In this paper, we try to compute Chow rings of versal complete flag varieties corresponding to simple Lie groups, by using generalized Rost motives. As applications, we give new proofs of Totaro's results for the torsion indexes of simple Lie groups except for spin groups.

In this paper, we study the joint denseness of the Riemann zeta function and Hurwitz zeta functions with certain algebraic irrational and transcendental parameters on ℜ𝑠 > 1. We also provide evidence for the denseness of the Hurwitz zeta function with an algebraic irrational parameter on 1/2 < ℜ𝑠 < 1.

In this paper, we study restricted modules over a class of (1/2)ℤ-graded Lie algebras 𝔤 related to the Virasoro algebra. We in fact give the classification of certain irreducible restricted 𝔤-modules in the sense of determining each irreducible restricted module up to an irreducible module over a subalgebra of 𝔤 which contains its positive part. Several characterizations of these irreducible 𝔤-modules are given. By the correspondence between restricted modules over 𝔤 and modules over the vertex algebra associated to 𝔤, we get the classification of certain irreducible modules over vertex algebras associated to these 𝔤.

We show that the strong slope conjecture implies that the degree of the colored Jones polynomial detects all torus knots. As an application we obtain that an adequate knot that has the same colored Jones polynomial degrees as a torus knot must be a (2, 𝑞)-torus knot.

In this paper, we apply the methods of Maynard and Tao to the set of products of two distinct primes (𝐸₂-numbers). We obtain several results on the distribution of 𝐸₂-numbers and primes. Among others, the result of Goldston, Graham, Pintz and Yıldırım on small gaps between 𝑚 consecutive 𝐸₂-numbers is improved.

We study the eigenvalue problem of the elliptic operator which arises in the linearized model of the periodic oscillations of a homogeneous and isotropic elastic body. The square of the frequency agrees to the eigenvalue. Particularly, we deal with a thin rod with non-uniform connected cross-section in several cases of boundary conditions. We see that there appear many small eigenvalues which accumulate to 0 as the thinness parameter 𝜀 tends to 0. These eigenvalues correspond to the bending mode of vibrations of the thin body. We investigate the asymptotic behavior of these eigenvalues and obtain a characterization formula of the limit equation for 𝜀 → 0.

We give a definition of singular integral operators on Morrey–Banach spaces which include Orlicz–Morrey spaces and Morrey spaces with variable exponents. The main result of this paper ensures that the singular integral operator is well-defined on the Morrey–Banach spaces. Therefore, it provides a solid foundation for the study of singular integral operators on Morrey type spaces. As an application of our main result, we study commutators of singular integral operators on Morrey–Banach spaces.

We focus on the cohomology of the 𝑘-th nilpotent quotient of a free group. We describe all the group 2-, 3-cocycles in terms of the Massey product and give expressions for some of the 3-cocycles. We also give simple proofs of some of the results on the Milnor invariant and Johnson–Morita homomorphisms.

A left order of a countable group 𝐺 is called isolated if it is an isolated point in the compact space 𝐿𝑂(𝐺) of all the left orders of 𝐺. We study properties of a dynamical realization of an isolated left order. Especially we show that it acts on ℝ cocompactly. As an application, we give a dynamical proof of the Tararin theorem which characterizes those countable groups which admit only finitely many left orders. We also show that the braid group 𝐵₃ admits countably many isolated left orders which are not the automorphic images of the others.

The purpose of this paper is to present a variational formula of Schläfli type for the volume of a spherically faced simplex in the Euclidean space. It is described in terms of Cayley–Menger determinants and their differentials involved with hypersphere arrangements. We derive it as a limit of fundamental identities for hypergeometric integrals associated with hypersphere arrangements obtained by the authors in the preceding article.

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of 𝐿-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of 𝐿-functions and Fourier coefficients of Siegel modular forms.

Extending the results of the current authors [Doc. Math., 21 (2016), 1607–1643] and [Asian J. Math. to appear, arXiv:1404.2978], we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field 𝔽_𝑝. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.