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Vol. 64, no. 1 (2022)

Front matter  
Preface.  
On a differential intermediate value property.
Liouville closed $H$fields are ordered differential fields where the ordering
and derivation interact in a natural way and every linear differential equation
of order $1$ has a nontrivial solution. (The introduction gives a precise
definition.) For a Liouville closed $H$field $K$ with small derivation we
show that $K$ has the Intermediate Value Property for differential polynomials
if and only if $K$ is elementarily equivalent to the ordered differential field
of transseries. We also indicate how this applies to Hardy fields.

1–10 
A remark on uniform expansion.
For every $\mathcal{U} \subset
\mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^2)$ there is a measure of
finite support contained in $\mathcal{U}$ which is uniformly expanding.

11–21 
The space of
infinite partitions of $\mathbb{N}$ as a topological Ramsey
space.
The Ramsey theory of the space of equivalence relations with infinite
quotients defined on the set $\mathbb{N}$ of natural numbers is an interesting field
of research.
We view this space as a topological Ramsey space $(\mathcal{E}_\infty,\leq,
r)$ and present a game theoretic characterization of the Ramsey property of
subsets of $\mathcal{E}_{\infty}$.
We define a notion of coideal and consider the Ramsey property of subsets of
$\mathcal{E}_\infty$ localized on a coideal
$\mathcal{H}\subseteq \mathcal{E}_{\infty}$. Conditions a coideal $\mathcal{H}$ should
satisfy to make the structure
$(\mathcal{E}_{\infty},\mathcal{H},\leq, r)$ a Ramsey space are presented. Forcing notions
related to a coideal $\mathcal{H}$ and their main properties are analyzed.

23–48 
Threshold Ramsey multiplicity for odd cycles.
The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any
twocoloring of the edges of the complete graph $K_n$ contains a monochromatic
copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum
number of monochromatic copies of $H$ taken over all twoedgecolorings of
$K_{r(H)}$.
The study of this concept was first proposed by Harary and Prins almost fifty
years ago.
In a companion paper, the authors have shown that there is a positive
constant $c$ such that the threshold Ramsey multiplicity for a path or even
cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the
value of $c$. Here, using different methods, we show that the same result
also holds for odd cycles with $k$ vertices.

49–68 
New bounds on Cantor maximal operators.
We prove $L^p$ bounds for the maximal operators associated to an
Ahlforsregular variant of fractal percolation. Our bounds improve upon those
obtained by I. Łaba and M. Pramanik and in some cases are sharp up to the
endpoint. A consequence of our main result is that there exist Ahlforsregular
Salem Cantor sets of any dimension $> 1/2$ such that the associated maximal
operator is bounded on $L^2(\mathbb{R})$. We follow the overall scheme of
Łaba–Pramanik for the analytic part of the argument, while the probabilistic
part is instead inspired by our earlier work on intersection properties of
random measures.

69–86 
Families of convex tilings.
We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such
tilings come in continuous families obtained by moving tile edges parallel
to themselves (keeping edge directions fixed). We study how the tile
shapes and areas change in these families. In particular we show that if
$R$ is convex, the tile shapes can be arbitrarily prescribed (up to
homothety). We also show that the tile areas and tile “orientations”
determine the tiling. We associate to a tiling an underlying bipartite
planar graph $\mathcal{G}$ and its corresponding Kasteleyn
matrix $K$. If $\mathcal{G}$ has
quadrilateral faces, we show that $K$ is the differential of the map from
edge intercepts to tile areas, and extract some geometric and probabilistic
consequences.

87–101 
Higher Fano manifolds.
We address in this paper Fano manifolds with positive higher Chern characters,
which are expected to enjoy stronger versions of several of the nice
properties of Fano manifolds. For instance, they should be covered by
higher dimensional rational varieties, and families of higher Fano
manifolds over higher dimensional bases should admit meromorphic sections
(modulo the Brauer obstruction). Aiming at finding new examples of higher
Fano manifolds, we investigate positivity of higher Chern characters of
rational homogeneous spaces. We determine which rational homogeneous
spaces of Picard rank 1 have positive second Chern character, and show
that the only rational homogeneous spaces of Picard rank 1 having
positive second and third Chern characters are projective spaces and
quadric hypersurfaces. We also classify Fano manifolds of large index
having positive second and third Chern characters. We conclude by
discussing conjectural characterizations of projective spaces and complete
intersections in terms of these higher Fano conditions.

