Hirokazu Anai

SyNRAC: a Maple-package for solving real algebraic constraints

In this paper we present a maple-package, named SyNRAC, for solving real algebraic constraints derived from various engineering problems. Our main tool is real quantifier elimination and we focus on its application to robust control design problems.

The Maple package SyNRAC and its application to robust control design

We have been developing a toolbox on Maple, called SyNRAC, for solving real algebraic constraints derived from various engineering problems. Its main tools are real quantifier elimination and the simplification of quantifier-free formulas. We illustrate algorithms implemented in SyNRAC, give some examples of how its commands are used, and present its application to design problems in systems and control theory.

Development of SyNRAC

We present newly implemented functions in SyNRAC, which is a Maple package for solving real algebraic constraints derived from various engineering problems. The current version of SyNRAC has added quantifier elimination (QE) by cylindrical algebraic decomposition (CAD), a general QE procedure. We also show a visualization tool for representing the possble region of an output quantifier-free formula for the two-dimensional case.

Exact parameter determination for Parkinson's disease diagnosis with PET using an algebraic approach

The mechanism of Parkinsons disease can be investigated at the molecular level by using radio-tracers. The concentration of dopamine in the brain can be observed by using a radio-tracer, 6-[18F]fluorodopa (FDOPA), with positron emission tomography (PET), and the dopamine kinetics can be described as compartmental models for tissues of the brain. The models for FDOPA kinetics are solved explicitly, but the solution shows a complicated form including several convolutions over time domain. Owing to the complicated form of the solution, graphical analyses such as Logan or Patlak analysis have been utilized as conventional methods over past decades. Because some kinetic constants for Parkinsons disease are estimated in the graphical analyses with the slope or intercept of the line obtained under various assumptions, only a limited set of parameters have approximately been estimated. We have analysed the compartmental models by using the Laplace transformation of differential equations and by algebraic computation with the aid of Grobner base constructions. We have obtained a rigorous solution with respect to the kinetic constants over the Laplace domain. Here, we first derive a rigorous solution for the parameters, together with a discussion about the merits of the derivation. Next, we describe a procedure to determine the kinetic constants with the observed time-radioactivity curves. Last, we discuss the feasibility of our method, especially as a criterion for diagnosing Parkinsons disease.

An Algebraic-Numeric Algorithm for the Model Selection in Kinetic Networks

We propose a novel algorithm to select a model that is consistent with the time series of observed data. In the first step, the kinetics for describing a biological phenomenon is expressed by a system of differential equations, assuming that the relationships between the variables are linear. Simultaneously, the time series of the data are numerically fitted as a series of exponentials. In the next step, both the system of differential equations with the kinetic parameters and the series of exponentials fitted to the observed data are transformed into the corresponding system of algebraic equations, by the Laplace transformation. Finally, the two systems of algebraic equations are compared by an algebraic approach. The present method estimates the models consistency with the observed data and the determined kinetic parameters. One of the merits of the present method is that it allows a kinetic model with cyclic relationships between variables that cannot be handled by the usual approaches. The plausibility of the present method is illustrated by the actual relationships between specific leaf area, leaf nitrogen and leaf gas exchange with the corresponding simulated data.

Derivation of rigorous conditions for high cell-type diversity by algebraic approach

The development of a multicellular organism is a dynamic process. Starting with one or a few cells, the organism develops into different types of cells with distinct functions. We have constructed a simple model by considering the cell number increase and the cell-type order conservation, and have assessed conditions for cell-type diversity. This model is based on a stochastic Lindenmayer system with cell-to-cell interactions for three types of cells. In the present model, we have successfully derived complex but rigorous algebraic relations between the proliferation and transition rates for cell-type diversity by using a symbolic method: quantifier elimination (QE). Surprisingly, three modes for the proliferation and transition rates have emerged for large ratios of the initial cells to the developed cells. The three modes have revealed that the equality between the development rates for the highest cell-type diversity is reduced during the development process of multicellular organisms. Furthermore, we have found that the highest cell-type diversity originates from order conservation.