Let $G$ be the group $SL(2,\mathbb{R})$, $P$ the parabolic subgroup in $G$ consisting of upper triangular matrices, and $\Gamma$ a cocompact lattice in $G$. The right homogeneous action of $P$ on $\Gamma\backslash G$ defines the orbit foliation $\mathcal{F}_P$. We compute the leafwise cohomology ring $H^*(\mathcal{F}_P)$ by exploiting non-abelian harmonic analysis on $G$.

Our aim in this paper is to establish the vanishing Sobolev integrability for fractional maximal functions and Riesz potentials in Morrey-Orlicz spaces. The size of the exceptional sets is evaluated by using the Hausdorff measure. In the borderline case we discuss Trudinger-type exponential integrability for fractional maximal functions and Riesz potentials. We also discuss the sharpness of our results.

We consider quasilinear ordinary differential equations which are related to pseudo-Laplace operators, and have critical coefficients. We determine asymptotic forms of slowly decaying positive solutions of these equations.

It is known that for every spatial complete graph on $n\ge 7$ vertices, the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots is congruent to $r_{n}$ modulo $(n-5)!$, where $r_{n} = (n-5)!/2$ if $n=8k,8k+7$, and $0$ if $n\neq 8k,8k+7$. In particular the case of $n=7$ is famous as the Conway–Gordon $K_{7}$ theorem. In this paper, conversely, we show that every integer $(n-5)! q + r_{n}$ is realized as the summation of the second coefficients of the Conway polynomials over the Hamiltonian knots in some spatial complete graph on $n$ vertices.

We study trajectories for Sasakian magnetic fields on a real hypersurface of type (A) in a complex projective space which can be seen as trajectories for some Kähler magnetic fields on the ambient space. In particular, when a real hypersurface is of type $({\rm A}_2)$, since the set of congruence classes of trajectories for a given Sasakian magnetic field is shown as an upper half disk in a Euclidean plane, in general, we investigate how the set of congruence classes of such special trajectories is included in this upper half disk.

The purpose of this note is to relate certain ring-theoretic properties of rings in mixed and positive characteristics that are related to each other by a tilting operation used in perfectoid geometry. To this aim, we exploit the multiplicative structure of the ``monoidal map", which is constructed on arbitrary $p$-adically complete rings.

Recently, Kaneko and Tsumura introduced multiple $\widetilde{T}$-values, a certain type of poly-Euler numbers and the zeta function, called the $\lambda$ function. Each of them has formulas similar to multiple zeta values, poly-Bernoulli numbers and the zeta functions of Arakawa-Kaneko type. The main object of this paper is special value of the $\lambda$ function at positive integer points. Some evaluation formulas and the duality type relations are shown. Combining the results by Kaneko and Tsumura, we also obtain relations among multiple $\widetilde{T}$-values.

In this paper, second order nonlinear totally characteristic type partial differential equations in the complex domain are discussed, and the local uniqueness of the solution is proved in a very wide class of functions. The result is applied to the problem of analytic continuation of local holomorphic solutions of the equation.

An integral binary quartic form is said to be locally soluble (resp.\ soluble) if the corresponding genus one curve has a rational point over $\mathbb{Q}_v$ for every place $v$ of $\mathbb{Q}$ (resp.\ over $\mathbb{Q}$). We consider the proportion of soluble integral binary quartic forms in locally soluble forms. Bhargava showed the proportion is positive when one considers all binary quartics, and Bhargava–Ho proved the proportion is zero for a subfamily. In this paper, we estimate the proportions for some other subfamilies. It relies on results for elliptic curves $y^2=x^3-n^2x$ by Heath-Brown, Xiong–Zaharescu and Smith.