We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimension-one simplices satisfying a certain restriction on an
$n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.

Inspired by Aldous' conjecture for
the spectral gap of the interchange process and its recent
resolution by Caputo, Liggett, and Richthammer, we define
an associated order $\prec$ on the irreducible representations of $S_n$. Aldous'
conjecture is equivalent to certain representations being comparable
in this order, and hence determining the ``Aldous order'' completely is a
generalized question. We show a few additional entries for this order.

In 1961, J. Barrett showed that if the first conjugate point
$\eta_1(a)$ exists for the differential equation $(r(x)y'')''=
p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first
systems-conjugate point $\widehat\eta_1(a)$. The aim of this note is to
extend this result to the general equation with middle term
$(q(x)y')'$ without further restriction on $q(x)$, other than
continuity.

We show that there
is a bijective correspondence between the polystable parabolic
principal $G$-bundles and solutions of the Hermitian-Einstein
equation.

We study the existence of continuity points for mappings
$f\colon X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are
fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are
quasicontinuous, where $X$ is a Baire space and $Z$
is a metric space. For the factor $Y$, we consider two
infinite ``point-picking'' games $G_1(y)$ and $G_2(y)$ defined respectively
for each $y\in Y$ as follows: in the $n$-th
inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then
Player II picks a point $y_n\in D_n$;
II wins if $y$ is in the closure of ${\{y_n:n\in\mathbb N\}}$, otherwise
I wins. It is shown that
(i) $f$ is
cliquish
if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $
f$ is quasicontinuous if
the $x$-sections of $f$ are continuous and the set of $y\in Y$
such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially
a result of Debs and item (ii) indicates that
the problem of Talagrand on separately continuous maps has a positive answer for a wide
class of ``small'' compact spaces.

A vector measure result is used to study the complementation of the
space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly
compact operators, $CC(X,Y)$ of completely continuous operators, and
$U(X,Y)$ of unconditionally converging operators.
Results of Kalton and Emmanuele concerning the complementation of
$K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment
of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.

Let $\sigma_{\mathbb Z}(k)$ be the smallest $n$ such that there exists an
identity
\[
(x_1^2 + x_2^2 + \cdots + x_k^2) \cdot (y_1^2 + y_2^2 + \cdots + y_k^2)
= f_1^2 + f_2^2 + \cdots + f_n^2,
\]
with $f_1,\dots,f_n$ being polynomials with integer coefficients in
the variables $x_1,\dots,x_k$ and $y_1,\dots,y_k$. We prove that
$\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$.

In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's
and Schaefer's fixed point theorems are employed in the analysis.
The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions.
We employ Liapunov's direct method to obtain such an a priori bound.
In the process, we compare these theorems in terms of assumptions and outcomes.

We give sufficient conditions for the following problem: given a
topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and
a continuous map $f$ from $X$ to $Y$, is it possible, by applying to
$f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not
meet $Z$? We also give a relative variant: if $f(X')$ does not meet
$Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged
on $X'$. We also develop a variant for continuous sections of
fibrations and discuss some applications to matrix perturbation
theory.

We study an even order system boundary value problem with
periodic boundary conditions. By establishing
the existence of a positive eigenvalue of an associated linear system
Sturm-Liouville problem, we obtain new conditions for the boundary
value problem to have a positive solution. Our major tools are the
Krein-Rutman theorem for linear spectra and the fixed point index theory
for compact operators.

An explicit construction of a pre-quantum line bundle for the moduli
space of flat $G$-bundles over a Riemann surface is given, where $G$
is any non-simply connected compact simple Lie group. This work helps
to explain a curious coincidence previously observed between
Toledano Laredo's work classifying central extensions of loop groups
$LG$ and the author's previous work on the obstruction to
pre-quantization of the moduli space of flat $G$-bundles.

In this paper, we discuss monotonicity formulae of various entropy functionals under various
rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue
of a family of geometric operators $-4\Delta + kR$ is monotonic along the
normalized Ricci flow for all $k\ge 1$ provided the initial manifold has
nonpositive total scalar curvature.

Product type equivalence relations are hyperfinite measured
equivalence relations, which, up to orbit equivalence, are generated
by product type odometer actions. We give a concrete example of a
hyperfinite equivalence relation of non-product type, which is the
tail equivalence on a Bratteli diagram.
In order to show that the equivalence relation constructed is not of
product type we will use a criterion called property A. This
property, introduced by Krieger for non-singular transformations, is
defined directly for hyperfinite equivalence relations in this paper.

In this paper we extend K. Rubin's methods to prove the Gras conjecture
for abelian extensions of a given imaginary quadratic field $k$ and
prime numbers $p$ that divide the number of roots of unity in $k$.

Several measures for the density of sets of integers have been proposed,
such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and
investigate some of its properties.

We introduce semi-invariant Riemannian submersions from almost
Hermitian manifolds onto Riemannian manifolds. We give examples,
investigate the geometry of foliations that arise from the
definition of a Riemannian submersion, and find necessary sufficient
conditions for total manifold to be a locally product Riemannian
manifold. We also find necessary and sufficient conditions for a
semi-invariant submersion to be totally geodesic. Moreover, we
obtain a classification for semi-invariant submersions with totally
umbilical fibers and show that such submersions put some
restrictions on total manifolds.

In this paper we study several non-Riemannian quantities in Finsler
geometry. These non-Riemannian quantities play an important role in
understanding the geometric properties of Finsler metrics. In
particular, we study a new non-Riemannian quantity defined by the
S-curvature. We show some relationships among the flag curvature,
the S-curvature, and the new non-Riemannian quantity.

Müntz-Legendre
polynomials $L_n(\Lambda;x)$ associated with a
sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the
system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in
$L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s
are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$
zeros $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$.
First we prove the following global bound for the smallest zero,
$$
\exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.
$$
An important consequence is that if the associated Müntz space is
non-dense in $L_2[0,1]$, then
$$
\inf_{n}x_{n,n}\geq
\exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,
$$
so
the elements $L_n(\Lambda;x)$ have no zeros close to 0.

Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,
$$
\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n
(2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,
$$
where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$.

We give easy proofs that (a) the Continuum Hypothesis implies that if
the product of $X$ with every Lindelöf space is Lindelöf, then $X$ is
a $D$-space, and (b) Borel's Conjecture implies every Rothberger space
is Hurewicz.

It was claimed by Halmos in 1944 that if $G$ is a
Hausdorff locally compact topological abelian
group and if the character group of $G$ is torsion
free, then $G$ is divisible.
We prove that such a claim is false by
presenting a family of counterexamples.
While other counterexamples are known,
we also present a family of stronger counterexamples,
showing that even if one assumes that the character
group of $G$ is both torsion free and divisible,
it does not follow that $G$ is divisible.

By exploring the relations among functional equations, harmonic analysis and representation theory,
we give a unified and very accessible approach to solve three important functional equations -
the d'Alembert equation, the Wilson equation, and the d'Alembert long equation -
on compact groups.