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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How do fractional tensor products work?
In this blog post, Terry Tao discusses the $n$-fold tensor product of a one-dimensional vector space $V^L$ ($L$ is just a non-numeric label, not an exponent). He claims that
With a bit of ...
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Imposing boundary conditions AND self-similarity on a PDE
I have a PDE in the form
$$u_t=F(u,u_x)$$
where the unknown is $u(x,t)$ on say $\mathbb R\times[0,\infty)$. $F$ is very nonlinear so I was told to assume self-similarity in the form
$$u(x,t)=t\tilde u(...
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"Axioms" of dimensional analysis
I wondered if there are any theoretical backgound or "formalization" of dimensional analysis. I had an attempt on doing this, by providing some axioms and then "deriving" how ...
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Rigorously distinguishing torque from work, or, a more accurate algebraic structure for dimensional analysis
The algebraic structure underlying dimensional analysis is commonly said to be a finitely generated Abelian group, whose generating set is the set of base units (e.g. length, time, mass, charge, and ...
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How is the dimenionality of the nonlinear diffusion equation in Perona-Malik filter satisfied?
The diffusion equation is given by,
$\partial h /\partial t = \nabla \cdot (c \cdot \nabla h) --(1)$.
But in the Perona - Malik nonlinear diffusion equation the choice of c given by two forms PM1 : $...
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Buckingham-$\pi$ doesn't work on Lotka-Volterra
Given the Lotka-Volterra system:
$$\frac{dR}{dt}=aR-bRF$$
$$\frac{dF}{dt}=-cF+dRF$$
I think Buckingham-$\pi$ predicts that - since there are 7 variables (R, F, a, b, c, d, t), and 2 units (# of ...
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Does the Buckingham PI theorem require base units?
When implementing the Buckingham PI theorem, it is common to use basis dimensions such as [MASS, LENGTH, TIME], where each is a base SI unit ...
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Intuitive explanation of Buckingham Pi Theorem?
I'm currently taking a class about model building in applied math. We spent quite a bit of time on Buckingham $\Pi$ theorem. I understand it on a shallow level. I know how to solve the homework and ...
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Proving the product rule using dimensional analysis
$$\frac{d}{dx}(fg) = \text{?}$$
$$\frac{d}{dx}(fg) \color{red}\neq \frac{df}{dx} \cdot \frac{dg}{dx} \tag{units don't work}$$
$$\frac{d}{dx}(fg) = \quad f'(x)g(x) \quad \text{or} \quad g'(x)f(x) \tag{...
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Formal dimensional analysis?
Dimensional analysis is a method that I know from physics, where quantities are "annotated" with a "dimension". E.g. rather than writing $$4\cdot 5 = 20$$ we write $$4 m \cdot 5 s = 20 m\cdot s$$
The ...
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Finding difference of 2 n-dimensional vectors
I have been meaning to find the difference between $2$ $n$-dimensional vectors. I am looking for a convex function (anti-commutative) that finds a signed difference between the two vectors (in the ...
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Intuition behind similarity solution for Heat equation
The heat equation is $$\frac{\partial\theta}{\partial t}=\kappa\frac{\partial^2\theta}{\partial x^2}.$$ From this, one can look at dimensions on both sides, and conclude that the quantity $$\eta=\frac{...
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Nondimensionalization of an ODE with an extra parameter
I have the following coupled ODEs, which describe a system of inter-specific interactions within an ecosystem:
$$\frac{dx}{dt} = rx\left(1 - \frac{x}{K}\right) -bxy$$
$$\frac{dx}{dt}= -cy + dxy$$
...
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Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?
As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64)
There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
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What is the process of nondimensionalizing an equation?
Question: I need to scale time by $\frac{1}{I}$ and species by $P$
for the following equation
$\frac{dS}{dt}=I(1-\frac{S}{P})-\frac{ES}{P}$
where
P - Size of the source pool of species on the ...
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