In this paper we study domains, Scott
domains, and the existence of measurements. We
use a space created by D.~K. Burke to show that
there is a Scott domain $P$ for which $\max(P)$ is
a $G_\delta$-subset of $P$ and yet no measurement
$\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also
correct a mistake in the literature asserting that
$[0, \omega_1)$ is a space of this type. We show
that if $P$ is a Scott domain and $X \subseteq
\max(P)$ is a $G_\delta$-subset of $P$, then $X$
has a $G_\delta$-diagonal and is weakly
developable. We show that if $X \subseteq
\max(P)$ is a $G_\delta$-subset of $P$, where
$P$ is a domain but perhaps not a Scott domain,
then $X$ is domain-representable,
first-countable, and is the union of dense,
completely metrizable subspaces. We also
show that there is a domain $P$ such that
$\max(P)$ is the usual space of countable
ordinals and is a $G_\delta$-subset of $P$ in
the Scott topology. Finally we show that the
kernel of a measurement on a Scott domain can
consistently be a normal, separable,
non-metrizable Moore space.

As an analog of a well-known theorem on the bilinear
fractional integral on $\mathbb{R}^{n}$ by Kenig and Stein,
we establish the similar boundedness
property for a bilinear fractional integral on a compact Lie group. Our
result is also a generalization of our recent theorem
about the
bilinear fractional integral on torus.

We consider a class of
strongly $q$-log-convex polynomials based on a triangular recurrence
relation with linear coefficients, and we show that the Bell
polynomials, the Bessel polynomials, the Ramanujan polynomials and
the Dowling polynomials are strongly $q$-log-convex. We also prove
that the Bessel transformation preserves log-convexity.

We establish the existence of power series in $\mathbb{C}^N$ with the property
that the subsequences of the sequence of partial sums uniformly
approach any holomorphic function on any well chosen compact subset
outside the set of convergence of the series. We also show that, in a
certain sense, most series enjoy this property.

Let $RG$ denote the group ring of the group $G$ over
the ring $R$. Using an isomorphism between $RG$ and a
certain ring of $n \times n$ matrices in conjunction with other
techniques, the structure of the unit group of the group algebra
of the dihedral group of order $8$ over any
finite field of chracteristic $2$ is determined in
terms of split extensions of cyclic groups.

A continuum is said to be Suslinian if it does not
contain uncountably many
mutually exclusive non-degenerate subcontinua. Fitzpatrick and
Lelek have shown that a metric Suslinian continuum $X$ has the
property that the set of points at which $X$ is connected im
kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua
and obtain a number of corollaries. In particular, we prove that a homogeneous,
non-degenerate, Suslinian continuum is a simple closed curve and that each separable,
non-degenerate, homogenous, Suslinian continuum is metrizable.

This paper deals with the analytic solvability of a special class of
complex vector fields defined on the real plane, where they are
tangent to
a closed real curve, while off the real curve, they are elliptic.

A classical question for a Toeplitz matrix with given symbol is how to
compute asymptotics for the determinants of its reductions to finite
rank. One can also consider how those asymptotics are affected when
shifting an initial set of rows and columns (or, equivalently,
asymptotics of their minors). Bump and Diaconis
obtained a formula for such shifts involving Laguerre polynomials and
sums over symmetric groups. They also showed how the Heine identity
extends for such minors, which makes this question relevant to Random
Matrix Theory. Independently, Tracy and Widom
used the Wiener-Hopf factorization to
express those shifts in terms of products of infinite matrices. We
show directly why those two expressions are equal and uncover some
structure in both formulas that was unknown to their authors. We
introduce a mysterious differential operator on symmetric functions
that is very similar to vertex operators. We show that the
Bump-Diaconis-Tracy-Widom identity is a differentiated version of the
classical Jacobi-Trudi identity.

It is shown that it follows from PFA
that there is no
compact scattered space of height greater than $\omega$
in which the sequential order and the scattering heights coincide.

Let $L$ be a finite distributive lattice. Let
$\operatorname{Sub}_0(L)$ be the lattice
$$
\{S\mid S\text{ is a sublattice of }L\}\cup\{\emptyset\}
$$
and let $\ell_*[\operatorname{Sub}_0(L)]$ be the length of the shortest maximal chain in $\operatorname{Sub}_0(L)$. It is proved that if $K$ and $L$ are non-trivial finite distributive lattices, then
$$
\ell_*[\operatorname{Sub}_0(K\times L)]=\ell_*[\operatorname{Sub}_0(K)]+\ell_*[\operatorname{Sub}_0(L)].
$$
A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.

We show that closed $\widetilde{\mathbb{SL}} \times \mathbb{E}^n$-manifolds
are topologically rigid if $n\geq 2$, and are rigid up to
$s$-cobordism, if $n=1$.

Let $T_{n}$ denote the $n$-th
Chebyshev polynomial of the first kind,
and let $U_{n}$ denote the $n$-th
Chebyshev polynomial of the second kind.
We give an explicit formula for the resultant
$\operatorname{res}( T_{m}, T_{n} )$.
Similarly, we give a formula for
$\operatorname{res}( U_{m}, U_{n} )$.

We give a new factorisation of the classical Dynkin operator,
an element of the integral group ring of the symmetric group that
facilitates projections of tensor powers onto Lie powers.
As an application we show that the iterated Lie power $L_2(L_n)$ is
a module direct summand of the Lie power $L_{2n}$ whenever the
characteristic of the ground field does not divide $n$. An explicit
projection of the latter onto the former is exhibited in this case.

Let $X$ be a separable non-reflexive Banach space. We show that there
is no Borel class which contains the set of norm-attaining functionals
for every strictly convex renorming of $X$.

We give some extension to theorems of Jiménez López and Soler López concerning the topological characterization for limit sets of continuous flows on closed orientable surfaces.

In this paper, we apply the saddle-point method in conjunction with
the theory of the Nörlund-Rice integrals to derive precise
asymptotic formula for the generalized Li coefficients established
by Omar and Mazhouda.
Actually, for any function $F$ in the Selberg class
$\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have
$$
\lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),
$$
with
$$
c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda
Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},
$$
where $\gamma$ is the Euler's constant and the notation is as below.

Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que
$p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q}
(\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le
$2$-groupe de classes de $k$. A. Fröhlich a
démontré que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article,
nous donnons une extension de ce résultat, en démontrant que le
rang de $\operatorname{Cl}_2(k)$ est toujours supérieur ou égal à $2$. Nous
démontrons aussi, que la valeur $2$ est optimale pour une famille
infinie de corps $k$.

We shall present examples of Schauder bases in the preduals to the
hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$,
$\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite
(respectively, purely infinite) setting, these systems form Schauder bases
in any associated separable symmetric space of measurable operators
(respectively, in any non-commutative $L^p$-space).

Two theorems regarding the
asymptotic behavior of evolution families are established in
terms of the solutions of a certain Lyapunov operator equation.

Klep and Velu\v{s}\v{c}ek generalized the Krull--Baer theorem for
higher level preorderings to the non-commutative setting. A $n$-real valuation
$v$ on a skew field $D$ induces a group homomorphism $\overline{v}$. A section
of $\overline{v}$ is a crucial ingredient of the construction of a complete
preordering on the base field $D$ such that its projection on the residue skew
field $k_v$ equals the given level $1$ ordering on $k_v$. In the article we give
a proof of the existence of the section of $\overline{v}$, which was left as an
open problem by Klep and Velu\v{s}\v{c}ek, and thus
complete the generalization of the Krull--Baer theorem for preorderings.