Narrow Results

In this paper, we develop a dynamic control to investigate the hospitals’ quality improvement and congestion reduction, along with corresponding knowledge accumulations. The main features of our work are: (i) developing a dynamic analysis model of the hospitals’ quality improvement and congestion reduction, considering knowledge accumulations in health care markets; (ii) the demand function for medical treatment depends on treatment quality, congestion degree, price, and traveling cost; (iii) each hospital’s instantaneous costs of quality improvement and congestion reduction depend on treatment quality, quality-improving investment, corresponding knowledge accumulation as well as congestion degree, congestion-reducing investment and the corresponding knowledge accumulation; (iv) the hospitals’ dynamic competitions are not only in treatment quality and congestion degree but also in treatment prices. Our results show that (i) there exists a unique saddle-stable steady-state equilibrium under hospital optimum and social optimum; (ii) knowledge accumulation and the complementarity or substitutability affect the hospitals’ decision behavior; (iii) whether the price and the hospitals’ investments in quality improvement and congestion reduction are higher or lower under hospital optimum than that under social optimum depends on the parameter regions of complementarity or substitutability.

Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson–Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count framed twisted quiver sheaves with the moment map relation and directly generalize the ADHM invariants of Diaconescu.

Let $v_d({\mathbb{P}}^2)\subset |{{𝒪}}_{{\mathbb{P}}^2}(d)|$ denote the $d$-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of $v_d({\mathbb{P}}^2)$ by embedding the blow-up ${\mathrm{\text{bl}}}_{v_d({\mathbb{P}}^2)}|{{𝒪}}_{{\mathbb{P}}^2}(d)|$ into a suitable moduli space of Bridgeland semistable objects on ${\mathbb{P}}^2$.

We apply equivariant localization to the theory of $Z$-stability and $Z$-critical metrics on a Kähler manifold $(X,\alpha )$, where $\alpha$ is a Kähler class. We show that the invariants used to determine $Z$-stability of the manifold, which are integrals over test configurations, can be written as a product of equivariant classes, hence equivariant localization can be applied. We also study the existence of $Z$-critical Kähler metrics in $\alpha$, whose existence is conjectured to be equivalent to $Z$-stability of $(X,\alpha )$. In particular, we study a class of invariants that give an obstruction to the existence of such metrics. Then we show that these invariants can also be written as a product of equivariant classes. From this we give a new, more direct proof of an existing result: the former invariants determining $Z$-stability on a test configuration are equal to the latter invariants related to the existence of $Z$-critical metrics on the central fibre of the test configuration. This provides a new approach from which to derive the $Z$-critical equation.

In this paper, we consider a general static, spherically symmetric background in the quadratic beyond Horndeski theory and analyze the behavior of linear perturbations in both parity odd and parity even sectors. We derive a full set of stability conditions for an arbitrary static, spherically symmetric solution which guarantees absence of ghosts, gradient instabilities, tachyons and superluminal modes in both sectors.

We investigate synchronization between two unidirectionally linearly coupled chaotic multifeedback Mackey–Glass systems and find the existence and stability conditions for complete synchronization. Numerical simulations fully support the theory. We also present generalization of the approach to the wider class of nonlinear systems.

Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

We compare inviscid stability conditions obtained by Lewicka for large-amplitude shock wave patterns with "slow eigenvalue", or low-frequency, stability conditions obtained by Lin and Schecter through a vanishing viscosity analysis of the Dafermos regularization. Under the structural condition that scattering coefficients for each component wave are positive, we show that BV and L¹ inviscid stability are equivalent to respective versions of low-frequency Dafermos-regularized stability. When scattering coefficients appear with different signs, the conditions are in general distinct. We give various examples demonstrating this phenomenon and indicating the subtle role of cancellation in linearized behavior in the presence of negative scattering coefficients.