Prove using the principle of mathematical induction:

(i) The number of diagonals of a convex polygon with n vertices is n(n − 3)/2, for n ≥ 4,

(ii) 2n < n! for all n > k > 0, discover the value of k before doing induction

This exercise requires the use of technology.

Four sectors of the U.S. economy are (1) livestock and livestock products, (2) other agricultural products, (3) forestry and fishery products, and (4) agricultural, forestry, and fishery services. Suppose that in 1977 the input-output table involving these four sectors was as follows (all figures are in millions of dollars).

Determine how these four sectors would react to an increase in demand for livestock (Sector 1) of $1,000 million, how they would react to an increase in demand for other agricultural products (Sector 2) of $1,000 million, and so on. (Round your answers to two decimal places. Let the columns of the matrix be given in millions of dollars.)

Answer is a 4x4 Matrix and is NOT 0.182 or 0.878 for the first box in the matrix answer

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.]

Minimize c = 2x − 2y subject to

c=

(x, y)=

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