332,297 research outputs found
Strictification of weakly stable type-theoretic structures using generic contexts
We present a new strictification method for type-theoretic structures that
are only weakly stable under substitution. Given weakly stable structures over
some model of type theory, we construct equivalent strictly stable structures
by evaluating the weakly stable structures at generic contexts. These generic
contexts are specified using the categorical notion of familial
representability. This generalizes the local universes method of Lumsdaine and
Warren.
We show that generic contexts can also be constructed in any category with
families which is freely generated by collections of types and terms, without
any definitional equality. This relies on the fact that they support
first-order unification. These free models can only be equipped with weak
type-theoretic structures, whose computation rules are given by typal
equalities. Our main result is that any model of type theory with weakly stable
weak type-theoretic structures admits an equivalent model with strictly stable
weak type-theoretic structures
Randomness and semigenericity
Let L contain only the equality symbol and let L^+ be an arbitrary finite
symmetric relational language containing L . Suppose probabilities are defined
on finite L^+ structures with ''edge probability'' n^{- alpha}. By T^alpha, the
almost sure theory of random L^+-structures we mean the collection of
L^+-sentences which have limit probability 1. T_alpha denotes the theory of the
generic structures for K_alpha, (the collection of finite graphs G with
delta_{alpha}(G)=|G|- alpha. | edges of G | hereditarily nonnegative.)
THEOREM: T_alpha, the almost sure theory of random L^+-structures is the same
as the theory T_alpha of the K_alpha-generic model. This theory is complete,
stable, and nearly model complete. Moreover, it has the finite model property
and has only infinite models so is not finitely axiomatizable
Stable and Efficient Structures for the Content Production and Consumption in Information Communities
Real-world information communities exhibit inherent structures that
characterize a system that is stable and efficient for content production and
consumption. In this paper, we study such structures through mathematical
modelling and analysis. We formulate a generic model of a community in which
each member decides how they allocate their time between content production and
consumption with the objective of maximizing their individual reward. We define
the community system as "stable and efficient" when a Nash equilibrium is
reached while the social welfare of the community is maximized. We investigate
the conditions for forming a stable and efficient community under two
variations of the model representing different internal relational structures
of the community. Our analysis results show that the structure with "a small
core of celebrity producers" is the optimally stable and efficient for a
community. These analysis results provide possible explanations to the
sociological observations such as "the Law of the Few" and also provide
insights into how to effectively build and maintain the structure of
information communities.Comment: 21 page
Mesa-type patterns in the one-dimensional Brusselator and their stability
The Brusselator is a generic reaction-diffusion model for a tri-molecular
chemical reaction. We consider the case when the input and output reactions are
slow. In this limit, we show the existence of -periodic, spatially bi-stable
structures, \emph{mesas}, and study their stability. Using singular
perturbation techniques, we find a threshold for the stability of mesas.
This threshold occurs in the regime where the exponentially small tails of the
localized structures start to interact. By comparing our results with Turing
analysis, we show that in the generic case, a Turing instability is followed by
a slow coarsening process whereby logarithmically many mesas are annihilated
before the system reaches a steady equilibrium state. We also study a
``breather''-type instability of a mesa, which occurs due to a Hopf
bifurcation. Full numerical simulations are shown to confirm the analytical
results.Comment: to appear, Physica
Origami Multistabilty: From Single Vertices to Metasheets
We explore the surprisingly rich energy landscape of origami-like folding
planar structures. We show that the configuration space of rigid-paneled
degree-4 vertices, the simplest building blocks of such systems, consists of at
least two distinct branches meeting at the flat state. This suggests that
generic vertices are at least bistable, but we find that the nonlinear nature
of these branches allows for vertices with as many as five distinct stable
states. In vertices with collinear folds and/or symmetry, more branches emerge
leading to up to six stable states. Finally, we introduce a procedure to tile
arbitrary 4-vertices while preserving their stable states, thus allowing the
design and creation of multistable origami metasheets.Comment: For supplemental movies please visit
http://www.lorentz.leidenuniv.nl/~chen/multisheet
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