On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

In this paper, we are interested in developing polynomial decomposition techniques to reformulate real valued multivariate polynomials into difference-of-sums-of-squares (namely, D-SOS) and difference-of-convex-sums-of-squares (namely, DC-SOS). Firstly, we prove that the set of D-SOS and DC-SOS polynomials are vector spaces and equivalent to the set of real valued polynomials. Moreover, the problem of finding D-SOS and DC-SOS decompositions are equivalent to semidefinite programs (SDP) which can be solved to any desired precision in polynomial time. Some important algebraic properties and the relationships among the set of sums-of-squares (SOS) polynomials, positive semidefinite (PSD) polynomials, convex-sums-of-squares (CSOS) polynomials, SOS-convex polynomials, D-SOS and DC-SOS polynomials are discussed. Secondly, we focus on establishing several practical algorithms for constructing D-SOS and DC-SOS decompositions for any polynomial without solving SDP. Using DC-SOS decomposition, we can reformulate polynomial optimization problems in the realm of difference-of-convex (DC) programming, which can be handled by efficient DC programming approaches. Some examples illustrate how to use our methods for constructing D-SOS and DC-SOS decompositions. Numerical performance of D-SOS and DC-SOS decomposition algorithms and their parallelized methods are tested on a synthetic dataset with 1750 randomly generated large and small sized sparse and dense polynomials. Some real-world applications in higher order moment portfolio optimization problems, eigenvalue complementarity problems, Euclidean distance matrix completion problems, and Boolean polynomial programs are also presented.Comment: 47 pages, 19 figure

Drude weight and dc-conductivity of correlated electrons

The Drude weight $D$ and the dc-conductivity $\sigma_{dc} (T)$ of strongly correlated electrons are investigated theoretically. Analytic results are derived for the homogeneous phase of the Hubbard model in $d = \infty$ dimensions, and for spinless fermions in this limit with $1/d$-corrections systematically included to lowest order. It is found that $\sigma_{dc}(T)$ is finite for all $T > 0$, displaying Fermi liquid behavior, $\sigma_{dc} \propto 1/T^2$, at low temperatures. The validity of this result for finite dimensions is examined by investigating the importance of Umklapp scattering processes and vertex corrections. A finite dc-conductivity for $T > 0$ is argued to be a generic feature of correlated lattice electrons in not too low dimensions.Comment: 15 pages, uuencoded compressed PS-fil

Dynamical symmetry breaking as the origin of the zero-dcdc-resistance state in an acac-driven system

Under a strong $ac$ drive the zero-frequency linear response dissipative resistivity $\rho_{d}(j=0)$ of a homogeneous state is allowed to become negative. We show that such a state is absolutely unstable. The only time-independent state of a system with a $\rho_{d}(j=0)<0$ is characterized by a current which almost everywhere has a magnitude $j_{0}$ fixed by the condition that the nonlinear dissipative resistivity $\rho_{d}(j_{0}^{2})=0$. As a result, the dissipative component of the $dc$ electric field vanishes. The total current may be varied by rearranging the current pattern appropriately with the dissipative component of the $dc$-electric field remaining zero. This result, together with the calculation of Durst \emph{et. al.}, indicating the existence of regimes of applied $ac$ microwave field and $dc$ magnetic field where $\rho_{d}(j=0)<0$, explains the zero-resistance state observed by Mani \emph{et. al.} and Zudov \emph{et. al.}.Comment: Published versio

Isolated output for class-D dc amplifiers

Transformer-coupled output stage is used with pulse-width modulated class-D dc amplifiers. Circuit is comprised of two channels corresponding to negative and positive input signals. Amplitude of secondary-current triangular pulse is function of duration of driving pulse. Therefore, circuit converts pulse-width modulated driving signal to pulse-amplitude modulated signal