Questions tagged [hyperspace]

For questions related to hyperspace (an $n$-dimensional Euclidian space with $n > 3$).

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projections from a plane in $\mathbb{R}^6$ onto three orthogonal planes

Let $\Pi_1=\operatorname{span}\{e_1,e_2\},\Pi_2=\operatorname{span}\{e_3,e_4\},\Pi_3=\operatorname{span}\{e_5,e_6\}$ be orthogonal 2D planes in $\mathbb{R}^6$. Let $U$ be arbitrary 2D plane in $\...
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sum of projected area and a generalization of Oppenheim's inequality

From this post: In $\mathbb R^4$, let $U$ be a 2D plane, let $\pi_1$ be the projection from $U$ onto $xy$-plane and $\pi_2$ be the projection from $U$ onto $zw$-plane, then $\det\pi_1+\det\pi_2\le1$, ...
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Projections onto orthocomplement 2D planes in 4D

From this post, It is possible to go another route and generalize. The orthogonal projection $\pi_2:U\to\Pi_2$ may be restricted to $\Pi_1$, and it has a "hypervolume distortion factor" $\...
hbghlyj's user avatar
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Construct perspective projection of rotating tesseract by perpendicular lines intersecting ellipse

The contruction was used in two different sources on the web: a Geogebra resource and a video using inRm3D so I think it must be documented and proved somewhere, but I didn't find any. Here is the ...
hbghlyj's user avatar
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Topology on the set of all separable Hausdorff topological spaces

It is well-known (1, 2) that the cardinality of a separable Hausdorff topological space is at most $2^{2^{\aleph_0}}$. Therefore, the collection $\mathcal{A}$ of all (homeomorphism classes of) ...
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The rotation of a solid object in 3D can be described by a single 3D vector. Is the same true for higher dimensions?

If a solid ball (like the Earth) is rotating in 3D space, you can point a single 3D vector out of the North Pole (according to the right hand rule), with the length of that vector proportional to the ...
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Book about explanation of 4th dimension using analogy with 2d object interacting with 3d

I'm searching for a book about the explanation of 4th and higher dimensions using an analogy with 2d space creatures (flatland) with 3d dimension (as humans). I need to to add and update Polish ...
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How can I prove $M+t$ is a hyperplane if $M$ is a maximal subspace

Let $M$ be a non-empty proper subset of a vector space $X$ over $\mathbb R$ and $t$ belongs to $X$, then $M$ is a maximal subspace if and only if $t+M$ is a hyperplane and $t$ belongs to $t+M$.
John MATH's user avatar
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Nuclear norm of matrix comprised of points sampled uniformly at random from the hypersphere?

In a subfield of machine learning called "self-supervised learning", many methods constrain network representations to lie on the hypersphere. I want to understand how such representations ...
Rylan Schaeffer's user avatar
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Relationship between trace norm (a.k.a. Schatten-1 norm) of a matrix and the vector norm of the matrix's row average?

I'm trying to understand whether a connection exists between two seemingly different optimization problems in machine learning. Setup: Suppose I have $N$ points $x_1, ..., x_N \in \mathbb{R}^D$, where ...
Rylan Schaeffer's user avatar
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Singular values of uniform random points on hypersphere?

This question is motivated by a self-supervised learning problem in machine learning, but I'll try to strip out as many unnecessary details as possible. In this setting, we have large datasets and we ...
Rylan Schaeffer's user avatar
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Maximum number of points on the surface of a 4D sphere

1000 alien spaceships meet in a 4-dimensional battlefield. At an agreed time (ignoring relativistic effects on their clocks) every spaceship fires its laser to the spaceship which is closest (assume ...
Rüdi Jehn's user avatar
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Two 4-d hyperspheres intersect in a sphere

One hypersphere at $x^2 + y^2 + z^2 + w^2 = 1$ intersects another hypersphere at $(x - 1)^2 + y^2 + z^2 + w^2 = 1$. (EDIT to address comments) The intersection results in a (3-d) sphere with radius $ \...
RTF's user avatar
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How many different 3D parallelepiped are in 4D hyper-parallelepiped?

Let's say I have three 3D linearly independent vectors, these vectors form a parallelepiped. This parallelepiped has 6 faces, but only 3 of them are "unique" (the other three can be obtained ...
narger's user avatar
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What is the determinant of the Jacobian of four-dimensional hyperspherical coordinates?

I am trying to do a four-dimensional change of variables problem, and I am working with hyperspherical coordinates. For $(x,y,z,w)=\varphi(\rho, \varphi, \theta, \lambda)$, I have $x= \rho \sin\varphi ...
ThePevster's user avatar
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Theorem 16 Corollary 1, Section 3.5 of Hoffman’s Linear Algebra

If $W$ is a $k$-dimensional subspace of an $n$-dimensional vector space $V$, then $W$ is the intersection of $(n- k)$ hyperspaces in $V$. Rephrasing problem to my taste: If $W$ is a $k$-dimensional ...
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