Show that log(r), where this is log base 10, is irrational when r is a positive rational that is not an integral power of 10 I have already proven that e^r is irrational for all rational numbers r
Let m, n be in N, with m, n >= 1 and n odd. Let S_m = 1^n + 2^n + 3^n + ... + m^n. Prove that S_m is divisible by 1+2+...+m.
Remainders of Euclidean Algorithms
Let b = r_0, r_1, r_2, ... be the successive remainders in the Euclidean Algorithm applied to a and b. Show that every 2 steps reduces the remainder by at least one half. In other words, verify that r_{i+2} < (1/2)r_{i}, for every i = 0,1,2,.... Conclude that the Euclidean algorithm terminates in at most 2log_{2}(b) steps, where ...continues
Coding Theory : Vectors and Generator Matrices
Please see the attached file for the fully formatted problems. 1(i) Explain what is meant by (a) a linear code over Fq, (b) the weight w(u) of a vector u and the distance d(u, v) between vectors u and v. (c) Define the weight tu(C) and the minimal distance d(C) of a code C. Prove that w(C) = d(C) if C is linear. (ii) Give ...continues
Please see the attached file for the fully formatted problems. (a) Explain what is meant by (i) a linear code over Fq (ii) the weight w(x) of a vector x (iii) the weight w(C) of a code. Prove that,... (b) Prove that w(C) = d(C) if C is a linear code. (c) Define F-linear equivalence of codes. State the three row and two ...continues
4 (i) Let C be a linear code in IF. Explain what is meant when we say that C is cyclic. Give also the algebraic characterisation of cyclic codes using the ring .... (ii) Explain why the cyclic codes in R are in 1-1 correspondence with the monic polynomials in IFq[xJ that divide — 1. Give the definition of the generator polynom ...continues
Coding Theory : Linear [n,k,d] Code, Coset Leaders and Vector Weight
Let x+C be a coset, and assume x+c, x+e have weight less or equal than t.... See attached file for full problem description.
Coding Theory : Sphere Packing and Coset Leaders
Explain what is meant by the sphere St () with centre i and radius / t in the vector space F. Show that... Let C be a linear [ri, k, dj-code over Fq and set t [i]. Show that... for all distinct elements 7 and of C. Hence show that... Give the definition of a perfect code. Give the definition of a coset leader/’Let C and t be ...continues
Palindromic Polynomials with Z Module Coefficients
I need to do some research on the properties of palindromic polynomials with Z(n) coefficients. I would like information/explanation of polynomials with Z(n) coefficients. I would like to see examples of polynomials with Z(1), Z(2), Z(3), Z(4), Z(5) and in general Z(n) coefficients. Also, I would like to see some examples of ...continues
What's the 'English' meaning of the word 'consecutive'?
