For $b \in Lip(\mathbf{R}^{n})$, the Calderón type commutator for the Littlewood-Paley operator with variable kernel is defined by $$\mu_{\Omega,1;b}(f)(x) = \left( \int_{0}^{\infty}\left| \frac{1}{t^2}\int_{\vert x-y\vert \leq t}\frac{\Omega(x, x-y)}{\vert x - y \vert ^{n-1}} (b(x) - b(y))f(y) dy\right|^{2} \frac{dt}{t}\right)^{1/2}.$$ By giving a method based on Littlewood-Paley theory, Fourier transform and the spherical harmonic development, we prove the $L^{2}$ norm inequalities for the rough operators $\mu_{\Omega,1;b}$ with $\Omega(x,z^{\prime}) \in L^{\infty}(\mathbf{R}^{n} \times L^{q}(S^{n-1})\left(q \gt \frac{2(n-1)}{n} \right)$ satisfying certain cancellation conditions.

In this paper, we study proper holomorphic self-mappings of generalized complex ellipsoids and generalized Hartogs triangles. By making use of our previous result on the holomorphic automorphism group of a generalized complex ellipsoid and Monti-Morbidelli's result on the extendability of a local CR-diffeomorphism between open subsets contained in the strictly pseudoconvex part of the boundary of a generalized complex ellipsoid, we obtain natural generalizations of some results due to Landucci, Chen-Xu and Zapalowski.

We study the orbital instability of solitary waves for a derivative nonlinear Schrödinger equation with a general nonlinearity. We treat a borderline case between stability and instability, which is left as an open problem by Liu, Simpson and Sulem (2013). We give a sufficient condition for instability of a two-parameter family of solitary waves in a degenerate case by extending the results of Ohta (2011), and verify this condition for some cases.

In this paper, we prove an addition type formula for the double cotangent function. Furthermore, we see that the addition theorem of the usual cotangent function, the reciprocity laws of (classical and higher) Dedekind sums, Lerch's functional equation and Ramanujan's formula can be deduced from it.

In this paper, we investigate the existence, uniqueness of almost automorphic in one-dimensional distribution mild solution for semilinear stochastic differential equations driven by Lévy noise. The semigroup theory, fixed point theorem and stochastic analysis technique are the main tools in carrying out proof. Finally, we give one example to illustrate the main findings.

In this paper, we obtain some vanishing and finiteness theorems for $L^{p}$ $p$-harmonic 1-forms on a locally conformally flat Riemmannian manifolds which satisfies an integral pinching condition on the traceless Ricci tensor, and for which the scalar curvature satisfies pinching curvature conditions or the first eigenvalue of the Laplace-Beltrami operator of $M$ is bounded by a suitable constant.

Let $G$ be an algebraic group over $\mathbf{C}$ corresponding a compact simply connected Lie group. When $H^{*}(G)$ has $p$-torsion, we see $ρ^{*}_{CH} : CH^{*}(BG) → CH^{*}(BT)^{W_{G}(T)}$ is always not surjective. We also study the algebraic cobordism version $ρ^{*}_{Ω}$. In particular when $G = Spin(7)$ and $p = 2$, we see each Griffiths element in $CH^{*}(BG)$ is detected by an element in $Ω^{*}(BT)$.

In this paper, we prove that if $X$ is a Riemann surface of infinite analytic type and $[\mu]_T$ is any element of Teichmüller space, then there exists $\mu _{1} \in [\mu]_{T}$ so that $[\mu_1]_{B}$ is an infinitesimal Strebel point.

We consider the following system of coupled nonlinear Schrödinger equations $$\left\{ \begin{array}\\-\Delta u + a(x)u = \vert u \vert^{p-2}u + \lambda(x)v, \quad x \in \mathbf{R}^{N},\\ -\Delta v + b(x)v = \vert v \vert^{2^{*}-2}v + \lambda(x)u, \quad x \in \mathbf{R}^{N},\\ u, v \in H^{1} (\mathbf{R}^{N}), \end{array} \right.$$ where $N \geq 3, 2 \lt p \lt 2^{*}, 2^{*} = 2N / (N - 2)$ is the Sobolev critical exponent, $a, b, \lambda \in C(\mathbf{R}^{N}, \mathbf{R}) \cap L^{\infty} (\mathbf{R}^{N}, \mathbf{R})$ and $a(x)$, $b(x)$ and $\lambda(x)$ are asymptotically periodic, and can be sign-changing. By using a new technique, we prove the existence of a ground state of Nehari type solution for the above system under some mild assumptions on $a, b$ and $\lambda$. In particular, the common condition that $\vert\lambda(x)\vert \lt \sqrt{a(x)b(x)}$ for all $x \in \mathbf{R}^{N}$ is not required.

In this note, we prove an index theorem on Galois coverings for Heisenberg elliptic (but not elliptic) differential operators, which is analogous to Atiyah's $Γ$-index theorem. This note also contains an example of Heisenberg differential operators with non-trivial $Γ$-index.

For any prime $p \geq 5$, we show that generic hypersurface $X_{p} \subset \mathbf{P}^{p}$ defined over $\mathbf{Q}$ admits a non-trivial rational dominant self-map of degree $> 1$, defined over ${\mathbf{\bar {Q}}}$. A simple arithmetic application of this fact is also given.

We study the Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one. A consequence of the main results is that the isomorphism class of a certain moduli space of hyperbolic curves of genus one over a sub-$p$-adic field is completely determined by the isomorphism class of the étale fundamental group of the moduli space over the absolute Galois group of the sub-$p$-adic field. We also prove related results in absolute anabelian geometry.

We classify $(ρ,τ)$-quasi Einstein solitons with $(a,τ)$-concurrent vector fields. We also give a necessary and sufficient condition for a submanifold to be a $(ρ,τ)$-quasi Einstein soliton in a Riemannian manifold equipped with an $(a,τ)$-concurrent vector field.