Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

We consider the family of Hecke triangle groups $$ \Gamma _{w} = \langle S, T_w\rangle $$ Γ w = ⟨ S , T w ⟩ generated by the Möbius transformations $$ S : z\mapsto -1/z $$ S : z ↦ - 1 / z and $$ T_{w} : z \mapsto z+w $$ T w : z ↦ z + w with $$ w > 2.$$ w > 2 . In this case, the corresponding hyperbolic quotient $$ \Gamma _{w}\backslash {\mathbb {H}}^2 $$ Γ w \ H 2 is an infinite-area orbifold. Moreover, the limit set of $$ \Gamma _w $$ Γ w is a Cantor-like fractal whose Hausdorff dimension we denote by $$ \delta (w). $$ δ ( w ) . The first result of this paper asserts that the twisted Selberg zeta function $$ Z_{\Gamma _{ w}}(s, \rho ) $$ Z Γ w ( s , ρ ) , where $$ \rho : \Gamma _{w} \rightarrow \mathrm {U}(V) $$ ρ : Γ w → U ( V ) is an arbitrary finite-dimensional unitary representation, can be realized as the Fredholm determinant of a Mayer-type transfer operator. This result has a number of applications. We study the distribution of the zeros in the half-plane $$\mathrm {Re}(s) > \frac{1}{2}$$ Re ( s ) > 1 2 of the Selberg zeta function of a special family of subgroups $$( \Gamma _w^N )_{N\in {\mathbb {N}}} $$ ( Γ w N ) N ∈ N of $$\Gamma _w$$ Γ w . These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $$X_w^N = \Gamma _w^N \backslash {\mathbb {H}}^2$$ X w N = Γ w N \ H 2 . We show that the classical Selberg zeta function $$Z_{\Gamma _w}(s)$$ Z Γ w ( s ) can be approximated by determinants of finite matrices whose entries are explicitly given in terms of the Riemann zeta function. Moreover, we prove an asymptotic expansion for the Hausdorff dimension $$\delta (w)$$ δ ( w ) as $$w\rightarrow \infty $$ w → ∞ .

Discrete Quantitative Nodal Theorem

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. A special case of this result will assert that if the second and third eigenvalues of the Laplacian are at least $\varepsilon$ apart, then the subgraphs induced by the positive and negative supports of the eigenvector belonging to $\lambda_2$ are not only connected, but edge-expanders (in a weighted sense, with expansion depending on $\varepsilon$).

Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian

We show a Gromov-Hausdorff approximation to the product of the standard spheres for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.

On the multiplicity of the second eigenvalue of the Laplacian in non simply connected domains – with some numerics –

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

Analogue of Brauer’s conjecture for the signless Laplacian of cographs

In this paper, we consider the class of cographs and its subclasses, namely, threshold graphs and anti-regular graphs. In 2011 H. Bai confirmed the Grone – Merris conjecture about the sum of the first k eigenvalues of the Laplacian of an arbitrary graph. As a variation of the Grone – Merris conjecture, A. Brouwer put forward his conjecture about an upper bound for this sum. Although the latter conjecture was confirmed for many graph classes, however, it remains open. By analogy to Brouwer’s conjecture, in 2013 F. Ashraf et al. put forward a conjecture about the sum of k eigenvalues of the signless Laplacian, which was also confirmed for some graph classes but remains open. In this paper, an analogue of the Brouwer’s conjecture is confirmed for the graph classes under our consideration for the eigenvalues of their signless Laplacian for some natural k which does not exceed the order of the considered graphs.

Eigenvalues of the Laplacian Operator and Neighborhood Enlargement for Compact Manifolds

In this investigation and under the conception of measure concentration phenomenon we found that the enlargement of the neighborhood for an n – dimensional compact Riemannian manifolds (M,g) relative to the eigenvalues λ of the Laplace operator ∆ on (M,g). And we found that r~1/√λ. تناول هذا البحث تجسيم الجوار لمتعدد الطيات المتراص في البعد n وذلك باستخدام مفهوم تركيز الحجم. كما وجدنا أن نصف القطر r لتجسيم الجوار لمتعدد الطيات يرتبط مع القيم الذاتية λ لمؤثر لابلاس Δ على متعدد الطيات. الكلمات المفتاحية: نصف قطر تجسيم الجوار، متباينات متساوي المقاييس، تركيز الحجم، مؤثر لابلاس، القيم الذاتية لمؤثر لابلاس.