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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18 views

What is the basis of the vector space of polynomials of any degree? [duplicate]

Could we say that the set $\{1,x,x^2,x^3,x^4,\cdots\}$ is a basis of the vector space of polynomials ? Note that, for example the set $\{e_1,e_2,e_3,\cdots\}$ such that $e_n=(\delta_{n,k})_{k \in \...
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How to calculate the center of an n-cube (hypercube) given the minimum values and maximum values in each dimension?

I am working with a box-embedding-based method in machine learning, and for a particular problem, I have two objects that are represented as two $n$-cubes in $n$-dimension ($n$ might be $16, 32, 64, .....
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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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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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Derivating $z(x)=y(x+\Delta x)-y(x)$

I have two real functions, $z$ and $y$, initially defined for discrete domain. The function $z$ is related to $y$ and to $\Delta x$ by $$z(x)=y(x+\Delta x)-y(x).$$ These functions are dimensioneless,...
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Dimensional analysis of a very simple utility function

Suppose we have a utility function $u:\mathbb{R}_+\rightarrow\mathbb{R}$ defined by $u(x)=x$, where $x$ stands for quantity of apples. Suppose we measure the quantity of apples in kg. How can we ...
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understanding the loss function and dimension of X

Let $P_X$ denote the data distribution, and $P_G$ denotes the encoded training distribution, where $Q$ is the encoders and $G$ is the decoders, $X ∼ P_X$ and $Z ∼ Q(Z|X).$ Considering the sparsity of ...
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Dimensional analysis of an integral

First of all, I am not a mathematician and my dimensional analysis skills are almost non-existent; so I express my apologies in advance if this question is silly. I am reading a fluid dynamics paper ...
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Can we always non-dimensionalize ODEs?

I'm trying to practice my ability to non-dimensionalize equations, and I am interested whether we can always non-dimensionalize certain differential equations. Take the system of equations: \begin{...
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Help non-dimensionalizing predator-prey ODE system

I am learning how to non-dimensionalize ODE systems, and I am struggling with a part of the predator-prey non-dimensionalization exercise here: http://wp.auburn.edu/radich/wp-content/uploads/2014/08/...
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other decomposition for rectangular matrix?

Do we have an iteration method which could do matrix decomposition? Currently which method is the best to speed up the matrix decomposition? thanks
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low rank approximation problem

I'm a new learner for low-rank approximation, may I know what is the current problem for low-rank approximation, so that i could explore it. thanks
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Appling Laplace in equations with units

Suppose that I have an equation $$\omega \cdot f(x)=g(x)$$ where $\omega$ is a constant with dimension, say, a velocity (and $f,g$ have both their own units, say $f(x)$ with unit $\frac{1}{s}$ ...
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How can one imagine the lattice packing of (hyper)spherical particles, in n-dimensional space? like we have the lattices in 3D?

We can put 4 unit circles inside a square of length 4 units, such that all the circles will touch the boundary and also adjacent circles, and there will be a hollow in between the 4 circles. So we can ...
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What, if anything, is a square radian?

My 1st year Mathematics BSc course notes on circular motion use \begin{align} \frac{d}{dt}(\sin\theta) &=\frac{d}{dt}(\sin(\omega t))\tag{1.1}\\ &=\omega\cos(\omega t),\tag{1.2}\\ \frac{d^2}{...
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Isn't $\overrightarrow{\omega} t=\theta$ equating a vector to a scalar?

Unless I'm misreading them somehow, my 1st year Mathematics BSc course notes use $$\overrightarrow{\omega}\cdot t=\theta,\tag{1}$$ where $\overrightarrow{\omega}$ is angular velocity, a vector; $t$ ...
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Fluid Mechanics. How to nondimensionalize a variable with a power law index.

can anyone help me in nondimensionalizing these equation? I start from the equation in the first line. Using the nondimensional variable from the third line, I get the half-completed equation in the ...
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Volume of $n$-hyper-sphere derivation

I am trying to prove the curse of dimensionality and on Wikipedia https://en.wikipedia.org/wiki/Curse_of_dimensionality The volume of Hypersphere is given as $\frac{2r^d \pi^{d/2}}{d\;\Gamma (d/2)}$ ...
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How far will a cup sink under the following conditions?

A cup made of $3/16$-inch-thick glass has an inside radius of $3$ inches and a total height of $6$ inches (including the bottom thickness of glass). The glass has a density of $165$ $lb/ft^3$. The jar ...
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How to achieve (approx) unit scaling of a non-linear diffusion (heat) equation with a wildly varying diffusion coefficient?

I have numerical issues with a poorly scaled one-dimensional non-linear diffusion equation in physical co-ordinates $$ \frac{\partial{u}}{\partial{t}}(x,t) = \frac{\partial}{\partial{x}}\left(D(u) \...
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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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176 views

Why are there factors of $2 \pi r$ in this volume integral?

I have a question regarding the solution to part of this homework question: An infinite filled cylinder of radius $a$ contains a 3D charge density $\rho$. A thin-walled hollow cylinder of radius $b ...
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How to find the box counting dimension of line segment [0,1]?

