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1. Stated SL(n$$n$$)‐skein modules and algebras

   https://doi.org/10.1112/topo.12350  

   Lê, Thang_T Q; Sikora, Adam S (September 2024, Journal of Topology)

   Abstract We develop a theory of stated SL()‐skein modules, , of 3‐manifolds marked with intervals in their boundaries. These skein modules, generalizing stated SL(2)‐modules of the first author, stated SL(3)‐modules of Higgins', and SU(n)‐skein modules of the second author, consist of linear combinations of framed, oriented graphs, called ‐webs, with ends in , considered up to skein relations of the ‐Reshetikhin–Turaev functor on tangles, involving coupons representing the anti‐symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3‐manifold along a disk resulting in a 3‐manifold yields a homomorphism for all . That result allows to analyze the skein modules of 3‐manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces , in whose case, is an algebra, denoted by . One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on . (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary every splitting homomorphism is injective and that is a free module with a basis induced from the Kashiwara–Lusztig canonical bases. Additionally, we show that a splitting of a thickened bigon near a marking defines a right ‐comodule structure on , or dually, a left ‐module structure. Furthermore, we show that the skein algebra of surfaces glued along two sides of a triangle is isomorphic with the braided tensor product of Majid. These results allow for geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. Building upon the above results, we prove that the factorization homology with coefficients in the category of representations of is equivalent to the category of left modules over for surfaces with . We also establish isomorphisms of our skein algebras with the quantum moduli spaces of Alekseev–Schomerus and with the internal algebras of the skein categories for these surfaces and . 

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2. Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)

   https://doi.org/10.1093/imrn/rnaa218  

   Ford, Kevin (August 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We show that for some even $$k\leqslant 3570$$ and all  $$k$$ with $442720643463713815200|k$, the equation $$\phi (n)=\phi (n+k)$$ has infinitely many solutions $$n$$, where $$\phi $$ is Euler’s totient function. We also show that for a positive proportion of all $$k$$, the equation $$\sigma (n)=\sigma (n+k)$$ has infinitely many solutions $$n$$. The proofs rely on recent progress on the prime $$k$$-tuples conjecture by Zhang, Maynard, Tao, and PolyMath. 

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3. Preparation and Characterization of N , N′ ‐Dialkyl‐1,3‐propanedialdiminium Chlorides, N , N′ ‐Dialkyl‐1,3‐propanedialdimines, and Lithium N , N′ ‐Dialkyl‐1,3‐propanedialdiminates

   https://doi.org/10.1002/hlca.202200152  

   Schmit, Claire E.; Girolami, Gregory S. (May 2023, Helvetica Chimica Acta)
4. Subset Sum in Time 2^{n/2}/poly(n)

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.39  

   Chen, Xi; Jin, Yaonan; Randolph, Tim; Servedio, Rocco A. (January 2023, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   A major goal in the area of exact exponential algorithms is to give an algorithm for the (worst-case) n-input Subset Sum problem that runs in time 2^{(1/2 - c)n} for some constant c > 0. In this paper we give a Subset Sum algorithm with worst-case running time O(2^{n/2} ⋅ n^{-γ}) for a constant γ > 0.5023 in standard word RAM or circuit RAM models. To the best of our knowledge, this is the first improvement on the classical "meet-in-the-middle" algorithm for worst-case Subset Sum, due to Horowitz and Sahni, which can be implemented in time O(2^{n/2}) in these memory models [Horowitz and Sahni, 1974]. Our algorithm combines a number of different techniques, including the "representation method" introduced by Howgrave-Graham and Joux [Howgrave-Graham and Joux, 2010] and subsequent adaptations of the method in Austrin, Kaski, Koivisto, and Nederlof [Austrin et al., 2016], and Nederlof and Węgrzycki [Jesper Nederlof and Karol Wegrzycki, 2021], and "bit-packing" techniques used in the work of Baran, Demaine, and Pǎtraşcu [Baran et al., 2005] on subquadratic algorithms for 3SUM. 

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5. Fast Flow Reconstruction via Robust Invertible n × n Convolution

   https://doi.org/10.3390/fi13070179  

   Truong, Thanh-Dat; Duong, Chi Nhan; Tran, Minh-Triet; Le, Ngan; Luu, Khoa (July 2021, Future Internet)

   Flow-based generative models have recently become one of the most efficient approaches to model data generation. Indeed, they are constructed with a sequence of invertible and tractable transformations. Glow first introduced a simple type of generative flow using an invertible 1×1 convolution. However, the 1×1 convolution suffers from limited flexibility compared to the standard convolutions. In this paper, we propose a novel invertible n×n convolution approach that overcomes the limitations of the invertible 1×1 convolution. In addition, our proposed network is not only tractable and invertible but also uses fewer parameters than standard convolutions. The experiments on CIFAR-10, ImageNet and Celeb-HQ datasets, have shown that our invertible n×n convolution helps to improve the performance of generative models significantly. 

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