arXiv subjects

Ziyu Gan

Publications and source records attributed to Ziyu Gan.

3 recordsLinked to original sources

arXiv2026

Extremizers for a trilinear Stein-Weiss inequality with nonnegative weights

We study extremizers for a trilinear Stein-Weiss inequality on $\mathbb{R}^n$. Within the known boundedness region, we prove attainment under two additional assumptions: all six weight exponents are nonnegative, and at least one pair of Lebesgue exponents is admissible. The proof combines symmetric decreasing rearrangement with a logarithmic radial reduction to a translation-invariant bilinear operator on $\mathbb{R}$ whose kernel belongs to $L^1\left(\mathbb{R}^2\right)$. A common-scale compactness argument rules out relative separation of the two arguments and yields norm attainment. We then derive the Euler-Lagrange system. In the fully symmetric case, every normalized nonnegative extremizing triple is diagonal. Finally, we establish the origin-centered Kelvin invariance of the resulting scalar equation at the scaling exponent and record the unweighted conformal example.

math.AP↗

arXiv2026

Agent Memory: Characterization and System Implications of Stateful Long-Horizon Workloads

LLM agents are increasingly deployed on long-horizon tasks requiring sustained reasoning over extended interaction histories. Realizing this at scale requires agents to persistently store, retrieve, and update their own memory across sessions. A rich ecosystem of agent memory systems has emerged spanning flat retrieval, LLM-mediated extraction, consolidating fact stores, and agentic control flows. Yet, their system-level behavior remains uncharacterized. We present the first systems characterization of agent memory. First, we introduce a system-oriented taxonomy classifying agent memory systems along four axes. Second, we build a phase-aware profiling harness attributing cost to construction, retrieval, and generation. Third, we characterize ten representative systems across two benchmark suites, uncovering how design choices shift cost across the write and read paths. Finally, we derive 10 system recommendations covering construction scheduling, capability floors, amortization via query volume, freshness-latency tradeoffs, and fleet-scale management.

cs.AI↗

arXiv2025

Locally uniform ellipticity of the fractional Hessian operators

In [1], Caffarelli-Charro introduced a fractional Monge-Ampère operator. Later, Wu [17] generalized it to a fractional analogue of $k$-Hessian operators and proved the strict ellipticity for $k=2$. In this paper, we introduce a fractional analogue of general Hessian operators and prove the stability. We also show that the fractional analogue $k$-Hessian operators defined in [17] are strictly elliptic with respect to convex solutions for all $2 \leq k \leq n$. Furthermore, we provide a new proof for the case $k=2$ without the convexity condition.

math.AP↗