arXiv subjects

Rongwei Yang

Publications and source records attributed to Rongwei Yang.

At least 19 recordsLinked to original sources

arXiv2026

Reducibility of linear representations, free ideals, and Kippenhahn's conjecture

This paper presents a comprehensive study of the characteristic polynomial of matrix tuples and finite dimensional group representations. Among other things, several key concepts are introduced, including the minimal polynomial, spectral index, spectral stability, and characteristic graph. Notably, this framework provides a complete resolution to Kippenhahn's conjecture, settling a long-standing and influential problem in the theory of matrix tuples.

math.RT↗

arXiv2026

Diophantine analysis and the Braid group ${\bf B}_3$

Given a finite dimensional representation $\pi$ of a finitely generated group $G=\langle g_1, \ldots, g_n\rangle$, the associated characteristic polynomial is defined as $Q_\pi(z):=\det(z_0I+z_1\pi(g_1)+\cdots +z_n\pi(g_n))$, and it is known to contain a good amount of structural information about $G$ and $\pi$. This paper is a part of an ongoing project to investigate the number-theoretic properties of the algebraic varieties (called {\em eigensurfaces}) $\{z\in \mathbb{C}^{n+1}: Q_\pi(z)=0\}$. Its focus is the distribution of prime triples in the eigensurface $S:=\{z\in \mathbb{C}^3: (z_0+z_1+z_2)^2+z_0z_1=0\}$ associated with the braid group ${\bf B}_3$ and its reduced Burau representation. We prove that such triples occur with higher frequency on $S$ than in the ambient lattice, revealing an unexpected connection between group representation theory and analytic number theory.

math.NT↗

arXiv2025

The Density of Primes in the Eigensurface of ${\bf S}_3$

The Prime Number Theorem asserts that the density of primes less than or equal to $N$ is asymptotically equal to $1/\log N$. The density of prime triples in coprime triples in $\mathbb{Z}^3_+$ is determined to be $3\zeta (3)/\log N$, where $\zeta$ is the Riemann zeta function. In this paper, we prove that the density of prime triples in coprime triples in the surface $S=\{z_0^{2} - z_1^{2} + z_2^{2} - z_0z_2=0\}$ is greater than $3\zeta (3)/\log N$, meaning that $S$ meets primes more frequently. This surface is the eigensurface of the symmetric group ${\bf S}_3$ with respect to an irreducible representation.

math.NT↗

arXiv2025

Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space

This paper investigates the spectral properties of Toeplitz operators on the Bergman space of unit disk. We present an integral representation of $ T^*_{z^m}$, which establishes a connection between the Bergman functions and the solutions of PDE theory. In fact, by leveraging the Poincar\'e theorem in difference equations and the solution forms of differential equations, this paper describes the kernels of certain Toeplitz operators with harmonic polynomial symbols, and further gives the sufficient conditions for the connectedness of the spectra of these Toeplitz operators. The spectral properties of $ T_\varphi$ with $\varphi (z) =\overline{z}^{m} + \alpha z^m + \beta$ are characterized, such as $\sigma(T_\varphi)= \overline{\varphi (\mathbb {D})}$, Fredholm index of $T_\varphi$ can only be one of $m,-m$ and $0$, $T_\varphi$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators.

math.FA↗

arXiv2025

Spectral Invariants of Complex Solvable Lie Algebras: Nilradical Weights and Hyperplane Arrangements

A central open problem in Lie theory is the classification of finite-dimensional complex solvable Lie algebras, but classification up to isomorphism becomes increasingly difficult as the dimension grows and the number of non-isomorphic families explodes. This motivates a classification method by studying coarser spectral invariants arising from the characteristic polynomial of the adjoint representation. We prove that the characteristic polynomial is determined by the generalized weights of the adjoint action on the nilradical. This addresses the problem of expressing the spectral index in Lie-theoretic terms, gives sharp bounds in terms of the nilradical and quotient dimensions, and characterizes spectral equivalence. We then resolve a second problem concerning the higher Betti numbers of the eigenvariety complement. We endow the distinct non-$z_0$ factors of the characteristic polynomial with a natural matroid structure, which we call the spectral matroid, and use the Orlik--Solomon algebra of the associated hyperplane arrangement to determine the Betti numbers and Poincar\'{e} polynomial combinatorially and to prove log-concavity of the Betti sequence. Finally, we apply our theory to solvable Lie algebras with abelian nilradical to obtain explicit characteristic polynomials and spectral-equivalence criteria, including examples of non-isomorphic algebras with identical characteristic polynomials.

math.RT↗

arXiv2025

Spectral test of reducibility for Matrix tuples

If a tuple of matrices has a common invariant subspace, its projective joint spectrum has an algebraic component. In general, the converse is not true, and there might be algebraic components in the projective joint spectrum without corresponding common invariant subspaces. In this paper we give necessary and sufficient conditions for the occurrence of such correspondence.

