AI 中文总结
研究非负整数梅森表示，开发相关函数与映射，用梅森组合学重建有限域BBS单孤子族的全局整数值行波剖面，证明整数窗口计数定理，还构造了特定行波τ函数，展示数表示组合学对可积系统解重建的作用。
AI 中文摘要
我们研究非负整数的梅森表示及其分解为二进制和非二进制部分。非二进制值构成A055938，其后续结构直接证明了A055938与A080578之间的关系，这在A080578的OEIS条目中被记录为猜想。二进制部分的计数函数与移位的康诺利序列相关。我们开发了父映射、截断块、截断余数和与此表示相关的梅森τ函数。父映射与最低位数字的删除共轭。梅森τ行的差值恢复父迭代和计数函数，并给出数字重建和对角τ缺陷的公式。最后，我们重新审视有限域BBS的已知有限深度单孤子族。梅森组合学直接用于重建全局整数值行波剖面并证明整数窗口计数定理。模3约简产生相应的有限域行波解。我们还构造了一个整数值行波τ函数，其前端值由梅森τ行给出。所得构造展示了数表示的组合学如何直接进入可积系统解的重建。
英文摘要
We study the Mersenne representation of nonnegative integers and its decomposition into the binary and nonbinary sides. The nonbinary values form A055938, and their successor structure gives a direct proof of the relation between A055938 and A080578 that is recorded as conjectural in the OEIS entry for A080578. The binary-side counting function is identified with a shifted Conolly sequence. We then develop the parent map, truncation blocks, truncation remainders, and the Mersenne tau function associated with this representation. The parent map is conjugate to deletion of the lowest digit. Differences of the Mersenne tau rows recover the parent iterates and the counting function, and they give formulas for digit reconstruction and for a diagonal tau defect. Finally, we revisit a known finite-depth one-soliton family of the finite-field BBS. The Mersenne combinatorics is used directly to reconstruct a global integer-valued traveling-wave profile and to prove an integer window-counting theorem. Reduction modulo 3 yields the corresponding finite-field traveling-wave solutions. We also construct an integer-valued traveling-wave tau function whose front values are given by the Mersenne tau rows. The resulting construction shows how the combinatorics of a number representation can enter directly into the reconstruction of solutions of an integrable system.
Comments34 pages, 2 figures, 1 table
