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He now focuses on a general Hamiltonian system, i.e., a system of N material points with an arbitrary interaction potential.
Instead of concentrating on a specific gas model, Boltzmann here assumes a much more general model with an arbitrary interaction potential $V(\vec{q}_1,\ldots,\vec{q}_N)$.
Recent studies have provided evidence of arbitrary interaction in infancy.
In the present study we focused on a phenomenon of arbitrary interaction of multimodal information, the 'audiovisual tau effect' [19].
As such, the audio-visual tau effect appears to be a typical example of arbitrary interaction in which auditory "temporal" interval alters visual "spatial" interval.
Another study found that 10-month-old infants could use the consequences of arbitrary interaction of audiovisual information to orient their head and gaze towards an audiovisual target, whereas infants under 8 months of age could not [12].
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The present results also demonstrated that arbitrary interactions between audition and vision were possible in infants around 5 8 months of age.
Thus, there is a possibility that the audiovisual tau effect, which is one example of arbitrary interactions, occurs during the first half of the first year of life.
Previous studies have shown that arbitrary interactions among different modalities emerge during the second half of the first year of life [12], [16], [17].
These interactions have been classified into arbitrary crossmodal interactions [5], [11], [12] (i.e. arbitrary interactions for simplicity), referring to the phenomenon whereby a cross-modal manipulation in one dimension (temporal or spatial) alters the perceptual experience of multimodal stimuli in the other dimension (spatial or temporal).
One example of arbitrary interactions is the audiovisual stream/bounce illusion [13] [15] whereby the temporal congruence of audiovisual signals alters the perception of the spatial dimension of motion (i.e. motion trajectories).
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