1.2Evaluate and Simplify Algebraic Expressions
... all like terms are combined. Like terms are terms that have the same variable parts. (Constant terms are also considered like terms.) The distributive property allows you to combine like terms by adding coefficients. ...
... all like terms are combined. Like terms are terms that have the same variable parts. (Constant terms are also considered like terms.) The distributive property allows you to combine like terms by adding coefficients. ...
A Contraction Theorem for Markov Chains on General State Spaces
... The proof of Theorem 1.1 is in principle quite easy and could probably be used as a home assignment for graduate students. Thus, let > 0 and η > 0 be given. The idea is simply to show that if 0 is chosen sufficiently small - and much smaller then , and we start our Markov chain in K(0 ), then t ...
... The proof of Theorem 1.1 is in principle quite easy and could probably be used as a home assignment for graduate students. Thus, let > 0 and η > 0 be given. The idea is simply to show that if 0 is chosen sufficiently small - and much smaller then , and we start our Markov chain in K(0 ), then t ...
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF
... are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like ...
... are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like ...
LECTURE 2 1. Finitely Generated Abelian Groups We discuss the
... Theorem 1.5. If A is a finitely generated torsion-free abelian group that has a minimal set of generators with q elements, then A is isomorphic to the free abelian group of rank q. Proof. By induction on the minimal number of generators of A. If A is cyclic (that is, generated by one non-zero elemen ...
... Theorem 1.5. If A is a finitely generated torsion-free abelian group that has a minimal set of generators with q elements, then A is isomorphic to the free abelian group of rank q. Proof. By induction on the minimal number of generators of A. If A is cyclic (that is, generated by one non-zero elemen ...
