1,339 research outputs found
Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane
We complete the complexity classification by degree of minimizing a
polynomial over the integer points in a polyhedron in . Previous
work shows that optimizing a quadratic polynomial over the integer points in a
polyhedral region in can be done in polynomial time, while
optimizing a quartic polynomial in the same type of region is NP-hard. We close
the gap by showing that this problem can be solved in polynomial time for cubic
polynomials.
Furthermore, we show that the problem of minimizing a homogeneous polynomial
of any fixed degree over the integer points in a bounded polyhedron in
is solvable in polynomial time. We show that this holds for
polynomials that can be translated into homogeneous polynomials, even when the
translation vector is unknown. We demonstrate that such problems in the
unbounded case can have smallest optimal solutions of exponential size in the
size of the input, thus requiring a compact representation of solutions for a
general polynomial time algorithm for the unbounded case
The Euclidean distance degree of an algebraic variety
The nearest point map of a real algebraic variety with respect to Euclidean
distance is an algebraic function. For instance, for varieties of low rank
matrices, the Eckart-Young Theorem states that this map is given by the
singular value decomposition. This article develops a theory of such nearest
point maps from the perspective of computational algebraic geometry. The
Euclidean distance degree of a variety is the number of critical points of the
squared distance to a generic point outside the variety. Focusing on varieties
seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic
Wilson lines and Chern-Simons flux in explicit heterotic Calabi-Yau compactifications
We study to what extent Wilson lines in heterotic Calabi-Yau
compactifications lead to non-trivial H-flux via Chern-Simons terms. Wilson
lines are basic ingredients for Standard Model constructions but their induced
H-flux may affect the consistency of the leading order background geometry and
of the two-dimensional worldsheet theory. Moreover H-flux in heterotic
compactifications would play an important role for moduli stabilization and
could strongly constrain the supersymmetry breaking scale. We show how to
compute H-flux and the corresponding superpotential, given an explicit complete
intersection Calabi-Yau compactification and choice of Wilson lines. We do so
by classifying special Lagrangian submanifolds in the Calabi-Yau, understanding
how the Wilson lines project onto these submanifolds, and computing their
Chern-Simons invariants. We illustrate our procedure with the quintic
hypersurface as well as the split-bicubic, which can provide a potentially
realistic three generation model.Comment: 41 pages, 7 figures. v2: Minor corrections, published versio
Multi-scale approach for strain-engineering of phosphorene
A multi-scale approach for the theoretical description of deformed
phosphorene is presented. This approach combines a valence-force model to
relate macroscopic strain to microscopic displacements of atoms and a
tight-binding model with distance-dependent hopping parameters to obtain
electronic properties. The resulting self-consistent electromechanical model is
suitable for large-scale modeling of phosphorene devices. We demonstrate this
for the case of an inhomogeneously deformed phosphorene drum, which may be used
as an exciton funnel
Dualities in Convex Algebraic Geometry
Convex algebraic geometry concerns the interplay between optimization theory
and real algebraic geometry. Its objects of study include convex semialgebraic
sets that arise in semidefinite programming and from sums of squares. This
article compares three notions of duality that are relevant in these contexts:
duality of convex bodies, duality of projective varieties, and the
Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the
optimal value of a polynomial program is an algebraic function whose minimal
polynomial is expressed by the hypersurface projectively dual to the constraint
set. We give an exposition of recent results on the boundary structure of the
convex hull of a compact variety, we contrast this to Lasserre's representation
as a spectrahedral shadow, and we explore the geometric underpinnings of
semidefinite programming duality.Comment: 48 pages, 11 figure
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