4,825 research outputs found
On the maximum values of the additive representation functions
Let and be sets of nonnegative integers. For a positive integer
let denote the number of representations of as the sum of two
terms from . Let and
\displaystyle d_{A,B}(x) = \max_{\hbox{t: a_{t} \le xb_{t} \le
x}}|a_{t} - b_{t}|. In this paper we study the connection between ,
and . We improve a result of Haddad and Helou about the
Erd\H{o}s - Tur\'an conjecture
Generalization of a theorem of Erdos and Renyi on Sidon Sequences
Erd\H os and R\'{e}nyi claimed and Vu proved that for all and for
all , there exists and a sequence of integers
such that the number of ordered representations of any number as a sum of
elements of is bounded by , and such that .
We give two new proofs of this result. The first one consists of an explicit
construction of such a sequence. The second one is probabilistic and shows the
existence of such a that satisfies ,
improving the bound obtained by Vu.
Finally we use the "alteration method" to get a better bound for
, obtaining a more precise estimate for the growth of
sequences.Comment: 12 pages, no figure
Analysis of an epidemic model with awareness decay on regular random networks
The existence of a die-out threshold (different from the classic disease-invasion one) defining a region of slow extinction of an epidemic has been proved elsewhere for susceptible-aware-infectious-susceptible models without awareness decay, through bifurcation analysis. By means of an equivalent mean-field model defined on regular random networks, we interpret the dynamics of the system in this region and prove that the existence of bifurcation for of this second epidemic threshold crucially depends on the absence of awareness decay. We show that the continuum of equilibria that characterizes the slow die-out dynamics collapses into a unique equilibrium when a constant rate of awareness decay is assumed, no matter how small, and that the resulting bifurcation from the disease-free equilibrium is equivalent to that of standard epidemic models. We illustrate these findings with continuous-time stochastic simulations on regular random networks with different degrees. Finally, the behaviour of solutions with and without decay in awareness is compared around the second epidemic threshold for a small rate of awareness decay
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