14,829 research outputs found
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
Inspired by some intriguing examples, we study uniform association schemes
and uniform coherent configurations, including cometric Q-antipodal association
schemes. After a review of imprimitivity, we show that an imprimitive
association scheme is uniform if and only if it is dismantlable, and we cast
these schemes in the broader context of certain --- uniform --- coherent
configurations. We also give a third characterization of uniform schemes in
terms of the Krein parameters, and derive information on the primitive
idempotents of such a scheme. In the second half of the paper, we apply these
results to cometric association schemes. We show that each such scheme is
uniform if and only if it is Q-antipodal, and derive results on the parameters
of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We
revisit the correspondence between uniform indecomposable three-class schemes
and linked systems of symmetric designs, and show that these are cometric
Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class
schemes in terms of only a few parameters, and show that any strongly regular
graph with a ("non-exceptional") strongly regular decomposition gives rise to
such a scheme. Hemisystems in generalized quadrangles provide interesting
examples of such decompositions. We finish with a short discussion of
five-class schemes as well as a list of all feasible parameter sets for
cometric Q-antipodal four-class schemes with at most six fibres and fibre size
at most 2000, and describe the known examples. Most of these examples are
related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions,
April 201
Fully differential NNLO computations with MATRIX
We present the computational framework MATRIX which allows us to evaluate
fully differential cross sections for a wide class of processes at hadron
colliders in next-to-next-to-leading order (NNLO) QCD. The processes we
consider are and hadronic reactions involving Higgs and
vector bosons in the final state. All possible leptonic decay channels of the
vector bosons are included for the first time in the calculations, by
consistently accounting for all resonant and non-resonant diagrams, off-shell
effects and spin correlations. We briefly introduce the theoretical framework
MATRIX is based on, discuss its relevant features and provide a detailed
description of how to use MATRIX to obtain NNLO accurate results for the
various processes. We report reference predictions for inclusive and fiducial
cross sections of all the physics processes considered here and discuss their
corresponding uncertainties. MATRIX features an automatic extrapolation
procedure that allows us, for the first time, to control the systematic
uncertainties inherent to the applied NNLO subtraction procedure down to the
few permille level (or better).Comment: 76 pages, 2 figures, 11 table
Driving Markov chain Monte Carlo with a dependent random stream
Markov chain Monte Carlo is a widely-used technique for generating a
dependent sequence of samples from complex distributions. Conventionally, these
methods require a source of independent random variates. Most implementations
use pseudo-random numbers instead because generating true independent variates
with a physical system is not straightforward. In this paper we show how to
modify some commonly used Markov chains to use a dependent stream of random
numbers in place of independent uniform variates. The resulting Markov chains
have the correct invariant distribution without requiring detailed knowledge of
the stream's dependencies or even its marginal distribution. As a side-effect,
sometimes far fewer random numbers are required to obtain accurate results.Comment: 16 pages, 4 figure
Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems
2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12
The PDF4LHC report on PDFs and LHC data: Results from Run I and preparation for Run II
The accurate determination of the Parton Distribution Functions (PDFs) of the
proton is an essential ingredient of the Large Hadron Collider (LHC) program.
PDF uncertainties impact a wide range of processes, from Higgs boson
characterisation and precision Standard Model measurements to New Physics
searches. A major recent development in modern PDF analyses has been to exploit
the wealth of new information contained in precision measurements from the LHC
Run I, as well as progress in tools and methods to include these data in PDF
fits. In this report we summarise the information that PDF-sensitive
measurements at the LHC have provided so far, and review the prospects for
further constraining PDFs with data from the recently started Run II. This
document aims to provide useful input to the LHC collaborations to prioritise
their PDF-sensitive measurements at Run II, as well as a comprehensive
reference for the PDF-fitting collaborations.Comment: 55 pages, 13 figure
Accurate Prediction of Core Properties for Chiral Molecules using Pseudo Potentials
Pseudo potentials (PPs) constitute perhaps the most common way to treat relativity, often in a formally non-relativistic framework, and reduce the electronic
structure to the chemically relevant part. The drawback is that orbitals obtained
in this picture (called pseudo orbitals (POs)) show a reduced nodal structure
and altered amplitude in the vicinity of the nucleus, when compared to the
corresponding molecular orbitals (MOs). Thus expectation values of operators
localized in the spatial core region that are calculated with POs, deviate significantly from the same expectation values calculated with all-electron (AE)
MOs. This study describes the reconstruction of AE MOs from POs, with a
focus on POs generated by energy consistent pseudo Hamiltonians. The method
reintroduces the nodal structure into the POs, thus providing an inexpensive
and easily implementable method that allows to use nonrelativistic, efficiently
calculated POs for good estimates of expectation values of core-like properties.
The discussion of the method proceeds in two parts: Firstly, the reconstruction scheme is developed for atomic cases. Secondly, the scheme is discussed in
the context of MO reconstruction and successfully applied to numerous numerical examples.
Starting from the equations of the state-averaged multi-configuration self-
consistent field method, used for the generation of energy consistent pseudo
potentials, the electronic spectrum of the many-electron Hamiltonian is linked
to the spectrum of the effective one-electron Fock operator by means of various
models systems. This relation and the Topp–Hopfield–Kramers theorem, are
used to show the shape-consistency of energy-consistent POs for atomic systems.
