International Journal of Mathematics

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group $C^{\ast }$-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce $(n,d)$-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for $(n,d)$-matrix Gabor frames.

We introduce a new class of entire functions ${ℰ}$ which consists of all $F_0\in {𝒪}(ℂ)$ for which there exists a sequence $(F_n)\in {𝒪}(ℂ)$ and a sequence $({\lambda }_n)\in ℂ$ satisfying $F_n(z)={\lambda }_{n+1}{e}^{F_{n+1}(z)}$ for all $n\ge 0$. This new class is closed under the composition and it is dense in the space of all nonvanishing entire functions. We prove that every closed set $V\subset ℂ$ containing the origin and at least one more point is the set of singular values of some locally univalent function in ${ℰ}$, hence, this new class has nontrivial intersection with both the Speiser class and the Eremenko–Lyubich class of entire functions. As a consequence, we provide a new proof of an old result by Heins which states that every closed set $V\subset ℂ$ is the set of singular values of some locally univalent entire function. The novelty of our construction is that these functions are obtained as a uniform limit of a sequence of entire functions, the process under which the set of singular values is not stable. Finally, we show that the class ${ℰ}$ contains functions with an empty Fatou set and also functions whose Fatou set is nonempty.

Let $G$ be a discrete group and ${𝒞}$ be an additive spherical $G$-fusion category. We prove that the state sum 3-dimensional HQFT derived from ${𝒞}$ is isomorphic to the surgery 3-dimensional HQFT derived from the $G$-center of ${𝒞}$.

For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G\subset \mathrm{\text{Aut}}(X,L)$ we define a $G$-invariant version of the $\delta$-threshold. We prove that for a Fano variety $(X,-K_X)$ and a connected subgroup $G\subset \mathrm{\text{Aut}}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.

We define the conformal change and elementary deformation in generalized complex geometry. We apply Swann’s twist construction to generalized (almost) complex and Hermitian structures obtained by these operations and establish conditions for the Courant integrability of the resulting twisted structures. We associate to any appropriate generalized Kähler manifold $(M,G,{𝒥})$ with a Hamiltonian Killing vector field a new generalized Kähler manifold, depending on the choice of a pair of non-vanishing functions and compatible twist data. We study this construction when $(M,G,{𝒥})$ is toric, with emphasis on the four-dimensional case, and we apply it to deformations of the standard flat Kähler metric on ${ℂ}^n$, the Fubini–Study metric on $ℂP^2$ and the admissible Kähler metrics on Hirzebruch surfaces. As a further application, we recover the K/K (Kähler/Kähler) correspondence, by specializing to ordinary Kähler manifolds.

A Kähler metric $g$ with integral Kähler form is said to be partially regular if the partial Bergman kernel associated to $mg$ is a positive constant for all integer $m$ sufficiently large. The aim of this paper is to prove that for all $n\ge 2$ there exists an $n$-dimensional complex manifold equipped with strictly partially regular and cscK metric $g$. Further, for $n\ge 3$, the (constant) scalar curvature of $g$ can be chosen to be zero, positive or negative.

Using a result of Longo and Xu, we show that the anomaly arising from a cyclic permutation orbifold of order 3 of a holomorphic conformal net ${𝒜}$ with central charge $c=8k$ depends on the “gravitational anomaly” $k(mod3)$. In particular, the conjecture that holomorphic permutation orbifolds are non-anomalous and therefore a stronger conjecture of Müger about braided crossed $S_n$-categories arising from permutation orbifolds of completely rational conformal nets are wrong. More generally, we show that cyclic permutations of order $n$ are non-anomalous if and only if $3\nmid n$ or $24|c$. We also show that all cyclic permutation gaugings of $\mathrm{\text{Rep}}({𝒜})$ arise from conformal nets.

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial $P_K(u)$ for each 2-bridge link $K$, and each zero of this polynomial gives a set of arc coloring vectors on the diagram of $K$ satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of $K$. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever