Solve the LP problem. If no optimal solution exists because there is no Solution Set, enter EMPTY. If no optimal solution exists because the region is unbounded, enter UNBOUNDED. Note that an unbounded region can still have an optimal solution while a bounded region is guaranteed to have optimal solutions. HINT [See Example 1.]

Maximize and minimize p = x + 2y subject to

Minimum

P=

(x, y)=

Maximum

P=

(x, y)=

In a (2, 5) Shamir secret sharing scheme with modulus 19, two of the shares are (0, 11) and (1, 8). Another share is (5, k), but the value of k is unreadable. Find the correct value of k

Distribute 13 indistinguishable balls in 6 distinguishable urns. What is the number of distributions in which the first three cells contain together AT LEAST 10 balls?

What would be the answer if the balls were distinguishable?

Consider the sine-Gordon equation (SGE)
θxt =sinθ, (1)

which governs a function θ(x,t). For any given λ denote the following B ̈acklund trans- formation by BTλ:

1
θ −θx=2λsin 2(θ +θ) , (2a)

2 1
θ +θt=λsin 2(θ −θ) , (2b)

1. (a) Given a solution θ(x, t) of the SGE, show that θ(x, t) also satisfies the SGE. Hint: Try calculating the t derivative of Equation (2a) and the x-derivative of Equation (2b) and then taking a sum or difference.

Prove or disprove each of the following statements.

(a) There exists a prime number x such that x + 16 and x + 32 are also prime numbers.

(b) ∀a, b, c, m ∈ Z +, if a ≡ b (mod m), then c a ≡ c b (mod m).

(c) For any positive odd integer n, 3|n or n 2 ≡ 1 (mod 12).

(d) There exist 100 consecutive composite integers.

Prove the following: theorem: every topological group is completely regular. Proof. Let V0 be a neighborhood of the identity elemetn e, in the topological group G. In general, coose Vn to be a neighborhood of e such that Vn.VncVn-1. Consider the set of all dyadic rationals p, that is all ratinal number of the form k/sn, with k and n inegers. FOr each dyadic rational p in (0,1], define an open set U(p) inductively as foloows: U(1)=V0 and

The surface area of a right-circular cone of radius r and height h is S=πrr2+h2−−−−−−√, and its volume is V=1/3πr2h.

(a) Determine h and r for the cone with given surface area S=4 and maximal volume V.
h=  , r=

(b) What is the ratio h/r for a cone with given volume V=4 and minimal surface area S?
hr=

(c) Does a cone with given volume V and maximal surface area exist?
A. yes
B. no

Let A ⊆ C be infinite and denote by A' the set of all the limit points of A.

Prove that if z ∈ A' then there is a non-trivial sequence of elements in A that converges to z

Let A be an m × n matrix and B be an m × p matrix. Let C =[A | B] be an m×(n + p) matrix.

(a) Show that R(C) = R(A) + R(B), where R(·) denotes the range of a matrix.

(b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).

Consider the following linear program:

   MAX Z = 25A + 30B

   s.t. 12A + 15B ≤ 300

   8A + 7B ≤ 168

  10A + 14B ≤ 280

   Solve this linear program graphically and determine the optimal quantities of A, B, and the    value of Z. Show the optimal area.

Hello,

In your own words, please if you were to these topics

Counted systems and integers

Fractions, decimals and percentages

Powers and laws of indices

Counting using the product rule

what kind of difficulties you might face when teaching these and the implications in classroom practice

how would you teach it,

why do you think it would be hard for students to learn.

I want this to be about 300 words essay.

Please answer this in essay-based format

in your own words please

Contradiction proof conception

Prove: If A is true, then B is true

Contradiction: If A is true, then B is false.

so we suppose B is false and follow the step to prove. At the end we get if A is true then B is true so contradict our assumption

However,

Theorem: Let (xn) be a sequence in R. Let L∈R. If every subsequence of (xn) has a further subsequence that converges to L, then (xn) converges to L.

Proof:  Assume, for contradiction, that (xn) doesn't converge to L

**my question is that at the end we will get no subsequence of (Xnk) converges to L (Xnk is a subsequence of Xn)****

why this is contradiction? This only shows the Contrapositive side of the theorem

show that for any two vectors u and v in an inner product space

||u+v||^2+||u-v||^2=2(||u||^2+||v||^2)

give a geometric interpretation of this result fot he vector space R^2

How can I do a chi square hypothesis test whether the provided data follows f-distribution?
I need to know how to set up the hypothesis test and how to check if the data follows f-distribution.
Thank you