We derive some Euler type double integral representations for hypergeometric functions in two variables. In the first part of this paper we deal with Horn's H₂ function, in the second part with Olsson's F_P function. Our double integral representing the F_P function is compared with the formula for the same integral representing an H₂ function by M. Yoshida (Hiroshima Math. J. 10 (1980), 329-335) and M. Kita (Japan. J. Math. 18 (1992), 25-74). As specified by Kita, their integral is defined by a homological approach. We present a classical double integral version of Kita's integral, with outer integral over a Pochhammer double loop, which we can evaluate as H₂ just as Kita did for his integral. Then we show that shrinking of the double loop yields a sum of two double integrals for F_P.

For some monoids, we give a method of composing invertibility preserving maps associated to ‘partial involutions'. Also, we define the notion of ‘determinants for finite dimensional algebras over a field'. As examples, we give invertibility preserving maps for Clifford algebras into a field and determinants for Clifford algebras into a field, where we assume that the algebras are generated by less than or equal to five generators over the field. On the other hand, ‘determinant formulas for Clifford algebras' are known. We understand these formulas as an expression that connects invertibility preserving maps for Clifford algebras and determinants for Clifford algebras. As a result, we have a better sense of determinant formulas. In addition, we show that there is not such a determinant formula for Clifford algebras generated by greater than five generators.

The asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by Shirai and Takahashi (J. Funct. Anal. 205 (2003), 414-463) in terms of the Fredholm determinant. In the subclass, the obtained leading orders are the same as the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point fields associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of states are also investigated in the case where the single site potentials have compact supports. The same method also applies to another well studied example, the zeros of a Gaussian random analytic function.

We define two parabolic flows on compact almost Hermitian manifolds, which coincide with the pluriclosed flow and the Hermitian curvature flow, respectively, on compact Hermitian manifolds with pluriclosed metric. We study the relation between these two parabolic evolution equations.

The non-existence of CR submanifolds of maximal CR dimension with umbilical shape operator in holomorphic statistical manifolds is proven. Our results are a generalization of the known results in the theory of CR submanifolds in complex space forms. Statistical manifolds in this paper are considered as manifolds consisting of certain probability density functions. In this setting we have two shape operators in the distinguished normal vector field direction with respect to the affine connection of the ambient space, and the one with respect to the dual connection. After obtaining the fundamental equations for CR submanifolds in holomorphic statistical manifolds, we examine umbilical dual shape operators.

We describe a finite presentation of T_g,r for g ≥ 3. Here T_g,r is the universal central extension of the mapping class group of the surface of genus g with r-boundaries. We also investigate the cases of g = 2 and 3, and give an application.

A classical congruence of Eisenstein and Lerch for the Fermat quotient with base 2 is generalized by Skula and Dobson. We give an alternative proof for the general congruence using a ‘Fermat quotient' associated to a unit of an abelian number field.

The manifold M of star-shaped curves in Rⁿ is considered via the theory of connections on vector bundles, and cyclic D-modules. The appropriate notion of an ‘integral curve' (i.e. certain admissible deformations) on M is defined, and the resulting space of admissible deformations is classified via iso-spectral flows, which are shown to be described by equations from the n-KdV (Korteweg-de Vries) hierarchy.

Let X be a normal, separated and integral scheme of finite type over Z and M a set of closed points of X. To a Galois cover X˜ of X unramified over M, we associate a quandle whose underlying set consists of points of X˜ lying over M. As the limit of such quandles over all étale Galois covers and all étale abelian covers, we define topological quandles Q(X, M) and Q^ab(X, M), respectively. Then we study the problem of reconstruction. Let K be Q or a quadratic field, O_K its ring of integers, X = Spec O_K \ {p} the complement of a closed point such that π₁(X)^ab is infinite, and M a set of primes with Dirichlet density one. Using results from p-adic transcendental number theory, we show that K, p and the projection M → Spec Z can be recovered from the topological quandle Q(X, M) or Q^ab(X, M).

For a non-square positive integer d with 4 ∤ d, put ω(d) :=(1 + √d)/2 if d is congruent to 1 modulo 4 and ω(d) := √d otherwise. Let a₁, a₂, . . . , a_ℓ-1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a₁, a₂, . . . , a_[ℓ/2] is the primary symmetric part of the simple continued fraction expansion of ω(d). A notion of ‘ELE type' for a finite sequence was introduced in Kawamoto et al (Comment. Math. Univ. St. Pauli 64(2) (2015), 131-155). The aims of this paper are to introduce a notion of ‘pre-ELE type' for a finite sequence and to give a way of constructing primary symmetric parts of ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields with period ℓ of minimal type for each even ℓ ≥ 6.

We report new hypergeometric constructions of rational approximations to Catalan's constant, log 2 and π², their connection with already known ones, and underlying ‘permutation group' structures. Our principal arithmetic achievement is a new partial irrationality result for the values of Riemann's zeta function at odd integers.

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound Ric_N ≥ K with K > 0 for the negative effective dimension N < 0. We analyze two one-dimensional examples of constant curvature Ric_N ≡ K with finite and infinite total volumes. We also discuss when the first non-zero eigenvalue of the Laplacian takes its minimum under the same condition Ric_N ≥ K > 0, as a counterpart to the classical Obata rigidity theorem. Our main theorem shows that, if N < −1 and the minimum is attained, then the manifold splits off the real line as a warped product of hyperbolic nature.