Constructing elliptic curves over finite fields with prescribed torsion
... a. Determine a minimal element C ′ of ρ(C, k). b. If v(C ′ ) < v(C), then set ϕ ← ϕ ◦ φ(C ′ , C), C ← C ′ , and k ← 0. c. Set k ← k + 1. 3. Output C1 = C and ϕ. We note that the output curve C1 is birationally equivalent to C0 via the map ϕ, and it is clearly R-optimal. Moreover, v(C1 ) ≤ v(C) for a ...
... a. Determine a minimal element C ′ of ρ(C, k). b. If v(C ′ ) < v(C), then set ϕ ← ϕ ◦ φ(C ′ , C), C ← C ′ , and k ← 0. c. Set k ← k + 1. 3. Output C1 = C and ϕ. We note that the output curve C1 is birationally equivalent to C0 via the map ϕ, and it is clearly R-optimal. Moreover, v(C1 ) ≤ v(C) for a ...
Artin E. Galois Theo..
... have only the trivial solution. If they have only the trivial solution, then the column vectors are independent. It follows that the original n equations in n unknowns will have a unique solution if they have any solution, since the difference, term by term, of two distinct solutions would be a non- ...
... have only the trivial solution. If they have only the trivial solution, then the column vectors are independent. It follows that the original n equations in n unknowns will have a unique solution if they have any solution, since the difference, term by term, of two distinct solutions would be a non- ...
Document
... Theorem: Let K be a differential field and f be from K . Then an elementary extension of the field K which has the same field of constants as K and contains an element g such that g’ = f exists if and only if there exist constants c1..cn from K and functions u0,..,un from K such that f= u0’+ i=1,.. ...
... Theorem: Let K be a differential field and f be from K . Then an elementary extension of the field K which has the same field of constants as K and contains an element g such that g’ = f exists if and only if there exist constants c1..cn from K and functions u0,..,un from K such that f= u0’+ i=1,.. ...
Augmented precision square roots, 2
... The unevaluated sum yh +y` is called an expansion. Several algorithms have been suggested for performing arithmetic on such expansions [3], [9]–[11]. In the first part of this paper, we briefly recall two wellknown error-free transforms used later on in the paper. Then, we analyze an augmented-preci ...
... The unevaluated sum yh +y` is called an expansion. Several algorithms have been suggested for performing arithmetic on such expansions [3], [9]–[11]. In the first part of this paper, we briefly recall two wellknown error-free transforms used later on in the paper. Then, we analyze an augmented-preci ...
Homology With Local Coefficients
... or a uniformoperatorringfor {G.}, it is also one forthe inducedsystem. It followsfrom(b) that,if R' is the coveringspace of R corresponding to the subgroupF' of F, the inducedsystemis simple. Thus any systemin R can be consideredas the continuousimage of a simplesystemin some coveringspace. It is na ...
... or a uniformoperatorringfor {G.}, it is also one forthe inducedsystem. It followsfrom(b) that,if R' is the coveringspace of R corresponding to the subgroupF' of F, the inducedsystemis simple. Thus any systemin R can be consideredas the continuousimage of a simplesystemin some coveringspace. It is na ...
Script 2013W 104.271 Discrete Mathematics VO (Gittenberger)
... A the adjacency matrix and D the degree matrix (with on its diagonal: d(v1 ), d(v2 ), . . . , d(vn )). The number of spanning trees is: | det((D − A)0 )|, in which (D − A)0 is the matrix D − A with one row and one column deleted. In the case that G is not connected, the same principle can be applied ...
... A the adjacency matrix and D the degree matrix (with on its diagonal: d(v1 ), d(v2 ), . . . , d(vn )). The number of spanning trees is: | det((D − A)0 )|, in which (D − A)0 is the matrix D − A with one row and one column deleted. In the case that G is not connected, the same principle can be applied ...
Review/Outline Frobenius automorphisms Other roots of equations Counting irreducibles
... quintics are also primitive. In other words, α is a primitive root in F32 . And any (non-zero) element of F32 is of the form αt for some t in the range 1 ≤ t ≤ 31. Thus, we might try plugging α, α2 , α3 , etc into Q(x) to see whether we get 0. That is, replace x by x2 , x3 , x4 , etc and reduce modu ...
... quintics are also primitive. In other words, α is a primitive root in F32 . And any (non-zero) element of F32 is of the form αt for some t in the range 1 ≤ t ≤ 31. Thus, we might try plugging α, α2 , α3 , etc into Q(x) to see whether we get 0. That is, replace x by x2 , x3 , x4 , etc and reduce modu ...
Solvable Groups
... hei = H0 ⊆ H1 ⊆ · · · ⊆ Hn = G such that, for each i, the subgroup Hi is normal in Hi+1 and the quotient group Hi+1 /Hi is Abelian. An Abelian group G is solvable; the chain of subgroups hei ⊂ G satisfies the definition. Also, the symmetric groups S3 and S4 are solvable by considering the chains hei ...
... hei = H0 ⊆ H1 ⊆ · · · ⊆ Hn = G such that, for each i, the subgroup Hi is normal in Hi+1 and the quotient group Hi+1 /Hi is Abelian. An Abelian group G is solvable; the chain of subgroups hei ⊂ G satisfies the definition. Also, the symmetric groups S3 and S4 are solvable by considering the chains hei ...
The Proof Complexity of Polynomial Identities
... such that the equations c = a + b and d = a0 · b0 hold in R). Convention: 1. When speaking about equational proofs over some ring R we refer to the systems P(R). 2. Associativity of addition allows us to identify (a+b)+c with a + (b + c), orP simply a + b + c. We can also abbreviate n a1 + · · · + a ...
... such that the equations c = a + b and d = a0 · b0 hold in R). Convention: 1. When speaking about equational proofs over some ring R we refer to the systems P(R). 2. Associativity of addition allows us to identify (a+b)+c with a + (b + c), orP simply a + b + c. We can also abbreviate n a1 + · · · + a ...
Interactive Formal Verification (L21) 1 Sums of Powers, Polynomials
... by (simp only: l1 l2 gauss) ...
... by (simp only: l1 l2 gauss) ...
Chap 6
... In other words, no matter which elements in each sector we choose, the sums and products are equivalent. This allows us to compute sums and products by choosing any representatives we want. Example 6.2. Referring to the equivalence classes for m = 5 of Example 6.1 and the letters of Theorem 5.3, tak ...
... In other words, no matter which elements in each sector we choose, the sums and products are equivalent. This allows us to compute sums and products by choosing any representatives we want. Example 6.2. Referring to the equivalence classes for m = 5 of Example 6.1 and the letters of Theorem 5.3, tak ...