This paper studies the unitary diagonalization of matrices over formal power series rings. Our main result shows that a normal matrix is unitarily diagonalizable if and only if its minimal polynomial completely splits over the ring and the associated spectral projections have entries in the ring. Building on this characterization, we develop an algorithm for deciding the unitary diagonalizability of matrices over regular local rings of algebraic varieties. A central ingredient of the algorithm is a decision procedure for determining whether a polynomial splits over a formal power series ring; we establish this using techniques from prime decomposition and the relative smoothness of integral closures in ramification theory.

Algebraic boundaries of convex semi-algebraic sets are closely related to polynomial optimization problems. Building upon Rainer Sinn’s work, we refine the stratification of iterated singular loci to a Whitney (a) stratification, which gives a list of candidates of varieties whose dual is an irreducible component of the algebraic boundary of the dual convex body. We also present an algorithm based on Teissier’s criterion to compute Whitney (a) stratifications, which employs conormal spaces and prime decomposition.

Multimodal named entity recognition (MNER) is a crucial task in social systems of artificial intelligence that requires precise identification of named entities in sentences using both visual and textual information. Previous methods have focused on capturing fine-grained visual features and developing complex fusion procedures. However, these approaches overlook the heterogeneity gap and loss of original modality uniqueness that may occur during fusion, leading to incorrect entity identification. This article proposes a novel approach for MNER called a gated disentangled network with cross-modality consensus fusion (GDN-CMCF) to address the above challenges. Specifically, to eliminate cross-modality variation, we propose a cross-modality consensus fusion module that generates a consensus representation by learning inter-and intramodality interactions with a designed commonality constraint. We then introduce a gated disentanglement module to separate modality-relevant features from support and auxiliary modalities, which further filters out extraneous information while retaining the uniqueness of unimodal features. Experimental results on two real public datasets are provided to verify the effectiveness of our proposed GDN-CMCF. The source code of this article can be found at https://github.com/HaoDavis/ GDN-CMCF.