In order to ease notation, we introduce an additional state N+1.
To ease notation, let begin{aligned} alpha =Pi /Ntext end{aligned}where (N) is the number of consumers.
We make the following assumptions on the utility functions (deleting superscripts to ease notation if a condition holds for both partners): Assumption For arbitrary values of l,q,Q we have that 1. lim x ↓ 0 u x ( l, q, Q ) = + ∞ for leisure (x = l), each private good (x = q k ) and each public good (x = Q k ), 2.
Before we define it, we make an auxiliary definition to ease notation: For λ>0, define Notice that for every clump size distribution Ψ, we have Ψ(0)=0 and, hence, 𝒟(0)=𝒞𝒫(0) and 𝒟(1)=∑ i =1∞𝒞𝒫(i).
In the statement of Theorem 1, the value (b=frac{1}{2}) was taken to ease the notation.
It is clear that (mathcal{A}_{1}), (mathcal{A}_{2}), (mathcal{A}_{3}) depend upon (delta > 0), but to ease the notation we will suppress this dependence.
In order to ease the notation, if no confusion arises, we drop · t from the above product, and we simply write mV and mX instead of m· t V and m· t X, respectively.
To ease the notations we write and The following relation holds: (3.33).
The reason why the reset time is the integer number [ t + τ sep ] instead of the real t + τ sep is that it eases the notations and proofs.
To ease the notations, we shall also say that the single parent of a tree node is its principal parent.
Furthermore, ten "vicariance" (for ease of notation, here we still keep the word "vicariance" for the isolation of populations of the same species) and seven dispersal events are identified within species-specific clades, which might play a role in promoting intraspecific divergence.
