Let Top and Diff be the categories of topological and diffeological spaces,respectively. By using an adjunction between Top and Diff we show that the full subcategory NG of Top consisting of numerically generated spaces is complete, cocomplete and cartesian closed. In fact, NG can be embedded into Diff as a cartesian closed full subcategory. It follows then that the category NG₀ of numerically generated pointed spaces is complete,cocomplete and monoidally closed with respect to the smash product. These features of NG₀ are used to establish a simple but flexible method for constructing generalized homology and cohomology theories by using the notion of enriched bifunctors.

The first aim of this paper is to show the second main theorem for holomorphic maps from a compact Riemann surface into the complex projective space which is ramified over hypersurfaces in subgeneral position. We then use it to study the ramification over hypersurfaces of the generalized Gauss map of complete regular minimal surfaces in R^m with finite total curvature, sharing hypersurfaces in subgeneral position. The results generalize our previous results [Thai and Thoan, Vietnam J. Math. 2017, doi:10.1007/s10013-017-0259-6].

We define the geometric simpleness for toroidal groups. We give an example of quasi-abelian variety which is geometrically simple, but not simple. We show that any quasi-abelian variety is isogenous to a product of geometrically simple quasi-abelian varieties. We also show that the Q-extension of the ring of all endomorphisms of a geometrically simple quasi-abelian variety is a division algebra over Q.

Ohno's relation is a well-known relation among multiple zeta values. In this paper,we prove an Ohno-type relation for finite multiple zeta values, which is conjectured by Kaneko. As a corollary, we give an alternative proof of the sum formula for finite multiple zeta values, which was first proved by Saito and Wakabayashi.

We prove the global existence and uniqueness of the strong solution pair (u, v) to the initial-boundary value problem for coupled equations of an m-Laplacian-type diffusion equation and a nonlinear wave equation. The interaction of the two equations is given through nonlinear source terms f (u, v) and g(u, v). To derive the required a priori estimates we employ a ‘loan' method. The estimation of the L^∞-norm of solutions of the nonlinear parabolic equation due to Moser's iteration method is a key step of our argument.

An identity that relates the Fourier transform of a complex power of homogeneous polynomial functions on a real vector space with a complex power of homogeneous polynomial functions on the dual vector space is called a local functional equation. A rich source of polynomials satisfying local functional equations is the theory of prehomogeneous vector spaces. Almost all known examples of local functional equations are of this type. However, recently, local functional equations of non-prehomogeneous type have been found. In this paper we present new examples of non-prehomogeneous polynomials satisfying a local functional equation. More precisely, we prove a local functional equation for the polarization of an arbitrary homaloidal polynomial, and calculate the associated b-function identities explicitly.

We present explicit formulas for all finite multiple zeta values by introducing a multiple generalization of Bernoulli polynomials associated with finite multiple zeta values. Furthermore we show that these values are also described by special values of multiple zeta functions and multiple star analogues of the Hurwitz zeta function.

Following the method of Arakawa, we express the special values of an L-function originally introduced by Arakawa and Hashimoto and later generalized by Ibukiyama and Saito at non-positive integers by finite sums of products of Bernoulli polynomials. As a corollary, we prove an infinite family of identities expressing finite sums of products of Bernoulli polynomials by generalized Bernoulli numbers.

A spectral theory of linear operators on a rigged Hilbert space is applied to Schrödinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach to resonances (generalized eigenvalues) for both classes of potentials without using any spectral deformation techniques. Generalized eigenvalues for one-dimensional Schrödinger operators (ordinary differential operators) are investigated in detail. A certain holomorphic function D(λ) is constructed so that D(λ) = 0 if and only if λ is a generalized eigenvalue. It is proved that D(λ) is equivalent to the analytic continuation of the Evans function. In particular, a new formulation of the Evans function and its analytic continuation is given.

The notion of absolute continuity for positive operators was studied by T. Ando,where parallel sums for such operators played an important role. On the other hand, a theory for parallel sums for densely defined positive self-adjoint operators (or more generally positive forms) was developed in our previous work. Based on this theory, we will investigate the notion of absolute continuity in such unbounded cases.

We present a proof of Hobson's formula with the use of the Euler operator and its commutation relation with the Laplacian.

In pursuit of the number-theoretic nature of a given set, one defines an arithmetic function and considers its average behavior in view of the fact that independent values are rather singular. We are interested in the asymptotic formula for the summatory function of an arithmetic function which is given as the coefficients of the product of two generating Dirichlet series, i.e. they are convolutions of the respective coefficients. Our main purpose is to elucidate the far-reaching theorem of Lau in the light of the Stieltjes resultant and to give some applications which involve possible logarithmic singularities.

Let Z(s) be the Selberg zeta-function associated with a compact Riemann surface. We obtain a bound Z(1/2 + it) ≪ exp(ct/log t) which allows us to improve error terms in asymptotic formulas related to the number of zeros of Z(s) derivative.