The tensor product $T(G_{1},G_{2},G_{3})$ of graphs $G_{1}$, $G_{2}$ and $G_{3}$ is defined by \[ VT(G_{1},G_{2},G_{3}) = V(G_{1}) \times V(G_{2}) \times V(G_{3}) \] and \[ ET(G_{1},G_{2},G_{3}) = \{ [(u_{1},u_{2},u_{3}), (v_{1},v_{2},v_{3})]: |\{ i: (u_{i},v_{i}) \in E(G_{i}) \}| \geq 2 \}. \] From this definition, it is easy to see that the preimage of the direct product of two independent sets of two factors under projections is an independent set of $T(G_{1}, G_2, G_3)$. So \[ \alpha T(G_{1},G_{2},G_{3}) \geq \max \{ \alpha(G_{1}) \alpha(G_{2}) |G_{3}|, \alpha(G_{1}) \alpha(G_{3}) |G_{2}|, \alpha(G_{2}) \alpha(G_{3}) |G_{1}| \}. \] In this paper, we prove that the equality holds if $G_{1}$ and $G_{2}$ are vertex-transitive graphs and $G_{3}$ is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of $T(G_{1}$, $G_{2}$, $G_{3})$ is determined.

Let $r$, $v$, $n$ be positive integers. This paper investigate the number of solutions $s_{r,v}(n)$ of the following infinite Diophantine equations \[ n = 1^{r} \cdot |k_{1}|^{v} + 2^{r} \cdot |k_{2}|^{v} + 3^{r} \cdot |k_{3}|^{v} + \cdots \] for $\boldsymbol{k} = (k_1,k_2,k_3,\ldots) \in \mathbb{Z}^{\infty}$. For each $(r,v) \in \mathbb{N} \times \{1,2\}$, a generating function and some asymptotic formulas of $s_{r,v}(n)$ are established.

Let $\mathcal{C}$ be a $(d+2)$-angulated category. In this note, we show that if $\mathcal{C}$ is locally finite, then $\mathcal{C}$ has Auslander-Reiten $(d+2)$-angles. This extends a result of Xiao and Zhu for triangulated categories.

We verify that the Albanese fibration of the Cartwright-Steger surface is stable, answering a problem left open in [4].

Let $X$ be a smooth complex projective curve of genus $g \geq 3$. Let $\mathbf{M}_2$ be the moduli space of semistable rank $2$ Higgs bundles with trivial determinant over $X$. We construct a desingularization $\mathbf{S}$ of $\mathbf{M}_2$ as a closed subvariety of a moduli space. We prove that $\mathbf{S}$ is a nonsingular variety containing the stable locus of $\mathbf{M}_2$ as an open dense subvariety. On the other hand, there is another desingularization $\mathbf{K}$ of $\mathbf{M}_2$ obtained from Kirwan's algorithm. We show that $\mathbf{S}$ can be obtained after two blow-downs of $\mathbf{K}$.

Let $M$ be a nondoubling parabolic manifold with ends. In this study, we will investigate the $L^p$ boundedness of the discrete square function in terms of the Littlewood-Paley decomposition. It should be pointed out that, in our setting, the doubling condition of the underlying space and the regularity estimate of the kernel are missing.

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $\mathscr{C}^{\omega}$ hypersurface $M^5 \subset \mathbb{C}^3$ in coordinates $(z, \zeta, w = u+iv)$: \[ u = F(z,\zeta,\overline{z},\overline{\zeta}). \] The Gaussier-Merker model $u = \frac{z \overline{z} + \frac{1}{2} z^2 \overline{\zeta} + \frac{1}{2} \overline{z}^2 \zeta}{1 - \zeta \overline{\zeta}}$ was shown by Fels-Kaup 2007 to be locally CR-equivalent to the light cone $\{ x_1^2 + x_2^2 - x_3^2 = 0 \}$. Another representation is the tube $u = \frac{(\operatorname{Re}z)^2}{1 - \operatorname{Re} \zeta}$. The Gaussier-Merker model has $7$-dimensional rigid automorphisms group.

Inspired by Alexander Isaev, we study rigid biholomorphisms: \[ (z,\zeta,w) \longmapsto (f(z,\zeta), g(z,\zeta), \rho w + h(z,\zeta)) =: (z',\zeta',w'). \] The goal is to establish the Poincaré-Moser complete normal form: \[ u = \frac{z\overline{z} + \frac{1}{2} z^2 \overline{\zeta} + \frac{1}{2} \overline{z}^2 \zeta}{1 - \zeta \overline{\zeta}} + \sum_{\substack{a,b,c,d \in \mathbb{N} \\ a+c \geq 3}} G_{a,b,c,d} z^a \zeta^b \overline{z}^c \overline{\zeta}^d \] with $0 = G_{a,b,0,0} = G_{a,b,1,0} = G_{a,b,2,0}$ and $0 = G_{3,0,0,1} = \operatorname{Im} G_{3,0,1,1}$.

This paper studies the properties of the parabolic free-boundary problem arising from pricing of American volatility options in mean-reverting volatility processes. When the volatility index follows the mean-reverting square root process (MRSRP), we derive a closed-form pricing formula for the perpetual American power volatility option. Moreover, an artificial neural network (ANN) approach is extended to find an approximate solution of the free boundary problem arising from pricing the perpetual American option. The comparison results demonstrates that the ANN provides an accurate approach to approximate solution for the free boundary problem.

Under investigation in this work is the coupled Sasa-Satsuma equation, which can describe the propagations of two optical pulse envelopes in birefringent fibers. The Riemann-Hilbert problem for the equation is formulated on the basis of the corresponding $5 \times 5$ matrix spectral problem, which allows us to present a suitable representation for the solution of the equation. Then the Deift-Zhou steepest descent method is used to analyze the long time behavior of the coupled Sasa-Satsuma equation.

This paper is concerned with the following Euler-Lagrange system \[ \begin{cases} \frac{d}{dt} \mathcal{L}_v(t,u(t),\dot{u}(t)) = \mathcal{L}_x(t,u(t),\dot{u}(t)) \quad \textrm{for a.e. $t \in [-T,T]$}, \\ u(-T) = u(T), \\ \mathcal{L}_v(-T,u(-T),\dot{u}(-T)) = \mathcal{L}_v(T,u(T),\dot{u}(T)), \end{cases} \] where Lagrangian is given by $\mathcal{L} = F(t,x,v) + V(t,x) + \langle f(t), x \rangle$, growth conditions are determined by an anisotropic G-function and some geometric conditions at infinity. We consider two cases: with and without forcing term $f$. Using a general version of the mountain pass theorem and Ekeland's variational principle we prove the existence of at least two nontrivial periodic solutions in an anisotropic Orlicz-Sobolev space.