Define contact of order >=n (n integer >=1) for regular curves in R^3 with a common point p and prove that
a. The notion of contact of order >=n is invariant by diffeomorphisms.
b.Two curves have contact of order >=1 at p if and only if they are tangent at p.
Define contact of order >=n (n integer >=1) for regular curves in R^3 with a common point p and prove that a. The notion of contact of order >=n is invariant by diffeomorphisms. b.Two curves...
Suppose that p is prime. For a regular p-gon to be Euclidean constructible, then the roots of r- 1 must be constructible. The roots of zp-4z?-2+ . . . +z+1 together with 1-1 form a regular p-gon. They would need to be constructible. Since z-1 +···+ z + 1 is irreducible, that means the degree of a root of this is p-1. Using this prove that p 22 + 1 for some nonnegative integer n Suppose that p is prime....
Let P(n) be some propositional function. In order to prove P(n) is true for all positive integers, n, using mathematical induction, which of the following must be proven? OP(K), where k is an arbitrary integer with k >= 1 If P(k) is true, then P(k+1) is true, where k is an arbitrary integer with k >= 1 P(O) P(k+1), where k is an arbitrary integer with k>= 1
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
Prove that if an integer n is not divisible by 3, then n^2=3k+1 for some integer k. Note: “not divisible by 3” means either “n=3m+1 for some integer m” or “n=3m+2 for some integer m”.
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
Let (dkdk−1⋯d0)3 be the base 3 representation of integer n ≥ 0. Prove that n is odd if and only if an odd number of the base 3 digits dk, dk−1, . . . , d0 are odd.
1 point Prove the following statement: If n2 is even, then n is even. Order each of the following sentences so that they form a logical proof. Proof by Contrapositive: Choose from these sentences: Your Proof: Suppose n is odd. Then by definitionn 2k +1 for some integer k Required to show if n is not even (odd), then n is not even (odd). Thus n2(2k1)2. n24k2 4k1. 22(22+2k) +1 Thus n2 (an integer) +1 and by definition is odd....
Theorem 16.1. Let p be a prime number. Suppose r is a Gaussian integer satisfying N(r) = p. Then r is irreducible in Z[i]. In particular, if a and b are integers such that a² +62 = p, then the Gaussian integers Ea – bi and £b£ai are irreducible. Exercise 16.1. Prove Theorem 16.1. (Hint: For the first part, suppose st is a factorization of r. You must show that this factorization is trivial. Apply the norm to obtain p=...
4. The Pell sequence P, P... is defined by P-1, P-2, and P-P-2+2P- for n 3. Define T E C(R") by T(z, y)-(y,x + 2y). (a) Prove that T"(0,1) (PP+) for every integer n1 (b) Find the eigenvalues of T. (c) Find a basis of R2 consisting of eigenvectors of T. (d) Use the solution to part (c) to compute T (0,1. Hint: Write (0,1) as a linear combination of eigenvectors.] (e) Use your answers to parts (a) and (d)...
EXERCISE 4.16. Prove that every compact regular surface has a point of positive Gaussian curvature. HINT: LetpES be a point of maximum distance to the origin. By applying Exercise 1.43 on page 32 to a normal section, conclude that the normal cur- vature of S at p in every direction is where r is the distance from p to the origin. EXERCISE 1.43 Let γ: 1 → Rn be a regular curve. Assume that the function t Iy(t) has a...