Express all solutions to the following equations as sets of integers: (a) 7x ≡ 12 mod 13 (b) 10x ≡ 4 mod 6 (c) 6x ≡ 8 mod 12

In: Advanced Math

- Seven thieves try to share a hoard of gold bars equally between themselves. Unfortunately, six bars are left over, and in the fight over them, one thief is killed. The remaining six thieves, still unable to share (all) the bars equally since two are left over, again fight, and another is killed. When the remaining five share (all) the bars, one bar is left over, and it is only after yet another thief is killed that an equal sharing is possible. What is the minimum number of bars which allows this to happen? [Hint: Be carefully to check that the conditions for the Chinese Remainder Theorem apply before using it.]

In: Advanced Math

real Analysis

1).
Let C be the intersection of all the Cn's. You must show C is
closed, measurable, has positive measure, and contains no interval,
i.e., if x is in C, every epsilon neighborhood of x contains points
not in C.Analysis
i should have note 0<alpha<1

Start with the interval [0,1] and remove the middle open
interval of length alpha/3 to form C1. Cn is then formed by
removing the middle open interval of length alpha/3^n from each
closed interval of C(n-1). Let C be the intersection of all the
Cn's. You must show C is closed, measurable, has positive measure,
and contains no interval, i.e., if x is in C, every epsilon
neighborhood of x contains points not in C.

In: Advanced Math

Differential Geometry

Open & Closed Sets, Continuity

(1) Prove (2,4) is open

(2) Prove [2,4) is not open

(3) Prove [2,4] is closed

In: Advanced Math

Solve the following differential equations:

1.) y"(x)+ y(x)=4e^x ; y(0)=0, y'(0)=0

2.) x"(t)+3x'(t)+2x(t)=4t^2 ; x(0)=0, x'(0)=0

In: Advanced Math

A Cartesian vector can be thought of as representing magnitudes along the x-, y-, and z-axes multiplied by a unit vector (i, j, k). For such cases, the dot product of two of these fectors {a} and {b} corresponds to the product of their magnitudes and the cosine of the angle between their tails as in {a}⋅ {b} = abcos(theta)

The cross product yields another vector, {c} = {a} × {b} , which is perpendicular to the plane defined by {a} and {b} such that its direction is specified by the right-hand rule. Develop and M-file function that is passed two such vectors and returns Theta, {c} and the magnitude of {c}, and generates a three-dimensional plot of the three vectors {a}, {b}, and {c} with their origins at zero. Use dashed lines for {a} and {b} and a solid line for {c}. Test your function using the following cases:

A. a = [ 6 4 2 ]; b = [ 2 6 4 ];

B. a = [ 3 2 -6 ]; b = [ 4 -3 1];

C. a = [ 2 -2 1 ]; b = [ 4 2 -4 ];

D. a = [ -1 0 0 ]; b = [ 0 -1 0 ];

I know how to find theta, {c}, and the magnitude of {c}, I just don't know how to plot a 3-dimensional graph so if someone could help me with that part of the code for MATLAB

In: Advanced Math

Assume A = R and the relation R ⊆ A × A such that for x, y, ∈ R, xRy if and only if sin2 x + cos2 y = 1. Prove that R is an equivalence relation and for any fixed x ∈ R, find the equivalence class x

In: Advanced Math

prove or disppprove.

Suppose A & B are sets.

(1) A function f has an inverse iff f is a bijection.

(2) An injective function f:A->A is surjective.

(3) The composition of bijections is a bijection.

In: Advanced Math

A tank originally contains 100 gal of fresh water. Then water containing

1 |

2 |

lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of

6 gal/min,

with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. (Round your answer to two decimal places.)

In: Advanced Math

\int \frac{x^6+5x^4+4x-3}{\left(x^2+3x+5\right)\left(x^2+7x+12\right)}dx

In: Advanced Math

Use laplace transforms to solve the following problem.

y' + 20y = 6sin 2x; y(0) = 6

In: Advanced Math

Find all possible homomorphisms between Z and Z5.

In: Advanced Math

Can you supply examples of logical operations you apply to your everyday experience?

In: Advanced Math

Use matlab code for bisection method and regula falsi. Thank you!

In: Advanced Math

- An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.

Medium Cost Per Ad # Reached Exposure Quality

TV 700 8000 35

Radio 240 3000 50

Newspaper 360 5000 25

If the number of TV ads cannot exceed the number of radio ads by more than 3, and if the advertising budget is $25,000. Develop the linear programming problem and solve the model that will maximize the number reached and achieve an exposure quality if at least 1000. (Hint: You need 3 decision variables to develop this LP problem.)

In: Advanced Math