# Tagged Questions

In general topology, dimension theory studies various notions of dimension defined for topological spaces, for example Lebesgue covering dimension, small and large inductive dimension or Hausdorff dimension. In commutative algebra, dimension can be defined for commutative rings.

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### Is it possible to construct a smooth curve with fractional Hausdorff dimension?

It is known that fractal curves have fractional Hausdorff dimension. These curves are not smooth and have undefined length. However, is the converse true? If a curve has a fractional Hausdorff ...
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### Where do extra dimensions in gradient come from?

The gradient of a scalar function $f\colon \mathbb{R}^n \to \mathbb{R}$ is a vector-valued function $\nabla f\colon \mathbb{R}^n \to \mathbb{R}^n$. Since applying a function can't increase ...
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### rectangular paddock, dimensions, maximise area it encloses

Having trouble trying to work out a question which involves finding a function to graph evidence of the correct answer, any advice would be greatly appreciated. I am struggling with part 'b' a lot, ...
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### A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
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### How many minimally sides are needed to fully enclose a volume in an $n$-dimensional spaece?

Imagine I have some unbendable*, but cuttable metal-plates. One side is silver, the other one is blue. Now I want to use them to build somekind of case (with no opening) that is fully blue. It just ...
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### Box-Counting Dimension with finite resolution

Does the method of determining dimension of a shape via the Box-Counting dimension (Minkowski–Bouligand dimension) have to be performed on fractals (objects that look the same at all scales), or can ...
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### Multi-dimensional naughts and crosses: victory for first player?

In a $2$-dimensional checkerboard which is $2\times2$, the player going first necessarily achieves $2$ in a row. In a $3$-dimensional board which is $3\times3\times3$, a player going first using a ...
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### Maximal subset with finite Assouad-Nagata dimension

Given some space $X$ with non-finite Assouad-Nagata dimension. Is it possible for a subset $Y \subset X$ with finite Assouad-Nagata dimension to exist such that $Y$ is maximal in the sense that if any ...
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### Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in Mathematics):...
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### Length of a vector in 3 dimension spaces

while I'm studying principal-component analysis, I have misunderstanding about principal component, I was think that it's orthogonal and has a single point,which I mean by single points is if we have ...
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### Explicit formula for IFS fractal dimesnion

Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or ...
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### Integral over Fractals with respect to fractal dimension

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the ...
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### Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
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### A question on Lebesgue's covering dimension

Roughly, a compact, Hausdorff space $X$ has covering dimension $\leqslant n$ if each finite cover $\mathcal{U}$ of $X$ can be refined by a cover $\mathcal{V}$ such that each point $x\in K$ belongs to ...
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### Wheel Graphs and Dimension of Embeddings

I'd like to preface this by saying this is the tip of the iceberg for an optional question for a summer REU program application, so if you think asking this question is in bad taste, let me know and I ...