291 research outputs found
Computational protein design with backbone plasticity
The computational algorithms used in the design of artificial proteins have become increasingly sophisticated in recent years, producing a series of remarkable successes. The most dramatic of these is the de novo design of artificial enzymes. The majority of these designs have reused naturally occurring protein structures as âscaffoldsâ onto which novel functionality can be grafted without having to redesign the backbone structure. The incorporation of backbone flexibility into protein design is a much more computationally challenging problem due to the greatly increase search space but promises to remove the limitations of reusing natural protein scaffolds. In this review, we outline the principles of computational protein design methods and discuss recent efforts to consider backbone plasticity in the design process
Irredundant Triangular Decomposition
Triangular decomposition is a classic, widely used and well-developed way to
represent algebraic varieties with many applications. In particular, there
exist sharp degree bounds for a single triangular set in terms of intrinsic
data of the variety it represents, and powerful randomized algorithms for
computing triangular decompositions using Hensel lifting in the
zero-dimensional case and for irreducible varieties. However, in the general
case, most of the algorithms computing triangular decompositions produce
embedded components, which makes it impossible to directly apply the intrinsic
degree bounds. This, in turn, is an obstacle for efficiently applying Hensel
lifting due to the higher degrees of the output polynomials and the lower
probability of success. In this paper, we give an algorithm to compute an
irredundant triangular decomposition of an arbitrary algebraic set defined
by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant
triangular decomposition, we were able to give intrinsic degree bounds for the
polynomials appearing in the triangular sets and apply Hensel lifting
techniques. Our decomposition algorithm is randomized, and we analyze the
probability of success
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Generalised resultants, dynamic polynomial combinants and the minimal design problem
The theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems (DAP), which provides the unifying description of the dynamic, as well as static pole and zero dynamic assignment problems in linear systems. The assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original DAP. This paper demonstrates the origin of dynamic polynomial combinants from linear systems, examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree, and examines their spectral assignment. Central to this study is the link of dynamic combinants to the theory of generalised resultants, which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterisation of combinants and respective generalised resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, which is referred to as the dynamic combinant minimal design (DCMD) problem. An algorithmic approach based on rank tests of Sylvester matrices is given, which produces the minimal order and degree solution in a finite number of steps. Such solutions provide low bounds for the respective dynamic assignment control problems
Toric Border Bases
We extend the theory and the algorithms of Border Bases to systems of Laurent
polynomial equations, defining "toric" roots. Instead of introducing new
variables and new relations to saturate by the variable inverses, we propose a
more efficient approach which works directly with the variables and their
inverse. We show that the commutation relations and the inversion relations
characterize toric border bases. We explicitly describe the first syzygy module
associated to a toric border basis in terms of these relations. Finally, a new
border basis algorithm for Laurent polynomials is described and a proof of its
termination is given for zero-dimensional toric ideals
Deciding the consistency of non-linear real arithmetic constraints with a conflict driven search using cylindrical algebraic coverings
We present a new algorithm for determining the satisfiability of conjunctions
of non-linear polynomial constraints over the reals, which can be used as a
theory solver for satisfiability modulo theory (SMT) solving for non-linear
real arithmetic. The algorithm is a variant of Cylindrical Algebraic
Decomposition (CAD) adapted for satisfiability, where solution candidates
(sample points) are constructed incrementally, either until a satisfying sample
is found or sufficient samples have been sampled to conclude unsatisfiability.
The choice of samples is guided by the input constraints and previous
conflicts.
The key idea behind our new approach is to start with a partial sample;
demonstrate that it cannot be extended to a full sample; and from the reasons
for that rule out a larger space around the partial sample, which build up
incrementally into a cylindrical algebraic covering of the space. There are
similarities with the incremental variant of CAD, the NLSAT method of Jovanovic
and de Moura, and the NuCAD algorithm of Brown; but we present worked examples
and experimental results on a preliminary implementation to demonstrate the
differences to these, and the benefits of the new approach
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The ERES method for computing the approximate GCD of several polynomials
The computation of the greatest common divisor (GCD) of a set of polynomials has interested the mathematicians for a long time and has attracted a lot of attention in recent years. A challenging problem that arises from several applications, such as control or image and signal processing, is to develop a numerical GCD method that inherently has the potential to work efficiently with sets of several polynomials with inexactly known coefficients. The presented work focuses on: (i) the use of the basic principles of the ERES methodology for calculating the GCD of a set of several polynomials and defining approximate solutions by developing the hybrid implementation of this methodology. (ii) the use of the developed framework for defining the approximate notions for the GCD as a distance problem in a projective space to develop an optimization algorithm for evaluating the strength of different ad-hoc approximations derived from different algorithms. The presented new implementation of ERES is based on the effective combination of symbolicânumeric arithmetic (hybrid arithmetic) and shows interesting computational properties for the approximate GCD problem. Additionally, an efficient implementation of the strength of an approximate GCD is given by exploiting some of the special aspects of the respective distance problem. Finally, the overall performance of the ERES algorithm for computing approximate solutions is discussed
Cylindrical Algebraic Sub-Decompositions
Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic
geometry, used primarily for eliminating quantifiers over the reals and
studying semi-algebraic sets. In this paper we introduce cylindrical algebraic
sub-decompositions (sub-CADs), which are subsets of CADs containing all the
information needed to specify a solution for a given problem.
We define two new types of sub-CAD: variety sub-CADs which are those cells in
a CAD lying on a designated variety; and layered sub-CADs which have only those
cells of dimension higher than a specified value. We present algorithms to
produce these and describe how the two approaches may be combined with each
other and the recent theory of truth-table invariant CAD.
We give a complexity analysis showing that these techniques can offer
substantial theoretical savings, which is supported by experimentation using an
implementation in Maple.Comment: 26 page
Static and dynamic analysis of linear elastic systems on non-prismatic three dimensional beam elements
Bibliography: leaves 100-103.A computer programme NONPRI, has been developed for the analysis of three dimensional skeletal assemblages consisting of non-prismatic members. It is capable of static and dynamic analysis of structures consisting of members whose constitutive relationship is linear elastic. The finite element formulation is based on the family of quadratic isoparametric finite elements. The three noded space frame element is quite versatile in that it can account for shear as well as flexural 9 axial and torsional deformation effects making it suitable for thin and thick beam analysis and for cases where the axial and torsional deformations are relevant. The element can be degenerated to a truss/frame transition element (3 translational degrees of freedom at each node - rotations ignored) and further degenerated to become a truss element. Furthermore, the element internal node is defined to lie at an arbitrary position inside the element. Thus, this flexibility in the non-prismatic element formulation makes it very powerful in practical analysis problems. An out-of-core solution technique is used for the equations of static analysis bearing in mind the capability for solving large structural systems. An in-core solution technique is used for the equations of dynamic analysis bearing now in mind that these equations represent an iterative process which can otherwise become computationally very expensive
Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]
We provide an irreducibility test in the ring K[[x]][y] whose complexity is
quasi-linear with respect to the valuation of the discriminant, assuming the
input polynomial F square-free and K a perfect field of characteristic zero or
greater than deg(F). The algorithm uses the theory of approximate roots and may
be seen as a generalization of Abhyankhar's irreducibility criterion to the
case of non algebraically closed residue fields. More generally, we show that
we can test within the same complexity if a polynomial is pseudo-irreducible, a
larger class of polynomials containing irreducible ones. If is
pseudo-irreducible, the algorithm computes also the valuation of the
discriminant and the equisingularity types of the germs of plane curve defined
by F along the fiber x=0.Comment: 51 pages. Title modified. Slight modifications in Definition 5 and
Proposition 1
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