Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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2answers
13 views

Jacobian matrix with two equations

Evaluate the Jacobian for: $$f(x,y)=(x^2+x+y, yx+x^2)$$ at the point $(1,2)$.
1
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1answer
8 views

Equivalence Class.

Let R be the relation of congruence mod 4 on Z: a R b if a - b = 4k, for some k in Z. What integers are in the equivalence class of 31? How many distinct equivalence classes are there? What are ...
1
vote
1answer
15 views

What are the limits for this joint pdf?

I'm given equation that the joint pdf is $(2/3)(x + 2y)$ when $0 < x, y < 1$ and we want to find the probability that $X < 1/3 + Y$. I understand how to do the actual math part, and that I ...
1
vote
1answer
18 views

Repeatedly rolling an $n$-sided die

Suppose I roll an $n$-sided die once. Now you repeatedly roll the die until you roll a number at least as large as I rolled. What is the expected number of rolls you have to make? I know the answer ...
0
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1answer
42 views

complex inner product from the real

Let $V$ be a real inner product space. If $W=V\times V$ with the operations $(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2)$ and $(\alpha +i\beta)(u,v)=(\alpha u-\beta v,\alpha v+\beta u)$, where $u, ...
0
votes
1answer
16 views

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
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1answer
8 views

${\bf E}[Y]$ of a joint distribution

So, I have that a joint pdf is given by the formula: $$ 5e^{-5x} / x, \quad 0 < y < x < \infty $$ and I have to find the $Cov(X,Y)$. I know that $Cov(X,Y) = {\bf E}[XY] - {\bf E}[X]{\bf ...
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3answers
53 views

If $d=\gcd(a+b,a^2+b^2)$, with $\gcd(a,b)=1$, then $d=1$ or $2$

Suppose $\gcd(a,b)=1$. Let $d=\gcd(a+b,a^2+b^2)$. I want to prove that $d$ equals $1$ or $2$. I get that $d\mid2ab$ but I can't find a linear combination that will give me some help to use the fact ...
1
vote
1answer
23 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
2
votes
1answer
45 views

what to do when the multivariable second derivative test is inconclusive?

What do we do when the second derivative test fails? How do we approach it, and is there a general method to further find whether a critical point is a maximum, minimum or a saddle point? For ...
3
votes
2answers
35 views

$\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = sec(u)$ substitution: $$ \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x = \int \frac{\sqrt{sec^2(u) - 1}}{sec(u)}tan(u)sec(u) \mathrm{d}u = \int tan^2(u) \mathrm{d}u = tan(u) - u + C = ...
1
vote
1answer
28 views

feedback on my answer regarding set intersections.

Prove or find a counter-example to the claim that for all sets $A,B,C$ if $A\cap B = B \cap C = A \cap C = \emptyset$, then $A \cap B \cap C=\emptyset $. the above statement is not true so i need a ...
1
vote
0answers
29 views

Logical expression

Please can someone give me feedback on my answer to the question below. Question. Surf the internet and find a theorem of number theory. State the claim of the theorem, and then express it in logical ...
0
votes
0answers
13 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
0
votes
3answers
33 views

proving or providing counter example in disrete mathematics

Prove or find a counterexample: The product of any three consecutive natural numbers is divisible by 6. if we take a few consecutive natural numbers such as 1 ,2 ,3. and multiply i get 6 which is ...
1
vote
1answer
570 views

Solving a recurrence realtion using backward substitution.

So I've been trying my best to do this, and I have made some good progress, I just need to know if what I have done is correct and if not, what the hell am I doing wrong? :P I start off with this ...
0
votes
2answers
32 views

Intermediate value theorem problem

Problem: The equation $x=-5\cos(x)$ has at least $3$ distinction solutions. Use the intermediate value theorem to show that this is true. I drew the function,but I don't know what to do next.
0
votes
1answer
27 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
1
vote
1answer
9 views

finding conditional expectation under binomial distribution.

Suppose X and Y independent and are both binomial random variables with parameter N, p Compute E(X|X+Y).
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1answer
8 views

Solving a recurrence relation using forward substitution.

How can I solve this? $$T(n)=3T\left(\frac{n}{4}\right),$$ for $n>1$, $n$ a power of $4$, and $T(1)=3$.
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votes
1answer
19 views

Simplifying Trig Identity

I have an equation I have been given to solve, I know how to start but I do not know what to do after I use the Trig Identities. Any help? Here is what I was given $$ \frac{\cos(A + B) + \cos(A - ...
1
vote
1answer
20 views

Find the conditional expectation of N given that there were exactly 2 heads in the first 3 tosses.

We have two biased coins. The first one yields heads with probability 0.1 and the second one yields heads with probability 0.9. We choose one of the two coins randomly (with probability 0.5 each; we ...
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votes
0answers
9 views

linear algebra - fourier coefficients of piecewise

Find fourier coefficients of given function: f(t) = {-1 if t $\leq$ 0; 1 if t > 0} so do I do this? $a_{0} = \int_{a+-\pi}^{a+\pi}1$, $a_{k} = \int_{a+-\pi}^{a+\pi}1*cos(kx)$, $b_{k} = ...
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0answers
5 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...
0
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0answers
20 views

For which values of $t$, is $x$ moving to the left or the right?

