This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives M_η over a number field F—one via ramification conditions on ℓ-adic realizations, another one via the K-theory of proper regular models—both agree with motivic cohomology of η_!* M_η[1]. Here η_!* is constructed by a limiting process in terms of intermediate extension functors j_!* defined in analogy to perverse sheaves.

Let 0 < α ≤ 1 and let $\boldsymbol{b}_\alpha^{2}$ be a Hilbert space of all square integrable solutions of a parabolic equation (∂_t + (−Δ)^α)u = 0 on the upper half space. We study the Toeplitz operators on $\boldsymbol{b}_\alpha^{2}$, which we characterize to be of Schatten class whose exponent is smaller than 1. For the proof, we use an atomic decomposition theorem of parabolic Bergman functions. Generalizations to Schatten class operators for Orlicz type and Herz type are also discussed.

In the paper we prove a normality criterion for a family of meromorphic functions which involves sharing of a non-zero finite value by certain differential polynomials generated by the members of the family.

We consider the rotation number ρ(t) of a diffeomorphism f_t = R_t $\circ$ f, where R_t is the rotation by t and f is an orientation preserving C^∞ diffeomorphism of the circle S¹. We shall show that if ρ(t) is irrational $$\limsup_{t'\to t}(ρ(t′) − ρ(t)) / (t′ − t) ≥ 1.$$

In this paper, we study the asymptotic behavior of solutions of a class of nonlinear neutral impulsive delay differential equations with forced term of the form $$\left\{\begin{array}{lll}[x(t)+c(t)x(t-\tau)]' +p(t)f(x(t-\delta))=q(t),& t\geq t_0, t\neq t_k,\\ x(t_k)=b_kx(t_k^-)+(1-b_k)\int_{t_k-\delta}^{t_k}p(s+\delta)f(x(s))ds\\ \qquad +(b_k-1)\int_{t_k}^\infty q(s)ds,& k\in{\mathbf{Z_+}}\end{array}\right.$$ Sufficient conditions are obtained for every solution of the equations that tends to a constant as t → ∞.

Following the idea of P. Schmutz Schaller, we shall consider a parametrization of the Teichmüller space $\mathscr{T}_2$ of compact Riemann surfaces of genus two. In the first part of this paper, we calculate the coordinates of 4 kinds of surface uniformized by Fuchsian groups whose fundamental regions can be the regular octagon. In the second part, we give a characterization of $\mathscr{T}_2$ in R⁷.

This paper addresses FMRA and MRA in the setting of reducing subspaces of L²(R^d). We prove that an FMRA must be contained in some MRA, and obtain a sufficient condition for an MRA to contain no FMRA other than itself.

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a germ of a normal isolated singularity coincides with the number of essential components of the exceptional set in any resolution of this singularity. We propose a program for an affirmative solution of the Nash problem for special types of normal isolated hypersurface singularities. We illustrate this program by giving an affirmative solution of the Nash problem for the rational double point E₆. We also prove some results on the algebraic structure of the space of k-jets of an arbitrary hypersurface singularity and apply them to the specific case of E₆.