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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 the Degree of Polynomials Computing Square Roots Mod p

   https://doi.org/10.4230/LIPIcs.CCC.2024.25  

   Kedlaya, Kiran S; Kopparty, Swastik (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   For an odd prime p, we say f(X) ∈ F_p[X] computes square roots in F_p if, for all nonzero perfect squares a ∈ F_p, we have f(a)² = a. When p ≡ 3 mod 4, it is well known that f(X) = X^{(p+1)/4} computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified (p-1)/2 evaluations (up to sign) of the polynomial f(X). On the other hand, for p ≡ 1 mod 4 there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in F_p. We show that for all p ≡ 1 mod 4, the degree of a polynomial computing square roots has degree at least p/3. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99% of the squares also has degree almost p/3. In the other direction, Agou, Deliglése, and Nicolas [Agou et al., 2003] showed that for infinitely many p ≡ 1 mod 4, the degree of a polynomial computing square roots can be as small as 3p/8. 

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3. 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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4. Shape allophiles improve entropic assembly

   https://doi.org/10.1039/C5SM01351H  

   Harper, Eric S.; Marson, Ryan L.; Anderson, Joshua A.; van Anders, Greg; Glotzer, Sharon C. (January 2015, Soft Matter)

   We investigate a class of “shape allophiles” that fit together like puzzle pieces as a method to access and stabilize desired structures by controlling directional entropic forces. Squares are cut into rectangular halves, which are shaped in an allophilic manner with the goal of re-assembling the squares while self-assembling the square lattice. We examine the assembly characteristics of this system via the potential of mean force and torque, and the fraction of particles that entropically bind. We generalize our findings and apply them to self-assemble triangles into a square lattice via allophilic shaping. Through these studies we show how shape allophiles can be useful for assembling and stabilizing desired phases with appropriate allophilic design. 

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5. Student reasoning about the least-squares problem in inquiry-oriented linear algebra

   Wawro, M.; Park, M.; Zandieh, M.; Bettersworth, Z.; & Lee, I. (January 2023, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Cook, S.; Katz, B.; Moore-Russo, D. (Ed.)

   The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

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