math.FA↗

arXiv2025

Relative Reality

The ``Hard Problem" of consciousness refers to a long-standing enigma about how qualia emerge from physical processes in the brain. Building on insights from the development of non-Euclidean geometry, this paper seeks to present a structured and logically coherent theory of qualia to address this problem. The proposed theory starts with a definition on what it means for an entity to be non-physical. A postulate about awareness is posed and utilized to rigorously prove that qualia are non-physical and thoughts are qualia. Then the paper introduces a key concept: relative reality, meaning that perceptions of reality are relative to the observer and time. The concept is analyzed through a mathematical model grounded in Hilbert space theory. The model also sheds new light on cognitive science and physics. In particular, the Schr\"{o}dinger equation can be derived easily through this model. Moreover, this model shows that eigenstates also exist for classical energy-conserving systems. Analyses on the G. P. Thomson experiment and the classical harmonic oscillator are made to illustrate this finding. The insight gained sheds new light on the Bohr-Einstein debate concerning the interpretation of quantum mechanics. At last, the paper proposes a postulate about qualia force and demonstrates that it constitutes a fundamental part of absolute reality, much like the four fundamental forces in nature.

physics.gen-ph↗

arXiv2024

LLM-POTUS Score: A Framework of Analyzing Presidential Debates with Large Language Models

Large language models have demonstrated remarkable capabilities in natural language processing, yet their application to political discourse analysis remains underexplored. This paper introduces a novel approach to evaluating presidential debate performances using LLMs, addressing the longstanding challenge of objectively assessing debate outcomes. We propose a framework that analyzes candidates' "Policies, Persona, and Perspective" (3P) and how they resonate with the "Interests, Ideologies, and Identity" (3I) of four key audience groups: voters, businesses, donors, and politicians. Our method employs large language models to generate the LLM-POTUS Score, a quantitative measure of debate performance based on the alignment between 3P and 3I. We apply this framework to analyze transcripts from recent U.S. presidential debates, demonstrating its ability to provide nuanced, multi-dimensional assessments of candidate performances. Our results reveal insights into the effectiveness of different debating strategies and their impact on various audience segments. This study not only offers a new tool for political analysis but also explores the potential and limitations of using LLMs as impartial judges in complex social contexts. In addition, this framework provides individual citizens with an independent tool to evaluate presidential debate performances, which enhances democratic engagement and reduces reliance on potentially biased media interpretations and institutional influence, thereby strengthening the foundation of informed civic participation.

cs.CL↗

arXiv2024

Pluriharmonic solutions to Yang-Mills equations: a $C^*$-algebras approach

This partially expository paper provides a view of Yang-Mills equations from the perspective of complex variables, operator theory, and $C^{*}$-algebras. Through operator-valued pluriharmonic and skew-Hermitian differential forms, it constructs a new class of instanton solutions. Furthermore, it provides a complex variable version of the Yang-Mills Lagrangian and the Belavin-Polyakov-Schwartz-Tyupkin instanton.

math-ph↗

arXiv2023

The Characteristic Polynomial of Projections

This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered.

math.RT↗

arXiv2023

Joint spectrum, group representations, and Julia set

The first half of this mostly expository note reviews some notions of joint spectrum of linear operators, and it gives a new characterization of amenable groups in terms of projective spectrum. The second half revisits an application of projective spectrum to the study of self-similar group representations made in [16]. In the case $\pi$ is the Koopman representation of the infinite dihedral group $D_\infty$ on the binary tree, it shows that the projective spectrum of $D_\infty$ coincides with the Julia set of a rational map $F_\pi: \mathbb{P}^2\to \mathbb{P}^2$ derived from the self-similarity of $\pi$. This improves the main result in [16].

math.FA↗

arXiv2021

DGEM: A New Dual-modal Graph Embedding Method in Recommendation System

In the current deep learning based recommendation system, the embedding method is generally employed to complete the conversion from the high-dimensional sparse feature vector to the low-dimensional dense feature vector. However, as the dimension of the input vector of the embedding layer is too large, the addition of the embedding layer significantly slows down the convergence speed of the entire neural network, which is not acceptable in real-world scenarios. In addition, as the interaction between users and items increases and the relationship between items becomes more complicated, the embedding method proposed for sequence data is no longer suitable for graphic data in the current real environment. Therefore, in this paper, we propose the Dual-modal Graph Embedding Method (DGEM) to solve these problems. DGEM includes two modes, static and dynamic. We first construct the item graph to extract the graph structure and use random walk of unequal probability to capture the high-order proximity between the items. Then we generate the graph embedding vector through the Skip-Gram model, and finally feed the downstream deep neural network for the recommendation task. The experimental results show that DGEM can mine the high-order proximity between items and enhance the expression ability of the recommendation model. Meanwhile it also improves the recommendation performance by utilizing the time dependent relationship between items.

