Wilf-Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf-Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore-Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic expressions, which extend multivariate hypergeometric terms to the additive setting. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and motivated by the data-driven prediction problem in quantum mechanics. Caro and Ruiz analysed the question of uniqueness for Hamiltonians of the form $-Delta + V$ with an electric potential $V = V(mathrm{t}, mathrm{x})$ that depends on the time and space variables. In this context, they proved that uniqueness holds in dimension $n geq 2$ whenever the potentials are bounded and have super-exponential decay at infinity. Although their result does not seem to be optimal, one would expect at least some degree of exponential decay to be necessary for the potentials. However, in this paper, we show that by restricting the analysis to Hamiltonians with time-independent electric potentials, namely $V = V(mathrm{x})$, uniqueness can be established for bounded integrable potentials exhibiting only super-linear decay at infinity, in any dimension $n geq 2$. This surprising improvement is possible because, unlike Caro and Ruiz's approach, our argument avoids the use of complex geometrical optics (CGO). Instead, we rely on the construction of stationary states at different energies -- this is possible because the potential does not depend on time. These states will have an explicit leading term, given by a Herglotz wave, plus a correction term that will vanish as the energy grows. Besides the significant relaxation of decay assumptions on the potential, the avoidance of CGO solutions is important in its own right, since such solutions are not readily available in more complicated geometric settings.
We identify a binary sequence S = (sn)infinity n=0 with the 2-adic integer GS(2) = & sum;infinity n=0 sn2n. In the case that GS(2) is algebraic over Q of degree d >= 2, we prove that the Nth 2-adic complexity of S is at least Nd + O(1), where the implied constant depends only on the minimal polynomial of GS(2). This result is an analog of the bound of M & eacute;rai and the second author on the linear complexity of automatic sequences, that is, sequences with algebraic GS(X) over the rational function field F2(X). We further discuss the most important case d = 2 in both settings and explain that the intersection of the set of 2-adic algebraic sequences and the set of automatic sequences is the set of (eventually) periodic sequences. Finally, we provide some experimental results supporting the conjecture that 2-adic algebraic sequences can have also a desirable Nth linear complexity and automatic sequences a desirable Nth 2-adic complexity, respectively.<br /> (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
