We wish to model a population of two fish species by a system of differential eqns. x′ = f(x,y), y′ = g(x,y).

Provide a system of differential eqns. that describe the following assumptions:
(a)(5) A predator-prey system in which prey can grow to no limits, but predators die out unless they find prey to eat.
(b)(5) Two species which individually would die out but collaboration enhances survival of BOTH species.
(c)(5) A predator-prey system in which prey can grow with no limits, but predators die out unless they find prey to eat. In addition the PREDATORS are being fished at a constant rate of 100 per units of time.
(d) (5) A predator- prey system in which the prey species is subject to environmental limits to growth(if the population is small it increases at a rate proportional to its size, but for populations > 1000 the rate becomes negative). The predator species dies out unless they have prey to eat.

9. Let S = {[ x y]; in R2 : xy ≥ 0} . Determine whether S is a subspace of R2.

(A) S is a subspace of R2.

(B) S is not a subspace of R2 because it does not contain the zero vector.

(C) S is not a subspace of R2 because it is not closed under vector addition.

(D) S is not a subspace of R2 because it is not closed under scalar multiplication.

Develop a function DNF that takes as input a proposition and returns as output a logically equivalent proposition in DNF. Hint: Adapt the CNF function.

Explain the basic concepts ordinary and partial 2 differential equations (ODEs,PDEs), order general and particular solutions initial value problems (IVPs)Give examples ?

Problem 7.6A. Unmatched Elegance Gift Shop sells cards, supplies, and various holiday greeting cards. Sales to retail customers are subject to an 8 percent sales tax. The firm sells it merchandise for cash; to customers using bank credit cards, such as MasterCard and VISA; and to customers using American Express. The bank credit cards charge a 2 percent fee. American Express charges a 3 percent fee. Unmatched Elegance Gift Shop also grants trade discounts to certain wholesale customers who place large orders. These orders are not subject to sales tax. During February 2016, Unmatched Elegance Gift Shop engaged in the following transactions:

Instructions

1. Open the general ledger accounts indicated below and enter the balances a of February 1, 2016.
2. Record the transactions in a general journal. Use 10 as the journal page number.
3. Post the entries from the general journal to the appropriate accounts in the general ledger.

General Ledger Accounts

101                  Cash, $23,230 Dr.                              401                  Sales

121                  Accounts Receivable                         521                  Credit Card Expense

222                  Sales Tax Payable

Analyze: What was the total credit card expense incurred in February?

y"+y=-1,y(0)=0,y(Pi/2)=0
I want to solve this equation by details
(by properties of the green function only)

Large semitrailer trucks cost ​$75 comma 000 each. A trucking company buys such a truck and agrees to pay for it by a loan that will be amortized with 8 semiannual payments at 5​% compounded semiannually. Complete an amortization schedule for the first four payments of the loan.

y'=2t-y Given: y(0)= -1 and h=0.1 estimate y(0.4) using (AB4) adam bashforth and (AM4) adam moulton using RK4 method

Prove the following stronger variant of Proposition 7.4. Suppose C is collection of connected subsets of a metric space X and B ∈ C. Show, if for each A ∈ C, A ∩ B not equal ∅, then Γ = ∪{C : C ∈ C} is connected. [Suggestion: Consider the collection D = {C ∪ B : C ∈ C}].

say R₁, R₂,...., R_n are commutative rings with unity. Show that U(R₁ + R₂ +.... + R_n) = U( R₁) + U(R₂)+ .... U(R_n). Where U - is the units of the ring.

Prove that for any integer n the expression
n7/7 + n5/5 + 23n/35
is whole integer.
(Hint: Note that the problem can be state in a following equivalent form: 35 | (5n7 +7n5 +23n);
even further, by the previews theorem, it would be enough to show that (5n7 + 7n5 + 23n) is
divisible by 5 and 7.)

Q3 [17% ] Let X be a set and A a σ-algebra of subsets of X.

(a) What does it mean for a function f : X → R to be measurable? [2%] If f and g are measurable, show that the function f − g is also measurable. [6%]

(b) Let (fn) be a sequence of measurable functions.

(i) What does it mean to say that (fn) converges pointwise to a function f? [2%]

(ii) If (fn) converges pointwise to f, show that f is a measurable function.