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1. Random Interpolating Sequences in the Polydisc and the Unit Ball

   https://doi.org/10.1007/s40315-022-00448-2  

   Dayan, Alberto; Wick, Brett D.; Wu, Shengkun (March 2023, Computational Methods and Function Theory)

   Abstract We study almost surely separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0–1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov–Sobolev spaces $$B_{2}^{\sigma }\left( \mathbb {B}_{d}\right) $$ B 2 σ B d , in the range $$0 < \sigma \le 1 / 2$$ 0 < σ ≤ 1 / 2 . For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) and its multiplier algebra $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) : in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) -interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $$0-1$$ 0 - 1 law for random interpolating sequences for $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) . 

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2. SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES

   https://doi.org/10.1017/s1446788718000198  

   LIU, WENCAI (August 2019, Journal of the Australian Mathematical Society)

   In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence $${\mathcal{D}}$$ , we obtain a sharp criterion such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ for a certain one-parameter family of $$\unicode[STIX]{x1D713}$$ . Also, under a minor condition on pseudo-absolute-value sequences $${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$$ , we obtain a sharp criterion on a general sequence $$\unicode[STIX]{x1D713}(n)$$ such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ . 

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3. On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

   https://doi.org/10.1007/s00229-025-01656-5  

   Gibara, Ryan; Kangasniemi, Ilmari; Shanmugalingam, Nageswari (July 2025, Manuscripta mathematica)

   We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure μ with μ(X) = ∞ and 0 < μ(B(x, r)) < ∞ for all x ∈ X and r ∈ (0, ∞). Our objective is to understand the relationship between the Dirichlet space D^(1,p)(X), defined using upper gradients, and the Newton-Sobolev space N^(1,p)(X)+ℝ, for 1 ≤ p < ∞. We show that when X is of uniformly locally p-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space ℍⁿ with n ≥ 2, these two spaces coincide precisely when 1 ≤ p ≤ n-1. We also provide additional characterizations of when a function in D^(1,p)(X) is in N^(1,p)(X)+ℝ in the case that the two spaces do not coincide. 

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4. An inequality for the normal derivative of the Lane–Emden ground state

   https://doi.org/10.1515/acv-2022-0005  

   Frank, Rupert L.; Larson, Simon (May 2022, Advances in Calculus of Variations)

   Abstract We consider Lane–Emden ground states with polytropic index 0 ≤ q - 1 ≤ 1 {0\leq q-1\leq 1} , that is, minimizers of the Dirichlet integral among L q {L^{q}} -normalized functions.Our main result is a sharp lower bound on the L 2 {L^{2}} -norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality.Our bound holds for arbitrary bounded open Lipschitz sets Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} , without assuming convexity. 

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5. Hermite Interpolation With Error Correction: Fields of Zero or Large Characteristic and Large Error Rate

   https://doi.org/10.1145/3452143.3465525  

   Kaltofen, Erich L.; Pernet, Clément; Yang, Zhi-Hong (July 2021, ISSAC '21: Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation)

   null (Ed.)

   Multiplicity code decoders are based on Hermite polynomial interpolation with error correction. In order to have a unique Hermite interpolant one assumes that the field of scalars has characteristic 0 or >= k+1, where k is the maximum order of the derivatives in the list of values of the polynomial and its derivatives which are interpolated. For scalar fields of characteristic k+1, the minimum number of values for interpolating a polynomial of degree <= D is D+1+2E(k+1) when <= E of the values are erroneous. Here we give an error-correcting Hermite interpolation algorithm that can tolerate more errors, assuming that the characteristic of the scalar field is either 0 or >= D+1. Our algorithm requires (k+1)D + 1 - (k+1)k/2 + 2E values. As an example, we consider k = 2. If the error ratio (number of errors)/(number of evaluations) <= 0.16, our new algorithm requires ceiling( (4+7/17) D - (1+8 /17) ) values, while multiplicity decoding requires 25D+25 values. If the error ratio is <= 0.2, our algorithm requires 5D-2 evaluations over characteristic 0 or >= D+1, while multiplicity decoding for an error ratio 0.2 over fields of characteristic 3 is not possible for D >= 3. Our algorithm is based on Reed-Solomon interpolation without multiplicities, which becomes possible for Hermite interpolation because of the high redundancy necessary for error-correction. 

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