227,982 research outputs found
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
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
The strongly regular (45,12,3,3) graphs
Using two backtrack algorithms based on dierent techniques, designed and implemented independently, we were able to determine up to isomorphism all strongly regular graphs with parameters v = 45, k = 12, λ = μ = 3. It turns out that there are 78 such graphs, having automorphism groups with sizes ranging from 1 to 51840
Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems
2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Efficient Prediction Designs for Random Fields
For estimation and predictions of random fields it is increasingly
acknowledged that the kriging variance may be a poor representative of true
uncertainty. Experimental designs based on more elaborate criteria that are
appropriate for empirical kriging are then often non-space-filling and very
costly to determine. In this paper, we investigate the possibility of using a
compound criterion inspired by an equivalence theorem type relation to build
designs quasi-optimal for the empirical kriging variance, when space-filling
designs become unsuitable. Two algorithms are proposed, one relying on
stochastic optimization to explicitly identify the Pareto front, while the
second uses the surrogate criteria as local heuristic to chose the points at
which the (costly) true Empirical Kriging variance is effectively computed. We
illustrate the performance of the algorithms presented on both a simple
simulated example and a real oceanographic dataset
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
RTL2RTL Formal Equivalence: Boosting the Design Confidence
Increasing design complexity driven by feature and performance requirements
and the Time to Market (TTM) constraints force a faster design and validation
closure. This in turn enforces novel ways of identifying and debugging
behavioral inconsistencies early in the design cycle. Addition of incremental
features and timing fixes may alter the legacy design behavior and would
inadvertently result in undesirable bugs. The most common method of verifying
the correctness of the changed design is to run a dynamic regression test suite
before and after the intended changes and compare the results, a method which
is not exhaustive. Modern Formal Verification (FV) techniques involving new
methods of proving Sequential Hardware Equivalence enabled a new set of
solutions for the given problem, with complete coverage guarantee. Formal
Equivalence can be applied for proving functional integrity after design
changes resulting from a wide variety of reasons, ranging from simple pipeline
optimizations to complex logic redistributions. We present here our experience
of successfully applying the RTL to RTL (RTL2RTL) Formal Verification across a
wide spectrum of problems on a Graphics design. The RTL2RTL FV enabled checking
the design sanity in a very short time, thus enabling faster and safer design
churn. The techniques presented in this paper are applicable to any complex
hardware design.Comment: In Proceedings FSFMA 2014, arXiv:1407.195
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