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1. 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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2. 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)

   We describe convenient preparations ofN,N′‐dialkyl‐1,3‐propanedialdiminium chlorides,N,N′‐dialkyl‐1,3‐propanedialdimines, and lithiumN,N′‐dialkyl‐1,3‐propanedialdiminates in which the alkyl groups are methyl, ethyl, isopropyl, ortert‐butyl. For the dialdiminium salts, the N₂C₃backbone is always in thetrans‐s‐transconfiguration. Three isomers are present in solution except for thetert‐butyl compound, for which only two isomers are present; increasing the steric bulk of theN‐alkyl substituents shifts the equilibrium away from the (Z,Z) isomer in favor of the (E,Z), and (E,E) isomers. For the neutral dialdimines, crystal structures show that the methyl and isopropyl compounds adopt the (E,Z) form, whereas thetert‐butyl compound is in the (E,E) form. In aprotic solvents all four dialdimines (as well as the lithium dialdiminate salts) adoptcis‐s‐cisconformations in which there presumably is either an intramolecular hydrogen bond (or a lithium cation) between the two nitrogen atoms.

    

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3. 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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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. Hidden U(N) symmetry behind $$ \mathcal{N} $$ = 1 superamplitudes

   https://doi.org/10.1007/JHEP11(2023)215  

   Delgado, Antonio ; Martin, Adam ; Wang, Runqing ( November 2023 , Journal of High Energy Physics)

   In this paper we develop a Young diagram approach to constructing higher dimensional operators formed from massless superfields and their superderivatives in$$ \mathcal{N} $$$N$= 1 supersymmetry. These operators are in one-to-one correspondence with non-factorizable terms in on-shell superamplitudes, which can be studied with massless spinor helicity techniques. By relating all spin-helicity variables to certain representations under a hidden U(N) symmetry behind the theory, we show each non-factorizable superamplitude can be identified with a specific Young tableau. The desired tableau is picked out of a more general set of U(N) tensor products by enforcing the supersymmetric Ward identities. We then relate these Young tableaux to higher dimensional superfield operators and list the rules to read operators directly from Young tableau. Using this method, we present several illustrative examples.

    

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