Strong multiplicity one for Siegel cusp forms of degree two

We prove strong multiplicity one results for Siegel eigenforms of degree two for the symplectic group Sp 4 ⁡ ( ℤ ) {\operatorname{Sp}_{4}(\mathbb{Z})} .

A conjectural refinement of strong multiplicity one for GL(n)

Given a pair of distinct unitary cuspidal automorphic representations for GL([Formula: see text]) over a number field, let [Formula: see text] denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this paper, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin–Selberg products imply lower bounds on the size of [Formula: see text]. We also obtain further results for GL(3).

Analytic Tools

This chapter goes to the transcendental level, i.e., take an embedding ι‎ : E → ℂ, and extend the ground field to ℂ. The entirety of this chapter works over ℂ and therefore suppresses the subscript ℂ. It begins with the cuspidal parameters and the representation 𝔻λ‎ at infinity. Next, the chapter defines the square-integrable cohomology as well as the de Rham complex. Finally, cuspidal cohomology is addressed. Here, the chapter looks at the cohomological cuspidal spectrum and the consequence of multiplicity one and strong multiplicity one. It also shows the character of the component group I, before dropping the assumption that we are working over ℂ and go back to our coefficient system 𝓜̃λ‎,E defined over E.

Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph

For a given graph $G$ and an associated class of real symmetric matrices whose diagonal entries are governed by the adjacencies in $G$, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdière in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with $G$, denoted by $q(G)$. The graphs for which $q(G)$ is at least the number of vertices of $G$ less one are characterized.

The Random Bit Complexity of Mobile Robots Scattering

We consider the problem of scattering n robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that n log n random bits are necessary to scatter n robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in o(log log n) rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal (O(n log n) random bits are used) and time optimal (O(log log n) rounds are used). This improves the time complexity of previous results in the same setting by a log n factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes the time complexity gap with and without strong multiplicity detection (that is, O(1) time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach a constant value as desired otherwise).

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions. These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.