Ex 3. Consider the following definitions:

Definition: Let a and b be integers. A linear combination of a and b is an expression of the form ax + by, where x and y are also integers. Note that a linear combination of a and b is also an integer.

Definition: Given two integers a and b we say that a divides b, and we write a|b, if there exists an integer k such that b = ka. Moreover, we write a - b if a does not divide b.

For each proof state clearly which technique you used (direct proof, proof by contrapositive, proof by contradiction). Even if you are not able to prove some of the following claims, you can still use them in the proof of the following ones, if needed.

(a) Given the above definition, is it true that a|0 for all a in Z? Is it true that 0|a for all a in Z? Is it true that a|a for all a in Z? Explain your answers.

(b) Prove that if a and b are two integers such that b≠0 and a|b, then |a| ≤ |b|.

(c) Prove that if a, b and c are three integers such that c|a and c|b then c divides any linear combination of a and b.

(d) Let a be a natural number and b be an integer. If a|(b + 1) and a|(b − 1), then a = 1 or a = 2. (Hint: you may use a clever linear combination...)

(e) Prove that if a and b are two integers with a ≥ 2, then a - b or a - b + 1

How important is the use of an appropriate system of symbols to the development of a branch of mathematics?

June Watson is contributing ​$3,500 each year to a Roth IRA. The IRA earns 3.3% per year. How much will she have at the end of 35​years?

Solve the following optimization problem (Be sure to include the statement of the optimization problem and a graph of the feasible in your solution):

Jamie has joined a building contest. A dog shape requires 3 small blocks and one large block to build. A robot shape requires 5 small bricks and 5 large bricks to build. Jamie has a supply of 240 small bricks and 100 large bricks.

If a dog is worth 2 points and a robot is worth 7 points, how many shapes of each type should Jamie build to maximize the points?

Solve the following differential equations:

1. x"(t)+ x(t)=6sin(2t) ; x(0)=3, x'(0)=1

2.y"(t)- y(t)=4cos(t) ; y(0)+0 , y'(0)=1

Given a second order ode m y’’  + c y’ + k y = 0 with  m, c and k all positive.  (like a mass‐spring system with damping) Argue that the solution will always be damped; the exponential portion can never be positive regardless of the particular m, c and k.

For each of the following data sets, write a system of equations to determine the coefficients of the natural cubic spline passing through the given points.

x| 3    4 6

------------------

y| 10 15 35

QUESTION 3 USE LINGO OR MANUAL LP) Chemco produces three products: A, B, and C. They can sell up to 30 pounds of each product at the following prices (per pound): product A, $10; product B, $12; product C, $20. Chemco purchases raw material at $5/lb. Each pound of raw material can be used to produce either 1 lb of A or 1 lb of B. For a cost of $3/lb processed, product A can be converted to .6 lb of product B and .4 lb of product C. For a cost of $2/lb processed, product B can be converted to .8 lb of product C. Formulate an LP whose solution will tell Chemco how to maximize their profit. Solve using using any method (LINGO, SOLVER OR MANUAL LP).

Solve the initial value problem

y′=(2cos(2x))/(3+2y), y(0)=−1

and determine where the solution attains its maximum value (for 0≤x≤1.697).

Enclose arguments of functions in parentheses. For example, sin(2x).

Y(x)=?

x=?

M r. and Mrs. Fox have each contributed $1235.00 per year for the last eight years intoRRSP accounts earning 4.3% compounded annually. Suppose they leave theiraccumulated contributions for another five years in the RRSP at the same rate of interest.

a) How much will Mr. and Mrs. Fox have in total in their RRSP accounts?

b) How much did the Fox's contribute?

c) How much will be interest?

There are 4 mathematicians m1;m2;m3;m4 and 4 computer scientists c1; c2; c3; c4. mi and ci are enemies for each i = 1; 2; 3; 4 (i.e. m1 and c1 are enemies, m2 and c2 are enemies etc.). By the end of part (d), we ought to know how many ways there are to line up these 8 people so that no enemies are next to each other.

(a) How many ways are there to line up the 8 people with no restriction?

(b) How many ways are there to line up the 8 people such that m1 and c1 ARE next to each other? Hint: there are 2 ways to arrange m1 and c1 between themselves. Then once we have done that, we can imagine them as “glued together". So there are now 7 objects to permute (6 people and 1 glued pair).

(c) How many ways are there to line up the 8 people so that m1 and c1 ARE next to each other AND m2 and c2 are next to each other? Hint: use the gluing idea again.

(d) For i = 1; 2; 3; 4 let Ai represent the set of permutations of the people where mi and ci are next to each other (note in part (b), you found |A1|. Use inclusion-exclusion to find the number of bad permutations |A1 U A2 U A3 U A4|. Then conclude the number of good permutations.

Need the math explanation

1. The value of a weight vector is given as (w1=3, w2=-2, w0=1) for a linear model with soft threshold (sigmoid) function f(x). Define a decision boundary, where the values of the feature vector x result in f(x)=0.5. Plot the decision boundary in two dimensions.

2. Generating training samples: In two dimensional feature space x: (x1, x2,1), generate 20 random samples, for different values of (x1,x2), that belong to two different classes C1 (1) and C2 (0). The label of each feature vector is assigned so that the samples are linearly separable, i.e., can be separated by a linear model with a soft threshold (sigmoid) function. Plot the samples you generate in a two dimensional plane of (x1,x2). Hint: You may construct an underlying linear model to cut the plane in two halves. Then generate random samples at either side with proper labels.

3. Construct a quadratic error function using a learn model with a soft threshold (sigmoid) function for augmented feature vectors in n+1 dimensions. Derive a gradient decent algorithm for learning the weights. Write a program using either Matlab or Python to learn the weights using the training samples you generate from Prob. 2. Plot the resulting decision boundary.

4. Consider a linear combination of three radial basis functions. Draw a network structure for the model. Write a (pseudo) algorithm for learning the parameters of the model. (You determine what error function to use, what training samples to use, and write iterative equations for learning the parameters.)

Please show how you got to answer!

Write a linear program for the following problem. (Do not solve.)
A ship is transporting rice and wheat from California to Alaska. It has three cargo holds with the following capacities:

• The forward cargo hold can carry at most 10,000 tons, and at most 400,000 cubic feet

. • The middle cargo hold can carry at 5,000 tons, and at most 250,000 cubic feet.

• The aft cargo hold can carry at most 12,000 tons, and at most 600,000 cubic feet.

In addition, for the ship to be balanced, each cargo hold must be filled to the same fraction of its total capacity, with respect to tonnage.

A ton of wheat takes up 44.7 cubic feet and can be sold at a profit of $20; a ton of rice takes up 40.9 cubic feet and can be sold at a profi t of $18.

The goal is to maximize the profit from the ship’s cargo.

Find ? as a function of ? if 81?″−90?′+29?=0, ?(3)=8,?′(3)=1

y=?