Consider the map f(x) =x^2+k .Find the values of k for which the map f has
a) two fixed points
b) only one fixed point
c) no fixed points
For what values of k there will be an attracting fixed point of the map?

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a subgroup of G such that K ⊂ H Suppose that H is also a normal subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b) Show that G/H is isomorphic to (G/K)/(H/K).

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only if every infinite subset of E has a point of accumulation that belongs to E.

Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real numbers is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E.

Must use Theorem 4.21 to prove Corollary 4.22 and there should be no mention of closed and bounded in the proof. The proof should start with,

[E closed and bounded] iff [E has the BW Property]

1. Using the simplex method, maximize revenue given the following
   1. R = 40q + 20z
   2. q + z ≤ 50
   3. 15q + 20z ≤ 200
   4. q,z ≥ 0

Give Mathematical definitions of periodic 2and periodic 3 cycles of a map. Show these graphically for a map

What is the solution to xy''+(1-x)y'+y=0 using the Frobenius method?

1. If you wanted to find the difference in Elementary Statistics grades between students who transferred to CSULB from a community college and students who entered CSULB straight out of high school, what test statistic would you use?

2. If you wanted to find the difference in grades among students who took Elementary statistics in their Freshman year, Sophomore year, Junior year, or Senior year in college, what test statistic would you use?

3. If you wanted to see if there is a difference among students who took Elementary statistics in their Freshman year, Sophomore year, Junior year, or Senior year in college, and whether their age at the time affects their grade, what test statistic would you use?

1. Let Q1=y(1.1), Q2=y(1.2), Q3=y(1.3) where y=y(x) solves y′′−2y′+ 5y= 0, y(0) = 1, y′(0) = 2.

2. Let Q1=y(1.1), Q2=y(1.2) , Q3=y(1.3) where y=y(x) solves y′′−2y′+ 5y=e^x cos(2x) , y(0) = 1, y′(0) = 2

Let Q= ln(3 +|Q1|+ 2|Q2|+ 3|Q3|). Then T= 5 sin2(100Q)

Please show all steps and thank you!!!!!

A binary string is a “word” in which each “letter” can only be 0 or 1

Prove that there are 2^n different binary strings of length n.

Note:

• Your goal is to produce a properly constructed proof by induction, but this does not mean you have to follow
• Mathematical induction, step-by-step..
• Write the statement with n replaced by k
• Write the statement with n replaced by k+1.
• Identify the connection between the kth statement and the (k+1)th statement.
• Complete the induction step by assuming that the n=k version of the statement is true, and using this assumption to prove that the n=k+1 version of the statement is true.
• Complete the induction proof by proving the base case.
• point-by-point.
• ANOTHER IMPORTANT NOTE:
• Make sure you are addressing the given statement: "there are 2ⁿ different binary strings of length n” !!! So your solution should primarily be discussing binary strings. Yes, the fact that 2^k+1 = 2^k∙2 will be useful in your solution, but it should not be the sole content of your solution, as by itself that equality is a fact about exponents, not a fact about binary strings.

Provide the general expression of a quadratic form whose solution set is two intersecting and no-overlapping straight lines.

Let

x0< x1< x2. Show that there is a unique polynomial P(x) of degree at most 3 such that

P(xj) =f(xj) j= 0,1,2, and P′(x1) =f′(x1) Give an explicit formula for P(x).

maybe this is a Hint using the Hermit Polynomial:

P(x) = a0 +a1(x-x0)+a2(x-x0)^2+a3(x-x0)^2(x-x1)

Hi. I have two questions about the linear algebra.

1. Consider the following sets:

(a) The set f all diagonal 3*3 matrices

(b) The set of all vectors in R^4 whose entries sum to 0.

For the cases where the set is a vectors space, give the dimension and a basis

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2. Let L be the set of all linear transforms from R^3 to R^2

(a) Verify that L is a vector space.

(b) Determine the dimension of L and give a basis for L.

Will thumb up for both answers. Thank you so much.

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].”