arXiv subjects

Kan Hu

Publications and source records attributed to Kan Hu.

18 recordsLinked to original sources

arXiv2026

Classification and enumeration of skew morphisms of skew-type four on cyclic $2$-groups

A skew morphism on a finite group $A$ is a permutation $φ$ on $A$ that fixes the identity element of $A$ and for which there exists an integer-valued function $π:A\to\mathbb{Z}_{|φ|}$ such that $φ(xy)=φ(x)φ^{π(x)}(y)$ for all $x,y\in A$. The kernel of $φ$ is the subgroup $\Kerφ=\{x\in A\mid π(x)=1\}$, and the index $[A:\Kerφ]$ is called the skew-type of $φ$. In this paper we construct, classify and enumerate the skew morphisms of skew-type four on cyclic $2$-groups. Our main results give explicit formulas for all such skew morphisms and closed-form expressions for their numbers.

math.CO↗

arXiv2026

Classification of regular Cayley maps of skew-type three on semidihedral groups

It is well known that every regular Cayley map $M = \CM(G,X,p)$ on a finite group $G$ with respect to an inverse-closed generating set $X$ of $G$ and a specified cyclic permutation $p$ on $X$ corresponds to a skew morphism $φ$ on $G$ such that the restriction of $φ$ to $X$ is $p$. The skew-type of the map $M$ is defined as the index $[G:\Ker φ]$, which equals the number of distinct values in $\mathbb{Z}_{|φ|}$ taken by the associated power function $π$ of the skew morphism $φ$. In this paper, we develop a covering theory of skew morphisms and as an application we provide a classification of regular Cayley maps of skew-type three on the semidihedral groups.

math.CO↗

arXiv2025

TD3-Sched: Learning to Orchestrate Container-based Cloud-Edge Resources via Distributed Reinforcement Learning

Resource scheduling in cloud-edge systems is challenging as edge nodes run latency-sensitive workloads under tight resource constraints, while existing centralized schedulers can suffer from performance bottlenecks and user experience degradation. To address the issues of distributed decisions in cloud-edge environments, we present TD3-Sched, a distributed reinforcement learning (DRL) scheduler based on Twin Delayed Deep Deterministic Policy Gradient (TD3) for continuous control of CPU and memory allocation, which can achieve optimized decisions for resource provisioning under dynamic workloads. On a realistic cloud-edge testbed with SockShop application and Alibaba traces, TD3-Sched achieves reductions of 17.9% to 38.6% in latency under same loads compared with other reinforcement-learning and rule-based baselines, and 16% to 31.6% under high loads. TD3-Sched also shows superior Service Level Objective (SLO) compliance with only 0.47% violations. These results indicate faster convergence, lower latency, and more stable performance while preserving service quality in container-based cloud-edge environment compared with the baselines.

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arXiv2025

A characterization of nilpotent bicyclic groups

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group theory, with applications extending to symmetric embeddings of the complete bipartite graphs. A classical result by Douglas establishes that every bicyclic group is supersolvable. More recently, Fan and Li (2018) proved that every finite $(m,n)$-bicyclic group is abelian if and only if $\gcd(m,ϕ(n))=\gcd(n,ϕ(m))=1$, where $ϕ$ is Euler's totient function. In this paper we generalize this result further and show that every $(m,n)$-bicyclic group is nilpotent if and only if $\gcd(n,ϕ(\mathrm{rad}(m)))=\gcd(m,ϕ(\mathrm{rad}(n)))=1$, where $\mathrm{rad}(m)$ denotes the radical of $m$ (the product of its distinct prime divisors).

math.GR↗

arXiv2024

LSRAM: A Lightweight Autoscaling and SLO Resource Allocation Framework for Microservices Based on Gradient Descent

Microservices architecture has become the dominant architecture in cloud computing paradigm with its advantages of facilitating development, deployment, modularity and scalability. The workflow of microservices architecture is transparent to the users, who are concerned with the quality of service (QoS). Taking Service Level Objective (SLO) as an important indicator of system resource scaling can effectively ensure user's QoS, but how to quickly allocate end-to-end SLOs to each microservice in a complete service so that it can obtain the optimal SLO resource allocation scheme is still a challenging problem. Existing microservice autoscaling frameworks based on SLO resources often have heavy and complex models that demand substantial time and computational resources to get a suitable resource allocation scheme. Moreover, when the system environment or microservice application changes, these methods require significant time and resources for model retraining. In this paper, we propose LSRAM, a lightweight SLO resource allocation management framework based on the gradient descent method to overcome the limitation of existing methods in terms of heavy model, time-consuming, poor scalability, and difficulty in retraining. LSRAM has two stages: at stage one, the lightweight SLO resource allocation model from LSRAM can quickly compute the appropriate SLO resources for each microservice; at stage two, LSRAM's SLO resource update model enables the entire framework to quickly adapt to changes in the cluster environment (e.g. load and applications). Additionally, LSRAM can effectively handle bursty traffic and highly fluctuating load application scenarios. Compared to state-of-the-art SLO allocation frameworks, LSRAM not only guarantees users' QoS but also reduces resource usage by 17%.

