@article{63078,
abstract = {{For a finite group $G$, we describe the asymptotic growth of the number of
connected components of Hurwitz spaces of marked $G$-covers (of both the affine
and projective lines) whose monodromy classes are constrained in a certain way,
as the number of branch points grows to infinity. More precisely, we compute
both the exponent and (in many cases) the coefficient of the leading monomial
in the count of components containing covers whose monodromy group is a given
subgroup of $G$. By the work of Ellenberg, Tran, Venkatesh and Westerland, this
asymptotic behavior is related to the distribution of field extensions
of~$\mathbb{F}_q(T)$ with Galois group $G$.}},
author = {{Seguin, Beranger Fabrice}},
issn = {{0021-2172}},
journal = {{Israel Journal of Mathematics}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Counting Components of Hurwitz Spaces}}},
doi = {{10.1007/s11856-025-2848-5}},
year = {{2025}},
}
@article{63262,
abstract = {{AbstractRadially symmetric global unbounded solutions of the chemotaxis system $$\left\{ {\matrix{{{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = \Delta v - \mu + u,} \hfill & {\mu = {1 \over {|\Omega |}}\int_\Omega {u,} } \hfill \cr } } \right.$$
{
u
t
=
∇
⋅
(
D
(
u
)
∇
u
)
−
∇
⋅
(
u
S
(
u
)
∇
v
)
,
0
=
Δ
v
−
μ
+
u
,
μ
=
1
|
Ω
|
∫
Ω
u
,
are considered in a ball Ω = BR(0) ⊂ ℝn, where n ≥ 3 and R > 0.Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)m−1 and S(ξ) = (ξ + 1)−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling $$m + \lambda < 1 - {2 \over n}$$
m
+
λ
<
1
−
2
n
, a considerably large set of initial data u0 is found to enforce a complete mass aggregation in infinite time in the sense that for any such u0, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying $${1 \over C} \cdot {(t + 1)^{{1 \over \lambda }}} \le ||u( \cdot ,t)|{|_{{L^\infty }(\Omega )}} \le C \cdot {(t + 1)^{{1 \over \lambda }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$
1
C
⋅
(
t
+
1
)
1
λ
≤
|
|
u
(
⋅
,
t
)
|
|
L
∞
(
Ω
)
≤
C
⋅
(
t
+
1
)
1
λ
f
o
r
a
l
l
t
>
0
as well as $$||u( \cdot \,,t)|{|_{{L^1}(\Omega \backslash {B_{{r_0}}}(0))}} \to 0\,\,\,{\rm{as}}\,\,t \to \infty \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$
|
|
u
(
⋅
,
t
)
|
|
L
1
(
Ω
\
B
r
0
(
0
)
)
→
0
as
t
→
∞
for all
r
0
∈
(
0
,
R
)
with some C > 0.}},
author = {{Winkler, Michael}},
issn = {{0021-2172}},
journal = {{Israel Journal of Mathematics}},
number = {{1}},
pages = {{93--127}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Complete infinite-time mass aggregation in a quasilinear Keller–Segel system}}},
doi = {{10.1007/s11856-024-2618-9}},
volume = {{263}},
year = {{2024}},
}
@article{63352,
author = {{Lankeit, Johannes and Winkler, Michael}},
issn = {{0021-2172}},
journal = {{Israel Journal of Mathematics}},
number = {{1}},
pages = {{249--296}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory}}},
doi = {{10.1007/s11856-019-1900-8}},
volume = {{233}},
year = {{2019}},
}