103–125 
Wave maps into the sphere.
In this note we discuss some geometric analogs of the classical harmonic
functions on $\mathbb{R}^n$ and their associated evolutions.

127–135 
Asymptotic mean value formulas for parabolic nonlinear equations.
In this paper we characterize viscosity solutions to nonlinear parabolic
equations (including parabolic Monge–Ampère equations) by asymptotic mean
value formulas. Our asymptotic mean value formulas can be interpreted from
a probabilistic point of view in terms of dynamic programming principles
for certain twoplayer, zerosum games.

137–164 
On fibrations and
measures of irrationality of hyperKhäler manifolds.
We prove some results on the fibers and images of rational maps from a
hyperKähler manifold. We study in particular the minimal genus of
fibers of a fibration into curves. The last section of this paper is
devoted to the study of the rational map defined by a linear system on a
hyperKähler fourfold satisfying numerical conditions similar to those
considered by O'Grady in his study of fourfolds numerically equivalent to
$\mathrm{K}3^{[2]}$. We extend his results to this more general context.

165–197 
On conformally
compact Einstein manifolds.
We survey some of the recent developments in the study of the compactness and
uniqueness problems for some classes of conformally compact Einstein manifolds.

199–213 
Front matter  
Preface.  
Univariate
rational sums of squares.
Given rational univariate polynomials $f$ and $g$ such that $\gcd (f,g)$ and
$f/\gcd(f,g)$ are relatively prime, we show that $g$ is nonnegative at all
the real roots of $f$ if and only if $g$ is a sum of squares of rational
polynomials modulo $f$. We complete our study by exhibiting an algorithm
that produces a certificate that a polynomial $g$ is nonnegative at the
real roots of a nonzero polynomial $f$ when the above assumption is
satisfied.

215–237 
A note on Bernstein–Sato ideals.
We define the Bernstein–Sato ideal associated to a tuple of ideals and we
relate it to the jumping points of the corresponding mixed multiplier ideals.

239–246 
On
essential selfadjointness of singular
Sturm–Liouville operators.
Considering singular Sturm–Liouville differential expressions of the type
\[
\tau_{\alpha} = (d/dx)x^{\alpha}(d/dx) + q(x),
\quad x \in (0,b), \, \alpha \in \mathbb{R},
\]
we employ some Sturm comparisontype results in the spirit of Kurss to derive
criteria for $\tau_{\alpha}$ to be in the limitpoint and limitcircle case
at $x=0$. More precisely, if $\alpha \in \mathbb{R}$ and, for $0 < x$
sufficiently small,
\[
q(x) \geq [(3/4)(\alpha/2)]x^{\alpha2},
\]
or, if $\alpha\in (\infty,2)$ and there exist $N\in\mathbb{N}$ and $\varepsilon>0$
such that, for $0 < x$ sufficiently small,
\begin{align*}
& q(x)\geq[(3/4)(\alpha/2)]x^{\alpha2}  (1/2) (2  \alpha) x^{\alpha2}
\sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{1}
\\
& \quad\quad\quad +[(3/4)+\varepsilon] x^{\alpha2}[\ln_{1}(x)]^{2},
\end{align*}
then $\tau_{\alpha}$ is nonoscillatory and in the limitpoint case at $x=0$.
Here iterated logarithms for $0 < x$ sufficiently small are of the form
\[
\ln_1(x) = \ln(x) = \ln(1/x),
\qquad
\ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}.
\]
Analogous results are derived for $\tau_{\alpha}$ to be in the limitcircle
case at $x=0$.
We also discuss a multidimensional application to partial differential
expressions of the type
\[
 \operatorname{div} x^{\alpha} \nabla + q(x),
\quad \alpha \in \mathbb{R}, \, x \in B_n(0;R) \backslash \{0\},
\]
with $B_n(0;R)$ the open ball in $\mathbb{R}^n$, $n\in \mathbb{N}$, $n \geq 2$,
centered at $x=0$ of radius $R \in (0, \infty)$.

