subsequent enhancements developed to address various LTSF tasks. Then, we summarize the publicly available LTSF datasets and relevant evaluation metrics. Furthermore, we provide valuable insights into the best practices and techniques for effectively training transformers in the context of time-series analysis. Lastly, we propose potential research directions in this rapidly evolving field

Strong Association Between Two Polymorphisms on 15q25.1 and Lung Cancer Risk: A Meta-Analysis

Background: The association between polymorphisms on 15q25.1 and lung cancer has been widely evaluated; however, the studies have yielded contradictory results. We sought to investigate this inconsistency by performing a comprehensive meta-analysis on two polymorphisms (CHRNA3 gene: rs1051730 and AGPHD1 gene: rs8034191) on 15q25.1. Methods: Data were extracted from 15 and 14 studies on polymorphisms rs1051730 and rs8034191 involving 12301/14000 and 14075/12873 lung cancer cases/controls, respectively. The random-effects model was applied, addressing heterogeneity and publication bias. Results: The two polymorphisms followed Hardy-Weinberg equilibrium for all studies (P\u3e0.05). For rs1051730-G/A, carriers of A allele had a 36% increased risk for lung cancer (95% confidence interval [CI]: 1.27–1.46; P\u3c0.0005), without heterogeneity (P = 0.258) or publication bias (PEgger = 0.462). For rs8034191-T/C, the allelic contrast indicated that C allele conferred a 23% increased risk for lung cancer (95% CI: 1.08–1.4; P = 0.002), with significant heterogeneity (P\u3c0.0005), without publication bias (PEgger = 0.682). Subgroup analyses suggested that the between-study heterogeneity was derived from ethnicity, study design, matched information, and lung cancer subtypes. For example, the association of polymorphisms rs1051730 and rs8034191 with lung cancer was heterogeneous between Caucasians (OR = 1.32 and 1.22; 95% CI: 1.25–1.44 and 1.05–1.42; PP = 0.237 and 0.934, respectively) under the allelic model, and this association was relatively strengthened under the dominant model. There was no observable publication bias for both polymorphisms. Conclusions: Our findings demonstrated that CHRNA3 gene rs1051730-A allele and AGPHD1 gene rs8034191-T allele might be risk-conferring factors for the development of lung cancer in Caucasians, but not in East-Asians