cs.DC↗

arXiv2024

MSARS: A Meta-Learning and Reinforcement Learning Framework for SLO Resource Allocation and Adaptive Scaling for Microservices

Service Level Objectives (SLOs) aim to set threshold for service time in cloud services to ensure acceptable quality of service (QoS) and user satisfaction. Currently, many studies consider SLOs as a system resource to be allocated, ensuring QoS meets the SLOs. Existing microservice auto-scaling frameworks that rely on SLO resources often utilize complex and computationally intensive models, requiring significant time and resources to determine appropriate resource allocation. This paper aims to rapidly allocate SLO resources and minimize resource costs while ensuring application QoS meets the SLO requirements in a dynamically changing microservice environment. We propose MSARS, a framework that leverages meta-learning to quickly derive SLO resource allocation strategies and employs reinforcement learning for adaptive scaling of microservice resources. It features three innovative components: First, MSARS uses graph convolutional networks to predict the most suitable SLO resource allocation scheme for the current environment. Second, MSARS utilizes meta-learning to enable the graph neural network to quickly adapt to environmental changes ensuring adaptability in highly dynamic microservice environments. Third, MSARS generates auto-scaling policies for each microservice based on an improved Twin Delayed Deep Deterministic Policy Gradient (TD3) model. The adaptive auto-scaling policy integrates the SLO resource allocation strategy into the scheduling algorithm to satisfy SLOs. Finally, we compare MSARS with state-of-the-art resource auto-scaling algorithms that utilize neural networks and reinforcement learning, MSARS takes 40% less time to adapt to new environments, 38% reduction of SLO violations, and 8% less resources cost.

cs.DC↗

arXiv2024

On exact products of two dihedral groups

An exact product of two finite groups $H$ and $K$ is a finite group $X$ which contains $H$ and $K$ as subgroups, satisfying $X=HK$ and $H\cap K=\{1_X\}$. In this paper, we provide a classification of the exact products of two dihedral groups of orders $2m$ and $2n$ for all odd numbers $m,n\geq 3$.

math.GR↗

arXiv2023

Cyclic complementary extensions and skew-morphisms

A cyclic complementary extension of a finite group $A$ is a finite group $G$ which contains $A$ and a cyclic subgroup $C$ such that $A\cap C=\{1_G\}$ and $G=AC$. For any fixed generator $c$ of the cyclic factor $C=\langle c\rangle$ of order $n$ in a cyclic complementary extension $G=AC$, the equations $cx=φ(x)c^{Π(x)}$, $x\in A$, determine a permutation $φ:A\to A$ and a function $Π:A\to\mathbb{Z}_n$ on $A$ characterized by the properties: (a) $φ(1_A)=1_A$ and $Π(1_A)\equiv1\pmod{n}$; (b) $φ(xy)=φ(x)φ^{Π(x)}(y)$ and $Π(xy)\equiv\sum_{i=1}^{Π(x)}Π(φ^{i-1}(y))\pmod{n}$, for all $x,y\in A$. The permutation $φ$ is called a skew-morphism of $A$ and has already been extensively studied. One of the main contributions of the present paper is the recognition of the importance of the function $Π$, which we call the extended power function associated with $φ$. We show that {\em every} cyclic complementary extension of $A$ is determined and can be constructed from a skew-morphism $φ$ of $A$ and an extended power function $Π$ associated with $φ$. As an application, we present a classification of cyclic complementary extensions of cyclic groups obtained using skew-morphisms which are group automorphisms.

math.GR↗

arXiv2023

Classification of cyclic groups underlying only smooth skew morphisms

A skew morphism of a finite group $A$ is a permutation $φ$ of $A$ fixing the identity element and for which there is an integer-valued function $π$ on $A$ such that $φ(ab)=φ(a)φ^{π(a)}(b)$ for all $a, b \in A$. A skew morphism $φ$ of $A$ is smooth if the associated power function $π$ is constant on the orbits of $φ$, that is, $π(φ(a))\equivπ(a)\pmod{|φ|}$ for all $a\in A$. In this paper we show that every skew morphism of a cyclic group of order $n$ is smooth if and only if $n=2^en_1$, where $0 \le e \le 4$ and $n_1$ is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.

math.GR↗

arXiv2020

A Survey of Non-Volatile Main Memory Technologies: State-of-the-Arts, Practices, and Future Directions

Non-Volatile Main Memories (NVMMs) have recently emerged as promising technologies for future memory systems. Generally, NVMMs have many desirable properties such as high density, byte-addressability, non-volatility, low cost, and energy efficiency, at the expense of high write latency, high write power consumption and limited write endurance. NVMMs have become a competitive alternative of Dynamic Random Access Memory (DRAM), and will fundamentally change the landscape of memory systems. They bring many research opportunities as well as challenges on system architectural designs, memory management in operating systems (OSes), and programming models for hybrid memory systems. In this article, we first revisit the landscape of emerging NVMM technologies, and then survey the state-of-the-art studies of NVMM technologies. We classify those studies with a taxonomy according to different dimensions such as memory architectures, data persistence, performance improvement, energy saving, and wear leveling. Second, to demonstrate the best practices in building NVMM systems, we introduce our recent work of hybrid memory system designs from the dimensions of architectures, systems, and applications. At last, we present our vision of future research directions of NVMMs and shed some light on design challenges and opportunities.

