Let u(x, y) be the harmonic function in the unit disk with the boundary values u(x, y) = x^2 on {x^2 + y^2 = 1}. Find its Rayleigh–Ritz approximation of the form x^2 +C1*(1−x^2 −y^2).

Let R be the relation on the set of people given by aRb if a and b have at least one parent in common. Is R an equivalence relation?

(Equivalence Relations and Partitions)

The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod ? is an
equivalence relation for an integer ? ≥ 2.
a) What are the equivalence classes for R given a fixed integer ? ≥ 2?
b) We denote the set of equivalence classes you found in (a) by Z_5. Even though elements of Z_5 are
sets, it turns out that we can define addition and multiplication in the expected ways: [?] + [?] = [? + ?] and [?] ⋅ [?] = [??]
Construct the addition and multiplication tables for Z_4 and Z_5. Record sums and products in the
form [r], where 0 ≤ r ≤ 3 (or 4, respectively).
c) Let [?], [?] ∈ Z_10. If [?][?] = [0], does it follow that [?] = [0] or [?] = [0]?
d) How would you answer the question from (c) for Z_11, Z_12?, Z_13?
e) For which integers ? ≥ 2 is the following statement true?
“Let [?], [?] ∈ Z_5. If [?][?] = [0], then [?] = [0] or [?] = [0].”

please tell me how to write proofs for diophantine equation

5. Of the 9-letter passwords formed by rearranging the letters AAAABBCCC (4 A’s, 2 B’s, and 3 C’s), I select one at random. Determine the following probabilities.

(a) Prob (my word is a palindrome and has no two C’s next to each other).

(b) Prob (my word has two C’s next to each other and the other C not next to them).

(c) Prob (my word has the three C’s next to each other and the B’s apart from each other). Leave your probabilities with factorials in them.

Let . If we use Accelerated Newton-Raphson method to approximate the root of the equation , which of the following(s) is/are ture:

(I)  is multiple root of order

(II) Accelerated Newton-Raphson formula is :

(III) The sequence  obtained by the Accelerated Newton-Raphson method converge to the root  quadratically.

Find the constants a, b, and c such that the function f(z)=ax^2-bxy+2y^2+i(x^2+cxy+dy^2) is entire?

f\left(z\right)=ax^2-bxy+2y^2+i\left(x^2+cxy+dy^2\right)

Write out the first 4 terms of y1 and y2:

( x+2 ) y" - 3y' + 2xy = 0 ; x0 = 3

For countable set G, H = {f : N → G : f is injective} where N is the natural numbers. What is the cardinality of H. Prove it.

The line k goes through the point Q(-3,5) and is perpendicular to the line g: x - 3y - 22 = 0. Where do the angle bisectors of lines g and k intersect the line AB when A = (-3,3) and B = (10,3)?

For the following relations on S = {0,1,2,3}, specify which of the properties (R), (AR), (AS), and (T) the relations satisfy.

(R) = reflexive

(AR) = anti reflexive

(AS) = anti symmetric

(T) = transitive

b.) (m,n) in the domain of R2 if m - n is even

c.) (m,n) in the domain of R3 if m less than or equal to n

d.) (m,n) in the domain of R4 if m + n is less than or equal to 4

e.) (m,n) in the domain of R5 if max{m,n} = 3

y''-3y'+2y=1+cost+e^-t

NOL Carryback and Carryforward, Valuation Account versus No Valuation Account)

(LO 3) Spamela Hamderson Inc. reports the following pretax income (loss) for both financial reporting purposes and tax purposes. (Assume the carryback provision is used for a net operating loss.)

The tax rates listed were all enacted by the beginning of 2015.

Instructions

(a)  

Prepare the journal entries for the years 2015-2018 to record income tax expense (benefit) and income taxes payable (refundable) and the tax effects of the loss carryback and carryforward, assuming that at the end of 2017 the benefits of the loss carryforward are judged more likely than not to be realized in the future.

(b)  

Using the assumption in (a), prepare the income tax section of the 2017 income statement beginning with the line “Operating loss before income taxes.”

(c)  

Prepare the journal entries for 2017 and 2018, assuming that based on the weight of available evidence, it is more likely than not that one-fourth of the benefits of the loss carryforward will not be realized.

(d)  

Using the assumption in (c), prepare the income tax section of the 2017 income statement beginning with the line “Operating loss before income taxes.”