Related papers: Any three eigenvalues do not determine a triangle

We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in \cite{MS3}.

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if…

We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of…

Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the…

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of…

Gromov used convex integration to prove that any closed orientable three-manifold equipped with a volume form admits three divergence-free vector fields which are linearly independent at every point. We provide an alternative proof of this…

We consider a $3$-dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is…

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the…

A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a…

In the paper we study the discrete spectrum of a pair of quantum two-dimensional waveguides having common boundary in which a window of finite length is cut out. We study the phenomenon of new eigenvalues emerging from the threshold of the…

We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…

Let $M$ be a compact 3-manifold with a triangulation $\tau$. We give an inequality relating the Euler characteristic of a surface $F$ normally embedded in $M$ with the number of normal quadrilaterals in $F$. This gives a relation between a…

To our knowledge there are no complete results expressed in terms of eigenfunctions (even not strictly proved mathematically) related to the system of three or more charged quantum particles. For the system of the three such identical…

Bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$ are investigated. We impose the Neumann boundary condition on the two concentric windows of the radii $a$ and $ b$ located…

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open sets of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it…

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by…

We analyze the behavior of the eigenvalues of the following non local mixed problem $\left\{ \begin{array}{rcll} (-\Delta)^{s} u &=& \lambda_1(D) \ u &\inn\Omega,\\ u&=&0&\inn D,\\ \mathcal{N}_{s}u&=&0&\inn N. \end{array}\right $ Our goal…

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any dimension n, but should be easy to demonstrate…

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These…

We derive eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with block-diagonal Schur complement matrices. This analysis applies to an arbitrary number of blocks and accounts for the case where…