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1. On Multilinear Forms: Bias, Correlation, and Tensor Rank

   Bhrushundi, Abhishek; Harsha, Prahladh; Hatami, Pooya; Kopparty, Swastik; Kumar, Mrinal (August 2020, Leibniz international proceedings in informatics)

   Byrka, Jaroslaw; Meka, Raghu (Ed.)

   In this work, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over F₂. We show the following results for multilinear forms and tensors. Correlation bounds. We show that a random d-linear form has exponentially low correlation with low-degree polynomials. More precisely, for d = 2^{o(k)}, we show that a random d-linear form f(X₁,X₂, … , X_d) : (F₂^{k}) ^d → F₂ has correlation 2^{-k(1-o(1))} with any polynomial of degree at most d/2 with high probability. This result is proved by giving near-optimal bounds on the bias of a random d-linear form, which is in turn proved by giving near-optimal bounds on the probability that a sum of t random d-dimensional rank-1 tensors is identically zero. Tensor rank vs Bias. We show that if a 3-dimensional tensor has small rank then its bias, when viewed as a 3-linear form, is large. More precisely, given any 3-dimensional tensor T: [k]³ → F₂ of rank at most t, the bias of the 3-linear form f_T(X₁, X₂, X₃) : = ∑_{(i₁, i₂, i₃) ∈ [k]³} T(i₁, i₂, i₃)⋅ X_{1,i₁}⋅ X_{2,i₂}⋅ X_{3,i₃} is at least (3/4)^t. This bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds. In particular, we use this approach to give a new proof that the finite field multiplication tensor has tensor rank at least 3.52 k, which is the best known rank lower bound for any explicit tensor in three dimensions over F₂. Moreover, this relation between bias and tensor rank holds for d-dimensional tensors for any fixed d. 

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2. A lower bound on end-periodic stretch factors

   https://doi.org/10.1090/proc/17304  

   Loving, Marissa; Wu, Chenxi (July 2025, Proceedings of the American Mathematical Society)

   Given an end-periodic homeomorphism $f:S\to <\#comment/>S$we give a lower bound on the Handel–Miller stretch factor of $f$in terms of thecore characteristic of $f$, which is a measure of topological complexity for an end-periodic homeomorphism. We also show that the growth rate of this bound is sharp. 

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3. From antiferromagnetic and hidden order to Pauli paramagnetism in U M ₂ Si ₂ compounds with 5 f electron duality

   https://doi.org/10.1073/pnas.2005701117  

   Amorese, Andrea; Sundermann, Martin; Leedahl, Brett; Marino, Andrea; Takegami, Daisuke; Gretarsson, Hlynur; Gloskovskii, Andrei; Schlueter, Christoph; Haverkort, Maurits W.; Huang, Yingkai; et al (December 2020, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Using inelastic X-ray scattering beyond the dipole limit and hard X-ray photoelectron spectroscopy we establish the dual nature of the U 5 f electrons in U M 2 S i 2 (M = Pd, Ni, Ru, Fe), regardless of their degree of delocalization. We have observed that the compounds have in common a local atomic-like state that is well described by the U 5 f 2 configuration with the Γ 1 ( 1 ) and Γ 2 quasi-doublet symmetry. The amount of the U 5 f 3 configuration, however, varies considerably across the U M 2 S i 2 series, indicating an increase of U 5f itineracy in going from M = Pd to Ni to Ru and to the Fe compound. The identified electronic states explain the formation of the very large ordered magnetic moments in U P d 2 S i 2 and U N i 2 S i 2 , the availability of orbital degrees of freedom needed for the hidden order in U R u 2 S i 2 to occur, as well as the appearance of Pauli paramagnetism in U F e 2 S i 2 . A unified and systematic picture of the U M 2 S i 2 compounds may now be drawn, thereby providing suggestions for additional experiments to induce hidden order and/or superconductivity in U compounds with the tetragonal body-centered T h C r 2 S i 2 structure. 

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4. Counting points on smooth plane quartics

   https://doi.org/10.1007/s40993-022-00397-8  

   Costa, Edgar; Harvey, David; Sutherland, Andrew V. (March 2023, Research in Number Theory)

   Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over  $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest. 

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5. Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

   https://doi.org/10.1017/fmp.2021.16  

   Chen, Xuwen; Holmer, Justin (January 2022, Forum of Mathematics, Pi)

   Abstract We consider the $$\mathbb {T}^{4}$$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U - V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U - V estimates then seamlessly conclude the $$H^{1}$$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $$H^{1}$$ uniqueness for the $$ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS. 

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