# Questions tagged [dimensional-analysis]

The study of the relationships between physical quantities by identifying their units of measure and fundamental dimensions. It is used to convert from one set of units to others such as from miles per hour to meters per second, or from calories per slice of cake to kilocalories per whole cake.

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### Interpretation of constants in ODE models

Full disclosure: I have asked a couple of questions over the last few days, but I'm still having some problems with describing parameters in ODE models in words. In the model: \begin{equation} \frac{...
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### Non-dimensionalise this ODE

I have the ODE: \begin{equation} \frac{dx}{dt} = ax - \frac{b}{b + x} x \end{equation} where $a$ has units $[1/t]$ and $b$ has units $[x/t]$. $x$, in this example, is a resource, and the last term ...
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### How does multiplication work when a 3D array is dotted with a 1D array?

EDIT: cleaned up as suggested by comments I know the shortcut to seeing if a matrix multiplication will work out okay: But how does this rule generalize when you have multidimensional arrays? I ran ...
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### What is the physical dimension of a reciprocal vector? [duplicate]

The question is of general interest for dimensional analysis. I am trying to formalize the dimensionality of vector spaces. Consider the usual orthonormal basis of the Euclidean space $e_1, e_2, e_3$...
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### Mathematical demonstration of the distance concentration in high dimensions

I know that in high-dimensional space, the distance between almost all pairs of points has almost the same value ("Distance Concentration"). See Aggarwal et al. 2001, On the Surprising Behavior of ...
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### Finding corresponding coordinates in multidimensional scaling (MDS)

In multidimensional scaling (MDS), a series of k-dimensional coordinates is discovered from an N x N distance matrix. For k = ...
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### Calculate Minkowski dimension for a square

How can i calculate the Minkowski dimension for a ordinary square? I saw the few examples where dimension is calculated with help fractal dimension - fractal dimension formula In this case D = 2, ...
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### Dimensional analysis of Stokes equation $\mathrm{div} \sigma = f$.

Consider the stationary (i.e, independent of time) Stokes equations $$\mathrm{div}~ \sigma = f$$ where $\sigma$ is the stress tensor, $f$ is the external force. Denote by $M,L,T$ the mass, length, ...
I've always worked with units using some rules that were never strictly defined, such as $m×m=m^2$ or 'you can't add meters with seconds' for example, so i' ve wondered if there's some kind of ...