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Solving polynomial equations for chemical problems using Gröbner bases
 MOLECULAR PHYSICS
, 2004
"... We explain the Gröbner basis method for solving simultaneous polynomial equations in several variables and describe some applications to chemical kinetics, stereochemistry, compartmental analysis and other chemical topics. ..."
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We explain the Gröbner basis method for solving simultaneous polynomial equations in several variables and describe some applications to chemical kinetics, stereochemistry, compartmental analysis and other chemical topics.
Right: Sketch of the
"... . The dark broken lines are in correspondence to surface ridges ( ), while the smaller dots correspond to surface vertices ( ). The larger nodes are shocks, the interior links have arrows to indicate flow (all ’s here), the hashed sheets are hyperlinks ( ; not all shown). Left: The fo ..."
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. The dark broken lines are in correspondence to surface ridges ( ), while the smaller dots correspond to surface vertices ( ). The larger nodes are shocks, the interior links have arrows to indicate flow (all ’s here), the hashed sheets are hyperlinks ( ; not all shown). Left: The for a truncated tetrahedron consists of 8 nodes, 7 links and 9 hyperlinks.
From the Infinitely Large to the Infinitely Small: Applications of Medial Symmetry Representations of Shape
, 2005
"... In this chapter we will cover a wide spectrum of applications of medial symmetries of shape from the infinitely large toward the infinitely small. Our journey starts with a dynamic model of the formation and evolution of galaxies. We then move on to the description of geographical information at the ..."
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In this chapter we will cover a wide spectrum of applications of medial symmetries of shape from the infinitely large toward the infinitely small. Our journey starts with a dynamic model of the formation and evolution of galaxies. We then move on to the description of geographical information at the scale of regions of planet Earth. Next is the representation of cities, buildings, and archaeological artefacts, followed by the perception of gardens, and the generation of virtual plants. Having reached the scale of human activities, we consider the perception and generation of artistic creations, the study of motion and the generation of animated virtual objects, and the representation of geometrically complex systems in machining, metal forging, object design. We then move inside the human body itself with applications in medical imaging and biology, followed by the representation of molecular structures. Our final stop is to consider the abstract scale of the perception of visual information.
Partitioning Multivariate Polynomial Equations via Vertex Separators for Algebraic Cryptanalysis and Mathematical Applications
"... Abstract. We present a novel approach for solving systems of polynomial equations via graph partitioning. The concept of a variablesharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the system of equations is actually two separate systems that can be s ..."
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Abstract. We present a novel approach for solving systems of polynomial equations via graph partitioning. The concept of a variablesharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the system of equations is actually two separate systems that can be solved individually. This can provide a significant speedup in computing the solution to the system, but is unlikely to occur either randomly or in applications. However, by deleting a small number of vertices on the graph, the variablesharing graph could be disconnected in a balanced fashion, and in turn the system of polynomial equations are separated into smaller ones of similar sizes. In graph theory terms, this process is equivalent to finding balanced vertex partitions with minimumweight vertex separators. The techniques of finding these vertex partitions are discussed, and experiments are performed to evaluate its practicality for general graphs and systems of polynomial equations. Applications of this approach to the QUAD family of stream ciphers, algebraic cryptanalysis of the stream cipher Trivium and its variants, as well as some mathematical problems in game theory and computational algebraic geometry are presented. In each of these cases, the systems of polynomial equations involved are wellsuited to our graph partitioning method, and constructive results are discussed.
Computational Study of 3D Affine Coordinate Transformation Part I. 3point Problem
"... e mail: J.awange curtin.edu.au In case of considerable nonlinearity e.g. in geodesy, photogrammetry, robotics, it is difficult to find proper initial values to solve the parameter estimation problem of 3D affine transformation with 9 parameters via linearization and/or iteration. In this paper we ..."
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e mail: J.awange curtin.edu.au In case of considerable nonlinearity e.g. in geodesy, photogrammetry, robotics, it is difficult to find proper initial values to solve the parameter estimation problem of 3D affine transformation with 9 parameters via linearization and/or iteration. In this paper we develop a symbolic numeric method to achieve the solution without initial guess. Our method employs explicit analytical expressions developed by the computer algebra technique Dixon resultant as well as by reduced Grobner basis for solving 3 points problem. Criteria for the proper selection of the 3 points from the N ones, is also given. Numerical illustration is presented with real world geodetic coordinates representing Hungarian Datum. For systems of algebraic equations, NSolve computes a numerical Grö bner basis using an efficient monomial ordering, then uses eigensystem methods to extract numerical roots.
Algebraic Biology 2005 1 Symbolic Calculation in the Life Sciences — Some Trends and Prospects
"... I discuss the literature and benefits of symbolic calculation in the life sciences; some ways to develop the field; and a way to model sequential processes. ..."
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I discuss the literature and benefits of symbolic calculation in the life sciences; some ways to develop the field; and a way to model sequential processes.