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Explicit Factorization of $x^{p+1}-1$ over $\mathbb{Z}_{p^e}$: A Structural Approach via Dickson Polynomials
arXiv:2604.19038v2 Announce Type: replace Abstract: Let $p$ be an odd prime. The factorization of the polynomial $x^{p+1}-1$ over the integer residue ring $\mathbb{Z}_{p^e}$ is pivotal for constructing cyclic codes with Hermitian symmetry, a critical resource for Linear Complementary Dual (LCD) codes and Entanglement-Assisted Quantum Error-Correcting Codes (EAQECC). Traditionally, lifting factorizations relies on the generic Hensel's Lemma, masking the underlying algebraic structure.
Weighted hp-Uniform Decompositions for H^k-Type Tensor-Product Spaces in Arbitrary Dimension
arXiv:2606.05615v1 Announce Type: new Abstract: We establish weighted hp-uniform vertex-patch decompositions in arbitrary space dimension d >= 1 for tensor-product discretizations of H^k-type conforming and nonconforming spaces, with arbitrary fixed Sobolev order k >= 1, on fitted interface meshes. The cells are coordinate-compatible cuboids, the local spaces are Q_{p_K}(K) with arbitrary elementwise degrees satisfying p_K >= 2k-1, and the coefficient may have arbitrarily large jumps across...
Bregman meets L\'evy: Stochastic mirror descent with heavy-tailed noise in continuous and discrete time
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Empirical Approximation of $L_p$ Norms
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Tight Long-Term Tail Decay of (Clipped) SGD in Non-Convex Optimization
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Scale-Invariant Neural Network Optimization: Norm Geometry and Heavy-Tailed Noise
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Efficient Mean Curvature Computation on High-Dimensional Data Manifolds
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Lean 4 Machine-Verified Proof of P = NP via the Pedigree Polytope Membership Problem
arXiv:2606.03194v1 Announce Type: new Abstract: The Membership Problem for Pedigree Polytope (M3P) asks, given $X\in\mathbb{Q}^{\binom{n}{3}}$, whether $X\in\mathrm{conv}(P_n)$, where $P_n$ is the set of all pedigrees. A pedigree is a structured encoding of a Hamiltonian cycle construction in $K_n$. We establish that M3P is solvable in strongly polynomial time via a recursively constructed layered network $(N_k, R_k, \mu)$ and a multicommodity flow problem MCF$(k)$. The necessary and...