We formalize AI alignment as a multi-objective optimization problem called $\langle M,N,\varepsilon,δ\rangle$-agreement, in which a set of $N$ agents (including humans) must reach approximate ($\varep...

…A Method for Training New Moms… Produced By: Yes, And! Theatrical Company in partnership with Virginia Repertory Theatre At: Theatre Gym at VaRep, 114 West Broad Street, RVA 23220 Performances: May 15-31, 2025 Ticket Prices: $40 general admission Info: Email: yesandrva@gmail.com; Website: yesandrva.org; Facebook: www.facebook.com/yesandrva/ A Theater Review by Julinda D. Lewis Some definitions you might need to know before seeing…

面白い妄想ですね。最近の実験結果（Nature 2025や関連理論）と照らし合わせると、かなり的を射たアナロジーだと思います。 閉じ込め水の事実確認 ナノ閉じ込め水（特に1–2 nm程度の準2次元水）では、確かに極端な誘電異方性が観測されています： - 面内誘電率 \(\varepsilon_\parallel \approx 1000\)（強誘電体並みの巨大値） - 面外誘電率 \(\varep...

**Authors:** Ethan N. W. Epperly The randomly pivoted Cholesky algorithm is one of the leading methods for computing a low-rank approximation to a large positive-semidefinite matrix. However, while it consistently achieves accuracy comparable to or better than competing methods of its type in experiments, its theoretical analysis lags somewhat behind other methods. This paper closes this gap, proving that randomly pivoted Cholesky produces an approximation with expected error within a $1+\varepsilon$ factor of the optimal rank-$r$ approximation in $\mathcal{O}(r/\varepsilon + r\sqrt{\log r})$ steps. This result nearly matches the optimal complexity $Θ(r/\varepsilon)$ for any low-rank approximation method based on a partial Cholesky decomposition (also known as a column Nyström approximation). The paper also presents bounds on the randomly pivoted Cholesky trace and spectral-norm errors that hold with high probability. The mathematical argument is largely due to GPT 5.6-Sol (Pro), with some refinements by the author.

**Authors:** Nicolas Menand, Erik Waingarten We provide a local computation algorithm to approximate the top eigenvector $x \in \mathbb{R}^n$ of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries between $-1$ and $1$, building on the work of Swartworth and Woodruff [SODA 25] who show how to approximate the eigenvalues up to additive-$\varepsilon n$ error using $\tilde{O}(1/\varepsilon^4)$ queries. Our local computation algorithm has a preprocessing complexity of $\tilde{O}(1/\varepsilon^4)$ and per-coordinate query complexity of $\tilde{O}(1/\varepsilon^2)$ for an additive-$\varepsilon n$ approximation whenever {$|λ_{\min}(A)| = O(λ_{\max}(A))$. When $λ_{\min}(A)$ greatly exceeds $λ_{\max}(A)$, our complexity degrades to at most $\tilde{O}(1/\varepsilon^{6.\overline{6}})$ in preprocessing and $\tilde{O}(1/\varepsilon^{3.\overline{3}})$ per query. Furthermore, we show a lower bound of $Ω(n/\varepsilon^2)$ on the total number of queries needed to output an approximately top eigenvector (implying that the per-coordinate query complexity of $Ω(1/\varepsilon^2)$ is necessary). As an application, we use our algorithm to provide local computation algorithms for the sparsest-cut and max-cut problems in the dense graph model of Goldreich, Goldwasser, Ron [JACM 98]. By accessing the top eigenvectors (of an approximate normalized adjacency), we implement local versions of Cheeger's inequality and Trevisan's algorithm [SICOMP 12] to obtain "square-root-opt" approximations in polynomial time (as opposed to exponential-in-$\text{poly}(1/\varepsilon)$ time which is incurred in Goldreich, Goldwasser, Ron.

Sommaire rédigé par l’IA Chambre commerciale, Cour de cassation, le 9 mai 2007, n° 06-10.928 Date de la décision: May 9, 2007 Dans cette affaire, le tribunal avait ouvert une procédure de redressement judiciaire à l'égard de la société Varep informatique, en fixant un délai pour établir la liste des créances. Le receveur des impôts a demandé un prolongement de ce délai.

**Authors:** Sándor Kisfaludi-Bak, Leonidas Theocharous The presence of obstacles has a major impact on distance computation, motion planning, and visibility. While these problems are well studied in the plane, our understanding in three and higher dimensions is still limited. We investigate how different obstacle properties affect the induced geodesic metric in three-dimensional space. A finite metric space is said to be approximately realizable by a collection of obstacles if, for any $\varepsilon>0$, it can be embedded into the free space around the obstacles with geodesic distance and worst-case distortion $1+\varepsilon$. We focus on three key properties-convexity, disjointness, and fatness-and analyze how omitting each of them influences realizability. Our main result shows that if fatness is dropped, then every finite metric space can be realized with distortion $1+\varepsilon$ using convex, pairwise disjoint obstacles in $\mathbb{R}^3$, even if all obstacles are congruent equilateral triangles. Moreover, if we enforce fatness but drop convexity or disjointness, the same realizability still holds. Our results have important implications on the approximability of TSP with Obstacles, a natural variant of TSP introduced recently by Alkema et al. (ESA 2022). Specifically, we use the recent results of Banerjee et al. on TSP in doubling spaces (FOCS 2024) and of Chew et al. on distances among obstacles (Information Processing Letters 2002) to show that TSP with Obstacles admits a PTAS if the obstacles are convex, fat, and pairwise disjoint. If any of these three properties is dropped, then our results, combined with the APX-hardness of Metric TSP, demonstrate that TSP with Obstacles is APX-hard.