247–269 
Inequivalent representations of the dual space.
We show that there exist inequivalent representations of the dual space of
$\mathbb{C}[0,1]$ and of $L_p[\mathbb{R}^n]$ for $p \in [1,\infty)$. We
also show how these inequivalent representations reveal new and important
results for both the operator and the geometric structure of these spaces.
For example, if $\mathcal{A}$ is a proper closed subspace of
$\mathbb{C}[0,1]$, there always exists a closed subspace $\mathcal{A}^\bot$
(with the same definition as for $L_2[0,1]$) such that $\mathcal{A} \cap
\mathcal{A}^\bot = \{0\}$ and $\mathcal{A} \oplus \mathcal{A}^\bot =
\mathbb{C}[0,1]$. Thus, the geometry of $\mathbb{C}[0,1]$ can be viewed
from a completely new perspective. At the operator level, we prove that
every bounded linear operator $A$ on $\mathbb{C}[0,1]$ has a uniquely
defined adjoint $A^*$ defined on $\mathbb{C}[0,1]$, and both can be
extended to bounded linear operators on $L_2[0,1]$. This leads to a polar
decomposition and a spectral theorem for operators on the space. The same
results also apply to $L_p[\mathbb{R}^n]$. Another unexpected result is a
proof of the Baire one approximation property (every closed densely defined
linear operator on $\mathbb{C}[0,1]$ is the limit of a sequence of bounded
linear operators). A fundamental implication of this paper is that the use
of inequivalent representations of the dual space is a powerful new tool
for functional analysis.

271–280 
Normal form
transformations for modulated deepwater gravity waves.
Modulation theory is a wellknown tool to describe the longtime evolution
and stability of smallamplitude, oscillating solutions to dispersive
nonlinear partial differential equations. There have been a number of
approaches to deriving envelope equations for weakly nonlinear waves. Here
we review a systematic method based on Hamiltonian transformation theory
and averaging Hamiltonians. In the context of the modulation of two or
threedimensional deepwater surface waves, this approach leads to a Dysthe
equation that preserves the Hamiltonian character of the water wave
problem. An explicit calculation of the thirdorder Birkhoff normal form
that eliminates all nonresonant cubic terms yields a nonperturbative
procedure for the reconstruction of the free surface. We also present new
numerical simulations of this weakly nonlinear approximation using the
version with exact linear dispersion. We compare them against computations
from the full water wave system and find very good agreement.

281–308 
Positivities in
Hall–Littlewood expansions and related plethystic operators.
The Hall–Littlewood polynomials $\mathbf{H}_\lambda = Q'_\lambda[X;q]$ are an
important symmetric function basis that appears in many contexts. In this
work, we give an accessible combinatorial formula for expanding the related
symmetric functions $\mathbf{H}_\alpha$ for any composition $\alpha$, in terms of
the complete homogeneous basis. We do this by analyzing Jing's operators,
which extend to give nice expansions for the related symmetric functions
$\mathbf{C}_\alpha$ and $\mathbf{B}_\alpha$ which appear in the formulation of the
Compositional Shuffle Theorem. We end with some consequences related to
eigenoperators of the modified Macdonald basis.

309–331 
Homogeneous weight enumerators over integer residue rings and failures of the MacWilliams identities.
The MacWilliams identities for the homogeneous weight enumerator over
$\mathbb{Z}/m\mathbb{Z}$ do not hold for composite $m \geq 6$. For such
$m$, there exist two linear codes over $\mathbb{Z}/m\mathbb{Z}$ that have
the same homogeneous weight enumerator, yet whose dual codes have different
homogeneous weight enumerators.

333–353 
Strongly aperiodic SFTs on
hyperbolic groups: where to find them and why we love them.
D. B. Cohen, C. GoodmanStrauss, and the author
[Ergodic Theory Dynam. Systems 42 (2022), no. 9, 2740–2783]
proved that a hyperbolic group admits an SA SFT if and
only if it has at most one end. This paper has two distinct parts: the
first is a conversation explaining what an SA SFT is and how it may be of
use, while in the second part I attempt to explain both old and new ideas that go
into that proof. References to specific claims
in the work cited above are given, with the hope that any interested reader may
be able to find the details there more accesible after reading this
exposition.

