Geolocation with FDOA Measurements via Polynomial Systems and RANSAC

The problem of geolocation of a transmitter via time difference of arrival (TDOA) and frequency difference of arrival (FDOA) is given as a system of polynomial equations. This allows for the use of homotopy continuation-based methods from numerical algebraic geometry. A novel geolocation algorithm employs numerical algebraic geometry techniques in conjunction with the random sample consensus (RANSAC) method. This is all developed and demonstrated in the setting of only FDOA measurements, without loss of generality. Additionally, the problem formulation as polynomial systems immediately provides lower bounds on the number of receivers or measurements required for the solution set to consist of only isolated points.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Bounds on the number of real solutions to polynomial equations

We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.Comment: 5 page

Numerical algebraic geometry for model selection and its application to the life sciences

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to non-linearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data is available. Here, we consider polynomial models (e.g., mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometric structures relating models and data, and we demonstrate its utility on examples from cell signaling, synthetic biology, and epidemiology.Comment: References added, additional clarification

Interactome comparison of human embryonic stem cell lines with the inner cell mass and trophectoderm

Networks of interacting co-regulated genes distinguish the inner cell mass (ICM) from the differentiated trophectoderm (TE) in the preimplantation blastocyst, in a species specific manner. In mouse the ground state pluripotency of the ICM appears to be maintained in murine embryonic stem cells (ESCs) derived from the ICM. This is not the case for human ESCs. In order to gain insight into this phenomenon, we have used quantitative network analysis to identify how similar human (h)ESCs are to the human ICM. Using the hESC lines MAN1, HUES3 and HUES7 we have shown that all have only a limited overlap with ICM specific gene expression, but that this overlap is enriched for network properties that correspond to key aspects of function including transcription factor activity and the hierarchy of network modules. These analyses provide an important framework which highlights the developmental origins of hESCs

Adhesion of Polymer Vesicles

The adhesion and bending modulus of polybutadiene-poly(ethylene oxide) block copolymer vesicles made from a bidisperse mixture of polymers is measured using micropipette aspiration. The adhesion energy between biotinylated vesicles and avidin beads is modeled by incorporating the extension of the adhesive ligands above the surface brush of the vesicle according to the blob model of bidisperse polymer mixtures of Komura and Safran assuming the polymer brush at the surface of the vesicle is compact. The same model accurately reproduces the scaling of the bending modulus with polymer composition