Algebraic group actions and quotients - IMJ-PRG
... (iv) If W1 , W2 are disjoint closed G-invariant subsets of X, then π(W1 ) and π(W2 ) are disjoint closed subsets of X. A good quotient is a categorical quotient. We will often say that Y is a good quotient of X by G and use the following notation : Y = X//G. Lemma 2.13. Let G be an algebraic group a ...
... (iv) If W1 , W2 are disjoint closed G-invariant subsets of X, then π(W1 ) and π(W2 ) are disjoint closed subsets of X. A good quotient is a categorical quotient. We will often say that Y is a good quotient of X by G and use the following notation : Y = X//G. Lemma 2.13. Let G be an algebraic group a ...
Topic 3
... Alisha and Brianna completed magic number puzzles for homework. Although their puzzles had the same steps, the two girls got different results. The work for each puzzle is shown side-by-side. For each set of puzzles, tell whose result is correct, then explain why that result is correct. Your explana ...
... Alisha and Brianna completed magic number puzzles for homework. Although their puzzles had the same steps, the two girls got different results. The work for each puzzle is shown side-by-side. For each set of puzzles, tell whose result is correct, then explain why that result is correct. Your explana ...
Divided powers
... the functor from A-algebras to A-modules which maps an A-algebra B to the Amodule homomorphisms !HomA (M, E(B))! from M to E(B) is representable. That is, there is a A-algebra !Γ(M )!, and for every A-algebra B a canonical bijection !ΨM (B) : HomA -alg (Γ(M ), B) → HomA (M, E(B))! from the A-algebra ...
... the functor from A-algebras to A-modules which maps an A-algebra B to the Amodule homomorphisms !HomA (M, E(B))! from M to E(B) is representable. That is, there is a A-algebra !Γ(M )!, and for every A-algebra B a canonical bijection !ΨM (B) : HomA -alg (Γ(M ), B) → HomA (M, E(B))! from the A-algebra ...
Algebraic Expressions
... Every expression is made up of terms. A term is a known number, a variable, or the product of a known number and variable(s). The expression 2x 1 5 has two terms: 2x and 5. A term that is a known number without variables is called a constant. The expression 2x 1 5 has one constant: 5. A term that in ...
... Every expression is made up of terms. A term is a known number, a variable, or the product of a known number and variable(s). The expression 2x 1 5 has two terms: 2x and 5. A term that is a known number without variables is called a constant. The expression 2x 1 5 has one constant: 5. A term that in ...
Nominal Monoids
... the syntactic monoid of a language, or more generally, to act on any monoid, goes back to nominal sets. The theory of nominal sets originates from the work of Frankel in 1922, further developed by Mostowski in the 1930s. At that time, nominal sets were used to prove independence of the axiom of choi ...
... the syntactic monoid of a language, or more generally, to act on any monoid, goes back to nominal sets. The theory of nominal sets originates from the work of Frankel in 1922, further developed by Mostowski in the 1930s. At that time, nominal sets were used to prove independence of the axiom of choi ...
Problem Shortlist with Solutions - International Mathematical Olympiad
... Answer. The sets A for which pA is maximal are the sets the form {d, 5d, 7d, 11d} and {d, 11d, 19d, 29d}, where d is any positive integer. For all these sets pA is 4. Solution. Firstly, we will prove that the maximum value of pA is at most 4. Without loss of generality, we may assume that a1 < a2 < ...
... Answer. The sets A for which pA is maximal are the sets the form {d, 5d, 7d, 11d} and {d, 11d, 19d, 29d}, where d is any positive integer. For all these sets pA is 4. Solution. Firstly, we will prove that the maximum value of pA is at most 4. Without loss of generality, we may assume that a1 < a2 < ...
M13/08
... least one tracial state on the noncommutative solenoid AαS . Moreover, this tracial state is unique if, and only if α is not periodic. Moreover, since noncommutative solenoids carry an ergodic action of the compact groups Sp , if one chooses any continuous length function on Sp , then one may employ ...
... least one tracial state on the noncommutative solenoid AαS . Moreover, this tracial state is unique if, and only if α is not periodic. Moreover, since noncommutative solenoids carry an ergodic action of the compact groups Sp , if one chooses any continuous length function on Sp , then one may employ ...
MATH20302 Propositional Logic
... Remark: Following the usual convention in mathematics we will use symbols such as p, q, respectively s, t, not just for individual propositional variables, respectively propositional terms, but also as variables ranging over propositional variables, resp. propositional terms, (as we did just above). ...
... Remark: Following the usual convention in mathematics we will use symbols such as p, q, respectively s, t, not just for individual propositional variables, respectively propositional terms, but also as variables ranging over propositional variables, resp. propositional terms, (as we did just above). ...
algebraic density property of homogeneous spaces
... Thus besides finite subgroups we are left to consider the one-dimensional reductive subgroups that include C∗ (which can be considered to be the diagonal subgroup since all tori are conjugated) and its finite extensions. The normalizer of C∗ which is its extension by Z2 generated by ...
... Thus besides finite subgroups we are left to consider the one-dimensional reductive subgroups that include C∗ (which can be considered to be the diagonal subgroup since all tori are conjugated) and its finite extensions. The normalizer of C∗ which is its extension by Z2 generated by ...
My Slides - Department of Computer Sciences
... • The halting sequence gives the number of steps Turing machine Tk executes on input I j – If machine Tk does not halt on input I j , then define the number of steps to be the symbol ∞ • The busy beaver sequence gives the maximal number of steps an n state Turing machine can make on an initially bla ...
... • The halting sequence gives the number of steps Turing machine Tk executes on input I j – If machine Tk does not halt on input I j , then define the number of steps to be the symbol ∞ • The busy beaver sequence gives the maximal number of steps an n state Turing machine can make on an initially bla ...