16,098 research outputs found
Natural maps between CAT(0) boundaries
It is shown that certain natural maps between the ideal, Gromov, and end
boundaries of a complete CAT(0) space can fail to be either injective or
surjective. Additionally the natural map from the Gromov boundary to the end
boundary of a complete CAT(-1) space can fail to be either injective or
surjective.Comment: 8 page
Optimal Switching for Hybrid Semilinear Evolutions
We consider the optimization of a dynamical system by switching at discrete
time points between abstract evolution equations composed by nonlinearly
perturbed strongly continuous semigroups, nonlinear state reset maps at mode
transition times and Lagrange-type cost functions including switching costs. In
particular, for a fixed sequence of modes, we derive necessary optimality
conditions using an adjoint equation based representation for the gradient of
the costs with respect to the switching times. For optimization with respect to
the mode sequence, we discuss a mode-insertion gradient. The theory unifies and
generalizes similar approaches for evolutions governed by ordinary and delay
differential equations. More importantly, it also applies to systems governed
by semilinear partial differential equations including switching the principle
part. Examples from each of these system classes are discussed
Instanton constituents in the O(3) model at finite temperature
It is shown that instantons in the O(3) model at finite temperature consist
of fractional charge constituents and the (topological) properties of the
latter are discussed.Comment: 5 pages, 12 plots in 3 figure
Rough CAT(0) spaces
We investigate various notions of rough CAT(0). These conditions define
classes of spaces that strictly include the union of all Gromov hyperbolic
length spaces and all CAT(0) spaces.Comment: Corrected typos and updated Corollaries 3.22 and 3.2
Strain localization in a shear transformation zone model for amorphous solids
We model a sheared disordered solid using the theory of Shear Transformation
Zones (STZs). In this mean-field continuum model the density of zones is
governed by an effective temperature that approaches a steady state value as
energy is dissipated. We compare the STZ model to simulations by Shi, et
al.(Phys. Rev. Lett. 98 185505 2007), finding that the model generates
solutions that fit the data,exhibit strain localization, and capture important
features of the localization process. We show that perturbations to the
effective temperature grow due to an instability in the transient dynamics, but
unstable systems do not always develop shear bands. Nonlinear energy
dissipation processes interact with perturbation growth to determine whether a
material exhibits strain localization. By estimating the effects of these
interactions, we derive a criterion that determines which materials exhibit
shear bands based on the initial conditions alone. We also show that the shear
band width is not set by an inherent diffusion length scale but instead by a
dynamical scale that depends on the imposed strain rate.Comment: 8 figures, references added, typos correcte
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