Computability of Heyting algebras and Distributive Lattices
... successor relation is easily seen to be computable in the standard ordering 0 < 1 < 2 < . . ., since y is a successor of x when y = x + 1. However, one can construct a more complicated ordering such as 18 < 100 < 34 < 2 < . . ., where it is more difficult to determine if one element is a successor o ...
... successor relation is easily seen to be computable in the standard ordering 0 < 1 < 2 < . . ., since y is a successor of x when y = x + 1. However, one can construct a more complicated ordering such as 18 < 100 < 34 < 2 < . . ., where it is more difficult to determine if one element is a successor o ...
Commutative Algebra I
... The ring Z[i] was used in a paper of Gauss (1828), in which he proved that non-unit elements in Z[i] can be factored uniquely into product of “prime” elements, which is a central property of ordinary integers. He then used this property to prove results on ordinary integers. For example, it is possi ...
... The ring Z[i] was used in a paper of Gauss (1828), in which he proved that non-unit elements in Z[i] can be factored uniquely into product of “prime” elements, which is a central property of ordinary integers. He then used this property to prove results on ordinary integers. For example, it is possi ...
Number Theory The Greatest Common Divisor (GCD) R. Inkulu http
... Note that pairwise relatively prime integers must be mutually relatively prime, but the converse is not necessarily true. ...
... Note that pairwise relatively prime integers must be mutually relatively prime, but the converse is not necessarily true. ...
Rational points on Shimura curves and Galois representations Carlos de Vera Piquero
... ingredients in this thesis is a cyclic Galois covering of Shimura curves over Q XD,` −→ XD associated with an odd prime ` dividing D. It was firstly introduced by Jordan in [Jor81, Chapter 5], and its maximal étale quotient is referred to2 as the Shimura covering of XD at `. Here, XD,` /Q is the co ...
... ingredients in this thesis is a cyclic Galois covering of Shimura curves over Q XD,` −→ XD associated with an odd prime ` dividing D. It was firstly introduced by Jordan in [Jor81, Chapter 5], and its maximal étale quotient is referred to2 as the Shimura covering of XD at `. Here, XD,` /Q is the co ...
Minimal ideals and minimal idempotents
... (1) We work in the semigroup (S =X X, ◦) of Example 1.6, for an arbitrary set X. Check which of the following subsets of S are left resp. right ideals. (a) I = {f ∈ S : ranf ⊆ Y } where Y is a non-empty subset of X (b) J = {f ∈ S : f is not one-one} (c) L = {f ∈ S : f is not onto}. (2) Assume X is a ...
... (1) We work in the semigroup (S =X X, ◦) of Example 1.6, for an arbitrary set X. Check which of the following subsets of S are left resp. right ideals. (a) I = {f ∈ S : ranf ⊆ Y } where Y is a non-empty subset of X (b) J = {f ∈ S : f is not one-one} (c) L = {f ∈ S : f is not onto}. (2) Assume X is a ...
Stanford University Educational Program for Gifted Youth (EPGY
... 6. Find all integers n ≥ 1 so that n3 −1 is prime. Hint: n3 −1 = (n2 +n+1)(n−1). 7. Show that if ac | bc, then a | b. 8. (a) Prove that the product of three consecutive integers is divisible by 6. (b) Prove that the product of four consecutive integers is divisible by 24. (c) Prove that the product ...
... 6. Find all integers n ≥ 1 so that n3 −1 is prime. Hint: n3 −1 = (n2 +n+1)(n−1). 7. Show that if ac | bc, then a | b. 8. (a) Prove that the product of three consecutive integers is divisible by 6. (b) Prove that the product of four consecutive integers is divisible by 24. (c) Prove that the product ...
Lecture 12 CS 282 - Computer Science Division
... By previous results, each of the n+1 terms on the right is less than 1/(n+1), so P(A) < 1. So the predicate “there exists a real number b, the encoding of B such that G(b) =F(B)< 0” is recursively undecidable. Now suppose G(x) 2 R, then so is |G(x)|-G(x) 2 R. We cannot tell if F(x) is zero if we can ...
... By previous results, each of the n+1 terms on the right is less than 1/(n+1), so P(A) < 1. So the predicate “there exists a real number b, the encoding of B such that G(b) =F(B)< 0” is recursively undecidable. Now suppose G(x) 2 R, then so is |G(x)|-G(x) 2 R. We cannot tell if F(x) is zero if we can ...
FINITE SIMPLICIAL MULTICOMPLEXES
... An element m ∈ Γ is called a maximal facet if it does not exist a ∈ Γ with a > m; in other words, if m is maximal with respect to ”≤”. We denote M(Γ) the set of maximal facets of Γ. If a ∈ Γ is a face, the dimension of a is the number dim(a) = |a| − 1. The dimension of Γ is the number dim(Γ) = max{d ...
... An element m ∈ Γ is called a maximal facet if it does not exist a ∈ Γ with a > m; in other words, if m is maximal with respect to ”≤”. We denote M(Γ) the set of maximal facets of Γ. If a ∈ Γ is a face, the dimension of a is the number dim(a) = |a| − 1. The dimension of Γ is the number dim(Γ) = max{d ...
