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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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Dimensional properties derived from PCA eigenvectors

Background Let's assume I'm using principal component analysis to carry out clustering of a 2-d data set, using a non-normalized covariance matrix to carry out the operation. I then solve for the ...
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Non-dimensionalising an equation containing an exponent

I am being asked to write the non-dimensional version of the equation $ae^{bx}+c=x^2$. I understand the process that one would use to non-dimensionalise an equation, but in this case it was the ...
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$\dim(\ker(ab)) = \dim(\ker(a)) + \dim(\ker(b))$

Can somebody help me please so solve a self-study problem $V$ is a vector space over $F$ and $a,b$ are in $L(V)$. Suppose $\ker(a)$ and $\ker(b)$ are finite dimensional and b is surjective. WTS: $\...
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Non-dimensionalisation and Taylor expansion

I need to expand an equation, of the form $$\dot{r} = \gamma(a,\mu) F_1 + g(\mu,\ell,h,R) F_2$$ in powers of $\epsilon = a/\ell$. So I thoughts I would non-dimensionalize it first. I know that $$\...
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Inputting the bounding area of a 4-sphere to output the radius

If you rearrange the formula used to find the bounding area of a 4-sphere to instead calculate the radius, would the input for the area have to be 3 dimensional?
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Counter-proof of orthogonality of random points in a higher-dimensional unit sphere

I seek to provide a counter-proof against the following statement about the unit sphere in a N-dimensional space, with a large value for N. Statement: Two randomly selected points on the surface of ...
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159 views

Probability of two random points being orthogonal in higher-dimensional unit sphere

I understand that most points will be close to surface due to volume concentration. Also I also understand the concentration of volume near the equator, relative to any specific point (North pole). ...
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Physical dimensions in math

I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need: Scalar multiplication over the ...
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Linear Discriminant Analysis project data on sum of eigenvectors

I know in Linear Discriminant Analysis (LDA) we project the data on a specific eigenvector or a matrix of eigenvectors to reduce the dimensionality of the data and separate classes. But what happens ...
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34 views

Inconsistency in units in differential equations for 2d projectile trajectory

I am to solve the following two coupled second order differential equations involving the motion of a projectile. For the $y''(t)$ differential equation, I do not understand why the "$g/m$" term is ...
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68 views

Dimension of Holder space

$\textbf{Definition}$ Let $ \Omega $ be open in $\mathbb{R}^n$ and $\alpha \in (0,1]$. \begin{align*} \textrm{For } \alpha\in(0,1],& \newline\\ &[u]_{\alpha,\Omega}:=\sup_{x\neq y}\frac{\...
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Determining points on a n-spiral

I'm trying to generate points on a n-dimensional helix. I'm using a fix angle and maintaining the radius equals to 1. Is there a formula/pattern to determine the points in more than 3 dimensions? ...
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78 views

Show that $f:T^2 \rightarrow T^2$ is topological transitive

What is given: Let K be a irrational number and $f:T^2 \rightarrow T^2$ be the homeomorphism of the 2-torus given by $f(x,y)=(x+K,x+y)$. The exercise: Show that for every non-empty, open, $f$-...
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How to obtain this equation of hyperspectral image restoration using total variation in the following equation?

While reading the paper "Total-Variation-Regularized Low-Rank Matrix Factorization for Hyperspectral Image Restoration" I came through the following equation. Can someone help me how the last equation ...
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Dimensions of parameters of a differential equation and change of variable

The original equation is $$ \frac{dN}{dt} = rN\left(1-\frac{N}{K}-\frac{a}{1+bN}\right),\quad N(0)>0, $$ Where $r,a,b,k$ are all positive parameters and a refers to the Allee effect(though Im not ...
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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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How to prove the relationship between krasnoselskii's genus and the dimenson of a vector space?

I have to work with this definition for Genus: Let us denote by $U$ the class of all closed subsets $A ⊂ X- \{0\}$ that are symmetric with respect to the origin, that is, $u ∈ A$ implies $−u ∈ A$. ...
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Why does the conversion factor method works? (dimensional analysis).

