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Nearly a billion dollars in pro bono payola buys a whole lot of silence.

Nearly a billion dollars in pro bono payola buys a whole lot of silence.

Nearly a billion dollars in pro bono payola buys a whole lot of silence.

In 4 giorni è stato prenotato il 50% dei posti disponibili agli eventi

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**Authors:** Huanyi Zhou Graph tensors extend the matrix multiplication tensor by placing a two-party identity tensor on every edge of a graph and keeping one tensor leg at every vertex. We study the asymptotic rank of the tensor associated with the complete graph. For every integer $n\geq 2$, over the complex numbers, we prove \\[ \left\lfloor\frac{n^2}{4}\right\rfloor \leq \log_2\widetilde R\bigl(\mathsf{GT}_2(K_n)\bigr) \leq \frac{n^2}{4}. \\] The upper bound matches the maximum-cut lower bound for every even $n$ and differs from it by only $1/4$ for every odd $n$. In particular, the exponent per edge is below $2/3$ for every $n\geq5$, answering a question of Christandl, Vrana, and Zuiddam. Our proof extends the polynomial multiplication approach underlying OpenAI's bound $ω\leq9/4$. The main step separates any fixed number of shared tensor legs simultaneously, using only linearly many direct-sum copies in the number of blocks. This yields an entropy inequality for higher-order tensors. We apply it to the tensor for multiplying $n-1$ polynomials, whose rank is linear in the input length. Discrete concavity and a shifted tripling inequality then force a lower bound on a symmetrized spectral profile. Comparing this growth with polynomial interpolation gives the stated upper bound while preserving all $n$ tensor legs. We also prove an operational barrier for these higher-order polynomial constructions. For $n\geq3$, the specified formal reduction rules cannot certify an exponent below $n^2/4$ for $\mathsf{GT}_2(K_n)$, even after taking tensor powers and allowing subexponential overhead and common auxiliary factors. This extends the symmetric three-leg barrier of Amanov, van den Berg, and Zuiddam and explains the remaining gap for odd complete graphs within this method.

そういえば、大昔のbiglobeのパスワードリセットって、契約書類に書いてある初期パスワードに戻るような実装だった覚えがあるけど、今はさすがに違う……？(やめろ

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Anàlisi complexa · Teorema dels residus Càlcul d’una integral impròpia Mètode dels residus i aplicacions a la física 1. Enunciat En la teoria de la corrent electrònica entre dos gasos electrònics bidimensionals apareix la integral \[ I=\int_{-\infty}^{+\infty} \frac{dx}{ \bigl[(x-\varepsilon_1)^2+a^2\bigr] \bigl[(x-\varepsilon_2)^2+a^2\bigr] }. \] On \(\varepsilon_1,\varepsilon_2\in\mathbb{R}\) i \(a>0\). Calculeu \(I\) mitjançant el mètode dels residus. 2. Definició de la funció complexa Considerem la funció complexa…

Camp vectorial: divergència, rotacional i integral de línia Estudi del camp, potencial i aplicació dels teoremes vectorials Enunciat Sigui el camp vectorial \[ \mathbf f(x,y,z) = \bigl(y^2,\,2xy,\,-2z\bigr). \] Es demana: Calcular \(\operatorname{div}\mathbf f\), \(\operatorname{rot}\mathbf f\), \(\nabla(\operatorname{div}\mathbf f)\) i \(\Delta(\mathbf f\cdot\mathbf f)\). Determinar si \(\mathbf f\) és conservatiu. Calcular \[ \iiint_V \operatorname{div}\mathbf f\,dV, \] on \(V\) és el sòlid limitat per…