355–373 
Stabilizing radial basis
functions techniques for a local boundary integral method.
Radial basis functions (RBFs) have been gaining popularity recently in the
development of methods for solving partial differential equations (PDEs)
numerically. These functions have become an extremely effective tool for
interpolation on scattered node sets in several dimensions.
One key issue with infinitely smooth RBFs is the choice of a suitable value for
the shape parameter $\varepsilon$, which controls the flatness of the
function. It is observed that best accuracy is often achieved when
$\varepsilon$ tends to zero. However, the system of discrete equations from
interpolation matrices becomes illconditioned.
A few numerical algorithms have been presented that are able to stably compute
an interpolant, even in the increasingly flat basis function limit, such as the
RBFQR method and the RBFGA method.
We present these techniques in the context of boundary integral methods to
improve the solution of PDEs with RBFs. These stable calculations open up
new opportunities for applications and developments of local integral
methods based on local RBF approximations.
Numerical results for a small shape parameter that stabilizes the error are
presented. Accuracy and comparisons are also shown for elliptic PDEs.

375–396 
A rich and complex
dynamics emerges between the subthalamic nucleus and the
globus pallidus externa in the basal ganglia.
The oscillatory nature of basal ganglia activity is highly tied with some movement
disorders. We study, through bifurcation analysis and computer simulations,
the appearance of abnormal oscillations in an already proposed reduced
neural circuit model of the subthalamic nucleus and globus pallidus loop.
The results show that the model exhibits stable steady states associated to
normal activity and oscillatory activity corresponding to the often termed
“tremor frequency” oscillations due to their coherence with parkinsonian
tremor observed in patients with Parkinson's disease.

397–412 
Nondegenerate extensions of
neargroup braided fusion categories.
This is a study of weakly integral braided fusion categories with elementary
fusion rules to determine which possess nondegenerately braided extensions
of theoretically minimal dimension, or equivalently in this case, which
satisfy the minimal modular extension conjecture. We classify neargroup
braided fusion categories satisfying the minimal modular extension
conjecture; the remaining Tambara–Yamagami braided fusion categories
provide arbitrarily large families of braided fusion categories with
identical fusion rules violating the minimal modular extension conjecture.
These examples generalize to braided fusion categories with the fusion
rules of the representation categories of extraspecial $p$groups for any
prime $p$, which possess a minimal modular extension only if they arise as
the adjoint subcategory of a twisted double of an extraspecial $p$group.

413–438 
Arithmetic properties of generalized Fibonacci numbers.
We present a survey of results concerning arithmetic properties of
generalized Fibonacci sequences and certain Diophantine equations involving
terms from that family of numbers. Most of these results have been recently
obtained by the research groups in number theory at the Universities of
Cauca (in Popayán) and of Valle (in Cali), Colombia, lead by the first two
authors.

439–460 
Homogeneous Einstein manifolds.
This survey builds on the two surveys by Wang and Lauret, written 10–15 years
ago, to give the current state of affairs regarding homogeneous Einstein
spaces.

461–485 
Mathematical model for the aquatic stage of Aedes aegypti
considering variable egghatching rate and interspecific competition
between larval stages.
Mosquitoborne diseases like dengue, Zika, and chikungunya, carried by Aedes
aegypti mosquitoes, pose significant health threats. Controlling these
diseases primarily involves reducing mosquito populations. Current
approaches rely on simplistic measures to decide on actions like mosquito
fogging or larval control. Adult mosquito numbers are often estimated from
aquatic populations due to easier counting. While these methods have
provided some assistance, there's a need for improvement. Existing risk and
population models overlook the various developmental stages of Aedes
aegypti and their complex interactions. In this work, several mathematical
models for the life cycle of the Aedes aegypti in its aquatic phase are
proposed. They consider the different aquatic developmental stages and
differ in how the competition between the stages occurs. Then, all the
models are discriminated against experimental data to select the one with
the best predictive power. The chosen model will help estimate the adult
mosquito population with a greater degree of precision as well as when they
will emerge. It will also help design better control strategies and better
risk indices.

487–510 
Published by the Unión Matemática Argentina
ISSN 16699637 (online), ISSN 00416932 (print)
Registro Nacional de la Propiedad Intelectual nro. 180.863
Contact: revuma(at)criba.edu.ar