Find five positive natural numbers u, v, w, x, y such that there is no subset with a sum divisible by 5

We say a graph is k-regular if every vertex has degree exactly k. In each of the following either give a presentation of the graph or show that it does not exist. 1) 3-regular graph on 2018 vertices. 2) 3-regular graph on 2019 vertices.

Consider a function f(x) which satisfies the following properties:

1. f(x+y)=f(x) * f(y)

2. f(0) does not equal to 0

3. f'(0)=1

Then:

a) Show that f(0)=1. (Hint: use the fact that 0+0=0)

b) Show that f(x) does not equal to 0 for all x. (Hint: use y= -x with conditions (1) and (2) above.)

c) Use the definition of the derivative to show that f'(x)=f(x) for all real numbers x

d) let g(x) satisfy properties (1)-(3) above and let k(x) =f(x)/g(x). Show that k(x) is defined for all x and find k'(x). Use this to discover the relationship between f(x) and g(x)

e) Can you think of a function that satisfies (1)-(3)? Could there be more than one such function? Explain why or why not.

f) What do you think would happen if you changed condition (3) so that f'(0)=a for some a>0, rather than 1? Would you still find a familiar function?

DIFFERENTIAL EQUATIONS
Situation
Describe in your own words the method you would use to find the Laplace transform of the first derivative, that is,. Give 2 examples.

Situacion:

Describa en sus propias palabras el método que usted utilizaría para hallar la transformada de Laplace de la primera derivada, o sea, . De 2 ejemplos.

Prove or disprove that 3|(n 3 − n) for every positive integer n.

Let Q1=y(2) Q2 =y(3) where y=y(x) solves

y'+2xy=2x^3     y(0)=1

dy/dx= (2x-y)/(x+4y)     y(1)=1

y'+ycotx=y^3sin^3x      y(pi/2)=1

(y^2-2x)dx+2xy dy=0    y(1)=1

2x^2y;+4xy=3sinx    y(2pi)=0

y"+2y^-1 (y')^2=y'

The Weigelt Corporation has three branch plants with excess production capacity. Fortunately, the corporation has a new product ready to begin production, and all three plants have this capability, so some of the excess capacity can be used in this way. This product can be made in three sizes--large, medium, and small--that yield a net unit profit of $420, $360, and $300, respectively. Plants 1, 2, and 3 have the excess capacity to produce 750, 900, and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved.

The amount of available in-process storage space also imposes a limitation on the production rates of the new product. Plants 1, 2, and 3 have 13,000, 12,000, and 5,000 square feet, respectively, of in-process storage space available for a day's production of this product. Each unit of the large, medium, and small sizes produced per day requires 20, 15, and 12 square feet, respectively.

Sales forecasts indicate that if available, 900, 1,200, and 750 units of the large, medium, and small sizes, respectively, would be sold per day.

At each plant, some employees will need to be laid off unless most of the plant’s excess production capacity can be used to produce the new product. To avoid layoffs if possible, management has decided that the plants should use the same percentage of their excess capacity to produce the new product.

Management wishes to know how much of each of the sizes should be produced by each of the plants to maximize profit.

1. Formulate the dual of the above problem and solve it.

2. Weigelt.lp

   /* objective function */

   Max: 420 L1 + 360 M1 + 300 S1 + 420 L2 + 360 M2 + 300 S2 + 420 L3 + 360 M3 + 300 S3;

   /* Constraints */

   /* capacity */

   L1 + M1 + S1 <= 750;

   L2 + M2 + S2 <= 900;

   L3 + M3 + S3 <= 450;

   /* square footage */

   20 L1 + 15 M1 + 12 S1 <= 13000;

   20 L2 + 15 M2 + 12 S2 <= 12000;

   20 L3 + 15 M3 + 12 S3 <= 5000;

   /* sales */

   L1 + L2 + L3 <= 900;

   M1 + M2 + M3 <= 1200;

   S1 + S2 + S3 <= 750;

   /* same percentage of capacity */

   900 L1 + 900 M1 + 900 S1 - 750 L2 - 750 M2 - 750 S2 = 0;

   450 L1 + 450 M1 + 450 S1 - 750 L3 - 750 M3 - 750 S3 = 0;

   * Formulate the dual of the above problem and solve it. I just need help with the dual formulation of this LP solution.

7.3.2 in Strogatz

"Using numerical integration, compute the limit cycle of Exercise 7.3.1 and verify that it lies in the trapping region you constructed."

Exercise 7.3.1 refers to the system "x'=x-y-x(x^2+5y^2), y'=x+y-y(x^2+y^2)"

(Advanced Calculus and Real Analysis) - Cantor set, Cantor function

* (a) Define the Cantor function.

(b) Prove that the Cantor function is non-decreasing.

(Advanced Calculus and Real Analysis) - Cantor set, Lebesgue outer measure

* (a) Define the Cantor set.

(b) Show that the Cantor set P has the Lebesgue outer measure zero.

(c) Find the Lebesgue outer measure of the set L in the construction of the Cantor set.

Consider a finite set A = {a1,...,an}, and let P(x) be some statement.

– Explain why ∀x(x∈A → P(x)) is equivalent to P(a1)∧P(a2)∧...∧P(an).

– Explain why ∃x(x∈A∧P(x)) is equivalent to P(a1)∨P(a2)∨...∨P(an).

(It may help to think about some special cases like n = 2, n = 3. A formal proof is not required here.)

Solve (1+e^x)dy/dx+(e^x)y=3x^2+1

Solve (x^3+y^3)dx+3xy^2 dy = 0

Solve (y-cos y)dx + (xsiny+x)dy = 0

Solve (1+ln x +y/x)dx = (1-lnx)dy

Solve (y^2+yx)dx - x^2dy =0

Solve (x^2+2y^2)dx = xydy

Solve Bernoulli's Equation x dy/dx + 2y = (x^4)(e^x)(y^2)

Solve Bernoulli's Equation (1+x^2) dy/dx = 2xy +(e^x)(y^2)

Solve IVP (3e^(x^2))dy + (xy^2)dx=0 ; y(1) = 2

Solve IVP dy/dx -2xy = e^(x^2) ; y(0)=0

Solve IVP (x^2+y^2)dx+(2xy)dy=0; y(1)=1

6. Mixture Problem

Initially 40 lb of salt is dissolved in a large tank holding 100 gallons of water. A pure water is pumped into the tank at a rate of 2 gal/min and the well-stirred solution is then pumped out at the same rate.

a. Determine the amount of salt in the tank at any time and How much salt is present at t  50 mi

b. At what time the concentration of the salt be 0.2 lb/gal?

c. At what time the concentration of the salt be 0.01 lb/gal?

Allocating the Transaction Price. HeavyEQ produces large conveyor belt systems for heavy manufacturing. HeavyEQ signs a $2 million fixed-price contract under which it makes three promises:

• Install a conveyor belt system: fair value $1.6 million
• Service the system over a five-year period: fair value $0.6 million
• Provide a warranty assuring that the conveyer belt meets the contract specification at the time of sale: fair value $0.2 millioN

REQUIRED

a. Allocate the transaction price to the performance obligations.

b. Reallocate the transaction price under the notion that HeavyEQ has no reasonable basis for determining the fair value of the servicing because the conveyor system is of such a unique nature that the servicing activities are highly variable and uncertain