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

Mutations that Separate the Functions of the Proofreading Subunit of the Escherichia coli Replicase

The dnaQ gene of Escherichia coli encodes the Ɛ subunit of DNA polymerase III, which provides the 3\u27 - 5\u27 exonuclease proofreading activity of the replicative polymerase. Prior studies have shown that loss of Ɛ leads to high mutation frequency, partially constitutive SOS, and poor growth. In addition, a previous study from our laboratory identified dnaQ knockout mutants in a screen for mutants specifically defective in the SOS response after quinolone (nalidixic acid) treatment. To explain these results, we propose a model whereby, in addition to proofreading, Ɛ plays a distinct role in replisome disassembly and/or processing of stalled replication forks. To explore this model, we generated a pentapeptide insertion mutant library of the dnaQgene, along with site-directed mutants, and screened for separation of function mutants. We report the identification of separation of function mutants from this screen, showing that proofreading function can be uncoupled from SOS phenotypes (partially constitutive SOS and the nalidixic acid SOS defect). Surprisingly, the two SOS phenotypes also appear to be separable from each other. These findings support the hypothesis that Ɛ has additional roles aside from proofreading. Identification of these mutants, especially those with normal proofreading but SOS phenotype(s), also facilitates the study of the role of e in SOS processes without the confounding results of high mutator activity associated with dnaQ knockout mutants

Meta SOS - A Maude Based SOS Meta-Theory Framework

Meta SOS is a software framework designed to integrate the results from the meta-theory of structural operational semantics (SOS). These results include deriving semantic properties of language constructs just by syntactically analyzing their rule-based definition, as well as automatically deriving sound and ground-complete axiomatizations for languages, when considering a notion of behavioural equivalence. This paper describes the Meta SOS framework by blending aspects from the meta-theory of SOS, details on their implementation in Maude, and running examples.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690

Predicting sinusoidal obstruction syndrome after allogeneic stem cell transplantation with the EASIX biomarker panel

No biomarker panel is established for prediction of sinusoidal obstruction syndrome/veno-occlusive disease (SOS/VOD), a major complication of allogeneic stem cell transplantation (alloSCT). We compared the potential of the Endothelial Activation and Stress Index (EASIX), based on lactate dehydrogenase, creatinine, and thrombocytes, with that of the SOS/VOD CIBMTR clinical risk score to predict SOS/VOD in two independent cohorts. In a third cohort, we studied the impact of endothelium-active prophylaxis with pravastatin and ursodeoxycholic acid (UDA) on SOS/VOD risk. The cumulative incidence of SOS/VOD within 28 days after alloSCT in the training cohort (Berlin, 2013-2015, n=446) and in the validation cohort (Heidelberg, 2002-2009, n=380) was 9.6% and 8.4%, respectively. In both cohorts, EASIX assessed at the day of alloSCT (EASIX-d0) was significantly associated with SOS/VOD incidence (p<0.0001), overall survival (OS) and non-relapse mortality (NRM). In contrast, the CIBMTR score showed no statistically significant association with SOS/VOD incidence, and did not predict OS and NRM. In patients receiving pravastatin/UDA, the cumulative incidence of SOS/VOD was significantly lower at 1.7% (p<0.0001, Heidelberg, 2010-2015, n=359) than in the two cohorts not receiving pravastatin/UDA. The protective effect was most pronounced in patients with high EASIX-d0. The cumulative SOS/VOD incidence in the highest EASIX-d0 quartiles were 18.1% and 16.8% in both cohorts without endothelial prophylaxis as compared to 2.2% in patients with pravastatin/UDA prophylaxis (p<0.0001). EASIX-d0 is the first validated biomarker for defining a subpopulation of alloSCT recipients at high risk for SOS/VOD. Statin/UDA endothelial prophylaxis could constitute a prophylactic measure for patients at increased SOS/VOD risk

A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Boundary Yang-Baxter equation in the RSOS/SOS representation

We construct and solve the boundary Yang-Baxter equation in the RSOS/SOS representation. We find two classes of trigonometric solutions; diagonal and non-diagonal. As a lattice model, these two classes of solutions correspond to RSOS/SOS models with fixed and free boundary spins respectively. Applied to (1+1)-dimenional quantum field theory, these solutions give the boundary scattering amplitudes of the particles. For the diagonal solution, we propose an algebraic Bethe ansatz method to diagonalize the SOS-type transfer matrix with boundary and obtain the Bethe ansatz equations.Comment: 30 pages, 5 figures, uses Latex with eepic.sty and epic.sty. Paper substantially expanded; section on SOS model is revised and a new section on the Bethe ansatz equation is adde

Analysis of Second-order Statistics Based Semi-blind Channel Estimation in CDMA Channels

The performance of second order statistics (SOS) based semi-blind channel estimation in long-code DS-CDMA systems is analyzed. The covariance matrix of SOS estimates is obtained in the large system limit, and is used to analyze the large-sample performance of two SOS based semi-blind channel estimation algorithms. A notion of blind estimation efficiency is also defined and is examined via simulation results.Comment: To be presented at the 2005 Conference on Information Sciences and System