I uploaded this question and no one has answered because there were some flaws in the question but I have the text now. Please can some one explain how we arrive on example of line segment to have a ...
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Why is length often a unitless quantity in mathematics?

This is something that's bothered me for a long time: Why is length often not given any units (e.g. inches or meters) in mathematics? The unit circle is said to have a "radius of 1," with no unit ...
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Why does this temperature conversion procedure work?

Recently while playing with conversion between different temperature scales I found a quite interesting and simple procedure for conversion from one scale to another. This is as follows: Suppose we ...
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Dimensional Analysis help

So a few people have tried explaining how to do dimensional analysis to me and I think I may understand how to but I am not 100% sure. Unsolved This is what I belive is correct My Answer Full ...
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The intuition behind volume and surface area of an $n$-ball

I'm a non-mathematician and am trying to understand intuitively what happens to the volume and surface area of an n-ball as the size and dimensionalty increases. I've tried looking at similar topics ...
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Is $180^\circ = 3.14..$? Correlating the circumference and diameter in terms of degrees

I'm trying to correlate $\pi$ to the circumference and diameter of a unit circle. Consider a unit circle and represent it in terms of an n-gon. Join each vertex to its center. The angle subtended is $\...
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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, ...
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Is there some kind of rigorous construction of units and Dimension analysis?

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 ...
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Euler Method for Dimensionless System

Consider the following system of couple ODEs $$ \begin{align} \frac{\partial n_A}{\partial \tau}&= \tilde{f}(n_B)-\delta_An_A\\ \frac{\partial n_B}{\partial \tau}&= \tilde{g}(n_A)-\delta_Bn_B. ...
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Hey, I am having trouble with Standard Deviation formula for normalization.and whats going in this question

I have been trying to solve this question for a couple of hours now. I did not understand the normal Standard Deviation formula for normalization. Can you guys help me what is happening in this ...
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What is the shape that have the biggest surface area with hypervolume of 1 in $N$ dimensional space?

There is a theorm saying that from all shapes in $N$ dimensional space with hypervolume of 1 (in arbitrary units) $N$ dimensional ball is the one with the smallest surface area ($N$ dimensional sphere)...
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Solving Inviscid Burgers' using Similarity

I want to solve the inviscid Burgers' equation: $$\begin{equation} \frac{\partial u}{\partial t} + u\frac{\partial u }{\partial x} = 0 \end{equation}$$ I want to reduce the PDE to an ODE by saying ...
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Do all physical equations have a polynomial and a non-polynomial part?

Let $Q,Q_1,...,Q_n$ be $n+1$ physical quantities associated with some physical dimensions. Assume, the physical equation $$Q = f(Q_1,...,Q_n) $$ is true. While discussing methods of dimensional ...
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Why does numerator differ from denominator when nondimensionalizing second order differential equation

I have been looking at nondimensionalizing differential equations, in particular second order differential equations. But I am having trouble understanding the rationale for treating the numerator and ...
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physical dimensions as derived objects?

In a physics course, dimensions (such as energy, length, duration), are taken as given. There are then certain algebraic rules associated to them: e.g. we can add energy with energy, and multiply ...
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48 views

Why does this dimensional analysis argument work for the harmonic oscillator?

In the wikipedia page on dimensional analysis, an example is given about the period of a harmonic oscillator. The rough idea of the argument is: We have the following variables and dimensions, ...
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Dimensional analysis on $v=kF^xl^ym^t$

The velocity V of a wave moving in a stretched string can be written as, ( $v=kF^xl^ym^t$ ). where $F$ is the tension in the string, $l$ is it's length and m is the mass of the string and $k$ is a ...
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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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Does it make sense to compare 'area' of hyper-sphere of different dimensions?

A recent question on puzzling relied on hyper-sphere having maximal area at dimension 7. Which is shown here. What I can't shake off is the feeling that one can't compare the surface of unit spheres ...
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38 views

Finding a similarity solution for a mass in free fall, but with air friction

I'd like to ask about using dimensional analysis to model the projectile of a mass that's thrown vertically upward. In many similar problems, air resistance is ignored- but in this particular case, ...
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Celsius to Fahrenheit formula to calculate range of temperatures

Let's assume the maximum temperature during a month is $86 F ^\circ$ of all days and the minimum temperature during a month of all days is $74 F ^ \circ $. The range of temperature is the difference ...
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Dimension Analysis of Units in a 2nd order ODE

I am a bit stuck for part 1 of this exercise with unit analysis and keep on going around in circles by substituting these units in the ODE. If someone could answer this question ASAP, it would be ...
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Scaling in heat transfer PDE and dimensionless group

I found the following in my Scaling in Heat Transfer notes: Rod Conduction A rod of length $L$ initially at $T_0$ then (at $t = 0$) one end is raised to $T_1$. Find $T(x, t)$. $$T_t = \kappa T_{x x}$$...

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