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1. 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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2. Beyond N = ∞ in large N conformal vector models at finite temperature

   https://doi.org/10.1007/JHEP08(2024)219  

   Diatlyk, Oleksandr; Popov, Fedor K; Wang, Yifan (August 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We investigate finite-temperature observables in three-dimensional largeNcritical vector models taking into account the effects suppressed by$$ \frac{1}{N} $$ $\frac{1}{N}$. Such subleading contributions are captured by the fluctuations of the Hubbard-Stratonovich auxiliary field which need to be handled with care due to a subtle divergence structure which we clarify. The examples we consider include the scalarO(N) model, the Gross-Neveu model, the Nambu-Jona-Lasinio model and the massless Chern-Simons Quantum Electrodynamics. We present explicit results for the free energy density to the subleading order in$$ \frac{1}{N} $$ $\frac{1}{N}$, which captures the thermal one-point function of the stress-energy tensor to this order. We also include the dependence on a chemical potential. We determine the Wilson coefficient in the thermal effective action that is sensitive to global symmetry for the first time directly in interacting CFTs, which produces a symmetry-resolved asymptotic density of states. We further provide a formula from diagrammatics for the one-point functions of general single-trace higher-spin currents. We observe that in most cases considered, these subleading effects lift the apparent degeneracies between observables in different models at infiniteN, while in special cases the discrepancies only start to appear at the next-to-subleading order. 

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3. Pyridine vs. Thiazole in Cyclometalated N^C^N Ni(II) Complexes

   https://doi.org/10.3390/inorganics13020041  

   Kletsch, Lukas; Jordan, Rose; Strippel, Julian; Vicic, David A; Klein, Axel (February 2025, Inorganics)

   Six N^C^N cyclometalated Ni(II) complexes [Ni(N^C^N)Cl] or [Ni(N^C^N’)Br] with symmetric N^C^N or non-symmetric N^C^N’ ligands in which the peripheral N-groups were varied with pyridine (Py), 4-thiazole (4Tz), 2-thiazole (2Tz), and 2-benzothiazole (2Btz) complementing the previously reported complexes with di(2-pyridyl)phenide ligands [Ni(Py(Ph)Py)X] X = Cl or Br. The non-symmetric [Ni(N^C^N’)Br] complexes were synthesized from NiBr2 and N^CH^N’ protoligands through base-assisted nickelation, while the symmetric [Ni(N^C^N)Cl] complexes were received from the N^C(Cl)^N protoligands and [Ni(COD)2] (COD = 1,5-cyclooctadiene). Introduction of 4Tz on both sides shifted the electrochemical gap ΔEexp = Eox–Ered and the long wavelength UV-vis absorption maxima of the complexes to higher energies, while 2Tz leads to a shift to lower energies. When introducing only one 4Tz or 2Tz as peripheral groups, the remaining PhPy moiety dominates the electronic properties and electrochemistry and photophysics are very similar to the Py(Ph)Py derivatives. In contrast to this, introduction of 2Btz shifts both values to lower energies, regardless of one or two 2Btz groups and the 2Btz moiety dominates the character of the frontier molecular orbitals of the complexes, as DFT calculations show. Long-wavelength UV-vis absorptions vary from 416 to 443 nm, and their energies correlate well with the first reduction potentials. Negishi-type C–C cross-coupling reactions gave total yields ranging from 1 to 60% and cross-coupling yields from 1 to 44%. The reactivities correlate roughly with the first reduction potentials. Facilitated reduction (E around –2 or higher) goes generally along with improved performance, making the thiazole-containing complexes interesting candidates for such catalysis. 

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4. N-Degron Pathways in Plastids

   https://doi.org/10.1016/j.tplants.2019.06.013  

   Bouchnak, Imen; van Wijk, Klaas J. (October 2019, Trends in Plant Science)
5. Large N limit and 1/N expansion of invariant observables in O(N) linear $$\sigma $$-model via SPDE

   https://doi.org/10.1007/s00440-025-01361-0  

   Shen, Hao; Zhu, Rongchan; Zhu, Xiangchan (April 2025, Probability Theory and Related Fields)