Let Γ be a co-compact Fuchsian group of isometries on the Poincaré disk D and ∆ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction f of ∆, equivariant by Γ with real eigenvalue λ = −s(1 − s), where s = 1 2 + it, admits an integral representation by a distribution D f,s (the Helgason distribution) which is equivariant by Γ and supported at infinity ∂D = S 1. The geodesic flo...

We investigate L(R) → L∞(Rn) dispersive estimates for the Schrödinger operator H = −∆ + V when there is an eigenvalue at zero energy in even dimensions n ≥ 6. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator Ft satisfying ‖Ft‖L1→L∞ . |t|2− n 2 for |t| > 1 such that ‖ePac − Ft‖L1→L∞ . |t| 1−n 2 , for |t| > 1. With stronger dec...

This paper gives an overview of the eigenvalue problems encountered in areas of data mining that are related to dimension reduction. Given some input high-dimensional data, the goal of dimension reduction is to map them to a lowdimensional space such that certain properties of the initial data are preserved. Optimizing the above properties among the reduced data can be typically posed as a trac...

We investigate L(R) → L∞(Rn) dispersive estimates for the Schrödinger operator H = −∆ + V when there is an eigenvalue at zero energy and n ≥ 5 is odd. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator Ft satisfying ‖Ft‖L1→L∞ . |t|2− n 2 for |t| > 1 such that ‖ePac − Ft‖L1→L∞ . |t| 1−n 2 , for |t| > 1. With stronger decay condi...

We show that the size of the Jordan blocks with eigenvalue one of the monodromy at infinity is estimated in terms of the weights of the cohomology of the total space and a general fiber. Let f : X → S be a morphism of complex algebraic varieties with relative dimension n. Assume S is a smooth curve. Let U be a dense open subvariety of S such that the H(Xs,Q) for s ∈ U form a local system (which...

How close is the Dirichlet-to-Neumann (DtN) map to square root of corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot techniques involved can be traced back newly rediscovered manuscript Hörmander from 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates spectral asymptotics...

‎In this paper‎, ‎we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated‎, ‎self-adjoint boundary conditions and we show that such SLP have finite spectrum‎. ‎Also for a given matrix eigenvalue problem $HX=lambda VX$‎, ‎where $H$ is a block tridiagonal matrix and $V$ is a block diagonal matrix‎, ‎we find a sixth order boundary value problem of Atkin...

Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let φ : M → N be a surjective harmonic morphism. We prove that if N admits a subharmonic function with finite Dirichlet integral which is not harmonic, and φ has finite energy, then φ is a constant map. Similarly, if f is a subharmonic function on N which is not harmonic and such that |df | is bounded, and ...

We calculate the eigenvalue ρ of the multiplication mapping R on the Cayley-Dickson algebra An. If the element in An is composed of a pair of alternative elements in An−1, half the eigenvectors of R in An are still eigenvectors in the subspace which is isomorphic to An−1. The invariant under the reciprocal transformation An × An ∋ (x, y) 7→ (−y, x) plays a fundamental role in simplifying the fu...

A new quaternion based method is presented which recovers albedo for shape reconstruction from three color images for three directions of white light source. In the conventional approaches, red, green and blue components are considered separately so there is a loss of information. The albedo maps recovered are not accurate enough to recover shape. Property of quaternion is that a color can then...