Reversed dimensional analysis in psychophysics

Abstract

It is argued that dimensional analysis (from physics) can be applied in psychophysics in the same manner as in physics, provided that the dimensions of sensations are interpreted in the sense of reversed dimensional analysis, which allows for extracting dimensions from data. A dimension obtained by this method, referred to as phenomenological dimension, is similar to a fractal dimension. The examples discussed show that, if the dimension of sensation in Stevens' psychophysical law for time duration (Y 5 CF n , n Å1.15) is interpreted as a phenomenological dimension, such a dimension can be traced even in short-term memory data. It is suggested that dimensions obtained via reversed dimensional analysis may serve as a link between physical and psychological variables and as a basis for developing the notion of dimension in psychology.

How to Prevent Bubbles in Microfluidic Channels

“…Based on Young’s equation and Laplace pressure, we obtain the following equilibrium equation: where γ sl is the solid–liquid surface tension, γ sv the solid–vapor surface tension, γ lv the liquid–vapor surface tension, r the radius of the pit, γ the surface tension of water, d the deviation distance, θ the static contact angle between water and the wall, h the height of the channel (3D model of the channel and the schematic of h can be found in the Supporting Information, Figure S1), f ww the static friction of the channel wall at h against water, which has a maximum value (noted as f wwmax ) intrinsically determined by the wall and fluid property, the average line static friction along the l h (marked in Figure S1), v the flow rate, and C i some constant ( i = 1, 2). The left and the right sides of eq have the same physical dimension MT –2 . A bubble is formed in the pit only if eq can be satisfied by parameter values of point K (marked in Figure c).…”

“…The left and the right sides of eq 4 have the same physical dimension MT −2 . 25 A bubble is formed in the pit only if eq 4 can be satisfied by parameter values of point K (marked in Figure 1c). The derivation of eq 4 can be further found in the Supporting Information.…”

An algebraic–analytic framework for measurement theory

“…However, since the stem's annular crosssection is homogeneous and its inner diameter is to a very high degree of accuracy constant along its length, the geometric equation V 0 = ' 0 A applies, where quantity A denotes the stem's constant inner cross-section area with the obvious dimension L 2 . Owing to this simple geometric relationship and the manufacturer's calibration, the 27 Concretely, at room temperature and at 1 atm normal pressure at sea level we have a = 182 Â 10 À6 K À1 . marked scale on the thermomenter's stem allows the user to read the mercury column's variable length directly as variable temperature vales.…”

“…An important question to ask is whether dimensional analysis readily extends to non-physical quantities. Using the idea of fractal dimension, Marinov in [27] develops a theory of dimensions for psychophysical and psychological quantities. Discussion of general forms of order-invariance under scale transformation semigroups can be found in [12].…”

“…27 Taking inspiration from the construction of the algebra for the gas-in-a-vessel system, we see that the quantity algebra tailored for the description of a mercury-in-glass thermometer has the quotient Banach algebra form…”