The category $\mathbf {W}$ is comprised of archimedean $\ell $-groups $G$ with distinguished weak unit and unit-preserving $\ell $-group homomorphisms. For $G \in \mathbf {W}$ there is the canonical Yosida representation $G \leq D(\mathcal {Y}G)$ as extended-real valued functions, with the unit $e_G$ represented as the constant $1$ function. We define $a \in G$ to be kernel-maximal if the $\mathbf {W}$-kernel generated by $a$ in $G$ is all of $G$, and we define $a$ to be Yosida invertible if there is some $b \in G$ with $ab = e_G$ in the partial multiplication inherited from $D(\mathcal {Y}G)$. The main theorem is that $G$ is isomorphic to $C(\mathcal {L})$ for some (identifiable) locale $\mathcal {L}$ iff $G$ is divisible, uniformly complete, and every kernel-maximal element of $G$ is Yosida invertible.

For $p\in (0,1)$, let $Q_p$ space be the space of all analytic functions on the unit disk $\mathbb {D}$ such that $\vert f'(z) \vert ^2 (1-\vert z\vert ^2)^p\, dA(z)$ is a $p$-Carleson measure. We prove that Wolff's ideal theorem on $H^\infty {(\mathbb {D})}$ can be extended to the Banach algebra $H^{\infty }(\mathbb {D})\cap Q_{p}$, and also to the multiplier algebra on $Q_p$ spaces.

We investigate the dynamics and stability behavior of the solutions of the second order nonlinear difference equation $$ x_{n+1}=ax_{n}+\frac {bx_{n}+cx_{n-1}}{d+ex_{n}x_{n-1}} $$ where the parameters $a$, $b$, $c$, $d$ and $e$ are positive real numbers and the initial conditions $x_{-1}$ and $x_{0}$ are positive real numbers.

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb {Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over $\mathbb {Q}$. In particular, we obtain infinitely many new examples of the phenomenon studied by Jones. While doing this, we also find, up to three possible exceptions, all integers $a$ such that all iterates of the quadratic polynomial ${(x+a)^2-a-1}$ are irreducible over $\mathbb {Q}$, which answers a question posed in by Ayad and McQuillan, except for three values of $a$. Finally, we make a conjecture that suggests a necessary and sufficient condition for the stability of any monic, post-critically finite quadratic polynomial over any field of characteristic $\neq 2$.

We derive several more accurate operator Bellman and Aczel inequalities. Among other inequalities, it is shown that if $\Phi :\mathbb {B}( \mathscr {H} )\to \mathbb {B}( \mathscr {H})$ is a unital positive linear map, $A,B\in \mathbb {B}( \mathscr {H})$ are two contraction operators, and we take $0\le \lambda ,v\le 1$ and $p>1$, then $$ \Phi ( {{( I-A )}^{\frac {1}{p}}}{{\nabla }_{\lambda }}{{( I-B )}^{\frac {1}{p}}} ) \le {{( \Phi ( I-A{{\nabla }_{\lambda }}B ) )}^{\frac {1}{p}},} $$ which nicely improves the operator Bellman inequality.

For a continuum $X$, let $C_{n}(X)$ be the hyperspace of nonempty closed subsets of $X$ with at most $n$ components. In this paper we prove that if $X $ is a fan and $n\neq 2$, then $C_{n}(X)$ is a cone if and only if $X$ is a cone.

We study a class of nonlinear fractional Schrodinger equations: \[ (-\Delta )^{s}u+V(x)u=f(x,u) \text { in } \mathbb{R} ^{N}, \] where $s\in (0,1)$, $N>2s$, $(-\Delta )^{s}$ stands for the fractional Laplacian. By using a variational approach, we establish the existence of at least one nontrivial solution for the above equation with a general potential $V(x)$ which is allowed to be sign-changing and a sublinear nonlinearity $f(x,u)$. Moreover, by using variational methods and the Moser iteration technique, we prove the existence of infinitely many solutions with $V(x)$ is a nonnegative potential and the nonlinearity $f(x,u)$ is locally sublinear with respect to $u$.

We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders. Our results are highly dependent on the computer software package Magma, although alternatives would probably work fine. A computer approach is required because our results require a listing of all maximal orders; the sheer number of maximal orders grows exponentially quickly and determining them all is not obtainable in polynomial time. When discriminants are large enough, the determination of a single maximal order is also exponentially hard. We indicate the limits as these problems arise.

