# Tagged Questions

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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### Jacobian matrix with two equations

Evaluate the Jacobian for: $$f(x,y)=(x^2+x+y, yx+x^2)$$ at the point $(1,2)$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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. ...
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### 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 ...
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### 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?
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### 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 ...
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### $A$ is an $n \times n$ invertible matrix, prove that $f(\mathbf u, \mathbf v)= \mathbf u^TAA^T \mathbf v$ defines an inner product on $\mathbb R^n$

I have difficulty especially proving that $f(\mathbf v, \mathbf v) \geq 0$ for all $\mathbf v$. Thanks
### 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 ...
### 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 ...