131 research outputs found
Models of discretized moduli spaces, cohomological field theories, and Gaussian means
We prove combinatorially the explicit relation between genus filtrated
-loop means of the Gaussian matrix model and terms of the genus expansion of
the Kontsevich--Penner matrix model (KPMM). The latter is the generating
function for volumes of discretized (open) moduli spaces
given by for
. This generating function therefore enjoys
the topological recursion, and we prove that it is simultaneously the
generating function for ancestor invariants of a cohomological field theory
thus enjoying the Givental decomposition. We use another Givental-type
decomposition obtained for this model by the second authors in 1995 in terms of
special times related to the discretisation of moduli spaces thus representing
its asymptotic expansion terms (and therefore those of the Gaussian means) as
finite sums over graphs weighted by lower-order monomials in times thus giving
another proof of (quasi)polynomiality of the discrete volumes. As an
application, we find the coefficients in the first subleading order for
in two ways: using the refined Harer--Zagier recursion and
by exploiting the above Givental-type transformation. We put forward the
conjecture that the above graph expansions can be used for probing the
reduction structure of the Delgne--Mumford compactification of moduli spaces of punctured Riemann surfaces.Comment: 36 pages in LaTex, 6 LaTex figure
The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion
We introduce the boundary length and point spectrum, as a joint
generalization of the boundary length spectrum and boundary point spectrum in
arXiv:1307.0967. We establish by cut-and-join methods that the number of
partial chord diagrams filtered by the boundary length and point spectrum
satisfies a recursion relation, which combined with an initial condition
determines these numbers uniquely. This recursion relation is equivalent to a
second order, non-linear, algebraic partial differential equation for the
generating function of the numbers of partial chord diagrams filtered by the
boundary length and point spectrum.Comment: 16 pages, 6 figure
Partial chord diagrams and matrix models
In this article, the enumeration of partial chord diagrams is discussed via
matrix model techniques. In addition to the basic data such as the number of
backbones and chords, we also consider the Euler characteristic, the backbone
spectrum, the boundary point spectrum, and the boundary length spectrum.
Furthermore, we consider the boundary length and point spectrum that unifies
the last two types of spectra. We introduce matrix models that encode
generating functions of partial chord diagrams filtered by each of these
spectra. Using these matrix models, we derive partial differential equations -
obtained independently by cut-and-join arguments in an earlier work - for the
corresponding generating functions.Comment: 42 pages, 14 figure
Topology of RNA-RNA interaction structures
The topological filtration of interacting RNA complexes is studied and the
role is analyzed of certain diagrams called irreducible shadows, which form
suitable building blocks for more general structures. We prove that for two
interacting RNAs, called interaction structures, there exist for fixed genus
only finitely many irreducible shadows. This implies that for fixed genus there
are only finitely many classes of interaction structures. In particular the
simplest case of genus zero already provides the formalism for certain types of
structures that occur in nature and are not covered by other filtrations. This
case of genus zero interaction structures is already of practical interest, is
studied here in detail and found to be expressed by a multiple context-free
grammar extending the usual one for RNA secondary structures. We show that in
time and space complexity, this grammar for genus zero
interaction structures provides not only minimum free energy solutions but also
the complete partition function and base pairing probabilities.Comment: 40 pages 15 figure
Topological recursion for Gaussian means and cohomological field theories
We introduce explicit relations between genus-filtrated s-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces M_(g,s)^(disc) (discrete volumes). Using these relations, we express Gaussian means in all orders of the genus expansion as polynomials in special times weighted by ancestor invariants of an underlying cohomological field theory. We translate the topological recursion of the Gaussian model into recurrence relations for the coefficients of this expansion, which allows proving that they are integers and positive. We find the coefficients in the first subleading order for M_(g,1) for all g in three ways: using the refined Harer–Zagier recursion, using the Givental-type decomposition of the KPMM, and counting diagrams explicitly
Ice-lens formation and geometrical supercooling in soils and other colloidal materials
We present a new, physically-intuitive model of ice-lens formation and growth
during the freezing of soils and other dense, particulate suspensions.
Motivated by experimental evidence, we consider the growth of an ice-filled
crack in a freezing soil. At low temperatures, ice in the crack exerts large
pressures on the crack walls that will eventually cause the crack to split
open. We show that the crack will then propagate across the soil to form a new
lens. The process is controlled by two factors: the cohesion of the soil, and
the geometrical supercooling of the water in the soil; a new concept introduced
to measure the energy available to form a new ice lens. When the supercooling
exceeds a critical amount (proportional to the cohesive strength of the soil) a
new ice lens forms. This condition for ice-lens formation and growth does not
appeal to any ad hoc, empirical assumptions, and explains how periodic ice
lenses can form with or without the presence of a frozen fringe. The proposed
mechanism is in good agreement with experiments, in particular explaining
ice-lens pattern formation, and surges in heave rate associated with the growth
of new lenses. Importantly for systems with no frozen fringe, ice-lens
formation and frost heave can be predicted given only the unfrozen properties
of the soil. We use our theory to estimate ice-lens growth temperatures
obtaining quantitative agreement with the limited experimental data that is
currently available. Finally we suggest experiments that might be performed in
order to verify this theory in more detail. The theory is generalizable to
complex natural-soil scenarios, and should therefore be useful in the
prediction of macroscopic frost heave rates.Comment: Submitted to PR
Three dimensional adaptive mesh refinement on a spherical shell for atmospheric models with lagrangian coordinates
One of the most important advances needed in global climate models is the development of atmospheric General Circulation Models (GCMs) that can reliably treat convection. Such GCMs require high resolution in local convectively active regions, both in the horizontal and vertical directions. During previous research we have developed an Adaptive Mesh Refinement (AMR) dynamical core that can adapt its grid resolution horizontally. Our approach utilizes a finite volume numerical representation of the partial differential equations with floating Lagrangian vertical coordinates and requires resolving dynamical processes on small spatial scales. For the latter it uses a newly developed general-purpose library, which facilitates 3D block-structured AMR on spherical grids. The library manages neighbor information as the blocks adapt, and handles the parallel communication and load balancing, freeing the user to concentrate on the scientific modeling aspects of their code. In particular, this library defines and manages adaptive blocks on the sphere, provides user interfaces for interpolation routines and supports the communication and load-balancing aspects for parallel applications. We have successfully tested the library in a 2-D (longitude-latitude) implementation. During the past year, we have extended the library to treat adaptive mesh refinement in the vertical direction. Preliminary results are discussed. This research project is characterized by an interdisciplinary approach involving atmospheric science, computer science and mathematical/numerical aspects. The work is done in close collaboration between the Atmospheric Science, Computer Science and Aerospace Engineering Departments at the University of Michigan and NOAA GFDL.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/58181/2/jpconf7_78_012072.pd
Hydrogen bond rotations as a uniform structural tool for analyzing protein architecture
Proteins fold into three-dimensional structures, which determine their diverse functions. The conformation of the backbone of each structure is locally at each Cα effectively described by conformational angles resulting in Ramachandran plots. These, however, do not describe the conformations around hydrogen bonds, which can be non-local along the backbone and are of major importance for protein structure. Here, we introduce the spatial rotation between hydrogen bonded peptide planes as a new descriptor for protein structure locally around a hydrogen bond. Strikingly, this rotational descriptor sampled over high-quality structures from the protein data base (PDB) concentrates into 30 localized clusters, some of which correlate to the common secondary structures and others to more special motifs, yet generally providing a unifying systematic classification of local structure around protein hydrogen bonds. It further provides a uniform vocabulary for comparison of protein structure near hydrogen bonds even between bonds in different proteins without alignment
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