cs.DC↗

arXiv2018

Smooth skew-morphisms of the dihedral groups

A skew-morphism $φ$ of a finite group $A$ is a permutation on $A$ such that $φ(1)=1$ and $φ(xy)=φ(x)φ^{π(x)}(y)$ for all $x,y\in A$ where $π:A\to\mathbb{Z}_{|φ|}$ is an integer function. A skew-morphism is smooth if $π(φ(x))=π(x)$ for all $x\in A$. The concept of smooth skew-morphisms is a generalization of that of $t$-balanced skew-morphisms. The aim of the paper is to develop a general theory of smooth skew-morphisms. As an application we classify smooth skew-morphisms of the dihedral groups.

math.GR↗

arXiv2018

Complete regular dessins and skew-morphisms of cyclic groups

A dessin is a 2-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph $K_{m,n}$, called $(m,n)$-complete regular dessins. The purpose is to establish a rather surprising correspondence between $(m,n)$-complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group $A$ is a bijection $φ\colon A\to A$ that satisfies the identity $φ(xy)=φ(x)φ^{π(x)}(y)$ for some function $π\colon A\to\mathbb{Z}$ and fixes the neutral element of~$A$. We show that every $(m,n)$-complete regular dessin $\mathcal{D}$ determines a pair of reciprocal skew-morphisms of the cyclic groups $\mathbb{Z}_n$ and $\mathbb{Z}_m$. Conversely, $\mathcal{D}$ can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers $m$ and $n$ for which there exists, up to interchange of colours, exactly one $(m,n)$-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order $m$ and $n$ is abelian, which eventually comes down to the condition $\gcd(m,ϕ(n))=\gcd(ϕ(m),n)=1$, where $ϕ$ is Euler's totient function.

math.CO↗

arXiv2018

Abelian regular coverings of the quaternion hypermap

A hypermap is an embedding of a connected hypergraph into an orientable closed surface. A covering between hypermaps is a homomorphism between the embedded hypergraphs which extends to an orientation-preserving covering of the supporting surfaces. A covering of a hypermap onto itself is an automorphism, and a hypermap is regular if its automorphism group acts transitively on the brins. Depending on the algebraic theory of regular hypermaps and hypermap operations, the abelian regular coverings over the quaternion hypermap are investigated. We define normalized multicyclic coverings between regular hypermaps, generalizing almost totally branched coverings studied in [K. Hu, R. Nedela, N.-E Wang, Branched cyclic regular coverings over platonic maps, European J. Combin. 36 (2014) 531--549]. It is shown that the covering transformation group of a normalized multicyclic covering is a nilpotent group with bounded class. As an application the abelian normalized bicyclic coverings over the quaternion hypemap are classified. In particular, those coverings which possess various level of external symmetry or fulfil certain smoothness conditions are explicitly determined.

math.CO↗

arXiv2018

Complete regular dessins of odd prime power order

A dessin is a $2$-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the edges. In this paper we employ group-theoretic method to determine and enumerate the isomorphism classes of regular dessins with the complete bipartite underlying graphs of odd prime power order.

math.CO↗

arXiv2018

Nilpotent groups of class two which underly a unique regular dessin

A dessin is an embedding of connected bipartite graph into an oriented closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges. In the present paper regular dessins with a nilpotent automorphism group are investigated, and attention are paid on those with the highest level of external symmetry. Depending on the algebraic theory of dessins and using group-theoretical methods, we present a classification of nilpotent groups of class two which underly a unique regular dessin.

math.GR↗

arXiv2018

Regular dessins uniquely determined by a nilpotent automorphism group

It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic $p$-groups $G$, i.e., the order of the automorphism group of $G$ attains Hall's upper bound. Maximally automorphic $p$-groups of nilpotency class three are classified.

math.GR↗

arXiv2015

Totally symmetric dessins with nilpotent automorphism groups of class three

A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. An automorphism of a dessin is a permutation of the edges of the underlying graph which preserves the colouring of the vertices and extends to an orientation-preserving self-homeomorphism of the supporting surface. A dessin is regular if its automorphism group is transitive on the edges, and a regular dessin is totally symmetric if it is invariant under all dessin operations. Thus totally symmetric dessins possesses the highest level of external symmetry. In this paper we present a classification of totally symmetric dessins with a nilpotent automorphism group of class three

math.GT↗

arXiv2015

Nilpotent dessins: Decomposition theorem and classification of the abelian dessins

A map is a 2-cell decomposition of an orientable closed surface. A dessin is a bipartite map with a fixed colouring of vertices. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges, and a regular dessin is symmetric if it admits an additional external symmetry transposing the vertex colours. Regular dessins with nilpotent automorphism groups are investigated. We show that each such dessin is a parallel product of regular dessins whose automorphism groups are the Sylow subgroups. Regular and symmetric dessins with abelian automorphism groups are classified and enumerated.

math.CO↗