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1. Distributional stability of the Szarek and Ball inequalities

   https://doi.org/10.1007/s00208-023-02669-9  

   Eskenazis, Alexandros; Nayar, Piotr; Tkocz, Tomasz (June 2024, Mathematische Annalen)

   Abstract We prove an extension of Szarek’s optimal Khinchin inequality (1976) for distributions close to the Rademacher one, when all the weights are uniformly bounded by a$$1/\sqrt{2}$$ $1/\sqrt{2}$fraction of their total$$\ell _2$$ ${\ell }_2$-mass. We also show a similar extension of the probabilistic formulation of Ball’s cube slicing inequality (1986). These results establish the distributional stability of these optimal Khinchin-type inequalities. The underpinning to such estimates is the Fourier-analytic approach going back to Haagerup (1981). 

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2. Einstein–Klein–Gordon system via Cauchy-characteristic evolution: computation of memory and ringdown tail

   https://doi.org/10.1088/1361-6382/adaf6f  

   Ma, Sizheng; Nelli, Kyle C; Moxon, Jordan; Scheel, Mark A; Deppe, Nils; Kidder, Lawrence E; Throwe, William; Vu, Nils L (February 2025, Classical and Quantum Gravity)

   Abstract Cauchy-characteristic evolution (CCE) is a powerful method for accurately extracting gravitational waves at future null infinity. In this work, we extend the previously implemented CCE system within the numerical relativity code SpECTRE by incorporating a scalar field. This allows the system to capture features of beyond-general-relativity theories. We derive scalar contributions to the equations of motion, Weyl scalar computations, Bianchi identities, and balance laws at future null infinity. Our algorithm, tested across various scenarios, accurately reveals memory effects induced by both scalar and tensor fields and captures Price’s power-law tail ( $u^{-l-2}$) in scalar fields at future null infinity, in contrast to the $t^{-2l-3}$tail at future timelike infinity. 

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3. Programming emergent symmetries with saddle-splay elasticity

   https://doi.org/10.1038/s41467-019-13012-9  

   Xia, Yu; DeBenedictis, Andrew A.; Kim, Dae Seok; Chen, Shenglan; Kim, Se-Um; Cleaver, Douglas J.; Atherton, Timothy J.; Yang, Shu (November 2019, Nature Communications)

   Abstract The director field adopted by a confined liquid crystal is controlled by a balance between the externally imposed interactions and the liquid’s internal orientational elasticity. While the latter is usually considered to resist all deformations, liquid crystals actually have an intrinsic propensity to adopt saddle-splay arrangements, characterised by the elastic constant$${K}_{24}$$ $K_{24}$. In most realisations, dominant surface anchoring treatments suppress such deformations, rendering$${K}_{24}$$ $K_{24}$immeasurable. Here we identify regimes where more subtle, patterned surfaces enable saddle-splay effects to be both observed and exploited. Utilising theory and continuum calculations, we determine experimental regimes where generic, achiral liquid crystals exhibit spontaneously broken surface symmetries. These provide a new route to measuring$${K}_{24}$$ $K_{24}$. We further demonstrate a multistable device in which weak, but directional, fields switch between saddle-splay-motivated, spontaneously-polar surface states. Generalising beyond simple confinement, our highly scalable approach offers exciting opportunities for low-field, fast-switching optoelectronic devices which go beyond current technologies. 

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4. An Efficient ZK Compiler from SIMD Circuits to General Circuits

   https://doi.org/10.1007/s00145-024-09531-4  

   Bui, Dung; Chu, Haotian; Couteau, Geoffroy; Wang, Xiao; Weng, Chenkai; Yang, Kang; Yu, Yu (January 2025, Journal of Cryptology)

   Abstract We propose a generic compiler that can convert any zero-knowledge (ZK) proof for SIMD circuits to general circuits efficiently, and an extension that can preserve the space complexity of the proof systems. Our compiler can immediately produce new results improving upon state of the art.By plugging in our compiler to Antman, an interactive sublinear-communication protocol, we improve the overall communication complexity for general circuits from$$\mathcal {O}(C^{3/4})$$ $O\left(C^{3/4}\right)$to$$\mathcal {O}(C^{1/2})$$ $O\left(C^{1/2}\right)$. Our implementation shows that for a circuit of size$$2^{27}$$ $2^{27}$, it achieves up to$$83.6\times $$ $83.6\times$improvement on communication compared to the state-of-the-art implementation. Its end-to-end running time is at least$$70\%$$ $70\%$faster in a 10Mbps network.Using the recent results on compressed$$\varSigma $$ $\Sigma$-protocol theory, we obtain a discrete-log-based constant-round zero-knowledge argument with$$\mathcal {O}(C^{1/2})$$ $O\left(C^{1/2}\right)$communication and common random string length, improving over the state of the art that has linear-size common random string and requires heavier computation.We improve the communication of a designatedn-verifier zero-knowledge proof from$$\mathcal {O}(nC/B+n^2B^2)$$ $O(nC/B+n^2{B}^2)$to$$\mathcal {O}(nC/B+n^2)$$ $O(nC/B+n^2)$.To demonstrate the scalability of our compilers, we were able to extract a commit-and-prove SIMD ZK from Ligero and cast it in our framework. We also give one instantiation derived from LegoSNARK, demonstrating that the idea of CP-SNARK also fits in our methodology. 

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5. The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

   https://doi.org/10.1007/s00526-021-01938-2  

   Banerjee, Agnid; Danielli, Donatella; Garofalo, Nicola; Petrosyan, Arshak (April 2021, Calculus of Variations and Partial Differential Equations)

   Abstract We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$ ${\left|y\right|}^a$for$$a \in (-1,1)$$ $a\in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$ ${({\partial }_t-{\Delta }_x)}^s$for$$s \in (0,1)$$ $s\in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ $a=0$). 

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