4,060 research outputs found
Complexity Analysis of Balloon Drawing for Rooted Trees
In a balloon drawing of a tree, all the children under the same parent are
placed on the circumference of the circle centered at their parent, and the
radius of the circle centered at each node along any path from the root
reflects the number of descendants associated with the node. Among various
styles of tree drawings reported in the literature, the balloon drawing enjoys
a desirable feature of displaying tree structures in a rather balanced fashion.
For each internal node in a balloon drawing, the ray from the node to each of
its children divides the wedge accommodating the subtree rooted at the child
into two sub-wedges. Depending on whether the two sub-wedge angles are required
to be identical or not, a balloon drawing can further be divided into two
types: even sub-wedge and uneven sub-wedge types. In the most general case, for
any internal node in the tree there are two dimensions of freedom that affect
the quality of a balloon drawing: (1) altering the order in which the children
of the node appear in the drawing, and (2) for the subtree rooted at each child
of the node, flipping the two sub-wedges of the subtree. In this paper, we give
a comprehensive complexity analysis for optimizing balloon drawings of rooted
trees with respect to angular resolution, aspect ratio and standard deviation
of angles under various drawing cases depending on whether the tree is of even
or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns
out that some are NP-complete while others can be solved in polynomial time. We
also derive approximation algorithms for those that are intractable in general
Simultaneous Embeddings with Few Bends and Crossings
A simultaneous embedding with fixed edges (SEFE) of two planar graphs and
is a pair of plane drawings of and that coincide when restricted to
the common vertices and edges of and . We show that whenever and
admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve
with few bends and every pair of edges has few crossings. Specifically: (1) if
and are trees then one bend per edge and four crossings per edge pair
suffice (and one bend per edge is sometimes necessary), (2) if is a planar
graph and is a tree then six bends per edge and eight crossings per edge
pair suffice, and (3) if and are planar graphs then six bends per edge
and sixteen crossings per edge pair suffice. Our results improve on a paper by
Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a
paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge
pair suffice.Comment: Full version of the paper "Simultaneous Embeddings with Few Bends and
Crossings" accepted at GD '1
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Polyhedra of small order and their Hamiltonian properties
We describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up to 12 vertices and up to 26 edges, simplical polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, and bipartite polyhedra with up to 22 vertices.The results of the enumeration were used to systematically search for certain minimal non-Hamiltonian polyhedra. In particular, the smallest polyhedra satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given
Semiclassical Quantisation of Finite-Gap Strings
We perform a first principle semiclassical quantisation of the general
finite-gap solution to the equations of a string moving on R x S^3. The
derivation is only formal as we do not regularise divergent sums over stability
angles. Moreover, with regards to the AdS/CFT correspondence the result is
incomplete as the fluctuations orthogonal to this subspace in AdS_5 x S^5 are
not taken into account. Nevertheless, the calculation serves the purpose of
understanding how the moduli of the algebraic curve gets quantised
semiclassically, purely from the point of view of finite-gap integration and
with no input from the gauge theory side. Our result is expressed in a very
compact and simple formula which encodes the infinite sum over stability angles
in a succinct way and reproduces exactly what one expects from knowledge of the
dual gauge theory. Namely, at tree level the filling fractions of the algebraic
curve get quantised in large integer multiples of hbar = 1/lambda^{1/2}. At
1-loop order the filling fractions receive Maslov index corrections of hbar/2
and all the singular points of the spectral curve become filled with small
half-integer multiples of hbar. For the subsector in question this is in
agreement with the previously obtained results for the semiclassical energy
spectrum of the string using the method proposed in hep-th/0703191.
Along the way we derive the complete hierarchy of commuting flows for the
string in the R x S^3 subsector. Moreover, we also derive a very general and
simple formula for the stability angles around a generic finite-gap solution.
We also stress the issue of quantum operator orderings since this problem
already crops up at 1-loop in the form of the subprincipal symbol.Comment: 53 pages, 22 figures; some significant typos corrected, references
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Understanding molecular representations in machine learning: The role of uniqueness and target similarity
The predictive accuracy of Machine Learning (ML) models of molecular
properties depends on the choice of the molecular representation. Based on the
postulates of quantum mechanics, we introduce a hierarchy of representations
which meet uniqueness and target similarity criteria. To systematically control
target similarity, we rely on interatomic many body expansions, as implemented
in universal force-fields, including Bonding, Angular, and higher order terms
(BA). Addition of higher order contributions systematically increases
similarity to the true potential energy and predictive accuracy of the
resulting ML models. We report numerical evidence for the performance of BAML
models trained on molecular properties pre-calculated at electron-correlated
and density functional theory level of theory for thousands of small organic
molecules. Properties studied include enthalpies and free energies of
atomization, heatcapacity, zero-point vibrational energies, dipole-moment,
polarizability, HOMO/LUMO energies and gap, ionization potential, electron
affinity, and electronic excitations. After training, BAML predicts energies or
electronic properties of out-of-sample molecules with unprecedented accuracy
and speed
3nj Morphogenesis and Semiclassical Disentangling
Recoupling coefficients (3nj symbols) are unitary transformations between
binary coupled eigenstates of N=(n+1) mutually commuting SU(2) angular momentum
operators. They have been used in a variety of applications in spectroscopy,
quantum chemistry and nuclear physics and quite recently also in quantum
gravity and quantum computing. These coefficients, naturally associated to
cubic Yutsis graphs, share a number of intriguing combinatorial, algebraic, and
analytical features that make them fashinating objects to be studied on their
own. In this paper we develop a bottom--up, systematic procedure for the
generation of 3nj from 3(n-1)j diagrams by resorting to diagrammatical and
algebraic methods. We provide also a novel approach to the problem of
classifying various regimes of semiclassical expansions of 3nj coefficients
(asymptotic disentangling of 3nj diagrams) for n > 2 by means of combinatorial,
analytical and numerical tools
Dynamics of Black Hole Pairs I: Periodic Tables
Although the orbits of comparable mass, spinning black holes seem to defy
simple decoding, we find a means to decipher all such orbits. The dynamics is
complicated by extreme perihelion precession compounded by spin-induced
precession. We are able to quantitatively define and describe the fully three
dimensional motion of comparable mass binaries with one black hole spinning and
expose an underlying simplicity. To do so, we untangle the dynamics by
capturing the motion in the orbital plane. Our results are twofold: (1) We
derive highly simplified equations of motion in a non-orthogonal orbital basis,
and (2) we define a complete taxonomy for fully three-dimensional orbits. More
than just a naming system, the taxonomy provides unambiguous and quantitative
descriptions of the orbits, including a determination of the zoom-whirliness of
any given orbit. Through a correspondence with the rationals, we are able to
show that zoom-whirl behavior is prevalent in comparable mass binaries in the
strong-field regime. A first significant conclusion that can be drawn from this
analysis is that all generic orbits in the final stages of inspiral under
gravitational radiation losses are characterized by precessing clovers with few
leaves and that no orbit will behave like the tightly precessing ellipse of
Mercury. The gravitational waveform produced by these low-leaf clovers will
reflect the natural harmonics of the orbital basis -- harmonics that,
importantly, depend only on radius. The significance for gravitational wave
astronomy will depend on the number of windings the pair executes in the
strong-field regime and could be more conspicuous for intermediate mass pairs
than for stellar mass pairs.Comment: 19 pages, lots of figure
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