Rationality and the Tangent Function
... We close this section with a seemingly unrelated application. Let X be the space obtained from the unit square [0, 1]2 ⊂ R2 by deleting all points with both coordinates rational except (0, 0) and (1, 1). The Baire category theorem implies the existence of a smooth path in X from (0, 0) to (1, 1). Fo ...
... We close this section with a seemingly unrelated application. Let X be the space obtained from the unit square [0, 1]2 ⊂ R2 by deleting all points with both coordinates rational except (0, 0) and (1, 1). The Baire category theorem implies the existence of a smooth path in X from (0, 0) to (1, 1). Fo ...
4. Rings 4.1. Basic properties. Definition 4.1. A ring is a set R with
... Much of the basic material on groups just carries over to rings (or other algebraic structures) in a very straightforward way. We already defined subrings. If R, R0 are rings, then a map ϕ : R → R0 is called a homomorphism if ϕ(a + b) = ϕ(a) + ϕ(b), ϕ(ab) = ϕ(a)ϕ(b), ϕ(1) = 10 . Equivalently, we ask ...
... Much of the basic material on groups just carries over to rings (or other algebraic structures) in a very straightforward way. We already defined subrings. If R, R0 are rings, then a map ϕ : R → R0 is called a homomorphism if ϕ(a + b) = ϕ(a) + ϕ(b), ϕ(ab) = ϕ(a)ϕ(b), ϕ(1) = 10 . Equivalently, we ask ...
How to get the Simplified Expanded Form of a polynomial, I
... this, and the following few, lessons we will see how to write any given polynomial in SEF. To do this we will repeatedly use the fact that multiplication distributes over addition: recall the distributive law a(b + c) = ab + ac, (b + c)a = ba + ca This law can be used in two directions. When we go f ...
... this, and the following few, lessons we will see how to write any given polynomial in SEF. To do this we will repeatedly use the fact that multiplication distributes over addition: recall the distributive law a(b + c) = ab + ac, (b + c)a = ba + ca This law can be used in two directions. When we go f ...
Factorization Methods: Very Quick Overview
... ax3 + bx + c. Each elliptic curve, along with a special point at infinity which we designate O, has an associated group (defined over any field). The group action is defined as follows: to calculate X + Y , draw a line between X and Y ; usually the line will touch the curve in a third point Z; the s ...
... ax3 + bx + c. Each elliptic curve, along with a special point at infinity which we designate O, has an associated group (defined over any field). The group action is defined as follows: to calculate X + Y , draw a line between X and Y ; usually the line will touch the curve in a third point Z; the s ...
Full text
... aV2 (72 -1) / 4. Within the first pair of parentheses is the greatest root of the Pell recurrence, rn+2-2rn+l-rn = 0, while within the second pair is the opposite of the remaining root of the Pell recurrence. This allows us to obtain sum formulas specific for Pell and Pell-Lucas numbers, thanks to ( ...
... aV2 (72 -1) / 4. Within the first pair of parentheses is the greatest root of the Pell recurrence, rn+2-2rn+l-rn = 0, while within the second pair is the opposite of the remaining root of the Pell recurrence. This allows us to obtain sum formulas specific for Pell and Pell-Lucas numbers, thanks to ( ...
Dimension theory
... PM (t) = f (t) = �(Mn )tn is a polynomial in t with coefficients in Z. Now suppose that s ≥ 1 and consider the last generator xs ∈ Aks . Multiplication by xs gives a graded map M → M of degree ks . Let K be the kernel and L the cokernel of this map. Then, for each n ≥ 0 we get an exact sequence: ...
... PM (t) = f (t) = �(Mn )tn is a polynomial in t with coefficients in Z. Now suppose that s ≥ 1 and consider the last generator xs ∈ Aks . Multiplication by xs gives a graded map M → M of degree ks . Let K be the kernel and L the cokernel of this map. Then, for each n ≥ 0 we get an exact sequence: ...
HOMEWORK SOLUTIONS Homework 1: 1. show if a|b and c|d then
... 3. If we just look at the possibilities for the combinations of even and odd numbers we see there are four possibilities: i) a,b,c are all even, ii) a even, b,c odd, iii) b even, a,c odd or iv) c even, a,b odd. The cases ii) and iii) are the same with the roles of a and b switched. The only even pri ...
... 3. If we just look at the possibilities for the combinations of even and odd numbers we see there are four possibilities: i) a,b,c are all even, ii) a even, b,c odd, iii) b even, a,c odd or iv) c even, a,b odd. The cases ii) and iii) are the same with the roles of a and b switched. The only even pri ...
Solutions to Some Review Problems for Exam 3 Recall that R∗, the
... this will yield |aHa−1 | = |H|. Consequently, if H is the only subgroup of G of order n, we get aHa−1 = H for all a ∈ G, and so H is normal in G. 9. Prove that R∗ is isomorphic to R+ × {1, −1} by defining an explicit function from one to the other, and showing that it is a group homomorphism, 1-1, ...
... this will yield |aHa−1 | = |H|. Consequently, if H is the only subgroup of G of order n, we get aHa−1 = H for all a ∈ G, and so H is normal in G. 9. Prove that R∗ is isomorphic to R+ × {1, −1} by defining an explicit function from one to the other, and showing that it is a group homomorphism, 1-1, ...
ALGORITHMS FOR D-FINITE FUNCTIONS 1. Introduction A function
... explicit expression f (x) = · · · , this is not always the case for D-finite functions. Instead, D-finite functions are specified implicitly through the differential equation they satisfy, plus an appropriate number of initial values. The situation is similar to the treatment of algebraic numbers: s ...
... explicit expression f (x) = · · · , this is not always the case for D-finite functions. Instead, D-finite functions are specified implicitly through the differential equation they satisfy, plus an appropriate number of initial values. The situation is similar to the treatment of algebraic numbers: s ...
Sample Problems
... 1. The numbers 1, 2, . . . , 49 are placed in a 7 × 7 table. We then add the numbers in each row and each column. Among these 14 sums we have a even numbers and b odd numbers. Is it possible that a = b? 2. The numbers a1 , a2 , . . . , a108 are written on a circle such that the sum of any 20 consecu ...
... 1. The numbers 1, 2, . . . , 49 are placed in a 7 × 7 table. We then add the numbers in each row and each column. Among these 14 sums we have a even numbers and b odd numbers. Is it possible that a = b? 2. The numbers a1 , a2 , . . . , a108 are written on a circle such that the sum of any 20 consecu ...
