Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most \(n+d\). Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most \(n^{O(d)}\) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

     

Fluctuation Bounds in the Exponential Bricklayers Process

This paper is the continuation of our earlier paper (Balázs et al. in Ann. Inst. Henri Poincaré Probab. Stat. 48(1):151–187, 2012), where we proved t ^1/3-order of current fluctuations across the characteristics in a class of one dimensional interacting systems with one conserved quantity. We also claimed two models with concave hydrodynamic flux which satisfied the assumptions which made our proof work. In the present note we show that the totally asymmetric exponential bricklayers process also satisfies these assumptions. Hence this is the first example with convex hydrodynamics of a model with t ^1/3-order current fluctuations across the characteristics. As such, it further supports the idea of universality regarding this scaling.     

Time-optimal coherence-order-selective transfer of in-phase coherence in heteronuclear IS spin systems

Based on principles of geometric optimal control theory, coherence transfer building blocks can be derived which achieve optimal sensitivity. Here, experimental pulse sequences are presented that achieve the best possible coherence-order-selective in-phase transfer (S(-)-->I(-)) for a heteronuclear 2-spin system for any given mixing time in the absence of relaxation. For short mixing times, the optimal experiment improves the sensitivity of isotropic mixing by up to 12.5%.     

Modeling of acoustic wave propagation in time-domain using the discontinuous Galerkin method – A comparison with measurements

Simulations of acoustic wave propagation in time-domain are presented. In the simulations, the discontinuous Galerkin method for spatial derivatives and the low-storage Runge–Kutta approach for time derivatives are used. Three different simulation cases are studied. First, the directivity of loudspeaker is simulated. In the second case, acoustic wave propagation in free space is studied using a short pulse. In the last case, acoustic wave scattering from a metallic cylinder is simulated. All simulation results are compared with measurement results. The measurements for the acoustic wave scattering from the metallic cylinder are made in 2D planes using an automated measurement system. Comparison between the simulation and measurement results are made both temporally and spatially and a good agreement between the simulation and measurement results is found. The results suggest that the discontinuous Galerkin method coupled with the low-storage Runge–Kutta approach is a viable tool for modeling acoustic wave propagation in the time-domain.     

Theory of parametric amplification in superlattices

We consider a high-frequency response of electrons in a single miniband of superlattice subject to dc and ac electric fields. We show that Bragg reflections in miniband result in a parametric resonance which is detectable using ac probe field. We establish theoretical feasibility of phase-sensitive THz amplification at the resonance. The parametric amplification does not require operation in conditions of negative differential conductance. This prevents a formation of destructive domains of high electric field inside the superlattice.     

On the Lieb-Thirring constantsL γ,1 for γ≧1/2

LetE _i(H) denote the negative eigenvalues of the one-dimensional Schrödinger operatorHu??u″?Vu,V≧0, onL ₂(∝). We prove the inequality (1) $$\mathop \sum \limits_i |E_i (H)|^{ \gamma } \leqq L_{\gamma ,1} \mathop \smallint \limits_\mathbb{R} V^{\gamma + 1/2} (x)dx,$$ for the “limit” case γ=1/2. This will imply improved estimates for the best constantsL _γ,1 in (1) as 1/2<γ<3/2.     

Entropic Contributions to Sodium Solvation and Solvent Stabilization upon Electrochemical Sodium Deposition from Diglyme and Propylene Carbonate Electrolytes

The formation of an appropriate solid electrolyte interphase (SEI) at the anode of a sodium battery is crucially dependent on the electrochemical stability of solvent and electrolyte at the redox potential of Na/Na⁺ in the respective system. In order to determine entropic contributions to the relative stability of the electrolyte solution, we measure the reaction entropy of Na metal deposition for diglyme (DG) and propylene carbonate (PC) based electrolyte solutions by electrochemical microcalorimetry at single electrodes. We found a large positive reaction entropy for Na⁺ deposition in DG of Δ_R 234 J mol⁻¹ K⁻¹ (c.f.: Δ_R 83 J mol⁻¹ K⁻¹), which signals substantial entropic destabilization of Na⁺ in DG by about 0.73 eV, thus increasing the stability of solvent and electrolyte relative to Na⁺ reduction. We attribute this strong entropic destabilization to a highly negative solvation entropy of Na⁺, due to the low dielectric constant and high freezing entropy of DG.     

The Newton-Kleinman method for the optimal control of stable weakly regular linear systems

We give an algorithm to find the approximate solution of the linear-quadratic optimal control problem for stable weakly regular linear systems. This algorithm can be understood as a generalization of the Newton-Kleinman method known from the finite-dimensional theory. The central characteristic of our approach is the possibility to solve problems with unbounded control and observation operators, which is motivated by partial differential equations with boundary control and observation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)