Algebra Notes
... and therefore it is possible to construct cos(2π/7). A true fact about the number cos(2π/7) (which we will not prove) is that it is a root of the polynomial 8x3 + 4x2 − 4x − 1 ∈ Q[x]. Using the same technique as in the last section, we can check that this polynomial has no roots in Q so is irreducib ...
... and therefore it is possible to construct cos(2π/7). A true fact about the number cos(2π/7) (which we will not prove) is that it is a root of the polynomial 8x3 + 4x2 − 4x − 1 ∈ Q[x]. Using the same technique as in the last section, we can check that this polynomial has no roots in Q so is irreducib ...
The Number of t-Cores of Size n
... multipartition of x into t partitions must have at least t − x zero partitions. We will count the number of multipartitions lying in M (x, t) that have exactly a non-zero partitions. Firstly, let us count the size of M 0 (x, a), the multipartitions of x into a partitions, none of which is zero. If a ...
... multipartition of x into t partitions must have at least t − x zero partitions. We will count the number of multipartitions lying in M (x, t) that have exactly a non-zero partitions. Firstly, let us count the size of M 0 (x, a), the multipartitions of x into a partitions, none of which is zero. If a ...
Solving Quadratic Equations
... Step #1: Write the equation in standard form: polynomial = 0 NOTE: If the equation is already in the form factored polynomial = 0, this step is unnecessary. Step #2: Factor the polynomial expression so it appears as factor · factor · factor = 0. Step #3: Set each factor containing a variable equal t ...
... Step #1: Write the equation in standard form: polynomial = 0 NOTE: If the equation is already in the form factored polynomial = 0, this step is unnecessary. Step #2: Factor the polynomial expression so it appears as factor · factor · factor = 0. Step #3: Set each factor containing a variable equal t ...
The Rational Numbers - Stony Brook Mathematics
... One student presented the following question: Based on the definition, should constants be considered irreducible polynomials? Since any polynomial f(x)=c, where c0, is invertible, we can consider all such polynomials as being irreducible. Based on our intuition, it might seem fair to suggest that ...
... One student presented the following question: Based on the definition, should constants be considered irreducible polynomials? Since any polynomial f(x)=c, where c0, is invertible, we can consider all such polynomials as being irreducible. Based on our intuition, it might seem fair to suggest that ...
1 Theorems
... • Let f, g : X → Y be continuous maps. f is Homotopic to g (written f g̃) if there exists a map F : X × I → Y such that F (t, 0) = f and F (t, 1) = g • Homotopy Equivalent (Homotopy Type): Two spaces X and Y are said to be homotopy equivalent if there exist maps f : X → Y and g : Y → X with the prop ...
... • Let f, g : X → Y be continuous maps. f is Homotopic to g (written f g̃) if there exists a map F : X × I → Y such that F (t, 0) = f and F (t, 1) = g • Homotopy Equivalent (Homotopy Type): Two spaces X and Y are said to be homotopy equivalent if there exist maps f : X → Y and g : Y → X with the prop ...
Solutions - Mu Alpha Theta
... 30. Note that differentiation of a function reverses its “parity,” that is, the derivative of an even function is odd, and vice versa. Thus, f is odd and f is even. From this, the sum of the missing entries is 6 0 0 18 84 -60 . 31. If (7, 2) is a point on g, then the corresponding poi ...
... 30. Note that differentiation of a function reverses its “parity,” that is, the derivative of an even function is odd, and vice versa. Thus, f is odd and f is even. From this, the sum of the missing entries is 6 0 0 18 84 -60 . 31. If (7, 2) is a point on g, then the corresponding poi ...
an implicit function theorem for nondifferentiable mappings 141
... us to obtain existence theorems for periodic solutions and solutions of boundary value problems for ordinary differential equations. Generally implicit function theorems are not related to the compactness of operators, since they can be proved by Banach's contraction principle. However, operators de ...
... us to obtain existence theorems for periodic solutions and solutions of boundary value problems for ordinary differential equations. Generally implicit function theorems are not related to the compactness of operators, since they can be proved by Banach's contraction principle. However, operators de ...
