First, starting at the end, numerals are just symbols with pre-determined fixed values. In other words, the idea of “two-ness” or “three-ness” is encapsulated by the numerals “2” and “3.” With that in mind, it’s not such a big leap to understand what a variable is: Just...
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Polynomial is derived from the Greek words "poly" meaning many and "nominal" meaning terms. A polynomial is a mathematical expression that contains two or more algebraic expressions such as constants, variables, and exponents that are combined by addition, subtraction, multiplication, and division. ...
The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.InequalityWordsInterval Notation a<x<ba<x<b all real numbers between a and b, not including a and b (a,b)(a,b) x>ax>a All real ...
Let P=⟨X:R⟩ be a group presentation. Recall from Sect. 1.2.2 that relations (l,r)∈R may involve words in (X∪X−1)∗ that are not freely reduced. However, to reduce notational clutter, we shall suppress mention of the free reduction map ρ:(X∪X−1)∗→F(X) in ...
//P describes the finite field specialization that was used. //The function HeadOfLocalModule has many parameters to fine-tune the //computation. > L; Graded module of dimension 8 over an algebra with generator degrees [ 0, 0, -1, -1, 1, 1 ] over Multivariate rational function field ...
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So the normal words of F〈X〉 modulo I are those monomials that do not contain a LM (gij) as a factor. Now the theorem follows by Proposition 6.1.1.□ Corollary 6.2.2 The map i: L → U(L) is injective. Proof. By Theorem 6.2.1 the elements x¯i for i≥ 1 are linearly ind...
Part 2. Prove that \left\|AB\right\|\leq \left\|A\right\|\cdot\left\|B\right\| for any operator norm or the Frobenius norm. In other words, any operator norm (or the Frobenius norm) is mutually consistent with itself. Proof. For operator norm, \begin{aligned} \left\|A\right\|...
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