# Questions tagged [projection]

This tag is for questions relating to "Projection", which is nothing but the shadow cast by an object. An everyday example of a projection is the casting of shadows onto a plane. Projection has many application in various areas of Mathematics (such as Euclidean geometry, linear algebra, topology, category theory, set theory etc.) as well as Physics.

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### Galerkin Methods

Let $(\varphi _n)_{n\in \mathbb{N}}$ be a basis of an Hilbert space $H$. Define $H_N=\operatorname{span}\{\varphi _1,\ldots ,\varphi _N\}$. Suppose that there exits $a:H\times H \to \mathbb{R}$, a ...
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### Euclidean projection on convex set of positive semidefinite matrices

Define the Euclidean projection for a convex set $C$ as follows $$\pi_C(y) := \min_{x \in C} \| y - x \|_2^2$$ How would we find the projection map when $C$ is the cone of positive semidefinite ...
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### Optimal (in terms of remaining vector lengths) 2-dimensional projection plane of $n$ $d$-dimensional unit vectors

I have a finite number of $n$ unit vectors in $\mathbb{R}^{d}$. I would like to find a two-dimensional projection plane such that each vector has a length larger than 0 in the projection. Moreover, I ...
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### projection of a non-zero mean Gaussian vector into a Ball

Let $d$ denote the dimension, $\mathbf{B}_d$ denote the ball of radius one in $\mathbb{R}^d$. For $x\in \mathbb{R}^d$ let $\Pi_{\mathbf{B}_d}(x) = \frac{x}{\max\{1,\|x\|_2\}}$. Consider a fixed vector ...
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### projection of the data along the 1st k principal components

I'm a final year maths undergrad doing a course in multivariate data analysis, but I'm really struggling with the linear algebra. In particular the “projection of the data along the 1st k principal ...
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### What is a symplectic structure on a smooth vector bundle

We just had the definition of a symplectic structure on a vector bundle in the lecture and I am having trouble understanding it Definition: Let $\pi : E \to M$ be a smooth vector bundle. Then a ...
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### How to calculate the mass of a 3-D sphere collapsed into 2-D plane?

My sphere has density, $p(R)=(R+10)^-2$. If I collapse this sphere into a $2D$ plane, let's say it forms a $2D$ ellipse as a result, how will I calculate the mass of this $2D$ ellipse? in what ...
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### Projecting a point onto a convex set given by Log-Sum-Exp

Motivated by a wish to encode signal temporal logic specs (with linear predicates) as optimization problems w/o mixed integer approaches, I've been attempting to find a way to define the projection ...
1 vote
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### Orthogonal projection of an ellipsoïd from N to 2 dimensional space

Suppose we have a $N\times N$ symmetric-positive-definite matrix $A$, representing an ellipsoïd in $N$ dimensional space. How to find the matrix $A_{xy}$ corresponding to orthogonal projection of ...
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### Existence of well defined map $R/J\to R/I$ implies $J\cong M\subseteq I$?

I'm currently trying to prove by myself some proposition related to Hopf-Galois theory (from the paper "Galois Correspondences for Hopf Bigalois Extensions", by Peter Schauenburg). The ...
1 vote
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### Notation for inner product between scalar-valued and vector-valued functions

Consider $\mathbf{f}:\mathcal{S}\rightarrow\mathbb{R}^{n}$ and $\left(\phi_k\right)_{k=1}^{\infty}$, where $\phi:\mathcal{S}\rightarrow\mathbb{C}$, and $\mathcal{S}\subset\mathbb{R}^{n}$. My intent is ...
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### Spectrum of general projection and orthogonal projection

I am trying to think about this, but I seem to be stuck. Suppose $P$ is a projection on a Hilbert space $\mathcal{H}$. If I am just talking about a general projection, where I only know that $P^2=P$, ...
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### Looking for an example for the projection's theorem on an inner product space?

I'm looking for an example of a non-empty, non-convex and complete subset $C$ of an incomplete inner product space $E$ such that if we apply the projection's theorem on $C$ it gives several (maybe ...
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### Calibrating a pinhole camera (finding $z_0$)

A pinhole camera is a very simple theoretical device for generating perspective images on a plane that a distance $z_0$ from the pinhole (a point) and whose normal vector is the direction vector at ...
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### Locating vertices of a known triangle in $3D$ from a single image

Suppose you have a labelled triangle with known side lengths, and you take one image of this triangle using a known pinhole camera (i.e. the focal length is known), from a point with known coordinates,...
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### Locating a point in $3D$ using two perspective images

Suppose I am given a point $P(x, y, z)$ where $x, y, z$ are unknown. I take two images (perspective projections) of this point using a simple pinhole camera with an unknown focal length $f$, from two ...
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### Are the projections of a direct sum continuous?

Let $(V, \lVert \cdot \rVert)$ be a normed vector space, and let $X, Y \subseteq V$ be linear subspaces such that $X + Y$ is a direct sum (that is, $X \cap Y = \{0\}$). Since the sum is direct, every ...
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### Numerically approximating projection onto an infinite-dimensional Hilbert-space

We have the following problem that we want to model numerically. We would be glad for any references, since we could not find much useful information on these kind of problems and since we do not come ...
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### Proof that dual of $L^p$ is $L^q$

Im working through a proof which shows that for $\frac{1}{p}+\frac{1}{q}$ the map $\Phi:L^q(\Omega;\mathbb{K})\to (L^p(\Omega;\mathbb{K}))^*$, $g\to \Phi(g)[f]:=\int_{\Omega}\overline{g}f\space d\mu$ ...