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

Quantum planes and quantum cylinders from Poisson homogeneous spaces

Quantum planes and a new quantum cylinder are obtained as quantization of Poisson homogeneous spaces of two different Poisson structures on classical Euclidean group E(2).Comment: 13 pages, plain Tex, no figure

Polaronic transport induced by competing interfacial magnetic order in a La0.7_{0.7}Ca0.3_{0.3}MnO3_{3}/BiFeO3_{3} heterostructure

Using ultrafast optical spectroscopy, we show that polaronic behavior associated with interfacial antiferromagnetic order is likely the origin of tunable magnetotransport upon switching the ferroelectric polarity in a La$_{0.7}$Ca$_{0.3}$MnO$_{3}$/BiFeO$_{3}$ (LCMO/BFO) heterostructure. This is revealed through the difference in dynamic spectral weight transfer between LCMO and LCMO/BFO at low temperatures, which indicates that transport in LCMO/BFO is polaronic in nature. This polaronic feature in LCMO/BFO decreases in relatively high magnetic fields due to the increased spin alignment, while no discernible change is found in the LCMO film at low temperatures. These results thus shed new light on the intrinsic mechanisms governing magnetoelectric coupling in this heterostructure, potentially offering a new route to enhancing multiferroic functionality

Antisite effect on ferromagnetism in (Ga,Mn)As

We study the Curie temperature and hole density of (Ga,Mn)As while systematically varying the As-antisite density. Hole compensation by As-antisites limits the Curie temperature and can completely quench long-range ferromagnetic order in the low doping regime of 1-2% Mn. Samples are grown by molecular beam epitaxy without substrate rotation in order to smoothly vary the As to Ga flux ratio across a single wafer. This technique allows for a systematic study of the effect of As stoichiometry on the structural, electronic, and magnetic properties of (Ga,Mn)As. For concentrations less than 1.5% Mn, a strong deviation from Tc ~ p^0.33 is observed. Our results emphasize that proper control of As-antisite compensation is critical for controlling the Curie temperatures in (Ga,Mn)As at the low doping limit.Comment: 10 pages, 7 figure

An Empirical Charge Transfer Potential with Correct Dissociation Limits

The empirical valence bond (EVB) method [J. Chem. Phys. 52, 1262 (1970)] has always embodied charge transfer processes. The mechanism of that behavior is examined here and recast for use as a new empirical potential energy surface for large-scale simulations. A two-state model is explored. The main features of the model are: (1) Explicit decomposition of the total system electron density is invoked; (2) The charge is defined through the density decomposition into constituent contributions; (3) The charge transfer behavior is controlled through the resonance energy matrix elements which cannot be ignored; and (4) A reference-state approach, similar in spirit to the EVB method, is used to define the resonance state energy contributions in terms of "knowable" quantities. With equal validity, the new potential energy can be expressed as a nonthermal ensemble average with a nonlinear but analytical charge dependence in the occupation number. Dissociation to neutral species for a gas-phase process is preserved. A variant of constrained search density functional theory is advocated as the preferred way to define an energy for a given charge.Comment: Submitted to J. Chem. Phys. 11/12/03. 14 pages, 8 figure

Flux through a hole from a shaken granular medium

We have measured the flux of grains from a hole in the bottom of a shaken container of grains. We find that the peak velocity of the vibration, vmax, controls the flux, i.e., the flux is nearly independent of the frequency and acceleration amplitude for a given value of vmax. The flux decreases with increasing peak velocity and then becomes almost constant for the largest values of vmax. The data at low peak velocity can be quantitatively described by a simple model, but the crossover to nearly constant flux at larger peak velocity suggests a regime in which the granular density near the container bottom is independent of the energy input to the system.Comment: 14 pages, 4 figures. to appear in Physical Review

Impurity Band Conduction in a High Temperature Ferromagnetic Semiconductor

The band structure of a prototypical dilute ferromagnetic semiconductor, Ga$_{1-x}$Mn$_{x}$As, is studied across the phase diagram via optical spectroscopy. We prove that the Fermi energy ($E_{F}$) resides in a Mn induced impurity band (IB). This conclusion is based upon careful analysis of the frequency and temperature dependence of the optical conductivity ($\sigma_{1}(\omega,T)$). From our analysis of $\sigma_{1}(\omega,T)$ we infer a large effective mass ($m^*$) of the carriers, supporting the view that conduction occurs in an IB. Our results also provide useful insights into the transport properties of Mn-doped GaAs.Comment: 4 pages, 4 figure

Stochastic Ballistic Annihilation and Coalescence

We study a class of stochastic ballistic annihilation and coalescence models with a binary velocity distribution in one dimension. We obtain an exact solution for the density which reveals a universal phase diagram for the asymptotic density decay. By universal we mean that all models in the class are described by a single phase diagram spanned by two reduced parameters. The phase diagram reveals four regimes, two of which contain the previously studied cases of ballistic annihilation. The two new phases are a direct consequence of the stochasticity. The solution is obtained through a matrix product approach and builds on properties of a q-deformed harmonic oscillator algebra.Comment: 4 pages RevTeX, 3 figures; revised version with some corrections, additional discussion and in RevTeX forma