In this paper, we present our optimality results on optimal control problems for ordinary differential equations on Riemannian manifolds. For the problems with free states at the terminal time, we obtain the first and second-order necessary conditions, dynamical programming principle, and their relations. Then, we consider the problems with the initial and final states satisfying some inequality-type and equality-type constraints, and establish the corresponding first and second-order necessary conditions of optimal pairs in the sense of either spike or convex variations. For each of the above results concerning second-order optimality conditions, the curvature tensor of the underlying manifold plays a crucial role.

We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables, which is the crucial step to derive macroscopic free energy. It is at the level of kernel expansion, or equivalently the free energy, that the symmetries of the interacting rigid molecules can be fully recognized. Thus, writing down the form of expansion consistent with the symmetries is significant. Symmetries of two types are considered. First, we examine the symmetry of an interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, with the linearly independent terms identified. Then, we study the molecular symmetry characterized by a point group in $O(3).$ The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.

In this paper, we propose and develop a saturation value total variation (SV-TV) regularization model for simultaneously image denoising and luminance adjustment. The idea is to propose a variational approach containing an energy functional to adjust the luminance between image patches, and the noise of the image can be removed. In the proposed model, we establish the adjustment term based on the concept of structure, luminance, and contrast similarity, and we make use of the SV-TV regularization to remove the noise simultaneously. We present an efficient and effective algorithm with convergence guaranteed to solve the proposed minimization model. Experimental results are presented to show the effectiveness of the proposed model compared with existing methods.

In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods possibly with variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update of the separated primal variable, which would allow a more flexible and larger convergence region of the dual variable compared with that of standard deterministic or stochastic ADMM. This new stochastic optimization algorithm is shown to have ergodic converge in expectation with $\mathcal{O}(1/T)$ convergence rate, where $T$ denotes the number of outer iterations. Our preliminary experiments indicate the proposed algorithm is very effective for solving separable optimization problems from big-data applications. Finally, 3-block extensions of the algorithm and its variant of an accelerated stochastic augmented Lagrangian method are discussed in the appendix.

In this paper, we develop a fully discrete scheme to solve the confined ternary blended polymers (TBP) model with four order parameters based on the stabilized-scalar auxiliary variable (S-SAV) approach in time and the Fourier spectral method in space. Then, theoretical analysis is given for the scheme based on the backward differentiation formula. The unconditional energy stability and mass conservation are derived. Rigorous error analysis is carried out to show that the fully discrete scheme converges with order $\mathcal{O}(\tau^2+h^m)$ in the sense of the $L^2$ norm, where $\tau$ is the time step, $h$ is the spatial step, and $m$ is the regularity of the exact solution. Finally, some numerical results are given to demonstrate the theoretical analysis. Moreover, the phase separation of two kinds of polymer particles, namely, Ashura and Janus core-shell particles, is presented to show the morphological structures.

This paper deals with the four-component Keller-Segel-Stokes model of coral fertilization $$\begin{cases} n_t+u\cdot \nabla n=\Delta n^m-\nabla\cdot(nS(x,n,c)\cdot \nabla c)-n\rho, \\ c_t+u\cdot \nabla c=\Delta c-c+\rho, \\\rho_t+u\cdot \nabla\rho=\Delta\rho-n\rho, \\ u_t+\nabla P=\Delta u+(n+\rho)\nabla\phi, ~\nabla\cdot u=0\end{cases}$$ in a bounded and smooth domain $Ω⊂\mathbb{R}^3$ with zero-flux boundary for $n,$ $c,$ $\rho$ and no-slip boundary for $u,$ where $m>0,$ $\phi∈W^{2,∞}(Ω),$ and $S: \overline{Ω}×[0,∞)^ 2→\mathbb{R}^{3×3}$ is given sufficiently smooth function such that $|S(x,n,c)|≤S_0(c)(n+1)^{−α}$ for all $(x,n,c)∈\overline{Ω}×[0,∞)^2$ with $α≥0$ and some nondecreasing function $S_0 : [0,∞)\mapsto [0,∞).$ It is shown that if $m> 1−α$ for $0≤α≤\frac{2}{3},$ or $m≥ \frac{1}{3}$ for $α>\frac{2}{3},$ then for any reasonably regular initial data, the corresponding initial-boundary value problem admits at least one globally bounded weak solution which stabilizes to the spatially homogeneous equilibrium $(n_∞,\rho_∞,\rho_∞,0)$ in an appropriate sense, where $n_∞ := \frac{1}{|Ω|}\{\int_Ω n_0−\int_Ω\rho_0\}_+$ and $\rho_∞ :=\frac{1}{|Ω|}\{\int_Ω\rho_0−\int_Ω n0\}_+.$ These results improve and extend previously known ones.

We study the multistationarity for the reaction networks with one dimensional stoichiometric subspaces, and we focus on the networks admitting finitely many positive steady states. We provide a necessary condition for a network to admit multistationarity in terms of the stoichiometric coefficients, which can be described by “arrow diagrams”. This necessary condition is not sufficient unless there exist two reactions in the network such that the subnetwork consisting of the two reactions admits at least one and finitely many positive steady states. We also prove that if a network admits at least three positive steady states, then it contains at least three “bi-arrow diagrams”. More than that, we completely characterize the bi-reaction networks that admit at least three positive steady states.