Related papers: Spectral and Tiling properties of the Unit Cube

We study the stable cohomology groups of the mapping class groups of surfaces with twisted coefficients given by the $d^{th}$ exterior powers of the first rational homology of the unit tangent bundles of the surfaces…

We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of…

We extend the notion of a dyadic grid of cubes in Euclidean space to include infinite dyadic cubes. These `tops' of a dyadic grid form a tiling of Euclidean space which is subject to the constraints similar to those arising in tiling…

We study the existence of Gabor orthonormal bases with window the characteristic function of the set W=[0,a] U [b+a, b+1] of measure 1, with a, b>0. By the symmetries of the problem, we can restrict our attention to the case a<=1/2. We…

In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check.…

We consider incomplete tilings of the equilateral triangle of edge length n that is subdivided into n^2 regular equilateral smaller unit triangles. Pairs of the unit triangles that share a side may be converted into lozenges, leaving some…

We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{dx^2}+ix^3+i\alpha x$ for non-negative values of the parameter $\alpha$, by getting the exponential growth rate of $\|\Pi_n(\alpha)\|$, where $\Pi_n(\alpha)$ is…

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$…

Let $X = S \times E$ be the product of a K3 surface $S$ and an elliptic curve $E$. Reduced stable pair invariants of $X$ can be defined via (1) cutting down the reduced virtual class with incidence conditions or (2) the Behrend function…

Topological insulators (TIs) are renowned for their remarkable electronic properties: quantised bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations…

Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…

For a field $E$ of characteristic different from $2$ and cohomological $2$-dimension one, quadratic forms over the rational function field $E(X)$ are studied. A characterisation in terms of polynomials in $E[X]$ is obtained for having that…

We study the quantal energy spectrum of triangular billiards on a spherical surface. Group theory yields analytical results for tiling billiards while the generic case is treated numerically. We find that the statistical properties of the…

We show in a unified manner that the factorization method describes completely the $L^2$-eigenspaces associated to the discrete part of the spectrum of the twisted Laplacian on constant curvature Riemann surfaces. Subclasses of two variable…

We prove that if a tile in $\mathbb Z^d$ has prime size $p$, then it must be spectral. The proof is by contradiction, it is simply shown that the tiling complement of such a tile can not annihilate all $p$-subgroups. In addition, with a…

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that…

In this paper we study the category of graded modules for the current algebra associated to $\mathfrak{sl}_2$. The category enjoys many nice properties, including a tilting theory which was established in previous work of the authors. We…

We prove the every spectral set in $\mathbb{Z}_{p^2qr}$ tiles, where $p$, $q$ and $r$ are primes. Combining this with a recent result of Malikiosis we obtain that Fuglede's conjecture holds for $\mathbb{Z}_{p^2qr}$.

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for…

We prove that any incomplete system of complex exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-\pi,\pi)$ is a subset of some complete and minimal system of exponentials. In addition, we prove analogous statement for systems of reproducing…