We construct two infinite families of curves with high rank. The first is defined by the equation $y^2=x(x-a^2)(x-b^2)+a^2b^2$, where $a,b \in \mathbb {Q}(t)$. The second family arises from a system of rational cuboids, i.e., a rectangular box for which the lengths of the edges and face diagonals are all rational. We create a second family with defining equation $y^2=(x-a^2)(x-b^2)(x-c^2)+a^2b^2c^2$, where $a,b,c \in \mathbb {Q}(t)$ are the edge lengths of a rational cuboid. We show that the rank of both families is $\geq 5$ over $\mathbb Q(t)$. We conclude by studying corresponding families of curves defined by other known solutions to the rational cuboid problem, and find some specific examples of curves from these various families with high rank.

Knebusch and Kaiser introduced the notion of Kronecker function ring of a ring extension with respect to a star operation to generalize the classical notion of Kronecker function ring. Let $\star $ be a star operation on the extension $R\subseteq S$. Let $Kr ( \star)$ be the set of all quotients ${f}/{g}\in S(X)$ such that $f, g\in S[X]$, $c_{S}(g) = S$ and $(c_{R}(f)H)^{\star } \subseteq (c_{R}(g)H)^{\star }$ for some finitely generated $S$-regular $R$-submodule $H$ of $S$, where for each ring $A$ such that $R\subseteq A \subseteq S$, $c_{A}(g)$ denotes the content of $g\in S[X]$. The ring $Kr (\star )$ is a Kronecker subring of $S(X)$ by a theorem of Knebusch and Kaiser. We study properties of the ring $Kr (\star )$ in pullback diagrams.

The Jacobi--Trudi identity associates a symmetric function to any integer sequence. Let $\Gamma _{(t|X)}$ be the vertex operator defined by $\Gamma _{(t|X)} s_\alpha =\sum _{n \in \mathbb{Z} } s_{(n,\alpha )} [X] t^n$. We provide a combinatorial proof for the identity $\Gamma _{(t|X)} s_\alpha = \sigma [tX] s_{\alpha }[x-1/t] $ due to Thibon et al. We include an overview of all the combinatorial ideas behind this beautiful identity, including a combinatorial description for the expansion of $s_{(n,\alpha )} [X] $ in the Schur basis, for any integer value of $n$.

Hassett spaces are moduli spaces of weighted stable pointed curves. We consider such spaces of curves of genus $0$ with weights all $\frac {1}{2}$. These spaces are interesting, since they are isomorphic to $\overline{M}_{0,n}$ but have different universal families and different intersection theory. We develop a closed formula for intersections of $\psi $-classes on such spaces. In our main result, we encode the formula for top intersections in a generating function obtained by applying a differential operator to the Witten potential.

By using the idea of $q$-calculus, we generalize the conic domain and define various subclasses of analytic functions which map the open unit disk $$ \mathbb {U}= \{ z:z\in \mathbb {C} \text { and } \lvert z \rvert \lt 1 \} $$ onto this generalized conic domain. We study here some conspicuous results such as: coefficients estimates, sufficient condition and convolution properties for these newly defined classes. We consider various corollaries and consequences of our main results. Most of our results are connected with earlier works related to this field.

We prove existence of propagators for a time-dependent Schrodinger equation with a new class of softened Coulomb potentials, which we allow to be time dependent, in the context of time-dependent density functional theory. We compute explicitly the Fourier transform of these new potentials, and provide an alternative proof for the Fourier transform of the Coulomb potential using distribution theory. Finally, we show the new potentials are dilatation analytic, and discuss various properties and applications related to the essential spectrum of the Hamiltonian associated to these potentials.

We prove that there are no Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb {C}^{m+2})$ if the normal Jacobi operator is $\mathfrak {D}$-parallel or $\mathfrak {D}^\perp $-parallel with respect to the generalized Tanaka--Webster connection and the Hopf principal curvature is invariant along the Reeb flow.

For the two-dimensional steady zero-pressure adiabatic flow, the Riemann problem with delta initial data is investigated and the global existence of generalized solution is established in four cases. Particularly, in solutions, a special type of nonclassical wave called the delta contact discontinuity with Dirac delta functions developing in both state variables is found. Furthermore, we show that the constructed generalized solutions are stable by the perturbation of initial data.

We establish Minkowski and Brunn-Minkowski inequalities for differences of quermassintegrals and dual quermassintegrals of convex and star bodies, respectively.

We present a family of nonstationary interpolatory subdivision schemes which reproduces high-order exponential polynomials. First, by extending the classical $\fbox {D-D}$ interpolatory schemes, we present the explicit expression of the symbols that identify a family of the nonstationary interpolatory subdivision schemes. These schemes can allow reproduction of more exponential polynomials, and represent exactly circular shapes, parts of conics which are important analytical shapes in geometric modeling. Furthermore, a rigorous analysis regarding the smoothness of the new nonstationary interpolatory schemes is provided. Next, based on the recursive formulas of the symbols, a repeated local operation is proposed for rapidly computing new points from the old points. Finally, two examples are presented to illustrate the performance of the new schemes.