arXiv subjects

Bruce Fang

Publications and source records attributed to Bruce Fang.

9 recordsLinked to original sources

arXiv2025

Benford Behavior in Stick Fragmentation Problems

Benford's law is the statement that in many real-world data sets, the probability of having digit $d$ in base $B$, where $1 \leq d \leq B$, as the first digit is $\log_{B}\left(\tfrac{d+1}{d}\right)$. We sometimes refer to this as weak Benford behavior, and we say that a data set exhibits strong Benford behavior in base $B$ if the probability of having significand at most $s$, where $s \in [1,B)$, is $\log_{B}(s)$. We examine Benford behaviors in the stick fragmentation model. Building on the work on the 1-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high-dimensional stick fragmentation model to the 1-dimensional model and provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior.

math.PR↗

arXiv2025

Benford behavior resulting from stick and box fragmentation processes

Benford's law is the statement that in many real world data sets, the probability of having digit $d$ in base $B$ as the first digit is \log_{B}\!\left(\frac{d+1}{d}\right) for all $1 \leq d \leq B$. We sometimes refer to this as weak Benford behavior, and we say that a data set satisfies strong Benford behavior in base $B$ if the probability of having significand at most $s$ is \log_{B}\!\left(s\right) for all $1 \leq s < B$, . We examine Benford behaviors in two different probabilistic models: stick and box fragmentation models. Building on the work arXiv:1309.5603 on the single proportion stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the multi-proportion stick fragmentation model to the single proportion model. We then provide a necessary and sufficient condition for the lengths of the stick fragments to converge to strong Benford behavior along with a quantification of the discrepancy from uniform distribution on $[0,1]$ in terms of irrationality exponent. Then we answer a conjecture of arXiv:2304.08335 on the high-dimensional box fragmentation model. Using tools from Fourier analysis and order statistics, we prove that under some mild conditions, faces of any arbitrary dimension of the box have total volume converging to strong Benford behavior.

math.PR↗

arXiv2025

Domain-Adversarial Transfer Learning for Fault Root Cause Identification in Cloud Computing Systems

This paper addresses the challenge of fault root cause identification in cloud computing environments. The difficulty arises from complex system structures, dense service coupling, and limited fault information. To solve this problem, an intelligent identification algorithm based on transfer learning is proposed. The method introduces a shared feature extraction module and a domain adversarial mechanism to enable effective knowledge transfer from the source domain to the target domain. This improves the model's discriminative ability and generalization performance in the target domain. The model incorporates a pseudo-label selection strategy. When labeled samples are lacking in the target domain, high-confidence predictions are used in training. This enhances the model's ability to recognize minority classes. To evaluate the stability and adaptability of the method in real-world scenarios, experiments are designed under three conditions: label scarcity, class imbalance, and heterogeneous node environments. Experimental results show that the proposed method outperforms existing mainstream approaches in several key metrics, including accuracy, F1-Score, and AUC. The model demonstrates stronger discriminative power and robustness. Notably, under extreme class imbalance and significant structural differences in the target domain, the model still maintains high performance. This validates the effectiveness and practical value of the proposed mechanisms in complex cloud computing systems.

cs.DC↗

arXiv2025

Collaborative Multi-Agent Reinforcement Learning Approach for Elastic Cloud Resource Scaling

This paper addresses the challenges of rapid resource variation and highly uncertain task loads in cloud computing environments. It proposes an optimization method for elastic cloud resource scaling based on a multi-agent system. The method deploys multiple autonomous agents to perceive resource states in parallel and make local decisions. While maintaining the distributed nature of the system, it introduces a collaborative value function to achieve global coordination. This improves the responsiveness of resource scheduling and enhances overall system performance. To strengthen system foresight, a lightweight state prediction model is designed. It assists agents in identifying future workload trends and optimizes the selection of scaling actions. For policy training, the method adopts a centralized training and decentralized execution reinforcement learning framework. This enables agents to learn effectively and coordinate strategies under conditions of incomplete information. The paper also constructs typical cloud scenarios, including multi-tenancy and burst traffic, to evaluate the proposed method. The evaluation focuses on resource isolation, service quality assurance, and robustness. Experimental results show that the proposed multi-agent scaling strategy outperforms existing methods in resource utilization, SLA violation control, and scheduling latency. The results demonstrate strong adaptability and intelligent regulation. This provides an efficient and reliable new approach to solving the problem of elastic resource scaling in complex cloud platforms.

