Is homophily in social and economic networks driven by a taste for homogeneity (preferences) or by a higher probability of meeting individuals with similar attributes (opportunity)? This paper studies identification and estimation of an iterative network game that distinguishes between these two mechanisms. Our approach enables us to assess the counterfactual effects of changing the meeting protocol between agents. As an application, we study the role of preferences and meetings in shaping classroom friendship networks in Brazil. In a network structure in which homophily due to preferences is stronger than homophily due to meeting opportunities, tracking students may improve welfare. Still, the relative benefit of this policy diminishes over the school semester.

In this paper, our aim is to prove the existence of normalized ground state for the following Schrödinger systems with potentials $$\begin{cases} -Δu_1+V_1(x)u_1+λ_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ -Δu_2+V_2(x)u_2+λ_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ 0<u_1,u_2\in H^1(\mathbb{R}^N), N\geq 1,\\ \int_{\mathbb{R}^N}u_1^2 \mathrm{d} x=a_1, \int_{\mathbb{R}^N}u_2^2 \mathrm{d} x=a_2. \end{cases}$$ The potentials $V_1(x),V_2(x)$ are general such that $\inf \text{ess}~σ(-Δ+V_ι)>-\infty$, which are allowed to be singular at some points. And the nonlinearities $G(u_1,u_2)$ are considered of the form $$ \begin{cases} G(u_1, u_2):=\sum_{i=1}^{\ell}\frac{μ_i}{p_i}|u_1|^{p_i}+\sum_{j=1}^{m}\frac{ν_j}{q_j}|u_2|^{q_j}+\sum_{k=1}^{n}β_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}},~~\ell,m,n\in \mathbb{N}^+_0, μ_i, ν_j,β_k>0, ~2<r_{1,k}+r_{2,k}, p_i, q_j<2+\frac{4}{N}, ~r_{1,k}, r_{2,k}>1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{cases} $$ Under the mass sub-critical assumption, the normalized ground states are obtained as the minimum of the functional $J$ on the manifold $S_{a_1,a_2}$. Since the functional is not weak lower semi-continuous, to prove the minimizing problem is achievable, the key step is establishing the strict sub-additive inequality. Among its main ingredients is the study of the sharp decay of the positive solutions and the interaction estimates.

Given a family of smooth immersions $F_t: M^n\to N^{n+1}$ of closed hypersurfaces in a locally symmetric Riemannian manifold $N^{n+1}$ with bounded geometry, moving by the mean curvature flow, we show that at the first finite singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a recent result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of N. Le in the Euclidean case (arXiv: math.DG/1002.4669v2).

In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first $p$-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type theorem when its Euler characteristic is negative.