18.906 Problem Set 8 Due Wednesday, April 11 in class
... classes. Recall that SO(n) is the group of n × n orthogonal matrices with determinant +1, and SO(n) is connected. 1. Use Question 4 on problem set 6 to show that H̃p (K(A, m); Q) = 0 for all finite abelian groups A and all m > 0. Use the Serre spectral sequence and Postnikov towers to conclude that ...
... classes. Recall that SO(n) is the group of n × n orthogonal matrices with determinant +1, and SO(n) is connected. 1. Use Question 4 on problem set 6 to show that H̃p (K(A, m); Q) = 0 for all finite abelian groups A and all m > 0. Use the Serre spectral sequence and Postnikov towers to conclude that ...
CPSC 121 - MATHEMATICAL PROOFS 1. Proofs in General
... Definition. The floor function assigns to the real number x the largest integer that is less than or equal to x. In other words, the floor function rounds a real number down to the nearest integer. The value of the floor function is denoted by bxc. Definition. The ceiling function assigns to the rea ...
... Definition. The floor function assigns to the real number x the largest integer that is less than or equal to x. In other words, the floor function rounds a real number down to the nearest integer. The value of the floor function is denoted by bxc. Definition. The ceiling function assigns to the rea ...
MS COMPREHENSIVE EXAM DIFFERENTIAL EQUATIONS SPRING 2007
... This exam has two parts, (A) ordinary differential equations and (B) partial differential equations. Choose three problems in part (A) and four problems in part (B). Mark clearly the problems you choose and show the details of your work. ...
... This exam has two parts, (A) ordinary differential equations and (B) partial differential equations. Choose three problems in part (A) and four problems in part (B). Mark clearly the problems you choose and show the details of your work. ...
PATTERNS, CONTINUED: VERIFYING FORMULAS
... A technique called Mathematic al Induction can be used to verify a formula or expression for the nth term in a pattern. This method is particularly useful when it is easy to describe the pattern recursively; that is, whenever it is easy to describe the procedure for going from the nth term to the ( ...
... A technique called Mathematic al Induction can be used to verify a formula or expression for the nth term in a pattern. This method is particularly useful when it is easy to describe the pattern recursively; that is, whenever it is easy to describe the procedure for going from the nth term to the ( ...
Lecture #36 The Connection Between Atomic Structure and
... Concentrated NaOH is then added until the solubility of Mg(OH)2 is 0.001 times that in H2O alone. (Ignore the change in volume resulting from the addition of NaOH.) The solubility product Ksp of Mg(OH)2 is 1.2 x 10-11 at 25oC. Calculate the concentration of hydroxide ion in the solution after the ad ...
... Concentrated NaOH is then added until the solubility of Mg(OH)2 is 0.001 times that in H2O alone. (Ignore the change in volume resulting from the addition of NaOH.) The solubility product Ksp of Mg(OH)2 is 1.2 x 10-11 at 25oC. Calculate the concentration of hydroxide ion in the solution after the ad ...
Knapsack problem
The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a mass and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography and applied mathematics.The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. It is not known how the name ""knapsack problem"" originated, though the problem was referred to as such in the early works of mathematician Tobias Dantzig (1884–1956), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined.