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1. Polynomials from Combinatorial -theory

   https://doi.org/10.4153/S0008414X19000464  

   Monical, Cara; Pechenik, Oliver; Searles, Dominic (February 2021, Canadian Journal of Mathematics)

   null (Ed.)

   Abstract We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $$K$$ -theoretic deformation of the quasi-key basis and also a lift of the $$K$$ -analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $$K$$ -analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $$K$$ -theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $$K$$ -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations. 

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2. Decompositions of Grothendieck Polynomials

   https://doi.org/10.1093/imrn/rnx207  

   Pechenik, Oliver; Searles, Dominic (September 2017, International Mathematics Research Notices)

   null (Ed.)

   Abstract We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $$K$$-theory of flag varieties (in type $$A$$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $$K$$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $$n$$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $$\beta$$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $$K$$-theory, and we state our results in this more general context. 

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3. K-theoretic Catalan functions

   https://doi.org/10.1016/j.aim.2022.108421  

   Blasiak, Jonah; Morse, Jennifer; Seelinger, George H (August 2022, Advances in Mathematics)

   We prove that the K-k-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the K-k-Schur functions as Schubert representatives for K-homology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the K-k-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for K-k-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a K-theoretic analog of the Peterson isomorphism. 

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4. Hamiltonians that realize perfect quantum state transfer and early state exclusion

   https://doi.org/10.1007/s11128-025-04667-z  

   Bailey, Rachel; Costa, Sara; Derevyagin, Maxim; Findley, Caleb; Zuang, Kai (February 2025, Quantum Information Processing)

   In this paper, we show how to construct XX Hamiltonians that realize perfect quantum state transfer and also have the property that the overlap of the time evolved state with the initial state is zero for some time before the transfer time. If the latter takes place, we call it an early exclusion state.We also show that in some cases, early state exclusion is impossible. The proofs rely on properties of Krawtchouk and Chebyshev polynomials. 

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5. Stable super-resolution of images: theoretical study

   https://doi.org/10.1093/imaiai/iaaa029  

   Eftekhari, Armin; Bendory, Tamir; Tang, Gongguo (November 2020, Information and Inference: A Journal of the IMA)

   null (Ed.)

   Abstract We study the ubiquitous super-resolution problem, in which one aims at localizing positive point sources in an image, blurred by the point spread function of the imaging device. To recover the point sources, we propose to solve a convex feasibility program, which simply finds a non-negative Borel measure that agrees with the observations collected by the imaging device. In the absence of imaging noise, we show that solving this convex program uniquely retrieves the point sources, provided that the imaging device collects enough observations. This result holds true if the point spread function of the imaging device can be decomposed into horizontal and vertical components and if the translations of these components form a Chebyshev system, i.e., a system of continuous functions that loosely behave like algebraic polynomials. Building upon the recent results for one-dimensional signals, we prove that this super-resolution algorithm is stable, in the generalized Wasserstein metric, to model mismatch (i.e., when the image is not sparse) and to additive imaging noise. In particular, the recovery error depends on the noise level and how well the image can be approximated with well-separated point sources. As an example, we verify these claims for the important case of a Gaussian point spread function. The proofs rely on the construction of novel interpolating polynomials—which are the main technical contribution of this paper—and partially resolve the question raised in Schiebinger et al. (2017, Inf. Inference, 7, 1–30) about the extension of the standard machinery to higher dimensions. 

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