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Although the head is more closely represented as an ellipsoid than a sphere, calculation in ellipsoidal coordinates is difficult. This paper presents a four shell ellipsoidal model, employing multipole expansion in ellipsoidal... more
Although the head is more closely represented as an ellipsoid than a sphere, calculation in ellipsoidal coordinates is difficult. This paper presents a four shell ellipsoidal model, employing multipole expansion in ellipsoidal coordinates, for EEG, MEG, and evoked potential applications. Computational detail and insight into efficient calculation of the Lamé functions of the first and second kind are provided to demonstrate feasibilty. The Lamé function of the second kind, derived from the Lamé function of the first kind, can be computed at higher degrees by means of partial fraction expansion.
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It is shown that every regular Krein-Feller eigenvalue problem can be transformed to a semidefinite Sturm-Liouville problem introduced by Atkinson. This makes it possible to transfer results between the corresponding theories. In... more
It is shown that every regular Krein-Feller eigenvalue problem can be transformed to a semidefinite Sturm-Liouville problem introduced by Atkinson. This makes it possible to transfer results between the corresponding theories. In particular, Prüfer angle methods become available for Krein-Feller problems.
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A four-shell ellipsoidal model had been developed for EEG and MRI applications, using a multipole expansion in ellipsoidal coordinates. The model provides accurate distribution of potentials and surface current density generated by a... more
A four-shell ellipsoidal model had been developed for EEG and MRI applications, using a multipole expansion in ellipsoidal coordinates. The model provides accurate distribution of potentials and surface current density generated by a dipole placed in the inner most ellipsoid. The dipole can be anywhere in any direction in the ellipsoid. Surface Laplacian data obtained from EEG measurements are fit to the surface current density from each of the models in a least squares sense for comparisons of the resulting dipole locations and orientations. The dipole location estimated from somatosensory evoked potentials is compared to the location of activation in functional MRI experiments using somatosensory stimulation
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ABSTRACT In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (X1,X2,…,Xn)(X1,X2,…,Xn) without the assumption of continuity and symmetry for each component... more
ABSTRACT In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (X1,X2,…,Xn)(X1,X2,…,Xn) without the assumption of continuity and symmetry for each component XiXi:P(|1n∑i=1nXi|≤|∑i=1nαiXi|)≥12 , where allαi≥0 are special weights with∑i=1nαi=1.
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It is well known that i.i.d. (independent and identically distributed) normal random variables are transformed into i.i.d. normal random variables by any orthogonal transformation. Less well known are nonlinear transformations with the... more
It is well known that i.i.d. (independent and identically distributed) normal random variables are transformed into i.i.d. normal random variables by any orthogonal transformation. Less well known are nonlinear transformations with the above-mentioned property. In this work we present nonlinear transformations preserving normality, which are more general than the existing ones in the literature.
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ABSTRACT It is shown that the main theorem of Arslan’s paper (Theorem 2, 2011), as stated, is incorrect. Under additional conditions, we present a short proof of the corrected version of the theorem. We also give a proof of a theorem of... more
ABSTRACT It is shown that the main theorem of Arslan’s paper (Theorem 2, 2011), as stated, is incorrect. Under additional conditions, we present a short proof of the corrected version of the theorem. We also give a proof of a theorem of Rao and Shanbhag (1991) [2], employed by Arslan, without the use of the Kolmogorov Consistency Theorem.