In many textbooks, the unit factor method for converting units is described in this way: In dimensional analysis, a ratio which converts one unit of measure into another without changing the ...
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Reducing or compressing the dimensionality of an sparse adjacency matrix m x n (m = 25000)

here's the problem in details: so for natural language processing, we have a computational model that explicates how world knowledge and linguistic experience are integrated at the level of ...
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What is the reason behind writing the symbol for the unit of area i.e. square meter, as $m^2$?

What I mean to say is as follows: Measuring the area of a surface is determining its ratio to a chosen surface called the unit of area and the chosen unit of area is a square whose side is a unit of ...
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Is there a way to non-dimensionalize curvature of a generic curve?

For a generic curve in 2D space, is there a way to take the curvature of the line at a point (where $\kappa$ is one over the radius of the osculating circle) and non-dimensionalize it? I am looking ...
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PDE Similarity Solutions Boundary Value Problem and Solution: Explanation of Steps Requested

I have the following similarity solutions problem and solution: Problem $u_t = ku_{xx}$ for all $x > 0$, with $u_x (0, t) = 1$, $u(x, t) \to 0$ as $x \to \infty$, and $u(x, 0) = 0$ for $x &...
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Constraints of $n$-dimensional cylinder inside sphere

In three dimensions, maximising volume of cylinder inside a sphere (denote $B_3(R)$ , wo.l.o.g centered around the origin) is straightforward. We get constraints to the radius of the cylinder via good ...
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Dimensional Analysis Conversion

I am new to learning about dimensional analysis and I am confused with a question. I am supposed to convert 45 km/day to inches/week. I would like to have all work shown. The work I have so far is ...
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Dimensionality of the Poincare section?

Consider the Poincare map $P$, for which we place the Poincare section transverse to the trajectories and it intersects at various points which forms a set of discrete points. For $\dot{x} = f(x)$ ...
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Similarity Solutions Example

An example shows the PDE $$\dfrac{\partial{T}}{\partial{t}} = \alpha \dfrac{\partial^2{T}}{\partial{y^2}}$$ It says we can non-dimensionalize the system with the transformations $$u = \dfrac{T - ...
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Introducing Dimensionless Variables into PDE and Performing Change of Variables

I have the following example in my notes: Suppose we want to know how long it takes to cool down a long metal cylinder of radius $a$, with thermal diffusivity $\alpha$, initial with a uniform ...
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How to find the dimensional analysis for gaussian integrals?

Testing several alternatives For the Gaussian integral $$\int_{-\infty}^{\infty}{e^{-\alpha x^2}}dx,$$ use the three easy-cases tests to evaluate the following candidates for its value. (...
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How to use dimensional analysis to find these integrals?

Use dimensional analysis to find $\displaystyle \int_0^∞ \mathrm{e}^{-ax} \,\mathrm{d}x$ and $\displaystyle \int\frac{\mathrm{d}x}{x^2 + a^2}$. A useful result is$$ \int \frac{\mathrm{d}x}{x^2 + 1} = ...
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Units of coefficients in Ornstein Uhlenbeck process

The Ornstein-Uhlenbeck process is defined as $$ \frac{d}{dt}x(t) = -\frac{1}{\mu}x + \sqrt{\frac{2\sigma^2}{\mu}}\xi(t) $$ where $\xi(t)$ is a unit white noise. What are the units the parameters? ...
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Dimensions of integral

Consider the famous work equation due to a continuous charge distribution: $W=\frac{\varepsilon_{0}}{2}\left ( \int_{volume \space space}\left \| \vec{E} \right \|^{2}.d \tau+\oint_{S}V.\vec{E}.d ...
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What is the dimension of a heap?

Heap is a kind of data structure used in computer science. But what is it dimension? An array can be 1-D 2-D...etc Young's tableaux is 2-D But what about heaps?
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In dimensional consistency checks is $\mathrm{M}=\sqrt{3}\mathrm{M}$ true?

