L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension $2$. In this note, instead of open covers consisting of displaceable sets in their work, we consider open covers constituting of topological disks and give a necessary and sufficient condition that Poisson bracket invariants of these covers are positive.

The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of connected sums, which was recently invented by Seki and the author. Interestingly, the connected sum involves the hypergeometric function in its connector. Moreover, we introduce two kinds of $q$-analogues of the one-variable multiple polylogarithms and generalize the duality to them.

We study anisotropic weighted Sobolev spaces defined on a half-space. These spaces are important in the study of singular elliptic operators which degenerate at the boundary. We prove trace properties and density of smooth functions. Hardy inequalities are studied in detail.

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$. For certain quadratic and biquadratic extensions $\mathfrak{F}/K$, we prove a simple exact formula for the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $\mathfrak{F}_\infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also exhibit computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$; here $K_n$ denotes the $n$-th layer of $K_\infty/K$.

Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in $4$-space. A finite type invariant theory for ribbon knotted surfaces was developed by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to $w_k$-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to $w_k$-equivalence. All results have direct corollaries for ribbon knotted surfaces.

We prove that oriented and standard shadowing properties are equivalent for topological flows with finite singularities that are Lyapunov stable or backward Lyapunov stable. Moreover, we prove that the direct product $\phi_1 \times \phi_2$ of two topological flows has the oriented shadowing property if $\phi_1$ with finite singularities has the oriented shadowing property, while $\phi_2$ has the limit set consisting of finite singularities that are Lyapunov stable or backward Lyapunov stable.

We determine all possible configurations for the unit signature rank for real biquadratic fields relative to the signature ranks of their quadratic subfields, providing infinite families for each possibility, and make some additional remarks for higher rank real multiquadratic fields.

In a previous paper, we introduced functions $\zeta_r(s,a,\Delta;q)$, which are $q$-analogues of zeta functions associated with root systems $\Delta$, and their $p$-deformations $\zeta_r(s,a,\beta,\Delta;p,q)$. It is known that zeta functions of root systems satisfy certain functional relations including Witten's volume formula. In this paper, we consider $q$-extensions of these functional relations for $\Delta=\Delta(A_2), \ \Delta(A_3), \ \Delta(B_2), \ \Delta(G_2)$. We also generalize expressions of ``Weyl group symmetric'' linear combination of $\zeta_r(s,a,\beta,\Delta;p,q)$ for $\Delta=\Delta(A_1), \ \Delta(A_2), \ \Delta(A_3)$, which were obtained in the previous paper, to arbitrary root systems $\Delta$ by using an identity due to Macdonald.

First we give a classification of Hopf real hypersurfaces with $\eta$-parallel shape operator in the complex quadric $Q^{m}$, $m \geq 3$. As an application of this result, we give another classification of cyclic parallel structure Jacobi operator on Hopf real hypersurfaces in $Q^{m}$.

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 $\left[ b_{0},b_{1},b_{2},\ldots ,b_{n},\ldots \right]$ 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.

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.

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 $G$.