T-theory
Encyclopedia
T-theory is a branch of discrete mathematics
dealing with analysis of tree
s and discrete metric spaces.
, a recipient of the Nobel Prize in Chemistry
, in the late seventies. He was trying to fit twenty distinct t-RNA
molecule
s of the E. Coli
bacterium into a tree.
One of the most important concepts of T-theory is the tight span
of a metric space. If X is a metric space, the tight span T(X) of X is, up to isomorphism, the unique minimal injective metric space
that contains X. John Isbell was the first to discover the tight span in 1964, which he called the injective envelope. Dress independently constructed the same construct, which he called the tight span.
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
dealing with analysis of tree
Tree (graph theory)
In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
s and discrete metric spaces.
General history
As per Andreas Dress, T-theory originated from a question raised by Manfred EigenManfred Eigen
Manfred Eigen is a German biophysical chemist who won the 1967 Nobel Prize in Chemistry for work on measuring fast chemical reactions.-Career:...
, a recipient of the Nobel Prize in Chemistry
Nobel Prize in Chemistry
The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...
, in the late seventies. He was trying to fit twenty distinct t-RNA
Transfer RNA
Transfer RNA is an adaptor molecule composed of RNA, typically 73 to 93 nucleotides in length, that is used in biology to bridge the three-letter genetic code in messenger RNA with the twenty-letter code of amino acids in proteins. The role of tRNA as an adaptor is best understood by...
molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s of the E. Coli
Escherichia coli
Escherichia coli is a Gram-negative, rod-shaped bacterium that is commonly found in the lower intestine of warm-blooded organisms . Most E. coli strains are harmless, but some serotypes can cause serious food poisoning in humans, and are occasionally responsible for product recalls...
bacterium into a tree.
One of the most important concepts of T-theory is the tight span
Tight span
In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes...
of a metric space. If X is a metric space, the tight span T(X) of X is, up to isomorphism, the unique minimal injective metric space
Injective metric space
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces...
that contains X. John Isbell was the first to discover the tight span in 1964, which he called the injective envelope. Dress independently constructed the same construct, which he called the tight span.
Application areas
- Phylogenetic analysis, which is used to create phylogenetic treePhylogenetic treeA phylogenetic tree or evolutionary tree is a branching diagram or "tree" showing the inferred evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical and/or genetic characteristics...
s. - Online algorithmOnline algorithmIn computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from...
s - k-server problemK-server problemThe k-server problem is a problem of theoretical computer science in the category of online algorithms, one of two abstract problems on metric spaces that are central to the theory of competitive analysis...
Recent developments
- Bernd SturmfelsBernd SturmfelsBernd Sturmfels is a Professor of Mathematics and Computer Science at the University of California, Berkeley.He received his PhD in 1987 from the University of Washington and the Technische Universität Darmstadt...
, Professor of Mathematics and Computer Science at BerkeleyUniversity of California, BerkeleyThe University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, and Josephine Yu classified six-point metrics using T-theory.