When checking an equation with dimensional consistency, if I get $\mathrm{M}=\sqrt{3}\mathrm{M}$ should I be worried? or the coefficients don't matter because we are concerned about dimensions?
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Interpreting what it means when the units, $\frac{\text{success rate}}{\text{time}}\cdot \text{currency}$

You are applying for a \$1000 scholarship and your time is worth \$10 an hour. If the chance of success is $1 -(1/x)$ from $x$ hours of writing, when should you stop? My way if thinking eventually ...
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Solving a second order differential equation numerically, by making it dimensionless

$$\textbf{I have been given the following:}$$ This is an example solution of a second order ODE, specifically that of a Harmonic Oscillator $$ {\text{d}v\over\text{d}t}=-\omega^2x-kv, $$ where $v=$d$...
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Dimensional analysis / units for best fit curves

I am doing a lot of curve fitting from experimental and modelling data. If I have a polynomial of the form which takes a temperature T and outputs a pressure Pa ...
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What's the use of scaling here?

Consider the following exercise: Suppose that a mathematical modelling is given by the following equations: \begin{align} \dfrac{\partial u}{\partial x} + \dfrac{\partial w}{\partial z} = 0, ...
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Second derivative of $\phi$ in nondimensionalization problem.

I am reading a section in my differential equations textbook and am trying to fill in the left out steps in their explananation of how to nondimensionalize the second order system $$mr \ddot{\phi}=-b\...
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Can the dimension of a space or quantity be complex?

I was wondering if the notion of a quantity having complex dimension makes sense mathematically? It's maybe just a bad question; I can't attach any physical meaning or intuition to, for example, a ...
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Reducing the number of parameters of an ODE system through nondimensionalization

I am trying to reduce the number of parameters of the ODE system: \begin{align} \label{eq:Iprime} I'(t)&= \sigma B-\mu I\\ B'_1(t)&=r_1 B_1\left(1-\frac{B_1 }{K_1}\right)-d_1IB_1-m (B_1-B_2) ...
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Units of parameters in differential equation

Suppose I have the system $$I'(t) = \frac{\alpha\beta I}{r+\alpha I} (1-I) - \mu I$$ where $I$ is a unitless quantity giving the portion of a population (in this case, total infected/total population, ...
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71 views

Can a one-dimensional shape have volume? [closed]

Today, I decided I would try to generalize a formula to find area and volume of a hypercube. The formulae I came up with are as follows: $n$ = number of dimensions $L$ = side length Surface area = $...
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Does the absolute value operator pick up dimension?

My specific question is whether the quantity $\rho = k\times | xyz|$, has the dimension of $k$, or it has the dimension of $k\times xyz$?
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Finding the maxima of a given function

Please look into the following question. I have solved Part (a). For part (b) I found $f_{\pi_{2}}$ and $f_{\pi_{3}}$, where $f_{\pi_{2}}$ stands for the partial of $f$ with respect to $\pi_{2}$. I ...
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Are one-dimensional maps still called the same if they involve multiple functions instead of one recurrent function?

Is the definition of a one-dimensional function necessarily something generated by x=f(x) or can a series of functions f,g,etc. be applied? Or is that called a different name?
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Why the objective function in optimization does not follow dimensional rule?

I have seen few studies where the objective function tries to minimize two different variables (let's say $a$ and $b$) that are of different dimensions using a superposition rule. For example, this is ...
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Why are radians dimensionless? [duplicate]

According to https://en.wikipedia.org/wiki/Dimensionless_quantity, "A dimensionless quantity is a quantity to which no physical dimension is applicable." The article then explains, a few sentences ...
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Can you mix units of $t$ and $t^2$ when constructing a shape?

Let's say I have a length (possibly a radius), we'll call it $y$ and is of unit $t$. My data suggests to me that the distance around this shape (possibly a circumference), is $y^2$, which means the ...
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Best way to non-dimensionalise equation of vertical motion with damping and external Force

Trying to non-dimensionalise the following equation and wondering what the best way would be? Any ideas would be appreciated. Thanks
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59 views

Cancelling units of empirical equation

OK this is probably a simple question, but I can't seem to find a good answer. I know that in dimensional analysis the units on both sides of the equation must be equivalent; I suppose the simplest ...