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In linear algebra, the trace of a square matrix A, denoted tr (A), 1 is the sum of the elements on its main diagonal, . It is only defined for a square matrix (n × n). In mathematical physics, if tr (A) = 0, the matrix is said to be traceless. This misnomer is widely used, as in the definition of Pauli matrices.
Trace class. In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra.
Written in. PHP, HTML and Adobe Flash. Cool Math Games (branded as Coolmath Games) [ a ] is an online web portal that hosts HTML and Flash web browser games targeted at children and young adults. Cool Math Games is operated by Coolmath LLC and first went online in 1997 with the slogan: "Where logic & thinking meets fun & games.".
Trace operator. A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red). In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential ...
Define. By convexity and monotonicity of trace functions, is convex, and so for all , which is, and, in fact, the right hand side is monotone decreasing in . Taking the limit yields, which with rearrangement and substitution is Klein's inequality: Note that if is strictly convex and , then is strictly convex.
Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L, is a K - linear transformation of this vector space into itself. The trace, TrL/K (α), is defined as the trace (in the linear algebra sense) of this linear transformation.
A framed trace diagram is a trace diagram together with a partition of the degree-1 vertices V 1 into two disjoint ordered collections called the inputs and the outputs. The "graph" underlying a trace diagram may have the following special features, which are not always included in the standard definition of a graph:
In mathematics, Abel's identity (also called Abel's formula[ 1 ] or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. The relation can be generalised to n th-order ...
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