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Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem description
# Problem setup:
In a long-range dispersal model, a cluster of species can grow from a single seed. The cluster expands into space with long-range power-law interactions. In one dimension, the asymptotic size of the cluster, denoted by , satisfies a self-consistent equation:
\begin{equation}
\frac{1}{\ell^{\mu}(t)} \int_0^t \ell(\tau) \ell(t - \tau) d\tau = K.
\end{equation}
Here, is the power-law index of the long-range interaction, and is a constant independent of time.
# Main problem:
Consider the specific scenario where , and let us define and . Expand in terms of for large and retain terms up to constant order in . That is, you should fix any corrections such as and if they are present; constant corrections can be ignored.
Plain-text mathematical notation (without MathML)
# Problem setup:
In a long-range dispersal model, a cluster of species can grow from a single seed. The cluster expands into space with long-range power-law interactions. In one dimension, the asymptotic size of the cluster, denoted by ℓ(t), satisfies a self-consistent equation:
\begin{equation}
\frac{1}{\ell^{\mu}(t)} \int_0^t \ell(\tau) \ell(t - \tau) d\tau = K.
\end{equation}
Here, μ is the power-law index of the long-range interaction, and K is a constant independent of time.
# Main problem:
Consider the specific scenario where μ=2, and let us define φ=log₂ℓ and z=log₂t. Expand φ in terms of z for large t and retain terms up to constant order in z. That is, you should fix any polylog(z) corrections such as log₂(z) and log₂²(z) if they are present; constant corrections can be ignored.Original LaTeX notation
# Problem setup:
In a long-range dispersal model, a cluster of species can grow from a single seed. The cluster expands into space with long-range power-law interactions. In one dimension, the asymptotic size of the cluster, denoted by $\ell(t)$, satisfies a self-consistent equation:
\begin{equation}
\frac{1}{\ell^{\mu}(t)} \int_0^t \ell(\tau) \ell(t - \tau) d\tau = K.
\end{equation}
Here, $\mu$ is the power-law index of the long-range interaction, and $K$ is a constant independent of time.
# Main problem:
Consider the specific scenario where $\mu = 2$, and let us define $\varphi = \log_2 \ell$ and $z = \log_2 t$. Expand $\varphi$ in terms of $z$ for large $t$ and retain terms up to constant order in $z$. That is, you should fix any $\text{polylog}(z)$ corrections such as $\log_2(z)$ and $\log^2_2(z)$ if they are present; constant corrections can be ignored.code template
Code
import sympy as sp
z, K = sp.symbols('z K')
def answer(z, K):
r"""
Return the expression of $\varphi$ in Sympy format.
Inputs
----------
z: sympy.Symbol, defined as $\log_2 t$
K: sympy.Symbol, constant from the self-consistent equation
$$\frac{1}{\ell^{\mu}(t)} \int_0^t \ell(\tau) \ell(t - \tau) d\tau = K$$
Outputs
----------
varphi: sympy.Expr, expansion of $\varphi$ in terms of $z$ for large $t$
and retain terms up to constant order in $z$
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
varphi = ... # SymPy expression in terms of z (and possibly K),
# retaining z, log_2(z), log_2(z)**2 terms; omit constants
# ---------------------------------------------------------------
return varphiDiscussion
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initial import