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Note: we define a location in an expression as a leaf node in the syntax tree. Variable binding occurs when that location is below the node n. In the lambda calculus, x is a bound variable in the term M = λx. T and a free variable in the term T. We say x is bound in M and free in T. If T contains a subterm λx. U then x is rebound in this term.
The variables corresponding to the columns of the identity matrix are called basic variables while the remaining variables are called nonbasic or free variables. If the values of the nonbasic variables are set to 0, then the values of the basic variables are easily obtained as entries in b {\displaystyle \mathbf {b} } and this solution is a ...
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
All other variables are called free. For example, in the expression λy.x x y, y is a bound variable and x is a free variable. Also a variable is bound by its nearest abstraction. In the following example the single occurrence of x in the expression is bound by the second lambda: λx.y (λx.z x).
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected ...
The characteristic polynomial of a square matrix is an example of application of Vieta's formulas. The roots of this polynomial are the eigenvalues of the matrix . When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain – up to their sign – the coefficients of the characteristic polynomial, which are ...
The square matrix of second partial derivatives of a function of several variables: Householder matrix: The matrix of a reflection with respect to a hyperplane passing through the origin: Jacobian matrix: The matrix of the partial derivatives of a function of several variables: Moment matrix: Used in statistics and Sum-of-squares optimization ...
The matrix form of the separation of variables is the Kronecker sum. As an example we consider the 2D discrete Laplacian on a regular grid : L = D x x ⊕ D y y = D x x ⊗ I + I ⊗ D y y , {\displaystyle L=\mathbf {D_{xx}} \oplus \mathbf {D_{yy}} =\mathbf {D_{xx}} \otimes \mathbf {I} +\mathbf {I} \otimes \mathbf {D_{yy}} ,\,}