6.s897 Algebra and Computation Lecture 24

6.s897 Algebra and Computation Lecture 4 2.1 Computational Model

We now dicuss algorithms for performing basic operations with polynomials, such as polynomial multiplication, division with remainder, and evaluation at multiple points. Most of today focuses on efficiently multiplying two degree n polynomials. We will see a simple O(n log n) Fourier transformation (FT) based algorithm under sufficiently nice conditions and then show how this gives rise to a ge...

6.s897 Algebra and Computation Lecture 3

1. (Identity) There exists e ∈ G such that for all a ∈ G, a · e = a. 2. (Associativity) For all a, b, c ∈ G, a · (b · c) = (a · b) · c. Definition 2 (Group) A monoid (G, ·) is a group iff for all a ∈ G, there exists an element b ∈ G such that a · b = e. We say a group (G, ·) is commutative or Abelian iff for all a, b ∈ G, a · b = b · a. Definition 3 (Ring) For a set R and binary operators · and...

6.s897 Algebra and Computation Lecture 12

Given an n-bit integer N , output YES if n is prime and NO otherwise. This is one of the most basic questions about numbers, with the following history. • By definition Prime ∈ coNP, because the prime decomposition is a short certificate for a number that is not prime. • [Pratt'75] showed that Prime ∈ NP. The Pratt certificate of a number N being prime, is by looking at all prime factor q of N ...

Algebra and Computation Course Instructor : V . Arvind Lecture 24 : The Hidden Subgroup Problem

Decreased levels of pNR1 S897 protein in the cortex of neonatal Sprague Dawley rats with hypoxic-ischemic or NMDA-induced brain damage

Our objective was to investigate the protein level of phosphorylated N-methyl-D-aspartate (NMDA) receptor-1 at serine 897 (pNR1 S897) in both NMDA-induced brain damage and hypoxic-ischemic brain damage (HIBD), and to obtain further evidence that HIBD in the cortex is related to NMDA toxicity due to a change of the pNR1 S897 protein level. At postnatal day 7, male and female Sprague Dawley rats ...