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1. A circle method approach to K‐multimagic squares

   https://doi.org/10.1112/jlms.70290  

   Flores, Daniel (September 2025, Journal of the London Mathematical Society)

   Abstract In this paper, we investigate‐multimagic squaresof order . These are magic squares that remain magic after raising each element to the th power for all . Given , we consider the problem of establishing the smallest integer for which there existnontrivial‐multimagic squares of order . Previous results on multimagic squares show that for large . We use the Hardy–Littlewood circle method to improve this to The intricate structure of the coefficient matrix poses significant technical challenges for the circle method. We overcome these obstacles by generalizing the class of Diophantine systems amenable to the circle method and demonstrating that the multimagic square system belongs to this class for all . 

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2. On Suboptimality of Least Squares with Application to Estimation of Convex Bodies

   Kur, Gil; Rakhlin, Alexander; Guntuboyina, Adityanand (October 2020, Proceedings of Thirty Third Conference on Learning Theory, PMLR)

   Abernethy, Jacob; Agarwal, Shivani (Ed.)

   We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dimension d \geq 6. Specifically, we establish that Least Squares is mimi- max sub-optimal, and achieves a rate of n^{-2/(d-1)} whereas the minimax rate is n^{-4/(d+3)}. 

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3. Sum of squares generalizations for conic sets

   https://doi.org/10.1007/s10107-022-01831-6  

   Kapelevich, Lea; Coey, Chris; Vielma, Juan Pablo (June 2022, Mathematical Programming)

   Abstract Polynomial nonnegativity constraints can often be handled using thesum of squarescondition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and Yildiz (Papp D in SIAM J O 29: 822–851, 2019), using the sum of squares cone directly in an interior point algorithm. Beyond nonnegativity, more complicated polynomial constraints (in particular, generalizations of the positive semidefinite, second order and$$\ell _1$$ ${\ell }_1$-norm cones) can also be modeled through structured sum of squares programs. We take a different approach and propose using more specialized cones instead. This can result in lower dimensional formulations, more efficient oracles for interior point methods, or self-concordant barriers with smaller parameters. 

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4. The Effect of Estimation Methods on SEM Fit Indices

   https://doi.org/10.1177/0013164419885164  

   Shi, Dexin; Maydeu-Olivares, Alberto (November 2019, Educational and Psychological Measurement)

   We examined the effect of estimation methods, maximum likelihood (ML), unweighted least squares (ULS), and diagonally weighted least squares (DWLS), on three population SEM (structural equation modeling) fit indices: the root mean square error of approximation (RMSEA), the comparative fit index (CFI), and the standardized root mean square residual (SRMR). We considered different types and levels of misspecification in factor analysis models: misspecified dimensionality, omitting cross-loadings, and ignoring residual correlations. Estimation methods had substantial impacts on the RMSEA and CFI so that different cutoff values need to be employed for different estimators. In contrast, SRMR is robust to the method used to estimate the model parameters. The same criterion can be applied at the population level when using the SRMR to evaluate model fit, regardless of the choice of estimation method. 

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5. Rapid simulation of two-dimensional spectra with correlated anisotropic dimensions

   https://doi.org/10.1063/5.0200042  

   Srivastava, Deepansh J; Baltisberger, Jay H; Grandinetti, Philip J (April 2024, The Journal of Chemical Physics)

   A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman–Solum–Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique’s efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt. 

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