Abstract:
The modern Phillips curve is about the relationship between inflation and unemployment and has been the center of a fierce debate in economics over fifty years. This paper reports empirical evidence that uncovers some of its mysteries. The rate of inflation and the unemployment rate are closely related to business cycles. What is of interest is that no two business cycles are exactly alike; however, all business cycles are essentially alike [1,2]. Each expansion is ended by a recession induced by adverse shocks. The U.S. economy suffers from adverse shocks all the time, but not every shock gives rise to a recession. Why do adverse shocks often induce a recession after an expansion that has lasted for a substantial duration? That is, why are double-dip recessions so rare? Here do we find important evidence in the Phillips curve that may help answer this question. We also discuss some issues related to the monetary policy and raise a few open questions about the relationship between unemployment and business cycles.

The modern Phillips curve is about the
relationship between the average rates of inflation and unemployment. We will
provide additional empirical evidence in the US economy from 1948:01 to 2013:03
that helps demonstrate why such a relationship has been built on a wrong
methodology, as revealed in Ma [1]. An erroneous approach can lead to a misunderstanding of business
cycles and a wrongful implementation of monetary policy. In particular, the way
how the two rates may evolve is now at a critical moment for the Fed to
decide if an exit from its quantitative easing should be initiated.

Abstract:
For a cyclic group $A$ and a connected Lie group $G$ with an $A$-module structure (with the additional conditions that $G$ is compact and the $A$-module structure on $G$ is 1-semisimple if $A\cong\ZZ$), we define the twisted Weyl group $W=W(G,A,T)$, which acts on $T$ and $H^1(A,T)$, where $T$ is a maximal compact torus of $G_0^A$, the identity component of the group of invariants $G^A$. We then prove that the natural map $W\backslash H^1(A,T)\to H^1(A,G)$ is a bijection, reducing the calculation of $H^1(A,G)$ to the calculation of the action of $W$ on $T$. We also prove some properties of the twisted Weyl group $W$, one of which is that $W$ is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.

Abstract:
We prove that Ad-semisimple conjugacy classes in a connected Lie group $G$ are closed embedded submanifolds of $G$. We also prove that if $\alpha:H\to G$ is a homomorphism of connected Lie groups such that the kernel of $\alpha$ is discrete in $H$, then for an Ad-semisimple conjugacy class $C$ in $G$, every connected component of $\alpha^{-1}(C)$ is a conjugacy class in $H$. Corresponding results for adjoint orbits in real Lie algebras are also proved.

Abstract:
We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is H\"{o}lder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to $f$ is conjugate to $f$ by a homeomorphism which is H\"{o}lder on the whole manifold.

Abstract:
By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group $G$ on a smooth or analytic manifold $M$ with a rigid $\mathrm{A}$-structure $\sigma$. It generalizes Gromov's centralizer and representation theorems to the case where $R(G)$ is split solvable and $G/R(G)$ has no compact factors, strengthens a special case of Gromov's open dense orbit theorem, and implies that for smooth $M$ and simple $G$, if Gromov's representation theorem does not hold, then the local Killing fields on $\widetilde{M}$ are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of $\mathrm{Iso}(M)$ for simply connected compact analytic $M$ with unimodular $\sigma$, (2) three results illustrating the phenomena that if $G$ is split solvable and large then $\pi_1(M)$ is also large, and (3) two fixed point theorems for split solvable $G$ and compact analytic $M$ with non-unimodular $\sigma$.

Abstract:
We prove that for any $s,t\ge0$ with $s+t=1$ and any $\theta\in\mathbb{R}$ with $\inf_{q\in\mathbb{N}}q^{\frac{1}{s}}\|q\theta\|>0$, the set of $y\in\mathbb{R}$ for which $(\theta,y)$ is $(s,t)$-badly approximable is 1/2-winning for Schmidt's game. As a consequence, we remove a technical assumption in a recent theorem of Badziahin-Pollington-Velani on simultaneous Diophantine approximation.

Abstract:
We prove that for any pair $(s,t)$ of nonnegative numbers with $s+t=1$, the set of two-dimensional $(s,t)$-badly approximable vectors is winning for Schmidt's game. As a consequence, we give a direct proof of Schmidt's conjecture using his game.

Abstract:
The speed tracking control problem of permanent magnet synchronous motors with parameter uncertainties and load torque disturbance is addressed. Fuzzy logic systems are used to approximate nonlinearities, and an adaptive backstepping technique is employed to construct controllers. The proposed controller guarantees the tracking error convergence to a small neighborhood of the origin and achieves the good tracking performance. Simulation results clearly show that the proposed control scheme can track the position reference signal generated by a reference model successfully under parameter uncertainties and load torque disturbance without singularity and overparameterization.

Abstract:
An adaptive fuzzy control method is developed to control chaos in the permanent magnet synchronous motor drive system via backstepping. Fuzzy logic systems are used to approximate unknown nonlinearities, and an adaptive backstepping technique is employed to construct controllers. The proposed controller can suppress the chaos of PMSM and track the reference signal successfully. The simulation results illustrate its effectiveness. 1. Introduction Permanent magnet synchronous motors (PMSMs) are intensively used in industrial applications due to their high speed, high efficiency, high power density, and large torque to inertia ratio. Then, it is still a challenging problem to control the PMSM to get the perfect dynamic performance, because the dynamic model of PMSM is nonlinear, multivariable and even experiencing Hopf bifurcation, limit cycles, and chaotic attractors with systemic parameters falling into a certain area [1]. The chaotic behavior in PMSM is undesirable since it can extremely destroy the stabilization of the motor or even induce drive system collapse. Chaos in the PMSM and its control have been an active research area in the field of nonlinear control of electric motors [2]. Up to now, some control methods, such as OGY method [3], feedback linearization [4], time delay feedback control [5–7], sliding model control [8], adaptive control method [9, 10], backstepping method [11–14], and dynamic surface control [9] are successfully used to control or suppress chaos in PMSM. However, the existing control methods also have some disadvantages. The OGY method requires a variable system parameter which is usually unavailable in the control of the PMSM. The employed method of feedback linearization requires the exact mathematical model; so the controller requires the desired dynamics to replace the system at the axis stator currents. The time delay feedback control was successfully implemented to control the PMSM, but it is difficult to determine the time delay for TDFC method given a special target and is not suitable when the desired target is not the equilibrium or an unstable periodic orbit of the system. Chattering phenomenon and high heat loss in electrical power circuits are the drawbacks of the sliding mode control. Backstepping is a newly developed technique to control the nonlinear systems with parameter uncertainty, particularly those systems in which the uncertainty does not satisfy matching conditions. Though the conventional backstepping is successfully applied to the control of PMSM drivers recently, it usually makes the designed