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1. Sparse Polynomial Hermite Interpolation

   https://doi.org/10.1145/3476446.3535501  

   Kaltofen, Erich L. (July 2022, ISSAC '22 Proc. 2022 ACM Internat. Symp. Symbolic Algebraic Comput.)

   Amir Hashemi (Ed.)

   We present Hermite polynomial interpolation algorithms that for a sparse univariate polynomial f with coefficients from a field compute the polynomial from fewer points than the classical algorithms. If the interpolating polynomial f has t terms, our algorithms, require argument/value triples (w^i, f(w^i), f'(w^i)) for i=0,...,t + ceiling( (t+1)/2 ) - 1, where w is randomly sampled and the probability of a correct output is determined from a degree bound for f. With f' we denote the derivative of f. Our algorithms generalize to multivariate polynomials, higher derivatives and sparsity with respect to Chebyshev polynomial bases. We have algorithms that can correct errors in the points by oversampling at a limited number of good values. If an upper bound B >= t for the number of terms is given, our algorithms use a randomly selected w and, with high probability, ceiling( t/2 ) + B triples, but then never return an incorrect output. The algorithms are based on Prony's sparse interpolation algorithm. While Prony's algorithm and its variants use fewer values, namely, 2t+1 and t+B values f(w^i), respectively, they need more arguments w^i. The situation mirrors that in algebraic error correcting codes, where the Reed-Solomon code requires fewer values than the multiplicity code, which is based on Hermite interpolation, but the Reed-Solomon code requires more distinct arguments. Our sparse Hermite interpolation algorithms can interpolate polynomials over finite fields and over the complex numbers, and from floating point data. Our Prony-based approach does not encounter the Birkhoff phenomenon of Hermite interpolation, when a gap in the derivative values causes multiple interpolants. We can interpolate from t+1 values of f and 2t-1 values of f'. 

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2. Teichmüller extremal maps on infinite Riemann surfaces

   https://doi.org/10.54330/afm.178987  

   Šarić, Dragomir (January 2026, Annales Fennici Mathematici)

   Let \(X=\mathbb{D}/\Gamma\) be a Riemann surface with \(\Gamma\) of the first kind. We establish a necessary and sufficient criterion for \([f]\in T(X)\) to have a Teichmüller-type extremal map. 

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3. Assessing the durability of biopolymer-treated soil under weather cycles

   https://doi.org/10.1007/s11440-026-03059-6  

   Ullah, Shafi; Schossler, Rodrigo Teixeira; Hegarty, Bridget; Yu, Xiong (June 2026, Acta Geotechnica)

   Abstract Biopolymers, as eco-friendly alternatives to traditional inorganic chemicals or synthetic polymers, have significant potential applications for sustainable soil stabilization and ground improvements. This study investigated the weather durability of xanthan gum (XG), an emerging biopolymer, and sticky rice (SR), a biopolymer used in construction for millennia. The performance of biopolymer-treated soils was evaluated through laboratory experiments focusing on their behavior under wet–dry (W-D) and freeze–thaw (F-T) cycles. The basis of comparison includes untreated soil as the baseline, and soil treated with lime, a conventional non-renewable chemical soil stabilizer. The underlying mechanisms for their performance were assessed by microstructural analysis using Scanning Electron Microscopy (SEM) and elemental composition assessment via Energy-Dispersive X-ray Spectroscopy (EDS). Results indicated that soil treated with biopolymers demonstrated different resistance to weather-induced degradation. In terms of susceptibility to W-D cycles, SR-treated soil exhibited higher resilience under W-D cycles, remaining intact after eight cycles with only 1.1% mass loss by the sixth cycle; while XG-treated soil lost 14% mass after the first W-D cycle and collapsed in W-D cycles. In terms of susceptibility to F-T cycles, SR-treated soils showed a 27.2% decrease in unconfined compressive strength (UCS) after six F-T cycles, while XG-treated soils experienced 18.1% reduction of UCS. The observed behaviors are attributed to the interactions between SR or XG with water, i.e., hydrophobicity versus hydrophilicity. SEM and EDS analyses revealed SR’s hydrophobic coating and XG’s fibrous network in the microstructure of treated soils as potential underlying mechanisms for observed behaviors. To leverage the resilience of SR in W-D cycles and XG in F-T cycles, a hybrid SR-XG treatment is recommended to utilize the complementary advantages of these biopolymers to provide sustainable soil improvements across diverse climate cycles. 

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4. Numerical approximation of nonlinear SPDE’s

   https://doi.org/10.1007/s40072-022-00271-9  

   Ondreját, Martin; Prohl, Andreas; Walkington, Noel J. (September 2022, Stochastics and Partial Differential Equations: Analysis and Computations)

   Abstract The numerical analysis of stochastic parabolic partial differential equations of the form $$\begin{aligned} du + A(u)\, dt = f \,dt + g \, dW, \end{aligned}$$ d u + A ( u ) d t = f d t + g d W , is surveyed, where A is a nonlinear partial operator and W a Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory. 

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5. Primitivity rank for random elements in free groups

   https://doi.org/10.1515/jgth-2021-0150  

   Kapovich, Ilya (March 2022, Journal of Group Theory)

   Abstract For a free group F r F_{r} of finite rank r ≥ 2 r\geq 2 and a non-trivial element w ∈ F r w\in F_{r} , the primitivity rank π ⁢ ( w ) \pi(w) is the smallest rank of a subgroup H ≤ F r H\leq F_{r} such that w ∈ H w\in H and 𝑤 is not primitive in 𝐻 (if no such 𝐻 exists, one puts π ⁢ ( w ) = ∞ \pi(w)=\infty ).The set of all subgroups of F r F_{r} of rank π ⁢ ( w ) \pi(w) containing 𝑤 as a non-primitive element is denoted by Crit ⁡ ( w ) \operatorname{Crit}(w) .These notions were introduced by Puder (2014).We prove that there exists an exponentially generic subset V ⊆ F r V\subseteq F_{r} such that, for every w ∈ V w\in V , we have π ⁢ ( w ) = r \pi(w)=r and Crit ⁡ ( w ) = { F r } \operatorname{Crit}(w)=\{F_{r}\} . 

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