Related papers: A linear programming approach to Fuglede's conject…

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $\Omega$ is spectral if and only if it can tile the space by…

The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that…

This study explores the properties of the function which can tile the field $\mathbb{Q}_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling $\mathbb{Q}_p$ is by translation uniformly locally constancy.…

In relation to Fuglede's conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of $\mathbb{R}$ that are in duality. In the terminology…

We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point.…

We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with…

We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region {\Omega} for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth…

We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^n q$, where $p$, $q$ are distinct primes and $n\geq1$. This means that a subset $E$ of such a group $G$ tiles the group by translation…

The scope of this text is to study a process that induces another proof of the Spectral Embedding Theorem: that any densely defined symmetric operator can be extended by a multiplication operator through an embedding of the Hilbert space…

We describe computer programs accompanying our paper and show that running them suffices to verify Fuglede's conjecture in $\mathbb{Z}_2^5$ and $\mathbb{Z}_2^6$.

Recent methods developed by Tao \cite{tao}, Kolountzakis and Matolcsi \cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in $\RR^5$ Tao produced a spectral set which is not a tile,…

Given bounded selfadjoint operators $A$ and $B$ acting on a Hilbert space $\mathcal{H}$, consider the linear pencil $P(\lambda)=A+\lambda B$, $\lambda\in\mathbb{R}$. The set of parameters $\lambda$ such that $P(\lambda)$ is a positive…

In this note we give a counterexample to a conjecture proposed by Ciliberto about special linear systems of P^n through multiple base points.

The noncommutative spectral action extends our familiar notion of commutative spaces, using the data encoded in a spectral triple on an almost commutative space. Varying a rather simple action, one can derive all of the standard model of…

We study spectral theory for bounded Borel subsets of $\br$ and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator…

Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis…

We show that in a vector space over Z_3, the union of any four linear bases is an additive basis, thus proving the Additive Basis Conjecture for p=3, and providing an alternative proof of the weak 3-flow conjecture.

In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic…

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells the relation of range and variance of exponents i.e. elements…

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if $A$ generates a polynomially bounded $n$-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_k;…