Finite Fields
... • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn w ...
... • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn w ...
Non-Commutative Arithmetic Circuits with Division
... the inverse of the generic n × n matrix X requires n nested inversions, namely has height n. Our lower bound on formula size of matrix inverse is obtained by showing that a formula of size s can compute a function of height at most logarithmic in s. This is obtained via a general balancing procedure ...
... the inverse of the generic n × n matrix X requires n nested inversions, namely has height n. Our lower bound on formula size of matrix inverse is obtained by showing that a formula of size s can compute a function of height at most logarithmic in s. This is obtained via a general balancing procedure ...
MA2215: Fields, rings, and modules
... 1. Clearly, it is enough to check it for f(x) = xk , since every polynomial is a linear combination of these, and if x − a divides each of the summands, it divides the whole sum too. But xk − ak = (x − a)(xk−1 + xk−2 a + . . . + xak−2 + ak−1 ). The statement about the roots is clear: f(x) = q(x)(x − ...
... 1. Clearly, it is enough to check it for f(x) = xk , since every polynomial is a linear combination of these, and if x − a divides each of the summands, it divides the whole sum too. But xk − ak = (x − a)(xk−1 + xk−2 a + . . . + xak−2 + ak−1 ). The statement about the roots is clear: f(x) = q(x)(x − ...
IDEAL CONVERGENCE OF BOUNDED SEQUENCES 1
... Some objects related to an ideal can be quite complex, for example its quotient Boolean algebra P (ω) /I (see e.g. [7]). In Section 3 we show how the BolzanoWeierstrass property distinguishes quotient Boolean algebras. In Section 5 we also investigate splitting families of quotient Boolean algebras. ...
... Some objects related to an ideal can be quite complex, for example its quotient Boolean algebra P (ω) /I (see e.g. [7]). In Section 3 we show how the BolzanoWeierstrass property distinguishes quotient Boolean algebras. In Section 5 we also investigate splitting families of quotient Boolean algebras. ...
Non-commutative arithmetic circuits with division
... Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division. It is chiefly interested in commutative polynomials or rational functions, defined over a set of multiplicatively commuting variables (see th ...
... Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division. It is chiefly interested in commutative polynomials or rational functions, defined over a set of multiplicatively commuting variables (see th ...
Noncommutative Monomial Symmetric Functions.
... Parts of the composition Ie conjugate to a composition I can be read from the diagram of the composition I from left to right and from bottom to top. A partition is a composition with weakly decreasing parts, i.e. λ = (λ1 , . . . , λn ) with λ1 ≥ λ2 ≥ . . . ≥ λn The number of times an integer i occu ...
... Parts of the composition Ie conjugate to a composition I can be read from the diagram of the composition I from left to right and from bottom to top. A partition is a composition with weakly decreasing parts, i.e. λ = (λ1 , . . . , λn ) with λ1 ≥ λ2 ≥ . . . ≥ λn The number of times an integer i occu ...