Diophantine
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A formal proof of the Ramanujan--Nagell theorem in Lean 4
arXiv:2604.09808v2 Announce Type: replace-cross Abstract: We present a complete formalization, in the Lean interactive theorem prover with the Mathlib library, of the Ramanujan--Nagell theorem: the only integer solutions to the Diophantine equation $x^2 + 7 = 2^n$ are $(n,x) \in \{(3,\pm1),(4,\pm3),(5,\pm5),(7,\pm11),(15,\pm181)\}$. The formalization includes all dependencies, notably the computation of the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{-7})$, its class number,...
CRB-Optimal Arrays and Waveforms in Active Sensing: Role of Redundancy and Spatial Covariance of Array Geometry
Announce Type: cross Abstract: This paper characterizes the performance limits of optimal array designs using orthogonal and coherent waveforms for both linear and planar arrays. For orthogonal waveforms, we show that the single-target Cram\'er-Rao Bound (CRB) depends on the sum of the so-called spatial variances of the transmit (Tx) and receive (Rx) arrays, or equivalently, the spatial variance of the sum co-array weighted by the multiplicities of the virtual sensors. This reveals that...