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1. Mimetic finite difference operators and higher order quadratures

   https://doi.org/10.1007/s13137-023-00230-z  

   Srinivasan, Anand; Dumett, Miguel; Paolini, Christopher; Miranda, Guillermo F.; Castillo, José E. (December 2023, GEM - International Journal on Geomathematics)

   Abstract Mimetic finite difference operators$$\textbf{D}$$ $D$,$$\textbf{G}$$ $G$are discrete analogs of the continuous divergence (div) and gradient (grad) operators. In the discrete sense, these discrete operators satisfy the same properties as those of their continuum counterparts. In particular, they satisfy a discrete extended Gauss’ divergence theorem. This paper investigates the higher-order quadratures associated with the fourth- and sixth- order mimetic finite difference operators, and show that they are indeed numerical quadratures and satisfy the divergence theorem. In addition, extensions to curvilinear coordinates are treated. Examples in one and two dimensions to illustrate numerical results are presented that confirm the validity of the theoretical findings. 

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2. Computing Proximity Operators of Scale and Signed Permutation Invariant Functions

   https://doi.org/10.1007/s10957-025-02871-6  

   Jia, Jianqing; Prater-Bennette, Ashley; Shen, Lixin (January 2026, Journal of Optimization Theory and Applications)

   Abstract This paper investigates the computation of proximity operators for scale and signed permutation invariant functions. A scale invariant function remains unchanged under uniform scaling, while a signed permutation invariant function retains its structure despite permutations and sign changes applied to its input variables. Noteworthy examples include the$$\ell _0$$ ${\ell }_0$function, the ratio of$$\ell _1/\ell _2$$ ${\ell }_1/{\ell }_2$, and its square, with their proximity operators being particularly crucial in sparse signal recovery. We delve into the properties of scale and signed permutation invariant functions, delineating the computation of their proximity operators into three sequential steps: the$${\varvec{w}}$$ $w$-step,r-step, andd-step. These steps collectively form a procedure termed as WRD, with the$${\varvec{w}}$$ $w$-step being of utmost importance and requiring careful treatment. Leveraging this procedure, we present a method for explicitly and efficiently computing the proximity operator of$$(\ell _1/\ell _2)^2$$ ${({\ell }_1/{\ell }_2)}^2$and introduce an algorithm for the proximity operator of$$\ell _1/\ell _2$$ ${\ell }_1/{\ell }_2$. Numerical experiments on sparse signal recovery corroborate the analysis and show that first-order methods equipped with these proximity operators outperform$$\ell _1$$ ${\ell }_1$-based baselines in reconstruction accuracy. 

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3. Endpoint Estimates for Haar Shift Operators with Balanced Measures

   https://doi.org/10.1007/s12220-025-02080-7  

   Conde_Alonso, José M; Wagner, Nathan A (September 2025, The Journal of Geometric Analysis)

   Abstract We prove$$\textrm{H}^1$$ ${\text{H}}^1$and$$\textrm{BMO}$$ $\text{BMO}$endpoint inequalities for generic cancellative Haar shifts defined with respect to a possibly non-homogeneous Borel measure$$\mu $$ $\mu$satisfying a weak regularity condition. This immediately yields a new, highly streamlined proof of the$$L^p$$ $L^p$-results for the same operators due to López-Sanchez, Martell, and Parcet [6]. We also prove regularity properties for the Haar shift operators on the natural martingale Lipschitz spaces defined with respect to the underlying dyadic system, and proving that the class of measures that we consider is sharp. 

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4. Comma Selection Outperforms Plus Selection on OneMax with Randomly Planted Optima

   https://doi.org/10.1007/s00453-025-01330-y  

   Jorritsma, Joost; Lengler, Johannes; Sudholt, Dirk (December 2025, Algorithmica)

   Abstract Evolutionary algorithms (EAs) are general-purpose optimisation algorithms that maintain a population (multiset) of candidate solutions and apply variation operators to create new solutions called offspring. A new population is typically formed using one of two strategies: a$$(\mu +\lambda )$$ EA (plus selection) keeps the best$$\mu $$search points out of the union of$$\mu $$parents in the old population and$$\lambda $$offspring, whereas a$$(\mu ,\lambda )$$ EA (comma selection) discards all parents and only keeps the best$$\mu $$out of$$\lambda $$offspring. Comma selection may help to escape from local optima, however when and how it is beneficial is subject to an ongoing debate. We propose a new benchmark function to investigate the benefits of comma selection: the well known benchmark functionOneMaxwith randomly planted local optima, generated by frozen noise. We show that comma selection (the$${(1,\lambda )}$$ EA) is faster than plus selection (the$${(1+\lambda )}$$ EA) on this benchmark, in a fixed-target scenario, and for offspring population sizes $$\lambda $$for which both algorithms behave differently. For certain parameters, the$${(1,\lambda )}$$ EAfinds the target in$$\Theta (n \ln n)$$evaluations, with high probability (w.h.p.), while the$${(1+\lambda )}$$ EAw.h.p. requires$$\omega (n^2)$$evaluations. We further show that the advantage of comma selection is not arbitrarily large: w.h.p. comma selection outperforms plus selection at most by a factor of$$O(n \ln n)$$for most reasonable parameter choices. We develop novel methods for analysing frozen noise and give powerful and general fixed-target results with tail bounds that are of independent interest. 

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5. DHR bimodules of quasi-local algebras and symmetric quantum cellular automata

   https://doi.org/10.4171/QT/216  

   Jones, Corey (October 2024, Quantum Topology)

   For a net of C*-algebras on a discrete metric space, we introduce a bimodule version of the DHR tensor category and show that it is an invariant of quasi-local algebras under isomorphisms with bounded spread. For abstract spin systems on a latticeL\subseteq \mathbb{R}^{n}satisfying a weak version of Haag duality, we construct a braiding on these categories. Applying the general theory to quasi-local algebrasAof operators on a lattice invariant under a (categorical) symmetry, we obtain a homomorphism from the group of symmetric QCA to\mathbf{Aut}_{\mathrm{br}}(\mathbf{DHR}(A)), containing symmetric finite-depth circuits in the kernel. For a spin chain with fusion categorical symmetry\mathcal{D}, we show that the DHR category of the quasi-local algebra of symmetric operators is equivalent to the Drinfeld center\mathcal{Z}(\mathcal{D}). We use this to show that, for the double spin-flip action\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\curvearrowright \mathbb{C}^{2}\otimes \mathbb{C}^{2}, the group of symmetric QCA modulo symmetric finite-depth circuits in 1D contains a copy ofS_{3}; hence, it is non-abelian, in contrast to the case with no symmetry. 

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