The Spontaneous Pricing Order in the Noncooperative Game of Monopolistic Manufacturer and Many Retailers

We consider the collective pricing orders in a minimum supply chain that is composed of a monopolistic manufacturer and many retailers that belong to the same chain store firm. The retailers have the freedom to raise or lower the local price. The chain store firm sets up the commercial rules for local retail stores to maximize its total payoff. The monopolistic manufacturer firm controls the total quantity supplied for the market to achieve maximum benefits. We applied the two dimensional Ising model in statistical physics to map the collective distribution of microscopic strategy of local retailers into the macroscopic total payoff of the chain store firm. The local stores choose to raise the price or lower the price based their own mind when the supply in market surpasses the demand. When the supply in market is far less than the demand, the stores synchronously raise prices, even though a local store only have the incomplete information of their nearest neighboring supermarket. We find the critical equation for the balance point between the action of supplier and the action of chain store management based on game theory and statistical physics. The critical equation can identify the Nash equilibrium point of the non-cooperative game between the manufacturer and the chain-store seller, and reveal different levels of collective operations. This statistical physics method also holds for more complicate supply chains and economic systems.

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

In this paper, we classified the paracontact metric κ , μ -manifold satisfying the Miao-Tam critical equation with κ > − 1 . We proved that it is locally isometric to the product of a flat n + 1 -dimensional manifold and an n -dimensional manifold of negative constant curvature − 4 .

Construction of solutions for a critical problem with competing potentials via local Pohozaev identities

In this paper, we consider the following critical equation: [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if [Formula: see text], [Formula: see text] has a stable critical point [Formula: see text] with [Formula: see text] and [Formula: see text], then the above equation has infinitely many positive solutions, where [Formula: see text] is the unique positive solution of [Formula: see text] with [Formula: see text]. Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606–2633], it implies that the role of stable critical points of [Formula: see text] in constructing bump solutions is more important than that of [Formula: see text] and that [Formula: see text] can influence the sign of [Formula: see text], i.e. [Formula: see text] can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of [Formula: see text] which include the case of a saddle point.

Mass and Extremals Associated with the Hardy–Schrödinger Operator on Hyperbolic Space

We consider the Hardy–Schrödinger operator {L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space {\mathbb{B}^{n}} ({n\geq 3}). Here {V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., {V_{2}(r)\sim\frac{1}{r^{2}}}. As in the Euclidean setting, {L_{\gamma}} is positive definite whenever {\gamma<\frac{(n-2)^{2}}{4}}, in which case we exhibit explicit solutions for the critical equation {L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in {\mathbb{B}^{n},} where {0\leq s<2}, {2^{*}(s)=\frac{2(n-s)}{n-2}}, and {V_{2^{*}(s)}} is a weight that behaves like {\frac{1}{r^{s}}} around 0. In dimensions {n\geq 5}, the equation {L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in a domain Ω of {\mathbb{B}^{n}} away from the boundary but containing 0 has a ground state solution, whenever {0<\gamma\leq\frac{n(n-4)}{4}}, and {\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}. On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

Superconducting Quantum Critical Phenomena

When the superconducting transition temperature [Formula: see text] sufficiently approaches zero, quantum fluctuations are expected to be overwhelmingly amplified around zero temperature so that the mean-field approximation may break down. This implies that quantum critical phenomena may emerge in highly underdoped and overdoped regions, where the transition temperature [Formula: see text] is sufficiently low. By using Gor’kov’s Green function method, we propose a superconducting quantum critical equation (SQCE) for describing such critical phenomena. For two-dimensional (2D) overdoped materials, SQCE shows that the transition temperature [Formula: see text] and the zero-temperature superfluid phase stiffness [Formula: see text] will obey a two-class scaling combined by linear and parabolic parts, which agrees with the existing experimental investigation [I. Božović et al., Dependence of the critical temperature in overdoped copper oxides on superfluid density, Nature 536 (2016) 309–311]. For three-dimensional (3D) overdoped materials, SQCE predicts that the two-class scaling will be replaced by the linear scaling. Furthermore, we show that SQCE can be applied into highly underdoped region by using Anderson’s non-Fermi liquid model.