arXiv subjects

Jiechen Zhang

Publications and source records attributed to Jiechen Zhang.

5 recordsLinked to original sources

arXiv2026

Extremal Spanning Trees in Product Grid Graphs

We study how fixed-volume spanning-tree extremality changes when product-grid boundary factors are free, periodic, or mixed. In two dimensions, extremality depends sharply on the boundary type. The free/free and periodic/periodic products both obey a closest-to-square principle: among fixed-area rectangles, $P_r\square P_s$ and $C_r\square C_s$ are maximized by the closest-to-square admissible side lengths. The mixed free/periodic cylinder $P_r\square C_s$ is different: closest-to-square fails, and in the divisor-rich case the optimizing cyclic circumference has scale $N^{1/3}$ when the area is $N=rs$. In arbitrary dimension we prove pairwise balancing theorems for pure free products and pure periodic products, and then strengthen them by a heat-trace Schur-concavity theorem in logarithmic side lengths. At perfect-power volume this gives the unique maximizers $P_n^{\square d}$ and, for $n\ge3$, $C_n^{\square d}$. These product-grid comparisons motivate perfect-power conjectures for connected induced lattice subgraphs and periodic analogues.

math.CO↗

arXiv2026

The Balancing Theorem for Spanning Trees of Rectangular Grid Graphs

We compare spanning-tree counts of rectangular grid graphs at fixed area. The main result is a balancing theorem: if $AB=ab$ and $A\le a\le b\le B$, then the more balanced rectangle has larger spanning-tree count, and in fact \[ \log\tau(a,b)-\log\tau(A,B) \ge \operatorname{arsinh}(1)\bigl((A+B)-(a+b)\bigr). \] The universal constant $\operatorname{arsinh}(1)=\log(1+\sqrt{2})\approx0.88137$ is optimal. Thus the log-ratio between equal-area rectangles grows at least linearly in the reduction of the sum of side lengths. The proof starts from the Laplacian product formula, passes to hyperbolic coordinates, and gives an exact finite-size identity for the log-gain of a balancing move. A monotone trapezoidal estimate gives the optimal main term, while a positive monotone residual gives strictness. The same decomposition identifies the closest-to-square divisor rectangle as the unique fixed-area maximizer and recovers the bounded-aspect rectangular asymptotic through the logarithmic corner term, including the square-grid case.

math.CO↗

arXiv2026

Threshold Rules for the Classical Prophet Inequality

This note records a common threshold/surplus decomposition for single-threshold stopping rules in the classical prophet inequality. The same decomposition is used to certify several deterministic thresholds, including the median, half-mean, and balanced-surplus thresholds, and to give an averaged certificate for randomized thresholds distributed as the maximum.

math.PR↗

arXiv2026

Explicit Universal Bounds for Cumulants via Moments

We establish explicit, universal, and distribution-free bounds for the $n$-th cumulant, $κ_n(X)$, of a scalar random variable, controlled solely by an $n$-th order absolute moment functional $M_n(X)$. The bounds take the form $\lvertκ_n(X)\rvert \le C_n M_n(X)$. Our principal contribution is the derivation of coefficients satisfying $C_n \sim (n-1)!/ρ^{\,n}$, which offers an exponential improvement over classical bounds where the coefficients grow superexponentially (on the order of $n^n$). We present a hierarchy of refinements where the rate parameter $ρ$ increases as the functional $M_n(X)$ incorporates more structural information. The most general bound uses the raw moment $M_n(X)=\mathsf{E}[\lvert X\rvert^n]$ with rate $ρ=\ln 2 \approx 0.693$. Using the central moment $M_n(X)=\mathsf{E}[\lvert X-\mathsf{E}[X]\rvert^n]$ improves the rate to $ρ_{\mathrm{cen}} \approx 1.146$, while assuming symmetry yields even higher rates. The proof is elementary, combining the moment-cumulant partition formula with a uniform moment-product inequality. We further prove that while these bounds are not attainable whenever the relevant coefficient is positive, they are asymptotically efficient given the limited information of a single moment. The utility of the bounds is demonstrated through an application to standardized cumulants of independent sums.

math.PR↗

arXiv2024

Sam2Rad: A Segmentation Model for Medical Images with Learnable Prompts

Foundation models like the segment anything model require high-quality manual prompts for medical image segmentation, which is time-consuming and requires expertise. SAM and its variants often fail to segment structures in ultrasound (US) images due to domain shift. We propose Sam2Rad, a prompt learning approach to adapt SAM and its variants for US bone segmentation without human prompts. It introduces a prompt predictor network (PPN) with a cross-attention module to predict prompt embeddings from image encoder features. PPN outputs bounding box and mask prompts, and 256-dimensional embeddings for regions of interest. The framework allows optional manual prompting and can be trained end-to-end using parameter-efficient fine-tuning (PEFT). Sam2Rad was tested on 3 musculoskeletal US datasets: wrist (3822 images), rotator cuff (1605 images), and hip (4849 images). It improved performance across all datasets without manual prompts, increasing Dice scores by 2-7% for hip/wrist and up to 33% for shoulder data. Sam2Rad can be trained with as few as 10 labeled images and is compatible with any SAM architecture for automatic segmentation.

cs.CV↗