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 study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.

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.

This note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps.

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.