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1. Twisted Mahler Discrete Residues

   https://doi.org/10.1093/imrn/rnae238  

   Arreche, Carlos_E; Zhang, Yi (October 2024, International Mathematics Research Notices)

   Abstract Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p> 1$. Here we develop a notion of $$\lambda $$-twisted Mahler discrete residues for $$\lambda \in \mathbb{Z}$$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$p^{\lambda } g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations. 

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2. Counting differentials with fixed residues

   https://doi.org/10.1007/s11005-025-01940-1  

   Chen, Dawei; Prado, Miguel (June 2025, Letters in Mathematical Physics)
3. Computing discrete residues of rational functions

   https://doi.org/10.1145/3666000.3669676  

   Arreche, Carlos E; Sitaula, Hari (July 2024, ACM)

   In 2012 Chen and Singer introduced the notion of discrete residues for rational functions as a complete obstruction to rational summability. More explicitly, for a given rational function f(x), there exists a rational function g(x) such that f(x) = g(x+1) - g(x) if and only if every discrete residue of f(x) is zero. Discrete residues have many important further applications beyond summability: to creative telescoping problems, thence to the determination of (differential-)algebraic relations among hypergeometric sequences, and subsequently to the computation of (differential) Galois groups of difference equations. However, the discrete residues of a rational function are defined in terms of its complete partial fraction decomposition, which makes their direct computation impractical due to the high complexity of completely factoring arbitrary denominator polynomials into linear factors. We develop a factorization-free algorithm to compute discrete residues of rational functions, relying only on gcd computations and linear algebra. 

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4. Incorporation of Azapeptoid Residues Into Collagen

   https://doi.org/10.1002/pep2.70008  

   Wright, Madison M; Rathman, Benjamin M; Del_Valle, Juan R (September 2025, Peptide Science)

   Collagen, the major structural protein in connective tissue, adopts a right‐handed triple helix composed of peptide chains featuring repeating Gly‐Xaa‐Yaa tripeptide motifs. While the cyclic residues proline (Pro) and hydroxyproline (Hyp) are prevalent in the Xaa and Yaa positions due to their PPII‐favoring conformational properties, diverse acyclic peptoid (N‐alkylated Gly) residues can also stabilize the collagen fold. Here, we investigated the effects of N‐aminoglycine (aGly) derivatives—so‐called “azapeptoid” residues—on the thermal stability of collagen mimetic peptides (CMPs). Substitution of Pro at the central Xaa11 position with aGly resulted in destabilization of the triple helix, yet the introduction of select N′‐alkyl groups (isopropyl, butyl) partially restored thermal stability. Moreover, the N‐amino group of azapeptoid residues enhanced thermal CMP stability relative to an unsubstituted Gly analog. Kinetic studies revealed that the introduction of the hydrazide bonds in aGly and (iPr)aGly CMPs did not significantly impact triple helix refolding rates. Their modular late‐stage derivatization and tunable properties highlight azapeptoid residues as potentially valuable tools for engineering CMPs and probing the structural determinants of collagen folding. 

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5. Residues, Grothendieck Polynomials, and K-Theoretic Thom Polynomials

   https://doi.org/10.1093/imrn/rnac345  

   Rimányi, Richárd; Szenes, András (December 2022, International Mathematics Research Notices)

   Abstract Grothendieck polynomials were introduced by Lascoux and Schützenberger and play an important role in K-theoretic Schubert calculus. In this paper, we give a new definition of double stable Grothendieck polynomials based on an iterated residue operation. We illustrate the power of our definition by calculating the Grothendieck expansion of K-theoretic Thom polynomials of $${\mathcal {A}}_{2}$$ singularities. We present this expansion in two versions: one displays its stabilization property, while the other displays its expected finiteness property. 

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