We compute the divisor of the modular equation on the modular curve $\Gamma_0(N) \setminus \mathbb{H}^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $\Gamma_0(N)$ of genus zero. We also introduce the notions and properties of $\Gamma$-equivalence and $\Gamma$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2,3,4,5$.

We say a monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree $n \geq 2$ is monogenic if $f(x)$ is irreducible over $\mathbb{Q}$ and $\{ 1, \theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring of integers of $\mathbb{Q}(\theta)$, where $f(\theta) = 0$. In this article, we investigate when a pair of polynomials $f(x) = x^n-a$ and $g(x) = x^m-b$ has the property that $f(x)$ and $f(g(x))$ are monogenic.

Let $R$ be a noncommutative prime ring, with maximal symmetric ring of quotients $Q_{ms}(R)$ and extended centriod $C$, and let $\tau$ be an anti-automorphism of $R$. An additive map $\delta \colon R \to Q_{ms}(R)$ is called a Jordan $\tau$-derivation if $\delta(x^2) = \delta(x) x^{\tau} + x\delta(x)$ for all $x \in R$. In 2015 Lee and the author proved that any Jordan $\tau$-derivation of $R$ is X-inner if either $R$ is not a GPI-ring or $R$ is a PI-ring except when $\operatorname{char}R = 2$ and $\dim_C RC = 4$. In the paper we prove that, when $R$ is a prime GPI-ring but is not a PI-ring, any Jordan $\tau$-derivation is X-inner if either $\tau$ is of the second kind or both $\operatorname{char}R \neq 2$ and $\tau$ is of the first kind with $\operatorname{deg} \tau^{2} \neq 2$.

In this note we provide a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96, improving the previous known bound of 72.

In this paper, the iterated commutators of multilinear maximal square function and pointwise multiplication with functions in Lipschitz spaces are studied. Some new estimates for the iterated commutators with kernels satisfying some Dini type conditions on Lebesgue spaces, homogenous Lipschitz spaces and homogenous Triebel-Lizorkin spaces will be given, respectively.

We define a parabolic flow of almost balanced metrics. We show that the flow has a unique solution on compact almost Hermitian manifolds.

In this article, we consider the following problem \[ \begin{cases} (-\Delta)^s u = \alpha u^+ - \beta u^{-} + f(u) + h &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^n \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n \gt 2s$, $0 \lt s \lt 1$, $(\alpha,\beta) \in \mathbb{R}^2$, $f \colon \mathbb{R} \to \mathbb{R}$ is a bounded and continuous function and $h \in L^2(\Omega)$. We prove the existence results in two cases: first, the nonresonance case where $(\alpha,\beta)$ is not an element of the Fučik spectrum. Second, the resonance case where $(\alpha,\beta)$ is an element of the Fučik spectrum. Our existence results follows as an application of the saddle point theorem. It extends some results, well known for Laplace operator, to the nonlocal operator.

In this paper, we study the large time behavior of a thermoelastic Timoshenko system with memory effects where the thermoelastic coupling is acting on shear force instead of the bending moment. We establish the existence of finite-dimensional global attractor and an exponential attractor.

This paper deals with the 2D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that $\overline{\rho} + \|\nabla d_0\|_{L^2}$ is suitably small with large velocity, which extends the recent work [Discrete Contin. Dyn. Syst. 37 (2017), 4907--4922] and [Methods Appl. Anal. 22 (2015), 201--220] to the case of variable viscosity. Furthermore, the exponential decay of the solution is also obtained.

This paper is concerned with the asymptotic behavior of the classical solutions of a Landau-Lifshitz-Schrödinger-type problem with initial-boundary values when the parameter $\varepsilon$ goes to zero. We establish several uniform estimates of $u_{\varepsilon}$ by a conservation result and the standard parabolic method. Based on these results, we obtain parabolic behavior in the dissipative case and non-parabolic behavior of the semi-classical limits of those solutions respectively.

In this paper, we are concerned with the multiplicity of solutions for a class of quasilinear elliptic equation arising from plasma physics. By using a dual approach, the existence of infinitely many small solutions are obtained. As a main novelty with respect to some previous results, we assume the potential $V$ may changes sign and do not require any condition at infinity on the nonlinear term.

In this paper, a family of non-stationary combined ternary $5$-point subdivision schemes with multiple variable parameters is proposed. The construction of the scheme is based on the generalized ternary subdivision scheme of order $4$, which is built upon refinement of a family of generalized B-splines, using the variable displacements. For such a non-stationary scheme, we study its smoothness and get that it can generate $C^{2}$ interpolating limit curves and $C^{4}$ approximating limit curves. Besides, we investigate the exponential polynomial generation/reproduction property and approximation order. It can generate/reproduce certain exponential polynomials with suitable choices of the variable parameters, and reach approximation order $5$.

A parametric finite horizon optimal economic growth problem is solved by using the maximum principle for optimal control problems with state constraints in the book by R. Vinter [Optimal Control, Birkhäuser, Boston, 2000; Theorem 9.3.1]. From the obtained results it follows that if the total factor productivity is relatively small, then the expansion of the production facility does not lead to a higher total consumption satisfaction of the society.