51,586 research outputs found
The isomorphism problem for Coxeter groups
By a recent result obtained by R. Howlett and the author considerable
progress has been made towards a complete solution of the isomorphism problem
for Coxeter groups. In this paper we give a survey on the isomorphism problem
and explain in particular how the result mentioned above reduces it to its
`reflection preserving' version. Furthermore we desrcibe recent developments
concerning the solution of the latter.Comment: 15 pages, 0 figures, to appear in 'The Coxeter Legacy: Reflections
and Projections', Fields Institute Communication
Faster Longest Common Extension Queries in Strings over General Alphabets
Longest common extension queries (often called longest common prefix queries)
constitute a fundamental building block in multiple string algorithms, for
example computing runs and approximate pattern matching. We show that a
sequence of LCE queries for a string of size over a general ordered
alphabet can be realized in time making only
symbol comparisons. Consequently, all runs in a string over a general
ordered alphabet can be computed in time making
symbol comparisons. Our results improve upon a solution by Kosolobov
(Information Processing Letters, 2016), who gave an algorithm with running time and conjectured that time is possible. We
make a significant progress towards resolving this conjecture. Our techniques
extend to the case of general unordered alphabets, when the time increases to
. The main tools are difference covers and the
disjoint-sets data structure.Comment: Accepted to CPM 201
The deformed Hermitian-Yang-Mills equation in geometry and physics
We provide an introduction to the mathematics and physics of the deformed
Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler
manifolds which plays an important role in mirror symmetry. We discuss the
physical origin of the equation, and some recent progress towards its solution.
In dimension 3 we prove a new Chern number inequality and discuss the
relationship with algebraic stability conditions.Comment: 20 page
Towards the Final Fate of an Unstable Black String
Black strings, one class of higher dimensional analogues of black holes, were
shown to be unstable to long wavelength perturbations by Gregory and Laflamme
in 1992, via a linear analysis. We revisit the problem through numerical
solution of the full equations of motion, and focus on trying to determine the
end-state of a perturbed, unstable black string. Our preliminary results show
that such a spacetime tends towards a solution resembling a sequence of
spherical black holes connected by thin black strings, at least at intermediate
times. However, our code fails then, primarily due to large gradients that
develop in metric functions, as the coordinate system we use is not well
adapted to the nature of the unfolding solution. We are thus unable to
determine how close the solution we see is to the final end-state, though we do
observe rich dynamical behavior of the system in the intermediate stages.Comment: 17 pages, 7 figure
Near-Optimal Computation of Runs over General Alphabet via Non-Crossing LCE Queries
Longest common extension queries (LCE queries) and runs are ubiquitous in
algorithmic stringology. Linear-time algorithms computing runs and
preprocessing for constant-time LCE queries have been known for over a decade.
However, these algorithms assume a linearly-sortable integer alphabet. A recent
breakthrough paper by Bannai et.\ al.\ (SODA 2015) showed a link between the
two notions: all the runs in a string can be computed via a linear number of
LCE queries. The first to consider these problems over a general ordered
alphabet was Kosolobov (\emph{Inf.\ Process.\ Lett.}, 2016), who presented an
-time algorithm for answering LCE queries. This
result was improved by Gawrychowski et.\ al.\ (accepted to CPM 2016) to time. In this work we note a special \emph{non-crossing} property
of LCE queries asked in the runs computation. We show that any such
non-crossing queries can be answered on-line in time, which
yields an -time algorithm for computing runs
Robust Adaptive Beamforming for General-Rank Signal Model with Positive Semi-Definite Constraint via POTDC
The robust adaptive beamforming (RAB) problem for general-rank signal model
with an additional positive semi-definite constraint is considered. Using the
principle of the worst-case performance optimization, such RAB problem leads to
a difference-of-convex functions (DC) optimization problem. The existing
approaches for solving the resulted non-convex DC problem are based on
approximations and find only suboptimal solutions. Here we solve the non-convex
DC problem rigorously and give arguments suggesting that the solution is
globally optimal. Particularly, we rewrite the problem as the minimization of a
one-dimensional optimal value function whose corresponding optimization problem
is non-convex. Then, the optimal value function is replaced with another
equivalent one, for which the corresponding optimization problem is convex. The
new one-dimensional optimal value function is minimized iteratively via
polynomial time DC (POTDC) algorithm.We show that our solution satisfies the
Karush-Kuhn-Tucker (KKT) optimality conditions and there is a strong evidence
that such solution is also globally optimal. Towards this conclusion, we
conjecture that the new optimal value function is a convex function. The new
RAB method shows superior performance compared to the other state-of-the-art
general-rank RAB methods.Comment: 29 pages, 7 figures, 2 tables, Submitted to IEEE Trans. Signal
Processing on August 201
On the maximal sum of exponents of runs in a string
A run is an inclusion maximal occurrence in a string (as a subinterval) of a
repetition with a period such that . The exponent of a run
is defined as and is . We show new bounds on the maximal sum of
exponents of runs in a string of length . Our upper bound of is
better than the best previously known proven bound of by Crochemore &
Ilie (2008). The lower bound of , obtained using a family of binary
words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the
maximal sum of exponents of runs in a string of length is smaller than Comment: 7 pages, 1 figur
Focused Local Search for Random 3-Satisfiability
A local search algorithm solving an NP-complete optimisation problem can be
viewed as a stochastic process moving in an 'energy landscape' towards
eventually finding an optimal solution. For the random 3-satisfiability
problem, the heuristic of focusing the local moves on the presently
unsatisfiedclauses is known to be very effective: the time to solution has been
observed to grow only linearly in the number of variables, for a given
clauses-to-variables ratio sufficiently far below the critical
satisfiability threshold . We present numerical results
on the behaviour of three focused local search algorithms for this problem,
considering in particular the characteristics of a focused variant of the
simple Metropolis dynamics. We estimate the optimal value for the
``temperature'' parameter for this algorithm, such that its linear-time
regime extends as close to as possible. Similar parameter
optimisation is performed also for the well-known WalkSAT algorithm and for the
less studied, but very well performing Focused Record-to-Record Travel method.
We observe that with an appropriate choice of parameters, the linear time
regime for each of these algorithms seems to extend well into ratios -- much further than has so far been generally assumed. We discuss the
statistics of solution times for the algorithms, relate their performance to
the process of ``whitening'', and present some conjectures on the shape of
their computational phase diagrams.Comment: 20 pages, lots of figure
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