For a positive integer $m$ and a real number $\alpha$, the Sós permutation of degree $m$ is defined to be the permutation which sorts the $m$ fractional parts of the sequence $\alpha, 2\alpha, \ldots, m\alpha$ in increasing order. It has been reported that the inverse of a Sós permutation satisfies a mod-$m$-congruential recurrence as an $m$-term sequence. This paper proves the converse: A permutation satisfying the mod-$m$-congruential recurrence is always the inverse of a Sós permutation. This implies that if a permutation's set of the first-order differences modulo the degree $m$ is a singleton or a set of two successive integers, then the permutation is a shift of the inverse of a Sós permutation, where shift means (the result of) an operation that adds a constant in modulo $m$ to each of the $m$ terms of a given permutation. As an application of these facts, we present a stand-alone combinatorial algorithm that lifts the set of the inverses of the Sós permutations of a given degree $m$ to the set of the same kind of degree $m + 1$ without referring the underlying the Sós permutations or the Farey sequence associated to them.
We study isoparametric hypersurfaces, whose principal curvatures are all constant, in the pseudo-Riemannian space forms. In this paper, we investigate three topics. Firstly, according to Petrov's classification theorem, we give a classification of hypersurfaces of index 2 with respect to a pair of a shape operator and a metric. Therefore, we can define types of isoparametric hypersurfaces of index 2 concerning the classification. Secondly, we give several examples of certain types. Thirdly, we show that there exist no isoparametric hypersurfaces of index 2 whose shape operators have complex principal curvatures in certain cases.
Let $k_1$ and $k_2$ be two multiplicatively independent positive integers with $k_1, k_2 \ge 2$, $S$ be a finite set of algebrac numbers and $\alpha$ be algebraic number $\alpha$ with $0 \lt |\alpha| \lt 1$. In this paper, we give the sufficient conditions that the infinite products $\prod_{y \ge 0} \left( 1 + \sum^{k_1-1}_{i=1} \tau(i,y) \alpha^{ik_1{}^y} \right)$ and $\prod_{y \ge 0} \left( 1 + \sum^{k_2-1}_{j=1} \delta(j,y) \alpha^{jk_2{}^y}\right)$ with $\tau(i,y), \delta(j,y) \in S$ $(1 \le i \le k_1-1, 1 \le j \le k_2-1, y \ge 0)$ are algebraically independent over $\mathbb{Q}$. This paper closely follows Nishioka's method.
Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between geometric data sets. In this paper, we begin by proving the non-separability of the observable distance between geometric data sets. We then extend the box distance between mm-spaces to that between geometric data sets and prove its completeness and non-separability.
In this article, we show new effective approximation measures for the shifted logarithm of algebraic numbers which belong to a number field of arbitrary degree. Our measures refine previous ones including those for usual logarithms. We adapt Pade approximants constructed by M. Kawashima and A. Poels in the rational case. Our key ingredient relies on the Poincare-Perron theorem which gives us asymptotic estimates at every archimedean place of the number field, that we apply to decide whether the shifted logarithm lies outside of the number field. We use Perron's second theorem and its modification due to M. Pituk to handle general cases.