Shape-consistency describes POs that follow distinct AOs exactly outside a core-radius r_core . In the cases presented here, shape-consistency holds to a high degree
and it follows that in atomic systems every PO has one distinct partner in the
set of AOs. The overlap integral between these two orbitals is close to one, as it
is determined mainly by the spatial orbital parts outside r_core . Expanding, e.g.,
a 5s PO in occupied AOs, the 5s AOs will have the highest contribution. The
POs itself contains contributions from high-energy unoccupied AOs as well (e.g.
15s), which damp the nodal structure of the POs near the nucleus. Consequently,
neglecting contributions from unoccupied orbitals in a projection of the POs
reintroduces the nodal structure.
This approach is not directly suitable for the reconstruction of MOs, as they
often need to be expanded in a full set of AOs at each atomic center, including all
unoccupied orbitals, to properly account for the electron density distribution in
the molecule. However, it is shown that the occupied MOs are well described by
occupied and low-energy unoccupied AOs only and a mapping of the POs onto
a basis containing only these orbitals reconstructs the nodal structure of the MO.
The approach uses only standard integrals available in most quantum chemistry
programs. The computational cost of these integrals scales with N^2 , where N is
the number of basis functions. The most time consuming step is a Gram-Schmidt
orthogonalization, which scales in this implementation with MN^2 , M being the
number of reconstructed orbitals.
The reconstruction method is subsequently tested: Valence orbitals of atomic,
closed-shell systems were reconstructed numerically exactly. The influence of
numerical parameters is investigated using the molecule BaF . It is shown that
the method is basis set dependent: One has to ensure that the PO basis can be
expanded exactly in the basis of AOs. Violating this rule of thumb may degrade
the quality of reconstructed orbitals. Additionally, the representation of MOs by
a linear combination of occupied and unoccupied AOs is investigated. For the
exemplary systems, the shells included in the fitting procedure of the PP were
sufficient.
Reconstruction of the alkaline earth monofluorides showed that periodic
trends can be reconstructed as well. Scaling of hyperfine structure parameters
with increasing atomic number is discussed. For hydrogenic atoms, the scaling should be linear, whereas small deviations from the linear behavior were
observed for molecules. The scaling laws computed from reconstructed and
reference orbitals were almost identical. In this context, the failure of commonly
used relativistic enhancement factors beyond atomic number 100 is discussed.
Applicability of the method is also tested on parity violating properties for which
the main contribution is generated by the valence orbitals near the nucleus.
Symmetry-independence of the method is shown by successful reconstruction of
orbitals of the tetrahedral PbCl_4 and chiral NWHClF. The reliable reconstruction
of chemical trends is shown with the help of the NWHClF derivatives NWHBrF
and NWHFI.
The study of chiral compounds as, e.g., NWHClF and its group 17 derivatives, which have been proposed as paradigm for the detection of parity-violation
in chiral molecules, remains of great importance. Especially the direct determination of absolute configuration of chiral centers is still non-trivial. The author
contributed to this field with a self-written molecular dynamics (MD) program
to simulate Coulomb explosions and thus to provide an insight especially into
the early explosion stages directly after an instantaneous multi-ionization of
the molecule CHBrClF, comparable to experiments using the Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) technique. An algorithm for
the determination of the investigated molecule’s absolute configuration from
time-of-flight data and detection locations of molecular fragments is included
in the program. The program was used to generate experiment-equivalent data
which allowed for the first time the investigation of non-racemic mixtures by
the analysis routines of the experiment. The MD program includes harmonic
and anharmonic bond potentials. A charge-exchange model can model partial
charges in early phases of the Coulomb explosion.
Furthermore, Born–Oppenheimer MD simulations and statistical models
are used to explain the relative abundance of products belonging to competing
reaction channels, as obtained by photoion coincidence measurements. Additionally, qualitative statements about reaction branching ratios are made by
comparing the partition functions of involved degrees of freedom. Analytic
equations for partition functions of simple models are used to provide a simple
formula allowing fast estimates of reaction branching ratios
Semidefinite programming and eigenvalue bounds for the graph partition problem
The graph partition problem is the problem of partitioning the vertex set of
a graph into a fixed number of sets of given sizes such that the sum of weights
of edges joining different sets is optimized. In this paper we simplify a known
matrix-lifting semidefinite programming relaxation of the graph partition
problem for several classes of graphs and also show how to aggregate additional
triangle and independent set constraints for graphs with symmetry. We present
an eigenvalue bound for the graph partition problem of a strongly regular
graph, extending a similar result for the equipartition problem. We also derive
a linear programming bound of the graph partition problem for certain Johnson
and Kneser graphs. Using what we call the Laplacian algebra of a graph, we
derive an eigenvalue bound for the graph partition problem that is the first
known closed form bound that is applicable to any graph, thereby extending a
well-known result in spectral graph theory. Finally, we strengthen a known
semidefinite programming relaxation of a specific quadratic assignment problem
and the above-mentioned matrix-lifting semidefinite programming relaxation by
adding two constraints that correspond to assigning two vertices of the graph
to different parts of the partition. This strengthening performs well on highly
symmetric graphs when other relaxations provide weak or trivial bounds
- …