Fock spaces corresponding to positive definite linear transformations

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt {\det A}} {\pi^n}e^{-{\rm Re}} dz$ is described in terms of the holomorphic--antiholomorphic decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal--Bargmann transform, which we also study in the Gaussian-measure setting

Direct Systems of Spherical Functions and Representations

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation theoretic construction $\phi (x) =$ where $e$ is a $K_\infty$--fixed unit vector for $\pi$. Specifically, we look at representations $\pi_\infty = \varinjlim \pi_n$ of $G_\infty$ where $\pi_n$ is $K_n$--spherical, so the spherical representations $\pi_n$ and the corresponding spherical functions $\phi_n$ are related by $\phi_n(x) = <e_n, \pi_n(x)e_n>$ where $e_n$ is a $K_n$--fixed unit vector for $\pi_n$, and we consider the possibility of constructing a $K_\infty$--spherical function $\phi_\infty = \lim \phi_n$. We settle that matter by proving the equivalence of condtions (i) $\{e_n\}$ converges to a nonzero $K_\infty$--fixed vector $e$, and (ii) $G_\infty/K_\infty$ has finite symmetric space rank (equivalently, it is the Grassmann manifold of $p$--planes in \F^\infty where $p < \infty$ and \F is $\R$, \C or \H). In that finite rank case we also prove the functional equation $\phi(x)\phi(y) = \lim_{n\to \infty} \int_{K_n}\phi(xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.Comment: 17 pages. New material added on the finite rank case

Psychiatric morbidity in epilepsy: a case controlled study of adults receiving disability benefits

To access publisher full text version of this article. Please click on the hyperlink in Additional Links fieldOBJECTIVE: To compare the prevalence of non-organic psychiatric disorders among disabled patients of normal intelligence with epilepsy with the prevalence of similar psychiatric disorders among age and sex matched disabled patients with other somatic diseases. METHODS: A case-control study was carried out in Iceland among people receiving disability benefits using information available at the State Social Security Institute. There were 344 patients with epilepsy in Iceland 16 to 66 years of age (inclusive) receiving disability benefits in 1995. By excluding mentally retarded patients, autistic patients, and patients with organic psychoses, 241 index cases with epilepsy qualified for the study. For each case two age and sex matched controls were selected from all patients receiving disability benefits who had cardiovascular diseases, respiratory diseases, or arthropathies. The same exclusion criteria were applied to the controls as the index cases. In both patient groups psychiatric diagnoses were classified into one of the four following categories: (1) psychotic illness; (2) neurotic illness or personality disorders; (3) alcohol or drug dependence or misuse; and (4) other mental disorders. RESULTS: Psychiatric diagnosis was present among 35% (85/241) of the cases compared with 30% (143/482) of the controls (p=0.15). There was a difference in the distribution of the two groups into different psychiatric categories (p=0.02). This was mainly due to an excess of men in the index group with psychosis, particularly schizophrenia or paranoid states. CONCLUSION: The results suggest that there is not a difference in the prevalence of non-organic psychiatric disorders among disabled patients of normal intelligence with epilepsy compared with patients with other disabling somatic diseases. However, the data indicate that when psychopathology is present disabled patients with epilepsy are more likely to have psychotic illness than the other disabled patients

Fast joint reconstruction of dynamic R2∗R_2^* and field maps in functional MRI.

Blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) is conventionally done by reconstructing T2 * -weighted images. However, since the images are unitless they are nonquantifiable in terms of important physiological parameters. An alternative approach is to reconstruct R2 * maps which are quantifiable and have comparable BOLD contrast as T2* -weighted images. However, conventional R2 * mapping involves long readouts and ignores relaxation during readout. Another problem with fMRI imaging is temporal drift/fluctuations in off-resonance. Conventionally, a field map is collected at the start of the fMRI study to correct for off-resonance, ignoring any temporal changes. Here, we propose a new fast regularized iterative algorithm that jointly reconstructs R2 * and field maps for all time frames in fMRI data. To accelerate the algorithm we linearize the MR signal model, enabling the use of fast regularized iterative reconstruction methods. The regularizer was designed to account for the different resolution properties of both R2 * and field maps and provide uniform spatial resolution. For fMRI data with the same temporal frame rate as data collected for T2 * -weighted imaging the resulting R2 * maps performed comparably to T2 * -weighted images in activation detection while also correcting for spatially global and local temporal changes in off-resonance.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86002/1/Fessler23.pd