The Wave Trace for Riemann Surfaces

We present a wave group version of the Selberg trace formula for an arbitrary surface of finite geometry. As an application we give a new lower bound on the number of resonances for hyperbolic surfaces. Motivated by recent results we formulate a conjecture on a lower bound for the counting function of resonances in a strip. Submitted: July 1998, revised: March 1999.     

Spectral multiplicity and splitting results for a class of qualitative matrices

This paper presents some results concerning the location and multiplicity of eigenvalues of sign symmetric matrices whose associated graphs are trees. In particular it extends previous spectral multiplicity and splitting results proved by others.     

Trace Formulas for Some Operators Related To Quadrature Domains in Riemann Surfaces

This note studies the trace formula of a class of pure operators A with finite rank self-commutators satisfying the condition that there is a finite dimensional subspace containing the image of the self-commutator and invariant with respect to A ^*. Besides, in this class, the spectrum of the operator A is covered by the projection of a union of quadrature domains in some Riemann surfaces.     

三对角与五对角Toeplitz矩阵求逆的算法

提出了一种求三对角与五对角Toeplitz矩阵逆的快速算法,其思想为先将Toeplitz矩阵扩展为循环矩阵,再快速求循环矩阵的逆,进而运用恰当矩阵分块求原Toeplitz矩阵的逆的算法.算法稳定性较好且复杂度较低.数值例子显示了算法的有效性和稳定性,并指出了算法的适用范围.     

Connecting optimization with spectral analysis of tri-diagonal matrices

Mathematical Programming - We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest...     

Trace formula in noncommutative geometry and the zeros of the Riemann zeta function

We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of Adele classes. This reduces the Riemann hypothesis to the validity of the trace formula and eliminates the parameter of our previous approach.     

Trace class multipliers and spectral variation of normal matrices

In this article we show how to estimate the trace multiplier norm of a rank 2 matrix. As an application, an alternative proof of a theorem of Holbrook et al. (Maximal spectral distance, Linear Algebra Appl., 249 (1996) 197–205) on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given.     

Remarks on a formula of Riemann for his zeta-function

Let $\text{h(s) = π}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext><mtext>s</mtext>}}\text{Γ(}\text{1}\text{2}\text{s) ζ(s)}$. A formula of Riemann [1; 2, 42.10] is $\text{h(s) = π}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext><mtext>s</mtext>}}\text{Γ(}\text{1}\text{2}\text{s)f}{}_1\text{(s) + π}{}^{\mathrm{<mtext>(?1\; +\; s</mtext><mtext>2</mtext>}}\text{Γ[(}\text{(1 ? s)}\text{2}\text{]f}{}_2\text{(s)}$, (0) where f₁ and f₂ are symmetrically related entire functions such that the first term on the right is close to h(s) in most of the right half-plane and the second term plays a similar role in the left half-plane. Here it will be shown that there is a one (complex) parameter set of such formulas all of which result from a simple decomposition of the path of integration of the Mellin transform of h(s) into two disjoint parts.     

Abel--Jacobi Mapping for Real Hyperelliptic Riemann Surfaces

We study the degrees of the Abel--Jacobi mapping on hyperelliptic Riemann surfaces of arbitrary genus and the restrictions of the corresponding mappings to the symmetric powers of the real locus of the given Riemann surface.     

Gromov Hyperbolicity of Riemann Surfaces

We study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its ＂building block components＂. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.     

Energy and Foliations on Riemann Surfaces

We define the energy of foliations on Riemann surfaces. We prove that meromorphic vector fields are critical points and we compute their energies using the Green’s function. We then generalize the results to principal circle bundles over Riemann surfaces.Mathematics Subject Classification (2000): 53C12, 53C15.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

Spectral analysis and the Riemann hypothesis

The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of the Dedekind zeta function of an algebraic number field K of degree n in an automorphic setting.

If K is a complex quadratic field, the torical forms are the functions defined on the modular surface X, such that the sum of this function over the “Gauss set” of K is zero, and Eisenstein series provide such torical forms.

In the case of a general number field, one can associate to K a maximal torus T of the general linear group G. The torical forms are the functions defined on the modular variety X associated to G, such that the integral over the subvariety induced by T is zero. Alternately, the torical forms are the functions which are orthogonal to orbital series on X.

We show here that the Riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms, constructed from Eisenstein series, the torical wave packets. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of critical zeros of the Dedekind zeta function of K.