67 research outputs found
Monotone graph limits and quasimonotone graphs
The recent theory of graph limits gives a powerful framework for
understanding the properties of suitable (convergent) sequences of
graphs in terms of a limiting object which may be represented by a symmetric
function on , i.e., a kernel or graphon. In this context it is
natural to wish to relate specific properties of the sequence to specific
properties of the kernel. Here we show that the kernel is monotone (i.e.,
increasing in both variables) if and only if the sequence satisfies a
`quasi-monotonicity' property defined by a certain functional tending to zero.
As a tool we prove an inequality relating the cut and norms of kernels of
the form with and monotone that may be of interest in its
own right; no such inequality holds for general kernels.Comment: 38 page
Counting Perfect Matchings and the Switch Chain
We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes
Peano's Existence Theorem revisited
We present new proofs to four versions of Peano's Existence Theorem for
ordinary differential equations and systems. We hope to have gained readability
with respect to other usual proofs. We also intend to highlight some ideas due
to Peano which are still being used today but in specialized contexts: it
appears that the lower and upper solutions method has one of its oldest roots
in Peano's paper of 1886
Remarks on -monotone operators
In this paper, we deal with three aspects of -monotone operators. First we
study -monotone operators with a unique maximal extension (called
pre-maximal), and with convex graph. We then deal with linear operators, and
provide characterizations of -monotonicity and maximal -monotonicity.
Finally we show that the Brezis-Browder theorem preserves -monotonicity in
reflexive Banach spaces.Comment: 15 page
Travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems
[[abstract]]This work investigates the existence and non-existence of travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems. Employing the cross iterative technique coupled with the explicit construction of upper and lower solutions in the theory of quasimonotone dynamical systems, we can find two threshold speeds c∗ and c∗ with c∗≥c∗>0. If the wave speed is greater than c∗, then we establish the existence of travelling wave solutions connecting two different equilibria. On the other hand, if the wave speed is smaller than c∗, we further prove the non-existence result of travelling wave solutions. Finally, several ecological examples including one-species, two-species and three-species models with various functional responses and time delays are presented to illustrate the analytical results.[[notice]]補æ£å®Œç•¢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[booktype]]紙本[[countrycodes]]GB
The effect of negative feedback loops on the dynamics of Boolean networks
Feedback loops in a dynamic network play an important role in determining the
dynamics of that network. Through a computational study, in this paper we show
that networks with fewer independent negative feedback loops tend to exhibit
more regular behavior than those with more negative loops. To be precise, we
study the relationship between the number of independent feedback loops and the
number and length of the limit cycles in the phase space of dynamic Boolean
networks. We show that, as the number of independent negative feedback loops
increases, the number (length) of limit cycles tends to decrease (increase).
These conclusions are consistent with the fact, for certain natural biological
networks, that they on the one hand exhibit generally regular behavior and on
the other hand show less negative feedback loops than randomized networks with
the same numbers of nodes and connectivity
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