Formulate the definitions of linear dependence and independence of a set of k vector functions f₁(x), ... , f_k(x) continuous on an interval I.

Solve the following problem using the simplex method. If the problem is two dimensional, graph the feasible region, and outline the progress of the algorithm.

Max               Z = 5X₁ + 3X₂ + 2X₃

Subject to    4X₁ + 5X₂ + 2X_{3 +} X4≤ 20

                     3X₁ + 4X₂ - X_{3 +} X4≤ 30

                      X₁, X2, X_3, X₄ ≥ 0

Assuming we have a case of influenza. Suppose the total cost of providing viraflu is $100 and the total cost of providing supportive care is $10. Suppose further that viraflu will result in a 0.5 QALY per person treated and providing supportive care alone results in 0.1 QALY.

1. What is the incremental cost-effectiveness of providing viraflu to persons with influenza relative to providing supportive care alone?

2. Suppose that the total cost of vaccinating an individual is $150 and that vaccination results in the gain of 0.75 QALY per person. Calculate the average cost-effectiveness of influenza vaccination and the incremental cost-effectiveness of influenza vaccination relative to treatment?

Find the solution to the heat equation in 3 dimensions on the half space z>0, with homogeneous Neumann boundary conditions at z=0.

. A routing transit number (RTN) is a bank code that appears in the bottom of checks. The most common form of an RTN has nine digits, where the last digit is a check digit. If d1d2 . . . d9 is a valid RTN, the congruence 3(d1 + d4 + d7) + 7(d2 + d5 + d8) + (d3 + d6 + d9) ≡ 0 (mod 10) must hold. (a) Show that the check digit of the RTN can detect all single errors. (b) Determine which transposition errors an RTN check digit can catch and which ones it cannot catch.

Give Examples (this is complex analysis):

(a.) First characterize open and closed sets in terms of their boundary points. Then give two examples of sets satisfying the given condition: one set that is bounded (meaning that there is some real number R > 0 such that |z| is greater than or equal to R for every z in S), and one that is not bounded. Give your answer in set builder notation. Finally, choose one of your two examples and prove that is neither open nor closed.

(b.) Give two examples of a function f: C→C that is continuous at z=0 but not differentiable at z=0 using the Cauchy-Riemann equations.

(c.) Find a cube root of -1, other than -1, in two ways: first, by using high school algebra (solve the equation z^3= -1 by factoring the polynomial z^3+1 as z+1 times a quadratic polynomial and then determine the roots of the quadratic polynomial) and second, by using the formula for computing nth roots of a complex number.

Recall the following definition: For X a topological space, and for A ⊆ X, we define the closure of A as cl(A) = ⋂{B ⊆ X : B is closed in X and A ⊆ B}. Let x ∈ X. Prove that x ∈ cl(A) if and only if every neighborhood of x contains a point from A. You may not use any definitions of cl(A) other than the one given.

Let X be a set with infinite cardinality. Define T_x = {U ⊆ X : U = Ø or X \ U is finite}. Prove that T_x is a topology on X. (T_x is called the Cofinite Topology or Finite Complement Topology.)

Using the method of recursion, compute y[n] for n = 0 to 20, when x[n]=u[n] and y[-1]=2:

?[? + 1] − 0.8?[?] = ?[?]

Find a closed-form expression for your result.

What are elliptic and hyperbolic geometries? Why were they developed? Provide references

Use Euclid’s algorithm to find integers x, y and d for which 3936 x + 1293 y = d is the smallest possible positive integer. Using your answers to this as your starting point, do the following tasks. (a) Find a solution of 3936 x ≡ d mod 1293. (b) Find an integer r that has the property that r ≡ d mod 1293 and r ≡ 0 mod 3936. (c) Find an integer R that has the property that R ≡ 126 mod 1293 and R ≡ 0 mod 3936. (d) Find an integer s that has the property that s ≡ d mod 3936 and s ≡ 0 mod 1293. (e) Find an integer S that has the property that S ≡ 573 mod 3936 and S ≡ 0 mod 1293. (f) Find an integer T that has the property that T ≡ 126 mod 1293 and T ≡ 573 mod 3936. (g) Is T the only number satisfying those two congruences; if not, which other numbers?

How many elements of order 2 are there in S5 and in S6? How many elements of order 2 are there in Sn? (abstract algebra)

Describe the inverse properties of real numbers and provide examples. What is useful about the inverse properties of real numbers?
Start with definitions for identity elements and then define additive and multiplicative inverses. Discuss why 0 and 1 are important in this context. Tell how and why we use inverses.
What is an example of an operation and its "additive inverse" that you use in everyday life? ( for example walking one block north and then one block south, to get back to where you started)
What is an example of an operation and its multiplicative inverse that you use in everyday life?( for example to convert from feet to inches multiply by 12. To convert inches to feet: divide by 12 or multiply by 1/ 12)

1. Use strong induction to show that every positive integer, n, can be written as a sum of powers of two: 2⁰, 2¹, 2², 2³, .....

Solve the following problems using the concepts and notation of set theory and vector spaces:

1. What is a Cartesian vector?
2. What is a real vector space?
3. Does the rational numbers fulfill the definition of a field?
4. Define linearly dependency for any set of vectors.
5. Assess the linearly dependency (or not) of the following vectors: u=<2,-1,1>, v=<3,-4,-2>, and w=<5,-10,-8>
6. Define basis