Lesson 8-5
... equation. A quadratic equation is an equation that can be written in the standard form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax2 + bx + c 0 = ax2 + bx + c ax2 + bx + c = ...
... equation. A quadratic equation is an equation that can be written in the standard form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. y = ax2 + bx + c 0 = ax2 + bx + c ax2 + bx + c = ...
Brauer-Thrall for totally reflexive modules
... (1.2) Theorem. If there exists a totally reflexive R-module without free summands, which is presented by a matrix that has a column or a row with only one non-zero entry, then that entry is an exact zero divisor in R. These two results—the latter of which is distilled from Theorem (5.3)—show that ex ...
... (1.2) Theorem. If there exists a totally reflexive R-module without free summands, which is presented by a matrix that has a column or a row with only one non-zero entry, then that entry is an exact zero divisor in R. These two results—the latter of which is distilled from Theorem (5.3)—show that ex ...
Cohomology of Categorical Self-Distributivity
... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...
... A quandle, X, is a set with a binary operation (a, b) 7→ a / b such that (I) For any a ∈ X, a / a = a. (II) For any a, b ∈ X, there is a unique c ∈ X such that a = c / b. (III) For any a, b, c ∈ X, we have (a / b) / c = (a / c) / (b / c). A rack is a set with a binary operation that satisfies (II) a ...
Solutions Sheet 7
... Solution: (a) If X were affine, it would be isomorphic to Spec OX (X). We claim that this is not the case. Similarly to the example of P1k in the lecture, we compute the ring of global sections OX (X) = OX (U1 ) ∩ OX (U2 ), where the intersection is as subrings of OX (U12 ) = k[X1 , X1−1 ]. This yie ...
... Solution: (a) If X were affine, it would be isomorphic to Spec OX (X). We claim that this is not the case. Similarly to the example of P1k in the lecture, we compute the ring of global sections OX (X) = OX (U1 ) ∩ OX (U2 ), where the intersection is as subrings of OX (U12 ) = k[X1 , X1−1 ]. This yie ...
A NEW EXAMPLE OF NON-AMORPHOUS ASSOCIATION
... [3] (see also [5]). Ito, Munemasa and Yamada constructed amorphous association schemes over Galois rings. Clearly, in an amorphous association scheme, every nontrivial relation is a strongly regular graph. A. V. Ivanov [8] conjectured the converse also holds, but later it was disproved by van Dam [1 ...
... [3] (see also [5]). Ito, Munemasa and Yamada constructed amorphous association schemes over Galois rings. Clearly, in an amorphous association scheme, every nontrivial relation is a strongly regular graph. A. V. Ivanov [8] conjectured the converse also holds, but later it was disproved by van Dam [1 ...
Rational Numbers – Fractions, Decimals and Calculators
... has been converted into 0.75, its decimal form. If the remainder was anything other than zero, we could continue the process followed in steps (3), (4), and (5). Each cycle through these steps adds one more digit to the decimal answer. If the decimal does not terminate, then – eventually – you would ...
... has been converted into 0.75, its decimal form. If the remainder was anything other than zero, we could continue the process followed in steps (3), (4), and (5). Each cycle through these steps adds one more digit to the decimal answer. If the decimal does not terminate, then – eventually – you would ...
