14,339 research outputs found
The FKG Inequality and Some Monotonicity Properties of Partial Orders
Let (a1 , . . . , am, b1, . . . , bn) be a random permutation of 1, 2, . . ., m + n. Let P be a partial order on the a’s and b’s involving only inequalities of the form ai \u3c aj or bi \u3c bj, and let P\u27 be an extension of P to include inequalities of the form ai \u3c bj; i.e, P\u27 = P ∪ P\u27\u27, where P\u27\u27 involves only inequalities of the form ai \u3c bj. We prove the natural conjecture of R. L. Graham, A. C. Yao, and F. F. Yao [SIAM J. Alg. Discr. Meth. 1 (1980), pp. 251–258] that in particular (*) Pr (a1 \u3c b1|P\u27) ≥ Pr (a1 \u3c b1|P). We give a simple example to show that the more general inequality (*) where P is allowed to contain inequalities of the form ai \u3c bj is false. This is surprising because as Graham, Yao, and Yao proved, the general inequality (*) does hold if P totally orders both the a’s and the b’s separately. We give a new proof of the latter result. Our proofs are based on the FKG inequality
Partial Strategyproofness: Relaxing Strategyproofness for the Random Assignment Problem
We present partial strategyproofness, a new, relaxed notion of
strategyproofness for studying the incentive properties of non-strategyproof
assignment mechanisms. Informally, a mechanism is partially strategyproof if it
makes truthful reporting a dominant strategy for those agents whose preference
intensities differ sufficiently between any two objects. We demonstrate that
partial strategyproofness is axiomatically motivated and yields a parametric
measure for "how strategyproof" an assignment mechanism is. We apply this new
concept to derive novel insights about the incentive properties of the
probabilistic serial mechanism and different variants of the Boston mechanism.Comment: Working Pape
On Monotonicity and Propagation of Order Properties
In this paper, a link between monotonicity of deterministic dynamical systems
and propagation of order by Markov processes is established. The order
propagation has received considerable attention in the literature, however,
this notion is still not fully understood. The main contribution of this paper
is a study of the order propagation in the deterministic setting, which
potentially can provide new techniques for analysis in the stochastic one. We
take a close look at the propagation of the so-called increasing and increasing
convex orders. Infinitesimal characterisations of these orders are derived,
which resemble the well-known Kamke conditions for monotonicity. It is shown
that increasing order is equivalent to the standard monotonicity, while the
class of systems propagating the increasing convex order is equivalent to the
class of monotone systems with convex vector fields. The paper is concluded by
deriving a novel result on order propagating diffusion processes and an
application of this result to biological processes.Comment: Part of the paper is to appear in American Control Conference 201
Monotone Control Systems
Monotone systems constitute one of the most important classes of dynamical
systems used in mathematical biology modeling.
The objective of this paper is to extend the notion of monotonicity to
systems with inputs and outputs, a necessary first step in trying to understand
interconnections, especially including feedback loops, built up out of monotone
components.
Basic definitions and theorems are provided, as well as an application to the
study of a model of one of the cell's most important subsystems.Comment: See http://www.math.rutgers.edu/~sontag/ for related wor
Oscillations in I/O monotone systems under negative feedback
Oscillatory behavior is a key property of many biological systems. The
Small-Gain Theorem (SGT) for input/output monotone systems provides a
sufficient condition for global asymptotic stability of an equilibrium and
hence its violation is a necessary condition for the existence of periodic
solutions. One advantage of the use of the monotone SGT technique is its
robustness with respect to all perturbations that preserve monotonicity and
stability properties of a very low-dimensional (in many interesting examples,
just one-dimensional) model reduction. This robustness makes the technique
useful in the analysis of molecular biological models in which there is large
uncertainty regarding the values of kinetic and other parameters. However,
verifying the conditions needed in order to apply the SGT is not always easy.
This paper provides an approach to the verification of the needed properties,
and illustrates the approach through an application to a classical model of
circadian oscillations, as a nontrivial ``case study,'' and also provides a
theorem in the converse direction of predicting oscillations when the SGT
conditions fail.Comment: Related work can be retrieved from second author's websit
Multi-Stability in Monotone Input/Output Systems
This paper studies the emergence of multi-stability and hysteresis in those
systems that arise, under positive feedback, starting from monotone systems
with well-defined steady-state responses. Such feedback configurations appear
routinely in several fields of application, and especially in biology.
Characterizations of global stability behavior are stated in terms of easily
checkable graphical conditions. An example of a signaling cascade under
positive feedback is presented.Comment: See http://www.math.rutgers.edu/~sontag for related work; to appear
in Systems and Control Letter
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