2,257 research outputs found
Deciding the Borel complexity of regular tree languages
We show that it is decidable whether a given a regular tree language belongs
to the class of the Borel hierarchy, or equivalently whether
the Wadge degree of a regular tree language is countable.Comment: 15 pages, 2 figure
Coarsening rates for the dynamics of slipping droplets
We derive reduced finite dimensional ODE models starting from one dimensional
lubrication equations describing coarsening dynamics of droplets in nanometric
polymer film interacting on a hydrophobically coated solid substrate in the
presence of large slippage at the liquid/solid interface. In the limiting case
of infinite slip length corresponding in applications to free films a
collision/absorption model then arises and is solved explicitly. The exact
coarsening law is derived for it analytically and confirmed numerically.
Existence of a threshold for the decay of initial distributions of droplet
distances at infinity at which the coarsening rates switch from algebraic to
exponential ones is shown
08271 Abstracts Collection -- Topological and Game-Theoretic Aspects of Infinite Computations
From June 29, 2008, to July 4, 2008, the Dagstuhl Seminar 08271 ``Topological and Game-Theoretic Aspects of Infinite Computations\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, many participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
Scale invariance and dynamic phase transitions in diffusion-limited reactions
Many systems that can be described in terms of diffusion-limited `chemical'
reactions display non-equilibrium continuous transitions separating active from
inactive, absorbing states, where stochastic fluctuations cease entirely. Their
critical properties can be analyzed via a path-integral representation of the
corresponding classical master equation, and the dynamical renormalization
group. An overview over the ensuing universality classes in single-species
processes is given, and generalizations to reactions with multiple particle
species are discussed as well. The generic case is represented by the processes
A A + A, and A -> 0, which map onto Reggeon field theory with the critical
exponents of directed percolation (DP). For branching and annihilating random
walks (BARW) A -> (m+1) A and A + A -> 0, the mean-field rate equation predicts
an active state only. Yet BARW with odd m display a DP transition for d <= 2.
For even offspring number m, the particle number parity is conserved locally.
Below d_c' = 4/3, this leads to the emergence of an inactive phase that is
characterized by the power laws of the pair annihilation process. The critical
exponents at the transition are those of the `parity-conserving' (PC)
universality class. For local processes without memory, competing pair or
triplet annihilation and fission reactions k A -> (k - l) A, k A -> (k+m)A with
k=2,3 appear to yield the only other universality classes not described by
mean-field theory. In these reactions, site occupation number restrictions play
a crucial role.Comment: 16 pages, talk given at 2003 German Physical Society Spring Meeting;
four figures and style files include
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
Characterizing the Shape of Activation Space in Deep Neural Networks
The representations learned by deep neural networks are difficult to
interpret in part due to their large parameter space and the complexities
introduced by their multi-layer structure. We introduce a method for computing
persistent homology over the graphical activation structure of neural networks,
which provides access to the task-relevant substructures activated throughout
the network for a given input. This topological perspective provides unique
insights into the distributed representations encoded by neural networks in
terms of the shape of their activation structures. We demonstrate the value of
this approach by showing an alternative explanation for the existence of
adversarial examples. By studying the topology of network activations across
multiple architectures and datasets, we find that adversarial perturbations do
not add activations that target the semantic structure of the adversarial class
as previously hypothesized. Rather, adversarial examples are explainable as
alterations to the dominant activation structures induced by the original
image, suggesting the class representations learned by deep networks are
problematically sparse on the input space
Integrable viscous conservation laws
We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven.
We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlevé I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe
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