New Algorithms for Computing Groebner Bases

In this thesis, we present new algorithms for computing Groebner bases. The first algorithm, G2V, is incremental in the same fashion as F5 and F5C. At a typical step, one is given a Groebner basis G for an ideal I and any polynomial g, and it is desired to compute a Groebner basis for the new ideal , obtained from I by joining g. Let (I : g) denote the colon ideal of I divided by g. Our algorithm computes Groebner bases for I, g and (I : g) simultaneously. In previous algorithms, S-polynomials that reduce to zero are useless, in fact, F5 tries to avoid such reductions as much as possible. In our algorithm, however, these \u27useless\u27 S-polynomials give elements in (I : g) and are useful in speeding up the subsequent computations. Computer experiments on some benchmark examples indicate that our algorithm is much more efficient (two to ten times faster) than F5 and F5C. Next, we present a more general algorithm that matches Buchberger\u27s algorithm in simplicity and yet is more flexible than G2V. Given a list of polynomials, the new algorithm computes simultaneously a Groebner basis for the ideal generated by the polynomials and a Groebner basis for the leading terms of the syzygy module of the polynomials. For any term order for the ideal, one may vary the term order for the syzygy module. Under one term order for the syzygy module, the new algorithm specializes to the G2V algorithm, and under another term order for the syzygy module, the new algorithm may be several times faster than G2V, as indicated by computer experiments on benchmark examples. Finally, we present a solid theoretical framework for G2V and GVW which makes the algorithm much more understandable. This theory also gives a major improvement of the GVW algorithm. A proof of termination is provided for all algorithms, and an argument is made that GVW computes the fewest number of generators for the signature based algorithms used by GVW and F5 (similarly for G2V and F5C)

Diagnostics for magnetically confined high-temperature plasmas

During the last 20 years, magnetically confined laboratory plasmas of steadily increasing temperatures and densities have been obtained, most notably in tokamak configurations, and now approach the conditions necessary to sustain a fusion reaction. Even more important to the goal of understanding the physics of such systems, remarkable advances in plasma diagnostics, the techniques for determining the properties of such plasmas, have accompanied these developments. More parameters can be determined with greater accuracy and finer spatial and temporal resolution. The magnetic configuration, the primary local thermodynamic quantities (density, temperature, and drift velocity), and other necessary quantities can now be measured with sufficient accuracy to determine particle and energy fluxes within the plasma and to characterize the basic transport processes. These plasmas are far from thermodynamic equilibrium. This deviation manifests itself in a variety of instabilities on several spatial and temporal scales, many of which are aptly described as turbulence. Many aspects of the turbulence can also be characterized. This article reviews the current state of diagnostics from an epistemoiogical perspective: the capabilities and limitations for measuring each important physical quantity are presented.Physic

A parallel Buchberger algorithm for multigraded ideals

We demonstrate a method to parallelize the computation of a Gr\"obner basis for a homogenous ideal in a multigraded polynomial ring. Our method uses anti-chains in the lattice $\mathbb N^k$ to separate mutually independent S-polynomials for reduction.Comment: 8 pages, 6 figure

Displacement Analysis of Under-Constrained Cable-Driven Parallel Robots

This dissertation studies the geometric static problem of under-constrained cable-driven parallel robots (CDPRs) supported by n cables, with n ≤ 6. The task consists of determining the overall robot configuration when a set of n variables is assigned. When variables relating to the platform posture are assigned, an inverse geometric static problem (IGP) must be solved; whereas, when cable lengths are given, a direct geometric static problem (DGP) must be considered. Both problems are challenging, as the robot continues to preserve some degrees of freedom even after n variables are assigned, with the final configuration determined by the applied forces. Hence, kinematics and statics are coupled and must be resolved simultaneously. In this dissertation, a general methodology is presented for modelling the aforementioned scenario with a set of algebraic equations. An elimination procedure is provided, aimed at solving the governing equations analytically and obtaining a least-degree univariate polynomial in the corresponding ideal for any value of n. Although an analytical procedure based on elimination is important from a mathematical point of view, providing an upper bound on the number of solutions in the complex field, it is not practical to compute these solutions as it would be very time-consuming. Thus, for the efficient computation of the solution set, a numerical procedure based on homotopy continuation is implemented. A continuation algorithm is also applied to find a set of robot parameters with the maximum number of real assembly modes for a given DGP. Finally, the end-effector pose depends on the applied load and may change due to external disturbances. An investigation into equilibrium stability is therefore performed

Algorithmic boundedness-from-below conditions for generic scalar potentials

Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results

Algorithmic Boundedness-From-Below Conditions for Generic Scalar Potentials

Checking that a scalar potential is bounded from below (BFB) is an ubiquitous and notoriously difficult task in many models with extended scalar sectors. Exact analytic BFB conditions are known only in simple cases. In this work, we present a novel approach to algorithmically establish the BFB conditions for any polynomial scalar potential. The method relies on elements of multivariate algebra, in particular, on resultants and on the spectral theory of tensors, which is being developed by the mathematical community. We give first a pedagogical introduction to this approach, illustrate it with elementary examples, and then present the working Mathematica implementation publicly available at GitHub. Due to the rapidly increasing complexity of the problem, we have not yet produced ready-to-use analytical BFB conditions for new multi-scalar cases. But we are confident that the present implementation can be dramatically improved and may eventually lead to such results.Comment: 27 pages, 2 figures; v2: added reference

New Calabi-Yau Manifolds with Small Hodge Numbers

It is known that many Calabi-Yau manifolds form a connected web. The question of whether all Calabi-Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of Calabi-Yau manifolds where the Hodge numbers (h^{11}, h^{21}) are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with \chi =-6 and manifolds with an attractive structure that may prove of interest for string phenomenology. We also examine the relation of some of these manifolds to the remarkable Gross-Popescu manifolds that have Euler number zero.Comment: 105 pages, pdflatex with about 70 pdf and jpeg figures. References corrected. Minor revisions to Fig1, and Table 9 extended to the range h^{11} + h^21 \leq 2