arXiv subjects

Ziyu Huang

Publications and source records attributed to Ziyu Huang.

40 records · Page 3Linked to original sources

arXiv2022

A classification of prime graphs of pseudo-solvable groups

The prime graph $Γ(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $Γ(G)$ if and only if $G$ has an element of order $pq$. Since their inception in the 1970s these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups such a classification was found in 2015. In this paper we go beyond solvable groups for the first time and characterize prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or $A_5$. The classification is based on two conditions: the vertices $\{2,3,5\}$ form a triangle in $\overlineΓ(G)$ or $\{p,3,5\}$ form a triangle for some prime $p\neq 2$.

math.GR↗

arXiv2022

On the Generation, Structure, and Symmetries of Minimal Prime Graphs

In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if and only if G contains an element of order pq. Since the discovery of a simple, purely graph theoretical characterization of the prime graphs of solvable groups in 2015 these graphs have been studied in more detail from a graph theoretic angle. In this paper we explore several new aspects of these graphs. We characterize regular reseminant graphs and study the automorphisms of reseminant graphs for arbitrary base graphs. We then study minimal prime graphs on larger vertex sets by a novel regular graph construction for base graphs and by proving results on prime graph properties under graph products. Lastly, we present the first new way, different from vertex duplication, to obtain a new minimal prime graph from a given minimal prime graph.

math.CO↗

arXiv2022

State-Density Flows of Non-Degenerate Density-Dependent Mean Field SDEs and Associated PDEs

In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,μ}$ with initial $(t,μ)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,μ}$ with initial $(t,x)\in[0,T]\times \mathbb{R}^d$, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional $V(t,x,μ)=\mathbb{E}[Φ(X^{t,x,μ}_T,m^{t,μ}(T,\cdot))]$. Our main result states that, under a non-degenerate condition and appropriate regularity assumptions on the coefficients, the function $V$ is the unique classical solution of a nonlocal partial differential equation of mean-field type. The proof depends heavily on the differential properties of the flow $μ\mapsto (m^{t,μ}, X^{t,x,μ})$ over $μ\in L^2(\mathbb{R}^d)$. We also give an example to illustrate the role of our main result. Finally, we give a discussion on the $L^1$ choice case in the initial $μ$.

math.PR↗

arXiv2021

Mean Field Games with Common Noises and Conditional Distribution Dependent FBSDEs

In this paper, we consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. We assume that the cost function satisfies a convexity and a weak monotonicity property. We use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE). We prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by Hu and Peng \cite{YH2}. We apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.

math.OC↗