Existence and uniqueness results are fundamental in mathematics because we often define objects by their properties, and then start calling them "the object", which is fantastically convenient.

But calling something "the object" only makes sense if there exists exactly one, and only one, object that satisfies the properties.

One particular context where these come up very explicitly is in solutions to differential equations, e.g. existence and uniqueness of solutions of partial differential equations.

Sequence of genes under a single promoter. For an example, see E. Coli K-12 MG1655 operon thrLABC.

A single operon may produce multiple different transcription units depending on certain conditions, see: operon vs transcription unit.

Lots of features, but slow because written in Python. A faster version may be csvtools. Also some annoyances like obtuse header handing and missing features like grep + cut in one go: csvgrep and select column in csvkit.

SGML predecessor.

Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.

The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.

The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.

The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument $x$ can now be an operator instead of just a number.

Examples:

An Erdős–Woods number is a specific type of integer related to the study of sets of consecutive integers and their relationships to prime numbers. More formally, an integer \( n \) has an Erdős–Woods number if there exists a positive integer \( k \) such that the set of integers: \[ \{ n, n+1, n+2, \ldots, n+k \} \] contains \( k \) or more prime numbers.

A multi-scenario demo.

Yuri Manin is a prominent Russian mathematician known for his work in the fields of algebraic geometry, number theory, and mathematical physics. Born on April 22, 1937, Manin made significant contributions to various areas, including the theory of motives, algebraic cycles, and arithmetic geometry. He is also recognized for his involvement in the development of noncommutative geometry and has written extensively on topics related to the foundations of mathematics.

"Algerian astrophysicists" refers to astrophysicists from Algeria or those who are of Algerian descent and work in the field of astrophysics. Astrophysicists study the universe, including the physical properties, behavior, and evolution of celestial objects and phenomena. Algeria has made contributions to science and astrophysics through its researchers and institutions, including participation in international collaborations and the development of local scientific capabilities.