950,772 research outputs found
Bicomplex quantum mechanics: I. The generalized Schr\"odinger equation
We introduce the set of bicomplex numbers which is a commutative
ring with zero divisors defined by where
$\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\
\bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates
and the moduli associated with the bicomplex numbers. Then we study the
bicomplex Schr\"odinger equation and found the continuity equations. The
discrete symmetries of the system of equations describing the bicomplex
Schr\"odinger equation are obtained. Finally, we study the bicomplex Born
formulas under the discrete symetries. We obtain the standard Born's formula
for the class of bicomplex wave functions having a null hyperbolic angle
Multidimensional Dynamical Systems Accepting the Normal Shift
The dynamical systems of the form \ddot\bold r=\bold F (\bold r,\dot\bold
r) in accepting the normal shift are considered. The concept of
weak normality for them is introduced. The partial differential equations for
the force field \bold F(\bold r,\dot\bold r) of the dynamical systems with
weak and complete normality are derived.Comment: AMS-TeX, ver. 2.1, IBM AT-386, size 16K (ASCII), short versio
Investigation of the neurovascular coupling in positive and negative BOLD responses in human brain at 7T
Decreases in stimulus-dependent blood oxygenation level dependent (BOLD) signal and their underlying neurovascular origins have recently gained considerable interest. In this study a multi-echo, BOLD-corrected vascular space occupancy (VASO) functional magnetic resonance imaging (fMRI) technique was used to investigate neurovascular responses during stimuli that elicit positive and negative BOLD responses in human brain at 7 T. Stimulus-induced BOLD, cerebral blood volume (CBV), and cerebral blood flow (CBF) changes were measured and analyzed in ‘arterial’ and ‘venous’ blood compartments in macro- and microvasculature. We found that the overall interplay of mean CBV, CBF and BOLD responses is similar for tasks inducing positive and negative BOLD responses. Some aspects of the neurovascular coupling however, such as the temporal response, cortical depth dependence, and the weighting between ‘arterial’ and ‘venous’ contributions, are significantly different for the different task conditions. Namely, while for excitatory tasks the BOLD response peaks at the cortical surface, and the CBV change is similar in cortex and pial vasculature, inhibitory tasks are associated with a maximum negative BOLD response in deeper layers, with CBV showing strong constriction of surface arteries and a faster return to baseline. The different interplays of CBV, CBF and BOLD during excitatory and inhibitory responses suggests different underlying hemodynamic mechanisms
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