Estimates from Below for the Spectral Function and for the Remainder in Local Weyl’s Law

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’s law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle. Our results develop and extend the unpublished thesis of A. Karnaukh [Ka]. The first author was supported by NSERC, FQRNT, Alfred P. Sloan Foundation Fellowship and Dawson Fellowship. The second author was supported by NSERC and FQRNT. Received: April 2006 Revision: October 2006 Accepted: October 2006     

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.     

The Nature of the vibronic intensity borrowing mechanism in the 1B3u(n)1Ag(X) transition of pyrazine

The transition intensities of the S₀S₁ gas phase absorption spectra of pyrazine-d₀ and -d₄ were carefully measured. A discussion of the various aspects of vibronic intensity borrowing for this unique case is presented using our new experimental results.     

Scalar Curvature and Q-Curvature of Random Metrics

We define a family of probability measures on the set of Riemannian metrics lying in a fixed conformal class, induced by Gaussian probability measures on the (logarithms of) conformal factors. We control the smoothness of the resulting metric by adjusting the decay rate of the variance of the random Fourier coefficients of the conformal factor. On a compact surface, we evaluate the probability of the set of metrics with non-vanishing Gauss curvature, lying in a fixed conformal class. On higher-dimensional manifolds, we estimate the probability of the set of metrics with non-vanishing scalar curvature (or Q-curvature), lying in a fixed conformal class.     

Spectra of elements in the group ring of SU(2)

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S² as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. Received June 1, 1998 / final version received September 8, 1998     

Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

Given a one-parameter familyf (x) of maps of the interval [0, 1], we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)f (x)=f(x) where 0<4 andf(x) is a functionC ³-near the quadratic mapx(1–x), and ii)f (x)=f(x) (mod 1) wheref isC ³,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].     

ISO 17025 quality system in a university environment

The experience of working under an ISO 17025 compliant quality system in a university environment is described in the example of the Testing Centre of University of Tartu. University-specific features of the quality system include that the Testing Centre acts as the “quality system server” for other units of the university and graduate students participate in the work carried out under the quality system. The benefits for the university and the graduate students are discussed. It is concluded that implementation of an ISO 17025 compatible quality system in a university is fully accomplishable and gives significant added value to the university by helping to destroy the ivory tower perception about universities and by introducing real-world flavour into the studies and broaden the minds (quality awareness) of the students.     

Linear Viscoelasticity of Liquid-Crystalline Polymers

A linear (small-amplitude) periodic shear deformation of anisotropic viscoelastic liquids obeying the Akay–Leslie rheological model is considered. The frequency dependences of the real and imaginary components of the complex shear modulus and complex normal-stress coefficient are determined. A comparison between calculation results and test data on the shear flow of poly(-benzylglutamate) in m-cresol is carried out. It is stated that, if the material is characterized by some initial orientation, both components of the complex shear modulus contain a multiplier which depends on the degree of the initial orientation and increases the values of the components compared with those for an initially isotropic material. The model predicts that, in a periodic shear flow, the components of shear and normal stresses are constant and, like the components of shear modulus, are independent of deformation frequency. If the parameter d ₀ of the Akay–Leslie model is equal to zero, the values of its other parameters can be determined from experimental results on periodic shear flow.     

High Energy Limits of Laplace-Type and Dirac-Type EigenFunctions and Frame Flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically. The first author was supported by NSERC, FQRNT and Dawson fellowship.     

On the Critical Line of Convex Co-Compact Hyperbolic Surfaces

Let ?? be a convex co-compact Fuchsian group. We formulate a conjecture on the critical line, i.e. what is the largest half-plane with finitely many resonances for the Laplace operator on the infinite-area hyperbolic surface ${X = \Gamma \backslash \mathbb{H}^2}$ . An upper bound depending on the dimension ?? of the limit set is proved which is in favor of the conjecture for small values of ?? and in the case when ???> 1/2 and ?? is a subgroup of an arithmetic group. New omega lower bounds for the error term in the hyperbolic lattice point counting problem are derived.