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We shall give some results for an integer divisible by its unitary totient. [ABSTRACT FROM AUTHOR]

We defined the notion of the quantum double inclusion [S. De, J. Math. Phys. 60, 071701 (2019)] associated with a finite-index and finite-depth subfactor, which is closely related to that of Ocneanu's asymptotic inclusion, and studied the quantum double inclusion associated with the Kac algebra subfactor RH ⊂ R, where H is a finite-dimensional Kac algebra acting outerly on the hyperfinite II1 factor R and RH denotes the fixed-point subalgebra. In this article, we analyze quantum double inclusions associated with the family of Kac algebra subfactors given by { R H ⊂ R ⋊ H ⋊ H * ⋊ ⋯ ︸ m times : m ≥ 1 }. For each m > 2, we construct a model N m ⊂ M for the quantum double inclusion of R H ⊂ R ⋊ H ⋊ H * ⋊ ⋯ ︸ m − 2 times , where N m = ((⋯ ⋊ H − 2 ⋊ H − 1 ) ⊗ ( H m ⋊ H m + 1 ⋊ ⋯ )) ′ ′ , M = (⋯ ⋊ H − 1 ⋊ H 0 ⋊ H 1 ⋊ ⋯ ) ′ ′ , and for any integer i, the notation Hi stands for H or H* according as i is odd or even. In this article, we give an explicit description of the subfactor planar algebra associated with N m ⊂ M (m > 2) which turns out to be a planar subalgebra of P * (m) ( H m ) (the adjoint of the m-cabling of the planar algebra of Hm). We then show that for each m > 2, the depth of N m ⊂ M is always two. Observing that N m ⊂ M is reducible for all m > 2, we study in great detail the weak Kac algebra structure of the relative commutant ( N m ) ′ ∩ M 2 . [ABSTRACT FROM AUTHOR]

We prove that the Kloosterman sum S(1, 1; c) changes sign infinitely often as c runs over squarefree numbers with at most 7 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler, Matomäki, and the author. [ABSTRACT FROM AUTHOR]

Two edges and in a graph are said to be adjacent if they have a vertex in common, and distance two apart if they are nonadjacent but both are adjacent to a common edge. An -edge-labeling of a graph is an assignment of nonnegative integers, called labels, to the edges of such that the difference between labels of any two adjacent edges is at least , and the labels of any two edges that are distance two apart are different. The span of an -edge-labeling of a graph is the difference between the maximum and minimum labels. The minimum span over all -edge-labelings of a graph is called the -edge-labeling number of , denoted by . For , the edge-path-replacement of a graph , denoted by , is a graph obtained by replacing each edge of with a path on vertices. This paper investigates the -edge-labeling number of the edge-path-replacement of a graph for . We get the following main results: Let be a graph with maximum degree and be an integer not less than , then if is odd, and otherwise . Let be a graph with maximum degree . Then when is even, and when is odd. [ABSTRACT FROM AUTHOR]

We construct the first II 1 factors having exactly two group measure space decompositions up to unitary conjugacy. Also, for every positive integer n , we construct a II 1 factor M that has exactly n group measure space decompositions up to conjugacy by an automorphism. [ABSTRACT FROM AUTHOR]

Let G be a bridgeless cubic graph. Consider a list of k 1-factors of G. Let [ABSTRACT FROM AUTHOR]

It is conjectured that for every pair ( ℓ , m ) of odd integers greater than 2 with m ≡ 1 ( mod ℓ ) , there exists a cyclic two-factorization of K ℓ m having exactly ( m − 1 ) / 2 factors of type ℓ m and all the others of type m ℓ . The authors prove the conjecture in the affirmative when ℓ ≡ 1 ( mod 4 ) and m ≥ ℓ 2 − ℓ + 1 . [ABSTRACT FROM AUTHOR]