Tensor Rank

The total number of contravariant and covariant indices of a tensor. The rank  of a tensor is independent of the number of dimensions  of the underlying space.

An intuitive way to think of the rank of a tensor is as follows: First, consider intuitively that a tensor represents a physical entity which may be characterized by magnitude and multiple directions simultaneously (Fleisch 2012). Therefore, the number of simultaneous directions is denoted  and is called the rank of the tensor in question. In -dimensional space, it follows that a rank-0 tensor (i.e., a scalar) can be represented by  number since scalars represent quantities with magnitude and no direction; similarly, a rank-1 tensor (i.e., a vector) in -dimensional space can be represented by  numbers and a general tensor by  numbers. From this perspective, a rank-2 tensor (one that requires  numbers to describe) is equivalent, mathematically, to an  matrix.

rankobject0scalar1vector2 matrixtensor

The above table gives the most common nomenclature associated to tensors of various rank. Some care must be exhibited, however, because the above nomenclature is hardly uniform across the literature. For example, some authors refer to tensors of rank 2 as dyads, a term used completely independently of the related term dyadic used to describe vector direct products (Kolecki 2002). Following such convention, authors also use the terms triad, tetrad, etc., to refer to tensors of rank 3, rank 4, etc.

Some authors refer to the rank of a tensor as its order or its degree. When defining tensors abstractly by way of tensor products, however, some authors exhibit great care to maintain the separation and distinction of these terms

https://mathworld.wolfram.com/TensorRank.html

https://en.wikipedia.org/wiki/Cantor%27s_theorem

https://en.wikipedia.org/wiki/Cantor%27s_theorem

https://en.wikipedia.org/wiki/Cantor%27s_theorem

https://en.wikipedia.org/wiki/Cantor%27s_theorem

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain.

The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken. This came after many false proofs and mistaken counterexamples in the preceding decades.

The Appel-Haken proof proceeds by analyzing a very large number of reducible configurations. This was improved upon in 1997 by Robertson, Sanders, Seymour, and Thomas who have managed to decrease the number of such configurations to 633 – still an extremely long case analysis. In 2005, the theorem was verified by Georges Gonthier using a general-purpose theorem-proving software.

https://en.wikipedia.org/wiki/Four_color_theorem

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain.

The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken. This came after many false proofs and mistaken counterexamples in the preceding decades.

The Appel-Haken proof proceeds by analyzing a very large number of reducible configurations. This was improved upon in 1997 by Robertson, Sanders, Seymour, and Thomas who have managed to decrease the number of such configurations to 633 – still an extremely long case analysis. In 2005, the theorem was verified by Georges Gonthier using a general-purpose theorem-proving software.

https://en.wikipedia.org/wiki/Four_color_theorem

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain.

The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken. This came after many false proofs and mistaken counterexamples in the preceding decades.

The Appel-Haken proof proceeds by analyzing a very large number of reducible configurations. This was improved upon in 1997 by Robertson, Sanders, Seymour, and Thomas who have managed to decrease the number of such configurations to 633 – still an extremely long case analysis. In 2005, the theorem was verified by Georges Gonthier using a general-purpose theorem-proving software.

https://en.wikipedia.org/wiki/Four_color_theorem