$x=t^2-2t$ , $y=3t-2$ . I've already found tangent lines for these parametric equations, but how can I determine when $x$ is moving to the left or the right? Is it the second derivative?
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votes
2answers
31 views

Find the coordinates of all points that satisfy certain conditions. [on hold]

Find the coordinates of all points whose distance from $(-3,6)$ is $\sqrt{13}$ and whose distance from $(2,7)$ is $\sqrt{13}$.
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votes
1answer
51 views

Number of surjective functions from $\{1,2,…,n\}$ to $\{a,b,c\}$

Ok so following questions are given in my text book Let $A = \{1, 2, 3,...., n\}$ and $B =\{a, b, c\}$ then the number of functions form $A$ to $B$ that are onto is. I have no idea how to find ...
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0answers
12 views

How to Find the inverse Fourier transform

How to find the inverse Fourier transform and how to transform the solution with refer to the Fourier transform table?
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3answers
42 views

Induction, show that something is smaller then …

I have to show the following by induction. $1 \cdot 2 \cdot 3 ... (n - 1) \leq (\frac{n}{2})^{n -1}$ As it is homework I "only" need a push in the right direction. my thought is that is something ...
0
votes
0answers
10 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
0
votes
2answers
85 views

Show that $ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$ is unbiased estimators of $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Show that $\hat{\theta}_2$ is ...
0
votes
0answers
37 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (ii) and the rest. ...
0
votes
3answers
37 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
0
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0answers
25 views

Simpson's Rule - Error of $O(h^5)$

The following is given: When deriving Simpson's Rule using the same approach as that of the Trapezoidal rule, it was stated that the method only generates an error term of $O(h^5)$ involving ...
1
vote
3answers
18 views

Sigma Algebra Measurable R.V

I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with ...
0
votes
2answers
21 views

Solving for x on unit circle equation

I have been given the equation $$\cos^2{x} + 2\sin{x}=2.$$ I have factored it, and the only answer I got was $x=\frac{\pi}{2}$. Is this correct or is there more than one answer? The interval is $0 ...
2
votes
2answers
25 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
0
votes
4answers
39 views

Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
2
votes
0answers
18 views

Maximum number of edges in a (n,n) bipartite graph, that doens't have a complete bipartite subgraph $K_{r,r}$

I need to prove that the maximum number of edges in a $n \times n$ bipartite graph, that doens't have a complete bipartite subgraph $K_{r,r}$ is lower bounded by $cn^{2-2/(r+1)}$ where c is a constant ...
0
votes
2answers
12 views

Proof that in any base $b$, the result of multipling two numbers of $k$ digits, doesn't recuire more than $2k$ digits

The proof that I came up whit is: Let, $c$ be $b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k r_k$ and $d = b^0 r_0'+ b^1 r_1'+b^2 r_2'+...+b^k r_k'$ then multipling both: $$(b^0 r_0+ b^1 r_1+b^2 r_2+...+b^k ...
2
votes
1answer
24 views

Calculate modulo of large numbers

I have $2^{2^n}+1$ and i want to calculate ($(2^{2^{^n}} +1 )\mod 19$). How can i do it if for example i choose $n = 19$. Can i use Fermat's Little Theorem?
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2answers
16 views

Proof that in any base, the sum of two numbers of fixed precision can't have carry more than 1

The best proof that I came with is, given any base $b$, let $c$ be the greatest number can be written whit $n$ digits. Then the number will be: $$c=b^0(b-1)+b^1(b-1)+\cdots +b^n(b-1)$$ Summing this ...
1
vote
3answers
87 views

how do i prove that $17^n-12^n-24^n+19^n \equiv 0 \pmod{35}$

How do i prove that 17n−12n−24n+19n≡0(mod35) for every possitive integer n. Can anyone give me a hint of how to start?
0
votes
1answer
19 views

Prove that Orthogonal Set Is Linearly Independent

Suppose that $V$ is an inner-product space; $(\space ,\space )$ is our inner-product. I have seen many proofs that go as follows: Let $\{x_1, x_2 ,\ldots, x_n\}$ be orthogonal. Set $a_1x_1 + a_2x_2 ...
1
vote
2answers
41 views
+50

Derivatives of component inverse functions

I might have missed the point of the following questions. Anyone kindly give a suggestion? Let $f:\mathbb{R}_\mathbf{x}^3\to\mathbb{R}_\mathbf{y}^3$ and ...
0
votes
1answer
27 views

$\left\{\frac{\pi}{6}+\frac{2K\pi}{3}\Big\vert K\in\mathbb {Z}\right\}\cap\left\{\frac{\pi}{3}+\frac{K\pi}{2}\Big\vert K\in\mathbb {Z}\right\}=$?

$$\left\{\frac{\pi}{6}+\frac{2K\pi}{3}\,\Big\vert\, K\in\mathbb {Z}\right\}\cap\left\{\frac{\pi}{3}+\frac{K\pi}{2}\,\Big\vert\vert\, K\in\mathbb {Z}\right\}=\varnothing$$ Is my answer right? If not, ...
1
vote
1answer
17 views

Solving a recurrence realtion using forward substitution.

I have to find $T(n) = 7 \cdot T\left(\frac{n}{7} \right)$ for $n>1$ when $n$ a power of $7$. So far I have: $$T(7) = 7\cdot T\left(\frac{7}{7}\right) = 7 \cdot T(1) = 7.$$ Then, $$T(49) = 49 ...
0
votes
1answer
23 views
1
vote
3answers
25 views

Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
2
votes
2answers
70 views

A solvable Lie-algebra of derived length 2 and nilpotency class $n$

Given a natural $n>2$, I want to show that there exists a lie algebra $g$ which is solvable of derived length 2, but nilpotent of degree $n$. I have seen a parallel idea in groups, but i can't see ...