6,052 articles
Transfinite induction
Transfinite induction, also called well-ordered induction, is a proof technique extending ordinary mathematical induction from the natural numbers to any well-ordered class, most prominently the ordinal numbers.
Transfinite number
A transfinite number, also called transfinity, is an infinite ordinal or cardinal number introduced by Georg Cantor in 1883, larger than every natural number yet exactly comparable.
Transform methods in probability
Transform methods in probability replace a probability distribution with a function such as its characteristic or Laplace transform, so sums of independent variables become products and moments follow by differentiation.
Transform of a random variable (generating function)
A transform of a random variable is a function, such as the moment-generating or characteristic function, that encodes its distribution so sums and scaling become simple algebra.
Transformation group
A transformation group is a set of invertible transformations of a space closed under composition, containing the identity and inverses, central to Klein's 1872 Erlangen Program and Lie's theory of continuous groups.
Transformation matrix
A transformation matrix represents a linear transformation between finite-dimensional vector spaces, converting coordinates under chosen bases; 4×4 versions combine rotation, scaling, and translation in computer graphics.
Transformation semigroup
A transformation semigroup is a set of functions from a set to itself that is closed under composition; with the identity function it is a transformation monoid.
Transience and recurrence of Lévy processes
Transience and recurrence of Lévy processes describes whether a Lévy process returns to bounded regions or drifts away, with exactly one of the two always holding.
Transitive closure
The transitive closure of a binary relation R is the smallest transitive relation that contains R. On a directed graph, it records which vertices are reachable from which.
Transitive group action
A transitive group action is a group action on a set with only one orbit, so some group element sends any point to any other.
Transitive relation
A transitive relation (also called the transitive property) is a binary relation in which aRb and bRc together imply aRc, forming the basis of equivalence relations and orderings.
Translation (geometry)
A translation in geometry is a transformation that slides every point of a figure the same distance in the same direction, without rotating or resizing it.
Transmission disequilibrium test
The transmission disequilibrium test (TDT) is a family-based genetic association test asking whether a heterozygous parent transmits an allele to an affected offspring more often than Mendelian expectation.
Transpose
The transpose, also called the matrix transpose or matrix transposition, is the matrix Aᵀ formed by exchanging rows and columns of A, an m×n matrix becoming n×m.
Transposition cipher
A transposition cipher, also known as a permutation cipher, encrypts by rearranging letters rather than replacing them, a classical method whose variants include columnar and rail fence.
Transseries
A transseries, also called a logarithmic-exponential series, is a formal asymptotic series built from real numbers and an infinitely large variable, introduced by Jean Écalle and used in asymptotic differential algebra, model theory, and physics.
Transversality condition
The transversality condition is a boundary condition that an extremal of a variational or optimal control problem must satisfy when an endpoint is not fixed.
Trapezohedron
The trapezohedron, also called deltohedron or antidipyramid, is a convex polyhedron with congruent kite faces, the dual of an antiprism; its pentagonal form is the standard ten-sided die.
Trapezoid
A trapezoid, also spelled trapazoid, is a quadrilateral with one pair of parallel sides, called bases, with an area of half the sum of the bases times the height.
Travelling salesman problem
The traveling salesman problem, or TSP, asks for the shortest route visiting each listed city exactly once and returning to the start; it is NP-hard.
Treataprost study
Treataprost study (NCT05306314), also called GMG.22.TRT.01, is a retrospective chart review by GMG Grand Medical of the herbal supplement Treataprost added to antibiotics and NSAIDs in 140 Turkish men with chronic prostatitis, status not yet recruiting or unknown.
Tree (descriptive set theory)
In descriptive set theory, a tree is a set of finite sequences closed under taking initial segments, the standard device for coding closed, analytic, and coanalytic sets of infinite sequences.
Tree (graph theory)
A tree, or tree graph, is a connected graph with no cycles, having exactly one path between any two vertices and n minus 1 edges on n vertices.
Tree automaton
A tree automaton is a finite-state machine that accepts trees rather than strings by labeling subtrees with states; it recognizes regular tree languages and underpins XML schema validation.
Tree decomposition
A tree decomposition of a graph maps vertices into bags arranged on a tree, covering every edge; the related measure treewidth was introduced by Robertson and Seymour in 1986.
Trefoil knot
The trefoil knot, or cloverleaf knot, is the simplest nontrivial knot in mathematics, obtained by joining the ends of an overhand knot; it is listed as 3₁.
Trend estimation
Trend estimation is the set of statistical methods for detecting and modeling the long-term direction of a time series, using tests like Mann–Kendall and decompositions like STL.
Triality
Triality is the phenomenon special to the Lie group Spin(8) in which its outer automorphisms, found by Élie Cartan in 1925, permute its three 8-dimensional representations.
Triangle
A triangle is a plane figure with three straight sides and three angles, the simplest polygon. Its angles sum to 180 degrees, and its rigidity underpins surveying and structural engineering.
Triangle inequality
The triangle inequality states that any side of a triangle is shorter than the sum of the other two sides, also known as the triangular inequality theorem.