M09/12
... For adequate L, M, L ≺ M means: For each M1 , M2 ∈ M and zero-sets Zi ⊇ Mi , there is an L ∈ L with L ⊆ Z1 ∩ Z2 . (“Adequate” refers to the filter F. If necessary, we shall say “F-adequate”.) (X, F) has the Topological Group Property T GP if [∀ adequate L ∃ adequate M z with L ≺ M]. Thus, (X, F) fai ...
... For adequate L, M, L ≺ M means: For each M1 , M2 ∈ M and zero-sets Zi ⊇ Mi , there is an L ∈ L with L ⊆ Z1 ∩ Z2 . (“Adequate” refers to the filter F. If necessary, we shall say “F-adequate”.) (X, F) has the Topological Group Property T GP if [∀ adequate L ∃ adequate M z with L ≺ M]. Thus, (X, F) fai ...
lecture 3
... algebras of Tarskian frames do not coincide x := 1; y := 2 and y := 2; x := 1 are equivalent in the relation algebra but not in the trace algebra Question Can we find algebras that are universal for the Tarskian trace and relation algebras? (i.e., that play the same role as the regular sets of ...
... algebras of Tarskian frames do not coincide x := 1; y := 2 and y := 2; x := 1 are equivalent in the relation algebra but not in the trace algebra Question Can we find algebras that are universal for the Tarskian trace and relation algebras? (i.e., that play the same role as the regular sets of ...
Definition of a quotient group. Let N ¢ G and consider as before the
... Expressed differently, we have [a] · [b] = [ab]. It is now easy to see that G/N is a group: (i) The associative law holds as it holds in G. Thus ([a] · [b]) · [c] = [ab] · [c] = [(ab)c] = [a(bc)] = [a] · [bc] = [a] · ([b] · [c]). (ii) The identity is [e] = N as [a] · [e] = [a · e] = [a] and [e] · [a ...
... Expressed differently, we have [a] · [b] = [ab]. It is now easy to see that G/N is a group: (i) The associative law holds as it holds in G. Thus ([a] · [b]) · [c] = [ab] · [c] = [(ab)c] = [a(bc)] = [a] · [bc] = [a] · ([b] · [c]). (ii) The identity is [e] = N as [a] · [e] = [a · e] = [a] and [e] · [a ...
TXProving the Pythagorean Theorem
... The Pythagorean theorem can be proved using altitudes and similar triangles. ...
... The Pythagorean theorem can be proved using altitudes and similar triangles. ...
On Two Function-Spaces which are Similar to L0
... Lorentz function spaces (see [10] for the original definition of A, ), it seems to be appropriateto denote the collection of all functions on (0, x ) with a finite distribution function by the symbol AO, and we have chosen to do so. In ?2 the Banach subspacesof Ao are investigated. It is proved that ...
... Lorentz function spaces (see [10] for the original definition of A, ), it seems to be appropriateto denote the collection of all functions on (0, x ) with a finite distribution function by the symbol AO, and we have chosen to do so. In ?2 the Banach subspacesof Ao are investigated. It is proved that ...
SHIMURA CURVES LECTURE 5: THE ADELIC PERSPECTIVE
... is not clear that any of the abelian varieties we’ve constructed actually have endomorphism ring isomorphic to O rather than merely containing it.) However, when we tried to show that V (O) parameterized all 2n-dimensional abelian varieties with O-QM we did not succeed (nor should we have!). By anal ...
... is not clear that any of the abelian varieties we’ve constructed actually have endomorphism ring isomorphic to O rather than merely containing it.) However, when we tried to show that V (O) parameterized all 2n-dimensional abelian varieties with O-QM we did not succeed (nor should we have!). By anal ...
On linear dependence in closed sets
... An example of a perfect Kronecker set of the Cantor type was given in Rudin [4]. Theorem 2 provides a positive answer to the question, raised by Kahane-Salem in [3], concerning the equivalence of Carleson-sets and Helson-sets. Both notions are equivalent to weak Kronecker sets. The following theorem ...
... An example of a perfect Kronecker set of the Cantor type was given in Rudin [4]. Theorem 2 provides a positive answer to the question, raised by Kahane-Salem in [3], concerning the equivalence of Carleson-sets and Helson-sets. Both notions are equivalent to weak Kronecker sets. The following theorem ...
16. Homomorphisms 16.1. Basic properties and some examples
... (F \ {0}, ·). Define the map ϕ(A) = det(A). In this example ϕ is a homomorphism thanks to the formula det(AB) = det(A) det(B). Note that while this formula holds for all matrices (not necessarily invertible ones), in the example we have to restrict ourselves to invertible matrices since the set M at ...
... (F \ {0}, ·). Define the map ϕ(A) = det(A). In this example ϕ is a homomorphism thanks to the formula det(AB) = det(A) det(B). Note that while this formula holds for all matrices (not necessarily invertible ones), in the example we have to restrict ourselves to invertible matrices since the set M at ...
7. Rationals
... common difference between their first and second components. In this section, we build the rationals as equivalence classes of an equivalence relations on ordered pairs of integers; the equivalence relation we will use identifies ordered pairs with a common quotient of their first and second compone ...
... common difference between their first and second components. In this section, we build the rationals as equivalence classes of an equivalence relations on ordered pairs of integers; the equivalence relation we will use identifies ordered pairs with a common quotient of their first and second compone ...