6,646 research outputs found
Formal matrix integrals and combinatorics of maps
This article is a short review on the relationship between convergent matrix
integrals, formal matrix integrals, and combinatorics of maps. We briefly
summarize results developed over the last 30 years, as well as more recent
discoveries. We recall that formal matrix integrals are identical to
combinatorial generating functions for maps, and that formal matrix integrals
are in general very different from convergent matrix integrals. Finally, we
give a list of the classical matrix models which have played an important role
in physics in the past decades. Some of them are now well understood, some are
still difficult challenges.Comment: few misprints corrected, biblio modifie
Subclass Discriminant Analysis of Morphological and Textural Features for HEp-2 Staining Pattern Classification
Classifying HEp-2 fluorescence patterns in Indirect Immunofluorescence (IIF) HEp-2 cell imaging is important for the differential diagnosis of autoimmune diseases. The current technique, based on human visual inspection, is time-consuming, subjective and dependent on the operator's experience. Automating this process may be a solution to these limitations, making IIF faster and more reliable. This work proposes a classification approach based on Subclass Discriminant Analysis (SDA), a dimensionality reduction technique that provides an effective representation of the cells in the feature space, suitably coping with the high within-class variance typical of HEp-2 cell patterns. In order to generate an adequate characterization of the fluorescence patterns, we investigate the individual and combined contributions of several image attributes, showing that the integration of morphological, global and local textural features is the most suited for this purpose. The proposed approach provides an accuracy of the staining pattern classification of about 90%
Probability Theory of Random Polygons from the Quaternionic Viewpoint
We build a new probability measure on closed space and plane polygons. The
key construction is a map, given by Knutson and Hausmann using the Hopf map on
quaternions, from the complex Stiefel manifold of 2-frames in n-space to the
space of closed n-gons in 3-space of total length 2. Our probability measure on
polygon space is defined by pushing forward Haar measure on the Stiefel
manifold by this map. A similar construction yields a probability measure on
plane polygons which comes from a real Stiefel manifold.
The edgelengths of polygons sampled according to our measures obey beta
distributions. This makes our polygon measures different from those usually
studied, which have Gaussian or fixed edgelengths. One advantage of our
measures is that we can explicitly compute expectations and moments for
chordlengths and radii of gyration. Another is that direct sampling according
to our measures is fast (linear in the number of edges) and easy to code.
Some of our methods will be of independent interest in studying other
probability measures on polygon spaces. We define an edge set ensemble (ESE) to
be the set of polygons created by rearranging a given set of n edges. A key
theorem gives a formula for the average over an ESE of the squared lengths of
chords skipping k vertices in terms of k, n, and the edgelengths of the
ensemble. This allows one to easily compute expected values of squared
chordlengths and radii of gyration for any probability measure on polygon space
invariant under rearrangements of edges.Comment: Some small typos fixed, added a calculation for the covariance of
edgelengths, added pseudocode for the random polygon sampling algorithm. To
appear in Communications on Pure and Applied Mathematics (CPAM
A PDE Approach to the Combinatorics of the Full Map Enumeration Problem: Exact Solutions and their Universal Character
Maps are polygonal cellular networks on Riemann surfaces. This paper
completes a program of constructing closed form general representations for the
enumerative generating functions associated to maps of fixed but arbitrary
genus. These closed form expressions have a universal character in the sense
that they are independent of the explicit valence distribution of the tiling
polygons. Nevertheless the valence distributions may be recovered from the
closed form generating functions by a remarkable {\it unwinding identity} in
terms of the Appell polynomials generated by Bessel functions. Our treatment,
based on random matrix theory and Riemann-Hilbert problems for orthogonal
polynomials reveals the generating functions to be solutions of nonlinear
conservation laws and their prolongations. This characterization enables one to
gain insights that go beyond more traditional methods that are purely
combinatorial. Universality results are connected to stability results for
characteristic singularities of conservation laws that were studied by
Caflisch, Ercolani, Hou and Landis as well as directly related to universality
results for random matrix spectra as described by Deift, Kriecherbauer,
McLaughlin, Venakides and Zhou
BFACF-style algorithms for polygons in the body-centered and face-centered cubic lattices
In this paper the elementary moves of the BFACF-algorithm for lattice
polygons are generalised to elementary moves of BFACF-style algorithms for
lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic
lattices. We prove that the ergodicity classes of these new elementary moves
coincide with the knot types of unrooted polygons in the BCC and FCC lattices
and so expand a similar result for the cubic lattice. Implementations of these
algorithms for knotted polygons using the GAS algorithm produce estimates of
the minimal length of knotted polygons in the BCC and FCC lattices
- …