410 research outputs found
Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer
We show that the reformulation of the geometric Robinson-Schensted-Knuth
(gRSK) correspondence via local moves, introduced in \cite{OSZ14} can be
extended to cases where the input matrix is replaced by more general polygonal,
Young-diagram-like, arrays of the form \polygon. We also show that a
rearrangement of the sequence of the local moves gives rise to a geometric
version of the polynuclear growth model (PNG). These reformulations are used to
obtain integral formulae for the Laplace transform of the joint distribution of
the point-to-point partition functions of the log-gamma polymer at different
space-time points. In the case of two points at equal time and space at
distance of order , we show formally that the joint law of the
partition functions, scaled by , converges to the two-point function
of the Airy processComment: 44 pages. Proposition 3.4 and Theorem 3.5 are now stated in a more
general form and some more minor changes are made (most of them following
suggestions by a referee). To appear at IMR
Melnikov theory to all orders and Puiseux series for subharmonic solutions
We study the problem of subharmonic bifurcations for analytic systems in the
plane with perturbations depending periodically on time, in the case in which
we only assume that the subharmonic Melnikov function has at least one zero. If
the order of zero is odd, then there is always at least one subharmonic
solution, whereas if the order is even in general other conditions have to be
assumed to guarantee the existence of subharmonic solutions. Even when such
solutions exist, in general they are not analytic in the perturbation
parameter. We show that they are analytic in a fractional power of the
perturbation parameter. To obtain a fully constructive algorithm which allows
us not only to prove existence but also to obtain bounds on the radius of
analyticity and to approximate the solutions within any fixed accuracy, we need
further assumptions. The method we use to construct the solution -- when this
is possible -- is based on a combination of the Newton-Puiseux algorithm and
the tree formalism. This leads to a graphical representation of the solution in
terms of diagrams. Finally, if the subharmonic Melnikov function is identically
zero, we show that it is possible to introduce higher order generalisations,
for which the same kind of analysis can be carried out.Comment: 30 pages, 6 figure
Convex hulls in concept induction
Classification learning is dominated by systems which induce large numbers of small axis-orthogonal decision surfaces. This strongly biases such systems towards particular hypothesis types but there is reason believe that many domains have underlying concepts which do not involve axis orthogonal surfaces. Further, the multiplicity of small decision regions mitigates against any holistic appreciation of the theories produced by these systems, notwithstanding the fact that many of the small regions are individually comprehensible. This thesis investigates modeling concepts as large geometric structures in n-dimensional space. Convex hulls are a superset of the set of axis orthogonal hyperrectangles into which axis orthogonal systems partition the instance space. In consequence, there is reason to believe that convex hulls might provide a more flexible and general learning bias than axis orthogonal regions. The formation of convex hulls around a group of points of the same class is shown to be a usable generalisation and is more general than generalisations produced by axis-orthogonal based classifiers, without constructive induction, like decision trees, decision lists and rules. The use of a small number of large hulls as a concept representation is shown to provide classification performance which can be better than that of classifiers which use a large number of small fragmentary regions for each concept. A convex hull based classifier, CH1, has been implemented and tested. CH1 can handle categorical and continuous data. Algorithms for two basic generalisation operations on hulls, inflation and facet deletion, are presented. The two operations are shown to improve the accuracy of the classifier and provide moderate classification accuracy over a representative selection of typical, largely or wholly continuous valued machine learning tasks. The classifier exhibits superior performance to well-known axis-orthogonal-based classifiers when presented with domains where the underlying decision surfaces are not axis parallel. The strengths and weaknesses of the system are identified. One particular advantage is the ability of the system to model domains with approximately the same number of structures as there are underlying concepts. This leads to the possibility of extraction of higher level mathematical descriptions of the induced concepts, using the techniques of computational geometry, which is not possible from a multiplicity of small regions
Fuzzy Interpolation Systems and Applications
Fuzzy inference systems provide a simple yet effective solution to complex non-linear problems, which have been applied to numerous real-world applications with great success. However, conventional fuzzy inference systems may suffer from either too sparse, too complex or imbalanced rule bases, given that the data may be unevenly distributed in the problem space regardless of its volume. Fuzzy interpolation addresses this. It enables fuzzy inferences with sparse rule bases when the sparse rule base does not cover a given input, and it simplifies very dense rule bases by approximating certain rules with their neighbouring ones. This chapter systematically reviews different types of fuzzy interpolation approaches and their variations, in terms of both the interpolation mechanism (inference engine) and sparse rule base generation. Representative applications of fuzzy interpolation in the field of control are also revisited in this chapter, which not only validate fuzzy interpolation approaches but also demonstrate its efficacy and potential for wider applications
The family Floer functor is faithful
Family Floer theory yields a functor from the Fukaya category of a symplectic
manifold admitting a Lagrangian torus fibration to a (twisted) category of
perfect complexes on the mirror rigid analytic space. This functor is shown to
be faithful by a degeneration argument involving moduli spaces of annuli.Comment: 70 pages, 24 figures. Final version, with substantially enhanced
exposition, accepted for publication at JEM
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