300 research outputs found
Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods
Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym-
metry consisting of only pentagons and hexagons faces. It has been the object
of interest for chemists and mathematicians due to its widespread application
in various fields, namely including electronic and optic engineering, medical sci-
ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity
of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751
isomers. The Fries and Clar numbers are stability predictors of a Fullerene
molecule. These number can be computed by solving a (possibly N P -hard)
combinatorial optimization problem. We propose several ILP formulation of
such a problem each yielding a solution algorithm that provides the exact value
of the Fries and Clar numbers. We compare the performances of the algorithm
derived from the proposed ILP formulations. One of this algorithm is used to
find the Clar isomers, i.e., those for which the Clar number is maximum among
all isomers having a given size. We repeated this computational experiment for
all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774
isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path
following (PDIPF) interior point (IP) algorithm for separable convex quadratic
minimum cost flow network problem. In each iteration of PDIPF algorithm, the
main computational effort is solving the underlying Newton search direction
system. We concentrated on finding the solution of the corresponding linear
system iteratively and inexactly. We assumed that all the involved inequalities
can be solved inexactly and to this purpose, we focused on different approaches
for distributing the error generated by iterative linear solvers such that the
convergences of the PDIPF algorithm are guaranteed. As a result, we achieved
theoretical bases that open the path to further interesting practical investiga-
tion
Exponentially many perfect matchings in cubic graphs
We show that every cubic bridgeless graph G has at least 2^(|V(G)|/3656)
perfect matchings. This confirms an old conjecture of Lovasz and Plummer.
This version of the paper uses a different definition of a burl from the
journal version of the paper and a different proof of Lemma 18 is given. This
simplifies the exposition of our arguments throughout the whole paper
Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau
3-folds starting with (almost) any deformation family of smooth weak Fano
3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau
3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We
pay particular attention to a subclass of weak Fano 3-folds that we call
semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing
theorems and enjoy certain topological properties not satisfied by general weak
Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike
Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to
compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.
We introduce some general methods to compute the basic topological invariants
of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a
small number of representative examples in detail. Similar methods allow the
computation of the topology in many other examples.
All the features of the ACyl Calabi-Yau 3-folds studied here find application
in arXiv:1207.4470 where we construct many new compact G_2-manifolds using
Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds
constructed from semi-Fano 3-folds are particularly well-adapted for this
purpose.Comment: 107 pages, 1 figure. v3: minor corrections, changed formattin
Positive del Pezzo Geometry
Real, complex, and tropical algebraic geometry join forces in a new branch of
mathematical physics called positive geometry. We develop the positive geometry
of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties.
Their connected components are derived from polyhedral spaces with Weyl group
symmetries. We study their canonical forms and scattering amplitudes, and we
solve the likelihood equations.Comment: 34 pages, 4 figure
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