Show that a graph T is a tree if and only if for every two vertices x, y ∈ V (T), there exists exactly one path from x to y.
In: Advanced Math
Draw a sketch of z ∈ C | Im((3 − 2i)z) > 6 .
2. Express z = −i − √ 3 in the form r cis θ where θ = Argz.
3. Use de Moivre’s theorem to find the two square roots of −4i.
In: Advanced Math
Perpetuity A pays $100 at the end of each year. Perpetuity B pays $25 at the end of each quarter. The present value of perpetuity A at the effective rate of interest is $2,000.
What is the present value of perpetuity B at the same annual effective rate of interest i?
In: Advanced Math
A stock expects to pay a dividend of $3.72 per share next year. The dividend is expected to grow at 25 percent per year for three years followed by a constant dividend growth rate of 4 percent per year in perpetuity. What is the expected stock price per share 5 years from today, if the required return is 12 percent?
In: Advanced Math
Solve the following differential equation using the power series method.
(1+x^2)y''-y'+y=0
In: Advanced Math
Residents were surveyed in order to determine which flowers to plant in the new public garden. A total of N people participated in the survey. Exactly 9/14 of those surveyed said that the color of the flower was important. Exactly 7/12 of those surveyed said that the smell of the flower was important. In total, 753 people said that both the color and smell were important. How many possible values are there for N?
Please explain all the steps clearly.
Thank you.
In: Advanced Math
Definition 2.3.2 in our book defines the power set of a set S, denoted by P(S), as the set of all subsets of S, that is P(S) = {A : A ⊆ S}. For example, P({1, 2}) = {∅, {1}, {2}, {1, 2}}, and P(∅) = {∅}. Consider the relation ⊆ on the power set P(Z), i.e. the is-a-subset-of relation defined on sets of integers. In other words, the objects we compare are sets of integers. (a) Prove or disprove: ⊆ is reflexive. (b) Prove or disprove: ⊆ is irreflexive. (c) Prove or disprove: ⊆ is symmetric. (d) Prove or disprove: ⊆ is antisymmetric. (e) Prove or disprove: ⊆ is transitive. (f) Is ⊆ on Z an equivalence relation? Is it a partial order? (g) Which other relation satisfies exactly the same properties?
In: Advanced Math
Let ∼ be the relation on P(Z) defined by A ∼ B if and only if there is a bijection f : A → B. (a) Prove or disprove: ∼ is reflexive. (b) Prove or disprove: ∼ is irreflexive. (c) Prove or disprove: ∼ is symmetric. (d) Prove or disprove: ∼ is antisymmetric. (e) Prove or disprove: ∼ is transitive. (f) Is ∼ an equivalence relation? A partial order?
In: Advanced Math
In: Advanced Math
In: Advanced Math
Period |
Payment |
Interest |
Balance |
Unpaid Balance |
$480,000 |
||||
1 |
||||
2 |
||||
3 |
In: Advanced Math
(3) Consider the linear map ? : R2 → R2 which sends (1, 0) ↦→ (−3, 5) and (0, 1) ↦→ (4, −1).
(a) What is the matrix of the transformation? What is the change of coordinates matrix? Do they agree? How come?
(b) Where does this transformation send the area between the vector (4, 2) and the x-axis? Explain algebraically and draw a picture.
(c) What is the image of the lower half plane under ?? Explain algebraically and draw a picture.
(d) What is the pre-image of the upper half plane under ?? Explain algebraically and draw a picture.
(e) What is the image of the unit circle under ?? Explain algebraically and draw a picture.
(f) Deduce whether or not linear transformations preserve angles/areas/curves/shapes, etc.
In: Advanced Math
Linear algebra confusion: We haven't learned about rank and
such:
For the following I am looking for examples or checking to see why
something is impossible.
a.) What does it mean when we have an inconsistent 2x3 linear
system
-What if we reversed it to 3x2? Even put other numbers in
place.
b.) Can a 2x3 linear system have a unique solution
-What if we reversed it to 3x2? Even put other numbers in
place.
c.) Can a 3x2 linear system have infinitely many soluitons
-What if we reversed it to 2x3? Even put other numbers in
place.
We had just one day of class and we didn't discuss: inconsistent,
unique, infinitely many solutions.
Where can I also reference to learn about this.
In: Advanced Math
2.
LetT:R3→R3suchthatT(1,1,—1)=2,1,0]andT(0,1,1)=1,1,1.
(a) Find T( 5,3-7).
(b) Can you find T (1,0,0) from the information given in part
(a)? Explain.
(c) Suppose you are now told that T(1,0,0) = 4,2,0. Find the
(standard) matrix of T and
compute T(x,y,z). Find a basis for range(T), the range of T.
In: Advanced Math
The metric space M is separable if it contains a countable dense
subset. [Note
the confusion of language: “Separable” has nothing to do with
“separation.”]
(a) Prove that R^m is separable.
(b) Prove that every compact metric space is separable.
In: Advanced Math