We introduce an analogy between the theory of autoionizing states as described by Fano and the tbeory of certain states ("pseudo-autoionizing states") of an irradiated atom. These "pseudo-autoionizing states" are used to obtain a number of qualitative results concerning resonant multiphoton ionization probabilities for several experimental arrangements. This method makes clear the importance of nonresonant processes in determining resonant multiphoton ionization line shapes.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. 1. Introduction. Let Mn be a compact oriented Riemannian n-manifold with Riemannian metric < , > and Riemannian connection V. We use APM for the space of p-forms, Tq,PM for the space of tensors of type (q,p), x(M) for the space of vector fields on M, d for exterior derivative, 8 for codifferential and A = d + 8d: APM--APM for the Laplace-Beltrami operator on p-forms. (As a general reference, see [4].) Let SpecPM={XERj There is 0 # co &APM for which Aco=Xco} be the spectrum of the Laplace-Beltrami operator. If X & SpecPM, then the multiplicity of X is the dimension of {co e APM IAo = Xco}. There has been a great deal of interest in the spectrum of a Riemannian manifold in terms of Minakshisundaram-Pleijel type expansions on both functions ([4]) and forms ([11]). The spectra of a Riemannian manifold has been used as the main tool in an analytic proof of the Atiyah-Singer Index theoreom ([1], [6], [7]). There is also interest, coming from physics, in expansions of functions in terms of eigenfunctions of the Laplacian ([2]) and expansions of forms in terms of eigenforms of the Laplacian ([3]). Unfortunately, SpecP(M) has actually been computed only in isolated cases. Speco(M) is known forflat tori, the Klein bottle, Sn, CPn and RP' as well as coverings, products and submersions with totally geodesic fibers [4] of the above. We know of no computations of SpecP (M) with p >0 except for the result of Lichnerowicz quoted in section 1, the fact that SpecP(M)=Spec'-P(M) and the fact, immediate from Hodge theory, that 0 & SpecP (M) if and only if 8, (M) 0 (and 0 has multiplicity /k (M)).In this note we announce that we have reduced the problem of computing SpecP (G) for an arbitrary compact semisimple Lie group (with the Killing form metric) to a completely algebraic problem. The solution to this algebraic problem is known in general and can be readily computed in any specific case although no general formula is known for an arbitrary group. This method also enables us to actually compute both the value of the eigenvalue, X, and its Manuscript

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Huntington's disease (HD) is an autosomal-dominant inherited neurodegenerative disorder that is caused by expansion of a CAG-repeat tract in the huntingtin gene and characterized by motor impairment, cognitive decline, and neuropsychiatric disturbances. Neuropathological studies show that disease progression follows a characteristic pattern of brain atrophy, beginning in the basal ganglia structures. The HD Regulatory Science Consortium (HD-RSC) brings together diverse stakeholders in the HD community—biopharmaceutical industry, academia, nonprofit, and patient advocacy organizations—to define and address regulatory needs to accelerate HD therapeutic development. Here, the Biomarker Working Group of the HD-RSC summarizes the cross-sectional evidence indicating that regional brain volumes, as measured by volumetric magnetic resonance imaging, are reduced in HD and are correlated with disease characteristics. We also evaluate the relationship between imaging measures and clinical change, their longitudinal change characteristics, and within-individual longitudinal associations of imaging with disease progression. This analysis will be valuable in assessing pharmacodynamics in clinical trials and supporting clinical outcome assessments to evaluate treatment effects on neurodegeneration.

Despite the heroic efforts of Laplace, Legendre, Maxwell, Hobson, and company, there are still new chapters to be written in the theory of spherical harmonics. This paper describes some of the special windfalls that result when one expands analytic functions. In technical language, we are concerned with Fourier series of analytic functions on SU(2).