cs.DC↗

arXiv2025

The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation

Inspired by the quantization of classical quantities and Rankin Selberg convolution, we study the anticommutator operation $\{\cdot, \cdot\}$, where $\{A,B\} = AB + BA$, applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the $k$-checkerboard ensemble, and the block $k$-circulant ensemble ($k$-BCE). Using combinatorial and topological techniques related to non-crossing and free matching properties of GOE and PTE, we obtain closed-form formulae for the moments of the limiting spectral distributions of $\{$GOE, GOE$\}$, $\{$PTE, PTE$\}$, $\{$GOE, PTE$\}$ and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, $\{$GOE, $k$-checkerboard$\}$ and $\{$$k$-checkerboard, $j$-checkerboard$\}$ exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of $\{$GOE, $k$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$ and 1 blip regime of size $\Theta(N^{3/2})$, the spectrum of $\{$$k$-checkerboard, $j$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$, 2 intermediary blip regimes of size $\Theta(N^{3/2})$, and 1 largest blip regime of size $\Theta(N^2)$. In both cases, with the appropriate weight function, we are able to isolate the largest regime for other regime(s) and analyze its moments and convergence results via combinatorics. We end with numerical computation of lower even moments of $\{$GOE, $k$-BCE$\}$ and $\{$$k$-BCE, $k$-BCE$\}$ based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of $\{$$k$-checkerboard, $j$-checkerboard$\}$.

math.PR↗

arXiv2024

A closed formula for linear recurrences with constant coefficients

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed formula for the coefficients of each polynomial $p_r(n)$ for any linear recurrence of this form.

math.CO↗

arXiv2024

Vacillating parking functions

For any integers $1\leq k\leq n$, we introduce a new family of parking functions called $k$-vacillating parking functions of length $n$. The parking rule for $k$-vacillating parking functions allows a car with preference $p$ to park in the first available spot in encounters among the parking spots numbered $p$, $p-k$, and $p+k$ (in that order and if those spots exists). In this way, $k$-vacillating parking functions are a modification of Naples parking functions, which allow for backwards movement of a car, and of $\ell$-interval parking functions, which allow a car to park in its preference or up to $\ell$ spots in front of its preference. Among our results, we establish a combinatorial interpretation for the numerator of the $n$th convergent of the continued fraction of $\sqrt{2}$, as the number of non-decreasing $1$-vacillating parking functions of length~$n$. Our main result gives a product formula for the enumeration of $k$-vacillating parking functions of length $n$ based on the number of $1$-vacillating parking functions of smaller length. We conclude with some directions for further research.

math.CO↗

arXiv2019

Gaps of Summands of the Zeckendorf Lattice

A beautiful theorem of Zeckendorf states that every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. Such decompositions exist more generally, and much is known about them. First, for any positive linear recurrence {Gn} the number of summands in the legal decompositions for integers in [Gn, Gn+1) converges to a Gaussian distribution. Second, Bower, Insoft, Li, Miller, and Tosteson proved that the probability of a gap between summands in a decomposition which is larger than the recurrence length converges to geometric decay. While most of the literature involves one-dimensional sequences, some recent work by Chen, Guo, Jiang, Miller, Siktar, and Yu have extended these decompositions to d-dimensional lattices, where a legal decomposition is a chain of points such that one moves in all d dimensions to get from one point to the next. They proved that some but not all properties from 1-dimensional sequences still hold. We continue this work and look at the distribution of gaps between terms of legal decompositions, and prove similar to the 1-dimensional cases that when d = 2 the gap vectors converge to a bivariate geometric random variable.

math.NT↗

arXiv2018

Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions

Several recent papers have considered the problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to rings of integers in quadratic number fields and polynomial rings over finite fields. We study the analogous problem in the Hurwitz quaternion order to see how non-commutativity affects the problem. We compute an exact formula for the density of a 3-term geometric-progression-free set of Hurwitz quaternions arising from a greedy algorithm and derive upper and lower bounds for the supremum of upper densities of 3-term geometric-progression-free sets of Hurwitz quaternions.

math.CO↗