cs.IR↗

arXiv2021

Maxwell's Equations in Complex Variables

This paper provides a view of Maxwell's equations from the perspective of complex variables. The study is made through complex differential forms and the Hodge star operator in $\mathbb{C}^2$ with respect to the Euclidean and the Minkowski metrics. It shows that holomorphic functions give rise to nontrivial solutions, and the inner product between the electric and the magnetic fields is considered in this case. Further, it obtains a simple necessary and sufficient condition regarding harmonic solutions to the equations. In the end, the paper gives an interpretation of the Lorenz gauge condition in terms of the codifferential operator.

math.AP↗

arXiv2020

Self-similarity and spectral dynamics

For a tuple $A= (A_0, A_1, \ldots , A_n)$ of elements in a unital Banach algebra $\mathcal{B}$, its \textit{projective (joint) spectrum} $p(A)$ is the collection of $z\in\mathbb{P}^{n}$ such that $A(z)=z_0A_0+z_1 A_1 + \ldots z_n A_n$ is not invertible. If the tuple $A$ is associated with the generators of a finitely generated group, then $p(A)$ is simply called the projective spectrum of the group. This paper investigates a connection between self-similar group representations and an induced polynomial map on the projective space that preserves the projective spectrum of the group. The focus is on two groups: the infinite dihedral group $D_\infty$ and the Grigorchuk group ${\mathcal G}$ of intermediate growth. The main theorem shows that for $D_\infty$ the Julia set of the induced rational map $F$ is equal to the union of the projective spectrum with the extended indeterminacy set. Moreover, the limit function of the iteration sequence $\{F^{\circ n}\}$ on the Fatou set is determined explicitly. The result has an application to the group ${\mathcal G}$ and gives rise to a conjecture about its associated Julia set.

math.FA↗

arXiv2020

Spectral invariants for finite dimensional Lie algebras

For a Lie algebra ${\mathcal L}$ with basis $\{x_1,x_2,\cdots,x_n\}$, its associated characteristic polynomial $Q_{\mathcal L}(z)$ is the determinant of the linear pencil $z_0I+z_1\text{ad} x_1+\cdots +z_n\text{ad} x_n.$ This paper shows that $Q_{\mathcal L}$ is invariant under the automorphism group $\text{Aut}({\mathcal L}).$ The zero variety and factorization of $Q_{\mathcal L}$ reflect the structure of ${\mathcal L}$. In the case ${\mathcal L}$ is solvable $Q_{\mathcal L}$ is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to $1$-dimensional extensions of nilpotent Lie algebras.

math.RT↗

arXiv2020

Hermitian geometry on resolvent set(I)

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its projective joint spectrum $P(A)$ is the collection of $z\in {\mathbb C}^n$ such that $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible. It is known that the ${\mathcal B}$-valued $1$-form $ω_A(z)=A^{-1}(z)dA(z)$ contains much topological information about the joint resolvent set $P^c(A)$. This paper studies geometric properties of $P^c(A)$ with respect to Hermitian metrics defined through the ${\mathcal B}$-valued {\em fundamental form} $Ω_A=-ω^*_A\wedge ω_A$ and its coupling with faithful states $ϕ$ on ${\mathcal B}$, i.e. $ϕ(Ω_A)$. The connection between the tuple $A$ and the metric is the main subject of this paper. In particular, it shows that the Kählerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede-Kadison determinant.

math.FA↗

arXiv2018

A brief survey on operator theory in $H^2(\mathbb D^2)$

This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc $H^2(\mathbb D^2)$. As an important component of multivariable operator theory, the theory in $H^2(\mathbb D^2)$ focuses primarily on two pairs of commuting operators that are naturally associated with invariant subspaces (or submodules) in $H^2(\mathbb D^2)$. Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. The theory in $H^2(\mathbb D^2)$ is motivated by and still tightly related to several other influential theories, namely Nagy-Foias theory on operator models, Ando's dilation theorem of commuting operator pairs, Rudin's function theory on $H^2(\mathbb D^n)$, and Douglas-Paulsen's framework of Hilbert modules. Due to the simplicity of the setting, a great supply of examples in particular, the operator theory in $H^2(\mathbb D^2)$ has seen remarkable growth in the past two decades. This survey is far from a full account of this development but rather a glimpse from the author's perspective. Its goal is to show an organized structure of this theory, to bring together some results and references and to inspire curiosity on new researchers.

math.FA↗

arXiv2018

Self-adjoint Elements in the Pseudo-unitary Group ${\bf U}\left(p,p\right)$

The pseudo-unitary group ${\bf U}\left(p,q\right)$ of signature $\left(p,q\right)$ is the group of matrices that preserve the indefinite pseudo-Euclidean metric on the vector space $\mathbb{C}^{p,q}$. The goal of this paper is to describe the set ${\bf U}_{s}\left(p,p\right)$ of Hermitian, or, self-adjoint elements in ${\bf U}\left(p,p\right)$.

math.RA↗