This paper is concerned with a semilinear elliptic equation with the Hardy-Leray potential. We employ the method of moving planes to prove the radial symmetry of positive solutions. Based on this result, we obtain the Liouville theorem in subcritical case. In addition, we find special radial solutions in critical case. All the properties above are similar to the corresponding results of the Lane-Emden equation.

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and $H_X({\mathbb R}^n)$ the Hardy space associated with $X$, and let $\alpha\in(0,n)$ and $\beta\in(1,\infty)$. In this article, assuming that the (powered) Hardy–Littlewood maximal operator satisfies the Fefferman–Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_{\alpha}$ can be extended to a bounded linear operator from $H_X({\mathbb R}^n)$ to $H_{X^{\beta}}({\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\subset \mathbb{R}^n$, $|B|^{\frac{\alpha}{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{\beta-1}{\beta}}$, where $X^{\beta}$ denotes the $\beta$-convexification of $X$. Moreover, under some different reasonable assumptions on both $X$ and another ball quasi-Banach function space $Y$, the authors also consider the mapping property of $I_{\alpha}$ from $H_X({\mathbb R}^n)$ to $H_Y({\mathbb R}^n)$ via using the extrapolation theorem. All these results have a wide range of applications.

Let $p$ be an odd prime number and $f$ a modular form. We consider the $\mathbb{F}_p$-valued Galois representation $\bar{\rho}_f$ attached to $f$ and its twist $\bar{\rho}_{f, D}$ by the quadratic character $\chi_D$ corresponding to a quadratic discriminant $D$. We define $K_{f, D}$ to be the field corresponding to the kernel of $\bar{\rho}_{f, D}$. In this article, we investigate the ideal class group $\mathrm{Cl}(K_{f, D})$ of the number field $K_{f, D}$ as a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-module. We give a condition which implies the existence of a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-equivariant surjective homomorphism from $\mathrm{Cl}(K_{f, D})\otimes \mathbb{F}_p$ to the representation space $M_{f, D}$ of $\bar{\rho}_{f, D}$, using Bloch and Kato's Selmer group of $\bar{\rho}_{f, D}$. We also give some numerical examples where we have such surjections by calculating the central value of the $L$-function of $f$ twisted by $\chi_D$ under Bloch and Kato's conjecture. Our main result in this paper is a partial generalization of the previous result of Prasad and Shekhar on elliptic curves to higher weight modular forms.

Let $\left[ b_{0},b_{1},b_{2},\ldots ,b_{n},\ldots \right]$ be the expansion of $\alpha \in \mathbb{R}\smallsetminus \mathbb{Q}$ in regular continued fraction, and let $p_{n}/q_{n}$ be the convergents of this continued fraction. It is known that the irrationality exponent of $\alpha$ is given by the formula $\mu \left( \alpha \right) =2+\limsup_{n\rightarrow \infty }\left( \log b_{n+1}/\log q_{n}\right)$. We prove that this formula remains valid for semi-regular continued fractions satisfying certain conditions. In particular, it remains valid for the nearest integer and singular continued fractions. We also show that it is no longer valid for the negative and farthest integer continued fractions. In application, examples of exact computation of irrationality exponents of semi-regular continued fractions are given.

Any rational number $r$ can be represented uniquely by a finite positive (resp. even) continued fraction in an essentially unique way. Thus positive continued fractions uniquely correspond to even continued fractions. In this paper, we give Algorithm PtoE and Algorithm EtoP ; Algorithm PtoE directly transforms positive continued fractions into even continued fractions and Algorithm EtoP directly transforms even continued fractions into positive continued fractions without knowing the value of them. These algorithms can be applied to infinite positive and even continued fractions, too. As an application, we give the even continued fraction expansions of $e$ and $\sqrt{e}$, where $e$ is Napier's constant.

We construct a totally complex immersion from almost complex submanifolds in $S^{6}$ into the associative Grassmann manifold. Also, as an analogy of almost complex immersions we construct a $CR$ immersion from $3$-dimensional $CR$ submanifolds in $S^{6}$ into the associative Grassmann manifold. Moreover, in the associative Grassmann manifold we show that many orbits of the isotropy group of the isometry group are the image of some $CR$ immersion.

In this paper, we study existence of positive and symmetric solutions for some boundary value problem on a spherical cap of unit ball. This problem has been studied mainly for the equation containing critical Sobolev exponent for three-dimensional space. So in this paper, we consider non-critical case for $n$-dimensional space. Then for sub-critical case, we can show the existence and non-existence result for such solution completely if the spherical cap is contained in the hemisphere.

Ishii and Oshiro introduced the notion of an $f$-twisted Alexander matrix, which is a quandle version of a twisted Alexander matrix of finitely presented groups. In this paper, we study the relation between $f$-twisted Alexander matrices of link quandles and quandle cocycle invariants. We show that certain information of the quandle cocycle invariant can be recovered from the $f$-twisted Alexander matrix. As an application, we show that an invariant obtained from $f$-twisted Alexander matrices is a strictly stronger invariant for oriented knots than the twisted Alexander polynomial.

We study the structure of the $(\mathfrak(g),K)$-modules of the principal series representations of $\mathrm{SL}(3,\mathbb{R})$ and $\mathrm{Sp}(2,\mathbb{R})$ induced from minimal parabolic subgroups, in the case when the infinitesimal character is nonsingular. The composition factors of these modules are known by Kazhdan-Lusztig-Vogan conjecture. In this paper, we give complete descriptions of the socle filtrations of these modules.

In this paper, we give a sufficient and necessary condition for a real quadratic order to have class number $1$. From this, and by using Lu's results, we obtain families of real quadratic orders with non-trivial class numbers.