Phaseless polynomial interpolation over Q is possible from n+2 adaptively chosen evaluation points, where n is an upper bound on the degree of the polynomial. However, known methods for choosing these points and reconstructing the polynomial require exponential time, whereas reconstruction from 2n+1 points can be performed in polynomial time. This gap raises two questions: how difficult is reconstruction from n+2 points, and how many points suffice for polynomial-time reconstruction? We address these questions by showing that (1) reconstruction from 2n−k points is possible in polynomial time for every fixed constant k; (2) the decision version of the reconstruction problem from (1+c)n+2 points is NP-complete for every fixed constant c∈[0,1); and (3) evaluation points ensuring unique reconstruction up to a global sign can be chosen adaptively in polynomial time. These results clarify the distinction between the information required for reconstruction and the computational cost of performing it.
Conjecture 2 of A. Hinrichs and J. Vybíral (2011) [2] asserts that every continuous, nonnegative, positive definite function f on Rd, normalized by f(0)=1, satisfies [f(xj−xk)]j,k=1n⪰11T/n. We give an explicit trigonometric polynomial on R7 and eight points for which this inequality fails. All hypotheses are verified directly: positive definiteness follows from nonnegative Fourier coefficients, and pointwise nonnegativity follows from interpolation on a cube. For an explicit vector of signs, the quadratic form is 22/5, whereas the proposed lower bound is 9/2. The resulting gap is exactly −1/10. The example is compatible with the established bound for functions of the form |g|2 with g positive definite. The solution was found with a single prompt by a colleague using his private license of ChatGPT 6. Our aim is to make the solution public, as well as to collect some ideas that emerged during the analysis of the solution.
Estimating population variance is a cornerstone of statistical inference, yet classical methods remain inefficient. Simple random sampling (SRS) demands impractically large samples for acceptable precision. Existing ranked set sampling (RSS) techniques exhibit critical flaws: Stokes [27] estimator is biased and yields only marginal small-sample gains, while the unbiased MacEachern et al. [19] estimator requires multiple cycles and, crucially, fails to extract the rich distributional information residing in sample extremes. This systematic underutilization defines the precise research gap we address. To bridge this gap, we introduce a new paradigm-extreme ranked set sampling (ERSS)-that strategically exploits these extremes. For normal and uniform distributions, we construct novel unbiased estimators for both single-cycle and multi-cycle designs, deriving their exact theoretical variances in closed form-a significant theoretical advance previously unavailable. Our numerical investigation confirms overwhelming superiority. The proposed ERSS estimators outperform SRS by up to 179% (normal) and 353% (uniform), dominate Stokes by 134% and 215%, and strictly surpass MacEachern whenever the set size satisfies n >= 4, delivering gains of 31-70% (normal) and 37-101% (uniform). Remarkably, for uniform data, our specialized range-based estimator, exploiting exact order-statistic properties, achieves as tounding gains of 2088% over SRS, 1232% over Stokes, and 1048% over MacEachern. We examine the implications of imperfect ranking and validate its practical utility through a real forest height application. This work fills the long-standing efficiency void, providing a transformative high-precision solution where conventional frameworks fundamentally fail.
We study the complexity of approximating the finite-dimensional vector space embedding ℓpm↪ℓqm for 2≤p<q≤∞ based on non-adaptive randomized algorithms that use up to n arbitrary linear functionals as information on a problem instance x∈Rm, where n≪m. We prove lower bounds on the non-adaptive randomized approximation error with a joint dependence on (n,m) matching previously known upper bounds.
