Explain graphically how to determine the shadow price of a binding constraint in an LP with two variables. Show also the formulation of your LP.

In the context of network models, briefly explain what a flow balance constraint is and write down one example for a network of your own choice.

Develop your own objective function and set of constraints for an LP with unbounded feasible region but finite optimal solution. Illustrate graphically.

A function f : X ------> Y between two topological spaces ( X , T_X ) and ( Y , T_Y ) is called a homeomorphism it has the following properties:

a) f is a bijection (one - to- one and onto )

b) f is continuous

c) the inverse fucntion f  ^-1 is continuous ( f is open mapping)

A function with these three properties is sometimes called bicontinuous . if such a function exists, we say X and Y are homeomorphic.

show that a complete metric space R is homeomorphic to the metric space ( 0 , 1 ), which is not complete . Metric defined as an ab solute value of the difference  

Create a grand strategy matrix for an organization of your choice. Ideally, the company you choose will be a familiar one and one to which you have easy access, such as your place of employment or a company close to where you live. You may use the same organization for other assessments in this course. You may wish to review the suggested readings listed in the Resources as well as do your own research into completing a grand strategy matrix to prepare for this assessment. Complete the following: •Answer the following questions: ◦Is your chosen organization in a weak or strong competitive position? ◦Is the market growth rapid or slow? •Complete a grand strategy matrix for your chosen organization using the Grand Strategy Matrix Template linked in the Resources under the Required Resources headings. •Compare whether the organization's strategies are aligned with the relevant quadrant strategies, and write a half-page summary on the differences and changes you identify. Make recommendations for changes to the organization's strategy based on your observations. Grand Strategy Matrix Rapid Market Growth Quadrant 2 Weak competitive position Quadrant 1 Strong competitive position Quadrant 3 Quadrant 4 Slow Market Growth

analyze the performance of a gasoline engine at various loads based on data given and to determine the torque curve, power curve and performance parameters of BMEP, volumetric efficiency, and Air/fuel ratio, plotting the data vs RPM. Making three graphs: one graphing Torque and Power.

Also address t

a. How do the torque curve and power curve compare (max values @ same RPM)?

b. Is volumetric efficiency constant, or varying with RPM..what is trend?

c. Is BMEP a constant? Does it compare more with torque or power output?

Use Cauchy-Riemann equations to show that the complex function f(z) = f(x + iy) = z(x + iy) is nowhere differentiable except at the origin z = 0.6 points) 2. Use Cauchy's theorem to evaluate the complex integral ekz -dz, k E R. Use this result to prove the identity 0"ck cos θ sin(k sin θ)de = 0

Sec 7.3

1. A chain saw requires 8 hours of assembly and a wood chiiper 3 hours. A max of 48 hours of assembly time is available. The profit is $190 on a chain saw and a $240 on a chipper. How many of each should be assembled for maximum profit? To attain the maximum profit, assemble ___ chain saws and ___ wood chippers.

2. Deluxe coffee is to be mixed with regular coffee to make atleast 51 pounds of a blended coffee. The mixture must contain at least 12 pounds of deluxe coffee. it costs $6 per pound and regular coffee $5 per pound. How many pounds of each kind of coffee should be used to minimize cost? Use___ pounds of deluxe coffee and ___pounds of regular coffee

The ”Brusselator” is a mathematical model for a class of oscillating chemical reactions

x'= 1−(b+ 1)x+a(x^2)y

y′= bx−a(x^2)y

where x,y ≥ 0 are the concentrations of the chemical and a and b are positive constants

(a) Find the value of x and y at the equilibrium point.

(b) Find values of a and b that give (i) a stable node (ii) an unstable node (iii) a stable spiral (iv) an unstable

spiral (v) a saddle point. (Not all are possible)

Prove Rolles Theorem

Prove that the function defined to be 1 on the Cantor set and 0 on the complement of the Cantor set is discontinuous at each point of the Cantor set and continuous at every point of the complement of the Cantor set.

Prove the Heine-Borel Theorem

5.How can you proof a proposition in the form of ∀x P(x) is NOT true.

6.a) Briefly explain what does it mean to say B is a subset of A? What is the procedure to prove that?
b) How many subsets of A are there, if |A| = n ?
c) Define an arbitrary set A (with |A|=4), list all the elements of the power set of A. (P(A))

3. Briefly explain how you can prove that two sets are equal.

find the dirichlet green function of laplace equation for the interior of a colander with radius a

Please find a solution to the following:

Δu=0, 1<r<4, 0≤θ<2π

u(1,θ)=cos5*θ, 0<θ<2π

u(4,θ)=sin4*θ, 0<θ<2π