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The Euler Characteristic V-E+F has an interesting history. It was initially stated that, for all polyhedra,
$$V(ertices)-E(dges)+F(aces)=2$$
and its proof was widely accepted, until people found counter-examples.
Imre Lakatos' book Proofs and Refutations has an imagined dialogue between t...
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, as demonstrated with the counterexample below.
One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, like e.g. Laurent Schwartz's theory of distributions.
The existence of smooth but non-analytic functions represents one...
The direct sum is an operation from abstract algebra, a branch of mathematics. For example, the direct sum
R
⊕
R
{\displaystyle \mathbf {R} \oplus \mathbf {R} }
, where
R
{\displaystyle \mathbf {R} }
is real coordinate space. Then
R
⊕
R
{\displaystyle \mathbf {R} \oplus \mathbf {R} }
is the Cartesian plane,
...
When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$.
Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$?
Thank you very much.
Edit: $A \times B$ is direct product and $A \oplus B$ is direct sum.
Edit: I am asking this qu...
