Great work on high dimensional video sequence modelling using RBM.To read... Learning Multilevel Distributed Representations for High-Dimensional Sequences.Ilya Sutskever and Geoffrey Hinton
Most physical quantities can be expressed in terms of combinations of five basic dimensions. These are mass (M), length (L), time (T), electrical current (I), and temperature , represented by the Greek letter theta ( q ). These five dimensions have been chosen as being basic because they are easy to measure in experiments. a) energy joule (J) kg·m 2 /s 2 b) force newton (N) kg·m/s 2 c) frequency hertz (Hz) (cycles)·s -1 d) power watt (W) J/s = kg·m 2 /s 3 e) charge coulomb (C) A·s The most basic consequence of dimensional analysis is: Only commensurable quantities (quantities with the same dimensions) may be compared, equated, added, or subtracted. However, One may take ratios of incommensurable quantities (quantities with different dimensions), and multiply or divide them. As a corollary of this requirement, it follows that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if m man , m rat and L man denote, respectively, the mass of some man, the mass of a rat and the length of that man, the expression m man + m rat is meaningful, but m man + L man is meaningless. However, m man / L 2 man is fine. Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions, i.e., the equation must be dimensionally homogeneous. Even when two physical quantities have identical dimensions, it may be meaningless to compare or add them. For example, although torque and energy share the dimension ML 2 /T 2 , they are fundamentally different physical quantities . Scalar arguments to transcendental functions such as exponential , trigonometric and logarithmic functions, or to inhomogeneous polynomials , must be dimensionless quantities . A simple example: period of a harmonic oscillator What is the period of oscillation T of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g ? The four quantities have the following dimensions: T ; m ; k ; and g . From these we can form only one dimensionless product of powers of our chosen variables, G 1 = T 2 k / m . The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables, but the group, G 1 , referred to means "collection" rather than mathematical group . They are often called dimensionless numbers as well. Note that no other dimensionless product of powers involving g with k, m, T, and g alone can be formed, because only g involves L . Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g : it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: , for some dimensionless constant κ. When faced with a case where our analysis rejects a variable (g, here) that we feel sure really belongs in a physical description of the situation, we might also consider the possibility that the rejected variable is in fact relevant, and that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here. When dimensional analysis yields a solution of problems where only one dimensionless product of powers is involved, as here, there are no unknown functions, and the solution is said to be "complete." A more complex example: energy of a vibrating wire Consider the case of a vibrating wire of length ℓ ( L ) vibrating with an amplitude A ( L ). The wire has a linear density ρ ( M / L ) and is under tension s ( ML / T 2 ), and we want to know the energy E ( ML 2 / T 2 ) in the wire. Let π 1 and π 2 be two dimensionless products of powers of the variables chosen, given by The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation where F is some unknown function, or, equivalently as where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f . But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ℓ , and so infer that E = ℓs . The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident. The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a dimensionless number such as the Reynolds number , which may be interpreted by dimensional analysis. 总结：找出无量纲的表达式是关键！
标签: Dimensional
