arXiv:2604. 08485v2 Announce Type: replace-cross Abstract: The purpose of this paper is two-fold.
arXiv:2604. 02995v3 Announce Type: replace-cross Abstract: We introduce the penalised Saito functional $\mathfrak S_{\lambda,\beta}(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ of $n$ lines and a prescribed pair $d_1+d_2=n-1$.
The article reports a machine‑checked Lean 4 formalization of Dong and Yang’s classification of optimal finite‑length $(n,4)$ binary block codes for binary symmetric channels. The authors used an AI tool to feed the paper’s proofs into Lean, verified the main theorem statements and accepted axioms, and documented corrections, simplifications, and discrepancies found during the formalization. The Lean code is publicly available on GitHub.
arXiv:2609. 20231v1 Announce Type: cross Abstract: Let \(\cthreshold(n)\) be the largest integer \(q\) such that every loopless digraph on \(n\) vertices with at most \(q\) arcs is isomorphic to a spanning subdigraph of a self-complementary digraph of order \(n\).
arXiv:2409. 15600v3 Announce Type: replace Abstract: A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general.
The paper investigates the polynomial coefficients of lightning self‑attention, treating them as coordinates of an algebraic variety. In the single‑token case it identifies the coefficient variety as a rank‑constrained Chow‑type variety and derives algebraic equations; for multiple tokens it shows that linear relations reduce the geometry to coefficients involving interactions between distinct tokens, characterized by a common linear factor and a low‑rank condition. The authors provide explicit families of determinantal, Veronese‑type, and Sylvester resultant‑based invariants, and in the rank‑one case give pencil and flattening equations that define the variety set‑theoretically, with small‑dimension computations confirming the theoretical generators.
arXiv:2607. 21651v1 Announce Type: new Abstract: We prove that the maximum of $n$ real numbers is exactly representable by a ReLU network with two hidden layers for every $n\le 10$.
arXiv:2605. 29151v2 Announce Type: replace-cross Abstract: We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli.
arXiv:2607. 02576v1 Announce Type: cross Abstract: Parameter-Efficient Fine-Tuning (PEFT) commonly adapts pretrained weights through low-rank updates, and recent methods further exploit the singular value decomposition (SVD) of the base weight for initialization or subspace selection.
arXiv:2608. 14396v1 Announce Type: cross Abstract: The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades.
arXiv:2609. 04583v1 Announce Type: new Abstract: The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty.
arXiv:2602. 16807v2 Announce Type: replace Abstract: A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior.
arXiv:2609.16412v1 Announce Type: cross Abstract: Whitney's theorem allows isomorphism testing for connected simple graphs, apart from $K_3$ and $K_{1,3}$, to be formulated as distinguishing their li...