S-Lemma with Equality and Its Applications

Let $f(x)=x^TAx+2a^Tx+c$ and $h(x)=x^TBx+2b^Tx+d$ be two quadratic functions having symmetric matrices $A$ and $B$. The S-lemma with equality asks when the unsolvability of the system $f(x)<0, h(x)=0$ implies the existence of a real number $\mu$ such that $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$. The problem is much harder than the inequality version which asserts that, under Slater condition, $f(x)<0, h(x)\le0$ is unsolvable if and only if $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$ for some $\mu\ge0$. In this paper, we show that the S-lemma with equality does not hold only when the matrix $A$ has exactly one negative eigenvalue and $h(x)$ is a non-constant linear function ($B=0, b\not=0$). As an application, we can globally solve $\inf\{f(x)\vert h(x)=0\}$ as well as the two-sided generalized trust region subproblem $\inf\{f(x)\vert l\le h(x)\le u\}$ without any condition. Moreover, the convexity of the joint numerical range $\{(f(x), h_1(x),\ldots, h_p(x)):~x\in\Bbb R^n\}$ where $f$ is a (possibly non-convex) quadratic function and $h_1(x),\ldots,h_p(x)$ are affine functions can be characterized using the newly developed S-lemma with equality.Comment: 34 page

A linear theory for control of non-linear stochastic systems

We address the role of noise and the issue of efficient computation in stochastic optimal control problems. We consider a class of non-linear control problems that can be formulated as a path integral and where the noise plays the role of temperature. The path integral displays symmetry breaking and there exist a critical noise value that separates regimes where optimal control yields qualitatively different solutions. The path integral can be computed efficiently by Monte Carlo integration or by Laplace approximation, and can therefore be used to solve high dimensional stochastic control problems.Comment: 5 pages, 3 figures. Accepted to PR

Joint measurement of complementary observables in moment tomography

Wigner and Husimi quasi-distributions, owing to their functional regularity, give the two archetypal and equivalent representations of all observable-parameters in continuous-variable quantum information. Balanced homodyning and heterodyning that correspond to their associated sampling procedures, on the other hand, fare very differently concerning their state or parameter reconstruction accuracies. We present a general theory of a now-known fact that heterodyning can be tomographically more powerful than balanced homodyning to many interesting classes of single-mode quantum states, and discuss the treatment for two-mode sources.Comment: 15 pages, 4 figures, conference proceedings for Quantum 2017 in Torin

Extracellular signal-regulated kinases mediate the enhancing effects of inflammatory mediators on resurgent currents in dorsal root ganglion neurons

Previously we reported that a group of inflammatory mediators significantly enhanced resurgent currents in dorsal root ganglion neurons. To understand the underlying intracellular signaling mechanism, we investigated the effects of inhibition of extracellular signal-regulated kinases and protein kinase C on the enhancing effects of inflammatory mediators on resurgent currents in rat dorsal root ganglion neurons. We found that the extracellular signal-regulated kinases inhibitor U0126 completely prevented the enhancing effects of the inflammatory mediators on both Tetrodotoxin-sensitive and Tetrodotoxin-resistant resurgent currents in both small and medium dorsal root ganglion neurons. U0126 substantially reduced repetitive firing in small dorsal root ganglion neurons exposed to inflammatory mediators, consistent with prevention of resurgent current amplitude increases. The protein kinase C inhibitor Bisindolylmaleimide I also showed attenuating effects on resurgent currents, although to a lesser extent compared to extracellular signal-regulated kinases inhibition. These results indicate a critical role of extracellular signal-regulated kinases signaling in modulating resurgent currents and membrane excitability in dorsal root ganglion neurons treated with inflammatory mediators. It is also suggested that targeting extracellular signal-regulated kinases-resurgent currents might be a useful strategy to reduce inflammatory pain

Chaotic universe in the z=2 Hovava-Lifshitz gravity

The deformed z=2 Horava-Lifshitz gravity with coupling constant w leads to a nonrelativistic "mixmaster" cosmological model. The potential of theory is given by the sum of IR and UV potentials in the ADM Hamiltonian formalism. It turns out that adding the UV-potential cannot suppress chaotic behaviors existing in the IR-potential.Comment: 7 pages, 5 figures, version to appear in PR