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:

This is good software.

If it only it were written in JavaScript instead of Haskell (!?), then Ciro might have used it as the basis for OurBigBook Markup.

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.

"Here be dragons" is a phrase that historically appeared on ancient maps to denote uncharted territories, suggesting that these areas were dangerous or unknown and should be approached with caution. The phrase symbolizes the fear of the unknown and the mysteries that lie beyond the familiar world. It has since evolved into a metaphor for uncertainty, risk, or areas of ignorance in various contexts, such as exploration, knowledge, and even personal endeavors.

A pyramidal alkene doesn't exist as a distinct category in traditional organic chemistry. However, the term might refer to alkenes that possess a certain spatial arrangement or stereochemistry. In organic chemistry, alkenes are compounds that contain at least one carbon-carbon double bond (C=C). They are typically characterized by a planar geometry around the double bond due to the sp² hybridization of the carbon atoms involved in the double bond, leading to a trigonal planar configuration.

A triple bond is a type of chemical bond that involves the sharing of three pairs of electrons between two atoms. This bond is stronger than a single bond (which shares one pair of electrons) and a double bond (which shares two pairs of electrons). In a triple bond, the two atoms involved each contribute three electrons, resulting in a total of six electrons being shared.

Discrete Mathematics and Theoretical Computer Science are two interrelated fields that contribute significantly to computer science as a whole. Here's a brief overview of each: ### Discrete Mathematics Discrete Mathematics is a branch of mathematics that deals with discrete (as opposed to continuous) objects. It includes a variety of topics that are foundational to computer science, including: 1. **Set Theory**: The study of sets, which are collections of objects.

The Hilbert–Samuel function is an important concept in commutative algebra and algebraic geometry, particularly in the study of the structure of space defined by ideals in rings and the geometry of schemes. It provides a way to measure the growth of the dimensions of the graded components of the quotient of a Noetherian ring by an ideal.